Entanglement and Complexity of Purification in (1+1)-dimensional free Conformal Field Theories
Hugo A. Camargo, Lucas Hackl, Michal P. Heller, Alexander Jahn, Tadashi Takayanagi, Bennet Windt
EEntanglement and Complexity of Purificationin (1+1)-dimensional free Conformal Field Theories
Hugo A. Camargo,
1, 2, ∗ Lucas Hackl,
3, 4, 5, † Michal P. Heller, ‡ Alexander Jahn, § Tadashi Takayanagi,
6, 7, 8, ¶ and Bennet Windt ∗∗ Max-Planck-Institut f¨ur Gravitationsphysik,Am M¨uhlenberg 1, 14476 Potsdam-Golm, Germany Dahlem Center for Complex Quantum Systems,Freie Universit¨at Berlin, Arnimallee 14, 14195 Berlin, Germany QMATH, Department of Mathematical Sciences, University of Copenhagen,Universitetsparken 5, 2100 Copenhagen, Denmark Max-Planck-Institut f¨ur Quantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany Munich Center for Quantum Science and Technology, Schellingstr. 4, 80799 M¨unchen, Germany Center for Gravitational Physics, Yukawa Institute for Theoretical Physics,Kyoto University, Kyoto 606-8502, Japan Inamori Research Institute for Science, 620 Suiginya-cho, Shimogyo-ku, Kyoto 600-8411 Japan Kavli Institute for the Physics and Mathematics of the Universe,University of Tokyo, Kashiwa, Chiba 277-8582, Japan Blackett Laboratory, Imperial College London, Prince Consort Road, SW7 2AZ, UK
Finding pure states in an enlarged Hilbert space that encode the mixed state of a quantum fieldtheory as a partial trace is necessarily a challenging task. Nevertheless, such purifications play thekey role in characterizing quantum information-theoretic properties of mixed states via entanglementand complexity of purifications. In this article, we analyze these quantities for two intervals in thevacuum of free bosonic and Ising conformal field theories using, for the first time, the most generalGaussian purifications. We provide a comprehensive comparison with existing results and identifyuniversal properties. We further discuss important subtleties in our setup: the massless limit ofthe free bosonic theory and the corresponding behaviour of the mutual information, as well as theHilbert space structure under the Jordan-Wigner mapping in the spin chain model of the Isingconformal field theory.
CONTENTS
I. Introduction 2II. Setup 3A. Klein-Gordon field 4B. Critical transverse field Ising model 6III. Mutual information 7IV. Gaussian purifications 8V. Entanglement of purification 10A. Definition and existing results 10B. Numerical studies using the most generalGaussian purifications 11VI. Complexity of purification 11A. Holographic predictions 12B. Definition and implementation 12 ∗ [email protected] † [email protected] ‡ [email protected]; On leave of absence from:
NationalCentre for Nuclear Research, Pasteura 7, 02-093 Warsaw, Poland § [email protected] ¶ [email protected] ∗∗ [email protected] C. Single interval in the vacuum 14D. Two adjacent intervals in the vacuum 15E. Single mode optimization for bosons 16F. Comparison with the Fisher-Rao distanceproposal 17VII. Comments 18A. Zero mode for free bosons 18B. Subsystems in Ising CFT vs. free fermions 20VIII. Discussion 20Acknowledgments 22A. Transverse field Ising model 221. Definition 222. Notions of locality 23B. Spin vs. Majorana Fermion 241. Partial traces and subsystems 242. Ising CFT representations 25C. Gaussian entanglement entropy 26D. Algorithm’s implementation 27References 28 a r X i v : . [ h e p - t h ] S e p I. INTRODUCTION
Understanding quantum information theoretic proper-ties of quantum field theories (QFTs) and, via hologra-phy, also of quantum gravity has been an enormouslyfruitful research front of the past two decades (as seen,for example, in [1–5]).The main player in this endeavour has been the notionof entanglement and its entropy S . Starting with a purestate | Ψ (cid:105) and a subsystem A (its complement denotedby ¯ A ), the entanglement entropy is defined as the vonNeumann entropy of the reduced density matrix ρ A =Tr ¯ A | Ψ (cid:105)(cid:104) Ψ | associated with A , specifically S ( A ) ≡ − Tr A ρ A log ρ A . (1)While entanglement entropy is very hard to calculatein a generic QFT, by now many results exist for freequantum fields, conformal field theories (primarily in twospatial dimensions) and strongly coupled QFTs with aholographic description. In the latter case, the entangle-ment entropy acquires a natural geometric descriptionin terms of the Bekenstein-Hawking entropy of certaincodimension-2 surfaces penetrating anti-de Sitter (AdS)geometries [11–14] and led to a wealth of results on quan-tum gravity in this setting.Complexity is another quantum information-theoreticnotion that made its appearance in the context of QFTsonly recently and is directly motivated by holography. Tothis end, it was observed in [15–20] that codimension-oneboundary-anchored maximal volumes and codimension-zero boundary-anchored causal diamonds have proper-ties expected from the hardness of preparing states usingtensor networks [21] in chaotic quantum many-body sys-tems.Subsequent articles starting with [22, 23] saw in thisconjecture a strong motivation to define the notion ofcomplexity in the realm of QFTs in a similar spirit inwhich pioneering works [24, 25] introduced the notionof entanglement entropy in the same context. The arti-cles [22, 23] were largely inspired by the continuous ten-sor network of cMERA [26] and viewed preparation of apure target state | ψ T (cid:105) in QFT as a unitary transforma-tion from some pure reference state | ψ R (cid:105)| ψ T (cid:105) = U | ψ R (cid:105) , (2)where the unitary U is obtained as a sequence of layersconstructed by exponentiation of more elementary Her-mitian operators O I U = P e − i (cid:82) dτ (cid:80) I O I Y I ( τ ) . (3)Following the approach of [27], which was originally de-vised to bound complexity of discrete quantum circuits, This approach assumes factorization of the Hilbert space between A and ¯ A . This is not the case for gauge theories, where morerefined approaches need to be invoked, see for example [6–10]. one can associate the cost of invocations of different gatesgenerated by O I as related to the infinitesimal parameter Y I ( τ ) dτ in the exponent. Translating this literally intoa mathematical formula would lead tocost L = (cid:90) dτ (cid:88) I | Y I | , (4)which is an integral over the circuit of a L norm of aformal vector formed from the parameters Y I . Complex-ity C arises then as the minimum of (4) subject to thecondition (2) C = min [cost] . (5)As anticipated already in [23], cost functions based on L norms such as (5) lead to challenging minimizationproblems. In the present work our rigorous results oncomplexity will be based on a particular choice of a L norm cost L = (cid:90) dτ (cid:115)(cid:88) I η IJ Y I Y J , (6)where, following [28, 29], η IJ is going to be a particularnon-negative definite constant matrix and O I are goingto be normalized accordingly. This preferred by us formof η IJ is naturally induced from the reference state | ψ R (cid:105) (bosonic systems) or from Lie algebra (fermionic sys-tems).The essence of recent progress on defining complexityin QFT using broadly defined approach of [27] lies inmaking educated choices for O I and | ψ R (cid:105) , which allowone to perform minimization encapsulated by Eq. (5). Inthe vast majority of cases, it was achieved by focusingon free QFTs and utilizing powerful toolkit of Gaussianstates and transformations [30].The discussion so far concerned pure states, i.e. , vonNeumann entropy as an entanglement measure betweena subregion and a complement in pure states and com-plexity as a way of quantifying hardness of preparationof pure states. Much less understood in the QFT contextare quantum information properties of mixed states andthe present paper concerns precisely this important sub-ject. Of interest to us will be entanglement of purification(EoP) [31] and complexity of purification (CoP) [32]. Wewill introduce these quantities in more detail in, respec-tively, sections V and VI. Here want to stress insteadthat the key motivating feature behind our work stemsfrom both of these quantities involving in their definitionscanning over purifications of mixed many-body states .Such purifications, i.e. , embedding a mixed state in anenlarged Hilbert space in which it arises as a reduced den-sity matrix, in the context we are interested in, i.e. , QFT Otherwise, EoP and CoP use regular notions of, respectively,entanglement entropy and pure state complexity, which is thereason why they already made an appearance in the text. physics, are clearly challenging to operate with. Earlierworks on EoP and CoP in high-energy physics includerespectively [33, 34] and [35] and focus on free QFTs inwhich mixed states of interest, such as vacuum reduceddensity matrices or thermal states, are Gaussian. Gaus-sian mixed states allow for purification to pure Gaussianstates, which underlied strategies employed in the afore-mentioned references. However, even purifications withinthe Gaussian manifold of states for large subsystems canbe challenging to operate with and the above works madeadditional choices in this respect.This is where the key novelty of our present work ap-pears, which is to consider the most general Gaussianpurifications. To this end, we will consider free QFTson a lattice and, whenever possible, encode reduced den-sity matrices in terms of corresponding quadratic corre-lations represented by covariance matrices. Consideringthe most general Gaussian purifications amounts then toembedding mixed state covariance matrices as parts oflarger covariance matrices corresponding to pure states.Utilizing efficient Gaussian techniques allows us to min-imize the two quantities of interest, EoP and CoP fora judicious choice of a definition of pure state complex-ity [29], over general purifications to a given number ofbosonic or fermionic modes.Our primary focus is on a particularly simple yet tellingsetup of two-intervals vacuum reduced density matricesin free QFTs with a vanishing or very small mass. Inquantum information context, such setup arose first instudies of mutual information (MI) defined using twosubsystems A and B as I ( A : B ) = S ( A ) + S ( B ) − S ( A ∪ B ) (7)in (1+1)-dimensional conformal field theories (CFTs),where A and B are two disjoint or adjacent intervals on aflat spatial slice as depicted by figure 1. MI will play animportant role in our studies providing us with a guid-ance regarding both the behaviour of EoP, as in [33], aswell as will help us to understand subtleties underlyingour models. Our studies will mostly concern scaling ofMI, EoP and MI with control parameters such as intervalsize, separation and, when present, system size and themass. While EoP turns out be such a ultraviolet finitequantity by itself, for CoP we will consider a combina-tion of single and two interval CoP results akin to (7) forwhich the leading ultraviolet divergences cancel and onlymilder divergences remain.Our paper is structured as follows. In section II, wereview the two models we consider, the Klein-Gordonfield in the massless limit and the critical transverse fieldIsing model, on a lattice paying a particular attention todescription of their ground states in terms of covariance One should also mention in this context [36], which, motivatedby holographic complexity proposals, explored properties of CoPin the setting of a single harmonic oscillator. dδ . . . . . . w B δw A δ . . . A subsystem B FIG. 1. The subsystem that defines reduced density matri-ces for our discretized bosonic and fermionic models in theirvacuum state consists of two intervals of a width of w A /δ and w B /δ sites and separated by a distance of d/δ sites, where δ is the lattice spacing. When d = 0, we will keep w A and w B generic. When d >
0, we will set for simplicity w A = w B ≡ w and the natural continuum combination is w/d . We will seethat numerically determined MI and EoP approach in thecontinuum limit functions of w/d . With CoP the situation ismore complicated, as it turns out to be ultraviolet divergentand brings in an additional dimensionful scale through theclass of reference states of interest | ψ R (cid:105) . matrices. In section III, we benchmark our abilities toreach continuum limit in lattice calculations by compar-ing the results of our numerics with existing analytic for-mulas for MI in the aforementioned two interval case. Insection IV, we discuss briefly the mathematics of purifi-cations of Gaussian states as seen by covariance matrices,which is the working horse behind most of the results re-ported in the present article. Subsequently, we use thismachinery to study EoP and CoP in the two-interval caseof figure 1, respectively, in sections V and VI. In sec-tion VII, we comment on two subtleties relevant for ourmodel, namely the zero mode when taking the masslesslimit for a bosonic theory and the different notions of lo-cality in the spin vs. fermion picture of the Ising model.We summarize our results and present an outlook in sec-tion VIII. We also provide an extensive appendix thatprovides further details regarding our methods. II. SETUP
In the present work we focus on two paradigmatic mod-els: the
Klein-Gordon field in the massless limit and the critical transverse field Ising model in 1+1 dimensions.For our numerical calculations, we discretize both theo-ries either on a lattice with N sites and periodic boundaryconditions( i.e. , we identify the sites N + j ≡ j ) or on aninfinite lattice. Both theories describe CFTs in the re-spective limits with central charge c = 1 (Klein-Gordon)and c = (Ising model). We will review the Hamil-tonians of both models and their ground states with aparticular focus on the covariance matrix formulation.The latter for free bosons will allow for an efficient cal-culation of EoP and CoP using Gaussian techniques. Forthe Ising model, we will discuss in detail how there aretwo distinct notions of locality associated to the spin andfermion formulation, respectively. Decompactified free boson CFT ( c = 1 ) Analytical predictions Gaussian numerics S ( A ) I ( A : B ) E P S ( A ) I ( A : B ) E P d = 0 c log w A w B ( w A + w B ) δ c log w A w B ( w A + w B ) δ [37] log w A w B ( w A + w B ) δ log w A w B ( w A + w B ) δ d (cid:28) w (cid:39) c log wδ [38] (cid:39) c log wd [39] (cid:39) c log wd [37] (cid:39) log wδ (cid:39) (cid:0) < . > . (cid:1) log wd (cid:39) log wd d (cid:29) w ∝ (cid:0) wd (cid:1) [40, 41] ∝ (cid:0) wd (cid:1) * < (cid:0) wd (cid:1) . < (cid:0) wd (cid:1) . Ising CFT ( c = ) Analytical predictions Gaussian numerics (fermions) S ( A ) I ( A : B ) E P S ( A ) I ( A : B ) E P d = 0 c log w A w B ( w A + w B ) δ c log w A w B ( w A + w B ) δ [37] log w A w B ( w A + w B ) δ log w A w B ( w A + w B ) δ d (cid:28) w (cid:39) c log wδ (cid:39) c log wd (cid:39) c log wd [37] (cid:39) log wδ (non-Gaussian setting) d (cid:29) w ∝ (cid:0) wd (cid:1) / [40, 41] ∝ (cid:0) wd (cid:1) / *TABLE I. Overview over known analytical results and numerical fits with approximate coefficients for entanglement entropy S ( A ), MI I ( A : B ), and EoP E P , all for an infinite-size system. Analytical entries marked with a star (*) are guesses based onanalogous behavior between MI and EoP. Twice the free fermion result in the d = 0 Ising is the c = 1 free Dirac fermion CFTquantity. CFT and complexity definition Single interval complexity / CoP Mutual complexity
Hologr. CFTs: subregion- C V ∝ wδ − π const.Hologr. CFTs: subregion- C V . ∝ wδ − (cid:0) wδ (cid:1) − π ∝ log w A w B ( w A + w B ) δ + π Hologr. CFTs: subregion- C A ∝ log (cid:16) (cid:96) CT L (cid:17) w δ − log (cid:16) (cid:96) CT L (cid:17) log (cid:0) wδ (cid:1) + π ∝ log (cid:16) (cid:96) CT L (cid:17) × log w A w B ( w A + w B ) δ − π Decomp. free boson CFT ( c = 1) (cid:16) f ( µ δ ) wδ + f ( mµ , µ δ ) log wδ + f ( mµ , µ δ ) (cid:17) f ( mµ , µ δ ) log w A w B ( w A + w B ) δ + f ( mµ , µ δ )Ising CFT ( c = ) (cid:0) . wδ + 0 . wδ + 0 . (cid:1) . w A w B ( w A + w B ) δ + 0 . L norm circuit complexity encapsulated by (59) and spatially disentangled reference states. The mutual complexityis defined differently for the holographic complexity proposals, see (49), and for our implementation of CoP, see (64). In thecase of the C A proposal, L is the AdS curvature radius and (cid:96) CT is an arbitrary length scale arising from counter-terms [45, 46].In the case of the bosonic calculation, µ is the reference state scale and functions f , f and f are defined in (63). A. Klein-Gordon field
We consider the well-known Klein-Gordon scalar fieldwith a mass m that we will later take to zero. Its dis-cretized Hamiltonian on a lattice with N sites isˆ H = δ N (cid:88) i =1 (cid:18) ˆ π i + m δ ˆ ϕ i + 1 δ ( ˆ ϕ i − ˆ ϕ i +1 ) (cid:19) , (8)where δ represents the lattice spacing. We thus have acircumference L = N δ. (9)We define canonical variablesˆ ξ ai ≡ ( ˆ ϕ i , ˆ π i ) , (10)where a = 1 ,
