Dynamics of quantum correlations for different types of noisy channels

Abstract

The effect of noise channels (amplitude damping, phase damping, depolarizing and phase flip) on the behaviors of quantum correlations such as quantum discord (QD), super quantum discord (SQD) within an atom-field interaction with environments are investigated. Generation and preservation of the quantum entanglement between the qubit and field have played an important role for both fundamental quantum theories and experiments. This topic motivates us to explore the role of noise channels on quantum correlations in the Jaynes–Cummings model. Our analytical results show that the entanglement and quantum correlation dynamics is strongly related to the decoherenc strength and type of noisy channel. One purpose of this work is to answer the following question: is it possible the effects of noise channels on entangled states lead to birth of entanglement and quantum correlations or do they just have a destructive effects? We show that, the noise of all above channels do not lead to a sudden vanishing of quantum correlations. But the noise of phase flip channel reproduces initial quantum correlations. Furthermore, we find that entanglement and QD are more fragile to environmental noises than the SQD. For this reason, it seems that our results can be beneficial to deeply understanding the weak measurement and quantum correlation in quantum information resource.

Appendix: a measures of quantum correlations

For a bipartite system AB quantum discord is given by Ollivier and Zurek (2001):

$$\begin{aligned} DQ(\rho _{AB})=I(\rho _{AB})- C(\rho _{AB}) \end{aligned}$$

(15)

the quantity $C(\rho _{AB})$ is defined as a measure of classical correlation Mazzola et al. (2010):

$$\begin{aligned} C(\rho _{AB})=S(\rho _{A})-\min _{{\varPi ^{B}_{j}} } S(\rho _{A|B}), \end{aligned}$$

(16)

where ${\{\varPi ^{B}_{j}}\}$ denotes a complete set of positive operator-valued measure (POVM) performed on the subsystem B, in such a way that $\sum _{j} \varPi ^{B}_{j} =1$. Where $\rho _{AB}$ denotes the bipartite density matrix of a composite system AB, $\rho _{A}$ and $\rho _{B}$ represent the density matrices of parts A and B. The quantity $S(\rho )= -tr \rho \log \rho$ refers to the Neumann entropy and $\rho _{A}= tr_{B} \rho _{AB}$ is the entropy of the reduced density matrix, where tr stands for the trace of matrix (Yan and Zhang 2014; Luo 2008; Datta 2009). The total correlation is quantified by the quantum mutual information $I(\rho _{AB})$:

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

(17)

The reduced matrix of $\rho _{A}$ and $\rho _{B}$ is given by:

$$\begin{aligned} \begin{array}{cc} S(\rho _{A}) = -(\rho _{11} + \rho _{22}) \log _{2}(\rho _{11} + \rho _{22})- (\rho _{33} + \rho _{44}) \log _{2}(\rho _{33} + \rho _{44})\\ S(\rho _{B}) = -(\rho _{11} + \rho _{33}) \log _{2}(\rho _{11} + \rho _{33}) - (\rho _{22} + \rho _{44}) \log _{2}(\rho _{22} + \rho _{44}) \end{array} \end{aligned}$$

(18)

The eigenvalues of the density matrix $S(\rho _{AB})= \varSigma ^{4}_{i=1} \epsilon _{i} \log ^{\epsilon _{i}}_{2}$ are given by:

$$\begin{aligned} \begin{array}{cc} \epsilon _{1}& = \frac{1}{2}[(\rho _{11} + \rho _{44})+\sqrt{(\rho _{11} - \rho _{44})^{2}+4 |\rho _{14}|^{2}}]\\ \epsilon _{2}& = \frac{1}{2} [(\rho _{11} + \rho _{44})-\sqrt{(\rho _{11} - \rho _{44})^{2}+4 |\rho _{14}|^{2}} ]\\ \epsilon _{3}& = \frac{1}{2} [(\rho _{22} + \rho _{33})+\sqrt{(\rho _{22} - \rho _{33})^{2}+4 |\rho _{23}|^{2}} ]\\ \epsilon _{4}& = \frac{1}{2} [(\rho _{22} + \rho _{33})-\sqrt{(\rho _{22} - \rho _{33})^{2}+4 |\rho _{23}|^{2}} ] \end{array} \end{aligned}$$

