Study on Non-Commutativity Measure of Quantum Discord

Abstract

In this paper, we are concerned with the non-commutativity measure of quantum discord. We first present an explicit expression of the non-commutativity measure of quantum discord in the two-qubit case. Then we compare the geometric quantum discords for two dynamic models with their non-commutativity measure of quantum discords. Furthermore, we show that the results conducted by the non-commutativity measure of quantum discord are different from those conducted by both or one of the Hilbert-Schmidt distance discord and trace distance discord. These intrinsic differences indicate that the non-commutativity measure of quantum discord is incompatible with at least one of the well-known geometric quantum discords in the quantitative and qualitative representation of quantum correlations.

1. Introduction

In recent years, because quantum information processing is superior to classical information processing, quantum information theory and technology have developed dramatically (cf., e.g., [1,2,3,4,5,6]). As an important resource in quantum computation, quantum correlations have been investigated extensively in the last decades. So far, many forms of quantum correlations have been proposed; for example, quantum discord [7], quantum deficit [8], quantum correlation derived from the distance between the reduced states [9]. Among various quantum correlations, quantum discord and its derived measures are important (cf., e.g., [3,5,6,7,8,10,11,12,13]). Most of them are not so hard to calculate and are more robust against the effects of decoherence [10,14]. Quantum discord was initially introduced by Ollivier and Zurek and by Henderson and Vedral [7,8]. In 2010, Dakic, Vedral and Brukner [12] find a “Necessary and Sufficient Condition for Nonzero Quantum Discord” (geometric quantum discord for the Hilbert-Schmidt norm). Since then, several equivalent measures have been introduced. Recently, the non-commutativity measure of quantum discord has been discussed in [13]. In this work, we first study the problem “how to give an explicit expression of the non-commutativity measure of quantum discord in the two-qubit case?” Then we compare the geometric quantum discords (the Hilbert-Schmidt distance discord and trace distance discord) for two dynamic models of open quantum systems with their non-commutativity measure of quantum discords. We select open quantum systems as our resource quantum systems since they are significant quantum systems and they can induce occurrence of decoherence which can cause decreasing of quantum correlations and may induce failure of the algorithms.

2. Non-Commutativity Measure of Quantum Discord and Geometric Quantum Discords

Consider a composite quantum system ${\mathcal{H}}_{AB}$, which consists of two subsystems A and B. Quantum discord is the difference of two natural quantum extensions of the classical mutual information. In [8], the authors pointed out that the quantum discord reaches zero for and only for the classical-quantum state. So we can look at the quantum discord as the ’distance’ between the state $\rho$ and the set of classical-quantum states. The state $\rho \in {\mathcal{H}}_{AB}$ is classical-quantum state if and only if $\rho$ can be written the following form (cf. [10]):

$$\rho =\sum _i{|i><i|}_A\otimes {\rho }_B^i,$$

where $\left\{|i{>}_A\right\}$ is any orthonormal basis of subsystem A and ${\rho }_B$ is a quantum state of subsystem B.

It is known that if $\rho \in {\mathcal{H}}_{AB}$ is a quantum state $\rho \in {\mathcal{H}}_{AB}$, then (cf. [13])

$$\rho =\sum _{ij}{E}_{ij}\otimes B_{ij},$$

where $E_{ij}={|i><j|}_A$ and $B_{ij}=<i_A\left|\rho \right|j_A>$. In [13], two non-commutativity measures are presented by:

$$D_{N1}^{{}^{\prime }}\left(\rho \right)=\sum _{\mathsf{\Omega }}\left|\right|[B_{ij},B_{i^{{}^{\prime }}{j}^{{}^{\prime }}}]{\left|\right|}_2,D_{N1}\left(\rho \right)=\sum _{\mathsf{\Omega }}\left|\right|[B_{ij},B_{i^{{}^{\prime }}{j}^{{}^{\prime }}}]{\left|\right|}_{Tr},$$

(1)

where $\mathsf{\Omega }$ is the set of all the possible pairs (regardless of the order), [·,·] denotes the commutator, $\left|\right|·{\left|\right|}_2$ is the Hilbert-Schmidt norm (i.e., ${∥A∥}_2=\sqrt{Tr\left(A^{†}A\right)}$) and $\left|\right|·{\left|\right|}_{Tr}$ is the trace norm (i.e., ${∥A∥}_{Tr}=Tr\left(\sqrt{A^{†}A}\right)$).

