Influence of Dzyaloshinshkii–Moriya interaction on quantum correlations in two-qubit Werner states and MEMS

Abstract

In this paper, we study the influence of Dzyaloshinskii–Moriya (DM) interaction on quantum correlations in two-qubit Werner states and maximally entangled mixed states (MEMS). We consider our system as a closed system of a qubit pair and one auxiliary qubit, which interact with any one of the qubit of the pair through DM interaction. We show that DM interaction, taken along any direction (x or y or z), does not affect two-qubit Werner states. On the other hand, the MEMS are affected by x and z components of DM interaction and remain unaffected by the y component. Further, we find that the state (i.e., probability amplitude) of auxiliary qubit does not affect the quantum correlations in both the states, and only DM interaction strength influences the quantum correlations. So one can avoid the intention to prepare the specific state of auxiliary qubit to manipulate the quantum correlations in both the states. We mention here that avoiding the preparation of state can contribute to cost reduction in quantum information processing. We also observe the phenomenon of entanglement sudden death in the present study.

Appendices

Appendix 1: Quantum discord

To make this paper self-contained, we briefly discuss quantum discord here [17]. It is known that a bipartite quantum state can carry classical and non-classical correlations. A theoretic measure of information is the mutual information in information theory. The mutual information stored in two classical distributions, equipped with two random variables $X$ and $Y$, assuming values $\{x_{i}\}$ and $\{y_{i}\}$ with probabilities $\{p_{i}\}$ and $\{p_{j}\}$, is defined as

$$\begin{aligned} I(X:Y)=H(X)+H(Y)-H(X,Y), \end{aligned}$$

(23)

or

$$\begin{aligned} I(X:Y)=H(X)-H(X|Y), \end{aligned}$$

(24)

where $H(X)=-\Sigma _{i}p_{i}\mathrm{log}_{2}p_{i}$ and $H(Y)=-\Sigma _{i}q_{i}\mathrm{log}_{2}q_{i}$ are the Shannon entropies associated with random variables $X$ and $Y$, respectively, $H(X,Y)$ is the joint entropy of $X$ and $Y$, and $H(X|Y)$ is the conditional entropy of $X$ and $Y$. From classical point of view, both the above Eqs. (23) and (24) are equivalent, but in quantum mechanical scenario both the expressions are different. Let us consider a bipartite quantum system equipped with parties A and B described by a composite density matrix $\rho ^{AB}$ with marginal density matrices $\rho ^{A}$ and $\rho ^{B}$. The quantum analogues of mutual information of Eqs. (23) and (24) is obtained by replacing the Shannon entropies by von Neumann entropies as

$$\begin{aligned} I(\rho ^{AB})=S(\rho ^{A})+S(\rho ^{B})-S(\rho ^{AB}) \end{aligned}$$

(25)

and

$$\begin{aligned} I(\rho ^{AB})=S(\rho ^{A})-S(\rho ^{A}|\rho ^{B}), \end{aligned}$$

(26)

where $S(\rho ^{A})=-Tr(\rho ^{A}\mathrm{log}_{2}\rho ^{A})$ and $S(\rho ^{B})=-Tr(\rho ^{B}\mathrm{log}_{2}\rho ^{B})$ are the von Neumann entropies of the subsystem A and B, respectively, and $S(\rho ^{AB})=Tr(\rho ^{AB})\mathrm{log}_{2}\rho ^{AB}$ is the joint von Neumann entropy of the composite quantum system AB. The quantity given by Eq.(26) is obscure, because the conditional entropy $S(\rho ^{A}|\rho ^{B})$ depends upon measurement performed on either one of the party A or B. So both the Eqs. (25) and (26) are not equivalent from quantum mechanical point of view. The von Neumann projective measurement performed on system A projects the system into a statistical ensemble $\{\rho ^{B},p_{k}^{A}\}$, such that

$$\begin{aligned} \rho ^{AB}\longmapsto \rho _{k}^{B}=\frac{(A_{k}\otimes I_{B})\rho ^{AB}(A_{k}\otimes I_{B})}{p_{k}}, \end{aligned}$$

(27)

where

$$\begin{aligned} p_{k}=Tr \left[ (A_{k}\otimes I_{B})\rho ^{AB} (A_{k} \otimes I_{B}) \right] \end{aligned}$$

(28)

$$\begin{aligned} A_{k}=UM_{k}U^{\dagger }, \nonumber \\ M_{k}=|k\rangle \langle k|, k=0,1 \end{aligned}$$

(29)

It is known that any unitary can be written up to a constant phase as

$$\begin{aligned} U=tI+y.\sigma , \end{aligned}$$

(30)

with $t\in R, \, y=(y_{1},y_{2},y_{3})\in R^{3}, \, \sigma$ is the Pauli vector of qubit and

$$\begin{aligned} t^{2}+y_{1}^2+y_{2}^2+y_{3}^2=1. \end{aligned}$$

Now Eq. (26) becomes, with respect to the projective measurement $A_{k}$, as

$$\begin{aligned} I (\rho ^{AB}|A_{k}):=S(\rho ^{B})-S(\rho ^{AB}|A_{k}), \end{aligned}$$

(31)

where $S(\rho ^{AB}|A_{k}):=\Sigma p_{k}S(\rho _{k}^{B})$ is the conditional quantum entropy based on the measurement $A_{k}$. Classical correlations is defined as the maximization of $I(\rho ^{Ab}|A_{k})$ over all possible measurements $A_{k}$ and is given by

$$\begin{aligned} C(\rho ^{AB}):=\mathrm{sup}_{\{A_{k}\}}I(\rho ^{AB}|A_{k}). \end{aligned}$$

(32)

