Extracting Bayesian networks from multiple copies of a quantum system

Abstract

Despite their theoretical importance, dynamic Bayesian networks associated with quantum processes are currently not accessible experimentally. We here describe a general scheme to determine the multi-time path probability of a Bayesian network based on local measurements on independent copies of a composite quantum system combined with postselection. We further show that this protocol corresponds to a non-projective measurement. It thus allows the investigation of the multi-time properties of a given local observable while fully preserving all its quantum features.Dynamic Bayesian networks offer a powerful framework to analyze conditional dependencies in a set of timedependent random quantities. In this approach, relationships between dynamical variables are specified through conditional probabilities evaluated via Bayes' rule [1][2][3][4]. They have found widespread application in statistics, engineering and computer science to model time series in probabilistic models. Hidden Markov models and Kalman filters are special cases of such networks [1-4]. In the past decade, Bayesian networks have been successfully employed to investigate the nonequilibrium thermodynamics of small, composite systems, both in the classical [5][6][7][8][9][10][11][12] and quantum [13][14][15][16] regimes. They have, in particular, been used to obtain fluctuation theorems, fundamental generalizations of the second law that characterize fluctuations of the entropy production arbitrarily far from equilibrium [17], for multipartite systems [5][6][7][8][9][10][11][12][13][14][15][16].An interesting property of dynamic Bayesian networks is that they allow to specify the local dynamics of a composite quantum system conditioned on its global state. The Bayesian network formalism thus preserves all the quantum features of the system, especially quantum correlations [18] and quantum coherence [19]. As a result, it permits to go beyond the standard two-projectivemeasurement scheme [20][21][22], which, owing to its projective nature, destroys off-diagonal density matrix elements. This characteristic has recently been exploited to derive fully quantum fluctuation theorems that not only account for the quantum nonequilibrium dynamics of a driven system [23], as in the two-projective-measurement approach, but also fully capture both quantum correlations and quantum coherence at arbitrary times [13][14][15]. However, while a number of methods to implement the two-projective-measurement approach (and its variants) have been both theoretically developed [24][25][26][27] and experimentally demonstrated [28][29][30][31][32][33][34], to date, no such protocol exists for dynamic Bayesian networks.

Energy conservation and fluctuation theorem are incompatible for quantum work

“…This conflict between the two manifestations of the energy conservation law, A 1 and A 2 , opens up a completely new perspective on the inadequacy, for coherent states, of a large class of schemes, that includes the TPM scheme, the scheme we introduced in Appendix C, and the scheme proposed in Ref. [21] (O ′ ρ does not depend on ρ and is empty for all these schemes).…”

Quantum mean-square predictors and thermodynamics

“…They also cannot be associated to the outcomes of a single experiment. But, as some of us have recently shown, they can be obtained from post-selection on measurements in two identical copies [28]. Finally, non-negative probabilities can also be obtained assuming the initial and final energy measurements are statistically independent (i.e., determined from two separate experiments) [29].…”