Asymptotic state transformations of continuous variable resources
Giovanni Ferrari, Ludovico Lami, Thomas Theurer, Martin B. Plenio
AAsymptotic state transformations of continuous variable resources
Giovanni Ferrari,
1, 2
Ludovico Lami, ∗ Thomas Theurer, and Martin B. Plenio Dipartimento di Fisica e Astronomia Galileo Galilei,Universit`a degli studi di Padova, via Marzolo 8, 35131 Padova, Italy Institut f¨ur Theoretische Physik und IQST, Universit¨at Ulm,Albert-Einstein-Allee 11, D-89069 Ulm, Germany
We prove that strongly superadditive monotones can be used to bound asymptotic state trans-formation rates in continuous variable resource theories. This removes the need for asymptoticcontinuity, which is typically lost in infinite-dimensional settings. We consider three applications,to the resource theories of (I) optical nonclassicality, (II) entanglement, and (III) quantum thermo-dynamics. In cases (II) and (III), the employed monotones are the squashed entanglement and thefree energy, respectively. For case (I), we consider the measured relative entropy of nonclassicalityand prove it to be strongly superadditive. Our technique then yields computable upper boundson asymptotic transformation rates including those achievable under linear optical elements. Weconclude by applying our findings to the problem of cat state manipulation.
Introduction. — In recent years, the paradigm of quan-tum resource theories has established itself as the mainframework to analyze and assess the operational useful-ness of quantum resources [1–3]. The general settinginvolves two sets of objects that are considered easilyaccessible: free states and free operations. Once thesehave been identified, the resource content of a state isdetermined by its transformation properties under freeoperations [3, Section V]. In the long-established tradi-tion of classical [4, 5] as well as quantum [6–8] informa-tion theory, in this work we consider ultimate limitationson those transformation properties, and thus look at theasymptotic setting. Namely, we study free approximateconversion of a large number of copies of the initial state ρ into as many copies of the target state σ as possible,under the constraint that the approximation error van-ishes asymptotically. The resulting transformation rate R ( ρ → σ ) can be turned into a whole family of resourcequantifiers: for a fixed resourceful state σ (respectively, ρ ), the function R ( · → σ ) (respectively, R ( ρ → · ) − ) isa resource quantifier with a solid operational interpreta-tion. In entanglement theory, for example, consideringfree all those transformations that can be implementedwith local operations assisted by classical communication(LOCC) and choosing as fixed states Bell pairs, the aboveprocedure leads to the distillable entanglement and theentanglement cost, respectively [8, Section XV].Since exact computations of asymptotic transforma-tion rates are often challenging, it is important to boundthem. If G is a resource monotone, i.e., a function fromquantum states to the nonnegative real numbers thatdoes not increase under free operations, the inequality R ( ρ → σ ) ≤ G ( ρ ) G ( σ ) holds if G is (i) additive on multiplecopies of a state, and (ii) asymptotically continuous [9–11] (see also [3, Section VI.A.5]). While (i) can be en-forced by regularization [3, Section VI.A.4], ensuring (ii)is more challenging. Indeed, although asymptotic con-tinuity holds for many monotones in finite-dimensionalsystems, it typically breaks down in infinite dimensions. Infinite-dimensional quantum systems, especially thosein the form of quantum harmonic oscillators, are how-ever ubiquitous in physics. The common harmonic ap-proximation of physical systems close to equilibrium andthe optical modes that underlie the flourishing field ofcontinuous-variable (CV) quantum technologies [12–14]are prime examples. And indeed, in the CV setting manymonotones – especially those based on entropic quanti-ties – are discontinuous everywhere. A weaker versionof asymptotic continuity can be restored by imposing anenergy constraint [15–17], yet doing so still does not re-sult in any bound on the transformation rates, becausethe free operations employed are a priori not required tobe (uniformly) energy-constrained.Here, we devise a general way to circumvent this prob-lem and establish rigorous bounds on transformationrates valid both for finite- and infinite-dimensional quan-tum resource theories. Our approach relies on monotones G that satisfy, in addition to (i), also (ii’) lower semicon-tinuity, which is much weaker than (ii), and (iii) strongsuperaddivity, i.e., G ( ρ AB ) ≥ G ( ρ A ) + G ( ρ B ). We showhow (i), (ii’), and (iii) combined imply the sought gen-eral bound R ( ρ → σ ) ≤ G ( ρ ) G ( σ ) on the transformation rate(Theorem 1).We then study three main applications, to the resourcetheories of: (I) optical nonclassicality [18–23]; (II) quan-tum entanglement [6–8]; and (III) quantum thermody-namics [24, 25]. Each of these applications rests upon adifferent strongly superadditive monotone, namely (I) themeasured relative entropy of nonclassicality, which weintroduce here, (II) the squashed entanglement [26–30],and (III) the free energy [24]. While strong subadditivitywas known to hold for the latter two monotones, one ofour technical contributions is to show that the same istrue for the measured relative entropy of nonclassicality.Additional technical details on the proofs of our resultsare given in the Supplemental Material (SM) [31]. Notation. — Let (cid:83) be a family of quantum systems thatis closed under tensor products and contains the trivial a r X i v : . [ qu a n t - ph ] S e p system, 1, with Hilbert space C . To define a quantumresource theory (QRT) over (cid:83) , for every two systems A, B ∈ (cid:83) we fix a set (cid:70) ( A → B ) of free operations ,i.e., quantum channels, from A to B [3, Definition 1].These are completely positive and trace preserving mapsΛ : T ( H A ) → T ( H B ), with T ( H ) denoting the space oftrace class operators on the Hilbert space H . We requirethe identity to be free, and free operations to be closedunder parallel and sequential composition. The set of freestates on A ∈ (cid:83) is then identified with (cid:70) A .. = (cid:70) (1 → A ).A function G defined on all states of all systems A ∈ (cid:83) and taking on values in the extended positive reals[0 , + ∞ ] is called a monotone if it is nonincreasing underfree operations, i.e., G (Λ( ρ )) ≤ G ( ρ ) for every state ρ on A and every free operation Λ ∈ (cid:70) ( A → B ). A mono-tone G is called: (a) faithful , if G ( ρ ) = 0 if and onlyif ρ is free; (b) convex , if G (cid:16)P j p j ρ j (cid:17) ≤ P j p j G ( ρ j )for all ensembles { p j , ρ j } ; (c) lower semicontinuous , iflim inf ρ → ρ G ( ρ ) ≥ G ( ρ ), where the limit is with re-spect to the trace norm; (d) strongly superadditive , if G ( ρ AB ) ≥ G ( ρ A ) + G ( ρ B ) for all states ρ AB on AB with local reductions ρ A .. = Tr B ρ AB and ρ B .. = Tr A ρ AB ;(e) additive , if G ( ρ A ⊗ σ B ) = G ( ρ A )+ G ( σ B ) for all states ρ A and σ B ; and (f) weakly additive , if that holds at leastwhen A is a copy of B and ρ = σ .We will use such monotones to bound transformationrates, which capture the intuitive notion of maximal yieldof copies of a target state σ B that can be obtained percopy of an input state ρ A by means of free operationswith asymptotically vanishing error. Formally, the trans-formation rate R ( ρ A → σ B ) is given by R ( ρ A → σ B ) .. = sup (cid:26) r : lim n →∞ inf Λ n (cid:13)(cid:13)(cid:13) Λ n (cid:0) ρ ⊗ nA (cid:1) − σ ⊗b rn c B (cid:13)(cid:13)(cid:13) = 0 (cid:27) , (1)where the infimum is taken over free operations Λ n ∈ (cid:70) (cid:0) A n → B b rn c (cid:1) , with the quantum system A n beingformed by n copies of A .While in (S22) we considered the total error, as is com-monly done, an alternative idea is to look instead at themaximum error over single output copies. We define theresulting maximal asymptotic transformation rate by e R ( ρ A → σ B ) .. = sup (cid:26) r : lim n inf Λ n max j (cid:13)(cid:13)(cid:13)(cid:0) Λ n (cid:0) ρ ⊗ nA (cid:1)(cid:1) j − σ B (cid:13)(cid:13)(cid:13) = 0 (cid:27) , (2)where Ω j denotes the reduced state of Ω on the j th sub-system. The data processing inequality for the tracenorm [32] implies that R ( ρ A → σ B ) ≤ e R ( ρ A → σ B ). Fur-thermore, we note that the maximal asymptotic transfor-mation rate is a relevant quantity when the output copiesof σ B are distributed to non-interacting parties.An important role in our paper is played by CV quan-tum systems, which model finite ensembles of bosonic(e.g., optical) modes. The Hilbert space correspond-ing to m modes is H m .. = L ( R m ). The annihilation and creation operators a j and a † j ( j = 1 , . . . , m ) sat-isfy the canonical commutation relations [ a j , a † k ] = δ jk ,[ a j , a k ] = 0 [33, 34]. Single-mode Fock states are de-fined by | k i .. = ( k !) − / ( a † ) k | i , where k ∈ N and | i is the vacuum state . For α ∈ C , the coherent state | α i is given by | α i .. = e −| α | / P ∞ k =0 α k √ k ! | k i [35–40]. In amultimode system, coherent states are parametrized byvectors α ∈ C m , with | α i .. = N j | α j i .The QRT of optical nonclassicality [18–23] is con-structed on the family (cid:83) CV of all CV quantum sys-tems. It is based on the premise that statistical mix-tures of coherent states are easy to synthesize, hence free,and “classical”, as they most closely approximate classi-cal electromagnetic waves. On the other hand, opera-tionally, nonclassical states, such as Fock states [41, 42],squeezed states [43–47], cat states [48–55], or NOONstates [56, 57], play an increasingly central role in ap-plications. Formally, for an m -mode system A we set (cid:70) A = C m .. = conv {| α ih α | : α ∈ C m } , where conv de-notes the closed convex hull. Since we are interestedin ultimate limitations on transformation rates, we deemfree all quantum channels with an m -mode input and an m -mode output that map C m to C m . These so-called classical channels comprise, but are possibly not limitedto, channels that can be obtained through passive lin-ear optics, destructive measurements, and feed-forwardof measurement outcomes [19, 20]. Results. — We are now in position to present our gen-eral bounds on transformation rates.
Theorem 1.
For a given QRT, let G be a monotonethat is strongly superadditive, weakly additive, and lowersemicontinuous. Then, for all states ρ A , σ B , it holds that R ( ρ A → σ B ) ≤ e R ( ρ A → σ B ) ≤ G ( ρ A ) G ( σ B ) , (3) whenever the rightmost side is well defined.Proof. It suffices to show that e R ( ρ A → σ B ) ≤ G ( ρ A ) G ( σ B ) . Forany sequence of free operations Λ n ∈ (cid:70) (cid:0) A n → B b rn c (cid:1) satisfying lim inf n →∞ (cid:13)(cid:13)(cid:13) Λ n (cid:0) ρ ⊗ nA (cid:1) − σ ⊗b rn c B (cid:13)(cid:13)(cid:13) = 0, it holdsthat G ( ρ A ) = lim inf n →∞ n G (cid:0) ρ ⊗ nA (cid:1) ≥ lim inf n →∞ n G (cid:0) Λ n (cid:0) ρ ⊗ nA (cid:1)(cid:1) ≥ lim inf n →∞ n b rn c X j =1 G (cid:16)(cid:0) Λ n (cid:0) ρ ⊗ nA (cid:1)(cid:1) j (cid:17) ≥ lim inf n →∞ b rn c n min j G (cid:16)(cid:0) Λ n (cid:0) ρ ⊗ nA (cid:1)(cid:1) j (cid:17) = lim inf n →∞ b rn c n G (cid:16)(cid:0) Λ n (cid:0) ρ ⊗ nA (cid:1)(cid:1) j n (cid:17) = r lim inf n →∞ G (cid:16)(cid:0) Λ n (cid:0) ρ ⊗ nA (cid:1)(cid:1) j n (cid:17) ≥ r G ( σ B ) . Here, 1 holds due to weak additivity, even without the liminf and for every n ; 2 comes from monotonicity; 3 fromstrong superadditivity; in 4 we constructed a sequenceof indices j n achieving the minimum; finally, 5 descendsfrom lower semicontinuity and the assumption on Λ n .Then a supremum over r yields the claim. The aboveargument generalizes that in [58, Theorem 4 and Re-mark 10].In what follows, we investigate some applications ofTheorem 1. (I) Optical nonclassicality. Over the past decades,there have been proposals to quantify the nonclassical-ity of quantum states of light, e.g., by their distancefrom the set of classical states [59–62], by the amountof noise needed in order to make them classical [63, 64],by their potential for entanglement generation [65–67] orfor metrological advantage [68], by the negativity [69, 70],the variances [20] or other features [71–76] of their phase-space distributions, or by the minimum number of su-perposed coherent states needed to reproduce the tar-get state [77]. Unfortunately, none of these monotonesappears to yield bounds on asymptotic transformationrates, for they fail to satisfy asymptotic continuity. Infact, to the extent of our knowledge, no rigorous boundson those rates are known for the resource theory of opti-cal nonclassicality [78].We therefore set out to pursue a different ap-proach. We start by fixing some terminology. Theclassical Kullback–Leibler divergence D KL ( p k q ) .. = P x p x log p x q x [79], defined for probability distributions p, q , can be extended to pairs of quantum states ρ, σ in many different ways. The approach taken byUmegaki leads to the modern definition of relative en-tropy , given by D ( ρ k σ ) .. = Tr[ ρ (log ρ − log σ )] [80].An alternative idea was pursued by Donald, who con-structed the measured relative entropy D M ( ρ k σ ) .. =sup M D KL ( P M ρ k P M σ ) [81–83]. Here, M = { E x } x is a (discrete) positive-operator valued measurement(POVM), i.e., a collection of positive semidefinite opera-tors E x ≥ P x E x = , and P M ρ ( x ) = Tr[ ρE x ]is the probability associated with the outcome x .In analogy to what was previously done for entangle-ment [84, 85], we can use these quantities to constructnonclassicality measures: the relative entropy of nonclas-sicality and the measured relative entropy of nonclassical-ity of an m -mode state ρ are defined by N r ( ρ ) .. = inf σ ∈ C m D ( ρ k σ ) , N Mr ( ρ ) .. = inf σ ∈ C m D M ( ρ k σ ) . (4)Note that our definition of N r differs from that of Mar-ian et al. [61], in that σ is allowed to be an arbitrary classical state, not necessarily Gaussian. It is easy to seethat N r and N Mr are faithful and convex nonclassicalitymonotones. Their regularized versions N ∞ r ( ρ ) .. = lim n →∞ N r ( ρ ⊗ n ) n , N M, ∞ r ( ρ ) .. = lim n →∞ N Mr ( ρ ⊗ n ) n , (5)where the limits exist by Fekete’s lemma [31, 86], areweakly additive nonclassicality monotones.It might not be clear at this point why to introduce N Mr alongside with N r , given that the former quantityinvolves one more nested optimization than the latter.However, we now show that its computation can be no-tably simplified. Theorem 2.
For all m -mode finite-entropy states ρ , itholds that N Mr ( ρ ) = sup L> (cid:26) Tr ρ log L − log sup α ∈ C m h α | L | α i (cid:27) , (6) where L ranges over all positive trace class operators on H m (equivalently, on all positive normalized states). The proof of Theorem 2 involves two main ingredi-ents. First, a generalization of the variational programfor D M put forth by Berta et al. [83, Lemma 1] to ourinfinite-dimensional setting. Second, an application ofSion’s minimax theorem [87] that allows us to exchangeinfimum and supremum in the resulting expression. Thislast step is technically challenging, as it involves a carefulchoice of topology on the domain of optimization [31].We now explore some important consequences of The-orem 2, where S ( ρ ) .. = − Tr[ ρ log ρ ] denotes the (vonNeumann) entropy. Theorem 3.
When computed on finite-entropy states, N Mr and N M, ∞ r are strongly superadditive, lower semi-continuous, and satisfy that N Mr ( ρ ) ≤ N M, ∞ r ( ρ ) ≤ N ∞ r ( ρ ) ≤ N r ( ρ ) . (7) Thus, if S ( ρ ) , S ( σ ) < ∞ then R ( ρ → σ ) ≤ e R ( ρ → σ ) ≤ N M, ∞ r ( ρ ) N M, ∞ r ( σ ) ≤ N r ( ρ ) N Mr ( σ ) , (8) provided that the ratios on the r.h.s. are well defined. To the best of our knowledge, (8) is the first explicitbound on asymptotic transformation rates in the con-text of CV nonclassicality. However, it would amountto a rather futile theoretical statement if not comple-mented with a systematic way of upper bounding the ra-tio N r ( ρ ) /N Mr ( σ ). Note that N r can be estimated fromabove by simply making suitable ansatzes in (4). Thea priori less trivial task of lower bounding N Mr can becarried out thanks to Theorem 2. Finally, note that (7)implies that both N ∞ r and N M, ∞ r are not only monotonicbut also faithful.As an immediate application of Theorem 3, we considerthe paradigmatic example of (Schr¨odinger) cat state ma-nipulation [55, 88–92]. For α ∈ C , cat states are definedby | ψ ± α i .. = c − / α ( | α i ± |− α i ), where c α is a normal-ization factor [48]. The transformations we look at are ψ + α → ψ + √ α (amplification) and ψ + √ α → ψ + α ⊗ ψ − α (sign-randomized dilution). A protocol for amplification usinglinear optical elements and quadrature measurements hasbeen designed by Lund et al. [88]. In the SM [31, § VI]we present an ameliorated version of it, together with asimple protocol for sign-randomized dilution. The corre-sponding lower bounds on rates are shown in Figure 1.The upper bound derived via our method is asymptot-ically tight for the dilution task, but not in the case ofamplification. This is due to the fact that our quantifiersall saturate to 1 for cat states with | α | → ∞ . . . . . . . | ↵ | T r a n s f o r m a t i o n r a t e R + ↵ ! + p ↵ Upper boundOur protocolLund et al.’s protocol0 . . . . . . . | ↵ | T r a n s f o r m a t i o n r a t e R + p ↵ ! + ↵ ⌦ ↵ Upper boundOur protocol
FIG. 1. Upper and lower bounds on asymptotic transforma-tion rates of Schr¨odinger cat states.
We complete the study of these measures by show-ing that both N r and N Mr are finite on finite-energystates , but can be infinite otherwise [31]. Some addi-tional bounds on N Mr are as follows. Proposition 4.
For any state ρ and for F = N Mr , N M, ∞ r it holds that − log k Q ρ k ∞ − m log ( π ) ≤ F ( ρ ) + S ( ρ ) ≤ S W ( ρ ) , (9) where Q ρ ( α ) = π h α | ρ | α i is the Husimi Q -function [93],and S W ( ρ ) .. = − R d m α Q ρ ( α ) log ( π m Q ρ ( α )) is theWehrl entropy [94]. These bounds take a simple form in the case of a Gaus-sian state, because both k Q ρ k ∞ and S W ( ρ ) can be easilyexpressed in terms of its covariance matrix [31]. (II) Entanglement. Our next application deals withthe QRT of entanglement [3, 6, 8, 95] in infinite dimen-sion [30, 96–103]. Formally, now (cid:83) is the family of all (finite- or infinite-dimensional) bipartite quantum sys-tems A : B , and (cid:70) ( A : B → A : B ) = LOCC( A : B → A : B ) is the set of LOCC protocols [104]. To the extent ofour knowledge, there is no available technique to deriveupper bounds on the transformation rate R ( ρ AB → σ AB )in terms of known monotones. As we show in the SM [31,§ II], even the energy-constrained version of asymptoticcontinuity recently established by Shirokov [15] is insuffi-cient, unless we also restrict to LOCCs that are uniformly constrained in energy.To apply our method, we need an entanglement mono-tone that obeys strong superadditivity. The squashedentanglement is a natural candidate [26–29]. Shi-rokov [30] has constructed two extensions of it to infinite-dimensional systems, denoted by E sq [30, Eq. (17)] and ˆE sq [30, Eq. (37)], respectively. They are both stronglysuperadditive, and they coincide, e.g., on all finite-entropy states. Moreover, ˆE sq is lower semicontinuouseverywhere. Applying Theorem 1 to ˆE sq , and exploit-ing [30, Propisition 3C], we deduce: Corollary 5.
In the QRT of entanglement, for all finite-entropy bipartite states ρ AB , σ AB it holds that R ( ρ AB → σ AB ) ≤ e R ( ρ AB → σ AB ) ≤ E sq ( ρ AB ) E sq ( σ AB ) . (10) (III) Thermodynamics. The last application of The-orem 1 deals with the QRT of thermodynamics. It isformally defined on a family (cid:83) of quantum systems A equipped with Hamiltonians (i.e., self-adjoint operators) H A that satisfy the Gibbs hypothesis, i.e., such thatTr e − βH A < ∞ for all inverse temperatures β > H AB = H A + H B for all A, B ∈ (cid:83) . Once a value of β > Gibbs states γ A .. = e − βHA Tr e − βHA ,while we consider free all Gibbs-preserving operations ,i.e., all those operations A → B that map γ A to γ B .The quantity G ( ρ A ) .. = β D ( ρ A k γ A ), which coincideswith the free energy difference between ρ A and γ A whenTr ρ A H A < ∞ [24], can be seen to be strongly superad-ditive, additive and lower semicontinuous. We deduce: Corollary 6.
In the QRT of thermodynamics, for allstates ρ A , σ B it holds that R ( ρ A → σ B ) ≤ e R ( ρ A → σ B ) ≤ D ( ρ A k γ A ) D ( σ B k γ B ) . (11)Let us stress that Corollary 6 extends the results ofBrand˜ao et al. [24], which are valid in finite-dimensionalsystems, to all quantum systems where a QRT of ther-modynamics can be constructed. Conclusions. —In this work, we provided a fully generalbound on asymptotic transformation rates valid both infinite as well as infinite-dimensional QRTs, which involvesresource monotones G that satisfy additivity on multi-ple copies of a state, lower semicontinuity, and strongsuperaddivity. We then applied our result to three ofthe most important QRTs involving infinite-dimensionalsystems, namely optical nonclassicality, quantum entan-glement, and quantum thermodynamics. To the bestof our knowledge, these are the first rigorous boundson infinite-dimensional asymptotic transformation ratesin these theories, thus marking a significant progress inour understanding of fundamental resource manipulationtasks. Acknowledgments. — GF and LL contributedequally to this paper. LL is grateful to Krishna Ku-mar Sabapathy and Andreas Winter for enlightening dis-cussions on the computation of the relative entropy ofnonclassicality for Fock states. LL, TT, and MBP aresupported by the ERC Synergy Grant BIOQ (grant no.319130). GF acknowledges the support received from theEU through the ERASMUS+ Traineeship program andfrom the Scuola Galileiana di Studi Superiori. ∗ [email protected][1] C. H. Bennett, Quantum Inf. Comput. , 460 (2004).[2] B. Coecke, T. Fritz, and R. W. Spekkens, Inf. Comput. , 59 (2016).[3] E. Chitambar and G. Gour, Rev. Mod. Phys. , 025001(2019).[4] C. E. Shannon, Bell Syst. Tech. J. , 379 (1948).[5] T. M. Cover and J. A. Thomas, Elements of InformationTheory , Wiley Series in Telecommunications and SignalProcessing (Wiley-Interscience, New York, NY, USA,2006).[6] C. H. Bennett, H. J. Bernstein, S. Popescu, andB. Schumacher, Phys. Rev. A , 2046 (1996).[7] M. B. Plenio and S. Virmani, Quantum Inf. Comput. ,1 (2007).[8] R. Horodecki, P. Horodecki, M. Horodecki, andK. Horodecki, Rev. Mod. Phys. , 865 (2009).[9] G. Vidal, J. Mod. Opt. , 355 (2000).[10] M. J. Donald, M. Horodecki, and O. Rudolph, J. Math.Phys. , 4252 (2002).[11] B. Synak-Radtke and M. Horodecki, J. Phys. A , L423(2006). [12] S. L. Braunstein and P. van Loock, Rev. Mod. Phys. , 513 (2005).[13] N. J. Cerf, G. Leuchs, and E. S. Polzik, Quantum in-formation with continuous variables of atoms and light (Imperial College Press, 2007).[14] C. Weedbrook, S. Pirandola, R. Garc´ıa-Patr´on, N. J.Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Rev.Mod. Phys. , 621 (2012).[15] S. M. E., Rep. Math. Phys. , 81 (2018).[16] M. E. Shirokov, Quantum Inf. Process. , 164 (2020).[17] M. E. Shirokov, Preprint arXiv:2007.00417 (2020).[18] J. Sperling and W. Vogel, Phys. Scr. , 074024 (2015).[19] K. C. Tan, T. Volkoff, H. Kwon, and H. Jeong, Phys.Rev. Lett. , 190405 (2017).[20] B. Yadin, F. C. Binder, J. Thompson,V. Narasimhachar, M. Gu, and M. S. Kim, Phys. Rev.X , 041038 (2018).[21] K. C. Tan and H. Jeong, AVS Quantum Sci. , 014701(2019).[22] L. Lami, B. Regula, R. Takagi, and G. Ferrari, PreprintarXiv:2009.11313 (2020).[23] B. Regula, L. Lami, R. Takagi, and G. Ferrari, PreprintarXiv:2009.11302 (2020).[24] F. G. S. L. Brand˜ao, M. Horodecki, J. Oppenheim, J. M.Renes, and R. W. Spekkens, Phys. Rev. Lett. ,250404 (2013).[25] J. Goold, M. Huber, A. Riera, L. del Rio, andP. Skrzypczyk, J. Phys. A , 143001 (2016).[26] R. R. Tucci, Preprint arXiv:quant-ph/9909041 (1999).[27] M. Christandl and A. Winter, J. Math. Phys. , 829(2004).[28] F. G. S. L. Brand˜ao, M. Christandl, and J. Yard, Com-mun. Math. Phys. , 805 (2011).[29] K. Li and A. Winter, Commun. Math. Phys. , 63(2014).[30] M. E. Shirokov, J. Math. Phys. , 032203 (2016).[31] See the SM, which contains the references [], for detailedproofs of the results discussed in the main text.[32] M. B. Ruskai, Rev. Math. Phys. , 1147 (1994).[33] A. S. Holevo, Probabilistic and Statistical Aspects ofQuantum Theory , Publications of the Scuola NormaleSuperiore (Scuola Normale Superiore, 2011).[34] A. Serafini,
Quantum Continuous Variables: A Primerof Theoretical Methods (CRC Press, Taylor & FrancisGroup, 2017).[35] E. Schr¨odinger, Naturwissenschaften , 664 (1926).[36] J. R. Klauder, Ann. Phys. (N. Y.) , 123 (1960).[37] R. J. Glauber, Phys. Rev. , 2766 (1963).[38] E. C. G. Sudarshan, Phys. Rev. Lett. , 277 (1963).[39] J. R. Klauder and E. C. G. Sudarshan, Fundamentalsof Quantum Optics (Benjamin, NY, 1968).[40] R. J. Glauber, “Optical coherence and photon statis-tics,” in
Optique Et ´Electronique Quantiques: QuantumOptics and Electronics , Quantum optics and electron-ics, edited by C. DeWitt-Morette, A. Blandin, andC. Cohen-Tannoudji (Gordon and Breach, New York,1965) p. 63.[41] E. Knill, R. Laflamme, and G. J. Milburn, Nature ,46 (2001).[42] M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov,Rev. Sci. Instrum. , 071101 (2011).[43] E. H. Kennard, Z. Phys. , 326 (1927).[44] D. F. Walls, Nature , 141 (1983). [45] R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz,and J. F. Valley, Phys. Rev. Lett. , 2409 (1985).[46] U. L. Andersen, T. Gehring, C. Marquardt, andG. Leuchs, Phys. Scr. , 053001 (2016).[47] R. Schnabel, Phys. Rep. , 1 (2017).[48] V. V. Dodonov, I. A. Malkin, and V. I. Man’ko, Physica , 597 (1974).[49] T. C. Ralph, A. Gilchrist, G. J. Milburn, W. J. Munro,and S. Glancy, Phys. Rev. A , 042319 (2003).[50] A. P. Lund, T. C. Ralph, and H. L. Haselgrove, Phys.Rev. Lett. , 030503 (2008).[51] J. Joo, W. J. Munro, and T. P. Spiller, Phys. Rev. Lett. , 083601 (2011).[52] A. Facon, E.-K. Dietsche, D. Grosso, S. Haroche, J.-M.Raimond, M. Brune, and S. Gleyzes, Nature , 262(2016).[53] J. B. Brask, I. Rigas, E. S. Polzik, U. L. Andersen, andA. S. Sørensen, Phys. Rev. Lett. , 160501 (2010).[54] S.-W. Lee and H. Jeong, Phys. Rev. A , 022326(2013).[55] D. V. Sychev, A. E. Ulanov, A. A. Pushkina, M. W.Richards, I. A. Fedorov, and A. I. Lvovsky, Nat. Pho-tonics , 379 (2017).[56] B. C. Sanders, Phys. Rev. A , 2417 (1989).[57] J. P. Dowling, Contemp. Phys. , 125 (2008).[58] L. Lami, IEEE Trans. Inf. Theory , 2165 (2020).[59] M. Hillery, Phys. Rev. A , 725 (1987).[60] V. V. Dodonov, O. V. Man’ko, M. V. I., andA. W¨unsche, J. Mod. Opt. , 633 (2000).[61] P. Marian, T. A. Marian, and H. Scutaru, Phys. Rev.A , 022104 (2004).[62] F. G. S. L. Brand˜ao and M. B. Plenio, Nat. Phys. ,873 EP (2008).[63] C. Lee, Phys. Rev. A , R2775 (1991).[64] N. L¨utkenhaus and S. M. Barnett, Phys. Rev. A ,3340 (1995).[65] J. K. Asb´oth, J. Calsamiglia, and H. Ritsch, Phys. Rev.Lett. , 173602 (2005).[66] W. Vogel and J. Sperling, Phys. Rev. A , 052302(2014).[67] N. Killoran, F. E. S. Steinhoff, and M. B. Plenio, Phys.Rev. Lett. , 080402 (2016).[68] H. Kwon, K. C. Tan, T. Volkoff, and H. Jeong, Phys.Rev. Lett. , 040503 (2019).[69] A. Kenfack and K. ˙Zyczkowski, J. Opt. B , 396 (2004).[70] K. C. Tan, S. Choi, and H. Jeong, Phys. Rev. Lett. , 110404 (2020).[71] W. Vogel, Phys. Rev. Lett. , 1849 (2000).[72] L. Di´osi, Phys. Rev. Lett. , 2841 (2000).[73] W. Vogel, Phys. Rev. Lett. , 2842 (2000).[74] T. Richter and W. Vogel, Phys. Rev. Lett. , 283601(2002).[75] M. Idel, D. Lercher, and M. M. Wolf, J. Phys. A ,445304 (2016).[76] M. Bohmann and E. Agudelo, Phys. Rev. Lett. ,133601 (2020).[77] C. Gehrke, J. Sperling, and W. Vogel, Phys. Rev. A , 052118 (2012).[78] The transformations considered in Yadin et al. [20, The-orems 2 and 3] are probabilistic but exact, and moreoversingle-shot rather than asymptotic. One could arguethat especially their zero-error nature somewhat limitstheir operational relevance in applications.[79] S. Kullback and R. A. Leibler, Ann. Math. Statist. , 79 (1951).[80] H. Umegaki, Kodai Math. Sem. Rep. , 59 (1962).[81] M. J. Donald, Commun. Math. Phys. , 13 (1986).[82] D. Petz, Commun. Math. Phys. , 123 (1986).[83] M. Berta, O. Fawzi, and M. Tomamichel, Lett. Math.Phys. , 2239 (2017).[84] V. Vedral, M. B. Plenio, M. A. Rippin, and P. L.Knight, Phys. Rev. Lett. , 2275 (1997).[85] V. Vedral and M. B. Plenio, Phys. Rev. A , 1619(1998).[86] M. Fekete, Math. Z. , 228 (1923).[87] M. Sion, Pacific J. Math. , 171 (1958).[88] A. P. Lund, H. Jeong, T. C. Ralph, and M. S. Kim,Phys. Rev. A , 020101 (2004).[89] A. Laghaout, J. S. Neergaard-Nielsen, I. Rigas,C. Kragh, A. Tipsmark, and U. L. Andersen, Phys.Rev. A , 043826 (2013).[90] D. V. Sychev, A. E. Ulanov, A. A. Pushkina, I. A. Fe-dorov, M. W. Richards, P. Grangier, and A. I. Lvovsky,AIP Conf. Proc. , 020018 (2018).[91] M. Wang, Z. Qin, M. Zhang, L. Zeng, X. Su, C. Xie,and K. Peng, in (2018) pp. 1–2.[92] C. Oh and H. Jeong, J. Opt. Soc. Am. B , 2933 (2018).[93] K. Husimi, Proc. Phys.-Math. Soc. Jpn. , 264 (1940).[94] A. Wehrl, Rep. Math. Phys. , 353 (1979).[95] C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher,J. A. Smolin, and W. K. Wootters, Phys. Rev. Lett. ,722 (1996).[96] J. Eisert and M. B. Plenio, Int. J. Quantum Inf. , 479(2003).[97] J. Eisert, C. Simon, and M. B. Plenio, J. Phys. A ,3911 (2002).[98] M. E. Shirokov, Mat. Sb. , 724 (2016).[99] R. F. Werner and M. M. Wolf, Phys. Rev. Lett. , 3658(2001).[100] G. Giedke, B. Kraus, M. Lewenstein, and J. I. Cirac,Phys. Rev. Lett. , 167904 (2001).[101] G. Giedke, J. Eisert, I. J. Cirac, and M. B. Plenio,Quantum Inf. Comput. , 211 (2003).[102] G. Adesso and F. Illuminati, J. Phys. A , 7821 (2007).[103] L. Lami, A. Serafini, and G. Adesso, New J. Phys. ,023030 (2018).[104] E. Chitambar, D. Leung, L. Manˇcinska, M. Ozols, andA. Winter, Commun. Math. Phys. , 303 (2014).[105] Wikipedia contributors, “Operator topolo-gies — Wikipedia, the free encyclopedia,” https://en.wikipedia.org/w/index.php?title=Operator_topologies&oldid=792420018 (2017),[Online; accessed 29-October-2019].[106] A. Winter, IEEE Trans. Inf. Theory , 2481 (1999).[107] E. B. Davies, Commun. Math. Phys. , 277 (1969).[108] A. Bach and U. L¨uxmann-Ellinghaus, Commun. Math.Phys. , 553 (1986).[109] M. Ohya and D. Petz, Quantum Entropy and Its Use ,Theoretical and Mathematical Physics (Springer BerlinHeidelberg, 2004).[110] G. Lindblad, Commun. Math. Phys. , 305 (1973).[111] F. Hiai and D. Petz, Comm. Math. Phys. , 99 (1991).[112] B. C. Hall, Quantum Theory for Mathematicians , Grad-uate Texts in Mathematics (Springer New York, 2013).[113] B. Levi, Rendic. Istit. Lombardo
XXXIX , 775 (1906).[114] W. Rudin,
Principles of mathematical analysis , Interna-tional series in pure and applied mathematics (McGraw-
Hill, 1964).[115] F. Hansen and G. K. Pedersen, Bull. Lond. Math. Soc. , 553 (2003).[116] Here, classical registers are thought of as special d -dimensional quantum systems X ( d < ∞ ) with theproperty that any free operation Λ ∈ (cid:70) ( A → XB )satisfies that (∆ X ⊗ I B ) ◦ Λ = Λ, with ∆ X ( · ) .. = P di =1 | i ih i | ( · ) | i ih i | being the dephasing map. In otherwords, if a quantum system plays the role of a classicalregister, then no state apart from those that are diag-onal in a fixed orthonormal basis {| i i} i =1 ,...,d is everaccessible with free operations.[117] This just means that it does not contain the fractions ∞∞ or . We instead convene that ∞ = 0 and ∞ = ∞ .[118] A. Winter, Preprint arXiv:1712.10267 (2017).[119] Classical information can be encoded reliably in a CVquantum system by using as bits the quasi-orthogonalstates | α i , |− α i , with | α | (cid:29) Information and Information Stability ofRandom Variables and Processes , edited by A. Feinstein,Holden-Day series in time series analysis (Holden-Day,1964).[121] K. P. Seshadreesan and M. M. Wilde, Phys. Rev. A ,042321 (2015).[122] R. E. Megginson, An introduction to Banach space the-ory , Graduate Texts in Mathematics No. 183 (SpringerScience & Business Media, 2012).[123] E. H. Lieb and M. B. Ruskai, Phys. Rev. Lett. , 434(1973).[124] E. H. Lieb and M. B. Ruskai, J. Math. Phys. , 1938(1973).[125] E. H. Lieb, Adv. Math. , 267 (1973).[126] G. Lindblad, Commun. Math. Phys. , 147 (1975).[127] A. Winter, Commun. Math. Phys. , 291 (2016).[128] LL acknowledges useful discussions with Andreas Win-ter and Krishna Kumar Sabapathy on the problem ofcalculating N r ( | n ih n | ).[129] S. Barnett and P. M. Radmore, Methods in TheoreticalQuantum Optics , Oxford Series in Optical and ImagingSciences (Clarendon Press, 2002).[130] M. S. Kim, W. Son, V. Buˇzek, and P. L. Knight, Phys.Rev. A , 032323 (2002). Supplemental Material:Asymptotic state transformations of continuous variable resources
In what follows, we will provide complete proofs of all the results stated in the main text.
