Quantum correlations in different density matrix representations of spin-1/2 open chain

TL;DR

This paper investigates quantum correlations in a spin-1/2 open chain using different density matrix representations, revealing how these bases influence entanglement and discord behaviors over chain length.

Contribution

It compares three density matrix bases in a spin chain, highlighting their effects on quantum correlations and identifying features like static discord and the 'echo' phenomenon.

Findings

Quantum discord is static in the $eta$-representation with zero entanglement.

In the $c$-representation, concurrence is limited to nearest neighbors, but discord extends further.

An 'echo' effect in discord evolution is observed, unlike in concurrence.

Abstract

We consider quantum correlations in a spin-1/2 open chain of $N$ nodes with the XY Hamiltonian using different bases for the density matrix representation and the initial state with a single polarized node. These bases of our choice are following: (i) the basis of eigenvectors of the fermion operators; this basis appears naturally through the Jordan-Wigner transformation (this representation of the density matrix is referred to as the $\beta$-representation), (ii) its Fourier representation ($c$-representation of the density matrix) and (iii) the basis of eigenvectors of the operators $I_{jz}$ (the $z$-projection of the $j$th spin, $j=1,...,N$). Although for the short chains (a few nodes) the qualitative behavior of the entanglement and the discord are very similar (the difference is quantitative), this is not valid for longer chains ($N\gtrsim 10$). In this case, there are qualitative and quantitative distinctions between the entanglement and the discord in all three cases. We underline three most important features: (i) the quantum discord is static in the $\beta$-representation, where the entanglement is identical to zero; (ii) in the $c$-representation, the concurrence may be non-zero only between the nearest neighbors (with a single exception), while the discord is nonzero between any two nodes; (iii) there is so-called "echo" in the evolution of the discord, which is not observed in the evolution of the concurrence. Using different bases, we may choose the preferable behavior of quantum correlations which allows a given quantum system to be more flexible in applications.