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arXiv:2211.06794 · Journal of Mathematical Physics

Matrix product states and the decay of quantum conditional mutual information

Pavel Svetlichnyy, Shivan Mittal, T. A. B. Kennedy

Published Feb 1, 2024• 5 citations
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Abstract

A uniform matrix product state defined on a tripartite system of spins, denoted by ABC, is shown to be an approximate quantum Markov chain when the size of subsystem B, denoted |B|, is large enough. The quantum conditional mutual information (QCMI) is investigated and proved to be bounded by a function proportional to exp(−q(|B| − K) + 2K ln |B|), with q and K computable constants. The properties of the bounding function are derived by a new approach, with a corresponding improved value given for its asymptotic decay rate q. We show the improved value of q to be optimal. Numerical investigations of the decay of QCMI are reported for a collection of matrix product states generated by selecting the defining isometry with respect to Haar measure.

DOI
10.1063/5.0152063

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A uniform matrix product state defined on a tripartite system of spins, denoted by ABC, is shown to be an approximate quantum Markov chain when the size of subsystem B, denoted |B|, is large enough. The quantum conditional mutual information (QCMI) is investigated and proved to be bounded by a function proportional to exp(−q(|B| − K) + 2K ln |B|), with q and K computable constants. The properties of the bounding function are derived by a new approach, with a corresponding improved value given for its asymptotic decay rate q. We show the improved value of q to be optimal. Numerical investigations of the decay of QCMI are reported for a collection of matrix product states generated by selecting the defining isometry with respect to Haar measure.
Quantum conditional mutual information (QCMI) is a fundamental measure of conditional independence, with central applications throughout quantum information theory and many-body physics. Despite its importance, finding a fully quantum Rényi generalization that satisfies all desirable structural properties has remained an open question. In this work, we introduce a quantum Rényi conditional mutual information possessing these desirable properties: it is monotone under local quantum channels, additive under tensor products, monotone in the Rényi parameter, and converges to the QCMI in the limit α→1α\to 1α→1. We establish bounds on this quantity in terms of relative entropies to the Petz-recovered state, showing that it provides a distinct measure of approximate conditional independence. As a consequence, we derive recovery-based upper and lower bounds on the QCMI in terms of quantum generalizations of the Kullback-Leibler divergence, improving upon the best known results in certain regimes. Finally, we establish bounds in terms of relative entropies to the sets of Markov and separable states and derive Rényi chain rules, which appear to be novel even in the classical setting.
The connection between quantum state recovery and quantum conditional mutual information (QCMI) is studied for the class of purely generated finitely correlated states (pgFCSs) of one-dimensional quantum spin chains. For a tripartition of the chain into two subsystems separated by a buffer region, it is shown that a pgFCS is an approximate quantum Markov chain (QMC) and, stronger, may be approximated by a quantum Markov chain in trace distance, with an error exponentially small in the buffer size. This implies that (1) a locally corrupted state can be approximately recovered by action of a quantum channel on the buffer system, and (2) QCMI is exponentially small in the size of the buffer region. Bounds on the exponential decay rate of QCMI and examples of quantum recovery channels are presented.
We establish improved locally Markovian properties of quantum Gibbs states at every fixed positive temperature. For finite-range interactions of bounded interaction degree, the conditional mutual information (CMI) decays exponentially with separation, with a prefactor exponential in the boundary size of only one of the two regions and independent of the size of the other region and the total system. This improves the exponential volume dependence of the local Markov bounds established by Chen and Rouzé \cite{Chen2025}. For exponentially and stretched-exponentially decaying interactions, we obtain stretched-exponential CMI decay with a boundary-dependent prefactor. For power-law interactions, we prove inverse-logarithmic or algebraic decay with polynomial region-size dependence. Our proof is static, in contrast to the dynamical approach of Chen and Rouzé. We compare the Gibbs state with a product state obtained by removing boundary interactions, reducing the CMI bound to a relative-entropy loss that we control using locality estimates.
A tripartite state ρ_ABC forms a Markov chain if there exists a recovery map R_B→_BC acting only on the B-part that perfectly reconstructs ρ_ABC from ρ_AB. To achieve an approximate reconstruction, it suffices that the conditional mutual information I(A:C|B)_ρ is small, as shown recently. Here we ask what conditions are necessary for approximate state reconstruction. This is answered by a lower bound on the relative entropy between ρ_ABC and the recovered state R_B→_BC(ρ_AB). The bound consists of the conditional mutual information and an entropic correction term that quantifies the disturbance of the B-part by the recovery map.
A Markov chain is a tripartite quantum state ρABCwhere there exists a recovery map RB→BCsuch that ρABC= RB→BC(ρAB). More generally, an approximate Markov chain ρABCis a state whose distance to the closest recovered state RB→BC(ρAB) is small. Recently it has been shown that this distance can be bounded from above by the conditional mutual information I(A : C|B)ρof the state. We improve on this connection by deriving the first bound that is tight in the commutative case and features an explicit recovery map that only depends on the reduced state pBC. The key tool in our proof is a multivariate extension of the Golden-Thompson inequality, which allows us to extend logarithmic trace inequalities from two to arbitrarily many matrices.
Abstract It is well known that the conditional mutual information of a quantum state is zero if, and only if, the quantum state is a quantum Markov chain. Replacing the Umegaki relative entropy in the definition of the conditional mutual information by the Belavkin–Staszewski (BS) relative entropy, we obtain the BS-conditional mutual information, and we call the states with zero BS-conditional mutual information Belavkin–Staszewski quantum Markov chains. In this article, we establish a correspondence which relates quantum Markov chains and BS-quantum Markov chains. This correspondence allows us to find a recovery map for the BS-entropy in the spirit of the Petz recovery map. Furthermore, we show that, over the set of BS-quantum Markov chains, this correspondence constitutes an entanglement-breaking map. Moreover, we prove a structural decomposition of the Belavkin–Staszewski quantum Markov chains and also study states for which the BS-conditional mutual information is only approximately zero. We subsequently extend the aforementioned correspondence, structural decomposition, and recovery map to arbitrary pairs of states and quantum channels. As an application of the correspondence, we find a family of states with non-vanishing conditional mutual information for which it decays superexponentially fast with the size of the middle system.