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Secure Quantum Communication Scheme for Six-Qubit Decoherence-Free States

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Abstract

Noise is currently unavoidable in quantum communication environments. Eavesdroppers can exploit this issue by disguising themselves as channel noise to avoid detection during eavesdropping checks performed by legitimate communicants. This paper first proposes a new coding function comprising eight unitary operations for two orthogonal bases for six-qubit decoherence-free states. Subsequently, based on the coding function, the first deterministic secure quantum communication (DSQC) scheme for quantum channels with collective noise is developed. The developed DSQC is robust against both collective-dephasing noise and collective-rotation noise Senders can choose one of six-qubit decoherence-free states to encode their two-bit message, and receivers simply conduct Bell measurement to obtain the message. Analyses conducted verify that the proposed scheme is both secure and robust.

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Authors and Affiliations

  1. Department of Computer Science and Information Engineering, National Cheng Kung University, No. 1, University Rd., Tainan City, 70101, Taiwan, Republic of China

    Chih-Lun Tsai & Tzonelih Hwang

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  1. Chih-Lun Tsai
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  2. Tzonelih Hwang
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Correspondence to Tzonelih Hwang.

Appendices

Appendix 1

$$\begin{array}{@{}rcl@{}} \mathrm{U}_{\mathrm{A}} &=& \mathrm{I}_{{64}\times {64}}\qquad \text{note: I is identity matrix}\\ \mathrm{U}_{B} &=& \\ \mathrm{U}_{\mathrm{A}} &+&[-\vert 000111\rangle \langle {000111}\vert {-\vert 001011}\rangle \langle {001011}\vert {-\vert 001101}\rangle \langle {001101}\vert {-\vert 001110}\rangle \langle {001110}\vert\\ &&{-\vert 010001}\rangle \langle {010001}\vert \\ &&{ -\vert 010010}\rangle \langle {010010}\vert {-\vert 010100}\rangle \langle {010100}\vert -2\vert 010110\rangle \langle {010110}\vert {-\vert 011000}\rangle \langle {011000}\vert \\ &&{-2\vert 011001}\rangle \langle {011001}\vert \end{array} $$
$$\begin{array}{@{}rcl@{}} &&{-2\vert 011100}\rangle \langle {011100}\vert {-2\vert 011111}\rangle \langle {011111}\vert {-\vert 100001}\rangle \langle {100001}\vert {-\vert 100011}\rangle \langle {100011}\vert\\ &&{-\vert 100100}\rangle \langle {100100}\vert \\ &&{ -\vert 100111}\rangle \langle {100111}\vert {-\vert 101000}\rangle \langle {101000}\vert {-\vert 101010}\rangle \langle {101010}\vert {-\vert 101110}\rangle \langle {101110}\vert\\ &&{+\vert 000111}\rangle \langle {001101}\vert \\ &&{ +\vert 001011}\rangle \langle {001110}\vert {-\vert 011101}\rangle \langle {011101}\vert {+\vert 001101}\rangle \langle {000111}\vert {+\vert 001110}\rangle \langle {001011}\vert\\ &&{+\vert 010001}\rangle \langle {010100}\vert \\ &&{ +\vert 010010}\rangle \langle {011000}\vert {+\vert 010011}\rangle \langle {010110}\vert {+\vert 010011}\rangle \langle {011001}\vert {+\vert 010100}\rangle \langle {010001}\vert\\ && {+\vert 011000}\rangle \langle {010010}\vert \\ &&{ +\vert 011100}\rangle \langle {010110}\vert {+\vert 011100}\rangle \langle {011001}\vert {+\vert 011001}\rangle \langle {011100}\vert {+\vert 011001}\rangle \langle {011111}\vert\\ &&{+\vert 011101}\rangle \langle {100011}\vert \\ &&{ +\vert 100001}\rangle \langle {100100}\vert {+\vert 100010}\rangle \langle {011100}\vert {+\vert 100010}\rangle \langle {011111}\vert {+\vert 100011}\rangle \langle {011101}\vert\\ &&{+\vert 100100}\rangle \langle {100001}\vert \\ &&{ +\vert 100111}\rangle \langle {101010}\vert {+\vert 101000}\rangle \langle {101110}\vert {+\vert 101010}\rangle \langle {100111}\vert {+\vert 101110}\rangle \langle {101000}\vert ] \end{array} $$
