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

Abstract

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.

Note on Connes spectral distances of qubits

“…It is easy to see that, for the diagonal states, namely x i = y i = 0 in (24), the optimal elements e for the Connes spectral distances can also be diagonal, this is similar to the result in Ref. [19].…”

“…Consider the states ρ 1 and ρ 2 corresponding to the Bloch vectors r 1 = (x 1 , y 1 , z 1 ) and r 2 = (x 2 , y 2 , z 2 ), respectively. Using the matrix representation (19), one can obtain…”