arXiv subjects

Bing-Sheng Lin

Publications and source records attributed to Bing-Sheng Lin.

13 recordsLinked to original sources

arXiv2026

Connes spectral distance on twisted fuzzy torus

In this paper, we study the Connes spectral distance between states on the fuzzy torus. We construct a Dirac operator by commutators and anticommutators. Based on this Dirac operator, we construct a spectral triple of the fuzzy torus. We study some properties of the spectral distance on the fuzzy torus. We find that there is a reciprocal Pythagorean theorem between the spectral distances. We construct a conditional expectation function of the optimal element which can lead to a contraction of the corresponding Lipschitz seminorm. We find that for any diagonal states, the corresponding optimal elements of spectral distances are also diagonal. We explicitly calculate the spectral distances of some simple states, including basic states and some simple mixed states. We find that there are some kinds of cyclic symmetry in both the optimal elements and the spectral distances between the diagonal states. Furthermore, we also construct a fuzzy torus with some type of conformal twist, and study the relation between conformal parameters and spectral distances.

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arXiv2026

A note on unitary invariance of Connes spectral distances of quantum states

In this paper, we study the properties of Connes spectral distances between quantum states under unitary transformations. We mainly focus on spectral triples with matrix algebras acting on finite dimensional Hilbert spaces. We prove that there are some finite spectral triples in which the Lipschitz seminorms are equal to the operator norms. We also explicitly construct some spectral triples in which the Connes spectral distances between quantum states are exactly the quantum trace distances. These results are helpful for us to better study the relationships among Connes spectral distance, quantum trace distance and other quantum distance measures. These concrete examples are significant for studies of geometric structures of finite spectral triples and mathematical relations of qubits and other quantum states in the framework of noncommutative geometry.

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arXiv2026

Deformation quantization for systems with second-class constraints in deformed fermionic phase space

In order to quantize systems involving second-class constraints, one should use Dirac bracket instead of Poisson bracket. Furthermore, one can specify a star product in which the term linear in $\hbar$ is proportional to the Dirac bracket. In this way an oscillator system in a deformed fermionic phase space is analyzed and the corresponding energy level and Wigner functions are evaluated according to scheme of deformation quantization. We also study the entanglement entropy induced by the deformation of the fermionic phase space.

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arXiv2024

Curvature, area and Gauss-Bonnet formula of the Moyal sphere

We studied some geometric properties of the Moyal sphere. Using the conformal metric of the sphere in ordinary space and the matrix basis, we calculated the scalar curvature, total curvature integral and area of the Moyal sphere. We found that when the noncommutative parameter approaches to 0, the scalar curvature and area of the Moyal sphere return to those of the ordinary sphere. As the noncommutative parameter increases, the area of the Moyal sphere will decrease and eventually approach to 0. We found that the total curvature integral of the two-dimensional Moyal sphere still satisfies the usual Gauss-Bonnet formula and does not depend on the noncommutative parameter. We also calculated the approximate expression of the conformal metric with a constant curvature and obtained the corresponding correction function. In addition, we studied a type of generalized deformed Moyal sphere with two noncommutative parameters and obtained similar results.

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arXiv2022

Connes spectral distances, quantum discord and coherence of qubits

We construct spectral triples of one- and two-qubit states using the Hilbert-Schmidt operatorial formulation, and study the Connes spectral distances. We also construct the Dirac operator corresponding to the normal quantum trace distances. Based on the Connes spectral distances, we propose some definitions of quantum discord and coherence measure of quantum states, and explicitly calculate the coherence of one-qubit states. We also study some simple cases about two-qubit states, and the corresponding spectral distances satisfy the Pythagoras theorem. These results are significant for studies on physical relations and geometric structures of qubits and other quantum states.

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arXiv2021

Connes spectral distance and nonlocality of generalized noncommutative phase spaces

We study the Connes spectral distance of quantum states and analyse the nonlocality of a 4D generalized noncommutative phase space. By virtue of the Hilbert-Schmidt operatorial formulation, we obtain the Dirac operator and construct a spectral triple corresponding to the noncommutative phase space. Based on the ball condition, we obtain some constraint relations about the optimal elements, and then calculate the Connes spectral distance between two Fock states. Due to the noncommutativity, the spectral distances between Fock states in generalized noncommutative phase spaces are shorter than those in normal phase spaces. This shortening of distances implies some type of nonlocality caused by the noncommutativity. These spectral distances in the 4D generalized noncommutative phase space are additive and satisfy the normal Pythagoras theorem. When the noncommutative parameters go to zero, the results return to those in normal quantum phase spaces.

