Gaussian optimizers and the additivity problem in quantum information theory
aa r X i v : . [ m a t h - ph ] M a r Gaussian optimizers and the additivityproblem in quantum information theory
A. S. HolevoSteklov Mathematical Institute, Moscow
Abstract
We give a survey of the two remarkable analytical problems ofquantum information theory. The main part is a detailed report of therecent (partial) solution of the quantum Gaussian optimizers problemwhich establishes an optimal property of Glauber’s coherent states –a particular instance of pure quantum Gaussian states. We elaborateon the notion of quantum Gaussian channel as a noncommutativegeneralization of Gaussian kernel to show that the coherent states,and under certain conditions only they, minimize a broad class ofthe concave functionals of the output of a Gaussian channel. Thus,the output states corresponding to the Gaussian input are “the leastchaotic”, majorizing all the other outputs. The solution, however, isessentially restricted to the gauge-invariant case where a distinguishedcomplex structure plays a special role.We also comment on the related famous additivity conjecture,which was solved in principle in the negative some five years ago.This refers to the additivity or multiplicativity (with respect to tensorproducts of channels) of information quantities related to the classicalcapacity of quantum channel, such as (1 → p )-norms or the minimalvon Neumann or R´enyi output entropies. A remarkable corollary ofthe present solution of the quantum Gaussian optimizers problem isthat these additivity properties, while not valid in general, do hold inthe important and interesting class of the gauge-covariant Gaussianchannels. ontents The quantum Gaussian optimizers problem is an analytical problem thatarose in quantum information theory at the end of past century, and whichhas an independent mathematical interest. Only recently a solution wasfound [23], [53] in a considerably common situation, while in full generalitythe problem still remains open. To explain the nature and the difficulty ofthe problem we start from the related classical problem of Gaussian maximiz-ers which has been studied rather exhaustively, see Lieb [50] and referencestherein. Consider an integral operator G from L p ( R s ) to L q ( R r ) given by aGaussian kernels (i.e. exponential of a quadratic form) with the ( q → p ) − norm k G k q → p = sup f =0 k Gf k p / k f k q = sup k f k q ≤ k Gf k p . (1)2nder certain broad enough assumptions concerning the quadratic formdefining the kernel, and also p and q , this operator is correctly defined, andthe supremum in (1) is attained on Gaussian f . Moreover, under some addi-tional restrictions any maximizer is Gaussian. As it is put in the title of thepaper [50]: “ Gaussian kernels have only Gaussian maximizers”.Knowledge that the maximizer is Gaussian can be used to compute exactvalue of the norm (1); in fact a starting point of the classical Gaussian max-imizers works were the result of K.I. Babenko [5] and a subsequent paperof Beckner [6] which established the best constant in the Hausdorff-Younginequality concerning the ( p → p ′ ) − norm, ( p − + ( p ′ ) − = 1 , < p ≤ , ofthe Fourier transform (which is apparently given by a degenerate imaginaryGaussian kernel).A difficulty in the optimization problem (1) is that it requires maximiza-tion of a convex function, so the general theory of convex optimization is notof great use here (it only implies that a maximizer of k Gf k p belongs to a faceof the convex set k f k q ≤ q → p ) − norms with respect to tensor products of the integraloperators.A notable application of these classical results to a problem in quantummathematical physics was Lieb’s solution [51] of Wehrl’s conjecture [63]. Let ρ be a density operator in a separable Hilbert space H representing state ofa quantum system; the “classical entropy” of the state ρ is defined as H cl ( ρ ) = − Z C p ρ ( z ) log p ρ ( z ) d zπ , where p ρ ( z ) = h z | ρ | z i is the diagonal value of the kernel of ρ in the systemof Glauber’s coherent vectors {| z i ; z ∈ C } [44], [33]. The conjecture wasthat H cl ( ρ ) has the minimal value if ρ is itself a coherent state i.e. projec-tor onto one of the coherent vectors. Lieb [51] used exact constants in theHausdorff-Young inequality for L p -norms of Fourier transform [5], [6] and theYoung inequality for convolution [6] to prove similar maximizer conjecturefor f ( x ) = x p and considered the limit lim p ↓ (1 − p ) − (1 − x p ) = − x log x . Throughout the paper the base of logarithm is a fixed number a >
1. In informationtheory the natural choice is a = 2, then all the entropic quantities are measured in “bits”. In analysis, they correspond to complex-parametrized Gaussian wavelets. Noticethat this is the only place in the present article where we formally used Dirac’s notations,uncommon among mathematicians. any functional of the form R C f ( p ρ ( z )) d zπ , where f ( x ) , x ∈ [0 ,
1] is a nonnegative concave function with f (0) = 0.In the language of quantum information theory, the affine map G : ρ → p ρ ( z ) , taking density operators ρ (quantum states) into probability densities p ρ ( z ) (classical states), is a “ quantum-classical channel” [39]. Moreover, ittransforms Gaussian density operators ρ (in the sense defined below in Sec.3.1 ) into Gaussian probability densities, and in this sense it is a “ Gaussianchannel” . From this point of view, Wehrl entropy H cl ( ρ ) is the outputentropy of the channel, and Lieb’s result says that it is minimized by pureGaussian states ρ . Moreover, the corresponding result for f ( x ) = x p can beinterpreted as “ Gaussian maximizer” statement for the norm k G k → p . Noticethat the case q = 1 , which is excluded in the classical problem for obviousreasons, appears and is the most relevant in the quantum (noncommutative)case.The quantum Gaussian optimizers problem described in the present paperrefers to Bosonic Gaussian channels – a noncommutative analog of GaussianMarkov kernels and, similarly, requires maximization of convex functions (orminimization of concave functions, such as entropy) of the output state ofthe channel, while the argument is the input state. A general conjectureis that the optimizers belongs to the class of pure Gaussian states. Theconjecture, first formulated in [42] in the context of quantum informationtheory, however natural it looks, resisted numerous attacks for several years.Among others, notable achievements were the exact solution for the classicalcapacity of pure loss channel [21] and a proof of additivity of the R´enyi en-tropies of integer orders p [24] for special channels models. Even restrictedto the class of Gaussian input states, the optimization problem turns out tobe nontrivial [56], [31]. There was some hope that in solving the problem,similarly to Wehrl’s conjecture, one could also use the classical “ Gaussianmaximizers” results. However the solution found recently by Giovannetti,Holevo, Garcia-Patron [23], and Mari, Giovannetti, Holevo [53] uses com-pletely different ideas based on a thorough study of structural properties ofquantum Gaussian channels. As it was mentioned, a solution of the classi-cal problem uses the Minkowski inequality and the implied multiplicativityof ( q → p )-norms. However, the noncommutative analog of the Minkowski’sinequality [12] is not powerful enough to guarantee the multipicativity of4orms (or additivity of the corresponding entropic quantities). Moreover,the related long-standing additivity problem in quantum information theory[34] was recently shown to have negative solution in general [26]. We showthat, remarkably, a solution of the quantum Gaussian optimizers problemgiven in [23] implies also a proof of the multipicativity/additivity propertyin the restricted class of gauge-covariant or contravariant quantum Gaussianchannels.It would then be interesting to investigate a possible development ofsuch an approach to obtain noncommutative generalizations of the classi-cal “ Gaussian maximizers” results for ( q → p ) − norms. Such generalizationcould shed a new light to the hypercontractivity problem for quantum dynam-ical semigroups and related noncommutative analogs of logarithmic Sobolevinequalities, see e.g. [62]. Let H be a separable complex Hilbert space, L ( H ) the algebra of all boundedoperators in H and T ( H ) the ideal of trace-class operators. The space T ( H )equipped with the trace norm k·k is Banach space, which is useful to consideras a noncommutative analog of the space L . The convex subset of T ( H ) S ( H ) = { ρ : ρ ∗ = ρ ≥ , Tr ρ = 1 } , is a base of the positive cone in T ( H ). Operators ρ from S ( H ) are called density operators or quantum states . The state space is a convex set withthe extreme boundary P ( H ) = (cid:8) ρ : ρ ≥ , Tr ρ = 1 , ρ = ρ (cid:9) . Thus extreme points of S ( H ) , which are called pure states , are one-dimensionalprojectors, ρ = P ψ for a vector ψ ∈ H with unit norm, see, e.g. [55].The class of maps we will be interested is a noncommutative analog ofMarkov maps (linear, positive, normalized maps) in classical analysis andprobability. Let H A , H B be the two Hilbert spaces, which will be called inputand output space, correspondingly. A map Φ : T ( H A ) → T ( H B ) is positive X ≥ X ] ≥
0, and it is completely positive [61], [54] if the mapsΦ ⊗ Id ( d ) are positive for all d = 1 , , . . . , where Id ( d ) is the identity map ofthe algebra L d = L ( C d ) of complex d × d − matrices. Equivalently, for everynonnegative definite block matrix [ X jk ] j,k =1 ,...,d the matrix [Φ[ X jk ]] j,k =1 ,...,d isnonnegative definite.A linear map Φ is trace-preserving if TrΦ[ X ] = Tr X for all X ∈ T ( H A ) . Definition
Quantum channel is a linear completely positive trace-preservingmap Φ : T ( H A ) → T ( H B ) . Letter A will be always associated with the input of the channel, while B with the output . Sometimes, to abbreviate notations,we will write simply Φ : A → B. (cid:4) Apparently, every channel is a positive map taking states into states:Φ[ S ( H A )] ⊆ S ( H B ) . Since T ( H ) is a base-normed space, this implies [17]that Φ is a bounded map from the Banach space T ( H A ) to T ( H B ) . The dualΦ ∗ of the map Φ is uniquely defined by the relationTrΦ[ X ] Y = Tr X Φ ∗ [ Y ]; X ∈ T ( H A ) , Y ∈ L ( H B ) , (2)and it is called dual channel . The dual channel is linear completely positive ∗− weakly continuous map from L ( H B ) to L ( H A ) , which is unital : Φ[ I H B ] = I H A . Here and in what follows I with possible index denotes the unit operatorin the corresponding Hilbert space.There are positive maps that are not completely positive, a basic exampleprovided by matrix transposition X → X ⊤ in a fixed basis.From the definition of complete positivity one easily derives [39] thatcomposition of channels Φ ◦ Φ defined asΦ ◦ Φ [ X ] = Φ [Φ [ X ]] , and naturally defined tensor product of channelsΦ ⊗ Φ = (Φ ⊗ Id ) ◦ (Id ⊗ Φ )are again channels. The notion of completely positive map was introduced by Stinespring [61] ina much wider context of C*-algebras. This allows also to cover the notion of6 ybrid channel where the input is quantum while the output is classical orvice versa. An example of such channel was mentioned in Sec. 1. We willnot pursue this topic further here, see [39], but only mention that completepositivity reduces to positivity in such cases.Motivated by the famous Naimark’s dilation theorem, Stinespring estab-lished a representation for completely positive maps of C*-algebras which inthe case of quantum channel reduces [39] to
Proposition 1
Let
Φ : A → B be a channel. There exist a Hilbert space H E and an isometric operator V : H A → H B ⊗ H E , such that Φ[ ρ ] = Tr E V ρV ∗ ; ρ ∈ T ( H A ) , (3) where Tr E denotes partial trace with respect to H E . The representation (3)is not unique, however any two representations with V : H A → H B ⊗ H E and V : H A → H B ⊗ H E are related via partial isometry W : H E → H E such that V = ( I B ⊗ W ) V and V = ( I B ⊗ W ∗ ) V . Consider a representation (3) for the channel Φ; the complementary chan-nel [37], [48] is then defined by the relation˜Φ[ ρ ] = Tr B V ρV ∗ ; ρ ∈ T ( H A ) . (4)From the relation between the different representations (3), it follows that thecomplementary channel is unique in the following sense: any two channels˜Φ , ˜Φ complementary to Φ are isometrically equivalent in the sense thatthere is a partial isometry W : H E → H E such that˜Φ [ ρ ] = W ˜Φ [ ρ ] W ∗ , ˜Φ [ ρ ] = W ∗ ˜Φ [ ρ ] W, (5)for all ρ. It follows that the initial projector W ∗ W satisfies ˜Φ [ ρ ] = W ∗ W ˜Φ [ ρ ] , i.e. its support contains the support of ˜Φ [ ρ ] , while the final projector W W ∗ has similar property with respect to ˜Φ [ ρ ] . The complementary to comple-mentary can be shown isometrically equivalent to the initial channel, so thatΦ , ˜Φ are called mutually complementary channels.In general, we will say that two density operators ρ and σ (possibly actingin different Hilbert spaces) are isometrically equivalent if there is a partialisometry W such that ρ = W σW ∗ , σ = W ∗ ρW. Apparently, this is the caseif and only if nonzero spectra (counting multiplicity) of the density operators ρ and σ coincide. We denote this fact with the notation ρ ∼ σ . We have justshown that ˜Φ [ ρ ] ∼ ˜Φ [ ρ ] for arbitrary ρ. emma 2 Let ˜Φ be a complementary channel (4 ), then Φ[ P ψ ] ∼ ˜Φ[ P ψ ] forall ψ ∈ H A . Proof.
