Energy-constrained diamond norms and their use in quantum information theory
aa r X i v : . [ qu a n t - ph ] J un Energy-constrained diamond norms and theiruse in quantum information theory
M.E. Shirokov ∗ Abstract
We consider the family of energy-constrained diamond norms onthe set of Hermitian-preserving linear maps (superoperators) betweenBanach spaces of trace class operators. We prove that any norm fromthis family generates the strong (pointwise) convergence on the set ofall quantum channels (which is more adequate for describing variationsof infinite-dimensional channels than the diamond norm topology).We obtain continuity bounds for information characteristics (inparticular, classical capacities) of energy-constrained quantum chan-nels (as functions of a channel) with respect to the energy-constraineddiamond norms which imply uniform continuity of these characteris-tics with respect to the strong convergence topology.
Contents ∗ Steklov Mathematical Institute, RAS, Moscow, email:[email protected] Continuity bounds for information characteristics of quan-tum channels with respect to the ECD-norms 8
The diamond-norm distance between two quantum channels is widely usedas a measure of distinguishability between these channels [21, Ch.9]. Butthe topology (convergence) generated by the diamond-norm distance on theset of infinite-dimensional quantum channels is too strong for analysis of realvariations of such channels. Indeed, for any sequence of unitary operators U n strongly converging to the unit operator I but not converging to I in theoperator norm the sequence of channels ρ U n ρU ∗ n does not converge to theidentity channel with respect to the diamond-norm distance. In general, thecloseness of two quantum channels in the diamond-norm distance means, byTheorem 1 in [11], the operator norm closeness of the corresponding Stine-spring isometries. So, if we use the diamond-norm distance then we take intoaccount only such perturbations of a channel that corresponds to uniform de-formations of the Stinespring isometry (i.e. deformations with small operatornorm). As a result, there exist quantum channels with close physical parame-ters (quantum limited attenuators) having the diamond-norm distance equalto 2 [25].To take into account deformations of the Stinespring isometry in thestrong operator topology one can consider the strong convergence topology on the set of all quantum channels defined by the family of seminormsΦ
7→ k Φ( ρ ) k , ρ ∈ S ( H A ) [9]. The convergence of a sequence of channelsΦ n to a channel Φ in this topology means thatlim n →∞ Φ n ( ρ ) = Φ ( ρ ) for all ρ ∈ S ( H A ) . In this paper we show that the strong convergence topology on the set ofquantum channels is generated by any of the energy-constrained diamondnorms on the set of Hermitian-preserving linear maps (superoperators) be-tween Banach spaces of trace class operators (provided the input Hamiltoniansatisfies the particular condition). 2he energy-constrained diamond norms turn out to be effective toolfor quantitative continuity analysis of information characteristics of energy-constrained quantum channels (as functions of a channel). They can be usedinstead of the diamond norm in standard argumentations, in particular, inthe Leung-Smith telescopic method used in [12] for continuity analysis ofcapacities of finite-dimensional channels.We obtain continuity bounds for information characteristics (in partic-ular, classical capacities) of energy-constrained quantum channels with re-spect to the energy-constrained diamond norms. They imply, by the above-mentioned correspondence between these norms and the strong convergence,the uniform continuity of these characteristics with respect to the strongconvergence topology.
Let H be a separable infinite-dimensional Hilbert space, B ( H ) the algebraof all bounded operators with the operator norm k · k and T ( H ) the Banachspace of all trace-class operators in H with the trace norm k·k . Let S ( H )be the set of quantum states (positive operators in T ( H ) with unit trace)[6, 21].Denote by I H the unit operator in a Hilbert space H and by Id H theidentity transformation of the Banach space T ( H ).If quantum systems A and B are described by Hilbert spaces H A and H B then the bipartite system AB is described by the tensor product of thesespaces, i.e. H AB . = H A ⊗H B . A state in S ( H AB ) is denoted ρ AB , its marginalstates Tr H B ρ AB and Tr H A ρ AB are denoted respectively ρ A and ρ B .The von Neumann entropy H ( ρ ) = Tr η ( ρ ) of a state ρ ∈ S ( H ), where η ( x ) = − x log x , is a concave lower semicontinuous function on S ( H ) takingvalues in [0 , + ∞ ] [14, 20]. We will use the binary entropy h ( x ) = η ( x ) + η (1 − x ) and the function g ( x ) = (1 + x ) h (cid:0) x x (cid:1) = ( x + 1) log( x + 1) − x log x .The quantum relative entropy for two states ρ and σ is defined as H ( ρ k σ ) = X i h i | ρ log ρ − ρ log σ | i i , where {| i i} is the orthonormal basis of eigenvectors of the state ρ and it isassumed that H ( ρ k σ ) = + ∞ if supp ρ is not contained in supp σ [14, 20].3he quantum mutual information (QMI) of a state ρ AB is defined as I ( A : B ) ρ = H ( ρ AB k ρ A ⊗ ρ B ) = H ( ρ A ) + H ( ρ B ) − H ( ρ AB ) , (1)where the second expression is valid if H ( ρ AB ) is finite [13]. Basic propertiesof the relative entropy show that ρ I ( A : B ) ρ is a lower semicontinuousfunction on the set S ( H AB ) taking values in [0 , + ∞ ].A quantum channel Φ from a system A to a system B is a completelypositive trace preserving superoperator T ( H A ) → T ( H B ), where H A and H B are Hilbert spaces associated with these systems [6, 21].Denote by F ( A, B ) the set of all quantum channels from A to B . Thereare different nonequivalent metrics on F ( A, B ). One of them is induced bythe diamond norm k Φ k ⋄ . = sup ρ ∈ S ( H AR ) k Φ ⊗ Id R ( ρ ) k of a Hermitian-preserving superoperator Φ : T ( H A ) → T ( H B ) [1]. Thelatter coincides with the norm of complete boundedness of the dual mapΦ ∗ : B ( H B ) → B ( H A ) to Φ [15]. The strong convergence topology on the set F ( A, B ) of quantum channelsfrom A to B is generated by the strong operator topology on the set ofall linear bounded operators from the Banach space T ( H A ) into the Ba-nach space T ( H B ) [16]. This topology is studied in detail in [9] where it isused for approximation of infinite-dimensional quantum channels by finite-dimensional ones. Separability of the set S ( H A ) implies that the strongconvergence topology on the set F ( A, B ) is metrisable (can be defined bysome metric). The convergence of a sequence { Φ n } of channels to a channelΦ in this topology means thatlim n →∞ Φ n ( ρ ) = Φ ( ρ ) for all ρ ∈ S ( H A ) . We will use the following simple observations easily proved by usingboundedness of the operator norm of all quantum channels.4 emma 1. [9]
The strong convergence topology on F ( A, B ) coincideswith the topology of uniform convergence on compact subsets of S ( H A ) . Lemma 2. [9]
The strong convergence of a sequence { Φ n } ⊂ F ( A, B ) toa channel Φ implies strong convergence of the sequence { Φ n ⊗ Id R } to thechannel Φ ⊗ Id R , where R is any system. It is the strong convergence topology that makes the set F ( A, B ) of allchannels topologically isomorphic to a subset of states of a composite system(generalized Choi-Jamiolkowski isomorphism).
Proposition 1. [9]
Let R ∼ = A and ω be a pure state in S ( H AR ) suchthat ω A is a full rank state in S ( H A ) . Then the map Φ Φ ⊗ Id R ( ω ) is a homeomorphism from the set F ( A, B ) equipped with the strong conver-gence topology onto the subset { ρ ∈ S ( H BR ) | ρ R = ω R } . If both systems A and B are infinite-dimensional then the set F ( A, B )is not compact in the strong convergence topology. Proposition 1 impliesthe following compactness criterion for subsets of quantum channels in thistopology.
Proposition 2. [9]
A subset F ⊆ F ( A, B ) is relatively compact in thestrong convergence topology if and only if there exists a full rank state σ in S ( H A ) such that { Φ( σ ) } Φ ∈ F is a relatively compact subset of S ( H B ) . Note that the ”only if” part of Proposition 2 is trivial, since the relativecompactness of F implies relative compactness of { Φ( σ ) } Φ ∈ F for arbitrarystate σ in S ( H A ) by continuity of the map Φ Φ( σ ).Proposition 2 makes it possible to establish existence of a channel withrequired properties as a limit point of a sequence of explicitly constructedchannels by proving relative compactness of this sequence. For example, bythis way one can generalize the famous Petz theorem for two non-faithful(degenerate) infinite rank states starting from the standard version of thistheorem (in which both states are assumed faithful) [18, the Appendix]. Let H A be a positive (unbounded) operator in H A with dense domain treatedas a Hamiltonian of quantum system A . Then Tr H A ρ is the (mean) energy5f a state ρ ∈ S ( H A ). Consider the family of norms k Φ k E ⋄ . = sup ρ ∈ S ( H AR ) , Tr H A ρ A ≤ E k Φ ⊗ Id R ( ρ ) k , E > E A , (2)on the set L ( A, B ) of Hermitian-preserving superoperators from T ( H A ) to T ( H B ), where E A is the infimum of the spectrum of H A and R is an infinite-dimensional quantum system. They can be called energy-constrained dia-mond norms (briefly, ECD-norms) . It is clear that formula (2) defines aseminorm on L ( A, B ), the implication k Φ k E ⋄ = 0 ⇒ Φ = 0 can be easilyshown by the ”convex mixture” arguments from the proof of part A of thefollowing
Proposition 3.
Let F ( A, B ) be the set of all channels from A to B . A) The convergence of a sequence { Φ n } of channels in F ( A, B ) to a chan-nel Φ with respect to any of the norms in (2) implies the strong convergenceof { Φ n } to Φ , i.e. for any E > E A the following implication holds lim n → + ∞ k Φ n − Φ k E ⋄ = 0 ⇒ lim n → + ∞ Φ n ( ρ ) = Φ ( ρ ) ∀ ρ ∈ S ( H A ) . (3)B) If the operator H A has discrete spectrum { E k } k ≥ of finite multiplicitysuch that E k → + ∞ as k → + ∞ then ” ⇔ ” holds in (3) for any E > E A . Remark 1.
The assertions of Proposition 3 remain valid for any operator-norm bounded subset of L ( A, B ) (instead of F ( A, B )). It follows, in partic-ular, that the strong operator topology on such subsets is generated by thesingle norm k · k E ⋄ provided that the corresponding operator H A satisfies thecondition of part B. Proof.