2. It is well-known that the Hamiltoniancan be diagonalized via Fourier transformations leading to N decoupled harmonic oscillators with frequencies ω k = (cid:114) m + 4 δ sin πkN . (11)The ground state | (cid:105) is Gaussian and fully characterizedby its covariance matrix G abij = (cid:104) | ˆ ξ ai ˆ ξ bj + ˆ ξ bj ˆ ξ ai | (cid:105) = N (cid:88) k =1 e i πkN ( i − j ) (cid:18) ω k ω k (cid:19) , (12)where a and b label the entries of the matrix. Continuumlimit on a circle requires taking N → ∞ keeping productof meaningful continuum quantities m L = m δ N fixed.Each value of this product corresponds to a different QFTas a continuum limit within the class of free Klein-Gordontheories. Furthermore, when considering subsystems, asdepicted in figure 1, continuum limit requires keepingratios of w δ and d δ to L fixed as N → ∞ . In practice,one takes N to be large but finite and requires that as N is increased well-defined quantities, for example theMI (7), stop changing significantly with N and stabilizein the vicinity of their QFT values.When w δL (cid:28) d δL (cid:28)
1, then the results of the cal-culations should be effectively indistinguishable from thesituation in which the spatial direction is a line. Themass m (cid:28) δ becomes then the only dimensionful pa-rameter in the continuum theory. Also, in this case thenumber k associated with discrete momenta in (11) getsincorporated into a continuum variable and a sum in (12)needs to be replaced by an appropriate integral, see forexample [29].We are particularly interested in the massless limit m →
0, where the Klein-Gordon field describes the CFTwith central charge c = 1. More precisely, the c = 1 CFTwith the periodic boundary conditions we imposed shouldbe regarded as a 1-parameter family of theories arisingin the path integral language the compactification of thebosonic field ϕ ( i.e. , periodically identified): ϕ + 2 πR ≡ ϕ. (13)The dimensionless parameter R is the radius of compact-ification in the field space and plays the role of a modulispecifying a particular c = 1 CFT. The scaling dimensionof the lightest operator is then given by∆ min = min (cid:18) R , R (cid:19) . (14)The above formula is a hint of an underlying dualitybetween theories with field complactification radia of R and R [47]. The massless limit of (8) corresponds tothe decompactification limit of compact free boson CFTs( R → ∞ ), which is a subtle limit since in light of (14) thegap in the operator spectrum approaches 0. While thislimit leads to correct correlation functions of vertex oper-ators or a single interval entanglement entropy, for otherquantities the situation is more complicated. In particu-lar, the modular invariant thermal partition function forthe free boson reads [47] Z mod − inv ∼ β/L ) / η ( i β/L ) , (15)whereas the free massive boson calculation for m L (cid:28) Z m L (cid:28) ∼ β m ) η ( i β/L ) . (16)In both expressions η is the Dedekind eta function definedas η ( i β/L ) = e − π β/L Π ∞ n =1 (1 − e − π n β/L ) . (17)The mismatch between the two calculations can be un-derstood using the representation of the partition func-tion on a circle as an Euclidean path integral on a torus. In the case of (15), the zero mode contribution is ne-glected, as its inclusion would lead to an infinite volumeterm coming from the integration over the field space.In the case of (15), the zero mode φ contribution to thepath integral is included and is finite, as it originates inthe path integral language from (cid:90) ∞−∞ dφ e − β L m φ ∼ m √ β L , (18)where the product β L is the torus spatial volume. Multi-plying the partition function (15) by the factor (18) leadsto (16), which explains the relation between the two par-tition functions. We will come back to these calculationsin section VII A, where we discuss the influence of thezero mode on MIdecay with separation between two in-tervals.In our studies, we will be using the free massive bosonsetup to extract the properties of the modular invariant c = 1 free boson CFT in the decompactification limit R → ∞ . From this perspective, the partition function ofinterest, i.e. , (15), can be indeed recovered from the mas-sive boson Gaussian calculation (16) by dividing it by theknown zero mode contribution (18). However, in the caseof other quantities calculated using Gaussian techniquesat non-vanishing mass the effect of the zero mode is notstraightforward to isolate. As we already mentioned, nu-merical studies showed that Gaussian calculations witha small mass reproduce the universal entanglement en-tropy result for a single interval [1]. Furthermore, onemay expect the two interval case at small separations tobe reliably described by the massive free boson calcula-tion, as the zero mode affects primarily the long distancephysics. As a result, these will be the mixed state setupsthat we will consider in our EoP and CoP explorations.On the other hand, the two interval case at large separa-tions is tricky and we will return to it in the case of MIin section VII A.Another subtlety that originates in the massless limitis that the ground state is only defined distribution-ally. The issue is best understood by diagonalizing theHamiltonian (8) by transforming to momentum modes˜ π k = √ N (cid:80) Nj =1 e − i πkjN π j and ˜ ϕ k = √ N (cid:80) Nj =1 e i πkjN ϕ j leading to ˆ H = 12 N (cid:88) k =1 (cid:16) δ | ˜ π k | + ω k δ | ˜ ϕ k | (cid:17) , (19)where we find N decoupled harmonic oscillators. Forthe oscillator with k = 0 (zero momentum mode), wehave ω = m , which vanishes in the massless limit. Con-sequently, the ground state of this mode approaches adelta distribution, which does not lie in Hilbert space.This leads to the divergence of certain terms in the co-variance matrix (12). However, we are still able to defineexpectation values of observables and entanglement mea-sures, such as the entanglement entropy, by computingthose quantities for finite m and generating numericallyresults for values of m gradually approaching 0. In sec-tion VII A, we will discuss the role of the zero mode forsuch calculations in more detail using MI as an example. B. Critical transverse field Ising model
We consider the transverse field Ising modelˆ H = − N (cid:88) i =1 (2 J ˆ S x i ˆ S x i +1 + h ˆ S z i ) (20)in the critical limit J = h , where ˆ S x , z i are spin- oper-ators in the direction x or z at position i in the chain, i.e. , related ˆ S x , z i = σ x , z i to the well-known Pauli matri-ces. The system consists of N spin- degrees of freedomarranged in a circle, i.e. , we assume periodic boundaryconditions with N + i ≡ i .The transverse field Ising model can be solved analyt-ically by employing the Jordan-Wigner transform [48], i.e. , eigenvalues and eigenvectors of the Hamiltonian ˆ H can be constructed in closed form. The transformation isbased on introducing fermionic creation and annihilationoperators ˆ f † i and ˆ f i . For the transformation, we write S ± i = S x i ± S y i as S + i = ˆ f † i exp − i π i − (cid:88) j =1 ˆ f † j ˆ f j , (21)which leads to the almost quadratic Hamiltonianˆ H = − N (cid:88) i =1 (cid:18) J (cid:104) ˆ f † i ( ˆ f i +1 + ˆ f † i +1 ) + h.c. (cid:105) + h ˆ f † i ˆ f i (cid:19) − J f † ( ˆ f N + ˆ f † N ) + h.c.]( ˆ P + 1) + h N , (22)where h. c. stands form Hermitian conjugation and ˆ P =exp (i π (cid:80) Nj =1 ˆ f † j ˆ f j ) is the parity operator.In this picture, the operators ˆ f † i and ˆ f i are fermioniccreation and annihilation operators, but with a differ-ent notion of locality than the spin operators appearingin (20). From the Jordan-Wigner transformation (21), itis clear that the fermionic operator on site i are local tothe whole region from site 1 to i in the fermionic picture,and vice versa. This ensures that bipartite entanglementof a connected region of sites is equivalent in the spinand the fermionic picture, because we can use transla-tional invariance to identify this region with the sites { , . . . , w } , for which the spin and the fermionic picturesare isomorphic, i.e. , the density operators are unitarilyequivalent leading to the same entanglement entropy.It is well-known [49] that the fermionic Hamilto-nian (22) can be written as a sum of two quadratic Hamil-tonians ˆ H ± of the formˆ H = ˆ H + (cid:80) + + ˆ H − (cid:80) − , (23) where (cid:80) ± represent orthogonal projectors onto theHilbert subspaces H ± of even and odd number of ex-citations, respectively, i.e. , H ± = span (cid:110) | n , . . . , n L (cid:105) (cid:12)(cid:12) e i π (cid:80) Ni =1 n i = ± (cid:111) (24)with ˆ f † i ˆ f i | n , . . . , n n (cid:105) = n i | n , . . . , n n (cid:105) . We can equiv-alently describe the Hamiltonian in terms of Majoronamodes ˆ ξ ai ≡ (ˆ q i , ˆ p i ) withˆ q i := 1 √ f † i + ˆ f i ) and ˆ p i := i √ f † i − ˆ f i ) , (25)which leads to the Hamiltonian given byˆ H = i N (cid:88) i =1 ( J ˆ p i ˆ q i +1 − h ˆ q i ˆ p i ) − hN − J ˆ p ˆ q N ( ˆ P + 1) . (26)If we define Majorana operators γ i − := ˆ q i and γ i := ˆ p i , (27)we can write the critical ( h = J ) model asˆ H ± = i J (cid:32) N − (cid:88) i =1 γ i γ i +1 ± γ γ N (cid:33) − JN . (28)This demonstrates explicitly that the two quadraticHamiltonians ˆ H ± in the fermionic picture only differby the type of boundary conditions in the two sectors,namely anti-periodic boundary conditions for ˆ H + andperiodic boundary conditions for ˆ H − .We are particularly interested in the ground state | (cid:105) of the critical model, which is completely characterizedby its covariance matrixΩ abij = (cid:104) | ˆ ξ ai ˆ ξ bj − ˆ ξ bj ˆ ξ ai | (cid:105) = (cid:88) κ ∈K + c κ ( i − j ) (cid:18) − (cid:19) , (29)where the canonical variables ˆ ξ ai are now defined us-ing (27) as ˆ ξ ai ≡ (ˆ q i , ˆ p i ) , (30) κ = πN (2 k + 1) with k = 0 , . . . , N − c κ ( j ) are given by c κ ( j ) = | cos κ | cos( κj ) − sin κ sin( κj )2 | cos κ | (31)for N being even .An important subtlety arises if we compute bipartiteentanglement of disconnected regions, because in this The formula for c κ ( j ) must be slightly adjusted for κ = π , whichoccurs for odd N , as explained in appendix A. case the entanglement entropies are different for the spinvs. fermion picture. This subtle fact has been recognizednumerous times in the literature [50–52] and plays animportant role when relating the lattice model with thecontinuum CFT. The key observation is that the canon-ical anticommutation relations induce a different notionof tensor product and partial trace for fermions [53]. In-terestingly, this different notion only affects the bipartiteentanglement entropy of disjoint regions, i.e. , the reducedstate in a subsystem consisting of two non-adjacent in-tervals on the circle will be different if we compute itusing the spin vs. fermion picture. We comment on thisin section VII B and review the respective literature inappendix B. In practical terms, this fact will lead us toapply our Gaussian numerics based on purifications onlyto the case when the two intervals are adjacent, i.e. , d = 0in figure 1.Finally, note that the c = 1 free Dirac fermion CFT canbe obtained from two copies of Ising model ( i.e. , Majo-rana fermion CFT) by imposing a different GSO projec-tion [47]. As a result, the discussion in the present sectionabout spatial locality and Gaussianity applies also to thefree Dirac fermion CFT. In effect, our fermionic Gaus-sian methods reproduce the properties of the free Diracfermion CFT only for a single subregion or adjacent sub-regions and the answers in these cases are simply given bytwice the answers for the corresponding Ising calculation.It is well-known that the Dirac fermion CFT is equivalentto the free compactified boson CFT at the compactifica-tion radius R = 1 (or equally R = 2) via the bosonizationprocedure [54, 55]. Let us also emphasize on this occasionthat modular invariance is a property that is not alwaysimposed in free fermion calculation available in the liter-ature (see [56] for a discussion of the modular invariancein the context of entanglement entropy in CFTs). Thissometimes leads to apparent tensions between CFT ex-pectations and free fermion results. We will come backto this point in the next section. III. MUTUAL INFORMATION
MI defined in (7) provides an important correlationmeasure between two subsystems A and B and belowwe summarize some of its properties. One reason to dothis is to test our ability to reproduce them using ournumerics before we apply it to a much less understoodcase of EoP and CoP. Another is to explore what kind ofbehaviour to expect from EoP and CoP.MI is generically a non-universal quantity in CFTs, asit is related to a four-point function of twist operatorsand the latter is spectrum-dependent [39].At large distances between the intervals, i.e. for d (cid:29) w in the notation of figure 1, the operator product expan-sion analysis predicts the following behaviour of MI I ( A : B ) ∼ |(cid:104) O min A O min B (cid:105)| ∼ (cid:16) wd (cid:17) − min , (32) where O min is the operator with lowest (but non-zero)conformal dimension and ∆ min = h min + ¯ h min [40, 41].At short separations, d (cid:28) w , one expects the followinguniversal result [39, 57] I ( A : B ) (cid:39) c w d . (33)For d = 0, one can use a universal, i.e. , only c -dependent,formula for a single interval entanglement entropy in thevacuum to arrive at a variant of (33) with d = δ . In ta-ble I we provided the form of (33) when the two intervalshave arbitrary lengths.Moving on to the two models we consider, for the freemassless scalar QFT one has continuous and gapless spec-trum of primary operators. As a result, the formula (32)does not apply as such and in section VII A we commenton a possible generalization. However, for a compacti-fied free boson CFT, see (13), the spectrum of operatorsdevelops a gap (14) and calculations of entanglement en-tropy in [58] do reproduce this behaviour.For c = 1 / d (cid:29) w , we expect I ( A : B ) ∼ ( w/d ) / , because O min is the spin operator σ ∼ ˆ S x i , see (20), which has h = ¯ h = , i.e. , ∆ min = .Furthermore, as the free Dirac fermion CFT is dual tothe free compact boson theory at R = 1 (or, equivalently, R = 2), according to (14) one expect MI to decay as 1 /d governed by the h = ¯ h = operator. Such an opera-tor would be natural to interpret as a product of two σ operators from each underlying Majorana fermion model.Let us also note that existence of the following formulafor MI for Dirac fermions [59] I ( A : B ) = 13 log ( d + w ) d (2 w + d ) (34)While this formula agrees at short distances with (33), atlarge distances it falls off as 1 /d rather than the afore-mentioned 1 /d predicted by bosonization. This is relatedto the fact that the calculation in [59] utilizes torus parti-tion function with the anti-periodic boundary conditionfor fermions. However, the modular invariant partitionfunction leading to (32) includes also contributions fromsectors in which fermions satisfy period boundary con-ditions. This is directly related to the discussion aboutreduced density matrices in the fermionic formulation ofthe Ising model mentioned in section II B and expandedlater in section VII B and appendix B.Having discussed analytic expectations, let us showhow our bosonic and fermionic Gaussian states numericsreproduces it. This should be regarded as a cross-checkof both our numerical lattice setup and its ability to re-produce features of the continuum limit. Also, it will il-lustrate to what extent considering a decompactified freeboson with the zero mode regulated via non-vanishingmass capture long ( d (cid:29) w ) and short distance ( d (cid:28) w )CFT expectations.First, we consider the behavior of MI for a free bosonicfield with central charge c = 1, shown in the first rowof figure 2. As anticipated in section II A, employingGaussian methods restricts us to the decompactified freeboson with non-vanishing mass.In the limit of small d/w , we find a logarithmic depen-dence similar to (33), specifically I ( A : B, w (cid:29) d ) = a + a log wd −
12 log(
L m ) . (35)Note the logarithmic Lm dependence already observedin [34]. The coefficient a , expected to be c/ ≡ / w and the size L of the periodic system are in-creased. However, we can bound it by considering thebehavior of I ( A : B ) and S ( A ∪ B ) separately. Estimat-ing a by a discrete derivative with respect to log( w/d ),we find that this estimate approaches 1 / I ( A : B ) data and from below for S ( A ∪ B ), as shownin the top-left corner of figure 2, suggesting an asymp-totic ∝ / w behavior identical to that of S ( A ). Ourdata, extending until N = 32000 and w = 2000 δ , yieldsa bound 0 . (cid:46) a (cid:46) .
40, consistent with our expecta-tions. All shown data uses a distance dδ = 1 of one latticesite, as lattice effects on the value of the a estimate werefound to be negligible.The behaviour at large d/w can be anticipated to besubtle in light of (32) as in the present case ∆ min → I ( A : B, w (cid:28) d ) = b + b (cid:16) wd (cid:17) b −
12 log( Lm ) , (36)one would expect b to vanish. The power b can beestimated by discretizing the derivatives in the expression b = 1 + dd log( w/d ) log d I ( A : B )d( w/d ) . (37)Indeed, we find its estimated value to gradually decreaseat large d/w as N is increased, though MI can be wellapproximated by a power law with b ≈ . < d/w < d/w <
50 range [34]. In the d/w → ∞ limit, we canbound b (cid:46) .
15. The coefficient of a potential logarith-mic growth I ( A : B ) ∼ b (cid:48) log( w/d ) in this limit can bebounded as b (cid:48) (cid:46) .