Let V : H A → H B ⊗ H E be the isometry from the representations(3), (4), then ρ BE = V P ψ V ∗ is a pure state in H B ⊗ H E , and the statementfollows from a basic result in quantum information theory (“Schmidt decom-position”): if ρ BE is a pure state in H B ⊗H E and ρ B = Tr E ρ BE , ρ E = Tr B ρ BE are its partial states, then ρ B ∼ ρ E (see e.g. Proposition 3 in [34])A different name for channel is dynamical map – in nonequilibrium quan-tum statistical mechanics they arise as irreversible evolutions of an openquantum system interacting with an environment [39]. Assume that there isa composite quantum system AD = BE in the Hilbert space H = H A ⊗ H D ≃ H B ⊗ H E , (6)which is initially prepared in the state ρ A ⊗ ρ D and then evolves accordingto the unitary operator U. Then the output state ρ B depending on the inputstate ρ A = ρ is Φ B [ ρ ] = Tr E U ( ρ ⊗ ρ D ) U ∗ , (7)while the output state of the “environment” E is the output of the channelΦ E [ ρ ] = Tr B U ( ρ ⊗ ρ D ) U ∗ . (8)If the initial state of D is pure, ρ D = P ψ D , then by introducing the isometry V : H A → H B ⊗ H D , which acts as V ψ = U ( ψ ⊗ ψ D ) , ψ ∈ H A , we see that the relations (7), (8) convert into (3), (4), and Φ E is just the com-plementary of Φ B . Notice also that both partial trace and unitary evolutionare completely positive operators, hence the maps (7), (8) are completelypositive; vice versa, any quantum channel has a representation of such aform, see, e.g. [39].Vast literature is devoted to study of quantum dynamical semigroups(noncommutative analog of Markov semigroups) and quantum Markov pro-cesses. Stinespring-type representation (3) underlies dilations of quantumdynamical semigroups to the unitary dynamics of open quantum system in-teracting with an environment [17], [35].8 .3 Entropic quantities and additivity
Consider the norm of the map Φ defined similarly to (1): k Φ k → p = sup X =0 k Φ[ X ] k p / k X k = sup k X k ≤ k Φ[ X ] k p , (9)where k·k p is the Schatten p − norm [55]. As shown in [4], k Φ k p → p = sup ρ ∈ S ( H A ) TrΦ[ ρ ] p = sup ψ ∈H A TrΦ[ P ψ ] p , (10)where the second equality follows from convexity of the function x p , p > p > ρ isdefined as R p ( ρ ) = 11 − p log Tr ρ p = p − p log k ρ k p , (11)Define the minimal output R´enyi entropy of the channel Φˇ R p (Φ) = inf ρ ∈ S ( H ) R p (Φ[ ρ ]) = p − p log k Φ k → p (12)and the minimal output von Neumann entropy ˇ H (Φ) = inf ρ ∈ S ( H ) H (Φ[ ρ ]) . (13)In the limit p → p → R p ( ρ ) = − Tr ρ log ρ = H ( ρ ) . In finite dimensions the set of quantum states is compact, hence by Dini’sLemma the minimal output R´enyi entropies converge to the minimal outputvon Neumann entropy .Multiplicativity of the norm (9) for some channels Φ , Φ , k Φ ⊗ Φ k → p = k Φ k → p · k Φ k → p (14)is equivalent to the additivity of the minimal output R´enyi entropiesˇ R p (Φ ⊗ Φ ) = ˇ R p (Φ ) + ˇ R p (Φ ) . (15) The corresponding statement is not valid for infinite-dimensional channels (even forclassical channels with countable set of states), M. E. Shirokov, private communication. H (Φ ⊗ Φ ) = ˇ H (Φ ) + ˇ H (Φ ) . (16)In finite dimensions, the validity of (15) for certain channels Φ , Φ and p close to 1 implies (16) for these channels.In the last two relations the inequality ≤ (similarly to the inequality ≥ in (14)) is obvious because the right-hand side is equal to the infimum overthe subset of product states ρ = ρ ⊗ ρ . On the other hand, existence of“entangled” pure states which are not reducible to product states, is thecause for possible violation of the equality for quantum channels. The practical importance of the additivity property (16) is revealed in con-nection with the notion of the channel capacity. To explain it we assumethat H A , H B are finite dimensional for the moment.For a quantum channel Φ, a noncommutative analog of the Shannoncapacity, which we call χ − capacity , is defined by C χ (Φ) = sup { π j ,ρ j } H Φ "X j π j ρ j − X j π j H (Φ[ ρ j ]) ! , (17)where the supremum is over all quantum ensembles , that is finite collections ofstates { ρ , . . . , ρ n } with corresponding probabilities { π , . . . , π n } . The quan-tity (17) is closely related to the capacity C (Φ) of quantum channel Φ fortransmitting classical information [34]. The classical capacity of a quantumchannel is defined as the maximal transmission rate per use of the channel,with coding and decoding chosen for increasing number n of independentuses of the channel Φ ⊗ n = Φ ⊗ · · · ⊗ Φ | {z } n such that the error probability goes to zero as n → ∞ (for a precise definitionsee [39]). A basic result of quantum information theory, HSW Theorem [32],says that such defined capacity C (Φ) is related to C χ (Φ) by the formula C (Φ) = lim n →∞ (1 /n ) C χ (Φ ⊗ n ) . C χ (Φ) is easily seen to be superadditive (i. e., C χ (Φ ⊗ Φ ) ≥ C χ (Φ ) + C χ (Φ ) ), one has C (Φ) ≥ C χ (Φ). However if the additivity C χ (Φ ⊗ Φ ) = C χ (Φ ) + C χ (Φ ) (18)holds for a given channel Φ = Φ and an arbitrary channel Φ , then C χ (Φ ⊗ n ) = nC χ (Φ) , (19)implying C (Φ) = C χ (Φ) . (20)The reason for possible violation of the equality here, as well as in the cases(14), (15), (16), is existence of entangled states, which are not reducible toproduct states, at the input of tensor product channel Φ ⊗ n . Thus it was natural to ask: does the the additivity property (16) holdsglobally, i.e. for tensor product of any pair of quantum channels Φ , Φ ?The problem can be traced back to [8], see also [34]. Quite remarkably,Shor [60], see also [19], had shown the equivalence of the global properties ofadditivity of the χ − capacity and of the minimal output entropy. Theorem 3 [60] The properties (18) and (16) are globally equivalent in thesense that if one of them holds for all channels Φ , Φ , then another is alsotrue for all channels. The additivity is proved rather simply for all classical channels (see e.g.[15]), but in the quantum case the question remained open for a dozen ofyears, and was ultimately solved in the negative.The detailed history of the problem up to 2006 can be found in [34], andhere we only sketch the basic steps and the final resolution. In [1] it wassuggested to approach the additivity property (16) via multiplicativity (14)of the (1 → p ) − norms (equivalent to additivity (15) of the minimal outputR´enyi entropies). The first explicit example where this property breaks for d = dim H ≥ p was transpose-depolarizing channel [64]:Φ( ρ ) = 1 d − (cid:2) I Tr ρ − ρ ⊤ (cid:3) , (21)11here ρ ∈ L d is a matrix and ρ ⊤ its transpose. In particular, (15) withΦ = Φ = Φ fails to hold for p ≥ , d = 3 (nevertheless, the addi-tivity of ˇ H (Φ) and of C χ (Φ) holds for this channel). Five years later cameimportant findings of Winter [65] and Hayden [28], see also [29], who showedexistence of a pair of channels breaking the additivity of the minimal outputR´enyi entropy for all values of the parameter p >
1. The method of theseand subsequent works is random choice of the channels, which for fixed di-mensions are parametrized by isometries V in the representation (3), as wellof the input states of the channels, combined with sufficiently precise proba-bilistic estimates for the norms (10). For finite dimensions the correspondingparametric sets are compact, and one usually takes the uniform distribution.Basing on this progress, Hastings [26] gave a proof of existence of channelsbreaking the additivity conjecture (16) corresponding to p = 1, in very highdimensions. Moreover, the probability of violation of the additivity tends to1 as the dimensionalities tend to infinity. Hastings gave only a sketch, andthe detailed proof following his approach was given by Fukuda, King andMoser [18], and further simplified by Brandao and M. Hordecki [10]. LaterSzarek et al. [3] proposed a proof related to the Dvoretzky-Mil’man theorem on almost Euclidean sections of high-dimensional convex bodies.Although, combined with theorem 3 this gives a definite negative answerto the additivity conjectures, several important issues remaine open. All theproofs use the technique of random unitary channels or random states and assuch are not constructive: they prove only existence of counterexamples butdo not allow to actually produce them. Attempts to give estimates for thedimensions in which nonadditivity can happen based on Hastings’ approachhas led to overwhelmingly high values: the detailed estimates made in [18]gave d ≈ breaking the additivity by a quantity of the order 10 − . Thebest result in this direction obtained in [7] states that “violations of theadditivity of the minimal output entropy, using random unitary channelsand a maximally entangled state state, can occur if and only if the outputspace has dimension at least 183. Almost surely, the defect of additivity isless than log 2, and it can be made as close as desired to log 2”.While this does not exclude possibility of better estimates, based per-haps on a different (but yet unknown) models, it casts doubt onto findingconcrete counterexamples by computer simulation of random channels. Fromthis point of view, the following explicit example given in [25] is of interest.12onsider the completely positive map ρ −→ Φ [ ρ ] = Tr P − ρP − , ρ ∈ T ( H ⊗ H ) , where P − is the projector onto the antisymmetric subspace H of H⊗H whichhas the dimensionality d ( d − , and the partial trace is taken with respect tothe second copy of H . Its restriction to the operators with support in thesubspace H is trace preserving, hence it is a channel. It can be shown [39]that Φ = ( d − ˜Φ ∗ where ˜Φ ∗ is the dual to the complementary of the channel(21). For this simple channel the minimal R´enyi entropies are nonadditivefor all p > d , but unfortunately it is not clear if itcould be extended to the most interesting range p ≥ . Coming back to arbitrary channels, it remains unclear what happensin small dimensions: perhaps the additivity still holds generically for someunknown reason, or its violation is so tiny that it cannot be revealed bynumerical simulations. This is indeed surprising in view of the fact that thephysical reason for nonadditivity is entanglement between the inputs of theparallel quantum channels, see [39] for more detail.On the other hand, these results stress the importance of continuing ef-forts to find special cases where the additivity holds for some reason, andcan be established analytically.A survey of the main classes of such “ additive” channels acting in finitedimensions was presented in [34]; below we briefly list the most importantclasses of channels Φ for which the additivity properties ( 16), (18) and (15)for p > and arbitrary Φ . • Qubit unital channels, i.e channels Φ : L → L satisfying Φ[ I ] = I [46]. Strikingly, there is still no analytical proof of the additivity fornonunital qubit channels, in spite of a convincing numerical evidence[27]. • Depolarizing channel in L d :Φ[ ρ ] = (1 − p ) ρ + p Id Tr ρ, ≤ p ≤ d d − , which is the only unitarily-covariant channel, and can be regarded asnoncommutative analog of completely symmetric channel in classicalinformation theory [15]. The additivity properties (16), (15), (18) wereproved by King [47]. 13 Entanglement-breaking channels. In finite dimensions these are chan-nels of the form Φ[ ρ ] = X j ρ B Tr ρM A , where { M A } is a resolution of the identity in H A : M A ≥ , P j M A = I A , and ρ B ∈ S ( H B ) (see [43]).For the finite-dimensional entanglement-breaking channels the additiv-ity of the minimal output von Neumann entropy and of the χ − capacitywas established by Shor [59] and the additivity of the minimal out-put R ´enyi entropies – by King [45]. The additivity properties ofentanglement-breaking channels were generalized to infinite dimensionsby Shirokov [58]. • Complementary channels.The additivity of the minimal output entropy is equivalent for a chan-nel Φ and its complementary ˜Φ, see Lemma 4 below. The class ofchannels complementary to entanglement-breaking contains the Schur-multiplication maps of matrices ρ = [ c jk ] j,k =1 ,...,d in L d :˜Φ[ ρ ] = [ γ jk c jk ] j,k =1 ,...,d , where [ γ jk ] j,k =1 ,...,d is a nonnegative definite matrix such that γ jj ≡ . For these channels, which are also called “Hadamard channels” theadditivity of the χ − capacity was also established [48].In the next Sections we consider Bosonic Gaussian channels which act ininfinite-dimensional spaces. One of the main goals of the present paper is toshow that the additivity holds for a wide class of gauge co- or contravariantGaussian channels, i.e. those which respect a fixed complex structure in theunderlying symplectic space. From now on we again allow the Hilbert spaces in question to be infinite-dimensional. Denote by F the class of real concave functions f on [0 , , suchthat f (0) = 0 . For any f ∈ F and for any density operator ρ we can considerthe quantity Tr f ( ρ ) = X j f ( λ j ) , λ j are the (nonzero) eigenvalues of the density operator ρ, countingmultiplicity. Note that this quantity is defined unambiguously with values in( −∞ , ∞ ] . This follows from the fact that f ( x ) ≥ cx, where c = f (1) , henceTr f ( ρ ) ≥ c Tr ρ = c. We also will use the fact that the functional ρ → Tr f ( ρ )is (strictly) concave on S ( H ) if f is (strictly) concave (see e.g. [11]).Denote by λ ↓ j ( ρ ) the eigenvalues of a density operator ρ, counting multi-plicity, arranged in the nonincreasing order. One says that density operator ρ majorizes density operator σ if k X j =1 λ ↓ j ( ρ ) ≥ k X j =1 λ ↓ j ( σ ) , k = 1 , , . . . A consequence of a well known result, see e.g. [11], is that this is the case ifand only if Tr f ( ρ ) ≤ Tr f ( σ ) for all f ∈ F . For a quantum channel Φ we introduce the quantityˇ f (Φ) = inf ρ ∈ S ( H ) Tr f (Φ[ ρ ]) = inf P ψ ∈ P ( H ) Tr f (Φ[ P ψ ]) , (22)where the second equality follows from the concavity of the functional ρ → Tr f (Φ[ ρ ]) on S ( H ) . Moreover, for strictly concave f, any minimizer is of theform P ψ for some vector ψ ∈ H . In particular, taking f ( x ) = − x log x and f ( x ) = − x p , we obtain ˇ f (Φ) =ˇ H (Φ) and ˇ f (Φ) = − k Φ k p → p . Lemma 4
For complementary channels, ˇ f (Φ) = ˇ f ( ˜Φ) . Hence k Φ k → p = k ˜Φ k → p , ˇ H (Φ) = ˇ H ( ˜Φ) , ˇ R p (Φ) = ˇ R p ( ˜Φ) , and the multiplicativity ( 14), aswell as the additivity of the minimal output entropies (16), (15) holds simul-taneously for pairs of channels Φ , Φ and ˜Φ , ˜Φ . Proof.
From Lemma 2, Φ[ P ψ ] and ˜Φ[ P ψ ] have identical nonzero spectrum(Φ[ P ψ ] ∼ ˜Φ[ P ψ ]) . Then Tr f (Φ[ P ψ ]) = Tr f ( ˜Φ[ P ψ ]) (23)since f (0) = 0 . Using second equality in (22) implies ˇ f (Φ) = ˇ f ( ˜Φ) . The statement about multiplicativity (additivity) then follows from thefact that the channel ˜Φ ⊗ ˜Φ is complementary to Φ ⊗ Φ .15 Quantum Gaussian systems
A real vector space Z equipped with a nondegenerate skew-symmetric form∆( z, z ′ ) is called symplectic space . In what follows Z is finite-dimensional,in which case its dimensionality is necessarily even, dim Z = 2 s [49]. A basis { e j , h j ; j = 1 , . . . , s } in which the form ∆( z, z ′ ) has the matrix∆ = diag (cid:20) − (cid:21) j =1 ,...,s (24)is called symplectic . The Weyl system in a Hilbert space H is a stronglycontinuous family { W ( z ); z ∈ Z } of unitary operators satisfying the Weyl-Segal canonical commutation relation (CCR) W ( z ) W ( z ′ ) = exp[ − i z, z ′ )] W ( z + z ′ ) . (25)Thus z → W ( z ) is a projective representation of the additive group of Z .We always assume that the representation is irreducible. The Stone-vonNeumann uniqueness theorem says that such a representation is unique upto unitary equivalence. It is well-known, see e.g. [55], that there is a familyof selfadjoint operators z → R ( z ) with a common essential domain D suchthat W ( z ) = exp i R ( z ) , moreover, for any symplectic basis { e j , h j ; j = 1 , . . . , s } R ( z ) = s X j =1 ( x j q j + y j p j )on D , where R ( e j ) = q j , R ( h j ) = p j , and [ x , y , . . . , x s , y s ] are coordinates ofvector z in the basis. Here the canonical observables q j , p j ; j = 1 , . . . , s areselfadjoint operators in H satisfying the Heisenberg CCR on D [ q j , p k ] ⊆ iδ jk I, [ q j , q k ] = 0 , [ p j , p k ] = 0 . (26)In physics the symplectic space is the phase space of the classical system (suchas electro-magnetic radiation modes in the cavity), the quantum version of16hich is described by CCR. Then s is number of degrees of freedom, or“normal modes” of the classical system.The state given by density operator ρ in H is called Gaussian , if its quantum characteristic function φ ( z ) = Tr ρW ( z )has the form φ ( z ) = exp (cid:18) i m ( z ) − α ( z, z ) (cid:19) , (27)where m is a real linear form and α is a real bilinear symmetric form on Z .A necessary and sufficient condition for (27) to define a state is nonnegativedefiniteness of the (complex) Hermitian form α ( z, z ′ ) − i ∆ ( z, z ′ ) on Z or,briefly: α ≥ i . (28)We will agree that the matrix of a bilinear form in fixed a symplectic baseis denoted by the same letter, then (28) can be understood as inequality forHermitian matrices, where α is real symmetric and ∆ is real skew-symmetric.A Gaussian state is pure if and only if α is a minimal solution of thisinequality, see e.g. [38]. Operator J in Z is called operator of complexstructure if J = − I, (29)where I is the identity operator in Z, and the bilinear form ∆( z, J z ′ ) is an(Euclidean) inner product in Z, i.e.∆( z, J z ′ ) = ∆( z ′ , J z ) (= − ∆( J z, z ′ )); (30)∆( z, J z ) ≥ , z ∈ Z. (31)The following characterization can be found in [16], [39]: Proposition 5
The minimal solutions of the inequality (28) are in one-to-one correspondence with the operators J of complex structure in Z given bythe relation α ( z, z ′ ) = 12 ∆( z, J z ′ ); z, z ′ ∈ Z. A complex-valued real-bilinear form β ( z, z ′ ) on Z will be called Hermitian if β ( z ′ , z ) = β ( z, z ′ ) .
