Tight continuity bounds for the quantum conditional mutual information ...
arXiv:1512.09047v2 [quant-ph] 2 Apr 2016
Tight continuity bounds for the quantum conditional mutual information, for the Holevo quantity and for capacities of a channel M.E. Shirokov∗
Abstract First we consider Fannes type and Winter type tight continuity bounds for the quantum conditional mutual information and their specifications. Then we analyse continuity of the Holevo quantity (as a function of an ensemble). We show that the Holevo quantity is continuous on the set of all ensembles of m states if and only if either m or the dimension of underlying Hilbert space is finite and obtain Fannes type tight continuity bound for the Holevo quantity in this case. In general case conditions for local continuity of the Holevo quantity and their corollaries (preserving of local continuity under quantum channels, stability with respect to perturbation of states) are considered. Winter’s type tight continuity bound for the Holevo quantity under the energy constraint on the average state of ensembles is obtained and applied to the system of quantum oscillators. The above results are used to obtain tight and close-to-tight continuity bounds for basic capacities of finite-dimensional quantum channels which essentially refine the Leung-Smith continuity bounds. ∗
Steklov Mathematical Institute, RAS, Moscow, email:[email protected]
1
1
Introduction
A quantitative analysis of continuity of basis characteristics of quantum systems and channels is a necessary technical tool in study of their information properties. It suffices to mention that the famous Fannes continuity bound for the von Neumann entropy and the Alicki-Fannes continuity bound for the conditional entropy are essentially used in the proofs of several important results of quantum information theory [12, 19, 26]. During the last decade many papers devoting to finding continuity bounds (estimates for variation) for different quantities have been appeared (see [2, 3, 4, 14, 21, 27] and the references therein). Although in many applications a structure of a continuity bound of a given quantity is more important than concrete values of its coefficients, a task of finding optimal values of these coefficients seems interesting from the both mathematical and physical points of view. This task can be formulated as a problem of finding so called ”tight” continuity bound, i.e. ε-achievable estimates for variations of a given quantity. The most known decision of this problem is the sharpest continuity bound for the von Neumann entropy obtained by Audenaert [2] (it refines the Fannes continuity bound mentioned above). Other result in this direction is the tight bound for the relative entropy difference via the entropy difference obtained by Reeb and Wolf [21]. Recently Winter presented tight continuity bound for the conditional entropy (improving the Alicki-Fannes continuity bound) and asymptotically tight continuity bounds for the entropy and for the conditional entropy in infinite-dimensional systems under energy constraint [27]. By using Winter’s technique a tight continuity bound for quantum conditional mutual information in infinite-dimensional tripartite systems under energy constraint on one subsystem is obtained in [24, the Appendix]. In this note we refine Fannes type and Winter type tight continuity bounds for the quantum conditional mutual information (obtained respectively in [23] and [24]). Then, by using specifications of these bounds for qc-states we obtain tight continuity bounds of the both types for the Holevo quantity of a discrete ensemble of quantum states. We analyse general continuity properties of the Holevo quantity (considered as a function of an ensemble). In particular, we show that local continuity the Holevo quantity is preserved by quantum channels (Proposition 4) and that the Holevo quantity is stable with respect to any perturbations of states provided their distribution has finite Shannon entropy (Corollary 4). 2
The above results are applied to obtain tight continuity bounds for the Holevo capacity and for the entanglement-assisted classical capacity of quantum channels with finite-dimensional output. The tight continuity bound for the quantum conditional mutual information and the Leung-Smith telescopic trick from [14] make possible to obtain a tight continuity bound for the output quantum mutual information for n-tensor power of a channel, which implies tight continuity bound for the quantum capacity and close-to-tight continuity bound for the classical capacity of quantum channels with finitedimensional output (they essentially refine the corresponding Leung-Smith continuity bounds [14]).
2
Preliminaries
Let H be a finite-dimensional or separable infinite-dimensional Hilbert space, B(H) the algebra of all bounded operators with the operator norm k · k and T(H) the Banach space of all trace-class operators in H with the trace norm k · k1. Let S(H) be the set of quantum states (positive operators in T(H) with unit trace) [12, 19, 26]. Denote by IH the identity operator in a Hilbert space H and by IdH the identity transformation of the Banach space T(H). A finite or countable collection {ρi } of states with a probability distribution P {pi } is conventionally called ensemble and denoted {pi , ρi }. The state . ρ¯ = i pi ρi is called average state of this ensemble. If quantum systems A and B are described by Hilbert spaces HA and HB then the bipartite system AB is described by the tensor product of these . spaces, i.e. HAB = HA ⊗HB . A state in S(HAB ) is denoted ωAB , its marginal states TrHB ωAB and TrHA ωAB are denoted respectively ωA and ωB . In this note a special role is plaid by so called qc-states having the form m X pi ρi ⊗ |iihi|, (1) ωAB = i=1
{pi , ρi }m i=1
where is an ensemble of m ≤ +∞ quantum states in S(HA ) and m {|ii}i=1 is an orthonormal basis in HB . We will say that ωAB is a qc-state determined by the ensemble {pi , ρi }m i=1 . The von Neumann entropy H(ρ) = Trη(ρ) of a state ρ ∈ S(H), where η(x) = −x log x, is a concave nonnegative lower semicontinuous function on S(H), it is continuous if and only if dim H < +∞ [18, 25]. 3
The concavity of the von Neumann entropy is supplemented by the inequality H(λρ + (1 − λ)σ) ≤ λH(ρ) + (1 − λ)H(σ) + h2 (λ),
λ ∈ [0, 1],
(2)
where h2 (λ) = η(λ) + η(1 − λ), valid for any states ρ and σ [19]. Audenaert obtained in [2] the sharpest continuity bound for the von Neumann entropy: |H(ρ) − H(σ)| ≤ ε log(d − 1) + h2 (ε) (3) for any ρ, σ ∈ S(H) such that ε = 12 kρ − σk1 ≤ 1 − 1/d, where d = dim H. This continuity bound is a refinement of the well known Fannes continuity bound [9]. The quantum conditional entropy H(A|B)ω = H(ωAB ) − H(ωB )
(4)
of a state ωAB with finite marginal entropies is essentially used in analysis of quantum systems [12, 26]. The function ωAB 7→ H(A|B)ω is continuous on S(HAB ) if and only if dim HA < +∞.1 The conditional entropy is concave and satisfies the following inequality H(A|B)λρ+(1−λ)σ ≤ λH(A|B)ρ + (1 − λ)H(A|B)σ + h2 (λ)
(5)
for any λ ∈ (0, 1) and any states ρAB and σAB . Inequality (5) follows from concavity of the entropy and inequality (2). Winter proved in [27] the following refinement of the Alicki-Fannes continuity bound for the conditional entropy (obtained in [1]): ε |H(A|B)ρ − H(A|B)σ | ≤ 2ε log d + (1 + ε)h2 (6) 1+ε for any states ρ, σ ∈ S(H) such that ε = 12 kρ − σk1 , where d = dim HA . He showed that this continuity bound is tight and that the factor 2 in (6) can be removed if ρ and σ are qc-states, i.e. states having form (1). 1
If dim HA < +∞ and dim HB = +∞ then formula (4) is not well defined for some states, but there is an alternative expression for H(A|B)ω (derived from the below formula (17)) which gives concave continuous function on S(HAB ) in this case [13].
4
Winter also obtained asymptotically tight continuity bounds for the entropy and for the conditional entropy for infinite-dimensional quantum states with bounded energy (see details in [27]). The quantum relative entropy for two states ρ and σ in S(H) is defined as follows X H(ρ kσ) = hi| ρ log ρ − ρ log σ |ii, where {|ii} is the orthonormal basis of eigenvectors of the state ρ and it is assumed that H(ρ kσ) = +∞ if suppρ is not contained in suppσ [18]. Several continuity bounds for the relative entropy are proved by Audenaert and Eisert [3, 4]. Tight bound for the relative entropy difference via the entropy difference is obtained by Reeb and Wolf [21].
3
Tight continuity bounds for the quantum conditional mutual information
The quantum mutual information of a state ωAB of a bipartite quantum system AB is defined as follows I(A : B)ω = H(ωAB kωA ⊗ ωB ) = H(ωA ) + H(ωB ) − H(ωAB ),
(7)
where the second expression is valid if H(ωAB ) is finite [17]. Basic properties of the relative entropy show that ω 7→ I(A : B)ω is a lower semicontinuous function on the set S(HAB ) taking values in [0, +∞]. It is well known that I(A : B)ω ≤ 2 min {H(ωA ), H(ωB )}
(8)
for any state ωAB and that I(A : B)ω ≤ min {H(ωA ), H(ωB )}
(9)
for any separable state ωAB [15, 26]. The quantum conditional mutual information of a state ωABC of a tripartite finite-dimensional system is defined as follows . I(A : B|C)ω = H(ωAC ) + H(ωBC ) − H(ωABC ) − H(ωC ).
5
(10)
This quantity plays important role in quantum information theory [8, 26], its nonnegativity is a basic result well known as strong subadditivity of von Neumann entropy [16]. If system C is trivial then (10) coincides with (7). In infinite dimensions formula (10) may contain the uncertainty ”∞−∞”. Nevertheless the conditional mutual information can be defined for any state ωABC by one of the equivalent expressions I(A : B|C)ω = sup [I(A : BC)QA ωQA − I(A : C)QA ωQA ] , QA = PA ⊗ IBC , (11) PA
I(A : B|C)ω = sup [I(B : AC)QB ωQB − I(B : C)QB ωQB ] , QB = PB ⊗IAC , (12) PB
where the suprema are over all finite rank projectors PA ∈ B(HA ) and PB ∈ B(HB ) correspondingly and it is assumed that I(X : Y )QX ωQX = λI(X : Y )λ−1 QX ωQX , where λ = TrQX ωABC [23]. It is shown in [23, Th.2] that expressions (11) and (12) define the same lower semicontinuous function on the set S(HABC ) possessing all basic properties of conditional mutual information valid in finite dimensions. If one of the marginal entropies H(ωA ) and H(ωB ) is finite then the above function is given respectively by the explicit formula2 I(A : B|C)ω = I(A : BC)ω − I(A : C)ω ,
(13)
I(A : B|C)ω = I(B : AC)ω − I(B : C)ω .
(14)
and By applying upper bound (8) to expressions (13) and (14) we see that I(A : B|C)ω ≤ 2 min {H(ωA ), H(ωB ), H(ωAC ), H(ωBC )}
(15)
for any state ωABC . The conditional quantum mutual information is not concave or convex but the following relation λI(A : B|C)ρ + (1 − λ)I(A : B|C)σ − I(A : B|C)λρ+(1−λ)σ ≤ h2 (λ) (16)
holds for λ ∈ (0, 1) and any states ρABC , σABC with finite I(A : B|C)ρ , I(A : B|C)σ . If ρABC , σABC are states with finite marginal entropies then (16) can be easily proved by noting that I(A : B|C)ω = H(A|C)ω − H(A|BC)ω , 2
The correctness of these formulas follows from upper bound (8).
6
(17)
and by using concavity of the conditional entropy and inequality (5). Validity of inequality (16) for any states ρABC , σABC with finite conditional mutual information can be proved by approximation (using the second part of Theorem 2 in [23]).
3.1
The Fannes type continuity bounds for I(A : B|C).
Property (16) makes possible to directly apply Winter’s modification of the Alicki-Fannes technic (cf.[1, 27]) to the conditional mutual information. Proposition 1. Let ρABC and σABC be states such that3 . D = max{I(A : B|C)τ− , I(A : B|C)τ+ } < +∞,
where τ± =
[σ − ρ]± . Tr[σ − ρ]±
Then |I(A : B|C)ρ − I(A : B|C)σ | ≤ Dε + 2g(ε), (18) . ε = (1 + ε) log(1 + ε) − ε log ε.4 where ε = 21 kρ − σk1 and g(ε) = (1 + ε)h2 1+ε If the function ω 7→ I(A : B|C)ω is either concave or convex on the convex hull of the states ρ, σ, τ+ , τ− then the factor 2 in the second term of (18) can be removed. Proof. Following [27] introduce the state ω ∗ = (1 + ε)−1 (ρ + [σ − ρ]+ ). Then 1 ε 1 ε ρ+ τ+ = ω ∗ = σ+ τ− , 1+ε 1+ε 1+ε 1+ε where τ+ = ε−1[σ − ρ]+ and τ− = ε−1 [σ − ρ]− are states in S(HABC ). By applying (16) to the above convex decompositions of ω ∗ we obtain (1 − p) [I(A : B|C)ρ − I(A : B|C)σ ] ≤ p I(A : B|C)τ− − I(A : B|C)τ+ + 2h2 (p) and
(1−p) [I(A : B|C)σ − I(A : B|C)ρ ] ≤ p I(A : B|C)τ+ − I(A : B|C)τ− +2h2 (p). where p = (18).
