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Quantum Probability and Randomness IV
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Quantum Probability and Randomness IV

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

Deadline for manuscript submissions: closed (1 June 2023) | Viewed by 28239

Special Issue Editors

Prof. Dr. Andrei Khrennikov

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Guest Editor
International Center for Mathematical Modeling in Physics and Cognitive Sciences, Linnaeus University, SE-351 95 Växjö, Sweden
Interests: quantum foundations; information; probability; contextuality; applications of the mathematical formalism of quantum theory outside of physics: cognition, psychology, decision making, economics, finances, and social and political sciences; p-adic numbers; p-adic and ultrametric analysis; dynamical systems; p-adic theoretical physics; utrametric models of cognition and psychological behavior; p-adic models in geophysics and petroleum research
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Karl Svozil

E-Mail Website
Guest Editor
Institute for Theoretical Physics, Vienna University of Technology Wiedner Hauptstrasse 8-10/136, A-1040 Vienna, Austria
Interests: quantum logic; automaton logic; conventionality in relativity theory; intrinsic embedded observers; physical (in)determinism; physical random number generators; generalized probability theory
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This is the fourth Special Issue devoted to the theme of “Quantum Probability and Randomness”; for the first three issues, see:

https://www.mdpi.com/si/entropy/Probability_Randomness

https://www.mdpi.com/si/entropy/Probability_Randomness_ii

https://www.mdpi.com/journal/entropy/special_issues/Probability_Randomness_iii

The previous Special Issues collected a sample of high-quality papers, both theoretical and experiment-related, written by experts in this area, which attracted considerable interest (including numerous downloads). This is why we have decided to proceed once again with this hot topic by considering structuring this theme into a regular series based on the Entropy journal.

The last few years have been characterized by tremendous developments in quantum information and probability and their applications, including quantum computing, quantum cryptography, and quantum random generators. Despite the successful development of quantum technology, its foundational basis is still not concrete and contains a few sandy and shaky slices. Quantum random generators are one of the most promising outputs of the recent quantum information revolution. Therefore, it is very important to reconsider the foundational basis of this project, starting with the notion of irreducible quantum randomness.

Quantum probabilities present a powerful tool to model uncertainty. Interpretations of quantum probability and foundational meanings of its basic tools, starting with the Born rule, are among the topics which will be covered in this Special Issue.

Recently, quantum probability has started to play an important role in a few areas of research outside quantum physics—in particular, the quantum probabilistic treatment of problems of the theory of decision-making under uncertainty. Such studies are also among the topics addressed in this Special Issue. 

The areas covered include:

  • Foundations of quantum information theory and quantum probability;
  • Quantum versus classical randomness and quantum random generators;
  • Generalized probabilistic models;
  • Quantum contextuality and generalized contextual models;
  • Bell’s inequality, entanglement, and randomness;
  • Quantum-like probabilistic modeling of the process of decision making under uncertainty;
  • Quantum probability and information in biology.

Of course, possible topics need not be restricted to the list above; any contribution directed to the improvement of quantum foundations, and the development of quantum probability and randomness is welcome.

Prof. Dr. Andrei Khrennikov
Prof. Dr. Karl Svozil
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • quantum foundation
  • quantum vs. classical probability and randomness
  • quantum information
  • Bell inequality
  • entanglement
  • contextuality
  • random generators
  • generalized probabilistic models

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Published Papers (15 papers)

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17 pages, 332 KiB  
Article
Probability Distributions Describing Qubit-State Superpositions
by Margarita A. Man’ko and Vladimir I. Man’ko
Entropy 2023, 25(10), 1366; https://doi.org/10.3390/e25101366 - 22 Sep 2023
Cited by 2 | Viewed by 1499
Abstract
We discuss qubit-state superpositions in the probability representation of quantum mechanics. We study probability distributions describing separable qubit states. We consider entangled states on the example of a system of two qubits (Bell states) using the corresponding superpositions of the wave functions associated [...] Read more.
We discuss qubit-state superpositions in the probability representation of quantum mechanics. We study probability distributions describing separable qubit states. We consider entangled states on the example of a system of two qubits (Bell states) using the corresponding superpositions of the wave functions associated with these states. We establish the connection with the properties and structure of entangled probability distributions. Full article
(This article belongs to the Special Issue Quantum Probability and Randomness IV)
44 pages, 1461 KiB  
Article
Systems of Precision: Coherent Probabilities on Pre-Dynkin Systems and Coherent Previsions on Linear Subspaces
by Rabanus Derr and Robert C. Williamson
Entropy 2023, 25(9), 1283; https://doi.org/10.3390/e25091283 - 31 Aug 2023
Cited by 1 | Viewed by 1371
Abstract
In the literature on imprecise probability, little attention is paid to the fact that imprecise probabilities are precise on a set of events. We call these sets systems of precision. We show that, under mild assumptions, the system of precision of a [...] Read more.
In the literature on imprecise probability, little attention is paid to the fact that imprecise probabilities are precise on a set of events. We call these sets systems of precision. We show that, under mild assumptions, the system of precision of a lower and upper probability form a so-called (pre-)Dynkin system. Interestingly, there are several settings, ranging from machine learning on partial data over frequential probability theory to quantum probability theory and decision making under uncertainty, in which, a priori, the probabilities are only desired to be precise on a specific underlying set system. Here, (pre-)Dynkin systems have been adopted as systems of precision, too. We show that, under extendability conditions, those pre-Dynkin systems equipped with probabilities can be embedded into algebras of sets. Surprisingly, the extendability conditions elaborated in a strand of work in quantum probability are equivalent to coherence from the imprecise probability literature. On this basis, we spell out a lattice duality which relates systems of precision to credal sets of probabilities. We conclude the presentation with a generalization of the framework to expectation-type counterparts of imprecise probabilities. The analogue of pre-Dynkin systems turns out to be (sets of) linear subspaces in the space of bounded, real-valued functions. We introduce partial expectations, natural generalizations of probabilities defined on pre-Dynkin systems. Again, coherence and extendability are equivalent. A related but more general lattice duality preserves the relation between systems of precision and credal sets of probabilities. Full article
(This article belongs to the Special Issue Quantum Probability and Randomness IV)
►▼ Show Figures

Figure 1

Figure 1
<p>Exemplary illustration of Venn diagram for four sets (By RupertMillard, CC BY-SA 3.0).</p> Full article ">Figure 2
<p>Illustration of the running example. The dark elements are contained in the pre-Dynkin system <math display="inline"><semantics> <mi mathvariant="script">D</mi> </semantics></math> on <math display="inline"><semantics> <mrow> <mi mathvariant="sans-serif">Ω</mi> <mo>=</mo> <mo>{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>}</mo> </mrow> </semantics></math>. The lower and upper coherent extension, respectively, the inner and outer extension, are denoted at the sides of the elements in the set system as shown in the example in the left upper corner. Elements in <math display="inline"><semantics> <mi mathvariant="script">D</mi> </semantics></math> possess a precise probability.</p> Full article ">Figure 3
<p>Galois connection between the lattice of pre-Dynkin systems and the set of credal sets. In the illustrated case, we have <math display="inline"><semantics> <mrow> <msubsup> <mi>m</mi> <mi>ψ</mi> <mo>∘</mo> </msubsup> <mrow> <mo>(</mo> <mi>Q</mi> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mi>m</mi> <mi>ψ</mi> <mo>∘</mo> </msubsup> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mi>ψ</mi> </msub> <mrow> <mo>(</mo> <msubsup> <mi>m</mi> <mi>ψ</mi> <mo>∘</mo> </msubsup> <mrow> <mo>(</mo> <mi mathvariant="script">A</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>, respectively, <math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mi>ψ</mi> </msub> <mrow> <mo>(</mo> <mi mathvariant="script">A</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>m</mi> <mi>ψ</mi> </msub> <mrow> <mo>(</mo> <msubsup> <mi>m</mi> <mi>ψ</mi> <mo>∘</mo> </msubsup> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mi>ψ</mi> </msub> <mrow> <mo>(</mo> <mi>Q</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>. The set containment on both sides follows from Proposition 2, Corollary 2 and Proposition 6.</p> Full article ">
57 pages, 732 KiB  
Article
Simultaneous Measurements of Noncommuting Observables: Positive Transformations and Instrumental Lie Groups
by Christopher S. Jackson and Carlton M. Caves
Entropy 2023, 25(9), 1254; https://doi.org/10.3390/e25091254 - 23 Aug 2023
Cited by 3 | Viewed by 1836
Abstract
We formulate a general program for describing and analyzing continuous, differential weak, simultaneous measurements of noncommuting observables, which focuses on describing the measuring instrument autonomously, without states. The Kraus operators of such measuring processes are time-ordered products of fundamental differential positive transformations [...] Read more.
We formulate a general program for describing and analyzing continuous, differential weak, simultaneous measurements of noncommuting observables, which focuses on describing the measuring instrument autonomously, without states. The Kraus operators of such measuring processes are time-ordered products of fundamental differential positive transformations, which generate nonunitary transformation groups that we call instrumental Lie groups. The temporal evolution of the instrument is equivalent to the diffusion of a Kraus-operator distribution function, defined relative to the invariant measure of the instrumental Lie group. This diffusion can be analyzed using Wiener path integration, stochastic differential equations, or a Fokker-Planck-Kolmogorov equation. This way of considering instrument evolution we call the Instrument Manifold Program. We relate the Instrument Manifold Program to state-based stochastic master equations. We then explain how the Instrument Manifold Program can be used to describe instrument evolution in terms of a universal cover that we call the universal instrumental Lie group, which is independent not just of states, but also of Hilbert space. The universal instrument is generically infinite dimensional, in which case the instrument’s evolution is chaotic. Special simultaneous measurements have a finite-dimensional universal instrument, in which case the instrument is considered principal, and it can be analyzed within the differential geometry of the universal instrumental Lie group. Principal instruments belong at the foundation of quantum mechanics. We consider the three most fundamental examples: measurement of a single observable, position and momentum, and the three components of angular momentum. As these measurements are performed continuously, they converge to strong simultaneous measurements. For a single observable, this results in the standard decay of coherence between inequivalent irreducible representations. For the latter two cases, it leads to a collapse within each irreducible representation onto the classical or spherical phase space, with the phase space located at the boundary of these instrumental Lie groups. Full article
(This article belongs to the Special Issue Quantum Probability and Randomness IV)
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Figure 1

