slug: ting-fullwood-wu-2026-three-weight-bayesian-inverse
bibkey: ting2026operational
doi: 10.48550/arXiv.2605.10375
url: https://arxiv.org/abs/2605.10375v1
triage: theorem
motivation_gids:
  - D5/S3/QuantumChannels/PauliThreeWeightBayesianInverseUniqueness.result

Three-weight Pauli channels have Bayesian inverses only at the maximally mixed state

Problem

O. Ting, J. Fullwood and Z. Wu, Operational time-reversal symmetry for unital qubit channels, arXiv:2605.10375v1, call a channel $\mathcal{F}$ a Bayesian inverse of $\mathcal{E}$ with respect to $\rho$ when $\{\mathcal{E}(\rho )\otimes 1,\mathcal{J}[\mathcal{F}]\}=\{1\otimes \rho ,\mathcal{J}[{\mathcal{E}}^{†}]\}$, $\mathcal{J}[\mathcal{N}]=(\mathrm{id}\otimes \mathcal{N})(\mathtt{SWAP})$. For Pauli channels $\mathcal{P}(A)={\sum }_{\mu }{p}_{\mu }{\sigma }_{\mu }A{\sigma }_{\mu }$ they settle the case of two non-zero weights and characterize the general case by three inequalities, and in Section V state that for exactly three non-zero weights they found no Bayesian inverse away from the maximally mixed state, which is “probably due to the fact that either the maximally mixed state is the only state for which a Bayesian inverse exists in such a case, or that such states are restricted to a measure-zero subset of the Bloch ball”. The verbatim statements are in the literature note.

Issue #13585 fixes the reading: for every probability vector with exactly three non-zero entries and every qubit density matrix $\rho$ (pure states included), a CPTP Bayesian inverse of $\mathcal{P}$ with respect to $\rho$ exists if and only if $\rho =1/2$; ${\mathcal{P}}^{†}=\mathcal{P}$ is part of the statement.

Motivation

A Bayesian inverse is the operational time reversal of a sequential measurement through the channel. The question is whether three-weight Pauli noise, the generic case between the two-weight channels (a coordinate axis of invertible states) and the four-weight channels such as depolarizing noise (a neighbourhood of the maximally mixed state, the paper’s Fig. 1), still admits any time-reversal symmetry away from the maximally mixed state. D5/S3/QuantumChannels/PauliThreeWeightBayesianInverseUniqueness.result shows it does not.

Gap

Section IV characterizes Bayesian-inverse existence by three Choi inequalities without evaluating the three-weight case. The Section V alternative is the question addressed here. The source and the preregistered full statement in #13585 specify all probability vectors with exactly three nonzero weights and all qubit density matrices. A bounded literature search found no separate settlement in the searched scope; it does not establish worldwide novelty or priority.

Route

1. The maximally mixed state. $\mathcal{P}$ is CPTP with Kraus operators $\sqrt{p_{\mu }}{\sigma }_{\mu }$ and fixes $1/2$, so $\mathcal{F}=\mathcal{P}$ satisfies the Bayes rule there. Cyclicity of trace and Hermiticity of the Pauli matrices prove ${\mathcal{P}}^{†}=\mathcal{P}$ for all matrix arguments.
2. The scaled candidate. Write $\rho =(1+r\cdot \sigma )/2$ with ${\sum }_i{r}_i^2\le 1$ and $s_i={\lambda }_i{r}_i$, where $\lambda =(p_0+p_1-p_2-p_3,p_0-p_1+p_2-p_3,p_0-p_1-p_2+p_3)$. For the canonical Choi convention $C_{(j_1,i_1),(j_2,i_2)}=\mathcal{F}(|i_1⟩⟨i_2|{)}_{j_1{j}_2}$, the expanded Jam Bayes rule is exactly $C+(VC+CV)/2=R$, where $V=1\otimes (s\cdot \sigma {)}^{\mathsf{T}}$ and $R=\{\rho \otimes 1,C(\mathcal{P})\}$. Since $V^2=|s{|}^{2}1$, the Sylvester identity gives $$DC=(1-|s{|}^2/2)R-(VR+RV)/2+VRV/2,D=1-|s{|}^2\ge 0.$$
3. A negative direction. If $p_k=0$, let $a_1,a_2,a_3$ be the other three weights, listed through the Pauli product table, and $h_i=(1-a_i)r_i$. The vector $w(j,i)=((1-h\cdot \sigma ){\sigma }_k{)}_{ji}$ is indexed by output $j$ and input $i$. For all four choices of $k$ it satisfies $$w^{\ast }(DC)w=-4[(1-|h{|}^2)B+KA],$$ where $A={\sum }_i{a}_i{h}_i^2$, $B={\sum }_i{a}_i^2(1-a_i)r_i^2$ and $K={\sum }_i{a}_i^2{r}_i^2$. The three active weights lie strictly between zero and one. If $r\ne 0$, then $B>0$ and $|h|<1$, while $K,A\ge 0$. The displayed value is therefore negative. Complete positivity makes $C$ positive semidefinite; scaling by $D\ge 0$ preserves positive semidefiniteness, a contradiction.

