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arXiv:1410.0664 · Communications in Mathematical Physics

Quantum Conditional Mutual Information and Approximate Markov Chains

Omar Fawzi, Renato Renner

Published Sep 9, 2015• 270 citations
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Abstract

A state on a tripartite quantum system $${A \otimes B \otimes C}$$ forms a Markov chain if it can be reconstructed from its marginal on $${A \otimes B}$$ by a quantum operation from B to $${B \otimes C}$$ . We show that the quantum conditional mutual information I(A : C|B) of an arbitrary state is an upper bound on its distance to the closest reconstructed state. It thus quantifies how well the Markov chain property is approximated.

DOI
10.1007/s00220-015-2466-x

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A state on a tripartite quantum system A⊗B⊗C{A \otimes B \otimes C}A⊗B⊗C forms a Markov chain if it can be reconstructed from its marginal on A⊗B{A \otimes B}A⊗B by a quantum operation from B to B⊗C{B \otimes C}B⊗C . We show that the quantum conditional mutual information I(A : C|B) of an arbitrary state is an upper bound on its distance to the closest reconstructed state. It thus quantifies how well the Markov chain property is approximated.
A central question in quantum information theory is to determine how well lost information can be reconstructed. Crucially, the corresponding recovery operation should perform well without knowing the information to be reconstructed. In this work, we show that the quantum conditional mutual information measures the performance of such recovery operations. More precisely, we prove that the conditional mutual information I ( A : C | B ) of a tripartite quantum state ρ ABC can be bounded from below by its distance to the closest recovered state R B → B C ( ρ A B ) , where the C -part is reconstructed from the B -part only and the recovery map R B → B C merely depends on ρ BC . One particular application of this result implies the equivalence between two different approaches to define topological order in quantum systems.
A tripartite state ρABC\rho _{ABC}ρABC​ forms a Markov chain if there exists a recovery map RB→BC\mathcal {R}_{B \rightarrow BC}RB→BC​ acting only on the B-part that perfectly reconstructs ρABC\rho _{ABC}ρABC​ from ρAB\rho _{AB}ρAB​ . To achieve an approximate reconstruction, it suffices that the conditional mutual information I(A:C∣B)ρI(A:C|B)_{\rho }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 ρABC\rho _{ABC}ρABC​ and the recovered state RB→BC(ρAB)\mathcal {R}_{B\rightarrow BC}(\rho _{AB})RB→BC​(ρAB​) . 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 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.
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.
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.