17n this way to every complex structure corresponds the family of pureGaussian states (27) with different values of m which are called the J - coherentstates. The state with m = 0 is called J - vacuum . Let ρ be a vacuum, thenany associated coherent state is of the form W ( z ′ ) ρ W ( z ′ ) ∗ , as follows fromthe relation W ( z ′ ) ∗ W ( z ) W ( z ′ ) = exp[ − i ∆( z, z ′ )] W ( z )and from nondegeneracy of the form ∆( z, z ′ ) due to which m ( z ) = ∆( z, z ′ m ).Operator S in Z is called symplectic if ∆( Sz, Sz ′ ) = ∆( z, z ′ ) for all z, z ′ ∈ Z. The unitary operators W ( Sz ) satisfy the CCR (25) hence by the Stone-von Neumann uniqueness theorem there is a unitary operator U S in H suchthat W ( Sz ) = U ∗ S W ( z ) U S , z ∈ Z. The map S → U S is a projective representation of the group of all symplectictransformations in Z, sometimes called “ metaplectic representation” [2] as itcan be extended to a faithful unitary representation of the metaplectic groupwhich is two-fold covering of the symplectic group.Similarly, T is antisymplectic if ∆( T z, T z ′ ) = − ∆( z, z ′ ) for all z, z ′ ∈ Z. There is an antiunitary operator U T in H such that W ( T z ) = U ∗ T W ( z ) U T , z ∈ Z. Let Z A , Z B be two symplectic spaces with the corresponding Weyl sys-tems. Consider a channel Φ : A −→ B . The channel is called Gaussian ifthe dual channel satisfiesΦ ∗ [ W B ( z )] = W A ( Kz ) exp (cid:20) il ( z ) − µ ( z, z ) (cid:21) , z ∈ Z B , (32)where K : Z B → Z A is a linear operator, l a linear form and µ is a realsymmetric form on Z B . In terms of characteristic functions of states, φ B ( z ) = φ A ( Kz ) exp (cid:20) il ( z ) − µ ( z, z ) (cid:21) . It follows that Gaussian channel maps Gaussian states into Gaussian states.A converse statement also holds true [16].A necessary and sufficient condition on parameters (
K, l, µ ) for completepositivity of the map Φ is (see [14]) nonnegative definiteness of the Hermitianform z, z ′ −→ µ ( z, z ′ ) − i B ( z, z ′ ) − ∆ A ( Kz, Kz ′ )]18n Z B , or, in matrix terms (if some bases are chosen in Z A , Z B ), µ ≥ i (cid:2) ∆ B − K t ∆ A K (cid:3) , (33)where t denotes transposition of a matrix. The proof using explicit construc-tion of the representation of type (7) is given in [14], see also [39]; below inProposition 11 below we give such a construction for an important particularclass of Gaussian channels.We call the Gaussian channel extreme if µ is a minimal solution of theinequality (33). This terminology stems from the fact that the minimality of µ is necessary and sufficient for the channel Φ to be an extreme point in theconvex set of all channels with fixed input and output spaces [38]. Additivity hypothesis for quantum Gaussian channels : The ad-ditivity properties (15), (16) hold for any pair of Gaussian channels Φ , Φ . Hypothesis of quantum Gaussian minimizers:
For any function f ∈ F the infimum in (22) is attained on a pure Gaussian state ρ. Any Gaussian channel has the covariance propertyΦ[ W A ( z ) ρW A ( z ) ∗ ] = W B ( K s z )Φ[ ρ ] W B ( K s z ) ∗ (34)where K s is the symplectic adjoint operator defined by the relation∆ B ( K s z A , z B ) = ∆ A ( z A , Kz B ) . It follows that the value Tr f (Φ[ ρ ]) is the same for all coherent states W ( z ) ρ W ( z ) ∗ associated with a vacuum state ρ . These two problems turn out to be closely related. In what follows wedescribe positive solution for both of them in a particular and importantclass of Gaussian channels with gauge symmetry. However both conjecturesremain open for general quantum Gaussian channels.
Given an operator of the complex structure J one defines in Z the Euclideaninner product j ( z, z ′ ) = ∆( z, J z ′ ) . Then one can define in Z the structure In quantum optics one speaks of quantum-limited channels [20]. s − dimensional unitary space Z in which i z corresponds to J z and the(Hermitian) inner product is j ( z , z ′ ) = 12 [∆( z, J z ′ ) + i ∆( z, z ′ )] = 12 [ j ( z, z ′ ) − ij ( z, J z ′ )] . From (29), (30) it follows that J is symplectic, that is ∆( J z, J z ′ ) =∆( z, z ′ ) for all z, z ′ ∈ Z . With every complex structure one can associate thecyclic one-parameter group of symplectic transformations (cid:8) e ϕJ ; ϕ ∈ [0 , π ) (cid:9) which we call the gauge group . Hence, by the Stone-von Neumann uniquenesstheorem, the gauge group in Z induces the one-parameter unitary group ofthe gauge transformations { U ϕ ; ϕ ∈ [0 , π ) } in H according to the formula W (e ϕJ z ) = U ∗ ϕ W ( z ) U ϕ . (35)For the future use it will be convenient to introduce the complex parametriza-tion of the Weyl operators by defining the displacement operators D ( z ) = W ( J z ) , z ∈ Z . (36)A state ρ is gauge invariant if ρ = U ϕ ρU ∗ ϕ for all ϕ , which is equivalentto the property φ ( z ) = φ (e ϕJ z ) of the characteristic function. In particular,Gaussian state (27) is gauge invariant if m ( z ) ≡ α ( z, z ′ ) = α ( J z, J z ′ ) . By introducing the Hermitian inner product in Z α ( z , z ′ ) = 12 [ α ( z, z ′ ) − iα ( z, J z ′ )] , we have α ( z , z ) = α ( z, z ) since α ( z, J z ′ ) is skew-symmetric; moreover, thecondition (28) is equivalent to nonnegative definiteness of the Hermitian form α ( z , z ′ ) − j ( z , z ′ ) on Z : α ≥ j . (37)This follows from application of the following Lemma to the form z, z ′ −→ β ( z, z ′ ) = α ( z, z ′ ) − i z, z ′ ) . The relation (37) can be considered as the inequality for the matrices of theform, provided a basis is chosen in Z . In an orthonormal basis, j = I is theunit matrix. In accordance with convention accepted in mathematical physics, the inner productis complex linear with respect to z ′ and anti-linear with respect to z . emma 6 Let β ( z, z ′ ) be a bilinear complex-valued Hermitian form on realvector space Z, satisfying β ( J z, J z ′ ) = β ( z, z ′ ) , where J is a linear operatorsuch that J = − I. Then β ( z, z ′ ) is nonnegative definite i.e. X jk ¯ c j c k β ( z j , z k ) ≥ for any finite collection { z j } ⊂ Z and any { c j } ⊂ C , if and only if Re β ( z, z ) ± Im β ( z, J z ) ≥ for all z ∈ Z. (39) Proof. (39)= ⇒ (38): We have β ( z, z ′ ) = Re β ( z, z ′ ) + i Im β ( z, z ′ ) , whereIm β ( z, z ′ ) is skew-symmetric, hence Im β ( z, z ) = 0 . By using the fact that β ( J z, z ′ ) = − β ( z, J z ′ ) we obtain that also Re β ( z, J z ′ ) is skew-symmetric,hence Re β ( z, J z ) = 0 . ThusRe β ( z, z ) ± Im β ( z, J z ) = β ( z, z ) ∓ iβ ( z, J z ) . Now introduce complexification z ↔ z by letting J z ↔ i z and define twoHermitian forms on the complexification Z of Z : β ∓ ( z , z ′ ) = β ( z, z ′ ) ∓ iβ ( z, J z ′ ) . (40)Then β − is sesquilinear i.e. complex linear with respect to z ′ and anti-linear with respect to z , while β + is anti-sesquilinear. From (40), ( 39), β ∓ ( z , z ) = Re β ( z, z ) ± Im β ( z, J z ) ≥ z ∈ Z , (41)hence by (anti-)sesquilinearity X jk ¯ c j c k β ∓ ( z j , z k ) ≥ . By adding the two inequalities corresponding to plus and minus, we get (38).Conversely, (38)= ⇒ (39): Applying (38) to the collection { z j , J z j } ⊂ Z, { c j , ± ic j } ⊂ C we obtain X jk ¯ c j c k [ β ( z j , z k ) ± iβ ( z j , J z k )] ≥ , hence the forms (40) are nonnegative definite. By (anti-)sesquilinearity ofthese forms, this is equivalent to (41) i.e. (39).21ssume that in Z A , Z B operators of complex structure J A , J B are fixed,and let U Aφ , U Bφ be the corresponding gauge operators in H A , H B acting ac-cording (35). Channel Φ : A → B is called gauge-covariant , ifΦ[ U Aφ ρ (cid:0) U Aφ (cid:1) ∗ ] = U Bφ Φ[ ρ ] (cid:0) U Bφ (cid:1) ∗ (42)for all input states ρ and all φ ∈ [0 , π ] . For the Gaussian channel (32) withparameters (
