Commun. Math. Phys. 364, 83–121 (2018) Communications in Digital Object Identifier (DOI) https://doi.org/10.1007/s00220-018-3249-y Mathematical Physics

Capacity Estimates via Comparison with TRO Channels

Li Gao1 , Marius Junge1, Nicholas LaRacuente2 1 Department of Mathematics, University of Illinois, Urbana, IL 61801, USA. E-mail: [email protected]; [email protected] 2 Department of Physics, University of Illinois, Urbana, IL 61801, USA. E-mail: [email protected]

Received: 1 November 2017 / Accepted: 27 July 2018 Published online: 8 September 2018 – © Springer-Verlag GmbH Germany, part of Springer Nature 2018

Abstract: A ternary ring of operators (TRO) in finite dimensions is an operator space as an orthogonal sum of rectangular matrices. TROs correspond to quantum channels that are diagonal sums of partial traces, we call TRO channels. TRO channels have simple, single-letter entropy expressions for quantum, private, and classical capacity. Using operator space and interpolation techniques, we perturbatively estimate capacities, capacity regions, and strong converse rates for a wider class of quantum channels by comparison to TRO channels.

1. Introduction

Channel capacity, introduced by Shannon in his foundational paper [45], is the ultimate rate at which information can be reliably transmitted over a communication channel. During the last decades, Shannon’s theory on noisy channels has been adapted to the framework of quantum physics. A quantum channel has various capacities depending on different communication tasks, such as quantum capacity for transmitting qubits, and private capacity for transmitting classical bits with physically ensured security. The coding theorems, which characterize these capacities by entropic expressions, were major successes in quantum information theory (see e.g. [56]). For instance, the quantum capacity Q(N ) of a channel N , by Lloyd–Shor–Devetak Theorem [14,34,46], is given by

(1) ⊗k Q (N ) ( )  Q(N ) = lim , Q 1 (N ) = max H(N (ρ)) − H(id ⊗ N (ρ AA )), k→∞ k ρ (1) where H(ρ) =−tr(ρ log ρ)is the entropy function, and the maximum runs over all pure  bipartite states ρ AA . Nevertheless, the capacities for many channels are computationally intractable due to regularization, the limit in which one takes the entropic expression (1) 84 L. Gao, M. Junge, N. LaRacuente over asymptotically many uses of the channel. Regularization is in general unavoidable, because the capacity of a combination of mutiple quantum channels may exceeds the sum of their individual capacities [13,48,50]. This phenomenon, called “super-additivity”, also exists for classical and private capacities [19,23,33]. Devetak and Shor in [16] consider degradable channels, for which the receiver can fully reproduce the information lost to the environment by “degrading” the received output through another channel. Degradable channels are additive, admitting the trivial regularization Q = Q(1) and a simple “single-letter” formula for quantum capacity. Several different methods have been introduced to give upper bounds on particular or general channels (e.g. [27,49,51,53,54]). Little is known about the exact value of quantum capacity beyond degradable cases. In addition, it is desirable to know that whether the strong converse theorem holds for quantum channels. That is, for quantum, private, and classical communication, quantum channels leave open the question whether there is a sharp trade off between the transmission rate and transmission accuracy, or there could exist an intermediate regime in which errors are necessary but few. In this paper, we give capacities estimate for quantum channels via their Stinespring dilation. We briefly explain our main idea in the following. A quantum channel N is a completely positive trace preserving (CPTP) map that sends densities (positive trace 1 operators) from one Hilbert space HA to another HB. N admits a Stinespring dilation as follows ∗ N (ρ) = trE (VρV ), where V : HA → HB ⊗ HE is a partial isometry and HE is the Hilbert space of the environment. We call the range ran(V ) ⊂ HB ⊗ HE Stinespring space of N .The capacities of a channel are actually determined by its Stinespring space, more precisely ∼ the operator space structure by viewing ran(V ) ⊂ HB ⊗ HE = B(HE , HB ) as operators from HE to HB. This perspective was previously used in [2] to understand Hastings’ counterexamples for additivity of minimal output entropy. A ternary ring of operators (TRO) is a closed operator subspace X closed under the triple product x, y, z ∈ X ⇒ xy∗z ∈ X. TROs were first introduced by Hestenes [24], and pursued by many others (see e.g. [30,63]). In finite dimensions, TROs are always diagonal sums of rectangular matri- ⊕ ⊗ ces i Mni 1mi (with multiplicities mk), and the quantum channel induced by them are diagonal sums of partial traces (Proposition 1). These simple channels have well- understood capacities [20] and the strong converse property. Let N be a channel and its Stinespring space ran(V ) be a TRO in B(HE , HB ). We consider the channel   ∗ N f (ρ) = trE (1 ⊗ f )VρV , for which the Stinepsring dilation is modified by multiplying a operator f on the en- vironment HE . With certain assumptions on f , N f is also a quantum channel and we proved that the capacity of N f is comparable to the original N in the following way,

Q(N ) ≤ Q(N f ) ≤ Q(N ) + τ(f log f ) (2) τ( ) = 1 ( ) where Q is the quantum capacity in 1 and f log f |E| trE f log f is a normalized entropy of f . This is a perturbation estimate from TRO channels, which have clear single- letter capacity formulae. One main class of our examples are random unitary channels Capacity Estimates via Comparison with TRO Channels 85 arising from group representations. Let G be a finite group and u : G → B(H) be a (projective) unitary representation. For probability distributions f on G, we define the random unitary  1  N (ρ) = f (g)u(g)ρu(g)∗, N (ρ) = u(g)ρu(g)∗. f |G| g g

N N ( 1 , ··· , 1 ) Here is a special case of f with f being the uniform distribution |G| |G| on G, and its capacity Q(N ) is given by the logarithm of the largest dimensions in the irreducible decomposition of u. The inequality implies that

Q(N ) ≤ Q(N f ) ≤ Q(N ) +log|G|−H( f ), (3)  where |G| is the order of G and H( f ) =− f (g) log f (g) is the Shannon entropy. The key inequality in our argument is the following “comparison property”: for any positive operators σ and ρ,

1 1 1 1 −  −  −  −  N (σ) 2p N (ρ)N (σ) 2p p ≤ N (σ) 2p N f (ρ)N (σ) 2p p 1 1 −  −  ≤ f p,τ N (σ) 2p N (ρ)N (σ) 2p p, (4)

1 1 = (| |p) p = 1 (| |p) p where a p tr a is the Schatten p-norm, f p,τ |E| trE f is the p- 1 1 = norm of normalized trace and p + p 1. The “local comparison property” is actually an inequality of sandwiched Rényi relative entropy introduced in [36,61]. The sandwiched Rényi relative entropies are used to prove the strong converse for entanglement-assisted communication [22], and to give upper bounds on the strong converse of classical com- munication [61] and quantum communication [53]. More recently, relative entropy of entanglement is shown to be a private converse rate [39], and later extended to a private strong converse rate via its sandwiched Rényi analogs [60]. Based on their results, we find that our capacity upper bound (2) are also strong converse rates for both quantum and private communication. Our method is compatible with the strong converse bounds [8,10,53,55] and gives estimate with a simple correction term. We organize this work as follows. Section 2 recalls the concept of TROs from operator algebras and proves the “local comparison theorem”. Section 3 is devoted to applications on estimating capacities, capacity regions and strong converse rates. Section 5 discusses examples from group representations. We provide an appendix describing the complex interpolation technique used in our argument.

2. TRO Channels and Local Comparison Property

2.1. Channels and Stinespring spaces. We denote by B(H) the bounded operators on a Hilbert space H. We restrict ourselves to finite dimensional Hilbert spaces and write |H| for the dimension of H. The standard n-dimensional Hilbert space is denoted by Cn and n × n matrix space is Mn.Astate on H is given by a density operator ρ in B(H), i.e. ρ ≥ 0, tr(ρ) = 1, where “tr” is the matrix trace. The physical systems and their Hilbert spaces are indexed by capital letters as A, B, ···. We use superscripts to track AB multipartite states and their reduced densities, i.e. for a bipartite state ρ on HA ⊗ HB, A AB ρ = trB(ρ ) presents its reduced density matrix on A.Weuse1A (resp. 1n)forthe 86 L. Gao, M. Junge, N. LaRacuente identity operator in B(HA) (resp. Mn), and idA (resp. idn) for the identity map on B(HA) (resp. Mn). Let N : B(HA) → B(HB) be a quantum channel (CPTP map) with Stinespring ∗ dilation given by N (ρ) = trE (VρV ), where HE is a Hilbert space, V : HA → HB ⊗ HE is an isometry and trE stands for the trace on HE . The complementary channel of N is E E ∗ N : B(HA) → B(HE ), N (ρ) = trB(VρV ). (5) This dilation (5) is not unique, but different ones are related by partial isometries between the environment systems. Given an orthonormal basis {|ei } of HE and its dual basis { |} ∗ ⊗ ei in HE , one can identify the tensor product Hilbert space HB HE with the operators B(HE , HB ) as follows,   |h= |hi ⊗|ei →h = |hi ⊗ei |, |hi ∈HB. i i

This identification depends on the choice of the basis {|ei } but is unique up to a unitary equivalence. It acts as a partial trace on pure bipartite states, ∗ ∗ trE (|hk|) = hk , trB(|hk|) = k h. (6) Throughout this paper we will use “bra-ket” notation for vectors and dual vectors. The Stinespring space X = ran(V ) then becomes an operator subspace of B(HE , HB).Note that the Hilbert space X is isomorphic to the input system HA by V . We can identify |h with V |h and denote the operator analog to V |h∈HB ⊗ HE by h as follows,

|h∈HA ←→ V |h∈HB ⊗ HE ←→ h ∈ B(HE , HB ). With this notation, the channel and its complementary channel can be viewed as the restriction of partial traces on the Stinespring space as

∗ E ∗ N (|hk|) = hk , N (|hk|) = k h, for |h, |k∈HA. (7)

2.2. TROs and TRO channels. Let us recall that a ternary ring of operators (TRO) X between Hilbert spaces H and K is a closed subspace of B(H, K ) stable under the triple product x, y, z ∈ X ⇒ xy∗z ∈ X. ATROX is a corner of its linking C∗-algebra A(X) introduced in [9],     xy∗ z L(X) X A(X) = span{ ∗ ∗ | x, y, z, u,v,w ∈ X}= ∗ . w v u X R(X)

The two diagonal blocks are C∗-algebras, ∗ ∗ L(X) = span{ xy | x, y ∈ X}⊂B(K ), R(X) = span{x y | x, y ∈ X}⊂B(H). They together with A(X) play an important role in the study of TROs (see again e.g. [30]). In particular, X is a natural L(X)-R(X) bimodule L(X)X = X , XR(X) = X. Capacity Estimates via Comparison with TRO Channels 87

In finite dimensions, TROs are direct sums of rectangular matrices with multiplicity. ⊕ ( ⊗ ) Namely, a TRO X is isomorphic to i Mni ,mi 1li , where li is the multiplicity of ith diagonal block Mni ,mi . In this situation, L( ) =⊕( ⊗ ) ⊂ , R( ) =⊕( ⊗ ) ⊂ . X i Mni 1li Mn X i Mmi 1li Mm

In most of our discussions, the multiplicities li are irrelevant and we may simple write =∼ ⊕ X i Mni ,mi . Proposition 1. Let N be a quantum channel with its Stinespring space X being a T RO. Then N is a direct sum of partial traces and the ranges ran(N ) = L(X), ran(N E ) = R(X).

