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14 pages, 596 KiB

Open AccessArticle

A Study of Spin 1 Unruh–De Witt Detectors

by F. M. Guedes, M. S. Guimaraes, I. Roditi and S. P. Sorella

Universe 2024, 10(8), 307; https://doi.org/10.3390/universe10080307 - 24 Jul 2024

Cited by 1 | Viewed by 458

A study of the interaction of spin 1 Unruh–De Witt detectors with a relativistic scalar quantum field is presented here. After tracing out the field modes, the resulting density matrix for a bipartite qutrit system is employed to investigate the violation of the [...] Read more.

A study of the interaction of spin 1 Unruh–De Witt detectors with a relativistic scalar quantum field is presented here. After tracing out the field modes, the resulting density matrix for a bipartite qutrit system is employed to investigate the violation of the Bell–CHSH inequality. Unlike the case of spin

1 / 2

, for which the effects of the quantum field result in a decrease in the size of violation, in the case of spin 1, both a decrease or an increase in the size of the violation may occur. This effect is ascribed to the fact that Tsirelson’s bound is not saturated in the case of qutrits. Full article

(This article belongs to the Section Field Theory)

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19 pages, 731 KiB

Open AccessArticle

Correlations in the EPR State Observables

by Daniel F. Orsini, Luna R. N. Oliveira and Marcos G. E. da Luz

Entropy 2024, 26(6), 476; https://doi.org/10.3390/e26060476 - 30 May 2024

Viewed by 555

Abstract

The identification and physical interpretation of arbitrary quantum correlations are not always effortless. Two features that can significantly influence the dispersion of the joint observable outcomes in a quantum bipartite system composed of systems I and II are: (a) All possible pairs of [...] Read more.

The identification and physical interpretation of arbitrary quantum correlations are not always effortless. Two features that can significantly influence the dispersion of the joint observable outcomes in a quantum bipartite system composed of systems I and II are: (a) All possible pairs of observables describing the composite are equally probable upon measurement, and (b) The absence of concurrence (positive reinforcement) between any of the observables within a particular system; implying that their associated operators do not commute. The so-called EPR states are known to observe (a). Here, we demonstrate in very general (but straightforward) terms that they also satisfy condition (b), a relevant technical fact often overlooked. As an illustration, we work out in detail the three-level systems, i.e., qutrits. Furthermore, given the special characteristics of EPR states (such as maximal entanglement, among others), one might intuitively expect the CHSH correlation, computed exclusively for the observables of qubit EPR states, to yield values greater than two, thereby violating Bell’s inequality. We show such a prediction does not hold true. In fact, the combined properties of (a) and (b) lead to a more limited range of values for the CHSH measure, not surpassing the nonlocality threshold of two. The present constitutes an instructive example of the subtleties of quantum correlations. Full article

(This article belongs to the Special Issue Quantum Probability and Randomness V)

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25 pages, 518 KiB

Open AccessFeature PaperArticle

The Fuzzy Bit

by Milagrosa Aldana and María Antonia Lledó

Symmetry 2023, 15(12), 2103; https://doi.org/10.3390/sym15122103 - 23 Nov 2023

Viewed by 1478

Abstract

In this paper, the formulation of Quantum Mechanics in terms of fuzzy logic and fuzzy sets is explored. A result by Pykacz, which establishes a correspondence between (quantum) logics (lattices with certain properties) and certain families of fuzzy sets, is applied to the [...] Read more.

In this paper, the formulation of Quantum Mechanics in terms of fuzzy logic and fuzzy sets is explored. A result by Pykacz, which establishes a correspondence between (quantum) logics (lattices with certain properties) and certain families of fuzzy sets, is applied to the Birkhoff–von Neumann logic, the lattice of projectors of a Hilbert space. Three cases are considered: the qubit, two qubits entangled, and a qutrit ‘nested’ inside the two entangled qubits. The membership functions of the fuzzy sets are explicitly computed and all the connectives of the fuzzy sets are interpreted as operations with these particular membership functions. In this way, a complete picture of the standard quantum logic in terms of fuzzy sets is obtained for the systems considered. Full article

(This article belongs to the Section Physics)

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30 pages, 1352 KiB

Open AccessFeature PaperArticle

Work Fluctuations in Ergotropic Heat Engines

by Giovanni Chesi, Chiara Macchiavello and Massimiliano Federico Sacchi

Entropy 2023, 25(11), 1528; https://doi.org/10.3390/e25111528 - 9 Nov 2023

Cited by 1 | Viewed by 925

Abstract

We study the work fluctuations in ergotropic heat engines, namely two-stroke quantum Otto engines where the work stroke is designed to extract the ergotropy (the maximum amount of work by a cyclic unitary evolution) from a couple of quantum systems at canonical equilibrium [...] Read more.

