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Universal recoverability of quantum states in tracial von-Neumann algebras

Saptak Bhattacharya

TL;DR

The paper addresses universal approximate recoverability of quantum states under CPTP maps in infinite-dimensional settings by refining the data processing inequality for the sandwiched quasi-relative entropy $S_2$ on tracial von Neumann algebras. It introduces the Araki-Masuda $L^2$-norm framework and shows the Petz recovery map $\mathcal{R}$ is the appropriate adjoint to realize a universal bound, namely $4[1-F(\psi_A|\psi_{\mathcal{R}(\phi(A))})]^2 \leq ||A-\mathcal{R}(\phi(A))||_1^2 \leq [\mathcal{S}_2(A|B) - \mathcal{S}_2(\phi(A)|\phi(B))]$, with a complementary $L^2$ bound, thereby extending finite-dimensional results to the infinite-dimensional case. The work connects DPI, recoverability, and fidelity via the Araki-Masuda geometry and proves a fidelity-based recoverability bound using the Uhlmann fidelity between states, with the appendix establishing joint concavity of the fidelity on $C^*$-algebras. This universal bound enables robust state recovery guarantees in infinite-dimensional quantum information processing and deepens the understanding of Petz-type recovery in broader operator-algebraic settings.

Abstract

In this paper, we discuss a refinement of quantum data processing inequality for the sandwiched quasi-relative entropy $\mathcal{S}_2$ on a tracial von-Neumann algebra. The main result is a universal recoverability bound with the Petz recovery map, which was previously obtained in the finite dimensional setup.

Universal recoverability of quantum states in tracial von-Neumann algebras

TL;DR

The paper addresses universal approximate recoverability of quantum states under CPTP maps in infinite-dimensional settings by refining the data processing inequality for the sandwiched quasi-relative entropy on tracial von Neumann algebras. It introduces the Araki-Masuda -norm framework and shows the Petz recovery map is the appropriate adjoint to realize a universal bound, namely , with a complementary bound, thereby extending finite-dimensional results to the infinite-dimensional case. The work connects DPI, recoverability, and fidelity via the Araki-Masuda geometry and proves a fidelity-based recoverability bound using the Uhlmann fidelity between states, with the appendix establishing joint concavity of the fidelity on -algebras. This universal bound enables robust state recovery guarantees in infinite-dimensional quantum information processing and deepens the understanding of Petz-type recovery in broader operator-algebraic settings.

Abstract

In this paper, we discuss a refinement of quantum data processing inequality for the sandwiched quasi-relative entropy on a tracial von-Neumann algebra. The main result is a universal recoverability bound with the Petz recovery map, which was previously obtained in the finite dimensional setup.

Paper Structure

This paper contains 3 sections, 14 theorems, 80 equations.

Table of Contents

  1. introduction
  2. Main results
  3. Appendix

Key Result

Theorem 1

Let $B\in M_n(\mathbb{C})$ be a density matrix and let $\phi:M_n(\mathbb{C})\to M_k(\mathbb{C})$ be a quantum channel. Let $\mathcal{R}:M_k(\mathbb{C})\to M_n(\mathbb{C})$ be given by Then, for some density matrix $A\in M_n(\mathbb{C})$, $D(A|B)=D(\phi(A)|\phi(B))$ if and only if $\mathcal{R}(\phi(A))=A$.

Theorems & Definitions (29)

  • Theorem
  • Remark
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Corollary 1
  • Lemma 1
  • proof
  • Lemma 2
  • ...and 19 more