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Detection of polarization shift-keyed/switched/multiplexed quantum coherent states in M-ary photonic communication systems

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Abstract

In this work, we present the analysis of the quantum detection of information-carrying coherent states used in optical communications systems that employ the degrees of freedom of both the complex amplitude and the state of polarization of the laser field. In these quantum communication systems, the transmitter prepares a set of quantum states (a constellation of quantum composite coherent states) producing a four-dimensional (4D) M-ary modulation, that is sent through the optical channel, in diverse formats such as: a) polarization shift-keying (PolSK), b) polarization switching (PS), or c) polarization multiplexing (PM). At the receiver, measurements are performed to determine as accurately as possible which state was sent, with a discrimination method aided by any prior information. With this purpose, in this work we apply the square root method (SRM) for the received composite quantum state constellations and study their performance in error probability and mutual information for several 4D M-ary coherent state modulation formats, inspired by constellations commonly used in classical photonic communications. We arrive at closed form expressions both for the error probability and for the mutual information, for most of the modulation formats analyzed.

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Data availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Funding

This work has been partially supported by the Mexican National Council for Science and Technology (CONACYT.)

Author information

Authors and Affiliations

  1. Electronics and Telecommunications Department, Centro de Investigación Científica Y de Educación Superior de Ensenada, 22800, Ensenada, B.C, Mexico

    Arturo Arvizu-Mondragón & Ramón Muraoka-Espíritu

  2. AI Mexico, 22880, Ensenada, B.C, Mexico

    Francisco J. Mendieta-Jiménez

  3. Universidad Autónoma de Baja California, FIAD, 22800, Ensenada, B.C, Mexico

    César A. López-Mercado

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  1. Arturo Arvizu-Mondragón
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  2. Francisco J. Mendieta-Jiménez
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Appendices

Appendices

Appendix 1. PS-QPSK composite states.

The 8 composite states for PS-QPSK are expressed by the permutations of the tensor products between the 4 states for each SOP: \(\left|\Delta \right.>,\left|\mathrm{i}\Delta \right.>, \left|-\Delta \right.>, \left|-\mathrm{i}\Delta \right.>\) and vacuum: \(\left|0\right.>\), as:

$$ \begin{aligned} \left| {\gamma_{0} } \right\rangle & = \left| \Delta \right\rangle \otimes \left| 0 \right\rangle ,\;\left| {\gamma_{1} } \right\rangle = \left| {i\Delta } \right\rangle \otimes \left| 0 \right\rangle , \left| {\gamma_{2} } \right\rangle = \left| { - \Delta } \right\rangle \otimes \left| 0 \right\rangle ,\; \\ \left| {\gamma_{3} } \right\rangle & = \left| { - i\Delta } \right\rangle \otimes \left| 0 \right\rangle ,\;\left| {\gamma_{4} } \right\rangle = \left| 0 \right\rangle \otimes \left| \Delta \right\rangle ,\left| {\gamma_{5} } \right\rangle = \left| 0 \right\rangle \otimes \left| {i\Delta } \right\rangle ,\; \\ \left| {\gamma_{6} } \right\rangle & = \left| 0 \right\rangle \otimes \left| { - \Delta } \right\rangle \;{\text{and}}\;\left| {\gamma_{7} } \right\rangle = \left| 0 \right\rangle \otimes \left| { - i\Delta } \right\rangle \\ \end{aligned} $$
(A1.1)

Appendix 2. 4PS-QPSK composite states.

