Entanglement Rearrangement in Self-Consistent Nuclear Structure Calculations
IINT-PUB-20-029
Entanglement Rearrangement in Self-Consistent Nuclear Structure Calculations
Caroline Robin, ∗ Martin J. Savage, † and Nathalie Pillet Institute for Nuclear Theory, University of Washington, Seattle, WA 98195, USA. CEA, DAM, DIF, F-91297 Arpajon, France. (Dated: July 21, 2020)
Background:
Entanglement plays a central role in a diverse array of increasingly important research areas,including quantum computation, simulation, measurement, sensing, and communication. Extensive suites ofinvestigations have been performed to better understand entanglement in atomic and molecular quantum many-body systems, while the exploration of entanglement in the structure of nuclei and their reactions is presently inits infancy.
Purpose:
The goal of this work is to begin investigating the entanglement properties of nuclei from first prin-ciples nuclear many-body calculations. We attempt to identify common features and emergent structures ofentanglement that could ultimately lead to new and natural many-body schemes. With an eye toward quantumaccelerators in future hybrid-supercomputers, criteria for partitioning nuclear many-body calculations into quan-tum and classical components may provide advantages in future large-scale computations. Along the way we lookfor explanations of the relative success of phenomenological models such as the nuclear shell model, and for betterways to match to low-energy nuclear effective field theories and lattice QCD calculations to nuclear many-bodytechniques that are based upon entanglement.
Method:
We explore the entanglement between single-particle states in He and He. The patterns of entan-glement emerging from different single-particle bases are compared, and possible links with the convergence ofobservable are explored, in particular, ground-state energies. The nuclear wavefunctions are obtained by per-forming active-space no-core configuration-interaction calculations using a two-body nucleon-nucleon interactionderived from chiral effective field theory. Entanglement measures within single-particle bases exhibiting differentdegrees of complexity are determined, in particular, harmonic oscillator (HO), Hartree-Fock (HF), natural (NAT)and variational natural (VNAT) bases. Specifically, single-orbital entanglement entropy, two-orbital mutual in-formation and negativity are studied.
Results:
The entanglement structures in He and He are found to be more localized within NAT and VNATbases than within a HO basis for the optimal HO parameters we have worked with. In particular a core-valencestructure clearly emerges from the full no-core calculation of He. The two-nucleon mutual information showsthat the VNAT basis, which typically exhibits good convergence properties, effectively decouples the active andinactive spaces.
Conclusions:
Measures of one- and two-nucleon entanglement are found to be useful in analyzing the structure ofnuclear wave functions, in particular the efficacy of basis states, and may provide useful metrics toward developingmore efficient schemes for ab initio computations of the structure and reactions of nuclei, and quantum many-bodysystems more generally.
I. INTRODUCTION
Developing quantitative first-principles predictive capa-bilities for computing the structure and reactions of nu-clei remains a grand challenge in nuclear physics re-search. From a fundamental standpoint, nuclei emergefrom quantum chromodynamics (QCD) [1, 2] and thestandard model of electroweak interactions [3–5] at low-energies, and display a delicate balance between classicaland quantum physics. A compelling explanation remainsto be uncovered for why nuclei can be approximatelydescribed by collections of nucleons with a hierarchy oftwo-, three- and higher-body forces, rather than a singlecomposite of quarks and gluons. Lattice QCD [6–8] cal-culations have shown that this emergence persists over asignificant range of standard model parameters beyondthe physical light quark masses [9, 10]. ∗ [email protected] † [email protected] Nuclear structure calculations have advanced dramati-cally since the 1970’s by building interactions around theapproximate global chiral symmetries of QCD [11–13],by utilizing renormalization group techniques [14–17] toeffectively smooth the short-range nature of the nuclearinteractions (and electroweak or beyond standard modeloperators) in a way that is consistent with flowed nuclearmany-body wavefunctions, by major advances in high-performance computing, and advances in algorithms forcomputing quantities related to nuclear many-body sys-tems. Before these advances, it was thought that reliablecalculations would not be achievable because, for exam-ple, the repulsive-core of NN interactions coupled largenumbers of many-body states, rendering a converged di-agonalization of the many-body Hamiltonian impractical.These new capabilities are enabling precision calculationsof properties of light and medium nuclei, see for exampleRefs. [18–34].In ways closely resembling effective field theory (EFT)constructions in perturbative quantum field theories(QFTs), the development of low-energy effective theories a r X i v : . [ nu c l - t h ] J u l of nuclei that faithfully reproduce low-energy observablesand formulated in terms of effective Hamiltonians is con-ceptually well understood. Such constructions are com-plicated by the nonperturbative nature of the nuclear sys-tems, and the role of induced multi-nucleon forces, witha power-counting that is more complicated and less obvi-ous than EFTs, and invariance under systematic changesto the model space highlight the evolution of relevant op-erator structures, see for example, Refs. [17, 35, 36]. Thefaithful reproduction of results requires including all ofthe relevant low-energy degrees of freedom in the activemodel space, and short-distance operators alone cannotsubstitute. Therefore, using the appropriate effective sin-gle nucleon states, or states that are perturbatively close,that is to say that choosing a ”good” single nucleon basis,significantly impacts the cost of numerical computation(see, for example, Ref. [37]) and accuracy of results.Efforts to better understand the role of quantum in-formation and entanglement in quantum many-body sys-tems and quantum field theories have begun. These in-clude investigations of information (Shannon) and vonNeumann entropies to study the complexity of nuclearstates and chaotic behavior in nuclei [38–42]. In higherenergy processes, the role of entanglement in dynamicalprocesses related to QCD, such as fragmentation [43–45],heavy-ion collisions [46–49], and deep inelastic scatter-ing [50, 51] is being examined, and suggestive hints havebeen found in the results of experiment [52]. Recently,it has been shown that chiral symmetry and entangle-ment are interconnected in describing the decompositionof the nucleon spin [53]. Further, there are indicationsthat entanglement may play an important role in thepower counting hierarchy of