06. While these results are obtainedon a circle and extrapolated to a line, in section VII Awe will discuss the large-distance behavior of free bosondirectly on a line, where we will also consider possiblesub-logarithmic decay functions at large d/w . Such func-tional dependencies, resulting from subtle large distancebehavior for free, nearly massless bosons, will reappearin the context of EoP studies in section V.As we can only study the Ising CFT via a Gaussianfermionic model under the Jordan-Wigner transforma-tion for adjacent intervals, MI computed in this approachis only relevant for the d = 0 case, which which followsdirectly from the entanglement entropy formula for a sin-gle interval. These formulas are included in table I forcompleteness. IV. GAUSSIAN PURIFICATIONS
As discussed in section II, we focus on free theories astheir ground states are Gaussian states, so we have a pow-erful machinery at our disposal to analytically computethe entanglement entropy and other quantities analyti-cally from the covariance matrix of pure Gaussian states(see appendix C). Similar analytical formulas exist alsofor the L circuit complexity of interest. The primarygoal of this paper is to use this machinery to define andcompute similar quantities, such as entanglement entropyand complexity, for mixed states leading to the notions ofEoP [60] and CoP [32]. They are defined in the followingway:1. We start with a function f ( | ψ (cid:105) ) that is defined forarbitrary pure states | ψ (cid:105) .2. For a mixed state ρ A ∈ H A , we construct the purifi-cation | ψ (cid:105) on a larger Hilbert space H = H A ⊗H A (cid:48) ,such that ρ A = Tr H A (cid:48) | ψ (cid:105) (cid:104) ψ | . Of course, thereis quite some freedom of how large the purifyingHilbert space H A (cid:48) can be.3. The purification | ψ (cid:105) is not unique, but if we havefound one purification | ψ (cid:105) , we can construct anyother purification by acting with a unitary U = (cid:49) A ⊗ U A (cid:48) , where U A (cid:48) is an arbitrary unitary on thepurifying Hilbert space H A (cid:48) .4. We then define a new function F ( ρ A ) for the mixedstate to be given by F ( ρ A ) = min U = (cid:49) A ⊗ U B f ( U | ψ (cid:105) ) , (38) i.e. , we minimize the original quantity f ( | ψ (cid:105) ) de-fined for pure states over all purifications of themixed state ρ A .Note that there are some subtleties related to the factthat the purifying Hilbert space H A (cid:48) may not have a di-rect physical interpretation, e.g. , if H A represents a localsubsystem (region in real space) of a QFT, it is a priorinot clear what the physical meaning of H A (cid:48) is. Conse-quently, the function f needs to be defined in an appro-priate way that it can be meaningfully applied to arbi-trary extended Hilbert spaces H = H A ⊗H A (cid:48) . While thisis relatively straightforward for entanglement entropy,one needs to be a bit careful about circuit complexitythat is usually defined with respect to a reference statethat is chosen as spatially unenentagled with respect toa physical notion of locality. As explained in (51), onecan show that this can also be done for the purifyingHilbert space H A (cid:48) in such a way that the resulting CoPis actually independent of the notion of locality or, putdifferently, the outcome of the minimization procedurecan even be understood as equipping H A (cid:48) with a notionof locality.While both EoP and CoP, have been introduced pre-viously, their efficient evaluation has been an ongoing Large block width w , small distance d Small block width w , large distance d B o s o n i c M I ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ ●● ● d = δ , N = ■ d = δ , N = ◆ d = δ , N = ▲ d = δ , N = ● Expected limit d / w Δ S A ⋃ B / Δ l o g ( w / d ) Logarithmic coefficient, S A ⋃ B ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ ●● ● d = δ , N = ■ d = δ , N = ◆ d = δ , N = ▲ d = δ , N = ● Expected limit d / w Δ I ( A : B ) / Δ l o g ( w / d ) Logarithmic coefficient, I ( A:B ) ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ ●● ● w = δ , N = ■ w = δ , N = ◆ w = δ , N = ▲ w = δ , N = ● Expected limit w / d P o w e r e s t i m a t e b Decay power ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ ● w = δ , N = ■ w = δ , N = ◆ w = δ , N = ▲ w = δ , N = w / d Δ I ( A : B ) / Δ l o g ( w / d ) Logarithmic coefficient B o s o n i c E o P ●●●●●●●●●●● ■■■■■■■■■■■ ◆◆◆◆◆◆◆◆◆◆◆ ▲▲▲▲▲▲▲▲▲▲ ●● ● d = δ , N = ■ d = δ , N = ◆ d = δ , N = ▲ d = δ , N = ● Expected limit d / w Δ E P / Δ l o g ( w / d ) Logarithmic coefficient ●●●●●●●●●●●●●●●●●●●●● ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ ●● ● w = δ , N = ■ w = δ , N = ◆ w = δ , N = ▲ w = δ , N = ● Expected limit w / d P o w e r e s t i m a t e d Decay power ●●●●●●●●●●●●●●●●●●●●●●●●●●● ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ ● w = δ , N = ■ w = δ , N = ◆ w = δ , N = ▲ w = δ , N = w / d Δ E P / Δ l o g ( w / d ) Logarithmic coefficient
FIG. 2. Numerical data for MI and EoP in the regimes of small and large ratio of block width w to distance d on a periodicsystem of N sites of bosons.Expected limits and numerical estimates are in table I. The bosonic scale mL = mNδ is set to10 − for MI and 10 − for EoP, as the MI computation is more stable at small values of mL . problem for practical applications. The reason is that therequired minimization procedure must be generally per-formed numerically, while the dimension of the respectivemanifold over which one needs to optimize grows quicklywith the number of degrees of freedom. Therefore, EoPhas been only studied for small systems and often onlywith respect to certain subfamilies of states, while CoPwas exclusively studied via purifying individual degreesof freedom [35] rather than directly larger subsystems.The key ingredient that enables the progress of thepresent paper is that we can efficiently compute EoP andCoP for the family of Gaussian states. For this, we startwith a Gaussian mixed state ρ G A and compute a Gaussianpurification | ψ G (cid:105) . When performing our minimizationalgorithm, we only sample over Gaussian states, i.e. , wedefine the new function F G ( ρ G A ) = min U G = (cid:49) A ⊗ U G A (cid:48) f ( U G | ψ G (cid:105) ) . (39)Clearly, we must have F ( ρ G A ) ≤ F G ( ρ G A ), i.e. , F G ( ρ G A )is an upper bound for the true minimum. Moreover,it is reasonable to assume that for many quantities,such as EoP and CoP, we actually have the equality F ( ρ G A ) = F G ( ρ G A ). This was already conjectured in [33]and is further supported by [61]. In the case of CoP, thereis still limited progress in even defining circuit complexityfor non-Gaussian states, which means that it is naturalto only consider F G ( ρ G A ) to start with. In both cases,it is therefore a meaningful restriction to only considerGaussian purifications of Gaussian states.For Gaussian states, we can use the covariance matrix and linear complex structure formalism as explained inappendix C. Rather than working with Hilbert space vec-tors, which would live in an infinite dimensional Hilbertspace for bosons and a 2 N A -dimensional Hilbert space forfermions, we can fully characterize the Gaussian state bya 2 N A -by-2 N A dimensional matrix, where N A representsthe number of bosonic or fermionic degrees of freedom.We restrict to Gaussian states with z a = tr( ˆ ξ a ρ G A ), forwhich all relevant information is encoded in the so calledrestricted complex structure J A defined in (C7). For amixed Gaussian state J A has purely imaginary eigenval-ues ± i c i , where c i ∈ [1 , ∞ ) for bosons and c i ∈ [0 ,
1] forfermions. Only if all c i = 1, we have a pure state. Forevery mixed state ρ G A J A ≡ (cid:67) (cid:67) N A with (cid:67) i = c i (cid:18) − (cid:19) . (40)We can always purify such a state using a Hilbertspace H A (cid:48) with the same number of degrees of freedomas H A , i.e. , N A (cid:48) = N A . Then, there always exists a basisin the system A (cid:48) , such that the complex structure J ofthe purified state | ψ G (cid:105) takes the form [62] J ≡ (cid:67) (cid:83) (cid:67) N A (cid:83) N A ± (cid:83) (cid:67) ± (cid:83) N A (cid:67) N A , (41)0where (+) applies to bosons and ( − ) to fermions with (cid:67) i = c i (cid:18) (cid:19) and (cid:83) i = s i (cid:18) (cid:19) (42)and s i = (cid:112) c i − s i = (cid:112) − c i forfermions . From the perspective of Gaussian states, dif-ferent purifications of ρ G A only differ in the choice ofbasis of the purifying system B , for which J takes theabove standard form. Consequently, we can use the ac-tion of the respective Lie group G B (symplectic groupSp(2 N A (cid:48) , (cid:82) ) for bosons, orthogonal group O(2 N A (cid:48) , (cid:82) )for fermions) to transform J → M JM − with M = (cid:49) A ⊕ M A (cid:48) , where M A (cid:48) ∈ G A (cid:48) represented as 2 N A (cid:48) -by-2 N A (cid:48) matrix.As reviewed in appendix D, we optimize over all Gaus-sian purification by taking the natural geometry (Fubini-Study metric) of the state manifold into account. Us-ing the fact that this geometry is compatible with thegroup action, i.e. , the Fubini-Study metric on the mani-fold of purifications is left-invariant under the group ac-tion of G A (cid:48) , we do not need to recompute the metric atevery step, but can fix an orthonormal basis of Lie alge-bra generators equal to the dimension of the manifold.This enables us to efficiently perform a gradient descentsearch attuned the geometry of states, which scales poly-nomially in the number of degrees of freedom and en-ables us probe the field theory regime of our discretizedmodels, which has not been possible previously in thissetting. In particular, previous studies [33, 34] of EoPrestricted to special classes of Gaussian states (namely,real Gaussian wave functions generated by the subgroupGL( N A (cid:48) , (cid:82) ) ⊂ Sp(2 N A (cid:48) , (cid:82) )) for a small number of de-grees of freedom. Similarly, CoP has been almost exclu-sively studied by purifying individual degrees of freedom(mode-by-mode purifications [35]) rather than whole sub-systems for larger N A .For purifications of small subsystems, e.g. of 1 + 1 or2 + 2 sites, this optimization only takes a few secondson a desktop computer, and is still feasible within a fewhours for 10 + 10 sites, with efficiency depending on theoptimization function, the accuracy threshold, and thehardware on which the computation is performed. Forthe particular case of CoP the optimization procedurefor bosons was found to be an order of magnitude fasterthan the fermionic case for the same accuracy threshold,even for small subsystems. This implied that for largersubsystems, e.g. , of order of 10+10 sites, the optimizationparameters such as the gradient and function tolerancewere lowered without compromising the results. For in-stance, lowering the gradient and function tolerance bya couple of orders of magnitude resulted in changes inthe final value of the optimization in the third or fourthdecimal. An equivalent parametrization is given by c i = cosh 2 r i and s i =sinh 2 r i for bosons and c i = cos 2 r i and s i = sin 2 r i for fermions,as used in [61, 62]. V. ENTANGLEMENT OF PURIFICATION
We discuss our results for the EoP in bosonic andfermionic field theories using the purifications discussedin the previous section the algorithm described in ap-pendix D.
A. Definition and existing results
EoP is a measure of correlations, which include bothclassical and quantum ones, and can be regarded as amixed state generalization of entanglement entropy [60].When a mixed state ρ AB : H AB → H AB is given, wefirst purify it into a pure state | ψ (cid:105) ∈ H by extending theHilbert space H AB according to H AB = H A ⊗ H B → H = H A ⊗ H B ⊗ H A (cid:48) ⊗ H B (cid:48) (43)such that ρ AB = Tr A (cid:48) B (cid:48) ( | ψ (cid:105) (cid:104) ψ | ). The EoP E P ( ρ AB )is defined as the minimum of the entanglement entropy S ( A ∪ A (cid:48) ) = − Tr( ρ AA (cid:48) log ρ AA (cid:48) ) for the reduced densitymatrix ρ AA (cid:48) = Tr BB (cid:48) ( | ψ (cid:105) (cid:104) ψ | ) over all possible purifica-tions E P ( ρ AB ) = min | ψ (cid:105) [ S ( A ∪ A (cid:48) )] . (44)When ρ AB is pure, it simply reduces to the entanglemententropy as E P ( ρ AB ) = S ( A ) = S ( B ).EoP is relatively new to the QFT setting and its un-derstanding in this context is in development, whichadds a strong motivation for our paper. Our knowledgeabout this subject is based on a conjecture in hologra-phy, results governed by local conformal transformationsin CFTs and ab initio studies in free QFTs, which is theresearch direction the present work subscribes to. Belowwe briefly summarize the state of the art that sets thestage for the results of our research.In strongly-coupled CFTs, a holographic formulawhich computes EoP was proposed in [31, 63]. Analyti-cal calculations of EoP, based on the idea of path-integraloptimization for CFTs [64], were given in [37]. In par-ticular, when the subsystem A and B are adjacent in aCFT, both holographic and path-integral result predictthe universal formula E P = c w A w B ( w A + w B ) δ , (45)where the widths of A and B are w A and w B , respec-tively and δ is the lattice spacing. Exploratory numericalcalculations of EoP in a lattice regularization of (1+1)-dimensional free scalar field theory have been performedin [33, 34]. Below we would like to extend such compu-tations so that we can compare the result (45) with ourdiscretized numerical calculations, as well as understandbetter the long distance physics ( d (cid:29) w ) in the QFTlimit. The key technical difference on this front with1 ● ● ● ● ● ● ● ● ● ● ● ≃ log w A w B ( w A + w B ) δ w A / ( w A + w B ) B o s o n i c E P ● ● ● ● ● ● ● ● ● ● ● ≃ log w A w B ( w A + w B ) δ w A / ( w A + w B ) B o s o n i c E P FIG. 3. Bosonic ( c = 1, left) and fermionic/Ising spin EoP( c = , right) for two adjacent ( d = 0) subsystems A and B on w A + w B δ = 12 sites, with the continuum result (45) fora fitted lattice spacing (cid:15) plotted as a dashed curve. Totalsystem size N = 1200. Bosonic mass scale m L = 10 − . respect to [33, 34] is using bigger total system sizes, sig-nificantly bigger subsystems – both of which are desiredto be closer to the QFT limit – and the most generalGaussian purifications discussed in section IV. B. Numerical studies using the most generalGaussian purifications
Using the approach outlined in section IV and numeri-cal techniques explained in appendix D, we can now com-pute both bosonic and fermionic EoP for purifications onthe whole Gaussian manifold. In light of the discussion ofour models in section II, for bosons we expect the Gaus-sian ansatz to describe well the CFT properties when thetwo intervals are adjacent ( d = 0) or at small separation d (cid:28) w , as in these cases we do not expect the zero modeto be a significant contribution to the calculations weperform. For fermions, we expect the Gaussian ansatzto be appropriate for CFT calculations only when thetwo intervals are adjacent. Otherwise, the desired start-ing point of our calculations, spatially reduced densitymatrices for the Ising and Dirac fermion CFTs are non-Gaussian and our method is not applicable. In order tocomplete the picture, we will nevertheless provide resultsof our methods for bosons and fermions in the aforemen-tioned regimes, however, they are not supposed to beseen as CFT predictions based on lattice calculations.Starting with the adjacent intervals, our Gaussian lat-tice calculations perfectly reproduce the behavior (45) asshown in figure 3, in both our bosonic and fermionic (orequivalently, Ising spin) computations up to slight latticeeffects at small w A or w B .We move on to a more general case where the subsys-tem A and B are disjoint intervals in a free CFT. Weagain take the lengths of both intervals to be w and thedistance between them to be d . When d (cid:28) w , both holo-graphic [31, 63] and path-integral approaches [37] predictthe behavior E P = c wd , (46) which agrees with (45) under the replacement δ = d, w A = w B = w . On the other hand, no universal re-sults have been known for d (cid:29) w and one possibility is abehavior similar to MI described in section III.The numerical results for nearly a massless free scalarQFT are plotted in the second row of figure 2. As ex-pected, we find a logarithmic dependence on w/d whenit becomes large, given by E P ( w (cid:29) d ) = c + c log wd −
12 log( Lm ) , (47)with a convergence to c ≈ much faster than seen inMI. This result is consistent with the ∝ c log wd behaviorof (46).In the case of small w/d , we observe that the bosonicEoP behaves extremely similar to bosonic MI. Such anobservation for smaller subsystems and separations wasalready made in [34] and our results should be seen as acorroboration of this earlier finding. Given this similarityand our discussion of MI in section III, it should not comeas a surprise that a power-law fit to the bosonic EoP inthe regime of small w/d , E P ( w (cid:28) d ) = d + d (cid:16) wd (cid:17) d −
12 log( Lm ) , (48)is unstable as w/d →
0. The best we could do is to pro-vide the upper bound on the power, d (cid:46) .
15, which isconsistent with the absence of a long-distance power be-haviour. One should note that the power of such a quasi-power-law for EoP agrees well with the one extracted forMI, as can be seen by comparing the two rows of fig-ure 2 (right).
VI. COMPLEXITY OF PURIFICATION
In the present section we provide a comprehensive dis-cussion of CoP in the single and two adjacent intervalcase ( d = 0 in figure 1). We start by briefly reviewingthe relevant results of the holographic complexity pro-posals, as well as the studies of pure state complexity infree QFTs. These results will guide us in the choice ofa reference state and, also, in choosing the way to com-bine two and single interval CoPs to get a complexityanalogue of MI. Subsequently, we discuss the CoP re-sults obtained via optimization over the whole Gaussianmanifold. We focus on the single and adjacent intervalsto provide a clean message and we hope to report the d -dependence of CoP, which at least superficially seemsinvolved, in further work. We also compare some of ourresults with a simplified version of a single mode purifi-cations adopted in an earlier study of single interval CoPby [35] to avoid the technical problem our work addresses, i.e. , optimizing over the full manifold of Gaussian purifi-cations. Finally, we compare the properties of CoP withthe notion of mixed state complexity discussed in [65].2 A. Holographic predictions
Holographic complexity proposals relate novel grav-itational observables associated with picking a timeslice on the asymptotic boundary of solutions of AdSgravity with measures of hardness of preparing corre-sponding pure states in dual QFTs using limited re-sources. The first covariant notion is the spatial vol-ume of the boundary-anchored extremal (codimension-one) bulk time slice ( C V ) [17]. The second covariant no-tion is the spacetime volume ( i.e. , a codimension-zeroquantity) of the bulk causal development of such a timeslice ( C V . ) [20]. The third covariant notion is also of acodimension-zero type and is the bulk action evaluatedin the causally defined region ( C A ) [18, 19]. The firsttwo quantities are unique up to an overall normalization,whereas C A has an additional ambiguity related to thepresence of null boundaries [45, 46].While there is also another evidence in support of theassociation of C V , C V . and C A with complexity, an im-portant clue about the correctness of these conjecturescomes from free CFT calculations of complexity of purestates along the lines of [22, 23]. In particular, such freeCFT calculations are able to match the structure of lead-ing divergences of holographic complexity [22, 23, 28, 66]provided the reference state is taken to be a spatiallydisentangled state. Interestingly, in the case of bosoniccalculations of [22, 23] the scale entering the definition ofa spatially disentangled reference state can be linked, viathe leading divergence, both to the overall normalizationfreedom in the case of all three proposals, as well as to anadditional ambiguity appearing in the C A case. Further-more, the free boson CFT calculation in [29] explainedqualitative features of the holographic complexity excessin thermofield double states as compared to the vacuumcomplexity reported in [67].All three holographic complexity proposals acquirenatural generalizations for mixed states represented asspatial subregions of globally pure states [68–70]. In-stead of considering extremal volumes or causal devel-opments of a full Cauchy slice in the bulk, the mixedstate version of holographic complexity proposals usesthe corresponding notions applied to the relevant entan-glement wedge [71–73]. While there are certainly otherpossibilities regarding the kind of complexity the propos-als [68–70] represent, see [32] for a discussion of some ofthe available options, we will treat their properties as aguiding principle to study CoP in free CFTs.The results of these proposals applied to a single andtwo interval cases of interest can be found in table II.One can clearly see that the leading divergence of holo-graphic complexity is in the volume of the combined sub-regions and there can be also subleading logarithmic di-vergences . An earlier study of divergences encountered Note also that taking (cid:96) CT ∼ δ can enhance the leading diver- in the case of a single interval CoP in the vacuum of afree boson theory using restricted purifications is [35]. Inthe present work we lift the restriction on purificationswithin the Gaussian manifold of states, include also thecorresponding results for fermions and carefully resolvefinite contributions to CoP including their dependenceon the reference state scale and residual mass for bosons.The latter we achieve by considering the two adjacentintervals case.For two intervals it is interesting to define a better be-haved (less divergent) quantity in a manner similar to thedefinition of MI (7). Led by the form of leading diver-gences, as well as simplicity, the mutual complexity ∆ C was defined in [74] as the sum of contributions for eachindividual intervals and subtract from it the holographiccomplexity of the union∆ C = C ( A ) + C ( B ) − C ( A ∪ B ) . (49)The results in the two intervals setup at a vanishing sepa-rations are included in table II and motivated us to seekfor a logarithmic behaviour as a function of w A w B ( w A + w B ) δ also in the analogous setting in free CFTs. This is alsoreminiscent of the behaviour of MI and EoP at d = 0,see table I. B. Definition and implementation
CoP is defined in analogy with EoP as a measure ofcomplexity for mixed states with the use of a definitionof complexity for pure states minimized with respect toall purifications [32, 75]. This includes, in principle, pu-rifications which contain an arbitrary number of ancillagreater or equal to number of the degrees of freedom inthe subsystem.Given a mixed state in a Hilbert Space H A charac-terized by a density matrix ρ A , we define a new Hilbertspace H (cid:48) = H A ⊗ H A (cid:48) (50)with ancillary system A (cid:48) . There exist many purifications | ψ T (cid:105) ∈ H (cid:48) , such that ρ A = Tr H A (cid:48) ( | ψ T (cid:105) (cid:104) ψ T | ). In anal-ogy to the EoP, see (44), we define the CoP C P as theminimum of the complexity function C with respect to areference state | ψ R (cid:105) and to all purifications | ψ T (cid:105) : C P ( ρ A ) = min | ψ T (cid:105)∈H (cid:48) C ( | ψ T (cid:105) , | ψ R (cid:105) ) . (51)CoP inherits the richness of building blocks of complex-ity for pure states, such as dependence on the choice of areference state | ψ R (cid:105) as well as on the cost function whichevaluates the circuits built from the unitaries generated gence in the C A case by a logarithm of the cut-off and changesubleading divergence. | J (cid:105) can be efficiently charac-terized by their linear complex structure J ab . The lat-ter can be constructed from their two-point function C ab = (cid:104) ˆ ξ a ˆ ξ b (cid:105) , where we only consider Gaussian stateswith (cid:104) ˆ ξ a (cid:105) = 0. As shown in [28, 29] the geodesic distancebetween | J R (cid:105) and | J T (cid:105) within the Gaussian state mani-fold gives rise to a version of complexity based on a L cost function C ( | J T (cid:105) , | J R (cid:105) ) = (cid:115) tr (log ( − J T J R )) . (52)To relate with the discussion in the introduction, theabove definition of complexity corresponds to optimiza-tion with respect to the L cost function (6) with η IJ = 14 tr( K I G K (cid:124) J G − ) (53)where G ab = (cid:104) J R | ˆ ξ a ˆ ξ b + ˆ ξ b ˆ ξ a | J R (cid:105) for the reference state | ψ R (cid:105) . Due to the canonical anti-commutation relations,this normalization of the Lie algebra elements K I is in-dependent of the reference state for fermions (see [28]),but for bosons (53) implies that K I is normalized basedon the specific reference state, which was also referredto as equating reference and gate scale (as discussedin [22, 23]). In the above expression K I are the respectiveLie algebra elements (symplectic for bosons, orthogonalfor fermions) associated to the quadratic operators O I intheir fundamental representation acting on the classicalphase space.In the following, we focus on minimal purifications, i.e. , , purifications whose ancilla have the same numberof degrees of freedom as the reduced density matrix ofthe subsystem. Our focus on minimality comes as a re-sult of a number of numerical computations for the costfunction (52) which indicate that purifying the reduceddensity matrix with a larger number of ancilla does not lead to a lower CoP. It would be very interesting to ex-plore if this feature is special to the cost function andthe resulting complexity (52) we considered, however, weleave it for future investigations.When applying the closed form complexity for-mula (52) to Gaussian purifications | J T (cid:105) of some mixedstate ρ A , we need to think about what an appropriatereference state | J R (cid:105) can be. The two most immediateapplications are thermal states and mixed states result-ing from the reduction to spatial subsystem: • Thermal states.
