An Operational Approach to Quantum Probability

E. B. Davies, John T. Lewis

John M. Myers, F. Hadi Madjid

Introduction The use of parameters to describe an experimenter's control over the devices used in an experiment is familiar in quantum physics, for example in connection with Bell inequalities. Parameters are also interesting in a different but related context, as we noticed when we proved a formal separation in quantum mechanics between linear operators and the probabilities that these operators generate. In comparing an experiment against its description by a density operator and detection operators, one compares tallies of experimental outcomes against the probabilities generated by the operators but not directly against the operators. Recognizing that the accessibility of operators to experimental tests is only indirect, via probabilities, motivates us to ask what probabilities tell us about operators, or, put more precisely, “what combinations of a parameterized density operator and parameterized detection operators generate any given set of parametrized probabilities?” Here, we review and augment recent proofs that any given parameterized probabilities can be generated in very diverse ways, so that a parameterized probability measure, detached from any of the (infinitely many) parameterized operators that generate it, becomes an interesting object in its own right. By detaching a parameterized probability measure from the operators that may have led us to it, we (1) strengthen Holevo's bound on a quantum communication channel and (2) clarify a role for multiple levels of modeling in an example based on quantum key distribution. We then inquire into some parameterized probability measures generated by entangled states and into the topology of the associated parameter spaces; in particular we display some previously overlooked topological features of level sets of these probability measures.

Vladimir N. Chernega, Olga V. Man’ko, Vladimir I. Man’ko

Abstract A review of the approach where the states of quantum systems are identified with fair probability distributions is presented. The quantizer–dequantizer operators used to construct the invertible map of the density operators onto the probability distributions are applied to obtain the kinetic equations for probability distributions identified with the quantum system states. For qubit states, the von Neumann evolution equation for the density operator is explicitly given in the form of kinetic equation for the probability distribution. Simplest tomographic probability distributions describing the states of multimode quantum oscillator are constructed.

Stanley Gudder

Vladimir I. Man’ko, Olga V. Man’ko, Vladimir N. Chernega

Abstract The review of a new probability representation of quantum states is presented, where the states are described by conventional probability distribution functions. The invertible map of the probability distribution onto density operators in the Hilbert space is found using the introduced operators called a quantizer–dequantizer, which specify the invertible map of operators of quantum observables onto functions and a product of the operators onto an associative product (star product) of the functions. Examples of a quantum oscillator and a spin-1/2 particle are considered. The kinetic equations for probabilities, specifying the evolution of the states of a quantum system, which are equivalent to Schrödinger and von Neumann equations, are derived explicitly.

Enrico G. Beltrametti, Slawomir Bugajski