effective theories of nuclearforces [54]. In particular, it is found that entanglementpreserving low-energy strong interactions lead to en-hanced global emergent spin-flavor symmetries [54], suchas Wigner’s SU(4) symmetry for two light quarks [55] andSU(16) symmetry for three light quarks [56], consistentwith t-channel exchanges of the σ -field with I = J = 0.These are symmetries beyond those present in the QCDLagrange density, and also beyond those predicted in thelarge- N c limit of QCD [57, 58]. Exploiting such emergentsymmetries can be used to mitigate the sign problem inMonte Carlo studies of light nuclei, through a two-stepalgorithm, with and without the symmetry-violating in-teractions, using adiabatic projection techniques [59, 60].The connections between entanglement, symmetries andsign problems are manifest in these computations, and re-main to be better understood. The appearance of entan-glement hierarchy in nuclear effective field theories thatcould be more fundamental than the expansion parame-ters that have so far been identified, i.e. momentum andquark masses, motivates us, in part, to explore the entan-glement structure of nuclear many-body systems. This isto begin to establish a phenomenological features of theentanglement structure of nuclei, to attempt to identifya better organizational many-body scheme, and to possi-bly uncover a connection between the nuclear EFT and many-body entanglement structures.There has been remarkable progress during the lasttwenty years in understanding fundamental aspectsof entanglement in quantum many-body systems andquantum field theories. The concepts of bound- anddistillable-entanglement, of importance for quantumcommunication and also for understanding the natureof quantum systems, are two such developments. Use-ful measures of bi-partite and multi-partite entanglementhave been developed that have different sensitivities tothese forms of entanglement, for example, entanglemententropy, mutual information, negativity, log-negativity,tangle and concurrence, see, for example, Refs. [61–67].In atomic nuclei, various partitions can be applied to thewave function, providing information on the nature of en-tanglement between different components of the nucleus.For example, the nuclear wavefunction can be written asa superposition of tensor products of proton and neutronconfigurations, leading to a natural bi-partitioning of thenuclear state to investigate entanglement between protonand neutron subsystems. A first study of this type of en-tanglement was recently undertaken in inspiring work byGorton and Johnson [68, 69]. They explored the behaviorof the corresponding von Neumann entropy in an effort toidentify the Slater determinants that dominate the entan-glement between the two sectors. In the context of phe-nomenological shell-model calculations, they found thatthe entanglement entropy decreases with growing isospinasymmetry, and increases with excitation energy. Theyare moving toward identifying and using a ”weak entan-glement approximation” for computational purposes.It is interesting to also investigate the entanglement be-tween single nucleon states. Such entanglement is con-ceptually more challenging due to indistinguishability incollections of protons and collections of neutrons. Thisis a well-known and still debated issue [65, 70–72]. Itlies in the fact, that, when dealing with identical par-ticles, the Hilbert space formulation of the many-bodystate does not have a tensor-product structure, whichprevents partitioning of the system. One solution is towork in the Fock space formulation (occupation numberrepresentation) and define entanglement between single-particle states, rather than between single particles [70].In this formulation, entanglement naturally depends onthe single-particle basis used to define the system, andexcludes entanglement due to the antisymmetry of thewave function. This form of ”orbital entanglement” or”mode entanglement” has been investigated in atomicand molecular systems (see, for example, Refs.[73, 74]).The entanglement between single-particle states in Gein the framework of density matrix renormalization group(DMRG) using a phenomenological shell model interac-tion [75] is one of the few studies that have been per-formed on this topic in nuclei.In the present work, we investigate orbital entanglementin the context of configuration-interaction calculations oflight Helium nuclei, He and He, using an interactionderived from chiral effective field theory ( χ EFT). Sincepractical nuclear structure calculations typically requiretruncations of the many-body wavefunction, it is knownthat the nature of the underlying single-particle basis isimportant as it can potentially accelerate the convergenceof observables, such as energies and radii, with respect tothe size of the model space [76–80]. In this context, itis therefore interesting to explore the connection betweenthe quality of a single-nucleon basis and its entanglementproperties. In this work, we establish underlying pat-terns of entanglement between single-particle states inbases used for nuclear structure calculations, specificallythe harmonic oscillator (HO), Hartree-Fock (HF), ”Nat-ural” (NAT) and ”Variational Natural” (VNAT) bases.In particular, we explore relations between the conver-gence of the ground-state energy and the containment ofentanglement within the active model space. We focuson the distribution of single-nucleon-state entanglemententropy, two-nucleon-state mutual information and neg-ativity to reflect bound and distillable entanglement.
II. MEASURES OF ENTANGLEMENT IN HE The eigenstates of nuclei, denoted as | Ψ (cid:105) , can be writ-ten as linear combinations of Slater determinants | φ α (cid:105) of nucleon wavefunctions, which can be decomposed intoneutron ( ν ) and proton ( π ) components: | Ψ (cid:105) = (cid:88) α A α | φ α (cid:105) (1) ≡ (cid:88) α π α ν A α π α ν | φ α π (cid:105) ⊗ | φ α ν (cid:105) . (2)Each Slater determinant | φ α (cid:105) represents a configurationof nucleons in a basis of single-particle states { i } : | φ α (cid:105) = (cid:89) i ∈ α a † i | (cid:105) , (3)where | (cid:105) denotes the true particle vacuum. The ba-sis states are denoted by their quantum numbers { i } = { n i , l i , j i , m i , τ i } (principal quantum number, orbital an-gular momentum (AM), total AM, total AM projection,and isospin projection), and can be, for example, statesof a HO, or states associated with a self-consistent po-tential. We will refer to these basis states alternativelyas single-particle states, or orbitals . To easily access themeasures of entanglement we are focusing on, the expan-sion in Eq. (1) is rewritten in terms of the single-particleoccupation states. In this occupation number formalismthe Slater determinants read | φ α (cid:105) = | n α n α ...n αN (cid:105) ≡ N (cid:79) i =1 | n αi (cid:105) , (4) Note that the terminology used here differs from the one usedin quantum chemistry studies, such as Refs. [73, 74], in which”orbitals” can usually be