K, l, µ ) this reduces to l ( z ) ≡ , KJ B − J A K = 0 , µ ( z, z ′ ) = µ ( J B z, J B z ′ ) . The relation (32) for gauge-covariant Gaussian channel takes the formΦ ∗ [ D B ( z )] = D A ( Kz ) exp [ − µ ( z , z )] , z ∈ Z B , (43)where µ ≥ ±
12 [ j B − K ∗ j A K ] (44)The equivalence of (44) and (33) is obtained by applying the lemma 6 tothe Hermitian form β ( z, z ′ ) = µ ( z, z ′ ) − i B ( z, z ′ ) − ∆ A ( Kz, Kz ′ )] . Channel Φ : A → B is called gauge-contravariant , ifΦ[ U Aφ ρ (cid:0) U Aφ (cid:1) ∗ ] = (cid:0) U Bφ (cid:1) ∗ Φ[ ρ ] U Bφ (45)for all input states ρ and all φ ∈ [0 , π ] . For the Gaussian channel (32) withparameters (
K, l, µ ) this reduces to l ( z ) ≡ , KJ B + J A K = 0 , µ ( z, z ′ ) = µ ( J B z, J B z ′ ) . The relation (32) for gauge-contravariant Gaussian channel takes the formΦ ∗ [ D B ( z )] = D A ( − Λ Kz ) exp [ − µ ( z , z )] , z ∈ Z B , (46)where Λ is antilinear operator of complex conjugation, Λ = I, Λ s = − Λ in Z A such that Λ J A + J A Λ = 0 , and K = − Λ K is complex linear operator from Z B to Z A . Here µ ≥ ±
12 [ j B + K ∗ j A K ] . (47)The last condition is obtained by applying Lemma 6 to the Hermitian form β ( z, z ′ ) = µ ( z, z ′ ) − i B ( z, z ′ ) − ∆ A ( Kz, Kz ′ )]= i B ( z, z ′ ) + ∆ A ( K z, K z ′ )] . .3 Attenuators and amplifiers In what follows we restrict to channels that are gauge-covariant or contravari-ant with respect to fixed complex structures. Therefore, to be specific, weconsider vectors in Z as s − dimensional complex column vectors, where theoperator J acts as multiplication by i , the corresponding Hermitian innerproduct is j ( z , z ′ ) = z ∗ z ′ and the symplectic form is ∆( z, z ′ ) = 2Im z ∗ z ′ , where ∗ denotes Hermitian conjugation. The linear operators in Z commut-ing with J are represented by complex s × s − matrices. The gauge group actsin Z as multiplication by e iφ . Gaussian gauge-invariant states are describedby the modified characteristic function φ ( z ) = Tr ρD ( z ) = exp ( − z ∗ αz ) , (48)where α is a Hermitian correlation matrix satisfying α ≥ I / I , to which correspond the vacuum state ρ and the familyof coherent states { ρ z ; z ∈ Z } , such that ρ z = D ( z ) ρ D ( z ) ∗ . One hasTr ρ w D ( z ) = exp (cid:18) i Im w ∗ z − | z | (cid:19) , where | z | = z ∗ z . Let Z A , Z B be the input and output spaces of dimensionalities s A , s B . Wedenote by s A = dim Z A , s B = dim Z B the numbers of modes of the inputand output of the channel. The action of a Gaussian gauge-covariant channel(43) can be described asΦ ∗ [ D B ( z )] = D A ( Kz ) exp ( − z ∗ µz ) , z ∈ Z B , (49)where K is complex s B × s A − matrix, µ is Hermitian s B × s B − matrix satis-fying the condition (see [30]) µ ≥ ±
12 ( I B − K ∗ K ) , (50)where I B is the unit s B × s B − matrix. This follows from (44) by taking intoaccount that the matrix of the form j ( z , z ′ ) in an orthonormal basis is justthe unit matrix I of the corresponding size. Later we will need the following23 emma 7 The map (49) is injective if and only if KK ∗ > (in whichcase necessarily s B ≥ s A ). Proof.
Injectivity means that Φ[ ρ ] = Φ[ ρ ] implies ρ = ρ . But Φ[ ρ ] =Φ[ ρ ] is equivalent to Tr ρ Φ ∗ [ D B ( z )] = Tr ρ Φ ∗ [ D B ( z )] , i.e. Tr ρ D A ( Kz ) =Tr ρ D A ( Kz ) for all z ∈ Z B . By irreducibility of the Weyl system, this prop-erty is equivalent to Ran K = Z A , i.e. Ker K ∗ = { } or KK ∗ > . The channel (49) is extreme if µ is a minimal solution of the inequal-ity (50). Special cases of the maps (49) are provided by the attenuator and amplifier channels, characterized by matrix K fulfilling the inequali-ties, K ∗ K ≤ I and K ∗ K ≥ I respectively. We are particularly interested in extreme attenuator which corresponds to K ∗ K ≤ I B , µ = 12 ( I B − K ∗ K ) , (51)and extreme amplifier K ∗ K ≥ I B , µ = 12 ( K ∗ K − I B ) . (52)Denoting by ¯z the column vector obtained by taking the complex con-jugate of the elements of z , the action of the Gaussian gauge-contravariantchannel (46) is described asΦ ∗ [ D B ( z )] = D A ( − Kz ) exp ( − z ∗ µz ) , (53)where µ is Hermitian matrix satisfying the inequality µ ≥
12 ( I B + K ∗ K ) , (54)which follows from (47). Here ¯ z is the column vector consisting of complexconjugates of the components of z . These maps are extreme if µ = 12 ( I B + K ∗ K ) . (55)The following proposition generalizes to many modes the decompositionof one-mode channels the usefulness of which was emphasized and exploitedin the paper [20] (see also [13] on concatenations of one-mode channels): For Hermitian matrices
M, N, the strict inequality
M > N means that M − N ispositive definite. roposition 8 Any Gaussian gauge-covariant channel
Φ : A → B is aconcatenation Φ = Φ ◦ Φ of extreme attenuator Φ : A → B and extremeamplifier Φ : B → B .Any Gaussian gauge-contravariant channel Φ : A → B is a concatenationof extreme attenuator Φ : A → B and extreme gauge-contravariant channel Φ : B → B . Proof.
The concatenation Φ = Φ ◦ Φ of Gaussian gauge-covariant channelsΦ and Φ obeys the rule: K = K K , (56) µ = K ∗ µ K + µ . (57)By inserting relations µ = 12 ( I B − K ∗ K ) = 12 (cid:0) I B − | K | (cid:1) , µ = 12 ( K ∗ K − I B ) = 12 (cid:0) | K | − I B (cid:1) into (57) and using (56) we obtain | K | = K ∗ K = µ + 12 ( K ∗ K + I B ) ≥ (cid:26) I B K ∗ K (58)from the inequality (50). By using operator monotonicity of the square root,we have | K | ≥ I B , | K | ≥ | K | . The first inequality (58) implies that choosing K = | K | = r µ + 12 ( K ∗ K + I B ) (59)and the corresponding µ = (cid:0) | K | − I B (cid:1) , we obtain extreme amplifierΦ : B → B .Then with K = K | K | − (60)we obtain, taking into account the second inequality in (58) and also Lemma9 below, K K ∗ = K | K | − K ∗ = K (cid:20) µ +
12 ( K ∗ K + I ) (cid:21) − K ∗ ≤ I A , (61)which implies K ∗ K ≤ I A , hence K with the corresponding µ = ( I B − K ∗ K )give the quantum-limited attenuator.25 emma 9 Let M ≥ K ∗ K , then KM − K ∗ ≤ I A , where − means (generalized)inverse. Proof.
By the definition of the generalized inverse, u ∗ M − u = sup v : v ∈ Ran M ,v =0 | u ∗ v | v ∗ M v . By inserting K ∗ u in place of u and using Cauchy-Schwarz inequality in thenominator of the fraction, we obtain u ∗ KM − K ∗ u ≤ sup v : v ∈ Ran M ,v =0 u ∗ u v ∗ K ∗ K vv ∗ M v ≤ u ∗ u. In the case of contravariant channel the relations (56), (57) are replacedwith ¯K = K ¯K , (62) µ = K ∗ ¯ µ K + µ . (63)By substituting µ = 12 ( I − K ∗ K ) , µ = 12 ( K ∗ K + I B )into (63) and using (54) we obtain | K | = K ∗ K = µ + 12 ( K ∗ K − I B ) ≥ K ∗ K . (64)Taking K = | K | , µ = ( | K | + I B ) gives extreme gauge-contravariantchannel Φ : B → B . With ¯K = K | K | − (65)we obtain, by using Lemma 9, ¯K ¯K ∗ = K (cid:0) | K | − (cid:1) K ∗ = K (cid:20) µ + 12 ( K ∗ K − I B ) (cid:21) − K ∗ ≤ I A , (66)which implies K K ∗ ≤ I A , with the corresponding µ give the extreme at-tenuator Φ : A → B . 26 emark 10 In the case of gauge-covariant channel, the equality in (61 )shows that KK ∗ > implies K K ∗ > , while the inequality µ > ( K ∗ K − I B ) implies K K ∗ < I A . In the case of gauge-contravariant channel, the inequal-ity µ > ( I B + K ∗ K ) implies < K K ∗ < I A via (66). Proposition 11
The extreme attenuator with matrix K and extreme atten-uator with matrix ˜ K = √ I A − KK ∗ are mutually complementary.The extreme amplifier with matrix K and gauge-contravariant channelwith matrix ˜ K = p ¯K ¯K ∗ − I A are mutually complementary. Proof.