\(\widehat{x}\;\mathrm{and }\;\widehat{y}\) are unit vectors in the horizontal and vertical directions, respectively; the 16 composite states for 4PS-QPSK are expressed by the permutations of the fourfold tensor products between the 4 states for each SOP: \(\left|\Delta \right.>,\left|\mathrm{i}\Delta \right.>, \left|-\Delta \right.>, \left|-\mathrm{i}\Delta \right.>\) and vacuum: \(\left|0\right.>\), as:

$$ \begin{aligned} \left| {\gamma _{0} } \right\rangle & = ~\left| \Delta \right\rangle \hat{x}~ \otimes \left| 0 \right\rangle \hat{y} \otimes \frac{1}{{\sqrt 2 }}\left( {\left| 0 \right\rangle \hat{x} + \left| 0 \right\rangle \hat{y}} \right) \otimes \frac{1}{{\sqrt 2 }}\left( { - \left| 0 \right\rangle \hat{x} + \left| 0 \right\rangle \hat{y}} \right), \\ \left| {\gamma _{1} } \right\rangle & = ~\left| 0 \right\rangle \hat{x}~ \otimes \left| \Delta \right\rangle \hat{y} \otimes \frac{1}{{\sqrt 2 }}\left( {\left| 0 \right\rangle \hat{x} + \left| 0 \right\rangle \hat{y}} \right) \otimes \frac{1}{{\sqrt 2 }}\left( { - \left| 0 \right\rangle \hat{x} + \left| 0 \right\rangle \hat{y}} \right), \\ \left| {\gamma _{2} } \right\rangle & = ~\left| 0 \right\rangle \hat{x}~ \otimes \left| 0 \right\rangle \hat{y} \otimes \frac{1}{{\sqrt 2 }}\left( {\left| \Delta \right\rangle \hat{x} + \left| \Delta \right\rangle \hat{y}} \right) \otimes \frac{1}{{\sqrt 2 }}\left( { - \left| 0 \right\rangle \hat{x} + \left| 0 \right\rangle \hat{y}} \right), \\ \left| {\gamma _{3} } \right\rangle & = ~\left| 0 \right\rangle \hat{x}~ \otimes \left| 0 \right\rangle \hat{y} \otimes \frac{1}{{\sqrt 2 }}\left( {\left| 0 \right\rangle \hat{x} + \left| 0 \right\rangle \hat{y}} \right) \otimes \frac{1}{{\sqrt 2 }}\left( { - \left| \Delta \right\rangle \hat{x} + \left| \Delta \right\rangle \hat{y}} \right), \\ \left| {\gamma _{4} } \right\rangle & = ~\left| {i\Delta } \right\rangle \hat{x}~ \otimes \left| 0 \right\rangle \hat{y} \otimes \frac{1}{{\sqrt 2 }}\left( {\left| 0 \right\rangle \hat{x} + \left| 0 \right\rangle \hat{y}} \right) \otimes \frac{1}{{\sqrt 2 }}\left( { - \left| 0 \right\rangle \hat{x} + \left| 0 \right\rangle \hat{y}} \right), \\ \left| {\gamma _{5} } \right\rangle & = ~\left| 0 \right\rangle \hat{x}~ \otimes ~\left| {i\Delta } \right\rangle \hat{y} \otimes \frac{1}{{\sqrt 2 }}\left( {\left| 0 \right\rangle \hat{x} + \left| 0 \right\rangle \hat{y}} \right) \otimes \frac{1}{{\sqrt 2 }}\left( { - \left| 0 \right\rangle \hat{x} + \left| 0 \right\rangle \hat{y}} \right), \\ \left| {\gamma _{6} } \right\rangle & = ~\left| 0 \right\rangle \hat{x}~ \otimes \left| 0 \right\rangle \hat{y} \otimes \frac{1}{{\sqrt 2 }}\left( {~\left| {i\Delta } \right\rangle \hat{x} + ~\left| {i\Delta } \right\rangle \hat{y}} \right) \otimes \frac{1}{{\sqrt 2 }}\left( { - \left| 0 \right\rangle \hat{x} + \left| 0 \right\rangle \hat{y}} \right), \\ \left| {\gamma _{7} } \right\rangle & = ~\left| 0 \right\rangle \hat{x}~ \otimes \left| 0 \right\rangle \hat{y} \otimes \frac{1}{{\sqrt 2 }}\left( {\left| 0 \right\rangle \hat{x} + \left| 