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Quantum Computation and Information: Multi-Particle Aspects
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Quantum Computation and Information: Multi-Particle Aspects

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

Deadline for manuscript submissions: closed (30 June 2015) | Viewed by 88903

Special Issue Editors

Prof. Dr. Demosthenes Ellinas

E-Mail Website
Guest Editor
School of Electrical and Computer Engineering QLab, Technical University of Crete, 73100 Chania, Greece
Interests: mathematical physics; quantum information science; quantum computation; lie groups algebras and applications
Prof. Dr. Giorgio Kaniadakis

E-Mail Website
Guest Editor
Dipartimento di Scienza Applicata e Tecnologia, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
Interests: entropy; statistical physics; foundation of statistical mechanics; complex systems
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Jiannis Pachos

E-Mail Website
Guest Editor
Centre for Topological Quantum Information, University of Leeds, Leeds, UK
Interests: Topological quantum computation and topological systems
Dr. Antonio M. Scarfone

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Guest Editor
Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche (ISC-CNR), c/o DISAT, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Turin, Italy
Interests: nonextensive statistical mechanics; nonlinear fokker-planck equations; geometry information; nonlinear schroedinger equation; quantum groups and quantum algebras; complex systems
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This Special Issue mainly focuses on state-of-the-art advancements concerning multi-particles. These advancements are emerging in various research directions within the field of Quantum Computation and Information. In particular, the Issue will target works on research topics that blend multi-particle physical and mathematical modelling with techniques and concepts from quantum statistical mechanics. Within this framework, the problems addressed specifically concern various kinds of quantum computational and informational multi-particle algorithms and related matters, such as performance and design, as well as quantum walks, topological quantum computing, and multi-particle entanglement.

Prof. Dr. Demosthenes Ellinas
Prof. Dr. Giorgio Kaniadakis
Prof. Dr. Jiannis Pachos
Prof. Dr. Antonio Maria Scarfone
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 computation
  • quantum information
  • multi-particle models
  • quantum walks
  • topological quantum computing
  • multi-particle entanglement
  • quantum potential
  • quantum statistical mechanics

Published Papers (17 papers)

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Editorial

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180 KiB  
Editorial
Quantum Computation and Information: Multi-Particle Aspects
by Demosthenes Ellinas, Giorgio Kaniadakis, Jiannis K. Pachos and Antonio Maria Scarfone
Entropy 2016, 18(9), 339; https://doi.org/10.3390/e18090339 - 20 Sep 2016
Viewed by 4319
Abstract
This editorial explains the scope of the special issue and provides a thematic introduction to the contributed papers. Full article
(This article belongs to the Special Issue Quantum Computation and Information: Multi-Particle Aspects)

