Continuous variable teleportation with indefinite causal order

Abstract

Continuous variable teleportation with indefinite causal order is studied for an optical system. The relation between input and teleported states is derived. The performance of the teleportation is evaluated in terms of the fidelity. As an example, the teleportation of a coherent state, a squeezed-vacuum state and a Schrödinger-cat state is investigated. The result shows that the indefinite causal order can improve the fidelity of the continuous variable teleportation.

A Randomly modulated amplitude channel with indefinite causal order

First, we briefly review a quantum channel in which amplitude of an optical system is randomly modulated [24]. If the quantum channel displaces the amplitude of the optical system initialized in a quantum state $\hat{\rho }_{\text {in}}$ by a complex number $\beta$, the quantum channel yields the state $\hat{\rho }_{\text {out}}(\beta ) =\hat{D}(\beta )\hat{\rho }_{\text {in}}\hat{D}^{\dagger }(\beta )$ with the displacement operator $\hat{D}(\beta )=e^{\beta \hat{a}^{\dagger }-\beta ^{*}\hat{a}}$. Then, when such a displacement occurs with probability $p(\beta )$ which is normalized by $\int \frac{d^{2}\beta }{\pi }p(\beta )=1$, the output state of the optical system is given by $\hat{\rho }_{\text {out}}=\int \frac{d^{2}\beta }{\pi }p(\beta )\hat{D}(\beta ) \hat{\rho }_{\text {in}}\hat{D}^{\dagger }(\beta )$, where the free part of the time-evolution has been ignored. This result is equal to Eq. (1). The detailed derivation and its physical meaning have been discussed in Ref. [24]. Next, we suppose that two quantum channels $\mathcal {A}$ and $\mathcal {B}$ sequentially apply to an optical system. The channel $\mathcal {A}$ displaces amplitude of the optical system by $\alpha$ with probability $p(\alpha )$ and the channel $\mathcal {B}$ by $\beta$ with probability $p(\beta )$. It is assumed that which one of the two channels applies first to the system is determined by a state of a control qubit. For instance, when the control qubit is initialized in a computational state $\vert 0_{c}\rangle$ ($\vert 1_{c}\rangle$), the quantum channel $\mathcal {B}$ ($\mathcal {A}$) affects the system first. Then, if the control qubit is prepared in a superposition state $\vert \xi _{c}\rangle$, the state $\hat{D}(\alpha ,\beta )(\hat{\rho }_{\text {in}}\otimes \vert \xi _{c}\rangle \langle \xi _{c}\vert )\hat{D}^{\dagger }(\alpha ,\beta )$ is obtained with probability $p(\alpha )p(\beta )$, where $\hat{D}(\alpha ,\beta )=\hat{D}(\alpha )\hat{D}(\beta )\otimes \vert 0_{c}\rangle \langle 0_{c}\vert +\hat{D}(\beta )\hat{D}(\alpha )\otimes \vert 1_{c}\rangle \langle 1_{c}\vert$. Hence, the output state of the optical system and the control qubit becomes $\hat{W}_{\text {out}}=\int \frac{d^{2}\alpha }{\pi } \int \frac{d^{2}\beta }{\pi }p(\alpha )p(\beta ) \hat{D}(\alpha ,\beta )(\hat{\rho }_{\text {in}}\otimes \vert \xi _{c}\rangle \langle \xi _{c}\vert )\hat{D}^{\dagger }(\alpha ,\beta )$, which is equivalent to Eq. (5).