doubly occupied. where N is the total number of single-particle states, and n αi is the occupation number of state i in the configura-tion | φ α (cid:105) , i.e. (cid:26) n αi = 0 if i is empty in configuration | φ α (cid:105) n αi = 1 if i is occupied in configuration | φ α (cid:105) . The sum of the occupation numbers is equal to the totalnumber of nucleons, n + ....n N = A . The many-bodywavefunction can then be written as | Ψ (cid:105) = (cid:88) α A α | n α n α ...n αN (cid:105) , ≡ (cid:88) n ...n N A n ...n N | n n ...n N (cid:105) . (5)In defining the density matrix, it is convenient to considerthe Hilbert space of a nucleus in terms of three spaces, A , B and C , that span the entire space. For the purposes ofthis work, while more general assignments can be made,we assign A and B to be single-nucleon basis states, while C includes the remaining states. The nuclear wavefunc-tion yields a density operator ˆ ρ ABC . To determine two-nucleon measures of entanglement, the states in C aretraced over, to give ˆ ρ AB = Tr C [ˆ ρ ABC ]. Similarly, to de-termine single-nucleon measures of entanglement, the B space is traced over, ˆ ρ A = Tr B [ˆ ρ AB ]. The von Neumannentanglement entropy associated with a density matrixˆ ρ A is defined as S (ˆ ρ A ) = − Trˆ ρ A log ˆ ρ A . The mutualinformation (MI) between states A and B is defined as I ( A : B ) = S (ˆ ρ A ) + S (ˆ ρ B ) − S (ˆ ρ AB ), and representsa measure of classical and quantum correlations, boundentanglement and distillable entanglement. An upperlimit to the distillable entanglement in A and B is de-fined by the negativity N (ˆ ρ ) = ( || ˆ ρ Γ A AB || − /
2, and therelated logarithmic-negativity, E N (ˆ ρ AB ) = log || ˆ ρ Γ A AB || ,where || ˆ ρ Γ A AB || is the 1-norm of the partial transpose ofthe density matrix with respect to A . A. Methods
Many-body configurations | φ α (cid:105) are selected in terms ofa truncation in the number of shells of the single-particlebasis. That is, we include all possible configurations inan active model space containing a given number of ma-jor shells N tot , that is varied. The calculations are per-formed using a two-body interaction derived from χ EFT.In particular, the bare NNLO opt interaction [81], withcounterterms that have been fit in order to minimize ex-plicit three-body forces is used. Starting from a HO basis,the goal of our study is to investigate how entanglementevolves and rearranges while optimizing and modifyingthe single-particle states. In particular, measures of en-tanglement are investigated when the nuclear state | Ψ (cid:105) is expanded in1. a HO single-particle basis;2. a HF single-particle basis obtained by perform-ing an a priori Hartree-Fock calculation using theNNLO opt interaction;3. a ”natural” (NAT) basis that diagonalizes the one-body density matrix γ ij = (cid:104) Ψ | a † j a i | Ψ (cid:105) . This basisis obtained by performing a diagonalization of thetwo-body Hamiltonian matrix in the many-bodyconfiguration space spanned by a HO basis (with N tot shells). Once the expansion coefficients {A α } in Eq. (5) are obtained, the one-body density iscomputed and diagonalized to obtain the ”natural”single-particle states;4. what we will refer to as ”variational natural”(VNAT) basis. This basis is obtained by applyinga variational principle to the energy of the corre-lated state | Ψ (cid:105) , with respect to the single-particleorbitals. This leads to a non-linear equation wherethe one-nucleon states incorporate the effect of two-body correlations. Specifically, (cid:104) ˆ h ( γ ) , ˆ γ (cid:105) = ˆ G ( σ ) , (6)is solved. In Eq. (6), ˆ h ( γ ) is a general mean-fieldHamiltonian: h ij ( γ ) = K ij + (cid:88) kl (cid:104) ik | (cid:101) V NN | jl (cid:105) γ lk , (7)where K denotes the intrinsic kinetic energy and (cid:101) V NN the anti-symmetrized two-body interaction. σ denotes the two-body correlation matrix of thestate | Ψ (cid:105) : σ il,jk = (cid:104) Ψ | a † i a † j a k a l | Ψ (cid:105) − γ li γ kj + γ lj γ ki , (8)and G ( σ ) is the source term containing the effect oftwo-body correlations beyond the mean-field h ( γ ): G ( σ ) ij = 12 (cid:88) klm σ ki,lm (cid:104) kl | (cid:101) V NN | jm (cid:105)− (cid:88) klm (cid:104) ik | (cid:101) V NN | lm (cid:105) σ jl,km . (9)The single-particle states are taken as eigenfunc-tions of the one-body density γ which satisfies thevariational equation given in Eq. (6). As Eq. (6) iscoupled to Eq. (5), both equations are solved iter-atively until convergence of the system is achieved.More details on the practical procedure can befound in Ref. [76]. We note that this approach isusually called multi-configurational self-consistentfield (MCSCF) or multi-configurational Hartree-Fock (MCHF) in quantum chemistry. When the model space involves a truncation of the single-particle basis, the ordering of the orbitals matters. In thecalculations, the HO states are ordered by their quan-tum numbers by increasing values of N = 2 n + l anddecreasing angular momentum j . The HF states areordered by increasing single-particle energies, while theNAT and VNAT orbitals are ordered by decreasing oc-cupation numbers.For our calculations, all single-particle bases are ex-panded on a set of 7 HO shells with frequency (cid:126) Ω = 30MeV. This value was found to be the optimal frequency interms of energy minimization when expanding the wave-function on the HO basis.
B. Entanglement entropy of single-particle states
It is interesting to start by evaluating the entanglementof one single-nucleon state, or orbital, ( A ≡ i ) with therest of the basis ( B ∪ C ), within the nuclear ground state.This is achieved by calculating the single-orbital reduceddensity matrix ρ ( i ) , which can be obtained by performingpermutations in Eq. (5): | Ψ (cid:105) = (cid:88) n ...n i ...n N A n ...n i ...n N | n n ...n i ...n N (cid:105) = (cid:88) n ...n i ...n N A n ...n i ...n N × ϕ i ×| n n ...n i − n i +1 ...n N (cid:105) ⊗ | n i (cid:105)≡ (cid:88) n i ,BC A BCn i × ϕ i | BC (cid:105) ⊗ | n i (cid:105) , (10)where BC ≡ ( n n ...n i − n i +1 ...n N ) and ϕ i is the phaseresulting from the permutation.The one-orbital reduced density matrix ρ ( i ) n i ,n (cid:48) i (i.e. thematrix elements of ˆ ρ A ) becomes ρ ( i ) n i ,n (cid:48) i = (cid:88) BC (cid:104) BC | (cid:104) n i | Ψ (cid:105) (cid:104) Ψ | n (cid:48) i (cid:105) | BC (cid:105) . (11) ρ ( i ) is then simply a 2x2 matrix with elements that can bewritten in terms of the diagonal elements of one-nucleondensity matrix γ ii = (cid:104) Ψ | a † i a i | Ψ (cid:105) as ρ ( i ) = (cid:18) − γ ii γ ii (cid:19) , (12)in the occupation number basis, | n i (cid:105) = {| (cid:105) , | (cid:105)} . Thederivation of Eq. (12) is given in Appendix A. The single-orbital entanglement entropy, S (1) i , characterizing the en-tanglement between single-particle state i and the otherorbitals in the nucleus, is S (1) i = − Tr (cid:104) ρ ( i ) ln ρ ( i ) (cid:105) = − (cid:88) k =1 ω ( i ) k ln ω ( i ) k , (13)where ω ( i ) k are the eigenvalues of ρ ( i ) .
