Capacity bounds via operator space methods

Abstract

Quantum capacity, the ultimate transmission rate of quantum communication, is characterized by regularized coherent information. In this work, we reformulate approximations of the quantum capacity by operator space norms and give both upper and lower estimates on quantum capacity and potential quantum capacity using complex interpolation techniques from operator space theory. Upper bounds are obtained by a comparison inequality for Rényi entropies. Analyzing the maximally entangled state for the whole system and for error-free subsystems provides lower bounds for the "one-shot" quantum capacity. These two results combined give upper and lower bounds on quantum capacity for our "nice" classes of channels, which differ only up to a factor 2, independent of the dimension. The estimates are discussed for certain classes of channels, including group channels, generalized Pauli channels and other highdimensional channels.where σ AB = id A ⊗ Φ(ρ AA ) and the maximum runs over all pure bipartite state ρ AA . I c (A B) σ is the coherent information of bipartite σ given by H(σ B ) − H(σ AB ), with H(σ) = −tr(σ log σ) being the von Neumann entropy, and Q (1) is the "one-shot" quantum capacity. Let us also recall that the negative cb-entropy (also called the reverse coherent information) of a channel Φ is defined similarly as −S cb (Φ) = max ρ H(A) ρ − H(AB) ρ (see Section 2 for formal definitions). *

Semidefinite Programming Converse Bounds for Quantum Communication

“…Recently, the Rains information [24] was established to be a strong converse bound for quantum communication. There are other known upper bounds for quantum capacity [13][14][15][27][28][29][30][31] and most of them require specific settings to be computable and relatively tight.…”

A semidefinite programming upper bound of quantum capacity

“…Given an arbitrary quantum channel, the only known general computable upper bound is the partial transposition bound introduced in [4]. Other known upper bounds [5], [6], [7], [8], [9], [10], [11], [12] all require specific settings to be tight and computable. For example, the upper bound from no cloning argument [8], [9] only behaves well at very high noise levels.…”

Partially Coherent Direct Sum Channels

“…In the limiting cases |κ| = 1 (no noise) and κ = 0 (full dephasing) the maximization can be explicitly performed leading to the expected results of Eqs. (30) and (31), respectively. For all the other choices of κ we resort to numerical evaluation and report the obtained results in Fig.…”

“…Specifically, we present compact expressions for the quantum capacity and entanglement assisted quantum capacity of a new class of channels that we called Partially Coherent Direct Sum (PCDS) channels, a generalization of the direct sum (DS) channels described in [30]. This formalism appears in a variety of contexts [31,32,33,34,35,36,37], among which recently the "gluing" procedure in [38] derived as a generalization of the construction in [39]. DS channels were initially introduced in [30] in the context of the additivity conjecture for the classical capacity, later proven wrong [10].…”