A) The assumed density of the domain of H A in H A implies densityof the set S of states ρ with finite energy Tr H A ρ in S ( H A ). Hence, sincethe operator norm of all the superoperators Φ n − Φ is bounded, to provethe implication in (3) it suffices to show that lim n → + ∞ Φ n ( ρ ) = Φ ( ρ ) forany ρ ∈ S provided that lim n → + ∞ k Φ n − Φ k E ⋄ = 0. The value of Tr H A ρ (finite or infinite) is defined as sup n Tr P n H A ρ , where P n is thespectral projector of H A corresponding to the interval [0 , n ]. I am grateful to A.Winter who pointed me that formula (2) defines a real norm, seethe Note Added at the end of the paper. It means that the topology generated on the set F ( A, B ) by any of the norms in (2)coincides with the strong convergence topology on F ( A, B ). ρ be state in S and σ a state such that Tr H A σ < E . Then forsufficiently small p > ρ p . = (1 − p ) σ + pρ does notexceed E . Hence the left part of (3) implieslim n → + ∞ Φ n ( ρ p ) = Φ ( ρ p ) and lim n → + ∞ Φ n ( σ ) = Φ ( σ ) . It follows that Φ n ( ρ ) tends to Φ ( ρ ) as n → + ∞ .B) By the assumption H A = P + ∞ k =0 E Ak | τ k ih τ k | , where {| τ k i} + ∞ k =0 is theorthonormal basis of eigenvectors of H A corresponding to the nondecreasingsequence { E Ak } + ∞ k =0 of eigenvalues tending to + ∞ . Let P n = P n − k =0 | τ k ih τ k | be the projector on the subspace H n spanned by the vectors | τ i , ..., | τ n − i .Consider the family of seminorms q n (Φ) . = sup ρ ∈ S ( H n ⊗H R ) k Φ ⊗ Id R ( ρ ) k , n ∈ N , (4)on the set of Hermitian-preserving superoperators from T ( H A ) to T ( H B ).Note that the system R in (4) may be n -dimensional. Indeed, by convexityof the trace norm the supremum in (4) can be taken over pure states ρ in S ( H n ⊗ H R ). Since the marginal state ρ R of any such pure state ρ has rank ≤ n , by applying local unitary transformation of the system R we can putall these states into the set S ( H n ⊗ H ′ n ), where H ′ n is any n -dimensionalsubspace of H R .Let { Φ k } be a sequence of channels strongly converging to a channel Φ .By Lemmas 1 and 2 this implies that sup ρ ∈ C k (Φ k − Φ ) ⊗ Id R ( ρ ) k tendsto zero for any compact subset C of S ( H AR ). Since dim H R = n , the set S ( H n ⊗ H R ) is compact and we conclude thatlim k →∞ q n (Φ k − Φ ) = 0 ∀ n ∈ N . (5)Let E ≥ E A and ρ be a state in S ( H AR ) such that Tr H A ρ A ≤ E . Thenthe state ρ n = (1 − r n ) − P n ⊗ I R ρ P n ⊗ I R , where r n = Tr( I A − P n ) ρ A ,belongs to the set S ( H n ⊗ H R ). By using the inequality k ( I A − P n ) ⊗ I R ρ P n ⊗ I R k ≤ p Tr( I A − P n ) ⊗ I R ρ = √ r n easily proved via the operator Cauchy-Schwarz inequality (see the proof ofLemma 11.1 in [6]) and by noting that Tr H A ρ A ≤ E implies r n ≤ E/E An we obtain k ρ − ρ n k ≤ k ( I A − P n ) ⊗ I R ρ P n ⊗ I R k + 2 r n ≤ √ r n ≤ p E/E An .
7t follows that k Φ k − Φ k E ⋄ ≤ q n (Φ k − Φ ) + 8 p E/E An . Since E An → + ∞ as n → + ∞ , this inequality and (5) show that lim k →∞ k Φ k − Φ k E ⋄ = 0. (cid:3) In this section we show that the ECD-norms can be effectively used for quan-titative continuity analysis of information characteristics of energy-constrainedquantum channels (as functions of a channel). They allow (due to Proposi-tion 3) to prove uniform continuity of these characteristics with respect tothe strong convergence of quantum channels.In what follows we will consider quantum channels between given infinite-dimensional systems A and B . We will assume that the Hamiltonians H A and H B of these systems are densely defined positive operators and thatTr e − λH B < + ∞ for all λ > . (6)To formulate our main results introduce the function F H B ( E ) . = sup Tr H B ρ ≤ E H ( ρ ) , E ≥ E B , (7)where E B is the minimal eigenvalue of H B . Properties of this function aredescribed in Proposition 1 in [17], where it is shown, in particular, that F H B ( E ) = λ ( E ) E + log Tr e − λ ( E ) H B = o ( E ) as E → + ∞ , (8)where λ ( E ) is determined by the equality Tr H B e − λ ( E ) H B = E Tr e − λ ( E ) H B ,provided that condition (6) holds.It is well known that condition (6) implies continuity of the von Neumannentropy on the subset of S ( H B ) determined by the inequality Tr H B ρ ≤ E for any E ≥ E B and attainability of the supremum