bibkey: burri2025petztransmission
authors: Laura Burri
year: 2025
title: Entanglement fidelity of Petz decoder for one-shot entanglement transmission
doi: null
url: https://arxiv.org/abs/2502.17411v1
claim: Question 1 asks which of the actual Schumacher-Westmoreland and ordinary Petz decoders performs better; two source-faithful instances have opposite strict entanglement-fidelity comparisons.
strata_touched:
  - D5/S3/Quantum/Recovery/CanonicalEntanglementFidelity
  - D5/S3/Quantum/Recovery/BurriReplacementDecoder
  - D5/S3/Quantum/Recovery/BurriReverseSource
  - D5/S3/Quantum/Recovery/BurriReverseCompletion
  - D5/S3/Quantum/Recovery/BurriReverseFidelity
  - D5/S3/Quantum/Recovery/BurriDecoderComparison
license: citation-only
triage: anchor

SW and ordinary Petz need not have a uniform order

The source is Laura Burri’s arXiv:2502.17411v1, not a later journal text. Section 3.3, Question 1 asks: “Which performs better: the SW decoder or the Petz decoder?” Section 3.1 and Theorem 1 explicitly construct and discuss SW for every $(\rho ,\mathcal{N})\in \mathcal{S}(A)\times \mathrm{CPTP}(A,B)$. The domain therefore includes nonuniform input densities. The flat code states of the numerical examples do not restrict that domain.

Section 2.3 uses squared fidelity against a purification for entanglement fidelity. Appendix A.6 specifies the transposed product of the two positive roots; A.9 fixes the physical purification and ancilla relation; A.11–A.13 specify the actual Kraus construction and its composition with preparation. The ordinary Petz decoder is Section 3.2’s reference-state Petz map, including its trace-preserving completion. Both examples below have full output support, so the completion term is proved to vanish on every matrix.

The owning Scribe gives the data, normalization and exact source bridges. Its common fidelity uses $v(r,i)=(\sqrt{\rho }{)}_{i,r}$ and $F_e(\rho ,C)=⟨v|(\mathrm{id}\otimes C)(|v⟩⟨v|)|v⟩$. Tracing the reference is proved to give the input state $\rho$, including for complex Hermitian states. A proved tensor flip transports the reverse calculation to this same definition.

Every complex coisometric positive-polar completion has the same physical prepared decoder action in each instance. The statements retain the literal source environment, the full SVD and all zero coordinates. They do not assert equality of arbitrary off-support unitary completions.

Related primary results and scope

Sun Woo P. Kim, Optimal recovery for quantum error correction, arXiv:2603.06520v1, Proposition 3, Eqs. (20)–(22), and Corollary 4 already identify Petz posterior sampling and SW squared-posterior sampling for CSS codes under Pauli noise. This is related forward coverage; the paper’s recovery-optimality discussion distinguishes threshold optimality from finite-instance fidelity. Its supplement, Eqs. (S21)–(S25), constructs the SVD recovery and requires matching preparation and basis conventions before transferring a decoder identity.

Raban Iten, Joseph M. Renes and David Sutter, Pretty good measures in quantum information theory, arXiv:1608.08229v2, Section 3.3 after Eq. (54), give the pretty-good singlet-recovery optimality criterion $[{\tau }_{AC},I_A\otimes {\mathrm{tr}}_A\sqrt{{\tau }_{AC}}]=0$. That criterion alone does not prove strict loss of the actual SW construction.

The bounded standing check also read arXiv:2410.23622v5, Theorem 2 and Corollary 1; arXiv:2602.14732v1, Section 5.3, Theorem 29; and arXiv:2610.31501v1, Theorem 3.1. These checked passages do not furnish the exact two-instance no-order result. This is not an exhaustive citation search or a global novelty claim. The publisher article was inaccessible to that check; its wording is not authenticated here.

Question 2 separately compares a SW lower bound and a Petz Renyi-information expression. No answer to Question 2, uniform strict advantage, or complete classification is claimed.