I. PRELIMINARIES
The mathematical object associated with a quantum system is a Hilbert space H . In light of this fundamentalassociation, we will discuss several spaces of operators on H . We fix some notation below.• B sa ( H ): the Banach space of bounded self-adjoint operators on H , equipped with the operator norm;• K sa ( H ): the Banach space of self-adjoint compact operators on H , equipped with the operator norm;• K +sa ( H ): the cone of positive semidefinite (and hence self-adjoint) compact operators on H ;• T sa ( H ): the Banach space of self-adjoint trace class operators on H , equipped with the trace norm;• T +sa ( H ): the cone of positive semidefinite (and hence self-adjoint) trace class operators on H ;• D ( H ): the set of density operators (i.e., positive semidefinite trace class operators with trace 1) on H .One has that T sa ( H ) ⊆ B sa ( H ), with equality iff H is finite-dimensional. Also, the duality relation T sa ( H ) ∗ = B sa ( H )holds at the level of Banach spaces. We remind the reader that the dual of a Banach space X equipped with a norm k · k X is the vector space of all linear functionals ϕ : X → R such that k ϕ k X ∗ .. = sup k x k X ≤ | ϕ ( x ) | < ∞ , equippedwith the norm k · k X ∗ .States on a quantum system A with Hilbert space H A are represented by density operators ρ A ∈ D ( H A ). Quantumchannels from A to B , where A, B are quantum systems, are nothing but completely positive trace preserving mapsΛ : T sa ( H A ) → T sa ( H B ). For a quantum channel Λ : T sa ( H A ) → T sa ( H B ), the adjoint Λ † is the linear mapΛ † : B sa ( H B ) → B sa ( H A ) defined by Tr (cid:2) T A Λ † ( X B ) (cid:3) .. = Tr [Λ( T A ) X B ] for all T A ∈ T sa ( H A ) and X B ∈ B sa ( H B ).Among the simplest examples of quantum channels are quantum measurements, represented by positive operator-valued measures (POVM), i.e., finite collections M = { E x } x ∈ X of positive semidefinite (bounded) operators E x ≥ P x E x = . Any quantum measurement can be written as a trace-preserving mapby making use of classical flags {| φ x i} x ∈ X : ρ P x Tr[ ρE x ] ρ x ⊗ | φ x ih φ x | , where ρ x is the output state in case theoutcome x is measured. A. Operator topologies
When dealing with infinite-dimensional quantum systems, some basic notions of topology are needed. As it turnsout, there is a wealth of topologies that can be defined on infinite-dimensional Banach spaces, and in particular onthe operator spaces discussed above [105]. For the sake of readability, we provide here a quick guide:• the weak operator topology on B sa ( H ) (and hence on T sa ( H ) and D ( H )) is the coarsest topology that makesall functionals A
7→ h ψ | A | ψ i continuous, for all | ψ i ∈ H ;• the weak* topology on T sa ( H ) is the coarsest topology that makes all functionals T Tr[
T K ] continuous,for all K ∈ K sa ( H );• the weak topology on T sa ( H ) is the coarsest topology that makes all functionals T Tr[
T A ] continuous, forall A ∈ B sa ( H );• the trace norm topology on T sa ( H ) is the one induced by the trace norm k · k ;• the operator norm topology on B sa ( H ) is the one induced by the operator norm k · k ∞ .The role of the weak* topology on T sa ( H ) will play a special role for us (cf. Lemma S36). Remark S1.
The weak* topology is the topology induced by the Banach space K sa ( H ) on its dual K sa ( H ) ∗ = T sa ( H ).Therefore, by the Banach–Alaoglu theorem the unit ball B T sa ( H ) .. = { T ∈ T sa ( H ) : k T k ≤ } of T sa ( H ) is weak*compact.A notable consequence of the gentle measurement lemma [106, Lemma 9] is the following. Lemma S2 (‘SWOT’ convergence lemma [107, Lemma 4.3]) . For a net ( ω α ) α ⊆ T +sa ( H ) of positive semidefinite traceclass operators, if ω α wot −−→ α ω ∈ T +sa ( H ) in the weak operator topology, and moreover Tr ω α −→ α Tr ω , then ω α n −→ α ω innorm.Proof. The proof of [107] can be extended without difficulty to the case of nets.
Corollary S3.
The weak topology and the norm topology coincide on T +sa ( H ) . They also coincide with the weakoperator topology on D ( H ) .Proof. Two topologies are equal iff they have the same convergent nets. A weakly convergent net inside T +sa ( H )converges also with respect to the weak operator topology, and moreover also the sequence of traces converges to thecorresponding limit (because is a bounded operator). Hence, convergence in norm follows from Lemma S2. A net ofdensity operators that converges with respect to the weak operator topology has constant trace, hence we can againapply Lemma S2 and deduce convergence in norm. Remark S4.
The norm topology does not coincide with the weak operator topology on T +sa ( H ). For instance, thesequence of Fock states ( | n ih n | ) n converges to 0 in the weak operator topology, but it is not convergent in the normtopology (for instance because it is not of Cauchy type). B. Continuous-variable systems
Among all infinite-dimensional quantum systems, a central role is played by continuous-variable (CV) systems ,and here, perhaps most notably, by finite collections of harmonic oscillators. As mentioned in the main text, theHilbert space corresponding to an m -mode CV system is composed of all square-integrable complex-valued functionson the Euclidean space R m , denoted with H m = L ( R m ). Note that one can identify H m ’ H ⊗ m . The canonicaloperators x j and p j .. = − i ∂∂x j ( j = 1 , . . . , m ) satisfy the canonical commutation relations [ x j , x k ] ≡ ≡ [ p j , p k ]and [ x j , p k ] = iδ jk , with denoting the identity over H m . It is customary to define the annihilation and creationoperators by a j .. = x j + ip j √ , a † j .. = x j − ip j √ . (S1)In terms of a j , a † j , the canonical commutation relations take the form [ a j , a k ] ≡
0, [ a j , a † k ] = δ jk .On a single-mode system, Fock states are defined for k ∈ N by | k i .. = √ k ! ( a † ) k | i , where | i is the vacuumstate . For α ∈ C , the associated coherent state takes the form [35–38] | α i .. = e − | α | ∞ X k =0 α k √ k ! | k i . (S2)Extending these definitions to multimode systems is elementary. For k = ( k , . . . , k m ) (cid:124) ∈ N m , one sets | k i .. = N mj =1 | k j i ; analogously, for α = ( α , . . . , α m ) (cid:124) ∈ C m , a multimode coherent state is defined by | α i .. = N mj =1 | α j i .The displacement operators form a special family of unitary operators acting on H m . For α ∈ C m , they aredefined by (cid:68) ( α ) .. = exp hX mj =1 (cid:16) α j a † j − α ∗ j a j (cid:17)i . (S3)They satisfy the identity (cid:68) ( α ) (cid:68) ( β ) = e ( α (cid:124) β ∗ − α † β ) (cid:68) ( α + β ) , (S4)called the Weyl form of the canonical commutation relations , for all α, β ∈ C m , and they yield coherent statesupon acting on the vacuum, i.e., (cid:68) ( α ) | i = | α i ∀ α ∈ C m . (S5)For an arbitrary trace class operator T ∈ T sa ( H m ), its characteristic function χ T : C m → C is given by χ T ( α ) .. = Tr T (cid:68) ( α ) . (S6)As mentioned in the main text, in the QRT of nonclassicality one considers as free all those states that can beobtained as a statistical mixture of coherent states. Formally, an m -mode state σ is said to be classical if it can beexpressed as σ = R C m dµ ( α ) | α ih α | for a probability measure µ on C m . The set of m -mode classical states turns outto be given by [108] C m = conv {| α ih α | : α ∈ C m } , (S7)where conv denotes the closed convex hull. In our framework, we consider free any classical channel , that is, anycompletely positive and trace preserving map Λ : T sa ( H m ) → T sa ( H m ) with the property thatΛ( C m ) ⊆ C m . (S8) C. Entropy and relative entropies
The (von Neumann) entropy of some positive semidefinite trace class operator A ∈ T +sa ( H ) can be defined as S ( A ) .. = − Tr [ A log A ] . (S9)Note that this is a well-defined although possibly infinite quantity. One way to make sense of the expression (S9)is via the infinite sum S ( A ) = P i ( − a i log a i ), where A = P i a i | a i ih a i | is the spectral decomposition of A wherewe convene that 0 log a i −−−→ i →∞ A is trace class, the terms of the above sum are eventuallypositive. Hence, the sum itself can be assigned a well-defined value, possibly + ∞ . The relative entropy betweentwo positive A, B ∈ T +sa ( H ) is usually written as [80, 109] D ( A k B ) .. = Tr [ A (log A − log B )] . (S10)Again, the above expression is well defined and possibly infinite [110]. To see why, we represent it as the infinitesum D ( A k B ) .. = P i,j |h a i | b j i| ( a i log a i − a i log b j + log ( e )( b j − a i )) + log ( e ) Tr[ A − B ], where A = P i a i | a i ih a i | and B = P j b j | b j ih b j | are the spectral decompositions of A and B , respectively, and we assume that only terms with a i > b j > D ( A k B ) = + ∞ if there exists two indices i and j with a i > b j = 0, and h a i | b j i 6 = 0. As detailed in [110], the convexity of a a log a implies that all terms of the above infinite sum are non-negative, making the expression well defined. Inlight of the above discussion, it is not difficult to realize that a necessary condition for D ( A k B ) to be finite is thatsupp A ⊆ supp B . Thus, up to projecting everything onto a subspace we will often assume that B is faithful, i.e., that B > ρ and a measurement M = { E x } x ∈ X ,we define the associated outcome probability distribution on X as P M ρ ( x ) .. = Tr [ ρE x ]. Remembering that for twoclassical probability distributions p and q the Kullback–Leibler divergence is given by D KL ( p k q ) .. = P x p x (log p x − log q x ) [79], let us define the measured relative entropy between any two states ρ and σ as [81, 111] D M ( ρ k σ ) .. = sup M D KL (cid:0) P M ρ (cid:13)(cid:13) P M ρ (cid:1) . (S11)It is known that D M ( ρ k σ ) ≤ D ( ρ k σ ) for all pairs of states ρ, σ [81]. Recently, extending a result by Petz [82], Berta etal. have shown that for finite-dimensional systems equality holds if and only if [ ρ, σ ] = 0 [83]. The main tool employedby Berta et al. is a beautifully simple variational expression for the otherwise slightly intimidating expression (S11).In the rest of this subsection, we show how to extend their result to the infinite-dimensional case. Lemma S5.