2s 1d S ( ) i HO1s 2s1p
2s 1d S ( ) i HF1s 2s 1p
2s 1d S ( ) i NATorder of the calculation1s 1p
2s 1p
2s 1d (ordered by quantum numbers) S ( ) i single-particle state i VNATorder of the calculation1s 1p
2s 1d (ordered by decreasing S (1)i )single-particle state i FIG. 1: Single-orbital entanglement entropy S (1) i of theHO, HF, NAT and VNAT single-neutron states i = ( n i , l i , j i , m i , τ i = − /
2) in He, obtained with amodel space of 3 shells. The left panels shows S (1) ordered by the states in the HO basis, while the rightpanels correspond to ordering by decreasing values of S (1) , which coincides with the ordering of thecalculation (by occupation number) for the NAT andVNAT bases.The entanglement entropy acquires its maximumvalue, ln(2), when the single-particle state i has anoccupation number of , and vanishes when the stateis fully occupied or empty. Therefore, if the nuclearstate | Ψ (cid:105) reduces to a single Slater determinant (in thesingle-particle basis { i } ), the entanglement entropy S (1) is zero in that basis.Figure 1 shows the entanglement entropies S (1) of the active neutron single-particle states in the Heground state wavefunction obtained with the HO, HF,NAT and VNAT bases, with a model space of N tot = 3active shells. Due to the spherical symmetry, the entan-glement entropy of states with same nlj and differentAM projections m i are equal: S (1) nlj,m i ≡ S (1) nlj . Theentropies of the proton orbitals are very similar to theneutron ones and are not shown. To allow for a directcomparison, the states on the left panels have beenordered by quantum numbers (decreasing N = 2 l + 1and increasing j , corresponding to the ordering ofthe HO states). On the right panels the states areordered by decreasing values of S (1) . We note thatin the practical calculation with the NAT and VNATbases, the states are ordered by decreasing occupationnumbers, which naturally coincides with the ordering in decreasing S (1) . Typically, the entanglement entropy ofa state with given quantum numbers is smaller in theNAT and VNAT bases than in the HO basis, and, inparticular, the entropy of the VNAT 1 s and 2 s orbitalsare importantly decreased compared to the HO states.The HF states exhibit small entanglement entropy forthis truncation of the model space. We will commentmore extensively on the HF basis in the next paragraph.Figure 2 shows the evolution of the entanglemententropy S (1) nlj as the size of the model space is variedfrom two to seven major shells. In this figure the statesare ordered by quantum numbers. We note that dueto mixing of high-lying single-particle states duringthe self-consistent procedure, and in order to have aconsistent truncation between the different bases themodel space with ”6 major shells” includes the first114 single-particle states. This means that in the HObasis the 4 s subshell is included in that model space.Examining Fig. 2, in an active space comprising only N tot = 2 major shells, the single orbital entanglemententropy is underestimated. This is because such a smallmodel space cannot accommodate sufficient correlation,and the wave function resembles a Slater determinant.When increasing the number of active shells, the con-clusions drawn for Fig. 1 also apply. The entanglemententropy appears to stabilize somewhat more rapidly withincreasing size of the active space in the NAT and VNATbases, compared to the HO basis. This can be seen moreclearly in Table I, which shows the sum of one-orbitalentanglement entropy over the active single-particlestates S (1) tot = (cid:80) i S (1) i . In the HO basis, S (1) tot is found tofluctuate somewhat with increasing model space, evenfor modestly large numbers of shells. In the NAT andVNAT basis the total entropy starts to stabilize witha model space of 4 shells around a value of 1.0, and isfound to be systematically smaller compared with otherbases. In fact, it was shown in Ref. [82] that S (1) tot isminimized in the eigenbasis of the one-body density γ .Keep in mind that, in the present set of calculations, thefull configuration space is exhausted for an active spaceof 7 shells. In that case Eq. (6) is automatically fulfilled,and the NAT and VNAT bases coincide.Overall the HF basis exhibits non-convergent be-havior. The entanglement entropy of the lowest HFsingle-particle states (in terms of ordering by quantumnumbers) is consistently underestimated compared tothe other bases for model spaces of 2,3, and 4 shells.However a jump occurs when including a fifth shell.This can be somewhat understood from the compositionof the nuclear state, which exhibits a large 0p-0hcomponent ( ≈ − ≈
70% when the 5 th shell is included, andto ≈
53% when including the 6 th shell. The 3 p and4 s shells are pushed out to the end of the basis duringthe HF calculations, and are not present in the modelspace with 6 shells, contrarily to the other bases. Thisis also found when using softer interactions. The 0p-0h ordering by quantum numbers active space = 2 major shells (s,p) S ( ) n lj HOHFNATVNAT active space = 3 major shells (s,p,sd) S ( ) n lj active space = 4 major shells (s,p,sd,fp) S ( ) n lj active space = 5 major shells (s,p,sd,fp,sdg)0.41 S ( ) n lj active space = 6 major shells0.56 S ( ) n lj
1s 1p
2s 1d
3s 2d
4s 3d active space = 7 major shells S ( ) n lj FIG. 2: Single-orbital entanglement entropy S (1) nljm = S (1) nlj of the HO, HF, NAT and VNAT single-neutron states in He, for different sizes of the active model space.component then increases to 91% when the model spaceexhausts the full configuration space (7 shells), and theentanglement entropy of the lowest orbitals reduces dra-matically. These results are consistent with pathologiesin the convergence of the ground-state energy in theHF basis [80]. Of course, He is a light nucleus where amean field cannot be firmly established. Therefore, wewould have to perform calculations in heavier systemsin order to see if these pathologies persist. For thisreason, in the following discussions of entanglement, wewill focus on the HO, VNAT (and NAT) bases, and notconsider further the HF basis.By looking at the one-orbital entanglement entropiesit is difficult to distinguish the NAT and VNAT bases asthey show very similar profiles (see e.g.
Fig. 1). How-ever the convergence of the ground-state energy in Ta-ble II shows that the VNAT states are marginally ”bet-ter” than the NAT ones when comparing with the targetvalue . With the present truncation scheme, the NAT In the present calculations, we have used an underlying harmonic basis leads to the same energy as the HO basis. This isunderstood as the natural orbitals only mix HO statesthat are partially occupied. In other words they onlymix HO states that are within the active model space.Since the ground-state wavefunction (5) includes all con-figurations in the active space, the ground-state energy isinvariant. This is different with the VNAT basis, which,due to Eq. (6), mixes both active and inactive HO or-bitals. At this point, it is not obvious how measures ofentanglement could be used to distinguish the NAT andVNAT bases. As entanglement is derived from reduceddensities, it does not give information on the couplingbetween the active and inactive orbital spaces. We willattempt to address this issue in Section IV by consideringmeasures of two-orbital entanglement. oscillator frequency of (cid:126)
Ω = 30 MeV, which is found to be op-timal in terms of energy minimization using the HO basis [78].The ground-state energy obtained with the HO and NAT basesimportantly depends on this frequency, and the improvementobtained using the VNAT basis is found to be greater for otheroscillator frequencies. N tot HO HF NAT VNAT2 shells 0.596 0.270 0.596 0.4413 shells 1.143 0.487 0.929 0.7464 shells 1.065 0.686 0.928 1.0635 shells 1.348 2.327 1.036 1.0426 shells 1.264 3.434 0.972 0.9637 shells 1.217 1.069 1.006 1.006
TABLE I: Sum of single-orbital entanglement entropies S (1) tot = (cid:80) i S (1) i for the different bases and with differentnumber of shells N tot in the active model space. N tot HO HF NAT VNAT2 shells -19.15 -16.58 -19.15 -21.403 shells -23.29 -17.35 -23.29 -24.894 shells -25.72 -20.48 -25.72 -26.615 shells -26.88 -23.37 -26.88 -27.276 shells -27.44 -23.81 -27.44 -27.477 shells -27.50 -27.50 -27.50 -27.50