ε . 1+ε
These inequalities and nonnegativity of I(A : B|C) imply
3
[ω]+ and [ω]− are respectively positive and negative parts of an operator ω. Note that the function g(ε) is involved in the expression for entropy of Gaussian states [12, Ch.12]. 4
7
The last assertion of the proposition is easily derived from the above arguments. Proposition 1 implies the following refinement of Corollary 8 in [23]. . Corollary 1. If d = min{dim HA , dim HB } < +∞ then |I(A : B|C)ρ − I(A : B|C)σ | ≤ 2ε log d + 2g(ε)
(19)
for any states ρ, σ in S(HABC ), where ε = 21 kρ − σk1 . Continuity bound (19) is tight even for trivial C, i.e. in the case I(A : B|C) = I(A : B). Proof. Continuity bound (19) directly follows from Proposition 1 and upper bound (15). The tightness of this bound with trivial C can be shown by using the example from [27, Remark 3]. Let HA = HB = Cd , ρAB be a maximally entangled pure state and σAB = (1 − ε)ρAB + d2ε−1 (IAB − ρAB ). Then it is easy to see that 12 kρAB − σAB k1 = ε and that I(A : B)ρ − I(A : B)σ = H(σAB ) − H(ρAB ) = 2ε log d + h2 (ε) + O(ε/d2). Remark 1. By using Audenaert’s continuity bound (3) and Winter’s continuity bound (6) one can obtain via representation (17) with trivial C the following continuity bound |I(A : B)ρ − I(A : B)σ | ≤ ε log(d − 1) + 2ε log d + h2 (ε) + g(ε), for the quantum mutual information. Since h2 (ε) < g(ε) for ε > 0, this continuity bound is slightly better than (19) for d = 2. Proposition 1 implies the following specification of continuity bound (19) with trivial C for qc-states. Corollary 2. If ρAB and σAB are qc-states (i.e. states having form (1)) then |I(A : B)ρ − I(A : B)σ | ≤ ε log d + 2g(ε), (20) where d = min{dim HA , dim HB } and ε = 21 kρ − σk1 . If ρAB and σAB are qc-states determined by ensembles {pi , ρi } and {qi , σi } such that ρi ≡ σi then the factor 2 in the second term in (20) can be removed. The first term in (20) can be replaced by ε max S({γi− }), S({γi+}) , where γi± = (2ε)−1 (kpi ρi − qi σi k1 ± (pi − qi )) and S is the Shannon entropy. 8
Proof. Continuity bound (20) follows from Proposition 1 and upper bound (9), since in this case τ+ and τ− are qc-states as well (this was mentioned by Winter in [27]). If ρ and σ are qc-states determined by ensembles with the same sets of states then all the states ρ, σ, τ− , τ+ are qc-states determined by ensembles with the same set of states and hence the function ω 7→ I(A : B)ω is concave on the convex hull of these states [26, Th.13.3.3]. So, the assertion concerning possibility to remove the factor 2 in the second term in (20) follows from the last assertion of Proposition 1. Since τ± =
m X
i
i
[pi ρ − qi σ ]± ⊗ |iihi| and hence [τ± ]B =
i=1
m X
γi± |iihi|,
i=1
the last assertion of the corollary follows from Proposition 1 and upper bound (9). Consider the states ρABC =
m X
pi ρiAC
⊗ |iihi| and σABC =
i=1
m X
i qi σAC ⊗ |iihi|,
(21)
i=1
i m where {pi , ρiAC }m i=1 and {qi , σAC }i=1 are ensemble of m ≤ +∞ quantum states in S(HAC ) and {|ii}m i=1 is an orthonormal basis in HB . Such states are called qqc-states in [26]. It follows from (14) and upper bound (9) that
I(A : B|C)ρ ≤ I(AC : B)ρ ≤ max {H(ρAC ), H(ρB )} for any qqc-state ρABC . Since the states τ− and τ+ introduced in Proposition 1 are qqc-states as well, this proposition implies the following Corollary 3. If ρABC and σABC are states (21) then |I(A : B|C)ρ − I(A : B|C)σ | ≤ ε log d + 2g(ε),
(22)
. where d = min{dim HAC , dim HB } and ε = 21 kρ − σk1 . The first term in (22) can be replaced by ε max S({γi− }), S({γi+}) , where γi± = (2ε)−1 (kpi ρi − qi σi k1 ± (pi − qi )) and S is the Shannon entropy. Corollary 3 can be used in analysis of the loss of the Holevo information under action of a quantum channel. 9
3.2
The Winter type continuity bound for I(A : B|C).
If the both systems A and B are infinite-dimensional (and C is arbitrary) then the function I(A : B|C)ω is not continuous on S(HABC ) (only lower semicontinuous) and takes infinite values. Several conditions of local continuity of this function are presented in Corollary 7 in [23], which implies, in particular, that the function I(A : B|C)ω is continuous on subsets of tripartite states ωABC with bounded energy of ωA , i.e. subsets of the form . SE = {ωABC | TrHA ωA ≤ E },
(23)
where HA is the Hamiltonian of system A and E > 0, provided5 Tre−βHA < +∞ for all β > 0.
(24)
This condition discrete spectrum of finite multiplicity, i.e. P+∞ implies that HA has+∞ HA = n=1 En |nihn|, where {|ni}n=1 is an orthonormal basis of eigenvectors +∞ of HA corresponding to the nondecreasing P+∞ −βEnsequence {En }n=1 of eigenvalues is finite for all β > 0. We will (energy levels of HA ) such that n=1 e assume that E1 = 0 for simplicity. By condition (24) for any E > 0 the von Neumann entropy H(ρ) attains its unique maximum under the constraint TrHA ρ ≤ E at the Gibbs state γ(E) = [Tre−β(E)HA ]−1 e−β(E)HA , where β(E) is the solution of the equation TrHA e−βHA = ETre−βHA [25]. Winter’s type tight continuity bound for the function I(A : B|C)ω on the subset SE is presended in [24, the Appendix]. The following proposition contains refinement of this bound obtained by using Corollary 1. Proposition 2. Let ρ and σ be any states in S(HABC ) such that ε′ −ε TrHA ρA , TrHA σA ≤ E, 12 kρ − σk1 ≤ ε < ε′ ≤ 1 and δ = 1+ε ′ . Then |I(A : B|C)ρ − I(A : B|C)σ | ≤ (2ε′ + 4δ)H(γ(E/δ)) + 2g(ε′) + 4h2 (δ), (25) x where g(x) = (1 + x)h2 1+x . Continuity bound (25) is asymptotically tight (for large E) even for trivial C, i.e. in the case I(A : B|C) = I(A : B). Remark 2. Condition (24) implies lim δH(γ(E/δ)) = 0 [22, Pr.1]. δ→+0
Hence, Proposition 2 shows that the function ωABC 7→ I(A : B|C)ω is uniformly continuous on the set SE for any E > 0. 5
Since condition (24) guarantees continuity of the entropy H(ωA ) on the set SE [25].
10
Proof.6 Following the proofs of Lemmas 16,17 in [27] define the projector X . Pδ = |nihn| 0≤En ≤E/δ
in B(HA ) and consider the states ρδ =
Pδ ⊗ IBC ρ Pδ ⊗ IBC , TrPδ ρA
σδ =
Pδ ⊗ IBC σ Pδ ⊗ IBC . TrPδ σA
In the proof of Lemma 16 in [27] it is shown that H(ωA ) − [TrPδ ωA ]H(ωAδ ) ≤ δH(γ(E/δ)) + h2 (TrPδ ωA ), H(ωAδ ) ≤ H(γ(E/δ)) ,
TrPδ ωA ≥ 1 − δ,
(26) (27)
where ω = ρ, σ, and that log TrPδ ≤ H(γ(E/δ)) ,
1 kρδ 2
− σ δ k1 ≤ ε ′ .
(28)
By using (26) and (27) it is easy to derive from Lemma 9 in [24] that |I(A : B|C)ω − I(A : B|C)ωδ | ≤ 2δH(γ(E/δ)) + 2h2 (δ),
ω = ρ, σ.
By using (28) and applying Corollary 1 we obtain I(A : B|C)ρδ − I(A : B|C)σδ ≤ 2ε′ log TrPδ + 2g(ε′) ′
(29)
(30)
′
≤ 2ε H(γ(E/δ)) + 2g(ε ).
Since
|I(A : B|C)ρ − I(A : B|C)σ | ≤ I(A : B|C)ρδ − I(A : B|C)σδ
+ I(A : B|C)ρ − I(A : B|C)ρδ + |I(A : B|C)σ − I(A : B|C)σδ | ,
continuity bound (25) follows from (29) and (30). To show the asymptotical tightness of the continuity bound (25) for trivial C one can exploit the arguments and the example from Remark 19 in [27]. If we take τ B = γ(E)B in that example then we obtain I(A : B)ρ − I(A : B)σ ≥ 2εH(γ(E)). 6
The main part of this proof differs from the proof of Lemma 25 in [24] only by using Corollary 1 instead of Corollary 8 in [23]. We present it here for reader’s convenience, since modifications of this proof will be used in Section 4.
11
4
Tight continuity bounds for the Holevo quantity
The Holevo quantity of an ensemble {pi , ρi }m i=1 of m ≤ +∞ quantum states is defined as χ ({pi , ρi }m i=1 )
m
m
i=1
i=1
X . X pi H(ρi ), pi H(ρi k¯ ρ) = H(¯ ρ) − =
ρ¯ =
m X
pi ρi ,
i=1
where the second formula is valid if H(¯ ρ) < +∞. This quantity gives the upper bound for classical information which can be obtained by applying quantum measurements to an ensemble [11]. It plays important role in analysis of information properties of quantum systems and channels [12, 19, 26]. Let HA = H and {|ii}m i=1 be an orthonormal basis in a m-dimensional Hilbert space HB . Then χ({pi , ρi }m ˆ AB = ˆ , where ω i=1 ) = I(A : B)ω
m X
pi ρi ⊗ |iihi|.
(31)
i=1
If H(¯ ρ) and S({pi }m i=1 ) are finite (here S is the Shannon entropy) then (31) is directly verified by noting that H(ˆ ωA ) = H(¯ ρ), H(ˆ ωB ) = S({pi }m i=1 ) and Pm m H(ˆ ωAB ) = i=1 pi H(ρi ) + S({pi }i=1 ). The validity of (31) in general case can be easily shown by two step approximation using Theorem 1A in [23]. Denote by Em (HA ) the set of all ensembles consisting of m ≤ +∞ states in S(HA ). We will say that a sequence {{pni , ρni }}n ⊂ Em (HA ) converges to an ensemble {p0i , ρ0i } ∈ Em (HA ) if 7 lim pni ρni = p0i ρ0i
i = 1, m.
n→∞
(32)
We explore continuity of the function Em (HA ) ∋ {pi , ρi } 7→ χ({pi , ρi }) with respect to this convergence and obtain continuity bounds in two cases: A) either m or dim HA is finite; 7
B) dim HA = m = +∞.