Research

Jump to: Editorial

248 KiB  
Article
Payoffs and Coherence of a Quantum Two-Player Game in a Thermal Environment
by Jerzy Dajka, Marcin Łobejko and Jan Sładkowski
Entropy 2015, 17(11), 7736-7751; https://doi.org/10.3390/e17117736 - 13 Nov 2015
Cited by 5 | Viewed by 4047
Abstract
A two-player quantum game is considered in the presence of a thermal decoherence modeled in terms of a rigorous Davies approach. It is shown how the energy dissipation and pure decoherence affect the payoffs of the players of the (quantum version) of prisoner [...] Read more.
A two-player quantum game is considered in the presence of a thermal decoherence modeled in terms of a rigorous Davies approach. It is shown how the energy dissipation and pure decoherence affect the payoffs of the players of the (quantum version) of prisoner dilemma. The impact of the thermal environment on a coherence of game, as a quantum system, is also presented. Full article
(This article belongs to the Special Issue Quantum Computation and Information: Multi-Particle Aspects)
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2714 KiB  
Article
A Tale of Two Entangled Instabilities—The Dual Role of δ-O in HgBa2Can-1CunO2(n+1)+δ
by Itai Panas
Entropy 2015, 17(10), 6765-6782; https://doi.org/10.3390/e17106765 - 05 Oct 2015
Cited by 4 | Viewed by 4887
Abstract
Low-energy instabilities in the hole-doped cuprates include, besides short range antiferromagnetic fluctuations and superconductivity, also ubiquitous translational and rotational symmetry breakings. The overwhelming majority of interpretations of these possibly related properties rely on mappings onto three bands spanned by the three atomic orbitals [...] Read more.
Low-energy instabilities in the hole-doped cuprates include, besides short range antiferromagnetic fluctuations and superconductivity, also ubiquitous translational and rotational symmetry breakings. The overwhelming majority of interpretations of these possibly related properties rely on mappings onto three bands spanned by the three atomic orbitals Cu3d(x2−y2)(σ), O2px(σ), and O2py(σ), these three local orbitals spanning the Zhang–Rice band (ZRB), the lower Hubbard bands (LHB) and the upper Hubbard bands (UHB), respectively. Here we demonstrate by means of supercell Density Functional Theory (DFT) (a) how oxygen intercalation affects the structures of the buffer layers, and (b) how the attenuated crystal field pulls two additional oxygen bands in the CuO2 plane to the Fermi level. The self-consistent changes in electronic structure reflected in the corresponding changes in external potential comprise formal properties of the Hohenberg–Kohn theorems. Validation of present days’ approximate exchange-correlation potentials to capture these qualitative effects by means of supercell DFT is made by comparing computed doping dependent structural shifts to corresponding experimentally observed correlations. The simplest generalization of Bardeen–Cooper–Schrieffer (BCS) theory is offered to articulate high-critical temperature superconductivity (HTS) from a normal state where crystal field causes states related to two non-hybridizing bands to coalesce at EF. Full article
(This article belongs to the Special Issue Quantum Computation and Information: Multi-Particle Aspects)
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875 KiB  
Article
Quantum Secure Direct Communication Based on Dense Coding and Detecting Eavesdropping with Four-Particle Genuine Entangled State
by Jian Li, Zeshi Pan, Fengqi Sun, Yanhua Chen, Zheng Wang and Zuozhi Shi
Entropy 2015, 17(10), 6743-6752; https://doi.org/10.3390/e17106743 - 30 Sep 2015
Cited by 10 | Viewed by 4970
Abstract
A novel quantum secure direct communication protocol based on four-particle genuine entangled state and quantum dense coding is proposed. In this protocol, the four-particle genuine entangled state is used to detect eavesdroppers, and quantum dense coding is used to encode the message. Finally, [...] Read more.
A novel quantum secure direct communication protocol based on four-particle genuine entangled state and quantum dense coding is proposed. In this protocol, the four-particle genuine entangled state is used to detect eavesdroppers, and quantum dense coding is used to encode the message. Finally, the security of the proposed protocol is discussed. During the security analysis, the method of entropy theory is introduced, and two detection strategies are compared quantitatively by comparing the relationship between the maximal information that the eavesdroppers (Eve) can obtain, and the probability of being detected. Through the analysis we can state that our scheme is feasible and secure. Full article
(This article belongs to the Special Issue Quantum Computation and Information: Multi-Particle Aspects)
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313 KiB  
Article
Self-Similar Solutions of Rényi’s Entropy and the Concavity of Its Entropy Power
by Agapitos N. Hatzinikitas
Entropy 2015, 17(9), 6056-6071; https://doi.org/10.3390/e17096056 - 31 Aug 2015
Cited by 2 | Viewed by 4758
Abstract