Proof. We can decompose X as =⊕ , =∼ . X i Xi Xi Mni ,mi

Because Xi are mutually orthogonal subspaces in X, the channel N can be written as N (|  |) = ∗ =⊕ ∗, | =⊕| , | =⊕| , x y xy i xi yi x i xi y i yi where |xi , |yi ∈Xi . It is sufficient to see on each subspace Xi , N is a partial trace. =∼ =∼ Cni ⊗ Cmi Indeed, by identifying Xi Mni ,mi as Hilbert spaces, we know from (6) that N =⊕i Ni and N (|  |) = ∗ = ⊗ (|  |). xi yi xi yi idni trmi xi yi 

⊕ ⊗ Remark 1. To be precise, X may be of the form i Mni ,mi 1li with the multiplicity li for i-th block. Each direct summand Ni is a “generalized” partial traces as follows N (ρ ) = ( ⊗ (ρ )) ⊗ ω , i i idni trmi i lk ω = 1 N where lk l 1lk is the lk-dimensional completely mixed state. Namely, i is a partial k π N =⊕N trace plus a dummy state lk . The channel i i here is equivalent to the one in Proposition 5 without redundancy, in the sense that they can factor through each other. In most of situations they are equivalent and we will use the simpler identification =⊕ X i Mni ,mi . τ = tr B( ) Let |H| be the normalized trace on H . A positive operator f is a normalized density if τ(f ) = 1. Note that this normalization differs from the usual matrix trace—for instance, the identity operator 1 is a normalized density. This normalized trace is more natural in von Neumann algebras and will simplify our notations. Given a C∗-subalgebra M ⊂ B(H),theconditional expectation EM is the unique CPTP and unital map from B(H) onto M (M + C1 if nonunital) such that

τ(EM (x)y) = τ(xy) for x ∈ B(H), y ∈ M. (8)

We say a positive operator x is independent of M if EM (x) = τ(x)1, or equivalently τ(xy) = τ(x)τ(y), for all y ∈ M, 88 L. Gao, M. Junge, N. LaRacuente and x is strongly independent of M if all the powers xn are independent of M.Nowwe define our modified TRO channels. Let N : B(H ) → B(H ) be a quantum channel A B ∼ with its Stinespring space X being a TRO. With identification HA = {|x|x ∈ X} as Hilbert spaces,

∗ N (|xy|) = xy , x, y ∈ X ⊂ B(HE , HB ).

Then for any operator f on HE , we define the following map ∗ N f : B(HA) → B(HB), N f (|xy|) = xfy .

Clearly, N f = N when f = 1.

Proposition 2. Let N f be defined as above. Suppose f ∈ B(HE ) is operator indepen- dent of the right algebra R(X). Then EL(X) ◦ N f = τ(f )N . In particular, N f is a quantum channel if f is a normalized density independent of R(X), and its Stinespring isometry is given by  V f : HA → HB ⊗ HE , V f |x=|x f .

Proof. Let a ∈ L(X). By independence, we have that for any |x, |y,

1 1 τ (a N (|xy|)) = tr (ax f y∗) = tr (y∗ax f ) B f |B| B |B| E 1 1 = τ ( f )tr (y∗ax) = τ ( f )tr (axy∗) |B| E E |B| E B = τE ( f )τB(a N (|xy|)).

Thus,

EL(X) ◦ N f (|xy|) = τ(f )N (|xy|) holds for any rank one operator |xy| and by linearity for arbitrary operators. Note that for positive f ,   ∗ N f (|xy|) = xfy = N (|x f y f |).

Then it is sufficient to verify that V f is an isometry given τE ( f ) = 1. Indeed, we have  2 ∗ ∗ ∗ 2 |x f  = trB(xfx ) = trE (x xf) = τE ( f )trE (x x) = |x .

In the second last equality we use the assumption that f is independent of R(X). 

Definition 1. We say N is a TRO channel if its Stinespring space is a TRO X.Let f ∈ B(HE ) be a normalized density independent of R(X).Wesay f is a symbol of N and

∗ N f (|xy|) = xfy , x, y ∈ X is a modified TRO channel. Capacity Estimates via Comparison with TRO Channels 89

p 1/p 2.3. Local comparison property. Recall that for 1 ≤ p ≤∞, a p= tr(|a| ) 1 p represents the Schatten p-class Sp norm and f p,τ = τ(| f | ) p the L p norm with τ 1 1 = respect to normalized trace . We fix the relation p + p 1. Our key technical theorem is proved under the stronger independence assumption.

Theorem 1. Let N be a TRO channel with Stinespring space X. Suppose f is a symbol strongly independent of R(X). Then for any positive operators σ ∈ L(X) and ρ,

N (ρ) p≤ N f (ρ) p≤ f p,τ N (ρ) p, 1 1 1 1 1 1 −  −  −  −  −  −  σ 2p N (ρ)σ 2p p≤ σ 2p N f (ρ)σ 2p p≤ f p,τ σ 2p N (ρ)σ 2p p .

Proof. We give the proof for the second inequality. The argument for the first one is similar and easier. Let EL(X) : B(HB) → L(X) be the conditional expectation onto L(X). By definition EL(X) is a quantum channel. From Proposition 2 and the assumption of f , EL(X) ◦ N f = N and EL(X)(σ) = σ for σ ∈ L(X), since L(X) is the range of N . Then the first inequality of (1) is an direct consequence of data processing inequality of Rényi sandwiched relative entropy (see Sect. 3 and e.g. [36]). Let σ −1 be the inverse of σ on its support. Write ρ = ηη∗ with η ∈ B(H , X) ⊂ B(H , H ⊗ H ) for some A A B E ∼ Hilbert space HA. Denote by ηˆ the corresponding operator of η via B(HA, HB ⊗ HE ) = B(HA ⊗ HE , HB). We can write

1 1 1 −  −  −  1 2 σ 2p N (ρ)σ 2p = σ 2p η(ˆ 1 ⊗ f 2 ) . f p A S2p(HA⊗HE ,HB )

Here S2p(HA ⊗ HE , HB) is the Schatten p-class of operators B(HA ⊗ HE , HB).For the second inequality, it is sufficient to show that

1 1 −  1 1 −  σ 2p η(ˆ ⊗ 2 ) ≤ 2 σ 2p ηˆ . 1A f S2p(HA⊗HE ,HB ) f 2p,τ S2p(HA⊗HE ,HB )

We prove it by a complex interpolation argument (see “Appendix” for basic information about complex interpolation). Define the norms

1 x p,σ := σ p x p,

˜ and X p,σ as the space HA ⊗ X equipped with above norms. Theorem 4 in the Appendix ˜ verifies that X p,σ forms a interpolation family and in particular

˜ ˜ ˜ Xσ,2p =[X∞, Xσ,2] 1 . p

1 1 −  1 Now assuming that f 2 2p,τ σ 2p ηˆ 2p < 1, we have σ 2p ξ0 2p < 1 where − 1 ˜ ∼ ξ0 = σ 2 ηˆ in X = X ⊗ B(HA, C). Then there exists an analytic function ξ : S = ˜ {z| 0 ≤ Re(z) ≤ 1}→X such that ξ(1/p) = ξ0 and moreover

1 ξ(it) ∞< 1, σ 2 ξ(1+it) 2< 1. 90 L. Gao, M. Junge, N. LaRacuente

z pz Given this, we define another analytic function T (z) = σ 2 ξ(z)(1A ⊗ a ), where 1 = f 2 a 1 . Observe that f 2 2p,τ

it itp T (it) ∞ = σ 2 ξ(it)1A ⊗ a ∞= ξ(it) ∞ < 1, 1+it p(1+it) 1 p T (1+it) 2 = σ 2 ξ(1+it)(1 ⊗ a ) 2= σ 2 ξ(1+it)(1 ⊗ a 2 ) 2 1 p = σ 2 ξ(1+it) 2 a 2 < 1.

p The last equality uses the fact a 2 2= 1 and the assumption f is strongly independent of R(X). By complex interpolation (Theorem 3 in Appendix), we obtain

1 1 1 −  1 T (1/p) 2p= σ p ξ(1/p)1A ⊗ a 2= √ σ 2p η(ˆ 1A ⊗ f 2 ) 2p≤ 1, f 2p,τ which completes the proof.  Remark 2. In above theorem we actually obtain an inequality of Rényi sandwiched rel- ative entropy (see Sect. 3). As we argue in the proof, the lower bound follows from the data processing inequality. The other side gives a upper bound on the difference of Rényi relative entropy when the Stinespring isometry is modified by a operator f from the environment satisfying the independence condition. Note that we have to restrict σ in the left algebra L(X), because in the interpolation argument we have to ensure the range of the analytic map belongs to TRO subspace. The above result is a property which applies for every input ρ. We naturally consider the restrictions of TRO channels on subspaces. Recall that we use the notation h for the operators in B(HE , HB ) corresponding to the vector V |h∈HB ⊗ HE .

Definition 2. Let N : B(HA) → B(HB) be a quantum channel with Stinespring space Y . We call a normalized density f ∈ B(HE ) a symbol of N if f is independent of the C∗-algebra L(Y ) generated by YY∗ ={hk∗||h, |k∈Y };anadmissible symbol if f is strongly independent of L(Y ). For each symbol f , we define the modified channel N f as follows, ∗ ∼ N f (|xy|) = xfy , |x, |y∈HA = Y. (9) Remark 3. (a) Let L(Y ) be the C∗-algebra generated by {x∗ y | x, y ∈ Y } and R(Y ) be the C∗-algebra generated by {yx∗| y, x ∈ Y }. Then X = Y R(Y ) = L(Y )Y is a TRO and actually the smallest TRO containing Y . Therefore, every admissible symbol of N gives rise to a modified X-TRO channel M (|xy|) = xfy∗, x, y ∈ X and N is the ∼ f f restriction of M f on HA = Y . (b) Using this terminology every channel is a restriction of a TRO channel with a trivial symbol 1. However, the smallest TRO obtained from the minimal Stinespring dilation may produce a large left algebra L(Y ), which leads to ineffective capacity estimates. Our estimates are most useful when the TRO is much smaller than the full unitary group on the input space. The local comparison property automatically generalizes to the restrictions. Corollary 1. Let N be a quantum channel with Stinespring space X. Assume that f is an admissible symbol for N , then Capacity Estimates via Comparison with TRO Channels 91

(i) EL(X) ◦ N f = N ; (ii) for any positive operators σ ∈ L(X) and ρ,

1 1 1 1 1 1 −  −  −  −  −  −  σ 2p N (ρ)σ 2p p≤ σ 2p N f (ρ)σ 2p p≤ f p,τ σ 2p N (ρ)σ 2p p . The definition of symbols is compatible with tensor products.