Figure 1
<p>Schematic of a sequence of indirect, differential weak measurements; full understanding comes after reading <xref ref-type="sec" rid="sec2dot1-entropy-25-01254">Section 2.1</xref> and <xref ref-type="sec" rid="sec2dot2-entropy-25-01254">Section 2.2</xref>. A system in a state <inline-formula><mml:math id="mm927"><mml:semantics><mml:mrow><mml:mo>|</mml:mo><mml:mi>ψ</mml:mi><mml:mo>〉</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> is indirectly measured by a sequence of weak interactions <inline-formula><mml:math id="mm928"><mml:semantics><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mi>i</mml:mi><mml:mi>H</mml:mi><mml:mi>d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msup></mml:semantics></mml:math></inline-formula>, where each set of meters is observed after its interaction; that is, the system is continuously monitored. The incremental Kraus operator for the measurement at time <italic>t</italic>, given outcomes <inline-formula><mml:math id="mm929"><mml:semantics><mml:mrow><mml:mi>d</mml:mi><mml:msub><mml:mover accent="true"><mml:mi>W</mml:mi><mml:mo>→</mml:mo></mml:mover><mml:mi>t</mml:mi></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula>, is <inline-formula><mml:math id="mm930"><mml:semantics><mml:mrow><mml:msqrt><mml:mrow><mml:msup><mml:mi>d</mml:mi><mml:mi>n</mml:mi></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>d</mml:mi><mml:msub><mml:mover accent="true"><mml:mi>W</mml:mi><mml:mo>→</mml:mo></mml:mover><mml:mi>t</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msqrt><mml:mfenced separators="" open="〈" close="〉"><mml:mi>d</mml:mi><mml:msub><mml:mover accent="true"><mml:mi>W</mml:mi><mml:mo>→</mml:mo></mml:mover><mml:mi>t</mml:mi></mml:msub><mml:mo>|</mml:mo><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mi>i</mml:mi><mml:mi>H</mml:mi><mml:mi>d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msup><mml:mo>|</mml:mo><mml:mover accent="true"><mml:mn>0</mml:mn><mml:mo>→</mml:mo></mml:mover><mml:mspace width="0.166667em"/></mml:mfenced></mml:mrow></mml:semantics></mml:math></inline-formula>. Under the conditions outlined in <xref ref-type="sec" rid="sec2dot1-entropy-25-01254">Section 2.1</xref>, this Kraus operator is the differential positive transformation of Equation (<xref ref-type="disp-formula" rid="FD1-entropy-25-01254">1</xref>), that is, <inline-formula><mml:math id="mm931"><mml:semantics><mml:mrow><mml:msqrt><mml:mrow><mml:mi>d</mml:mi><mml:mi>μ</mml:mi><mml:mo>(</mml:mo><mml:mi>d</mml:mi><mml:msub><mml:mover accent="true"><mml:mi>W</mml:mi><mml:mo>→</mml:mo></mml:mover><mml:mi>t</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:msqrt><mml:mspace width="0.166667em"/><mml:msub><mml:mi>L</mml:mi><mml:mover accent="true"><mml:mi>X</mml:mi><mml:mo>→</mml:mo></mml:mover></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>d</mml:mi><mml:msub><mml:mover accent="true"><mml:mi>W</mml:mi><mml:mo>→</mml:mo></mml:mover><mml:mi>t</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>, with <inline-formula><mml:math id="mm932"><mml:semantics><mml:mrow><mml:msub><mml:mi>L</mml:mi><mml:mover accent="true"><mml:mi>X</mml:mi><mml:mo>→</mml:mo></mml:mover></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>d</mml:mi><mml:msub><mml:mover accent="true"><mml:mi>W</mml:mi><mml:mo>→</mml:mo></mml:mover><mml:mi>t</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:msup><mml:mover accent="true"><mml:mi>X</mml:mi><mml:mo>→</mml:mo></mml:mover><mml:mn>2</mml:mn></mml:msup><mml:mi>κ</mml:mi><mml:mspace width="0.166667em"/><mml:mi>d</mml:mi><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mover accent="true"><mml:mi>X</mml:mi><mml:mo>→</mml:mo></mml:mover><mml:mo>·</mml:mo><mml:msqrt><mml:mi>κ</mml:mi></mml:msqrt><mml:mspace width="0.166667em"/><mml:mi>d</mml:mi><mml:msub><mml:mover accent="true"><mml:mi>W</mml:mi><mml:mo>→</mml:mo></mml:mover><mml:mi>t</mml:mi></mml:msub></mml:mrow></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula>. The incremental Kraus operators “pile up” to become, at time <italic>T</italic>, the overall Kraus operator <inline-formula><mml:math id="mm933"><mml:semantics><mml:mrow><mml:msqrt><mml:mrow><mml:mi mathvariant="script">D</mml:mi><mml:mi>μ</mml:mi><mml:mo>[</mml:mo><mml:mi>d</mml:mi><mml:msub><mml:mover accent="true"><mml:mi>W</mml:mi><mml:mo>→</mml:mo></mml:mover><mml:mrow><mml:mo>[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:msub><mml:mo>]</mml:mo></mml:mrow></mml:msqrt><mml:mspace width="0.166667em"/><mml:mi>L</mml:mi><mml:mrow><mml:mo>[</mml:mo><mml:mi>d</mml:mi><mml:msub><mml:mover accent="true"><mml:mi>W</mml:mi><mml:mo>→</mml:mo></mml:mover><mml:mrow><mml:mo>[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:msub><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>, which is written as a time-ordered exponential in Equation (<xref ref-type="disp-formula" rid="FD2-entropy-25-01254">2</xref>). The overall Kraus operator gives the unnormalized final state at time <italic>T</italic>, as shown in the figure. The collection of Kraus operators at time <italic>T</italic>, for all Wiener outcome paths <inline-formula><mml:math id="mm934"><mml:semantics><mml:mrow><mml:mi>d</mml:mi><mml:msub><mml:mover accent="true"><mml:mi>W</mml:mi><mml:mo>→</mml:mo></mml:mover><mml:mrow><mml:mo>[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula>, defines an <italic>instrument</italic>, which can be analyzed on its own, independent of system states—simply omit <inline-formula><mml:math id="mm935"><mml:semantics><mml:mrow><mml:mo>|</mml:mo><mml:mi>ψ</mml:mi><mml:mo>〉</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> from the figure—a style of analysis we call <italic>instrument autonomy</italic>. The Kraus operators move across the manifold of an <italic>instrumental Lie group</italic>, which is generated by the measured observables. Placing the instrument within its instrumental Lie group and analyzing its evolution there is what we call the <italic>Instrument Manifold Program</italic>.</p> Full article ">
17 pages, 359 KiB  
Article
Joint Probabilities Approach to Quantum Games with Noise
by Alexis R. Legón and Ernesto Medina
Entropy 2023, 25(8), 1222; https://doi.org/10.3390/e25081222 - 16 Aug 2023
Cited by 2 | Viewed by 1388
Abstract
A joint probability formalism for quantum games with noise is proposed, inspired by the formalism of non-factorizable probabilities that connects the joint probabilities to quantum games with noise. Using this connection, we show that the joint probabilities are non-factorizable; thus, noise does not [...] Read more.
A joint probability formalism for quantum games with noise is proposed, inspired by the formalism of non-factorizable probabilities that connects the joint probabilities to quantum games with noise. Using this connection, we show that the joint probabilities are non-factorizable; thus, noise does not generically destroy entanglement. This formalism was applied to the Prisoner’s Dilemma, the Chicken Game, and the Battle of the Sexes, where noise is coupled through a single parameter μ. We find that for all the games except for the Battle of the Sexes, the Nash inequalities are maintained up to a threshold value of the noise. Beyond the threshold value, the inequalities no longer hold for quantum and classical strategies. For the Battle of the sexes, the Nash inequalities always hold, no matter the noise strength. This is due to the symmetry and anti-symmetry of the parameters that determine the joint probabilities for that game. Finally, we propose a new correlation measure for the games with classical and quantum strategies, where we obtain that the incorporation of noise, when we have quantum strategies, does not affect entanglement, but classical strategies result in behavior that approximates quantum games with quantum strategies without the need to saturate the system with the maximum value of noise. In this manner, these correlations can be understood as entanglement for our game approach. Full article
(This article belongs to the Special Issue Quantum Probability and Randomness IV)
►▼ Show Figures

