Negative Conditional Entropy of Post-Selected States

TL;DR

This paper introduces a quantum conditional entropy based on post-selection that can be negative, revealing that parts of a quantum system can contain less information than the entire system, with implications for quantum information theory.

Contribution

It defines a new quantum conditional entropy for post-selected states that generalizes classical conditional probabilities and allows for negative values, expanding understanding of quantum information.

Findings

Conditional entropy can be negative for post-selected states.

The definition aligns with quantum generalizations of classical probability.

Density operators are consistent with Dirac's quasiprobability formalism.

Abstract

We define a quantum entropy conditioned on post-selection which has the von Neumann entropy of pure states as a special case. This conditional entropy can take negative values which is consistent with part of a quantum system containing less information than the whole which can be in a pure state. The definition is based on generalised density operators for postselected ensembles. The corresponding density operators are consistent with the quantum generalisation of classical conditional probabilities following Dirac s formalism of quasiprobability distributions.