Bidirectional teleportation under correlated noise

Abstract

In this contribution, we investigate bidirectional quantum teleportation of single-qubit state, for the case in which qubits from the quantum channel of teleportation are distributed by typical correlated noisy channels such as bit-flip, phase-flip, depolarizing, and amplitude damping channels. Expressions of the negativity of the partially transposed density operator, fidelities of teleportation, and quantum Fisher information are evaluated, we found that all these quantities depend on factor noise and the correlation strength of the noisy channel. It is shown that the effect of the noise on entanglement, averaged fidelities of teleportation, and on quantum Fisher information could be noticeably reduced due to the existence of the noise channel correlations.

Graphic abstract

Appendix

To evaluate the expression of QFI of the parameter $\theta _A$, namely $\mathcal {I}_{\theta _A}$, we use the formula (Eq. 17):

$$\begin{aligned} \mathcal {I}_{\theta _A} = |\partial _{\theta } \overrightarrow{a}|^{2}+\frac{(\overrightarrow{a}.\partial _{\theta } \overrightarrow{a})^{2}}{1-|\overrightarrow{a}|^{2}}, \end{aligned}$$

(50)

whereas $\partial _{\theta }=\frac{\partial }{\partial _{\theta _A}}$.

The Bloch vectors of the teleported state (Eq. 24) are given as:

$$\begin{aligned} a_{x}^{(bf)}= & {} p_A {{\bar{p}}}_B\cos (\phi _A) \sin (\theta _A),\nonumber \\ a_{y}^{(bf)}= & {} - p_A {{\bar{p}}}_B\sin (\phi _A) \sin (\theta _A),\nonumber \\ a_{z}^{(bf)}= & {} p_A {{\bar{p}}}_B(\mathcal {E}^\mathrm{NPT}_{bf})\cos (\theta _A). \end{aligned}$$

(51)

The norm $|\overrightarrow{a}|$:

$$\begin{aligned} |\overrightarrow{a}|= (p_A {{\bar{p}}}_B)^2 \Bigr (\sin (\theta _A)^2 + (\mathcal {E}^\mathrm{NPT} \cos (\theta _A))^2\Bigl ) \end{aligned}$$

(52)

and

$$\begin{aligned} \partial _{\theta } \overrightarrow{a}= \left\{ \begin{array}{lll} {{\bar{p}}}_B \cos (\phi _A) (\partial _{\theta }p_A \sin (\theta _A)+p_A \cos (\theta _A)) \\ -{{\bar{p}}}_B \sin (\phi _A) (\partial _{\theta }p_A \sin (\theta _A)+p_A \cos (\theta _A)) \\ {{\bar{p}}}_B \mathcal {E}^\mathrm{NPT}(\partial _{\theta }p_A \cos (\theta _A)-p_A \sin (\theta _A) ) \end{array} \right. \nonumber \\ \end{aligned}$$

(53)

The expression of QFI is given as follows:

$$\begin{aligned} \mathcal {I}_{\theta _A} = \chi +\frac{v^2}{1-|\overrightarrow{a}|^{2}}, \end{aligned}$$

(54)

where

$$\begin{aligned} \chi= & {} {{\bar{p}}}_B \Bigr ( (\partial _{\theta }p_A \sin (\theta _A)+p_A \cos (\theta _A))^2 \nonumber \\&+( \mathcal {E}^\mathrm{NPT}(\partial _{\theta }p_A \cos (\theta _A)-p_A \sin (\theta _A)))^2 \Bigl ); \end{aligned}$$

(55)

and

$$\begin{aligned} v= & {} p_A {{\bar{p}}}_B^2 \Bigr ((\partial _{\theta }p_A \sin (\theta _A)^2 + p_A \cos (\theta _A)\sin (\theta _A))\nonumber \\&+(\mathcal {E}^\mathrm{NPT})^2 (\partial _{\theta }p_A \cos (\theta _A)^2 - p_A\cos (\theta _A)\sin (\theta _A)) \Bigl )\nonumber \\ \end{aligned}$$

(56)

Similarly, QFI of the parameter $\theta _B$ could be easily evaluated using the same method, QFI of $\theta _B$ is given as:

$$\begin{aligned} \mathcal {I}_{\theta _B} = \mathcal {T}+\frac{w^2}{1-|\overrightarrow{a}|^{2}}, \end{aligned}$$

(57)

where

$$\begin{aligned} \mathcal {T}= & {} {{\bar{p}}}_{A} \Bigr ( (\partial _{\theta }p_B \sin (\theta _B)+p_B \cos (\theta _B))^2 \nonumber \\&+( \mathcal {E}^\mathrm{NPT}(\partial _{\theta }p_B \cos (\theta _B)-p_B \sin (\theta _B)))^2 \Bigl );\qquad \end{aligned}$$

(58)

and

$$\begin{aligned} w= & {} p_B {{\bar{p}}}_{A}^2 \Bigr ((\partial _{\theta }p_B \sin (\theta _B)^2 + p_B \cos (\theta _B)\sin (\theta _B))\nonumber \\&+(\mathcal {E}^\mathrm{NPT})^2 (\partial _{\theta }p_B \cos (\theta _B)^2 - p_B\cos (\theta _B)\sin (\theta _B)) \Bigl )\nonumber \\ \end{aligned}$$

(59)

using the same method, one can easily evaluate the QFI for the other types of noise.