Sharp Pinsker-type trace-norm bounds on conditional mutual information from Petz reconstruction and the self-commutator (Zhang–Kim conjectures on approximate strong subadditivity)

For a tripartite quantum state $\rho_{ABC}$, Zhang (arXiv:1212.5023), reporting a conjecture of I. H. Kim, conjectured that the conditional mutual information lower-bounds the squared trace distance to the Petz reconstruction operators $MM^\dagger$ and $M^\dagger M$, $M=\rho_{AB}^{1/2}\rho_B^{-1/2}\rho_{BC}^{1/2}$, with the sharp Pinsker constant: $I(A\!:\!C|B)_\rho\geq\frac{1}{2\ln 2}\max\{\|\rho_{ABC}-MM^\dagger\|_1^2,\|\rho_{ABC}-M^\dagger M\|_1^2\}$, together with the implied commutator bound $I(A\!:\!C|B)_\rho\geq\frac{1}{8\ln 2}\|[M,M^\dagger]\|_1^2$ and a channel-level trace-norm-corrected monotonicity of relative entropy with the plain Petz map. The surrounding literature establishes: dimension-free closeness to some Markov state is impossible (Ibinson–Linden–Winter 2008); approximate recoverability with the sharp trace-norm constant holds for rotated Petz maps (Fawzi–Renner 2015 and successors, via measured-relative-entropy remainders and Pinsker's inequality); the relative-entropy strengthenings are refuted (Fawzi–Fawzi 2018), as is the plain-Petz fidelity form for general channels (Bhattacharya 2023). Whether the sharp-constant trace-norm statements hold for the plain ($t=0$) Petz reconstruction survives as a precise open core.

