Coherence dynamics induced by attenuation and amplification Gaussian channels

Abstract

Quantum Gaussian channels play a key role in quantum information theory. In particular, the attenuation and amplification channels are useful to describe noise and decoherence effects on continuous variables systems. They are directly associated with the beam splitter and two-mode squeezing operations, which have operational relevance in quantum protocols with bosonic models. An important property of these channels is that they are Gaussian completely positive maps, and the action on a general input state depends on the parameters characterizing the channels. In this work, we study the coherence dynamics introduced by these channels on input Gaussian states. We derive explicit expressions for the coherence depending on the parameters describing the channels. By assuming a displaced thermal state with initial coherence as input state, for the attenuation case it is observed a revival of the coherence as a function of the transmissivity coefficient, whereas for the amplification channel the coherence reaches asymptotic values depending on the gain coefficient. Further, we obtain the entropy production for these classes of operations, showing that it can be reduced by controlling the parameters involved. We write a simple expression for computing the entropy production due to the coherence for both channels. This can be useful to simulate many processes in quantum thermodynamics, as finite-time driving on bosonic modes.

Appendix: Entropy production

Although simple, here we derive the expression in Eq. (36). As mentioned in main text, the protocol is composed of two parts. The Gaussian input state passes through an attenuation (amplification) channel and then thermalizes with a Markovian thermal reservoir with inverse temperature $\beta$ and average number of photons $\bar{N}$. The entropy production of a unitary process followed by a thermalization can be written as [17]

$$\begin{aligned} \langle \Sigma \rangle =S(\rho _{\tau _{1}}||\rho _{\tau _{1}}^{\mathrm{eq,h}}) -S(\rho _{\tau _{2}}||\rho _{\tau _{2}}^{\mathrm{eq,h}}), \end{aligned}$$

(39)

where $\rho _{\tau _{1}}$is the state after the unitary transformation, $\rho _{\tau _{2}}$ is the state during the thermalization process, and $\rho _{\tau _{1}}^{\mathrm{eq,h}}$ is the reference thermal state associated with the thermal reservoir. As the reference state is thermal the relative entropy can be rewritten as $S(\rho _{t}||\rho _{t}^{\mathrm{eq,h}})=\beta \left[ {\mathcal {U}}(\rho _{t})-F_{t}^{\mathrm{eq}}\right] -S(\rho _{t})$, with ${\mathcal {U}}(\rho _{t})$ and $F_{t}^{\mathrm{eq}}$ the internal energy of the system and the free energy, respectively. By manipulating, we obtain

$$\begin{aligned} \langle \Sigma \rangle&=\beta \left[ {\mathcal {U}}(\rho _{\tau _{1}})-F_{\tau _{1}}^{\mathrm{eq}}\right] -S(\rho _{\tau _{1}})\\&\quad -\left\{ \beta \left[ {\mathcal {U}}(\rho _{\tau _{2}})-F_{\tau _{2}}^{\mathrm{eq}}\right] -S(\rho _{\tau _{2}})\right\} \\&=\beta \left[ {\mathcal {U}}(\rho _{\tau _{1}})-{\mathcal {U}}(\rho _{\tau _{2}})\right] -S(\rho _{\tau _{1}})+S(\rho _{\tau _{2}})\\&=-\beta \Delta {\mathcal {U}}_{\tau _{2},\tau _{1}}+\Delta S_{\tau _{2},\tau _{1}}. \end{aligned}$$

To complete, for Gaussian states the internal energy can be written in terms of the covariance matrix, i.e.,

$$\begin{aligned} {\mathcal {U}}(\rho _{t})=\frac{\hbar \omega }{4}\text {Tr}\left[ \sigma _{t}\right] . \end{aligned}$$

(40)

With this, for Gaussian states the entropy production can be completely characterized by the covariance matrix of the state.