This means that lim pni = p0i for all i and lim ρni = ρ0i for all i such that p0i 6= 0. n→∞
n→∞
12
4.1
Case A
This is exactly the case of global continuity of the Holevo quantity. Proposition 3. The function χ({pi , ρi }) is continuous on Em (HA ) if and only if either dim HA or m is finite. In this case |χ({pi , ρi }) − χ({qi , σi })| ≤ ε log min{dim HA , m} + 2g(ε)
(33)
for arbitrary ensembles {pi , ρi } and {qi , σi } consisting of m states in S(HA ), Pm 1 ε where ε = 2 i=1 kpi ρi − qi σi k1 and g(ε) = (1 + ε)h2 1+ε . If ρi ≡ σi then the factor 2 in the second term in (33) can be removed. The first term in (33) can be replaced by ε max S({γi− }), S({γi+}) , where γi± = (2ε)−1 (kpi ρi − qi σi k1 ± (pi − qi )) and S is the Shannon entropy. Proof. It is easy to see that the function χ({pi , ρi }) is not continuous on Em (HA ) if dim HA = m = +∞. By representation (31) continuity bound (33) and its specifications follow from Corollary 2. Remark 3. The continuity bound (33) is tight. Indeed, let {|ii}di=1 be . an orthonormal basis in HA = Cd and ρc = IA /d the chaotic state in S(HA ). For given ε ∈ (0, 1) consider the ensembles {pi , ρi }di=1 and {qi , σi }di=1 , where ρi = |iihi|, P σi = (1 − ε)|iihi| + ερc , pi = qi = 1/d for all i. Then it is easy to see that 12 m i=1 kpi ρi − qi σi k1 = ε(1 − 1/d), while concavity of the entropy implies χ({pi , ρi }) − χ({qi , σi }) = log d − log d + H(σi ) ≥ ε log d. Since in this case dim HA = m = d, we see that the main term in (33) is optimal. This example with d = 3 also shows that the second term in (33) can not be less than ε log 3/3 ≈ 0.53ε. In fact, Proposition 3 contains two estimates: the continuity bounds with the main term ε log dim HA depending only on the dimension of underlying Hilbert space HA and the continuity bound with the main term ε log m depending only on the size m of ensembles. Continuity bounds of the last type are sometimes called dimension-independent. Recently Audenaert obtained the following dimension-independent continuity bound for the Holevo quantity in the case pi ≡ qi [5, Th.15]: |χ({pi , ρi }) − χ({pi , σi })| ≤ t log(1 + (m − 1)/t) + log(1 + (m − 1)t), 13
where t = 21 maxi kρi − σi k1 is the maximal distance between corresponding states of ensembles. Proposition 3 in this case gives |χ({pi , ρi }) − χ({pi , σi })| ≤ ε log m + 2g(ε),
(34)
P where ε = 21 i pi kρi − σi k1 is the average distance between corresponding states of ensembles. Since ε ≤ t and g(x) is an increasing function on [0, 1], we may replace ε by t in (34). The following continuity bound for the Holevo quantity not depending on the size m of an ensemble is obtained by Oreshkov and Calsamiglia in [20]: |χ({pi , ρi }) − χ({qi , σi })| ≤ 2εK log(d − 1) + 2h2 (εK ),
εK ≤ (d − 1)/d,
where d = dim HA and εK is the Kantorovich distance between the ensemPm 1 bles {pi , ρi } and {qi , σi }. Since εK ≥ ε = 2 i=1 kpi ρi − qi σi k1 [20, Ap.B], Proposition 3 gives stronger continuity bound for the Holevo quantity for d > 2.
4.2
Case B
If dim HA = m = +∞ then the Holevo quantity is not continuous on Em (HA ) (only lower semicontinuous). Conditions for its local continuity are presented in the following proposition. Proposition 4. A) Let {{pni , ρni }}n ⊂ E∞ (HA ) be a sequence converging to an ensemble {p0i , ρ0i } ∈ E∞ (HA ) in the sense (32). Then lim χ({pni , ρni }) = χ({p0i , ρ0i }) < +∞
n→∞
provided one of the following conditions is valid: ! ! X X a) lim H pni ρni = H p0i ρ0i < +∞; n→∞
i
i
b) lim S ({pni }) = S {p0i } < +∞, where S is the Shannon entropy. n→∞
B) If (35) holds then
lim χ({pni , Φ(ρni )}) = χ({p0i , Φ(ρ0i )}) < +∞
n→∞
14
(35)
for arbitrary quantum channel Φ : S(HA ) → S(HA′ ). Proof. A) The both conditions follow from Theorem 1A in [23] and representation (31), since (32) implies convergence of the corresponding sequence n 0 {ˆ ωAB } to the state ω ˆ AB (this can be shown by proving the convergence in the weak operator topology and by using the result from [7]). B) This assertion follows from Theorem 1B in [23] and (31) . Condition b) in Proposition 4A implies the following observation which can be interpreted as stability of the Holevo quantity with respect to perturbation of states of a given ensemble. Corollary 4. Let {pi } be a probability distribution with finite Shannon entropy. Then lim χ({pi , ρni }) = χ({pi , ρ0i }) ≤ S({pi })
n→∞
(36)
for any sequences {ρn1 }, {ρn2 }, . . . converging respectively to states ρ01 , ρ02 , . . . By Corollary 4 the finiteness of S({pi }) guarantees the validity of (36) despite possible discontinuity of the entropy for the sequences {ρn1 }, {ρn2 }, . . . Condition a) in Proposition 4A shows, in particular, that for any E > 0 the Holevo quantity isPcontinuous on the set of all ensembles {pi , ρi } with . the average state ρ¯ = i pi ρi satisfying the inequality TrHA ρ¯ ≤ E provided the Hamiltonian HA satisfies condition (24). The following proposition gives Winter’s type continuity bound for the Holevo quantity under energy constraint on the average state of an ensemble. Proposition 5. Let {pi , ρi } and {qi , σi } be countable ensembles of states inPS(HA ) with the average states ρ¯ and σ¯ such that Tr¯ ρHA , Tr¯ σ HA ≤ E, ∞ 1 ε′ −ε ′ i=1 kpi ρi − qi σi k1 = ε < ε ≤ 1 and δ = 1+ε′ . Then 2 |χ({pi , ρi }) − χ({qi , σi })| ≤ (ε′ + 2δ)H(γ(E/δ)) + 2g(ε′) + 2h2 (δ), (37) ε = (1 + ε) log(1 + ε) − ε log ε. This continuity where g(ε) = (1 + ε)h2 1+ε bound is asymptotically tight (for large E). Remark 4. Condition (24) implies lim δH(γ(E/δ)) = 0 [22, Pr.1]. δ→+0
Hence, Proposition 5 shows that the Holevo quantity is uniformly continuous on the set of all ensembles {pi ,P ρi } having bounded energy of the average state with respect to the distance ∞ i=1 kpi ρi − qi σi k1 between ensembles {pi , ρi } and {qi , σi }. 15
Proof. By representation (31) continuity bound (37) means that |I(A : B)ρ − I(A : B)σ | ≤ (ε′ + 2δ)H(γ(E/δ)) + 2g(ε′) + 2h2 (δ)
(38)
for arbitrary qc-states ρAB and σAB such that TrHA ρA , TrHA σA ≤ E and kρ − σk1 = 2ε. Inequality (38) can be proved by repeating the arguments from the proof of Proposition 2 with trivial C with the following two ingredients: • since ρAB and σAB are qc-states, Lemma 1 below makes possible to replace (29) by the inequality |I(A : B)ω − I(A : B)ωδ | ≤ δH(γ(E/δ)) + h2 (δ),
ω = ρ, σ.
δ • since ρδAB and σAB are qc-states, Corollary 2 makes possible to replace (30) by the inequality I(A : B)ρδ − I(A : B)σδ ≤ ε′ H(γ(E/δ)) + 2g(ε′).
The asymptotic tightness of continuity bound (37) follows from the asymptotic tightness of the continuity bound in Corollary 5 (see Remark 5 below). Lemma 1. Let PA be a projector in B(HA ) and ωAB be a qc-state (1) with finite H(ωA ). Then − (1 − τ )H(˜ ωA ) ≤ I(A : B)ω − I(A : B)ω˜ ≤ H(ωA ) − τ H(˜ ωA ),
(39)
where τ = TrPA ωA and ω ˜ AB = τ −1 PA ⊗ IB ωAB PA ⊗ IB .8 Proof. The both inequalities in (39) are easily derived from the inequalities 0 ≤ I(A : B)ω − τ I(A : B)ω˜ ≤ H(ωA ) − τ H(˜ ωA ) (40) by using nonnegativity of I(A : B) and upper bound (9). Note that representation (31) remains valid for an ensemble {pi , ρi } of any positive trace class operators if we assume that H and I(A : B) are homogenuous extensions of the von Neumann entropy and of the quantum mutual information to the cones of all positive trace class operators and that 8
For arbitrary state ωAB double inequality (39) holds with additional factors 2 in the left and in the right sides (see Lemma 9 in [24]).
16
P χ ({pi , ρi }) = H(¯ ρ) − i pi H(ρi ) provided H(¯ ρ) < +∞. This shows that the double inequality (40) can be rewritten as follows 0 ≤ χ({pi , ρi }) − χ({pi , PA ρi PA }) ≤ H(¯ ρ) − H(PA ρ¯PA ). The first of these inequalities is easily derived from monotonicity of the quantum relative entropy and concavity of the function η(x) = −x log x. The second one follows from the definition of the Holevo quantity, since H(ρi ) ≥ H(PA ρi PA ) for all i [18]. Example: the system of quantum oscillators. By using Proposition 5 and the estimates from [27] one can obtain a continuity bound for the Holevo quantity of ensembles of states of the system composed of ℓ quantum oscillators under the energy constraint on the average state of ensembles. The Hamiltonian of such system has the form HA =
ℓ X
~ωi a+ i ai ,
i=1
where ai and a+ i are the annihilation and creation operators and ωi is a frequency of the i-th oscillator [12]. To be consistent with P our assumption E0 = 0 we will consider shifted Hamiltonian HA′ = HA − 21 ℓi=1 ~ωi IA and will assume that E(ρ) = TrHA′ ρ.9 Now we can repeat all the arguments from the proof of Lemma 18 in [27] by using Proposition 5 instead of Meta-Lemmas 16,17 and obtain the following Corollary 5. Let {pi , ρi } and {qi , σi } be countable ensembles of states of the quantum system composed of ℓ oscillators with the average states ρ¯ P∞ 1 ′ ′ and σ ¯ such that Tr¯ ρHA , Tr¯ σ HA ≤ E, 2 i=1 kpi ρi − qi σi k1 ≤ ε ≤ 1 . Then " ℓ # X 1+α E e |χ({pi , ρi }) − χ({qi , σi })| ≤ ε 1−α log ℓ~ω + 2α + 1 + ℓ log α(1−ε) i i=1
+ℓ
1+α 1−α
˜ 2 (ε) + 2h ˜ 2 (αε) + 2g + 2α h
1+α ε 1−α
,
˜ 2 (x) = h2 (x) for x ≤ 1/2 and h ˜ 2 (x) = 1 for x ≥ 1/2, where α ∈ (0, 21 ), h g(x) = (x + 1) log(x + 1) − x log x. 9
One can Pℓ take into account in Corollary 5 below that the real energy of ρ is equal to E(ρ) + 12 i=1 ~ωi .
17
Note that the main term in this continuity bound coincides with the main term in the continuity bound for the von Neumann entropy of states of the system of ℓ oscillators with the energy not exceeding E presented in Lemma 16 in [27]. Parameter α in Corollary 5 is a free parameter which can be used to optimize the continuity bound for given value of energy E. It is easy to see that for large energy E the optimal value of α is close to zero, so the main term in this continuity bound is approximately equal to ℓ X E log ℓ~ω εH(γ(E)) ≈ ε + 1 . i i=1
Remark 5. To show the asymptotical tightness of the continuity bound in Proposition 5 for large E it suffices to show this property for the continuity bound in Corollary 5. By the above note this can be done by finding for given ε > 0 and E > 0 two ensembles {pi , ρi } and {qi , σi } satisfying the condition of Corollary 5 such that |χ({pi , ρi }) − χ({qi , σi })| ≥ εH(γ(E)) .
(41)
Let {pi , ρi } be any pure state ensemble with the average state γ(E) and qi = pi , σi = (1 − ε)ρi + εγ(E) for all i. Then ∞ ∞ X X kpi ρi − qi σi k1 = εpi kρi − γ(E)k1 ≤ 2ε i=1
i=1
while (41) follows from concavity of the entropy.
5
Applications
Now we apply the results of the previous sections to obtain tight and closeto-tight continuity bounds for basic capacities of a quantum channel. We restrict attention to channels with finite-dimensional output.
