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Entanglement Sudden Death and Birth Effects in Two Qubits Maximally Entangled Mixed States Under Quantum Channels

Abstract

In the present article, the robustness of entanglement in two qubits maximally entangled mixed states (MEMS) have been studied under quantum decoherence channels. Here we consider bit flip, phase flip, bit-phase-flip, amplitude damping, phase damping and depolarization channels. To quantify the entanglement, the concurrence has been used as an entanglement measure. During this study interesting results have been found for sudden death and birth of entanglement under bit flip and bit-phase-flip channels. While amplitude damping channel produces entanglement sudden death and does not allow re-birth of entanglement. On the other hand, two qubits MEMS exhibit the robust character against the phase flip, phase damping and depolarization channels. The elegant behavior of all the quantum channels have been investigated with varying parameter of quantum state MEMS in different cases.

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Acknowledgments

The first author Kapil K. Sharma acknowledges support from the Laboratory of Information Technology situated at Joint Institute For Nuclear Research, Dubna, Moscow Oblast. The contribution of second author Vladimir P. Gerdt was partially supported by the RUDN University Program (5–100).

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Appendix: Operators for Quantum Channels

Appendix: Operators for Quantum Channels

Kraus Operators → E 1 E 2 E 3 E 4
Bit Flip \( \left [\begin {array}{cc} \sqrt {p} & 0\\ 0 & \sqrt {p} \end {array}\right ] \) \( \left [\begin {array}{cc} 0 & \sqrt {(1-p)}\\ \sqrt {(1-p)} & 0 \end {array}\right ] \)
Phase flip \(\left [\begin {array}{cc} \sqrt {p} & 0\\ 0 & \sqrt {p} \end {array}\right ]\) \(\left [\begin {array}{cc} \sqrt {(1-p)} & 0 \\ 0 & \sqrt {(1-p)} \end {array}\right ]\)
Bit Phase flip \(\left [\begin {array}{cc}\sqrt {p} & 0\\ 0 & \sqrt {p} \end {array}\right ]\) \(\left [\begin {array}{cc} 0 & -i\sqrt {(1-p)}\\ i\sqrt {(1-p)} & 0 \end {array}\right ]\)
Amplitude damping \( \left [\begin {array}{cc} 1 & 0\\ 0 & \sqrt {e^{-\gamma t}} \end {array}\right ]\) \(\left [\begin {array}{cc} 0 & \sqrt {1-e^{-\gamma t}}\\ 0 & 0 \end {array}\right ]\)
Phase damping \(\left [\begin {array}{cc} \sqrt {e^{-\gamma t}} & 0\\ 0 & \sqrt {e^{-\gamma t}} \end {array}\right ]\) \(\left [\begin {array}{cc} \sqrt {1-e^{-\gamma t}} & 0\\ 0 & 0 \end {array}\right ]\) 1
Depolarization \(\left [\begin {array}{cc} \sqrt {e^{-\gamma t}} & 0\\ 0 & \sqrt {e^{-\gamma t}} \end {array}\right ]\) \(\left [\begin {array}{cc} 0 & \sqrt {\frac {1}{3}(1-e^{-\gamma t})} \\ \sqrt {\frac {1}{3}(1-e^{-\gamma t})} & 0 \end {array}\right ]\) 2 3
$$ \begin{array}{@{}rcl@{}} \star^{1}=\left[ \begin{array}{cc} 0 & 0 \\ 0 & \sqrt{1-{e}^{-\gamma t}} \end{array}\right]; \star^{2}&=&\left[\begin{array}{cc} 0 & -i\sqrt{\frac{1}{3}(1-e^{-\gamma t})} \\ i\sqrt{\frac{1}{3}(1-e^{-\gamma t})} & 0 \end{array}\right];\\ \star^{3}&=&\left[\begin{array}{cc} \sqrt{\frac{1}{3}(1-e^{-\gamma t})} & 0 \\ 0 & -\sqrt{\frac{1}{3}(1-e^{-\gamma t})} \end{array}\right]. \end{array} $$
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Sharma, K.K., Gerdt, V.P. Entanglement Sudden Death and Birth Effects in Two Qubits Maximally Entangled Mixed States Under Quantum Channels. Int J Theor Phys 59, 403–414 (2020). https://doi.org/10.1007/s10773-019-04332-z

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