Dual quantum-correlation paradigms exhibit opposite statistical-mechanical properties

We report opposite statistical-mechanical behaviors of the two major paradigms in which quantum correlation measures are defined, viz., the entanglement-separability paradigm and the information-theoretic one. We show this by considering the ergodic properties of such quantum correlation measures in transverse quantum $XY$ spin-$\frac{1}{2}$ systems in low dimensions. While entanglement measures are ergodic in such models, the quantum correlation measures defined from an information-theoretic perspective can be nonergodic.

Figure 1

Ergodic entanglement measures versus nonergodic information-theoretic measures. Entanglement measures (concurrence and logarithmic negativity, in ebits) and information-theoretic quantum correlation measures (quantum discord and quantum work deficit, in bits) are plotted on the vertical axes against the dimensionless parameter $\tilde{\beta }$, for the infinite quantum transverse $XY$ spin chain. Concurrence ($C$, top left) and logarithmic negativity ($E_N$, top right) are ergodic for any value of initial transverse field $a$. In contrast, information-theoretic quantum correlations—quantum discord ($Q$, bottom left) and quantum work deficit ($▵$, bottom left)—are nonergodic for moderate values of the applied transverse field. The curves with black circles are for the canonical equilibrium state at large time, as functions of $\tilde{\beta }$. We have plotted the values of the corresponding physical quantity, for the time-evolved state with the initial state of the evolution being the canonical state at $\tilde{\beta }=20$, as horizontal lines for different values of the applied transverse field. Precisely, the applied fields used for the horizontal lines are $a/J=0.2$ (red squares), $a/J=0.6$ (blue diamonds), and $a/J=2$ (pink triangles). Note that $a/J$ is a dimensionless quantity. Here $\gamma =0.5$.

Figure 2

Quantum discord versus logarithmic negativity at large time. Quantum discord (in bits, left figure) and logarithmic negativity (in ebits, right figure) of the time-evolved state at $t\to \infty$ are plotted on the vertical axis, for the infinite quantum transverse $XY$ spin chain is plotted against the dimensionless transverse magnetic field parameter $a/J$, on the horizontal axis. The temperature of the initial canonical state of the evolution is taken such that $\tilde{\beta }=20$, and $\gamma =0.5$.

Figure 3

Reproducing the content of Fig. 1 for finite chain of 12 spins with periodic boundary conditions. All considerations remain the same as in Fig. 1.

Figure 4

Ergodicity versus nonergodicity of quantum correlation measures on $XY$ ladders. Quantum discord (left, in bits) and logarithmic negativity (right, in ebits) for a ladder of eight spins with periodic boundary condition. The considerations are the same as in Fig. 1, except for the orange upsidedown triangles, which are for $a/J=0.1$.

Figure 5

Ergodicity versus nonergodicity on 2D $XY$ models. Quantum discord (left, in bits) and logarithmic negativity (right, in ebits) for a 2D array of 12 spins with periodic boundary condition. All considerations are the same as in Fig. 1.