Our mixed state could be thethermal state ρ in a system H =: H A , which wepurify to H (cid:48) = H A ⊗ H A . Here, we can alwayschoose a spatially unentangled and pure referencestate, which we can extend to the purifying systemas | J R (cid:105) = | J R (cid:105) A ⊗ | J R (cid:105) A ∈ H (cid:48) . • Subsystems.
We consider a pure Gaussian state | ψ (cid:105) ∈ H = H A ⊗H B , which we reduce to some localsubsystem ρ A = Tr H B | ψ (cid:105) (cid:104) ψ | . In this subsystem,we have a pure and spatially unentangled Gaussianreference state | J R (cid:105) A , which we can extend to thepurifying system as | J R (cid:105) = | J R (cid:105) A ⊗ | J R (cid:105) A ∈ H (cid:48) .The spatially unentangled character of | J R (cid:105) is a choicemotivated by the fact that such a state is on one handtruly simple and, on the other, in the case of pure statecomplexity it reduces the kind of divergence encounteredin the holographic complexity proposals.In both scenarios outlined above, only the target state | J T (cid:105) is entangled across H A ⊗ H A (cid:48) , while the referencestate is a product state | J R (cid:105) = | J R (cid:105) A ⊗ | J R (cid:105) A . As thereis no a priori physical notion of locality in the ancillarysystem, we only require that | J R (cid:105) A is pure and Gaussian.We choose[ J R ] ≡ N A (cid:77) i =1 (cid:18) µ − µ (cid:19) , (bosons) (54a)[ J R ] ≡ N A (cid:77) i =1 (cid:18) − (cid:19) , (fermions) (54b)over spatially local sites i , where we only introduced areference scale µ for bosons .The optimization over all purification in (51) couldtherefore be equivalently performed over reference or tar-get state or even both. The minimum would always bethe same, which can be seen as follows. By construction,the complexity function is invariant under the action ofa single Gaussian unitary U acting on both states, i.e. ,we have C ( | J T (cid:105) , | J R (cid:105) ) = C ( U | J T (cid:105) , U | J R (cid:105) ) , (55) For fermions, the spatially unentangled vacuum is essentiallyunique if we require it to be translationally invariant over sitesand have the same parity as the vacuum of the Ising model. U is related to a group transformation M ab via U † ˆ ξ a U = M ab ˆ ξ b . In the case of Gaussian purifications,we optimize over all Gaussian purifications for the targetstate, i.e. , if we have found such a purification | J T (cid:105) , anyother purification is given by (cid:49) A ⊗ U A (cid:48) | J T (cid:105) . We thus find C P = min U A (cid:48) C ( (cid:49) A ⊗ U A (cid:48) | J T (cid:105) , | J R (cid:105) )= min V A (cid:48) C ( | J T (cid:105) , (cid:49) A ⊗ V A (cid:48) | J R (cid:105) )= min U A (cid:48) ,V A (cid:48) C ( (cid:49) A ⊗ U A (cid:48) | J T (cid:105) , (cid:49) A ⊗ V A (cid:48) | J R (cid:105) ) , (56)where the equalities follow from (55) and where both U A (cid:48) and V A (cid:48) are Gaussian unitarites on the system A (cid:48) . Inpractice, we can therefore start with a basis ˆ ξ A , such that[ J T ] A takes the mixed standard form (40). It can then bepurified so that the purification takes the standard formwith respect to the extended basis ˆ ξ (cid:48) = ( ˆ ξ A , ˆ ξ A (cid:48) ), J T ≡ (cid:67) · · · (cid:83) · · · · · · (cid:67) N A · · · (cid:83) N A ± (cid:83) · · · (cid:67) · · · · · · ± (cid:83) N A · · · (cid:67) N A , (57)as defined in (41). The reference state has the blockdiagonal form J R ≡ [ J R ] A ⊕ [ J R ] A (cid:48) ≡ (cid:18) [ J R ] A
00 [ J R ] A (cid:48) (cid:19) (58)as it is a product state. We have ( (cid:49) A ⊗ U A (cid:48) ) | J (cid:105) = | M JM − (cid:105) with M = (cid:49) A ⊕ M A (cid:48) , so the optimization gives C P = min M = (cid:49) A ⊕ M A (cid:48) (cid:114) log( M J T M − J R ) , (59)which shows explicitly that we can think of the optimiza-tion as either applied to target or reference state. Whenwe perform the optimization, it is actually advantageousto optimize over the reference state, as its stabilizer groupis larger ( i.e. , there are more group elements that pre-serve J R than J T ) and so we can identify a fewer numberof directions/parameters to optimize over. Our algorithmis described in more detail in appendix D and is one ofsubjects of the companion paper [61]. C. Single interval in the vacuum
The discussion in the previous section was very generaland here and in the following we want to apply them tofree CFTs on the lattice, as we did for EoP in section V.The first case to consider is a single interval in the vac-uum. This case appeared earlier in the context of theaforementioned mode-by-mode purifications in [35]. In the present section we readdress the same problem us-ing the most general purifications, whereas in later sec-tion VI E we reconsider the same problem using our sim-plified take on the mode-by-mode purifications to makefurther contact with [35]. In light of a general physicspicture where circuits acting on a spatially disentangledstate need to build entanglement at all scales to matchfeatures of CFT vacua, as well as explicit results in [35],one expects CoP to diverge in the continuum limit.For a single interval on a line, fermionic CoP will be afunction only of wδ as the system size N becomes large.Bosonic CoP, however, also contains two additional pa-rameters, the reference state scale µ , see (54a), and theeffective mass m δ . As changing µ → aµ is equivalentto rescaling the mass and lattice spacing according to m → m/a , δ → aδ , we can set µ = 1 in numerical cal-culations and restore it in analytical formulas containingthe (now unitless and independent) m and δ .We begin with the simpler case of fermionic CoP (atcentral charge c = ). Here, we find a relationship of theform lim N →∞ C P = e wδ + e log wδ + e . (60)Note that we consider the squared CoP. We test this func-tional form by computing the discrete derivative with re-spect to w/δ , expected to be described by the expression e + e δ/w . As figure 4 (top) shows, it is indeed perfectlylinear, allowing us to determine e = 0 . , e = 0 . , e = 0 .
103 (61)with the given three significant digits corresponding tothe numerical accuracy of the optimization algorithm.The bosonic case (with c = 1) is somewhat more com-plicated, as we must subtract terms in mµ and δµ thatdiverge in the continuum limit mµ , δµ →
0. However, westill see in figure 4 (bottom) that the functional form of(60) still holds, but with dependencieslim N →∞ C P = f ( µ δ ) wδ + f ( mµ , µ δ ) log wδ + f ( mµ , µ δ ) . (62)This form accurately describes the w dependence over alarge range of m/µ and µδ . The functions f to f areestimated as f ( mµ , µ δ ) = 0 . (cid:114) log( µ δ ) log mµ + 0 .
25 log mµ , (63a) f ( mµ , µ δ ) = 0 . − .
46 log mµ − .
17 log( µ δ ) , (63b) f ( µ δ ) = 0 .
22 + 0 .
25 log ( µ δ ) , (63c)in the region m/µ, µ δ (cid:28)
1. The leading divergencesin µ δ and mµ are visible in figure 5, where we plot C P (non-squared) and find linear divergences in log( µ δ ) andlog mµ , respectively. The dependence of the slope of thedivergence on w , given by ≈ √ w , is clearly visible in the5 ●●●●●●● ●● ● N = w / δ ● Linear coefficent δ / w D e r i v a t i v e Δ ( P ) / Δ ( w / δ ) ●●●●●●●●●● ■■■■■■■■■■ ◆◆◆◆◆◆◆◆◆◆ ▲▲▲▲▲▲▲▲▲▲ ●● ● m = - ■ m = - ◆ m = - ▲ m = - ● Linear coefficient δ / w D e r i v a t i v e Δ ( P ) / Δ ( w / δ ) Lattice spacing δ = ●●●●●●●●●● ■■■■■■■■■■ ◆◆◆◆◆◆◆◆◆◆ ▲▲▲▲▲▲▲▲▲▲ ●● ● m = - ■ m = - ◆ m = - ▲ m = - ● Linear coefficient δ / w D e r i v a t i v e Δ ( P ) / Δ ( w / δ ) Lattice spacing δ = - FIG. 4. Discrete derivative of fermionic (top) and bosonicsquared CoP (bottom) of a single interval of length (cid:96)/δ . m and δ are given in units where µ = 1. In both cases, thenumber of total sites is given by N = 100 w/δ . ●●●●●● ■■■■■■ ◆◆◆◆◆◆ ▲▲▲▲▲▲ ● w = δ ■ w = δ ◆ w = δ ▲ w = δ - - - - - m B o s o n i c C o P P ●●●●●● ■■■■■■ ◆◆◆◆◆◆ ▲▲▲▲▲▲ ● w = δ ■ w = δ ◆ w = δ ▲ w = δ - - - - - δ B o s o n i c C o P P FIG. 5. Divergences of bosonic CoP of a single interval of size w at fixed µ δ = 10 − and mµ → mµ = 10 − and µ δ → µ = 1. In both cases, theleading divergence is linear. While the µ δ divergence is w -dependent, the mµ one is not. µ δ case, while the mµ term diverges with a constant slope ≈ at mµ (cid:28)
1, consistent with the appearance of theseterms in f and f , respectively. Note that f and f are estimated from the setup of two adjacent intervals,analyzed in the next section, where the linear term f cancels. In particular, the square root term in f canbe seen in figure 6, where the leading divergence in mµ issubtracted.To corroborate this discussion, let us also note thatthe structure of the leading divergences in the two casesmatches the result of the vacuum complexity in freeCFTs, see [22, 23] for bosons and [28, 66] for fermions.In this pure state case analogy, the role of w is playedby the total system size measured in lattice units. Thepresence of the log µ δ contribution in the vacuum case ●●●●● ■■■■■ ◆◆◆◆◆ ▲▲▲▲▲ ▼▼▼▼▼ ● δ = ■ δ = - ◆ δ = - ▲ δ = - ▼ δ = - - m ( f - . l o g m ) ●●●●● ■■■■■ ◆◆◆◆◆ ▲▲▲▲▲ ▼▼▼▼▼ ● m = ■ m = - ◆ m = - ▲ m = - ▼ m = - - - - - δ ( f - . l o g m ) FIG. 6. Subleading contribution to constant term f ofbosonic CoP with respect to m/µ (left) and µ δ (right), with µ = 1. After subtracting the leading contribution 0 .
25 log mµ from f , the square of the remainder is linear in log mµ andlog( µ δ ) when both are small. comes from the ratio of the highest momentum frequencyof the order of the inverse lattice spacing to the referencestate scale and the overall coefficient in front of the wholedivergence is µ -independent. The logarithmic divergenceis present because the symplectic group is non-compact.For fermions, the group of transformations is compactand there is no logarithmic enhancement of the leadingdivergence. D. Two adjacent intervals in the vacuum
The next case to consider are two adjacent intervalsin the vacuum. This is basically the application of theformulas from the previous section with the twist thatit allows us to gain a better control over the finite term f ( mµ , µ δ ) in the bosonic single interval CoP (62).To this end, we are interested in a better behaved com-bination of complexities akin to (49). We take it to be∆ C (2) P ≡ C P ( A ) + C P ( B ) − C P ( A ∪ B ) , (64)where we put two on the LHS in the brackets to em-phasize that it does not denote taking a square. Therational behind this expression is that when one keeps µ δ fixed, the whole power-law divergent part cancels be-tween the three terms. Similar combinations to (64) inthe aforementioned context of pure state complexity inthermofield double states appear in [29]. Also, note thedifference with respect to holographic mutual complex-ity (49).Simple manipulations lead to the following result forthe Ising CFT ( c = ):lim N →∞ ∆ C (2) P = e log w A w B ( w A + w B ) δ + e . (65)For the free decompactified boson CFT ( c = 1), we canagain use the single-interval result (62) to arrive atlim N →∞ ∆ C (2) P = f ( mµ , µ δ ) log w A w B ( w A + w B ) δ + f ( mµ , µ δ ) , (66)6 ● ● ● ● ● ● ● ● ● ● ● ● ●● Numerical data w A / ( w A + w B ) Δ C P ( ) Fermions ● ● ● ● ● ● ● ● ●■ ■ ■ ■ ■ ■ ■ ■ ■● ● ● ● ● ● ● ● ●■ ■ ■ ■ ■ ■ ■ ■ ■● m = - , δ = - ■ m = - , δ = - ● m = - , δ = - ■ m = - , δ = - w A / ( w A + w B ) Δ C P ( ) Bosons
FIG. 7. Fermionic/Ising (left) and bosonic (right) CoP fortwo adjacent ( d = 0) subsystems A and B , in units of µ = 1.The expected analytical forms, (65) for fermions and (66) forbosons, are plotted as dashed curves. We consider w A + w B δ =14 total sites for fermions and 20 sites for bosons. The totalsystem size is set to N = 100 wδ for each block width w . but where the logarithmic coefficient and constant termnow depend on the divergences in mµ and µδ in preciselythe same way as the single-interval expression. The formof ∆ C (2) P for both bosons and fermions in shown in figure 7in terms of the ratio w A w A + w B . The qualitative behavior isthe same as for the EoP result shown in figure 3 and alsothe holographic complexity proposal results encapsulatedin table II. However, one should bear in mind a differentsubtraction of complexity with the latter case related tothe use of a L norm in our Gaussian studies. Note alsothat ∆ C (2) P is logarithmically ultraviolet divergent.Finally, it is interesting to observe that the mutualcomplexity depicted in figure 7 is positive, which indi-cates subadditivity of our CoP definition. This is in linewith an earlier observation in the case of two coupledharmonic oscillations in [36]. E. Single mode optimization for bosons
A natural hope in the context of Gaussian states is tosplit the problem of finding the CoP for a system with N A modes into N A problems for a single mode. In [35],the authors use a formula for the L norm complexity ofa single bosonic degree of freedom for certain Gaussianstates derived in [23], where they introduce two types of L norm bases (called the physical and the diagonal one).Note that the authors use the geodesic with respect tothe L norm, but compute its length with respect to the L , as it is difficult to prove that a path is minimal withrespect to the L norm (in particular, for several modes).When considering several modes of the free Klein-Gordonfield, we need to distinguish two settings reviewed previ-ously: • Thermal states.
Here, we can choose a basis ˆ ξ A ,such that both [ J T ] A of a mixed thermal Klein-Gordon state and [ J R ] A of a spatially unentangledvacuum decompose into 2-by-2 blocks, so it is easy to argue that the Gaussian CoP results from opti-mizing over individual modes. • Subsystems.
