Non-Asymptotic Classical Data Compression With Quantum Side Information

Abstract

In this paper, we analyze classical data compression with quantum side information (also known as the classical-quantum Slepian-Wolf protocol) in the so-called large and moderate deviation regimes. In the non-asymptotic setting, the protocol involves compressing classical sequences of finite length n and decoding them with the assistance of quantum side information. In the large deviation regime, the compression rate is fixed, and we obtain bounds on the error exponent function, which characterizes the minimal probability of error as a function of the rate. Devetak and Winter showed that the asymptotic data compression limit for this protocol is given by a conditional entropy. For any protocol with a rate below this quantity, the probability of error converges to one asymptotically and its speed of convergence is given by the strong converse exponent function. We obtain finite blocklength bounds on this function, and determine exactly its asymptotic value. In the moderate deviation regime for the compression rate, the latter is no longer considered to be fixed. It is allowed to depend on the blocklength n, but assumed to decay slowly to the asymptotic data compression limit. Starting from a rate above this limit, we determine the speed of convergence of the error probability to zero and show that it is given in terms of the conditional information variance. Our results complement earlier results obtained by Tomamichel and Hayashi, in which they analyzed the so-called small deviation regime of this protocol.

Reliability Function of Quantum Information Decoupling via the Sandwiched Rényi Divergence

“…Our results, along with [33], are given in terms of the sandwiched Rényi divergence [27,34], providing it with operational meanings in characterizing how fast the performance of quantum information tasks approach the perfect. This is in stark contrast to those of the previous ones [29][30][31][35][36][37], which are concerned with the strong converse exponents.…”

“…However, the reliability function is not known even for classical-quantum channels. Nevertheless, see References [25][26][27][28][29][30][31] for a partial list of the fruitful results for the strong converse exponent in the quantum setting, which characterizes how fast a quantum information task is getting the useless.…”

Properties of Scaled Noncommutative Rényi and Augustin Information

“…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].…”

Minimizing Quantum Renyi Divergences via Mirror Descent with Polyak Step Size

“…For example, the quantum Augustin and Rényi information determine the error exponent of communications over a classical-quantum channel using certain random codebooks and that of composite quantum hypothesis testing [DW17, CHT19, CGH18, MO17, MO21, HT16]. The quantum conditional Rényi entropies have profound applications to quantum cryptographic protocols, such as bounding the convergence rate of privacy amplification [Dup21], analyzing the security of quantum key distributions [Ren05, KRS09,Tom12], and bounding the error exponent of source coding with quantum side information [CHDH18,CHDH21]. The above-mentioned quantities share a common feature-they require optimizing some order-α quantum Rényi divergence over the set of all quantum states.…”