Figure 1

Figure 1
<p>Entanglement measure for quantum games with noise from joint probabilities. Corresponding to the classical strategies (red) and quantum strategies (blue).</p> Full article ">
57 pages, 592 KiB  
Article
Simultaneous Momentum and Position Measurement and the Instrumental Weyl-Heisenberg Group
by Christopher S. Jackson and Carlton M. Caves
Entropy 2023, 25(8), 1221; https://doi.org/10.3390/e25081221 - 16 Aug 2023
Cited by 5 | Viewed by 1466
Abstract
The canonical commutation relation, [Q,P]=iℏ, stands at the foundation of quantum theory and the original Hilbert space. The interpretation of P and Q as observables has always relied on the analogies that exist between the [...] Read more.
The canonical commutation relation, [Q,P]=iℏ, stands at the foundation of quantum theory and the original Hilbert space. The interpretation of P and Q as observables has always relied on the analogies that exist between the unitary transformations of Hilbert space and the canonical (also known as contact) transformations of classical phase space. Now that the theory of quantum measurement is essentially complete (this took a while), it is possible to revisit the canonical commutation relation in a way that sets the foundation of quantum theory not on unitary transformations but on positive transformations. This paper shows how the concept of simultaneous measurement leads to a fundamental differential geometric problem whose solution shows us the following. The simultaneous P and Q measurement (SPQM) defines a universal measuring instrument, which takes the shape of a seven-dimensional manifold, a universal covering group we call the instrumental Weyl-Heisenberg (IWH) group. The group IWH connects the identity to classical phase space in unexpected ways that are significant enough that the positive-operator-valued measure (POVM) offers a complete alternative to energy quantization. Five of the dimensions define processes that can be easily recognized and understood. The other two dimensions, the normalization and phase in the center of the IWH group, are less familiar. The normalization, in particular, requires special handling in order to describe and understand the SPQM instrument. Full article
(This article belongs to the Special Issue Quantum Probability and Randomness IV)
24 pages, 456 KiB  
Article
A Transverse Hamiltonian Approach to Infinitesimal Perturbation Analysis of Quantum Stochastic Systems
by Igor G. Vladimirov
Entropy 2023, 25(8), 1179; https://doi.org/10.3390/e25081179 - 8 Aug 2023
Cited by 1 | Viewed by 1105
Abstract
This paper is concerned with variational methods for open quantum systems with Markovian dynamics governed by Hudson–Parthasarathy quantum stochastic differential equations. These QSDEs are driven by quantum Wiener processes of the external bosonic fields and are specified by the system Hamiltonian and system–field [...] Read more.
This paper is concerned with variational methods for open quantum systems with Markovian dynamics governed by Hudson–Parthasarathy quantum stochastic differential equations. These QSDEs are driven by quantum Wiener processes of the external bosonic fields and are specified by the system Hamiltonian and system–field coupling operators. We consider the system response to perturbations in these operators and introduce a transverse Hamiltonian which encodes the propagation of the perturbations through the unitary system–field evolution. This approach provides an infinitesimal perturbation analysis tool which can be used for the development of optimality conditions in quantum control and filtering problems. As performance criteria, such settings employ quadratic (or more complicated) cost functionals of the system and field variables to be minimized over the energy and coupling parameters of system interconnections. We demonstrate an application of the transverse Hamiltonian variational approach to a mean square optimal coherent quantum filtering problem for a measurement-free field-mediated cascade connection of a quantum system with a quantum observer. Full article
(This article belongs to the Special Issue Quantum Probability and Randomness IV)
►▼ Show Figures

Figure 1

Figure 1
<p>The series connection of a quantum plant with a coherent quantum observer, mediated by the plant output field <span class="html-italic">Y</span> and affected by the environment through the input quantum Wiener processes <span class="html-italic">W</span> and <math display="inline"><semantics><mi>ω</mi></semantics></math>. The observer design objective is that the drift part of its output <math display="inline"><semantics><mi>η</mi></semantics></math> has to approximate the plant variables of interest in a mean square optimal fashion.</p> Full article ">
21 pages, 1029 KiB  
Article
Non-Kochen–Specker Contextuality
by Mladen Pavičić
Entropy 2023, 25(8), 1117; https://doi.org/10.3390/e25081117 - 26 Jul 2023
Cited by 2 | Viewed by 2145 | Correction
Abstract
Quantum contextuality supports quantum computation and communication. One of its main vehicles is hypergraphs. The most elaborated are the Kochen–Specker ones, but there is also another class of contextual sets that are not of this kind. Their representation has been mostly operator-based and [...] Read more.
Quantum contextuality supports quantum computation and communication. One of its main vehicles is hypergraphs. The most elaborated are the Kochen–Specker ones, but there is also another class of contextual sets that are not of this kind. Their representation has been mostly operator-based and limited to special constructs in three- to six-dim spaces, a notable example of which is the Yu-Oh set. Previously, we showed that hypergraphs underlie all of them, and in this paper, we give general methods—whose complexity does not scale up with the dimension—for generating such non-Kochen–Specker hypergraphs in any dimension and give examples in up to 16-dim spaces. Our automated generation is probabilistic and random, but the statistics of accumulated data enable one to filter out sets with the required size and structure. Full article
(This article belongs to the Special Issue Quantum Probability and Randomness IV)
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Figure 1