TABLE II: Ground-state energy of He (in MeV)obtained with the different bases and with differentnumber of shells N tot in the active space. All basesbeing expanded on 7 HO shells, they all lead to thesame energy when the active space comprises 7 shellsand exhausts the full configuration space. C. Two-orbital entanglement entropy and Mutualinformation
The mutual information (MI) within a pair of single-particle states (
A, B ) ≡ ( i, j ), can be determined fromthe two-orbital reduced density matrix obtained by trac-ing over the rest of the basis C , ρ ( ij ) n i n j ,n (cid:48) i n (cid:48) j = (cid:88) C (cid:104) C | (cid:104) n j n i | Ψ (cid:105) (cid:104) Ψ | n (cid:48) i n (cid:48) j (cid:105) | C (cid:105) , (14)(i.e., the matrix elements of ˆ ρ AB discussed earlier) wherethe nuclear wavefunction is structured as | Ψ (cid:105) = (cid:88) n ...n i ...n j ...n N A n ...n i ...n j ...n N × ϕ i ϕ j ×| n n ...n i − n i +1 ...n j − n j +1 ...n N (cid:105) ⊗ | n i n j (cid:105)≡ (cid:88) n i ,n j ,C A Cn i n j × ϕ i ϕ j | C (cid:105) ⊗ | n i n j (cid:105) . (15)As derived in Appendix B, in the basis | n i n j (cid:105) = {| (cid:105) , | (cid:105) , | (cid:105) , | (cid:105)} the two-orbital reduced densitymatrix becomes ρ ( ij ) = − γ ii − γ jj + γ ijij γ jj − γ ijij γ ji γ ij γ ii − γ ijij
00 0 0 γ ijij , (16) where γ ij = (cid:104) Ψ | a † j a i | Ψ (cid:105) denotes non-diagonal terms ofthe one-body density, and γ ijij = (cid:104) Ψ | a † i a † j a j a i | Ψ (cid:105) is anelement of the two-nucleon density. The two-orbital en-tanglement entropy becomes S (2) ij = − Tr (cid:104) ρ ( ij ) ln( ρ ( ij ) ) (cid:105) = − (cid:88) k =1 η ( ij ) k ln η ( ij ) k , (17)where η ( ij ) k are the eigenvalues of ρ ( ij ) . The MI betweenthese states becomes, I ( i : j ) = (cid:16) S (1) i + S (1) j − S (2) ij (cid:17) (1 − δ ij ) . (18)Consistent with works in quantum chemistry [73], afactor of (1 − δ ij ) has been introduced to ensure thevanishing of the entanglement of a single-particle statewith itself.Figure 3 shows the MI of two neutron orbitals (in3D) computed from He ground state wavefunctions inthe HO and VNAT bases using 5 active shells. In orderto analyse the results more closely, Fig. 4 shows theneutron-neutron and proton-neutron MI (in 2D) for thelowest single-particle states. For direct comparison, thesingle-nucleon states are ordered by quantum numbersin both figures.Let us first examine the neutron-neutron sector (Fig. 3and top panels of Fig. 4). In the HO basis, the mostimportant correlations appear between states of the1 s shell and states of the 2 s shell, with aligned AMprojections. To a lesser extent MI between 1 s and3 s shells, and between 1 s and 1 p / orbitals are alsoimportant. Remarkably, this large MI between the s states is suppressed by approximately an order ofmagnitude in the VNAT basis, compared to the HO one.This is observed systematically when varying the size ofthe model space, as shown in Table III. Interestingly,the MI between time-reversed states (with same nlj andopposite AM projection) of the 1 s , that could indicateisovector BCS-type neutron-neutron pairing, is weak inboth bases. N tot HO VNAT3 shells 0.11 0.00404 shells 0.071 0.00475 shells 0.15 0.00916 shells 0.12 0.00757 shells 0.089 0.0083
TABLE III: The neutron-neutron mutual information ofthe 1 s -2 s orbitals.In the proton-neutron sector (bottom panels of Fig. 4),the strength of the MI remains similar in the HO andFIG. 3: The mutual information within theneutron-neutron orbitals of He using 5-shell activespaces in the HO (upper panel) and VNAT (lowerpanel) bases. The states are ordered by quantumnumbers, and the vertical scales are the same.VNAT bases. This is likely because the VNAT basis isobtained via a unitary transformation of the HO basis,which does not mix proton and neutron states, and thus,does not capture proton-neutron correlations. The mostimportant couplings are of the type 1 s -1 s , 1 s -1 p / and1 p / -1 p / with aligned AM projection. These are re-lated to deuteron-type (J=1,T=0) correlations.Again we remind the reader that the orbitals resultingfrom the self-consistent VNAT calculation are in prac-tice ordered by occupation numbers. In that basis, the1 p / orbitals are adjacent to the 1 s / orbitals, and theneutron-neutron and proton-neutron MI, shown in Fig. 5with the occupation number ordering, becomes more lo-calized. HO - nnordering by quantum numbers
VNAT - nnordering by quantum numbers HO - pn
1s 1p
2s 1d VNAT - pn
1s 1p
2s 1d FIG. 4: Mutual information between two HO (left) andVNAT (right) states, obtained for a model space of N tot = 5 shells. Each pixel corresponds to the MIbetween states i = ( n i , l i , j i , m i , τ i ). The states areordered by quantum numbers N i = 1 n i + l i , j i and AMprojection m i = +1 / , − / , +3 / , − / VNAT - nnordering of the calculation
1s 1p
2s 1p VNAT - pnordering of the calculation
1s 1p
2s 1p FIG. 5: Neutron-neutron (left panel) andproton-neutron (right panel) MI obtained using VNATorbitals ordered with occupation numbers, for N tot = 5shells. In the right panel, the proton (neutron) statesare shown on the y (x) axis. D. Negativity
The negativity N ( ρ ( ij ) ) is a measure of entanglementthat provides an upper bound to the amount of distillableentanglement. It and its variants such as log-negativity,play a central role in, for example, quantum communica-tion. In the context of nuclear physics, as will be consid-ered in this section, the practical implications of negativ-ity in a nucleus are not immediately obvious. However,it is a distinct measure of entanglement beyond MI, andas such is expected to provide insight into nuclear struc-ture and reactions. Negativity is defined as the sum ofthe negative eigenvalues of the partially transposed two-orbital reduced density, ρ T ( ij ) n i n j ,n (cid:48) i n (cid:48) j = (cid:88) C (cid:104) C | (cid:104) n j n (cid:48) i | Ψ (cid:105) (cid:104) Ψ | n i n (cid:48) j (cid:105) | C (cid:105) . (19)In the basis | n i n j (cid:105) = {| (cid:105) , | (cid:105) , | (cid:105) , | (cid:105)} , this becomes ρ T ( ij ) = − γ ii − γ jj + γ ijij γ ji γ jj − γ ijij γ ii − γ ijij γ ij γ ijij . (20)As ρ T ( ij ) differs from ρ ( ij ) only through the non-diagonalelements γ ij , γ ji , and since γ ij = γ ji = 0 when i and j have different isospin projections, in the proton-neutroncase ρ T ( ij ) = ρ ( ij ) , which only has positive eigenvalues.Therefore the proton-neutron negativity vanishes. More-over, due to other symmetries, γ ij can only be non-zero if i and j have same AM, AM projection and parity. There-fore distillable entanglement could only arise between twosingle-particle states that have these same quantum num-bers, that is, between states that can mix though unitarytransformations of the single-particle basis. The negativ-ities within the neutron sector of He using the HO andVNAT bases with 5 active shells are shown in Fig.6. Astriking feature of the NAT and VNAT bases is that,as the one-body density matrix γ becomes diagonal, thenegativity vanishes identically. Note that the small neg-ativity found between the VNAT 1s and 2s shells of theorder of 10 − is due to the numerical precision of theself-consistent procedure.In the HO basis, the only non-negligible terms appearbetween s-shell orbitals for the case of He. We presentthe corresponding values in Table IV with increasing sizeof the model space. Generally, we observe larger valuesof the negativity between the 1 s orbitals and other ns shells, with the negativity decreasing with increasing n .Given the simple structure of the transposed two-orbitaldensity, shown in Eq. (20), a condition for the appear-ance of non-zero negativity for the case of an arbitrarysingle-particle basis can be easily derived. This condition(detailed in Appendix C) relates the non-diagonal ele-ments γ ij to the occupation numbers and diagonal termsof the two-body density: | γ ij | ≥ (cid:113) (1 − γ ii − γ jj ) γ ijij + ( γ ijij ) . (21) FIG. 6: The negativity within the neutron-neutronorbitals of He using 5-shell active spaces in the HO(upper panel) and VNAT (lower panel) bases. Thevertical scales are the same in both panels.