For the case of one mode see [13] or [39], Sec. 12.6.1. We sketchthe proof for several modes below. Define Z E ≃ Z A , Z D ≃ Z B , so that Z = Z A ⊕ Z D ≃ Z B ⊕ Z E In the case of attenuator consider the block unitary matrix in Z : V = (cid:20) K √ I A − KK ∗ √ I B − K ∗ K − K ∗ (cid:21) (67)which defines unitary dynamics U in H = H A ⊗ H D ≃ H B ⊗ H E by therelation U ∗ D BE ( z BE ) U = D AD ( Vz BE ). Here z BE = [ z B z E ] t , D BE ( z BE ) = D B ( z B ) ⊗ D E ( z E ), and the unitarity follows from the relation K p I B − K ∗ K = p I A − KK ∗ K . (68)Let ρ D = ρ be the vacuum state, ρ A = ρ an arbitrary state. Then theformulas (7), (8) define the mutually complementary extreme attenuators asdescribed in the first statement. The proof is obtained by computing thecharacteristic function of the output states for the channels. For the state ofthe composite system ρ BE = U ( ρ ⊗ ρ D ) U ∗ we have φ BE ( z BE ) = Tr U ( ρ ⊗ ρ D ) U ∗ [ D B ( z B ) ⊗ D E ( z E )]= Tr( ρ ⊗ ρ D ) U ∗ [ D B ( z B ) ⊗ D E ( z E )] U = Tr( ρ ⊗ ρ D ) h D A ( Kz B + ˜Kz E ) ⊗ D D ( p I B − K ∗ Kz B − K ∗ z E ) i = φ A ( Kz B + ˜Kz E ) exp (cid:20) − | p I B − K ∗ Kz B − K ∗ z E | (cid:21) . (69)By setting z E = 0 or z B = 0 we obtain φ B ( z B ) = φ A ( Kz B ) exp (cid:20) − z ∗ B ( I B − K ∗ K ) z B (cid:21) ,φ E ( z E ) = φ A ( ˜Kz E ) exp (cid:20) − z ∗ E KK ∗ z E (cid:21)
27s required.In the case of amplifier, set V = (cid:20) K −√ KK ∗ − I A Λ − Λ √ K ∗ K − I B Λ K ∗ Λ (cid:21) , where Λ is the operator of complex conjugation, anticommuting with mul-tiplication by i. By using the property ∆(Λ z, Λ z ′ ) = − ∆( z, z ′ ), we obtainthat V corresponds to a symplectic transformation in Z generating unitarydynamics U in H . Let again ρ be the vacuum state of the environment.Then the formulas (7), (8) define the mutually complementary channels asdescribed in the second statement of the Proposition, and the proof is similar.To show that V is a symplectic transformation, introduce the matrices Θ = (cid:20) I B − Λ (cid:21) , V = (cid:20) K √ KK ∗ − I A √ K ∗ K − I B K ∗ (cid:21) , Σ = (cid:20) I B − I A (cid:21) . Notice that V = ΘV Θ , and V ∗ ΣV = Σ , which means that V preservesthe indefinite Hermitian form σ ( z BE , z ′ BE ) = z ∗ B z ′ B − z ∗ E z ′ E . By taking intoaccount∆ BE ( Θ z BE , Θ z ′ BE ) = ∆ B ( z B , z ′ B ) − ∆ E ( z E , z ′ E ) = Im σ ( z BE , z ′ BE ) , we obtain∆ BE ( V z BE , V z ′ BE ) = Im σ ( V Θ z BE , V Θ z ′ BE ) = Im σ ( Θ z BE , Θ z ′ BE ) = ∆ BE ( z BE , z ′ BE ) , as required.Again, later we will need the following Lemma 12
Let Φ : A → B be an extreme attenuator with < K K ∗ < I A , then Φ [ P ψ ] = P ψ ′ (a pure state) if and only if P ψ is a coherent state. Proof.
According to Proposition 11, the complementary channel ˜Φ is anextreme attenuator with the matrix ˜ K = p I A − K K ∗ , such that < ˜K < I A . Its output is also pure, Φ [ P ψ ] = P ψ ′ E , as the outputs of complementarychannels have identical nonzero spectra by Lemma 2. Thus U ( ψ ⊗ ψ ) = ψ ′ ⊗ ψ ′ E , ψ ∈ H D is the vacuum vector and U is the unitary operator in H implementing the symplectic transformation corresponding to the unitary(67) in Z A ⊕ Z D ≃ Z B ⊕ Z E , with Z D ≃ Z B , Z E ≃ Z A Denoting by φ ( z ) = Tr P ψ D A ( z ) , φ ′ ( z B ) = Tr P ψ ′ D B ( z B ) , φ E ( z E ) = Tr P ψ ′ E D E ( z E )the quantum characteristic functions and using the relation (69), we havethe functional equation φ ′ ( z B ) φ E ( z E ) = φ ( K z B + ˜ Kz E ) exp (cid:20) − | p I B − K ∗ K z B − K ∗ z E | (cid:21) . (70)By letting z E = 0 , respectively z = 0 , we obtain φ ′ ( z B ) = φ ( K z B ) exp (cid:20) − | p I B − K ∗ K z B | (cid:21) ,φ E ( z E ) = φ ( ˜ Kz E ) exp (cid:20) − | K ∗ z E | (cid:21) , thus, after the change of variables z = K z B , z ′ = ˜ Kz E , and using ( 68), theequation (70) reduces to φ ( z ) φ ( z ′ ) = φ ( z + z ′ ) exp [Re z ∗ z ′ ] . The condition of the Lemma ensures that Ran K = Ran ˜ K = Z A . Substi-tuting ω ( z ) = φ ( z ) exp (cid:2) | z | (cid:3) , this becomes ω ( z ) ω ( z ′ ) = ω ( z + z ′ ) (71)for all z , z ′ ∈ Z A . The function ω ( z ) , as well as the characteristic function φ ( z ) , is continuous and satisfies ω ( − z ) = ω ( z ) . The only solution of (71)satisfying these conditions is the exponent ω ( z ) = exp [ i Im w ∗ z ] for somecomplex w . Thus φ ( z ) = exp (cid:20) i Im w ∗ z − | z | (cid:21) is the characteristic function of the coherent state ρ w / . .4 Gaussian optimizers The following basic result for one mode was obtained in [53]. Here we presenta complete proof in the multimode case, a sketch of which was given in [22].
Theorem 13 (i) Let Φ be a gauge covariant or contravariant channel andlet f be a real concave function on [0 , , such that f (0) = 0 , then Tr f (Φ[ ρ ]) ≥ Tr f (Φ[ ρ w ]) = Tr f (Φ[ ρ ]) (72) for all states ρ and any coherent state ρ w (the value on the right is thesame for all coherent states by the unitary covariance property of a Gaussianchannel (34)).(ii) Let f be strictly concave, then equality in (72) is attained only if ρ isa coherent state in the following cases:a) s B = s A and Φ is an extreme amplifier with µ = ( K ∗ K − I B ) > ;b) s B ≥ s A , the channel Φ is gauge-covariant with KK ∗ > and µ >
12 ( K ∗ K − I B ) ; (73) c) s B ≥ s A , the channel Φ is gauge-contravariant with KK ∗ > and µ > ( I B + K ∗ K ) . Proof. (i) We first prove the inequality (72) for strictly concave f. Then theinequality for arbitrary concave f follows by the monotone approximation f ( x ) = lim ε ↓ f ε ( x ) , since f ε ( x ) = f ( x ) − εx are strictly concave. Also, byconcavity, it is sufficient to prove (72) only for ρ = P ψ . By Proposition 8, Φ = Φ ◦ Φ where Φ : A → B is an extreme at-tenuator and Φ : B → B is either extreme amplifier or extreme gauge-contravariant channel. Any extreme attenuator maps vacuum state into vac-uum. Indeed,TrΦ [ ρ ] D B ( z ) = Tr ρ Φ ∗ [ D B ( z )]= Tr ρ D A ( Kz ) exp (cid:18) − z ∗ ( I B − K ∗ K ) z (cid:19) = exp (cid:18) − | z | (cid:19) = Tr ρ D B ( z ) . f (Φ[ ρ ]) = Tr f (Φ [ ρ ]) . Then it is sufficient to prove (72) for allextreme amplifiers and all extreme gauge-contravariant channels Φ . Indeed,assume that we have provedTr f (Φ [ P ψ ]) ≥ Tr f (Φ [ ρ ]) . (74)for any state vector ψ. Consider the spectral decomposition Φ [ P ψ ] = P j p j P φ j , where p j > , thenTr f (Φ[ P ψ ]) = Tr f (Φ [Φ [ P ψ ]]) (75) ≥ X j p j Tr f (Φ [ P φ j ]) (76) ≥ Tr f (Φ [ ρ ]) (77)= Tr f (Φ [Φ [ ρ ]]) = Tr f (Φ[ ρ ]) . (78)Then, according to the second statement of Proposition 11 and Lemma 2Tr f (Φ [ P ψ ]) = Tr f ( ˜Φ [ P ψ ]) , where Φ is an extreme amplifier and ˜Φ is an extreme gauge-contravariantchannel. Thus it is sufficient to prove (74) only for an extreme amplifierΦ : B → B, with Hermitian matrix K ≥ I B .The following result is based on a key observation by Giovannetti. Lemma 14
For an extreme amplifier Φ : B → B, with matrix K ≥ I B , there is an extreme attenuator Φ ′ such that for all ψ ∈ H B Φ ( P ψ ) ∼ (Φ ◦ Φ ′ ) ( P ψ ) . (79) Proof.
