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Quantum Entanglement

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Quantum Information".

Deadline for manuscript submissions: closed (17 July 2020) | Viewed by 31167

Special Issue Editors

Prof. Dr. Leong Chuan Kwek

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Guest Editor
Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, Singapore
Interests: foundation in quantum mechanics; quantum entanglement; photon–atom interactions; quantum synchronization; atomtronics; chip-based quantum cryptography; quantum devices; quantum metrology
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Marco Genovese

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Guest Editor
Istituto Nazionale di Ricerca Metrologica, Strada delle Cacce 91, 10135 Turin, Italy
Interests: experimental quantum; imaging metrology & sensing; quantum information processing; foundation in quantum mechanics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Quantum entanglement and, more in general, quantum correlations, represent a characteristic trait of quantum mechanics. In the recent years, quantum entanglement and its correlations have become a formidable tool for overcoming the classical limits in several fields, ranging from calculus and communication, to imaging and metrology.

We can quantify quantum entanglement and correlations in different ways. Some of these measures are hierarchically related each other. There have also been extensive applications of these measures to many fields, ranging from quantum optics to atomic and molecular physics.

This Special Issue hopes to present both theoretical and experimental works related to quantum entanglement and correlations. The Guest Editors welcome theoretical and experimental papers on all aspects of research on quantum correlations, ranging from purely abstract matter to commercial applications. Topics of interest include, but are not limited, to quantum technologies (including quantum information, quantum communication, quantum metrology and sensing, quantum imaging, and so forth), foundations of quantum mechanics, and new measures and applications of quantum correlations.

Prof. Leong Chuan Kwek
Prof. Marco Genovese
Guest Editors

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Research

15 pages, 1451 KiB  
Article
Strategies for Positive Partial Transpose (PPT) States in Quantum Metrologies with Noise
by Arunava Majumder, Harshank Shrotriya and Leong-Chuan Kwek
Entropy 2021, 23(6), 685; https://doi.org/10.3390/e23060685 - 28 May 2021
Viewed by 2853
Abstract
Quantum metrology overcomes standard precision limits and has the potential to play a key role in quantum sensing. Quantum mechanics, through the Heisenberg uncertainty principle, imposes limits on the precision of measurements. Conventional bounds to the measurement precision such as the shot noise [...] Read more.
Quantum metrology overcomes standard precision limits and has the potential to play a key role in quantum sensing. Quantum mechanics, through the Heisenberg uncertainty principle, imposes limits on the precision of measurements. Conventional bounds to the measurement precision such as the shot noise limit are not as fundamental as the Heisenberg limits, and can be beaten with quantum strategies that employ ‘quantum tricks’ such as squeezing and entanglement. Bipartite entangled quantum states with a positive partial transpose (PPT), i.e., PPT entangled states, are usually considered to be too weakly entangled for applications. Since no pure entanglement can be distilled from them, they are also called bound entangled states. We provide strategies, using which multipartite quantum states that have a positive partial transpose with respect to all bi-partitions of the particles can still outperform separable states in linear interferometers. Full article
(This article belongs to the Special Issue Quantum Entanglement)
►▼ Show Figures

Figure 1

Figure 1
<p>Classical (<b>A</b>) vs. Quantum strategy(Entangled assisted strategy) (<b>B</b>).</p> Full article ">Figure 2
<p>Two different strategies in which a state is imprinted with phase <math display="inline"><semantics> <mi>θ</mi> </semantics></math> for each qudit and undergoes (<b>A</b>) no noise, and (<b>B</b>) “global” noise.</p> Full article ">Figure 3
<p>Change of strategy from entangled assisted strategy to sequential ancilla assisted strategy where the second qudit acts as an ancilla. However the right one has two iterations and thus two phases are imprinted on the state, just as in the left one. The sequential strategy (with just two iterations) in the ancilla assisted case (<b>right</b>) gives a better QFI compared to the entangled assisted strategy with one iteration (<b>left</b>), the amount of resources being the same in the two strategies.</p> Full article ">Figure 4
<p>Two different strategies in which a state is imprinted with phase <math display="inline"><semantics> <mi>θ</mi> </semantics></math> in both the qudits and undergoes the two cases as in <a href="#entropy-23-00685-f002" class="html-fig">Figure 2</a> but now the output state at one stage is fed into the input for <span class="html-italic">m</span> iterations. The subfigure (<b>A</b>) shows iterations without noise while subfigure (<b>B</b>) involves noise after each iteration.</p> Full article ">Figure 5
<p>Two different strategies in which one qudit of a state is imprinted with phase <math display="inline"><semantics> <mi>θ</mi> </semantics></math> and undergoes the iterated cases as in <a href="#entropy-23-00685-f004" class="html-fig">Figure 4</a>. The (<b>A</b>) part is useless but the (<b>B</b>) with <span class="html-italic">m</span> = 2 is equivalent to <a href="#entropy-23-00685-f004" class="html-fig">Figure 4</a>A (<span class="html-italic">m</span> = 1) thus we only have advantage from <span class="html-italic">m</span> = 2 ∼ <a href="#entropy-23-00685-f004" class="html-fig">Figure 4</a>A (<span class="html-italic">m</span> = 1).</p> Full article ">Figure 6
<p>Quantum Fisher Information Vs Noise (P) plots (In each case blue, green and red are the 1st, 2nd and 3rd iterations respectively).</p> Full article ">Figure 6 Cont.
<p>Quantum Fisher Information Vs Noise (P) plots (In each case blue, green and red are the 1st, 2nd and 3rd iterations respectively).</p> Full article ">Figure 7
<p>This figure shows the comparison between the two strategies of <a href="#entropy-23-00685-f003" class="html-fig">Figure 3</a>. Dimension is varying along x-axis and QFI along y-axis. The Red bars correspond to the left strategy of <a href="#entropy-23-00685-f003" class="html-fig">Figure 3</a> and the blue ones correspond to the right one. Notice that the even dimensional states don’t provide any advantage of using sequential ancilla assisted strategy with two iterations instead of entangled assisted one with one iteration except <math display="inline"><semantics> <mrow> <mn>4</mn> <mo>×</mo> <mn>4</mn> </mrow> </semantics></math> dimensional optimal state (for <span class="html-italic">m</span> = 2).</p> Full article ">Figure 8
<p>One more iteration applied to both the cases of <a href="#entropy-23-00685-f003" class="html-fig">Figure 3</a>.</p> Full article ">Figure 9
<p>Comparison of two schemes in <a href="#entropy-23-00685-f003" class="html-fig">Figure 3</a> with multiple iterations for <math display="inline"><semantics> <mrow> <mn>3</mn> <mo>×</mo> <mn>3</mn> </mrow> </semantics></math> the PPT state using Equations (<a href="#FD22-entropy-23-00685" class="html-disp-formula">22</a>) and (<a href="#FD23-entropy-23-00685" class="html-disp-formula">23</a>) combined as they are considered to be equivalent.</p> Full article ">Figure 10
<p>Comparison of two schemes in <a href="#entropy-23-00685-f003" class="html-fig">Figure 3</a> with multiple iterations for <math display="inline"><semantics> <mrow> <mn>4</mn> <mo>×</mo> <mn>4</mn> </mrow> </semantics></math> the PPT state using Equations (<a href="#FD22-entropy-23-00685" class="html-disp-formula">22</a>) and (<a href="#FD23-entropy-23-00685" class="html-disp-formula">23</a>) combined as they are considered to be equivalent.</p> Full article ">
18 pages, 1488 KiB  
Article
Performance Improvement of Discretely Modulated Continuous-Variable Quantum Key Distribution with Untrusted Source via Heralded Hybrid Linear Amplifier
by Kunlin Zhou, Xuelin Wu, Yun Mao, Zhiya Chen, Qin Liao and Ying Guo
Entropy 2020, 22(8), 882; https://doi.org/10.3390/e22080882 - 12 Aug 2020
Viewed by 2623
Abstract
In practical quantum communication networks, the scheme of continuous-variable quantum key distribution (CVQKD) faces a challenge that the entangled source is controlled by a malicious eavesdropper, and although it still can generate a positive key rate and security, its performance needs to be [...] Read more.
In practical quantum communication networks, the scheme of continuous-variable quantum key distribution (CVQKD) faces a challenge that the entangled source is controlled by a malicious eavesdropper, and although it still can generate a positive key rate and security, its performance needs to be improved, especially in secret key rate and maximum transmission distance. In this paper, we proposed a method based on the four-state discrete modulation and a heralded hybrid linear amplifier to enhance the performance of CVQKD where the entangled source originates from malicious eavesdropper. The four-state CVQKD encodes information by nonorthogonal coherent states in phase space. It has better transmission distance than Gaussian modulation counterpart, especially at low signal-to-noise ratio (SNR). Moreover, the hybrid linear amplifier concatenates a deterministic linear amplifier (DLA) and a noiseless linear amplifier (NLA), which can improve the probability of amplification success and reduce the noise penalty caused by the measurement. Furthermore, the hybrid linear amplifier can raise the SNR of CVQKD and tune between two types of performance for high-gain mode and high noise-reduction mode, therefore it can extend the maximal transmission distance while the entangled source is untrusted. Full article
(This article belongs to the Special Issue Quantum Entanglement)
►▼ Show Figures