We study the class of self-similar probability density functions with finite mean and variance, which maximize Rényi’s entropy. The investigation is restricted in the Schwartz space S(Rd) and in the space of l-differentiable compactly supported functions Clc (Rd). Interestingly, the solutions of this [...] Read more.
We study the class of self-similar probability density functions with finite mean and variance, which maximize Rényi’s entropy. The investigation is restricted in the Schwartz space S(Rd) and in the space of l-differentiable compactly supported functions Clc (Rd). Interestingly, the solutions of this optimization problem do not coincide with the solutions of the usual porous medium equation with a Dirac point source, as occurs in the optimization of Shannon’s entropy. We also study the concavity of the entropy power in Rd with respect to time using two different methods. The first one takes advantage of the solutions determined earlier, while the second one is based on a setting that could be used for Riemannian manifolds. Full article
(This article belongs to the Special Issue Quantum Computation and Information: Multi-Particle Aspects)
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1675 KiB  
Article
Minimal Rényi–Ingarden–Urbanik Entropy of Multipartite Quantum States
by Marco Enríquez, Zbigniew Puchała and Karol Życzkowski
Entropy 2015, 17(7), 5063-5084; https://doi.org/10.3390/e17075063 - 20 Jul 2015
Cited by 12 | Viewed by 5419
Abstract
We study the entanglement of a pure state of a composite quantum system consisting of several subsystems with d levels each. It can be described by the Rényi–Ingarden–Urbanik entropy Sq of a decomposition of the state in a product basis, minimized over [...] Read more.
We study the entanglement of a pure state of a composite quantum system consisting of several subsystems with d levels each. It can be described by the Rényi–Ingarden–Urbanik entropy Sq of a decomposition of the state in a product basis, minimized over all local unitary transformations. In the case q = 0, this quantity becomes a function of the rank of the tensor representing the state, while in the limit q → ∞, the entropy becomes related to the overlap with the closest separable state and the geometric measure of entanglement. For any bipartite system, the entropy S1 coincides with the standard entanglement entropy. We analyze the distribution of the minimal entropy for random states of three- and four-qubit systems. In the former case, the distribution of the three-tangle is studied and some of its moments are evaluated, while in the latter case, we analyze the distribution of the hyperdeterminant. The behavior of the maximum overlap of a three-qudit system with the closest separable state is also investigated in the asymptotic limit. Full article
(This article belongs to the Special Issue Quantum Computation and Information: Multi-Particle Aspects)
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3877 KiB  
Article
The Critical Point Entanglement and Chaos in the Dicke Model
by Lina Bao, Feng Pan, Jing Lu and Jerry P. Draayer
Entropy 2015, 17(7), 5022-5042; https://doi.org/10.3390/e17075022 - 16 Jul 2015
Cited by 5 | Viewed by 5759
Abstract
Ground state properties and level statistics of the Dicke model for a finite number of atoms are investigated based on a progressive diagonalization scheme (PDS). Particle number statistics, the entanglement measure and the Shannon information entropy at the resonance point in cases with [...] Read more.
Ground state properties and level statistics of the Dicke model for a finite number of atoms are investigated based on a progressive diagonalization scheme (PDS). Particle number statistics, the entanglement measure and the Shannon information entropy at the resonance point in cases with a finite number of atoms as functions of the coupling parameter are calculated. It is shown that the entanglement measure defined in terms of the normalized von Neumann entropy of the reduced density matrix of the atoms reaches its maximum value at the critical point of the quantum phase transition where the system is most chaotic. Noticeable change in the Shannon information entropy near or at the critical point of the quantum phase transition is also observed. In addition, the quantum phase transition may be observed not only in the ground state mean photon number and the ground state atomic inversion as shown previously, but also in fluctuations of these two quantities in the ground state, especially in the atomic inversion fluctuation. Full article
(This article belongs to the Special Issue Quantum Computation and Information: Multi-Particle Aspects)
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268 KiB  
Article
Faster Together: Collective Quantum Search
by Demosthenes Ellinas and Christos Konstandakis
Entropy 2015, 17(7), 4838-4862; https://doi.org/10.3390/e17074838 - 10 Jul 2015
Cited by 3 | Viewed by 4056
Abstract
Joining independent quantum searches provides novel collective modes of quantum search (merging) by utilizing the algorithm’s underlying algebraic structure. If n quantum searches, each targeting a single item, join the domains of their classical oracle functions and sum their Hilbert spaces (merging), instead [...] Read more.