5.1
Tight continuity bounds for the Holevo capacity and for the entanglement-assisted classical capacity of a quantum channel
A quantum channel Φ from a system A to a system B is a completely positive trace preserving linear map T(HA ) → T(HB ), where HA and HB are Hilbert 18
spaces associated with the systems A and B. The Holevo capacity of a quantum channel Φ : A → B is defined as follows ¯ C(Φ) = sup χ({pi , Φ(ρi )}), (42) {pi ,ρi }∈E(HA )
where the supremum is over all ensembles of input states. This quantity is closely related to the classical capacity of a quantum channel (see Section 5.2 below). The classical entanglement-assisted capacity of a quantum channel determined an ultimate rate of transmission of classical information when an entangled state between the input and the output of a channel is used as an additional resource (see details in [12, 19, 26]). By the Bennett-ShorSmolin-Thaplyal theorem the classical entanglement-assisted capacity of a finite-dimensional quantum channel Φ : A → B is given by the expression Cea (Φ) =
sup I(Φ, ρ),
(43)
ρ∈S(HA )
in which I(Φ, ρ) is the quantum mutual information of the channel Φ at a state ρ defined as follows I(Φ, ρ) = I(B : R)Φ⊗IdR (ˆρ) ,
(44)
where HR ∼ = HA and ρˆ is a pure state in S(HAR ) such that ρˆA = ρ [6, 12, 26]. ¯ In analysis of variations of the capacities C(Φ) and Cea (Φ) as functions of a channel we will use two norms on the set of all completely positive maps Φ : T(HA ) 7→ T(HB ): the operator norm . kΦk =
sup
kΦ(ρ)k1
ρ∈T(HA ),kρk1 =1
and the diamond norm . kΦk⋄ =
sup
kΦ ⊗ IdR (ρ)k1 ,
ρ∈T(HAR ),kρk1 =1
which coincides with the norm of complete boundedness of the dual map Φ∗ : B(HB ) 7→ B(HA ) to the map Φ [12, 26]. Proposition 3 and Corollary 1 imply the following
19
Proposition 6. Let Φ and Ψ be quantum channels from A to B and ε g(ε) = (1 + ε)h2 1+ε . Then ¯ ¯ |C(Φ) − C(Ψ)| ≤ ε log dB + 2g(ε),
(45)
where ε = 21 kΦ − Ψk and dB = dim HB , and |Cea (Φ) − Cea (Ψ)| ≤ 2ε log d + 2g(ε),
(46)
where ε = 21 kΦ − Ψk⋄ and d = min{dim HA , dim HB }. The both continuity bounds (45) and (46) are tight. Proof. For given ensemble {pi , ρi } Proposition 3 shows that |χ({pi , Φ(ρi )}) − χ({pi , Ψ(ρi )})| ≤ ε log dB + 2g(ε), P where ε = 21 i pi kΦ(ρi ) − Ψ(ρi )k1 ≤ 21 kΦ − Ψk. This and (42) imply (45). Continuity bounds (46) is derived similarly from Corollary 1 and expression (43), since for any pure state ρˆAR in (44) we have kΦ ⊗ IdR (ˆ ρ) − Ψ ⊗ IdR (ˆ ρ)k1 ≤ kΦ − Ψk⋄ . To show the tightness of the both continuity bounds assume that HA = HB = Cd , Φ is the noiseless channel (i.e. Φ = IdCd ) and Ψ is the depolarizing channel: Ψ(ρ) = (1 − p)ρ + pd−1 ICd , p ∈ [0, 1]. Since
¯ C(Ψ) = log d + (1 − pc) log(1 − pc) + pc log(p/d), ¯ ¯ where c = 1 − 1/d [12], Cea (Φ) = 2C(Φ) = 2 log d and Cea (Ψ) ≤ 2C(Ψ), we have ¯ ¯ C(Φ) − C(Ψ) = pc log d + h2 (pc) + pc log c and Cea (Φ) − Cea (Ψ) ≥ 2pc log d + 2h2 (pc) + 2pc log c. These relations show the tightness of continuity bound (45) and (46), since it is easy to see that kΦ − Ψk ≤ kΦ − Ψk⋄ ≤ 2p.
20
5.2
Refinement of the Leung-Smith continuity bounds for classical and quantum capacities of a channel
By the Holevo-Schumacher-Westmoreland theorem the classical capacity of a finite-dimensional channel Φ : A → B is given by the expression (cf.[12, 26]) ¯ ⊗n ), C(Φ) = lim n−1 C(Φ
(47)
n→+∞
¯ where C(Φ) is the Holevo capacity defined in the previous subsection. By the Lloyd-Devetak-Shor theorem the quantum capacity of a finitedimensional channel Φ : A → B is given by the expression (cf.[12, 26]) ¯ ⊗n ), Q(Φ) = lim n−1 Q(Φ
(48)
n→+∞
. ¯ where Q(Φ) is the maximum of the coherent information Ic (Φ, ρ) = H(Φ(ρ))− b b is a complementary channel to Φ). H(Φ(ρ)) over all states ρ ∈ S(HA ) (Φ Leung and Smith obtained in [14] the following continuity bounds for classical and quantum capacities of a channel with finite-dimensional output |C(Φ) − C(Ψ)| ≤ 16ε log dB + 4h2 (2ε) ,
(49)
|Q(Φ) − Q(Ψ)| ≤ 16ε log dB + 4h2 (2ε) ,
(50)
where ε = 12 kΦ−Ψk⋄ and dB = dim HB .10 By using Winter’s tight continuity bound (6) for the conditional entropy (instead of the original Alicki-Fannes continuity bound) in the Leung-Smith proof one can replace the main terms in (49) and (50) by 4ε log dB . By using Corollary 1 and modifying the arguments of Leung and Smith one can replace the main terms in (49) and (50) by 2ε log dB (which gives tight continuity bound for the quantum capacity and close-to-tight continuity bound for the classical capacity). Proposition 7. Let Φ and Ψ be arbitrary channels from A to B. Then |C(Φ) − C(Ψ)| ≤ 2ε log dB + 2g(ε), |Q(Φ) − Q(Ψ)| ≤ 2ε log dB + 2g(ε), where ε = 21 kΦ − Ψk⋄ , dB = dim HB and g(ε) = (1 + ε)h2
ε 1+ε
The continuity bound for the quantum capacity Q is tight. 10
.
It is assumed that expressions (47) and (48) remain valid in the case dim HA = +∞.
21
The tight continuity bound (45) for the Holevo capacity shows that the continuity bound for the classical capacity C in Proposition 7 is close-to-tight (up to the factor 2 in the main term). Proof. The continuity bound for the classical capacity C is directly obtained by using Lemma 12 in [14], representation (31) and Lemma 2 below. To prove the continuity bound for the quantum capacity Q note that the coherent information can be represented as follows Ic (Φ, ρ) = I(B : R)Φ⊗IdR (ˆ ρ) − H(ρ) where ρˆ ∈ S(HAR ) is a purification a state ρ. Hence for arbitrary quantum ⊗n channels Φ and Ψ, arbitrary n and any state ρ in S(HA ) we have ρ) ρ) − I(B n : Rn )Ψ⊗n ⊗IdRn (ˆ Ic (Φ⊗n , ρ) − Ic (Ψ⊗n , ρ) = I(B n : Rn )Φ⊗n ⊗IdRn (ˆ ⊗n where ρˆ ∈ S(HAR ) is a purification of the state ρ. This representation, Lemma 2 below and Lemma 12 in [14] imply the continuity bound for the quantum capacity. To show the tightness of the continuity bound for the quantum capacity consider the family of erasure channels (1 − p)ρ 0 , p ∈ [0, 1]. (51) Φp (ρ) = 0 pTrρ
from d-dimensional system A to d + 1-dimensional system B. It is easy to show (see [12, Ch.10]) that Q(Φp ) = (1 − 2p) log d for p ≤ 1/2 and Q(Φp ) = 0 for p ≥ 1/2. Hence Q(Φ0 ) − Q(Φp ) = 2p log d for p ≤ 1/2. By noting that kΦ0 − Φp k⋄ ≤ 2p we see that continuity bound for the quantum capacity is tight (for large d). The following lemma is an I(A : B)-analog of Theorem 11 in [14]. It is proved by the same telescopic trick (by using the conditional mutual information instead of the conditional entropy). Lemma 2. Let Φ and Ψ be channels from arbitrary system A to a finite-dimensional system B, C be any system and n ∈ N. Then |I(B n : C)Φ⊗n ⊗IdC (ρ) − I(B n : C)Ψ⊗n ⊗IdC (ρ) | ≤ 2nε log dB + 2ng(ε), ⊗n for any state ρ ∈ S(HA ⊗ HC ), where ε = 12 kΦ − Ψk⋄ and dB = dim HB . This continuity bound is tight (for any given n and large dB ).
22
Proof. Following the proof of Theorem 11 in [14] introduce the states σk = Φ⊗k ⊗ Ψ⊗(n−k) ⊗ IdC (ρ),
k = 0, 1, ..., n.
We have |I(B n : C)σn
n X I(B n : C)σk − I(B n : C)σk−1 − I(B n : C)σ0 | = k=1 n X I(B n : C)σ − I(B n : C)σ . ≤ k k−1
(52)
k=1
By using the chain rule
I(X : Y Z) = I(X : Y ) + I(X : Z|Y ) we obtain for each k (cf.[14]) I(B n : C)σk − I(B n : C)σk−1 = I(B1 ...Bk−1 Bk+1 ...Bn : C)σk + I(Bk : C|B1 ...Bk−1 Bk+1 ...Bn )σk − I(B1 ...Bk−1 Bk+1 ...Bn : C)σk−1 − I(Bk : C|B1 ...Bk−1 Bk+1 ...Bn )σk−1 = I(Bk : C|B1 ...Bk−1 Bk+1 ...Bn )σk − I(Bk : C|B1 ...Bk−1 Bk+1 ...Bn )σk−1 , where it was used that TrBk σk = TrBk σk−1 . So, Corollary 1 gives I(B n : C)σ − I(B n : C)σ ≤ 2ε log dB + 2g(ε), k k−1
for any k, where ε = 21 kσk − σk−1 k1 ≤ 12 kΦ − Ψk⋄ . This and (52) imply the required inequality (since Φ⊗n ⊗ IdC (ρ) = σn and Ψ⊗n ⊗ IdC (ρ) = σ0 ). The tightness of the continuity bound in Lemma 2 for any given n can be easily shown by using the erasure channels (51) again. Indeed, let Φ = Φ0 , Ψ = Φp , C = A and ρ be a maximally entangled pure state in S(HAC ). Then I(B : C)Φ⊗IdC (ρ) = 2 log dA and by using inequality (16) it is easy to show that I(B : C)Ψ⊗IdC (ρ) ≤ 2(1 − p) log dA + h2 (p). So, we have I(B n : C n )Φ⊗n ⊗IdC n (ρ⊗n ) = nI(B : C)Φ⊗IdC (ρ) = 2n log dA , I(B n : C n )Ψ⊗n ⊗IdC n (ρ⊗n ) = nI(B : C)Ψ⊗IdC (ρ) ≤ 2n(1 − p) log dA + nh2 (p) 23
and hence I(B n : C n )Φ⊗n ⊗IdC n (ρ⊗n ) − I(B n : C n )Ψ⊗n ⊗IdC n (ρ⊗n ) ≥ 2np log dA − nh2 (p) Since dB = dA + 1 and kΦ − Ψk⋄ ≤ 2p, this shows the tightness of the continuity bound in Lemma 2 for large dB . I am grateful to A.Winter for valuable communication and the technical tricks from his recent paper [27] essentially used in this work. I am also grateful to A.S.Holevo and G.G.Amosov for useful discussion. The research is funded by the grant of Russian Science Foundation (project No 14-21-00162).
References [1] R.Alicki, M.Fannes, ”Continuity of quantum conditional information”, Journal of Physics A: Mathematical and General, V.37, N.5, L55-L57 (2004); arXiv: quant-ph/0312081. [2] K.M.R.Audenaert, ”A sharp continuity estimate for the von Neumann entropy”, J. Math. Phys. A: Math. Theor. 40(28), 8127-8136 (2007). [3] K.M.R. Audenaert, J. Eisert, ”Continuity bounds on the quantum relative entropy”, J. Math. Phys. 46, 102104 (2005); quant-ph/0503218. [4] K.M.R. Audenaert, J. Eisert, ”Continuity bounds on the quantum relative entropy-II”, J. Math. Phys. 52, 112201 (2011); arXiv:1105.2656. [5] K.M.R. Audenaert ”Quantum Skew Divergence”, J. Math. Phys. 55, 112202 (2014); arXiv:1304.5935. [6] C.H.Bennett, P.W.Shor, J.A.Smolin, A.V.Thapliyal ”Entanglementassisted classical capacity of noisy quantum channel”, Phys. Rev. Lett. V.83. 3081-3084 (1999). [7] G.F.Dell’Antonio ”On the limits of sequences of normal states”, Commun. Pure Appl. Math. V.20, 413-430, (1967). [8] I.Devetak, J.Yard, ”The operational meaning of quantum conditional information”, Phys. Rev. Lett. 100, 230501 (2008); arXiv:quant-ph/0612050. 24
[9] M. Fannes, ”A continuity property of the entropy density for spin lattice systems”, Commun. Math. Phys. V.31, 291-294 (1973). [10] O.Fawzi, R.Renner, ”Quantum conditional mutual information and approximate Markov chains”, arXiv:1410.0664. [11] A.S.Holevo, ”Bounds for the quantity of informa- tion transmitted by a quantum communication channel”, Probl. Inf. Transm. (USSR) V.9, 177-183 (1973). [12] A.S.Holevo ”Quantum systems, channels, information. A mathematical introduction”, Berlin, DeGruyter, 2012. [13] A.A.Kuznetsova, ”Quantum conditional entropy for infinite-dimensional systems”, Theory of Probability and its Applications, V.55, N.4, 709-717 (2011). [14] D.Leung, G.Smith, ”’Continuity of quantum channel capacities”, Commun. Math. Phys., V.292, 201-215 (2009). [15] N.Li, S.Luo, ”Classical and quantum correlative capacities of quantum systems”, Phys. Rev. A 84, 042124 (2011). [16] E.H.Lieb, M.B.Ruskai, ”Proof of the strong suadditivity of quantum mechanical entropy”, J.Math.Phys. V.14. 1938 (1973). [17] G.Lindblad ”Entropy, information and quantum measurements”, Comm. Math. Phys. V.33. 305-322 (1973). [18] G.Lindblad ”Expectation and Entropy Inequalities for Finite Quantum Systems”, Comm. Math. Phys. V.39. N.2. 111-119 (1974). [19] M.A.Nielsen, I.L.Chuang ”Quantum Computation and Quantum Information”, Cambridge University Press, 2000. [20] O.Oreshkov, J.Calsamiglia, ”Distinguishability measures between ensembles of quantum states”, Phys. Rev. A 79, 032336 (2009); arXiv:0812.3238. [21] D.Reeb, M.M.Wolf, ”Tight bound on relative entropy by entropy difference”, IEEE Trans. Inf. Theory 61, 1458-1473 (2015); arXiv:1304.0036. 25
[22] M.E.Shirokov, ”Entropic characteristics of subsets of states I”, Izvestiya: Mathematics, V.70, N.6, 1265-1292 (2006); arXiv:quant-ph/0510073. [23] M.E.Shirokov, ”Measures of quantum correlations dimensional systems”, arXiv:1506.06377 (v.2).