We study the work fluctuations in ergotropic heat engines, namely two-stroke quantum Otto engines where the work stroke is designed to extract the ergotropy (the maximum amount of work by a cyclic unitary evolution) from a couple of quantum systems at canonical equilibrium at two different temperatures, whereas the heat stroke thermalizes back the systems to their respective reservoirs. We provide an exhaustive study for the case of two qutrits whose energy levels are equally spaced at two different frequencies by deriving the complete work statistics. By varying the values of temperatures and frequencies, only three kinds of optimal unitary strokes are found: the swap operator

U_{1}

, an idle swap

U_{2}

(where one of the qutrits is regarded as an effective qubit), and a non-trivial permutation of energy eigenstates

U_{3}

, which indeed corresponds to the composition of the two previous unitaries, namely

U_{3} = U_{2} U_{1}

. While

U_{1}

and

U_{2}

are Hermitian (and hence involutions),

U_{3}

is not. This point has an impact on the thermodynamic uncertainty relations (TURs), which bound the signal-to-noise ratio of the extracted work in terms of the entropy production. In fact, we show that all TURs derived from a strong detailed fluctuation theorem are violated by the transformation

U_{3}

. Full article

(This article belongs to the Section Statistical Physics)

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0 pages, 1029 KiB

Open AccessArticle

Non-Kochen–Specker Contextuality

by Mladen Pavičić

Entropy 2023, 25(8), 1117; https://doi.org/10.3390/e25081117 - 26 Jul 2023

Cited by 1 | Viewed by 1762 | 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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7 pages, 274 KiB

Open AccessProceeding Paper

Qutrit-Based Orthogonal Approximations with Inverse-Free Quantum Gate Sets

by Anuradha Mahasinghe, Dulmi Fernando and Kaushika De Silva

Comput. Sci. Math. Forum 2023, 7(1), 12; https://doi.org/10.3390/IOCMA2023-14416 - 13 Jun 2023

Viewed by 981

Abstract

The efficient compiling of arbitrary single-qubit gates into a sequence of gates from a finite gate set is of fundamental importance in quantum computation. The exact bounds of this compilation are given by the Solovay–Kitaev theorem, which serves as a powerful tool in [...] Read more.

The efficient compiling of arbitrary single-qubit gates into a sequence of gates from a finite gate set is of fundamental importance in quantum computation. The exact bounds of this compilation are given by the Solovay–Kitaev theorem, which serves as a powerful tool in compiling quantum algorithms that require many qubits. However, the inverse closure condition it imposes on the gate set adds a certain complexity to the experimental compilation, making the process less efficient. This was recently resolved by a version of the Solovay–Kitaev theorem for inverse-free gate sets, yielding a significant gain. Considering the recent progress in the field of three-level quantum systems, in which qubits are replaced by qutrits, it is possible to achieve the quantum speedup guaranteed by the Solovay–Kitaev theorem simply from orthogonal gates. Nevertheless, it has not been investigated previously whether the condition of inverse closure can be relaxed for these qutrit-based orthogonal compilations as well. In this work, we answer this positively, by obtaining improved Solovay–Kitaev approximations to an arbitrary orthogonal qutrit gate, with an accuracy

ε

from a sequence of

O (l o g^{8.62} (1 / ε))

orthogonal gates taken from an inverse-free set. Full article

(This article belongs to the Proceedings of The 1st International Online Conference on Mathematics and Applications)

18 pages, 1602 KiB

Open AccessArticle

Pseudo-Qutrit Formed by Two Interacting Identical Spins (s = 1/2) in a Variable External Magnetic Field

by Yury Belousov, Igor Chernousov and Vladimir Man’ko

Entropy 2023, 25(5), 716; https://doi.org/10.3390/e25050716 - 26 Apr 2023

Cited by 1 | Viewed by 1338

Abstract

An analytical solution is obtained for the problem of two interacting, identical but separated spin 1/2 particles in a time-dependent external magnetic field, in a general case. The solution involves isolating the pseudo-qutrit subsystem from a two-qubit system. It is shown that the [...] Read more.