On simplifying, we get

$$\begin{aligned} C(\rho ^{AB}):=S(\rho ^{B})-\min _{\{A_{k}\}}S(\rho ^{B}|A_{k}). \end{aligned}$$

(33)

Quantum discord is given by the difference of mutual quantum information $I(\rho ^{AB})$ given in Eq. (25) and classical correlations $C(\rho ^{AB})$ given in Eq. (33). So we obtained the expression of quantum discord as

$$\begin{aligned} Q(\rho ^{AB})=I(\rho ^{AB})-C(\rho ^{AB}) =S(\rho ^{A})-S(\rho ^{AB})+\min _{\{{A_{k}}\}}S(\rho ^{B}|A_{k}). \end{aligned}$$

(34)

The calculation of quantum discord is based on complex maximization procedure and obtaining an analytic expression of $C(\rho _{AB})$ for general states is not an easy task. Here we write our X-structured density matrix as follows

$$\begin{aligned} \rho ^{AB}= \left[ \begin{array}{cccc} \rho _{11} &{}{\quad }0 &{}{\quad }0 &{}{\quad }\rho _{14} \\ 0 &{}{\quad }\rho _{22}&{}{\quad }\rho _{23}&{}{\quad }0 \\ 0&{}{\quad }\rho _{32}&{}{\quad }\rho _{33}&{}{\quad }0 \\ \rho _{41}&{}{\quad }0&{}{\quad }0&{}{\quad }\rho _{44} \\ \end{array} \right] . \ \ \ \end{aligned}$$

(35)

Here we use the method to calculate the quantities $Q(\rho ^{AB}), \, CC(\rho ^{AB})$ given by C. Z. Wang et al.[28], so we obtain $Q(\rho ^{AB}), \, CC(\rho ^{AB})$ as follows

$$\begin{aligned} CC(\rho ^{AB})&= \mathrm{max}(CC_{1},CC_{2}), \end{aligned}$$

(36)

$$\begin{aligned} QD(\rho ^{AB})&= \mathrm{min}(QD_{1},QD_{2}), \end{aligned}$$

(37)

where

$$\begin{aligned} CC_{j}&= H(\rho _{11}+\rho _{22})-D_{j}, \\ QD_{j}&= H(\rho _{11}+\rho _{33})+\Sigma _{k=1}^{4}\lambda _{k}\mathrm{Log}_{2}\lambda _{k}+D_{j}, \end{aligned}$$

and

$$\begin{aligned} D_{1} = H(\tau ), D_{2}&= -\Sigma _{j=1}^{4}\rho _{jj}\mathrm{log}_{2}\rho _{jj}-H(\rho _{11}+\rho _{33}), \\ H(\rho _{11}+\rho _{22})&= -(\rho _{11}+\rho _{22})\mathrm{log}_{2}(\rho _{11}+\rho _{22}) \\&\quad -\,[1-(\rho _{11}+\rho _{22})]\mathrm{log}_{2}[1-(\rho _{11}+\rho _{22})],\\ \tau&= \frac{1+\sqrt{[1-2(\rho _{33}+\rho _{44})]^{2}+4(|\rho _{14}|+|\rho _{23}|)^{2}}}{2}, \end{aligned}$$

where $\lambda _{k}$ are the eigenvalues of $\rho ^{AB}$ and $\rho _{33}, \, \rho _{44}, \, \rho _{14}$ and $\rho _{23}$ are the elements of X-structured density matrix given in Eq. (35).

Appendix 2: Mathematica code

Under this section, we provide the mathematica code to calculate the quantum discord for X-structured density matrices. The code is given below.

$m=\{\{\rho _{11},0,0,\rho _{14}\},\{0,\rho _{22},\rho _{23},0\},\{ 0,\rho _{32},\rho _{33},0\},\{\rho _{14},0,0,\rho _{44}\}\};$ $e$=Eigenvalues [m]; $\lambda _{1}=e[[1]];$ $\lambda _{2}=e[[2]];$ $\lambda _{3}=e[[3]];$ $\lambda _{4}=e[[4]];$ $H[x_{-}]:=$ Module[$\{ x1=x\},$ $-x1$ Log2[$x1$]-$(1-x1)$ Log2[$1-x1$]] Log2[$x_{-}$]=If[$x=0$,0,Log2[x]]; p=FullSimplify[(ComplexExpand [Abs$[m[[1]][[4]]]$+ComplexExpand [Abs$[m[[2]][[3]]])^{2}$]; $\tau =(1/2)*(1+\sqrt{(}(1-2(m[[3]][[3]]+m[[4]][[4]]))^2+4(p)))$; DD1=N[H[$\tau$]]; a=$-$m[[1]][[1]]Log2[m[[1]]m[[1]]]; b=$-$m[[2]][[2]] Log2[m[[2]][[2]]]; c=$-$m[[3]][[3]] Log2 [m[[3]][[3]]]; d=$-$m[[4]][[4]] Log2 [m[[4]][[4]]]; f=$-$H[m[[1]][[1]]]+m[[3]][[3]]; DD2=a+b+c+d+f; Q11=H[m[[1]][[1]]+m[[3]][[3]]]+$\lambda _{1}$ Log2 [$\lambda _{1}$]+$\lambda _{2}$ Log2 [$\lambda _{2}$]+$\lambda _{3}$ Log2 [$\lambda _{3}$]+$\lambda _{4}$ Log2 [$\lambda _{4}$]+DD1; Q22=H[m[[1]][[1]]+m[[3]][[3]]]+$\lambda _{1}$ Log2 [$\lambda _{1}$]+$\lambda _{2}$ Log2 [$\lambda _{2}$]+$\lambda _{3}$ Log2 [$\lambda _{3}$]+$\lambda _{4}$ Log2 [$\lambda _{4}$]+DD2; QD=N[Min[Q11,Q22]]; Quit[]