The Lean argument never divides by $D$ and includes pure states. As explanatory algebra, exactly three positive weights imply $|{\lambda }_i|<1$, hence $D>0$ throughout the closed Bloch ball. The same Sylvester identity then fixes the candidate uniquely and yields the equivalent expression $w^{\ast }Cw=-4[(1-|h{|}^2)B+KA]/D<0$. These strict-denominator and uniqueness consequences are not additional Lean conclusions.

Falsifier

The proof would fail if a three-weight channel had a Bayesian inverse at some $r\ne 0$, i.e. if the forced candidate’s Choi matrix were positive semidefinite there.

Evidence

The mathematical source is D5/S3/QuantumChannels/PauliThreeWeightBayesianInverseUniqueness.lean. Its public definitions are sigma, pauliChannel, jam, IsBayesianInverse and claim; result proves the complete claim. IsDensity and IsCPTP are owned by CoPRelativeQuantumnessRefutation, and pauliMatrix by StabilizerPairLocalUnitaryInequivalence. Density coordinates use bloch_geometry_of_isHermitian and norm_bloch_le_trace_of_posSemidef from ActualPureQubitGeometry, with EuclideanSpace.real_norm_sq_eq supplying the coordinate norm identity. The Choi matrix and its CP-to-PSD implication are MatrixMap.choi_matrix and MatrixMap.choi_PSD_iff_CP_map from FiniteKrausRepresentation; the Kraus construction uses FiniteKrausChannel.

#13585 reports numerical experiments using the paper’s coefficient formulas: 20 000 random three-weight channels and states (10% pure) gave negative candidate Choi eigenvalues; the reported Bayes-rule residual was below $10^{-15}$, and the witness-identity residual was at most $7\times 10^{-16}$. Its positive controls were depolarizing noise at small $r$ and three-weight channels at $\rho =1/2$. These are source-reported numerical results, not independent replications or substitutes for the universal proof.

Triage

Tier 1: an unresolved alternative in the cited paper. The complete Pauli statement is the conclusion of result; the associated unital-channel application below additionally uses the paper’s Proposition 1.

What the settlement shows

Proved by result: for every Pauli channel with exactly three non-zero weights, a Bayesian inverse with respect to a qubit state exists exactly at the maximally mixed state; this is the first alternative of the paper’s sentence, The singleton conclusion also gives the measure-zero alternative under the usual Lebesgue measure on the Bloch ball; that measure statement is not part of result.

Established inside the proof. The obstruction is a single explicit direction: the scaled Choi matrix $DC$ is negative on the vector $w(j,i)=((1-h\cdot \sigma ){\sigma }_k{)}_{ji}$ as soon as $r\ne 0$, with a closed-form value (witness_numerator, witness_strict). The corresponding divided Choi value uses the strict-denominator algebra described above.

Argued, not formalized.

• The number of non-zero weights orders the invertible states: one weight (a unitary channel) gives every state; two weights give a coordinate axis (the paper’s Section III); three weights give only $1/2$ (here); four positive weights give a neighbourhood of $1/2$, because the Choi matrix of $\mathcal{P}$ has eigenvalues $2p_{\mu }>0$ and the forced candidate depends continuously on $r$ (the original dossier reports 2 000 random four-weight channels positive semidefinite at small $r$; this numerical result is not independently replicated).
• At three weights the Choi matrix of $\mathcal{P}$ has a zero eigenvalue (the Bell vector of the missing Pauli), so the four-weight continuity argument is not available, and the explicit witness shows that the forced candidate leaves the positive cone for every $r\ne 0$.

Open. The paper’s inequalities for four weights are not evaluated in general, and unital channels are treated only through their associated Pauli channels.

Effect on the paper. The Section V alternative is settled in its first form: three-weight Pauli channels, and by the paper’s unitary-equivalence proposition every unital qubit channel whose associated Pauli channel has three non-zero weights, admit operational time reversal only at the maximally mixed state. Precisely, for $\mathcal{N}=\mathcal{U}\circ \mathcal{E}\circ \mathcal{V}$, Proposition 1 maps $\rho$ to ${\mathcal{V}}^{†}(\rho )$ and $\mathcal{F}$ to ${\mathcal{V}}^{†}\circ \mathcal{F}\circ {\mathcal{U}}^{†}$; unitary conjugation fixes $1/2$ in both directions. This transport is a cited application, not part of result. Operational reversal preserves the specified sequential measurement correlations; it gives no arbitrary-state physical noise inversion or resource bound. The paper’s two-weight theorem and general four-weight inequalities have their original scope; the three-weight conclusion does not strengthen them.

ASSUMED-UNVERIFIED

The bounded literature check does not establish exhaustive worldwide novelty, priority, or the absence of an independent proof.