By Proposition 11 and Lemma 2 Φ ( P ψ ) ∼ ˜Φ ( P ψ ) for all ψ ∈ H B , where ˜Φ is extreme contravariant channel with the matrix ˜ K = p ¯ K − I B . Define the transposition map T : B → B by the relation T [ D ( z )] = D ( − ¯z ) . The concatenation Φ =
T ◦ ˜Φ is a covariant Gaussian channel:Φ ∗ [ D ( z )] = ˜Φ ∗ ◦ T [ D ( z )] = D ( q K − I B z ) exp (cid:18) − z ∗ K z (cid:19) . Applying decomposition from Proposition 8, namely the relation ( 58), givesΦ = Φ ◦ Φ ′ , where Φ is the original amplifier, and Φ ′ : B → B is another31xtreme attenuator with matrix K = p I B − K − . This implies the relation(79).Lemma 14 and Lemma 4 implyTr f (Φ ( P ψ )) = Tr f ((Φ ◦ Φ ′ ) ( P ψ )) . (80)Again, consider the spectral decomposition of the density operatorΦ ′ ( P ψ ) = X j p ′ j P ψ j , p ′ j > . By concavity, Tr f ((Φ ◦ Φ ′ ) [ P ψ ]) ≥ X j p ′ j Tr f (cid:0) Φ [ P ψ j ] (cid:1) . (81)Since f is assumed strictly concave, then ρ → Tr f (Φ [ ρ ]) is strictly con-cave [11]. Assuming that P ψ is a minimizer for the functional (80), we con-clude that Φ [ P ψ j ] must all coincide, otherwise the above inequality would bestrict, contradicting the assumption. From Lemma 7 it follows that P ψ j = P ψ ′ for all j and for some ψ ′ ∈ H B , hence, assuming that P ψ is a minimizer, theoutput Φ [ P ψ ] = P ψ ′ is a pure state.Since K = p I B − K − , the condition of Lemma 12 is fulfilled if K > I B . In this case, if P ψ is a minimizer, the Lemma implies that P ψ is a coherentstate. Thus we obtain the inequality (74) for the amplifier Φ with K > I B and strictly concave f . In this way we also obtain the case a) of the “onlyif” statement (ii).In the case of amplifier Φ with K ≥ I B , we can take any sequence K ( n )2 > I B , K ( n )2 → K , and the corresponding amplifiers Φ ( n )2 . ThenTr f (Φ ( n )2 [ ρ ]) → Tr f (Φ [ ρ ]) for any concave polygonal function f on [0 , , such that f (0) = 0 , and any ρ ∈ S ( H A ) . This follows from the fact that anysuch function is Lipschitz, | f ( x ) − f ( y ) | ≤ κ | x − y | , and (cid:13)(cid:13)(cid:13) Φ ( n )2 [ ρ ] − Φ [ ρ ] (cid:13)(cid:13)(cid:13) → . It follows that (74) holds for all extreme amplifiers Φ in the case ofpolygonal concave functions f . For arbitrary concave f on [0 ,
1] there isa monotonously nondecreasing sequence of concave polygonal functions f m converging to f pointwise. Passing to the limit m → ∞ gives the inequal-ity (74) for arbitrary extreme amplifier, and hence, (72) holds for arbitraryGaussian gauge-covariant or contravariant channels.32ii) The “only if” statement in the cases b), c) are obtained from thedecomposition Φ = Φ ◦ Φ and the relations (75)-(77) by applying argumentsimilar to the case of extreme amplifier. Notice that the conditions on thechannel Φ imply that in the decomposition Φ = Φ ◦ Φ the attenuatorΦ is defined by the matrix K such that 0 < K K ∗ < I A (see Remark10). Applying the argument involving the relations (80)-(81) with strictlyconcave f to the relations (75)-(78), we obtain that for any pure minimizer P ψ of Tr f (Φ[ P ψ ]) the output of the extremal attenuator Φ [ P ψ ] is necessarilya pure state. Applying Lemma 18 to the attenuator Φ we conclude that P ψ is necessarily a coherent state. Proposition 15
For any p > and any Gaussian gauge-covariant or con-travariant channel Φ k Φ k → p = (TrΦ[ ρ ] p ) /p , (82)ˇ R p (Φ) = R p (Φ[ ρ ]) , (83)ˇ H (Φ) = H (Φ[ ρ ]) , (84) where ρ is the vacuum state.The multiplicativity property (14) holds for any two Gaussian gauge-covariant (contravariant) channels Φ and Φ , as well as the additivity ofthe minimal R´enyi entropy (15) and of the minimal von Neumann entropy(16). Proof.
The first statement follows from Theorem 13 by taking f ( x ) = − x p ,f ( x ) = − x log x. If Φ and Φ are both gauge-covariant (contravariant), then their tensorproduct Φ ⊗ Φ shares this property. The second statement then followsfrom the expressions (82) - (84 ) and the product property of the vacuumstate ρ = ρ (1)0 ⊗ ρ (2)0 , which follows from the definition.From the definitions of gauge-co/contravariant channels (49), ( 53), itfollows that the state Φ[ ρ ] is gauge-invariant Gaussian with the correlationmatrix µ + K ∗ K / . The spectrum of Φ[ ρ ] is computed explicitly leading tothe expressions [41] k Φ k → p = [det [( µ + K ∗ K / I B / p − ( µ + K ∗ K / − I B / p ]] − /p H (Φ) = tr g ( µ + ( K ∗ K − I B ) / , (85)where g ( x ) = ( x + 1) log( x + 1) − x log x and tr denotes trace of operatorsin Z . In the last case we used the formula for the entropy of Gaussian state(48) [40]: H ( ρ ) = tr g ( α − I / . We now turn to the classical capacity of the channel Φ. In infinite dimen-sions, there are two novel features as compared to the situation described inSec. 2.4. First, one has to extend the notion of ensemble to embrace continualfamilies of states. We call generalized ensemble an arbitrary Borel probabil-ity measure π on S ( H A ). The average state of the generalized ensemble π isdefined as the barycenter of the probability measure¯ ρ π = Z S ( H A ) ρ π ( dρ ) . The conventional ensembles correspond to finitely supported measures.Second, one has to consider the input constraints to avoid infinite valuesof the capacities. Let F be a positive selfadjoint operator in H A , whichusually represents energy in the system A . We consider the input states withconstrained energy: Tr ρF ≤ E, where E is a fixed positive constant. Sincethe operator F is usually unbounded, care should be taken in defining thetrace; we put Tr ρF = R ∞ λ dm ρ ( λ ) , where m ρ ( λ ) = Tr ρE ( λ ) , and E ( λ ) isthe spectral function of the selfadjoint operator F. Then the constrained χ − capacity is given by the following generalization of the expression (17 ): C χ (Φ , F, E ) = sup π :Tr¯ ρ π F ≤ E χ ( π ) , (86)where χ ( π ) = H (Φ[ ¯ ρ π ]) − Z S ( H A ) H (Φ[ ρ ]) π ( dρ ) (87)To ensure that this expression is defined correctly, certain additional condi-tions upon the channel Φ and the constraint operator F should be imposed(see [39], Sec. 11.5), which however are always fulfilled in the Gaussian casewe consider below. 34enote F ( n ) = F ⊗ I · · · ⊗ I + · · · + I ⊗ · · · ⊗ I ⊗ F, then the constrainedclassical capacity is given by the expression C (Φ , F, E ) = lim n →∞ n C χ (Φ ⊗ n , F ( n ) , nE ) . (88)Now let Φ be a Gaussian gauge-covariant channel, and consider gauge-invariant oscillator energy operator F = P s A j,k =1 ǫ jk a ∗ j a k , where ǫ = [ ǫ jk ] isa Hermitian positive definite matrix, a j = √ ( q j + ip j ) – the annihilationoperator for j -th mode. For any state ρ satisfying Tr ρF < ∞ , the firstmoments Tr ρa j and the second moments Tr ρa ∗ j a k , Tr ρa j a k are well defined.For gauge-invariant state Tr ρa j = 0 and Tr ρa j a k = 0 . For a Gaussian gauge-invariant state (48) α − I / (cid:2) Tr ¯ ρ π a ∗ j a k (cid:3) j,k =1 ,...,s , see e.g. [33]. Proposition 16
The constrained classical capacity of the Gaussian gauge-covariant channel Φ is C (Φ; F, E ) = C χ (Φ; F, E ) (89)= max ν : tr νǫ ≤ E tr g ( K ∗ ν K + µ + ( K ∗ K − I B ) / − tr g ( µ + ( K ∗ K − I B ) / . The optimal ensemble π which attains the supremum in (86) consists of co-herent states ρ z = D A ( z ) ρ D A ( z ) ∗ , z ∈ Z A distributed with gauge-invariantGaussian probability distribution Q ν ( d s z ) on Z A having zero mean and thecorrelation matrix ν which solves the maximization problem in (89). Proof.
Consider a Gaussian ensemble π ν consisting of coherent states ρ z = D A ( z ) ρ D A ( z ) ∗ , z ∈ Z A , with gauge-invariant Gaussian probability distribu-tion Q ν ( d s z ) on Z A having zero mean and some correlation matrix ν. It isdefined by the classical characteristic function Z Z A exp (2 i Im w ∗ z ) Q ν ( d s w ) = exp ( − z ∗ ν z ) . By using the covariance property (34) of Gaussian channel, we have H (Φ[ ρ z ]) = H (Φ[ D A ( z ) ρ D A ( z ) ∗ ]) = H (Φ[ ρ ]) = tr g ( µ + ( K ∗ K − I B ) / , z , and hence it gives the value of the integral termin (87). Integration of the characteristic functions of coherent states givesTr ¯ ρ π ν D A ( z ) = exp ( − z ∗ ( ν + I A / z ) . Then ν = (cid:2) Tr ¯ ρ π a ∗ j a k (cid:3) j,k =1 ,...,s A and Tr ¯ ρ π ν F = P sj,k =1 ǫ jk Tr ¯ ρ π ν a ∗ j a k = tr νǫ. The state Φ[ ¯ ρ π ν ] is gauge-invariant Gaussian with the correlation matrix K ∗ ( ν + I A / K + µ , hence it has the entropy tr g ( K ∗ ν K + µ +( K ∗ K − I B ) / . Thus for the Gaussian ensemble π ν χ ( π ν ) = tr g ( K ∗ ν K + µ + ( K ∗ K − I B ) / − tr g ( µ + ( K ∗ K − I B ) / . (90)Summarizing, we need to show C (Φ; F, E ) = C χ (Φ; F, E ) = sup ν : tr νǫ ≤ E χ ( π ν ) . (91)Let us denote by G the set of Gaussian gauge-invariant states in H A . Lemma 17 max ρ ( n ) :Tr ρ ( n ) F ( n ) ≤ nE H (cid:0) Φ ⊗ n (cid:2) ρ ( n ) (cid:3)(cid:1) ≤ n max ρ : ρ ∈G , Tr ρF ≤ E H (Φ [ ρ ]) . (92) Proof.
