Quantum Probabilities of Composite Events in Quantum Measurements With Multimode States

The problem of defining quantum probabilities of composite events for quantum decision theory and theory of quantum measurements is considered. These theories not merely are closely related with each other, but decision theory, actually, can be treated as a part of measurement theory. We show that the Luders 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 the 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. A special attention is payed to the application of the approach to systems with multimode states. The measurement of composite events including multimode states corresponds to the measurement under uncertainty, which is equivalent to decision making under uncertainty