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arXiv2020

Connes distance of $2D$ harmonic oscillators in quantum phase space

We study the Connes distance of quantum states of $2D$ harmonic oscillators in phase space. Using the Hilbert-Schmidt operatorial formulation, we construct a boson Fock space and a quantum Hilbert space, and obtain the Dirac operator and a spectral triple corresponding to a $4D$ quantum phase space. Based on the ball condition, we obtain some constraint relations about the optimal elements. We construct the explicit expressions of the corresponding optimal elements and then derive the Connes distance between two arbitrary Fock states of $2D$ quantum harmonic oscillators. We prove that these two-dimensional distances satisfy the Pythagoras theorem.

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arXiv2019

Induced entanglement entropy of harmonic oscillators in noncommutative phase space

We study the entanglement entropy of harmonic oscillators in noncommutative phase space. We propose a new definition of quantum Rényi entropy based on Wigner functions in noncommutative phase space. Using the Rényi entropy, we calculate the entanglement entropy of the ground state of the 2D isotropic harmonic oscillators. We find that for some values of the noncommutative parameters, the harmonic oscillators can be entangled in noncommutative phase space. This is a new entanglement-like effect caused by the noncommutativity of the phase space.

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arXiv2016

A groupoidification of the fermion algebra

In this paper, we consider the groupoidification of the fermion algebra. We construct a groupoid as the categorical analogues of the fermionic Fock space, and the creation and annihilation operators correspond to spans of groupoids. The categorical fermionic Fock states have some extra structures comparing with the normal forms. We also construct a 2-category of spans of groupoids corresponding to the fermion algebra. The relations of the morphisms in this 2-category are consistent with those in the graphical category which is represented by string diagrams.

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arXiv2013

A diagrammatic categorification of the fermion algebra

In this paper, we study the diagrammatic categorification of the fermion algebra. We construct a graphical category corresponding to the one-dimensional fermion algebra, and we investigate the properties of this category. The categorical analogues of the Fock states are some kind of 1-morphisms in our category, and the dimension of the vector space of 2-morphisms is exactly the inner product of the corresponding Fock states. All the results in our categorical framework coincide exactly with those in normal quantum mechanics.

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arXiv2009

Deformed squeezed states in noncommutative phase space

A deformed boson algebra is naturally introduced from studying quantum mechanics on noncommutative phase space in which both positions and momenta are noncommuting each other. Based on this algebra, corresponding intrinsic noncommutative coherent and squeezed state representations are constructed, and variances of single- and two-mode quadrature operators on these states are evaluated. The result indicates that in order to maintain Heisenberg's uncertainty relations, a restriction between the noncommutative parameters is required.

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arXiv2009

A new kind of representations on noncommutative phase space

We introduce new representations to formulate quantum mechanics on noncommutative phase space, in which both coordinate-coordinate and momentum-momentum are noncommutative. These representations explicitly display entanglement properties between degrees of freedom of different coordinate and momentum components. To show their potential applications, we derive explicit expressions of Wigner function and Wigner operator in the new representations, as well as solve exactly a two-dimensional harmonic oscillator on the noncommutative phase plane with both kinetic coupling and elastic coupling.

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arXiv2009

Deformation quantization for coupled harmonic oscillators on a general noncommutative space

Deformation quantization is a powerful tool to quantize some classical systems especially in noncommutative space. In this work we first show that for a class of special Hamiltonian one can easily find relevant time evolution functions and Wigner functions, which are intrinsic important quantities in the deformation quantization theory. Then based on this observation we investigate a two coupled harmonic oscillators system on the general noncommutative phase space by requiring both spatial and momentum coordinates do not commute each other. We derive all the Wigner functions and the corresponding energy spectra for this system, and consider several interesting special cases, which lead to some significant results.

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