$$\begin{array}{@{}rcl@{}} \mathrm{U}_{\mathrm{C}} &=& \\ \mathrm{U}_{\mathrm{A}} &+&[-\vert 000011\rangle \langle {000011}\vert {-\vert 001100}\rangle \langle {001100}\vert {-\vert 001101}\rangle \langle {001101}\vert {-\vert 001110}\rangle \langle {001110}\vert\\ &&{-\vert 010001}\rangle \langle {010001}\vert \\ &&{ -\vert 010010}\rangle \langle {010010}\vert {-\vert 010011}\rangle \langle {010011}\vert {-2\vert 010101}\rangle \langle {010101}\vert {-2\vert 011001}\rangle \langle {011001}\vert\\ &&{-\vert 011100}\rangle \langle {011100}\vert \\ &&{ -\vert 011001}\rangle \langle {011001}\vert {-2\vert 011100}\rangle \langle {011100}\vert {-2\vert 100000}\rangle \langle {100000}\vert {-\vert 1000010}\rangle \langle {100010}\vert\\ &&{-\vert 100011}\rangle \langle {100011}\vert \\ &&{ -\vert 100100}\rangle \langle {100100}\vert {-\vert 100111}\rangle \langle {100111}\vert {-\vert 101000}\rangle \langle {101000}\vert {-\vert 101001}\rangle \langle {101001}\vert\\ &&{-\vert 110010}\rangle \langle {110010}\vert \\ &&{ +\vert 000011}\rangle \langle {010010}\vert {+\vert 000111}\rangle \langle {001101}\vert {+ 2\vert 000111}\rangle \langle {010101}\vert {+\vert 000111}\rangle \langle {011100}\vert\\ &&{+\vert 001011}\rangle \langle {001101}\vert \\ &&{ + 2\vert 001011}\rangle \langle {011001}\vert {+\vert 001011}\rangle \langle {011100}\vert {+\vert 001100}\rangle \langle {010001}\vert {-\vert 001101}\rangle \langle {011100}\vert \\ &&{+\vert 001110}\rangle \langle {010011}\vert \\ &&{ +\vert 010001}\rangle \langle {001100}\vert {+\vert 010010}\rangle \langle {000011}\vert {+\vert 010011}\rangle \langle {001110}\vert {-\vert 011100}\rangle \langle {001101}\vert\\ &&{-\vert 011001}\rangle \langle {101000}\vert \\ &&{ +\vert 100010}\rangle \langle {100111}\vert {+\vert 100111}\rangle \langle {110010}\vert {+\vert 100100}\rangle \langle {101001}\vert {+\vert 100111}\rangle \langle {100010}\vert\\ &&{-\vert 101000}\rangle \langle {011001}\vert \\ &&{ +\vert 101001}\rangle \langle {100100}\vert {+\vert 101010}\rangle \langle {011001}\vert {+ 2\vert 101010}\rangle \langle {011100}\vert {+\vert 101010}\rangle \langle {101000}\vert\\ &&{+\vert 110010}\rangle \langle {100011}\vert \\ &&{ +\vert 101110}\rangle \langle {011001}\vert {+\vert 101110}\rangle \langle {100000}\vert {+\vert 101110}\rangle \langle {101000}\vert ] \end{array} $$
$$\begin{array}{@{}rcl@{}} \mathrm{U}_{\mathrm{D}} &=& \\ \mathrm{U}_{\mathrm{A}} &+&[-\vert 000011\rangle \langle {000011}\vert {+\vert 000011}\rangle \langle {011000}\vert {-\vert 000111}\rangle \langle {000111}\vert {-\vert 000111}\rangle \langle {011100}\vert \\ &&{-\vert 001011}\rangle \langle {001011}\vert \\ &&{ +\vert 001011}\rangle \langle {010011}\vert {-\vert 001100}\rangle \langle {001100}\vert {+\vert 001100}\rangle \langle {010100}\vert {+\vert 001101}\rangle \langle {000111}\vert\\ && {+ 2\vert 001101}\rangle \langle {010101}\vert \end{array} $$
$$\begin{array}{@{}rcl@{}} &&{+\vert 001101}\rangle \langle {011100}\vert {+\vert 001110}\rangle \langle {000111}\vert {+ 2\vert 001110}\rangle \langle {010110}\vert {+\vert 001110}\rangle \langle {011100}\vert\\ && {+\vert 010011}\rangle \langle {001011}\vert \\ &&{ -\vert 010011}\rangle \langle {010011}\vert {+\vert 010100}\rangle \langle {001100}\vert {-\vert 010100}\rangle \langle {010100}\vert {-2\vert 010101}\rangle \langle {010101}\vert\\ && {-2\vert 010110}\rangle \langle {010110}\vert \\ &&{ +\vert 011000}\rangle \langle {000011}\vert {-\vert 011000}\rangle \langle {011000}\vert {-\vert 011100}\rangle \langle {000111}\vert {-\vert 011100}\rangle \langle {011100}\vert\\ && {-\vert 011001}\rangle \langle {011001}\vert \\ &&{ -\vert 011001}\rangle \langle {101111}\vert {-\vert 011101}\rangle \langle {011101}\vert {+\vert 011101}\rangle \langle {110010}\vert {-2\vert 011111}\rangle \langle {011111}\vert\\ && {-2\vert 100000}\rangle \langle {100000}\vert \\ &&{ -\vert 100001}\rangle \langle {100001}\vert {+\vert 100001}\rangle \langle {101001}\vert {-\vert 100010}\rangle \langle {100010}\vert {+\vert 100010}\rangle \langle {101010}\vert\\ && {+\vert 100111}\rangle \langle {011001}\vert \\ &&{ + 2\vert 100111}\rangle \langle {011111}\vert {+\vert 100111}\rangle \langle {101111}\vert {+\vert 101000}\rangle \langle {011001}\vert {+ 2\vert 101000}\rangle \langle {100000}\vert\\ && {+\vert 101000}\rangle \langle {101111}\vert \\ &&{ +\vert 101001}\rangle \langle {100001}\vert {-\vert 101001}\rangle \langle {101001}\vert {+\vert 101010}\rangle \langle {100010}\vert {-\vert 101010}\rangle \langle {101010}\vert\\ && {-\vert 101110}\rangle \langle {011001}\vert \\ &&{ -\vert 101110}\rangle \langle {101110}\vert {-\vert 110010}\rangle \langle {110010}\vert {+\vert 110010}\rangle \langle {011101}\vert] \end{array} $$

