slug: chen-kato-brandao-2020-trace-norm-cmi-contraction bibkey: chen2020mpdoparent doi: null url: https://arxiv.org/abs/2010.14682v3 triage: theorem motivation_gids:

• D5/S3/Quantum/Information/ChenKatoBrandaoTraceNormCMIRefutation.result

A completely contractive data-processing inequality for the trace-norm CMI

Problem

C.-F. Chen, K. Kato and F. G. S. L. Brandão, Matrix Product Density Operators: when do they have a local parent Hamiltonian?, arXiv:2010.14682v3, Section III.C (PDF p. 17), Conjecture III.2:

For any channel $\mathcal{E}:C\to C^{\prime }$ with local contraction ratio ${\eta }_{1,C}:={\sup }_{{\rho }_C,{\rho }_C^{\prime }}\frac{\Vert {\mathcal{E}}_C[{\rho }_C]-{\mathcal{E}}_C[{\rho }_C^{\prime }]{\Vert }_1}{\Vert {\rho }_C-{\rho }_C^{\prime }{\Vert }_1}<1$ There exists a global constant $\eta <1$ such that for any tripartite system $ABC$ and any state ${\rho }_{ABC}$, it holds that $I_1(A:C^{\prime }|B{)}_{\mathcal{E}(\rho )}\le \eta I_1(A:C|B{)}_{\rho }$.

Here $I_1(A:C|B)=\Vert {\rho }_{ABC}-{\rho }_A\otimes {\rho }_{BC}{\Vert }_1-\Vert {\rho }_{AB}-{\rho }_A\otimes {\rho }_B{\Vert }_1$ (Definition 2, PDF p. 14). The verbatim texts are in the literature note. Issue #14728 preregisters the reading over finite-dimensional Kraus channels and density matrices, and the refutation.

Motivation

The source studies whether matrix product density operators are Gibbs states of quasi-local parent Hamiltonians through the decay of conditional mutual information. Conjecture III.1 (the entropic version, under trivial correctable algebra) and Conjecture III.2 (this trace-norm version, under a strict local trace-distance contraction) would each make one step of the generating channel shrink conditional correlations by a uniform factor and so give exponential decay.

Gap

The source proves contraction for channels with a forgetful component and for partially invariant channels in a strongly noisy regime, and does not know whether dimension factors are needed between ${\eta }_{1,C}$ and $\eta$. The repository refuted Conjecture III.1 in its dossier; that witness has local contraction ratio $1$ and does not meet the hypothesis of Conjecture III.2. The literature check in #14728 found no settlement; not-found-in-searched-scope.

Route

Let $\mathcal{M}$ act on a qubit $C$ with the four Kraus operators $K_k=\frac{1}{\sqrt{2}}|k⟩⟨v_k|$, $(v_0,v_1,v_2,v_3)=(|0⟩,|1⟩,|+⟩,|-⟩)$: it measures $Z$ or $X$ with probability $1/2$ each and records basis and outcome in $C^{\prime }={\mathbb{C}}^4$.

1. For density matrices with $\rho -{\rho }^{\prime }=\left(\begin{array}{cc}x& y\\ \bar{y}& -x\end{array}\right)$, $\Vert \rho -{\rho }^{\prime }{\Vert }_1=2\sqrt{x^2+|y{|}^2}$ and $\Vert \mathcal{M}(\rho )-\mathcal{M}({\rho }^{\prime }){\Vert }_1=|x|+|\mathrm{Re}y|$, so ${\eta }_{1,C}\le 1/\sqrt{2}$.
2. Take qubits $A,B$ and ${\rho }_{ABC}=\frac{1}{2}(|0⟩⟨0{|}_A\otimes {\Phi }_{BC}+|1⟩⟨1{|}_A\otimes {\Psi }_{BC})$ with $\Phi ,\Psi$ the projectors on $(|00⟩+|11⟩)/\sqrt{2}$ and $(|01⟩-|10⟩)/\sqrt{2}$. Then ${\rho }_{AB}=I/4={\rho }_A\otimes {\rho }_B$ before and after $\mathcal{M}$.
3. $I_1(A:C|B{)}_{\rho }=1$ and $I_1(A:C^{\prime }|B{)}_{\mathcal{M}(\rho )}=1$: after the measurement, in every recorded basis and outcome the conditional states of $B$ for $\Phi$ and for $\Psi$ are orthogonal, and the centred output $X$ satisfies $X^{†}=X$, $X^2=I/256$, so $16X$ is a unitary witnessing $\Vert X{\Vert }_1\ge 1$.
4. Any $\eta$ in the conjecture would satisfy $1\le \eta \cdot 1$.

