Quantum Sphere-Packing Bounds With Polynomial Prefactors

Abstract

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

Properties of Scaled Noncommutative Rényi and Augustin Information

“…The sphere-packing bound for constant composition codes were also proved by using E (2) 0 (s, P ) [92,51]. Comparing with Winter's result, the Petz's version is tighter than the log-Euclidean when R < C W by Golden-Thompson's inequality [93,94,95,34,51]. In the strong converse regime (R > C W ), Mosonyi and Ogawa [84] proved that the strong converse exponent is determined by the sandwiched version, see Eq.…”

“…We remark that the concavity property in s has a plethora of usefulness. For examples, it determines the convexity and decreases of the entropic quantities in R [19, p. 142], and it is indispensable in proving the saddle-point property in sphere-packing exponents [73,51,53], and the moderate deviations [65]. The properties of the auxiliary functions can also be derived via those of the Rényi and Augustin information.…”

“…Therefore, one of the main aims of the current paper is to investigate properties of the auxiliary functions using noncommutative measure theory. Moreover, the established results could be employed to perform refined analysis in quantum information processing tasks [51,52,53,54,55].…”

“…We will exploit these facts in our derivations later. 4 The auxiliary functions in different protocols are defined in a slightly different but similar way [50,51,52,53,54,55]. We refer the readers to Section 5 for further discussion.…”

“…optimality) was first studied by Winter [90], and he proved the bound with the log-Euclidean version. Recently, Dalai [91] and part of the present authors [51] established a sphere-packing bound for all codes with Petz's version when maximizing over all priors P as in the Eq. (1).…”