(19)

For the simplest case of two-qubit state described by the density matrix $\rho$, the analytical expression of the QD is specified by:

$$\begin{aligned} DQ(\rho _{AB}) = \min (Q_{1},Q_{2}), \end{aligned}$$

(20)

where,

$$\begin{aligned} Q_{j}= & H( \rho _{11} + \rho _{33} ) + \sum ^{4} _{i=1}\epsilon _{i} \log _{2} \epsilon _{i} + D_{j} ,\\ D_{1}= & H\left(\frac{1+\sqrt{[1-2(\rho _{33} + \rho _{44})]^{2}+4(|\rho _{14}|+|\rho _{23}|)^{2}}}{2}\right),\\ D_{2}= & -\sum _{i} \rho _{ii} \log _{2}^{\rho _{ii}} -H( \rho _{11} + \rho _{33} ),\\ H(x)= & -x \log _{2}{x}-(1-x)\log _{2}{({1-x})}. \end{aligned}$$

1.1 Super quantum discord

The weak measurement operators are given as (Wang et al. 2010; Datta et al. 2005; Jing and Yu 2017; Das and De 2016; Singh and Pati 2014):

$$\begin{aligned} \begin{array}{cc} P(\pm x)=\sqrt{\frac{1\mp \tanh x}{2}}\varPi _{0}+\sqrt{\frac{1\pm \tanh x }{2}}\varPi _{1} \end{array} \end{aligned}$$

(21)

where x is the strength parameter of measurement, $\varPi _{0}$ and $\varPi _{1}$ are orthogonal projectors that satisfy $\varPi _{0}+\varPi _{1}=I$. In addition, in the strong measurement limit we have the projective measurement operators $\lim _{x\rightarrow \infty } P(+x)= \varPi _{0}$ and $\lim _{x\rightarrow \infty }P(-x) = \varPi _{1}$.

If we replace all projection measurements with weak measurements in classical correlation and QD, it leads to a new type of quantum correlations called SQD. The super quantum discord denoted by $D_{w}(\rho _{AB})$ is defined as:

$$\begin{aligned} \begin{array}{cc} D_{w}(\rho _{AB})=S(\rho _{B})-S(\rho _{AB})+\min _ {{\{\varPi _{i}^{B}}\}} S_{w}(A|{P^{B}(x)}) \end{array} \end{aligned}$$

(22)

This is a positive quantity which follows from monotonicity of the mutual information, where the weak quantum conditional entropy is given by:

$$\begin{aligned} \begin{array}{cc} S_{w}(A|{P^{B}(x)})=P(+x) S(\rho _{A|P^{B}(+x)})+P(-x) S(\rho _{A|P^{B}(-x)}), \end{array} \end{aligned}$$

(23)

with

$$\begin{aligned} P(\pm x)=tr_{AB}[(I_{A}\otimes P^{B}(\pm x))\rho _{AB} (I_{A}\otimes P^{B}(\pm x))], \end{aligned}$$

and

$$\begin{aligned} \rho _{A|P^{B}(\pm x)}=\frac{tr_{B}[(I_{A}\otimes P^{B}(\pm x))\rho _{AB} (I_{A}\otimes P^{B}(\pm x))]}{tr_{AB}[(I_{A}\otimes P^{B}(\pm x))\rho _{AB} (I_{A}\otimes P^{B}(\pm x))}. \end{aligned}$$

where $I_{A}$ is the identity operator on the Hilbert space $\textit{H}_{A}$. Also, $P^{B}(\pm x)$ is the weak measurement operator performed on subsystem B.