Figure 1
<p>(<b>a</b>) Yu-Oh’s three-dim non-KS 13-16 non-KS NBMMPH ( [<a href="#B36-entropy-25-01117" class="html-bibr">36</a>] Figure 2); gray vertices that enlarge 13-16 to 25-16 are necessary for coordinatization and implementation; (<b>b</b>) Howard, Wallman, Veitech, and Emerson’s four-dim 30-108 non-KS NBMMPH ([<a href="#B4-entropy-25-01117" class="html-bibr">4</a>] Figure 2); (<b>c</b>) Cabello, Portillo, Solís, and Svozil’s five-dim 10-9 non-KS NBMMPH ([<a href="#B42-entropy-25-01117" class="html-bibr">42</a>] Figure 5a); the original symbols are presented in brackets (A,B).</p> Full article ">Figure 2
<p>(<b>a</b>) Distributions of critical four-dim non-KS NBMMPHs obtained from submaster 20-10, which was obtained from (Peres’) 24-24 supermaster (generated by vector components <math display="inline"> <semantics> <mrow> <mo>{</mo> <mn>0</mn> <mo>,</mo> <mo>±</mo> <mn>1</mn> <mo>}</mo> </mrow> </semantics> </math>) by <b>M1</b> (dots in red) and from submaster 58-51, itself obtained from the 60-72 supermaster (generated by vector components <math display="inline"> <semantics> <mrow> <mo>{</mo> <mn>0</mn> <mo>,</mo> <mo>±</mo> <mi>ϕ</mi> <mo>,</mo> <mi>ϕ</mi> <mo>−</mo> <mn>1</mn> <mo>}</mo> </mrow> </semantics> </math>, where <math display="inline"> <semantics> <mi>ϕ</mi> </semantics> </math> is the golden ratio: <math display="inline"> <semantics> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <msqrt> <mn>5</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> </semantics> </math>) by <b>M1</b> (in black); the abscissa is <span class="html-italic">l</span> (number of hyperedges); and the ordinate is <span class="html-italic">k</span> (number of vertices). The dots represent <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>k</mi> <mo>,</mo> <mi>l</mi> <mo>)</mo> </mrow> </semantics> </math>. Consecutive dots (same <span class="html-italic">l</span>) are shown as strips; (<b>b</b>) the smallest non-KS in the distributions: 4-3; (<b>c</b>) BMMPH 8-3—filled with 4-3—which one needs for obtaining the coordinatization and implementation of 4-3; (<b>d</b>) the 16-9 critical obtained from the 20-10 master; (<b>e</b>) the 16-9 critical obtained from the 58-51 master; (<b>f</b>) distributions of critical five-dim non-KS NBMMPHs obtained from submaster 66-50 which was obtained from the 105-136 supermaster (generated by vector components <math display="inline"> <semantics> <mrow> <mo>{</mo> <mn>0</mn> <mo>,</mo> <mo>±</mo> <mn>1</mn> <mo>}</mo> </mrow> </semantics> </math>); (<b>g</b>) the smallest critical; (<b>h</b>) a 16-9 critical for the sake of comparison with four-dim 16-9s; strings and coordinatizations are given in <a href="#app1-entropy-25-01117" class="html-app">Appendix A</a>.</p> Full article ">Figure 3
<p>(<b>a</b>) Distributions of 6-dim critical non-KS NBMMPHs obtained from two different submasters—see text; (<b>b</b>) the smallest critical non-KS NBMMPH obtained from the former class by <b>M3</b>; it has a parity proof; (<b>c</b>) an even smaller critical non-KS NBMMPH obtained from it by hand; it has a parity proof; (<b>d</b>) the smallest critical non-KS NBMMPH obtained from the latter class by <b>M1</b>; (<b>e</b>) distributions of 7-dim critical non-KS NBMMPHs—see text; (<b>f</b>) 14-8 non-KS NBMMPH, one of the smallest non-KS NBMMPHs obtained via <b>M3</b> from the smallest KS NBMMPH 34-14; (<b>g</b>) 31-13 also obtained from the 34-14 (no m = 1 vertices essential for criticality); (<b>h</b>,<b>i</b>) two 8-dim KS MMPHs with the smallest number of hyperedges (9); (<b>i</b>) serves us in generating the 15-9 non-KS NBMMPH in (<b>j</b>); (<b>h</b>–<b>j</b>) MMPHs have parity proofs; strings and coordinatizations are given in <a href="#app1-entropy-25-01117" class="html-app">Appendix A</a>.</p> Full article ">Figure 4
<p>(<b>a</b>) The 44-6 BMMPH and its critical subgraph 13-6 non-KS NBMMPH directly obtained from the supermaster via <b>M1</b>; (<b>b</b>) the critical nine-dim 19-8 obtained via <b>M3</b> from the master 47-16; (<b>c</b>) the critical ten-dim 18-9 non-KS NBMMPH obtained via <b>M3</b> from the 50-15 master; (<b>d</b>) the critical eleven-dim 19-8 non-KS NBMMPH obtained via <b>M3</b> from the 50-14 master. Strings and coordinatizations are given in <a href="#app1-entropy-25-01117" class="html-app">Appendix A</a>.</p> Full article ">Figure 5
<p>(<b>a</b>) Twelve-dim 19-9 critical non-KS NBMMPH directly obtained from the master 52-9 via <b>M3</b>; (<b>b</b>) 13-dim 19-8 critical non-KS NBMMPH obtained from the master 63-16, where the hyperedges do not form any loop with an order of three or higher; (<b>c</b>) 14-dim critical obtained from 66-15, where the maximal loop also has an order of 2; (<b>d</b>) 15-dim 25-8 critical from the 66-14 master; (<b>e</b>) 16-dim 22-9 critical from the 70-9 master, where all criticals are obtained via <b>M3</b>; all criticals and masters are given in the <a href="#app1-entropy-25-01117" class="html-app">Appendix A</a>.</p> Full article ">
11 pages, 295 KiB  
Article
Gas of Particles Obeying the Monotone Statistics
by Francesco Fidaleo
Entropy 2023, 25(7), 1095; https://doi.org/10.3390/e25071095 - 21 Jul 2023
Viewed by 1159
Abstract
The present note is devoted to the detailed investigation of a concrete model satisfying the block-monotone statistics introduced in a previous paper (joint, with collaborators) of the author. The model under consideration indeed describes the free gas of massless particles in a one-dimensional [...] Read more.
The present note is devoted to the detailed investigation of a concrete model satisfying the block-monotone statistics introduced in a previous paper (joint, with collaborators) of the author. The model under consideration indeed describes the free gas of massless particles in a one-dimensional environment. This investigation can have consequences in two fundamental respects. The first one concerns the applicability of the (block-)monotone statistics to concrete physical models, yet completely unknown. Since the formula for the degeneracy of the energy-levels of the one-particle Hamiltonian of a free particle is very involved, the second aspect might be related to the, highly nontrivial, investigation of the expected thermodynamics of the free gas of particles obeying the block-monotone statistics in arbitrary spatial dimensions. A final section contains a comparison between the various (block, strict, and weak) monotone schemes with the Boltzmann statistics, which describes the gas of classical particles. It is seen that the block-monotone statistics, which takes into account the degeneracy of the energy-levels, seems the unique one having realistic physical applications. Full article
(This article belongs to the Special Issue Quantum Probability and Randomness IV)
17 pages, 4797 KiB  
Article
Unstable Points, Ergodicity and Born’s Rule in 2d Bohmian Systems
by Athanasios C. Tzemos and George Contopoulos
Entropy 2023, 25(7), 1089; https://doi.org/10.3390/e25071089 - 20 Jul 2023
Cited by 5 | Viewed by 1318
Abstract
We study the role of unstable points in the Bohmian flow of a 2d system composed of two non-interacting harmonic oscillators. In particular, we study the unstable points in the inertial frame of reference as well as in the frame of reference of [...] Read more.
We study the role of unstable points in the Bohmian flow of a 2d system composed of two non-interacting harmonic oscillators. In particular, we study the unstable points in the inertial frame of reference as well as in the frame of reference of the moving nodal points, in cases with 1, 2 and multiple nodal points. Then, we find the contributions of the ordered and chaotic trajectories in the Born distribution, and when the latter is accessible by an initial particle distribution which does not satisfy Born’s rule. Full article
(This article belongs to the Special Issue Quantum Probability and Randomness IV)
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Figure 1
<p>(<b>a</b>) A nodal point-X-point complex and the deviations of the trajectories approaching the X-point. (<b>b</b>) The total potential close to a nodal point and its corresponding X-point (red dot) and Y-point (black dot). Both figures are drawn in the system <math display="inline"><semantics><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow></semantics></math> of the moving nodal point.</p> Full article ">Figure 2
<p>(<b>a</b>) A snapshot of the time-dependent Bohmian flow and the invariant curves of the Y-point at <math display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>1</mn></mrow></semantics></math> along with various Bohmian trajectories integrated from <math display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>0</mn></mrow></semantics></math> (green dots) up to <math display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>3</mn></mrow></semantics></math> (red dots) in the case of Equation (<a href="#FD6-entropy-25-01089" class="html-disp-formula">6</a>) (<math display="inline"><semantics><mrow><mi>a</mi><mo>=</mo><mi>b</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>c</mi><mo>=</mo><msqrt><mn>2</mn></msqrt><mo>/</mo><mn>2</mn></mrow></semantics></math>). The black dots correspond to <math display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>1</mn></mrow></semantics></math>, i.e., to the flow snapshot. We note that the flow changes in <math display="inline"><semantics><mrow><mi>t</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>3</mn><mo>]</mo></mrow></semantics></math>, but we still understand the form of the trajectories by reading the coordinates of the nodal point (<math display="inline"><semantics><msub><mi>x</mi><mi>N</mi></msub></semantics></math> red) and the Y-point (<math display="inline"><semantics><msub><mi>x</mi><mi>Y</mi></msub></semantics></math> green and <math display="inline"><semantics><mrow><msub><mi>y</mi><mi>N</mi></msub><mo>=</mo><msub><mi>y</mi><mi>Y</mi></msub></mrow></semantics></math> blue) for <math display="inline"><semantics><mrow><mi>t</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>3</mn><mo>]</mo></mrow></semantics></math>, as shown in (<b>b</b>). There, we see that <math display="inline"><semantics><mrow><msub><mi>y</mi><mi>N</mi></msub><mo>=</mo><msub><mi>y</mi><mi>Y</mi></msub></mrow></semantics></math> passes from <math display="inline"><semantics><mrow><mo>−</mo><mo>∞</mo></mrow></semantics></math> to <span class="html-italic">∞</span> (at <math display="inline"><semantics><mrow><mi>t</mi><mo>≃</mo><mn>1.84</mn></mrow></semantics></math>), <math display="inline"><semantics><msub><mi>x</mi><mi>N</mi></msub></semantics></math> changes its sign from negative to positive. (<b>c</b>) The Bohmian flow and the invariant curves of the Y-point at <math display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>2.5</mn></mrow></semantics></math>, where <math display="inline"><semantics><mrow><msub><mi>x</mi><mi>N</mi></msub><mo>&gt;</mo><mn>0</mn></mrow></semantics></math>. The stable/unstable invariant curves are calculated in positive/negative time <span class="html-italic">s</span>.</p> Full article ">Figure 3
