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11 pages, 654 KiB  
Article
Helstrom Bound for Squeezed Coherent States in Binary Communication
by Evaldo M. F. Curado, Sofiane Faci, Jean-Pierre Gazeau and Diego Noguera
Entropy 2022, 24(2), 220; https://doi.org/10.3390/e24020220 - 31 Jan 2022
Cited by 2 | Viewed by 2790
Abstract
In quantum information processing, using a receiver device to differentiate between two non-orthogonal states leads to a quantum error probability. The minimum possible error is known as the Helstrom bound. In this work, we study the conditions for state discrimination using an alphabet [...] Read more.
In quantum information processing, using a receiver device to differentiate between two non-orthogonal states leads to a quantum error probability. The minimum possible error is known as the Helstrom bound. In this work, we study the conditions for state discrimination using an alphabet of squeezed coherent states and compare them with conditions using the Glauber-Sudarshan, i.e., standard, coherent states. Full article
(This article belongs to the Special Issue Quantum Mechanics and Its Foundations II)
Show Figures

Figure 1

Figure 1
<p>Helstrom bound for <math display="inline"><semantics> <mrow> <mi>ν</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>φ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msup> <mi>φ</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>π</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>=</mo> <msup> <mi>θ</mi> <mo>′</mo> </msup> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> using different coherent states with PSK encoding as a function of the mean energy <span class="html-italic">n</span>. The black dotted curve represents the Glauber-Sudarshan coherent states treated in <span class="html-italic">Case 1</span>. The blue dashed curve depicts the squeezed <span class="html-italic">Case 2</span> with <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <msup> <mi>β</mi> <mo>′</mo> </msup> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>. The red curve pictures the optimal <span class="html-italic">Case 3</span>, i.e., with <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <msup> <mi>β</mi> <mo>′</mo> </msup> <mo>=</mo> <msub> <mi>β</mi> <mrow> <mi>P</mi> <mi>S</mi> <msub> <mi>K</mi> <mrow> <mi>o</mi> <mi>p</mi> <mi>t</mi> </mrow> </msub> </mrow> </msub> </mrow> </semantics></math> (<a href="#FD12-entropy-24-00220" class="html-disp-formula">12</a>). Notice that the curve of squeezed <span class="html-italic">Case 2</span> is below that of GS-CS <span class="html-italic">Case 1</span> for <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>≳</mo> <mn>0.25</mn> </mrow> </semantics></math> while the optimal <span class="html-italic">Case 3</span> stands below the other two curves for any value of <span class="html-italic">n</span>.</p>
Full article ">Figure 2
<p>Helstrom bound as function of <math display="inline"><semantics> <mi>φ</mi> </semantics></math> and <math display="inline"><semantics> <mi>θ</mi> </semantics></math>, for <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <msup> <mi>β</mi> <mo>′</mo> </msup> <mo>=</mo> <mn>1</mn> <mo>/</mo> <mn>3</mn> </mrow> </semantics></math>, (which corresponds to the optimal case of squeezed PSK) <math display="inline"><semantics> <mrow> <msup> <mi>φ</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>φ</mi> <mo>+</mo> <mi>π</mi> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msup> <mi>θ</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>θ</mi> </mrow> </semantics></math>. The brown-orange surface depicts the <math display="inline"><semantics> <mrow> <mi>ν</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> squeezed PSK case while the light blue surface pictures a general squeezed case with <math display="inline"><semantics> <mrow> <mi>ν</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>. One can easily see that the minimum value for Helstrom bound corresponds to <math display="inline"><semantics> <mrow> <mn>2</mn> <mi>φ</mi> <mo>−</mo> <mi>θ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 3