in
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[24] M.E.Shirokov, ”Squashed entanglement in infinite dimensions”, arXiv:1507.08964. [25] A.Wehrl, ”General properties of entropy”, Rev. Mod. Phys. 50, 221-250, (1978). [26] M.M.Wilde, ”From Classical arXiv:1106.1445 (v.6).
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[27] A.Winter, ”Tight uniform continuity bounds for quantum entropies: conditional entropy, relative entropy distance and energy constraints”, arXiv:1507.07775 (v.6), to appear in Commun.Math.Phys.
26
Tight continuity bounds for the quantum conditional mutual information, for the Holevo quantity and for capacities of a channel M.E. Shirokov∗
Abstract First we consider Fannes type and Winter type tight continuity bounds for the quantum conditional mutual information and their specifications. Then we analyse continuity of the Holevo quantity (as a function of an ensemble). We show that the Holevo quantity is continuous on the set of all ensembles of m states if and only if either m or the dimension of underlying Hilbert space is finite and obtain Fannes type tight continuity bound for the Holevo quantity in this case. In general case conditions for local continuity of the Holevo quantity and their corollaries (preserving of local continuity under quantum channels, stability with respect to perturbation of states) are considered. Winter’s type tight continuity bound for the Holevo quantity under the energy constraint on the average state of ensembles is obtained and applied to the system of quantum oscillators. The above results are used to obtain tight and close-to-tight continuity bounds for basic capacities of finite-dimensional quantum channels which essentially refine the Leung-Smith continuity bounds. ∗
Steklov Mathematical Institute, RAS, Moscow, email:[email protected]
1
1
Introduction
A quantitative analysis of continuity of basis characteristics of quantum systems and channels is a necessary technical tool in study of their information properties. It suffices to mention that the famous Fannes continuity bound for the von Neumann entropy and the Alicki-Fannes continuity bound for the conditional entropy are essentially used in the proofs of several important results of quantum information theory [12, 19, 26]. During the last decade many papers devoting to finding continuity bounds (estimates for variation) for different quantities have been appeared (see [2, 3, 4, 14, 21, 27] and the references therein). Although in many applications a structure of a continuity bound of a given quantity is more important than concrete values of its coefficients, a task of finding optimal values of these coefficients seems interesting from the both mathematical and physical points of view. This task can be formulated as a problem of finding so called ”tight” continuity bound, i.e. ε-achievable estimates for variations of a given quantity. The most known decision of this problem is the sharpest continuity bound for the von Neumann entropy obtained by Audenaert [2] (it refines the Fannes continuity bound mentioned above). Other result in this direction is the tight bound for the relative entropy difference via the entropy difference obtained by Reeb and Wolf [21]. Recently Winter presented tight continuity bound for the conditional entropy (improving the Alicki-Fannes continuity bound) and asymptotically tight continuity bounds for the entropy and for the conditional entropy in infinite-dimensional systems under energy constraint [27]. By using Winter’s technique a tight continuity bound for quantum conditional mutual information in infinite-dimensional tripartite systems under energy constraint on one subsystem is obtained in [24, the Appendix]. In this note we refine Fannes type and Winter type tight continuity bounds for the quantum conditional mutual information (obtained respectively in [23] and [24]). Then, by using specifications of these bounds for qc-states we obtain tight continuity bounds of the both types for the Holevo quantity of a discrete ensemble of quantum states. We analyse general continuity properties of the Holevo quantity (considered as a function of an ensemble). In particular, we show that local continuity the Holevo quantity is preserved by quantum channels (Proposition 4) and that the Holevo quantity is stable with respect to any perturbations of states provided their distribution has finite Shannon entropy (Corollary 4). 2
The above results are applied to obtain tight continuity bounds for the Holevo capacity and for the entanglement-assisted classical capacity of quantum channels with finite-dimensional output. The tight continuity bound for the quantum conditional mutual information and the Leung-Smith telescopic trick from [14] make possible to obtain a tight continuity bound for the output quantum mutual information for n-tensor power of a channel, which implies tight continuity bound for the quantum capacity and close-to-tight continuity bound for the classical capacity of quantum channels with finitedimensional output (they essentially refine the corresponding Leung-Smith continuity bounds [14]).
2
Preliminaries
Let H be a finite-dimensional or separable infinite-dimensional Hilbert space, B(H) the algebra of all bounded operators with the operator norm k · k and T(H) the Banach space of all trace-class operators in H with the trace norm k · k1. Let S(H) be the set of quantum states (positive operators in T(H) with unit trace) [12, 19, 26]. Denote by IH the identity operator in a Hilbert space H and by IdH the identity transformation of the Banach space T(H). A finite or countable collection {ρi } of states with a probability distribution P {pi } is conventionally called ensemble and denoted {pi , ρi }. The state . ρ¯ = i pi ρi is called average state of this ensemble. If quantum systems A and B are described by Hilbert spaces HA and HB then the bipartite system AB is described by the tensor product of these . spaces, i.e. HAB = HA ⊗HB . A state in S(HAB ) is denoted ωAB , its marginal states TrHB ωAB and TrHA ωAB are denoted respectively ωA and ωB . In this note a special role is plaid by so called qc-states having the form m X pi ρi ⊗ |iihi|, (1) ωAB = i=1
{pi , ρi }m i=1
where is an ensemble of m ≤ +∞ quantum states in S(HA ) and m {|ii}i=1 is an orthonormal basis in HB . We will say that ωAB is a qc-state determined by the ensemble {pi , ρi }m i=1 . The von Neumann entropy H(ρ) = Trη(ρ) of a state ρ ∈ S(H), where η(x) = −x log x, is a concave nonnegative lower semicontinuous function on S(H), it is continuous if and only if dim H < +∞ [18, 25]. 3
The concavity of the von Neumann entropy is supplemented by the inequality H(λρ + (1 − λ)σ) ≤ λH(ρ) + (1 − λ)H(σ) + h2 (λ),
λ ∈ [0, 1],
(2)
where h2 (λ) = η(λ) + η(1 − λ), valid for any states ρ and σ [19]. Audenaert obtained in [2] the sharpest continuity bound for the von Neumann entropy: |H(ρ) − H(σ)| ≤ ε log(d − 1) + h2 (ε) (3) for any ρ, σ ∈ S(H) such that ε = 12 kρ − σk1 ≤ 1 − 1/d, where d = dim H. This continuity bound is a refinement of the well known Fannes continuity bound [9]. The quantum conditional entropy H(A|B)ω = H(ωAB ) − H(ωB )
(4)
of a state ωAB with finite marginal entropies is essentially used in analysis of quantum systems [12, 26]. The function ωAB 7→ H(A|B)ω is continuous on S(HAB ) if and only if dim HA < +∞.1 The conditional entropy is concave and satisfies the following inequality H(A|B)λρ+(1−λ)σ ≤ λH(A|B)ρ + (1 − λ)H(A|B)σ + h2 (λ)
(5)
for any λ ∈ (0, 1) and any states ρAB and σAB . Inequality (5) follows from concavity of the entropy and inequality (2). Winter proved in [27] the following refinement of the Alicki-Fannes continuity bound for the conditional entropy (obtained in [1]): ε |H(A|B)ρ − H(A|B)σ | ≤ 2ε log d + (1 + ε)h2 (6) 1+ε for any states ρ, σ ∈ S(H) such that ε = 12 kρ − σk1 , where d = dim HA . He showed that this continuity bound is tight and that the factor 2 in (6) can be removed if ρ and σ are qc-states, i.e. states having form (1). 1
If dim HA < +∞ and dim HB = +∞ then formula (4) is not well defined for some states, but there is an alternative expression for H(A|B)ω (derived from the below formula (17)) which gives concave continuous function on S(HAB ) in this case [13].
4
Winter also obtained asymptotically tight continuity bounds for the entropy and for the conditional entropy for infinite-dimensional quantum states with bounded energy (see details in [27]). The quantum relative entropy for two states ρ and σ in S(H) is defined as follows X H(ρ kσ) = hi| ρ log ρ − ρ log σ |ii, where {|ii} is the orthonormal basis of eigenvectors of the state ρ and it is assumed that H(ρ kσ) = +∞ if suppρ is not contained in suppσ [18]. Several continuity bounds for the relative entropy are proved by Audenaert and Eisert [3, 4]. Tight bound for the relative entropy difference via the entropy difference is obtained by Reeb and Wolf [21].
3
Tight continuity bounds for the quantum conditional mutual information
The quantum mutual information of a state ωAB of a bipartite quantum system AB is defined as follows I(A : B)ω = H(ωAB kωA ⊗ ωB ) = H(ωA ) + H(ωB ) − H(ωAB ),
(7)
where the second expression is valid if H(ωAB ) is finite [17]. Basic properties of the relative entropy show that ω 7→ I(A : B)ω is a lower semicontinuous function on the set S(HAB ) taking values in [0, +∞]. It is well known that I(A : B)ω ≤ 2 min {H(ωA ), H(ωB )}
(8)
for any state ωAB and that I(A : B)ω ≤ min {H(ωA ), H(ωB )}
(9)
for any separable state ωAB [15, 26]. The quantum conditional mutual information of a state ωABC of a tripartite finite-dimensional system is defined as follows . I(A : B|C)ω = H(ωAC ) + H(ωBC ) − H(ωABC ) − H(ωC ).
5
(10)
This quantity plays important role in quantum information theory [8, 26], its nonnegativity is a basic result well known as strong subadditivity of von Neumann entropy [16]. If system C is trivial then (10) coincides with (7). In infinite dimensions formula (10) may contain the uncertainty ”∞−∞”. Nevertheless the conditional mutual information can be defined for any state ωABC by one of the equivalent expressions I(A : B|C)ω = sup [I(A : BC)QA ωQA − I(A : C)QA ωQA ] , QA = PA ⊗ IBC , (11) PA
I(A : B|C)ω = sup [I(B : AC)QB ωQB − I(B : C)QB ωQB ] , QB = PB ⊗IAC , (12) PB
where the suprema are over all finite rank projectors PA ∈ B(HA ) and PB ∈ B(HB ) correspondingly and it is assumed that I(X : Y )QX ωQX = λI(X : Y )λ−1 QX ωQX , where λ = TrQX ωABC [23]. It is shown in [23, Th.2] that expressions (11) and (12) define the same lower semicontinuous function on the set S(HABC ) possessing all basic properties of conditional mutual information valid in finite dimensions. If one of the marginal entropies H(ωA ) and H(ωB ) is finite then the above function is given respectively by the explicit formula2 I(A : B|C)ω = I(A : BC)ω − I(A : C)ω ,
(13)
I(A : B|C)ω = I(B : AC)ω − I(B : C)ω .