<p>The distance between chaotic Bohmian trajectory and the nodal point (blue curve of the upper part) and the Y-point (red curve of the upper part) and the corresponding stretching number <span class="html-italic">a</span> for <math display="inline"><semantics><mrow><mi>t</mi><mo>∈</mo><mo>[</mo><mn>300</mn><mo>,</mo><mn>400</mn><mo>]</mo></mrow></semantics></math>. We observe that most of the significant scattering events correspond to the close approaches to the nodal points (and their associated X-points).</p> Full article ">Figure 4
<p>The distribution of the points (at every <math display="inline"><semantics><mrow><mo>Δ</mo><mi>t</mi><mo>=</mo><mn>0.05</mn></mrow></semantics></math>) of 3000 trajectories when the initial distribution satisfies BR, up to <math display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>3000</mn></mrow></semantics></math>.</p> Full article ">Figure 5
<p>The colorplots of two locally ergodic–chaotic trajectories separately up to <math display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>2</mn><mo>×</mo><msup><mn>10</mn><mn>6</mn></msup></mrow></semantics></math> (<b>a</b>) on the left and (<b>b</b>) on the right of <span class="html-italic">y</span>-axis in the single node case.</p> Full article ">Figure 6
<p>5000 initial conditions, in the case of a single nodal point, distributed according to BR at <math display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>0</mn></mrow></semantics></math> (<b>a</b>) on the <math display="inline"><semantics><mrow><mi>x</mi><mo>−</mo><mi>y</mi></mrow></semantics></math> plane and (<b>b</b>) projected on <math display="inline"><semantics><mrow><msub><mi>P</mi><mn>0</mn></msub><mo>=</mo><msup><mrow><mo>|</mo><msub><mo>Ψ</mo><mn>0</mn></msub><mo>|</mo></mrow><mn>2</mn></msup></mrow></semantics></math>. Blue/red initial conditions produce chaotic/ordered Bohmian trajectories.</p> Full article ">Figure 7
<p>Colorplot of 5000 trajectories (in the case of a single nodal point) with <math display="inline"><semantics><mrow><msub><mi>P</mi><mn>0</mn></msub><mo>≠</mo><msup><mrow><mo>|</mo><msub><mo>Ψ</mo><mn>0</mn></msub><mo>|</mo></mrow><mn>2</mn></msup></mrow></semantics></math>, at <math display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>3000</mn></mrow></semantics></math>. It is very different from that of the BR distribution in <a href="#entropy-25-01089-f004" class="html-fig">Figure 4</a>.</p> Full article ">Figure 8
<p>The stable (blue) and unstable (red) asymptotic curves of the Y-point in the case of two nodal points for (<b>a</b>) t = 1.5 and (<b>b</b>) t = 1.8 in the case of Equation (<a href="#FD19-entropy-25-01089" class="html-disp-formula">19</a>) (<math display="inline"><semantics><mrow><mi>a</mi><mo>=</mo><mn>1.23</mn><mo>,</mo><mi>b</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>c</mi><mo>=</mo><msqrt><mn>2</mn></msqrt><mo>/</mo><mn>2</mn></mrow></semantics></math>). We observe the change in the behavior of the nodal points from attractors to repellers.</p> Full article ">Figure 9
<p>5000 initial conditions in the case of 2 nodal points distributed according to BR at <math display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>0</mn></mrow></semantics></math> (<b>a</b>) on the <math display="inline"><semantics><mrow><mi>x</mi><mo>−</mo><mi>y</mi></mrow></semantics></math> plane and (<b>b</b>) projected on <math display="inline"><semantics><mrow><msub><mi>P</mi><mn>0</mn></msub><mo>=</mo><msup><mrow><mo>|</mo><msub><mo>Ψ</mo><mn>0</mn></msub><mo>|</mo></mrow><mn>2</mn></msup></mrow></semantics></math>. Blue/red initial conditions produce chaotic/ordered Bohmian trajectories.</p> Full article ">Figure 10
<p>The colorplot of the points of 5000 trajectories (in the case of two nodal points) initially satisfying BR, up to <math display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>3000</mn></mrow></semantics></math>.</p> Full article ">Figure 11
<p>The colorplots of (<b>a</b>) the ordered and (<b>b</b>) of the chaotic trajectories in an initial BR distribution (in the case of two nodal points).</p> Full article ">Figure 12
<p>The colorplots of 5000 trajectories in two initial distributions with <math display="inline"><semantics><mrow><msub><mi>P</mi><mn>0</mn></msub><mo>≠</mo><msup><mrow><mo>|</mo><msub><mo>Ψ</mo><mn>0</mn></msub><mo>|</mo></mrow><mn>2</mn></msup></mrow></semantics></math>, up to <math display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>3000</mn></mrow></semantics></math>. The shape of (<b>a</b>) is different from that of (<b>b</b>) and they both are very different from that of the BR distribution (<a href="#entropy-25-01089-f010" class="html-fig">Figure 10</a>).</p> Full article ">Figure 13
<p>The Bohmian flow along with the stationary nodal points (red dots), the moving nodal points (black dots) and the Y-points (green dots) at <math display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>0.1</mn></mrow></semantics></math>.</p> Full article ">Figure 14
<p>Details of the collision between the moving nodal point 19 and the fixed nodal point 19 at <math display="inline"><semantics><mrow><mi>y</mi><mo>=</mo><mn>0</mn></mrow></semantics></math>. (<b>a</b>) Before the collision (<math display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>1.8</mn></mrow></semantics></math>), the nodal point 19 and a nearby green point <math display="inline"><semantics><msub><mi>Y</mi><mn>19</mn></msub></semantics></math> move toward the fixed point 16 together with the green point <math display="inline"><semantics><msub><mi>Y</mi><mn>19</mn></msub></semantics></math> on the left of 19 (see the arrows). (<b>b</b>) At <math display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><msub><mi>t</mi><mrow><mi>c</mi><mi>o</mi><mi>l</mi></mrow></msub><mo>=</mo><mn>1.8403</mn></mrow></semantics></math> we observe the collision. (<b>c</b>) After the collision (<math display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>1.87</mn></mrow></semantics></math>), the moving nodal point 19 and <math display="inline"><semantics><msub><mi>Y</mi><mn>19</mn></msub></semantics></math> are above point 16 but <math display="inline"><semantics><msub><mi>Y</mi><mn>19</mn></msub></semantics></math> is now on the right of the nodal point 19 and the green point <math display="inline"><semantics><msub><mi>Y</mi><mn>16</mn></msub></semantics></math> has moved to the left of 16.</p> Full article ">Figure 15
<p>The asymptotic curves of some Y-points of the upper left corner of <a href="#entropy-25-01089-f013" class="html-fig">Figure 13</a>, stable (blue) and unstable (red).</p> Full article ">Figure 16
<p>Colorplots of two different chaotic trajectories in the case of multiple nodal points up to <math display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>5</mn><mo>×</mo><msup><mn>10</mn><mn>6</mn></msup></mrow></semantics></math>: (<b>a</b>) <math display="inline"><semantics><mrow><mi>x</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>2.8</mn><mo>,</mo><mi>y</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>0.8</mn></mrow></semantics></math> and (<b>b</b>) <math display="inline"><semantics><mrow><mi>x</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mo>−</mo><mn>3</mn><mo>,</mo><mi>y</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mo>−</mo><mn>1</mn></mrow></semantics></math>. They are very similar, i.e., they are approximately ergodic.</p> Full article ">Figure 17
<p>10,000 initial conditions in the case of multiple nodal points distributed according to Born’s distribution at <math display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>0</mn></mrow></semantics></math>, chaotic (blue) and ordered (red).</p> Full article ">
12 pages, 251 KiB  
Article
Quantum Mechanical Approach to the Khintchine and Bochner Criteria for Characteristic Functions
by Leon Cohen
Entropy 2023, 25(7), 1042; https://doi.org/10.3390/e25071042 - 11 Jul 2023
Viewed by 968
Abstract
While it is generally accepted that quantum mechanics is a probability theory, its methods differ radically from standard probability theory. We use the methods of quantum mechanics to understand some fundamental aspects of standard probability theory. We show that wave functions and operators [...] Read more.
While it is generally accepted that quantum mechanics is a probability theory, its methods differ radically from standard probability theory. We use the methods of quantum mechanics to understand some fundamental aspects of standard probability theory. We show that wave functions and operators do appear in standard probability theory. We do so by generalizing the Khintchine and Bochner criteria for a complex function to be a characteristic function. We show that quantum mechanics clarifies these criteria and suggests generalizations of them. Full article
(This article belongs to the Special Issue Quantum Probability and Randomness IV)
23 pages, 2881 KiB  
Article
Differential Shannon Entropies Characterizing Electron–Nuclear Dynamics and Correlation: Momentum-Space Versus Coordinate-Space Wave Packet Motion
by Peter Schürger and Volker Engel
Entropy 2023, 25(7), 970; https://doi.org/10.3390/e25070970 - 23 Jun 2023
Cited by 3 | Viewed by 1685
Abstract
We calculate differential Shannon entropies derived from time-dependent coordinate-space and momentum-space probability densities. This is performed for a prototype system of a coupled electron–nuclear motion. Two situations are considered, where one is a Born–Oppenheimer adiabatic dynamics, and the other is a diabatic motion [...] Read more.
We calculate differential Shannon entropies derived from time-dependent coordinate-space and momentum-space probability densities. This is performed for a prototype system of a coupled electron–nuclear motion. Two situations are considered, where one is a Born–Oppenheimer adiabatic dynamics, and the other is a diabatic motion involving strong non-adiabatic transitions. The information about coordinate- and momentum-space dynamics derived from the total and single-particle entropies is discussed and interpreted with the help of analytical models. From the entropies, we derive mutual information, which is a measure for the electron–nuclear correlation. In the adiabatic case, it is found that such correlations are manifested differently in coordinate- and momentum space. For the diabatic dynamics, we show that it is possible to decompose the entropies into state-specific contributions. Full article
(This article belongs to the Special Issue Quantum Probability and Randomness IV)
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Figure 1