Joining independent quantum searches provides novel collective modes of quantum search (merging) by utilizing the algorithm’s underlying algebraic structure. If n quantum searches, each targeting a single item, join the domains of their classical oracle functions and sum their Hilbert spaces (merging), instead of acting independently (concatenation), then they achieve a reduction of the search complexity by factor O(√n). Full article
(This article belongs to the Special Issue Quantum Computation and Information: Multi-Particle Aspects)
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3729 KiB  
Article
Non-Abelian Topological Approach to Non-Locality of a Hypergraph State
by Vesna Berec
Entropy 2015, 17(5), 3376-3399; https://doi.org/10.3390/e17053376 - 15 May 2015
Cited by 4 | Viewed by 4834
Abstract
We present a theoretical study of new families of stochastic complex information modules encoded in the hypergraph states which are defined by the fractional entropic descriptor. The essential connection between the Lyapunov exponents and d-regular hypergraph fractal set is elucidated. To further [...] Read more.
We present a theoretical study of new families of stochastic complex information modules encoded in the hypergraph states which are defined by the fractional entropic descriptor. The essential connection between the Lyapunov exponents and d-regular hypergraph fractal set is elucidated. To further resolve the divergence in the complexity of classical and quantum representation of a hypergraph, we have investigated the notion of non-amenability and its relation to combinatorics of dynamical self-organization for the case of fractal system of free group on finite generators. The exact relation between notion of hypergraph non-locality and quantum encoding through system sets of specified non-Abelian fractal geometric structures is presented. Obtained results give important impetus towards designing of approximation algorithms for chip imprinted circuits in scalable quantum information systems. Full article
(This article belongs to the Special Issue Quantum Computation and Information: Multi-Particle Aspects)
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883 KiB  
Article
Stabilization Effects of Dichotomous Noise on the Lifetime of the Superconducting State in a Long Josephson Junction
by Claudio Guarcello, Davide Valenti, Angelo Carollo and Bernardo Spagnolo
Entropy 2015, 17(5), 2862-2875; https://doi.org/10.3390/e17052862 - 06 May 2015
Cited by 97 | Viewed by 5209
Abstract
We investigate the superconducting lifetime of a long overdamped current-biased Josephson junction, in the presence of telegraph noise sources. The analysis is performed by randomly choosing the initial condition for the noise source. However, in order to investigate how the initial value of [...] Read more.
We investigate the superconducting lifetime of a long overdamped current-biased Josephson junction, in the presence of telegraph noise sources. The analysis is performed by randomly choosing the initial condition for the noise source. However, in order to investigate how the initial value of the dichotomous noise affects the phase dynamics, we extend our analysis using two different fixed initial values for the source of random fluctuations. In our study, the phase dynamics of the Josephson junction is analyzed as a function of the noise signal intensity, for different values of the parameters of the system and external driving currents. We find that the mean lifetime of the superconductive metastable state as a function of the noise intensity is characterized by nonmonotonic behavior, strongly related to the soliton dynamics during the switching towards the resistive state. The role of the correlation time of the noise source is also taken into account. Noise-enhanced stability is observed in the investigated system. Full article
(This article belongs to the Special Issue Quantum Computation and Information: Multi-Particle Aspects)
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652 KiB  
Article
Multi-State Quantum Dissipative Dynamics in Sub-Ohmic Environment: The Strong Coupling Regime
by Luca Magazzù, Davide Valenti, Angelo Carollo and Bernardo Spagnolo
Entropy 2015, 17(4), 2341-2354; https://doi.org/10.3390/e17042341 - 17 Apr 2015
Cited by 23 | Viewed by 5599
Abstract
We study the dissipative quantum dynamics and the asymptotic behavior of a particle in a bistable potential interacting with a sub-Ohmic broadband environment. The reduced dynamics, in the intermediate to strong dissipation regime, is obtained beyond the two-level system approximation by using a [...] Read more.
We study the dissipative quantum dynamics and the asymptotic behavior of a particle in a bistable potential interacting with a sub-Ohmic broadband environment. The reduced dynamics, in the intermediate to strong dissipation regime, is obtained beyond the two-level system approximation by using a real-time path integral approach. We find a crossover dynamic regime with damped intra-well oscillations and incoherent tunneling and a completely incoherent regime at strong damping. Moreover, a nonmonotonic behavior of the left/right well population difference is found as a function of the damping strength. Full article
