Quantifying the non-Gaussian character of a quantum state by quantum relative entropy
aa r X i v : . [ qu a n t - ph ] J a n Quantifying the non-Gaussian character of a quantum state by quantum relative entropy
Marco G. Genoni,
1, 2
Matteo G. A. Paris,
1, 2, 3 and Konrad Banaszek Dipartimento di Fisica dell’Universit`a di Milano, I-20133, Milano, Italia CNISM, UdR Milano Universit`a, I-20133 Milano, Italia ISI Foundation, I-10133 Torino, Italia Institute of Physics, Nicolaus Copernicus University, PL-87-100 Toru´n, Poland (Dated: November 8, 2018)We introduce a novel measure to quantify the non-Gaussian character of a quantum state: the quantum relativeentropy between the state under examination and a reference Gaussian state. We analyze in details the propertiesof our measure and illustrate its relationships with relevant quantities in quantum information as the Holevobound and the conditional entropy; in particular a necessary condition for the Gaussian character of a quantumchannel is also derived. The evolution of non-Gaussianity (nonG) is analyzed for quantum states undergoingconditional Gaussification towards twin-beam and de-Gaussification driven by Kerr interaction. Our analysisallows to assess nonG as a resource for quantum information and, in turn, to evaluate the performances ofGaussification and de-Gaussification protocols.
PACS numbers: 03.67.-a, 03.65.Bz, 42.50.Dv
Introduction —The use of Gaussian states and operationsallows the implementation of relevant quantum informationprotocols including teleportation, dense coding and quantumcloning [1]. Indeed, the Gaussian sector of the Hilbert spaceplays a crucial role in quantum information processing withcontinuous variables (CV), especially for what concerns quan-tum optical implementations [2]. On the other hand, quan-tum information protocols required for long distance commu-nication, as for example entanglement distillation and entan-glement swapping, require nonG operations [3]. Besides, ithas been demonstrated that using nonG states and operationsteleportation [4, 5, 6] and cloning [7] of quantum states maybe improved. Indeed, de-Gaussification protocols for single-mode and two-mode states have been proposed [4, 5, 6, 8, 9]and realized [10]. From a more theoretical point of view, itshould be noticed that any strongly superadditive and continu-ous functional is minimized, at fixed covariance matrix (CM),by Gaussian states. This is crucial to prove extremality ofGaussian states and Gaussian operations [11, 12] for variousquantities such as channel capacities [13], multipartite entan-glement measures [14] and distillable secret key in quantumkey distribution protocols. Overall, nonG appears to be a re-source for CV quantum information and a question naturallyarises on whether a convenient measure to quantify the nonGcharacter of a quantum state may be introduced. Notice thatthe notion of nonG already appeared in classical statistics inthe framework of independent component analysis [15].The first measure of nonG of a CV state ̺ has been sug-gested in [16] based on the Hilbert-Schmidt (HS) distance be-tween ̺ and a reference Gaussian state. In turn, the HS-basedmeasure has been used to characterize the role of nonG as a re-source for teleportation [17, 18] and in promiscuous quantumcorrelations in CV systems [19]. Here we introduce a novelmeasure δ [ ̺ ] based on the quantum relative entropy between ̺ and a reference Gaussian state. The novel quantity is related toinformation measures and allows to assess nonG as a resourcefor quantum information as well as the performances of Gaus- sification and de-Gaussification protocols. In the following,after introducing its formal definition and showing that it canbe easily computed for any state, either single-mode or multi-mode, we analyze in details the properties of δ [ ̺ ] as well asits dynamics under Gaussification [29] and de-Gaussificationprotocols. Gaussian states —Let us consider a CV system made of d bosonic modes described by the mode operators a k , k =1 . . . d , with commutation relations [ a k , a † j ] = δ kj . A quan-tum state ̺ of d bosonic modes is fully described by its char-acteristic function χ [ ̺ ]( λ ) = Tr [ ̺ D ( λ )] where D ( λ ) = N dk =1 D k ( λ k ) is the d -mode displacement operator, with λ =( λ , . . . , λ d ) T , λ k ∈ C , and where D k ( λ k ) = exp { λ k a † k − λ ∗ k a k } is the single-mode displacement operator. The canoni-cal operators are given by q k = ( a k + a † k ) / √ and p k = ( a k − a † k ) / √ i with commutation relations given by [ q j , p k ] = iδ jk .Upon introducing the vector R = ( q , p , . . . , q d , p d ) T , theCM σ ≡ σ [ ̺ ] and the vector of mean values X ≡ X [ ̺ ] ofa quantum state ̺ are defined as σ kj = h R k R j + R j R k i −h R j ih R k i and X j = h R j i , where h O i = Tr [ ̺ O ] is the ex-pectation value of the operator O . A quantum state ̺ G is saidto be Gaussian if its characteristic function is Gaussian, that is χ [ ̺ G ]( Λ ) = exp n − Λ T σ Λ + X T Λ o , where Λ is the realvector Λ = ( Re λ , Im λ , . . . , Re λ d , Im λ d ) T . Once the CMand the vectors of mean values are given, a Gaussian state isfully determined. For a system of d bosonic modes the mostgeneral Gaussian state is described by d (2 d + 3) independentparameters. Non-Gaussianity of a CV state —The von Neumann en-tropy of a quantum state is defined as S ( ̺ ) = − Tr [ ̺ log ̺ ] .The von Neumann entropy is non-negative and equals zeroiff ̺ is a pure state. In order to quantify the nonG char-acter of a quantum state ̺ we employ the quantum relativeentropy (QRE) S ( ̺ k τ ) = Tr [ ̺ (log ̺ − log τ )] between ̺ and a reference Gaussian state τ . As for its classical coun-terpart, the Kullback-Leibler divergence, it can be demon-strated that ≤ S ( ̺ k τ ) < ∞ when it is definite, i.e. when supp ̺ ⊆ supp τ . In particular S ( ̺ k τ ) = 0 iff ̺ ≡ τ . Thisquantity, though not defining a proper metric in the Hilbertspace, has been widely used in different fields of quantuminformation as a measure of statistical distinguishability forquantum states [20, 21]. Therefore, given a quantum state ̺ with finite first and second moments, we define its nonG as δ [ ̺ ] = S ( ̺ k τ ) , where the reference state τ is is the Gaussianstate with X [ ̺ ] = X [ τ ] and σ [ ̺ ] = σ [ τ ] , i.e. the Gaussianstate with the same CM σ and the same vector X of the state ̺ . Finally, since τ is Gaussian, then log τ is a polynomial op-erator of the second order in the canonical variables which,together with the fact that τ and ρ have the same CM leads toTr [( τ − ̺ ) log τ ] = 0 [22], i.e. S ( ̺ k τ ) = S ( τ ) − S ( ̺ ) . Thuswe have δ [ ̺ ] = S ( τ ) − S ( ̺ ) (1) i.e. nonG is the difference between the von Neumann en-tropies of τ and ̺ . In turn, several properties of the nonGmeasure δ [ ̺ ] may be derived from the fundamental propertiesof QRE [20, 21]. In the following we summarize the relevantones by the following Lemmas: L1 : δ [ ̺ ] is a well defined non negative quantity, that is ≤ δ [ ̺ ] < ∞ and δ [ ̺ ] = 0 iff ̺ is a Gaussian state. Proof :Nonnegativity is guaranteed by the nonnegativity of the quan-tum relative entropy. Moreover, if δ [ ̺ ] = 0 then ̺ = τ andthus it is a Gaussian state. If ̺ is a Gaussian state, then it isuniquely identified by its first and second moments and thusthe reference Gaussian state τ is given by τ = ̺ , which, inturn, leads to δ [ ̺ ] = S ( ̺ k τ ) = 0 . L2 : δ [ ̺ ] is a continuous functional. Proof : It follows from thecontinuity of trace operation and QRE. L3 : δ [ ̺ ] is additive for factorized states: δ [ ̺ ⊗ ̺ ] = δ [ ̺ ] + δ [ ̺ ] . As a corollary we have that if ̺ is a Gaussian state,then δ [ ̺ ] = δ [ ̺ ] . Proof : The overall reference Gaussian stateis the tensor product of the relative reference Gaussian states, τ = τ ⊗ τ . The lemma thus follows from the additivity ofQRE and the corollary from L1. L4 : For a set of states { ̺ k } having the same first and sec-ond moments, then nonG is a convex functional, that is δ [ P k p k ̺ k ] ≤ P k p k δ [ ̺ i ] , with P k p k = 1 . Proof : Thestates ̺ k , having the same first and second moments, havethe same reference Gaussian state τ which in turn is thereference Gaussian state of the convex combination ̺ = P k p k ̺ k . Since conditional entropy S ( ̺ k τ ) is a jointly con-vex functional respect to both states, we have δ [ P k p k ̺ k ] = S ( P k p k ̺ k k τ ) ≤ P k p k S ( ̺ k k τ ) = P k p k δ [ ̺ k ] . (cid:3) Noticethat, in general, nonG is not convex, as it may easily provedupon considering the convex combination of two Gaussianstates with different parameters. L5 : If U b is a unitary evolution corresponding to a symplectictransformation in the phase space, i.e. if