Figure 1

Figure 1
<p>(Color online) Schematic of the entangled source in middle (ESIM) CVQKD using a hybrid amplifier. In the entanglement-based model, Alice detects one of the EPR states by heterodyne detector and the hybrid linear amplifier is installed before Bob uses either the homodyne or heterodyne detector to measure the other half of EPR states. Eve’s attack consists of two entangling cloner attacks on either side of the source. The yellow box of <math display="inline"><semantics> <msup> <mi>g</mi> <mover accent="true"> <mi>n</mi> <mo stretchy="false">^</mo> </mover> </msup> </semantics></math> shows the hybrid linear amplifier.</p> Full article ">Figure 2
<p>(Color online) Four-state discrete modulation in phase space.</p> Full article ">Figure 3
<p>(Color online) An EPR state <math display="inline"><semantics> <mrow> <mo stretchy="false">|</mo> <mi>λ</mi> <mo stretchy="false">〉</mo> </mrow> </semantics></math> sent through a Gaussian quantum channel with transmittance <math display="inline"><semantics> <msub> <mi>T</mi> <mn>1</mn> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>T</mi> <mn>2</mn> </msub> </semantics></math>, excess noise <math display="inline"><semantics> <mi>ϵ</mi> </semantics></math>, and detection-added noise <math display="inline"><semantics> <mi>χ</mi> </semantics></math> has been replaced by an EPR state <math display="inline"><semantics> <mrow> <mo stretchy="false">|</mo> <mi>ζ</mi> <mo stretchy="false">〉</mo> </mrow> </semantics></math> sent through a Gaussian quantum channel with transmittance <math display="inline"><semantics> <msub> <mi>η</mi> <mi>a</mi> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>η</mi> <mn>2</mn> </msub> </semantics></math>, excess noise <math display="inline"><semantics> <msub> <mi>ϵ</mi> <mrow> <mi>g</mi> <mi>n</mi> </mrow> </msub> </semantics></math>, and detection-added noise <math display="inline"><semantics> <msub> <mi>χ</mi> <mrow> <mi>g</mi> <mi>d</mi> </mrow> </msub> </semantics></math>, but without the hybrid amplifier.</p> Full article ">Figure 4
<p>(Color online) The heralded hybrid linear amplifier is applied at Bob side. The mode <math display="inline"><semantics> <msub> <mi>B</mi> <mn>1</mn> </msub> </semantics></math> first goes through into a beam splitter with transmissivity <math display="inline"><semantics> <msub> <mi>T</mi> <mi>g</mi> </msub> </semantics></math>, then the reflected mode goes through into the MB-NLA concatenated by a dual-heterodyne detection and NLA. Here, the dual-heterodyne detection is used to measure the <span class="html-italic">X</span> and <span class="html-italic">P</span> quadrature of the reflected mode, respectively. After that we set a DLA and an elector-optic modulator (EOM) to dispose the amplified signal pulse and output mode <math display="inline"><semantics> <msub> <mi>B</mi> <mn>2</mn> </msub> </semantics></math> by a beam splitter with transmissivity 99:1.</p> Full article ">Figure 5
<p>(Color online) The optimal value range for <math display="inline"><semantics> <msub> <mi>g</mi> <mrow> <mi>N</mi> <mi>L</mi> <mi>A</mi> </mrow> </msub> </semantics></math>. The X-coordination and Y-coordination represent gain value of NLA and secret key rate, respectively. The figure shows that the proposed scheme with different detector (homodyne detector and heterodyne detector) can obtain relative higher secrete key rate in the public area 1. Here, the distance be set as 15, 20, and 25 km. Moreover, the fixed parameter values are set with <math display="inline"><semantics> <mrow> <msub> <mi>g</mi> <mi>D</mi> </msub> <mo>=</mo> <mn>12</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>ε</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>V</mi> <mi>M</mi> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p>(Color online) The optimal value range for <math display="inline"><semantics> <msub> <mi>g</mi> <mrow> <mi>N</mi> <mi>L</mi> <mi>A</mi> </mrow> </msub> </semantics></math>. The X-coordination and Y-coordination represent the gain value of NLA and secret key rate, respectively. The figure shows that the proposed scheme with different detectors (homodyne detector and heterodyne detector) can obtain relative higher secrete key rate in the public area 2. Here, the distance be set as 30, 35, 40, and 45 km. Moreover, the fixed parameter values are set with <math display="inline"><semantics> <mrow> <msub> <mi>g</mi> <mi>D</mi> </msub> <mo>=</mo> <mn>12</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>ε</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>V</mi> <mi>M</mi> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>.</p> Full article ">Figure 7
<p>(Color online) The optimal value range for <math display="inline"><semantics> <msub> <mi>g</mi> <mrow> <mi>N</mi> <mi>L</mi> <mi>A</mi> </mrow> </msub> </semantics></math>. The X-coordination and Y-coordination represent gain value of NLA and secret key rate, respectively. The figure shows that the proposed scheme with different detector (homodyne detector and heterodyne detector) can obtain relative higher secrete key rate in the public area 3. Here, the distance be set as 90, 95, and 100 km. Moreover, the fixed parameter values are set with <math display="inline"><semantics> <mrow> <msub> <mi>g</mi> <mi>D</mi> </msub> <mo>=</mo> <mn>12</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>ε</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>V</mi> <mi>M</mi> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>.</p> Full article ">Figure 8
<p>(Color online) Considering the above three public areas denoted in <a href="#entropy-22-00882-f005" class="html-fig">Figure 5</a>, <a href="#entropy-22-00882-f006" class="html-fig">Figure 6</a> and <a href="#entropy-22-00882-f007" class="html-fig">Figure 7</a>, we get the overlapping region in this figure, which represents the optimal gain value range of proposed scheme.</p> Full article ">Figure 9
<p>(Color online) The performance of proposed schemes for different <math display="inline"><semantics> <msub> <mi>L</mi> <mn>1</mn> </msub> </semantics></math>. The fixed parameters are set with <math display="inline"><semantics> <mrow> <msub> <mi>g</mi> <mi>N</mi> </msub> <mo>=</mo> <mn>8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>g</mi> <mi>D</mi> </msub> <mo>=</mo> <mn>12</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>ε</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>V</mi> <mi>M</mi> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>. Here, the light blue solid lines and red solid lines express the scheme without hybrid amplifier, moreover dark blue solid lines and orange lines represent the scheme with hybrid amplifier.</p> Full article ">Figure 10
<p>(Color online) The performance of schemes with four-state untested source via hybrid amplifier and heterodyne detection in different <math display="inline"><semantics> <msub> <mi>L</mi> <mn>1</mn> </msub> </semantics></math>. The fixed parameters are set with <math display="inline"><semantics> <msub> <mi>g</mi> <mi>N</mi> </msub> </semantics></math> = 8, <math display="inline"><semantics> <mrow> <msub> <mi>g</mi> <mi>D</mi> </msub> <mo>=</mo> <mn>12</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>ε</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>V</mi> <mi>M</mi> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>.</p> Full article ">Figure 11
<p>(Color online) The performance of schemes with four-state untested source via hybrid amplifier and homodyne detection in different <math display="inline"><semantics> <msub> <mi>L</mi> <mn>1</mn> </msub> </semantics></math>. The fixed parameters are set with <math display="inline"><semantics> <mrow> <msub> <mi>g</mi> <mi>N</mi> </msub> <mo>=</mo> <mn>8</mn> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <msub> <mi>g</mi> <mi>D</mi> </msub> <mo>=</mo> <mn>12</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>ε</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>V</mi> <mi>M</mi> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>.</p> Full article ">Figure 12
<p>(Color online) Panel (<b>a</b>) shows the relationship between the transmission distance and secret key rate. It demonstrates the maximal transmission distance for the scheme with Gaussian modulation. Here, the parameters are set as <math display="inline"><semantics> <mrow> <msub> <mi>g</mi> <mi>N</mi> </msub> <mo>=</mo> <mn>8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>g</mi> <mi>D</mi> </msub> <mo>=</mo> <mn>12</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>ε</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>V</mi> <mi>M</mi> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>. The dot-dash lines in the figure represent Gaussian modulation with an untested source. Furthermore, the solid lines represent Gaussian modulation with untested source via hybrid amplifier. Panel (<b>b</b>) also shows the relationship between transmission distance and secret key rate. Here, the parameters are also set as <math display="inline"><semantics> <mrow> <msub> <mi>g</mi> <mi>N</mi> </msub> <mo>=</mo> <mn>8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>g</mi> <mi>D</mi> </msub> <mo>=</mo> <mn>12</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>ε</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>V</mi> <mi>M</mi> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>. In panel (<b>b</b>), the green, dark blue, red, and light blue solid lines represent the Gaussian modulation with untested source via hybrid amplifier. Furthermore, the red dash line and yellow solid line represent the four-state discrete modulation with untested source. The red dot-dash line and yellow dot-dash line denote our proposed scheme (four-state discrete modulation with untested source via hybrid amplifier).</p> Full article ">
35 pages, 791 KiB  
Article
Wave–Particle–Entanglement–Ignorance Complementarity for General Bipartite Systems
by Wei Wu and Jin Wang
Entropy 2020, 22(8), 813; https://doi.org/10.3390/e22080813 - 24 Jul 2020
Cited by 4 | Viewed by 2624
Abstract
Wave–particle duality as the defining characteristic of quantum objects is a typical example of the principle of complementarity. The wave–particle–entanglement (WPE) complementarity, initially developed for two-qubit systems, is an extended form of complementarity that combines wave–particle duality with a previously missing ingredient, quantum [...] Read more.
Wave–particle duality as the defining characteristic of quantum objects is a typical example of the principle of complementarity. The wave–particle–entanglement (WPE) complementarity, initially developed for two-qubit systems, is an extended form of complementarity that combines wave–particle duality with a previously missing ingredient, quantum entanglement. For two-qubit systems in mixed states, the WPE complementarity was further completed by adding yet another piece that characterizes ignorance, forming the wave–particle–entanglement–ignorance (WPEI) complementarity. A general formulation of the WPEI complementarity can not only shed new light on fundamental problems in quantum mechanics, but can also have a wide range of experimental and practical applications in quantum-mechanical settings. The purpose of this study is to establish the WPEI complementarity for general multi-dimensional bipartite systems in pure or mixed states, and extend its range of applications to incorporate hierarchical and infinite-dimensional bipartite systems. The general formulation is facilitated by well-motivated generalizations of the relevant quantities. When faced with different directions of extensions to take, our guiding principle is that the formulated complementarity should be as simple and powerful as possible. We find that the generalized form of the WPEI complementarity contains unequal-weight averages reflecting the difference in the subsystem dimensions, and that the tangle, instead of the squared concurrence, serves as a more suitable entanglement measure in the general scenario. Two examples, a finite-dimensional bipartite system in mixed states and an infinite-dimensional bipartite system in pure states, are studied in detail to illustrate the general formalism. We also discuss our results in connection with some previous work. The WPEI complementarity for general finite-dimensional bipartite systems may be tested in multi-beam interference experiments, while the second example we studied may facilitate future experimental investigations on complementarity in infinite-dimensional bipartite systems. Full article
(This article belongs to the Special Issue Quantum Entanglement)
►▼ Show Figures