in (7) at the Gibbs state γ B ( E ) . = e − λ ( E ) H B / Tr e − λ ( E ) H B [20]. So, we have F H B ( E ) = H ( γ B ( E )) forany E > E B .The function F H B is increasing and concave on [ E B , + ∞ ). Denote by b F H B any upper bound for F H B defined on [0 , + ∞ ) possessing the properties b F H B ( E ) > , b F ′ H B ( E ) > , b F ′′ H B ( E ) < E > b F H B ( E ) = o ( E ) as E → + ∞ . (10)At least one such function b F H B always exists. It follows from (8) that onecan use F H B ( E + E B ) in the role of b F H B ( E ).If B is the ℓ -mode quantum oscillator with the Hamiltonian H B = ℓ X i =1 ~ ω i (cid:0) a + i a i + I B (cid:1) , where a i and a + i are the annihilation and creation operators and ω i is thefrequency of the i -th oscillator [6, Ch.12] then one can show that F H B ( E ) ≤ b F ℓ,ω ( E ) . = ℓ log E + E ℓE ∗ + ℓ, (11)where E . = P ℓi =1 ~ ω i and E ∗ = hQ ℓi =1 ~ ω i i /ℓ and that upper bound (11)is ε -sharp for large E [19]. It is easy to see that the function b F ℓ,ω possessesproperties (9) and (10). So, it can be used in the role of b F H B in this case. In this subsection we obtain continuity bound for the function Φ χ (Φ( µ ))with respect to the ECD-norm, where χ (Φ( µ )) is the Holevo quantity of theimage of a given (discrete or continuous) ensemble µ of input states underthe channel Φ.Discrete ensemble µ = { p i , ρ i } is a finite or countable collection { ρ i } ofquantum states with the corresponding probability distribution { p i } . TheHolevo quantity of µ is defined as χ ( µ ) . = X i p i H ( ρ i k ¯ ρ ( µ )) = H ( ¯ ρ ( µ )) − X i p i H ( ρ i ) , where ¯ ρ ( µ ) = P i p i ρ i is the average state of µ and the second formula isvalid if H ( ¯ ρ ( µ )) < + ∞ . This quantity gives the upper bound for classicalinformation obtained by recognizing states of the ensemble by quantum mea-surements [5]. It plays important role in analysis of information propertiesof quantum systems and channels [6, 21]. We use the natural logarithm. µ is the barycenter of µ defined by theBochner integral ¯ ρ ( µ ) = Z ρµ ( dρ ) . The Holevo quantity of a generalized ensemble µ is defined as χ ( µ ) = Z H ( ρ k ¯ ρ ( µ )) µ ( dρ ) = H ( ¯ ρ ( µ )) − Z H ( ρ ) µ ( dρ ) , where the second formula is valid under the condition H ( ¯ ρ ( µ )) < + ∞ [8].The average energy of µ is given by the formula E ( µ ) = Tr H ¯ ρ ( µ ) = Z Tr Hρ µ ( dρ ) , where H is the Hamiltonian of the system (the integral is well defined, sincethe function ρ Tr Hρ is lower semicontinuous). For a discrete ensemble µ = { p i , ρ i } we have E ( µ ) = P i p i Tr Hρ i .For an ensemble µ of states in S ( H A ) its image Φ( µ ) under a quantumchannel Φ : A → B is defined as the ensemble corresponding to the measure µ ◦ Φ − on S ( H B ), i.e. Φ( µ )[ S B ] = µ [Φ − ( S B )] for any Borel subset S B of S ( H B ), where Φ − ( S B ) is the pre-image of S B under the map Φ. If µ = { p i , ρ i } then this definition implies Φ( µ ) = { p i , Φ( ρ i ) } . Proposition 4.
Let µ be a generalized ensemble of states in S ( H A ) withfinite average energy E ( µ ) . Let Φ and Ψ be channels from A to B such that Tr H B Φ( ¯ ρ ( µ )) , Tr H B Ψ( ¯ ρ ( µ )) ≤ E and k Φ − Ψ k E ( µ ) ⋄ ≤ ε . If the Hamiltonian H B satisfies conditions (6) then | χ (Φ( µ )) − χ (Ψ( µ )) | ≤ ε (2 t + r ε ( t )) b F H B (cid:18) Eεt (cid:19) + 2 g ( εr ε ( t )) + 2 h ( εt ) (12) for any t ∈ (0 , ε ] , where b F H B ( E ) is any upper bound for the function F H B ( E ) (defined in (7)) with properties (9) and (10), in particular b F H B ( E ) = F H B ( E + E B ) and r ε ( t ) = (1 + t/ / (1 − εt ) . The functions h ( x ) and g ( x ) are defined in Section 2. f B is the ℓ -mode quantum oscillator then | χ (Φ( µ )) − χ (Ψ( µ )) | ≤ ε (2 t + r ε ( t )) h b F ℓ,ω ( E ) − ℓ log( εt ) i + 2 g ( εr ε ( t )) + 2 h ( εt ) (13) for any t ∈ (0 , ε ] , where b F ℓ,ω ( E ) is defined in (11). Continuity bound (13)with optimal t is tight for large E . Remark 2.
Condition (10) implies that lim x → +0 x b F H B ( E/x ) = 0 . Hence,the right hand side of (12) tends to zero as ε → Remark 3.
It is easy to see that the right hand side of (12) attainsminimum at some optimal t = t ( E, ε ). It is this minimum that gives properupper bound for | χ (Φ( µ )) − χ (Ψ( µ )) | . Proof.
