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Reliability Function of Quantum Information Decoupling via the Sandwiched Rényi Divergence

Ke Li, Yongsheng Yao

TL;DR

The paper addresses the reliability function for catalytic quantum information decoupling, establishing an exact exponential rate below a critical decoupling cost and meaningful bounds above it. It achieves this via new tools: a convex-split bound expressed through the sandwiched Rényi divergence and exact exponents for smoothing max-information and conditional min-entropy, with the sandwiched Rényi quantities gaining a direct operational meaning. The results apply to three decoupling schemes (removing a subsystem, projective measurement, and random unitaries) and extend to quantum state merging through established task relations, highlighting deep connections between decoupling fidelity and information measures. The work advances the understanding of reliability functions in quantum information and provides a framework for applying sandwiched Rényi divergences to precise asymptotics in quantum tasks, with potential implications for channel simulation and entanglement-assisted protocols.

Abstract

Quantum information decoupling is a fundamental quantum information processing task, which also serves as a crucial tool in a diversity of topics in quantum physics. In this paper, we characterize the reliability function of catalytic quantum information decoupling, that is, the best exponential rate under which perfect decoupling is asymptotically approached. We have obtained the exact formula when the decoupling cost is below a critical value. In the situation of high cost, we provide meaningful upper and lower bounds. This result is then applied to quantum state merging, exploiting its inherent connection to decoupling. In addition, as technical tools, we derive the exact exponents for the smoothing of the conditional min-entropy and max-information, and we prove a novel bound for the convex-split lemma. Our results are given in terms of the sandwiched Rényi divergence, providing it with a new type of operational meaning in characterizing how fast the performance of quantum information tasks approaches the perfect.

Reliability Function of Quantum Information Decoupling via the Sandwiched Rényi Divergence

TL;DR

The paper addresses the reliability function for catalytic quantum information decoupling, establishing an exact exponential rate below a critical decoupling cost and meaningful bounds above it. It achieves this via new tools: a convex-split bound expressed through the sandwiched Rényi divergence and exact exponents for smoothing max-information and conditional min-entropy, with the sandwiched Rényi quantities gaining a direct operational meaning. The results apply to three decoupling schemes (removing a subsystem, projective measurement, and random unitaries) and extend to quantum state merging through established task relations, highlighting deep connections between decoupling fidelity and information measures. The work advances the understanding of reliability functions in quantum information and provides a framework for applying sandwiched Rényi divergences to precise asymptotics in quantum tasks, with potential implications for channel simulation and entanglement-assisted protocols.

Abstract

Quantum information decoupling is a fundamental quantum information processing task, which also serves as a crucial tool in a diversity of topics in quantum physics. In this paper, we characterize the reliability function of catalytic quantum information decoupling, that is, the best exponential rate under which perfect decoupling is asymptotically approached. We have obtained the exact formula when the decoupling cost is below a critical value. In the situation of high cost, we provide meaningful upper and lower bounds. This result is then applied to quantum state merging, exploiting its inherent connection to decoupling. In addition, as technical tools, we derive the exact exponents for the smoothing of the conditional min-entropy and max-information, and we prove a novel bound for the convex-split lemma. Our results are given in terms of the sandwiched Rényi divergence, providing it with a new type of operational meaning in characterizing how fast the performance of quantum information tasks approaches the perfect.

Paper Structure

This paper contains 17 sections, 16 theorems, 118 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and basic properties
  4. Quantum entropies and information divergences
  5. Problem Statement and Main Results
  6. Catalytic quantum information decoupling
  7. Main results
  8. Applications to quantum state merging
  9. Proof of the Characterization of Reliability Functions
  10. A convex-split lemma
  11. Smoothing of max-information and conditional min-entropy
  12. Proof of Theorem \ref{['thm:reliability-dec']}
  13. Proof of the relations between quantum tasks
  14. Proof of the relation between decouplings
  15. Proof of the relation between decoupling and quantum state merging
  16. ...and 2 more sections

Key Result

Lemma 1

For every Hilbert space $\mathcal{H}_A$ and $n\in\mathbb{N}$, there exists a universal symmetric state $\omega^{(n)}_{A^n} \in \mathcal{S}^{\rm{sym}}(A^n)$, such that for any $\rho_{A^n} \in \mathcal{S}^{\rm{sym}}(A^n)$ we have where $g_{n,|A|} \leq (n+1)^{\frac{(|A|+2)(|A|-1)}{2}}$ and $v(\omega^{(n)}_{A^n})$ denotes the number of different eigenvalues of $\omega^{(n)}_{A^n}$.

Figures (1)

  • Figure 1: Reliability function of quantum information decoupling. $E_u(r):=\sup_{s \geq 0} \{s(r-\frac{1}{2}I_{1+s}(R:A)_\rho)\}$ is the upper bound of Eq. (\ref{['eq:exp-dec-up']}). $E_l(r):=\sup_{0\leq s\leq1} \{s(r-\frac{1}{2}I_{1+s}(R:A)_\rho)\}$ is the lower bound of Eq. (\ref{['eq:exp-dec-low']}). The two bounds are equal in the interval $[0,R_\text{critical}]$, giving the exact reliability function. The reliability function equals $0$ when $r<\frac{1}{2}I(R:A)_\rho$ and it is strictly positive when $r>\frac{1}{2}I(R:A)_\rho$. Above the critical value $R_\text{critical}$, the upper bound $E_u(r)$ becomes larger than the lower bound and it is $\infty$ when $r>\frac{1}{2}I_{\rm{max}} (R:A)_\rho$, while the lower bound $E_l(r)$ becomes linear and reaches $\log |A|-\frac{1}{2}I_2(R:A)_\rho$ at $r=\log |A|$.

Theorems & Definitions (24)

  • Lemma 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Proposition 6
  • Theorem 7
  • Theorem 8
  • Definition 9
  • Definition 10
  • ...and 14 more