Proposition 3. Let N : B(HA) → B(HB) and M : B(HA ) → B(HB ) be two quan- tum channels. Let f be an admissible symbol for N and g be an admissible sym- bol for M. Then f ⊗ g is an admissible symbol for N ⊗ M. Moreover, we have (N ⊗ M) f ⊗g = N f ⊗ Mg. In particular, for any channel M the identity operator 1 is always an admissible symbol and (N ⊗ M) f ⊗1 = N f ⊗ M. Proof. Let Y N be the Stinespring spaces of N and Y M be the Stinespring spaces of M. Since f and g are admissible symbols for N and M respectively, there exists TROs N N M M X ⊂ B(HE , HB) containing Y and X ⊂ B(HE , HB ) containing Y such that f is strongly independent of R(XN ) and g is independent of R(XM). Then N M ∼ X ⊗ X ⊂ B(HE , HB ) ⊗ B(HE , HB ) = B(HE ⊗ HE , HB ⊗ HB ) is a TRO containing the Stinespring space Y N ⊗M = Y N ⊗ Y M. It is easy to see that N M N M f ⊗ g ∈ B(HE ) ⊗ B(HB) is independent of R(X ⊗ X ) = R(X ) ⊗ R(X ). Moreover, f ⊗ g is again a normalized density hence an admissible symbol for N ⊗M. Let V be Stinespring isometry of N and W of M.For|h0, |k0∈HA, |h1, |k1∈HA ,

(N ⊗ M) f ⊗g(|h0k0|⊗|h1k1|) = (h0 ⊗ h1)( f ⊗ g)(k0 ⊗ k1) = ( ∗) ⊗ ( ∗) = N (|  |) ⊗ M (|  |). h0 fk0 h1gk1 f h0 k0 g h1 k1 

3. Applications to Capacity Estimates 3.1. Entropic inequalities. Recall that the relative entropy D(ρ||σ)for two states ρ and σ is defined as,

tr(ρ log ρ − ρ log σ) if supp(ρ) ⊂ supp(σ) D(ρ||σ) = . +∞ else

AB For a bipartite state ρ , the coherent information Ic(AB)ρ, mutual information I (A : B)ρ are given by, AB B AB B I (AB)ρ = H(ρ ) − H(ρ ) = inf D(ρ ||1 ⊗ σ ), c σ A A B AB AB A B I (A : B)ρ = H(ρ ) + H(ρ ) − H(ρ ) = inf D(ρ ||ρ ⊗ σ ), σ B where the infimum runs over all states σ on HB. The sandwiched relative Rényi entropy Dp(ρ||σ) and conditional entropy were introduced in [36,61]. For 1 < p ≤∞, it can be written with Schatten p-norms as follows,

1 1  −  −  Dp(ρ||σ) = p log σ 2p ρσ 2p p (if finite) , lim Dp(ρ||σ) = D(ρ||σ), p→1 AB B Hp(A|B)ρ =−inf Dp(ρ ||1A ⊗ σ ), lim Hp(A|B)ρ = H(AB)ρ − H(B)ρ. σ B p→1 92 L. Gao, M. Junge, N. LaRacuente

The latter one connects to the vector-valued noncommutative L p-spaces introduced by Pisier (see [40]). Indeed, let 1 ≤ p, q ≤∞and fix 1/r =|1/p − 1/q|. For a bipartite operator ρ ∈ B(HA ⊗ HB),theSq (A, Sp(B)) norms are given as follows: for p ≤ q, 1/q +1/r = 1/p

ρ = ( ⊗ )ρ( ⊗ ) , Sq (A,Sp(B)) sup a 1B b 1B Sp(A⊗B) (10) ( ) ( )≤ a S2r A b S2r A 1 and for p ≥ q,1/p +1/r = 1/q,

ρ = η . Sq (A,Sp(B)) inf a S2r (HA) Sp(HA⊗HB ) b S2r (HA) (11) ρ=(a⊗1B )η(b⊗1B )

When ρ is positive, it is sufficient to choose a = b ≥ 0in(10) and (11), and then the S1(Sp) norm connects to the sandwiched Rényi conditional entropy as follows, −  ρ = ( | ) . p log S1(B,Sp(A)) Hp A B ρ

This observation enables us to translate norm estimates into entropic inequalities.

Corollary 2. Let N : B(HA ) → B(HB) be a channel and f be an admissible symbol AA of N . Let HA be an arbitrary Hilbert space and ρ be a bipartite state HA ⊗ HA . ωAB = ⊗ N (ρ AA ) ωAB = ⊗ N (ρ AA ) Denote f idA f and idA . Then the following inequalities hold ( ) − τ( ) ≤ ( ) ≤ ( ) (i) H AB ω f log f H AB ω f H AB ω; (  ) ≤ (  ) ≤ (  ) τ( ) (ii) Ic A B ω Ic A B ω f Ic A B ω + f log f ; ( ; ) ≤ ( ; ) ≤ ( ; ) τ( ) (iii) I A B ω I A B ω f I A B ω + f log f .

Proof. By Lemma 3, f ⊗ 1 is an admissible symbol for N ⊗ idA and (N ⊗ idA) f ⊗1 = N f ⊗ idA. The first inequality in Theorem 1 gives ω p≤ ω f p≤ f p,τ ω p. Then i) follows from taking logarithm and the limit

  lim p log ω p=− lim Hp(ω) =−H(ω), lim p log f p,τ = τ(f log f ). p→1+ p→1+ p→1+

For (ii), denote E := EL(X) the conditional expectation onto the the left algebra L(X ) = ran(N ) the left algebra. Then E ◦ N f = N by Proposition 2. Then (E ⊗ idA)(ω f ) = ω and the data processing inequality implies

(  ) ≥ (  ) . Ic A B ω f Ic A B ω f

For the other direction, applying Theorem 1,

1 1 −  −  ω = (σ 2p ⊗ )ω (σ 2p ⊗ ) f S1(B,Sp(A)) inf 1A f 1A p σ B 1 1 −  −  ≤ inf (σ 2p ⊗ 1A)ω f (σ 2p ⊗ 1A) p σ B ∈L(X) 1 1 −  −  ≤ f p,τ inf (σ 2p ⊗ 1A)ω(σ 2p ⊗ 1A) p . σ B ∈L(X) Capacity Estimates via Comparison with TRO Channels 93

Note that E(ω) = ω and by the data processing inequality,

1 1 −  −  ω = (σ 2p ⊗ )ω(σ 2p ⊗ ) S1(B,Sp(A)) inf 1A 1A p σ B 1 1 −  −  ≥ inf (E(σ) 2p ⊗ 1A)ω(E(σ) 2p ⊗ 1A) p σ B 1 1 −  −  ≥ inf (σ 2p ⊗ 1A)ω(σ 2p ⊗ 1A) p σ B ∈L(X) 1 1 −  −  ≥ inf (σ 2p ⊗ 1A)ω(σ 2p ⊗ 1A) p . σ B Thus, ω ≤ ω ≤ ω . S1(B,Sp(A)) f S1(B,Sp(A)) f p,τ S1(B,Sp(A)) (12) We obtain (ii) via the limit  ω =− ( | ) = (  ) . lim p log S1(B,Sp(A)) lim Hp A B ω Ic A B ω p→1+ p→1+ ( : ) = ( ) (  ) ωA = ωA Finally, (iii) is a consequence of ii) because I A B H A + Ic A B and f . 

Remark 4. (a) The term τ(f log f ) corresponds to a normalized entropy that differs from τ( ) = | |− ( 1 ) | | the usual entropy by a constant. Namely, f log f log E H |E| f , E is the 1 dimension of system, |E| f is a density operator of the matrix trace.  (b) The inequality (12) is of its own interests. It states that for any state ρ AA , N ⊗ (ρ) ≤ N ⊗ (ρ) idA S1(B,Sp(A)) f idA S1(B,Sp(A)) ≤ N ⊗ (ρ) . f p,τ idA S1(B,Sp(A))

3.2. Capacity bounds. The comparison property naturally generalizes to capacities of quantum channels. Let us recall the operational definitions of channel capacities. Let N : B(HA ) → B(HB) be a quantum channel and V ∈ B(HA , HB ⊗ HE ) be its Stinespring isometry. A quantum code C over N is a triple C = (m, E, D), which consists of an encoding E : Mm → B(HA ) and a decoding D : B(HB) → Mm as completely positive trace preserving maps. |C|=m is the size of the code. The quantum communication fidelity FQ of the code C is defined by

FQ(C, N ) =ψm|idm ⊗ (D ◦ N ◦ E)(|ψmψm|)|ψm ,  ψ = 1 ⊗ ⊗ where m m i, j eij eij is the maximally entangled state on Mm Mm.Arate triple (n, R, ) consists of the number n of channel uses, the rate R of transmission and the error ∈[0, 1].Wesayaratetriple(n, R, )is achievable on N for quantum communication if there exists a quantum code C of N ⊗n such that

log m ⊗ ≥ R and F (C, N n) ≥ 1 − . n Q 94 L. Gao, M. Junge, N. LaRacuente

Then quantum capacity Q(N ) is defined as Q(N ) = lim sup{R | (n, R, )achievable on N for quantum communication}. →0 n Similarly, one can define the classical capacity C(N ) and private classical capacity P(N ). The classical capacity C(N ) is the largest rate of classical bits that the channel N can reliably transmit from Alice to Bob. The private capacity P(N ) is still for trans- mitting classical information, but which would be indiscernible to a hypothetical eaves- dropper with complete access to the environment and unlimited computing resources. A classical communication code C = (m, E, D) consists of an encoding classical to E : m → ( ) quantum channel l1 B HB and a decoding quantum to classical measurement D : B( ) → m m HB l1 . Here the space l1 represents m-dimensional classical system and m C m is the size of the code . We denote the identity map on l1 as idm. Recall that for general mixed states ρ and σ, the fidelity is √ √ F(ρ, σ) = ρ σ 1 .

The classical communication fidelity FC and the private communication fidelity FP of a code C are given by

FC (C, N ) = F(idm ⊗ (D ◦ N ◦ E)(φm), φm), ∗ F (C, N ) = max F(id ⊗ D(1 ⊗ V (id ⊗ E(φ ))V ⊗ 1), φ ⊗ ρ E ), P ρ m m m m  φ = 1 ⊗ F where m m i eii eii is the maximal correlated state and in P the maximum runs E over states ρ on HE . The achievable triple for classical (resp. private) communication are defined similarly as for the quantum capacity but now with fidelity FC (resp. FP ). Then the classical capacity C(N ) and the private classical capacity P(N ) are C(N ) = lim sup{R | (n, R, )is achievable on N for classical communication}, →0 n P(N ) = lim sup{R | (n, R, )is achievable on N for private communication}. →0 n

The entanglement-assisted classical capacity CEA considers the improved rate with the assistance of (unlimited) pre-generated bipartite entanglement shared by Alice and Bob. We refer to [56] for the definition of CEA and more detailed discussions about C, P and Q. Thanks to the capacity theorems proved by Holevo [25,26], Schumacher and West- moreland [44], Bennett et al [5], Lloyd [34], Shor [46], and Devetak [14], these opera- tionally defined capacities are characterized by entropic expressions as follows,

1 ⊗k C(N ) = lim χ(N ), χ(N ) = max I (X; B)ω ; k→∞ k ρ XA

1 (1) ⊗k (1) Q(N ) = lim Q (N ), Q (N ) = max I (AB)ω ; k→∞ k ρ AA

1 (1) ⊗k (1) P(N ) = lim P (N ), P (N ) = max I (X; B)ω − I (X; E)ω ; k→∞ k ρ XA

CEA(N ) = max I (A; B)ω, ρ AA Capacity Estimates via Comparison with TRO Channels 95

(1) AA where the maximums in CEA and Q run over bipartite input states ρ and for χ and  P(1) classical-quantum ρ XA .Hereω always denotes the output of ρ. More precisely, AB AA for Q and CEA it represents ω = idA ⊗ N (ρ ) and for C and P,  ωXBE = |  |⊗ ρ A ∗ px x x V x V x  ρ XA = |  |⊗ρ A N where x px x x x and V is the Stinespring isometry of . In the four capacities above, only CEA admits a single-letter expression. The other three involve with the limits—the regularization over many uses of the channel. Motivated by the super-additive phenomenon of the “one-shot” expressions χ, Q(1) and P(1), Winter and Yang in [62] introduced the potential capacities χ (p), Q(p), P(p) as follows, ( ) ( ) ( ) ( ) χ p (N ) = sup χ(N ⊗ M) − χ(M), Q p (N ) = sup Q 1 (N ⊗ M) − Q 1 (M), M M ( ) ( ) ( ) P p (N ) = sup P 1 (N ⊗ M) − P 1 (M), M where the supremums runs over all channels M. Note that here we use different notations from [62] to save the subscript “p”forL p-norms and Rényi-type expressions. The potential capacity is always an upper bound for corresponding capacity and hence the one-shot expression. A channel N is strongly additive for χ (resp. Q(1) and P(1)) if χ(N ) = χ (p)(N ) (resp. Q(1)(N ) = Q(p)(N ) and P(1)(N ) = P(p)(N )). This means χ(N ⊗ M) = χ(N ) + χ(M) (similar for Q(1) and P(1)) for any M and hence χ(N ) = C(N ) (resp. Q(1) = Q and P(1) = P).

Proposition 4. χ,Q(1) and P(1) and their potential analogs are convex functions over channels.

