Approximate quantum Markov chains

Abstract

This book is an introduction to quantum Markov chains and explains how this concept is connected to the question of how well a lost quantum mechanical system can be recovered from a correlated subsystem. To achieve this goal, we strengthen the data-processing inequality such that it reveals a statement about the reconstruction of lost information. The main difficulty in order to understand the behavior of quantum Markov chains arises from the fact that quantum mechanical operators do not commute in general. As a result we start by explaining two techniques of how to deal with non-commuting matrices: the spectral pinching method and complex interpolation theory. Once the reader is familiar with these techniques a novel inequality is presented that extends the celebrated Golden-Thompson inequality to arbitrarily many matrices. This inequality is the key ingredient in understanding approximate quantum Markov chains and it answers a question from matrix analysis that was open since 1973, i.e., if Lieb's triple matrix inequality can be extended to more than three matrices. Finally, we carefully discuss the properties of approximate quantum Markov chains and their implications.

110 pages; PhD thesis, ETH Zurich; to appear as SpringerBriefs in Mathematical Physics; contains material from arXiv:1507.00303, arXiv:1509.07127, arXiv:1604.03023, and arXiv:1705.06749

References in corpus (6)

• Structure of states which satisfy strong subadditivity of quantum entropy with equality
• Renyi generalizations of the conditional quantum mutual information
• Conditional Mutual Information of Bipartite Unitaries and Scrambling
• Mixed s-sourcery: Building many-body states using bubbles of Nothing
• On Composite Quantum Hypothesis Testing
• A quasipolynomial-time algorithm for the quantum separability problem

Cited by in corpus (12)

• Improved thermal area law and quasi-linear time algorithm for quantum Gibbs states
• Generalised entropy accumulation
• Quantum many-body systems in thermal equilibrium
• Emergent classicality in general multipartite states and channels
• Quantum Brascamp-Lieb Dualities
• Entanglement monogamy via multivariate trace inequalities
• Recoverability from direct quantum correlations
• Conditional Independence of 1D Gibbs States with Applications to Efficient Learning
• Sample optimal tomography of quantum Markov chains
• One-Shot Distributed Source Simulation: As Quantum as it Can Get
• Virtual Quantum Markov Chains
• The Quantum Multiple-Access Channel with Cribbing Encoders