Clearly:

(1)

$D_{N1}\left(\rho \right)=0$ and $D_{N1}^{{}^{\prime }}\left(\rho \right)=0$ if and only if $\rho$ is a quantum-classical state.

(2)

$D_{N1}$ and $D_{N1}^{{}^{\prime }}$ are invariant under local unitary operations.

Moreover, we see that if $\rho ={\sum }_{ij}{A}_{ij}\otimes E_{ij}^{{}^{\prime }}$. According to Equation (1), we have:

$$D_N^{{}^{\prime }}\left(\rho \right)=\sum _{{\mathsf{\Omega }}^{{}^{\prime }}}\left|\right|[A_{ij},A_{i^{{}^{\prime }}{j}^{{}^{\prime }}}]{\left|\right|}_2,D_N\left(\rho \right)=\sum _{{\mathsf{\Omega }}^{{}^{\prime }}}\left|\right|[A_{ij},A_{i^{{}^{\prime }}{j}^{{}^{\prime }}}]{\left|\right|}_{Tr},$$

(2)

where $E_{ij}^{{}^{\prime }}={|i><j|}_B$, $A_{ij}=<i_B\left|\rho \right|j_B>$ and $\left\{|i{>}_B\right\}$ is any orthonormal basis of subsystem B, and $D_N\left(\rho \right)$ ( $D_N^{{}^{\prime }}\left(\rho \right))$ equals zero if and only if $\rho$ is a zero-discord state.

The Hilbert-Schmidt distance discord (cf., e.g., [12]) is defined by:

$$D_{Hs}\left(\rho \right)=\underset{\chi \in CQ}{min}{d}_{Hs}\left(\rho ,\chi \right),d_{Hs}\left(\rho ,\chi \right)={∥\rho -\chi ∥}_2^2,$$

where $CQ$ is the set of classical-quantum states. The trace distance discord (cf., e.g., [15]) is defined by:

$$D_{Tr}\left(\rho \right)=\underset{\chi \in CQ}{min}{d}_{Tr}\left(\rho ,\chi \right),d_{Tr}\left(\rho ,\chi \right)={∥\rho -\chi ∥}_{Tr},$$

Both the Hilbert-Schmidt distance discord and the trace distance discord are geometric quantum discords.

3. An Explicit Expression of The Non-Commutativity Measure of Quantum Discord in the Two-Qubit Case

Since the non-commutativity measures of quantum discord are invariant under local unitary operations, every state $\rho$ is locally unitary equivalent to:

$$\rho =\frac{1}{4}(I\otimes I+X\sigma \otimes I+I\otimes Y\sigma +\sum _i{c}_i{\sigma }_i\otimes {\sigma }_i),$$

(3)

where $X=(x_1,x_2.x_3),$ $Y=(y_1,y_2,y_3),$ $\sigma =\left(\begin{array}{c}{\sigma }_1\\ {\sigma }_2\\ {\sigma }_3\end{array}\right)$ in two qubits, and ${\sigma }_i$ $(i=1,2,3)$ are the three Pauli matrices (cf. [5]), which is called Bloch’s representation. Therefore, if the state $\rho$ satisfies Equation (3), we deduce that:

$$\begin{array}{c}{A}_{00}=\frac{1}{4}\left(\begin{array}{cc}1+x_3+y_{y}3+c_3& x_1-i{x}_2\\ x_1+i{x}_2& 1-x_3+y_{y}3-c_3\end{array}\right),A_{01}=\frac{1}{4}\left(\begin{array}{cc}{y}_1-i{y}_2& c_1-c_2\\ c_1+c_2& y_1-i{y}_2\end{array}\right),\hfill \\ A_{10}=\frac{1}{4}\left(\begin{array}{cc}{y}_1+i{y}_2& c_1+c_2\\ c_1-c_2& y_1+i{y}_2\end{array}\right),A_{11}=\frac{1}{4}\left(\begin{array}{cc}1+x_3-y_{y}3-c_3& x_1-i{x}_2\\ x_1+i{x}_2& 1-x_3-y_{y}3+c_3\end{array}\right).\hfill \end{array}$$