Proof. We provide a uniform argument using heralded channels. Given two channels N : B(  ) → B( ) M : (  ) → ( ) HA HB1 and B HA B HB2 with common input space, Φ : B(  ) → B( ⊕ ) let us define the heralded channel λ HA HB1 HB2 with a probability λ ∈[0, 1],   λN (ρ) 0 Φλ(ρ) = λ N (ρ) ⊕ (1 − λ)M(ρ) := . 0 (1 − λ)M(ρ)

The output signal is heralded because Bob knows which channel is used by measuring the corresponding block. Because of the block diagonal structure, it is not hard to see that

(1) Q (Φλ) = max λIc(AB1) + (1 − λ)Ic(AB2), ρ AA

χ(Φλ) = max λI (X; B1) + (1 − λ)I (X; B2). ρ XA

Note that the complementary channel of a heralded channel is again a heralded channel E E E of complementary channels, i.e. Φλ (ρ) = λ N (ρ) ⊕ (1 − λ)M (ρ). Then a similar formula holds for one-shot private capacity P(1),

(1) P (Φλ) = max λ(I (X; B1) − I (X; E1)) + (1 − λ)(I (X; B2) − I (X; E2)). ρ XA 96 L. Gao, M. Junge, N. LaRacuente

N M = Now if and have the same output space HB1 HB2 , then the convex combination λN + (1 − λ)M can be factorized through the heralded channel Φλ via a partial trace map. Therefore by data processing,

(1) (1) Q (λN + (1 − λ)M) ≤ Q (Φλ) = max λIc(AB1) + (1 − λ)Ic(AB2) ρ AA ( ) ( ) ≤ λQ 1 (N ) + (1 − λ)Q 1 (M).

Here the Q(1) can be replaced by χ and P(1). Moreover, the convexity of potential capacities follow from the convexity of their “one-shot” expressions.  We have seen that when the Stinespring space is TRO, the channel is a diagonal sum of partial traces. The capacity formulae of these channel follows from [20, Proposition 1]. N =⊕ ⊗ N Proposition 5. Let i idnk trmk be a direct sum of partial traces. Then is strongly additive for χ, Q(1) and P(1), and moreover  (1) (1) Q (N ) = P (N ) = log max ni ,χ(N ) = log( ni ), i  i (N ) = ( 2). CEA log ni i The next theorem provides the comparison property for capacities. Corollary 3. Let N be a channel and f be an admissible symbol for N . Then,

(i) C(N ) ≤ C(N f ) ≤ C(N ) + τ(f log f ); (ii) Q(N ) ≤ Q(N f ) ≤ Q(N ) + τ(f log f ); (iii) P(N ) ≤ P(N f ) ≤ P(N ) + τ(f log f ); (iv) CEA(N ) ≤ CEA(N f ) ≤ CEA(N ) + τ(f log f ). For (i), (ii) and (iii), the capacity can be replaced by corresponding one-shot expression and potential capacity.

(1) Proof. The inequalities for χ, Q and CEA follows from Corollary 2 by taking maxi- mum over all possible inputs. Note that the “one-shot” private capacity can be rewritten as  (1)(N ) = (  ) − ( ) (  ) , P max Ic A B ω p x Ic A B ωx ρ XA A x where the maximum runs over all states  ρ XA A = ( )|  |⊗ρ AA p x x x x x ρ AA ωAB = ⊗ and x are pure states. The coherent information is for the output x idA AA AB AA AA A AA N (ρ ) and ω = idA ⊗ N (ρ˜ ) where ρ˜ is any purification of ρ (so ρ˜ x  may not be the reduced density of ρ XA A). Applying Corollary 2 (ii) one have   (  ) − ( ) (  ) ≤ (  ) − ( ) (  ) τ( ). Ic A B ω f p x Ic A B ω f,x Ic A B ω p x Ic A B ωx + f log f x x Capacity Estimates via Comparison with TRO Channels 97

(1) Then the upper bound of P (N f ) follows and the lower bound is a consequence of the ⊗k lifting property EL(X) ◦ N f = N . For the regularization, note that by Lemma 3, f is an admissble symbol for N ⊗k. Therefore we have

(N ) = 1 (1)(N ⊗k) = 1 (1)(N ⊗k ) P f lim P lim P ⊗k k→∞ k f k→∞ k f 1 ( ) ⊗ ⊗ ⊗ ≤ lim [P 1 (N k) + τ k( f k log f k)] k→∞ k 1 ( ) ⊗ = lim (P 1 (N k) + kτ(f log f )) k→∞ k 1 ( ) ⊗ = lim P 1 (N k) + τ(f log f ) = P(N ) + τ(f log f ). k→∞ k

Similarly, for the potential capacities, we use that M ⊗ N f = (M ⊗ N )1⊗ f for an arbitrary channel M and τ((1 ⊗ f ) log 1(⊗ f )) = τ(f log f ). The arguments for classical capacity and quantum capacity are the same. 

The gap of upper and lower estimates are bounded uniformly by the term τ(f log f ). This can be viewed as a “first order” approximation of the capacity of N f by the entropic τ( ) = | |− ( 1 ) term f log f log E H |E| f . The next theorem is a formula of the negative cb-entropy introduced in [15]. The negative cb-entropy −Scb(N ) of a channel N : B(HA ) → B(HB) is defined as

−Scb(N ) = sup H(A)ω − H(AB)ω, ρ

AB AA AA where ω = idA ⊗N (ρ ) and the supremum runs over all pure bipartite states ρ . It was characterized in [15] as the derivative at p = 1 of the completely bounded norm from trace class to Schatten p class, d −S (M) = | = M : S (H  ) → S (H ) , (13) cb dp p 1 1 A p B cb and later rediscovered as “reverse coherent information” with an operational meaning in [21]. Recall that |A|:=dimHA denotes the dimension of a Hilbert space.

Theorem 2. Let N be a quantum channel and f be an admissible symbol for N . Suppose E that the complimentary channel N : B(HA ) → B(HE ) is unital up to a scalar, i.e. | | N E (  ) = A 1A |E| 1E . Then

|A| −S (N ) = log + τ(f log f ). cb f |E| ∼ ∼  Proof. Let HA = HA and denote eij the matrix units in B(HA) = Mm for m =|A |= |A|.Let{|hi } be an orthonormal basis of HA . For a channel M : B(HA ) → B(HB), its Choi matrix JM is given by  JM = eij ⊗ M(|hi h j |) ∈ B(HA ⊗ HB). i, j 98 L. Gao, M. Junge, N. LaRacuente

The completely bounded 1 → p norm of a map M is same with the vector-valued (∞, p) norm (defined in (11)) of its Choi matrix JM (see e.g. [18,41]),

M : (  ) → ( ) = . S1 HA Sp HB cb JM S∞(A,Sp(B))

In particular, for p =∞, S∞(A, S∞(B)) = B(HA ⊗ HB). The Choi matrix of N f is given by   ⊗ N (|  |) = ⊗ ( ∗) = ( ⊗ ) ∗ ∈ B( ⊗ ), eij f hi h j ei, j hi fhj W e11 f W HA HB ij i, j  B( , ) |  = ⊗ where hi are operators in HE HB corresponding to hi and W i ei1 hi . Since N E is unital up to a factor,    ∗ |A | N E ( |h h |) = h h = 1 . i i i i |E| E i i

|E| 1 ( ) 2 ∗ This implies that |A| W is an isometry. Wethen define the following -homomorphism

|E| |E| ∗ π : B(H ) → B(H ⊗ H ), π( f ) := JN = W(e ⊗ f )W . (14) E A B |A| f |A| 11 Note that tr( f ) = tr(π( f )), then p −1 p −1 p −1 p f p,τ =|E| trE (| f | ) =|E| trAB(π(| f | )) =|E| π(| f |) p. Therefore we get | | −1/p −1/p E ∗ f ,τ =|E| π( f ) =|E| W(e ⊗ f )W p p |A| 11 p =| |1−1/p| |−1 . E A JN f p  By the definition (10), we obtain a lower bound for the S∞(A , Sp(B)) norm, | | 1−1/p  −1/p A JN ( , ( )) ≥|A | JN = f ,τ . (15) f S∞ A Sp B f p |E| p For the upper bound, note that π is a ∗-homomorphism, then |E| |E| π( f ) ∞= JN ∞= N : S (H  ) → B(H ) . (16) |A| f |A| f 1 A B cb Now assume that X is a TRO containing N ’s Stinespring space and f is strongly ∗ independent R(X).LetM ⊂ B(HE ) be the C -subalgebra generated by f . Then for ∗ any operator g ∈ M,themapNg(|hk|) = hgk satisfies that

tr(Ng(ρ)) = τ(g)tr(N (ρ)).

Thus Ng : S1(HA ) → S1(HB) cb=|τ(g)|≤ g 1,τ .wehave |E| π : L (M,τ)→ S∞(A, S (B)) ≤ . 1 1 |A| Capacity Estimates via Comparison with TRO Channels 99

Note that for L∞ spaces,

π : M → B(HA ⊗ HB) ≤1, because π is a ∗-homomorphism. Then by Stein’s interpolation theorem (Theorem 4), | | E 1/p π : L (M,τ)→ S∞(A, S (B)) ≤( ) , (17) p p |A| combining (17) with (15), the upper and lower bounds coincide and give | | A 1−1/p JN ( , ( )) = ( ) f ,τ . f S∞ A Sp B |E| p

The assertion follows by differentiating the above equality at p = 1. Note that for all ≤ ≤∞ ⊗|  | 1 p , the maximal entangled state i eij hi h j is a norm attaining element. 

3.3. The capacity regions. The capacity regions of a quantum channel consider the trade offs between different resources in quantum information theory. The notion of a capacity region relies on the availability of quantum protocols, such as teleportation and dense coding, that exchange one type of resource for another. Based on research due to Devetak and Shor [16], Abeyesinghe et al [1], Collins and Popescu [11], and many others, Hsieh and Wilde introduced the two kinds of capacity regions: the quantum dynamic region CCQE and private dynamic region CRPS. The quantum dynamic region CCQE considers a combined version of classical communication “C”, quantum communication “Q” and entanglement generation “E”, while the private dynamic region CRPS, with the idea of the Collins-Popescu analogy [11], unifies the public classical communication “R”, private classical communication “P” and secret key distribution “S”. We refer to their papers [58,59] for the operational definitions of CCQE and CRPS. Here we state the capacity region theorems from [58,59]. Let N : B(HA ) → B(HB) be a quantum channel and V : HA → HB ⊗ HE be its Stinespring isometry. The quantum dynamic region CCQE(N ) is characterized as follows, ∞ 1 ( ) ( ) ( ) C (N ) = C 1 (N ⊗k), C 1 ≡ C 1 , CQE k CQE CQE CQE,ω k=1 ω

(1) ⊂ R3 where the overbar represents the closure of a set. The “one-shot” region CCQE (1) is the union of the “one-shot, one-state” regions CCQE,ω, which are the sets of all rate triples (C, Q, E) such that:

C +2Q ≤ I (AX; B)ω, Q + E ≤ I (ABX)ω, C + Q + E ≤ I (X; B)ω + I (ABX)ω.

The above entropy quantities are with respect to a classical-quantum state  ωXABE = ( )|  |X ⊗ ( ⊗ )ρ AA ( ⊗ ∗) pX x x x 1A V x 1A V x 100 L. Gao, M. Junge, N. LaRacuente

ρ AA and the states x are pure. Similarly, the private dynamic region is given by, ∞ 1 ( ) ( ) ( ) C (N ) = C 1 (N ⊗), C 1 ≡ C 1 . RPS k RPS RPS RPS,ω k=1 ω

(1) (N ) ⊂ R3 ( , , ) The “one-shot, one-state” region CRPS is the set of all triples R P S such that

R + P ≤ I (YX; B)ω , P + S ≤ I (Y ; B|X)ω − I (Y ; E|X)ω , R + P + S ≤ I (YX; B)ω − I (Y ; E|X)ω.