Let µ = { p i , ρ i } mi =1 be a discrete ensemble of m ≤ + ∞ statesand ˆ ρ AC = P mi =1 p i ρ i ⊗ | i ih i | the corresponding cq -state (here {| i i} mi =1 is anorthonormal basis in a m -dimensional Hilbert space H C ). Then Tr H A ˆ ρ A = E ( µ ) and hence k Φ ⊗ Id C ( ˆ ρ AC ) − Ψ ⊗ Id C ( ˆ ρ AC ) k ≤ k Φ − Ψ k E ( µ ) ⋄ . (14)So, in this case the assertions of the proposition follow from Proposition 7 in[19], since the left hand side of (14) coincides with2 D ( { p i , Φ( ρ i ) } , { p i , Ψ( ρ i ) } ) . = m X i =1 p i k Φ( ρ i ) − Ψ( ρ i ) k . Let µ be an arbitrary generalized ensemble. The construction from theproof of Lemma 1 in [8] gives the sequence { µ n } of discrete ensembles weakly converging to µ such that ¯ ρ ( µ n ) = ¯ ρ ( µ ) for all n . Since the assumptionTr H B Φ( ¯ ρ ( µ )) , Tr H B Ψ( ¯ ρ ( µ )) ≤ E implies H (Φ( ¯ ρ ( µ ))) , H (Ψ( ¯ ρ ( µ ))) < + ∞ ,Corollary 1 in [9] shows thatlim n →∞ χ (Φ( µ n )) = χ (Φ( µ )) and lim n →∞ χ (Ψ( µ n )) = χ (Ψ( µ )) . The weak convergence of a sequence { µ n } to a measure µ means thatlim n →∞ R f ( ρ ) µ n ( dρ ) = R f ( ρ ) µ ( dρ ) for any continuous bounded function f on S ( H ) [2]. µ follows from the validityof this inequality for all the discrete ensembles µ n proved before. Inequality(13) is a direct corollary of (12).The tightness of continuity bound (13) follows from the tightness of con-tinuity bound (22) for the Holevo capacity in this case (obtained from (13)).By Remark 2 Propositons 3 and 4 imply the following Corollary 1.
If the Hamiltonians H A and H B satisfy, respectively,the condition of Proposition 3B and condition (6) then for any general-ized ensemble µ of states in S ( H A ) with finite average energy the function Φ χ (Φ( µ )) is uniformly continuous on the sets F µ,E . = { Φ ∈ F ( A, B ) | Tr H B Φ( ¯ ρ ( µ )) ≤ E } , E > E B , with respect to the strong convergence topology. Remark 4.
The uniform continuity with respect to the strong conver-gence topology means uniform continuity with respect to any metric gener-ating this topology, in particular, with respect to any of the ECD-norms.
The following proposition is a corollary of Proposition 3B in [19] proved bythe Leung-Smith telescopic trick used in [12] and Winter’s technique from[23]. It gives tight continuity bounds for the function Φ I ( B n : C ) Φ ⊗ n ⊗ Id C ( ρ ) for any given n and a state ρ ∈ S ( H ⊗ nA ⊗ H C ) with respect to the ECD-normprovided that the marginal states ρ A , ..., ρ A n have finite energy. Proposition 5.
Let Φ and Ψ be channels from A to B , C be any systemand ρ a state in S ( H ⊗ nA ⊗ H C ) such that E A . = max ≤ k ≤ n { Tr H A ρ A k } isfinite. Let k Φ − Ψ k E A ⋄ ≤ ε , Tr H B Φ( ρ A k ) , Tr H B Ψ( ρ A k ) ≤ E k for k = 1 , n and ∆ n (Φ , Ψ , ρ ) . = (cid:12)(cid:12) I ( B n : C ) Φ ⊗ n ⊗ Id C ( ρ ) − I ( B n : C ) Ψ ⊗ n ⊗ Id C ( ρ ) (cid:12)(cid:12) . If the Hamiltonian H B satisfies conditions (6) then ∆ n (Φ , Ψ , ρ ) ≤ nε (2 t + r ε ( t )) b F H B (cid:18) Eεt (cid:19) + 2 ng ( εr ε ( t )) + 4 nh ( εt ) (15) for any t ∈ (0 , ε ] , where E = n − P nk =1 E k , b F H B ( E ) is any upper boundfor the function F H B ( E ) (defined in (7)) with properties (9) and (10), inparticular, b F H B ( E ) = F H B ( E + E ) and r ε ( t ) = (1 + t/ / (1 − εt ) . The functions h ( x ) and g ( x ) are defined in Section 2. f B is the ℓ -mode quantum oscillator then ∆ n (Φ , Ψ , ρ ) ≤ nε (2 t + r ε ( t )) h b F ℓ,ω ( E ) − ℓ log( εt ) i + 2 ng ( εr ε ( t )) + 4 nh ( εt ) , (16) where b F ℓ,ω ( E ) is defined in (11). Continuity bound (16) with optimal t istight for large E (for any given n ). Remark 5.
All the assertions of Proposition 5 remain valid for thequantum conditional mutual information (see Proposition 3B in [19]).Since condition (10) implies lim x → +0 x b F H B ( E/x ) = 0 , the right hand side of(15) tends to zero as ε →
0. Hence Propositons 3 and 5 imply the following
Corollary 2.