Figure 1
<p>Upper panels: Adiabatic potential curves <math display="inline"><semantics> <mrow> <msub> <mi>V</mi> <mi>n</mi> </msub> <mrow> <mo>(</mo> <mi>R</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> obtained for two choices of the screening parameter <math display="inline"><semantics> <msub> <mi>R</mi> <mi>c</mi> </msub> </semantics></math>. The left- and right-hand columns are associated with the cases of a BO motion <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> Å) and a diabatic motion <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> Å), respectively. The two lower rows show the electronic eigenfunctions <math display="inline"><semantics> <mrow> <msub> <mi>φ</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>r</mi> <mo>;</mo> <mi>R</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>φ</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>r</mi> <mo>;</mo> <mi>R</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> as indicated.</p> Full article ">Figure 2
<p>Nuclear density dynamics in the presence of weak (BO dynamics, left-hand column) and strong non-adiabatic coupling (diabatic dynamics, right-hand column). The upper panels show the nuclear densities in coordinate space and the lower panels in momentum space. While in the weakly coupled case the densities disperse quickly, the strongly coupled case shows quasi-harmonic-dynamics.</p> Full article ">Figure 3
<p>BO dynamics. The left-hand panels show the coordinate-space entropies for the nuclear (<b>upper panel</b>) and electronic (<b>middle panel</b>) degrees of freedom. Also displayed is the entropy of the coupled system (<b>lower panel</b>). The right-hand column contains the same functions derived from the momentum-space densities. In each case, the numerically determined functions are compared to the analytically ones. The dashed lines mark the times when the wave packet reaches the classical turning points of its motion.</p> Full article ">Figure 4
<p>BO-dynamics. The left-hand column shows the coordinate-space covariance, correlation and mutual information, as indicated. The respective curves obtained from the momentum-space densities are depicted in the right-hand column. In each case, numerically and analytically derived results are compared. The times when the turning points are reached are marked by the vertical lines.</p> Full article ">Figure 5
<p>BO-dynamics. (<b>a</b>) Shown are momentum-space densities <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>P</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> for different times as indicated. Abscissa and ordinate correspond to the nuclear and electronic momentum, respectively. Also shown is the MI <math display="inline"><semantics> <mrow> <msub> <mi>I</mi> <mi>π</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. The vertical red lines mark times when the MI exhibits extrema, and the black lines indicate the times when the classical turning points are reached. (<b>b</b>) Same as (a), but in coordinate space. The densities <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>(</mo> <mi>r</mi> <mo>,</mo> <mi>R</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> (abscissa <span class="html-italic">R</span>, ordinate <span class="html-italic">r</span>) are depicted for selected times.</p> Full article ">Figure 6
<p>Same as <a href="#entropy-25-00970-f003" class="html-fig">Figure 3</a> but for the strong coupling case.</p> Full article ">Figure 7
<p>Decomposition of the entropies in the strong coupling case: Nuclear (<b>upper panels</b>), electronic (<b>middle panels</b>) and total entropies (<b>lower panels</b>). The coordinate-space entropies (<math display="inline"><semantics> <mrow> <msubsup> <mi>S</mi> <mi>R</mi> <mrow> <mi>n</mi> <mi>u</mi> <mi>c</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>,</mo> <msubsup> <mi>S</mi> <mi>r</mi> <mrow> <mi>e</mi> <mi>l</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mi>S</mi> <mi>x</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>) and the contributions of the two adiabatic electronic states (<math display="inline"><semantics> <mrow> <msubsup> <mi>S</mi> <mrow> <mi>R</mi> <mo>,</mo> <mi>n</mi> </mrow> <mrow> <mi>n</mi> <mi>u</mi> <mi>c</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>,</mo> <msubsup> <mi>S</mi> <mrow> <mi>r</mi> <mo>,</mo> <mi>n</mi> </mrow> <mrow> <mi>e</mi> <mi>l</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mi>S</mi> <mrow> <mi>x</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>,</mo> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> </mrow> </semantics></math>) are shown, and also the sum (<math display="inline"><semantics> <mo>Σ</mo> </semantics></math>) of the state-specific entropies and the numerically exact curve. The right-hand column contains the respective quantities derived from the momentum-space density. The vertical dashed lines indicate the times when the classical turning points are reached.</p> Full article ">
21 pages, 453 KiB  
Article
Winning a CHSH Game without Entangled Particles in a Finite Number of Biased Rounds: How Much Luck Is Needed?
by Christoph Gallus, Pawel Blasiak and Emmanuel M. Pothos
Entropy 2023, 25(5), 824; https://doi.org/10.3390/e25050824 - 21 May 2023
Cited by 1 | Viewed by 2361
Abstract
Quantum games, such as the CHSH game, are used to illustrate the puzzle and power of entanglement. These games are played over many rounds and in each round, the participants, Alice and Bob, each receive a question bit to which they each have [...] Read more.
Quantum games, such as the CHSH game, are used to illustrate the puzzle and power of entanglement. These games are played over many rounds and in each round, the participants, Alice and Bob, each receive a question bit to which they each have to give an answer bit, without being able to communicate during the game. When all possible classical answering strategies are analyzed, it is found that Alice and Bob cannot win more than 75% of the rounds. A higher percentage of wins arguably requires an exploitable bias in the random generation of the question bits or access to “non-local“ resources, such as entangled pairs of particles. However, in an actual game, the number of rounds has to be finite and question regimes may come up with unequal likelihood, so there is always a possibility that Alice and Bob win by pure luck. This statistical possibility has to be transparently analyzed for practical applications such as the detection of eavesdropping in quantum communication. Similarly, when Bell tests are used in macroscopic situations to investigate the connection strength between system components and the validity of proposed causal models, the available data are limited and the possible combinations of question bits (measurement settings) may not be controlled to occur with equal likelihood. In the present work, we give a fully self-contained proof for a bound on the probability to win a CHSH game by pure luck without making the usual assumption of only small biases in the random number generators. We also show bounds for the case of unequal probabilities based on results from McDiarmid and Combes and numerically illustrate certain exploitable biases. Full article
(This article belongs to the Special Issue Quantum Probability and Randomness IV)
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<p>Two probability distributions for <math display="inline"><semantics> <msubsup> <mi>S</mi> <mn>1</mn> <mi>obs</mi> </msubsup> </semantics></math> generated by a Monte Carlo simulation of <math display="inline"><semantics> <mrow> <mn>10</mn> <mo>,</mo> <mn>000</mn> </mrow> </semantics></math> CHSH games of <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>500</mn> </mrow> </semantics></math> rounds each. The threshold of <math display="inline"><semantics> <mrow> <mn>2</mn> <mo>+</mo> <mi>η</mi> <mo>=</mo> <mn>2.25</mn> </mrow> </semantics></math> is shown in red. (<b>left</b>) The graph on the left-hand side was generated with Alice and Bob randomly picking elementary strategies with <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, while the regimes <math display="inline"><semantics> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </semantics></math> were generated by independent and unbiased coin tosses. The simulated probability is <math display="inline"><semantics> <mrow> <mi>P</mi> <mi>r</mi> <mo>{</mo> <msubsup> <mi>S</mi> <mn>1</mn> <mi>obs</mi> </msubsup> <mo>⩾</mo> <mn>2.25</mn> <mo>}</mo> <mo>=</mo> <mn>4.5</mn> <mo>%</mo> </mrow> </semantics></math> with a maximum value of <math display="inline"><semantics> <mrow> <msubsup> <mi>S</mi> <mrow> <mn>1</mn> <mo>,</mo> <mo movablelimits="true" form="prefix">max</mo> </mrow> <mi>obs</mi> </msubsup> <mo>=</mo> <mn>2.52</mn> </mrow> </semantics></math> observed in a single CHSH game. (<b>right</b>) The graph on the right-hand side has been generated with Alice and Bob randomly picking elementary strategies that win in the regime <math display="inline"><semantics> <mrow> <mi>x</mi> <mi>y</mi> <mo>=</mo> <mn>00</mn> </mrow> </semantics></math> as well as satisfy <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>. Here, all regimes <math display="inline"><semantics> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </semantics></math> were generated by independent, but biased coin tosses with <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>x</mi> <mi>y</mi> <mo>=</mo> <mn>00</mn> <mo>)</mo> <mo>=</mo> <mn>0.7</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>x</mi> <mi>y</mi> <mo>=</mo> <mn>01</mn> <mo>)</mo> <mo>=</mo> <mi>P</mi> <mo>(</mo> <mi>x</mi> <mi>y</mi> <mo>=</mo> <mn>10</mn> <mo>)</mo> <mo>=</mo> <mi>P</mi> <mo>(</mo> <mi>x</mi> <mi>y</mi> <mo>=</mo> <mn>11</mn> <mo>)</mo> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>. The simulated probability is <math display="inline"><semantics> <mrow> <mi>P</mi> <mi>r</mi> <mo>{</mo> <msubsup> <mi>S</mi> <mn>1</mn> <mi>obs</mi> </msubsup> <mo>⩾</mo> <mn>2.25</mn> <mo>}</mo> <mo>=</mo> <mn>14</mn> <mo>%</mo> </mrow> </semantics></math> with a maximum value of <math display="inline"><semantics> <mrow> <msubsup> <mi>S</mi> <mrow> <mn>1</mn> <mo>,</mo> <mo movablelimits="true" form="prefix">max</mo> </mrow> <mi>obs</mi> </msubsup> <mo>=</mo> <mn>2.86</mn> </mrow> </semantics></math> observed in a single CHSH game.</p> Full article ">
17 pages, 319 KiB  
Article
Dynamics of System States in the Probability Representation of Quantum Mechanics
by Vladimir N. Chernega and Olga V. Man’ko
Entropy 2023, 25(5), 785; https://doi.org/10.3390/e25050785 - 11 May 2023
Cited by 6 | Viewed by 1763
Abstract
A short description of the notion of states of quantum systems in terms of conventional probability distribution function is presented. The notion and the structure of entangled probability distributions are clarified. The evolution of even and odd Schrödinger cat states of the inverted [...] Read more.