The above entropic quantities are with respect to a classical-quantum state ωXYBE where  ωXYBE ≡ ( , )|  |X ⊗|  |Y ⊗ ( ρ A ∗). pX,Y x y x x y y V x,y V x N =∼ ⊕ Example 1. Let be a channel and its Stinespring space be a TRO X i Mni ,mi .We know from Proposition 5 that N =⊕ ⊗ i idni trmi as a direct sum of partial traces. The capacity regions of this class of channels are (N ) = (1) (N ) accessible. The quantum dynamic region regularizes CCQE CCQE , and it is characterized as a union of the following regions   C +2Q ≤ H({pλ,μ(i)}) +2 pλ,μ(i) log ni , Q + E ≤ pλ,μ(i) log ni , i i C + Q + E ≤ H({pλ,μ(i)}) + pλ,μ(i) log ni . i for all λ, μ ≥ 0. Here {pλ,μ(i)} is the probability distribution given by

2+λ+μ  2+λ+μ ( ) = 1+μ /( 1+μ ). pλ,μ i ni ni i

(N ) = (1) (N ) Similarly, for the public-private dynamic region, CRPS CRPS is the union of   R + P ≤ H({qλ,μ(i)}) + qλ,μ(i) log ni , P + S ≤ qλ,μ(i) log ni , i  i R + P + S ≤ H({qλ,μ(i)}) + qλ,μ(i). i for all λ, μ ≥ 0. Here {qλ,μ(i)} is the probability distribution given by

1+λ+μ  1+λ+μ ( ) = 1+μ /( 1+μ ). qλ,μ i ni ni i Capacity Estimates via Comparison with TRO Channels 101

In general it is difficult to completely characterize the capacity regions. Let us consider two cones,

W1 ={(C, Q, E)|2Q + C ≤ 0, Q + E ≤ 0, Q + E + C ≤ 0} and

W2 ={(R, P, S)| R + P ≤ 0, P + S ≤ 0, R + P + S ≤ 0}.

The first one is the resource trading off via teleportation, superdense coding and entan- glement distribution and the second is the cone obtained from secret key distribution, the one-time pad and private-to-public transmission (see [58,59]). We have for each sin- ρ XA ( ( ; ) , 1 ( ; | ) , gle input state a comparison property of the rate triple I X B ω 2 I A B X ω 1 − I (A; E|X)ω) and respectively (I (X, B)ω, I (Y ; B|X)ω, −I (Y ; E|X)ω) for each 2  ρ XY A .

Corollary 4. Let N be a channel and f be an admissible symbol for N . Denote the quantity τ := τ(f log f ). Then (N ) ⊂ (N ) ⊂ (N ) (τ, τ , τ ) (i) CCQE CCQE f CCQE + 2 2 ; (ii) CRPS(N ) ⊂ CRPS(N f ) ⊂ CRPS(N ) + (τ,τ,τ).

Proof. The argument for the two kinds of regions are similar. Here we give the proof for the private dynamic region CRPS and the argument for quantum dynamic region CCQE is similar. Let us assume  ωXY ABE = ( , )|  |X ⊗|  |Y ⊗ ( ⊗ )(ρ A A)( ⊗ ∗), f pX,Y x y x x y y 1 V f x,y 1 V f x

ρ A A ( f , f , f ) where x,y are pure states. We denote R P S for the rate triple ( ( ; ) , ( ; | ) , − ( ; | ) ). I X B ω f I Y B X ω f I Y E X ω f

By the entropic inequality (2),

( ; ) ≤ τ( ) ( ; ) . I X B ω f f log f + I X B ω1

f 1 From this, we may assume R = R + τ(f log f ) − α1 for some α1 ≥ 0. Similarly, we have

I (Y ; B|X)ω = H(Y |X)ω + H(B|X)ω − H(YB|X)ω f f  f  f = ( | ) ( ) (ωB ) − ( )( ( | = ) H Y X ω f + p x H f,x p x H Y X x  x x ( | ) (ωB )) + p y x H x,y, f y ≤ I (Y ; B|X)ω + τ(f log f ), 102 L. Gao, M. Junge, N. LaRacuente and

I (Y ; E|X)ω = H(Y |X)ω + H(E|X)σ, − H(YE|X)ω f f  f  f = ( | ) ( ) (ωE ) − ( )( ( | = ) H Y X ω f + p x H f,x p x H Y X x  x x ( | ) (ωE )) + p y x H x,y, f y ≥ I (Y ; E|X)ω − τ(f log f ).

This means

f 1 f 1 P = P + τ(f log f ) − α2 , S = S + τ(f log f ) − α3 for some α2,α3 ≥ 0. Now it is obvious that (−α1, −α2, −α3) ∈ W, then we have f f f 1 1 1 (R , P , S ) ∈ (τ,τ,τ)+ (R , P , S ) + W2. Taking the union for all ω,wehave

(1) (N ) ∈ (τ,τ,τ) (1) (N ) . CRPS f + CRPS + W2 + W2

For the cone W2,wehaveW2 + W2 = W2 and this concludes that (1) (N ) ⊂ (τ,τ,τ) (1) (N ). CRPS f + CRPS For regularization, we apply the above estimates to the tensor channel

1 (1) ⊗k 1 (1) ⊗k 1 (1) ⊗k C (N ) = C ((N ) ⊗k ) ⊂ (k(τ,τ,τ)+ C (N )) k RPS f k RPS f k RPS 1 ( ) ⊗ = (τ,τ,τ)+ C 1 (N k), k RPS which completes the proof. 

3.4. Strong converse rates. A “strong converse” means there is a sharp drop off for code fidelity above the optimal transmission rate. More generally, we will investigate rates above which the transmission only succeeds with arbitrarily small probability. We say r is a strong converse rate for classical (respectively, quantum, private) communication if for every sequence of achievable triple (n, Rn, n) of classical (resp. quantum, private) communication, we have

lim inf Rn > r ⇒ lim n = 1. n→∞ n→∞

We denote C† (resp. Q†, P†) as the smallest strong converse rate of classical commu- nication (resp. quantum, private communication). We say a channel N has (classical, quantum or private) strong converse if the smallest strong converse rate equals to the capacity C†(N ) = C(N ) (resp. Q†(N ) = Q(N ), P†(N ) = P(N )). Here we briefly review some strong converse bounds which we will use in this subsection. We will discuss more prior works on strong converse bounds in Sect. 3.5. Capacity Estimates via Comparison with TRO Channels 103

It was proved in [61] that for any channel N and 1 < p ≤∞, χ (N ⊗k) † p (N ) ≤ ,χ(N ) = ( ; )ω, C lim sup p max Ip X B (18) k→∞ k ρ XA where the sandwich Rényi mutual information is given by

AB A B Ip(A; B)ρ = inf Dp(ρ ||ρ ⊗ σ ). (19) σ B This bounds implies the strong converses of Hadamard channels and entanglement- breaking channels for classical communication. For private communication, the relative entropy of entanglement was shown to be an upper bound for the private capacity [39] and the strong converse capacity [60],

† P (N ) ≤ E R(N ), E R(N ) := max E R(idA ⊗ N (ρ)). (20) ρ AA

AB The relative entropy of entanglement E R(ρ) for a bipartite ρ is AB AB AB E R(ρ ) = inf D(ρ ||σ ), σ AB∈S(A:B) where S(A:B) stands for the separable states between A and B. Based on (20), the private strong converse for generalized dephasing channels was proved in [60]. For 1 < p ≤∞and 1/p +1/p = 1, we consider the Rényi coherent information of a channel for as an analog of (18) (1)(N ) = (  ) , (  ) =  ωBA . Q p max Ic,p A B ω Ic,p A B ω p log S1(B,Sp(A)) ω=id⊗N (ρ) The following is a folklore result which is probably known to experts but not stated explicitly in the literature. Proposition 6. Let N be a channel. For all 1 < p ≤∞,

(1) ⊗k Q p (N ) Q†(N ) ≤ lim sup . k→∞ k

= 1 (1)(N ⊗k) = M 1 1 = Proof. Denote Rp lim supk→∞ k Q p .Letm 2 and p + p 1. It is sufficient to show that for an arbitrary code C = (m, E, D) of N ,

− 1 1 ( ) F(C, N ) ≤ m p exp( Q 1 (N )). (21) p p C 1 |C | > Indeed, let n be a sequence of codes such that lim infn→∞ n log n Rp + ,

⊗ − 1 1 ( ) ⊗ 1 ( ) ⊗ (− 1 n ) F(C , N n) ≤ m p exp( Q 1 (N n)) = exp( (Q 1 (N n) − M)) ≤ 2 p , n p p p p for n large enough. To prove (21), we define ω = idm ⊗ (D ◦ N ◦ C)(|ψmψm|). Then the fidelity is given by

F(C, N ) = tr(ω|ψmψm|) ≤ ω Sm (Sm ) |ψmψm| M (Sm ) . 1 p m p 104 L. Gao, M. Junge, N. LaRacuente

Note that E and D are completely positive trace preserving maps and hence

E : m → (  ) = , D : ( ) → m = . S1 S1 HA cb 1 S1 HB S1 cb 1 This implies

( ) ( ) (p Q 1 (N )) (p Q 1 (N )) ω m ( m )≤ 2 p |ψ ψ | m ⊗ m = 2 p . S1 Sp m m S1 S1 For the second term, we use the interpolation relation (see “Appendix”) ( m ) =[ ( m), ( )] . Mm Sp Mm S1 Mm Mm 1 p We have

1 1 1 p p −  |ψψ| m = |ψ ψ | |ψ ψ | ≤ p , Mm (S  ) m m ( ) m m ( m ) m (22) p Mm Mm Mm S1 where we used the fact |ψ ψ | ( m )= 1/m.(22) is indeed an equality. Combining m m Mm S1 these two estimates, we obtain (21). 

The following lemma is an analog of Corollary 2 for Rényi p-information measure.

Lemma 1. Let N be a channel and f be an admissible symbol of N . Let HA be an  ρ AA ⊗  ωAB = ⊗ arbitrary Hilbert space and be a bipartite state HA HA . Denote f idA AA AB AA N f (ρ ) and ω = idA ⊗ N (ρ ). Then the following inequalities hold: (  ) ≤ (  ) ≤ (  )  ; (i) Ic,p A B ω1 Ic,p A B ω f Ic,p A B ω1 + p log f p,τ ( : ) ≤ ( : ) ≤ ( : )  ; (ii) Ip A B ω1 Ip A B ω f Ip A B ω1 + p log f p,τ  (iii) E R,p(ω1) ≤ E R,p(ω f ) ≤ E R,p(ω1) + p log f p,τ , (iv) E R(ω1) ≤ E R(ω f ) ≤ E R(ω1) + τ(f log f ).

Proof. Let X be a TRO containing N ’s Stinespring space and f is independent of R(X). All lower bounds follows from the factorization property EL(X) ◦ N f = N and data processing inequality, where EL(X) : B(HB) → L(X) is the conditional expectation onto the left algebra L(X). The upper estimate of (i) is a direct consequence of the vector-valued (1, p) norm inequality (12). Indeed, (  ) =  ω ≤  ω Ic,p A B ω f p log f S1(B,Sp(A)) p log f τ,p S1(B,Sp(A)) ≤   ω ≤  (  ) . p log f τ,p +p log S1(B,Sp(A)) p log f τ,p +Ic,p A B ω

For (ii), note that EL(X) ◦ N = N ,

idA ⊗ EL(X)(ω) = idA ⊗ (EL(X) ◦ N )(ρ) = idA ⊗ N (ρ) = ω.

Therefore, for the Rényi mutual information,

A B A B Ip(A; B)ω = inf Dp(ω||ω ⊗ σ ) ≥ inf Dp(ω||ω ⊗ EL(X)(σ )) σ B σ B A B A B ≥ inf Dp(ω||ω ⊗ σ ) ≥ inf Dp(ω||ω ⊗ σ ). σ B ∈L(X) σ B Capacity Estimates via Comparison with TRO Channels 105

( ; ) = (ω||ωA ⊗ σ B) σ B ∈ Hence, Ip A B ω infσ B ∈L(X) Dp where it suffices to consider L(X) for the infimum. Combined with the Theorem 1,wehave ( ; ) = (ω ||ωA ⊗ σ B) ≤ (ω ||ωA ⊗ σ B) Ip A B ω f inf Dp f inf Dp f σ B σ B ∈L(X) A B  ≤ inf Dp(ω||ω ⊗ σ ) + p log f τ,p σ B ∈L(X)  = Ip(A; B)ω + p log f τ,p .