U b = exp {− iH } with H at most bilinear in the field operator, then δ [ U b ̺U † b ] = δ [ ̺ ] . Proof : Let us consider ̺ ′ = U b ̺U † b , where U is atmost bilinear in the field-mode, then its CM transforms as σ [ ̺ ′ ] = Σ σ [ ̺ ]Σ T , Σ being the symplectic transformation as-sociated to U . At the same time the vector of mean valuessimply translates to X ′ = X + X . Since any Gaussian stateis fully characterized by its first and second moments, then thereference state must necessarily transform as τ ′ = U b τ U † b , i.e. with the same unitary transformation U . Lemma follows frominvariance of QRE under unitary transformations. L6 : nonG is monotonically decreasing under partial trace, thatis δ [ Tr B [ ̺ ]] ≤ δ [ ̺ ] . Proof : Let us consider ̺ ′ = Tr B [ ̺ ] . ItsCM is the submatrix of σ [ ̺ ] and its first moment vector isthe subvector of X [ ̺ ] corresponding to the relevant Hilbertspace. As before, also the new reference Gaussian state mustnecessarily transform as τ ′ = Tr B [ τ ] . QRE is monotonousdecreasing under partial trace and the lemma is proved. L7 : nonG is monotonically decreasing under Gaussian quan-tum channels, that is δ [ E G ( ̺ )] ≤ δ [ ̺ ] . Proof : Any Gaus-sian quantum channel can be written as E G ( ̺ ) = Tr E [ U b ( ̺ ⊗ τ E ) U † b ] , where U b is a unitary operation corresponding to anHamiltonian at most bilinear in the field modes and where τ E is a Gaussian state [23]. Then, by using lemmas L3, L5 andL6 we obtain δ [ E G ( ̺ )] ≤ δ [ U b ( ̺ ⊗ τ E ) U † b ] = δ [ ̺ ] . (cid:3) In turn,L7 provides a necessary condition for a channel to be Gaus-sian: given a quantum channel E , and a generic quantum state ̺ , if the inequality δ [ E ( ̺ )] ≤ δ [ ̺ ] is not fulfilled, the channelis nonG. Maximally non-Gaussian states —Let us now consider asingle mode ( d = 1 ) system and look for states with the max-imum amount of nonG at fixed average number of photons N = h a † a i . Since δ [ ̺ ] = S ( τ ) − S ( ̺ ) we have to maxi-mize S ( τ ) and, at the same time, minimize S ( ̺ ) . For a single-mode system the most general Gaussian state can be written as ̺ G = D ( α ) S ( ζ ) ν ( n t ) S † ( ζ ) D † ( α ) , D ( α ) being the displace-ment operator, S ( ζ ) = exp( ζa † − ζ ∗ a ) the squeezingoperator, α, ζ ∈ C , and ν ( n t ) = (1 + n t ) − [ n t / (1 + n t )] a † a a thermal state with n t average number of photons. Dis-placement and squeezing applied to thermal states increasethe overall energy, while entropy is an increasing monotonousfunction of the number of thermal photons n t and is invariantunder unitary operations, thus, at fixed energy, S ( τ ) is max-imized for τ = ν ( N ) . Therefore, the state with the maxi-mum amount of nonG must be a pure state (in order to have S ( ̺ ) = 0 ) with the same CM σ = ( N + ) I of the ther-mal state ν ( N ) . These properties individuate the superposi-tions of Fock states | ψ N i = P k α k | n + l k i where n ≥ , l k ≥ l k − + 3 or l k = 0 , with the constraint N = h a † a i , i.e n + P k | α k | l k = N = (det σ [ ν ( N )]) − . Theserepresent maximally nonG states, and include Fock states | ψ N i = | N i as a special case. Let us consider now d -mode quantum states with fixed average number of photons P dk =1 Tr [ a † k a k ̺ ] = N = P k n k . Also in this case maximallynonG states are pure states; the CM being equal to that of amultimode classical state τ = R ⊗ k ν ( m k ) R † , P k m k = N ,where we denote with R a generic set of symplectic passiveoperations ( e.g. beam splitter evolution) which do not increasethe energy. In order to maximize S ( τ ) = P k S ( ν ( m k )) wehave to choose m k = N/d for every k . As for example, fac-torized states of the form | Ψ N i = | ψ N/d i ⊗ d , whose referenceGaussian states are τ = [ ν ( N/d )] ⊗ d , are maximally nonGstates at fixed N . Of course for the multi-mode case there areother more complicated classes of maximally nonG states thatinclude also entangled pure states. Finally, we observe that themaximum value of nonG is a monotonous increasing functionof the number of photons N . Non-Gaussianity in quantum information —Gaussian statesare extremal for several functionals in quantum information[11]. In the following we consider two relevant examples,and show how extremality properties may be quantified by thenonG measure δ [ ̺ ] . Let us first consider a generic communi-cation