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Quantum channel estimations via indefinite causal order

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Abstract

In this paper, we consider the quantum switch, which is one of the indefinite causal structures, in the study of quantum channel estimation. We show that such an indefinite causal order cannot provide any advantage for estimating quantum channels when the Kraus operators of the channel are commutative. We investigate the effects of the quantum switch when studying unitary qubit channels and amplitude damping qubit channels with noncommutative Kraus operators.

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Acknowledgements

Z. Yin was partially supported by NSFC No. 12031004.

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Authors and Affiliations

  1. School of Mathematics and Information Sciences, Yantai University, Yantai, 264005, China

    Juan Gu

  2. School of Mathematics and Statistics, Central South University, Changsha, 410083, China

    Zhi Yin

  3. School of Mathematics, Harbin Institute of Technology, Harbin, 150006, China

    Longsuo Li

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  1. Juan Gu
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  2. Zhi Yin
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  3. Longsuo Li
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Correspondence to Juan Gu.

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Appendices

Appendix A: Generators of Lie algebras su(2) and su(4)

The generators of Lie algebra su(2) are given by

$$\begin{aligned} \eta _{1}=\left( \begin{array}{cc} 0&{}\quad 1\\ 1&{}\quad 0\nonumber \\ \end{array}\right) , \quad \eta _{2}=\left( \begin{array}{cc} 1&{}\quad 0\\ 0&{}\quad 1\nonumber \\ \end{array} \right) , \quad \eta _{3}=\left( \begin{array}{cc} 0&{}\quad -i\\ i&{}\quad 0 \end{array} \right) . \end{aligned}$$

And the generators of Lie algebra su(4) are given by

$$\begin{aligned} \eta _{1}= & {} \left( \begin{array}{cccc} 0&{}\quad 1&{}\quad 0&{}\quad 0\\ 1&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0\\ \end{array} \right) , \quad \eta _{2}=\left( \begin{array}{cccc} 0&{}\quad 0&{}\quad 1&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 1&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0\\ \end{array}\right) , \quad \eta _{3}=\left( \begin{array}{cccc} 0&{}\quad 0&{}\quad 0&{}\quad 1\\ 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 1&{}\quad 0&{}\quad 0&{}\quad 0\\ \end{array}\right) ,\\ \eta _{4}= & {} \left( \begin{array}{cccc} 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 1&{}\quad 0\\ 0&{}\quad 1&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0\\ \end{array}\right) , \eta _{5}= \left( \begin{array}{cccc} 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 1\\ 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 1&{}\quad 0&{}\quad 0\\ \end{array} \right) , \quad \eta _{6}= \left( \begin{array}{cccc} 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 1\\ 0&{}\quad 0&{}\quad 1&{}\quad 0\\ \end{array} \right) , \\ \eta _{7}= & {} \left( \begin{array}{cccc} 0&{}\quad -i&{}\quad 0&{}\quad 0\\ i&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0\\ \end{array} \right) , \eta _{8}= \left( \begin{array}{cccc} 0&{}\quad 0&{}\quad -i&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0\\ i&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0\\ \end{array} \right) , \eta _{9}= \left( \begin{array}{cccc} 0&{}\quad 0&{}\quad 0&{}\quad -i\\ 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0\\ i&{}\quad 0&{}\quad 0&{}\quad 0\\ \end{array} \right) , \\ \eta _{10}= & {} \left( \begin{array}{cccc} 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad -i&{}\quad 0\\ 0&{}\quad i&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0\\ \end{array} \right) , \eta _{11}= \left( \begin{array}{cccc} 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad -i\\ 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad i&{}\quad 0&{}\quad 0\\ \end{array} \right) , \quad \eta _{12}= \left( \begin{array}{cccc} 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad -i\\ 0&{}\quad 0&{}\quad i&{}\quad 0\\ \end{array} \right) ,\\ \eta _{13}= & {} \left( \begin{array}{cccc} 1&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad -1&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0\\ \end{array} \right) , \quad \eta _{14}=\frac{1}{\sqrt{3}} \left( \begin{array}{cccc} 1&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 1&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad -2&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0\\ \end{array} \right) , \\ \eta _{15}= & {} \frac{1}{\sqrt{6}} \left( \begin{array}{cccc} 1&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 1&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 1&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad -3\\ \end{array} \right) . \end{aligned}$$

