Multivariate Trace Inequalities

Abstract

Abstract:We prove several trace inequalities that extend the Golden-Thompson and the Araki-Lieb-Thirring inequality to arbitrarily many matrices. In particular, we strengthen Lieb's triple matrix inequality. As an example application of our four matrix extension of the Golden-Thompson inequality, we prove remainder terms for the monotonicity of the quantum relative entropy and strong sub-additivity of the von Neumann entropy in terms of recoverability. We find the first explicit remainder terms that are tight in the commutative case. Our proofs rely on complex interpolation theory as well as asymptotic spectral pinching, providing a transparent approach to treat generic multivariate trace inequalities.

Relativistic (or 2-Prover 1-Round) Zero-Knowledge Protocol for $$\mathsf {NP}$$ Secure Against Quantum Adversaries

“…We note that this result can be obtained as an application of the pinching inequality [Hay02,SBT16], but we provide a proof here for completeness.…”

Approximate Quantum Markov Chains

“…However, we show that these quasi-norms still satisfy an asymptotic convexity property for tensor products of operators in the following sense [142].…”

“…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 …”

“…Recently, an extension of the ALT inequality to arbitrarily many operators has been proven [142] which was further generalized in [69].…”

“…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.…”