III. ENTANGLEMENT IN HE He is a halo nucleus consisting of two protons and fourneutrons. As such, it provides a ”sandbox” in which totest basic aspects of entanglement in the context of thetraditional nuclear shell model, where the naive neutronconfiguration is (1 s / ) (1 p / ) while the naive protonconfiguration is (1 s / ) . The same numerical frameworkis used for He and He. Specifically, all possible config-urations (up to 6p-6h) in an active space comprising agiven number of shells N tot are included.Figure 7 shows the single-orbital entanglement entropy in He obtained with the HO and VNAT bases in a modelspace of N tot =4 shells. In the proton sector, the single-0 N tot s -2 s s -3 s s -3 s s -4 s s -4 s s -4 s × − - - - - -4 6.94 × − - - - - -5 1.12 × − × − × − - - -6 1.01 × − × − × − × − × − × − × − × − × − × − × − × − TABLE IV: The negativity between neutron states of the s shells (with same AM projection) in He in the HO basisfor different number of shells N tot in the active space. The negativity in the VNAT basis vanishes by definition. neutrons S ( ) n lj HOVNAT
1s 1p
2s 1d protons S ( ) n lj FIG. 7: The entanglement entropy of a single-neutron(top) and single-proton (bottom) HO and VNAT state,in He obtained with N tot =4 shells. N tot HO HF NAT VNAT2 shells 8.90 -2.99 8.90 -6.323 shells -6.52 -7.44 -6.52 -13.914 shells -15.98 -12.41 -15.98 -19.415 shells -20.30 -18.03 -20.30 -22.50
TABLE V: The ground-state energy of He (in MeV)obtained in the different bases with different numbers ofshells in the active space.orbital entropy profile resembles that obtained in He.However, there is a small increase in the entropy of thestates on the 1 s shell, due to a decrease of their occu-pation number from 0 .
95 to 0 .
92 (in the VNAT basis)through proton-neutron interactions. In the neutron sec-tor, the 1 p / and 1 p / sub-shells that the two extraneutrons are expected to primarily occupy appear as themost entangled orbitals. In particular, 1 p / is almostmaximally entangled, with occupation numbers of 0 . .
43 in the HO and VNAT bases, respectively. Theground state energy of He is given in Table V in the HOand VNAT bases as a function of the number of activeshells. Figure 8 shows the MI (in 3D) within two HO andVNAT neutron orbitals. In the HO basis, localized re-gions of MI are distributed between single-neutron stateswithin the active model space. In contrast, the two-neutron MI in the wavefunction in the VNAT basis islargely localized within the 1 p shell, pointing to an emerg-ing core-valence picture, where the two extra p -shell neu-trons decouple from the He core. To analyse the re-sults in more details, Fig. 9 shows the neutron-neutron(top), proton-proton (middle) and proton-neutron (bot-tom) MI in He . Generally the MI in the proton-proton and proton-neutron sector is weak compared tothe neutron-neutron sector, and this is particularly soin the VNAT basis. Both bases show strong neutron-neutron MI within 1 p / states, and to a lesser extentwithin the 1 p / , where the two halo neutrons primarilyreside. This is a clear signature of the important isovectorpairing correlations between these two neutrons, which isknown to be responsible for the binding of He. Withinthe 1 p / , couplings between states with different AMprojection e.g. with | m i | = 1 / | m j | = 3 /
2, areevident. In the HO basis, we observe important MI be-tween the 1 s and 2 s shells and also between the 1 p and 2 p shells. These couplings vanish in the VNAT basis. Thus,in that basis, He resembles much more a system of twoneutrons orbiting in the 1 p − shell on top of a He core, asalready seen in Fig. 8. Since the VNAT 1 s shell does notcouple to other neutron states, the single-orbital entropy S (1)1 s ≈ .
2, shown in Fig. 7, only results from interactionwith proton states (mostly 1 s and, to a lesser extent, 2 s states). This can be seen from the bottom panel of Fig. 9. IV. NAT VERSUS VNAT BASIS
In Section II, it was shown that the NAT and VNATbases typically present similar entanglement profiles.This is understood as the entanglement measures arebased on the computation of reduced density matrices,which cancel outside the active model spaces. However,the VNAT basis, which mixes both active and emptyHO single-particle orbitals, leads to a faster convergenceof the ground-state energy with respect to the size of themodel space (see Table V and [78]). This is to be com-pared with the NAT basis, which only mixes HO states1FIG. 8: The mutual information within two neutronorbitals in He with the HO (upper panel) and VNAT(lower panel) bases using N tot =4 active shells.in the active space, and thus does not improve the energyconvergence compared to the HO basis.In order to distinguish the NAT and VNAT bases, thecoupling between the active and inactive (empty) single-particle spaces needs to be quantified. To do that, weperform initial calculations of the NAT and VNAT statesin a model space of given N tot major shells, and, as asecond step, use these bases to perform one diagonal-ization of the two-body Hamiltonian in a configurationspace spanned by a larger single-particle basis, i.e. with N (cid:48) tot > N tot . One- and two-orbital entanglement mea-sures can then be calculated. Since the VNAT basismixes HO states from both active and inactive spaces,entanglement measures are expected to be weak betweensingle-particle states below and above N tot .As an example, Fig. 10 shows the neutron-neutron MI in HO - nn
VNAT - nn HO - pp
VNAT - pp HO - pnordering by quantum numbers
1s 1p
2s 1d VNAT - pnordering by quantum numbers
1s 1p
2s 1d FIG. 9: Mutual information of HO (left) and VNAT(right) single-particle states in He obtained for anactive space of N tot =4 shells. The top panels show theMI between neutron orbitals, the middle panels showthe MI between proton orbitals, and the bottom onesshow the proton-neutron MI. (The f shells are notshown as their MI is not visible). In the bottom panelsthe proton (neutron) states are on the y (x) axis. He obtained with N tot = 3 and N (cid:48) tot = 5. In the VNATbasis the couplings between the 1 p and 2 p shells are verysmall, even though the 2 p shell was absent from the ini-tial self-consistent calculation with N tot = 3. In the NATbasis however these couplings are sizeable. The same istrue for the MI between the 1 s -2 s and 3 s shells, and to alesser extent for the MI between the 1 d / -2 d / states. V. SUMMARY AND CONCLUSION