(14)
and By applying upper bound (8) to expressions (13) and (14) we see that I(A : B|C)ω ≤ 2 min {H(ωA ), H(ωB ), H(ωAC ), H(ωBC )}
(15)
for any state ωABC . The conditional quantum mutual information is not concave or convex but the following relation λI(A : B|C)ρ + (1 − λ)I(A : B|C)σ − I(A : B|C)λρ+(1−λ)σ ≤ h2 (λ) (16)
holds for λ ∈ (0, 1) and any states ρABC , σABC with finite I(A : B|C)ρ , I(A : B|C)σ . If ρABC , σABC are states with finite marginal entropies then (16) can be easily proved by noting that I(A : B|C)ω = H(A|C)ω − H(A|BC)ω , 2
The correctness of these formulas follows from upper bound (8).
6
(17)
and by using concavity of the conditional entropy and inequality (5). Validity of inequality (16) for any states ρABC , σABC with finite conditional mutual information can be proved by approximation (using the second part of Theorem 2 in [23]).
3.1
The Fannes type continuity bounds for I(A : B|C).
Property (16) makes possible to directly apply Winter’s modification of the Alicki-Fannes technic (cf.[1, 27]) to the conditional mutual information. Proposition 1. Let ρABC and σABC be states such that3 . D = max{I(A : B|C)τ− , I(A : B|C)τ+ } < +∞,
where τ± =
[σ − ρ]± . Tr[σ − ρ]±
Then |I(A : B|C)ρ − I(A : B|C)σ | ≤ Dε + 2g(ε), (18) . ε = (1 + ε) log(1 + ε) − ε log ε.4 where ε = 21 kρ − σk1 and g(ε) = (1 + ε)h2 1+ε If the function ω 7→ I(A : B|C)ω is either concave or convex on the convex hull of the states ρ, σ, τ+ , τ− then the factor 2 in the second term of (18) can be removed. Proof. Following [27] introduce the state ω ∗ = (1 + ε)−1 (ρ + [σ − ρ]+ ). Then 1 ε 1 ε ρ+ τ+ = ω ∗ = σ+ τ− , 1+ε 1+ε 1+ε 1+ε where τ+ = ε−1[σ − ρ]+ and τ− = ε−1 [σ − ρ]− are states in S(HABC ). By applying (16) to the above convex decompositions of ω ∗ we obtain (1 − p) [I(A : B|C)ρ − I(A : B|C)σ ] ≤ p I(A : B|C)τ− − I(A : B|C)τ+ + 2h2 (p) and
(1−p) [I(A : B|C)σ − I(A : B|C)ρ ] ≤ p I(A : B|C)τ+ − I(A : B|C)τ− +2h2 (p). where p = (18).
ε . 1+ε
These inequalities and nonnegativity of I(A : B|C) imply
3
[ω]+ and [ω]− are respectively positive and negative parts of an operator ω. Note that the function g(ε) is involved in the expression for entropy of Gaussian states [12, Ch.12]. 4
7
The last assertion of the proposition is easily derived from the above arguments. Proposition 1 implies the following refinement of Corollary 8 in [23]. . Corollary 1. If d = min{dim HA , dim HB } < +∞ then |I(A : B|C)ρ − I(A : B|C)σ | ≤ 2ε log d + 2g(ε)
(19)
for any states ρ, σ in S(HABC ), where ε = 21 kρ − σk1 . Continuity bound (19) is tight even for trivial C, i.e. in the case I(A : B|C) = I(A : B). Proof. Continuity bound (19) directly follows from Proposition 1 and upper bound (15). The tightness of this bound with trivial C can be shown by using the example from [27, Remark 3]. Let HA = HB = Cd , ρAB be a maximally entangled pure state and σAB = (1 − ε)ρAB + d2ε−1 (IAB − ρAB ). Then it is easy to see that 12 kρAB − σAB k1 = ε and that I(A : B)ρ − I(A : B)σ = H(σAB ) − H(ρAB ) = 2ε log d + h2 (ε) + O(ε/d2). Remark 1. By using Audenaert’s continuity bound (3) and Winter’s continuity bound (6) one can obtain via representation (17) with trivial C the following continuity bound |I(A : B)ρ − I(A : B)σ | ≤ ε log(d − 1) + 2ε log d + h2 (ε) + g(ε), for the quantum mutual information. Since h2 (ε) < g(ε) for ε > 0, this continuity bound is slightly better than (19) for d = 2. Proposition 1 implies the following specification of continuity bound (19) with trivial C for qc-states. Corollary 2. If ρAB and σAB are qc-states (i.e. states having form (1)) then |I(A : B)ρ − I(A : B)σ | ≤ ε log d + 2g(ε), (20) where d = min{dim HA , dim HB } and ε = 21 kρ − σk1 . If ρAB and σAB are qc-states determined by ensembles {pi , ρi } and {qi , σi } such that ρi ≡ σi then the factor 2 in the second term in (20) can be removed. The first term in (20) can be replaced by ε max S({γi− }), S({γi+}) , where γi± = (2ε)−1 (kpi ρi − qi σi k1 ± (pi − qi )) and S is the Shannon entropy. 8
Proof. Continuity bound (20) follows from Proposition 1 and upper bound (9), since in this case τ+ and τ− are qc-states as well (this was mentioned by Winter in [27]). If ρ and σ are qc-states determined by ensembles with the same sets of states then all the states ρ, σ, τ− , τ+ are qc-states determined by ensembles with the same set of states and hence the function ω 7→ I(A : B)ω is concave on the convex hull of these states [26, Th.13.3.3]. So, the assertion concerning possibility to remove the factor 2 in the second term in (20) follows from the last assertion of Proposition 1. Since τ± =
m X
i
i
[pi ρ − qi σ ]± ⊗ |iihi| and hence [τ± ]B =
i=1
m X
γi± |iihi|,
i=1
the last assertion of the corollary follows from Proposition 1 and upper bound (9). Consider the states ρABC =
m X
pi ρiAC
⊗ |iihi| and σABC =
i=1
m X
i qi σAC ⊗ |iihi|,
(21)
i=1
i m where {pi , ρiAC }m i=1 and {qi , σAC }i=1 are ensemble of m ≤ +∞ quantum states in S(HAC ) and {|ii}m i=1 is an orthonormal basis in HB . Such states are called qqc-states in [26]. It follows from (14) and upper bound (9) that
I(A : B|C)ρ ≤ I(AC : B)ρ ≤ max {H(ρAC ), H(ρB )} for any qqc-state ρABC . Since the states τ− and τ+ introduced in Proposition 1 are qqc-states as well, this proposition implies the following Corollary 3. If ρABC and σABC are states (21) then |I(A : B|C)ρ − I(A : B|C)σ | ≤ ε log d + 2g(ε),
(22)
. where d = min{dim HAC , dim HB } and ε = 21 kρ − σk1 . The first term in (22) can be replaced by ε max S({γi− }), S({γi+}) , where γi± = (2ε)−1 (kpi ρi − qi σi k1 ± (pi − qi )) and S is the Shannon entropy. Corollary 3 can be used in analysis of the loss of the Holevo information under action of a quantum channel. 9
3.2
The Winter type continuity bound for I(A : B|C).
If the both systems A and B are infinite-dimensional (and C is arbitrary) then the function I(A : B|C)ω is not continuous on S(HABC ) (only lower semicontinuous) and takes infinite values. Several conditions of local continuity of this function are presented in Corollary 7 in [23], which implies, in particular, that the function I(A : B|C)ω is continuous on subsets of tripartite states ωABC with bounded energy of ωA , i.e. subsets of the form . SE = {ωABC | TrHA ωA ≤ E },
(23)
where HA is the Hamiltonian of system A and E > 0, provided5 Tre−βHA < +∞ for all β > 0.
(24)
This condition discrete spectrum of finite multiplicity, i.e. P+∞ implies that HA has+∞ HA = n=1 En |nihn|, where {|ni}n=1 is an orthonormal basis of eigenvectors +∞ of HA corresponding to the nondecreasing P+∞ −βEnsequence {En }n=1 of eigenvalues is finite for all β > 0. We will (energy levels of HA ) such that n=1 e assume that E1 = 0 for simplicity. By condition (24) for any E > 0 the von Neumann entropy H(ρ) attains its unique maximum under the constraint TrHA ρ ≤ E at the Gibbs state γ(E) = [Tre−β(E)HA ]−1 e−β(E)HA , where β(E) is the solution of the equation TrHA e−βHA = ETre−βHA [25]. Winter’s type tight continuity bound for the function I(A : B|C)ω on the subset SE is presended in [24, the Appendix]. The following proposition contains refinement of this bound obtained by using Corollary 1. Proposition 2. Let ρ and σ be any states in S(HABC ) such that ε′ −ε TrHA ρA , TrHA σA ≤ E, 12 kρ − σk1 ≤ ε < ε′ ≤ 1 and δ = 1+ε ′ . Then |I(A : B|C)ρ − I(A : B|C)σ | ≤ (2ε′ + 4δ)H(γ(E/δ)) + 2g(ε′) + 4h2 (δ), (25) x where g(x) = (1 + x)h2 1+x . Continuity bound (25) is asymptotically tight (for large E) even for trivial C, i.e. in the case I(A : B|C) = I(A : B). Remark 2. Condition (24) implies lim δH(γ(E/δ)) = 0 [22, Pr.1]. δ→+0
Hence, Proposition 2 shows that the function ωABC 7→ I(A : B|C)ω is uniformly continuous on the set SE for any E > 0. 5
Since condition (24) guarantees continuity of the entropy H(ωA ) on the set SE [25].
10
Proof.6 Following the proofs of Lemmas 16,17 in [27] define the projector X . Pδ = |nihn| 0≤En ≤E/δ
in B(HA ) and consider the states ρδ =
Pδ ⊗ IBC ρ Pδ ⊗ IBC , TrPδ ρA
σδ =
Pδ ⊗ IBC σ Pδ ⊗ IBC . TrPδ σA
In the proof of Lemma 16 in [27] it is shown that H(ωA ) − [TrPδ ωA ]H(ωAδ ) ≤ δH(γ(E/δ)) + h2 (TrPδ ωA ), H(ωAδ ) ≤ H(γ(E/δ)) ,
TrPδ ωA ≥ 1 − δ,
(26) (27)
where ω = ρ, σ, and that log TrPδ ≤ H(γ(E/δ)) ,
1 kρδ 2
− σ δ k1 ≤ ε ′ .
(28)
By using (26) and (27) it is easy to derive from Lemma 9 in [24] that |I(A : B|C)ω − I(A : B|C)ωδ | ≤ 2δH(γ(E/δ)) + 2h2 (δ),
ω = ρ, σ.
By using (28) and applying Corollary 1 we obtain I(A : B|C)ρδ − I(A : B|C)σδ ≤ 2ε′ log TrPδ + 2g(ε′) ′
(29)
(30)
′
≤ 2ε H(γ(E/δ)) + 2g(ε ).
Since
|I(A : B|C)ρ − I(A : B|C)σ | ≤ I(A : B|C)ρδ − I(A : B|C)σδ
+ I(A : B|C)ρ − I(A : B|C)ρδ + |I(A : B|C)σ − I(A : B|C)σδ | ,
continuity bound (25) follows from (29) and (30). To show the asymptotical tightness of the continuity bound (25) for trivial C one can exploit the arguments and the example from Remark 19 in [27]. If we take τ B = γ(E)B in that example then we obtain I(A : B)ρ − I(A : B)σ ≥ 2εH(γ(E)). 6
The main part of this proof differs from the proof of Lemma 25 in [24] only by using Corollary 1 instead of Corollary 8 in [23]. We present it here for reader’s convenience, since modifications of this proof will be used in Section 4.
11
4
Tight continuity bounds for the Holevo quantity
The Holevo quantity of an ensemble {pi , ρi }m i=1 of m ≤ +∞ quantum states is defined as χ ({pi , ρi }m i=1 )
m
m
i=1
i=1
X . X pi H(ρi ), pi H(ρi k¯ ρ) = H(¯ ρ) − =
ρ¯ =
m X
pi ρi ,
i=1
where the second formula is valid if H(¯ ρ) < +∞. This quantity gives the upper bound for classical information which can be obtained by applying quantum measurements to an ensemble [11]. It plays important role in analysis of information properties of quantum systems and channels [12, 19, 26]. Let HA = H and {|ii}m i=1 be an orthonormal basis in a m-dimensional Hilbert space HB . Then χ({pi , ρi }m ˆ AB = ˆ , where ω i=1 ) = I(A : B)ω
m X
pi ρi ⊗ |iihi|.