channel where the letters from an alphabet are encodedonto a set of quantum states ̺ k with probabilities p k . The Holevo Bound represents the upper bound to the accessibleinformation, and is defined as χ ( ̺ ) = S ( ̺ ) − P k p k S ( ̺ k ) where ̺ = P k p k ̺ k is the overall ensemble sent throughthe channel. Upon fixing the CM (and the first moments)of ̺ we rewrite the Holevo bound as χ ( ̺ ) = S ( τ ) − δ [ ̺ ] − P k p k S ( ̺ k ) , where τ is the Gaussian reference of ̺ . Thishighlights the role of the nonG δ [ ̺ ] of the overall state in de-termining the amount of accessible information: at fixed CMthe most convenient encoding corresponds to a set of purestates ̺ k , S ( ̺ k ) = 0 , forming an overall Gaussian ensem-ble with the largest entropy. In other words, at fixed CM,we achieve the maximum value of χ upon encoding encodingsymbols onto the eigenstates of the corresponding Gaussianstate [24]. If the alphabet is encoded onto the eigenstates of agiven state ̺ , we have χ ( ̺ ) = S ( τ ) − δ [ ̺ ] . This suggests anoperational interpretation of nonG δ [ ̺ ] as the loss of informa-tion we get by encoding symbols on the eigenstates of ̺ ratherthan on those of its reference Gaussian state.Let us now consider the state ̺ AB describing two quan-tum systems A and B and define the conditional entropy S ( A | B ) = S ( ̺ AB ) − S ( ̺ B ) . Let us fix the CM of ̺ AB and thus also that of ̺ B , and consider the reference Gaus-sian states τ AB and τ B . We may write S ( A | B ) = S G ( A | B ) − ( δ [ ̺ AB ] − δ [ ̺ B ]) where S G ( A | B ) = S ( τ AB ) − S ( τ B ) , i.e. the conditional entropy evaluated for the reference Gaussianstates τ AB and τ B . Then, upon using L6 we have δ [ ̺ AB ] − δ [ ̺ B ] ≥ and thus S ( A | B ) ≤ S G ( A | B ) , i.e the maximumof conditional entropy at fixed CM is achieved by Gaussianstates. In classical information theory the conditional entropy H ( X | Y ) = H ( X, Y ) − H ( Y ) , where von Neumann entropiesare replaced by Shannon entropies of classical probability dis-tributions, is a positive quantity and may be interpreted [25]as the amount of partial information that Alice must send toBob so that he gains full knowledge of X given his previousknowledge from Y . When quantum systems are involved theconditional entropy may be negative, negativity being a suffi-cient condition for the entanglement of the overall state ̺ AB .This negative information may be seen as follows [26] for adiscrete variable quantum system. Given an unknown quan-tum state distributed over two systems, we can discriminatebetween two different cases: if S ( A | B ) ≥ , as in the classical case, it gives the amount of information that Alice should sendto Bob to give him the full knowledge of the overall state ̺ AB .When S ( A | B ) < Alice does not need to send any informa-tion to Bob and moreover they gain − S ( A | B ) ebits. If weconjecture that this interpretation can be extended to the CVcase, the relation S ( A | B ) ≤ S G ( A | B ) ensures that, at fixedCM, nonG states always perform better: Alice needs to sendless information, or, for negative values of the conditional en-tropy, more entanglement is gained. Moreover, since nega-tivity of conditional entropy is a sufficient condition for en-tanglement [27] we have that for any given bipartite quantumstate ̺ AB , if the conditional entropy of the reference Gaussianstate τ AB is negative, then ̺ AB is an entangled state. Thoughbeing a weaker condition than the negativity of S ( A | B ) , thisis a simple and easy computable test for entanglement whichis equivalent to evaluate the symplectic eigenvalues [28] of theinvolved Gaussian states. Gaussification and de-Gaussification protocols —Since theamount of nonG of a state affects its performances in quantuminformation protocols a question naturally arises on whetherthis may be engineered or modified at will. As concerns Gaus-sification, Lemma L7 assures that Gaussian maps do not in-crease nonG. In turn, the simplest example of Gaussificationmap is provided by dissipation in a thermal bath [16], whichfollows from bilinear interactions between the systems underinvestigation and the environment. On the other hand, a condi-tional iterative Gaussification protocol has been recently pro-posed [29] which cannnot be reduced to a trace-preservingGaussian quantum map. It requires only the use of