An analytical solution is obtained for the problem of two interacting, identical but separated spin 1/2 particles in a time-dependent external magnetic field, in a general case. The solution involves isolating the pseudo-qutrit subsystem from a two-qubit system. It is shown that the quantum dynamics of a pseudo-qutrit system with a magnetic dipole–dipole interaction can be described clearly and accurately in an adiabatic representation, using a time-dependent basis set. The transition probabilities between the energy levels for an adiabatically varying magnetic field, which follows the Landau–Majorana–Stuckelberg–Zener (LMSZ) model within a short time interval, are illustrated in the appropriate graphs. It is shown that for close energy levels and entangled states, the transition probabilities are not small and strongly depend on the time. These results provide insight into the degree of entanglement of two spins (qubits) over time. Furthermore, the results are applicable to more complex systems with a time-dependent Hamiltonian. Full article

(This article belongs to the Section Statistical Physics)

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24 pages, 417 KiB

Open AccessArticle

Hermitian and Unitary Almost-Companion Matrices of Polynomials on Demand

by Liubov A. Markovich, Agostino Migliore and Antonino Messina

Entropy 2023, 25(2), 309; https://doi.org/10.3390/e25020309 - 8 Feb 2023

Viewed by 1570

Abstract

We introduce the concept of the almost-companion matrix (ACM) by relaxing the non-derogatory property of the standard companion matrix (CM). That is, we define an ACM as a matrix whose characteristic polynomial coincides with a given monic and generally complex polynomial. The greater [...] Read more.

We introduce the concept of the almost-companion matrix (ACM) by relaxing the non-derogatory property of the standard companion matrix (CM). That is, we define an ACM as a matrix whose characteristic polynomial coincides with a given monic and generally complex polynomial. The greater flexibility inherent in the ACM concept, compared to CM, allows the construction of ACMs that have convenient matrix structures satisfying desired additional conditions, compatibly with specific properties of the polynomial coefficients. We demonstrate the construction of Hermitian and unitary ACMs starting from appropriate third-degree polynomials, with implications for their use in physical-mathematical problems, such as the parameterization of the Hamiltonian, density, or evolution matrix of a qutrit. We show that the ACM provides a means of identifying the properties of a given polynomial and finding its roots. For example, we describe the ACM-based solution of cubic complex algebraic equations without resorting to the use of the Cardano-Dal Ferro formulas. We also show the necessary and sufficient conditions on the coefficients of a polynomial for it to represent the characteristic polynomial of a unitary ACM. The presented approach can be generalized to complex polynomials of higher degrees. Full article

(This article belongs to the Special Issue Quantum Mechanics and Its Foundations III)

11 pages, 543 KiB

Open AccessArticle

Bypassing the Kochen–Specker Theorem: An Explicit Non-Contextual Statistical Model for the Qutrit

by David H. Oaknin

Axioms 2023, 12(1), 90; https://doi.org/10.3390/axioms12010090 - 15 Jan 2023

Cited by 1 | Viewed by 1562

Abstract

We describe an explicitly non-contextual statistical model of hidden variables for the qutrit, which fully reproduces the predictions of quantum mechanics, and thus, bypasses the constraints imposed by the Kochen–Specker theorem and its subsequent reformulations. We notice that these renowned theorems crucially rely [...] Read more.

We describe an explicitly non-contextual statistical model of hidden variables for the qutrit, which fully reproduces the predictions of quantum mechanics, and thus, bypasses the constraints imposed by the Kochen–Specker theorem and its subsequent reformulations. We notice that these renowned theorems crucially rely on the implicitly assumed existence of an absolute frame of reference with respect to which physically indistinguishable tests related by spurious gauge transformations can supposedly be assigned well-defined distinct identities. We observe that the existence of such an absolute frame of reference is not required by fundamental physical principles, and hence, assuming it is an unnecessarily restrictive demand. Full article

(This article belongs to the Section Mathematical Physics)

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19 pages, 5698 KiB

Open AccessArticle

Decoherence Effects in a Three-Level System under Gaussian Process

by Sultan M. Zangi, Atta ur Rahman, Zhao-Xo Ji, Hazrat Ali and Huan-Guo Zhang

Symmetry 2022, 14(12), 2480; https://doi.org/10.3390/sym14122480 - 23 Nov 2022

Cited by 8 | Viewed by 1656

Abstract

When subjected to a classical fluctuating field characterized by a Gaussian process, we examine the purity and coherence protection in a three-level quantum system. This symmetry of the three-level system is examined when the local random field is investigated further in the noiseless [...] Read more.