Figure 1

Figure 1
<p>Schematic two-beam interferometer. (<b>a</b>) two-beam interferometer without which-way detectors. The input beam is split into two beams by the beam splitter, phase-shifted by the phase shifters, and then recombined by the beam merger to produce the output beam; (<b>b</b>) two-beam interferometer with which-way detectors. Detectors are placed on the path of each beam to acquire which-way knowledge, adapted from Reference [<a href="#B5-entropy-22-00813" class="html-bibr">5</a>].</p> Full article ">Figure 2
<p>Schematic four-beam interferometer. Detectors are placed on the path of each beam to acquire which-way knowledge.</p> Full article ">Figure 3
<p>A schematic representation of the WPEI complementarity relations in general bipartite systems.</p> Full article ">Figure 4
<p>A schematic representation of the structure of a hierarchical bipartite system as a full binary tree.</p> Full article ">Figure 5
<p>The graphs of the purity <math display="inline"><semantics> <mi mathvariant="script">Q</mi> </semantics></math> and the concurrence <math display="inline"><semantics> <mi mathvariant="script">C</mi> </semantics></math> (divided by <math display="inline"><semantics> <msqrt> <mn>2</mn> </msqrt> </semantics></math>) as functions of <math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>z</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>z</mi> <mn>2</mn> </msub> <mrow> <mo>|</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p>The graphs of the predictability <math display="inline"><semantics> <mi mathvariant="script">P</mi> </semantics></math> and the visibility <math display="inline"><semantics> <mi mathvariant="script">V</mi> </semantics></math> as functions of <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>∈</mo> <mo>[</mo> <mo>−</mo> <mi>π</mi> <mo>,</mo> <mi>π</mi> <mo>]</mo> </mrow> </semantics></math> for the case with <math display="inline"><semantics> <mrow> <msub> <mi>z</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>z</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>e</mi> <mrow> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </mrow> </semantics></math>. The purity <math display="inline"><semantics> <mi mathvariant="script">Q</mi> </semantics></math> in this case is constant, with the value <math display="inline"><semantics> <mrow> <mi mathvariant="script">Q</mi> <mo>≈</mo> <mn>0.945</mn> </mrow> </semantics></math>.</p> Full article ">
24 pages, 10627 KiB  
Article
Quantum Correlation Dynamics in Controlled Two-Coupled-Qubit Systems
by Iulia Ghiu, Roberto Grimaudo, Tatiana Mihaescu, Aurelian Isar and Antonino Messina
Entropy 2020, 22(7), 785; https://doi.org/10.3390/e22070785 - 18 Jul 2020
Cited by 15 | Viewed by 3647
Abstract
We study and compare the time evolutions of concurrence and quantum discord in a driven system of two interacting qubits prepared in a generic Werner state. The corresponding quantum dynamics is exactly treated and manifests the appearance and disappearance of entanglement. Our analytical [...] Read more.
We study and compare the time evolutions of concurrence and quantum discord in a driven system of two interacting qubits prepared in a generic Werner state. The corresponding quantum dynamics is exactly treated and manifests the appearance and disappearance of entanglement. Our analytical treatment transparently unveils the physical reasons for the occurrence of such a phenomenon, relating it to the dynamical invariance of the X structure of the initial state. The quantum correlations which asymptotically emerge in the system are investigated in detail in terms of the time evolution of the fidelity of the initial Werner state. Full article
(This article belongs to the Special Issue Quantum Entanglement)
►▼ Show Figures