passive el-ements and on/off photodetectors. Given a bipartite pure statein the Schmidt form | ψ ( k ) i = P ∞ n =0 α ( k ) n,n | n, n i the state at k + 1 -th step of the protocol has the same Schmidt form with α ( k +1) n,n = 2 − n P nr =0 (cid:0) nr (cid:1) α ( k ) r,r α ( i ) n − r,n − r . We have consideredthe initial nonG superposition | ψ (0) i = (1 + λ ) − / ( | , i + λ | , i ) which is asymptotically driven towards the Gaussiantwin-beam state | ψ i = √ − λ P ∞ n =0 λ n | n, n i . We haveevaluated nonG at any step of the protocol, for every value of λ . Λ ∆ Λ ∆ FIG. 1: (Left): nonG after some steps of the conditional Gaussifi-cation protocol of Ref. [29] considering as the initial state the nonGsuperposition | ψ (0) i = (1 + λ ) − / ( | , i + λ | , i ) : black-solid:initial state; black-dashed : step 1; gray-solid: step 2; gray-dashed:step 3. (Right): black-solid: initial state; black-dashed : step 5; gray-solid : step 10; gray-dashed: step 20. Results are reported in Fig. 1: for the first steps, nonG de-creases monotonically for almost all values of λ (only at thethird step, for λ ≈ the state is more nonG than at the pre-vious steps). Notice that increasing the number of steps nonGmay also increase, e.g , for λ ≈ , δ reaches very high val-ues and the maximum value increases. On the other hand, theoverall effectiveness of the protocol is confirmed by our anal-ysis, since the range of values of λ for which δ ≈ increasesat each step of the protocol. In other words, though not be-ing a proper Gaussian map, the conditional protocol of [29]indeed provides an effective Gaussification procedure.Conditional de-Gaussification procedures have been re-cently proposed and demonstrated [5, 6, 8, 10]. Here we ratherconsider the unitary de-Gaussification evolution provided byself-Kerr interaction U γ = exp {− iγ ( a † a ) } [30, 31], whichdoes not correspond to a symplectic transformation and leadsto a nonG state even if applied to a Gaussian state. We haveevaluated the nonG of the state obtained from a coherent state | α i undergoing Kerr interaction. Results are reported in Fig. 2as a function of the average number of photons (up to pho-tons) and for different values of the coupling constant γ . As itis apparent from the plot, nonG is an increasing function of thenumber of photons and the Kerr coupling γ . For γ ≈ − ,the maximum nonG achievable at fixed energy is quite rapidlyachieved. For more realistic values of the nonlinear coupling, i.e. γ ≤ − nonG states may be obtained only for a largeaverage number of photons in the output state. In fact, to ob-tain entanglement, experimental realizations [30, 31] involvepulses with an average number of the order of photons,which are needed to compensate the almost vanishing smallKerr nonlinearities of standard glass fiber. We finally no- n5101520 ∆ FIG. 2: NonG of coherent states undergoing Kerr interaction as afunction of the average number of photons and for different valuesof the coupling constant γ . Black dashed line: γ = 10 − ; blackdotted: γ = 10 − ; solid black: γ = 10 − . The gray solid lines isthe maximum nonG at fixed number of photons. tice that a good measure for the nonG character of quantumstates allows us to define a measure of the nonG character ofa quantum operations. Let us denote by G the whole set ofGaussian states. A convenient definition for the nonG of of amap E reads as δ [ E ] = max ̺ ∈G δ [ E ( ̺ )] , where E ( ̺ ) denotesthe quantum state obtained after the evolution imposed by themap. Conclusions —We have introduced a novel measure to quantify the nonG character of a CV quantum state based onquantum relative entropy. We have analyzed in details theproperties owned by this measure and its relation with somerelevant quantities in quantum information. In particular, anecessary condition for the Gaussian character of a quantumchannel and a sufficient condition for entanglement of bipar-tite quantum states can be derived. Our measure is easilycomputable for any CV state and allows to assess nonG asa resource for quantum technology. In turn, we exploited ourmeasure to evaluate the performances of conditional Gaussi-fication towards twin-beam and de-Gaussification processesdriven by Kerr interaction.We thank M. Horodecki for discussions on conditional en-tropy in the CV regime. 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