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Photonics Research
Vol. 7, Issue 5, A7 (2019)

Limitations of teleporting a qubit via a two-mode squeezed state

Seok Hyung Lie and Hyunseok Jeong^*

Author Affiliations

Center for Macroscopic Quantum Control, Department of Physics and Astronomy, Seoul National University, Seoul 151-742, South Korea

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DOI: 10.1364/PRJ.7.0000A7 Cite this Article Set citation alerts

Seok Hyung Lie, Hyunseok Jeong. Limitations of teleporting a qubit via a two-mode squeezed state[J]. Photonics Research, 2019, 7(5): A7 Copy Citation Text

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Abstract

Recently, a teleportation scheme using a two-mode squeezed state to teleport a photonic qubit, so called a “hybrid” approach, has been suggested and experimentally demonstrated as a candidate to overcome the limitations of all-optical quantum information processing. We find, however, that there exists the upper bound of fidelity when teleporting a photonic qubit via a two-mode squeezed channel under a lossy environment. The increase of photon loss decreases this bound, and teleportation better than this limit is impossible even when the squeezing degree of the teleportation channel becomes infinity. Our result indicates that the hybrid scheme can be valid for fault-tolerant quantum computing only when the photon loss rate can be suppressed under a certain limit.

{| β ⟩}_{A, R} = \frac{1}{\sqrt{π}} \sum_{n = 0}^{\infty} {\hat{D}}_{A} (β) {| n, n ⟩}_{A, R},

(1)

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{| q ⟩}_{R, B} = \sqrt{1 - q^{2}} \sum_{n = 0}^{\infty} q^{n} {| n, n ⟩}_{R, B} .

(2)

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{| ψ ⟩}_{B} = \sqrt{\frac{1 - q^{2}}{π}} \sum_{n = 0}^{\infty} q^{n} {| n ⟩}_{B} {⟨ n |}_{A} {\hat{D}}_{A} (- β) {| ψ ⟩}_{A} .

(3)

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{| ψ_{out} (β) ⟩}_{B} = {\hat{T}}_{q}^{g} (β) {| ψ ⟩}_{A},

(4)

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{\hat{T}}_{q}^{g} (β) = \sqrt{\frac{1 - q^{2}}{π}} \sum_{n = 0}^{\infty} q^{n} {\hat{D}}_{B} (g β) {| n ⟩}_{B} {⟨ n |}_{A} {\hat{D}}_{A} (- β) .

(5)

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{Tr}_{E_{1} E_{2}} (U_{A E_{1}}^{BS} U_{B E_{2}}^{BS} | q ⟩ ⟨ q |_{A B} \otimes | 00 ⟩ {⟨ 00 |}_{E_{1} E_{2}} U_{{A E}_{1}}^{BS †} U_{{B E}_{2}}^{BS †}),

(6)

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ϕ_{channel} = (1 - q^{2}) \sum_{n, m} \sum_{\binom{k, l =}{\begin{matrix} \max {0, n - m} \end{matrix}}}^{n} c_{n m k l} q^{n + m} t^{2 (m - n) + 2 (k + l)} r^{4 n - 2 (k + l)} | k l ⟩ ⟨ m - n + k, m - n + l |,

(7)

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c_{n m k l} = \sqrt{(\begin{matrix} n \\ k \end{matrix}) (\begin{matrix} n \\ l \end{matrix}) (\begin{matrix} m \\ n - k \end{matrix}) (\begin{matrix} m \\ n - l \end{matrix})}

(8)

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ρ_{B, out} (β) = {\hat{D}}_{B} (g β) {⟨ β |}_{A R} (| ψ ⟩ {⟨ ψ |}_{A} \otimes ϕ_{R B, channel}) {| β ⟩}_{A R} {\hat{D}}_{B} .

(9)

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{\hat{T}}_{k l} (β) = \sqrt{\frac{1 - q^{2}}{π}} \sum_{n = \max (k, l)}^{\infty} q^{n} \sqrt{(\begin{matrix} n \\ k \end{matrix}) (\begin{matrix} n \\ l \end{matrix})} t^{2 n - (k + l)} r^{k + l} \hat{D} (g β) | n - l ⟩ ⟨ n - k | D (- β),

(10)

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ρ_{out} (β) = \sum_{k, l = 0}^{\infty} {\hat{T}}_{k l} (β) | ψ ⟩ ⟨ ψ | {\hat{T}}_{k l}^{†} (β) .

(11)

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\sum_{k, l = 0}^{\infty} \int d^{2} β {\hat{T}}_{k l}^{†} (β) {\hat{T}}_{k l} (β) = 1 .

(12)

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F = \sum_{k, l} F_{k l} = \sum_{k, l} \int d^{2} β {| ⟨ ψ | {\hat{T}}_{k l} (β) | ψ ⟩ |}^{2},

(13)

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F_{0 \to 0} = \frac{1 - q^{2}}{(1 + g^{2}) (1 - q^{2} r^{2}) - 2 g q (1 - r^{2})} .

(14)

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F_{0 \to 0}^{perfect} = \frac{1 - q^{2}}{1 - 2 g q + g^{2}} .

(15)

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F_{1 \to 1} = \frac{1 - q^{2}}{{(1 - q^{2} r^{4})}^{2} {[(1 + g^{2}) (1 - q^{2} r^{2}) - 2 g q t^{2}]}^{3}} \times {q t^{2} {q r^{2} [{(g - q)}^{2} + {(1 - g q)}^{2}] (3 + q^{2} r^{4}) + 2 (g - q) (1 - g q) (1 + 3 q^{2} r^{4})} [{(1 + g)}^{2} (1 - q^{2} r^{2}) - 2 g q t^{2}] + q^{2} t^{4} (1 + q^{2} r^{4}) {[(1 + g^{2}) (1 - q^{2} r^{2}) - 2 g q t^{2}]}^{2} + 2 {[1 - g q + q r^{2} (g - q)]}^{2} {[g - q + q r^{2} (1 - g q)]}^{2}},

(16)

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F_{1 \to 1}^{perfect} = \frac{1 - q^{2}}{{(1 - 2 g q + g^{2})}^{3}} [{(g - q)}^{2} {(1 - g q)}^{2} + g^{2} {(1 - q^{2})}^{2}] .

(17)

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F_{qubit} = F_{0 \to 0} F_{1 \to 1} .

(18)

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E_{N} (ρ) = \max {0, - \ln [{\tilde{ν}}_{-} (ρ)]},

(19)

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{\tilde{ν}}_{-} (ρ) = (1 - R) \cosh (2 ar \tanh q) + R - (1 - R) \sinh (2 ar \tanh q),

(20)

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F_{q = g \to 1} (R) = \frac{1 + R^{2}}{{(1 + R)}^{4}} .

(21)

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R_{classic} \approx 0.11 .

(22)

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References (35)

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Seok Hyung Lie, Hyunseok Jeong. Limitations of teleporting a qubit via a two-mode squeezed state[J]. Photonics Research, 2019, 7(5): A7

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