If the Hamiltonians H A and H B satisfy, respectively, thecondition of Proposition 3B and condition (6) then for any n ∈ N and anystate ρ in S ( H ⊗ nA ⊗ H C ) such that max ≤ k ≤ n { Tr H A ρ A k } < + ∞ the function Φ I ( B n : C ) Φ ⊗ n ⊗ Id C ( ρ ) is uniformly continuous on the sets F ρ,E . = ( Φ ∈ F ( A, B ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n n X k =1 Tr H B Φ( ρ A k ) ≤ E ) , E > E B , with respect to the strong convergence topology. When we consider transmission of classical information over infinite dimen-sional quantum channels we have to impose the energy constraint on statesused for coding information. For a single channel Φ : A → B the energyconstraint is determined by the linear inequalityTr H A ρ ≤ E, where H A is the Hamiltonian of the input system A . For n copies of thischannel the energy constraint is given by the inequalityTr H A n ρ ( n ) ≤ nE, (17) see Remark 4. ρ ( n ) is a state of the system A n ( n copies of A ) and H A n = H A ⊗ · · · ⊗ I + · · · + I ⊗ · · · ⊗ H A (18)is the Hamiltonian of the system A n [6, 7, 22].An operational definition of the classical capacity of an infinite-dimensionalenergy-constrained quantum channel can be found in [7]. If only nonentan-gled input encoding is used then the ultimate rate of transmission of classicalinformation trough the channel Φ with the constraint (17) on mean energyof a code is determined by the Holevo capacity C χ (Φ , H A , E ) = sup Tr H A ¯ ρ ≤ E χ ( { p i , Φ( ρ i ) } ) , ¯ ρ = X i p i ρ i , (19)(the supremum is over all input ensembles { p i , ρ i } such that Tr H A ¯ ρ ≤ E ).By the Holevo-Schumacher-Westmoreland theorem adapted to constrainedchannels ([7, Proposition 3]), the classical capacity of the channel Φ withconstraint (17) is given by the following regularized expression C (Φ , H A , E ) = lim n → + ∞ n − C χ (Φ ⊗ n , H A n , nE ) , where H A n is defined in (18). If C χ (Φ ⊗ n , H A n , nE ) = nC χ (Φ , H A , E ) for all n then C (Φ , H A , E ) = C χ (Φ , H A , E ) , (20)i.e. the classical capacity of the channel Φ coincides with its Holevo capacity.Note that (20) holds for many infinite-dimensional channels [6]. Recently itwas shown that (20) holds if Φ is a gauge covariant or contravariant Gaussianchannel and H A = P ij ǫ ij a † i a j is a gauge invariant Hamiltonian (here [ ǫ ij ] isa positive matrix) [3, 4].The following proposition presents estimates for differences between theHolevo capacities and between the classical capacities of channels Φ and Ψwith finite energy amplification factors for given input energy , i.e. such thatsup Tr H A ρ ≤ E H B Φ( ρ ) ≤ kE and sup Tr H A ρ ≤ E H B Ψ( ρ ) ≤ kE (21)for given E ≥ E A and finite k = k ( E ). Note that any channels produced ina physical experiment satisfy condition (21). The gauge invariance condition for H A can be replaced by the condition (18) in [4]. roposition 6. Let Φ and Ψ be quantum channels from A to B satis-fying condition (21) and k Φ − Ψ k E ⋄ ≤ ε . If the Hamiltonian H B satisfiesconditions (6) then | C χ (Φ , H A , E ) − C χ (Ψ , H A , E ) | ≤ ε (2 t + r ε ( t )) b F H B (cid:0) kEεt (cid:1) + 2 g ( εr ε ( t )) + 2 h ( εt ) (22) and | C (Φ , H A , E ) − C (Ψ , H A , E ) | ≤ ε (2 t + r ε ( t )) b F H B (cid:0) kEεt (cid:1) + 2 g ( εr ε ( t )) + 4 h ( εt ) (23) for any t ∈ (0 , ε ] , where r ε ( t ) = (1 + t/ / (1 − εt ) and b F H B ( E ) is any upperbound for the function F H B ( E ) (defined in (7)) with properties (9) and (10),in particular, b F H B ( E ) = F H B ( E + E B ) . If B is the ℓ -mode quantum oscillator then the right hand sides of (22)and (23) can be rewritten, respectively, as follows ε (2 t + r ε ( t )) h b F ℓ,ω ( kE ) − ℓ log( εt ) i + 2 g ( εr ε ( t )) + 2 h ( εt ) and ε (2 t + r ε ( t )) h b F ℓ,ω ( kE ) − ℓ log( εt ) i + 2 g ( εr ε ( t )) + 4 h ( εt ) , where b F ℓ,ω ( E ) is defined in (11). In this case continuity bound (22) withoptimal t is tight for large E while continuity bound (23) is close-to-tight(up to the factor in the main term).Proof. Inequality (22) follows from definition (19) and Proposition 4.To prove inequality (23) note that C χ (Φ ⊗ n , H A n , nE ) = sup χ ( { p i , Φ ⊗ n ( ρ i ) } ) , where the supremum is over all ensembles { p i , ρ i } of states in S ( H ⊗ nA ) withthe average state ¯ ρ such that Tr H A ¯ ρ A j ≤ E for all j = 1 , n . This can beeasily shown by using symmetry arguments and the following well knownproperty of the Holevo quantity:1 n n X j =1 χ (cid:0) { q ji , σ ji } i (cid:1) ≤ χ ( q ji n , σ ji ) ij The functions h ( x ) and g ( x ) are defined in Section 2. { q i , σ i } , ..., { q ni , σ ni } of discrete ensembles.Since condition (21) impliesTr H B Φ( ¯ ρ A j ) ≤ kE and Tr H B Ψ( ¯ ρ A j ) ≤ kE, j = 1 , n, for any ensemble { p i , ρ i } satisfying the above