Systems with stationary distribution of quantum correlations: open spin-1/2 chains with $XY$ interaction

Abstract

Although quantum correlations in a quantum system are characterized by the evolving quantities (which are entanglement and discord usually), we reveal such basis (i.e. the set of virtual particles) for the representation of the density matrix that the entanglement and/or discord between any two virtual particles in such representation are stationary. In particular, dealing with the nearest neighbor approximation, this system of virtual particles is represented by the $\beta$-fermions of the Jordan–Wigner transformation. Such systems are important in quantum information devices because the evolution of quantum entanglement/discord leads to the problems of realization of quantum operations. The advantage of stationary entanglement/discord is that they are completely defined by the initial density matrix and by the Hamiltonian governing the quantum dynamics in the system under consideration. Moreover, using the special initial condition together with the special system’s geometry, we construct large cluster of virtual particles with the same pairwise entanglement/discord. In other words, the measure of quantum correlations is stationary in this system and correlations are uniformly “distributed” among all virtual particles. As examples, we use both homogeneous and non-homogeneous spin-1/2 open chains with XY-interaction although other types of interactions might be also of interest.

Appendix: Minimization in Eq. (19)

Let us show that the minimum in Eq. (19) corresponds to $\eta =0$, similar to ref. [20]. Equations (21) and (22) at $\eta =0$ yield

$$\begin{aligned}&p_i(0) \equiv p_i|_{\eta =0}=\frac{1}{2}, \end{aligned}$$

(67)

$$\begin{aligned}&\theta _i(0)\equiv \theta _i|_{\eta =0}=2 \sqrt{\rho _{nn}\rho _{mm} +\frac{1}{4}(1-2\rho _{mm})^2},\quad i=0,1. \end{aligned}$$

(68)

Consequently, using the definition of $S_i$ given by Eq. (20), we conclude that $S_1|_{\eta =0}=S_0|_{\eta =0} \equiv S(\theta _0(0))$ and

$$\begin{aligned} (p_0S_0 +p_1 S_1)|_{\eta =0} = 2 p_0(0) S(\theta _0(0))= S(\theta _0(0)) = S\left( 2 \sqrt{\rho _{nn}\rho _{mm} +\frac{1}{4}(1-2\rho _{mm})^2} \right) \qquad \quad \end{aligned}$$

(69)

Similarly, Eqs. (21) and (22) at $\eta =1$ yield

$$\begin{aligned}&p_0(1)=1-\rho _{nn},\quad p_1(1)=\rho _{nn}, \end{aligned}$$

(70)

$$\begin{aligned}&\theta _0(1)= \frac{|1-2\rho _{mm}-\rho _{nn}|}{1-\rho _{nn}} \end{aligned}$$

(71)

$$\begin{aligned}&\theta _1(1)=1. \end{aligned}$$

(72)

Again, using the definition of $S_i$ given by Eq. (20) we have $S_1|_{\eta =1}=0$ and we can write

$$\begin{aligned} (p_0S_0 +p_1 S_1)|_{\eta =1} = p_0(1) S(\theta _0(1)) = (1-\rho _{nn}) S\left( \frac{|1-2\rho _{mm}-\rho _{nn}|}{1-\rho _{nn}}\right) . \end{aligned}$$

(73)

Thus we have to find the minimum of two quantities:

$$\begin{aligned} \min \left( S\left( 2 \sqrt{\rho _{nn}\rho _{mm} +\frac{1}{4}(1-2\rho _{mm})^2} \right) , (1-\rho _{nn}) S\left( \frac{|1-2\rho _{mm}-\rho _{nn}|}{1-\rho _{nn}}\right) \right) .\nonumber \\ \end{aligned}$$

(74)

Representing the ratio of these two quantities as a two-dimensional surface in the space of the parameters $\rho _{nn}$ and $\rho _{mm}$ ($\rho _{nn},\rho _{mm}\le 1, \rho _{nn}+\rho _{mm} \le 1$) we conclude that the first of them (corresponding to $\eta =0$) is always less than the second one. Consequently the minimum in Eq. (19) is always at $\eta =0$.