Figure 1

Figure 1
<p>Concurrence when the state at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> is the singlet state <math display="inline"><semantics> <mrow> <mrow> <mo stretchy="false">|</mo> <mspace width="0.166667em"/> </mrow> <msup> <mo>Ψ</mo> <mo>−</mo> </msup> <mrow> <mspace width="0.166667em"/> <mo>〉</mo> </mrow> </mrow> </semantics></math> in the Case 1 of Equation (<a href="#FD14-entropy-22-00785" class="html-disp-formula">14</a>)—<b>left</b>, and in the Case 2 of Equation (<a href="#FD15-entropy-22-00785" class="html-disp-formula">15</a>)—<b>right</b>.</p> Full article ">Figure 2
<p>Concurrence (left) and quantum discord (right) for the two-qubit system when the initial state is the Werner state (<a href="#FD17-entropy-22-00785" class="html-disp-formula">17</a>) in the Case 1 of Equation (<a href="#FD14-entropy-22-00785" class="html-disp-formula">14</a>) in terms of the parameter <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>∈</mo> <mo>[</mo> <mo>−</mo> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> <mo>,</mo> <mn>1</mn> <mo>]</mo> </mrow> </semantics></math> of the Werner state and <math display="inline"><semantics> <mrow> <msub> <mi>τ</mi> <mo>−</mo> </msub> <mo>=</mo> <mfrac> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msub> <mo>Γ</mo> <mo>−</mo> </msub> <mrow> <mo stretchy="false">|</mo> </mrow> </mrow> <mo>ℏ</mo> </mfrac> <mspace width="0.166667em"/> <mi>t</mi> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>Concurrence (left) and quantum discord (right) for the two-qubit system when the initial state is the Werner state (<a href="#FD17-entropy-22-00785" class="html-disp-formula">17</a>) in the Case 2 of Equation (<a href="#FD15-entropy-22-00785" class="html-disp-formula">15</a>) in terms of the parameter <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>∈</mo> <mo>[</mo> <mo>−</mo> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> <mo>,</mo> <mn>1</mn> <mo>]</mo> </mrow> </semantics></math> of the Werner state and <math display="inline"><semantics> <mrow> <msub> <mi>τ</mi> <mo>−</mo> </msub> <mo>=</mo> <mfrac> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msub> <mo>Γ</mo> <mo>−</mo> </msub> <mrow> <mo stretchy="false">|</mo> </mrow> </mrow> <mo>ℏ</mo> </mfrac> <mspace width="0.166667em"/> <mi>t</mi> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>Plot of <math display="inline"><semantics> <mi>α</mi> </semantics></math> in terms of <math display="inline"><semantics> <mrow> <mo stretchy="false">|</mo> <mi>μ</mi> <mo stretchy="false">|</mo> </mrow> </semantics></math> by using Equation (<a href="#FD30-entropy-22-00785" class="html-disp-formula">30</a>) for which the generalized Werner state <math display="inline"><semantics> <msubsup> <mi>η</mi> <mrow> <mi>μ</mi> <mo>,</mo> <mi>ν</mi> </mrow> <mrow> <mo>(</mo> <mi>α</mi> <mo>)</mo> </mrow> </msubsup> </semantics></math> is characterized by a vanishing concurrence.</p> Full article ">Figure 5
<p>Concurrence of the generalized Werner states <math display="inline"><semantics> <msubsup> <mi>η</mi> <mrow> <mi>μ</mi> <mo>,</mo> <mi>ν</mi> </mrow> <mrow> <mo>(</mo> <mi>α</mi> <mo>)</mo> </mrow> </msubsup> </semantics></math> in terms of <math display="inline"><semantics> <mi>α</mi> </semantics></math> and <math display="inline"><semantics> <mrow> <mo stretchy="false">|</mo> <mi>μ</mi> <mo stretchy="false">|</mo> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p>Concurrence (black, solid) and quantum discord (red, dashed) when the state at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> is the Werner state (<a href="#FD17-entropy-22-00785" class="html-disp-formula">17</a>) in the Case 1 of Equation (<a href="#FD14-entropy-22-00785" class="html-disp-formula">14</a>) in terms of <math display="inline"><semantics> <mrow> <msub> <mi>τ</mi> <mo>−</mo> </msub> <mo>=</mo> <mfrac> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msub> <mo>Γ</mo> <mo>−</mo> </msub> <mrow> <mo stretchy="false">|</mo> </mrow> </mrow> <mo>ℏ</mo> </mfrac> <mspace width="0.166667em"/> <mi>t</mi> </mrow> </semantics></math> for: (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.25</mn> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.55</mn> </mrow> </semantics></math>.</p> Full article ">Figure 7
<p>Concurrence (black, solid) and quantum discord (red, dashed) when the state at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> is the Werner state (<a href="#FD17-entropy-22-00785" class="html-disp-formula">17</a>) in the Case 2 of Equation (<a href="#FD15-entropy-22-00785" class="html-disp-formula">15</a>) in terms of <math display="inline"><semantics> <mrow> <msub> <mi>τ</mi> <mo>−</mo> </msub> <mo>=</mo> <mfrac> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msub> <mo>Γ</mo> <mo>−</mo> </msub> <mrow> <mo stretchy="false">|</mo> </mrow> </mrow> <mo>ℏ</mo> </mfrac> <mspace width="0.166667em"/> <mi>t</mi> </mrow> </semantics></math> for: (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.25</mn> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.55</mn> </mrow> </semantics></math>, (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.582</mn> </mrow> </semantics></math>, (<b>d</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics></math>.</p> Full article ">Figure 8
<p>Fidelity when the initial state is the singlet state, i.e., <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> in Case 1—<b>left</b> and in Case 2—<b>right</b>.</p> Full article ">Figure 9
<p>Fidelity versus dimensionless time <math display="inline"><semantics> <mrow> <msub> <mi>τ</mi> <mo>−</mo> </msub> <mo>=</mo> <mfrac> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msub> <mo>Γ</mo> <mo>−</mo> </msub> <mrow> <mo stretchy="false">|</mo> </mrow> </mrow> <mo>ℏ</mo> </mfrac> <mspace width="0.166667em"/> <mi>t</mi> </mrow> </semantics></math> when the initial state is the Werner state in Equation (<a href="#FD17-entropy-22-00785" class="html-disp-formula">17</a>) for <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.25</mn> </mrow> </semantics></math> (black, solid), <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.55</mn> </mrow> </semantics></math> (red, dashed), and <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics></math> (blue, dot-dashed) in Case 1—<b>left</b> and Case 2—<b>right</b>.</p> Full article ">Figure A1
<p>Concurrence (black, solid) and quantum discord (red, dashed) when the initial state is the Werner state (<a href="#FD17-entropy-22-00785" class="html-disp-formula">17</a>) in the case of constant fields, when <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, for: (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.55</mn> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.75</mn> </mrow> </semantics></math>, (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics></math>.</p> Full article ">Figure A2
<p>Concurrence (black, solid) and quantum discord (red, dashed) when the initial state is the Werner state (<a href="#FD17-entropy-22-00785" class="html-disp-formula">17</a>) in the case of constant fields, when <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, for: (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.55</mn> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.75</mn> </mrow> </semantics></math>, (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics></math>.</p> Full article ">
12 pages, 623 KiB  
Article
Nonclassical Effects Based on Husimi Distributions in Two Open Cavities Linked by an Optical Waveguide
by Abdel-Baset A. Mohamed and Hichem Eleuch
Entropy 2020, 22(7), 767; https://doi.org/10.3390/e22070767 - 13 Jul 2020
Cited by 3 | Viewed by 2096
Abstract
Nonclassical effects are investigated in a system formed by two quantum wells, each of which is inside an open cavity. The cavities are spatially separated, linked by a fiber, and filled with a linear optical medium. Based on Husimi distributions (HDs) and Wehrl [...] Read more.
Nonclassical effects are investigated in a system formed by two quantum wells, each of which is inside an open cavity. The cavities are spatially separated, linked by a fiber, and filled with a linear optical medium. Based on Husimi distributions (HDs) and Wehrl entropy, we explore the effects of the physical parameters on the generation and the robustness of the mixedness and HD information in the phase space. The generated quantum coherence and the HD information depend crucially on the cavity-exciton and fiber cavity couplings as well as on the optical medium density. The HD information and purity are lost due to the dissipation. This loss may be inhibited by increasing the optical susceptibility as well as the couplings of the exciton-cavity and the fiber-cavity. These parameters control the regularity, amplitudes, and frequencies of the generated mixedness. Full article
(This article belongs to the Special Issue Quantum Entanglement)
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Figure 1

