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Capacity Estimates Via Comparison with TRO Channels 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.
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