Unitary invariant discord as a measure of bipartite quantum correlations in an $N$-qubit quantum system

TL;DR

This paper introduces a new measure of quantum correlations called the unitary invariant discord, which remains unchanged under system evolutions and is based on eigenvalues, with practical modifications and applications to quantum state transfer.

Contribution

It proposes a novel $SU(2^N)$-invariant quantum discord measure, simplifies its calculation, and explores its properties and relation to quantum state transfer in multi-qubit systems.

Findings

The invariant discord reaches maximum for pure states.

A geometric measure of the invariant discord is derived.

The measure relates to quantum state transfer in spin chains.

Abstract

We introduce a measure of quantum correlations in the $N$-qubit quantum system which is invariant with respect to the $SU(2^N)$ group of transformations of this system. This measure is a modification of the quantum discord introduced earlier and is referred to as the unitary or $SU(2^N)$-invariant discord. Since the evolution of a quantum system is equivalent to the proper unitary transformation, the introduced measure is an integral of motion and is completely defined by eigenvalues of the density matrix. As far as the calculation of the unitary invariant discord is rather complicated computational problem, we propose its modification which may be found in a simpler way. The case N=2 is considered in details. In particular, it is shown that the modified SU(4)-invariant discord reaches the maximum value for a pure state. {A geometric measure of the unitary invariant discord of an $N$-qubit state is introduced and a simple formula for this measure is derived, which allows one to consider this measure as a witness of quantum correlations.} The relation of the unitary invariant discord with the quantum state transfer along the spin chain is considered. We also compare the modified SU(4)-invariant discord with the geometric measure of SU(4)-invariant discord of the two-qubit systems in the thermal equilibrium states governed by the different Hamiltonians.