If we consider a mixed Gaus-sian state ρ A resulting from restricting the Klein-Gordon vacuum to a region, it is typically not pos-sible to bring both [ J T ] A and [ J R ] A of a spatiallyunentangled vacuum into block diagonal form, so itwill not suffice to optimize over individual modes.Let us emphasize that in both cases, we have a mode-by-mode purification with respect to the standard form(41), but it is the reference state that will only be atensor product over these modes in the case of thermalstates, but not for local subsystems. However, we mayencounter situations where the standard decompositionof the mixed target state also approximately decomposesthe reference state into individual modes.We consider a single bosonic mode with a pure Gaus-sian reference state and a mixed Gaussian target statethat both do not have ϕπ -correlations, i.e. , (cid:104) ˆ ϕ i ˆ π j (cid:105) = 0.We extend this system to a system of two bosonic modes H (cid:48) = H A ⊗H A (cid:48) to have the extended reference state | J R (cid:105) and the purified target state | J T (cid:105) , such that the respec-tive complex structures are given by J T ≡ λ √ λ − − λ √ λ − √ λ − λ √ λ − − λ (67) J R ≡ µ − µ ν − µ , (68)where λ ∈ [1 , ∞ ) is the same as c i for several degreesof freedom in (40), µ is the reference state frequency forthe original single mode, and ν is a parameter in thereference state, for which we will minimize the complexityfunctional (52). The latter is given for us by C ( λ, µ, ν ) = 12 (cid:115) log (cid:18) ω + µ (cid:19) + log (cid:18) ω − µ (cid:19) , (69)where we defined the variables ω ± := µ (cid:16) λ ( µ + ν ) ± (cid:112) λ ( µ + ν ) − µν (cid:17) . (70)In order to find the minimum of C P = min ν C ( λ, µ, ν ),we need to solve a transcendental equation for ν . Thiscan be done numerically for any given value of c and µ in a very efficient manner. Unfortunately there is noclosed analytic expression for C P of a single mode, but wereduced the problem of a single mode (with vanishing ϕπ -correlations in reference and target states) to a problemthat is much simpler than the optimization over a largemanifold. Can we extend this to larger systems?As derived in [22, 28] for the perspective of the L norm, the optimal geodesic from reference to target state7gives (52), so we only need to worry about what goes intothis formula. We consider a mixed state of a region A with N A sites. With respect to the local basisˆ ξ A = ( ˆ ϕ A , . . . , ˆ ϕ AN A , ˆ π A , . . . , ˆ π AN A ) , (71)the covariance matrix of reference and target state willhave the following forms[ G T ] A ≡ (cid:18) G ϕϕ G ππ (cid:19) , [ G R ] A ≡ (cid:18) µ (cid:49) µ (cid:49) (cid:19) . (72)If the Gaussian target state were pure, it is well-knownthat we can find symplectic transformation M = (cid:18) O O (cid:19) , (73)where O is an orthogonal matrix, such that ˜ G T = M G T M (cid:124) is diagonal, while G R = M G R M (cid:124) is preserved.As soon as G T represents a mixed Gaussian state, therestill exists a symplectic transformation M , such that˜ G T = M G T M (cid:124) is diagonal, but now M will not be ofthe form (73) anymore and will thus not preserve G R , i.e. , M G R M (cid:124) (cid:54) = G R . However, we could pretend to ap-proximate the true M by only diagonalizing G ϕϕ with arespective O , such that ˜ G ϕϕ = OG ϕϕ O (cid:124) is diagonal. Wethen apply the respective M of the form (73) and pre-tend that also ˜ G ππ = OG ππ O (cid:124) is diagonal, i.e. , we dropthe off-diagonal terms which we hope to be sufficientlysmall. With this assumption, we can apply a mode-by-mode optimization based on (69), such that C P ( | J T (cid:105) , | J R (cid:105) ) ≈ (cid:115)(cid:88) i min ν C ( λ i , µ, ν ) , (74)where λ i is extracted from the diagonal entries of ˜ G ϕϕ and ˜ G ππ . For a pure target state, (74) becomes an equal-ity, where both sides match the regular complexity (52).We consider the restriction of the Klein-Gordon vac-uum to the subsystem of a single interval as explored insection VI C. In this setup, we can compare the approx-imate single mode optimization with the full optimiza-tion. While the full optimization takes several hours ona regular desktop computer, our approximate scheme ofoptimizing (69) over individual modes only takes a fewseconds. Figure 8 shows how the single mode optimiza-tion is almost indistinguishable from the full optimizationfor small µL , but the approximation becomes increas-ingly worse for larger µL . In [35], the authors perform asimilar calculation for the L norm, which they refer toas mode-by-mode purifications . The difference between As pointed out at the beginning of this section, any Gaussian pu-rification is a mode-by-mode purifications, but what the authorsof [35] mean is that they only optimize over the purifications ina specific way, as if reference and target state would decomposeinto the same individual modes as an approximation. μ L = - μ L = μ L = μ L = w / L C o m p l e x i t y o f p u r i f i c a t i o n P FIG. 8. Comparison of CoP obtained using the Gaussian opti-mization algorithm (solid) and the Single Mode Optimizationapproximation (dashed) for the single interval case for massscale mL = 10 − , N = 100, and for µL = 10 − , , , our single mode optimization and what the authors in[35] do is two-fold: First, we optimize over the L norm,while they consider the L norm. Second, we change thetarget state by hand to decompose into a product overmodes in the same basis as the reference state and thenperform the optimization semi-analytically for individualmodes, i.e. , we optimize for each mode independently.In contrast, the authors [35] do not change reference ortarget state, but only consider a subset of possible purifi-cations, i.e. , they evaluate the full complexity functionand optimize over a restricted subset of parameters (oneparameter per mode). As they do not change the targetstate, they cannot evaluate the complexity for individ-ual modes, so would need in principle to optimize overall parameters simultaneously, but find good convergencewhen optimizing over O (1) parameters at once. Clearly,the approximation in [35] and our single mode optimiza-tion work because the for small µ L reference and targetstates are close to decomposable over individual modes.This will not be the case for generic subsystems (suchas two intervals), general states (such as those with ϕπ -correlations) and fermionic systems (which cannot be de-composed into single mode squeezings), in which case ourfull optimization algorithm is required. F. Comparison with the Fisher-Rao distanceproposal
In [65], the authors propose a measure of bosonic mixedstate complexity based on the Fisher-Rao distance, whichcan be defined on the manifold (cid:80) ( N ) of 2 N × N realand positive definite matrices, of which bosonic covari-ance matrices are a subset of. Without the need for anypurifications, the proposal for the complexity of mixedstates is formally equivalent to (52), where here G and G are taken to be the covariance matrices of the mixedtarget and reference state, respectively. The motivationfor this definition is that the Fisher-Rao distance function d ( G, G ) := (cid:112) tr (log( G G − ) ) / √ μ L = / μ L = μ L = μ L = w / L C o m p l e x i t y o f p u r i f i c a t i o n P FIG. 9. Comparison of CoP for a single interval obtainedusing the Gaussian optimization algorithm (solid) and theFisher-Rao distance function (dashed). The data are gener-ated for mL = 10 − and N = 100. measures the geodesic distance in the manifold of co-variance matrices. It is important to highlight that theauthors in [65] focus on bosonic Gaussian states occur-ring in the Hilbert space of harmonic lattices, and hencethe proposal of the Fisher-Rao distance function shouldbe thought of as applicable in principle only to bosonicstates, although one could conjecture that a simliar for-mula should be applicable to fermionic Gaussian states.Nonetheless, it is interesting to compare the propertiesof such distance function with the bosonic CoP measurearising from the Gaussian optimization procedure devel-oped in this paper.For the single interval case, there is in fact a note-worthy qualitative and quantitative agreement betweenthe two, as shown in figure 9. The Fisher-Rao distancefunction and the CoP measure offer a comparable mea-sure of the complexity of the mixed state associated tothe single interval, which is remarkable given the factthat the Fisher-Rao distance function is a geodesic dis-tance being evaluated on the manifold of mixed stateson the Hilbert space H A , whereas CoP is a geodesic dis-tance on the manifold of pure states on the larger Hilbertspace H = H A ⊗ H A (cid:48) and these two need not be thesame or even comparable to one another, as explainedon figure 12. This comparison seems to work better,the smaller the parameter µL is taken to be. As ourstudies indicate, for two adjacent intervals the distinc-tion between the Fisher-Rao distance and CoP deviatesignificantly from each other, even though the qualita-tive behaviour remains comparable. VII. COMMENTS
In our considerations, two important subtleties of freeQFTs, known to the literature, played a key role andinfluenced the vacuum subregions we could consider tomake genuine QFT predictions. They are the zero modein the case of free boson QFTs and spatial locality ofdisjoint intervals under the Jordan-Wigner mapping be-tween the Ising model and the free Majorana fermion the-ory. Below we provide an additional discussion of these m δ = - m δ = - m δ = - m δ = - m δ = - m δ = - d / w M u t u a l i n f o r m a t i o n I ( A : B ) Mass dependence ( line ) ●● Line, m δ = - Line, m δ = - Periodic, N = N = ● Expected limit w / d P o w e r e s t i m a t e b Decay power
FIG. 10. MI on an infinite line in the small m δ limit. Shownare the general mass dependence (left) as well as the decaypower in the d/w → ∞ limit compared to the periodic systemin the large N limit (right). two important points. A. Zero mode for free bosons
The presence of the zero mode is a known subtlety ofthe free boson theory in the massless limit, as we dis-cussed in section II A. The simplest way to deal with itis to keep the mass term in the Hamiltonian (8) nonzeroand try to numerically approach the limit m →
0, whichis precisely the strategy we adopted in the calculations ofthe bosonic EoP and CoP. Given the scarceness of othermethods to shed light on EoP and CoP in QFTs, it isimportant to learn about the role of zero mode in better-understood problems.To some degree we already explored this issue in sec-tion II A when understanding what needs to be done inorder to reproduce the modular invariant thermal parti-tion function (15) from the Gaussian calculation in themassive theory (16). Here we will address another quan-tity, which is the two interval vacuum MI reviewed insection III. Fitting a power law for the data on a peri-odic chain of bosons does not lead to convergence of theestimated power to a nonzero value in the d/w → ∞ limit (see table I), implying a decay with slower asymp-totic functional dependency. Indeed, the massless limitof our study can be thought of as a decompactificationlimit of the free boson theory on a circle in the field space(which can be seen as another way of dealing with thezero mode), as reviewed in section II A, and the pre-dicted large distance behaviour in this case is not of apower-law type. In our periodic setup we considered thelimit of a large number of sites N with the dimensionlessscale mL = mN δ kept constant and small. In this limit,the mass dependence of both MI and EoP is accuratelydescribed by an additive − log( Lm ) term (as alreadynoted in [34]), so that the dependence on w/d can bestudied independently of the mass.However, one may alternatively consider free bosonson an infinite line, i.e. , taking the limit m δ → N → ∞ . As the scale m δ N formally di-9 m δ = - m δ = - m δ = - m δ = - w / d P o w e r e s t i m a t e g Power - logarithmic fit m δ = - m δ = - m δ = - m δ = - w / d P o w e r e s t i m a t e h Power - log - log fit FIG. 11. Power-logarithmic and power-double-logarithmicfits of bosonic MI on an infinite line at large d/w and smalleffective mass mδ . Block width set to w/δ = 10. Plotted arethe estimated coefficients g and h from (77) and (78). verges in the first limit, the resulting mass dependenceis qualitatively different from the periodic setup. To in-vestigate the behavior in this case, we adapted our nu-merical method to free bosons on an infinite line andcomputed MI at extended precision at small m δ . Com-parable studies were performed earlier in [76] and the au-thors reported a power-law fall-off of the form (cid:0) wd (cid:1) . ,see also (36). The value of I ( A : B ) on the line changesonly very slowly with mδ , as shown in figure 10 (left).For d/w (cid:28) mδ , this mass dependence can be expressedby a constant offset I ( A : B ) = f MI (cid:18) dw (cid:19) + 12 log (cid:18) log 1 mδ (cid:19) , (76)where the factor of 1 / I ( A : B ) prevents this depen-dence persisting at finite m δ , and thus I ( A : B ) beginsto decay exponentially as d/w (cid:29) m δ . Apart from thedifferent mass dependence, the periodic and line setupin their respective limits yield equivalent results. As weshow in figure 10 (right), the estimated decay power fromboth limits exactly matches, vanishing as d/w → ∞ .As the line setup is more efficient at probing large val-ues of d/w due to an absence of finite-size effects, weuse this setup to test for functional dependencies slowerthan the previously considered power law and logarith-mic functions. As the functional dependence of MI andEoP match in this limit, we expect the results to extendto both measures. In particular, we consider the power-logarithmic asymptotics f MI ( dw ) ∼ g − g (cid:18) log dw (cid:19) g as dw → ∞ , (77)as well as a power-double-logarithmic one, f MI ( dw ) ∼ h − h (cid:18) log log dw (cid:19) h as dw → ∞ . (78) In figure 11 coefficients from both fits are shown. Whilethe power e of the power-logarithmic fit converges to avalue e (cid:46) . d/w → ∞ , mδ → f clearly convergesto a value f (cid:46) . d/w ; while the functional dependencein this range can be well approximated by a power law,as also shown in [34], the apparent power law is not sta-ble as d/w → ∞ . Curiously, the authors of [76] analyzein the same setup another measure of entanglement –the logarithmic negativity [78] – and our extended preci-sion calculations in this case reproduce the exponentialfall-off as reported in [76] for the same range of massesconsidered in figure 11. This shows that not all non-localquantities are affected by the zero mode problem.An additional insight about the expected behaviourof MI in the case of the free boson CFT in the decom-pactification limit R → ∞ can be obtained from havinganother look at the modular invariant partition functionon the circle (15). The partition function provides theinformation about the density of states, which, via thestate-operator correspondence, describes also the densityof operators in the spectrum. The latter quantity, in con-junction with (32), will provide an indication on whatto expect from the two interval case in the large separa-tion limit for the decompactified free boson CFT. To thisend, the partition function of a CFT with a continuousspectrum of operators (for a decompactified free bosontheory vertex operators are labelled with a continuousindex) can be written as Z ∼ (cid:90) ∞ d ∆ ρ (∆) e − π ( β/L ) ∆ e π ( β/L ) , (79)where the second exponent comes from the Casimir en-ergy and ∆ is the scaling dimension of operators in thetheory. Note that descendent operators appear in thesum only for ∆ >
1. The power law multiplying theCasimir contribution in the low temperature limit of (15)points to the density of operators behaving in a powerlaw fashion in the vicinity of ∆ = 0. In particular, thebehavior Z (cid:12)(cid:12)(cid:12) β/L (cid:29) ∼ ( β/L ) − α e π ( β/L ) (80)can be explained by ρ (∆ ≈ ∼ ∆ α − . (81)Note that for a free decompactified boson α = 12 . (82)In this case, the density of states can be easily understoodto be given by ∆ − / upon noting that the operators of0interest are vertex operators : e i ν φ : specified by a realnumber ν . The scaling dimension of vertex operatorsis given by ∆ ∼ ν . The density of operators is uniformwhen parametrized by ν and viewing it as a function of ∆brings in the Jacobian ∼ ∆ − / , which gives (82).Now we can come back to MI at large separation. Sincethe formula (32) incorporates the exchange of a single op-erator, in the absence of a gap in the spectrum one needsto sum over the continuum of light operators with theirdensity given by (81). Following [41] we can schemati-cally write I ( A : B ) (cid:12)(cid:12)(cid:12) w (cid:28) d ∼ (cid:90) ∞ d ∆ ∆ α − (cid:16) wd (cid:17) ( c T T O ∆ ) + . . . , (83)where c T T O ∆ is the three-point function coefficient be-tween two twist fields and a primary with dimension ∆and the ellipsis denote contributions with higher powersof (cid:0) wd (cid:1) . At the present moment we do not have controlover the two kinds of contributions. However, neglectingthe additional contributions and assuming that c T T O ∆ has a power-law dependence on ∆ for small scaling di-mensions c T T O ∆ ? ∼ ∆ κ , (84)the long distance behaviour of MI becomes I ( A : B ) (cid:12)(cid:12)(cid:12) w (cid:28) d ? ∼ (cid:18) log dw (cid:19) − α − κ . (85)Let us re-stress that the above equation is based on un-verified assumptions and the correct answer is likely tobe more involved, yet in principle calculable. However,what (85) indicates is that MI may decay much moreslowly with distance than a simple power law, as is alsoshown by our numerical results. In particular, the fit-ting ansatz corresponding to (85) is (77) consistent with − α − κ (cid:46) . dw → ∞ . It is unclear atthe moment if this is a feature of regularizing the zeromode via introducing a non-vanishing mass, or if thebehaviour (77) or (78) that we see using our Gaussiannumerics persists in the free boson CFT in the decom-pactification limit. B. Subsystems in Ising CFT vs. free fermions