Figure 1
<p>Husimi distribution <math display="inline"><semantics> <mrow> <mi>H</mi> <mo>(</mo> <mi>θ</mi> <mo>,</mo> <mi>ϕ</mi> <mo>)</mo> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mo>|</mo> <mn>110</mn> <mo>〉</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>ϵ</mi> <mo>=</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math> for the cases <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <msub> <mi>λ</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>χ</mi> <mo>,</mo> <mi>κ</mi> <mo>,</mo> <msub> <mi>t</mi> <mi>p</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mo>.</mo> <mn>25</mn> <mi>π</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<b>a</b>), <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <msub> <mi>λ</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>χ</mi> <mo>,</mo> <mi>κ</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>t</mi> <mi>m</mi> </msub> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<b>b</b>), <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <msub> <mi>λ</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>χ</mi> <mo>,</mo> <mi>κ</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>t</mi> <mi>m</mi> </msub> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<b>c</b>) and <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <msub> <mi>λ</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>χ</mi> <mo>,</mo> <mi>κ</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>.</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>t</mi> <mi>m</mi> </msub> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<b>d</b>).</p> Full article ">Figure 2
<p>Husimi distribution <math display="inline"><semantics> <mrow> <mi>H</mi> <mo>(</mo> <mi>θ</mi> <mo>)</mo> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <msub> <mi>t</mi> <mi>m</mi> </msub> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <msub> <mi>λ</mi> <mi>A</mi> </msub> <mo>,</mo> <msub> <mi>λ</mi> <mi>B</mi> </msub> <mo>,</mo> <msub> <mi>λ</mi> <mi>f</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>.</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<b>a</b>) and <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <msub> <mi>λ</mi> <mi>A</mi> </msub> <mo>,</mo> <msub> <mi>λ</mi> <mi>B</mi> </msub> <mo>,</mo> <msub> <mi>λ</mi> <mi>f</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mn>0</mn> <mo>.</mo> <mn>2</mn> <mo>,</mo> <mn>0</mn> <mo>.</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<b>b</b>) for the cases: <math display="inline"><semantics> <mrow> <mi>χ</mi> <mo>=</mo> <mi>κ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> (solid curves), <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>χ</mi> <mo>,</mo> <mi>κ</mi> <mo>)</mo> <mo>=</mo> <mo>(</mo> <mn>2</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </semantics></math> (dashed curves), <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>χ</mi> <mo>,</mo> <mi>κ</mi> <mo>)</mo> <mo>=</mo> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>.</mo> <mn>025</mn> <mo>)</mo> </mrow> </semantics></math> (dash-dot curves) and <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>χ</mi> <mo>,</mo> <mi>κ</mi> <mo>)</mo> <mo>=</mo> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>.</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math> (doted curves).</p> Full article ">Figure 3
<p>The Wehrl entropy (WE) for <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <msub> <mi>λ</mi> <mi>A</mi> </msub> <mo>,</mo> <msub> <mi>λ</mi> <mi>B</mi> </msub> <mo>,</mo> <mi>ϵ</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mo>)</mo> </mrow> </mrow> </semantics></math> for the different cases: <math display="inline"><semantics> <mrow> <mi>χ</mi> <mo>=</mo> <mi>κ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> (solid curves), <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>χ</mi> <mo>,</mo> <mi>κ</mi> <mo>)</mo> <mo>=</mo> <mo>(</mo> <mn>2</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </semantics></math> (dash curves), <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>χ</mi> <mo>,</mo> <mi>κ</mi> <mo>)</mo> <mo>=</mo> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>.</mo> <mn>025</mn> <mo>)</mo> </mrow> </semantics></math> (dashed-doted curves) and <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>χ</mi> <mo>,</mo> <mi>κ</mi> <mo>)</mo> <mo>=</mo> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>.</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math> (doted curves) for <math display="inline"><semantics> <mrow> <msub> <mi>λ</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> in (<b>a</b>) and <math display="inline"><semantics> <mrow> <msub> <mi>λ</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>2</mn> </mrow> </semantics></math> in (<b>b</b>).</p> Full article ">
9 pages, 275 KiB  
Article
In Praise of Quantum Uncertainty
by Eliahu Cohen and Avishy Carmi
Entropy 2020, 22(3), 302; https://doi.org/10.3390/e22030302 - 6 Mar 2020
Cited by 7 | Viewed by 3553
Abstract
Quantum uncertainty has a tremendous explanatory power. Coherent superposition, quantum equations of motion, entanglement, nonlocal correlations, dynamical nonlocality, contextuality, discord, counterfactual protocols, weak measurements, quantization itself, and even preservation of causality can be traced back to quantum uncertainty. We revisit and extend our [...] Read more.
Quantum uncertainty has a tremendous explanatory power. Coherent superposition, quantum equations of motion, entanglement, nonlocal correlations, dynamical nonlocality, contextuality, discord, counterfactual protocols, weak measurements, quantization itself, and even preservation of causality can be traced back to quantum uncertainty. We revisit and extend our previous works, as well as some other works of the community, in order to account for the above claims. Special emphasis is given to the connection between uncertainty and nonlocality, two notions which evolved quite independently and may seem distinct but, in fact, are tightly related. Indeterminism, or more precisely, locally consistent indeterminism, should be understood as the enabler of most quantum phenomena (and possibly all of them). Full article
(This article belongs to the Special Issue Quantum Entanglement)
9 pages, 244 KiB  
Article
A Private Quantum Bit String Commitment
by Mariana Gama, Paulo Mateus and André Souto
Entropy 2020, 22(3), 272; https://doi.org/10.3390/e22030272 - 27 Feb 2020
Cited by 2 | Viewed by 3023
Abstract
We propose an entanglement-based quantum bit string commitment protocol whose composability is proven in the random oracle model. This protocol has the additional property of preserving the privacy of the committed message. Even though this property is not resilient against man-in-the-middle attacks, this [...] Read more.
We propose an entanglement-based quantum bit string commitment protocol whose composability is proven in the random oracle model. This protocol has the additional property of preserving the privacy of the committed message. Even though this property is not resilient against man-in-the-middle attacks, this threat can be circumvented by considering that the parties communicate through an authenticated channel. The protocol remains secure and private (but not composable) if we realize the random oracles as physical unclonable functions (PUFs) in the so-called bad PUF model. Full article
(This article belongs to the Special Issue Quantum Entanglement)
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Figure 1

Figure 1
<p>Commitment functionality.</p> Full article ">Figure 2
<p>EPR pair source functionality.</p> Full article ">Figure 3
<p>Random oracle functionality.</p> Full article ">Figure 4
<p>Conditions for the constructability of the resource <math display="inline"> <semantics> <msub> <mi mathvariant="script">F</mi> <mrow> <mi>C</mi> <mi>O</mi> <mi>M</mi> </mrow> </msub> </semantics> </math> from the resources <math display="inline"> <semantics> <msub> <mi mathvariant="script">F</mi> <mrow> <mi>E</mi> <mi>P</mi> <mi>R</mi> </mrow> </msub> </semantics> </math> and <math display="inline"> <semantics> <msub> <mi mathvariant="script">F</mi> <mrow> <mi>R</mi> <mi>O</mi> </mrow> </msub> </semantics> </math> (<b>a</b>) corresponds to the soundness property by showing the equivalence between the ideal commitment functionality <math display="inline"> <semantics> <msub> <mi mathvariant="script">F</mi> <mrow> <mi>C</mi> <mi>O</mi> <mi>M</mi> </mrow> </msub> </semantics> </math> and the protocol for honest parties (Alice and Bob behave according to <math display="inline"> <semantics> <msub> <mi>π</mi> <mi>A</mi> </msub> </semantics> </math> and <math display="inline"> <semantics> <msub> <mi>π</mi> <mi>B</mi> </msub> </semantics> </math>, respectively); (<b>b</b>,<b>c</b>) correspond to security against dishonest Bob and Alice, respectively. Since the algorithm they follow is unknown, <math display="inline"> <semantics> <msub> <mi>π</mi> <mi>A</mi> </msub> </semantics> </math> and <math display="inline"> <semantics> <msub> <mi>π</mi> <mi>B</mi> </msub> </semantics> </math> are removed from the respective real system, while the simulators <math display="inline"> <semantics> <msub> <mi>σ</mi> <mi>A</mi> </msub> </semantics> </math> and <math display="inline"> <semantics> <msub> <mi>σ</mi> <mi>B</mi> </msub> </semantics> </math> are respectively added to the ideal system.</p> Full article ">
29 pages, 1577 KiB  
Article
Theory of Quantum Path Entanglement and Interference with Multiplane Diffraction of Classical Light Sources
by Burhan Gulbahar
Entropy 2020, 22(2), 246; https://doi.org/10.3390/e22020246 - 21 Feb 2020
Cited by 2 | Viewed by 3955
Abstract
Quantum history states were recently formulated by extending the consistent histories approach of Griffiths to the entangled superposition of evolution paths and were then experimented with Greenberger–Horne–Zeilinger states. Tensor product structure of history-dependent correlations was also recently exploited as a quantum computing resource [...] Read more.
Quantum history states were recently formulated by extending the consistent histories approach of Griffiths to the entangled superposition of evolution paths and were then experimented with Greenberger–Horne–Zeilinger states. Tensor product structure of history-dependent correlations was also recently exploited as a quantum computing resource in simple linear optical setups performing multiplane diffraction (MPD) of fermionic and bosonic particles with remarkable promises. This significantly motivates the definition of quantum histories of MPD as entanglement resources with the inherent capability of generating an exponentially increasing number of Feynman paths through diffraction planes in a scalable manner and experimental low complexity combining the utilization of coherent light sources and photon-counting detection. In this article, quantum temporal correlation and interference among MPD paths are denoted with quantum path entanglement (QPE) and interference (QPI), respectively, as novel quantum resources. Operator theory modeling of QPE and counterintuitive properties of QPI are presented by combining history-based formulations with Feynman’s path integral approach. Leggett–Garg inequality as temporal analog of Bell’s inequality is violated for MPD with all signaling constraints in the ambiguous form recently formulated by Emary. The proposed theory for MPD-based histories is highly promising for exploiting QPE and QPI as important resources for quantum computation and communications in future architectures. Full article
(This article belongs to the Special Issue Quantum Entanglement)
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Graphical abstract