Let ρ ∈ D ( H ) be a density operators on a (possibly infinite-dimensional) Hilbert space H , and let σ ∈ T +sa ( H ) be positive semidefinite and nonzero. Then D M ( ρ k σ ) = sup h ∈ B sa ( H ) (cid:8) Tr ρh − log Tr σ h (cid:9) (S12)= sup h ∈ B sa ( H ) (cid:8) Tr ρh + log ( e ) (cid:0) − Tr σ h (cid:1)(cid:9) (S13)= sup <δ For the case where the measurements in (S11) are restricted to be projective (i.e., M = { E x } x ∈ X with E x a projector for all x , and P x E x = ), the expression in (S12) has been obtained already by Petz [109,Proposition 7.13]). Remark S7. Let us highlight the main differences and similarities between the above six variational expressions.• We see immediately that they can be grouped in pairs: (S12) and (S13); (S14) and (S15); finally, (S16) and (S17).The two expressions in each pair involve an optimization over exactly the same set, and differ only by theobjective function, which contains a − log x in (S12), (S14), and (S16), and its linearized version log ( e )(1 − x )in (S13), (S15), and (S17).• The programs in (S14) and (S15) contain an optimization over all bounded operators L that are also boundedaway from 0, i.e., such that L ≥ δ for some δ > 0, where is the identity on H .• In the programs (S16) and (S17) we instead removed this latter constraint, and optimized only on positiveoperators L > 0. Of course, this is a priori not the same: in infinite dimension, it can happen — e.g., for anystrictly positive density operator — that L > L ≥ δ > L is possibly unbounded from below, it may happen that Tr[ ρ log L ] = −∞ . This is not a problem, because we always have that Tr[ σL ] > − log Tr[ σL ] < + ∞ ; therefore,the first addend is the only one that may diverge, and no uncertainties of the form −∞ + ∞ can arise in theobjective function. Proof of Lemma S5. Following the above observations, we divide the proof in several smaller steps.1. Let us start by showing that (S12) is equivalent to (S13), (S14) to (S15), and (S16) to (S17). We only presentthe argument for the equivalence between (S12) and (S13), as the others are entirely analogous. First, from theinequality log x ≤ log ( e )( x − 1) we see thatTr ρh − log Tr σ h ≥ Tr ρh + log ( e ) (cid:0) − Tr σ h (cid:1) for any h . At the same time, the expression (S12) is manifestly invariant under transformations of the type h h + λI for any λ ∈ R . So, we can always choose a λ in both expressions such that Tr σ h = 1, thussaturating the aforementioned inequality.2. Now, observe that (S12) is equivalent to (S14), upon a change in parametrization h = log L . In fact, log L is bounded if and only if L itself is bounded and moreover L ≥ δ > 0. This implies that the variationalexpressions in (S12), (S13), (S14), and (S15) all coincide.3. We now show that they also coincide with those in (S16) and (S17). Clearly, since the optimization in (S16)is over a larger set than that in (S14), its value cannot decrease. Therefore, to prove equality we only have toprove that sup <δ L > 0, and let us show how to construct a family of bounded L δ ≥ δ > δ → + { Tr ρ log L δ − log Tr σL δ } = Tr ρ log L − log Tr σL . (S18)Since the expression Tr ρ log L − log Tr σL is clearly scale-invariant in L , i.e., it takes the same value for L and λL , for all λ > 0, we can assume without loss of generality that L ≤ / 2. For 0 < δ ≤ / 2, set L δ .. = L + δ ≥ δ .Using the spectral theorem for bounded operators [112, Theorem 7.12], we can find a projection-valued measure µ on [0 , / 2] such that L = R / λdµ ( λ ) and therefore L δ = R / ( λ + δ ) dµ ( λ ). Defining the real-valued measure µ ρ on [0 , / 2] such that µ ρ ( X ) = Tr[ ρµ ( X )] for all measurable sets X ⊆ [0 , / ρ ( − log L )] = Z / ( − log λ ) dµ ρ ( λ ) , Tr [ ρ ( − log L δ )] = Z / ( − log ( λ + δ )) dµ ρ ( λ ) . Since the functions λ 7→ − log ( λ + δ ) are pointwise monotonically decreasing in δ , converge pointwise to λ 7→ − log λ , and all the functions involved are nonnegative, we can apply Levi’s monotone convergencetheorem [113] (see also [114, Theorem 11.28]) and conclude thatlim δ → + Tr [ ρ ( − log L δ )] = lim δ → + Z / ( − log ( λ + δ )) dµ ρ ( λ ) = Z / ( − log λ ) dµ ρ ( λ ) = Tr [ ρ ( − log L )] . On the other hand, clearly Tr[ σL δ ] = Tr[ σL ] + δ converges to Tr[ σL ] > δ → + . This proves (S18), andthus allows us to conclude that the optimizations in (S12)–(S17) all coincide.4. We now show that the variational program in (S15) actually yields the measured relative entropy D M ( ρ k σ ). Tobegin, we prove that in (S15) we can restrict L to be of the form L = I + R , with rk R < ∞ , without changingthe value of the supremum. To this end, pick L such that 1 /m ≤ L ≤ m for some m > 0, and consider anarbitrary (cid:15) > 0. Construct a finite-dimensional projector P such that k ρ − P ρP k , k σ − P σP k ≤ (cid:15) . Then,Tr ρ log L + log ( e ) (1 − Tr σL ) ≤ Tr P ρP log L + log ( e ) (1 − Tr P σP L ) + (cid:15) (log m + m log ( e )) ≤ Tr ρ log ( P LP + − P ) + log ( e ) (1 − Tr P σP L ) + (cid:15) (log m + m log ( e )) ≤ Tr ρ log ( P LP + − P ) + log ( e ) (1 − Tr σ ( P LP + − P )) + (cid:15) (log m + ( m + 1) log ( e )) . Here, 1 follows because k log L k ∞ ≤ log m and k L k ∞ ≤ m (where k · k ∞ is the operator norm), in 2 weapplied the operator Jensen inequality [115] to the operator-concave function log , and 3 is an application ofthe estimate Tr[ σ ( − P )] = Tr[ σ − P σP ] ≤ k σ − P σP k ≤ (cid:15) . We see that up to introducing an arbitrarilysmall error we can substitute L P LP + − P = + R , where rk R ≤ rk P < ∞ .Now, let R be of finite rank, and denote with R = P Nn =1 λ n P n its spectral decomposition. Then L = + R = P Nn =0 (1 + λ n ) P n , where P = − P Nn =1 P n and λ = 0, and consequentlyTr[ ρ log L ] + log ( e ) (1 − Tr[ σL ])= log ( e )(1 − Tr σ ) + N X n =0 (log (1 + λ n ) Tr[ ρP n ] − log ( e ) λ n Tr[ σP n ]) ≤ log ( e )(1 − Tr σ ) + N X n =1 (cid:18) Tr[ ρP n ] log Tr[ ρP n ]Tr[ σP n ] − log ( e ) (Tr[ ρP n ] − Tr[ σP n ]) (cid:19) ≤ log ( e )(1 − Tr σ ) + N X n =0 (cid:18) Tr[ ρP n ] log Tr[ ρP n ]Tr[ σP n ] − log ( e ) (Tr[ ρP n ] − Tr[ σP n ]) (cid:19) = N X n =0 Tr[ ρP n ] log Tr[ ρP n ]Tr[ σP n ] = D KL (cid:0) P M ρ (cid:13)(cid:13) P M σ (cid:1) ≤ D M ( ρ k σ ) . Here, the inequality in 4 comes from the estimate a log (1 + x ) − log ( e ) bx ≤ a log ab − log ( e )( a − b ), (which canbe proven simply by maximisation in x ), while 5 is a consequence of the fact that a log ab − log ( e )( a − b ) ≥ a, b ≥ 0. In 6, we introduced the measurement M .. = { P x } x ∈{ ,...,N } .The converse is proved with exactly the same argument put forth by Berta et al. in the proof of [83, Lemma 1].Namely, let M = { E x } x ∈ X be a quantum measurement. If there exists x ∈ X such that Tr[ σE x ] = 0 < Tr[ ρE x ],then on the one hand clearly D M ( ρ k σ ) ≥ D KL (cid:0) P M ρ (cid:13)(cid:13) P M σ (cid:1) = + ∞ . On the other, we see that the kernels of ρ and σ obey ker( σ ) (cid:42) ker( ρ ), i.e., there exists a pure state | ψ i ∈ ker( σ ) \ ker( ρ ). Setting L = λψ + − ψ andletting λ → + ∞ proves that the variational program in (S15) is unbounded from above, as it should be.We now consider the case where Tr[ σE x ] = 0 only when also Tr[ ρE x ] = 0. Introduce the set e X .. = { x ∈ X : Tr[ ρE x ] Tr[ σE x ] > } , and write: D KL (cid:0) P M ρ (cid:13)(cid:13) P M σ (cid:1) = X x ∈ e X Tr[ ρE x ] (log Tr[ ρE x ] − log Tr[ σE x ])= Tr (cid:20) ρ X x ∈ e X p E x log (cid:18) Tr[ ρE x ]Tr[ σE x ] · (cid:19) p E x (cid:21) ≤ Tr (cid:20) ρ log (cid:18)X x ∈ e X Tr[ ρE x ]Tr[ σE x ] E x (cid:19)(cid:21) = Tr [ ρ log L ] + log ( e ) (1 − Tr[ σL ]) , where 7 is again an application of the operator Jensen inequality [115] to the operator-concave function log ,and in 8 we defined L .. = P x Tr[ ρE x ]Tr[ σE x ] E x , so that Tr[ σL ] = 1. Remark S8. The programs (S13), (S15), and (S17) are all well defined also for σ = 0. They yield D M ( ρ k 0) = + ∞ ,as it should be. D. Quantum Resource Theories We now introduce a general notion of quantum resource theory. Note that our definition is slightly different fromthat in the recent review by Chitambar and Gour [3, Definition 1], in that we require also parallel composition (i.e.,tensor product) of free operations to be free. Definition S9. A quantum resource theory (QRT) is a pair (cid:82) = ( (cid:83) , (cid:70) ) , where (cid:83) is a family of (possiblyinfinite-dimensional) quantum systems that is closed under tensor products, in the sense that A, B ∈ (cid:83) implies that AB .. = A ⊗ B ∈ (cid:83) ; and contains the trivial system with Hilbert space C , while (cid:70) , called the set of free operations,is a mapping that assigns to every pair of systems A, B ∈ (cid:83) a set of channels from system A to B . Such a set will bedenoted with (cid:70) ( A → B ) [116]. We will require that the following two consistency conditions are satisfied:(i) for all A ∈ (cid:83) , the identity is a free operation on A , in formula I A ∈ (cid:70) ( A → A ) ;(ii) free operations are closed under sequential compositions, namely, if A, B, C ∈ (cid:83) and Λ ∈ (cid:70) ( A → B ) , Γ ∈ (cid:70) ( B → C ) , then also Γ ◦ Λ ∈ (cid:70) ( A → C ) ;(iii) free operations are closed under parallel compositions, namely, if for j = 1 , one chooses A j , B j ∈ (cid:83) and Λ j ∈ (cid:70) ( A j → B j ) , then also Λ ⊗ Λ ∈ (cid:70) ( A ⊗ A → B ⊗ B ) . Given a QRT (cid:82) as above, one defines the set of free states on the system A ∈ (cid:83) as (cid:70) S ( A ) .. = (cid:70) (1 → A ) . (S19)Clearly, if partial traces are free, then Tr A (cid:70) S ( AB ) ⊆ (cid:70) S ( B ). A central role in our paper is played by resourcequantifiers, i.e., monotones. We define them as follows. Definition S10. Let (cid:82) = ( (cid:83) , (cid:70) ) be a resource theory. A mapping G assigning to each A ∈ (cid:83) a function G A : D ( H A ) → [0 , + ∞ ] on the set of states on A that takes on values in the extended reals [0 , + ∞ ] is called a resourcemonotone — or simply a monotone — if(i) G B (Λ( ρ )) ≤ G A ( ρ ) holds for all states ρ on A ∈ (cid:83) and for all free operations Λ ∈ (cid:70) ( A → B ) , where B ∈ (cid:83) isarbitrary;(ii) G A ( σ ) = 0 for all σ ∈ (cid:70) S ( A ) , with (cid:70) S ( A ) defined by (S19) .A monotone G is said to be:(a) faithful , if G A ( ρ ) = 0 implies that ρ ∈ (cid:70) S ( A ) ;(b) convex , if all functions G A are convex, i.e., G A ( P i p i ρ i ) ≤ P i p i G ( ρ i ) for all A ∈ (cid:83) and all statistical ensembles { p i , ρ i } on A ;(c) lower semicontinuous , if G A is lower semicontinuous as a function on D ( H A ) for all A ∈ (cid:83) , i.e., if lim n →∞ k ρ n − ρ k = 0 for a sequence of states on A implies that lim inf n →∞ G A ( ρ n ) ≥ G ( ρ ) ;(d) strongly superadditive , if G AB ( ρ AB ) ≥ G A ( ρ A ) + G B ( ρ B ) holds for all A, B ∈ (cid:83) and for all states ρ AB ∈ D ( H AB ) , with AB = A ⊗ B ;(e) superadditive , if G AB ( ρ A ⊗ σ B ) ≥ G A ( ρ A ) + G B ( σ B ) for all A, B ∈ (cid:83) and for all states ρ A ∈ D ( H A ) and σ B ∈ D ( H B ) ;(f) weakly superadditive , if G A ...A n ( ρ ⊗ n ) ≥ nG A ( ρ ) for all A ∈ (cid:83) and all states ρ ∈ D ( H A ) , where A . . . A n denotes the joint system formed by n copies of A ;(g) additive , if G AB ( ρ A ⊗ σ B ) = G A ( ρ A )+ G B ( σ B ) for all A, B ∈ (cid:83) and for all states ρ A ∈ D ( H A ) and σ B ∈ D ( H B ) ;(h) weakly additive , if G A ...A n ( ρ ⊗ n ) = nG A ( ρ ) for all A ∈ (cid:83) and all states ρ ∈ D ( H A ) , where A . . . A n denotesthe joint system formed by n copies of A . Remark S11. In Definition S10, (d) ⇒ (e) ⇒ (f) and (g) ⇒ (h). Remark S12. Any monotone is automatically invariant under free unitaries whose inverse is also free. Remark S13. The notions of strongly subadditive, subadditive, or upper semicontinuous monotone are obtained byreversing the inequalities and exchanging lim inf with lim sup in (c), (d), and (e) of Definition S10. Note. In what follows, with a slight abuse of notation we will often drop the subscript of G specifying the system itrefers to, and think of a monotone G as a function defined directly on the collection of states on all possible systems A ∈ (cid:83) .It turns out that any monotone G can be made weakly additive by a procedure known as “regularization”. Definition S14. Let ( (cid:83) , (cid:70) ) be a QRT equipped with a monotone G . Then the functions G ↓ , ∞ ( ρ ) .. = lim inf n →∞ n G (cid:0) ρ ⊗ n (cid:1) , (S20) G ↑ , ∞ ( ρ ) .. = lim sup n →∞ n G (cid:0) ρ ⊗ n (cid:1) (S21) are called the lower and upper regularizations of G . On the domain of states ρ such that G ↓ , ∞ ( ρ ) = G ↑ , ∞ ( ρ ) = .. G ∞ ( ρ ) one can speak of a unique regularization G ∞ . The following result is immediate from the definition. Lemma S15. Let ( (cid:83) , (cid:70) ) be a QRT equipped with a monotone G . Then the lower and upper regularizations G ↓ , ∞ and G ↑ , ∞ given by Definition S14 are also monotones. Moreover, G ∞ is weakly additive on its domain, i.e., G ↓ , ∞ ( ρ ) = G ↑ , ∞ ( ρ ) for a state ρ implies that G ∞ ( ρ ⊗ n ) ≡ n G ( ρ ) for all n ∈ N + . Proof. Let us start by showing that, e.g., G ↓ , ∞ is a monotone. Since parallel composition of free operations is free,for all ρ ∈ D ( H A ) and for all Λ ∈ (cid:70) ( A → B ), with A, B ∈ (cid:83) , we obtain that G ↓ , ∞ (Λ( ρ )) = lim inf n →∞ n G (cid:0) Λ( ρ ) ⊗ n (cid:1) = lim inf n →∞ n G (cid:0) Λ ⊗ n (cid:0) ρ ⊗ n (cid:1)(cid:1) ≤ lim inf n →∞ n G (cid:0) ρ ⊗ n (cid:1) = G ↓ , ∞ ( ρ ) . Moreover, if ρ is free, also ρ ⊗ n is so, and hence G ↓ , ∞ ( ρ ) = 0 as well. This proves the first claim.Now, by definition G ↓ , ∞ ( ρ ) = G ↑ , ∞ ( ρ ) implies that the sequence (cid:0) G ( ρ ⊗ k ) (cid:1) k ∈ N + has a limit. If that is the case,then clearly G ∞ ( ρ ⊗ n ) = lim k →∞ k