There is a subtle difference for computations of MIand EoP between the XY spin model and the Majo-rana fermion model. The spin model in the continuumlimit becomes the genuine c = 1 / A and B are adjacent. However, when A and B areseparated, the non-trivial topology in the replica com-putation of entanglement entropy leads to a differencebetween the spin and fermion calculation as the simplechoice of subsystems does not respect the projections.This is consistent with our numerical results for Majo-rana fermions and adjacent intervals shown in figure 3.We also note that MI and EoP are smaller in the fermionmodel calculations without projects compared with thosein spin model calculations. Refer to appendix B for moredetails. VIII. DISCUSSION
In this manuscript, we have a presented a systematicand comprehensive analysis of EoP and CoP that char-acterize mixed states. We computed the EoP betweentwo blocks of widths w A and w B at distance d in one-dimensional periodic systems at large size for both criti-cal bosons and fermions, the latter of which are equiva-lent to a discretized Ising CFT while d = 0. Furthermore,we compared these results with the well-studied MI. At d = 0, our data shows I ( A : B ) = c w A w B ( w A + w B ) δ , (86a) E P = c w A w B ( w A + w B ) δ , (86b)confirming previous expectations through analyticalmethods [37, 38]. For d >
0, we considered the sym-metric setup w A = w B = w in the two limits d/w (cid:28) d/w (cid:29)
1. In the former limit, our data is consistentwith I ( A : B ) ∝
13 log wd , (87a) E P ∝
16 log wd , (87b)for our bosonic model. The latter limit shows a sub-polynomial (logarithmic or double-logarithmic) decay ofboth I ( A : B ) and E P at large d/w . In summary, MIand EoP show the same scaling at large distance d . Thisresult is consistent with the observation in [34] that EoPappears to weight quantum and classical correlations dif-ferently from MI, leading to different qualitative behav-ior only when both become relevant, i.e. , at small dis-tances. Indeed, this is the regime in which our numerical1results show new model-dependent features that distin-guish both measures. For both the periodic and infiniteline setup, two-interval MI and EoP are divergent in thezero-mass limit; this divergence can be regulated by a log( mδN ) and log log( mδ ) term, respectively. Let usalso emphasize that the large distance behaviour of MI inthe free boson case at small masses is very subtle and isdescribed by the fall-off slower than any power-law, con-trary to earlier studies in the literature. We discussedthis at length in section VII A.In the studies of CoP, our only guidance were the pre-dictions of holographic complexity proposals for subre-gions, as summarized in table II. It is interesting to note,that our studies reproduce qualitatively terms presentin the holographic results. In particular, for appropri-ately defined mutual complexity, all complexity notionswe considered lead to an analogous dependence on thesizes of two adjacent intervals to the one seen in for en-tanglement measures in (86). What is also worth a sepa-rate remark is an intricate dependence of the subleadingdivergent and the finite term in the bosonic CoP on thereference state scale µ and the mass m acting as the zeromode regulator, see (63).The other important set of CoP results has to do witha comparison with an earlier study of CoP using a re-stricted set of Gaussian purifications in [35], as well aswith the Fisher-Rao distance introduced in a related con-text in [65]. As we saw respectively in sections VI Eand VI F, both of these notions can reproduce qualita-tive behaviour of the CoP, but they can also exhibit sig-nificant deviations from the numerical answer predictedby the full Gaussian optimization. This indicates that ingeneral one indeed needs to optimize over as large setsof states in the Hilbert state as possible to reproduce thetrue CoP. Of course, even if the reduced density matrix isGaussian, this does not imply that the optimal circuit isnecessarily such. Generalizing our study to non-Gaussianstates is an outstanding challenge and in the last para-graph of this discussion, we sketch what we believe mightbe an interesting and workable example.In the context of CoP, we also want to offer an in-tuitive, yet rigorous interpretation of the setup that wewere using. Formula (52) was derived in [28, 29] basedon a certain metric on the group manifold, which coin-cided with the geodesic distance on the Gaussian statemanifold (Fubini-Study metric), as studied in [22]. Notethat there is an ongoing debate what the most appropri-ate notion of distance is, both on the manifold of statesand the Lie group. While the L norm appears to havesimilar properties as the different holographic complex-ity proposals, only the L norm induced by a certainRiemannian metric can be analytically minimized. Thisis the key reason why we focused on the L norm in thepresent manuscript, as most other notions of distancesare intractable in the setup we are considering (manydegrees of freedom, most general Gaussian gates, analyt-ical optimization over all trajectories from reference totarget state, numerical optimization of all possible pu- pure U A (cid:48) | ψ T (cid:105) AA (cid:48) C P | ψ R (cid:105) AA (cid:48) mixed | ψ R (cid:105) A ( ρ T ) A FIG. 12. We sketch how the manifold of mixed states onHilbert space H A is related to the manifold of pure stateson the larger Hilbert space H = H A ⊗ H A (cid:48) . We indicatethe manifold (red line) of all possible purifications | ψ T (cid:105) AA (cid:48) related by Gaussian unitaries U A (cid:48) . The CoP C P is given bythe geodesic distance (blue) between the purified referencestate | ψ R (cid:105) AA (cid:48) . rifications). Figure 12 provides a visualization of thisinterpretation: CoP becomes in essence another type ofminimization, namely finding the minimal distance be-tween the unique reference state | ψ R (cid:105) to the the set of allpossible purifications (or all Gaussian purifications) thatis fully determined by the single mixed state, whose CoPwe are computing.Let us emphasize that there have been several ap-proaches to define and compute complexity for mixedstates, which predominantly focus again on Gaussianstates. On the one hand, there are approaches [32, 35, 36]based on purifications, to which a notion of pure statecomplexity is applied. This provides an elegant way tocarry any definition of pure state complexity over to ar-bitrary mixed states. Our present draft uses the samephilosophy, but performs the required optimization overall possible (Gaussian) purifications numerically. On theother hand, there are approaches [65] to define mixedstate complexity directly on the set of mixed states. Here,one introduces additional non-unitary gates that allowto change the spectrum of the density operator, so thatunitarily inequivalent mixed states can be reached. Theresulting geodesic distance agrees with the L norm whenrestricted to pure states and the procedure can be under-stood as measuring the geodesic distance on the manifoldof (Gaussian) mixed states equipped with the Fisher in-formation geometry. Remarkably, the resulting analyt-ical formula in terms of covariance matrices for bosonsagrees with the one (51) derived for pure states. Onecan expect the same result to also hold for fermions. Inparticular, we can use these gates to transform a purereference state into a mixed target state. Finally, a com-plementary approach is based on path integral optimiza-tion [64, 79], which uses gates being exponentials of theenergy-momentum tensor operators with complex coeffi-2cients and so both unitary, as well as Hermitian opera-tors [80]. In the case of free CFTs, these are Gaussiangates, however, in the interacting cases they are not.An important feature of our works stems from the factthat the Ising model in the spin picture (rather thanthe fermionic picture) leads to genuinely non-Gaussianmixed states if we restrict to disconnected regions, i.e. ,the spectrum of the reduced density operator to suchsubsystem cannot be reproduced by bosonic or fermionicmixed Gaussian states. This may open the window tostudy non-Gaussian circuit complexity (of purification)in a genuine QFT limit. For this, we propose to con-sider two individual separated sites in the spin picture ofthe critical Ising model. This leads to a mixed state ina system of two qubits, i.e. , in a four-dimensional com-plex Hilbert space. We can now study CoP in this setupas a function of the separation between the two sites .While these are ideas for future work, the implementedalgorithm to optimize over all possible purifications (inthis cases non-Gaussian ones) can be used, once an ap-propriate notion of non-Gaussian circuit complexity (forgeneric gates) is defined. In this sense, we believe thatour considerations of CoP in the context of the criticalIsing model CFT presents a stepping stone to exploregenuine non-Gaussian circuit complexity in the contextof QFTs. ACKNOWLEDGMENTS
We thank J. Eisert, R. Jefferson, R. Myers, N. Shiba,K. Tamaoka, and K. Umemoto for helpful conversationsand J. Knaute and V. Svensson for comments on thedraft. The Gravity, Quantum Fields and Informationgroup at AEI is supported by the Alexander von Hum-boldt Foundation and the Federal Ministry for Educationand Research through the Sofja Kovalevskaja Award. HCis partially supported by the Konrad-Adenauer-Stiftungthrough their Sponsorship Program for Foreign Studentsand by the International Max Planck Research School forMathematical and Physical Aspects of Gravitation, Cos-mology and Quantum Field Theory. AJ has been sup-ported by the FQXi as well as the Perimeter Institutefor Theoretical Physics. Research at Perimeter Instituteis supported by the Government of Canada through theDepartment of Innovation, Science, and Economic De-velopment, and by the Province of Ontario through theMinistry of Research and Innovation. TT is supportedby Inamori Research Institute for Science and World Pre-mier International Research Center Initiative (WPI Ini-tiative) from the Japan Ministry of Education, Culture,Sports, Science and Technology (MEXT). TT is also sup-ported by JSPS Grant-in-Aid for Scientific Research (A) Of course, one should in principle consider more sites to be closerto continuum, but this will very quickly make the optimizationproblem intractable.
No.16H02182 and by JSPS Grant-in-Aid for ChallengingResearch (Exploratory) 18K18766. We are also grate-ful to the long term workshop Quantum Informationand String Theory (YITP-T-19-03) held at Yukawa In-stitute for Theoretical Physics, Kyoto University, wherethis work was initiated.
Appendix A: Transverse field Ising model
We review the construction of fermionic Hamiltoniandescribing the transverse field Ising model. In particular,we explain how we decompose the fermionic Hamilto-nian into two quadratic parts and how we compute itsbipartite entanglement entropy using standard Gaussiantechniques.
1. Definition
We consider the transverse field Ising model in onedimension with N sites arranged in circle. We denotesites as i = 1 , . . . , N , with the site N + 1 is identifiedwith the site 1. For simplicity, we will assume that N isan even integer. The Hamiltonian is given byˆ H = − N (cid:88) i =1 (2 J ˆ S x i ˆ S x i +1 + h ˆ S z i ) , (A1)where ˆ S x i , ˆ S y i and ˆ S z i are the standard Pauli operators.This model can be solved by applying the following steps.We follow the conventions introduced in [49, 81, 82]. (1) Jordan-Wigner transformation. We expressthe spin operators ˆ S x i and ˆ S y i in terms of ladder opera-tors ˆ S ± i = ˆ S x i ± i ˆ S y i . We then perform a Jordan-Wignertransformation by expressing all spin operators in termsof fermionic creation and annihilation operators ˆ f † i andˆ f i , namely ˆ S + i = ˆ f † i exp − i π i − (cid:88) j =1 ˆ f † j ˆ f j . (A2)The resulting Hamiltonian is then given byˆ H = − N (cid:88) i =1 (cid:20) J (cid:16) ˆ f † i ( ˆ f i +1 + ˆ f † i +1 ) + h.c. (cid:17) + h ˆ f † i ˆ f i (cid:21) − J (cid:104) ˆ f † ( ˆ f N + ˆ f † N ) + h.c. (cid:105) ( ˆ P + 1) + hN , (A3)where ˆ P tot = e i π ˆ N is the parity operator with ˆ N = (cid:80) Ni =1 ˆ f † i ˆ f i . The last term is a boundary term. (2) Quadratic Hamiltonians. Because of the opera-tor ˆ P in (A3), the Hamiltonian is not quadratic in ˆ f † i andˆ f i . The non-quadratic term containing ˆ P distinguishesthe sectors of even and odd eigenvalues of the number op-erator ˆ N . The Hilbert space can be decomposed as direct3sum H = H + ⊕H − where H + and H − are the eigenspacesof the number parity operator ˆ P = e i π ˆ N with eigenvalues ±
1. The projectors onto these eigenspaces are given byˆ P ± = 12 ( (cid:49) ± ˆ P ) . (A4)We can diagonalize ˆ H XY over H ± individually by apply-ing the Fourier transformationsˆ c κ = 1 √ N N (cid:88) j =1 e i κj ˆ f j , (A5)where κ ∈ K ± with K + = (cid:8) πN + πkN (cid:12)(cid:12) k ∈ (cid:90) , − N ≤ k < N (cid:9) , (A6) K − = (cid:8) πkN (cid:12)(cid:12) k ∈ (cid:90) , − N ≤ k < N (cid:9) . (A7)The resulting Hamiltonian ˆ H = ˆ H + ˆ P + + ˆ H − ˆ P − is com-posed of the quadratic piecesˆ H ± = (cid:88) κ ∈K ± > (cid:104) a κ (cid:16) ˆ c † κ ˆ c κ + ˆ c †− κ ˆ c − κ − (cid:17) − b κ (cid:16) i ˆ c † κ ˆ c †− κ − i ˆ c − κ ˆ c κ (cid:17)(cid:105) (A8)with the parameters defined as a κ = − J cos( κ ) − h , b κ = J sin( κ ) . (A9) (3) Diagonalizing Hamiltonian. At this point,we only need to perform individual fermionic two-modesqueezing transformations mixing the mode pair ( κ, − κ )ˆ η κ = u κ ˆ c κ − v ∗ κ ˆ c †− κ . (A10)with transformation coefficients explicitly given by (cid:15) κ = (cid:112) h + 2 hJ cos( κ ) + J ,u κ = (cid:15) κ + a κ (cid:112) (cid:15) κ ( (cid:15) κ + a κ ) , v κ = ib κ (cid:112) (cid:15) κ ( (cid:15) κ + a κ ) . (A11)As fermionic Bogoliubov coefficients, u κ and v κ satisfy | u κ | + | v κ | = 1. The cases κ ∈ { , π } , for which theHamiltonian is already diagonal, can be treated by defin-ing the special coefficients u = v π = v − π = 0 , u π = u − π = 1 , v = i , (A12)which leads to the identification ˆ η = ˆ c † and ˆ η π = ˆ c †− π .After performing this last transformation, the quadraticpieces take the diagonal formˆ H ± = (cid:88) κ ∈K ± (cid:15) κ (cid:0) ˆ η † κ ˆ η κ − (cid:1) , (A13)With this in hand, we can analyze efficiently the entangle-ment structure of eigenstates. The relevant informationis fully contained in the transformation from the (local) fermionic operators ( ˆ f i , ˆ f † i ) to the (non-local) operators(ˆ η κ , ˆ η † κ ). This allows us to define and compute the co-variance matrix of an eigenstate |{ N κ }(cid:105) as [82]Ω abij ≡ (cid:104) | ˆ ξ a ˆ ξ b − ˆ ξ b ˆ ξ a | (cid:105) = 1 N (cid:88) κ ∈K ± c κ ( i − j ) (cid:18) N κ − N κ (cid:19) (A14)with the single function c κ ( j ) given by c κ ( j ) = 2( | v κ | − | u κ | ) cos( κj ) + 2Im( u κ v κ ) sin( κj ) . (A15)The numbers N κ ∈ {− , } are given by N κ = ( − n κ ,where we have the regular occupation numbers n κ aseigenvalues of ˆ η † κ ˆ η κ . For states with an even numberof excitations, i.e. , where (cid:80) κ N κ is even, we need to use κ ∈ K + , while for an odd number of excitations, we use κ ∈ K − .
2. Notions of locality
First, let us note that the transverse field Ising modelis invariant under lattice translations, which implies thatwe can choose the reference point i = 1 of our Jordan-Wigner transformation without loss of generality. Ittherefore suffices to consider intervals that start at i = 1.The Jordan-Wigner transformation (A2) does not onlyprovide an isomorphism between operators, but it alsopreserves bipartite entanglement of connected interval, i.e. , the entanglement entropy associated to a region con-sisting of adjacent sites R = (1 , . . . , wδ ) is the same re-gardless if we use the tensor product structure inducedby the spin operators or the fermionic creation and anni-hilation operators. This is a consequence of the remark-able fact that, despite the Jordan-Wigner transformationbeing non-local, the operator ˆ S ± i only depends on cre-ation and annihilation operators ˆ f † j and ˆ f j in the range1 ≤ j ≤ i . Therefore, the subalgebras generated eitherby ˆ S σi (with σ ∈ { x , y , z } ) in the spin formulation or byˆ f i and ˆ f † i in the fermionic formulation will both probethe same observables.This is no longer true if we consider regions con-sisting two non-adjacent intervals, such as R =(1 , . . . , wδ , d + wδ , . . . , d +2 wδ ) for d > w >
0. In thiscase, the spin operators ˆ S σi on sites ( d + wδ , . . . , d +2 wδ ) willdepend on the fermionic operators associated to all sites(1 , . . . , d +2 wδ ) as seen from (A2), which in particular in-cludes the sites ( wδ +1 , . . . , w + dδ ). Consequently, comput-ing the bipartite entanglement entropy associated to theregion R will be different depending on if we define thesubsystem in the spin picture vs. the fermionic picture.We will discuss this issue in more detail in appendix B.4 Appendix B: Spin vs. Majorana Fermion
In this section, we highlight the known subtlety dueto the different notion of locality in the fermionic modelcompared to the spin model.
