Communications in Mathematical Physics

, Volume 352, Issue 1, pp 37–58 | Cite as

Multivariate Trace Inequalities

Open Access


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.


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Authors and Affiliations

  1. 1.Institute for Theoretical PhysicsETH ZurichZurichSwitzerland
  2. 2.Institute for Quantum Information and MatterCaltechCaliforniaUSA
  3. 3.School of PhysicsThe University of SydneySydneyAustralia

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