<p>Helstrom bound for <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>ν</mi> <mo>=</mo> <mn>0.7</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>φ</mi> <mo>=</mo> <msup> <mi>θ</mi> <mo>′</mo> </msup> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msup> <mi>φ</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>π</mi> </mrow> </semantics></math>. Blue flat surface: Glauber-Sudarshan alphabet. Brown-orange curved surface: squeezed coherent state alphabet with <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <msup> <mi>β</mi> <mo>′</mo> </msup> </mrow> </semantics></math>.</p>
Full article ">Figure 4
<p>Helstrom bound as function of <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <msup> <mi>β</mi> <mo>′</mo> </msup> </mrow> </semantics></math> for different values of <math display="inline"><semantics> <mi>ν</mi> </semantics></math> while <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>φ</mi> <mo>=</mo> <mi>θ</mi> <mo>=</mo> <msup> <mi>θ</mi> <mo>′</mo> </msup> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msup> <mi>φ</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>π</mi> </mrow> </semantics></math>. The Glauber-Sudarshan case (<math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <msup> <mi>β</mi> <mo>′</mo> </msup> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>) for <math display="inline"><semantics> <mrow> <mi>ν</mi> <mo>=</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math> is also plotted for comparison. The optimal <math display="inline"><semantics> <mi>β</mi> </semantics></math> depends on <math display="inline"><semantics> <mi>ν</mi> </semantics></math> and thus is not given by (<a href="#FD12-entropy-24-00220" class="html-disp-formula">12</a>) in the general case, since for <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <msup> <mi>β</mi> <mo>′</mo> </msup> <mo>&gt;</mo> <mn>4</mn> <mo>/</mo> <mn>5</mn> </mrow> </semantics></math> the Glauber-Sudarshan coherent states present a lower Helstrom bound than the PSK case. The optimal squeezed PSK states (<math display="inline"><semantics> <mrow> <mi>ν</mi> <mo>=</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math>) yields the lowest possible Helstrom bound (<math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <msup> <mi>β</mi> <mo>′</mo> </msup> <mo>=</mo> <mn>1</mn> <mo>/</mo> <mn>3</mn> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>).</p>
Full article ">Figure 5
<p>Helstrom bounds for two squeezed states distinguished by their energy channel shares (<math display="inline"><semantics> <mrow> <mi>ν</mi> <mo>=</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>ν</mi> <mo>=</mo> <mn>1</mn> <mo>/</mo> <mn>5</mn> </mrow> </semantics></math>) as functions of arbitrary squeezing parameters <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>,</mo> <msup> <mi>β</mi> <mo>′</mo> </msup> </mrow> </semantics></math>, with <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>φ</mi> <mo>=</mo> <mi>θ</mi> <mo>=</mo> <msup> <mi>θ</mi> <mo>′</mo> </msup> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msup> <mi>φ</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>π</mi> </mrow> </semantics></math>.</p>
Full article ">Figure 6
<p>Helstrom bounds for two squeezed states distinguished by their energy channel shares (<math display="inline"><semantics> <mrow> <mi>ν</mi> <mo>=</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>ν</mi> <mo>=</mo> <mn>4</mn> <mo>/</mo> <mn>5</mn> </mrow> </semantics></math>) as functions of arbitrary squeezing parameters <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>,</mo> <msup> <mi>β</mi> <mo>′</mo> </msup> </mrow> </semantics></math>, with <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>φ</mi> <mo>=</mo> <mi>θ</mi> <mo>=</mo> <msup> <mi>θ</mi> <mo>′</mo> </msup> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msup> <mi>φ</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>π</mi> </mrow> </semantics></math>.</p>
Full article ">
30 pages, 460 KiB  
Article
Differential Geometric Aspects of Parametric Estimation Theory for States on Finite-Dimensional C-Algebras
by Florio M. Ciaglia, Jürgen Jost and Lorenz Schwachhöfer
Entropy 2020, 22(11), 1332; https://doi.org/10.3390/e22111332 - 23 Nov 2020
Cited by 7 | Viewed by 2779
Abstract
A geometrical formulation of estimation theory for finite-dimensional C-algebras is presented. This formulation allows to deal with the classical and quantum case in a single, unifying mathematical framework. The derivation of the Cramer–Rao and Helstrom bounds for parametric statistical models with [...] Read more.