In this work we have explored measures of entanglement,entanglement entropy, mutual information and negativ-ity in He and He in a selection of bases and a chiralinteraction. The nuclear shell model, that successfullydescribes some features of nuclei, and has formed thebasis for a large selection of increasingly sophisticateddescriptions of nuclei and their interactions, is in some2
2p 3s 2d
2p 3s 2d
FIG. 10: Neutron-neutron MI for NAT (upper) andVNAT (lower) single-particle states in He obtainedwith N tot = 3 and N (cid:48) tot = 5. The dashed lines show theinitial active space of N tot = 3 shells.ways pinned upon the fact that entanglement within anucleus is somewhat localized. In that model, the local-ization of entanglement manifests itself in employing anactive valence sector built upon an inert core, i.e. a tensorproduct system. While this is far from exact, residual in-teractions and sophisticated many-body techniques buildupon such tensor product starting points. In the presentcalculations of He, this core-valence structure emergesfrom the full 6-body computation. It is known that en-tanglement measures in systems with identical particlesare basis dependent, and we have shown that commonlyused bases employed for ab-initio nuclear structure calcu-lations support quite different entanglement structures.In particular, the widely used HO basis exhibits a some-what distributed two-nucleon entanglement structure forboth MI and negativity, and a somewhat larger single-orbital entanglement entropy. In contrast, the VNAT ba-sis exhibits a more compact two-nucleon MI and single- orbital entanglement entropy, and vanishing negativityby construction. The potential utility of MI, and moregenerally measures of entanglement, is made clearest incomparing He and He, where the additional two p-shellneutrons provide a substantial and structured MI withinthe p-shell.Studying the entanglement structure of nuclei may havefuture benefits when considering workflows for hybridclassical-quantum computations of nuclear structure andreactions. Elements of such computations with minimalor vanishing entanglement are amenable to classical com-putations, while those where entanglement is significantwill be computed using a quantum device. An optimalworkflow would have the intrinsically quantum aspectsof the computation performed using a quantum device,while those that are intrinsically classical would be bestperformed on a classical device. The entanglement mea-sures we have considered in this work could provide help-ful diagnostics in designing workflows for such hybridclassical-quantum computations.While we have not provided evidence, it is possible thatusing a basis for nuclear many-body computations thathas minimal support of entanglement entropy and two-nucleon MI and negativity may provide a better low-energy model to match to low-energy effective field theo-ries and also the results of lattice QCD calculations. Atthe physical point, the low-energy two-nucleon entangle-ment power is near minimal, and related to enhancedspin-flavor symmetries. It seems natural to preserve thisfeature during matching to nuclear many-body systemsin order to address more complex nuclear systems. Thispoint requires significantly more investigation before con-clusive statements can be made.Our investigations suggest that there maybe utility indesigning effective interactions that are organized by, tosome extent, entanglement. We have not addressed thispotential yet, but suggest that exploring the behaviorof entanglement induced by forces as a function of evo-lution under SRG flow, or renormalization group flows,more generally, has the potential to provide valuable in-sight.The results we have presented, represent some of the firststeps in an emerging line of investigation. We are encour-aged by the entanglement structures we have found, andintend to extend these studies to include multi-partite en-tanglement in nuclei and nuclear reactions, with a partic-ular focus on extracting further insights into clustering,three-nucleon and four-nucleon forces. It is plausible thatnuclei near the drip line exhibit entanglement structuresthat differ from those enjoying the valley of stability. Wewill be pursuing such systems in upcoming research. Inthe future these calculations of entanglement can also beused to improve our many-body scheme. In particular weplan to investigate a selection of orbitals based on one-or two-orbital entanglement measures (”`a la DMRG”)within the self-consistent procedure.3
ACKNOWLEDGMENTS
We thank Petr Navr´atil for graciously providing matrixelements of the chiral interactions. We would also like tothank Natalie Klco, Alessandro Roggero and GuillaumeHupin for a number of fruitful discussions. This workwas supported by the Institute for Nuclear Theoryunder US-DOE Grant DE-FG02-00ER41132 and byJINA-CEE under US-NSF Grant PHY-1430152. Thecomputational work for this project was partly carriedout on the Hyak High Performance Computing and DataEcosystem at the University of Washington, supported, in part, by the U.S. National Science FoundationMajor Research Instrumentation Award, Grant Number0922770, the Institute for Nuclear Theory and thePhysics Department at the University of Washington.This research also used resources of the National EnergyResearch Scientific Computing Center, a DOE Office ofScience User Facility supported by the Office of Scienceof the U.S. Department of Energy under Contract No.DE-AC02-05CH11231.
Note added : After completion of this work we becameaware of Ref. [83], in which single-orbital entangle-ment entropy and two-orbital mutual information intwo-nucleon systems are considered.
