Quantum probabilities of composite events in quantum measurements with multimode states

Abstract

The problem of defining quantum probabilities of composite events is considered. This problem is of high importance for the theory of quantum measurements and for quantum decision theory that 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 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 multi-mode states, such as atoms, molecules, quantum dots, or trapped Bose-condensed atoms with several coherent modes.

Inconclusive Quantum Measurements and Decisions under Uncertainty

“…Sometimes, one treats the Lüders probability of consecutive measurements as a conditional probability. This, however, is not justified from the theoretical point of view [21,23,30] and also contradicts experimental data [31,32]. But defining the quantum joint probability according to expression (Equation 8) contains no contradictions.…”

“…The event operator for an inconclusive event is not necessarily a projector, but a member of a positive operator-valued measure [7,[21][22][23].…”

“…The prospect operator is a member of a positive operator-valued measure, which implies that these operators satisfy the resolution of unity [21,23]. Since they contain random quantities b α , the corresponding random resolution has to be understood not as a direct equality between numbers, but, e.g., as the equality in mean [24].…”

“…This way makes it possible to introduce joint quantum probabilities for several measurements [21,23]. Contrary to this, considering operators on the same Hilbert space does not allow one to define joint probabilities for non-commuting operators.…”

“…A detailed mathematical theory of quantum measurements in the case of composite events has been developed in our previous papers [21][22][23]. In the present paper, we concentrate our attention on composite measurements including intermediate inconclusive events and on the application of this notion for characterizing decision making under risk and uncertainty.…”