Moderate deviation analysis for classical communication over quantum channels

2019

Quantum

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Given two pairs of quantum states, we want to decide if there exists a quantum channel that transforms one pair into the other. The theory of quantum statistical comparison and quantum relative majorization provides necessary and sufficient conditions for such a transformation to exist, but such conditions are typically difficult to check in practice. Here, by building upon work by Matsumoto, we relax the problem by allowing for small errors in one of the transformations. In this way, a simple sufficient condition can be formulated in terms of one-shot relative entropies of the two pairs. In the asymptotic setting where we consider sequences of state pairs, under some mild convergence conditions, this implies that the quantum relative entropy is the only relevant quantity deciding when a pairwise state transformation is possible. More precisely, if the relative entropy of the initial state pair is strictly larger compared to the relative entropy of the target state pair, then a transformation with exponentially vanishing error is possible. On the other hand, if the relative entropy of the target state is strictly larger, then any such transformation will have an error converging exponentially to one. As an immediate consequence, we show that the rate at which pairs of states can be transformed into each other is given by the ratio of their relative entropies. We discuss applications to the resource theories of athermality and coherence. * marco.tomamichel@uts.edu.au 1 A transformation is said to be bistochastic if it transforms probability distributions to probability distributions, while keeping the uniform distribution fixed.The majorization preorder is particularly relevant and useful because of a famous result by Hardy, Littlewood, and Polya, according to which the relation p 1 p 2 can be expressed in terms of a finite set of inequalities [22] of the form f i ( p 1 ) ≥ f i ( p 2 ), intuitively capturing the idea that p 1 is "less uniform" than p 2 . Such inequalities can be conveniently visualized by plotting the Lorenz curve of p 1 versus that of p 2 [32].As it involves the comparison of two probability distributions relative to a third one (i.e., the uniform distribution), the majorization preorder is naturally generalized by considering two pairs of probability distributions, that is, two dichotomies ( p 1 , q 1 ) and ( p 2 , q 2 ), where now q 1 and q 2 are arbitrary distributions. One then writes ( p 1 , q 1 ) ( p 2 , q 2 ) whenever there exists a stochastic transformation simultaneously mapping p 1 to p 2 and q 1 to q 2 . As a consequence of Blackwell's equivalence theorem [5], also the more general case of dichotomies is completely characterized by a finite set of simple inequalities, which directly reduce to those of Hardy, Littlewood and Polya if q 1 and q 2 are both taken to be uniform. Also in this more general scenario, a relative Lorenz curve can be associated to each dichotomy, and the preorder visualized accordingly [45].In relation to quantum information sciences, while the preorder of majoriz...

Section: Discussionmentioning

confidence: 72%

An information-theoretic treatment of quantum dichotomies

Buscemi

Sutter

2019

Quantum

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“…We note that the exact prefactor for general codes is still open even in the classical case. Finally, our refinement enables a moderate deviation analysis in c-q channels [29] (see also [30]).…”

Section: Discussionmentioning

confidence: 99%

“…(1) and (1), and discusses their generalizations to c-q channels. We remark that the established finite blocklength bounds and the polynomial prefactor are crucial for the analysis of coding performance in the medium error probability regime (more commonly known as moderate deviation analysis) [28,29,30], classical data compression with quantum side information [31,32], and joint source-channel coding with quantum side information [33].…”

Section: Introductionmentioning

confidence: 99%

Quantum Sphere-Packing Bounds With Polynomial Prefactors

Cheng

Hsieh

IEEE Trans. Inform. Theory

2019

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We study lower bounds on the optimal error probability in classical coding over classicalquantum channels at rates below the capacity, commonly termed quantum sphere-packing bounds. Winter and Dalai have derived such bounds for classical-quantum channels; however, the exponents in their bounds only coincide when the channel is classical. In this paper, we show that these two exponents admit a variational representation and are related by the Golden-Thompson inequality, reaffirming that Dalai's expression is stronger in general classical-quantum channels. Second, we establish a finite blocklength sphere-packing bound for classical-quantum channels, which significantly improves Dalai's prefactor from the order of subexponential to polynomial. Furthermore, the gap between the obtained error exponent for constant composition codes and the best known classical random coding exponent vanishes in the order of o(log n/n), indicating our sphere-packing bound is almost exact in the high rate regime. Finally, for a special class of symmetric classical-quantum channels, we can completely characterize its optimal error probability without the constant composition code assumption. The main technical contributions are two converse Hoeffding bounds for quantum hypothesis testing and the saddle-point properties of error exponent functions.The distributions p ρ,σ , q ρ,σ have the same mathematical properties as the density operators ρ, σ in some cases, and thus are useful in the sequel. First, one can verify that [62,46], D α (ρ σ) = D α (p ρ,σ q ρ,σ ) , ∀α ∈ [0, 1]. 7 Throughout this paper, we skip the dependence of the channel W in the reliability function and error-exponent functions. 9 42

“…The study of such non-asymptotic scenarios has recently garnered great interest in classical information theory (e.g., [16][17][18]) as well as in quantum information theory (e.g., [19][20][21][22][23][24][25][26][27][28][29][30][31]). Here we study the setting of entanglement distillation.…”

mentioning

confidence: 99%

Non-Asymptotic Entanglement Distillation

Fang

Wang

IEEE Trans. Inform. Theory

et al. 2019

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Non-asymptotic entanglement distillation studies the trade-off between three parameters: the distillation rate, the number of independent and identically distributed prepared states, and the fidelity of the distillation. We first study the one-shot ε-infidelity distillable entanglement under quantum operations that completely preserve positivity of the partial transpose (PPT) and characterize it as a semidefinite program (SDP). For isotropic states, it can be further simplified to a linear program. The one-shot ε-infidelity PPTassisted distillable entanglement can be transformed to a quantum hypothesis testing problem. Moreover, we show efficiently computable second-order upper and lower bounds for the non-asymptotic distillable entanglement with a given infidelity tolerance. Utilizing these bounds, we obtain the second order asymptotic expansions of the optimal distillation rates for pure states and some classes of mixed states. In particular, this result recovers the second-order expansion of LOCC distillable entanglement for pure states in [Datta/Leditzky, IEEE Trans. Inf. Theory 61:582, 2015]. Furthermore, we provide an algorithm for calculating the Rains bound and present direct numerical evidence (not involving any other entanglement measures, as in [Wang/Duan, Phys. Rev. A 95:062322, 2017]), showing that the Rains bound is not additive under tensor products. I. INDRODUCTION A. BackgroundQuantum entanglement is a striking feature of quantum mechanics and a key ingredient in many quantum information processing tasks, including teleportation [1], superdense coding [2], and quantum cryptography [3,4]. All these protocols necessarily rely on entanglement resources, especially the maximally entangled states. It is thus of great importance to develop entanglement distillation protocols to transform less useful entangled states into more suitable ones such as maximally entangled states. In general, the task of entanglement distillation aims at obtaining maximally entangled states from less-entangled bipartite states shared between two parties (Alice and Bob) and it allows them to perform local operations and classical communication (LOCC). The concept of distillable entanglement characterizes the rate at which one can asymptotically obtain maximally entangled states from a collection of identically and independently distributed (i.i.d) prepared entangled states by LOCC [5,6]. Distillation from non-i.i.d prepared states has also been considered recently [7]. Distillable entanglement is a fundamental entanglement measure which captures the resource character of entanglement. Up to now, how to calculate distillable entanglement for gen-* Electronic address: kun.fang-