Dependence of Bipartite Entanglement on Qubit-basis in the Amplitude-damping Environment

Xin-wen WANG; Hui-min ZHANG; Ke LIU; Yan LIU

doi:10.3788/gzxb20204910.1027002

Journals >Acta Photonica Sinica >Volume 49 >Issue 10 >Page 1027002 > Article

Acta Photonica Sinica
Vol. 49, Issue 10, 1027002 (2020)

Dependence of Bipartite Entanglement on Qubit-basis in the Amplitude-damping Environment

Xin-wen WANG^1、2, Hui-min ZHANG¹, Ke LIU¹, and Yan LIU¹

Author Affiliations

¹College of Physics and Electronic Engineering，Hengyang Normal University，Hengyang，Hunan 421002，China

²Hunan Provincial Key Laboratory of Intelligent Information Processing and Application，Hengyang，Hunan 421002，China

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DOI: 10.3788/gzxb20204910.1027002 Cite this Article

Xin-wen WANG, Hui-min ZHANG, Ke LIU, Yan LIU. Dependence of Bipartite Entanglement on Qubit-basis in the Amplitude-damping Environment[J]. Acta Photonica Sinica, 2020, 49(10): 1027002 Copy Citation Text

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Abstract

In order to enhance the intrinsic robustness or prolong the disentangling time of entangled states， the influence of qubi-basis on the robustness of two-qubit entanglement under local amplitude-damping environments is discussed. The decay law and lifetime of entanglement under various qubit-bases are displayed. The optimal qubit-bases in variety of cases are found. When an entangled state is prepared in an optimal qubit-basis， the entanglement decays most slowly and won't disappear in a finite time. The results imply that one could improve the robustness of entangled states by rotating the bases of qubits， which are expected to be helpful for increasing the qualities or efficiencies of entanglement-based quantum protocols.

Keywords

Amplitude-damping Bipartite entanglement Quantum optics Qubit-basis Robustness

\{\begin{array}{l} |ψ^{+} (θ, ϕ)〉 = c o s \frac{θ}{2} |0〉 + e^{i ϕ} s i n \frac{θ}{2} |1〉 \\ |ψ^{^{-}} (θ, ϕ)〉 = e^{- i ϕ} s i n \frac{θ}{2} |0〉 - c o s \frac{θ}{2} |1〉 \end{array}

（1）

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\{\begin{array}{l} {\hat{K}}_{0} = |0〉 〈0| + \sqrt[]{1 - d} |1〉 〈1| \\ {\hat{K}}_{1} = \sqrt[]{d} |0〉 〈1| \end{array}

（2）

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|Ψ〉 = α {|ψ^{+} (θ, ϕ)〉}_{1} {|ψ^{+} (θ, ϕ)〉}_{2} + β {|ψ^{-} (θ, ϕ)〉}_{1} {|ψ^{-} (θ, ϕ)〉}_{2}

（3）

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C (ρ) = m a x \{0, \sqrt[]{E_{1}} - \sqrt[]{E_{2}} - \sqrt[]{E_{3}} - \sqrt[]{E_{4}}\}

（4）

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ρ_{d} = \sum_{m, n = 0}^{1} {\hat{K}}_{m} \otimes {\hat{K}}_{n} |Ψ〉 〈Ψ| {\hat{K}}_{m}^{†} \otimes {\hat{K}}_{n}^{†}

（5）

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ρ_{d} = (\begin{matrix} W_{1} & W_{2} & W_{2} & W_{3} \\ W_{2}^{*} & W_{4} & W_{5} & W_{6} \\ W_{2}^{*} & W_{5} & W_{4} & W_{6} \\ W_{3}^{*} & W_{6}^{*} & W_{6}^{*} & W_{7} \end{matrix})

（6）

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\{\begin{array}{l} W_{1} = {|V_{0}|}^{2} + 2 d {|V_{1}|}^{2} + d^{2} {|V_{2}|}^{3} \\ W_{2} = \sqrt[]{1 - d} V_{0} V_{1}^{*} + d \sqrt[]{1 - d} V_{1} V_{2}^{*} \\ W_{3} = (1 - d) V_{0} V_{2}^{*} \\ W_{4} = (1 - d) {|V_{1}|}^{2} + d (1 - d) {|V_{2}|}^{2} \\ W_{5} = (1 - d) {|V_{1}|}^{2} \\ W_{6} = {(1 - d)}^{3 / 2} V_{1} V_{2}^{*} \\ W_{7} = {(1 - d)}^{2} {|V_{2}|}^{2} \\ V_{0} = α c o s^{2} \frac{θ}{2} + β e^{- i 2 ϕ} s i n^{2} \frac{θ}{2} \\ V_{1} = (α e^{i ϕ} - β e^{- i ϕ}) c o s \frac{θ}{2} s i n \frac{θ}{2} \\ V_{2} = α e^{i 2 ϕ} s i n^{2} \frac{θ}{2} + β c o s^{2} \frac{θ}{2} \end{array}

（7）

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C_{d} = m a x [0, 2 (1 - d) (|V_{0} V_{2} - V_{1}^{2}| - d {|V_{2}|}^{2})] = m a x [0, 2 (1 - d) (α β - d {|V_{2}|}^{2})]

（8）

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{|V_{2}|}^{2} = \frac{1}{4} {(1 - c o s θ)}^{2} + β^{2} c o s θ + \frac{1}{2} α β s i n^{2} θ c o s (2 ϕ)

（9）

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θ_{b} = a r c c o s \frac{1 - 2 β^{2}}{1 - 2 α β c o s (2 ϕ)}

（10）

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θ_{b} (ϕ = ϕ_{b}) = a r c c o s \frac{1 - 2 β^{2}}{1 + 2 α β}

（11）

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\{\begin{array}{l} |ψ^{+} (\frac{π}{2}, \frac{π}{2})〉 = \frac{1}{\sqrt[]{2}} (|0〉 + i |1〉) \\ |ψ^{-} (\frac{π}{2}, \frac{π}{2})〉 = \frac{- i}{\sqrt[]{2}} (|0〉 - i |1〉) \end{array}

（12）

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Xin-wen WANG, Hui-min ZHANG, Ke LIU, Yan LIU. Dependence of Bipartite Entanglement on Qubit-basis in the Amplitude-damping Environment[J]. Acta Photonica Sinica, 2020, 49(10): 1027002

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