Joint probability distribution of composite quantum systems

Vyacheslav I. Yukalov, Didier Sornette

The problem of defining quantum probabilities of composite events is considered. This problem is of great importance for the theory of quantum measurements and for quantum decision theory, which is a part of measurement theory. We show that the Lüders probability of consecutive measurements is a transition probability between two quantum states and that this probability cannot be treated as a quantum extension of the classical conditional probability. The Wigner distribution is shown to be a weighted transition probability that cannot be accepted as a quantum extension of the classical joint probability. We suggest the definition of quantum joint probabilities by introducing composite events in multichannel measurements. The notion of measurements under uncertainty is defined. We demonstrate that the necessary condition for mode interference is the entanglement of the composite prospect together with the entanglement of the composite statistical state. As an illustration, we consider an example of a quantum game. Special attention is paid to the application of the approach to systems with multimode states, such as atoms, molecules, quantum dots, or trapped Bose-condensed atoms with several coherent modes.

Leon W. Cohen

Stanley Gudder

Andreas Daffertshofer, Angel Ricardo Plastino, Angel Ricardo Plastino

A classical version of the no-cloning theorem is discussed. We show that an arbitrary probability distribution associated with a (source) system cannot be copied onto another (target) system while leaving the original distribution of the source system unperturbed. For classical dynamical systems such a perfect cloning process is not permitted by the Liouvillian (ensemble) evolution associated with the joint probability distribution of the composite source-target-copying machine system.

Kazimierz Urbanik

Gerd Niestegge

An axiomatic approach to the mathematical formalism of quantum mechanics, based upon a certain concept of conditional probability, has been proposed in two recent papers by the author. It leads to Jordan operator algebras and thus comes rather close to the standard Hilbert space model of quantum mechanics, but still includes the so-called exceptional Jordan algebras, for which a Hilbert space representation does not exist. This approach is now extended by defining a mathematical model of composite systems. Such a model is required for the study of the joint distribution of two quantum observables. A very general type of observables (not only the real-valued observables corresponding to the self-adjoint operators) is considered. The joint distribution is defined, using the concept of conditional probability, and exhibits a certain dependence on the succession of the observations which is different from the classical case and unknown so far in quantum mechanics. Finally, it turns out that, at least in the finite-dimensional case, a really satisfying model of the composite system exists only if each single system is modeled by a complex Jordan matrix algebra (or a direct sum), and the model then becomes the tensor product. This provides some reasoning why the exceptional Jordan algebras can be ruled out, why quantum mechanics needs the complex numbers and the complex Hilbert space, and why the tensor product is the right choice for the model of a composite system.