A geometrical formulation of estimation theory for finite-dimensional C-algebras is presented. This formulation allows to deal with the classical and quantum case in a single, unifying mathematical framework. The derivation of the Cramer–Rao and Helstrom bounds for parametric statistical models with discrete and finite outcome spaces is presented. Full article
(This article belongs to the Special Issue Quantum Statistical Decision and Estimation Theory)
7 pages, 341 KiB  
Article
Binary Communication with Gazeau–Klauder Coherent States
by Jerzy Dajka and Jerzy Łuczka
Entropy 2020, 22(2), 201; https://doi.org/10.3390/e22020201 - 10 Feb 2020
Cited by 4 | Viewed by 2385
Abstract
We investigate advantages and disadvantages of using Gazeau–Klauder coherent states for optical communication. In this short paper we show that using an alphabet consisting of coherent Gazeau–Klauder states related to a Kerr-type nonlinear oscillator instead of standard Perelomov coherent states results in lowering [...] Read more.
We investigate advantages and disadvantages of using Gazeau–Klauder coherent states for optical communication. In this short paper we show that using an alphabet consisting of coherent Gazeau–Klauder states related to a Kerr-type nonlinear oscillator instead of standard Perelomov coherent states results in lowering of the Helstrom bound for error probability in binary communication. We also discuss trace distance between Gazeau–Klauder coherent states and a standard coherent state as a quantifier of distinguishability of alphabets. Full article
Show Figures

Figure 1

Figure 1
<p>Trace distance between the Gazeau–Klauder coherent states <math display="inline"><semantics> <mrow> <mo>|</mo> <mi>J</mi> <mo>,</mo> <mn>0</mn> <mo>〉</mo> </mrow> </semantics></math> given by Equation (<a href="#FD3-entropy-22-00201" class="html-disp-formula">3</a>) and the Perelomov coherent states <math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> <mi>z</mi> <mo>=</mo> </mrow> <msqrt> <mi>J</mi> </msqrt> <mrow> <mo>〉</mo> </mrow> </mrow> </semantics></math> depicted for selected values of the rescaled susceptibility <math display="inline"><semantics> <mi>μ</mi> </semantics></math>.</p>
Full article ">Figure 2
<p>Helstrom bound <math display="inline"><semantics> <msub> <mi>P</mi> <mi>H</mi> </msub> </semantics></math> given by Equation (<a href="#FD19-entropy-22-00201" class="html-disp-formula">19</a>) depicted for selected values of <math display="inline"><semantics> <mi>μ</mi> </semantics></math>. For the sake of clarity, the range of <math display="inline"><semantics> <msub> <mi>P</mi> <mi>H</mi> </msub> </semantics></math> in the figure is limited to <math display="inline"><semantics> <mrow> <msub> <mi>P</mi> <mi>H</mi> </msub> <mo>≤</mo> <mn>0.1</mn> </mrow> </semantics></math>.</p>
Full article ">
4 pages, 1749 KiB  
Proceeding Paper
Squeezing-Enhanced Phase-Shift-Keyed Binary Communication in Noisy Channels
by Giovanni Chesi, Stefano Olivares and Matteo G. A. Paris
Proceedings 2019, 12(1), 58; https://doi.org/10.3390/proceedings2019012058 - 25 Nov 2019
Cited by 1 | Viewed by 995
Abstract
We address the use of squeezing in binary phase-shift-keyed (PSK) channels at fixed energy. In particular, we assess homodyne receivers against the Helstrom bound in the presence of phase noise. We also take into account possible imperfections in the generation of squeezing and [...] Read more.