(This article belongs to the Special Issue Quantum Computation and Information: Multi-Particle Aspects)
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233 KiB  
Article
A Simple Decoder for Topological Codes
by James Wootton
Entropy 2015, 17(4), 1946-1957; https://doi.org/10.3390/e17041946 - 01 Apr 2015
Cited by 20 | Viewed by 6013
Abstract
Here we study an efficient algorithm for decoding topological codes. It is a simple form of HDRG decoder, which could be straightforwardly generalized to complex decoding problems. Specific results are obtained for the planar code with both i.i.d. and spatially correlated errors. The [...] Read more.
Here we study an efficient algorithm for decoding topological codes. It is a simple form of HDRG decoder, which could be straightforwardly generalized to complex decoding problems. Specific results are obtained for the planar code with both i.i.d. and spatially correlated errors. The method is shown to compare well with existing ones, despite its simplicity. Full article
(This article belongs to the Special Issue Quantum Computation and Information: Multi-Particle Aspects)
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2145 KiB  
Article
Statistical Correlations of the N-particle Moshinsky Model
by Hsuan Tung Peng and Yew Kam Ho
Entropy 2015, 17(4), 1882-1895; https://doi.org/10.3390/e17041882 - 31 Mar 2015
Cited by 19 | Viewed by 5892
Abstract
We study the correlation of the ground state of an N-particle Moshinsky model by computing the Shannon entropy in both position and momentum spaces. We have derived the Shannon entropy and mutual information with analytical forms of such an N-particle Moshinsky [...] Read more.
We study the correlation of the ground state of an N-particle Moshinsky model by computing the Shannon entropy in both position and momentum spaces. We have derived the Shannon entropy and mutual information with analytical forms of such an N-particle Moshinsky model, and this helps us test the entropic uncertainty principle. The Shannon entropy in position space decreases as interaction strength increases. However, Shannon entropy in momentum space has the opposite trend. Shannon entropy of the whole system satisfies the equality of entropic uncertainty principle. Our results also indicate that, independent of the sizes of the two subsystems, the mutual information increases monotonically as the interaction strength increases. Full article
(This article belongs to the Special Issue Quantum Computation and Information: Multi-Particle Aspects)
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587 KiB  
Article
Generalized Remote Preparation of Arbitrary m-qubit Entangled States via Genuine Entanglements
by Dong Wang, Ross D. Hoehn, Liu Ye and Sabre Kais
Entropy 2015, 17(4), 1755-1774; https://doi.org/10.3390/e17041755 - 30 Mar 2015
Cited by 26 | Viewed by 5954
Abstract
Herein, we present a feasible, general protocol for quantum communication within a network via generalized remote preparation of an arbitrary m-qubit entangled state designed with genuine tripartite Greenberger–Horne–Zeilinger-type entangled resources. During the implementations, we construct novel collective unitary operations; these operations are tasked [...] Read more.
Herein, we present a feasible, general protocol for quantum communication within a network via generalized remote preparation of an arbitrary m-qubit entangled state designed with genuine tripartite Greenberger–Horne–Zeilinger-type entangled resources. During the implementations, we construct novel collective unitary operations; these operations are tasked with performing the necessary phase transfers during remote state preparations. We have distilled our implementation methods into a five-step procedure, which can be used to faithfully recover the desired state during transfer. Compared to previous existing schemes, our methodology features a greatly increased success probability. After the consumption of auxiliary qubits and the performance of collective unitary operations, the probability of successful state transfer is increased four-fold and eight-fold for arbitrary two- and three-qubit entanglements when compared to other methods within the literature, respectively. We conclude this paper with a discussion of the presented scheme for state preparation, including: success probabilities, reducibility and generalizability. Full article
(This article belongs to the Special Issue Quantum Computation and Information: Multi-Particle Aspects)
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657 KiB  
Article
Quantum Discord and Information Deficit in Spin Chains
by Norma Canosa, Leonardo Ciliberti and Raúl Rossignoli
Entropy 2015, 17(4), 1634-1659; https://doi.org/10.3390/e17041634 - 26 Mar 2015
Cited by 8 | Viewed by 5804
Abstract