G (cid:0) ρ ⊗ kn (cid:1) = n lim n →∞ kn G (cid:0) ρ ⊗ kn (cid:1) = n G ∞ ( ρ ) for all n ∈ N + .A useful fact that is slightly less obvious is as follows. Lemma S16. Let ( (cid:83) , (cid:70) ) be a QRT equipped with a monotone G that is weakly (respectively, strongly) superadditive.Then the regularization G ∞ in Definition S14 exists for all states ρ , i.e., G ↓ , ∞ ( ρ ) = G ↑ , ∞ ( ρ ) = .. G ∞ ( ρ ) for all ρ ∈ D ( H A ) with A ∈ (cid:83) . It is weakly additive (respectively, weakly additive and strongly superadditive), and satisfiesthat G ∞ ≥ G . If G was lower semicontinuous, then so is G ∞ . Remark S17. The above result is still valid if we replace superadditivity with subadditivity, lower semicontinuitywith upper semicontinuity, and reverse all inequalities. Proof of Lemma S16. Due to weak superadditivity, for all states ρ the sequence ( a n ) n ∈ N + defined by a n .. = G ( ρ ⊗ n ) issuperadditive, meaning that a n + m ≥ a n + a m . Therefore, by Fekete’s lemma [86] lim n →∞ a n n exists, and it satisfiesthat lim n →∞ a n n = sup n ∈ N + a n n . Therefore, G ∞ ( ρ ) = lim n →∞ n G (cid:0) ρ ⊗ n (cid:1) = sup n ∈ N + n G (cid:0) ρ ⊗ n (cid:1) is well defined for all ρ , and satisfies G ∞ ( ρ ) ≥ G ( ρ ). We already saw in Lemma S15 that it is a weakly additivemonotone, so it suffices to show that it is strongly superadditive if G was such. This is easy to establish: G ∞ ( ρ AB ) = lim n →∞ n G (cid:0) ρ ⊗ nAB (cid:1) ≥ lim n →∞ n (cid:0) G (cid:0) ρ ⊗ nA (cid:1) + G (cid:0) ρ ⊗ nB (cid:1)(cid:1) = lim n →∞ n G (cid:0) ρ ⊗ nA (cid:1) + lim n →∞ n G (cid:0) ρ ⊗ nB (cid:1) . To see that G ∞ is lower semicontinuous if so was G , just notice that G ∞ ( ρ ) = sup n ∈ N + n G ( ρ ⊗ n ) is the pointwisesupremum of lower semicontinuous functions and thus must itself be lower semicontinuous.We continue by recalling the definition of asymptotic transformation rate. Definition S18. Let ( (cid:83) , (cid:70) ) be a QRT. For any two systems A, B ∈ (cid:83) and any two states ρ A ∈ D ( H A ) and σ B ∈ D ( H B ) , the corresponding (standard) asymptotic transformation rate is given by R ( ρ A → σ B ) .. = sup ( r : lim n →∞ inf Λ n ∈ (cid:70) ( A n → B b rn c ) (cid:13)(cid:13)(cid:13) Λ n (cid:0) ρ ⊗ nA (cid:1) − σ ⊗b rn c B (cid:13)(cid:13)(cid:13) = 0 ) , (S22) where A n denotes the system composed of n copies of A . Any number r > in the set on the right-hand side of (S22) is called a (standard) achievable rate for the transformation ρ A → σ B . As discussed in the main text, the above definition captures the intuitive notion of maximum yield of copies of thetarget state σ B that can be obtained per copy of the initial state ρ A by means of free operations and with asymptoticallyvanishing error. In Definition S18, we have measured the error using the global trace distance. However, it is possibleand sometimes even reasonable to modify the error criterion. For instance, in a situation where the output copies aredistributed to noninteracting parties, what is relevant is the maximum local error rather than the global one. Thistrain of thought inspires the following definition. Definition S19. Let ( (cid:83) , (cid:70) ) be a QRT. For any two systems A, B ∈ (cid:83) and any two states ρ A ∈ D ( H A ) and σ B ∈ D ( H B ) , the corresponding maximal asymptotic transformation rate is given by e R ( ρ A → σ B ) .. = sup ( r : lim n →∞ inf Λ n ∈ (cid:70) ( A n → B b rn c ) max j =1 ,..., b rn c (cid:13)(cid:13)(cid:13)(cid:0) Λ n (cid:0) ρ ⊗ nA (cid:1)(cid:1) j − σ B (cid:13)(cid:13)(cid:13) = 0 ) , (S23) where for a state Ω ∈ D ( H B k ) defined on k copies of B we defined Ω j .. = Tr B k \ B j Ω ∈ D ( H B j ) as the reduced state onthe j th subsystem. Any number r > in the set on the right-hand side of (S22) is called a maximally achievablerate for the transformation ρ A → σ B . The following elementary inequality between the rates in Definitions S18 and S19 holds. Lemma S20. Let ( (cid:83) , (cid:70) ) be a QRT. For any two systems A, B ∈ (cid:83) and any two states ρ A ∈ D ( H A ) and σ B ∈ D ( H B ) ,it holds that R ( ρ A → σ B ) ≤ e R ( ρ A → σ B ) . (S24) Proof. For all n and all free operations Λ n ∈ (cid:70) (cid:0) A n → B b rn c (cid:1) , the data processing inequality for the trace norm [32]implies that max j =1 ,..., b rn c (cid:13)(cid:13)(cid:13)(cid:0) Λ n (cid:0) ρ ⊗ nA (cid:1)(cid:1) j − σ B (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13) Λ n (cid:0) ρ ⊗ nA (cid:1) − σ ⊗b rn c B (cid:13)(cid:13)(cid:13) . Therefore, a sequence of protocols that achieves a rate r in (S22) (i.e., that makes the global error vanish) achievesthe same rate in (S23) (because the maximum local error will also vanish). The claim follows. II. RESTRICTED ASYMPTOTIC CONTINUITY IN INFINITE DIMENSIONA. Abstract approach The traditional approach to the problem of bounding asymptotic transformation rates in infinite-dimensional QRTsmakes use of the notion of restricted asymptotic continuity [15–17, 30, 97]. Let us give a formal definition, takenfrom [30, Corollary 7]. Definition S21. Let ( (cid:83) , (cid:70) ) be a QRT equipped with a monotone G . For some B ∈ (cid:83) , fix a family of states (cid:84) = { (cid:84) B n } n ∈ N + , with (cid:84) B n ⊆ D ( H B n ) = (cid:68) (cid:0) H ⊗ nB (cid:1) . We say that G is asymptotically continuous on (cid:84) if for allsequence of states ( ρ n ) n ∈ N + and ( σ n ) n ∈ N + with ρ n , σ n ∈ (cid:84) B n and lim n →∞ k ρ n − σ n k = 0 it holds that lim n →∞ | G ( ρ n ) − G ( σ n ) | n = 0 . (S25)Formally, from the above definition the following can be deduced. Proposition S22. Let ( (cid:83) , (cid:70) ) be a QRT equipped with a weakly additive monotone G . Consider A, B ∈ (cid:83) , and assumethat there exists a family of states (cid:84) = { (cid:84) B n } n ∈ N + on which G is asymptotically continuous. Pick ρ A ∈ D ( H A ) and σ B ∈ D ( H B ) with σ ⊗ nB ∈ (cid:84) B n for all sufficiently large n , and consider the modified asymptotic transformation rate R (cid:84) ( ρ A → σ B ) .. = sup r : lim n →∞ inf Λ n ∈ (cid:70) ( A n → B b rn c ) Λ n ( ρ ⊗ nA ) ∈ (cid:84) Bn (cid:13)(cid:13)(cid:13) Λ n (cid:0) ρ ⊗ nA (cid:1) − σ ⊗b rn c B (cid:13)(cid:13)(cid:13) = 0 (S26) satisfies that R (cid:84) ( ρ A → σ B ) ≤ G ( ρ A ) G ( σ B ) , (S27) whenever the right-hand side is well defined [117].Proof. For any sequence of free operations Λ n ∈ (cid:70) (cid:0) A n → B b rn c (cid:1) , Λ n ( ρ ⊗ nA ) ∈ (cid:84) B n satisfyinglim inf n →∞ (cid:13)(cid:13)(cid:13) Λ n (cid:0) ρ ⊗ nA (cid:1) − σ ⊗b rn c B (cid:13)(cid:13)(cid:13) = 0, it holds that G ( ρ A ) = lim inf n →∞ n G (cid:0) ρ ⊗ nA (cid:1) ≥ lim inf n →∞ n G (cid:0) Λ n (cid:0) ρ ⊗ nA (cid:1)(cid:1) ≥ lim inf n →∞ (cid:18) n G (cid:16) σ ⊗b rn c B (cid:17) − n (cid:12)(cid:12)(cid:12) G (cid:0) Λ n (cid:0) ρ ⊗ nA (cid:1)(cid:1) − G (cid:16) σ ⊗b rn c B (cid:17)(cid:12)(cid:12)(cid:12)(cid:19) = lim inf n →∞ b rn c n G ( σ B )= r G ( σ B ) . B. A physically interesting case: energy-constrained asymptotic continuity The above definition may seem rather abstract. However, there is a physically very natural scenario where it canbe applied. Let us assume that a certain system B ∈ (cid:83) is equipped with a Hamiltonian (i.e., a self-adjoint operator) H B . Let us assume that the Hamiltonians add up without interaction terms upon taking multiple copies of B , informula H B n = H B ⊗ ⊗ ( n − B + B ⊗ H B ⊗ ⊗ ( n − B + . . . + ⊗ ( n − B ⊗ H B . Now, for a real number E , set (cid:84) EB n .. = { ρ ∈ D ( H B n ) : Tr [ ρ H B n ] ≤ nE } . (S28)Basically, we are considering states whose energy increases at most linearly in the number of systems n . When theset (S28) is chosen in Definition S21, the corresponding notion of restricted asymptotic continuity becomes a physicallyrelevant and indeed fruitful one. Definition S23. Let ( (cid:83) , (cid:70) ) be a QRT endowed with a monotone G . Let B ∈ (cid:83) be equipped with a Hamiltonian H B . If for all E the monotone G is asymptotically continuous on the set (cid:84) E defined by (S28) , then we say that itis asymptotically continuous in the presence of an energy constraint , or EC asymptotically continuous for short. In practice, the above definition just means that whenever ( ρ n ) n ∈ N + and ( σ n ) n ∈ N + are sequences of stateswith ρ n , σ n ∈ D ( H A n ), Tr[ ρ n H A n ] , Tr[ σ n H A n ] ≤ nE (for a fixed but arbitrary real number E ), and moreoverlim n →∞ k ρ n − σ n k = 0, then lim n →∞ | G ( ρ n ) − G ( σ n ) | n = 0 . (S29)This definition turns out to encompass a sufficiently wide set of monotones. For example, Shirokov has provedthat many important entanglement monotones are EC asymptotically continuous, with respect to several physicallyrelevant Hamiltonians [15–17, 30]. To deduce a useful result from Proposition S22 we need to fix some terminology. Definition S24 [118, p. 9] . Let A, B be two quantum system equipped with Hamiltonians H A , H B . A quantum channel Λ : T sa ( H A ) → T sa ( H B ) from A to B is called ( κ, δ ) -energy-limited if Λ † ( H B ) ≤ κH A + δ , with Λ † : B sa ( H B ) → B sa ( H A ) being the adjoint of Λ . The set of such channels will be denoted with EL κ,δ ( A → B ) , where the choice of theHamiltonians is not made explicit and assumed to be clear from the context. In such a setting, directly from Proposition S22 we deduce the following. Proposition S25. Let ( (cid:83) , (cid:70) ) be a QRT. Let H A , H B be two Hamiltonians on A, B ∈ (cid:83) , and let G be a weaklyadditive monotone that is EC asymptotically continuous. Then for all ρ A ∈ D ( H A ) and σ B ∈ D ( H B ) with finiteenergy (i.e., such that Tr[ ρ A H A ] , Tr[ σ B H B ] < ∞ ) the uniformly energy-constrained (UEC) asymptotic transformationrate defined by R UEC ( ρ A → σ B ) .. = sup <κ,δ< ∞ sup ( r : lim n →∞ inf Λ n ∈ (cid:70) ( A n → B b rn c ) ∩ EL κ,δ ( A n → B b rn c ) (cid:13)(cid:13)(cid:13) Λ n (cid:0) ρ ⊗ nA (cid:1) − σ ⊗b rn c B (cid:13)(cid:13)(cid:13) = 0 ) (S30) satisfies that R UEC ( ρ A → σ B ) ≤ G ( ρ A ) G ( σ B ) , (S31) whenever the right-hand side is well defined. Proof. Let E .. = max { Tr[ ρ A H A ] , Tr[ σ B H B ] } < ∞ . Fix arbitrary 0 < κ, δ < ∞ , and let the rate r be achievablein (S30) by means of a sequence of protocols Λ n ∈ (cid:70) (cid:0) A n → B b rn c (cid:1) ∩ EL κ,δ (cid:0) A n → B b rn c (cid:1) . Then for sufficientlylarge n it holds thatTr (cid:2) σ ⊗ nB H B n (cid:3) = n Tr[ σ B H B ] ≤ nE ≤ nE , Tr (cid:2) Λ n (cid:0) ρ ⊗ nA (cid:1) H B n (cid:3) = Tr (cid:2) ρ ⊗ nA Λ † n ( H B n ) (cid:3) ≤ Tr (cid:2) ρ ⊗ nA ( κH A n + δ ) (cid:3) = nκ Tr[ ρ A H A ] + δ ≤ nκE + δ ≤ nE , where we set E .. = max { κ, } ( E + 1). Since G is asymptotically continuous on the set (cid:84) E , and we have just shownthat σ ⊗ nB , Λ n (cid:0) ρ ⊗ nA (cid:1) ∈ (cid:84) E , then we can apply Proposition S22 and conclude the proof.The reason why we would like to improve Proposition S25 is twofold. First, it only allows us to bound the standardasymptotic transformation rate (Definition S18), while we have seen that in certain settings the relevant quantityis the maximal asymptotic transformation rate (Definition S19). Secondly, it takes only into account sequences ofprotocols (Λ n ) n that are uniformly energy-constrained, meaning that the output energy is at most E out ≤ κE in + δ ,with κ and δ fixed for the whole sequence. If each Λ n is ( κ n , δ n )-energy-limited for each n , but lim sup n →∞ κ n = + ∞ or lim sup n →∞ δ n n , the above method does not seem to tell us much about the corresponding rate, even when the initialand final states have a fixed (and finite) energy. Therefore, for instance, a sequence of free operations on CV systemswhere each Λ n involves either (a) a squeezing whose intensity increases with n and tends to ∞ in the limit n → ∞ ; or(b) a displacement unitary whose parameter α n is superlinear in n , are excluded from the bound in Proposition S25.We will see that our method eliminates the need for both of these requirements, and instead provides ultimatebounds on maximal (instead of standard) asymptotic transformation rates, in a setting where the free protocolsemployed are otherwise totally unconstrained. III. PROOF OF THEOREMS 2 AND 3A. Elementary properties of nonclassicality monotones based on relative entropies Throughout this section, we complete the study of the nonclassicality monotones introduced in the main text, givingalso a complete proof of Theorems 2 and 3. We start by recalling the main definitions. Definition S26. The QRT of nonclassicality is constructed as in Definition S9 by letting (cid:83) be the family of allCV quantum systems [119]. The set of free operations (cid:70) is composed by all classical channels , that is, all quantumchannels Λ : T sa ( H m ) → T sa ( H m ) with the property that Λ( C m ) ⊆ C m , where C m .. = conv {| α ih α | : α ∈ C m } is theset of classical states . Remark S27. The set of classical states and that of classical channels are both convex. Definition S28. Let ρ ∈ D ( H m ) be an m -mode state. The relative entropy of nonclassicality and the measuredrelative entropy of nonclassicality of ρ are defined by N r ( ρ ) .. = inf σ ∈ C m D ( ρ k σ ) , N Mr ( ρ ) .. = inf σ ∈ C m D M ( ρ k σ ) . (4) Their lower and upper regularizations as given by Definition S14 are denoted with N ↓ , ∞ r , N ↑ , ∞ r and N M, ↓ , ∞ r , N M, ↓ , ∞ r ,respectively. Clearly, all of these measures are invariant under the action of displacements (S3), because these form a group ofclassical operations. In fact, they are also invariant under any passive quantum operation, i.e., under any unitary U on H m such that h U, P j a † j a j i = 0.If the reader is worried by the proliferation of regularized measures in Definition