Appendix 2

$$\begin{array}{@{}rcl@{}} \mathrm{U}_{E} &=&U_{\mathrm{A}} {= I}_{{64}\times {64}} \\ \mathrm{U}_{F} &=& \\ \mathrm{U}_{E} &+&[-\vert 000011\rangle \langle 000011\vert {-\vert }000101\rangle \langle 000101\vert {-\vert }001001\rangle \langle 001001\vert {-2\vert }001011\rangle \langle 001011\vert\\ &&{-\vert }001110\rangle \langle 001110\vert \\ &&{ -\vert }010001\rangle \langle 010001\vert {-2\vert }010101\rangle \langle 010101\vert {-\vert }010110\rangle \langle 010110\vert {-\vert }011010\rangle \langle 011010\vert\\ &&{-\vert }011100\rangle \langle 011100\vert \\ &&{ -\vert }011001\rangle \langle 011001\vert {-\vert }011011\rangle \langle 011011\vert {-\vert }011111\rangle \langle 011111\vert {-2\vert }100000\rangle \langle 100000\vert\\ &&{-\vert }100100\rangle \langle 100100\vert \\ &&{ -\vert }100111\rangle \langle 100111\vert {-2\vert }101010\rangle \langle 101010\vert {-\vert }101100\rangle \langle 101100\vert {-\vert }110000\rangle \langle 110000\vert\\ &&{-\vert }110010\rangle \langle 110010\vert \\ &&{ +\vert }000011\rangle \langle 000101\vert +{\vert }000101\rangle \langle 000011\vert \!+\!{\vert }001001\rangle \langle 010001\vert \!+\!{\vert }001101\rangle \langle 001011\vert\\ &&+{\vert }001101\rangle \langle 010101\vert \\ &&{ +\vert }001110\rangle \langle 010110\vert +{\vert }010001\rangle \langle 001001\vert \!+\!{\vert }010011\rangle \langle 001011\vert \!+\!{\vert }010011\rangle \langle 010101\vert\\ &&+{\vert }010110\rangle \langle 001110\vert \\ &&{ +\vert }011010\rangle \langle 011100\vert +{\vert }011100\rangle \langle 011010\vert \!+\!{\vert }011001\rangle \langle 011011\vert \!+\!{\vert }011011\rangle \langle 011001\vert\\ &&+{\vert }011111\rangle \langle 100111\vert \\ &&{ +\vert }100010\rangle \langle 100000\vert +{\vert }100010\rangle \langle 101010\vert \!+\!{\vert }100100\rangle \langle 101100\vert \!+\!{\vert }100111\rangle \langle 011111\vert\\ &&+{\vert }101000\rangle \langle 100000\vert \\ &&{ +\vert }101000\rangle \langle 101010\vert +{\vert }101100\rangle \langle 100100\vert \!+\!{\vert }110000\rangle \langle 110010\vert \!+\!{\vert }110010\rangle \langle 110000\vert ]\\ \mathrm{U}_{G} &=& \\ \mathrm{U}_{E} &+&[-2\vert 001011\rangle \langle 001011\vert {-\vert }001101\rangle \langle 001101\vert {-\vert }010011\rangle \langle 010011\vert {-\vert }010110\rangle \langle 010110\vert \end{array} $$