condition, continuity bound(23) is obtained by using the representations χ ( { p i , Λ ⊗ n ( ρ i ) } ) = I ( B n : C ) Λ ⊗ n ⊗ Id C (ˆ ρ ) , Λ = Φ , Ψ , where ˆ ρ AC = X i p i ρ i ⊗| i ih i | , Proposition 5 and the corresponding analog of Lemma 12 in [12].If B is the ℓ -mode quantum oscillator and b F H B = b F ℓ,ω then we can esti-mate the right hand sides of (22) and (23) from above by using the inequality b F ℓ,ω ( E/x ) ≤ b F ℓ,ω ( E ) − ℓ log x valid for any positive E and x ≤ ℓ -mode quantum oscillator A to B = A and Ψis the completely depolarizing channel with the vacuum output state. Thesechannels satisfy condition (21) with k = 1 and | C χ (Φ , H A , E ) − C χ (Ψ , H A , E ) | = sup Tr H A ρ ≤ E H ( ρ ) = F H A ( E ) . (24)By Lemma 2 in [19] in the case b F H B = b F ℓ,ω the main term of (22) can bemade not greater than ε [ b F ℓ,ω ( E ) + o ( b F ℓ,ω ( E ))] for large E by appropriatechoice of t . So, the tightness of continuity bound (22) follows from (24),since k Φ − Ψ k E ⋄ ≤ k Φ − Ψ k ⋄ = 2 and lim E → + ∞ ( b F ℓ,ω ( E ) − F H A ( E )) = 0.The above example also shows that the main term in continuity bound(23) is close to the optimal one up to the factor 2, since C (Φ , H A , E ) = C χ (Φ , H A , E ) and C (Ψ , H A , E ) = C χ (Ψ , H A , E ) for the channels Φ and Ψ. (cid:3) The operational definition of the entanglement-assisted classical capacityof an infinite-dimensional energy-constrained quantum channel is given in [7,10]. By the Bennett-Shor-Smolin-Thaplyal theorem adapted to constrainedchannels ([10, Theorem 1]) the entanglement-assisted classical capacity of aninfinite-dimensional channel Φ with the energy constraint (17) is given bythe expression C ea (Φ , H A , E ) = sup Tr H A ρ ≤ E I (Φ , ρ ) , I (Φ , ρ ) = I ( B : R ) Φ ⊗ Id R (ˆ ρ ) , ˆ ρ A = ρ, rank ˆ ρ = 1, is the quantum mutualinformation of the channel Φ at a state ρ . Proposition 5 with n = 1 impliesthe following Proposition 7.
Let Φ and Ψ be quantum channels from A to B and k Φ − Ψ k E ⋄ ≤ ε , where k . k E ⋄ is the ECD-norm defined in (2). A) If the Hamiltonian H A satisfies condition (6) then | C ea (Φ , H A , E ) − C ea (Ψ , H A , E ) | ≤ ε (2 t + r ε ( t )) b F H A (cid:0) Eεt (cid:1) + 2 g ( εr ε ( t )) + 4 h ( εt ) (25) for any t ∈ (0 , ε ] , where r ε ( t ) = (1 + t/ / (1 − εt ) and b F H A ( E ) is any upperbound for the function F H A ( E ) (defined in (7)) with properties (9) and (10),in particular, b F H A ( E ) = F H A ( E + E A ) .If A is the ℓ -mode quantum oscillator then the right hand side of (25)can be rewritten as follows ε (2 t + r ε ( t )) h b F ℓ,ω ( E ) − ℓ log( εt ) i + 2 g ( εr ε ( t )) + 4 h ( εt ) , where b F ℓ,ω ( E ) is defined in (11). In this case continuity bound (25) withoptimal t is tight for large E . B) If the channels Φ and Ψ satisfy condition (21) and the Hamiltonian H B satisfies condition (6) then | C ea (Φ , H A , E ) − C ea (Ψ , H A , E ) | ≤ ε (2 t + r ε ( t )) b F H B (cid:0) kEεt (cid:1) + 2 g ( εr ε ( t )) + 4 h ( εt ) (26) for any t ∈ (0 , ε ] , where b F H B ( E ) is any upper bound for the function F H B ( E ) with properties (9) and (10), in particular, b F H B ( E ) = F H B ( E + E B ) .If B is the ℓ -mode quantum oscillator then the right hand side of (26)can be rewritten as follows ε (2 t + r ε ( t )) h b F ℓ,ω ( kE ) − ℓ log( εt ) i + 2 g ( εr ε ( t )) + 4 h ( εt ) , where b F ℓ,ω ( E ) is defined in (11). In this case continuity bound (26) withoptimal t is tight for large E . arbitrary channels Φ and Ψ. Proof.
A) Let H R ∼ = H A and H R be an operator in H R unitarily equivalentto H A . For any state ρ satisfying the condition Tr H A ρ ≤ E there exists apurification ˆ ρ ∈ S ( H AR ) such that Tr H R ˆ ρ R ≤ E . Since I (Φ , ρ ) = I ( B : R ) σ and I (Ψ , ρ ) = I ( B : R ) ς , where σ = Φ ⊗ Id R ( ˆ ρ ) and ς = Ψ ⊗ Id R ( ˆ ρ ) are states in S ( H BR ) such thatTr H R σ R , Tr H R ς R ≤ E and k σ − ς k ≤ k Φ − Ψ k E ⋄ , Proposition 2 in [19] withtrivial C shows that the value of | I (Φ , ρ ) − I (Ψ , ρ ) | is upper bounded by theright hand side of (25).B) Continuity bound (26) is obtained similarly, since in this case we haveTr H B σ B , Tr H B ς B ≤ kE .The specifications concerning the cases when either A or B is the ℓ -modequantum oscillator follow from the inequality b F ℓ,ω ( E/x ) ≤ b F ℓ,ω ( E ) − ℓ log x valid for any positive E and x ≤ ℓ -mode quantum oscillator A to B = A and Ψ is the completely depolarizing channel with the vac-uum output state. It suffices to note that C ea (Φ , H A , E ) = 2 F H A ( E ) and C ea (Ψ , H A , E ) = 0 and to repeat the arguments from the proof of Proposi-tion 6. (cid:3) If a function b F H satisfies condition (10) then lim x → +0 x b F H ( E/x ) = 0. So,Propositions 3, 6 and 7 imply the following observations.
Corollary 3.
