Approximate reversibility in the context of entropy gain, information gain, and complete positivity

Francesco Buscemi, Siddhartha Das, Mark M. Wilde

Published 2016-01-06Version 1

There are several inequalities in physics which limit how well we can process physical systems to achieve some intended goal, including the second law of thermodynamics, entropy bounds in quantum information theory, and the uncertainty principle of quantum mechanics. Recent results provide physically meaningful enhancements of these limiting statements, determining how well one can attempt to reverse an irreversible process. In this paper, we apply and extend these results to give strong enhancements to several entropy inequalities, having to do with entropy gain, information gain, and complete positivity of physical evolutions. Our first result is a remainder term for the entropy gain of a quantum channel. This result implies that a small increase in entropy under the action of a unital channel is a witness to the fact that the channel's adjoint can be used as a recovery channel to undo the action of the original channel. We apply this result to pure-loss, quantum-limited amplifier, and phase-insensitive quantum Gaussian channels, showing how a quantum-limited amplifier can serve as a recovery from a pure-loss channel and vice versa. Our second result regards the information gain of a quantum measurement, both without and with quantum side information. We find here that a small information gain implies that it is possible to undo the action of the original measurement if it is efficient. The result also has operational ramifications for the information-theoretic tasks known as measurement compression without and with quantum side information. We finally establish that the reduced dynamics of a system-environment interaction are approximately CPTP if and only if the data processing inequality holds approximately.