We examine the behavior of quantum correlations of spin pairs in a finite anisotropic XY spin chain immersed in a transverse magnetic field, through the analysis of the quantum discord and the conventional and quadratic one-way information deficits. We first provide a brief [...] Read more.
We examine the behavior of quantum correlations of spin pairs in a finite anisotropic XY spin chain immersed in a transverse magnetic field, through the analysis of the quantum discord and the conventional and quadratic one-way information deficits. We first provide a brief review of these measures, showing that the last ones can be obtained as particular cases of a generalized information deficit based on general entropic forms. All of these measures coincide with an entanglement entropy in the case of pure states, but can be non-zero in separable mixed states, vanishing just for classically correlated states. It is then shown that their behavior in the exact ground state of the chain exhibits similar features, deviating significantly from that of the pair entanglement below the critical field. In contrast with entanglement, they reach full range in this region, becoming independent of the pair separation and coupling range in the immediate vicinity of the factorizing field. It is also shown, however, that significant differences between the quantum discord and the information deficits arise in the local minimizing measurement that defines them. Both analytical and numerical results are provided. Full article
(This article belongs to the Special Issue Quantum Computation and Information: Multi-Particle Aspects)
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6666 KiB  
Article
Do Transitive Preferences Always Result in Indifferent Divisions?
by Marcin Makowski, Edward W. Piotrowski and Jan Sładkowski
Entropy 2015, 17(3), 968-983; https://doi.org/10.3390/e17030968 - 02 Mar 2015
Cited by 11 | Viewed by 5556
Abstract
The transitivity of preferences is one of the basic assumptions used in the theory of games and decisions. It is often equated with the rationality of choice and is considered useful in building rankings. Intransitive preferences are considered paradoxical and undesirable. This problem [...] Read more.
The transitivity of preferences is one of the basic assumptions used in the theory of games and decisions. It is often equated with the rationality of choice and is considered useful in building rankings. Intransitive preferences are considered paradoxical and undesirable. This problem is discussed by many social and natural scientists. A simple model of a sequential game in which two players choose one of the two elements in each iteration is discussed in this paper. The players make their decisions in different contexts defined by the rules of the game. It appears that the optimal strategy of one of the players can only be intransitive (the so-called relevant intransitive strategy)! On the other hand, the optimal strategy for the second player can be either transitive or intransitive. A quantum model of the game using pure one-qubit strategies is considered. In this model, an increase in the importance of intransitive strategies is observed: there is a certain course of the game where intransitive strategies are the only optimal strategies for both players. The study of decision-making models using quantum information theory tools may shed some new light on the understanding of mechanisms that drive the formation of types of preferences. Full article
(This article belongs to the Special Issue Quantum Computation and Information: Multi-Particle Aspects)
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196 KiB  
Article
Probabilistic Three-Party Sharing of Operation on a Remote Qubit
by Chuanmei Xie, Yimin Liu, Hang Xing and Zhanjun Zhang
Entropy 2015, 17(2), 841-851; https://doi.org/10.3390/e17020841 - 12 Feb 2015
Cited by 15 | Viewed by 4712
Abstract
A probabilistic tripartite single-qubit operation sharing scheme is put forward by utilizing a two-qubit and a three-qubit non-maximally entangled state as quantum channels. Some specific comparisons between our scheme and another probabilistic scheme are made. It is found that, if the product of [...] Read more.
A probabilistic tripartite single-qubit operation sharing scheme is put forward by utilizing a two-qubit and a three-qubit non-maximally entangled state as quantum channels. Some specific comparisons between our scheme and another probabilistic scheme are made. It is found that, if the product of the two minimal coefficients characterizing channel entanglements is greater than 3/16, our scheme is more superior than the other one. Nonetheless, the price is that more classical and quantum resources are consumed, and the operation difficulty is rather increased. Moreover, some important features of the scheme, such as its security, probability and sharer symmetry, are revealed through concrete discussions. Additionally, the experimental feasibility of our scheme is analyzed and subsequently confirmed according to the current experimental techniques. Full article
(This article belongs to the Special Issue Quantum Computation and Information: Multi-Particle Aspects)
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