Appendix B: Coefficients of \(\rho _\theta \) in (29)

Recall that

$$\begin{aligned} \rho _{\theta }= & {} \mathcal {S}(\mathcal {U}_{1\theta },\mathcal {U}_{2\theta })(\rho ;|c\rangle ) = \begin{pmatrix} pE&{}\sqrt{p(1-p)}G\\ \sqrt{p(1-p)}H&{}(1-p)M\\ \end{pmatrix},\\ E= & {} \frac{1}{4} \{ 2+2(2q-1)\cos 2\theta +\sqrt{q(1-q)}(2-2\cos 4\theta ) \} |0\rangle \langle 0|\\{} & {} + \frac{1}{4} \{ 4\sqrt{q(1-q)}\cos 2\theta -2i[(1-2q)\sin 2\theta +\sqrt{q(1-q)}\sin 4\theta ] \} |0\rangle \langle 1|\\{} & {} + \frac{1}{4} \{ 4\sqrt{q(1-q)}\cos 2\theta +2i[(1-2q) \sin 2\theta +\sqrt{q(1-q)}\sin 4\theta ] \} |1\rangle \langle 0|\\{} & {} + \frac{1}{4} \{ 2+2(1-2q)\cos 2\theta +\sqrt{q(1-q)}(2\cos 4\theta -2)\}|1\rangle \langle 1|,\\ G= & {} \frac{1}{4} \{[3q-1+2\cos 2\theta +(q-1)\cos 4\theta +\sqrt{q(1-q)}(1-\cos 4\theta )]\\{} & {} -i[(2q-2)\sin 2\theta +(1-q)\sin 4\theta +\sqrt{q(1-q)}(\sin 4\theta -2\sin 2\theta )]\}|0\rangle \langle 0|\\{} & {} +\frac{1}{4} \{[\sqrt{q(1-q)}(3+\cos 4\theta )+q(1-\cos 4\theta )]\\{} & {} -i[-q\sin 4\theta +2(1-q)\sin 2\theta +\sqrt{q(1-q)}(\sin 4\theta +2\sin 2\theta )]\}|0\rangle \langle 1|\nonumber \\{} & {} +\frac{1}{4} \{[(1-q)(\cos 4\theta -1)+\sqrt{q(1-q)}(3+\cos 4\theta )]\\{} & {} -i[2q\sin 2\theta -(1-q)\sin 4\theta -\sqrt{q(1-q)}(\sin 4\theta +2\sin 2\theta )]\}|1\rangle \langle 0|\\{} & {} +\frac{1}{4} \{[2\cos 2\theta +2-3q+(\sqrt{q(1-q)}-q)\cos 4\theta -\sqrt{q(1-q)}]\\{} & {} -i[(2q-2\sqrt{q(1-q)})\sin 2\theta +(\sqrt{q(1-q)}-q)\sin 4\theta ]\} |1\rangle \langle 1|,\\ H= & {} \frac{1}{4}\{[3q-1+2\cos 2\theta +(q-1)\cos 4\theta +\sqrt{q(1-q)}(1-\cos 4\theta )]\\{} & {} +i[(2q-2)\sin 2\theta +(1-q)\sin 4\theta +\sqrt{q(1-q)}(\sin 4\theta -2\sin 2\theta )]\}|0\rangle \langle 0|\\{} & {} +\frac{1}{4} \{[(1-q)(\cos 4\theta -1)+\sqrt{q(1-q)}(3+\cos 4\theta )]\\{} & {} +i[2q\sin 2\theta -(1-q)\sin 4\theta -\sqrt{q(1-q)}(\sin 4\theta +2\sin 2\theta )]\}|0\rangle \langle 1|\\{} & {} +\frac{1}{4}\{[q(1-\cos 4\theta )+\sqrt{q(1-q)}(3+\cos 4\theta )]\\{} & {} +i[2(1-q)\sin 2\theta +\sqrt{q(1-q)}(2\sin 2\theta +\sin 4\theta )-q\sin 4\theta ]\}|1\rangle \langle 0|\\{} & {} +\frac{1}{4} \{[2\cos 2\theta +2-3q-\sqrt{q(1-q)}+(\sqrt{q(1-q)}-q)\cos 4\theta ]\\{} & {} +i[(2q-2\sqrt{q(1-q)})\sin 2\theta +(\sqrt{q(1-q)}-q)\sin 4\theta ] \} |1\rangle \langle 1|,\\ M= & {} \frac{1}{4} \{2+2(2q-1)\cos 2\theta \} |0\rangle \langle 0|\\{} & {} +\frac{1}{4} \{[(2q-1)(1-\cos 4\theta )+4\sqrt{q(1-q)}\cos 2\theta ]-i[(1-2q)\sin 4\theta \\{} & {} +4\sqrt{q(1-q)}\sin 2\theta ]\}|0\rangle \langle 1|+\frac{1}{4} \{[(2q-1)(1-\cos 4\theta )+4\sqrt{q(1-q)}\cos 2\theta ]\\{} & {} +i[(1-2q)\sin 4\theta +4\sqrt{q(1-q)}\sin 2\theta ]\}|1\rangle \langle 0|\\{} & {} +\frac{1}{4} \{2+2(1-2q)\cos 2\theta \} |1\rangle \langle 1|. \end{aligned}$$