Now we consider the upper bound for Rényi relative entropy of entanglement E R,p. σ AB = ( )σ A ⊗ σ B Note that for a separable state i p i i i ,  ⊗ E (σ AB) = ( )σ A ⊗ E (σ B) idA L(X) p i i L(X) i i is again a separable state in B(HA)⊗L(X) ⊂ B(HA ⊗ HB). Let us denote S(HA : L(X)) for separable states in B(HA) ⊗ L(X). Then

E R,p(ω) = inf Dp(ω||σ) ≥ inf Dp(ω||idA ⊗ EL(X)(σ)) σ∈S(A:B) σ∈S(A:B)

≥ inf Dp(ω||σ) ≥ inf Dp(ω||σ). σ∈S(A:L(X)) σ∈S(A:B)

Thus, E R,p(ω) = infσ∈S(A:L(X)) Dp(ω||σ). Again by Theorem 1,

E R,p(ω f ) = inf Dp(ω f ||σ) ≤ inf Dp(ω f ||σ) σ∈S(A:B) σ∈S(A:L(X))   ≤ inf Dp(ω||σ)+ p log f τ,p= E R,p(ω) + p log f τ,p . σ∈S(A:L(X)) (iv) follows from (iii) by taking the limit p → 1+.  Let us denote as the regularization of p-Rényi coherent information and p-Rényi Helevo information as follows, ⊗k (1) ⊗k χp(N ) Q p (N ) C p(N ) := lim sup , Q p(N ) := lim sup . k→∞ k k→∞ k N N =⊕ ⊗ Example 2. Let be a TRO channel and assume that i idni trmi as a direct sum of partial traces. It is not hard to calculate that  χ (N ) = , (1)(N ) = (N ) = . p log ni Q p E R,p log max ni i i

Note that all these terms are additive. Let M =⊕j id  ⊗tr  be another TRO channels. n j m j Then

N ⊗ M =⊕i, j id  ⊗ tr  . ni n j mi m j is again of orthogonal sum of partial traces. Apply the above formulae for N ⊗ M,we obtain    χ (N ⊗ M) = (  ) = ( ) (  ) = χ (N ) χ (M), p log ni n j log ni +log n j p + p i, j i j (1)(N ⊗ M) =  =  = (1)(N ) (1)(M). Q p max log ni n j max log ni +maxlog n j Q p + Q p i, j i j 106 L. Gao, M. Junge, N. LaRacuente

Hence, the regularizations are trivial and we have  C p(N ) = log ni , Q p(N ) = log max ni . i i

By Proposition 5, we know that TRO channels has strong converse for classical, quantum and private communication.

The next corollary is the comparison property for information measures of channels.

Corollary 5. Let N be a channel and f be an admissible symbol of N .  (i) C p(N ) ≤ C p(N f ) ≤ C p(N ) + p log f p,τ ;  (ii) Q p(N ) ≤ Q p(N f ) ≤ Q p(N ) + p log f p,τ ; (iii) E R(N ) ≤ E R(N f ) ≤ E R(N ) + τ(f log f );

Proof. (iii) is a direct consequence of Lemma 1 part (iii). Here we show the upper estimates for C p and the argument for Q p is similar. Taking the supremum of all inputs  ρ XA for (1), we have

 χp(N f ) ≤ χp(N ) + p log f τ,p .

By regularization,

(N ) = 1χ (N ⊗k) ≤ 1(χ (N ⊗k)  ⊗k ) C p f lim p lim p + p log f τ k ,p k→∞ k f k→∞ k   1 ⊗k   ≤ lim χp(N ) + kp log f τ,p = C p(N ) + p log f τ,p k→∞ k where we used the facts that ⊗k = k χ (N ⊗k) = χ (N ). f τ n,p f τ,p and p k p 

We know from Example 2 that for a TRO channel N ,  † † C p(N ) = C (N ) = log( ni ), Q p(N ) = Q (N ) = E R(N ) i † = P (N ) = log(max ni ). i

Then we have the following estimates of strong converse capacities.

Corollary 6. Let N be a TRO channel and f be an admissible symbol of N . Assume N =⊕ ⊗ that i idni trmi as a direct sum of partial traces.   † (i) log( ni ) ≤ C (N f ) ≤ log( ni ) + τ(f log f ) ; i i † † (ii) log(max ni ) ≤ Q (N f ) ≤ P (N f ) ≤ log(max ni ) + τ(f log f ). i i Capacity Estimates via Comparison with TRO Channels 107

Proof. The lower bounds again follows from the factorization property. We show (ii) for the quantum strong converse capacity Q† and others are similar. First, the lower bounds † † = Q (N ) ≤ Q (N f ) follows from the factorization property. For the upper estimate, we apply Proposition 6 and Theorem 5 part (ii). †  Q (N f ) ≤ Q p(N f ) ≤ Q p(N ) + p log f p,τ . (23) We know from Example 2 that for a TRO channel N , † Q p(N ) = Q(N ) = Q (N ) = log(max ni ). i Then take the limit p → 1+ of (23), we obtain †  Q (N f ) ≤ lim Q p(N f ) ≤ lim (Q(N ) + p log f p,τ ) = Q(N ) + τ(f log f ). p→1+ p→1+  Remark 5. A channel N has the strong converse theorem for quantum communication if

lim Q p(N ) = Q(N ). p→1 Wedo not know for general N whether this holds. For this reason, we need the assumption that N is a TRO channel in the above corollary.

3.5. Relations to other capacity bounds. Here we briefly review the prior works on strong converses and quantum and private capacities [10,27,35,53–55,60], and discuss how our method also applies to some of known strong converse bounds. Let Q(N ) (resp. Q†(N )) denote the quantum capacity (resp. the smallest quantum strong converse rates) of a quantum channel N . In recent works, Quantum capacity and strong converse rates has been also been considered with assisted protocols. The LOCC- assisted quantum capacity Q↔(N ) [4] is defined as the maximum rate at which qubits can be communicated reliably, when the sender and receiver are allowed to use arbitrary local operations and classical communication (LOCC) between every use of the channel N . A larger set of operation than the LOCC is the positive partial transpose preserving (PPT-preserving) channels. The PPT-assisted quantum capacity Q PPT,↔(N ) is defined for the setting that the sender and receiver are allowed to use a PPT-preserving channel between every use of N . The corresponding smallest strong converse rates are denoted by Q↔,†(N ) and Q PPT,↔,†(N ). Recall that the Rains relative entropy [3,42,43] and max-Rains relative entropy of a bipartite state ρ AB is defined as AB AB R(A; B)ρ = min D(ρ ||σ ), σ AB∈PPT(A:B) AB AB Rmax(A; B)ρ = min D∞(ρ ||σ ), σ AB∈PPT(A:B)  AB AB AB where PPT (A : B) is the set {σ | σ ≥ 0, TB(σ ) 1≤ 1} and TB is the partial transpose on B. It was proved in [53] that the Rains information R(N ) of a channel is a quantum strong converse rate, †  Q (N ) ≤ R(N ), R(N ) := sup R(A ; B)N (ρ). ρ A A 108 L. Gao, M. Junge, N. LaRacuente

Later [8], based on the work [55], showed that the max Rains information bounds the PPT-assisted strong converse rate

PPT,↔,†  Q (N ) ≤ Rmax(N ), Rmax(N ) := sup Rmax(A ; B)N (ρ). (24) ρ A A

Before these works, the partial transpose bound QΘ was proved in [27],

† Q (N ) ≤ QΘ (N ), QΘ (N ) := log T ◦ N  where ·  denotes the diamond norm. More recently it has been improved in [35] that

PPT,↔,† Q (N ) ≤ QΘ (N ).

Both QΘ and Rmax are efficiently computable via semi-definite program (SDP), and it was shown in [55] that Rmax ≤ QΘ . For private communication, we have used the result in [60] that

† P (N ) ≤ E R(N ).

Recall that max relative entropy of entanglement E R,max is the p-Rényi version of E R for p =∞defined as follows, AB E R,max(N ) := max E R,max(idA ⊗ N (ρ)), E R,max(ρ ) ρ AA AB AB := inf D∞(ρ ||σ ). σ AB∈S(A:B) The work [10] established the bound ↔,† P (N ) ≤ E R,max(N ), (25) where P↔,† is the smallest two way LOPC (local operation and public communication)- assisted strong converse rate. Also in the recent works [10,52,57], the squashed entan- glement Esq(N ) in ([52]) were shown to be an upper bound for Q↔(N ) and P↔(N ), and the entanglement cost EC (N ) in [7] were shown to be the strong converse bounds for Q↔,†(N ) and P↔,†(N ). Among the capacity bounds mentioned above, our comparison method is compatible for R, Rmax, E R,max as well as E R used in the Sect. 3.4. More precisely, using the identical argument for E R in Lemma 1 and Corollary 5,

R(N ) ≤ R(N f ) ≤ R(N ) + τ(f log f ), Rmax(N ) ≤ Rmax(N f ) ≤ Rmax(N ) +log f ∞, E R,max(N ) ≤ E R,max(N f ) ≤ E R,max(N ) +log f ∞ . On the other hand, we know by Example 2 that when N is a TRO channel,

Q(N ) ≤ R(N ) ≤ Rmax ≤ E R,max = Q(N ). N =⊕ ⊗ Assume that i idni trmi ,wehave

log(max ni ) ≤ R(N f ) ≤ E R(N f ) ≤ log(max ni ) + τ(f log f ), i i log(max ni ) ≤ Rmax(N f ) ≤ E R,max(N f ) ≤ log(max ni ) +log f ∞ . i i Capacity Estimates via Comparison with TRO Channels 109

Recall that for the folklore bound in Proposition 6, we proved

† log(max ni ) ≤ Q (N f ) ≤ inf Q p(N f ) ≤ log(max ni ) + τ(f log f ). i p>1 i

Thus the upper estimate “log(maxi ni ) + τ(f log f )” bounds R(N f ), E R(N f ) and inf p Q p(N f ). In general we do not know a relation between inf p Q p(N f ) and R(N f ) or between inf p Q p(N f ) and E R(N f ). Also, it is not clear whether Rmax(N f ) ≤ log maxi ni + τ(f log f ), because the correction term log f ∞>τ(f log f ) when f = 1. Note that E R and E R,max are not known to be efficiently computable via semi- definite programs and R(N ) involves a “max–min” optimization. The above inequalities gives estimates of these information measures with a simple correction term by the en- tropy of the symbol f .

