trureturing — Discovering truth

bibkey: chen2020mpdoparent authors: Chi-Fang Chen, Kohtaro Kato and Fernando G. S. L. Brandão year: 2020 title: “Matrix Product Density Operators: when do they have a local parent Hamiltonian?” doi: null url: https://arxiv.org/abs/2010.14682v3 claim: “Conjecture III.1 asserts a uniform completely contractive data-processing bound for conditional mutual information under every channel with scalar whole-space correctable algebra. Conjecture III.2 asserts the analogous uniform bound for the trace-norm CMI of Definition 2 under every channel with local trace-norm contraction ratio below one.” strata_touched:

• D5/S3/Quantum/Information/ChenKatoBrandaoCMIRefutation
• D5/S3/Quantum/Information/ChenKatoBrandaoTraceNormCMIRefutation license: citation-only triage: anchor

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Source: https://arxiv.org/abs/2010.14682v3

The arXiv abstract page lists the arXiv-issued DOI https://doi.org/10.48550/arXiv.2010.14682 and no published journal DOI. Version 3 is the source version. Section II.D, PDF p. 11, equation (22) defines the correctable algebra. Section III.C, PDF p. 16, contains Proposition III.3 and Conjecture III.1; Proposition III.4 follows on p. 17.

The correctable-algebra definition reads:

For a given channel $\mathcal{E}:A\to A^{\prime }$, we define the correctable algebra $\mathcal{A}(\mathcal{E})\subset \mathcal{B}({\mathcal{H}}_A)$ as $\mathcal{A}(\mathcal{E}):=\mathrm{Alg}\{O_A\mid [O_A,E_a^{†}{E}_b]=0,\forall a,b\}$.

Conjecture III.1 reads:

For any channel $\mathcal{E}:C\to C^{\prime }$ with trivial correctable algebra $\mathcal{A}(\mathcal{E})=\mathbb{C}I$, there exists a constant $\eta <1$ such that for any tripartite system $ABC$ and any state ${\rho }_{ABC}$, it holds that $I(A:C^{\prime }|B{)}_{\mathcal{E}(\rho )}\le \eta I(A:C|B{)}_{\rho }$.

Proposition III.3 reads:

Let $\mathcal{E}:C\to C^{\prime }$ be a CPTP map which has trivial correctable algebra. Consider a tripartite system $A\otimes B\otimes C$. Then, for any state ${\rho }_{ABC}$ with $I(A:C|B{)}_{\rho }>0$, we have $I(A:C^{\prime }|B{)}_{\mathcal{E}(\rho )}<I(A:C|B{)}_{\rho }$.

Section I, PDF p. 3, equation (1) reads:

The CMI $I(A:C|B{)}_{\rho }$ is a function defined for a tripartite state ${\rho }_{ABC}$ as $I(A:C|B{)}_{\rho }=S(AB{)}_{\rho }+S(BC{)}_{\rho }-S(B{)}_{\rho }-S(ABC{)}_{\rho }$, where $S(A{)}_{\rho }=-\mathrm{tr}{\rho }_A{\log }_2{\rho }_A$ is the von Neumann entropy of the reduced state on $A$.

Definition 2 (Section III.A.1, PDF p. 14) reads:

$I_1(A:C|B):=\Vert {\rho }_{ABC}-{\rho }_A\otimes {\rho }_{BC}{\Vert }_1-\Vert {\rho }_{AB}-{\rho }_A\otimes {\rho }_B{\Vert }_1$

Conjecture III.2 (Section III.C, PDF p. 17) reads:

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 $I_1(A:C^{\prime }|B{)}_{\mathcal{E}(\rho )}\le \eta I_1(A:C|B{)}_{\rho }$.

It is followed by: “We do not know if extra constants or factor of dimension $d_C$ should be present between ${\eta }_{1,C}$ and $\eta$ like the crude bound in Theorem III.2.”

The source’s Proposition III.4 is conditional on Conjecture III.1. A refutation of that conjecture removes this route to unconditional decay; it does not refute exponential decay for every Y-shaped MPDO.

Bény, Kempf and Kribs, Quantum Error Correction of Observables, https://arxiv.org/abs/0705.1574, DOI https://doi.org/10.1103/PhysRevA.76.042303, Theorem 9, supplies the support-sensitive criterion $[P{E}_a^{†}{E}_{b}P,O]=0$ for correctability on a projected support. The support projection $P$ is part of that criterion; a correctable observable on a chosen support need not be correctable on the whole input space. This distinguishes recovery of selected preparations from the whole-space algebra in equation (22).