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arXiv:1504.07251 · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences

Universal recovery map for approximate Markov chains

David Sutter, Omar Fawzi, Renato Renner

Published Feb 1, 2016• 56 citations
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Abstract

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.

DOI
10.1098/rspa.2015.0623

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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 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 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 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 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.
The data processing inequality states that the quantum relative entropy between two states ρ and σ can never increase by applying the same quantum channel N to both states. This inequality can be strengthened with a remainder term in the form of a distance between ρ and the closest recovered state (R∘N)(ρ), where R is a recovery map with the property that σ=(R∘N)(σ). We show the existence of an explicit recovery map that is universal in the sense that it depends only on σ and the quantum channel N to be reversed. This result gives an alternate, information-theoretic characterization of the conditions for approximate quantum error correction.
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.