0 \right\rangle \hat{y}} \right) \otimes \frac{1}{{\sqrt 2 }}\left( { - ~\left| {i\Delta } \right\rangle \hat{x} + ~\left| {i\Delta } \right\rangle \hat{y}} \right), \\ \left| {\gamma _{8} } \right\rangle & = ~\left| { - \Delta } \right\rangle \hat{x}~ \otimes \left| 0 \right\rangle \hat{y} \otimes \frac{1}{{\sqrt 2 }}\left( {\left| 0 \right\rangle \hat{x} + \left| 0 \right\rangle \hat{y}} \right) \otimes \frac{1}{{\sqrt 2 }}\left( { - \left| 0 \right\rangle \hat{x} + \left| 0 \right\rangle \hat{y}} \right), \\ \left| {\gamma _{9} } \right\rangle & = \left| 0 \right\rangle \hat{x}~ \otimes \left| { - \Delta } \right\rangle \hat{y} \otimes \frac{1}{{\sqrt 2 }}\left( {\left| 0 \right\rangle \hat{x} + \left| 0 \right\rangle \hat{y}} \right) \otimes \frac{1}{{\sqrt 2 }}\left( { - \left| 0 \right\rangle \hat{x} + \left| 0 \right\rangle \hat{y}} \right), \\ \left| {\gamma _{{10}} } \right\rangle & = ~\left| 0 \right\rangle \hat{x}~ \otimes \left| 0 \right\rangle \hat{y} \otimes \frac{1}{{\sqrt 2 }}\left( {\left| { - \Delta } \right\rangle \hat{x} + \left| { - \Delta } \right\rangle \hat{y}} \right) \otimes \frac{1}{{\sqrt 2 }}\left( { - \left| 0 \right\rangle \hat{x} + \left| 0 \right\rangle \hat{y}} \right), \\ \left| {\gamma _{{11}} } \right\rangle & = \left| 0 \right\rangle \hat{x}~ \otimes \left| 0 \right\rangle \hat{y} \otimes \frac{1}{{\sqrt 2 }}\left( {\left| 0 \right\rangle \hat{x} + \left| 0 \right\rangle \hat{y}} \right) \otimes \frac{1}{{\sqrt 2 }}\left( { - \left| { - \Delta } \right\rangle > \hat{x} + \left| { - \Delta } \right\rangle > \hat{y}} \right), \\ \left| {\gamma _{{12}} } \right\rangle & = ~\left| { - i\Delta } \right\rangle \hat{x}~ \otimes \left| 0 \right\rangle \hat{y} \otimes \frac{1}{{\sqrt 2 }}\left( {\left| 0 \right\rangle \hat{x} + \left| 0 \right\rangle \hat{y}} \right) \otimes \frac{1}{{\sqrt 2 }}\left( { - \left| 0 \right\rangle \hat{x} + \left| 0 \right\rangle \hat{y}} \right), \\ \left| {\gamma _{{13}} } \right\rangle & = \left| 0 \right\rangle \hat{x}~ \otimes \left| { - i\Delta } \right\rangle \hat{y} \otimes \frac{1}{{\sqrt 2 }}\left( {\left| 0 \right\rangle \hat{x} + \left| 0 \right\rangle \hat{y}} \right) \otimes \frac{1}{{\sqrt 2 }}\left( { - \left| 0 \right\rangle \hat{x} + \left| 0 \right\rangle \hat{y}} \right), \\ \left| {\gamma _{{14}} } \right\rangle & = ~\left| 0 \right\rangle \hat{x}~ \otimes \left| 0 \right\rangle \hat{y} \otimes \frac{1}{{\sqrt 2 }}\left( {\left| { - i\Delta } \right\rangle \hat{x} + \left| { - i\Delta } \right\rangle \hat{y}} \right) \otimes \frac{1}{{\sqrt 2 }}\left( { - \left| 0 \right\rangle \hat{x} + \left| 0 \right\rangle \hat{y}} \right), \\ \left| {\gamma _{{15}} } \right\rangle & = ~\left| 0 \right\rangle \hat{x}~ \otimes \left| 0 \right\rangle \hat{y} \otimes \frac{1}{{\sqrt 2 }}\left( {\left| 0 \right\rangle \hat{x} + \left| 0 \right\rangle \hat{y}} \right) \otimes \frac{1}{{\sqrt 2 }}\left( { - \left| { - i\Delta } \right\rangle \hat{x} + \left| { - i\Delta } \right\rangle \hat{y}} \right) \\ \end{aligned} $$
(A2.1)