Graphical abstract
Full article ">Figure 1
<p>(<b>a</b>) System model of the free propagating light with velocity <span class="html-italic">c</span> in the <span class="html-italic">z</span>-direction and MPD through <span class="html-italic">N</span> planes, where <span class="html-italic">j</span>th plane includes <math display="inline"><semantics> <msub> <mi>S</mi> <mi>j</mi> </msub> </semantics></math> slits at positions <math display="inline"><semantics> <msub> <mi>X</mi> <mrow> <mi>j</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>i</mi> <mo>∈</mo> <mo>[</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>S</mi> <mi>j</mi> </msub> <mo>]</mo> </mrow> </semantics></math> and interplane distance of <math display="inline"><semantics> <msub> <mi>L</mi> <mrow> <mi>j</mi> <mo>,</mo> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </semantics></math>. (<b>b</b>) Example of three plane diffractions (<math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>) with two slits for the first and second planes showing all the possible seven types of histories composed of diffractions or projections <math display="inline"><semantics> <msub> <mi>P</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>P</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>P</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </semantics></math>, and <math display="inline"><semantics> <msub> <mi>P</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </semantics></math> through slits and measurements <math display="inline"><semantics> <msub> <mi>M</mi> <mn>1</mn> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>M</mi> <mn>2</mn> </msub> </semantics></math>, and <math display="inline"><semantics> <msub> <mi>M</mi> <mn>3</mn> </msub> </semantics></math> on the planes. There are <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mi>p</mi> </msub> <mo>≡</mo> <msubsup> <mo>∏</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi>S</mi> <mi>j</mi> </msub> <mo>=</mo> <mn>2</mn> <mo>×</mo> <mn>2</mn> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> paths detected on the third plane.</p> Full article ">Figure 2
<p>(<b>a</b>) The violation of Leggett–Garg Inequality (LGI) with the setup of two planes with triple slits where the event set at time <math display="inline"><semantics> <msub> <mi>t</mi> <mn>1</mn> </msub> </semantics></math> is <math display="inline"><semantics> <mfenced separators="" open="[" close="]"> <msub> <mi>P</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mfenced> </semantics></math>, <math display="inline"><semantics> <mfenced separators="" open="[" close="]"> <msub> <mi>P</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mfenced> </semantics></math>, and <math display="inline"><semantics> <mfenced separators="" open="[" close="]"> <msub> <mi>P</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>3</mn> </mrow> </msub> </mfenced> </semantics></math> and, at time <math display="inline"><semantics> <msub> <mi>t</mi> <mn>2</mn> </msub> </semantics></math>, are <math display="inline"><semantics> <mfenced separators="" open="[" close="]"> <msub> <mi>P</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mfenced> </semantics></math>, <math display="inline"><semantics> <mfenced separators="" open="[" close="]"> <msub> <mi>P</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mfenced> </semantics></math>, <math display="inline"><semantics> <mfenced separators="" open="[" close="]"> <msub> <mi>P</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>3</mn> </mrow> </msub> </mfenced> </semantics></math>, and <math display="inline"><semantics> <mfenced separators="" open="[" close="]"> <msub> <mi>M</mi> <mn>2</mn> </msub> </mfenced> </semantics></math> and ambiguous measurement setups by closing (<b>b</b>) the third, (<b>c</b>) the second, and (<b>d</b>) the first slits on the first plane.</p> Full article ">Figure 3
<p>Setup for constructive and destructive interferences in time for the probabilities to diffract through each plane showing the history states (<b>a</b>) <math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msubsup> <mo>Ψ</mo> <mrow> <mn>3</mn> </mrow> <mi>a</mi> </msubsup> <mrow> <mo stretchy="false">)</mo> </mrow> <mo>≡</mo> <mfenced separators="" open="[" close="]"> <msub> <mi mathvariant="bold">P</mi> <mrow> <mn>3</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mfenced> <mo>⊙</mo> <mfenced separators="" open="[" close="]"> <msub> <mi mathvariant="bold">P</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mfenced> <mo>⊙</mo> <mfenced separators="" open="(" close=")"> <mfenced separators="" open="[" close="]"> <msub> <mi mathvariant="bold">P</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mfenced> <mo>+</mo> <mfenced separators="" open="[" close="]"> <msub> <mi mathvariant="bold">P</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mfenced> </mfenced> <mo>⊙</mo> <mfenced separators="" open="[" close="]"> <msub> <mi>ρ</mi> <mn>0</mn> </msub> </mfenced> </mrow> </semantics></math> as the superposition of <math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msubsup> <mo>Ψ</mo> <mrow> <mn>3</mn> </mrow> <mi>b</mi> </msubsup> <mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msubsup> <mo>Ψ</mo> <mrow> <mn>3</mn> </mrow> <mi>c</mi> </msubsup> <mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msubsup> <mo>Ψ</mo> <mrow> <mn>3</mn> </mrow> <mi>b</mi> </msubsup> <mrow> <mo stretchy="false">)</mo> </mrow> <mo>≡</mo> <mfenced separators="" open="[" close="]"> <msub> <mi mathvariant="bold">P</mi> <mrow> <mn>3</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mfenced> <mo>⊙</mo> <mfenced separators="" open="[" close="]"> <msub> <mi mathvariant="bold">P</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mfenced> <mo>⊙</mo> <mfenced separators="" open="[" close="]"> <msub> <mi mathvariant="bold">P</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mfenced> <mo>⊙</mo> <mfenced separators="" open="[" close="]"> <msub> <mi>ρ</mi> <mn>0</mn> </msub> </mfenced> </mrow> </semantics></math>, and (<b>c</b>) <math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msubsup> <mo>Ψ</mo> <mrow> <mn>3</mn> </mrow> <mi>c</mi> </msubsup> <mrow> <mo stretchy="false">)</mo> </mrow> <mo>≡</mo> <mfenced separators="" open="[" close="]"> <msub> <mi mathvariant="bold">P</mi> <mrow> <mn>3</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mfenced> <mo>⊙</mo> <mfenced separators="" open="[" close="]"> <msub> <mi mathvariant="bold">P</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mfenced> <mo>⊙</mo> <mfenced separators="" open="[" close="]"> <msub> <mi mathvariant="bold">P</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mfenced> <mo>⊙</mo> <mfenced separators="" open="[" close="]"> <msub> <mi>ρ</mi> <mn>0</mn> </msub> </mfenced> </mrow> </semantics></math>. The targeted scenario with <span class="html-italic">classically counterintuitive</span> nature where a specific example of interference pattern (represented as the number of lambs denoting the number of photons for a practical counting experiment) for the cases of (<b>d</b>) two slits on PL-1 both open and (<b>e</b>) only the second slit open. The operation of closing the first slit decreases the number of photons diffracted through PL-2 while counterintuitively increases the number of photons through PL-3 since we classically expect a decrease. This scenario shows the interference of histories at two different time instants for PL-2 and PL-3 with firstly constructive and then destructive effects, respectively.</p> Full article ">Figure 4
<p>The layouts used in (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>i</mi> <msub> <mi>m</mi> <mn>1</mn> </msub> </mrow> </semantics></math> and (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>i</mi> <msub> <mi>m</mi> <mn>2</mn> </msub> </mrow> </semantics></math>, where for <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>i</mi> <msub> <mi>m</mi> <mn>2</mn> </msub> </mrow> </semantics></math>, the fixed values of the parameters are <math display="inline"><semantics> <mrow> <msub> <mi>σ</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>200</mn> </mrow> </semantics></math> (<math display="inline"><semantics> <mi mathvariant="sans-serif">μ</mi> </semantics></math>m), <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mn>01</mn> </msub> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> (ns), <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mn>12</mn> </msub> <mo>=</mo> <mn>0.2</mn> </mrow> </semantics></math> (ns), <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mn>23</mn> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math> (ns), <math display="inline"><semantics> <mrow> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>25</mn> </mrow> </semantics></math> (<math display="inline"><semantics> <mi mathvariant="sans-serif">μ</mi> </semantics></math>m), <math display="inline"><semantics> <mrow> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>35</mn> </mrow> </semantics></math> (<math display="inline"><semantics> <mi mathvariant="sans-serif">μ</mi> </semantics></math>m), and <math display="inline"><semantics> <mrow> <msub> <mi>β</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>45</mn> </mrow> </semantics></math> (<math display="inline"><semantics> <mi mathvariant="sans-serif">μ</mi> </semantics></math>m) in addition to the fixed values of the slit positions on the first plane. The practical measurement setups to be utilized in future experiments are illustrated for the probabilities (<b>c</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mrow> <mo stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">}</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math> and (<b>d</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mrow> <mo stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">}</mo> </mrow> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math>. The measurement planes count the detected number of photons compared with the number of photons emitted by the source in unit time.</p> Full article ">Figure 5
<p>(<b>a</b>) LGI violation (<math display="inline"><semantics> <mrow> <msub> <mi>K</mi> <mi>A</mi> </msub> <mspace width="0.166667em"/> <mo>−</mo> <mspace width="0.166667em"/> <msub> <mi>K</mi> <mi>V</mi> </msub> </mrow> </semantics></math>) and signaling (<math display="inline"><semantics> <mrow> <msub> <mi>K</mi> <mi>V</mi> </msub> <mspace width="0.166667em"/> <mo>−</mo> <mn>1</mn> <mspace width="0.166667em"/> </mrow> </semantics></math>) for varying <math display="inline"><semantics> <msub> <mi>D</mi> <mi>s</mi> </msub> </semantics></math>, where <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mn>01</mn> </msub> <mo>=</mo> <mn>0.2</mn> </mrow> </semantics></math> (ns), <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mn>12</mn> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math> (ns), <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>x</mi> <mo>=</mo> <mn>7</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>15</mn> </mrow> </semantics></math> (<math display="inline"><semantics> <mi mathvariant="sans-serif">μ</mi> </semantics></math>m), <math display="inline"><semantics> <mrow> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>30</mn> </mrow> </semantics></math> (<math display="inline"><semantics> <mi mathvariant="sans-serif">μ</mi> </semantics></math>m), and <math display="inline"><semantics> <mrow> <msub> <mi>σ</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>130</mn> </mrow> </semantics></math> (<math display="inline"><semantics> <mi mathvariant="sans-serif">μ</mi> </semantics></math>m) and (<b>b</b>) the corresponding dichotomic sign assignments for ambiguous measurements maximizing the violation for each <math display="inline"><semantics> <msub> <mi>D</mi> <mi>s</mi> </msub> </semantics></math>.</p> Full article ">Figure 6