$$\begin{array}{c}{[A_{00},A_{01}]}^{†}[A_{00},A_{01}]=\frac{1}{64}\left(\begin{array}{cc}{u}^2+K_{+}^2{(c_1+c_2)}^2& 2c_1{c}_2{K}_{+}(x_1-i{x}_2)\\ 2c_1{c}_2{K}_{+}(x_1+i{x}_2)& c_1^2{y}_2^2+c_2^2{y}_1^2+K_{+}^2{(c_1-c_2)}^2\end{array}\right),\hfill \\ {[A_{00},A_{10}]}^{†}[A_{00},A_{10}]=\frac{1}{64}\left(\begin{array}{cc}{u}^2+K_{+}^2{(c_1-c_2)}^2& -2c_1{c}_2{K}_{+}(x_1-i{x}_2)\\ -2c_1{c}_2{K}_{+}(x_1+i{x}_2)& c_1^2{y}_2^2+c_2^2{y}_1^2+K_{+}^2{(c_1+c_2)}^2\end{array}\right),\hfill \\ {[A_{00},A_{11}]}^{†}[A_{00},A_{11}]=\frac{1}{16}\left(\begin{array}{cc}(x_1^2+x_2^2)c_3^2& 0\\ 0& (x_1^2+x_2^2)c_3^2\end{array}\right),\hfill \\ {[A_{01},A_{10}]}^{†}[A_{01},A_{10}]=\frac{1}{16}\left(\begin{array}{cc}{c}_1^2{c}_2^2& 0\\ 0& c_1^2{c}_2^2\end{array}\right),\hfill \\ {[A_{01},A_{11}]}^{†}[A_{01},A_{11}]=\frac{1}{64}\left(\begin{array}{cc}{u}^2+K_{-}^2{(c_1+c_2)}^2& -2c_1{c}_2{K}_{-}(x_1-i{x}_2)\\ -2c_1{c}_2{K}_{-}(x_1+i{x}_2)& c_1^2{y}_2^2+c_2^2{y}_1^2+K_{-}^2{(c_1-c_2)}^2\end{array}\right),\hfill \\ {[A_{10},A_{11}]}^{†}[A_{10},A_{11}]=\frac{1}{64}\left(\begin{array}{cc}{u}^2+K_{-}^2{(c_1-c_2)}^2& 2c_1{c}_2{K}_{-}(x_1-i{x}_2)\\ 2c_1{c}_2{K}_{-}(x_1+i{x}_2)& c_1^2{y}_2^2+c_2^2{y}_1^2+K_{-}^2{(c_1+c_2)}^2\end{array}\right).\hfill \end{array}$$

Finally, we obtain:

$$\begin{array}{c}{D}_N\left(\rho \right)=\frac{{\sum }_{i=1}^4{\mu }_i+4|c_1{c}_2|+|c_3|\sqrt{x_1^2+x_2^2}}{8},\hfill \\ D_N^{{}^{\prime }}\left(\rho \right)=\frac{\sqrt{u^2+K_{-}^2(c_1^2+c_2^2)}+\sqrt{u^2+K_{+}^2(c_1^2+c_2^2)}+|c_1{c}_2|+|c_3|\sqrt{x_1^2+x_2^2}}{2\sqrt{2}}\hfill \end{array}$$

in Equation (2), where:

$$K_{\pm }=c3\pm x3,u=\sqrt{c_1^2{y}_2^2+c_2^2{y}_1^2},$$

$${\mu }_{1,2}=\sqrt{|u^2+K_{+}^2\pm 2c_1{c}_2{K}_{+}\sqrt{K_{+}^2+(x_1^2+x_2^2)}|},$$

$${\mu }_{3,4}=\sqrt{|u^2+K_{-}^2\pm 2c_1{c}_2{K}_{-}\sqrt{K_{-}^2+(x_1^2+x_2^2)}|}.$$

4. Comparisons between Two Geometric Quantum Discords for Two Dynamic Models of Open Quantum Systems and Corresponding Non-Commutativity Measure of Quantum Discords