S28, they should not be. In fact,we will show that the regularizations are unique in all physically interesting cases. The first step is as follows. Lemma S29. The quantities N r and N Mr are faithful and convex nonclassicality monotones. They obey the inequality N r ≥ N Mr . Moreover, N r is strongly subadditive.Proof. The argument is completely standard. The inequality N r ≥ N Mr is obvious, and follows from the same relationbetween the relative entropy and its measured version. Since both D ( ·k· ) and D M ( ·k· ) obey the data processinginequality, for every classical channel Λ : T sa ( H m ) → T sa ( H m ) we obtain that N r (Λ( ρ )) = inf σ ∈ C m D (cid:0) Λ( ρ ) (cid:13)(cid:13) σ (cid:1) ≤ inf σ ∈ Λ( C m ) D (cid:0) Λ( ρ ) (cid:13)(cid:13) σ (cid:1) = inf σ ∈ C m D (cid:0) Λ( ρ ) (cid:13)(cid:13) Λ( σ ) (cid:1) ≤ inf σ ∈ C m D ( ρ k σ ) = N r ( ρ ) , N Mr . This proves monotonicity.Convexity descends from the fact that both N r and N Mr are defined as the infimum of a jointly convex function ona convex domain. For example, N r (cid:16)X i p i ρ i (cid:17) = inf σ ∈ C m D (cid:16)X i p i ρ i (cid:13)(cid:13) σ (cid:17) = inf { σ i } i ⊆ C m D (cid:16)X i p i ρ i (cid:13)(cid:13) X i p i σ i (cid:17) ≤ inf { σ i } i ∈ C m X i p i D ( ρ i k σ i )= X i p i inf σ i ∈ C m D ( ρ i k σ i )= X i p i N r ( ρ i ) . The proof for N Mr is entirely analogous.Faithfulness follows, e.g., from Pinsker’s inequality D KL ( p k q ) ≥ log ( e ) k p − q k [120], which implies that D M ( ρ k σ ) = sup M D KL (cid:0) P M ρ (cid:13)(cid:13) P M σ (cid:1) ≥ 12 log ( e ) sup M (cid:13)(cid:13) P M ρ − P M σ (cid:13)(cid:13) = 12 log ( e ) k ρ − σ k , where in the last line we used the elementary fact that the trace distance is achieved by the (binary) measurement { Π , − Π } , with Π being the projector onto the positive subspace of ρ − σ .To prove the strong subadditivity of N r , just notice that for all ( m + n )-mode CV systems AB it holds that N r ( ρ A ⊗ σ B ) = inf σ AB ∈ C m + n D ( ρ A ⊗ σ B k σ AB ) ≤ inf σ A ⊗ σ B ∈ C m + n D ( ρ A ⊗ σ B k σ A ⊗ σ B )= inf σ A ∈ C m , σ B ∈ C n { D ( ρ A k σ A ) + D ( ρ B k σ B ) } = N r ( ρ A ) + N r ( ρ B ) , where in the third line we used the identity [109, Eq. (5.22)]. Corollary S30. The functions N M, ↓ , ∞ r , N M, ↓ , ∞ r are nonclassicality monotones. The regularization N ↓ , ∞ r = N ↑ , ∞ r = .. N ∞ r is unique and is a weakly additive nonclassicality monotone; it satisfies that N ∞ r ≤ N r .Proof. Follows directly from Lemma S29 and Lemma S16. B. The long march towards Theorem 2: the monotone Γ We now set out to prove Theorem 2. To this end, we first formalize the definition of the quantity that appears onthe right-hand side of (6). Definition S31. For an arbitrary m -mode state ρ , let us construct the quantity Γ( ρ ) .. = sup h ∈ B sa ( H m ) (cid:26) Tr[ ρh ] − log sup α ∈ C m h α | h | α i (cid:27) (S32)= sup h ∈ B sa ( H m ) (cid:26) Tr[ ρh ] + log ( e ) (cid:18) − sup α ∈ C m h α | h | α i (cid:19)(cid:27) (S33)Note that since 2 h > 0, there must exist some α ∈ C m such that h α | h | α i > 0. Moreover, the two programs in (S32)and (S33) are equivalent, as can be verified by following the same strategy as in step 1 of the proof of Lemma S5.This ensures that Γ is indeed well defined. Let us now establish some of its basic properties.3 Lemma S32. For an m -mode state ρ , we have that Γ( ρ ) = sup <δ The argument proceeds exactly as in steps 1–3 of the proof of Lemma S5.We deduce the following elementary but important properties of the function Γ. Proposition S33. The function Γ in Definition S31 is a convex, lower semicontinuous, strongly superadditive non-classicality monotone. It holds that Γ( ρ ) ≤ N Mr ( ρ ) for all states ρ .Proof. First of all, Γ is convex and lower semicontinuous because it is the pointwise supremum of convex-linearand lower semicontinuous functions ρ Tr[ ρh ] − log sup α ∈ C m h α | h | α i (cf. Definition S31). To see that it is anonclassicality monotone, consider ρ ∈ D ( H m ) and a classical channel Λ : T sa ( H m ) → T sa ( H m ), and writeΓ (Λ( ρ )) = sup The regularization Γ ∞ ( ρ ) .. = lim n →∞ n Γ( ρ ⊗ n ) exists and is unique for all states ρ . It is a lowersemicontinuous, weakly additive, and strongly superadditive nonclassicality monotone, and it satisfies that Γ ∞ ≥ Γ .Proof. Follows by combining Lemma S16 and Proposition S33. C. Proof of Theorems 2 and 3 In order to prove Theorem 2, and from there deduce Theorem 3, we need two more technical lemmata. The first onetells us that provided a state ρ has finite entropy, which will most definitely be the case in all situations of physicalinterest, we can take the operator L in the variational program for N Mr to be not only bounded but also trace class. Lemma S35. On an m -mode system, let e C m .. = conv ( C m ∪ { } ) (S39)5 denote the set of subnormalized classical states. Then, the measured relative entropy of nonclassicality admits thevariational expressions N Mr ( ρ ) = inf σ ∈ C m sup The cone C + m .. = { λσ : λ ≥ , σ ∈ C m } ⊂ T +sa ( H m ) (S49) generated by the set of classical states is closed with respect to the weak* topology on T sa ( H m ) . Therefore, the set e C m = conv ( C m ∪ { } ) of subnormalized classical states, defined in (S39) , is weak*-compact.Proof. Remember by Remark S1 that we can think of T sa ( H m ) as the dual space to K sa ( H m ), the set of compactoperators on H m . We now show that C + m is in fact the dual of a set S ⊆ K sa ( H m ) of compact operators, i.e., C + m = S ∗ .. = { T ∈ T sa ( H m ) : Tr[ T K ] ≥ ∀ K ∈ S } . Dual sets turn out to be automatically weak*-closed. This can be seen, e.g., in the case of S ∗ , by noting that it canbe written as the intersection S ∗ = \ K ∈ S { T ∈ T sa ( H m ) : Tr[ T K ] ≥ } = \ K ∈ S ϕ − K ([0 , ∞ )) , where ϕ K : T sa ( H m ) → R is defined by ϕ K ( T ) .. = Tr[ T K ]. Since the maps ϕ K are weak*-continuous by definition,each set ϕ − K ([0 , ∞ )) is weak*-closed, and therefore so is their intersection S ∗ .7From now on, for the sake of readability we write everything for single-mode systems only. Set S .. = ( n X µ,ν =1 ψ ∗ µ ψ ν e | α µ − α ν | λ a † a (cid:68) ( α µ − α ν ) λ a † a : n ∈ N + , ψ ∈ C n , α ∈ C n , λ ∈ [0 , ) , where (cid:68) is the displacement operator (S3). Note that every operator in S is a finite linear combination of operatorsof the form λ a † a (cid:68) ( α µ − α ν ) λ a † a , which are clearly compact (in fact, even trace class) as long as λ ∈ [0 , | β ih β | ∈ S ∗ for every β ∈ C , because h β | n X µ,ν =1 ψ ∗ µ ψ ν e | α µ − α ν | λ a † a (cid:68) ( α µ − α ν ) λ a † a ! | β i = n X µ,ν =1 ψ ∗ µ ψ ν e | α µ − α ν | h β | λ a † a (cid:68) ( α µ − α ν ) λ a † a | β i = n X µ,ν =1 ψ ∗ µ ψ ν e | α µ − α ν | e − (1 − λ ) | β | h λβ | (cid:68) ( α µ − α ν ) | λβ i = e − (1 − λ ) | β | n X µ,ν =1 ψ ∗ µ ψ ν e λ (( α µ − α ν ) β ∗ − ( α µ − α ν ) ∗ β ) = e − (1 − λ ) | β | n X µ,ν =1 ψ ∗ µ e λ ( α µ β ∗ − α ∗ µ β ) ψ ν e λ ( α ∗ ν β − α ν β ∗ ) = e − (1 − λ ) | β | (cid:12)(cid:12)(cid:12)X nµ =1 ψ ∗ µ e λ ( α µ β ∗ − α ∗ µ β ) (cid:12)(cid:12)(cid:12) ≥ , where in 1 we used (S5) and in 2 the Weyl form (S4) of the canonical commutation relations multiple times. Since S ∗ is convex and weak*-closed, and hence in particular closed with respect to the trace norm topology, we see that C = conv {| β ih β | : β ∈ C } ⊆ S ∗ . Noting that S ∗ is a cone, i.e., it is closed under multiplication by nonnegativescalars, we conclude that in fact C +1 ⊆ S ∗ .Let us now prove the opposite inclusion, again in the single-mode case. Pick T ∈ T sa ( H ) such that Tr[ T K ] ≥ K ∈ S ; then 0 ≤ lim inf λ → − n X µ,ν =1 ψ ∗ µ ψ ν e | α µ − α ν | Tr h T λ a † a (cid:68) ( α µ − α ν ) λ a † a i ≤ n X µ,ν =1 ψ ∗ µ ψ ν e | α µ − α ν | lim λ → − Tr h T λ a † a (cid:68) ( α µ − α ν ) λ a † a i = n X µ,ν =1 ψ ∗ µ ψ ν e | α µ − α ν | Tr [ T (cid:68) ( α µ − α ν )]= n X µ,ν =1 ψ ∗ µ ψ ν e | α µ − α ν | χ T ( α µ − α ν )for all α ∈ C n and ψ ∈ C n , where the function χ T : C → C defined by χ T ( α ) = Tr[ T (cid:68) ( α )] is the character-istic function (S6) of T . To prove 3, since (cid:68) ( α µ − α ν ) is bounded (actually, unitary) it suffices to show thatlim λ → − (cid:13)(cid:13)(cid:13) λ a † a T λ a † a − T (cid:13)(cid:13)(cid:13) = 0 for all trace class T . To see this, we decompose T = T + − T − into its positive andnegative parts T ± ≥ 0, which are also trace class operators. Note thatlim λ → − Tr λ a † a T ± = lim λ → − ∞ X n =0 λ n h n | T ± | n i = ∞ X n =0 h n | T ± | n i , and therefore, by the gentle measurement lemma [106, Lemma 9] (see also [121, Lemma 9.4.2]),lim λ → − (cid:13)(cid:13)(cid:13) T ± − λ a † a T ± λ a † a (cid:13)(cid:13)(cid:13) = 0 , λ → − (cid:13)(cid:13)(cid:13) λ a † a T λ a † a − T (cid:13)(cid:13)(cid:13) ≤ lim λ → − (cid:13)(cid:13)(cid:13) λ a † a T + λ a † a − T + (cid:13)(cid:13)(cid:13) + lim λ → − (cid:13)(cid:13)(cid:13) λ a † a T − λ a † a − T − (cid:13)(cid:13)(cid:13) = 0 . We have just established that, for all α ∈ C n , the matrix (cid:16) e | α µ − α ν | χ T ( α µ − α ν ) (cid:17) µ,ν =1 ,...,n is positive semidefinite.This is known [74] to imply that T = λσ for some λ ≥ σ , i.e., T ∈ C + m .This latter claim can be also verified as follows. Applying the classical Bochner theorem, we see that the function C α ϕ T ( α ) .. = χ T ( α ) e | α | is the Fourier transform of a positive measure. Since ϕ T is well-known to be theFourier transform of the P -function [108, Lemma 1], we conclude that the P -function of T is non-negative, i.e., T isa non-negative multiple of a classical state.We conclude that C +1 = S ∗ , and hence that C +1 is weak*-closed. The exact same argument in fact shows that C + m is weak*-closed for any finite number of modes m . Since the unit ball B m .. = { T ∈ T sa ( H m ) : k T k ≤ } of T sa ( H m ) = K sa ( H m ) ∗ is weak*-compact by the Banach–Alaoglu theorem [122, Thm. 2.6.18], e C m = conv ( C m ∪ { } ) = C + m ∩ B m is the intersection of a weak*-closed and a weak*-compact set, and hence it is itself weak*-compact.We are finally ready to present our proof of Theorem 2. Theorem 2. Let ρ be an m -mode state with finite entropy S ( ρ ) < ∞ . Then, it holds that N Mr ( ρ ) = Γ( ρ ) = sup Note that Γ is given by either of the variational expressions (S32)–(S37). Let us rewrite (S41) as N Mr ( ρ ) = inf σ ∈ e C m sup 0; by definition of weak* topologyit is also weak*-continuous (because L is also compact);(iv) F ρ ( σ, · ) is a concave function on { L ∈ T sa ( H m ) : L > } for all σ ∈ e C m , because log is operator concave; itis also upper semicontinuous with respect to the trace norm topology, because Tr ρ log L = − S ( ρ ) − D ( ρ k L ),and L D ( ρ k L ) is lower semicontinuous with respect to the weak topology [109, Corollary 5.12(i)] and hence(Corollary S3) with respect to the trace norm topology, too.Since all assumptions of Sion’s minimax theorem [87] are satisfied, we can exchange infimum and supremum, and9write N Mr ( ρ ) = sup 0, 4 is proved by scale invariance of the expression on the sixth line exactlyas in step 1 of the proof of Lemma S5, in 5 we extended the supremum to all 0 < L ∈ B sa ( H m ), and finally 6 holdsthanks to Lemma S32. Since Proposition S33 establishes that Γ ≤ N Mr on all states, we have actually proved that N Mr ( ρ ) = sup When computed on finite-entropy states, N Mr is strongly superadditive and lower semicontinuous.Hence, its regularization N M, ∞ r exists on all finite-entropy states, and there satisfies that Γ ∞ = N M, ∞ r ≥ N Mr = Γ ; itis also strongly superadditive, lower semicontinuous, and moreover weakly additive and faithful.Proof. Thanks to the above Theorem 2, the function N Mr inherits all properties of Γ, as established in Proposition S33and Corollary S34, on the whole set of finite-entropy states. Faithfulness follows from the inequality N M, ∞ r ≥ N Mr and from the fact that N Mr itself is faithful (Lemma S29).We now restate Theorem 3, and prove the remaining claims. Theorem 3. When computed on finite-entropy states, N Mr and N M, ∞ r are strongly superadditive, lower semicontin-uous, and satisfy that N Mr ( ρ ) ≤ N M, ∞ r ( ρ ) ≤ N ∞ r ( ρ ) ≤ N r ( ρ ) . (7)Thus, if S ( ρ ) , S ( σ ) < ∞ then R ( ρ → σ ) ≤ e R ( ρ → σ ) ≤ N M, ∞ r ( ρ ) N M, ∞ r ( σ ) ≤ N r ( ρ ) N Mr ( σ ) , (8)provided that the ratios on the right-hand side are well defined. Proof. The properties of N Mr and N M, ∞ r follow from Corollary S37. From there we also see that N M, ∞ r ( ρ ) ≥ N Mr ( ρ )holds for all finite-entropy states ρ (it is in fact an elementary consequence of strong superadditivity). The inequality N ∞ r ≤ N r holds on all states, as established in Corollary S30 (it follows from the weak subadditivity of N r ). Moreover,regularizing the inequality N Mr ≤ N r (Lemma S29) we also see that N M, ∞ r ≤ N ∞ r . This completes the proof of (7).0To establish (8), we apply Theorem 1 to the lower semicontinuous, weakly additive, and strongly superadditivenonclassicality monotone Γ ∞ (Corollary S34): R ( ρ → σ ) ≤ e R ( ρ → σ ) ≤ Γ ∞ ( ρ )Γ ∞ ( σ ) = N M, ∞ r ( ρ ) N M, ∞ r ( σ ) , where the last equality follows from Corollary S37 and from the fact that S ( ρ ) , S ( σ ) < ∞ . Finally, the last estimatein (8) is a simple application of (7). IV. BOUNDING OUR NONCLASSICALITY MONOTONES We now come to the general problem of estimating the value of the nonclassicality monotones that we have intro-duced. A. Estimates based on the energy Our first concern is to show that the monotones N r , N Mr and their regularizations take on finite values on physicallyrelevant states. Proposition S38. Let ρ be an m -mode state with finite mean photon number E .. = Tr ρ (cid:16)P mj =1 a † j a j (cid:17) < ∞ . Then N Mr ( ρ ) ≤ N M, ∞ r ( ρ ) ≤ N ∞ r ( ρ ) ≤ N r ( ρ ) ≤ m g ( E/m ) , (S50) where g ( x ) .. = ( x + 1) log ( x + 1) − x log x .Proof. It is well known that the entropy of an m -mode state with finite mean photon number E is at most mg ( E/m ),which indeed corresponds to the entropy of the thermal state with the same energy. Hence, ρ has finite entropy, sothat (7) holds. Thus, we only have to show that N r ( ρ ) ≤ mg ( E/m ). For an arbitrary ν ≥ 0, let τ ν .. = 11 + ν ∞ X n =0 (cid:18) ν ν (cid:19) n | n ih n | = 11 + ν (cid:18) ν ν (cid:19) a † a (S51)be the single-mode thermal state of mean photon number ν . It is well known that τ ν ∈ C , and hence τ ⊗ mν ∈ C m , forall ν ∈ [0 , ∞ ). Therefore, N r ( ρ ) ≤ inf ν ≥ D (cid:0) ρ (cid:13)(cid:13) τ ⊗ mν (cid:1) = inf ν ≥ (cid:26) − S ( ρ ) + m log (1 + ν ) − E log (cid:18) ν ν (cid:19)(cid:27) = − S ( ρ ) + m g ( E/m ) , where we used the variational representation g ( x ) = inf ν ≥ (cid:26) log (1 + ν ) − x log (cid:18) ν ν (cid:19)(cid:27) , whose proof is elementary.At this point the reader may wonder, whether N r and N Mr can take the value + ∞ at all. We now set out to showthat this may indeed be the case. By Proposition S38, any state with this property must have infinite mean photonnumber. Proposition S39. There exists a single-mode (infinite-energy) state ρ ∈ D ( H ) such that N Mr ( ρ ) = N r ( ρ ) = + ∞ ,i.e., D ( ρ k σ ) = D M ( ρ k σ ) = + ∞ for all classical states σ ∈ C m — including those of infinite energy! Proof. Let ρ .. = 6 π X n n + 1) | n ih n | (S52)be a modified “Basel-type state”. Then, because of Lemma S33, we see that N r ( ρ ) ≥ N Mr ( ρ ) ≥ Γ( ρ ) = sup h ∈ B sa ( H m ) (cid:26) Tr[ ρh ] − log sup α ∈ C h α | h | α i (cid:27) . Now, set h N .. = P Nn =0 n | n ih n | . Observe thatTr[ ρ h N ] = 2 π N X n =0 n ( n + 1) −−−−→ N →∞ + ∞ , while sup α ∈ C h α | h N | α i = sup α ∈ C h α | (cid:18)X Nn =0 n/ | n ih n | (cid:19) | α i = sup α ∈ C X Nn =0 n/ | α | n +1 e −| α | (2 n )! ≤ X Nn =0 n/ sup α ∈ C | α | n +1 e −| α | (2 n )!