A short description of the notion of states of quantum systems in terms of conventional probability distribution function is presented. The notion and the structure of entangled probability distributions are clarified. The evolution of even and odd Schrödinger cat states of the inverted oscillator is obtained in the center-of-mass tomographic probability description of the two-mode oscillator. Evolution equations describing the time dependence of probability distributions identified with quantum system states are discussed. The connection with the Schrödinger equation and the von Neumann equation is clarified. Full article
(This article belongs to the Special Issue Quantum Probability and Randomness IV)
20 pages, 923 KiB  
Article
Visualizing Quantum Circuit Probability: Estimating Quantum State Complexity for Quantum Program Synthesis
by Bao Gia Bach, Akash Kundu, Tamal Acharya and Aritra Sarkar
Entropy 2023, 25(5), 763; https://doi.org/10.3390/e25050763 - 7 May 2023
Viewed by 3364
Abstract
This work applies concepts from algorithmic probability to Boolean and quantum combinatorial logic circuits. The relations among the statistical, algorithmic, computational, and circuit complexities of states are reviewed. Thereafter, the probability of states in the circuit model of computation is defined. Classical and [...] Read more.
This work applies concepts from algorithmic probability to Boolean and quantum combinatorial logic circuits. The relations among the statistical, algorithmic, computational, and circuit complexities of states are reviewed. Thereafter, the probability of states in the circuit model of computation is defined. Classical and quantum gate sets are compared to select some characteristic sets. The reachability and expressibility in a space-time-bounded setting for these gate sets are enumerated and visualized. These results are studied in terms of computational resources, universality, and quantum behavior. The article suggests how applications like geometric quantum machine learning, novel quantum algorithm synthesis, and quantum artificial general intelligence can benefit by studying circuit probabilities. Full article
(This article belongs to the Special Issue Quantum Probability and Randomness IV)
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<p>Growth of the number of programs with qubit count and circuit depth for two types of gate sets: (i) <math display="inline"><semantics> <mrow> <mo>[</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>]</mo> </mrow> </semantics></math> qubits: <math display="inline"><semantics> <mrow> <mo>{</mo> <mi mathvariant="monospace">X</mi> <mo>,</mo> <mi mathvariant="monospace">CCX</mi> <mo>}</mo> </mrow> </semantics></math>, (ii) <math display="inline"><semantics> <mrow> <mo>[</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>]</mo> </mrow> </semantics></math> qubits: <math display="inline"><semantics> <mrow> <mo>{</mo> <mi mathvariant="monospace">H</mi> <mo>,</mo> <mi mathvariant="monospace">S</mi> <mo>,</mo> <mi mathvariant="monospace">CX</mi> <mo>}</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>{</mo> <mi mathvariant="monospace">H</mi> <mo>,</mo> <mi mathvariant="monospace">T</mi> <mo>,</mo> <mi mathvariant="monospace">CX</mi> <mo>}</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>{</mo> <mi mathvariant="monospace">P</mi> <mo>(</mo> <mo>ß</mo> <mo>/</mo> <mn mathvariant="monospace">4</mn> <mo>)</mo> <mo>,</mo> <mi mathvariant="monospace">RX</mi> <mo>(</mo> <mo>ß</mo> <mo>/</mo> <mn mathvariant="monospace">2</mn> <mo>)</mo> <mo>,</mo> <mi mathvariant="monospace">CX</mi> <mo>}</mo> </mrow> </semantics></math>.</p> Full article ">Figure 2
<p>Expressibility and Reachability for gate set <math display="inline"><semantics> <mrow> <mo>{</mo> <mi mathvariant="monospace">X</mi> <mo>,</mo> <mi mathvariant="monospace">CCX</mi> <mo>}</mo> </mrow> </semantics></math> on 4 qubits and of circuit depth from 0 to 3.</p> Full article ">Figure 3
<p>Expressibility and Reachability for gate set <math display="inline"><semantics> <mrow> <mo>{</mo> <mi mathvariant="monospace">H</mi> <mo>,</mo> <mi mathvariant="monospace">S</mi> <mo>,</mo> <mi mathvariant="monospace">CX</mi> <mo>}</mo> </mrow> </semantics></math> on 4 qubits and of circuit depth from 0 to 3.</p> Full article ">Figure 4
<p>Expressibility and Reachability for gate set <math display="inline"><semantics> <mrow> <mo>{</mo> <mi mathvariant="monospace">H</mi> <mo>,</mo> <mi mathvariant="monospace">T</mi> <mo>,</mo> <mi mathvariant="monospace">CX</mi> <mo>}</mo> </mrow> </semantics></math> on 4 qubits and of circuit depth from 0 to 3.</p> Full article ">Figure 5
<p>Expressibility and Reachability for gate set <math display="inline"><semantics> <mrow> <mo>{</mo> <mi mathvariant="monospace">P</mi> <mo>(</mo> <mo>ß</mo> <mo>/</mo> <mn mathvariant="monospace">4</mn> <mo>)</mo> <mo>,</mo> <mi mathvariant="monospace">RX</mi> <mo>(</mo> <mo>ß</mo> <mo>/</mo> <mn mathvariant="monospace">2</mn> <mo>)</mo> <mo>,</mo> <mi mathvariant="monospace">CX</mi> <mo>}</mo> </mrow> </semantics></math> on 4 qubits and of circuit depth from 0 to 3.</p> Full article ">Figure 6
<p>Approximation of circuit probability of states on 4 qubits for <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math> using two gate sets (<b>a</b>) <math display="inline"><semantics> <mrow> <mo>{</mo> <mi mathvariant="monospace">X</mi> <mo>,</mo> <mspace width="3.33333pt"/> <mi mathvariant="monospace">CCX</mi> <mo>}</mo> </mrow> </semantics></math> and (<b>b</b>) <math display="inline"><semantics> <mrow> <mo>{</mo> <mi mathvariant="monospace">H</mi> <mo>,</mo> <mspace width="3.33333pt"/> <mi mathvariant="monospace">T</mi> <mo>,</mo> <mspace width="3.33333pt"/> <mi mathvariant="monospace">CX</mi> <mo>}</mo> </mrow> </semantics></math>.</p> Full article ">Figure 7
<p>Expressibility and Reachability for gate set <math display="inline"><semantics> <mrow> <mo>{</mo> <mi mathvariant="monospace">P</mi> <mo>(</mo> <mo>ß</mo> <mo>/</mo> <mn mathvariant="monospace">4</mn> <mo>)</mo> <mo>,</mo> <mi mathvariant="monospace">RX</mi> <mo>(</mo> <mo>ß</mo> <mo>/</mo> <mn mathvariant="monospace">2</mn> <mo>)</mo> <mo>,</mo> <mi mathvariant="monospace">CX</mi> <mo>}</mo> </mrow> </semantics></math> on 5 qubits and of circuit depth from 0 to 3 on the IBM T-topology.</p> Full article ">Figure 8
<p>Expressibility and Reachability for gate set <math display="inline"><semantics> <mrow> <mo>{</mo> <mi mathvariant="monospace">P</mi> <mo>(</mo> <mo>ß</mo> <mo>/</mo> <mn mathvariant="monospace">4</mn> <mo>)</mo> <mo>,</mo> <mi mathvariant="monospace">RX</mi> <mo>(</mo> <mo>ß</mo> <mo>/</mo> <mn mathvariant="monospace">2</mn> <mo>)</mo> <mo>,</mo> <mi mathvariant="monospace">CX</mi> <mo>}</mo> </mrow> </semantics></math> on 5 qubits and of circuit depth from 0 to 3 on the IBM L-topology.</p> Full article ">Figure 9
<p><math display="inline"><semantics> <msub> <mi>M</mi> <mrow> <mi>c</mi> <mi>i</mi> <mi>r</mi> <mi>c</mi> </mrow> </msub> </semantics></math> and their comparison for gate set <math display="inline"><semantics> <mrow> <mo>{</mo> <mi mathvariant="monospace">P</mi> <mo>(</mo> <mo>ß</mo> <mo>/</mo> <mn mathvariant="monospace">4</mn> <mo>)</mo> <mo>,</mo> <mi mathvariant="monospace">RX</mi> <mo>(</mo> <mo>ß</mo> <mo>/</mo> <mn mathvariant="monospace">2</mn> <mo>)</mo> <mo>,</mo> <mi mathvariant="monospace">CX</mi> <mo>}</mo> </mrow> </semantics></math> on 5 qubits and of circuit depth from 0 to 3 on the IBM L-topology and T-topology.</p> Full article ">
37 pages, 2262 KiB  
Article
Non-Local Parallel Processing and Database Settlement Using Multiple Teleportation Followed by Grover Post-Selection
by Francisco Delgado and Carlos Cardoso-Isidoro
Entropy 2023, 25(2), 376; https://doi.org/10.3390/e25020376 - 18 Feb 2023
Cited by 2 | Viewed by 2092
Abstract
Quantum information applications emerged decades ago, initially introducing a parallel development that mimicked the approach and development of classical computer science. However, in the current decade, novel computer-science concepts were rapidly extended to the fields of quantum processing, computation, and communication. Thus, areas [...] Read more.
Quantum information applications emerged decades ago, initially introducing a parallel development that mimicked the approach and development of classical computer science. However, in the current decade, novel computer-science concepts were rapidly extended to the fields of quantum processing, computation, and communication. Thus, areas such as artificial intelligence, machine learning, and neural networks have their quantum versions; furthermore, the quantum brain properties of learning, analyzing, and gaining knowledge are discussed. Quantum properties of matter conglomerates have been superficially explored in such terrain; however, the settlement of organized quantum systems able to perform processing can open a new pathway in the aforementioned domains. In fact, quantum processing involves certain requisites as the settlement of copies of input information to perform differentiated processing developed far away or in situ to diversify the information stored there. Both tasks at the end provide a database of outcomes with which to perform either information matching or final global processing with at least a subset of those outcomes. When the number of processing operations and input information copies is large, parallel processing (a natural feature in quantum computation due to the superposition) becomes the most convenient approach to accelerate the database settlement of outcomes, thus affording a time advantage. In the current study, we explored certain quantum features to realize a speed-up model for the entire task of processing based on a common information input to be processed, diversified, and finally summarized to gain knowledge, either in pattern matching or global information availability. By using superposition and non-local properties, the most valuable features of quantum systems, we realized parallel local processing to set a large database of outcomes and subsequently used post-selection to perform an ending global processing or a matching of information incoming from outside. We finally analyzed the details of the entire procedure, including its affordability and performance. The quantum circuit implementation, along with tentative applications, were also discussed. Such a model could be operated between large processing technological systems using communication procedures and also on a moderately controlled quantum matter conglomerate. Certain interesting technical aspects involving the non-local control of processing via entanglement were also analyzed in detail as an associated but notable premise. Full article