Appendix 3. PM-QPSK composite states.

The 16 composite states for PM-QPSK are expressed by the permutations of the tensor inter-products between the states for each SOP: \(\left|\Delta \right.>,\left|\mathrm{i}\Delta \right.>, \left|-\Delta \right.>, \left|-\mathrm{i}\Delta \right.>\), as:

$$ \begin{aligned} \left| {\gamma_{0} } \right\rangle & = \left| \Delta \right\rangle \otimes \left| \Delta \right\rangle ,\;\left| {\gamma_{1} } \right\rangle = \left| {i\Delta } \right\rangle \otimes \left| \Delta \right\rangle ,\;\left| {\gamma_{2} } \right\rangle = \left| { - \Delta } \right\rangle \otimes \left| \Delta \right\rangle , \\ \left| {\gamma_{3} } \right\rangle & = \left| { - i\Delta } \right\rangle \otimes \left| \Delta \right\rangle ,\;\left| {\gamma_{4} } \right\rangle = \left| \Delta \right\rangle \otimes \left| {i\Delta } \right\rangle ,\;\left| {\gamma_{5} } \right\rangle = \left| {i\Delta } \right\rangle \otimes \left| {i\Delta } \right\rangle , \\ \left| {\gamma_{6} } \right\rangle & = \left| { - \Delta } \right\rangle \otimes \left| {i\Delta } \right\rangle ,\;\left| {\gamma_{7} } \right\rangle = \left| { - i\Delta } \right\rangle \otimes \left| {i\Delta } \right\rangle ,\;\left| {\gamma_{8} } \right\rangle = \left| \Delta \right. > \otimes \left| { - \Delta } \right\rangle , \\ \left| {\gamma_{9} } \right\rangle & = \left| {i\Delta } \right\rangle \otimes \left| { - \Delta } \right\rangle ,\;\left| {\gamma_{10} } \right\rangle = \left| { - \Delta } \right\rangle \otimes \left| { - \Delta } \right\rangle ,\;\left| {\gamma_{11} } \right\rangle = \left| { - \Delta } \right\rangle \otimes \left| { - \Delta } \right\rangle , \\ \left| {\gamma_{12} } \right\rangle & = \left| \Delta \right. > \otimes \left| { - i\Delta } \right\rangle ,\;\left| {\gamma_{13} } \right\rangle = \left| {i\Delta } \right. > \otimes \left| { - i\Delta } \right\rangle ,\;\left| {\gamma_{14} } \right\rangle = \left| { - \Delta } \right\rangle \otimes \left| { - i\Delta } \right\rangle \;and\; \\ \left| {\gamma_{15} } \right\rangle & = \left| { - i\Delta } \right\rangle > \otimes \left| { - i\Delta } \right\rangle \\ \end{aligned} $$
(A3.1)

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Arvizu-Mondragón, A., Mendieta-Jiménez, F.J., López-Mercado, C.A. et al. Detection of polarization shift-keyed/switched/multiplexed quantum coherent states in M-ary photonic communication systems. Quantum Inf Process 21, 345 (2022). https://doi.org/10.1007/s11128-022-03687-3

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