Schr ¨ odinger bridges for discrete-time, classical and quantum Markovian evolutions
Michele Pavon, Francesco Ticozzi · 2010
The theory of Schr¨ odinger bridges for diffusion processes is extended to discrete-time Markov chains, and to some problems for quantum discrete-time processes. Taking into account the past-future lack of symmetry of the discrete-time setting, results bear a striking resemblance to the classical ones. In particular, the solution of the path space maximum entropy problems is always obtained from the model by means of a suitable multiplicative functional transformation. Index Terms—Markov chain, maximum entropy problem, Schr¨ odinger bridge, time reversal evolution, space-time har- monic function, quantum operation. odinger considered, and basically solved, the following abstract probabilistic problem: Suppose a large number N of independent Brownian particles have been observed to have density ρ0(x) at time t0 and density ρ1(x) at some later time t1. Suppose the latter density considerably differs from what is predicted by the law of large numbers. It is apparent that the particles have been transported in an unlikely way. But of the many unlikely ways in which this could have happened, which one is the most likely? In modern terminology, this is a problem of large deviations of the empirical distribution. Using a coarse graining approach, Schrcomputed the most likely endpoint distribution under the prior tran- sition density of the Brownian motion p(s,x,t,y) and with the prescribed marginals. It turned out that the solution, namely the bridge from ρ0 to ρ1 over Brownian motion, has at each time a density q that factors as q(x,t )= ϕ(x,t)ˆ ϕ(x,t), where ϕ and ˆ ϕ are, in the language of Doob, a p-harmonic and a p-coharmonic functions, respectively. The existence and uniqueness of such a pair (ϕ, ˆ ϕ) satisfying the factorization above and the boundary conditions was guessed by Schron the basis of his intuition. He was later shown to be quite right in various degrees of generality by Fortet (9), Beurling (4), Jamison (11), F¨