= X Nn =0 n/ sup t ≥ t n e − t (2 n )!= X Nn =0 n/ n n e n (2 n )! −−−−→ N →∞ const < ∞ , where the evaluation of the limit is made possible by the fact that2 n/ n n e n (2 n )! ∼ n/ √ π n/ = 1 √ π n/ by Stirling’s formula, in the sense that the ratio between the left-hand and the right-hand sides tends to 1 as n → ∞ .We conclude that N r ( ρ ) ≥ N Mr ( ρ ) ≥ Γ( ρ ) ≥ lim N →∞ (cid:26) Tr[ ρ h N ] − log sup α ∈ C h α | h N | α i (cid:27) = + ∞ , as claimed. B. Estimates based on the Wehrl entropy The next result gives another independent upper bound for the relative entropy of nonclassicality. Proposition 4. For any state ρ it holds that − log k Q ρ k ∞ − S ( ρ ) − m log π ≤ N Mr ( ρ ) ≤ N M, ∞ r ( ρ ) ≤ S W ( ρ ) − S ( ρ ) , (9)where the Husimi Q -function Q : C m → C is defined by Q ρ ( α ) .. = π m h α | ρ | α i [93], and S W ( ρ ) .. = − Z d m α Q ρ ( α ) log ( π m Q ρ ( α )) (S53)is the Wehrl entropy [94].2 Proof. Let us start by proving that N Mr ( ρ ) ≤ S W ( ρ ) − S ( ρ ). We can restrict without loss of generality to those statessuch that S W ( ρ ) < ∞ . Since Wehrl has proved that S W > S [94], this also implies that S ( ρ ) < ∞ . We are thereforein the situation of Theorem 2, so that we can write N Mr ( ρ ) = sup ω ∈ D ( H m ) (cid:26) Tr ρ log ω − log sup α ∈ C m h α | ω | α i (cid:27) = sup ω ∈ D ( H m ) (cid:26) − S ( ρ ) − D ( ρ k ω ) − log sup α ∈ C m h α | ω | α i (cid:27) = sup ω ∈ D ( H m ) {− S ( ρ ) − D ( ρ k ω ) − log ( π m k Q ω k ∞ ) } ≤ sup ω ∈ D ( H m ) {− S ( ρ ) − D KL ( Q ρ k Q ω ) − log ( π m k Q ω k ∞ ) } = sup ω ∈ D ( H m ) (cid:26) − S ( ρ ) + S W ( ρ ) + Z d m αQ ρ ( α ) log Q ω ( α ) − log k Q ω k ∞ (cid:27) ≤ S W ( ρ ) − S ( ρ ) . (S54)Here, 1 is just Theorem 2, in 2 we introduced the notation k f k ∞ .. = sup α ∈ C m | f ( α ) | for a function f : C m → C , in 3we applied the data processing inequality [123–126] (see also [109, Proposition 5.23(iv)]) to the quantum-to-classicalchannel ρ Q ρ , which physically corresponds to a heterodyne detection [34, 5.4.2], and finally in 4 we noted that Q ω ( α ) ≤ k Q ω k ∞ and remembered that Q ρ is a probability density function.Since N Mr ( ω ) ≤ S W ( ω ) − S ( ω ) whenever ω has finite entropy, setting ω = ρ ⊗ n yields N M, ∞ r ( ρ ) = lim n →∞ n N Mr ( ρ ⊗ n ) ≤ lim n →∞ n (cid:0) S W ( ρ ⊗ n ) − S ( ρ ⊗ n ) (cid:1) = S W ( ρ ) − S ( ρ ) , where in the last step we used the additivity of both the von Neumann and the Wehrl entropies.To prove the lower bound on N Mr , we start from the first line of (S54) and simply choose ω = ρ in the variationalprogram: N Mr ( ρ ) = sup ω ∈ D ( H m ) (cid:26) Tr ρ log ω − log sup α ∈ C m h α | ω | α i (cid:27) ≥ Tr ρ log ρ − log ( π m k Q ρ k ∞ ) = − S ( ρ ) − m log π − log k Q ρ k ∞ . This completes the proof.We can immediately draw some interesting consequences concerning Gaussian states. Following the conventionsof the excellent monograph by Serafini [34], for an m -mode state ρ we set s j .. = Tr ρR j , with j = 1 , . . . , m and R .. = ( x , p . . . , x m , p m ) (cid:124) , and define the quantum covariance matrix by V jk .. = Tr ρ { R j , R k } − s j s k . Gaussianstates are those whose characteristic function (S6) is a multivariate Gaussian, and are uniquely characterized by thevector s and the quantum covariance matrix V . Corollary S40. Let ρ be an arbitrary m -mode Gaussian state with quantum covariance matrix V . Then 12 log det( V + ) − S ( ρ ) − m ≤ N Mr ( ρ ) ≤ N M, ∞ r ( ρ ) ≤ 12 log det( V + ) − S ( ρ ) + m log ( e ) . Proof. One just needs to remember that the Husimi function Q ρ of a Gaussian state ρ with quantum covariancematrix V is a Gaussian with (classical) covariance matrix ( V + ) / σ = V and σ m = in [34,Eq. (5.139)]). This implies immediately that k Q ρ k ∞ = π − m (cid:0) det (cid:0) V + (cid:1)(cid:1) − / , and that the Wehrl entropy of ρ satisfies S W ( ρ ) = − Z d m α Q ρ ( α ) log ( π m Q ρ ( α )) = 12 log det( V + ) + m log ( e ) . (S55)This concludes the proof.3 C. Approximation by spectral truncation We now study the problem of approximating N Mr by truncating the input state. We state first a useful lemma,whose proof follows closely that of [127, Lemma 7], with some adaptations made to fit our infinite-dimensional case.In what follows, for a trace class operator X ∈ T sa ( H ) with decomposition X = X + − X − into positive and negativeparts, we denote with | X | .. = X + + X − its absolute value. Lemma S41. Let ρ, σ ∈ D ( H m ) be two m -mode states, and set (cid:15) .. = k ρ − σ k . Assume that the operator | ρ − σ | has finite mean photon number E .. = Tr | ρ − σ | (cid:16)P nj =1 a † j a j (cid:17) < ∞ . Then, for F = N Mr , N ∞ r , N r it holds that | F ( ρ ) − F ( σ ) | ≤ m (cid:15) g (cid:18) Em (cid:15) (cid:19) + g ( (cid:15) ) , (S56) where g ( x ) = (1 + x ) log (1 + x ) − x log x . Remark S42. Recently, Shirokov [15] has put forward a more general technique that allows to obtain generalcontinuity results for relative entropy distance measures in infinite-dimension, thus removing the need to make anyassumption concerning the operator | ρ − σ | . While theoretically superior, his bounds are less tight and ultimately notsuited for our practical purposes. Proof. We start with the case where F = N r . Here we actually prove that | N r ( ρ ⊗ τ ) − N r ( σ ⊗ τ ) | ≤ m (cid:15) g (cid:18) Em (cid:15) (cid:19) + g ( (cid:15) ) , (S57)for all n -mode auxiliary states τ . Call h ( p ) .. = − p log p − (1 − p ) log (1 − p ) the binary entropy function. Usingthe convexity of N r (Lemma S29) together with [109, Proposition 5.24], it is not difficult to observe, as done byWinter [127, Lemma 7], that pN r ( ρ ) + (1 − p ) N r ( ρ ) − h ( p ) ≤ N r ( pρ + (1 − p ) ρ ) ≤ pN r ( ρ ) + (1 − p ) N r ( ρ ) . (S58)We now construct two states δ, δ ∈ D ( H m ) such that ρ − σ = (cid:15) ( δ − δ ) , | ρ − σ | = (cid:15) ( δ + δ ) . In particular, the mean photon number of δ satisfies thatTr δ (cid:16)X nj =1 a † j a j (cid:17) ≤ (cid:15) Tr | ρ − σ | (cid:16)X nj =1 a † j a j (cid:17) ≤ E(cid:15) . Now, set ω .. = 11 + (cid:15) ρ + (cid:15) (cid:15) δ = 11 + (cid:15) σ + (cid:15) (cid:15) δ . Then, on the one hand N r ( ω ⊗ τ ) ≤ 11 + (cid:15) N r ( σ ⊗ τ ) + (cid:15) (cid:15) N r ( δ ⊗ τ ) ≤ 11 + (cid:15) N r ( σ ⊗ τ ) + (cid:15) (cid:15) (cid:18) m g (cid:18) Em (cid:15) (cid:19) + N r ( τ ) (cid:19) . Here, the estimate in 1 comes from convexity (S58), while that in 2 is an application of the subadditivity of N r (Lemma S29) together with Proposition S38. On the other hand, we can write N r ( ω ⊗ τ ) = N r (cid:18) 11 + (cid:15) ρ ⊗ τ + (cid:15) (cid:15) δ ⊗ τ (cid:19) ≥ 11 + (cid:15) N r ( ρ ⊗ τ ) + (cid:15) (cid:15) N r ( δ ⊗ τ ) − h (cid:18) (cid:15) (cid:15) (cid:19) ≥ 11 + (cid:15) N r ( ρ ⊗ τ ) + (cid:15) (cid:15) N r ( τ ) − h (cid:18) (cid:15) (cid:15) (cid:19) , (S59)4where the inequality in 3 is the lower bound in (S58), and that in 4 holds because of the monotonicity of N r underthe classical operation of tracing away subsystems. Putting all together we see that N r ( ρ ⊗ τ ) − N r ( σ ⊗ τ ) ≤ m (cid:15) g (cid:18) Em (cid:15) (cid:19) + (1 + (cid:15) ) h (cid:18) (cid:15) (cid:15) (cid:19) = m (cid:15) g (cid:18) Em (cid:15) (cid:19) + g ( (cid:15) ) . Together with the corresponding inequality with ρ and σ exchanged, this yields (S57), and in particular proves (S56)for F = N r .Now, again borrowing a telescopic argument from [127], for all n ∈ N + we have that (cid:12)(cid:12) N r ( ρ ⊗ n ) − N r ( σ ⊗ n ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)X n − k =0 (cid:16) N r (cid:16) ρ ⊗ ( n − k ) ⊗ σ ⊗ k (cid:17) − N r (cid:16) ρ ⊗ ( n − k − ⊗ σ ⊗ ( k +1) (cid:17)(cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X n − k =0 (cid:12)(cid:12)(cid:12) N r (cid:16) ρ ⊗ ( n − k ) ⊗ σ ⊗ k (cid:17) − N r (cid:16) ρ ⊗ ( n − k − ⊗ σ ⊗ ( k +1) (cid:17)(cid:12)(cid:12)(cid:12) = X n − k =0 | N r ( ρ ⊗ τ k ) − N r ( σ ⊗ τ k ) |≤ k (cid:18) m (cid:15) g (cid:18) Em (cid:15) (cid:19) + g ( (cid:15) ) (cid:19) , where in the last line we applied (S57). Diving by k and taking the limit for k → ∞ we see that | N ∞ r ( ρ ) − N ∞ r ( σ ) | ≤ m (cid:15) g (cid:18) Em (cid:15) (cid:19) + g ( (cid:15) ) , which proves (S56) also when F = N ∞ r .The case of F = N Mr can be tackled with exactly the same techniques, because N Mr obeys an inequality analogousto (S58). In turn, this is a consequence of the fact that the classical Kullback–Leibler divergence satisfies the estimatesin [109, Proposition 5.24]. Remark S43. We have not been able to establish (S56) also for the remaining case of F = N M, ∞ r , essentially becausewe lack a statement similar to (S57) for N Mr . In turn, this is due to the fact that this latter quantity is not subadditive— in fact, it is strongly superadditive!The application of Lemma S41 that is of interest to us is as follows. Corollary S44. Let ρ, σ ∈ D ( H m ) be two m -mode states, and set (cid:15) .. = k ρ − σ k . Assume that Tr ρ (cid:16)P mj =1 a † j a j (cid:17) ≤ E and also Tr σ (cid:16)P mj =1 a † j a j (cid:17) ≤ E . Then, for F = N Mr , N ∞ r , N r it holds that | F ( ρ ) − F ( σ ) | ≤ m (cid:15) g (cid:18) Em (cid:15) (cid:19) + g ( (cid:15) ) , (S60) where again g ( x ) = (1 + x ) log (1 + x ) − x log x . In particular, denoting with ρ = P k p k | e k ih e k | the spectral decom-position of ρ , the sequence of spectral truncations ρ n .. = (cid:16)P k ≤ n p k (cid:17) − P k ≤ n p k | e k ih e k | satisfies that F ( ρ ) = lim n →∞ F ( ρ n ) . (S61) Proof. Thanks to Lemma S41, in order to prove (S60) it suffices to show that Tr | ρ − σ | (cid:16)P mj =1 a † j a j (cid:17) ≤ E . Indeed,if ρ = P k p k | e k ih e k | and σ = P k q k | e k ih e k | then | ρ − σ | = X k | p k − q k | | e k ih e k | ≤ X k ( p k + q k ) | e k ih e k | = ρ + σ , so that Tr | ρ − σ | (cid:16)X mj =1 a † j a j (cid:17) ≤ Tr( ρ + σ ) (cid:16)X mj =1 a † j a j (cid:17) ≤ E . To deduce (S61), note that [ ρ, ρ n ] = 0, with (cid:15) n .. = k ρ − ρ n k −−−→ n →∞ 0. Also, for sufficiently large n the meanphoton number of ρ n is at most twice that of ρ (call it E ), so that | F ( ρ ) − F ( ρ n ) | ≤ m (cid:15) n g (cid:18) Em (cid:15) n (cid:19) + g ( (cid:15) n ) −−−→ n →∞ , where we used the well-known fact that lim (cid:15) → + (cid:15) g ( δ/(cid:15) ) = 0 for all δ > V. NONCLASSICALITY OF NOTABLE STATES In what follows, we calculate our monotones on some states of physical interest. A. Symmetries A notion that we will often exploit is that of symmetry. Its implications for the variational program in Theorem 2are as follows. Proposition S45. Let Λ : T sa ( H m ) → T sa ( H m ) be a classical operation on an m -mode system, and let ρ ∈ D ( H m ) be an invariant state, in formula Λ( ρ ) = ρ . Then we have that N Mr ( ρ ) = inf σ ∈ Λ( C m ) D M ( ρ k σ ) (S62) If S ( ρ ) < ∞ , then it also holds that N Mr ( ρ ) = sup We start with (S62), which follows from general and well-known arguments. We have that N Mr ( ρ ) = inf σ ∈ C m D M ( ρ k σ ) ≥ inf σ ∈ C m D M (Λ( ρ ) k Λ( σ )) = inf σ ∈ C m D M ( ρ k Λ( σ )) = inf σ ∈ Λ( C m ) D M ( ρ k σ ) , where 1 holds because of the monotonicity under channels of D M . Clearly, since restricting the infimum can onlyincrease the value of the program, it also holds that N Mr ( ρ ) ≤ inf σ ∈ Λ( C m ) D M ( ρ k σ ). This proves (S62)To prove (S63), we go back to (S38). Assuming that Γ((Λ( ρ )) = Γ( ρ ) = N Mr ( ρ ), as implied by Theorem 2, thederivation in (S38) also shows that we can in fact restrict L to belong to Λ † ( B sa ( H m )).The above result is particularly useful when the state ρ under examination is invariant under a group action. Corollary S46. Let U : G → B sa ( H m ) be a unitary representation of a compact