Determination of Entropy Production for a Quantum System in the Presence of an Auxiliary System

Abstract

A thorough understanding of entropy production, which can be used as a natural quantifier of the degree of irreversibility of a process, is both fundamentally relevant and technologically desirable. Recently, Esposito et al. (J. Phys. 12, 013013 2010), have derived an exact expression for the entropy production in terms of correlations between a system and its reservoir. However, evaluation of the mentioned correlations is virtually impossible to access in real world situations. Here, it is demonstrated that how we can overcome this problem by considering an auxiliary system and using the monogamy relations. To this aim, we consider a system A that is initially correlated with an auxiliary system B, which in turn interacts with the environment E. In the presence of the auxiliary system B, the amount of created classical and quantum correlations between the system A and the environment can be obtained by using existed monogamy properties and consequently the associated entropy production is determined. In addition, another situation is investigated in this work where only the system A is interacting with the environment E and the auxiliary system B evolves free of any direct interaction. As an example, this scenario is illustrated by considering the systems A and B as two-level systems undergoing an amplitude damping process and generalized amplitude damping process.

Appendix: Kraus Operators

Here, we summarize the calculations that lead to the Kraus operators of (16) and (20). At first, we remind that the Kraus operators of amplitude damping channel are given as [17]

$$ \begin{array}{@{}rcl@{}} \mathcal{M}_{1}=\left( \begin{array}{cc} 1 & 0\\\\ 0 & \sqrt{1-\eta} \end{array} \right),\qquad \mathcal{M}_{2}=\left( \begin{array}{cc} 0 & \sqrt{\eta}\\\\ 0 & 0 \end{array} \right). \end{array} $$

(28)

If we write the initial density matrix as

$$ \begin{array}{@{}rcl@{}} \rho(0)=\left( \begin{array}{cc} \rho_{00}(0) & \rho_{01}(0) \\\\ \rho^{*}_{01}(0) & 1-\rho_{00}(0) \end{array} \right), \end{array} $$

(29)

so, the density matrix evolves as

$$ \begin{array}{@{}rcl@{}} \rho(t)=\sum\limits_{i=1}^{2} \mathcal{M}_{i}\rho(0) \mathcal{M}_{i}^{\dag}=\left( \begin{array}{cc} \rho_{00}(0)+(1-\rho_{00}(0))\eta & \sqrt{1-\eta}\rho_{01}(0) \\\\ \sqrt{1-\eta}\rho^{*}_{01}(0) & (1-\eta)(1-\rho_{00}(0)) \end{array} \right). \end{array} $$

(30)

It can be easily checked that this channel can also be generated by the evolution given by the master equation

$$ \begin{array}{@{}rcl@{}} \frac{\partial \rho(t)}{\partial t}=\gamma \left( \sigma_{-} \rho(t) \sigma_{+}-\frac{1}{2}\{\sigma_{+}\sigma_{-},\rho(t)\}\right), \end{array} $$

(31)

when η(t) = 1 − e^−γt. It should be noted that here the rate of dissipation, i.e. γ, is time-independent. In a similar way, we want to determine the Kraus operators corresponding to the channel which is generated by the master equation in (13) with the time-dependent decay rate γ(t) as given in (14). To this aim, we define the new Kraus operators as

$$ \begin{array}{@{}rcl@{}} M_{1}(t)=\left( \begin{array}{cc} 1 & 0\\\\ 0 & \sqrt{1-q(t)} \end{array} \right),\qquad M_{2}(t)=\left( \begin{array}{cc} 0 & \sqrt{q(t)}\\\\ 0 & 0 \end{array} \right), \end{array} $$

(32)

where the coefficient q(t) is calculated as

$$ \begin{array}{@{}rcl@{}} q(t)=1-e^{-{{\int}_{0}^{t}}\gamma(t^{\prime}) dt^{\prime}}=1-e^{-\lambda t}\left[\cosh(\frac{dt}{2})+\frac{\lambda}{d}\sinh(\frac{dt}{2})\right]^{2}. \end{array} $$

(33)

It is clear that, when the decay rate is independent of time, i.e. γ(t^′) = γ, we have q(t) = η(t) = 1 − e^−γt.

Moreover, the Kraus operators corresponding to generalized amplitude damping channel are obtained in the same way as given in (20).