Appendix A: Single-orbital reduced density
The single-orbital reduced density matrix is ρ ( i ) n i ,n (cid:48) i = (cid:88) BC (cid:104) Ψ | BC (cid:105) | n (cid:48) i (cid:105) (cid:104) n i | (cid:104) BC | Ψ (cid:105) , (A1)where BC ≡ ( n n ...n i − n i +1 ...n N ). Each fixed state i can be occupied or empty so that we have a basis {| n i (cid:105)} = {| (cid:105) ; | (cid:105) = a † i | (cid:105)} . In this basis there are 4 matrix elements: ρ ( i )1 , = (cid:88) BC (cid:104) Ψ | BC (cid:105) | (cid:105) (cid:104) | (cid:104) BC | Ψ (cid:105) = (cid:88) BC (cid:104) Ψ | BC (cid:105) a † i | (cid:105) (cid:104) | a i (cid:104) BC | Ψ (cid:105) = (cid:88) BC (cid:104) Ψ | BC (cid:105) a † i (cid:88) n i | n i (cid:105) (cid:104) n i | (cid:124) (cid:123)(cid:122) (cid:125) ˆ1 a i (cid:104) BC | Ψ (cid:105) − (cid:88) BC (cid:104) Ψ | BC (cid:105) a † i | (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) (cid:104) | a i (cid:104) BC | Ψ (cid:105) = (cid:104) Ψ | a † i a i | Ψ (cid:105) = γ ii , (A2) ρ ( i )0 , = (cid:88) BC (cid:104) Ψ | BC (cid:105) | (cid:105) (cid:104) | (cid:104) BC | Ψ (cid:105) = (cid:88) BC (cid:88) i (cid:104) Ψ | BC (cid:105) | n i (cid:105) (cid:104) n i | (cid:104) BC | Ψ (cid:105) − (cid:88) BC (cid:104) Ψ | BC (cid:105) | (cid:105) (cid:104) | (cid:104) BC | Ψ (cid:105) = (cid:104) Ψ | Ψ (cid:105) − ρ ( i )1 , = 1 − (cid:104) Ψ | a † i a i | Ψ (cid:105) = 1 − γ ii , (A3)and ρ ( i )1 , = ρ ( i )0 , = 0 due to conservation of particle number. Appendix B: Two-orbital reduced density
The two-orbital reduced density matrix is ρ ( ij ) n i n j ,n (cid:48) i n (cid:48) j = (cid:88) C (cid:104) Ψ | C (cid:105) | n (cid:48) i n (cid:48) j (cid:105) (cid:104) n j n i | (cid:104) C | Ψ (cid:105) , (B1)where | C (cid:105) = | n n ...n i − n i +1 ...n j − n j +1 ...n N (cid:105) . There are 4 states for the basis | n i n j (cid:105) that we denote: | (cid:105) ≡ | (cid:105) , | (cid:105) ≡ | (cid:105) , | (cid:105) ≡ | (cid:105) , | (cid:105) ≡ | (cid:105) . (B2)4The matrix elements of ρ ( ij ) are then ρ ( ij ) , ≡ ρ ( ij )11 , = (cid:88) C (cid:104) Ψ | C (cid:105) | (cid:105) (cid:104) | (cid:104) C | Ψ (cid:105) = (cid:88) C (cid:104) Ψ | C (cid:105) a † i a † j | (cid:105) (cid:104) | a j a i (cid:104) C | Ψ (cid:105) = (cid:88) C (cid:104) Ψ | C (cid:105) a † i a † j (cid:88) n i n j | n i n j (cid:105) (cid:104) n j n i | (cid:124) (cid:123)(cid:122) (cid:125) ˆ1 a j a i (cid:104) C | Ψ (cid:105) − (cid:88) C (cid:104) Ψ | C (cid:105) a † i a † j | (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) (cid:104) | a j a i (cid:104) C | Ψ (cid:105)− (cid:88) C (cid:104) Ψ | C (cid:105) a † i a † j | (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) (cid:104) | a j a i (cid:104) C | Ψ (cid:105) − (cid:88) C (cid:104) Ψ | C (cid:105) a † i a † j | (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) (cid:104) | a j a i (cid:104) C | Ψ (cid:105) = (cid:104) Ψ | a † i a † j a j a i | Ψ (cid:105) = γ ijij , (B3) ρ ( ij ) , ≡ ρ ( ij )10 , = (cid:88) C (cid:104) Ψ | C (cid:105) | (cid:105) (cid:104) | (cid:104) C | Ψ (cid:105) = (cid:88) C (cid:104) Ψ | C (cid:105) a † i | (cid:105) (cid:104) | a i (cid:104) C | Ψ (cid:105) = (cid:88) C (cid:104) Ψ | C (cid:105) a † i (cid:88) n i n j | n i n j (cid:105) (cid:104) n j n i | (cid:124) (cid:123)(cid:122) (cid:125) ˆ1 a i (cid:104) C | Ψ (cid:105) − (cid:88) C (cid:104) Ψ | C (cid:105) a † i | (cid:105) (cid:104) | a i (cid:104) C | Ψ (cid:105)− (cid:88) C (cid:104) Ψ | C (cid:105) a † i | (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) (cid:104) | a i (cid:104) C | Ψ (cid:105) − (cid:88) C (cid:104) Ψ | C (cid:105) a † i | (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) (cid:104) | a i (cid:104) C | Ψ (cid:105) = (cid:104) Ψ | a † i a i | Ψ (cid:105) − (cid:88) C (cid:104) Ψ | C (cid:105) a † i a † j | (cid:105) (cid:104) | a j a i (cid:104) C | Ψ (cid:105) = (cid:104) Ψ | a † i a i | Ψ (cid:105) − (cid:104) Ψ | a † i a † j a j a i | Ψ (cid:105) = γ ii − γ ijij . (B4)Similarly, ρ ( ij ) , ≡ ρ ( ij )01 , = (cid:104) Ψ | a † j a j | Ψ (cid:105) − (cid:104) Ψ | a † i a † j a j a i | Ψ (cid:105) = γ jj − γ ijij , (B5)and, ρ ( ij ) , ≡ ρ ( ij )00 , = (cid:88) C (cid:104) Ψ | C (cid:105) | (cid:105) (cid:104) | (cid:104) C | Ψ (cid:105) = (cid:88) C (cid:104) Ψ | C (cid:105) (cid:88) n i n j | n i n j (cid:105) (cid:104) n j n i | (cid:124) (cid:123)(cid:122) (cid:125) ˆ1 a j a i (cid:104) C | Ψ (cid:105) − (cid:88) C (cid:104) Ψ | C (cid:105) | (cid:105) (cid:104) | (cid:104) C | Ψ (cid:105)− (cid:88) C (cid:104) Ψ | C (cid:105) | (cid:105) (cid:104) | (cid:104) C | Ψ (cid:105) − (cid:88) C (cid:104) Ψ | C (cid:105) | (cid:105) (cid:104) | (cid:104) C | Ψ (cid:105) = (cid:104) Ψ | Ψ (cid:105) − ρ ( ij ) , − ρ ( ij ) , − ρ ( ij ) ,
4= 1 − (cid:104) Ψ | a † j a j | Ψ (cid:105) − (cid:104) Ψ | a † i a i | Ψ (cid:105) + (cid:104) Ψ | a † i a † j a j a i | Ψ (cid:105) = 1 − γ jj − γ ii + γ ijij . (B6)Finally, the non-zero off-diagonal elements are ρ ( ij ) , ≡ ρ ( ij )01 , = (cid:88) C (cid:104) Ψ | C (cid:105) | (cid:105) (cid:104) | (cid:104) C | Ψ (cid:105) = (cid:88) C (cid:104) Ψ | C (cid:105) a † i | (cid:105) (cid:104) | a j (cid:104) C | Ψ (cid:105) = (cid:104) Ψ | a † i a j | Ψ (cid:105) = γ ji , (B7)and, ρ ( ij ) , ≡ ρ ( ij )10 , = (cid:104) Ψ | a † j a i | Ψ (cid:105) = γ ij . (B8)All other matrix elements cancel due to particle-number conservation.5 Appendix C: Condition for non-zero two-orbital negativity
The negativity N ( ij ) is defined as the sum of the negative eigenvalues of the partially transposed two-orbital density,given in Eq. (20), which has the following structure ρ T ( ij ) = M M M M M M , (C1)with M = 1 − γ ii − γ jj + γ ijij , M = γ jj − γ ijij , M = γ ii − γ ijij (C2) M = γ ijij , M = M = γ ji = γ ij . (C3)The eigenvalues of this matrix are λ = M , λ = M (C4) λ = 12 (cid:18) M + M − (cid:113) M + 4 M M − M M + M (cid:19) (C5) λ = 12 (cid:18) M + M + (cid:113) M + 4 M M − M M + M (cid:19) . (C6) • M and M are eigenvalues of the two-orbital density in Eq. (16), thus M and M are positive by definition,and thus λ ≥ • λ , λ ≥
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