1. Partial traces and subsystems
The inequivalence between spin and Majorana entan-glement can also be seen in the behavior of partial tracesunder an explicit mapping between both models, theJordan-Wigner transformation. Explicitly, it relates spinand fermionic operators viaˆ f j = (cid:32) j − (cid:89) i =1 Z i (cid:33) X j + i Y j √ , (B1)ˆ f † j = (cid:32) j − (cid:89) i =1 Z i (cid:33) X j − i Y j √ , (B2)where we defined the k -site Pauli operators on N totalsites as X j = ( (cid:49) ) ⊗ ( j − ⊗ σ x ⊗ ( (cid:49) ) ⊗ ( N − j ) , (B3) Y j = ( (cid:49) ) ⊗ ( j − ⊗ σ y ⊗ ( (cid:49) ) ⊗ ( N − j ) , (B4) Z j = ( (cid:49) ) ⊗ ( j − ⊗ σ z ⊗ ( (cid:49) ) ⊗ ( N − j ) . (B5)Alternatively, we can write the Jordan-Wigner trans-formation in terms of 2 N real Majorana operators γ j ,related to the standard fermionic operators by ˆ f j =( γ j − +i γ j ) / √
2. It then takes the form γ j − = (cid:32) j − (cid:89) i =1 Z i (cid:33) X j , (B6) γ j = (cid:32) j − (cid:89) i =1 Z i (cid:33) Y j . (B7)Consider the mapping between basis states under thistransformation. The basis decomposition of a pure state, | ψ (cid:105) = (cid:88) n ∈{ , } × N T n | n (cid:105) | n (cid:105) . . . | n N (cid:105) N , (B8)differs between fermions and spins: For the former, thebasis states | n k (cid:105) k = ( ˆ f † k ) n k | (cid:105) k (with a local Fock vac-uum | (cid:105) k ) only commute for b k = 0, while for the latter( | (cid:105) , | (cid:105) ) = ( |↑(cid:105) , |↓(cid:105) ) are simply commuting spin states.Thus, choosing a different ordering under the Jordan-Wigner transformation leads to different fermionic states.Its entanglement entropies S ( A ) = − Tr A ( ρ A log ρ A ),computed from the spectrum of the reduced density ma-trix ρ A = Tr ¯ A ρ , are then generally different as well. Wecan easily see that a partial trace over fermionic sites, unlike spin degrees of freedom, is not commuting:Tr j Tr k ρ = (cid:104) | (1 + ˆ f j )(1 + ˆ f k ) ρ (1 + ˆ f † k )(1 + ˆ f † j ) | (cid:105)(cid:54) = Tr k Tr j ρ = (cid:104) | (1 + ˆ f k )(1 + ˆ f j ) ρ (1 + ˆ f † j )(1 + ˆ f † k ) | (cid:105) . (B9)Now consider a total system of N sites separated into | A | sites in A and | ¯ A | sites in the complement region ¯ A . Inthe simplest case, A and ¯ A are connected regions and ¯ A begins at the first site of the given ordering. By definingthe trace over a subsystem asTr B = Tr B | B | Tr B | B |− . . . Tr B Tr B , (B10) i.e. , tracing out sites in their reverse order, the reduceddensity matrix ρ A is equivalent in both fermions andspins, i.e. , ρ A = Tr ¯ A | ψ (cid:105) (cid:104) ψ | = ( (cid:104) | ¯ A N + (cid:104) | ¯ A N ) . . . ( (cid:104) | ¯ A + (cid:104) | ¯ A ) | ψ (cid:105) + H . c . . (B11)We call this setup, where creation operators appear inproducts with increasing site index and annihilation op-erators in reverse order, the canonical ordering . Allother subsystem choices can be brought into this formby permutation of indices. While a reordering of modesleaves spin states invariant due to their commuting ba-sis, fermionic permutations change the equivalence underJordan-Wigner transformations: Entanglement entropies S ( A ) are only equivalent between spins and fermions forregions A in the canonical ordering.This equivalence can be extended to connected regions A of fermionic Gaussian states. Consider a first Jordan-Wigner transformation where A is in the canonical or-dering, and a second one where spin indices are cyclicallypermuted as i → ( i + 1) mod N . We denote the Majo-rana operators for both transformations as γ k and ˜ γ k ,respectively. They are related as follows: γ ≡ X → X ≡ − i ˜ γ ˜ γ ˜ γ ,γ ≡ Y → Y ≡ − i ˜ γ ˜ γ ˜ γ ,γ ≡ Z X → Z X ≡ − i ˜ γ ˜ γ ˜ γ ,. . .γ N − ≡ Z . . . Z N − Y N − → Z . . . Z N − Y N ≡ − i ˜ γ ˜ γ ˜ γ N ,γ N − ≡ Z . . . Z N − X N → X Z . . . Z N ≡ − i ˜ γ ˜ γ ˆ P tot ˜ γ ,γ N ≡ Z . . . Z N − Y N → Y Z . . . Z N ≡ − i ˜ γ ˜ γ ˆ P tot ˜ γ . (B12)Here, we defined the total parity operator ˆ P tot = (cid:81) i Z i ≡ ( − i) N (cid:81) k ˜ γ k . As Gaussian states are fully character-ized by their covariance matrix entries Γ j,k , we considerhow they change between both Jordan-Wigner transfor-mations. For parity-even states, γ k → − i ˜ γ ˜ γ ˜ γ ˜ k with5˜ k = ( k + 1) mod 2 N , so we findΓ + ij = i2 (cid:104) ψ + | [ γ i , γ j ] | ψ + (cid:105) → ˜Γ + i,j = i2 (cid:104) ψ + | [˜ γ i , ˜ γ j ] | ψ + (cid:105) , (B13)where | ψ + (cid:105) is a Gaussian state vector with ˆ P tot | ψ + (cid:105) = | ψ + (cid:105) . In matrix form, this can be written asΓ + → ˜Γ + = (cid:32) +[1 , , [3 , N ] Γ +[3 , N ] , [1 , Γ +[3 , N ] , [3 , N ] (cid:33) , (B14)where Γ +[ i,j ] , [ k,l ] is the sub matrix consisting of the rowsfrom i , i + 1 up to row j and columns from k , k + 1up to l . For a parity-odd Gaussian state vector | ψ − (cid:105) with P tot | ψ − (cid:105) = − | ψ − (cid:105) , however, the rows and columnscorresponding to the first two Majorana modes are sign-flipped:Γ − → ˜Γ − = (cid:32) − Γ − [1 , , [3 , N ] − Γ − [3 , N ] , [1 , Γ − [3 , N ] , [3 , N ] (cid:33) . (B15)In consequence, only parity-even fermionic Gaussianstates are equivalent under different Jordan-Wignertransformations related by cyclic permutations, whileparity-odd ones acquire a sign flip in the two-point cor-relations related to modes that are moved from the endto the beginning of the fermionic ordering. Fortunately,this sign flip does not affect fermionic entanglement en-tropies S ( A ), which are computed from the spectrum ofthe submatrix of Γ corresponding to sites in A , denotedΓ A .As the eigenvalue spectrum is not affected by change ofsign across entire rows and columns of a matrix, ˜Γ | ˜ A andΓ | A have the same eigenvalues, and hence, S ( A ) = S ( ˜ A )independent of parity. Note that once we consider in-dex transpositions more general than cyclic permuta-tions, the relative fermionic ordering is broken and evenGaussian states are no longer equivalent under differentJordan-Wigner transformations.
2. Ising CFT representations
Here, we would like to discuss subtle differences be-tween the spin model and Majorana fermion model whenwe calculate MI and EoP. Refer also to [83] for an ear-lier detailed analysis on this problem, which is essentiallyequivalent to our argument below.First let us remember that the Ising spin CFT is de-fined from Majorana fermion CFT by the GSO projec-tion [55]. Explicitly the modular invariant torus partitionfunction of the Ising CFT is schematically written as Z Ising = Tr NS (cid:20) − F (cid:21) + Tr R (cid:20) − ( − F (cid:21) , (B16) where the first trace is taken for the NS sector ( H + sec-tor), i.e. , the anti-periodic boundary condition is imposedfor the free fermion on a circle. Also − F is the re-striction to the even fermion number state. The secondone is for the R sector ( H − sector), i.e. , the periodicboundary condition is imposed for the free fermion on acircle. Also − ( − F is the restriction to the odd fermionnumber state. Note that the spin operator σ is includedin the R sector of the Majorana fermion model.In the calculation of entanglement entropy S ( A ∪ B )or its Renyi entropy in Ising CFT, we need to performthis GSO projection of the Majorana fermion along theinterval C between A and B . This is illustrated in theupper pictures in figure 13 by having in mind the cal-culation of Tr[ ρ AB ] as an example, which is essentiallythe 2nd Renyi entropy and which is equivalent to a toruspartition function (B16). If we write the fermion numberon this interval C as F C ( i.e. , F C = (cid:80) i ∈ C ˆ f † i ˆ f i ), then thewave function for the calculation of Renyi entropy shouldbe | Ψ (cid:105) Ising = | Ψ (cid:105) Majorana + ( − F C | Ψ (cid:105) Majorana , (B17)which is depicted in the lower pictures in figure 13 Thisprocedure leads to the spin operator in the spectrum andleads to the expected behavior of MI I ( A : B ) ∼ ( w/d ) / when d (cid:29) w . However, this form (B17) is not includedin our Gaussian ansatz of the numerical calculation. TheGaussian ansatz only takes into account the first termwhich ignores the twisted sector of Majorana fermionnamely the spin operator sector ( i.e. , R sector). Thiscauses the absence of the spin operator excitation andexplains our numerical result I ( A, B ) ∼ ( w/d ) for theMajorana fermion model when d (cid:29) w .Moreover, this difference between the XY spin modeland the Majorana fermion model leads to a significantlydifferent behavior of MI when d (cid:28) w . To see this, asa toy model which mimics the calculation of MI in XYspin model at d = 1, we would like to analyze a threequbit system, whose spins are called A, C and B , fromthe left to right. We consider the following spin state forthis ABC system: | Ψ( x, y, θ ) (cid:105) ACB = 1 √ U ( x ) AC · U ( y ) CB · ( | (cid:105) A | θ (cid:105) C | (cid:105) B + | (cid:105) A | θ (cid:105) C | (cid:105) B ) , (B18)where | θ (cid:105) = cos θ | (cid:105) + sin θ | (cid:105) . The 4 × U ( x ) is defined by | (cid:105) → cos x | (cid:105) − sin x | (cid:105) , | (cid:105) → − sin x | (cid:105) + cos x | (cid:105) , | (cid:105) → cos x | (cid:105) − sin x | (cid:105) , | (cid:105) → − sin x | (cid:105) + cos x | (cid:105) . (B19)Note that U ( x ) AC and U ( y ) CB mimic the entanglementbetween nearest neighbor sites due to the standard inter-actions of spin system.6 ψ ψCut A B A B BA σ σCut
Cut CutA B A B BA [1] NS-sector[2] R-sector
CutA BA B | Ψ 〉 = A B A BCut + ρ AB =Tr 𝑨𝑨𝑨𝑨 [ | Ψ 〉〈 Ψ |]= + FIG. 13. The replica calculation of Tr[ ρ AB ] (left) and thewave function for the GSO projected Majorana fermion CFTwhich is equivalent to the Ising CFT (right). If we consider this state from the viewpoint of Majo-rana fermions, then we need to take into account thatthe fermion in C and the fermion in B do anti-commute.This happens only if there is a fermion in both B and C site. Therefore we have the following rule | p (cid:105) C | q (cid:105) B = ( − ab | q (cid:105) B | p (cid:105) C , (B20)where p = 0 , q = 0 ,
1. We act this rule to | Ψ( x, y, θ ) (cid:105) ACB and obtain the wave function for the Ma-jorana fermion, which is written as | Ψ f ( x, y, θ ) (cid:105) ABC .We compare MI I s ( A : B ) obtained from the spin wavefunction | Ψ( x, y, θ ) (cid:105) ACB , assuming the spin at B andthat at C do commute, with the fermionic one I f ( A, B )calculated from | Ψ f ( x, y, θ ) (cid:105) ABC in figure 14. In the nextsubsection we will explain the same difference from theviewpoint of orderings of traces.First of all, even at x = y = 0 (no nearest neighborentanglement), we find that I f ( A : B ) can be reducedfrom I s ( A : B ). The reduction is maximized at θ = π/ I f ( A : B ) /I s ( A : B ) = 1 /
2. By turningon the neighbor entanglement, we can reduce the fermionMI more. In particular, we find at x = y = π and θ = π we obtain I f ( A, B ) = 0. In summary, we always have I f ( A : B ) /I s ( A : B ) ≤ i.e. , the fermion MI is smallerthan that of spin.Let us consider this difference between the spin calcu-lation and fermion calculation from the viewpoint of two dimensional CFTs. Remember again the computation ofTr ρ AB . In the c = 1 / C . This GSO projection on C picks upthe periodic or anti-periodic boundary condition for thefermion ˆ f † i depending on the even or odd value of thetotal fermion number in C . This projection is automati-cally performed in the spin system calculation. Howeverif we just focus on the Majorana fermion system and ig-nore this phase factor, we do not find the correct GSOprojection or equally the boundary condition to definethe correct CFT partition function. Indeed, in the limit d (cid:28) w we are interested, the entropy S ( A ∪ B ) signifi-cantly depends on this boundary condition of the fermionon C because the size of C is very small. In summary, cal-culations in the spin system correspond to the CFT cal-culations of entanglement entropy and related quantitiessuch as MI and EoP for standard choices of subsystems,while those in the naive Majorana fermion calculationswithout the GSO projection do not. Appendix C: Gaussian entanglement entropy
As explained in section III and V, important measuresof quantum correlations, such as MI and EoP, are con-structed from the bipartite entanglement entropy. Whilethis is hard to evaluate for general quantum states | ψ (cid:105) , itis well-known that the entanglement entropy of a Gaus-sian state | ψ (cid:105) can be computed analytically based on theentanglement entropy. Let us therefore briefly review therespective formulas.A bosonic or fermionic system with N degrees of free-dom is characterized by the linear observables ˆ ξ a ≡ (ˆ q , ˆ p , . . . , ), as introduced in section II. Mathematically,we refer ˆ ξ a forms a basis of the classical phase space V . They satisfy the canonical commutation or anti-commutation relations, namely[ ˆ ξ a , ˆ ξ b ] = iΩ ab , (C1) { ˆ ξ a , ˆ ξ b } = G ab , (C2)where Ω ab is a symplectic form and G ab a positive def-inite metric. For a normalized quantum state | ψ (cid:105) with (cid:104) ψ | ˆ ξ a | ψ (cid:105) = 0, we compute its two-point function as C ab = (cid:104) ψ | ˆ ξ a ˆ ξ b | ψ (cid:105) . (C3)We can always decompose C ab into C ab = 12 ( G ab + iΩ ab ) , (C4)7 θ M u t u a l i n f o r m a t i o n I ( A : B ) I s ( A:B ) I f ( A:B ) FIG. 14. The first graph plots the value of I s ( A : B ) (blue)and I f ( A : B ) (orange) as a function of θ at x = y = 0. Thesecond graph is a 3D plot of the ratio I f ( A : B ) /I s ( A : B )as a function x (horizontal) and y (depth) at θ = π/ I f ( A : B ) /I s ( A : B ) asa function x (horizontal) and y (depth) at θ = π/ where G ab is a symmetric positive definite metric andΩ ab is non-degenerate antisymmetric symplectic form.Here, Ω ab is fixed by (C1) for bosons, while G ab is sim-ilarly fixed by (C2) for fermions. Consequently, onlythe respective other piece, i.e. , G ab for bosons and Ω ab for fermions, which are often called the bosonic andfermionic covariance matrix , respectively, will depend onthe state | ψ (cid:105) . We can define the linear map J ab = G ac Ω − cb . (C5)We refer to the state | ψ (cid:105) as Gaussian if and only if J squares to minus identity, i.e. , | ψ (cid:105) is pure Gaussian state ⇐⇒ J = − (cid:49) , (C6)in which case J is called linear complex structure .The entanglement entropy S ( A ) of a Gaussian state | ψ (cid:105) with complex structure J can be efficiently computed from the restriction J A of J to the respective subsystem A . More precisely, if we have H = H A ⊗ H B with phasespace decomposition V = A ⊕ B . The restriction J A = J [1 , N A ] , [1 , N A ] (C7)represents then the 2 N A -by-2 N A sub matrix of J asso-ciated to the subspace A ⊂ V . While J associated to apure Gaussian state has eigenvalues ± i, its restriction J A will have eigenvalues ± λ i with λ i ∈ [0 , ∞ ) for bosons and λ i ∈ [0 ,
1] for fermions. The entanglement entropy canbe computed from these eigenvalues using the formulas S ( A ) = (cid:40) tr (cid:0) (cid:49) A − i J A log (cid:12)(cid:12) (cid:49) A − i J A (cid:12)(cid:12)(cid:1) (bosons) − tr (cid:0) (cid:49) A − i J A log (cid:12)(cid:12) (cid:49) A − i J A (cid:12)(cid:12)(cid:1) (fermions) , (C8)where we wrote covariant matrix equations, which areequivalent to replacing J A by its eigenvalues and the traceby a sum over them. Such analytical formulas for Gaus-sian entanglement entropy were first derived in [84, 85]based on the covariance matrices and later also phrasedin terms of linear complex structures [86–88].Note that the definition of entanglement is a subtle is-sue. Usually, one decomposes the Hilbert space into aregular tensor product H = H A ⊗ H B , such that oper-ators O A and O B that are only supported in one of thesubsystems are given as O A = ˆ A ⊗ (cid:49) and O B = (cid:49) ⊗ ˆ B ,leading to [ O A , O B ] = 0. For fermions, we would liketo define a similar structure, but independent operatorsshould anticommute, i.e. , {O A , O B } = 0. This requires aslightly different definition of “fermionic tensor product”,as recognized and discussed in [50–52, 89]. We discusssome of the resulting subtleties in the next section. Appendix D: Algorithm’s implementation