Background

Let $S(\rho)=-\mathrm{Tr}(\rho\log\rho)$ denote the von Neumann entropy (logarithms base 2) and $S(\rho\|\sigma)=\mathrm{Tr}\,\rho(\log\rho-\log\sigma)$ the quantum relative entropy, defined when the support of $\rho$ lies in that of $\sigma$. The strong subadditivity (SSA) inequality of Lieb and Ruskai states $S(\rho_{ABC})+S(\rho_B)\leq S(\rho_{AB})+S(\rho_{BC})$ for any state on $H_A\otimes H_B\otimes H_C$, so the conditional mutual information $I(A\!:\!C|B)_\rho=S(\rho_{AB})+S(\rho_{BC})-S(\rho_{ABC})-S(\rho_B)$ is nonnegative. A tripartite state with $I(A\!:\!C|B)=0$ is called a quantum Markov chain in order $A\!-\!B\!-\!C$; by results of Hayden–Jozsa–Petz–Winter and Petz's characterization of equality in monotonicity of relative entropy, this holds exactly when there is a recovery channel $\mathcal{R}_{B\to BC}$ with $\rho_{ABC}=(\mathrm{id}_A\otimes\mathcal{R})(\rho_{AB})$, where the canonical choice is the Petz recovery map $\Phi^\dagger_\sigma=\mathrm{Ad}_{\sigma^{1/2}}\circ\Phi^\dagger\circ\mathrm{Ad}_{\Phi(\sigma)^{-1/2}}$ (here $\mathrm{Ad}_X(Y)=XYX^\dagger$ and $\Phi^\dagger$ is the adjoint of the channel $\Phi$). Zhang's paper makes this concrete through the operator $M=\rho_{AB}^{1/2}\,\rho_B^{-1/2}\,\rho_{BC}^{1/2},$ for which he proves $I(A\!:\!C|B)_\rho=0$ if and only if $\rho_{ABC}=MM^\dagger=M^\dagger M$; note that the reconstructions $MM^\dagger$ and $M^\dagger M$ are always states (trace-one positive semidefinite operators), but for positive CMI need not be quantum Markov chains. Throughout, $\|X\|_1$ denotes the trace norm (sum of singular values) and $[M,M^\dagger]=MM^\dagger-M^\dagger M$ the self-commutator. Two neighboring facts motivate the problem. First, classically (all marginals commuting, e.g. $\rho_{ABC}=\sum_{ijk}p_{ijk}|ijk\rangle\langle ijk|$), one has the identity $I(A\!:\!C|B)=S(\rho\|MM^\dagger)$, and Pinsker's inequality $S(\rho\|\sigma)\geq\frac{1}{2\ln 2}\|\rho-\sigma\|_1^2$ then gives $I(A\!:\!C|B)\geq\frac{1}{2\ln 2}\|\rho_{ABC}-MM^\dagger\|_1^2$: a robust, dimension-free SSA with the sharp Pinsker constant. Second, such a dimension-free statement cannot hold with an arbitrary Markov state in place of $MM^\dagger$: Ibinson, Linden and Winter (2008) showed that quantum CMI is only a lower bound on the relative-entropy distance to the nearest quantum Markov chain, with a gap that can grow with dimension (via tripartite states whose $AC$-marginal is the antisymmetric state — the pure two-qudit state whose coefficient matrix in a product basis is $(e_{ij}-e_{ji})/\sqrt{d(d-1)}$, which is maximally far from separable), so the robustness question must be formulated through the reconstruction operators themselves. A decade of subsequent work clarified the landscape. Fawzi and Renner (2015) proved the dimension-free recoverability bound $I(A\!:\!C|B)\geq\min_{\mathcal{R}}S_{1/2}\big(\rho_{ABC}\big\|(\mathrm{id}\otimes\mathcal{R})(\rho_{AB})\big)$, where $S_{1/2}(\rho\|\sigma)=-2\log F(\rho,\sigma)$ is a Rényi divergence built from the fidelity $F(\rho,\sigma)=\|\sqrt\rho\sqrt\sigma\|_1$. Refinements (Sutter–Fawzi–Renner; Wilde; Junge–Renner–Sutter–Wilde–Winter; Sutter–Berta–Tomamichel; Sutter–Tomamichel–Harrow) show it for a universal rotated Petz map $\mathcal{R}_t(\xi)=\sigma^{(1+it)/2}\,\Phi^\dagger\big(\Phi(\sigma)^{-(1-it)/2}\,\xi\,\Phi(\sigma)^{-(1+it)/2}\big)\,\sigma^{(1-it)/2}$ and upgrade the fidelity to the measured relative entropy $D_{\mathcal{M}}(\rho\|\sigma)=\max_{\{M_i\}}\sum_i\big(\mathrm{Tr}\,M_i\rho\big)\log\frac{\mathrm{Tr}\,M_i\rho}{\mathrm{Tr}\,M_i\sigma}$ (the classical relative entropy of the best post-measurement distributions), with remainder terms tight in the commutative case. Two consequences calibrate what is known: Fawzi–Renner's own Eq. (6) gives $\|\rho_{ABC}-\widehat\rho\|_1^2\leq 4\ln 2\,I(A\!:\!C|B)$ for their reconstruction $\widehat\rho$, and combining the measured-relative-entropy remainder $I(A\!:\!C|B)\geq D_{\mathcal{M}}(\rho_{ABC}\|\tilde\rho)$ with classical Pinsker applied to the maximizing (Helstrom) measurement gives $\|\rho_{ABC}-\tilde\rho\|_1^2\;\leq\;2\ln 2\;I(A\!:\!C|B)_\rho$ for a rotated reconstruction $\tilde\rho$ — the conjectured sharp constant, but for a rotated rather than the plain ($t=0$) Petz reconstruction $MM^\dagger$. Conversely, the analogous strengthenings with the honest relative entropy are false: Fawzi and Fawzi (2018) exhibited states with $I(A\!:\!C|B)<\min_{\mathcal{R}}S\big(\rho_{ABC}\big\|(\mathrm{id}\otimes\mathcal{R})(\rho_{AB})\big)$, refuting the Winter–Li question, and Bhattacharya (2023) refuted the plain-Petz fidelity version of the strengthened data-processing inequality for general channels. Whether the sharp-constant trace-norm statements hold for the plain Petz reconstruction remains open, supported by the (unpublished) numerical evidence of I. H. Kim reported in Zhang's paper and by Li–Winter's survey.