It is known that a real physical system is never perfectly closed. Environment acting on the quantum system are regarded as noise. In mathematics, it means that there exist a series of operators (referred to as { $E_u$ }) acting on the density matrix. In this way, quantum operations can be represented in an elegant form known as the operator-sum (Kraus) representation, and by [16] we see that:

$${\sum }_{u,v}{E}_{u,v}^{†}{E}_{u,v}=I,E_{u,v}=E_{u_A}\otimes E_{v_B}.$$

Thus, the related density matrix can be written in the following form:

$$\rho \left(t\right)={\sum }_{u,v}{E}_{u,v}\rho \left(0\right)E_{u,v}^{†}.$$

(4)

Dephasing channel. It is known that the dephasing channel is the only channel that has possibility of having no energy penalty when quantum information loses. If two qubits pass through the dephasing channel respectively, the Hamiltonian ([16]):

$$H=\hslash q{a}^{†}a(b^{†}+b)$$

where a and $a^{†}$, and b and $b^{†}$ are the annihilation operator and creation operator of A and B respectively, and the Kraus operators are:

$$E_0=\left(\begin{array}{cc}1& 0\\ 0& \sqrt{1-\hslash q}\end{array}\right)E_1=\left(\begin{array}{cc}0& 0\\ 0& \sqrt{\hslash q},\end{array}\right)$$

(5)

$$E_0=E_{0_A}=E_{0_B},E_1=E_{1_A}=E_{1_B}$$

where $q=1-e^{-2\gamma t}$ is photon scattering rate for the system, and $\gamma$ is phase damping dissipation rate. In this paper, we suppose $\hslash =1$.

For the Hilbert-Schmidt distance discord and trace distance discord, the dephasing channel has an important property: the freezing phenomenon (or the semi-freezing phenomenon). However, the non-commutativity measures of quantum discord is different. Actually, if we consider the initial state:

$${\rho }_1\left(0\right)=\frac{1}{4}(I\otimes I+\sum _i{c}_i\left(0\right){\sigma }_i\otimes {\sigma }_i)$$

with Equations (4) and (5), we have:

$${\rho }_1\left(t\right)=\frac{1}{4}(I\otimes I+\sum _i{c}_i\left(t\right){\sigma }_i\otimes {\sigma }_i)$$

where:

$$c_1\left(t\right)=c_1\left(0\right)e^{-2\gamma t},c_2\left(t\right)=c_2\left(0\right)e^{-2\gamma t},c_3\left(t\right)=c_3\left(0\right).$$

In Figure 1, we plot the time evolution of four different quantum discords for $c_1\left(0\right)=0.6,$ $c_2\left(0\right)=0,$ $c_3\left(0\right)=0.2$. The plot clearly shows that the Hilbert-Schmidt distance discord and trace distance discord have the sharp transition when $t^{{}^{\prime }}=0.5493/\gamma$. If $t<t^{{}^{\prime }}$, their values keep constant. If we change the initial condition (mainly about $c_3$), we can increase the time of the values keeping constant. However, the non-commutativity measures of quantum discord do not have this feature. They are strictly monotonous decreasing and strictly convex with time growing. Moreover, no matter what value of $c_3$ (except zero) could be, they still remain strictly monotonous decreasing.

Multimode vacuum field coupling the qubits. Consider a system consists of two qubits, with $\omega$ being the transition frequency and two-level energy separated by the energy gap $\hslash \omega$. The qubits are coupled to a multimode radiation field whose modes are initially in the vacuum state $|0>$. The evolution system in time is governed by the following master equation (cf., e.g., [17,18,19]):

$$\frac{\partial \rho }{\partial t}=-i\omega \sum _{i=1}^2[{\sigma }_3^i,\rho ]-i\sum _{i\ne j}^2{\xi }_{ij}[{\sigma }_{+}^i{\sigma }_{-}^j,\rho ]+\frac{1}{2}\sum _{i,j=1}^2{\zeta }_{ij}(2{\sigma }_{-}^j\rho {\sigma }_{+}^i-\{{\sigma }_{+}^j{\sigma }_{-}^i,\rho \})$$