Our minimization algorithm has the goal to minimize afunction f ( J ), where J is the linear complex structure ofa pure Gaussian state. Intuitively, we would like to applygradient descent, i.e. , solve the differential equation˙ x µ = − (cid:88) j G µν ( x ) ∂f∂x ν ( x ) , (D1)where G µν is the matrix representation of the Rieman-nian metric on the manifold of states. Using coordinatehas two disadvantages: First, in general it will be diffi-cult or even impossible to find a global coordinate systemdepending on the topology of manifold (in particular, forfermions the topology is non-trivial). Second, the matrix G µν ( x ) will depend on the position and would need to beevaluated at every step, which will slow down the com-putation. Our approach based on the natural Lie groupparametrization avoids both of these disadvantages.Our parametrization is defined relative to an initialstate | J (cid:105) . We then choose a basis of Lie algebra gener-8ators (Ξ µ ) ab satisfying the conditions { Ξ µ , J } = 0 and tr(Ξ µ G Ξ (cid:124) ν g ) = δ µν , K µ Ω = − Ω K (cid:124) µ , (bosons) tr(Ξ µ G Ξ (cid:124) ν g ) = − δ ij , K i G = − GK (cid:124) i , (fermions) (D2)where G ab = (cid:104) J |{ ˆ ξ a , ˆ ξ b }| J (cid:105) and Ω ab = (cid:104) J | [ ˆ ξ a , ˆ ξ b ] | J (cid:105) .One can show [90] that the dimension of this spaceis N ( N + 1)-dimensional for bosons and N ( N − x i , we use the matrix M ab to parametrize all Gaussianstates J = M J M − connected to J . The gradient de-scent equation for M reduces then to dMdt = − M (cid:32)(cid:88) i X µ ( M )Ξ µ (cid:33) . (D3)where the gradient vector is computed as [90] F µ ( M ) = − dds (cid:12)(cid:12)(cid:12)(cid:12) s =0 f (cid:0) M e s Ξ µ J e − s Ξ µ M − (cid:1) . (D4)In our algorithm, we discretize (D3). Starting from the identity M = (cid:49) , we perform individual steps as M n +1 = M n e (cid:15)K n ≈ M n (cid:18) (cid:49) + (cid:15) K n (cid:49) − (cid:15) K n (cid:19) , (D5)where K n = (cid:80) µ F µ ( M n )Ξ µ / (cid:107)F(cid:107) . Here, we approxi-mate the exponential for small (cid:15) in such a way that M n +1 is ensured to lie in the symplectic or orthogonal group forbosons or fermions, respectively, which e (cid:15)K n ≈ (cid:49) + (cid:15)K n would not achieve. At each step, we choose 0 < (cid:15) < f ( M n +1 ) < f ( M n ), which is alwayspossible to achieve for sufficiently small (cid:15) .Our algorithm circumvents the disadvantages of a co-ordinate parametrization by distinguishing between stateand tangent space. While we use M to parametrizeour state | J (cid:105) with J = M J M − , we construct an or-thonormal basis of Lie algebra generators K i which canbe identified with the respective tangent vector of thecurve γ ( s ) = M e s Ξ µ at the point M . This allows us inparticular that our Riemannian metric on the manifoldof Gaussian states is left-invariant, such that the so con-structed tangent vectors are orthonormal at every point.We therefore do not need to compute its matrix represen-tation G µν ( x ), but can instead work with orthonormalframes at every point M . [1] H. Casini and M. Huerta, “Entanglement entropy in freequantum field theory,” J. Phys. A , 504007 (2009),arXiv:0905.2562 [hep-th].[2] Daniel Harlow, “Jerusalem Lectures on Black Holes andQuantum Information,” Rev. Mod. Phys. , 015002(2016), arXiv:1409.1231 [hep-th].[3] Mukund Rangamani and Tadashi Takayanagi, Holo-graphic Entanglement Entropy , Vol. 931 (Springer, 2017)arXiv:1609.01287 [hep-th].[4] Leonard Susskind, “Three Lectures on Complexity andBlack Holes,” (2018) arXiv:1810.11563 [hep-th].[5] Matthew Headrick, “Lectures on entanglement en-tropy in field theory and holography,” (2019),arXiv:1907.08126 [hep-th].[6] P.V. Buividovich and M.I. Polikarpov, “Entanglemententropy in lattice gauge theories,” PoS
CONFINE-MENT8 , 039 (2008), arXiv:0811.3824 [hep-lat].[7] William Donnelly, “Decomposition of entanglement en-tropy in lattice gauge theory,” Phys. Rev. D , 085004(2012), arXiv:1109.0036 [hep-th].[8] Horacio Casini, Marina Huerta, and Jose AlejandroRosabal, “Remarks on entanglement entropy for gaugefields,” Phys. Rev. D , 085012 (2014), arXiv:1312.1183[hep-th].[9] Sudip Ghosh, Ronak M Soni, and Sandip P. Trivedi, “OnThe Entanglement Entropy For Gauge Theories,” JHEP , 069 (2015), arXiv:1501.02593 [hep-th].[10] Sinya Aoki, Takumi Iritani, Masahiro Nozaki, Tokiro Nu-masawa, Noburo Shiba, and Hal Tasaki, “On the defini-tion of entanglement entropy in lattice gauge theories,”JHEP , 187 (2015), arXiv:1502.04267 [hep-th].[11] Shinsei Ryu and Tadashi Takayanagi, “Holographic derivation of entanglement entropy from AdS/CFT,”Phys. Rev. Lett. , 181602 (2006), arXiv:hep-th/0603001.[12] Veronika E. Hubeny, Mukund Rangamani, and TadashiTakayanagi, “A Covariant holographic entanglement en-tropy proposal,” JHEP , 062 (2007), arXiv:0705.0016[hep-th].[13] Aitor Lewkowycz and Juan Maldacena, “General-ized gravitational entropy,” JHEP , 090 (2013),arXiv:1304.4926 [hep-th].[14] Xi Dong, Aitor Lewkowycz, and Mukund Rangamani,“Deriving covariant holographic entanglement,” JHEP , 028 (2016), arXiv:1607.07506 [hep-th].[15] Leonard Susskind, “Entanglement is not enough,”Fortsch. Phys. , 49–71 (2016), arXiv:1411.0690 [hep-th].[16] Leonard Susskind, “Computational Complexity andBlack Hole Horizons,” Fortsch. Phys. , 44–48 (2016),[Fortsch. Phys.64,24(2016)], arXiv:1403.5695 [hep-th].[17] Douglas Stanford and Leonard Susskind, “Complexityand Shock Wave Geometries,” Phys. Rev. D , 126007(2014), arXiv:1406.2678 [hep-th].[18] Adam R. Brown, Daniel A. Roberts, Leonard Susskind,Brian Swingle, and Ying Zhao, “Holographic Complex-ity Equals Bulk Action?” Phys. Rev. Lett. , 191301(2016), arXiv:1509.07876 [hep-th].[19] Adam R. Brown, Daniel A. Roberts, Leonard Susskind,Brian Swingle, and Ying Zhao, “Complexity, action,and black holes,” Phys. Rev. D , 086006 (2016),arXiv:1512.04993 [hep-th].[20] Josiah Couch, Willy Fischler, and Phuc H. Nguyen,“Noether charge, black hole volume, and complexity,” JHEP , 119 (2017), arXiv:1610.02038 [hep-th].[21] Roman Orus, “Advances on Tensor Network Theory:Symmetries, Fermions, Entanglement, and Holography,”Eur. Phys. J. B , 280 (2014), arXiv:1407.6552 [cond-mat.str-el].[22] Shira Chapman, Michal P. Heller, Hugo Marrochio, andFernando Pastawski, “Toward a Definition of Complexityfor Quantum Field Theory States,” Phys. Rev. Lett. ,121602 (2018), arXiv:1707.08582 [hep-th].[23] Ro Jefferson and Robert C. Myers, “Circuit complex-ity in quantum field theory,” JHEP , 107 (2017),arXiv:1707.08570 [hep-th].[24] Luca Bombelli, Rabinder K. Koul, Joohan Lee, andRafael D. Sorkin, “A Quantum Source of Entropy forBlack Holes,” Phys. Rev. D , 373–383 (1986).[25] Mark Srednicki, “Entropy and area,” Phys. Rev. Lett. , 666–669 (1993), arXiv:hep-th/9303048.[26] Jutho Haegeman, Tobias J. Osborne, Henri Verschelde,and Frank Verstraete, “Entanglement Renormalizationfor Quantum Fields in Real Space,” Phys. Rev. Lett. ,100402 (2013), arXiv:1102.5524 [hep-th].[27] Michael A. Nielsen, Mark R. Dowling, Mile Gu, and An-drew C. Doherty, “Quantum computation as geometry,”Science , 1133–1135 (2006), arXiv:quant-ph/0603161[quant-ph].[28] Lucas Hackl and Robert C. Myers, “Circuit com-plexity for free fermions,” JHEP , 139 (2018),arXiv:1803.10638 [hep-th].[29] Shira Chapman, Jens Eisert, Lucas Hackl, Michal P.Heller, Ro Jefferson, Hugo Marrochio, and Robert C.Myers, “Complexity and entanglement for ther-mofield double states,” SciPost Phys. , 034 (2019),arXiv:1810.05151 [hep-th].[30] Christian Weedbrook, Stefano Pirandola, Ra´ul Garc´ıa-Patr´on, Nicolas J. Cerf, Timothy C. Ralph, Jeffrey H.Shapiro, and Seth Lloyd, “Gaussian quantum informa-tion,” Reviews of Modern Physics , 621–669 (2012).[31] Tadashi Takayanagi and Koji Umemoto, “Entangle-ment of purification through holographic duality,” Na-ture Phys. , 573–577 (2018), arXiv:1708.09393 [hep-th].[32] Cesar A. Ag´on, Matthew Headrick, and Brian Swingle,“Subsystem Complexity and Holography,” JHEP , 145(2019), arXiv:1804.01561 [hep-th].[33] Arpan Bhattacharyya, Tadashi Takayanagi, and KojiUmemoto, “Entanglement of Purification in Free ScalarField Theories,” JHEP , 132 (2018), arXiv:1802.09545[hep-th].[34] Arpan Bhattacharyya, Alexander Jahn, TadashiTakayanagi, and Koji Umemoto, “Entanglement ofPurification in Many Body Systems and SymmetryBreaking,” Phys. Rev. Lett. , 201601 (2019),arXiv:1902.02369 [hep-th].[35] Elena Caceres, Shira Chapman, Josiah D. Couch, Juan P.Hernandez, Robert C. Myers, and Shan-Ming Ruan,“Complexity of Mixed States in QFT and Holography,”JHEP , 012 (2020), arXiv:1909.10557 [hep-th].[36] Hugo A. Camargo, Pawel Caputa, Diptarka Das,Michal P. Heller, and Ro Jefferson, “Complexity asa novel probe of quantum quenches: universal scalingsand purifications,” Phys. Rev. Lett. , 081601 (2019),arXiv:1807.07075 [hep-th].[37] Pawel Caputa, Masamichi Miyaji, Tadashi Takayanagi,and Koji Umemoto, “Holographic Entanglement of Pu- rification from Conformal Field Theories,” Phys. Rev.Lett. , 111601 (2019), arXiv:1812.05268 [hep-th].[38] Christoph Holzhey, Finn Larsen, and Frank Wilczek,“Geometric and renormalized entropy in conformal fieldtheory,” Nucl. Phys. B , 443–467 (1994), arXiv:hep-th/9403108.[39] Pasquale Calabrese and John L. Cardy, “Entanglemententropy and quantum field theory,” J. Stat. Mech. ,P06002 (2004), arXiv:hep-th/0405152 [hep-th].[40] John Cardy, “Some results on the mutual informationof disjoint regions in higher dimensions,” J. Phys. A46 ,285402 (2013), arXiv:1304.7985 [hep-th].[41] Tomonori Ugajin, “Mutual information of excited statesand relative entropy of two disjoint subsystems in CFT,”JHEP , 184 (2017), arXiv:1611.03163 [hep-th].[42] Raimond Abt, Johanna Erdmenger, Haye Hinrichsen,Charles M. Melby-Thompson, Ren´e Meyer, ChristianNorthe, and Ignacio A. Reyes, “Topological Complex-ity in AdS /CFT ,” Fortsch. Phys. , 1800034 (2018),arXiv:1710.01327 [hep-th].[43] Raimond Abt, Johanna Erdmenger, Marius Gerber-shagen, Charles M. Melby-Thompson, and ChristianNorthe, “Holographic Subregion Complexity from Kine-matic Space,” JHEP , 012 (2019), arXiv:1805.10298[hep-th].[44] Roberto Auzzi, Stefano Baiguera, Andrea Legramandi,Giuseppe Nardelli, Pratim Roy, and Nicol`o Zenoni,“On subregion action complexity in AdS and in theBTZ black hole,” JHEP , 066 (2020), arXiv:1910.00526[hep-th].[45] Luis Lehner, Robert C. Myers, Eric Poisson, andRafael D. Sorkin, “Gravitational action with null bound-aries,” Phys. Rev. D , 084046 (2016), arXiv:1609.00207[hep-th].[46] Alan Reynolds and Simon F. Ross, “Divergences in Holo-graphic Complexity,” Class. Quant. Grav. , 105004(2017), arXiv:1612.05439 [hep-th].[47] P. Di Francesco, P. Mathieu, and D. Senechal, ConformalField Theory , Graduate Texts in Contemporary Physics(Springer-Verlag, New York, 1997).[48] Pascual Jordan and Eugene Paul Wigner, “¨uber daspaulische ¨aquivalenzverbot,” in
The Collected Works ofEugene Paul Wigner (Springer, 1993) pp. 109–129.[49] Lev Vidmar and Marcos Rigol, “Generalized gibbs en-semble in integrable lattice models,” Journal of Statis-tical Mechanics: Theory and Experiment , 064007(2016).[50] Pawe(cid:32)l Caban, Krzysztof Podlaski, Jakub Rembielinski,Kordian A Smolinski, and Zbigniew Walczak, “En-tanglement and tensor product decomposition for twofermions,” Journal of Physics A: Mathematical and Gen-eral , L79 (2005).[51] Mari-Carmen Banuls, J Ignacio Cirac, and Michael MWolf, “Entanglement in fermionic systems,” Physical Re-view A , 022311 (2007).[52] Nicolai Friis, Antony R Lee, and David Edward Br-uschi, “Fermionic-mode entanglement in quantum infor-mation,” Physical Review A , 022338 (2013).[53] Andrea Coser, Erik Tonni, and Pasquale Calabrese,“Spin structures and entanglement of two disjoint in-tervals in conformal field theories,” Journal of Statisti-cal Mechanics: Theory and Experiment , 053109(2016).[54] S. Elitzur, E. Gross, E. Rabinovici, and N. Seiberg, “As- pects of Bosonization in String Theory,” Nucl. Phys. B , 413–432 (1987).[55] Paul H. Ginsparg, “APPLIED CONFORMAL FIELDTHEORY,” in Les Houches Summer School in Theoreti-cal Physics: Fields, Strings, Critical Phenomena (1988)pp. 1–168, arXiv:hep-th/9108028.[56] Sagar Fakirchand Lokhande and Sunil Mukhi, “Modu-lar invariance and entanglement entropy,” JHEP , 106(2015), arXiv:1504.01921 [hep-th].[57] Pasquale Calabrese and John Cardy, “Entanglement en-tropy and conformal field theory,” J. Phys. A , 504005(2009), arXiv:0905.4013 [cond-mat.stat-mech].[58] Pasquale Calabrese, John Cardy, and Erik Tonni, “En-tanglement entropy of two disjoint intervals in confor-mal field theory,” J. Stat. Mech. , P11001 (2009),arXiv:0905.2069 [hep-th].[59] H. Casini, C. D. Fosco, and M. Huerta, “Entangle-ment and alpha entropies for a massive Dirac field intwo dimensions,” J. Stat. Mech. , P07007 (2005),arXiv:cond-mat/0505563 [cond-mat].[60] Barbara M. Terhal, Micha(cid:32)l Horodecki, Debbie W. Leung,and David P. DiVincenzo, “The entanglement of purifi-cation,” Journal of Mathematical Physics , 4286–4298(2002).[61] Bennet Windt, Alexander Jahn, Jens Eisert, and LucasHackl, “Local optimization on pure gaussian state man-ifolds,” unpublished (2020).[62] Lucas Hackl and Robert H Jonsson, “Minimal energy costof entanglement extraction,” Quantum , 165 (2019).[63] Phuc Nguyen, Trithep Devakul, Matthew G. Halbasch,Michael P. Zaletel, and Brian Swingle, “Entangle-ment of purification: from spin chains to hologra-phy,” Journal of High Energy Physics (2018),10.1007/jhep01(2018)098.[64] Pawel Caputa, Nilay Kundu, Masamichi Miyaji, TadashiTakayanagi, and Kento Watanabe, “Anti-de SitterSpace from Optimization of Path Integrals in ConformalField Theories,” Phys. Rev. Lett. , 071602 (2017),arXiv:1703.00456 [hep-th].[65] Giuseppe Di Giulio and Erik Tonni, “Complexity ofmixed Gaussian states from Fisher information geome-try,” (2020), arXiv:2006.00921 [hep-th].[66] Rifath Khan, Chethan Krishnan, and Sanchita Sharma,“Circuit Complexity in Fermionic Field Theory,” Phys.Rev. D , 126001 (2018), arXiv:1801.07620 [hep-th].[67] Shira Chapman, Hugo Marrochio, and Robert C. Myers,“Complexity of Formation in Holography,” JHEP , 062(2017), arXiv:1610.08063 [hep-th].[68] Mohsen Alishahiha, “Holographic Complexity,” Phys.Rev. D , 126009 (2015), arXiv:1509.06614 [hep-th].[69] Dean Carmi, Robert C. Myers, and Pratik Rath,“Comments on Holographic Complexity,” JHEP , 118(2017), arXiv:1612.00433 [hep-th].[70] Omer Ben-Ami and Dean Carmi, “On Volumes of Sub-regions in Holography and Complexity,” JHEP , 129(2016), arXiv:1609.02514 [hep-th].[71] Bartlomiej Czech, Joanna L. Karczmarek, FernandoNogueira, and Mark Van Raamsdonk, “The GravityDual of a Density Matrix,” Class. Quant. Grav. ,155009 (2012), arXiv:1204.1330 [hep-th].[72] Aron C. Wall, “Maximin Surfaces, and the StrongSubadditivity of the Covariant Holographic Entangle-ment Entropy,” Class. Quant. Grav. , 225007 (2014),arXiv:1211.3494 [hep-th]. [73] Matthew Headrick, Veronika E. Hubeny, AlbionLawrence, and Mukund Rangamani, “Causality & holo-graphic entanglement entropy,” JHEP , 162 (2014),arXiv:1408.6300 [hep-th].[74] Mohsen Alishahiha, Komeil Babaei Velni, and M. RezaMohammadi Mozaffar, “Black hole subregion actionand complexity,” Phys. Rev. D , 126016 (2019),arXiv:1809.06031 [hep-th].[75] Henry Stoltenberg, “Properties of the (Un)Complexityof Subsystems,” Phys. Rev. D , 126012 (2018),arXiv:1807.05218 [quant-ph].[76] S. Marcovitch, A. Retzker, M.B. Plenio, and B. Reznik,“Critical and noncritical long-range entanglement inKlein-Gordon fields,” Phys. Rev. A , 012325 (2009),arXiv:0811.1288 [quant-ph].[77] H. Casini and M. Huerta, “Entanglement and alphaentropies for a massive scalar field in two dimen-sions,” J. Stat. Mech. , P12012 (2005), arXiv:cond-mat/0511014.[78] G. Vidal and R.F. Werner, “Computable measureof entanglement,” Phys. Rev. A , 032314 (2002),arXiv:quant-ph/0102117.[79] Pawel Caputa, Nilay Kundu, Masamichi Miyaji, TadashiTakayanagi, and Kento Watanabe, “Liouville Actionas Path-Integral Complexity: From Continuous Ten-sor Networks to AdS/CFT,” JHEP , 097 (2017),arXiv:1706.07056 [hep-th].[80] Hugo A. Camargo, Michal P. Heller, Ro Jefferson, andJohannes Knaute, “Path integral optimization as cir-cuit complexity,” Phys. Rev. Lett. , 011601 (2019),arXiv:1904.02713 [hep-th].[81] Lev Vidmar, Lucas Hackl, Eugenio Bianchi, and Mar-cos Rigol, “Volume law and quantum criticality in theentanglement entropy of excited eigenstates of the quan-tum ising model,” Physical review letters , 220602(2018).[82] Lucas Hackl, Lev Vidmar, Marcos Rigol, and Euge-nio Bianchi, “Average eigenstate entanglement entropyof the xy chain in a transverse field and its universalityfor translationally invariant quadratic fermionic models,”Physical Review B , 075123 (2019).[83] Andrea Coser, Erik Tonni, and Pasquale Calabrese,“Spin structures and entanglement of two disjoint inter-vals in conformal field theories,” J. Stat. Mech. ,053109 (2016), arXiv:1511.08328 [cond-mat.stat-mech].[84] Rafael D Sorkin, “On the entropy of the vacuum out-side a horizon,” in Tenth International Conference onGeneral Relativity and Gravitation (held in Padova, 4-9July, 1983), Contributed Papers , Vol. 2 (1983) pp. 734–736, arXiv:1402.3589.[85] Ingo Peschel, “Calculation of reduced density matricesfrom correlation functions,” Journal of Physics A: Math-ematical and General , L205 (2003).[86] Eugenio Bianchi, Lucas Hackl, and Nelson Yokomizo,“Entanglement entropy of squeezed vacua on a lattice,”Physical Review D , 085045 (2015).[87] Lev Vidmar, Lucas Hackl, Eugenio Bianchi, and MarcosRigol, “Entanglement entropy of eigenstates of quadraticfermionic hamiltonians,” Physical review letters ,020601 (2017).[88] Lucas Fabian Hackl, “Aspects of gaussian states: Entan-glement, squeezing and complexity,” (2018).[89] Szil´ard Szalay, Zolt´an Zimbor´as, Mih´aly M´at´e, Gergely1