The generalized Bloch expression of \(\rho _\theta \) is given by

$$\begin{aligned} \rho _{\theta }=\frac{1}{4}1\hspace{-2.22214pt}{\textrm{l}}_{4}+\frac{1}{2} \sum _{i=1}^{15} \omega _i \eta _i, \end{aligned}$$

with coefficients

$$\begin{aligned} \omega _{1}&=2p\sqrt{q(1-q)}\cos 2\theta , \\ \omega _{2}&=\frac{\sqrt{p(1-p)}}{2}[3q-1+\sqrt{q(1-q)}+2\cos 2\theta +(q-1-\sqrt{q(1-q)})\cos 4\theta ],\\ \omega _{3}&=\frac{\sqrt{p(1-p)}}{2}[q+3\sqrt{q(1-q)}-(q-\sqrt{q(1-q)})\cos 4\theta ],\\ \omega _{4}&=\frac{\sqrt{p(1-p)}}{2}[q-1+3\sqrt{q(1-q)}+(1-q+\sqrt{q(1-q)})\cos 4\theta ],\\ \omega _{5}&=\frac{\sqrt{p(1-p)}}{2}[2-3q-\sqrt{q(1-q)}+2\cos 2\theta +(\sqrt{q(1-q)}-q)\cos 4\theta ],\\ \omega _{6}&=\frac{1-p}{2}[2q-1-(2q-1)\cos 4\theta +4\sqrt{q(1-q)}\cos 2\theta ],\\ \omega _{7}&=p[(1-2q)\sin 2\theta +\sqrt{q(1-q)}\sin 4\theta ],\\ \omega _{8}&=\frac{\sqrt{p(1-p)}}{2}[(2q-2-2\sqrt{q(1-q)})\sin 2\theta +(1-q+\sqrt{q(1-q)})\sin 4\theta ],\\ \omega _{9}&=\frac{\sqrt{p(1-p)}}{2}[(2-2q+2\sqrt{q(1-q)})\sin 2\theta +(\sqrt{q(1-q)}-q)\sin 4\theta ],\\ \omega _{10}&=\frac{\sqrt{p(1-p)}}{2}[(2q-2\sqrt{q(1-q)})\sin 2\theta -(1-q+\sqrt{q(1-q)})\sin 4\theta ],\\ \omega _{11}&=\frac{\sqrt{p(1-p)}}{2}[(2q-2\sqrt{q(1-q)})\sin 2\theta +(\sqrt{q(1-q)}-q)\sin 4\theta ],\\ \omega _{12}&=\frac{1-p}{2}[(1-2q)\sin 4\theta +4\sqrt{q(1-q)}\sin 2\theta ],\\ \omega _{13}&=p[(2q-1)\cos 2\theta +\sqrt{q(1-q)}(1-\cos 4\theta )],\\ \omega _{14}&=\frac{1}{\sqrt{3}}-\frac{1}{\sqrt{3}}(1-p)[2+(2q-1)\cos 2\theta ],\\ \omega _{15}&=\frac{1}{\sqrt{6}}-\frac{\sqrt{6}}{3}(1-p)[1+(1-2q)\cos 2\theta ]. \end{aligned}$$

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Gu, J., Yin, Z. & Li, L. Quantum channel estimations via indefinite causal order. Quantum Inf Process 22, 369 (2023). https://doi.org/10.1007/s11128-023-04118-7

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