(6)

where ${\sigma }_{\pm }^i$ are the raising and lowering operators and ${\sigma }_3^i$ is the energy operator (Pauli operator) of the ith qubit. The spontaneous decay rates of the qubits caused by the vacuum field ${\gamma }^{{}^{\prime }}={\xi }_{ii}$ coupling with the qubits. If $i\ne j$, ${\xi }_{ij}$ and ${\zeta }_{ij}$ in Equation (6) are described by the collective damping and the “dipole-dipole” interaction, and take the forms:

$$\begin{array}{c}\hfill {\xi }_{ij}=\frac{3}{2}{\gamma }^{{}^{\prime }}[\frac{sin\left(k{r}_{ij}\right)}{k{r}_{ij}}+\frac{cos\left(k{r}_{ij}\right)}{{\left(k{r}_{ij}\right)}^2}-\frac{sin\left(k{r}_{ij}\right)}{{\left(k{r}_{ij}\right)}^3}]\\ \hfill {\zeta }_{ij}=\frac{3}{4}{\gamma }^{{}^{\prime }}[-\frac{cos\left(k{r}_{ij}\right)}{k{r}_{ij}}+\frac{sin\left(k{r}_{ij}\right)}{{\left(k{r}_{ij}\right)}^2}+\frac{cos\left(k{r}_{ij}\right)}{{\left(k{r}_{ij}\right)}^3}]\end{array}$$

where $k=\frac{2\pi }{\lambda }$ is the wave vector with $\lambda$ being the atomic resonant wavelength and $r_{ij}=|r_i-r_j|$ is the distance between the qubits, we assume that the atomic dipole moments are parallel to each other and are polarized in the direction perpendicular to the interatomic axis.

For this model, we consider initial state:

$${\rho }_2\left(0\right)=|\mathsf{\Psi }><\mathsf{\Psi }|,|\mathsf{\Psi }>=\sqrt{\alpha }|11>+\sqrt{1-\alpha }|00>.$$

Then ${\rho }_2\left(t\right)$ in the standard basis $\left\{\right|11>,|10>,|01>,|00>\}$ take the following forms:

$$\begin{array}{c}{\rho }_{11}^{\left(2\right)}\left(t\right)=\alpha e^{-2{\gamma }^{{}^{\prime }}t}\hfill \\ {\rho }_{13,41}^{\left(2\right)}\left(t\right)=\sqrt{\alpha (1-\alpha )}{e}^{-{\gamma }^{{}^{\prime }}t}\hfill \\ {\rho }_{22,33}^{\left(2\right)}\left(t\right)=a_1[e^{-{\xi }_{12}^{+}t}-e^{-2{\gamma }^{{}^{\prime }}t}]+a_2[e^{-{\xi }_{12}^{-}t}-e^{-2{\gamma }^{{}^{\prime }}t}]\hfill \\ {\rho }_{23,32}^{\left(2\right)}\left(t\right)=a_1[e^{-{\xi }_{12}^{+}t}-e^{-2{\gamma }^{{}^{\prime }}t}]+a_2[e^{-{\xi }_{12}^{-}t}-e^{-2{\gamma }^{{}^{\prime }}t}]\hfill \\ {\rho }_{44}^{\left(2\right)}\left(t\right)=1-{\rho }_{11}^2\left(t\right)-{\rho }_{22}^2\left(t\right)-{\rho }_{33}^2\left(t\right)\hfill \end{array}$$

where ${\xi }_{12}^{\pm }={\gamma }^{{}^{\prime }}\pm {\xi }_{12},a_{1,2}=\alpha {\xi }_{12}^{\pm }/{\xi }_{12}^{\mp }$. In Figure 2, we show the change of the discord with respect to $\gamma t$ and $\alpha$ as ${\xi }_{ij}=0$, where the values are normalized.

For this model, the trace distance discord is symmetrical about $\alpha$ = 0.5. Moreover, compared to the trace distance discord, we find that the non-commutativity measure of quantum discord and the Hilbert-Schmidt distance discord have more similar properties at different $\alpha$ values. On the other hand, the Hilbert-Schmidt distance discord will revive after $\alpha$ large enough. In this aspect, the non-commutativity measure of quantum discord are closer to the trace distance discord.

5. Conclusions

In conclusion, we present an explicit expression of the non-commutativity measure of quantum discord in the two-qubit case. We also compare the non-commutativity measure of quantum discord with the geometric quantum discord in the models of two qubits passing through the dephasing channel and the multimode vacuum field coupling the qubits, respectively. Our study shows that the non-commutativity measures of quantum discord lose some important features of the geometric quantum discords, such as the freezing phenomenon in the dephasing channel and the revival in the multimode vacuum field coupling the qubits.