Quantum counterparts of Schrodinger's classical bridge problem have been around for the better part of half a century. During that time, several quantum approaches to this multifaceted classical problem have been introduced. In the present work, we unify, extend, and interpret several such approaches through a classical large deviations perspective. To this end, we consider time-symmetric ensembles that are pre- and post-selected before and after a Markovian experiment is performed. The Schrodinger bridge problem is that of finding the most likely joint distribution of initial and final outcomes that is consistent with obtained endpoint results. The derived distribution provides quantum Markovian dynamics that bridge the observed endpoint states in the form of density matrices. The solution retains its classical structure in that density matrices can be expressed as the product of forward-evolving and backward-evolving matrices. In addition, the quantum Schrodinger bridge allows inference of the most likely distribution of outcomes of an intervening measurement with unknown results. This distribution may be written as a product of forward- and backward-evolving expressions, in close analogy to the classical setting, and in a time-symmetric way. The derived results are illustrated through a two-level amplitude damping example.
Quantum counterparts of Schrodinger's classical bridge problem have been around for the better part of half a century. During that time, several quantum approaches to this multifaceted classical problem have been introduced. In the present work, we unify, extend, and interpret several such approaches through a classical large deviations perspective. To this end, we consider time-symmetric ensembles that are pre- and post-selected before and after a Markovian experiment is performed. The Schrodinger bridge problem is that of finding the most likely joint distribution of initial and final outcomes that is consistent with obtained endpoint results. The derived distribution provides quantum Markovian dynamics that bridge the observed endpoint states in the form of density matrices. The solution retains its classical structure in that density matrices can be expressed as the product of forward-evolving and backward-evolving matrices. In addition, the quantum Schrodinger bridge allows inference of the most likely distribution of outcomes of an intervening measurement with unknown results. This distribution may be written as a product of forward- and backward-evolving expressions, in close analogy to the classical setting, and in a time-symmetric way. The derived results are illustrated through a two-level amplitude damping example.
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¨
We consider a quantum system with a time-independent Hamiltonian parametrized by a set of unknown parameters $\ensuremath{\alpha}$. The system is prepared in a general quantum state by an evolution operator that depends on a set of unknown parameters P. After the preparation, the system evolves in time, and it is characterized by a time-dependent observable O(t). We show that it is possible to obtain closed-form expressions for the gradients of the distance between O(t) and a calculated observable with respect to $\ensuremath{\alpha}, P$, and all elements of the system density matrix, whether for pure or mixed states. These gradients can be used in projected gradient descent to infer $\ensuremath{\alpha}, P$, and the relevant density matrix from dynamical observables. We combine this approach with random phase wave function approximation to obtain closed-form expressions for gradients that can be used to infer population distributions from averaged time-dependent observables in problems with a large number of quantum states participating in dynamics. The approach is illustrated by determining the temperature of molecular gas (initially, in thermal equilibrium at room temperature) from laser-induced time-dependent molecular alignment.
We study three definitions of large deviation in many-body quantum dynamics: (i) via the full distribution of an extensive observable, (ii) via the distribution of measurement outcomes (from a continuous monitoring of the observable) over a time interval t≤tmax, and (iii) via the distribution of expectation values over t≤tmax. In generic systems without conservation laws, the large deviation function (i) reaches its longtime limit at t∼O(1), independently of system size N. (ii) and (iii) reach their longtime limit at t∼eN and t∼exp(eN), respectively. Before that, there is a {\it sharp} frontier between the explored and unexplored outcomes/expectation values; their distribution equals the longtime limit truncated at values that drift with tmax. We propose that the evolution of these values with tmax provides a measure of quantum complexity.
The classical probability theory initiated by Kolmogorov and its quantum counterpart, pioneered by von Neumann, were created at about the same time in the 1930s, but development of the quantum theory has trailed far behind. Although highly appealing, the quantum theory has a steep learning curve, requiring tools from both probability and analysis and a facility for combining the two viewpoints. This book is a systematic, self-contained account of the core of quantum probability and quantum stochastic processes for graduate students and researchers. The only assumed background is knowledge of the basic theory of Hilbert spaces, bounded linear operators, and classical Markov processes. From there, the book introduces additional tools from analysis, and then builds the quantum probability framework needed to support applications to quantum control and quantum information and communication. These include quantum noise, quantum stochastic calculus, stochastic quantum differential equations, quantum Markov semigroups and processes, and large-time asymptotic behavior of quantum Markov semigroups.
Our account begins in 1931/1932, when one of the fathers of quantum theory, Erwin Schrodinger, proposed a Gedankenexperiment to comprehend the propagation of a cloud of Brownian particles. He proposed [1], [2] the inference problem to identify the most likely evolution of a large collection of particles as they traverse between two endpoints in time- their distribution being measured and available at the two endpoints. This random evolution, now known as the Schrodinger bridge, entails a flow of one-time distributions that interpolate the known initial and final marginals.
Wheeler's delayed-choice experiments and the time-symmetric formalisms of quantum measurement have long fueled a debate over whether time symmetry in quantum theory entails retrocausality. We argue that the debate conflates two distinct structures, and that the framework for separating them is the classical theory of conditioning: Doob h-transforms, Markov bridges, and enlargement of filtrations on the probabilistic side, and the forward-state/backward-effect (two-state-vector) formalism of pre- and post-selected systems on the quantum side. We sharpen Wheeler's participatory universe, delayed choice, and ``It from Bit'' into three theses and build a correspondence dictionary that maps each to a precise statement in one or both formalisms. The dictionary is anchored by two structurally isomorphic results: an operational proposition that a future measurement choice sorts the past ensemble into subensembles without disturbing any earlier marginal, and its classical counterpart, the disintegration of an unconditioned law into future-conditioned bridges -- both a single fact: marginalizing over the final measurement leaves earlier marginals fixed, by completeness of the POVM together with trace preservation on the quantum side and by the tower property of conditional expectation on the classical. The conditional calculus is time-symmetric; its causal structure is not. Delayed choice is thereby a selection effect rather than retrocausation, and the apparent ``pull from the future'' is a bridge drift. We state explicitly what the framework does not do -- it neither derives the Born rule nor solves the measurement problem -- and delineate the legitimate scope of participatory language in quantum mechanics.