Generalized conditional entropy in bipartite quantum systems

TL;DR

This paper explores a generalized form of quantum conditional entropy in bipartite systems, linking it to entanglement and quantum discord, and provides analytical solutions for specific cases and states.

Contribution

It introduces a unified analysis of conditional entropy for various concave entropic forms, revealing universal measurement strategies for certain states and deriving explicit formulas for linear entropy.

Findings

Minimum conditional entropy relates to entanglement of formation.

Explicit formulas for linear entropy conditional entropy in qudit-qubit states.

Measurements minimizing von Neumann and linear entropies often coincide in X states.

Abstract

We analyze, for a general concave entropic form, the associated conditional entropy of a quantum system A+B, obtained as a result of a local measurement on one of the systems (B). This quantity is a measure of the average mixedness of A after such measurement, and its minimum over all local measurements is shown to be the associated entanglement of formation between A and a purifying third system C. In the case of the von Neumann entropy, this minimum determines also the quantum discord. For classically correlated states and mixtures of a pure state with the maximally mixed state, we show that the minimizing measurement can be determined analytically and is universal, i.e., the same for all concave forms. While these properties no longer hold for general states, we also show that in the special case of the linear entropy, an explicit expression for the associated conditional entropy can be obtained, whose minimum among projective measurements in a general qudit-qubit state can be determined analytically, in terms of the largest eigenvalue of a simple 3x3 correlation matrix. Such minimum determines the maximum conditional purity of A, and the associated minimizing measurement is shown to be also universal in the vicinity of maximal mixedness. Results for X states, including typical reduced states of spin pairs in XY chains at weak and strong transverse fields, are also provided and indicate that the measurements minimizing the von Neumann and linear conditional entropies are typically coincident in these states, being determined essentially by the main correlation. They can differ, however, substantially from that minimizing the geometric discord.