Let F ( A, B ) be the set of all quantum channels from A to B equipped with the strong convergence topology (described in Section 3). A) If the Hamiltonians H A and H B satisfy, respectively, the condition ofProposition 3B and condition (6) then the functions Φ C χ (Φ , H A , E ) , Φ C (Φ , H A , E ) and Φ C ea (Φ , H A , E ) are uniformly continuous on any subset of F ( A, B ) consisting of channelswith bounded energy amplification factor (for the given input energy E ). B) If the Hamiltonian H A satisfies conditions (6) then the function Φ C ea (Φ , H A , E ) is uniformly continuous on F ( A, B ) . see Remark 4. ote Added: After posting the first version of this paper I was informedby A.Winter that he and his colleagues independently have come to thesame ”energy bounded” modification of the diamond norm. I am gratefulto A.Winter for sending me a draft of their paper [24], which complementsthe present paper by detailed study of the ECD-norm from the physicalpoint of view and by continuity bounds for the quantum and private classicalcapacities of energy-constrained channels. I hope it will appear soon.I am grateful to A.Winter for valuable communication, in particular, forhis talk about drawbacks of the diamond-norm topology in infinite dimen-sions motivating this work. I am also grateful to A.S.Holevo and G.G.Amosovfor useful discussion. Special thanks to A.V.Bulinski for essential suggestionsused in preparing this paper.The research is funded by the grant of Russian Science Foundation (projectNo 14-21-00162).
References [1] D.Aharonov, A.Kitaev, N.Nisan, ”Quantum circuits with mixedstates”, in: Proc. 30th STOC, pp. 20-30, ACM Press, 1998;arXiv:quant-ph/9806029.[2] P.Billingsley, ”Convergence of probability measures”, John Willey andSons. Inc., New York-London-Sydney-Toronto, 1968.[3] V.Giovannetti, A.S.Holevo, R.Garcia-Patron, ”A solution of Gaussianoptimizer conjecture for quantum channels”, Commun. Math. Phys.,V.334, N.3, 1553-1571 (2015); arXiv:1312.2251.[4] A.S.Holevo, ”On the constrained classical capacity of infinite-dimensional covariant quantum channels”, J. Math. Phys., V.57, N.1,015203, 11 pp (2016); arXiv:1409.8085.[5] A.S.Holevo, ”Bounds for the quantity of information transmitted bya quantum communication channel”, Probl. Inf. Transm. (USSR) V.9,177-183 (1973).[6] A.S.Holevo, ”Quantum systems, channels, information. A mathematicalintroduction”, Berlin, DeGruyter, 2012.197] A.S.Holevo, ”Classical capacities of quantum channels with constrainedinputs”, Probability Theory and Applications. V.48. N.2. 359-374(2003).[8] A.S.Holevo, M.E.Shirokov ”Continuous ensembles and the χ -capacity ofinfinite dimensional channels”, Theory of Probability and its Applica-tions, V.50, N.1, P.86-98 (2005); arXiv:quant-ph/0408176.[9] A.S.Holevo, M.E.Shirokov ”On approximation of infinite dimensionalquantum channels”, Problems of Information Transmission. 2008. V.44.N.2. P.3-22; arXiv: quant-ph/0711.2245.[10] A.S.Holevo, M.E.Shirokov, ”On classical capacities of infinite-dimensional quantum channels”, Problems of Information Transmission,V.49, N.1, 15-31 (2013); arXiv:1210.6926.[11] D.Kretschmann, D.Schlingemann, R.F.Werner, ”A Continuity Theoremfor Stinespring’s Dilation”, arXiv:0710.2495.[12] D.Leung, G.Smith, ”Continuity of quantum channel capacities”, Com-mun. Math. Phys., V.292, 201-215 (2009).[13] G.Lindblad ”Entropy, information and quantum measurements”,Comm. Math. Phys. V.33. 305-322 (1973).[14] G.Lindblad ”Expectation and Entropy Inequalities for Finite QuantumSystems”, Comm. Math. Phys. V.39. N.2. 111-119 (1974).[15] V.I.Paulsen, ”Completely Bounded Maps and Operator Algebras”,Cambridge Studies in Advanced Mathematics, Cambridge UniversityPress, Cambridge, 2002.[16] M.Reed, B.Simon, ”Methods of Modern Mathematical Physics. Vol I.Functional Analysis”, Academic Press Inc., 1980.[17] M.E.Shirokov, ”Entropic characteristics of subsets of states I”, Izvestiya:Mathematics, V.70, N.6, 1265-1292 (2006); arXiv:quant-ph/0510073.[18] M.E.Shirokov, ”Reversibility conditions for quantum channels andtheir applications”, Sb. Math., V.204, N.8, 1215-1237 (2013);arXiv:1203.0262. 2019] M.E.Shirokov, ”Tight continuity bounds for the quantum conditionalmutual information, for the Holevo quantity and for capacities of quan-tum channels”, arXiv:1512.09047 (v.7).[20] A.Wehrl, ”General properties of entropy”, Rev. Mod. Phys. 50, 221-250,(1978).[21] M.M.Wilde, ”From Classical to Quantum Shannon Theory”,arXiv:1106.1445 (v.6).[22] M.M.Wilde, H.Qi, ”Energy-constrained private and quantum capacitiesof quantum channels”, arXiv:1609.01997.[23] A.Winter, ”Tight uniform continuity bounds for quantum entropies:conditional entropy, relative entropy distance and energy constraints”,Comm. Math. Phys., V.347, N.1, 291-313 (2016); arXiv:1507.07775.[24] A.Winter, et al.et al.