Generalized conditional entropy optimization for qudit-qubit states

TL;DR

This paper presents a general approximate method for minimizing the conditional entropy in qudit-qubit systems, linking it to quantum discord and providing a geometric interpretation, applicable to various entropic forms.

Contribution

It introduces a unified quadratic form approach for entropy minimization in qudit-qubit states, extending to general entropies and offering a geometric perspective.

Findings

The method simplifies entropy minimization to diagonalizing a 3x3 matrix.

It provides a geometric picture using a correlation ellipsoid.

The approach enables easy estimation of quantum discord.

Abstract

We derive a general approximate solution to the problem of minimizing the conditional entropy of a qudit-qubit system resulting from a local projective measurement on the qubit, which is valid for general entropic forms and becomes exact in the limit of weak correlations. This entropy measures the average conditional mixedness of the post-measurement state of the qudit, and its minimum among all local measurements represents a generalized entanglement of formation. In the case of the von Neumann entropy, it is directly related to the quantum discord. It is shown that at the lowest non-trivial order, the problem reduces to the minimization of a quadratic form determined by the correlation tensor of the system, the Bloch vector of the qubit and the local concavity of the entropy, requiring just the diagonalization of a $3\times 3$ matrix. A simple geometrical picture in terms of an associated correlation ellipsoid is also derived, which illustrates the link between entropy optimization and correlation access and which is exact for a quadratic entropy. The approach enables a simple estimation of the quantum discord. Illustrative results for two-qubit states are discussed.