4. Examples

4.1. Random unitary. Random unitary channels are convex combinations of unitary conjugation maps. We observe that the random unitary gives a class of TRO-channels if the unitaries form a projective unitary representation of a group. Let G be a finite group and T be the unit complex scalars. We write 1 as the identity element of G. A projective unitary representation of G is a map u from G into the unitary group U(H) of some Hilbert space H such that

u(g)u(h) = σ(g, h)u(gh), g, h ∈ G, where σ(g, h) is a function σ : G×G → T. Basically, a projective unitary representation is a representation up to a phase factor, or into the quotient U(H)/T. Because the laws of group multiplication, the function σ satisfies the following conditions (i) σ(g, 1) = σ(1, g) = 1 ; (ii) σ(g, g)σ(gg, g) = σ(g, gg)σ(g, g) ; for all g, g, g ∈ G. Suppose |G|=n and dimH = m are finite. We define a m- dimensional channel N : B(H) → B(H) as follows

1  N (ρ) = u(g)ρu(g)∗. n g

Its Stinespring isometry is given by

1  V : H → H ⊗ l (G), V |h= u(g)|h⊗|g, 2 n g wherel2(G) is the Hilbert space spanned by the canonical orthogonal basis {|g|g ∈ G}. N The Stinespring space X , as a subspace of operators B(l2(G), H),is  N X = ran(V ) ={ u(g)|h⊗g|||h∈H}. g 110 L. Gao, M. Junge, N. LaRacuente

N We claim that X is a TRO space. Let |h1, |h2, |h3 be vectors in H and h, h1, h2 be N the corresponding operators in X ⊂ B(l2(G), H)  ∗ = ( )|  |  | ∗( ) ( )|  | h1h2h3 u g h1 g g h2 u g u g h3 g , ,  gg g ∗   = u(g)|h1h2|u (g)u(g )|h3g | ,  g g   = nN (|h1h2|)u(g )|h3g | g      = u(g ) nN(|h1h2|)|h3 g |. g

In the last step, we use the fact N is the conditional expectation onto the commutant  u(G) ⊂ B(H). Indeed, for g0 ∈ G,   1 1 − − N (ρ)u(g ) = u(g)ρu(g)∗u(g ) = u(g)ρu(g)∗u(g 1)∗σ(g , g 1) 0 n 0 n 0 0 0 g g  1 − − − = u(g)ρu(g 1g)∗σ(g 1, g)σ(g , g 1) n 0 0 0 0 g  1 − − − − = u(g )u(g 1g)ρu(h−1g)∗σ(g , g 1)σ(g 1, g)σ (g , g 1) n 0 0 0 0 0 0 0 g  1 − − = u(g )u(g 1g)ρu(g 1g)∗ = u(g )N (ρ). n 0 0 0 0 g

Here we use the conditions of the phase factor σ,

σ( , −1 )σ( −1, )σ( , −1) = σ( , )σ( , −1)σ( , −1) g0 g0 g g0 g g0 g0 1 g g0 g0 g0 g0 = σ( , )σ( , −1)σ( , −1) = . 1 g g0 g0 g0 g0 1

N ∗ Thus, we verify that X is a tenary ring of operator in B(l2(G), H).TheleftC -algebra L(XN ) = ran(N ) is exactly the commutant u(G). For the right C∗-algebra R(XN ),  ∗ = | −1 | | ( −1)∗ ( )|  h1h2 gg0 g h1 u gg0 u g h2 , g g0 = | −1 | |σ( , −1) ( )∗|  gg0 g h1 g h u g0 h2 , g g0  =  | ( )∗| ( σ( , −1)| −1 |). h1 u g0 h2 g g0 gg0 g g0 g

This gives an element in the σ-twisted right regular representation πσ : G → B(l2(G)),

ρ ( )| =σ( , −1)| −1. σ g0 g g g0 gg0 Capacity Estimates via Comparison with TRO Channels 111

N Thus, R(X ) ⊂ ρσ (G) as a subalgebra. The diagonal matrices l∞(G) is independent ρ ( ) = ( )|  | ( ) = of σ G . Indeed, for f g f g g g be a diagonal matrix in l∞ G and x α( )ρ ( ) π ( ) g g σ g be a element in σ G

1 tr( fx) = tr( f )α(1) = tr( f )τ(x). n  Given a normalized density f ∈ l∞(G)( f (g) =|G|, f ≥ 0), the channel N f is

1  N (ρ) = f (g)u(g)ρu(g)∗. f |G| g

Note that both N and its complementary channel

1  N E : B(H) → B(l (G)), N E (ρ) = tr(u(g)ρu(h)∗)|gh| 2 |G| g,h∈G are unital. Hence, Theorem 2 applies here

1 Q(N ) ≥−S (N ) = log m − log |G| + τ(f log f ) = log m − H( f ), f cb f |G|

1 ( 1 ) where |G| f is a probability distribution on G and H |G| f is the Shannon entropy. ( ) =⊕ ⊗ Assume that u G k Mni 1mi . mi ’s are the dimensions of the irreducible decom- position of u and ni ’s are corresponding multiplicities. Then  C(N ) = log ni , Q(N ) = P(N ) = log max ni . i i

Combined with the Corollary 3, we know that for all normalized densities f ∈ l∞(G),   † log ni ≤ C(N f ) ≤ C (N f ) ≤ log ni + τ(f log f ), i i 1 { m − H( f ), ( n )}≤Q(N ) ≤ P(N ) max log | | log max i f f G i ≤ log max ni + τ(f log f ), i and the second upper bound also bounds the strong converse Q† and P†. We illustrates the above estimates with the example of left regular representation. Recall that the left regular representation of a group G is

λ : G → l2(G), λ(g)|h=|gh.

The following channel arose from the left regular representation were discussed in [12],

1  N : B(l (G)) → B(l (G)), N (ρ) = f (g)λ(g)ρλ(g)∗, f 2 2 f |G| g∈G 112 L. Gao, M. Junge, N. LaRacuente

( ) = 1 where f is a normalised density in l∞ G . In particular, for f |G| being the uniform distribution, we have N is the conditional expectation onto the commutant λ(G) = {x |xλ(g) = λ(g)x for all g ∈ G}. By Peter–Weyl theorem (cf. Theorem 1.12 of [31]), λ( ) =∼ ⊕ ⊗ G k Mnk 1nk where nk’s are dimensions of irreducible representations of G and the direct sum runs over all irreducible representations up to unitary equivalence. Thus, the above estimates give

max{τ(f log f ), log(max ni )}≤Q(N f ) ≤ P(N f ) ≤ log(max ni ) + τ(f log f ), i i (26) where maxi ni is the largest degree of irreducible representations of G. We know that maxi ni = 1 if and only if G is commutative, and in this situation N f is a Hadamard channel (see Sect. 4.2) and the above estimates become a equality

Q(N f ) = τ(f log f ). N (1)(N E )> For noncommutative G, f are in general non-degradable because Q f 0for N E the complementary channel f . The estimate (26) is nearly optimal for groups that the largest degree of irreducible representations is small. For example, the dihedral groups D2n is defined by the relation n 2 −1 D2n =r, s|r = 1, s = 1, sr = r s.

The maximum degree of irreducible representations of D2n is always 2. Then our upper and lower bound differs up to 1 qubit (1) max{1,τ(f log f )}≤Q (N f ) ≤ Q(N f ) ≤ P(θ f ) ≤ τ(f log f ) +1, (27) while the term τ(f log f ) varying between 0 to log 2n. The similar estimate can be obtained for semi-direct products Zn  Z p for which p is a prime number and the largest degree of irreducible representations is p. Another example of projective representation is the generalized Pauli channels. Recall that the d-dimensional Pauli matrix are defined as 2πi Xe = exp ( )e , Ze = e j d j j j+1 d where (e j )1≤ j≤d is the standard basis for C . X, Z are unitary satisfying the relation 2πi XZ = e d ZX and {X j Z k |0 ≤ j, k ≤ d − 1 } gives a projective representation of Zd × Zd . Given a probability distribution f on Zd × Zd , the generalized Pauli channels is defined as  j k j k ∗ M f (ρ) = f ( j, k)X Z ρ(X Z ) . 0≤ j,k≤d−1 = 1 Recall that the entanglement assisted quantum capacity Q EA 2 CEA is super-additive and a strong converse upper bound for Q† [22]. For generalized Pauli channel, the Theorem 2 gives 2 −Scb(N f ) = log d − log d + τ(f log f ) =−log dτ(f log f ), Capacity Estimates via Comparison with TRO Channels 113 which leads a capacity bounds via Q EA as follows,

† 1 1 Q(N f ) ≤ Q (N f ) ≤ Q EA(N f ) ≤ (−Scb(N f ) +logd ≤ τ(f log f ). 2 2

† Thus, for Pauli channels, this bound is better than the comparison bounds Q (N f ) ≤ τ(f log f ).Thed-dimensional deporalizing channel Dp can be written as a Pauli chan- nels  1 1 − p ∗ D (ρ) = pρ + (1 − p) = pρ + X j Z kρ(X j Z k) . p d d2 j,k

Nevertheless, our method is not optimal comparing to the bound in [47,49].

4.2. Generalized dephasing channels. Generalized dephasing channels are also called Schur multipliers in the literature (see e.g. [38]). They are special cases of Hadamard channels which are known to be degradable, hence the quantum capacity and private capacity does not require regularization, and Q(1) = Q = P. Our estimates here recovers the quantum capacity formula in [12] in a different way, but both approaches are based on the unfortunately unpublished joint work [28]. Our approach provides a new proof of Q = Q(p) for these particular Schur multipliers. This is already known [62] thanks to the fact that Hadamard channels are strongly additive for Q(1). The Schur multiplication (or Hadamard product) of matrices is given by

(aij) ∗ (bij) = (aij · bij).

It is a well-known fact (see [38]) that the multiplier map for a given matrix a = (aij),

Ma(b) = a ∗ b for b = (bij) ∈ Mn, is completely positive if and only if a is positive. Ma is trace preserving if and only if aii = 1for1≤ i ≤ n. Let G be a finite group with order |G|=n. A function f : G → C is positive definite , , ··· , ∈ ( ( −1 ))k if for any finite sequence g1 g2 gk G, the matrix f g j gi i, j=1 is positive. Consider the Schur multiplier

 −1  M f : B(l2(G)) → B(l2(G)), M f (|gg |) = f (g g1)|gg |.

M f is completely positive if f is positive definite and f (1) = 1. In particular, the 1, if g = 1 function δ(g) = gives the completely dephasing channel 0, otherwise.    Mδ( ag,g |gg |) = agg|gg|. g,g g

This is a TRO channel as its Stinespring dilation is given by

V : l2(G) → l2(G) ⊗ l2(G), V |g=|g⊗|g. 114 L. Gao, M. Junge, N. LaRacuente

The Stinespring space ran(V ), via the identification |g⊗|g←→|gg|,isthe ( ) ⊂ B( ( )) M diagonal matrices l∞ G l2 G . f can be viewed as a modified channel of = ( −1 )|  | symbol f g,g f g g g g as follows,    M f (|gg |) =|gg| f |g g |.

Such an operator f ∈ B(l2(G)) is belongs to the right regular representation and is strongly independent to l∞(G). Note that Mδ is a channel with commutative range hence has C(Mδ) = log |G|, Q(Mδ) = P(Mδ) = 0. Therefore, for any Schur multiplier given by positive definite functions, Theorem 3 combined gives 1 Q(M ) ≤ P(M ) ≤ τ(f log f ) = log |G|−H( f ). f f |G|

(1) Recall that the negative cb-entropy −Scb is a lower bound for Q for unital channels N . Then Theorem 2 gives the lower bound via −Scb(M f ) = τ(f log f ) hence we have the formula 1 Q(M ) = P(M ) = log |G|−H( f ), f f |G| which recovers the formula from [12]inadifferentway.

Example 3. The qubit example is the dephasing channel. Let 0 ≤ q ≤ 1 be the dephasing parameter, we have   Φ ( ab ) = aqb. q cd qc d   10 The channel can also be expressed using the Pauli matrix Z = , 0 −1

1 − q 1 − q Φq (ρ) = (1 − )ρ + Zρ Z. 2 2 1 q  This corresponds to G = Z for ρ = 1+qX = in our setting. The formula for 2 q 1 (Φ ) = − ( 1+q ) = τ(ρ ρ) the quantum capacity is Q q log 2 H 2 log . When the dimension d > 2, not every generalized dephasing channel can be expressed via positive definite functions.

