Quantum Correlations of Two Relativistic Spin-$\frac {1}{2}$ Particles Under Noisy Channels

Abstract

We study the quantum correlation dynamics of bipartite spin-$\frac {1}{2}$ density matrices for two particles under Wigner rotations induced by Lorentz transformations which is transmitted through noisy channels. We compare quantum entanglement, geometric discord(GD), and quantum discord (QD) for bipartite relativistic spin-$\frac {1}{2}$ states under noisy channels. We find out QD and GD tend to death asymptotically but a sudden change in the decay rate of the entanglement occurs under noisy channels. Also, bipartite relativistic spin density matrices are considered as a quantum channel for teleportation one-qubit state under the influence of depolarizing noise and compare fidelity for various velocities of observers.

Appendix A

Wigner representation for spin-$\frac {1}{2}$: It follows [35] that the effect of an arbitrary Lorentz transformation unitarily implemented as U(Λ) on single-particle states is as follow

$$\begin{array}{@{}rcl@{}} U({\Lambda})(|p,\sigma\rangle)=\sqrt{\frac{({\Lambda} p)^{0}}{p^{0}}}{\Sigma}_{\sigma^{\prime}}D_{\sigma^{\prime} \sigma}(W({\Lambda},p))(|{\Lambda} p,\sigma^{\prime}\rangle) \end{array} $$

(7.55)

where

$$W({\Lambda},p)=L^{-1}({\Lambda} p){\Lambda} L(p)$$

is the Wigner rotation [36]. The Wigner rotation is an element of the spacial rotation group S O(3) or subgroup of the homogeneous Lorentz group since it leaves the rest momentum k ^ν unchanged:

$$W^{\mu}_{\nu}k^{\nu}=k^{\mu}.$$

This subgroup is called (Wigner's) little group. We will consider two reference frames in this work: one is the rest frame S and the other is the moving frame $S^{\prime }$ in which a particle whose four-momentum p in S is seen as boosted with the velocity $\vec {\upsilon }.$ By setting the boost and particle moving directions in the rest frame to be $\hat {\upsilon }$ with $\hat {e}$ as the normal vector in the boost direction and $\hat {p_{1,(2)}}$, respectively, and $\hat {n}=\hat {e}\times \hat {p_{1,(2)}}$ the Wigner representation for spin-$\frac {1}{2}$ is found as [13],

$$\begin{array}{@{}rcl@{}} D^{\frac{1}{2}}(W({\Lambda},p_{1,(2)}))=\cos \frac{\alpha_{p_{1(2)}}}{2}+i\sin\frac{\alpha_{p_{1(2)}}}{2}(\vec{\sigma}.\hat{n}), \end{array} $$

(7.56)

where

$$\begin{array}{@{}rcl@{}} \cos\frac{\alpha_{p_{1(2)}}}{2}=\frac{\cosh\frac{\alpha}{2}\cosh\frac{\delta}{2}+\sinh\frac{\alpha}{2}\sinh\frac{\delta}{2}(\hat{e}.\hat{p}_{1(2)})}{\sqrt{\left[\frac{1}{2}+\frac{1}{2}\cosh\alpha\cosh\delta+\frac{1}{2}\sinh\alpha\sinh\delta(\hat{e}.\hat{p}_{1(2)})\right]}} \end{array} $$

(7.57)

$$\begin{array}{@{}rcl@{}} \sin\frac{\alpha_{p_{1(2)}}}{2}\hat{n}=\frac{\sinh\frac{\alpha}{2}\sinh\frac{\delta}{2}(\hat{e}\times\hat{p}_{1(2)})}{\sqrt{\left[\frac{1}{2}+\frac{1}{2}\cosh\alpha\cosh\delta+\frac{1}{2}\sinh\alpha\sinh\delta(\hat{e}.\hat{p}_{1(2)})\right]}} \end{array} $$

(7.58)

and

$$\cosh\alpha=\gamma=\frac{1}{\sqrt{1-\beta^{2}}},\,\,\, \cosh\delta=\frac{E}{m},\,\,\, \beta=\frac{\upsilon}{c}$$

$$\left( \cos\frac{\alpha_{p_{1(2)}}}{2}\right)^{2}+\left( \sin\frac{\alpha_{p_{1(2)}}}{2}\hat{n}\right)^{2}=1.$$

Note, in this paper we consider the boost in the x direction and the momentum vector in the z direction. Then after some calculation we get the Wigner representation $D^{\frac {1}{2}}(W({\Lambda },p_{1,(2)}))$ for spin-$\frac {1}{2}$ as:

$$\begin{array}{@{}rcl@{}} D^{\frac{1}{2}}(W({\Lambda},p_{1,(2)}))=\left( \begin{array}{cc} \cos\frac{\alpha_{p_{1(2)}}}{2}& -\sin\frac{\alpha_{p_{1(2)}}}{2} \\ \sin\frac{\alpha_{p_{1(2)}}}{2}& \cos\frac{\alpha_{p_{1(2)}}}{2} \\ \end{array} \right). \end{array} $$

(7.59)