<p>(<b>a</b>) Maximum LGI violation (<math display="inline"><semantics> <mrow> <msub> <mi>K</mi> <mi>A</mi> </msub> <mo>−</mo> <msub> <mi>K</mi> <mi>V</mi> </msub> </mrow> </semantics></math>) and the corresponding amount of signaling (<math display="inline"><semantics> <mrow> <msub> <mi>K</mi> <mi>V</mi> </msub> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math>) for varying <math display="inline"><semantics> <msub> <mi>σ</mi> <mn>0</mn> </msub> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>x</mi> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mn>01</mn> </msub> <mo>=</mo> <msub> <mi>t</mi> <mn>12</mn> </msub> </mrow> </semantics></math> and (<b>b</b>) the corresponding values of <math display="inline"><semantics> <msub> <mi>β</mi> <mn>1</mn> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>β</mi> <mn>2</mn> </msub> </semantics></math>, and <math display="inline"><semantics> <msub> <mi>D</mi> <mi>s</mi> </msub> </semantics></math> maximizing the violation for each <math display="inline"><semantics> <msub> <mi>σ</mi> <mn>0</mn> </msub> </semantics></math> assuming fully coherent sources. Maximum violation for varying <math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </semantics></math> pairs for fully coherent sources where (<b>c</b>) <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>x</mi> <mo>=</mo> <mn>7</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mn>01</mn> </msub> <mo>=</mo> <msub> <mi>t</mi> <mn>12</mn> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math> (ns) at the maximizing <math display="inline"><semantics> <mrow> <msub> <mi>σ</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>30</mn> </mrow> </semantics></math> (<math display="inline"><semantics> <mi mathvariant="sans-serif">μ</mi> </semantics></math>m), (<b>d</b>) <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>x</mi> <mo>=</mo> <mn>7</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mn>01</mn> </msub> <mo>=</mo> <msub> <mi>t</mi> <mn>12</mn> </msub> <mo>=</mo> <mn>0.2</mn> </mrow> </semantics></math> (ns) at <math display="inline"><semantics> <mrow> <msub> <mi>σ</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>230</mn> </mrow> </semantics></math> (<math display="inline"><semantics> <mi mathvariant="sans-serif">μ</mi> </semantics></math>m), and (<b>e</b>) <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>x</mi> <mo>=</mo> <mn>11</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mn>01</mn> </msub> <mo>=</mo> <msub> <mi>t</mi> <mn>12</mn> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math> (ns) at <math display="inline"><semantics> <mrow> <msub> <mi>σ</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>150</mn> </mrow> </semantics></math> (<math display="inline"><semantics> <mi mathvariant="sans-serif">μ</mi> </semantics></math>m). It is observed that there is a large set of slit pairs and beam width resulting in LGI violation reaching <math display="inline"><semantics> <mrow> <mo>≈</mo> <mn>0.4</mn> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>x</mi> <mo>=</mo> <mn>7</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>≈</mo> <mn>0.23</mn> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>x</mi> <mo>=</mo> <mn>11</mn> </mrow> </semantics></math>, respectively, while there are local peaks for <math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </semantics></math> pairs for all cases. Increasing <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mn>01</mn> </msub> <mo>,</mo> <msub> <mi>t</mi> <mn>12</mn> </msub> </mrow> </semantics></math> values expands the <math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </semantics></math> pairs for similar values of violations. (<b>f</b>) The comparison of the spatial coherence diameters <math display="inline"><semantics> <msub> <mi>D</mi> <mi>c</mi> </msub> </semantics></math> with the diffraction setup diameters <math display="inline"><semantics> <msub> <mi>D</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>D</mi> <mn>2</mn> </msub> </semantics></math> for the first and second planes, respectively, where the targeted case is <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>x</mi> <mo>=</mo> <mn>11</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mn>01</mn> </msub> <mo>=</mo> <msub> <mi>t</mi> <mn>12</mn> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math> (ns), i.e., analyzed as the red curve in <a href="#entropy-22-00246-f006" class="html-fig">Figure 6</a>a, and (<b>g</b>) the corresponding LGI violation curve plotted again by emphasizing the coherence including the peak points.</p> Full article ">Figure 7
<p>(<b>a</b>) <math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>ψ</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>0</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mspace width="0.166667em"/> <mo>+</mo> <mspace width="0.166667em"/> <msub> <mi>ψ</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <msup> <mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> </mrow> </semantics></math> compared with <math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>ψ</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <msup> <mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>ψ</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>0</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <msup> <mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> </mrow> </semantics></math> for diffraction through the layer PL-2, (<b>b</b>) <math display="inline"><semantics> <mrow> <munder> <mo movablelimits="false" form="prefix">max</mo> <msub> <mi>x</mi> <mn>3</mn> </msub> </munder> <mfenced separators="" open="{" close="}"> <mrow> <mo>|</mo> </mrow> <msub> <mi>ψ</mi> <mrow> <mn>3</mn> <mo>,</mo> <mn>0</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>3</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mspace width="0.166667em"/> <mo>+</mo> <mspace width="0.166667em"/> <msub> <mi>ψ</mi> <mrow> <mn>3</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>3</mn> </msub> <mo stretchy="false">)</mo> </mrow> <msup> <mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> <mo>−</mo> <msup> <mrow> <mo>|</mo> <msub> <mi>ψ</mi> <mrow> <mn>3</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>3</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> </mfenced> </mrow> </semantics></math> for varying <math display="inline"><semantics> <msub> <mi>X</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </semantics></math> on PL-3 such that destructive interference is maximized for each <math display="inline"><semantics> <msub> <mi>X</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </semantics></math> with respect to <math display="inline"><semantics> <msub> <mi>x</mi> <mn>3</mn> </msub> </semantics></math> while <math display="inline"><semantics> <mrow> <msub> <mi>X</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>≈</mo> <mn>140</mn> <mspace width="0.166667em"/> <mi mathvariant="sans-serif">μ</mi> </mrow> </semantics></math>m maximizes the destructive interference, (<b>c</b>) <math display="inline"><semantics> <msub> <mi>X</mi> <mrow> <mn>3</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </semantics></math> maximizing the destructive interference for varying <math display="inline"><semantics> <msub> <mi>X</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </semantics></math>, (<b>d</b>) the comparison of <math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>ψ</mi> <mrow> <mn>3</mn> <mo>,</mo> <mn>0</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>3</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mspace width="0.166667em"/> <mo>+</mo> <mspace width="0.166667em"/> <msub> <mi>ψ</mi> <mrow> <mn>3</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>3</mn> </msub> <mo stretchy="false">)</mo> </mrow> <msup> <mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>ψ</mi> <mrow> <mn>3</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>3</mn> </msub> <mo stretchy="false">)</mo> </mrow> <msup> <mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> </mrow> </semantics></math> on PL-3 for specific <math display="inline"><semantics> <mrow> <msub> <mi>X</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>≈</mo> <mn>140</mn> <mspace width="0.166667em"/> <mi mathvariant="sans-serif">μ</mi> </mrow> </semantics></math>m showing the destructive interference maximized with <math display="inline"><semantics> <mrow> <msub> <mi>X</mi> <mrow> <mn>3</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>≈</mo> <mn>143</mn> <mspace width="0.166667em"/> <mi mathvariant="sans-serif">μ</mi> </mrow> </semantics></math>m, and (<b>e</b>) the marked regions satisfy the counterintuitive scenario in (<a href="#FD53-entropy-22-00246" class="html-disp-formula">53</a>)–(<a href="#FD55-entropy-22-00246" class="html-disp-formula">55</a>) for varying <math display="inline"><semantics> <msub> <mi>X</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </semantics></math> with the corresponding <math display="inline"><semantics> <msub> <mi>X</mi> <mrow> <mn>3</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </semantics></math> pair in <a href="#entropy-22-00246-f007" class="html-fig">Figure 7</a>c. Constructive and destructive interferences are observed for diffraction through PL-2 and PL-3, respectively, with different kinds of correlation of the paths at different times as a proof-of-concept numerical simulation of <span class="html-italic">quantum path interference (QPI) in time</span> between the two paths. (<b>f</b>) The comparison of setup diameters on the second and third planes, i.e., <math display="inline"><semantics> <msub> <mi>D</mi> <mn>2</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>D</mi> <mn>3</mn> </msub> </semantics></math>, respectively, with the spatial coherence diameters <math display="inline"><semantics> <mrow> <msub> <mi>D</mi> <mi>c</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mn>12</mn> </msub> <mo>,</mo> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>D</mi> <mi>c</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mn>23</mn> </msub> <mo>,</mo> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math>, respectively, in the targeted range of <math display="inline"><semantics> <mrow> <msub> <mi>X</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>∈</mo> <mrow> <mo>[</mo> <mn>140</mn> <mo>,</mo> <mn>170</mn> <mo>]</mo> </mrow> </mrow> </semantics></math> (<math display="inline"><semantics> <mi mathvariant="sans-serif">μ</mi> </semantics></math>m) in <a href="#entropy-22-00246-f007" class="html-fig">Figure 7</a>e.</p> Full article ">Figure 8
<p>(<b>a</b>) The conventional modeling for the spatial coherence of light sources based on double-slit diffraction [<a href="#B43-entropy-22-00246" class="html-bibr">43</a>], where <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>θ</mi> <mspace width="0.166667em"/> <mo>Δ</mo> <mi>s</mi> <mo>≤</mo> <mi>λ</mi> </mrow> </semantics></math> is required for the fringes to be observed determining the spatial coherence diameter (<math display="inline"><semantics> <msub> <mi>D</mi> <mi>c</mi> </msub> </semantics></math>); (<b>b</b>) free-space propagation of Gaussian beam, where <math display="inline"><semantics> <msub> <mi>D</mi> <mi>c</mi> </msub> </semantics></math> is approximated as the <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>/</mo> <msup> <mi>e</mi> <mn>2</mn> </msup> </mrow> </semantics></math> intensity beamwidth of <math display="inline"><semantics> <mrow> <mn>2</mn> <mspace width="0.166667em"/> <msqrt> <mn>2</mn> </msqrt> <mspace width="0.166667em"/> <msub> <mi>σ</mi> <mn>0</mn> </msub> </mrow> </semantics></math> with the standard deviation of <math display="inline"><semantics> <msub> <mi>σ</mi> <mi>D</mi> </msub> </semantics></math>. The descriptions of the calculation of the setup diameters on the planes to include the slits are denoted by <math display="inline"><semantics> <msub> <mi>D</mi> <mi>j</mi> </msub> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>j</mi> <mo>∈</mo> <mo>[</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>]</mo> </mrow> </semantics></math> with respect to the location and the standard deviation of the source on the previous plane (<math display="inline"><semantics> <msub> <mi>σ</mi> <mn>0</mn> </msub> </semantics></math> for the first plane and <math display="inline"><semantics> <msub> <mi>β</mi> <mrow> <mi>j</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </semantics></math> for the <span class="html-italic">j</span>th plane) for (<b>c</b>) LGI violation numerical analysis <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>i</mi> <msub> <mi>m</mi> <mn>1</mn> </msub> </mrow> </semantics></math> with two planes of triple slits on each plane and (<b>d</b>) interference in time scenario <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>i</mi> <msub> <mi>m</mi> <mn>2</mn> </msub> </mrow> </semantics></math> with three planes.</p> Full article ">
8 pages, 473 KiB  
Article
Non-Monogamy of Spatio-Temporal Correlations and the Black Hole Information Loss Paradox
by Chiara Marletto, Vlatko Vedral, Salvatore Virzì, Enrico Rebufello, Alessio Avella, Fabrizio Piacentini, Marco Gramegna, Ivo Pietro Degiovanni and Marco Genovese
Entropy 2020, 22(2), 228; https://doi.org/10.3390/e22020228 - 18 Feb 2020
Cited by 7 | Viewed by 3003
Abstract
Pseudo-density matrices are a generalisation of quantum states and do not obey monogamy of quantum correlations. Could this be the solution to the paradox of information loss during the evaporation of a black hole? In this paper we discuss this possibility, providing a [...] Read more.
Pseudo-density matrices are a generalisation of quantum states and do not obey monogamy of quantum correlations. Could this be the solution to the paradox of information loss during the evaporation of a black hole? In this paper we discuss this possibility, providing a theoretical proposal to extend quantum theory with these pseudo-states to describe the statistics arising in black-hole evaporation. We also provide an experimental demonstration of this theoretical proposal, using a simulation in optical regime, that tomographically reproduces the correlations of the pseudo-density matrix describing this physical phenomenon. Full article
(This article belongs to the Special Issue Quantum Entanglement)
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Figure 1