We address the use of squeezing in binary phase-shift-keyed (PSK) channels at fixed energy. In particular, we assess homodyne receivers against the Helstrom bound in the presence of phase noise. We also take into account possible imperfections in the generation of squeezing and the effect of losses during propagation. We find that squeezing is a useful resource if its amplitude is below a given threshold depending on the energy of the signals and on the properties of the channel. Squeezing enhancement is present also when phase-noise becomes large. Full article
(This article belongs to the Proceedings of 11th Italian Quantum Information Science conference (IQIS2018))
Show Figures

Figure 1

Figure 1
<p>(<b>a</b>) The Helstrom bound for DSSs as function of the squeezing fraction <math display="inline"> <semantics> <mi>β</mi> </semantics> </math> and the channel energy <span class="html-italic">N</span>. The plane corresponds to the Helstrom bound for coherent states and the solid line to the threshold <math display="inline"> <semantics> <mrow> <msub> <mi>β</mi> <mi>th</mi> </msub> <mrow> <mo>(</mo> <mi>N</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math>. (<b>b</b>) The homodyne-detection error probability <math display="inline"> <semantics> <msub> <mi>P</mi> <mi>err</mi> </msub> </semantics> </math> as a function of <math display="inline"> <semantics> <mi>β</mi> </semantics> </math> for different values of the channel energy <span class="html-italic">N</span>. The plane corresponds to the minimum error probability achievable using only coherent states and homodyne detection and the solid line to the threshold <math display="inline"> <semantics> <mrow> <msub> <mi>β</mi> <mi>th</mi> </msub> <mrow> <mo>(</mo> <mi>N</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math>. Figures adapted from [<a href="#B9-proceedings-12-00058" class="html-bibr">9</a>].</p>
Full article ">Figure 2
<p>Comparison between the error probability <math display="inline"> <semantics> <msub> <mi>P</mi> <mi>err</mi> </msub> </semantics> </math> (solid lines) and the Helstrom bound (dashed lines) as functions of the noise parameter <math display="inline"> <semantics> <mi>σ</mi> </semantics> </math> for the DSS and the coherent state. We set <math display="inline"> <semantics> <mrow> <mi>β</mi> <mo>=</mo> <msub> <mi>β</mi> <mi>opt</mi> </msub> <mrow> <mo>(</mo> <mi>N</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>N</mi> <mo>/</mo> <mrow> <mo>(</mo> <mn>2</mn> <mi>N</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics> </math>. The shaded region refers to the range of the noise parameter values such that the homodyne probability with DSS is below the Helstrom bound with coherent states. Figures adapted from [<a href="#B9-proceedings-12-00058" class="html-bibr">9</a>].</p>
Full article ">Figure 3
<p>(<b>a</b>) Error probability <math display="inline"> <semantics> <msub> <mi>P</mi> <mi>err</mi> </msub> </semantics> </math> of the homodyne receiver as a function of <math display="inline"> <semantics> <mi>β</mi> </semantics> </math> and the purity <math display="inline"> <semantics> <mi>μ</mi> </semantics> </math> for different values of the noise parameter <math display="inline"> <semantics> <mi>σ</mi> </semantics> </math>. (<b>b</b>) Threshold value of the squeezing fraction <math display="inline"> <semantics> <msub> <mi>β</mi> <mi>th</mi> </msub> </semantics> </math> as a function of the purity <math display="inline"> <semantics> <mi>μ</mi> </semantics> </math> for different values of <math display="inline"> <semantics> <mi>σ</mi> </semantics> </math>. The shaded regions refer to the pairs of parameters <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>μ</mi> <mo>,</mo> <mi>β</mi> <mo>)</mo> </mrow> </semantics> </math> for which DSSs outperform coherent states. Note that <math display="inline"> <semantics> <mrow> <msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mi>N</mi> <mo>)</mo> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>≤</mo> <mi>μ</mi> <mo>≤</mo> <mn>1</mn> </mrow> </semantics> </math>. In both the panels we set <math display="inline"> <semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics> </math>. Figures adapted from [<a href="#B9-proceedings-12-00058" class="html-bibr">9</a>].</p>
Full article ">
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