Falsifier

An error in the local contraction bound, in either trace norm, or a reading of ${\eta }_{1,C}$ or $I_1$ different from the source. Positive controls in #14728: a partially depolarizing channel lowers $I_1$ of the same state to $0.5$ and $0$, and over 300 random states the ratio under $\mathcal{M}$ stays at most $0.6406$.

Evidence

D5/S3/Quantum/Information/ChenKatoBrandaoTraceNormCMIRefutation.lean defines localContraction, I1 and claim on finite Kraus channels (of_kraus) and the repository trace norm (FiniteTraceDistance.traceNorm), and proves result : ¬ claim. It reuses the partial traces of PartialTraceMutualInformation, whose density_posSemidef this delivery makes public, and the trace-norm lemmas traceNorm_eq_max_re_tr_U and traceNorm_add_le. The axiom closure of result is exactly propext, Classical.choice and Quot.sound; there is no sorry or native_decide. The module statement is sha256:59c0c2ef1556cef08c3d28b2e9a0d2132ea16db116307491c77525f16db3bfca; result is sha256:3e1ffcaec7645dee9cd57abe3f1ee71b15962ee7a0f8203526c77dfb12a8d185 and claim is sha256:65766632e3a552683ceeba34ba75bcb76a28891dafd6e5e8ca3b035f9d137dc0. The Freeze event is sha256:a44281242aa538a7d14e9a3f945880d1e54bde9f8049bb7442d1a42203b37866.

Triage

Tier 1: a named conjecture of a 2020 paper, retained in its 2023 revision, preregistered in #14728 before any Lean. theorem; resolution refuted. result is content (the local contraction bound over all pairs of qubit states, the input value $I_1=1$ and the output bound $I_1\ge 1$); admission basis open-problem-resolution; utility kind=certified-instance; basis=refutes. There is no digestion atom.

What the refutation shows

• Proved by result: a qubit channel can contract every trace distance by the factor $1/\sqrt{2}$ and leave the trace-norm CMI of a state with entangled side information undiminished ($I_1=1$ before and $I_1\ge 1$ after). A strict local contraction ratio does not control conditional correlations through a uniform global factor.
• Mechanism: the Lean derivation checks the four-outcome channel and the centred-output identities; the rest of this item is explanatory. Conditioned on the classical flag $A$, the pair $BC$ is in the Bell state ${\Phi }^{+}$ (for $A=0$) or ${\Psi }^{-}$ (for $A=1$); the $BC$ marginal $(\Phi^++\Psi^-)/2=(I\otimes I-Y\otimes Y)/4 =\tfrac12(P_+\otimes P_-+P_-\otimes P_+)$,$P_\pm=(I\pm Y)/2$, is separable. For each of the four real measurement vectors of $\mathcal{M}$ the conditional $B$ states of the two branches are orthogonal, so $A$ stays perfectly correlated with $B{C}^{\prime }$ although the channel forgets part of $C$. The identity $v^{T}Jv=0$ for every real $v$, $J=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$, explains this for any real measurement and is not formalized.
• Computed, not formalized (#14728): for ${\mathcal{M}}_p$ measuring $Z$ with probability $p$ the local ratio is $\sqrt{p^2+(1-p{)}^2}<1$ for every $0<p<1$ and $I_1$ is still preserved; for the full-rank states $(1-\epsilon )\rho +\epsilon I/8$ the trace-norm ratio after/before is numerically $1$ at $\epsilon =10^{-1},\dots ,10^{-4}$ (NumPy); analytically, both centred matrices scale by $1-\epsilon$ because $\mathcal{M}$ maps $I/2$ to $I/4$, so the ratio is exactly $1$ and a restriction to faithful states does not rescue the conjecture. The same state and channel keep the entropic CMI at one bit while the channel has trivial correctable algebra; for the faithful perturbations the entropic CMI strictly decreases, with numerical ratios $0.911$, $0.984$, $0.9976$, $0.99968$ at the same $\epsilon$.
• Refuted together with the conjecture: a strict contraction with $\eta <1$ depending on the dimensions or on the channel. The witness fixes $\dim A=\dim B=\dim C=2$, $\dim C^{\prime }=4$ and one channel, and has $I_1=1$ before and $I_1\ge 1$ after, so no $\eta <1$ works for it.
• Open: the trace-norm analogue of Proposition III.4 for MPDO generated by channels with ${\eta }_{1,C}<1$, which this counterexample does not decide.

ASSUMED-UNVERIFIED

The literature check in #14728 is bounded and does not establish worldwide novelty or priority; the journal status of the source was not checked beyond its arXiv record.