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arXiv:2212.10710 · Journal of Mathematical Physics

Quantum vs classical birth and death processes; exactly solvable examples

Ryu Sasaki B

Published Feb 1, 2025• 3 citations
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Abstract

A coinless quantisation procedure of continuous and discrete time Birth and Death (BD) processes is presented. The quantum Hamiltonian H is derived by similarity transforming the matrix L describing the BD equation in terms of the square root of the stationary (reversible) distribution. The quantum and classical systems share the entire eigenvalues and the eigenvectors are related one to one. When the birth rate B(x) and the death rate D(x) are chosen to be the coefficients of the difference equation governing the orthogonal polynomials of Askey scheme, the quantum system is exactly solvable. The eigenvectors are the orthogonal polynomials themselves and the eigenvalues are given analytically. Many examples are periodic since their eigenvalues are all integers, or all integers for integer parameters. The situation is very similar to the exactly solvable one dimensional quantum mechanical systems. These exactly solvable BD processes contain many adjustable free parameters which could be helpful for various simulation purposes.

DOI
10.1063/5.0161779

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A coinless quantisation procedure of continuous and discrete time Birth and Death (BD) processes is presented. The quantum Hamiltonian H is derived by similarity transforming the matrix L describing the BD equation in terms of the square root of the stationary (reversible) distribution. The quantum and classical systems share the entire eigenvalues and the eigenvectors are related one to one. When the birth rate B(x) and the death rate D(x) are chosen to be the coefficients of the difference equation governing the orthogonal polynomials of Askey scheme, the quantum system is exactly solvable. The eigenvectors are the orthogonal polynomials themselves and the eigenvalues are given analytically. Many examples are periodic since their eigenvalues are all integers, or all integers for integer parameters. The situation is very similar to the exactly solvable one dimensional quantum mechanical systems. These exactly solvable BD processes contain many adjustable free parameters which could be helpful for various simulation purposes.
Many examples of exactly solvable birth and death processes, a typical stationary Markov chain, are presented together with the explicit expressions of the transition probabilities. They are derived by similarity transforming exactly solvable ‘matrix’ quantum mechanics, which is recently proposed by Odake and the author. The (q-)Askey-scheme of hypergeometric orthogonal polynomials of a discrete variable and their dual polynomials play a central role. The most generic solvable birth/death rates are rational functions of q x (x being the population) corresponding to the q-Racah polynomial. 1
A coinless quantisation procedure of general reversible Markov chains on graphs is presented. A quantum Hamiltonian H is obtained by a similarity transformation of the fundamental transition probability matrix K in terms of the square root of the reversible distribution. The evolution of the classical and quantum Markov chains is described by the solutions of the eigenvalue problem of the quantum Hamiltonian H. About twenty plus exactly solvable Markov chains based on the hypergeometric orthogonal polynomials of Askey scheme, derived by Odake-Sasaki, would provide a good window for scrutinising the quantum/classical contrast of Markov chains. Among them five explicit examples, related to the Krawtchouk, Hahn, q-Hahn, Charlier and Meixner, are demonstrated to illustrate the actual calculations.
Many examples of exactly solvable birth and death processes, a typical stationary Markov chain, are presented together with the explicit expressions of the transition probabilities. They are derived by similarity transforming exactly solvable “matrix” quantum mechanics, which is recently proposed by Odake and the author [S. Odake and R. Sasaki, J. Math. Phys. 49, 053503 (2008)]. The (q-) Askey scheme of hypergeometric orthogonal polynomials of a discrete variable and their dual polynomials play a central role. The most generic solvable birth/death rates are rational functions of qx (with x being the population) corresponding to the q-Racah polynomial.
Abstract 15 exactly solvable inhomogeneous (spinless) fermion systems on 1D lattices are constructed explicitly based on the discrete orthogonal polynomials of the Askey scheme, e.g. Krawtchouk, Hahn, Racah, Meixner, and q-Racah polynomials. The Schrödinger and Heisenberg equations are solved explicitly, as the entire set of the eigenvalues and eigenstates are known explicitly. The ground-state two-point correlation functions are derived explicitly. The multipoint correlation functions are obtained by Wick’s theorem. 15 corresponding exactly solvable XX spin systems are also displayed. They all have nearest-neighbor interactions. The exact solvability of the Schrödinger equation means that the corresponding Fokker–Planck equation is also exactly solvable. This leads to 15 exactly solvable birth and death fermions and 15 birth and death spin models. These provide plenty of material for calculating interesting quantities, e.g. entanglement entropy, etc.
Various examples of exactly solvable ‘discrete’ quantum mechanics are explored explicitly with emphasis on shape invariance, Heisenberg operator solutions, annihilation-creation operators, the dynamical symmetry algebras and coherent states. The eigenfunctions are the (q-) Askey-scheme of hypergeometric orthogonal polynomials satisfying difference equation versions of the Schrödinger equation. Various reductions (restrictions) of the symmetry algebra of the Askey-Wilson system are explored in detail.
This paper underscores the intimate connection between the quantum walks generated by certain semi-infinite spin chain Hamiltonians and classical birth and death processes. It is observed that transition amplitudes between single excitation states of the spin chains have an expression in terms of orthogonal polynomials which is analogous to the Karlin-McGregor representation formula of the transition probability functions for classes of birth and death processes. As an application, we present a characterization of spin systems for which the probability to return to the point of origin at some time is 1 or almost 1.