When subjected to a classical fluctuating field characterized by a Gaussian process, we examine the purity and coherence protection in a three-level quantum system. This symmetry of the three-level system is examined when the local random field is investigated further in the noiseless and noisy regimes. In particular, we consider fractional Gaussian, Gaussian, Ornstein–Uhlenbeck, and power law noisy regimes. We show that the destructive nature of the Ornstein–Uhlenbeck noise toward the symmetry of the qutrit to preserve encoded purity and coherence remains large. Our findings suggest that properly adjusting the noisy parameters to specifically provided values can facilitate optimal extended purity and coherence survival. Non-vanishing terms appear in the final density matrix of the single qutrit system, indicating that it is in a strong coherence regime. Because of all of the Gaussian noises, monotonic decay with no revivals has been observed in the single qutrit system. In terms of coherence and information preservation, we find that the current qutrit system outperforms systems with multiple qubits or qutrits using purity and von Neumann entropy. A comparison of noisy and noiseless situations shows that the fluctuating nature of the local random fields is ultimately lost when influenced using the classical Gaussian noises. Full article

(This article belongs to the Special Issue Advances in Quantum Information)

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13 pages, 502 KiB

Open AccessArticle

Quantum Tomography of Two-Qutrit Werner States

by Haigang Wang and Kan He

Photonics 2022, 9(10), 741; https://doi.org/10.3390/photonics9100741 - 8 Oct 2022

Cited by 3 | Viewed by 1573

Abstract

In this article, we introduce a framework for two-qutrit Werner states tomography with Gaussian noise. The measurement scheme is based on the symmetric, informationally complete positive operator-valued measure. To make the framework realistic, we impose the Gaussian noise on the measured states numbers. [...] Read more.

In this article, we introduce a framework for two-qutrit Werner states tomography with Gaussian noise. The measurement scheme is based on the symmetric, informationally complete positive operator-valued measure. To make the framework realistic, we impose the Gaussian noise on the measured states numbers. Through numerical simulation, we successfully reconstructed the two-qutrit Werner states in various experimental scenarios and analyzed the optimal scenario from four aspects: fidelity, purity, entanglement, and coherence. Full article

(This article belongs to the Special Issue Photonic State Tomography: Methods and Applications)

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13 pages, 3297 KiB

Open AccessArticle

Realization of Quantum Swap Gate and Generation of Entangled Coherent States

by Ziqiu Zhang, Xi Jiang and Shiqing Tang

Symmetry 2022, 14(9), 1951; https://doi.org/10.3390/sym14091951 - 19 Sep 2022

Cited by 3 | Viewed by 2094

Abstract

The cross fusion of quantum mechanics and information science forms quantum information science. Quantum logic gates and quantum entanglement are very important building blocks in quantum information processing. In this paper, we propose one-step schemes for realizing quantum swap gates and generating two-mode [...] Read more.

The cross fusion of quantum mechanics and information science forms quantum information science. Quantum logic gates and quantum entanglement are very important building blocks in quantum information processing. In this paper, we propose one-step schemes for realizing quantum swap gates and generating two-mode entangled coherent states via circuit QED. In our scheme, due to the adiabatic elimination of the excited state of the qutrit under the condition of large detuning, the decoherence of the spontaneous emission of the qutrit can be ignored. The fidelity of the quantum swap gate remains at a very high level. In addition, we also explore the nonclassical properties of two-mode entangled coherent states prepared in our scheme by addressing the second-order correlation function and intermodal squeezing. In particular, two classes of entangled coherent states demonstrate distinct entanglement and nonclassical behavior. Full article

(This article belongs to the Special Issue Advances in Quantum Information)

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10 pages, 888 KiB

Open AccessFeature PaperArticle

Character Varieties and Algebraic Surfaces for the Topology of Quantum Computing

by Michel Planat, Marcelo M. Amaral, Fang Fang, David Chester, Raymond Aschheim and Klee Irwin

Symmetry 2022, 14(5), 915; https://doi.org/10.3390/sym14050915 - 30 Apr 2022

Cited by 5 | Viewed by 3172

Abstract

It is shown that the representation theory of some finitely presented groups thanks to their

S L_{2} (C)

character variety is related to algebraic surfaces. We make use of the Enriques–Kodaira classification of algebraic surfaces and related topological tools to [...] Read more.

