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Resolvability of classical-quantum channels

Masahito Hayashi, Hao-Chung Cheng, Li Gao

Abstract

Channel resolvability concerns the minimum resolution for approximating the channel output. We study the resolvability of classical-quantum channels in two settings, for the channel output generated from the worst input, and form the fixed independent and identically distributed (i.i.d.) input. The direct part of the worst-input setting is derived from sequential hypothesis testing as it involves of non-i.i.d.~inputs. The strong converse of the worst-input setting is obtained via the connection to identification codes. For the fixed-input setting, while the direct part follows from the known quantum soft covering result, we exploit the recent alternative quantum Sanov theorem to solve the strong converse.

Resolvability of classical-quantum channels

Abstract

Channel resolvability concerns the minimum resolution for approximating the channel output. We study the resolvability of classical-quantum channels in two settings, for the channel output generated from the worst input, and form the fixed independent and identically distributed (i.i.d.) input. The direct part of the worst-input setting is derived from sequential hypothesis testing as it involves of non-i.i.d.~inputs. The strong converse of the worst-input setting is obtained via the connection to identification codes. For the fixed-input setting, while the direct part follows from the known quantum soft covering result, we exploit the recent alternative quantum Sanov theorem to solve the strong converse.

Paper Structure

This paper contains 11 sections, 11 theorems, 126 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Formulation of Channel Resolvability and our results
  3. Classical Channel Resolvability
  4. Direct part for CQ-Channel Resolvability
  5. Strong converse for Channel Resolvability with fixed input
  6. Quantum Sanov theorem
  7. Strong converse of fixed-input resolvability
  8. Strong converse for channel resolvability of worst input
  9. Identification codes
  10. Strong converse for Channel Resolvability
  11. Discussion

Key Result

Theorem 1

For any classical-quantum channel $W:\mathcal{X} \to \mathcal{S(H)}$, the resolvability rate for the worst-input setting, defined in MA2 and eq:def:worst-input, is given by

Figures (3)

  • Figure 1: A schematic diagram of channel resolvability for the fixed-input case. The goal is to employ $nR$-bit uniform randomness and an optimal codebook of size $2^{nR}$ to simulate the target state $W^{\otimes n}(p^{\otimes n})$, which is generated from a fixed i.i.d. input distribution $p^{\otimes n}$. The minimum asymptotic rate $\liminf_{n\to \infty} \frac{1}{n} \log M$ for an $\epsilon$-approximation is $R_{\epsilon}(p,W)$ defined in \ref{['eq:def:fixed-input']}.
  • Figure 2: A schematic diagram of channel resolvability for the worst-input case. The goal is to $nR$-bit uniform randomness and an optimal codebook of size $2^{nR}$ to simulate the target state $W^{\otimes n}(p^n)$, which is generated from a worst (probably non-i.i.d.) input distribution $p^{n} \in \mathcal{P}(\mathcal{X}^n)$. The minimum asymptotic rate $\liminf_{n\to \infty} \frac{1}{n} \log M$ for an $\epsilon$-approximation is $R_{\epsilon}(W)$ defined in \ref{['eq:def:worst-input']}.
  • Figure 3: An illustration of the separation for \ref{['NBSA']} via the channel given in \ref{['table:separation']}. The upper curve is the channel capacity $1 + \epsilon \log \epsilon + (1-\epsilon )\log (1-\epsilon)$; the lower dotted curve is always $0$. The two curves coincide only when $\epsilon = 1/2$.

Theorems & Definitions (17)

  • Theorem 1: Worst-Input Setting
  • Theorem 2: Fixed-Input Setting
  • Example 3
  • Lemma 4: hbook
  • Lemma 5
  • proof
  • Lemma 6: 4069150
  • Theorem 7: Direct Part of Worst-Input Setting
  • proof
  • Proposition 8: another
  • ...and 7 more