Connes distance function on fuzzy sphere and the connection between geometry and statistics

Devi, Yendrembam Chaoba; Prajapat, Shivraj; Mukhopadhyay, Aritra K.; Chakraborty, Biswajit; Scholtz, F. G.

doi:10.1063/1.4918648

Cited by 6 publications

(5 citation statements)

References 29 publications

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“…[17] Scholtz and his collaborators have done many works on the studies of Connes spectral distances in Moyal plane and fuzzy sphere. [18][19][20] They developed the Hilbert-Schmidt operatorial formulation, and obtained the distances of harmonic oscillator states and also coherent states. Kumar et al used Dirac eigen-spinor method to compute spectral distances in doubled Moyal plane.…”

Section: Introductionmentioning

confidence: 99%

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

Lin¹,

Heng²

2021

Chinese Phys. B

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We study the Connes distance of quantum states of two-dimensional (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 four-dimensional (4D) quantum phase space. Based on the ball condition, we obtain some constraint relations about the optimal elements. We construct 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. These results are significant for the study of geometric structures of noncommutative spaces, and it can also help us to study the physical properties of quantum systems in some kinds of noncommutative spaces.

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Section: Introductionmentioning

confidence: 99%

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

Lin¹,

Heng²

2021

Chinese Phys. B

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“…In a noncommutative space, a pure state is the analog of a traditional point in a normal commutative space, and the Connes spectral distance between pure states corresponds to the geodesic distance between points [3]. The Connes spectral distances in some kinds of noncommutative spaces have already been studied in the literatures [4][5][6][7][8][9][10][11][12][13][14][15][16]. For example, Dai et.…”

Section: Introductionmentioning

confidence: 99%

“…al. obtained the spectral distance between coherent states in the so-called double Moyal plane [7]. Scholtz and his collaborators have studied the Connes spectral distances of harmonic oscillator states and also coherent states in Moyal plane and fuzzy space [11][12][13]. In the present work, we will study the Connes spectral distance between qubits which can be represented by fermionic Fock states in phase spaces.…”

Section: Introductionmentioning

confidence: 99%

Note on Connes spectral distances of qubits

Lin¹,

Chen²,

Heng³

2022

Preprint

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By virtue of the mathematical tools in noncommutative geometry, we study the Connes spectral distances between one-and two-qubit states. We construct a spectral triple corresponding to the 2D fermionic phase spaces, and calculate the Connes spectral distance between one qubits. Based on the Connes spectral distance, we define a coherence measure of quantum states, and 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. We find that the Connes spectral distances are different from quantum trace distances. The Connes spectral distance can be considered as a significant supplement to the trace distances in quantum information sciences. These results are significant for the study of physical relations and geometric structures of qubits and other quantum states.

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“…One can calculate some kinds of distance measures between the states, such as the Connes spectral distance [25]. The Connes spectral distances in some kinds of noncommutative spaces have already been studied in the literatures [26][27][28][29][30][31][32][33][34][35][36][37]. For example, Cagnache et.…”

Section: Introductionmentioning

confidence: 99%

“…They explicitly computed Connes spectral distance between the pure states which corresponding to eigenfunctions of the quantum harmonic oscillators. Scholtz and his collaborators have developed the Hilbert-Schmidt operatorial formulation [32][33][34], they studied the Connes spectral distances of harmonic oscillator states and also coherent states in Moyal plane and fuzzy space. Barrett et.…”

Section: Introductionmentioning

confidence: 99%

Connes spectral distance and nonlocality of generalized noncommutative phase spaces

Lin¹,

Heng²

2021

Preprint

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We study the Connes spectral distance of quantum states and analyse the nonlocality of the 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 space are shorter than those in normal phase space. This shortening of distances implies some kind of nonlocality caused by the noncommutativity. We also find that these spectral distances in 4D generalized noncommutative phase space are additive and satisfy the normal Pythagoras theorem. When the noncommutative parameters equal zero, the results return to those in normal quantum phase space.

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Connes distance function on fuzzy sphere and the connection between geometry and statistics

Cited by 6 publications

References 29 publications

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

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

Note on Connes spectral distances of qubits

Connes spectral distance and nonlocality of generalized noncommutative phase spaces

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