(31)
i=1
If H(¯ ρ) and S({pi }m i=1 ) are finite (here S is the Shannon entropy) then (31) is directly verified by noting that H(ˆ ωA ) = H(¯ ρ), H(ˆ ωB ) = S({pi }m i=1 ) and Pm m H(ˆ ωAB ) = i=1 pi H(ρi ) + S({pi }i=1 ). The validity of (31) in general case can be easily shown by two step approximation using Theorem 1A in [23]. Denote by Em (HA ) the set of all ensembles consisting of m ≤ +∞ states in S(HA ). We will say that a sequence {{pni , ρni }}n ⊂ Em (HA ) converges to an ensemble {p0i , ρ0i } ∈ Em (HA ) if 7 lim pni ρni = p0i ρ0i
i = 1, m.
n→∞
(32)
We explore continuity of the function Em (HA ) ∋ {pi , ρi } 7→ χ({pi , ρi }) with respect to this convergence and obtain continuity bounds in two cases: A) either m or dim HA is finite; 7
B) dim HA = m = +∞.
This means that lim pni = p0i for all i and lim ρni = ρ0i for all i such that p0i 6= 0. n→∞
n→∞
12
4.1
Case A
This is exactly the case of global continuity of the Holevo quantity. Proposition 3. The function χ({pi , ρi }) is continuous on Em (HA ) if and only if either dim HA or m is finite. In this case |χ({pi , ρi }) − χ({qi , σi })| ≤ ε log min{dim HA , m} + 2g(ε)
(33)
for arbitrary ensembles {pi , ρi } and {qi , σi } consisting of m states in S(HA ), Pm 1 ε where ε = 2 i=1 kpi ρi − qi σi k1 and g(ε) = (1 + ε)h2 1+ε . If ρi ≡ σi then the factor 2 in the second term in (33) can be removed. The first term in (33) can be replaced by ε max S({γi− }), S({γi+}) , where γi± = (2ε)−1 (kpi ρi − qi σi k1 ± (pi − qi )) and S is the Shannon entropy. Proof. It is easy to see that the function χ({pi , ρi }) is not continuous on Em (HA ) if dim HA = m = +∞. By representation (31) continuity bound (33) and its specifications follow from Corollary 2. Remark 3. The continuity bound (33) is tight. Indeed, let {|ii}di=1 be . an orthonormal basis in HA = Cd and ρc = IA /d the chaotic state in S(HA ). For given ε ∈ (0, 1) consider the ensembles {pi , ρi }di=1 and {qi , σi }di=1 , where ρi = |iihi|, P σi = (1 − ε)|iihi| + ερc , pi = qi = 1/d for all i. Then it is easy to see that 12 m i=1 kpi ρi − qi σi k1 = ε(1 − 1/d), while concavity of the entropy implies χ({pi , ρi }) − χ({qi , σi }) = log d − log d + H(σi ) ≥ ε log d. Since in this case dim HA = m = d, we see that the main term in (33) is optimal. This example with d = 3 also shows that the second term in (33) can not be less than ε log 3/3 ≈ 0.53ε. In fact, Proposition 3 contains two estimates: the continuity bounds with the main term ε log dim HA depending only on the dimension of underlying Hilbert space HA and the continuity bound with the main term ε log m depending only on the size m of ensembles. Continuity bounds of the last type are sometimes called dimension-independent. Recently Audenaert obtained the following dimension-independent continuity bound for the Holevo quantity in the case pi ≡ qi [5, Th.15]: |χ({pi , ρi }) − χ({pi , σi })| ≤ t log(1 + (m − 1)/t) + log(1 + (m − 1)t), 13
where t = 21 maxi kρi − σi k1 is the maximal distance between corresponding states of ensembles. Proposition 3 in this case gives |χ({pi , ρi }) − χ({pi , σi })| ≤ ε log m + 2g(ε),
(34)
P where ε = 21 i pi kρi − σi k1 is the average distance between corresponding states of ensembles. Since ε ≤ t and g(x) is an increasing function on [0, 1], we may replace ε by t in (34). The following continuity bound for the Holevo quantity not depending on the size m of an ensemble is obtained by Oreshkov and Calsamiglia in [20]: |χ({pi , ρi }) − χ({qi , σi })| ≤ 2εK log(d − 1) + 2h2 (εK ),
εK ≤ (d − 1)/d,
where d = dim HA and εK is the Kantorovich distance between the ensemPm 1 bles {pi , ρi } and {qi , σi }. Since εK ≥ ε = 2 i=1 kpi ρi − qi σi k1 [20, Ap.B], Proposition 3 gives stronger continuity bound for the Holevo quantity for d > 2.
4.2
Case B
If dim HA = m = +∞ then the Holevo quantity is not continuous on Em (HA ) (only lower semicontinuous). Conditions for its local continuity are presented in the following proposition. Proposition 4. A) Let {{pni , ρni }}n ⊂ E∞ (HA ) be a sequence converging to an ensemble {p0i , ρ0i } ∈ E∞ (HA ) in the sense (32). Then lim χ({pni , ρni }) = χ({p0i , ρ0i }) < +∞
n→∞
provided one of the following conditions is valid: ! ! X X a) lim H pni ρni = H p0i ρ0i < +∞; n→∞
i
i
b) lim S ({pni }) = S {p0i } < +∞, where S is the Shannon entropy. n→∞
B) If (35) holds then
lim χ({pni , Φ(ρni )}) = χ({p0i , Φ(ρ0i )}) < +∞
n→∞
14
(35)
for arbitrary quantum channel Φ : S(HA ) → S(HA′ ). Proof. A) The both conditions follow from Theorem 1A in [23] and representation (31), since (32) implies convergence of the corresponding sequence n 0 {ˆ ωAB } to the state ω ˆ AB (this can be shown by proving the convergence in the weak operator topology and by using the result from [7]). B) This assertion follows from Theorem 1B in [23] and (31) . Condition b) in Proposition 4A implies the following observation which can be interpreted as stability of the Holevo quantity with respect to perturbation of states of a given ensemble. Corollary 4. Let {pi } be a probability distribution with finite Shannon entropy. Then lim χ({pi , ρni }) = χ({pi , ρ0i }) ≤ S({pi })
n→∞
(36)
for any sequences {ρn1 }, {ρn2 }, . . . converging respectively to states ρ01 , ρ02 , . . . By Corollary 4 the finiteness of S({pi }) guarantees the validity of (36) despite possible discontinuity of the entropy for the sequences {ρn1 }, {ρn2 }, . . . Condition a) in Proposition 4A shows, in particular, that for any E > 0 the Holevo quantity isPcontinuous on the set of all ensembles {pi , ρi } with . the average state ρ¯ = i pi ρi satisfying the inequality TrHA ρ¯ ≤ E provided the Hamiltonian HA satisfies condition (24). The following proposition gives Winter’s type continuity bound for the Holevo quantity under energy constraint on the average state of an ensemble. Proposition 5. Let {pi , ρi } and {qi , σi } be countable ensembles of states inPS(HA ) with the average states ρ¯ and σ¯ such that Tr¯ ρHA , Tr¯ σ HA ≤ E, ∞ 1 ε′ −ε ′ i=1 kpi ρi − qi σi k1 = ε < ε ≤ 1 and δ = 1+ε′ . Then 2 |χ({pi , ρi }) − χ({qi , σi })| ≤ (ε′ + 2δ)H(γ(E/δ)) + 2g(ε′) + 2h2 (δ), (37) ε = (1 + ε) log(1 + ε) − ε log ε. This continuity where g(ε) = (1 + ε)h2 1+ε bound is asymptotically tight (for large E). Remark 4. Condition (24) implies lim δH(γ(E/δ)) = 0 [22, Pr.1]. δ→+0
Hence, Proposition 5 shows that the Holevo quantity is uniformly continuous on the set of all ensembles {pi ,P ρi } having bounded energy of the average state with respect to the distance ∞ i=1 kpi ρi − qi σi k1 between ensembles {pi , ρi } and {qi , σi }. 15
Proof. By representation (31) continuity bound (37) means that |I(A : B)ρ − I(A : B)σ | ≤ (ε′ + 2δ)H(γ(E/δ)) + 2g(ε′) + 2h2 (δ)
(38)
for arbitrary qc-states ρAB and σAB such that TrHA ρA , TrHA σA ≤ E and kρ − σk1 = 2ε. Inequality (38) can be proved by repeating the arguments from the proof of Proposition 2 with trivial C with the following two ingredients: • since ρAB and σAB are qc-states, Lemma 1 below makes possible to replace (29) by the inequality |I(A : B)ω − I(A : B)ωδ | ≤ δH(γ(E/δ)) + h2 (δ),
ω = ρ, σ.
δ • since ρδAB and σAB are qc-states, Corollary 2 makes possible to replace (30) by the inequality I(A : B)ρδ − I(A : B)σδ ≤ ε′ H(γ(E/δ)) + 2g(ε′).
The asymptotic tightness of continuity bound (37) follows from the asymptotic tightness of the continuity bound in Corollary 5 (see Remark 5 below). Lemma 1. Let PA be a projector in B(HA ) and ωAB be a qc-state (1) with finite H(ωA ). Then − (1 − τ )H(˜ ωA ) ≤ I(A : B)ω − I(A : B)ω˜ ≤ H(ωA ) − τ H(˜ ωA ),
(39)
where τ = TrPA ωA and ω ˜ AB = τ −1 PA ⊗ IB ωAB PA ⊗ IB .8 Proof. The both inequalities in (39) are easily derived from the inequalities 0 ≤ I(A : B)ω − τ I(A : B)ω˜ ≤ H(ωA ) − τ H(˜ ωA ) (40) by using nonnegativity of I(A : B) and upper bound (9). Note that representation (31) remains valid for an ensemble {pi , ρi } of any positive trace class operators if we assume that H and I(A : B) are homogenuous extensions of the von Neumann entropy and of the quantum mutual information to the cones of all positive trace class operators and that 8
For arbitrary state ωAB double inequality (39) holds with additional factors 2 in the left and in the right sides (see Lemma 9 in [24]).
16
P χ ({pi , ρi }) = H(¯ ρ) − i pi H(ρi ) provided H(¯ ρ) < +∞. This shows that the double inequality (40) can be rewritten as follows 0 ≤ χ({pi , ρi }) − χ({pi , PA ρi PA }) ≤ H(¯ ρ) − H(PA ρ¯PA ). The first of these inequalities is easily derived from monotonicity of the quantum relative entropy and concavity of the function η(x) = −x log x. The second one follows from the definition of the Holevo quantity, since H(ρi ) ≥ H(PA ρi PA ) for all i [18]. Example: the system of quantum oscillators. By using Proposition 5 and the estimates from [27] one can obtain a continuity bound for the Holevo quantity of ensembles of states of the system composed of ℓ quantum oscillators under the energy constraint on the average state of ensembles. The Hamiltonian of such system has the form HA =
ℓ X
~ωi a+ i ai ,
i=1
where ai and a+ i are the annihilation and creation operators and ωi is a frequency of the i-th oscillator [12]. To be consistent with P our assumption E0 = 0 we will consider shifted Hamiltonian HA′ = HA − 21 ℓi=1 ~ωi IA and will assume that E(ρ) = TrHA′ ρ.9 Now we can repeat all the arguments from the proof of Lemma 18 in [27] by using Proposition 5 instead of Meta-Lemmas 16,17 and obtain the following Corollary 5. Let {pi , ρi } and {qi , σi } be countable ensembles of states of the quantum system composed of ℓ oscillators with the average states ρ¯ P∞ 1 ′ ′ and σ ¯ such that Tr¯ ρHA , Tr¯ σ HA ≤ E, 2 i=1 kpi ρi − qi σi k1 ≤ ε ≤ 1 . Then " ℓ # X 1+α E e |χ({pi , ρi }) − χ({qi , σi })| ≤ ε 1−α log ℓ~ω + 2α + 1 + ℓ log α(1−ε) i i=1
+ℓ
1+α 1−α
˜ 2 (ε) + 2h ˜ 2 (αε) + 2g + 2α h
1+α ε 1−α
,
˜ 2 (x) = h2 (x) for x ≤ 1/2 and h ˜ 2 (x) = 1 for x ≥ 1/2, where α ∈ (0, 21 ), h g(x) = (x + 1) log(x + 1) − x log x. 9
One can Pℓ take into account in Corollary 5 below that the real energy of ρ is equal to E(ρ) + 12 i=1 ~ωi .
17
Note that the main term in this continuity bound coincides with the main term in the continuity bound for the von Neumann entropy of states of the system of ℓ oscillators with the energy not exceeding E presented in Lemma 16 in [27]. Parameter α in Corollary 5 is a free parameter which can be used to optimize the continuity bound for given value of energy E. It is easy to see that for large energy E the optimal value of α is close to zero, so the main term in this continuity bound is approximately equal to ℓ X E log ℓ~ω εH(γ(E)) ≈ ε + 1 . i i=1
Remark 5. To show the asymptotical tightness of the continuity bound in Proposition 5 for large E it suffices to show this property for the continuity bound in Corollary 5. By the above note this can be done by finding for given ε > 0 and E > 0 two ensembles {pi , ρi } and {qi , σi } satisfying the condition of Corollary 5 such that |χ({pi , ρi }) − χ({qi , σi })| ≥ εH(γ(E)) .
