A Factor-Graph Representation of Probabilities in Quantum Mechanics

A factor-graph representation of quantum-mechanical probabilities is proposed. Unlike standard statistical models, the proposed representation uses auxiliary variables (state variables) that are not random variables.Comment: Proc. IEEE International Symposium on Information Theory (ISIT), Cambridge, MA, July 1-6, 201

Factor Graphs for Quantum Probabilities

A factor-graph representation of quantum-mechanical probabilities (involving any number of measurements) is proposed. Unlike standard statistical models, the proposed representation uses auxiliary variables (state variables) that are not random variables. All joint probability distributions are marginals of some complex-valued function $q$, and it is demonstrated how the basic concepts of quantum mechanics relate to factorizations and marginals of $q$.Comment: To appear in IEEE Transactions on Information Theory, 201

Erlangen Programme at Large 3.1: Hypercomplex Representations of the Heisenberg Group and Mechanics

In the spirit of geometric quantisation we consider representations of the Heisenberg(--Weyl) group induced by hypercomplex characters of its centre. This allows to gather under the same framework, called p-mechanics, the three principal cases: quantum mechanics (elliptic character), hyperbolic mechanics and classical mechanics (parabolic character). In each case we recover the corresponding dynamic equation as well as rules for addition of probabilities. Notably, we are able to obtain whole classical mechanics without any kind of semiclassical limit h->0. Keywords: Heisenberg group, Kirillov's method of orbits, geometric quantisation, quantum mechanics, classical mechanics, Planck constant, dual numbers, double numbers, hypercomplex, jet spaces, hyperbolic mechanics, interference, Segal--Bargmann representation, Schroedinger representation, dynamics equation, harmonic and unharmonic oscillator, contextual probabilityComment: AMSLaTeX, 17 pages, 4 EPS pictures in two figures; v2, v3, v4, v5, v6: numerous small improvement

Testing axioms for Quantum Mechanics on Probabilistic toy-theories

In Ref. [1] one of the authors proposed postulates for axiomatizing Quantum Mechanics as a "fair operational framework", namely regarding the theory as a set of rules that allow the experimenter to predict future events on the basis of suitable tests, having local control and low experimental complexity. In addition to causality, the following postulates have been considered: PFAITH (existence of a pure preparationally faithful state), and FAITHE (existence of a faithful effect). These postulates have exhibited an unexpected theoretical power, excluding all known nonquantum probabilistic theories. Later in Ref. [2] in addition to causality and PFAITH, postulate LDISCR (local discriminability) and PURIFY (purifiability of all states) have been considered, narrowing the probabilistic theory to something very close to Quantum Mechanics. In the present paper we test the above postulates on some nonquantum probabilistic models. The first model, "the two-box world" is an extension of the Popescu-Rohrlich model, which achieves the greatest violation of the CHSH inequality compatible with the no-signaling principle. The second model "the two-clock world" is actually a full class of models, all having a disk as convex set of states for the local system. One of them corresponds to the "the two-rebit world", namely qubits with real Hilbert space. The third model--"the spin-factor"--is a sort of n-dimensional generalization of the clock. Finally the last model is "the classical probabilistic theory". We see how each model violates some of the proposed postulates, when and how teleportation can be achieved, and we analyze other interesting connections between these postulate violations, along with deep relations between the local and the non-local structures of the probabilistic theory.Comment: Submitted to QIP Special Issue on Foundations of Quantum Informatio

Computation in Finitary Stochastic and Quantum Processes

We introduce stochastic and quantum finite-state transducers as computation-theoretic models of classical stochastic and quantum finitary processes. Formal process languages, representing the distribution over a process's behaviors, are recognized and generated by suitable specializations. We characterize and compare deterministic and nondeterministic versions, summarizing their relative computational power in a hierarchy of finitary process languages. Quantum finite-state transducers and generators are a first step toward a computation-theoretic analysis of individual, repeatedly measured quantum dynamical systems. They are explored via several physical systems, including an iterated beam splitter, an atom in a magnetic field, and atoms in an ion trap--a special case of which implements the Deutsch quantum algorithm. We show that these systems' behaviors, and so their information processing capacity, depends sensitively on the measurement protocol.Comment: 25 pages, 16 figures, 1 table; http://cse.ucdavis.edu/~cmg; numerous corrections and update

Interference Effects in Quantum Belief Networks

Probabilistic graphical models such as Bayesian Networks are one of the most powerful structures known by the Computer Science community for deriving probabilistic inferences. However, modern cognitive psychology has revealed that human decisions could not follow the rules of classical probability theory, because humans cannot process large amounts of data in order to make judgements. Consequently, the inferences performed are based on limited data coupled with several heuristics, leading to violations of the law of total probability. This means that probabilistic graphical models based on classical probability theory are too limited to fully simulate and explain various aspects of human decision making. Quantum probability theory was developed in order to accommodate the paradoxical findings that the classical theory could not explain. Recent findings in cognitive psychology revealed that quantum probability can fully describe human decisions in an elegant framework. Their findings suggest that, before taking a decision, human thoughts are seen as superposed waves that can interfere with each other, influencing the final decision. In this work, we propose a new Bayesian Network based on the psychological findings of cognitive scientists. We made experiments with two very well known Bayesian Networks from the literature. The results obtained revealed that the quantum like Bayesian Network can affect drastically the probabilistic inferences, specially when the levels of uncertainty of the network are very high (no pieces of evidence observed). When the levels of uncertainty are very low, then the proposed quantum like network collapses to its classical counterpart