(This article belongs to the Special Issue Quantum Probability and Randomness IV)
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<p>Classic (blue) and Quantum (brown) interactions between database settlement and parallel processing to first set a distributed database moved on one localized, temporary memory instance. Critical aspects for each approach are remarked in red.</p> Full article ">Figure 2
<p>General scheme of the procedure with the steps considered: (MT, PP, GAA, and FGP). Each state mentioned is the final state of each step. Note that local information can be integrated in each local processing in the PP step.</p> Full article ">Figure 3
<p>Further processing is performed by each receiver, subsequently translating the outcomes on a selected party, thereby setting a local database. Finally, a final processing or pattern matching can be performed using a subset of selected outcomes, either stochastically via measurement or using Grover’s selection.</p> Full article ">Figure 4
<p>Quantum circuit version for multiple teleportation algorithm including a configurable qubit for their control <math display="inline"><semantics> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msub> <mi>φ</mi> <mi>C</mi> </msub> <mrow> <mo stretchy="false">〉</mo> </mrow> </mrow> </semantics></math>. A possible ending measurement on the control qubit with outcome <math display="inline"><semantics> <mrow> <mo stretchy="false">|</mo> <mi>c</mi> <mo stretchy="false">〉</mo> </mrow> </semantics></math> will definitively teleport the state <math display="inline"><semantics> <mrow> <mo stretchy="false">|</mo> <mi>ψ</mi> <mo stretchy="false">〉</mo> </mrow> </semantics></math> in the qubit 0 on the qubit <math display="inline"><semantics> <mrow> <mn>2</mn> <mi>c</mi> </mrow> </semantics></math>. Otherwise, teleportation remains superpositioned on the even qubits.</p> Full article ">Figure 5
<p>Settlement of a database using multiple teleportation and post-processing. It is based on a detonating initial state on local receivers integrating local information and processing. A selection of some outcomes can be obtained using amplitude amplification. The final output state can be used to extract global information or to match an external state on the database.</p> Full article ">Figure 6
<p>(<b>a</b>) Repetitions <span class="html-italic">s</span> of Grover algorithm to extract the processing <math display="inline"><semantics> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msubsup> <mi>φ</mi> <mn>0</mn> <mi>k</mi> </msubsup> <mrow> <mo stretchy="false">〉</mo> </mrow> </mrow> </semantics></math> raising its amplitude <math display="inline"><semantics> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msub> <mi>α</mi> <mi>k</mi> </msub> <msup> <mrow> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> </mrow> </semantics></math>; (<b>b</b>) Number of repetitions <span class="html-italic">R</span> (orange dashed line) to raise <math display="inline"><semantics> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msub> <mi>α</mi> <mi>k</mi> </msub> <msup> <mrow> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> <mo>≈</mo> <mn>1</mn> </mrow> </semantics></math>, along with its associated logarithmic error <math display="inline"><semantics> <mrow> <mo>−</mo> <mo form="prefix">log</mo> <mo stretchy="false">(</mo> <msub> <mo>Δ</mo> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msub> <mi>α</mi> <mi>k</mi> </msub> <msup> <mrow> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </semantics></math> (black dots), compared with the theoretical values (red dashed line).</p> Full article ">Figure 7
<p>For <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <msup> <mn>10</mn> <mn>6</mn> </msup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msub> <mi mathvariant="script">A</mi> <mi>e</mi> </msub> <mrow> <mo stretchy="false">〉</mo> <mo>=</mo> </mrow> <mfrac> <mn>1</mn> <msqrt> <mi>b</mi> </msqrt> </mfrac> <msub> <mo>∑</mo> <mrow> <mi>k</mi> <mo>∈</mo> <mi>K</mi> </mrow> </msub> <mrow> <mo stretchy="false">|</mo> <msub> <mi>A</mi> <mi>k</mi> </msub> <mo stretchy="false">〉</mo> </mrow> </mrow> </semantics></math>: (<b>a</b>) Repetitions <span class="html-italic">s</span> and distance <span class="html-italic">D</span> evolution of Grover algorithm for several values of <span class="html-italic">b</span>; (<b>b</b>) Number of repetitions needed <span class="html-italic">R</span> (solid blue line) compared with the theoretical value <math display="inline"><semantics> <msub> <mi>R</mi> <mi>th</mi> </msub> </semantics></math> (dashed black line) along with <span class="html-italic">D</span> (dashed red line) as a function of <math display="inline"><semantics> <mfrac> <mi>b</mi> <mi>n</mi> </mfrac> </semantics></math>.</p> Full article ">Figure 8
<p>Distribution of distance <span class="html-italic">D</span> versus ratio <math display="inline"><semantics> <mfrac> <mi>b</mi> <mi>n</mi> </mfrac> </semantics></math> in log-scales with the number of repetitions <span class="html-italic">R</span> in colour in agreement with the colour scale below, for: (<b>a</b>) both evenly distributed initial and target states, (<b>b</b>) evenly distributed initial state and randomly distributed target state, and (<b>c</b>) randomly distributed initial state and evenly distributed target state. Distributions were obtained by selecting <math display="inline"><semantics> <mrow> <mn>10</mn> <mo>≤</mo> <mi>n</mi> <mo>≤</mo> <msup> <mn>10</mn> <mn>3</mn> </msup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>≤</mo> <mi>b</mi> <mo>≤</mo> <mi>n</mi> </mrow> </semantics></math>. Upper-right insets show the normal scales in comparison with <a href="#entropy-25-00376-f007" class="html-fig">Figure 7</a>.</p> Full article ">Figure 9
<p>Close-up to plots in <a href="#entropy-25-00376-f008" class="html-fig">Figure 8</a> in the region <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>/</mo> <mi>n</mi> <mo>≪</mo> <mn>1</mn> </mrow> </semantics></math> in normal scales for (<b>a</b>) evenly distributed initial and target states, (<b>b</b>) evenly distributed initial state and randomly distributed target state, and (<b>c</b>) randomly distributed initial state and evenly distributed target state.</p> Full article ">Figure 10
<p>(<b>a</b>) Measurement process for pattern matching represented by the external state <math display="inline"><semantics> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msup> <mi>φ</mi> <mo>∗</mo> </msup> <mrow> <mo stretchy="false">〉</mo> </mrow> </mrow> </semantics></math>. (<b>b</b>) Tree diagram for the measurement process matching the pattern <math display="inline"><semantics> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msup> <mi>φ</mi> <mo>∗</mo> </msup> <mrow> <mo stretchy="false">〉</mo> </mrow> </mrow> </semantics></math> on Bob<math display="inline"><semantics> <msub> <mrow/> <mi>k</mi> </msub> </semantics></math> and subsequently identifying an improbable control state <math display="inline"><semantics> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msub> <mover accent="true"> <mi>ϕ</mi> <mo stretchy="false">^</mo> </mover> <mi>C</mi> </msub> <mrow> <mo stretchy="false">〉</mo> </mrow> </mrow> </semantics></math>. The figure of interest is <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>≡</mo> <msup> <mi>P</mi> <mo>∗</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>P</mi> <mi>C</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math>. (<b>c</b>) Mean of <span class="html-italic">P</span>, and (<b>d</b>) Standard deviation of <span class="html-italic">P</span>, as function of <span class="html-italic">b</span> and <math display="inline"><semantics> <msup> <mi>θ</mi> <mo>∗</mo> </msup> </semantics></math>. In the last two cases, dashed lines correspond to one exact matching in the database subset.</p> Full article ">Figure 11
<p>(<b>a</b>) Effect of <math display="inline"><semantics> <msub> <mi>U</mi> <mi>e</mi> </msub> </semantics></math> on each <math display="inline"><semantics> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msub> <mi mathvariant="script">A</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">〉</mo> <mo>,</mo> <mi>k</mi> <mo>∈</mo> <mi>K</mi> </mrow> </mrow> </semantics></math> as a multidimensional rotation by angles <math display="inline"><semantics> <msub> <mi>δ</mi> <mi>e</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>χ</mi> <mi>e</mi> </msub> </semantics></math> with respect to itself and other <math display="inline"><semantics> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msubsup> <mi mathvariant="script">A</mi> <mi>k</mi> <mo>′</mo> </msubsup> <mrow> <mo stretchy="false">〉</mo> <mo>,</mo> </mrow> <msup> <mi>k</mi> <mo>′</mo> </msup> <mo>∈</mo> <mi>K</mi> </mrow> </semantics></math>. (<b>b</b>) Effect of <math display="inline"><semantics> <msub> <mi>U</mi> <mi>m</mi> </msub> </semantics></math> on each <math display="inline"><semantics> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msub> <mi mathvariant="script">A</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">〉</mo> </mrow> </mrow> </semantics></math> as a multidimensional rotation by angles <math display="inline"><semantics> <msub> <mi>δ</mi> <mi>m</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>χ</mi> <mi>m</mi> </msub> </semantics></math> with respect to itself and other <math display="inline"><semantics> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msubsup> <mi mathvariant="script">A</mi> <mi>k</mi> <mo>′</mo> </msubsup> <mrow> <mo stretchy="false">〉</mo> </mrow> </mrow> </semantics></math>. (<b>c</b>) Simplified Grover circuit considering the oracle <math display="inline"><semantics> <msub> <mi>U</mi> <mi>e</mi> </msub> </semantics></math> and the Grover diffusion operator <math display="inline"><semantics> <msub> <mi>U</mi> <mi>m</mi> </msub> </semantics></math>, both repeated <span class="html-italic">R</span> times after teleportation and processing, with a possible control state measurement at the end.</p> Full article ">Figure 12
<p>Pattern matching problem with translations. (<b>a</b>) Ten first images of an original dataset with the image on the left stored as local information, and those on the right obtained as an outcome by cyclic right-forward (<math display="inline"><semantics> <mrow> <mo stretchy="false">|</mo> <mn>0</mn> <mo stretchy="false">〉</mo> </mrow> </semantics></math>) or upward (<math display="inline"><semantics> <mrow> <mo stretchy="false">|</mo> <mn>1</mn> <mo stretchy="false">〉</mo> </mrow> </semantics></math>) translations in superposition obtained under parallel processing. (<b>b</b>) First five images in the subset of the previous dataset obtained by amplitude amplification on the cases with exactly 8 dark pixels.</p> Full article ">
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