Inferring topology of quantum phase space

Abstract

Does a semiclassical particle remember the phase space topology?We discuss this question in the context of the Berezin-Toeplitz quantization and quantum measurement theory by using tools of topological data analysis. One of its facets involves a calculus of Toeplitz operators with piecewise constant symbol developed in an appendix by Laurent Charles. 6 Inferring the nerve of a cover 14 7 Discussion and further directions 16 A Toeplitz operators with piecewise constant symbol (by Laurent Charles) 17

Finding self-similar behavior in quantum many-body dynamics via persistent homology

“…To accomplish this, simplicial complexes such as so-calledČech complexes, Vietoris-Rips complexes or alpha shapes [6,7] are employed. Besides the mathematical investigations on persistent homology, very fruitful applications to physical systems include studies in astrophysics and cosmology [8][9][10][11], physical chemistry [12], amorphous materials [13], quantum algorithms [14][15][16][17][18] and the theory of quantum phase space [19]. In particular, persistent homology has been successfully applied to the detection of equilibrium phase transitions in statistical mechanics [20] as well as to the identification of phases in lattice spin models [21].…”

Finding self-similar behavior in quantum many-body dynamics via persistent homology

“…To accomplish this, simplicial complexes such as so-called Čech complexes, Vietoris-Rips complexes or alpha shapes [7,8] are employed. Besides the mathematical investigations on persistent homology, very fruitful applications to physical systems include studies in astrophysics and cosmology [9][10][11][12], physical chemistry [13], amorphous materials [14], the vacua landscape in string theories [15] and the theory of quantum phase space [16]. Moreover, it has been shown that quantum algorithms can speed up persistent homology computations [17][18][19][20][21].…”

Entanglement Entropy and Berezin–Toeplitz Operators

“…In the context of Berezin-Toeplitz operators in Kähler manifolds, two recent papers on different but related subjects are [ZZ17] on partial Bergman kernels and [PU18] on the quantization of submanifolds. Finally, the appendix of [Pol18] is devoted to multiplicative properties of the Toeplitz operators with a characteristic function symbol. Here the scalar product ·, · of sections of L k is defined by integrating the pointwise scalar product against the Riemannian measure µ, the Riemannian metric g of M being determined by the curvature Θ(L) of the Chern connection of L, that is g(X, Y ) = −iΘ(L)(X, jY ) with j the complex structure.…”

“…In the context of Berezin-Toeplitz operators in Kähler manifolds, two recent papers on different but related subjects are [ZZ17] on partial Bergman kernels and [PU18] on the quantization of submanifolds. Finally, the appendix of [Pol18] is devoted to multiplicative properties of the Toeplitz operators with a characteristic function symbol.…”