$$\begin{array}{@{}rcl@{}} &&{-2\vert }011001\rangle \langle 011001\vert \\ &&{ -\vert }011010\rangle \langle 011010\vert {-\vert }0011011\rangle \langle 011011\vert {-2\vert }011100\rangle \langle 011100\vert {-\vert }011111\rangle \langle 011111\vert\\ && {-\vert }100010\rangle \langle 100010\vert \\ &&{ -\vert }101000\rangle \langle 101000\vert {-2\vert }101010\rangle \langle 101010\vert +{\vert }001101\rangle \langle 010110\vert +{\vert }001110\rangle \langle 001011\vert\\ &&{-\vert }001110\rangle \langle 011001\vert \\ &&{ -\vert }010011\rangle \langle 011010\vert +{\vert }010110\rangle \langle 001101\vert {-\vert }011010\rangle \langle 010011\vert {-\vert }011100\rangle \langle 001011\vert\\ &&+{\vert }011100\rangle \langle 011001\vert \\ &&{ +\vert }011001\rangle \langle 011100\vert {-\vert }011001\rangle \langle 101010\vert {-\vert }011011\rangle \langle 100010\vert +{\vert }011111\rangle \langle 101000\vert\\ &&{-\vert }100010\rangle \langle 011011\vert \\ &&{ -\vert }100111\rangle \langle 011100\vert +{\vert }100111\rangle \langle 101010\vert +{\vert }101000\rangle \langle 011111\vert ] \end{array} $$
$$\begin{array}{@{}rcl@{}} \mathrm{U}_{H} &=& \\ \mathrm{U}_{E} &+&[-\vert 001101\rangle \langle 001101\vert {-\vert }001110\rangle \langle 001110\vert {-\vert }010011\rangle \langle 010011\vert {-2\vert }010101\rangle \langle 010101\vert \\ &&{-2\vert }010101\rangle \langle 010110\vert \\ &&{ -2\vert }011001\rangle \langle 011001\vert {-2\vert }011010\rangle \langle 011010\vert {-\vert }011100\rangle \langle 011100\vert {-\vert }011001\rangle \langle 011001\vert\\ &&{-2\vert }011011\rangle \langle 011011\vert \\ &&{ -2\vert }011100\rangle \langle 011100\vert {-2\vert }011111\rangle \langle 011111\vert {-2\vert }100000\rangle \langle 100000\vert {-\vert }100010\rangle \langle 100010\vert \\ &&{-\vert }100111\rangle \langle 100111\vert \\ &&{ -\vert }101000\rangle \langle 101000\vert +{\vert }001101\rangle \langle 001110\vert +{\vert }001110\rangle \langle 001101\vert {-\vert }010011\rangle \langle 011100\vert\\ &&{-\vert }010110\rangle \langle 000111\vert \\ &&{ +\vert }010110\rangle \langle 001011\vert +{\vert }011010\rangle \langle 000111\vert {-\vert }011010\rangle \langle 001011\vert {-\vert }011100\rangle \langle 010011\vert\\ &&{-\vert }011001\rangle \langle 100010\vert \\ &&{ -\vert }011011\rangle \langle 101010\vert +{\vert }011011\rangle \langle 101110\vert +{\vert }011111\rangle \langle 101010\vert {-\vert }011111\rangle \langle 101110\vert\\ &&{-\vert }100010\rangle \langle 011001\vert \\ &&{ +\vert }100111\rangle \langle 101000\vert +{\vert }101000\rangle \langle 100111\vert ] \end{array} $$

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Tsai, CL., Hwang, T. Secure Quantum Communication Scheme for Six-Qubit Decoherence-Free States. Int J Theor Phys 57, 3808–3818 (2018). https://doi.org/10.1007/s10773-018-3893-1

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