Problem Statement

With the notation above — $\rho_{ABC}$ a state on finite-dimensional $H_A\otimes H_B\otimes H_C$, $M=\rho_{AB}^{1/2}\rho_B^{-1/2}\rho_{BC}^{1/2}$ (with $\rho_B^{-1/2}$ understood on the support of $\rho_B$), all entropies in bits, and $\|\cdot\|_1$ the trace norm — determine, proving or disproving each of the following dimension-free statements:

(i) Zhang's Eq. (3.2), originating in I. H. Kim's conjecture: $I(A\!:\!C|B)_\rho\ \geq\ \frac{1}{2\ln 2}\,\max\left\{\big\|\rho_{ABC}-MM^\dagger\big\|_1^{2},\ \big\|\rho_{ABC}-M^\dagger M\big\|_1^{2}\right\}\quad\text{for all tripartite states }\rho_{ABC};$

(ii) the self-commutator bound, Zhang's Eq. (3.3), which follows from (i) via the triangle inequality but may hold even if (i) fails: $I(A\!:\!C|B)_\rho\ \geq\ \frac{1}{8\ln 2}\,\big\|[M,M^{\dagger}]\big\|_1^{2}\quad\text{for all tripartite states }\rho_{ABC};$

(iii) the channel-level trace-norm-corrected monotonicity of relative entropy with the plain Petz recovery map $\Phi^\dagger_\sigma=\mathrm{Ad}_{\sigma^{1/2}}\circ\Phi^\dagger\circ\mathrm{Ad}_{\Phi(\sigma)^{-1/2}}$: $S(\rho\|\sigma)\ \geq\ S(\Phi(\rho)\|\Phi(\sigma))\ +\ \frac{1}{2\ln 2}\big\|\rho-\Phi^\dagger_\sigma\circ\Phi(\rho)\big\|_1^{2}\quad\text{for all channels }\Phi\text{ and states }\rho,\sigma\text{ with }\mathrm{supp}\,\rho\subseteq\mathrm{supp}\,\sigma;$ this strengthens (i): applied to the partial trace $\Phi=\mathrm{Tr}_C$ with $\rho=\rho_{ABC}$ and $\sigma=\rho_A\otimes\rho_{BC}$, the entropy difference on the left equals $I(A\!:\!C|B)_\rho$ and $\Phi^\dagger_\sigma\circ\Phi(\rho)=M^\dagger M$, while $\Phi=\mathrm{Tr}_A$ with $\sigma=\rho_{AB}\otimes\rho_C$ gives the $MM^\dagger$ half; and it is not refuted by the known counterexamples to the relative-entropy and fidelity strengthenings.

Since (iii) implies (i) and (i) implies (ii), a proof of (iii) settles (i) and (ii); a single explicit finite-dimensional state (and, for (iii), a channel) violating (i), (ii) or (iii) disproves that statement. A strictly weaker positive target, in the spirit of Kim's original unpublished conjecture, asks only for a dimension-free function $\Omega$ with $\Omega(0)=0$, $\Omega(x)>0$ for $x>0$, and $I(A\!:\!C|B)_\rho\geq\Omega\big(\max\{\|\rho-MM^\dagger\|_1^2,\|\rho-M^\dagger M\|_1^2\}\big)$ for the plain Petz reconstruction; settling this weaker target does not by itself settle the sharp-constant statements (i)–(iii). Zhang's suggested route to $[M,M^\dagger]$ through the theory of almost-commuting normal matrices (Huaxin Lin's theorem) is an available approach rather than an additional requirement.

References

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