(41)
Let {pi , ρi } be any pure state ensemble with the average state γ(E) and qi = pi , σi = (1 − ε)ρi + εγ(E) for all i. Then ∞ ∞ X X kpi ρi − qi σi k1 = εpi kρi − γ(E)k1 ≤ 2ε i=1
i=1
while (41) follows from concavity of the entropy.
5
Applications
Now we apply the results of the previous sections to obtain tight and closeto-tight continuity bounds for basic capacities of a quantum channel. We restrict attention to channels with finite-dimensional output.
5.1
Tight continuity bounds for the Holevo capacity and for the entanglement-assisted classical capacity of a quantum channel
A quantum channel Φ from a system A to a system B is a completely positive trace preserving linear map T(HA ) → T(HB ), where HA and HB are Hilbert 18
spaces associated with the systems A and B. The Holevo capacity of a quantum channel Φ : A → B is defined as follows ¯ C(Φ) = sup χ({pi , Φ(ρi )}), (42) {pi ,ρi }∈E(HA )
where the supremum is over all ensembles of input states. This quantity is closely related to the classical capacity of a quantum channel (see Section 5.2 below). The classical entanglement-assisted capacity of a quantum channel determined an ultimate rate of transmission of classical information when an entangled state between the input and the output of a channel is used as an additional resource (see details in [12, 19, 26]). By the Bennett-ShorSmolin-Thaplyal theorem the classical entanglement-assisted capacity of a finite-dimensional quantum channel Φ : A → B is given by the expression Cea (Φ) =
sup I(Φ, ρ),
(43)
ρ∈S(HA )
in which I(Φ, ρ) is the quantum mutual information of the channel Φ at a state ρ defined as follows I(Φ, ρ) = I(B : R)Φ⊗IdR (ˆρ) ,
(44)
where HR ∼ = HA and ρˆ is a pure state in S(HAR ) such that ρˆA = ρ [6, 12, 26]. ¯ In analysis of variations of the capacities C(Φ) and Cea (Φ) as functions of a channel we will use two norms on the set of all completely positive maps Φ : T(HA ) 7→ T(HB ): the operator norm . kΦk =
sup
kΦ(ρ)k1
ρ∈T(HA ),kρk1 =1
and the diamond norm . kΦk⋄ =
sup
kΦ ⊗ IdR (ρ)k1 ,
ρ∈T(HAR ),kρk1 =1
which coincides with the norm of complete boundedness of the dual map Φ∗ : B(HB ) 7→ B(HA ) to the map Φ [12, 26]. Proposition 3 and Corollary 1 imply the following
19
Proposition 6. Let Φ and Ψ be quantum channels from A to B and ε g(ε) = (1 + ε)h2 1+ε . Then ¯ ¯ |C(Φ) − C(Ψ)| ≤ ε log dB + 2g(ε),
(45)
where ε = 21 kΦ − Ψk and dB = dim HB , and |Cea (Φ) − Cea (Ψ)| ≤ 2ε log d + 2g(ε),
(46)
where ε = 21 kΦ − Ψk⋄ and d = min{dim HA , dim HB }. The both continuity bounds (45) and (46) are tight. Proof. For given ensemble {pi , ρi } Proposition 3 shows that |χ({pi , Φ(ρi )}) − χ({pi , Ψ(ρi )})| ≤ ε log dB + 2g(ε), P where ε = 21 i pi kΦ(ρi ) − Ψ(ρi )k1 ≤ 21 kΦ − Ψk. This and (42) imply (45). Continuity bounds (46) is derived similarly from Corollary 1 and expression (43), since for any pure state ρˆAR in (44) we have kΦ ⊗ IdR (ˆ ρ) − Ψ ⊗ IdR (ˆ ρ)k1 ≤ kΦ − Ψk⋄ . To show the tightness of the both continuity bounds assume that HA = HB = Cd , Φ is the noiseless channel (i.e. Φ = IdCd ) and Ψ is the depolarizing channel: Ψ(ρ) = (1 − p)ρ + pd−1 ICd , p ∈ [0, 1]. Since
¯ C(Ψ) = log d + (1 − pc) log(1 − pc) + pc log(p/d), ¯ ¯ where c = 1 − 1/d [12], Cea (Φ) = 2C(Φ) = 2 log d and Cea (Ψ) ≤ 2C(Ψ), we have ¯ ¯ C(Φ) − C(Ψ) = pc log d + h2 (pc) + pc log c and Cea (Φ) − Cea (Ψ) ≥ 2pc log d + 2h2 (pc) + 2pc log c. These relations show the tightness of continuity bound (45) and (46), since it is easy to see that kΦ − Ψk ≤ kΦ − Ψk⋄ ≤ 2p.
20
5.2
Refinement of the Leung-Smith continuity bounds for classical and quantum capacities of a channel
By the Holevo-Schumacher-Westmoreland theorem the classical capacity of a finite-dimensional channel Φ : A → B is given by the expression (cf.[12, 26]) ¯ ⊗n ), C(Φ) = lim n−1 C(Φ
(47)
n→+∞
¯ where C(Φ) is the Holevo capacity defined in the previous subsection. By the Lloyd-Devetak-Shor theorem the quantum capacity of a finitedimensional channel Φ : A → B is given by the expression (cf.[12, 26]) ¯ ⊗n ), Q(Φ) = lim n−1 Q(Φ
(48)
n→+∞
. ¯ where Q(Φ) is the maximum of the coherent information Ic (Φ, ρ) = H(Φ(ρ))− b b is a complementary channel to Φ). H(Φ(ρ)) over all states ρ ∈ S(HA ) (Φ Leung and Smith obtained in [14] the following continuity bounds for classical and quantum capacities of a channel with finite-dimensional output |C(Φ) − C(Ψ)| ≤ 16ε log dB + 4h2 (2ε) ,
(49)
|Q(Φ) − Q(Ψ)| ≤ 16ε log dB + 4h2 (2ε) ,
(50)
where ε = 12 kΦ−Ψk⋄ and dB = dim HB .10 By using Winter’s tight continuity bound (6) for the conditional entropy (instead of the original Alicki-Fannes continuity bound) in the Leung-Smith proof one can replace the main terms in (49) and (50) by 4ε log dB . By using Corollary 1 and modifying the arguments of Leung and Smith one can replace the main terms in (49) and (50) by 2ε log dB (which gives tight continuity bound for the quantum capacity and close-to-tight continuity bound for the classical capacity). Proposition 7. Let Φ and Ψ be arbitrary channels from A to B. Then |C(Φ) − C(Ψ)| ≤ 2ε log dB + 2g(ε), |Q(Φ) − Q(Ψ)| ≤ 2ε log dB + 2g(ε), where ε = 21 kΦ − Ψk⋄ , dB = dim HB and g(ε) = (1 + ε)h2
ε 1+ε
The continuity bound for the quantum capacity Q is tight. 10
.
It is assumed that expressions (47) and (48) remain valid in the case dim HA = +∞.
21
The tight continuity bound (45) for the Holevo capacity shows that the continuity bound for the classical capacity C in Proposition 7 is close-to-tight (up to the factor 2 in the main term). Proof. The continuity bound for the classical capacity C is directly obtained by using Lemma 12 in [14], representation (31) and Lemma 2 below. To prove the continuity bound for the quantum capacity Q note that the coherent information can be represented as follows Ic (Φ, ρ) = I(B : R)Φ⊗IdR (ˆ ρ) − H(ρ) where ρˆ ∈ S(HAR ) is a purification a state ρ. Hence for arbitrary quantum ⊗n channels Φ and Ψ, arbitrary n and any state ρ in S(HA ) we have ρ) ρ) − I(B n : Rn )Ψ⊗n ⊗IdRn (ˆ Ic (Φ⊗n , ρ) − Ic (Ψ⊗n , ρ) = I(B n : Rn )Φ⊗n ⊗IdRn (ˆ ⊗n where ρˆ ∈ S(HAR ) is a purification of the state ρ. This representation, Lemma 2 below and Lemma 12 in [14] imply the continuity bound for the quantum capacity. To show the tightness of the continuity bound for the quantum capacity consider the family of erasure channels (1 − p)ρ 0 , p ∈ [0, 1]. (51) Φp (ρ) = 0 pTrρ
from d-dimensional system A to d + 1-dimensional system B. It is easy to show (see [12, Ch.10]) that Q(Φp ) = (1 − 2p) log d for p ≤ 1/2 and Q(Φp ) = 0 for p ≥ 1/2. Hence Q(Φ0 ) − Q(Φp ) = 2p log d for p ≤ 1/2. By noting that kΦ0 − Φp k⋄ ≤ 2p we see that continuity bound for the quantum capacity is tight (for large d). The following lemma is an I(A : B)-analog of Theorem 11 in [14]. It is proved by the same telescopic trick (by using the conditional mutual information instead of the conditional entropy). Lemma 2. Let Φ and Ψ be channels from arbitrary system A to a finite-dimensional system B, C be any system and n ∈ N. Then |I(B n : C)Φ⊗n ⊗IdC (ρ) − I(B n : C)Ψ⊗n ⊗IdC (ρ) | ≤ 2nε log dB + 2ng(ε), ⊗n for any state ρ ∈ S(HA ⊗ HC ), where ε = 12 kΦ − Ψk⋄ and dB = dim HB . This continuity bound is tight (for any given n and large dB ).
22
Proof. Following the proof of Theorem 11 in [14] introduce the states σk = Φ⊗k ⊗ Ψ⊗(n−k) ⊗ IdC (ρ),
k = 0, 1, ..., n.
We have |I(B n : C)σn
n X I(B n : C)σk − I(B n : C)σk−1 − I(B n : C)σ0 | = k=1 n X I(B n : C)σ − I(B n : C)σ . ≤ k k−1
(52)
k=1
By using the chain rule
I(X : Y Z) = I(X : Y ) + I(X : Z|Y ) we obtain for each k (cf.[14]) I(B n : C)σk − I(B n : C)σk−1 = I(B1 ...Bk−1 Bk+1 ...Bn : C)σk + I(Bk : C|B1 ...Bk−1 Bk+1 ...Bn )σk − I(B1 ...Bk−1 Bk+1 ...Bn : C)σk−1 − I(Bk : C|B1 ...Bk−1 Bk+1 ...Bn )σk−1 = I(Bk : C|B1 ...Bk−1 Bk+1 ...Bn )σk − I(Bk : C|B1 ...Bk−1 Bk+1 ...Bn )σk−1 , where it was used that TrBk σk = TrBk σk−1 . So, Corollary 1 gives I(B n : C)σ − I(B n : C)σ ≤ 2ε log dB + 2g(ε), k k−1
for any k, where ε = 21 kσk − σk−1 k1 ≤ 12 kΦ − Ψk⋄ . This and (52) imply the required inequality (since Φ⊗n ⊗ IdC (ρ) = σn and Ψ⊗n ⊗ IdC (ρ) = σ0 ). The tightness of the continuity bound in Lemma 2 for any given n can be easily shown by using the erasure channels (51) again. Indeed, let Φ = Φ0 , Ψ = Φp , C = A and ρ be a maximally entangled pure state in S(HAC ). Then I(B : C)Φ⊗IdC (ρ) = 2 log dA and by using inequality (16) it is easy to show that I(B : C)Ψ⊗IdC (ρ) ≤ 2(1 − p) log dA + h2 (p). So, we have I(B n : C n )Φ⊗n ⊗IdC n (ρ⊗n ) = nI(B : C)Φ⊗IdC (ρ) = 2n log dA , I(B n : C n )Ψ⊗n ⊗IdC n (ρ⊗n ) = nI(B : C)Ψ⊗IdC (ρ) ≤ 2n(1 − p) log dA + nh2 (p) 23
and hence I(B n : C n )Φ⊗n ⊗IdC n (ρ⊗n ) − I(B n : C n )Ψ⊗n ⊗IdC n (ρ⊗n ) ≥ 2np log dA − nh2 (p) Since dB = dA + 1 and kΦ − Ψk⋄ ≤ 2p, this shows the tightness of the continuity bound in Lemma 2 for large dB . I am grateful to A.Winter for valuable communication and the technical tricks from his recent paper [27] essentially used in this work. I am also grateful to A.S.Holevo and G.G.Amosov for useful discussion. The research is funded by the grant of Russian Science Foundation (project No 14-21-00162).
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