We first prove thatsup ρ ( n ) :Tr ρ ( n ) F ( n ) ≤ nE H (Φ ⊗ n [ ρ ( n ) ]) ≤ n sup ρ :Tr ρF ≤ E H (Φ[ ρ ]) . (93)Indeed, denoting by ρ j the partial state of ρ ( n ) in the j − th tensor factorof H ⊗ nA and letting ¯ ρ = n P nj =1 ρ j , we have H (Φ ⊗ n [ ρ ( n ) ]) ≤ n X j =1 H (Φ[ ρ j ]) ≤ nH (Φ[ ¯ ρ ]) , where in the first inequality we used subadditivity of the quantum entropy,while in the second – its concavity. Moreover, Tr ¯ ρF = n Tr ρ ( n ) F ( n ) ≤ E, hence (93) follows.Using gauge covariance of the channel Φ, we can then reduce maximiza-tion in the right hand side of (93) to gauge-invariant states. Indeed, for agiven state ρ , satisfying the constraint Tr ρF ≤ E the averaging ρ av = 12 π Z π U ϕ ρU ∗ ϕ dϕ H (Φ[ ρ ]) ≤ H (Φ[ ρ av ]) by concavity of theentropy.Finally, we use the maximum entropy principle which says that amongstates with fixed second moments the Gaussian state has maximal entropy(see e.g. [39], Lemma 12.25). This proves (92).We havemax ρ : ρ ∈G , Tr ρF ≤ E H (Φ [ ρ ]) = max ν :tr νǫ ≤ E tr g ( K ∗ ν K + µ + ( K ∗ K − I B ) / . (94)Now let ν be the solution of the maximization problem in the righthand side.To prove (89) observe that nχ ( π ν ) ≤ nC χ (Φ , F, E ) ≤ C χ (Φ ⊗ n , F ( n ) , nE ) ≤ max ρ ( n ) :Tr ρ ( n ) F ( n ) ≤ nE H (cid:0) Φ ⊗ n (cid:2) ρ ( n ) (cid:3)(cid:1) − min ρ ( n ) H (cid:0) Φ ⊗ n (cid:2) ρ ( n ) (cid:3)(cid:1) . By using Lemma 17 and Proposition 15 we see that this is less than orequal to n (cid:20) max ρ : ρ ∈G , Tr ρF ≤ E H (Φ [ ρ ]) − H (Φ [ ρ ]) (cid:21) = nχ ( π ν ) , where the equality follows from (94) and (90).Thus C χ (Φ ⊗ n , F ( n ) , nE ) = nC χ (Φ , F, E ) and hence the constrained clas-sical capacity (88) of the Gaussian gauge-covariant channel is given by theexpression (89).Similar argument applies to Gaussian gauge-contravariant channel (53),giving the expression (89) with ǫ replaced by ¯ ǫ . Indeed, in this case the stateΦ[ ¯ ρ π ν ] is gauge-invariant Gaussian with the characteristic functionTrΦ[ ¯ ρ π ν ] D ( z ) = exp (cid:0) − ( Kz ) ∗ ( ν + I A / Kz − z ∗ µz (cid:1) = exp ( − ( Kz ) ∗ (¯ ν + I A / Kz − z ∗ µz ) , with the correlation matrix K ∗ (¯ ν + I A / K + µ . On the other hand, tr νǫ =tr¯ ν ¯ ǫ , so that redefining ¯ ν as ν , we get the statement.The maximization in (89) is a finite-dimensional optimization problemwhich is a quantum analog of “water-filling” problem in classical informationtheory, see e.g. [15, 40]. It can be solved explicitly only in some special cases,e.g. when K , µ , ǫ commute, and it is a subject of separate study.37 .6 The case of quantum-classical Gaussian channel Consider affine map which transforms quantum states ρ ∈ S ( H ) into prob-ability densities on Z ρ → p ρ ( z ) = Tr ρD ( z ) ρ D ( z ) ∗ , (95)where D ( z ) are the displacement operators, ρ is the vacuum state with thequantum characteristic function φ ( z ) ≡ Tr ρ D ( z ) = exp (cid:18) − z ∗ z (cid:19) , The function p ρ ( z ) is bounded by 1 and is indeed a continuous probabilitydensity, the normalization follows from the resolution of the identity Z Z D ( z ) ρ D ( z ) ∗ d s z π s = I. Proposition 18
Let f be a concave function on [0 , , such that f (0) = 0 , then for arbitrary state ρ Z Z f ( p ρ ( z )) d s z π s ≥ Z Z f ( p ρ w ( z )) d s z π s . (96) Proof.
For any c > c defined by the relationΦ c [ ρ ] = Z d s z π s c s Tr[ ρD ( c − z ) ρ D ∗ ( c − z )] ρ z . (97)The map (97) is a Gaussian gauge-covariant channel such thatΦ ∗ c [ D ( z )] = D ( c z ) exp (cid:20) − ( c + 1)2 | z | (cid:21) , cf. [23]. Therefore by Theorem 13,Tr f (Φ c [ ρ ]) ≥ Tr f (Φ c [ ρ w ]) (98)for all states ρ and any coherent state ρ w . We will prove the Proposition 18by taking the limit c → ∞ .
38n the proof we also use a simple generalization of the Berezin-Lieb in-equalities [9]: Z Z f ( p ( z )) d s z π s ≤ Tr f ( σ ) ≤ Z Z f (¯ p ( z )) d s z π s , (99)valid for any quantum state admitting the representation σ = Z Z p ( z ) ρ z d s z π s with a probability density p ( z ). In the right side of (99) ¯ p ( z ) = Tr σρ z . Inthe inequalities (99) one has to assume that f is defined on [0 , ∞ ) (in fact, p ( z ) can be unbounded). We shall assume this for a while.Taking σ = Φ c [ ρ ] , from (97) we have p ( z ) = 1 c s Tr ρD ( c − z ) ρ D ∗ ( c − z ) = 1 c s p ρ ( c − z ) . while ¯ p ( z ) = Tr ρ z Φ c [ ρ ] = Z Z p ( w )Tr ρ z ρ w d s w π s . (100)We use the well-known formula, see e.g. [44], [33],Tr ρ z ρ w = exp[ −| z − w | ] . By introducing the probability density of a normal distribution q c ( z ) = c s π s exp (cid:0) − c | z | (cid:1) tending to δ − function when c → ∞ and substituting this into (100), we have¯ p ( z ) = Z d s w p ( w ) q ( z − w )= Z d s w ′ p ρ ( w ′ ) q ( z − c w ′ )= 1 c s p ρ ∗ q c ( c − z ) . (101)With the change of the integration variable c − z → z , the inequalities (99)become Z Z f ( c − s p ρ ( z )) d s z π s ≤ c − s Tr f (Φ c [ ρ ]) ≤ Z C s f ( c − s p ρ ∗ q c ( z )) d s z π s , ρ = ρ w , we have Z Z f ( c − s p ρ w ( z )) d s z π s ≤ c − s Tr f (Φ c [ ρ w ]) ≤ Z Z f ( c − s p ρ w ∗ q c ( z )) d s z π s . Combining the last two displayed formulas with (98) we obtain Z Z g ( p ρ ( z )) d s z π s − Z C s g ( p ρ w ( z )) d s z π s ≥ Z Z g ( p ρ ( z )) d s z π s − Z Z g ( p ρ ∗ q c ( z )) d s z π s , (102)where we denoted g ( x ) = f ( c − s x ) , which is again a concave function. More-over, arbitrary concave polygonal function g on [0 , , satisfying g (0) = 0 , can be obtained in this way by defining f ( x ) = (cid:26) g ( c s x ) , x ∈ [0 , c − s ] g (1) + g ′ (1)( x − c − s ) , x ∈ [ c − s , ∞ ) , hence (102) holds for any such function. Then the right hand side of theinequality (102) tends to zero as c → ∞ . Indeed, for polygonal function | g ( x ) − g ( y ) | ≤ κ | x − y | , and the asserted convergence follows from theconvergence p ρ ∗ q c −→ p ρ in L : if p ( z ) is a bounded continuous probabilitydensity, then lim c →∞ Z Z | p ∗ q c ( z ) − p ( z ) | d s z = 0 . Thus we obtain (96) for the concave polygonal functions f. But for arbi-trary continuous concave f on [0 ,
1] there is a monotonously nondecreas-ing sequence of concave polygonal functions f n converging to f . ApplyingBeppo-Levy’s theorem, we obtain the statement. Consider a gauge-covariant channel Φ such that the matrices K ∗ K and µ commute (in particular, this condition is satisfied by extreme amplifiers andattenuators). These channels are diagonalizable in the following sense. Wehave K = V A K d V B , µ = V ∗ B µ d V B , V A , V B are unitaries and K d , µ d are diagonal (rectangular) matriceswith nonnegative values on the diagonal. Then K ∗ K = V ∗ B K d V B , andΦ[ ρ ] = U B Φ d [ U A ρU ∗ A ] U ∗ B , (103)where U A , U B are canonical unitary (“metaplectic” [2]) transformations act-ing on H A , H B such that U ∗ B D B ( z ) U B = D B ( V B z ) , U ∗ A D A ( z ) U A = D A ( V A z ) , To describe the action of “ diagonal” channel Φ d in more detail, we have toconsider separately the cases s A = s B , s A ≤ s B and s A > s B . In the case s A = s B we have K d = diag [ k j ] j =1 ,...,s B ; µ d = diag [ µ j ] j =1 ,...,s B . Then Φ d = ⊗ s B j =1 Φ j , where, in self-explanatory notations,Φ ∗ j [ D j ( z j )] = D j ( k j z j ) exp (cid:0) − µ j | z j | (cid:1) . (104)In the case s A < s B K d = (cid:20) diag [ k j ] j =1 ,...,s A (cid:21) where denotes block of zeroes of the size ( s B − s A ) × s A . ThenΦ d [ ρ ] = ⊗ s A j =1 Φ j [ ρ ] ⊗ ρ [ s A +1 ,...,s B ]0 , where for j = 1 , . . . , s A the one-mode channels Φ j are given by ( 104), and ρ [ s A +1 ,...,s B ]0 is the vacuum state of the modes s A + 1 , . . . , s B . In the case s A > s B K d = (cid:2) diag [ k j ] j =1 ,...,s B (cid:3) where denotes block of zeroes of the size s B × ( s A − s B ), andΦ d [ ρ ] = (cid:0) ⊗ s A j =1 Φ j (cid:1) [Tr s B +1 ,...,s A ρ ] , where Tr s B +1 ,...,s A denotes partial trace over the last s A − s B modes of theoperator ρ. There is a similar reduction to the diagonal form for gauge-contravariantchannels. 41
Acknowledgments
The author is grateful to M. E. Shirokov (Steklov Mathematical Institute)for comments and discussions. Thanks are due to David Ding (StanfordUniversity) for pointing out some typos. The work was supported by thegrant of Russian Scientific Foundation (project No 14-21-00162).
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