slug: burri-2025-sw-petz-decoder-comparison
bibkey: burri2025petztransmission
doi: null
url: https://arxiv.org/abs/2502.17411v1
triage: theorem
motivation_gids:
  - D5/S3/Quantum/Recovery/BurriDecoderComparison.result

Burri’s SW–Petz decoder comparison

Problem

Laura Burri, Entanglement fidelity of Petz decoder for one-shot entanglement transmission, arXiv:2502.17411v1, Section 3.3, Question 1 asks:

Which performs better: the SW decoder or the Petz decoder?

The source domain is every density state and CPTP noise channel (Section 3.1, Theorem 1). The question concerns the actual Appendix A SW decoder and ordinary Petz decoder, measured by canonical entanglement fidelity. It is separate from Question 2’s comparison with a lower bound.

Motivation

The question compares two concrete recovery constructions on the full source domain, while the numerical examples in the source use flat code states. A faithful answer must retain the physical isometry, source environment, SVD and completion data, and must compare both constructions through one canonical entanglement-fidelity definition.

Gap

Question 1 does not specify a uniform ordering. The bounded checked primary sources and the related literature in the literature entry do not provide this exact unrestricted two-instance comparison. This dossier does not claim an exhaustive citation search or global novelty.

Route

Use two source-faithful Appendix A instances with opposite strict comparisons. The common root vector is $v(r,i)=(\sqrt{\rho }{)}_{i,r}$ in reference-first order; its pure density is normalized and has channel-input marginal $\rho$. The reverse calculation is transported by an actual tensor-order reindexing of the vector, density and channel extension, including the reverse factor flip.

Falsifier

The settlement would fail if either encoded instance did not satisfy the source domain, if the ordinary Petz completion or the SW preparation were not proved for the literal physical construction, if the two fidelity expressions used different input states or roots, or if either strict inequality failed. A claim that one decoder uniformly dominates the other would also be incompatible with the two certified opposite witnesses.

Evidence

The owning Scribe contains the full instance data and cites the physical construction proofs. For the first instance, $A=\mathrm{Fin}2$, $B=\mathrm{Fin}1$, $E=A\times B$ and $\rho =\mathrm{diag}(2/3,1/3)$. The isometry sends $|i⟩$ to $|0{⟩}_B|i,0{⟩}_E$. The normalized correlated root vector supplies the fixed ancilla and $W=I$. SW has fidelity $11/27$, ordinary Petz has fidelity $1/3$, and their gap is $2/27$. This uses the existing BurriReplacementDecoder source, including its full-channel Petz identification and supported-action uniqueness.

For the second instance, $A=\mathrm{Fin}2$, $B=\mathrm{Fin}2\times \mathrm{Fin}3$, $E=A\times B$ and $\rho =I_2/2$. The positive $6\times 6$ matrix $Y$ gives physical Kraus operators $K_a(b,i)=Y((i,a),b)/\sqrt{31}$, retaining the full twelve-dimensional environment, with output $Y^2/62$. Ordinary Petz has fidelity $27/31$; the actual SW fidelity is strictly smaller by an exact radical sum-of-squares certificate. The seven BurriReverse* modules prove the isometry, state, marginals, positive roots, literal source matrix, full SVD with eighteen null coordinates, $W$, normalized ancilla, preparation identity and full ordinary Petz action.

forward_fidelity and reverse_fidelity prove equality to the same common fidelity at their respective different states. result quantifies over every complex $U^{\prime }$ with $U^{\prime }{U}^{\prime †}=I$ and $M{U}^{\prime }⪰0$ in each instance. The prepared source rows define a CPTP decoder on every physical matrix and have the stated strict comparison. No decoder action or fidelity reduction is a premise.

Triage

The settlement is proved for the two explicit witnesses: SW is strictly better than ordinary Petz in the replacement instance, and strictly worse in the reverse instance. Therefore neither decoder uniformly dominates the other on the source domain. The conclusion applies to every complex positive-polar completion covered by the theorem and leaves the physical support, source environment and completion quantifiers intact.

Sun Woo P. Kim, Optimal recovery for quantum error correction, arXiv:2603.06520v1, Proposition 3, Eqs. (20)–(22), and Corollary 4, preserves the related CSS/Pauli posterior-sampling comparison. Raban Iten, Joseph M. Renes and David Sutter, Pretty good measures in quantum information theory, arXiv:1608.08229v2, Section 3.3 after Eq. (54), supplies the pretty-good singlet-recovery commutator criterion. These are prior results, not discoveries of this formalization.

The result does not settle Question 2, classify all channels, assert uniform strict advantage, or claim that every input admits both directions.

ASSUMED-UNVERIFIED

The bounded literature check did not authenticate the inaccessible publisher article and did not constitute an exhaustive citation search. The new source-owner four-slot registrations remain unfinished in issue #22675; no new declared_validated registration is claimed. Global novelty and claims beyond the two kernel-certified instances remain unverified.