Quantum Conditional Mutual Information, Reconstructed States, and State Redistribution

Abstract

We give two strengthenings of an inequality for the quantum conditional mutual information of a tripartite quantum state recently proved by Fawzi and Renner, connecting it with the ability to reconstruct the state from its bipartite reductions. Namely, we show that the conditional mutual information is an upper bound on the regularized relative entropy distance between the quantum state and its reconstructed version. It is also an upper bound for the measured relative entropy distance of the state to its reconstructed version. The main ingredient of the proof is the fact that the conditional mutual information is the optimal quantum communication rate in the task of state redistribution. with SðXÞ ρ ≔ −trðρ X log ρ X Þ as the von Neumann entropy. It measures the correlations of subsystems C and R relative to subsystem B. The fact the classical CMI is non-negative is a simple consequence of the properties of the probability distributions; the same fact for the quantum CMI is equivalent to a deep result of quantum information theory-strong subadditivity of the von Neumann entropy [1]. Naturally, this led to a variety of applications in different areas, ranging from quantum information theory [2-4] to condensed matter physics [5][6][7].In the classical case, for every tripartite probability distribution p XYZ , IðX∶ ZjYÞ ¼ min where Sðp∥qÞ ≔ P i p i logðp i =q i Þ is the relative entropy and the minimum is taken over the set MC of all distributions q such that X − Y − Z form a Markov chain. Equivalently, the minimization in the right-hand side of Eq. (2) could be taken over Λ ⊗ id Z ðp YZ Þ, for reconstruction channels Λ∶ Y → YX. In particular, IðX∶ ZjYÞ ¼ 0 if, and only if, X − Y − Z form a Markov chain [which is equivalent to the existence of a channelThe class of tripartite quantum states ρ BCR satisfying IðC∶ RjBÞ ρ ¼ 0 has also been similarly characterized [8]: The B subsystem can be decomposed as B ¼ ⨁ k B L;k ⊗ B R;k (with orthogonal vector spaces B L;k ⊗ B R;k ) and the state written asfor a probability distribution fp k g and states ρ CB L;k ∈ C ⊗ B L;k and ρ B R;k R ∈ B R;k ⊗ R. States of this form are called quantum Markov because in analogy to Markov chains, conditioned on the outcome of the measurement onto fB L;k ⊗ B R;k g, the resulting state on C and R is product. Paralleling the classical case, ρ BCR is a quantum Markov state if, and only if, there exists a reconstruction channel Λ∶ B → BC such that Λ ⊗ id R ðρ BR Þ ¼ ρ BCR [9]. Having generalized the definition of CMI, can we also retain the above equivalence, with the set of quantum Markov states taking the role of Markov chains? Surprisingly, it turns out that this is not the case, [10] and it seems not to be possible to connect states that are close to Markov states with states of small CMI in a meaningful way (see, however, [2,11]). Nonetheless, it might be possible to relate states with small CMI with those that can be approximately reconstructed from their bipartite reductions, i.e., such that Λ ⊗ id R ðρ BR Þ ≈ ρ BCR . Indeed, several conje...

Quantum Information Processing with Finite Resources

“…The data-processing inequality can be strengthened by including a remainder term that characterizes how well the channel can be recovered. This has been shown by Fawzi and Renner [53] for the partial trace (see also [25,29] for refinements and simplifications of the proof). Recently these results were extended to general channels in [173] (see also [23]) and further refined in [149].…”

Multipartite quantum correlations and local recoverability

“…A similar operational interpretation applies to the original quantum discord quantity as well. Finally, proposition 7.1 gives another lower bound on the multipartite information gap by generalizing an approach recently outlined in [18].…”

“…First, it would be interesting if the inequality in (7.8) were true. It is true for classical systems, and to show it for quantum systems, one could consider extending the methods given in [18,Proposition 4] to this multipartite setting. Next, in light of the recent developments in [11][12][13], one could define a geometric CEMI as follows:…”

Approximate Quantum Markov Chains

“…More precisely, it was shown [28,53,85,142,145,153,175] that for any state ρ ABC there exists a recovery map R B→BC such that …”

“…In particular, if it is possible to relate the conditional mutual information with a measure of how well the C-system can be recovered by only acting on the B-system with a recovery map. The following theorem [28,53,85,142,145,153,175] shows that whenever the conditional mutual information I(A : C|B) ρ of a quantum state ρ ABC is small, then the Markov condition (5.1) approximately holds, i.e, there exists a recovery map from B to B ⊗ C that approximately reconstructs ρ ABC from ρ AB . This therefore justifies the definition of approximate quantum Markov chains as tripartite states ρ ABC such that the conditional mutual information I(A : C|B) ρ is small.…”