group G on the Hilbert space H m .Assume that U ( g ) maps coherent states to coherent states for all g ∈ G . Let ρ ∈ D ( H m ) be a finite-entropy state suchthat is invariant under G , i.e., such that U ( g ) ρU ( g ) † ≡ ρ for all g ∈ G . Then N Mr ( ρ ) = inf σ ∈ C Gm D M ( ρ k σ ) (S64)= sup We now apply the above theory to Fock-diagonal states on m -mode CV quantum systems. Denoting with {| n i} n ∈ N m the Fock basis, as usual, define the totally dephasing map ∆ by∆( ρ ) .. = X n ∈ N m | n ih n | ρ | n ih n | . (S67)6This is a classical channel because it is of the form (S66), for G = ( S ) × m ’ [0 , π ) m and U ( ϕ ) = e i P j ϕ j a † j a j . Inother words, ∆( ρ ) = 12 π Z π d m ϕ e i P j ϕ j a † j a j ρe − i P j ϕ j a † j a j . Clearly, the unitary e i P j ϕ j a † j a j , which is nothing but a phase space rotation, sends coherent states to coherent states.Applying Corollary S46 to any finite-entropy Fock-diagonal state ρ ∈ C FD1 then yields N Mr ( ρ ) = N M, ∞ r ( ρ ) = N ∞ r ( ρ ) = N r ( ρ ) = inf σ ∈ C FD m D ( ρ k σ ) = sup Let ρ be a single-mode Fock-diagonal state with finite rank. Let M .. = max { n : h n | ρ | n i 6 = 0 } . Thenin (S68) we can also take L to have the same support as that of ρ (and to be positive there only). In formula, N Mr ( ρ ) = sup L ∈ e B FDsa ( H ) Tr ρ log L − log sup α ∈ [ , √ M ] h α | L | α i , (S69) where e B FDsa ( H ) .. = { L ∈ B sa ( H ) : L = ∆( L ) , supp L = supp ρ, P ρ LP ρ > } , and P ρ : H → supp ρ is the projectoronto the finite-dimensional space supp ρ .Proof. We have that N Mr ( ρ ) = sup From Corollary S44 we know that N Mr ( ρ n ) −−−→ n →∞ N Mr ( ρ ) , (S70)where ρ n is the spectral truncation of the Fock-diagonal state ρ . Therefore, in principle we can use Proposition S47to approximate numerically N Mr ( ρ ) for any Fock-diagonal state ρ with arbitrary precision. Explicit estimates of theerror associated with each truncation can be deduced from Corollary S44.7 . . . . n N M r ( | n ih n | ) FIG. 1. The measured relative entropy of nonclassicality for Fock states | n ih n | , for different values of n . The simplest example of Fock diagonal states is naturally given by Fock states themselves [128]. Lemma S49. For a Fock state | n i we have that N Mr ( | n ih n | ) = N M, ∞ r ( | n ih n | ) = N ∞ r ( | n ih n | ) = N r ( | n ih n | ) = log (cid:18) n ! e n n n (cid:19) = 12 log (2 πn ) + O ( n − ) . (S71) Proof. The optimization in (S69) involves a single parameter and is thus elementary. To deduce the asymptoticexpansion on the righmost side, it suffices to apply Stirling’s formula.The function (S71) is plotted in Figure 1.Another example of Fock diagonal state is a noisy Fock state, e.g., a Fock state mixed with a certain amount ofthermal noise. These states, herafter called noisy Fock states , are defined by ρ n,ν ( p ) .. = p | n ih n | + (1 − p ) τ ν , (S72)where the thermal state τ ν is given in (S51). In principle, we can approximate the exact value of N Mr ( ρ n,ν ( p )) witharbitrary precision for any n and ν , as pointed out in Remark S48. Let us first consider the simpler case ν = 0, whichis a good approximation in certain regimes, e.g., optical frequencies at room temperature. The state then becomes ρ n, ( p ) = p | n ih n | + (1 − p ) | ih | , and thanks to Proposition S47 we can assume L to be in the form L = ‘ | n ih n | + | ih | (we already exploited the scale invariance). Now we have to perform just two nested optimizations over one realparameter each, that is, N Mr ( ρ n, ( p )) = sup ‘> ( p log ‘ − log max α ∈ [ , √ n ] e − α (cid:18) ‘α n n ! (cid:19)) . (S73)For n ≤ n -th order algebraic equation. For example, for n = 1 one simply finds β = √ p , ‘ = 1 / (1 − p ) and N Mr ( ρ , ( p )) = p +(1 − p ) log (1 − p ). The case of a nonzero temperature can be tackled by considering truncations of ρ and performingnumerical optimizations until some tolerance threshold is achieved. The results for different values of ν and n arereported in Figures 2 and 3.8 . . . . . . . p N M r ( ρ , ν ( p )) ν = 0 ν = 1 ν = 2 ν = 3 (a) 1 photon . . . . . . . p N M r ( ρ , ν ( p )) ν = 0 ν = 1 ν = 2 ν = 3 (b) 2 photons . . . . . . . p N M r ( ρ , ν ( p )) ν = 0 ν = 1 ν = 2 ν = 3 (c) 3 photons . . . . . . . p N M r ( ρ , ν ( p )) ν = 0 ν = 1 ν = 2 ν = 3 (d) 4 photons FIG. 2. Nonclassicality for noisy Fock states: varying ν at fixed n . . . . . . . . p N M r ( ρ n , ( p )) n = 1 n = 2 n = 3 n = 4 (a) ν =0 . . . . . . . p N M r ( ρ n , ( p )) n = 1 n = 2 n = 3 n = 4 (b) ν =1 . . . . . . . p N M r ( ρ n , ( p )) n = 1 n = 2 n = 3 n = 4 (c) ν =2 . . . . . . . p N M r ( ρ n , ( p )) n = 1 n = 2 n = 3 n = 4 (d) ν =3 FIG. 3. Nonclassicality for noisy Fock states: varying n at fixed ν . . 25 0 . . 75 1 1 . 25 1 . . . . . | α | Lower bound for N Mr ( | ψ + α ih ψ + α | )Upper bound for N r ( | ψ + α ih ψ + α | ) (a) Even cat state . 25 0 . . 75 1 1 . 25 1 . . . . . . | α | Lower bound for N Mr ( | ψ − α ih ψ − α | ) (b) Odd cat state FIG. 4. Bounds for the nonclassicality of a cat state, for different values of | α | C. Schr¨odinger cat states For α ∈ C , the associated Schr¨odinger cat states (or simply cat state ) is defined by [48] | ψ ± α i .. = 1 q (cid:0) ± e − | α | (cid:1) ( | α i ± |− α i ) . (S74)It is a nonclassical state for all α = 0. Since a phase space rotation acts as e iϕa † a | ψ ± α i = | ψ ± e iϕ α i , and all of ournonclassicality monotones are left invariant by such transformations, in what follows we can without loss of generalityassume that α ∈ R . Now, for a cat state with real α , we can consider the group G = Z and its representation U : G → B sa ( H ) given by the reflection with respect to the real and/or imaginary axis. Applying Corollary S46 tothis setting (with m = 1) shows immediately that (S64)–(S65) hold with C G and B G sa ( H ) being the sets of classicalstates and bounded operators that are invariant under reflections with respect to the real and/or imaginary axis. Alower bound for N Mr ( ψ ± α ) can be easily computed by setting a maximum rank for L in the second line of (S65) andthen optimizing numerically. When rk L ≤ 3, in order to preserve the symmetry, L must be supported on the subspace V = span( | α i , |− α i , | i ). Analogously, an upper bound for N r ( ψ ± α ) can be found with a classical σ belonging to V .In Figure 4 we report these two bounds for the even cat state ψ + α , and an analogous lower bound for N Mr ( | ψ − α ih ψ − α | ). D. Squeezed states A single-mode squeezed vacuum state is defined by [129, Eq. (3.7.5)] | ζ r,φ i = 1 p cosh( r ) ∞ X n =0 s(cid:18) nn (cid:19) (cid:18) − e iφ tanh( r ) (cid:19) n | n i . (S75)Since changing φ amounts to a simple rotation in phase space, and this cannot modify the value of any of ournonclassicality monotones, we will assume φ = 0 from now on. A squeezed state ζ r .. = ζ r, has always finite energy E ( ψ r ) = sinh ( r ), and hence we can use Proposition S38 to get the upper bound N r ( ζ r ) ≤ g (sinh ( r )) = 2 log cosh r − ( r ) log tanh( r ) . A second upper bound on N r can be found by considering a (classical) squeezed thermal state σ s = S ( s ) τ N ( s ) S † ( s ) = s π ( e s − Z + ∞−∞ dt e − t e s − | it ih it | , N ( s ) .. = e s − , s ≥ , . . . . r Lower bound for N Mr ( | ζ r ih ζ r | )Upper bound for N r ( | ζ r ih ζ r | ) FIG. 5. Bounds for the nonclassicality of a squeezed state, for different values of r . and plugging it in the infimum that defines N r (cf. (4)), i.e., N r ( ζ r ) ≤ inf s ≥ D ( ζ r || σ s ) = inf s ≥ (cid:18) log (1 + N ( s )) + 2 sinh ( r − s ) log (cid:18) N ( s ) (cid:19)(cid:19) . The last expression can be easily optimized numerically. A lower bound on N Mr can be found from Corollary S40. Allthese estimates are plotted in Figure 5. VI. ASYMPTOTIC TRANSFORMATION RATES IN THE QRT OF NONCLASSICALITY: EXAMPLES To get a feeling of how tight the estimates in Theorem 3 for asymptotic transformation rates in the QRT ofnonclassicality really are, we need to design distillation protocols that can provide lower bounds on those rates. Westart by fixing some notation. Consider a two-mode CV quantum system with annihilation operators a, b , and pick λ ∈ [0 , beam splitter with transmissivity λ is represented by the unitary U λ .. = e arccos √ λ ( a † b − ab † ) . (S76)Its action on operators and vectors is given by U λ (cid:16) ab (cid:17) U † λ = (cid:16) √ λ −√ − λ √ − λ √ λ (cid:17)(cid:16) ab (cid:17) , (S77) U λ (cid:68) (cid:16)(cid:16) αβ (cid:17)(cid:17) U † λ = (cid:68) (cid:16)(cid:16) √ λ √ − λ −√ − λ √ λ (cid:17)(cid:17) (cid:16) αβ (cid:17) . (S78)Therefore, thanks to (S5) we see that U λ | α i | β i = |√ λα + √ − λβ i |−√ − λα + √ λβ i . (S79) A. Cat state manipulation In the main text we mentioned some protocols to transform cat states, enlarging or reducing their amplitude α .Let us discuss them at length here. Hereafter we take without loss of generality α to be real. The first transformation2we consider is amplification: ψ + α → ψ + √ α . Lund et al. [88] have provided a protocol that achieves exact conversion oftwo copies of the initial state with probability P Lund (cid:16) ψ + α ⊗ ψ + α → ψ + √ α (cid:17) = e − α cosh(2 α ) sinh (cid:0) α / (cid:1) cosh ( α ) . (S80)Hence, R (cid:16) ψ + α → ψ + √ α (cid:17) ≥ P Lund (cid:16) ψ + α ⊗ ψ + α → ψ + √ α (cid:17) = e − α cosh(2 α ) sinh (cid:0) α / (cid:1) ( α ) . (S81)Mimicking the protocol of Lund et al. but employing slightly better (yet less realistic) measurements, we are able toobtain a better bound. Proposition S50. In the QRT of nonclassicality it is possible to achieve exact conversion ψ + α ⊗ ψ + α → ψ + √ α withprobability P our (cid:16) ψ + α ⊗ ψ + α → ψ + √ α (cid:17) = 12 tanh ( α ) . (S82) Therefore, R (cid:16) ψ + α → ψ + √ α (cid:17) ≥ 14 tanh ( α ) . (S83) Proof. Consider the following protocol. Apply a beam splitter with trasmissivity 1 / | ψ + α i | ψ + α i .Using (S79), we obtain that U / | ψ + α i | ψ + α i = 12 (1 + e − α ) (cid:16) | i |√ α i + | i |−√ α i + |√ α i | i + |−√ α i | i (cid:17) = p cosh(2 α )2 cosh( α ) (cid:16) | i | ψ + √ α i + | ψ + √ α i | i (cid:17) . Carrying out on the second mode the measurement {| χ ih χ | , − | χ ih χ |} , with | χ i .. = 1 √ α ) (cid:16)p cosh(2 α ) | i − | ψ + √ α i (cid:17) , yields h χ | U / | ψ + α ψ + α i = 1 √ α ) | ψ + √ α i , where the subscripts identify different modes. Computing the norm of the above vector yields (S82) and this inturn (S83).We now move on to cat state dilution. We consider the slightly simpler task of balanced dilution ψ + √ α → ψ + α ⊗ ψ − α . Proposition S51. In the QRT of nonclassicality it holds that R (cid:16) ψ + √ α → ψ + α ⊗ ψ − α (cid:17) ≥ sinh ( α )2 cosh(2 α ) . (S84) Proof. Consider the following protocol. Apply a beam splitter with trasmissivity 1 / | ψ + √ α i | i .Using one again (S79), we obtain that U / | ψ + √ α i | i = 1 p e − α ) (cid:16)(cid:16) e − α (cid:17) | ψ + α i | ψ + α i + (cid:16) − e − α (cid:17) | ψ − α i | ψ − α i (cid:17) . | ψ + α i and | ψ − α i , we obtainthat h ψ + α | U / | ψ + √ α , i = cosh( α ) p cosh(2 α ) | ψ ± α i , h ψ − α | U / | ψ + √ α , i = sinh( α ) p cosh(2 α ) | ψ ∓ α i . Computing the norms of the vectors on the right-hand side yields the estimates P our (cid:16) ψ + √ α → ψ + α (cid:17) = cosh ( α )cosh(2 α ) ,P our (cid:16) ψ + √ α → ψ − α (cid:17) = sinh ( α )cosh(2 α ) . Applying the above protocol to n copies of ψ + √ α yields, in the limit of large n , at least n sinh ( α )2 cosh(2 α ) copies of ψ + α ⊗ ψ − α .Hence, R (cid:16) ψ + √ α → ψ + α ⊗ ψ − α (cid:17) ≥ sinh ( α )2 cosh(2 α ) , which completes the proof.Finally, we upper bound the maximal asymptotic transformation rates of both amplification and dilution of catstates by means of the formula 8, and the numerical results reported in Figure 4. B. Fock state dilution Now we the results are ready to report an example in which the bound in Theorem 3 is (asymptotically) tight. Proposition S52. Let < p ≤ and n ≥ be fixed. Consider the transformation ρ n, ( p ) → | n − ih n − | , where thenoisy Fock state is defined in (S72) . It holds that p ≤ R ( ρ n, ( p ) → | n − ih n − | ) ≤ p log (cid:0) n ! e n n n (cid:1) log (cid:16) ( n − e n − ( n − n − (cid:17) −−−→ n →∞ p , (S85) with the upper bound being given by Theorem 3.Proof. We start with the lower bound. Consider the following protocol, implemented with only linear optics, destruc-tive measurements, and feed forward.(1) We send ρ n, ( p ) into a beam splitter with transmissivity λ whose second mode’s initial state is the vacuum.(2) We perform photon counting on the ancillary mode.(3) If we measure 0 photons, the output state of the remaining mode is ρ n, ( p ), with p .. = pλ n pλ n +1 − p . We restart withstep (1).(4) If we measure 1 photon, the output state of the remaining mode is | n − i , and we have succeeded.(5) If we measure 2 or more photons, the protocol is aborted.Using the well-known formula [130] U λ | n, i = λ n n X ‘ =0 ( − ‘ s(cid:18) n‘ (cid:19) (cid:18) − λλ (cid:19) ‘ | n − ‘, ‘ i , (S86)4a lengthy but straightforward calculation shows that the global probability of success of this protocol is P s ( n, p ; λ ) = pn (1 − λ ) λ n − (1 − λ n )( pλ n + 1 − p ) . (S87)Since we can take λ arbitrarily close to 1, we see that R ( ρ n, ( p ) → | n − ih n − | ) ≥ lim λ → − P s ( n, p ; λ ) = p , which proves the lower bound.As for the upper bound, using (S68) together with convexity and (S71), we see that N M, ∞ r ( ρ n, ( p )) = N M, ∞ r ( ρ n, ( p )) ≤ pN Mr ( | n ih n | ) = p log (cid:18) n ! e n n n (cid:19) . Leveraging once again (S71), this entails that R ( ρ n, ( p ) → | n − ih n − | ) ≤ p log (cid:0) n ! e n n n (cid:1) log (cid:16) ( n − e n − ( n − n − (cid:17) −−−→ n →∞ p .p .