Figure 1
<p>Experimental setup. A maximally entangled singlet state is generated by pumping a type-II <math display="inline"><semantics> <msub> <mrow> <mi>Beta</mi> <mo>−</mo> <mi>BaB</mi> </mrow> <mn>2</mn> </msub> </semantics></math><math display="inline"><semantics> <msub> <mi mathvariant="normal">O</mi> <mn>4</mn> </msub> </semantics></math> (BBO) crystal. Two polarization measurements, M1 and M2 (at times <math display="inline"><semantics> <msub> <mi>t</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>t</mi> <mn>2</mn> </msub> </semantics></math>, respectively) are performed in sequence on photon A, while a single measurement (M3) is carried on photon B. Correlations among them certify entanglement monogamy violation for the whole pseudo-density operator (PDO) <math display="inline"><semantics> <msub> <mi>R</mi> <mn>123</mn> </msub> </semantics></math> in Equation (<a href="#FD2-entropy-22-00228" class="html-disp-formula">2</a>), describing the scenario of the spatio-temporal multi-partite entanglement (outside and inside the black hole) considered.</p> Full article ">Figure 2
<p>Tomographic reconstruction of the real (panel <b>a</b>) and imaginary (panel <b>b</b>) part of the reduced pseudo-density operator <math display="inline"><semantics> <mrow> <msub> <mi>R</mi> <mn>12</mn> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <mrow> <mo>(</mo> <mi>I</mi> <mo>+</mo> <msub> <mi mathvariant="sans-serif">Σ</mi> <mn>12</mn> </msub> <mo>)</mo> </mrow> </mrow> </semantics></math>, describing the temporal correlations between qubits 1 and 2, compared with the corresponding theoretical expectations (panels <b>c</b> and <b>d</b>, respectively).</p> Full article ">Figure 3
<p>Tomographic reconstruction of the real (panel <b>a</b>) and imaginary (panel <b>b</b>) part of the reduced pseudo-density operator <math display="inline"><semantics> <mrow> <msub> <mi>R</mi> <mn>13</mn> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <mrow> <mo>(</mo> <mi>I</mi> <mo>−</mo> <msub> <mi mathvariant="sans-serif">Σ</mi> <mn>13</mn> </msub> <mo>)</mo> </mrow> </mrow> </semantics></math>, related to the spatially maximally entangled state within the black hole, compared with the corresponding theoretically-expected counterparts (panels <b>c</b> and <b>d</b>, respectively).</p> Full article ">
7 pages, 261 KiB  
Article
A Note on the Hierarchy of Quantum Measurement Incompatibilities
by Bao-Zhi Sun, Zhi-Xi Wang, Xianqing Li-Jost and Shao-Ming Fei
Entropy 2020, 22(2), 161; https://doi.org/10.3390/e22020161 - 30 Jan 2020
Cited by 1 | Viewed by 2335
Abstract
The quantum measurement incompatibility is a distinctive feature of quantum mechanics. We investigate the incompatibility of a set of general measurements and classify the incompatibility by the hierarchy of compatibilities of its subsets. By using the approach of adding noises to measurement operators, [...] Read more.
The quantum measurement incompatibility is a distinctive feature of quantum mechanics. We investigate the incompatibility of a set of general measurements and classify the incompatibility by the hierarchy of compatibilities of its subsets. By using the approach of adding noises to measurement operators, we present a complete classification of the incompatibility of a given measurement assemblage with n members. Detailed examples are given for the incompatibility of unbiased qubit measurements based on a semidefinite program. Full article
(This article belongs to the Special Issue Quantum Entanglement)
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