4.3. Small dimensional examples. We provide a concrete example in small dimensions to illustrate how to find TRO channels. Let |α|≤1 be a real number. Define the channel Φα : M4 → M3 as follows, ⎡ ⎤ ⎡ ⎤ a11 a12 a13 a14 a11 + a22 αa13 αa24 ⎢ a21 a22 a23 a24 ⎥ ⎣ ⎦ Φα(⎣ ⎦) = αa31 a33 0 . a31 a32 a33 a34 αa42 0 a44 a41 a42 a43 a44 Capacity Estimates via Comparison with TRO Channels 115

This channel is non-degradable since it traces out the first 2 × 2 block. We claim that

(1) (p) † (1) (p) Q (Φα) = Q (Φα) = Q (Φα) = P (Φα) = P (Φα) α † 1+ = P (Φα) = 1 − h( ), 2 where h(λ) =−λ log λ − (1 − λ) log(1 − λ) is the binary entropy function. Let us first consider the diagonal part of the channels. That is when α = 0, ⎛⎡ ⎤⎞ ⎡ ⎤ a11 a12 a13 a14 a11 + a22 00 ⎜⎢ a21 a22 a23 a24 ⎥⎟ ⎣ ⎦ Φ0 ⎝⎣ ⎦⎠ = 0 a33 0 . a31 a32 a33 a34 00a44 a41 a42 a43 a44 It is an orthogonal sum of partial trace maps hence the Stinespring space corresponds to aTRO.Let{ei } be the standard (computational) basis. The Stinespring isometry V0 of Φ0 is given by  3 4 V ( hi ei ) = h1e1 + h2e1 ⊗ e2 + h3e2 ⊗ e3 + h4e3 ⊗ e4 ∈ C ⊗ C . i ⎡ ⎤ h1 h2 00 ⎣ ⎦ The corresponding operators are 3 × 4 matrices, h = 00h3 0 . Then the Stine- 000h4 spring space X = M1,2 ⊕ C ⊕ C as a TRO. The left and right algebra are given by

L(X) = C ⊕ C ⊕ C , R(X) = M2 ⊕ C ⊕ C. ⎡ ⎤ 0010 ⎢ 0001⎥ Let S = ⎣ ⎦. One verifies that the only nontrivial ∗-subalgebra independent of 1000 0100 R(X) in M4 is

N ={β I + αS | α, β ∈ C}.

The normalized densities in N given by the one-parameter class {I + αS|−1 ≤ α ≤ 1}. Denote the symbol fα = I + αS. Note that f0 is the identity 1E . Φα is a modified TRO channel with symbol fα, ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 10α 0 h1 00 h1 h2 00 ⎢ α ⎥ ⎢ ⎥ Φ (|  |) = ∗ = ⎣ ⎦ 010 ⎢ h2 00⎥ . α h h hfh 00h3 0 ⎣ α ⎦ ⎣ ⎦ 010 0 h3 0 000h4 α 0 01 00h4   1+α 0 Via a change of basis, one can identify f = I ⊗ . Thus, for the entropy 2 01− α τ( ) = − ( 1+α ) Φ term we have f log f 1 h 2 . Since 0’s outputs are all diagonal, then

(p) (p) † † Q (Φ0) = P (Φ0) = Q (Φ0) = P (Φ0) = 0. 116 L. Gao, M. Junge, N. LaRacuente

− ( 1+α ) By our estimates, we justify that 1 h 2 is indeed an upper bound. On the other − ( 1+α ) hand, 1 h 2 is the quantum capacity of a qubit dephasing channel with parameter α,     a11 a12 a11 αa12 Ψα = , a21 a22 αa21 a22 which can be implemented in Φα by using the block input ⎡ ⎤ ⎡ ⎤ a11 0 a13 0 0000 ⎢ 0000⎥ ⎢ 0 a 0 a ⎥ ⎣ ⎦ or ⎣ 22 24 ⎦ . a31 0 a33 0 0000 0000 0 a42 0 a44 Hence, our upper bound is also achievable.

5. Summary and Discussion In finite dimensions, TROs are direct sums of rectangle matrix spaces. TRO channels, which have Stinespring spaces as TROs, have simple, single-letter, strongly-additive capacities and strong converse theorems. Based on the nice properties of TRO channels, we develop the quantum capacity estimates

Q(N ) ≤ Q(N f ) ≤ Q(N ) + τ(f log f ) for quantum channels N f whose Stinespring spaces are modified from the environ- ment. This construction includes examples such as random unitary channels arising from groups representation, for which our estimates gives new bounds. Our estimate can be viewed as a perturbation of TRO channels, which is tight when f is close to 1 and N f is close to a TRO channel. The work [51] established perburtative estimates for channels that are degradable up to a -error in the diamond norm. In general TRO channels need not to be degradable nor anti-degradable. Our angle is different from [51] and also extends to potential capacity, strong converse rates and capacity regions. The key inequality in our estimates is (Corollary 1),  Dp(N (ρ)||σ) ≤ Dp(N f (ρ)||σ) ≤ Dp(N (ρ)||σ)+ p log f p,τ which compares the sandwiched Rényi relative entropy of output states. We call this inequality a local comparison property as it holds for each input ρ. The comparison inequality passes to the capacities and several other information measures of channels such as (max) Rains information and (max) relative entropy of entanglement, which are known as strong converses bounds [8,10,53,55]. Based on that, the advantage of our strong converse estimates is of a simple explicit correction term, instead of an optimization expression. On the other hand, for modified TRO channels our method also gives perturbative estimates for those information measures of channels, among which the relative entropy of entanglement is not known to be efficiently computable. So far beyond the degradable channel, we know little about the exact value of quantum or private capacity and it is an open question whether the quantum and private strong converse theorem holds for all degradable channels. In Sect. 4.3, we provide a concrete non-degradable example in small dimensions for which our upper estimates are tight and obtaining a formula for quantum and private capacity as well as strong converse Capacity Estimates via Comparison with TRO Channels 117 rates and potential capacities. It indicates that there exists nontrivial non-degradable quantum channels with super-additive capacity formula as well as the strong converse. In other words, the degradablity may not capture the full class of quantum channels which admits single-letter formula for quantum (and private) capacity and strong converse. In a forthcoming paper, we have more higher dimensional but still concrete examples of non-degradable quantum channels that our estimate is tight and implies a super-additive capacity formula and strong converse. Nevertheless, that construction is more involved and beyond the scope of this paper. It will be interesting to know a (possibly algebraic) criterion under which our upper estimates are acheivable. This may shed a light on identifying a larger class of channels which has trackable quantum capacity and the strong converse.

6. Appendix: Complex Interpolation and Noncommutative L p Spaces In this Appendix, we briefly review the complex interpolation theory that is used in the proof of Theorem 1. The readers are referred to [6] for interpolation theory and [40]for information about vector-valued noncommutative L p spaces. Two Banach spaces X0 and X1 are compatible if there exists a Hausdorff topological vector space X such that X0, X1 ⊂ X as subspaces. The sum space X0 + X1 is a Banach space

X0 + X1 := {x ∈ X | x = x0 + x1 for some x0 ∈ X0, x1 ∈ X1}, equipped with the norm = ( ). x X0+X1 inf x0 X0 + x1 X1 x=x0+x1 Let S ={z|0 ≤ Re(z) ≤ 1} be the vertical strip of unit width on the complex plane, and let S0 denote its open interior {z|0 < Re(z)<1}. We denote by F(X0, X1) the space of all functions f : S → X0 + X1, which are bounded and continuous on S and analytic on S0, and moreover

{ f (it) | t ∈ R}⊂X0 , { f (1+it) | t ∈ R}⊂X1.

F(X0, X1) is again a Banach space with the norm = { ( ) , ( ) }. f F max sup f it X0 sup f 1+it X1 t∈R t∈R

The complex interpolation space (X0, X1)θ ,for0<θ<1, is the quotient space of F(X0, X1) given as follows,

(X0, X1)θ ={x ∈ X0 + X1 | x = f (θ), f ∈ F(X0, X1)}. The quotient norm is defined as

x θ = inf{ f F | f (θ) = x}. For example, the Schatten-p class is the interpolation space of bounded operators and trace class

Sp(H) = (B(H), S1(H)) 1 . p 118 L. Gao, M. Junge, N. LaRacuente

This generalizes to vector-valued noncommutative L p-space Sp(A, Sq (B)) (see [40]). In particular, for any 1 ≤ p, q ≤∞one has the relations

Sp(A, Sq (B)) =[S∞(A, Sq (B)), S1(A, Sq (B))] 1 , p Sp(A, Sq (B)) =[S∞(A, S∞(B)), S1(A, S1(B))] 1 . q The following Stein’s interpolation theorem (cf. [6]) is a key tool in our analysis.

Theorem 3. Let (X0, X1) and (Y0, Y1) be two compatible couples of Banach spaces. Let {Tz|z ∈ S}⊂B(X0 + X1, Y0 + Y1) be a bounded analytic family of maps such that

{Tit| t ∈ R}⊂B(X0, Y0), {T1+it| t ∈ R}⊂B(X1, Y1). Λ = Λ = Suppose 0 supt Tit B(X0,Y0) and 1 supt T1+it B(X1,Y1) are both finite, then for 0 <θ<1,Tθ is a bounded linear map from (X0, X1)θ to (Y0, Y1)θ and ≤ Λ1−θ Λθ . Tθ B((X0,X1)θ ,(Y0,Y1)θ ) 0 1 In particular, when T is a constant map, the above theorem implies

1−θ θ T B(( , )θ ,( , )θ )≤ T T . (28) X0 X1 Y0 Y1 B(X0,Y0) B(X1,Y1)

Let X ⊂ B(H, K ) be a TRO. Denote X p the closure of intersection X ∩ Sp(H, K ) in the Schatten p-class. TROs X and their corresponding subspaces X p in Sp(H, K ) are completely 1-complemented for all p ∈[1, ∞] (see [17,37]). That is, there exists a projection map P from B(H, K ) (resp. Sp(H, K )) onto X (resp. X p) such that idn ⊗ P is contractive for every n. A direct consequence is that X p are interpolation spaces of X1 and X = X∞,

X p = (X∞, X1) 1 . p In the proof of Theorem 1, we used the following simple application of Kosaki-type interpolation [32].

Theorem 4. Let X be a TRO. For a positive operator σ ∈ L(X) and 1 ≤ p ≤∞, define X p,σ as the space X equipped with the following norms,

1 x p,σ := σ p x p . Then

[X∞, X1,σ ] 1 = X p,σ . p

1 Proof. Let us first assume that σ is invertible. For x ∈ X such that σ p x p= 1, 1 1 we consider the polar decomposition σ p x = v|σ p x|:=vy, where v ∈ X is a par- tial isometry and y ∈ L(X). Then we define the analytic function x from the strip S ={z| 0 ≤ Re(z) ≤ 1} to X as follows,

− − 1 x(z) = σ zvy pz , x(1/p) = σ p vy = x. Capacity Estimates via Comparison with TRO Channels 119

Note that −it itp x(it) ∞ = σ vy ∞≤ 1, −1−it p(1+it) p p x(1+it) 1,σ = σσ vy 1= vy 1≤ vy p≤ 1.

1 ≤ σ p Therefore x [X∞,X1,σ ] 1 x p. On the other hand, suppose that we have an p analytic function x : S → X such that

sup{ x(it) ∞, σ x(1+it) 1}≤1. t  / / = ( , )≤ Recall that 1 p +1 p 1. For any a Sp K H 1, we claim / tr(σ 1 p xa) ≤ 1.  Indeed, consider the analytic function h(z) = tr(σ z x(z)a(z)) where a(z) = w|a|p (1−z). On the boundary of the strip S,

|h(it)|≤ x(it) ∞ a(it) 1≤ 1 , |h(1+it)|≤ σ x(1+it) 1 a(1+it) ∞≤ 1.

1 By the maximum principle, we obtain that |h(1/p)|=|tr(σ p xy)|≤1, which proves the claim. For noninvertible σ, one can repeat the argument for σ˜ = σ + δ1 with δ>0 and let δ go to 0.  Remark 6. The above interpolation relation can be generalized to two-sided densities. Let 0 ≤ θ ≤ 1. Given σ ∈ L(X), ρ ∈ R(X), one can define X p,θ as the corresponding space equipped with the norm,

θ 1−θ x p,θ,σ,ρ = σ p xρ p p .

These L p-spaces also interpolate [29],

[X∞, X1,θ,σ,ρ] 1 = X p,θ,σ,ρ. p

Acknowledgements. We thank Mark M. Wilde for helpful discussions and comments. We thank the anonymous referees for the careful reading and constructive suggestions.

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Communicated by M. M. Wolf