It is shown that the representation theory of some finitely presented groups thanks to their

S L_{2} (C)

character variety is related to algebraic surfaces. We make use of the Enriques–Kodaira classification of algebraic surfaces and related topological tools to make such surfaces explicit. We study the connection of

S L_{2} (C)

character varieties to topological quantum computing (TQC) as an alternative to the concept of anyons. The Hopf link H, whose character variety is a Del Pezzo surface

f_{H}

(the trace of the commutator), is the kernel of our view of TQC. Qutrit and two-qubit magic state computing, derived from the trefoil knot in our previous work, may be seen as TQC from the Hopf link. The character variety of some two-generator Bianchi groups, as well as that of the fundamental group for the singular fibers

{\tilde{E}}_{6}

and

{\tilde{D}}_{4}

contain

f_{H}

. A surface birationally equivalent to a

K_{3}

surface is another compound of their character varieties. Full article

(This article belongs to the Special Issue Topological Aspects of Quantum Gravity and Quantum Information Theory)

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13 pages, 379 KiB

Open AccessArticle

Device-Independent Certification of Maximal Randomness from Pure Entangled Two-Qutrit States Using Non-Projective Measurements

by Jakub J. Borkała, Chellasamy Jebarathinam, Shubhayan Sarkar and Remigiusz Augusiak

Entropy 2022, 24(3), 350; https://doi.org/10.3390/e24030350 - 28 Feb 2022

Cited by 8 | Viewed by 2354

Abstract

While it has recently been demonstrated how to certify the maximal amount of randomness from any pure two-qubit entangled state in a device-independent way, the problem of optimal randomness certification from entangled states of higher local dimension remains open. Here we introduce a [...] Read more.

While it has recently been demonstrated how to certify the maximal amount of randomness from any pure two-qubit entangled state in a device-independent way, the problem of optimal randomness certification from entangled states of higher local dimension remains open. Here we introduce a method for device-independent certification of the maximal possible amount of

2 {log}_{2} 3

random bits using pure bipartite entangled two-qutrit states and extremal nine-outcome general non-projective measurements. To this aim, we exploit a device-independent method for certification of the full Weyl–Heisenberg basis in three-dimensional Hilbert spaces together with a one-sided device-independent method for certification of two-qutrit partially entangled states. Full article

(This article belongs to the Section Quantum Information)

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18 pages, 362 KiB

Open AccessFeature PaperArticle

Symmetry-Induced Emergence of a Pseudo-Qutrit in the Dipolar Coupling of Two Qubits

by Yury Belousov, Vladimir I. Man’ko, Agostino Migliore, Alessandro Sergi and Antonino Messina

Entropy 2022, 24(2), 223; https://doi.org/10.3390/e24020223 - 31 Jan 2022

Cited by 2 | Viewed by 2217

Abstract

We investigate a system of two identical and distinguishable spins 1/2, with a direct magnetic dipole–dipole interaction, in an external magnetic field. Constraining the hyperfine tensor to exhibit axial symmetry generates the notable symmetry properties of the corresponding Hamiltonian model. In fact, we [...] Read more.

We investigate a system of two identical and distinguishable spins 1/2, with a direct magnetic dipole–dipole interaction, in an external magnetic field. Constraining the hyperfine tensor to exhibit axial symmetry generates the notable symmetry properties of the corresponding Hamiltonian model. In fact, we show that the reduction of the anisotropy induces the invariance of the Hamiltonian in the

3 \times 3

subspace of the Hilbert space of the two spins in which

{\hat{S}}^{2}

invariably assumes its highest eigenvalue of 2. By means of appropriate mapping, it is then possible to choose initial density matrices of the two-spin system that evolve in such a way as to exactly simulate the time evolution of a pseudo-qutrit, in the sense that the the actual two-spin system nests the subdynamics of a qutrit regardless of the strength of the magnetic field. The occurrence of this dynamic similitude is investigated using two types of representation for the initial density matrix of the two spins. We show that the qutrit state emerges when the initial polarizations and probability vectors of the two spins are equal to each other. Further restrictions on the components of the probability vectors are reported and discussed. Full article

(This article belongs to the Special Issue Quantum Information and Quantum Optics)

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