A review of various definitions of "compatibility" expressed in terms of ordinary probability, and a discussion of the occurrence of incompatibility (and the related phenomenon of interference) in non-quantal probabilistic systems.
Quantum instruments describe outcome probability as well as state change induced by measurement of a quantum system. Incompatibility of two instruments, i. e. the impossibility to realize them simultaneously on a given quantum system, generalizes incompatibility of channels and incompatibility of positive operator-valued measures (POVMs). We derive implications of instrument compatibility for the induced POVMs and channels. We also study relation of instrument compatibility to the concept of non-disturbance. Finally, we prove equivalence between instrument compatibility and postprocessing of certain instruments, which we term complementary instruments. We illustrate our findings on examples of various classes of instruments.
Abstract This chapter contains a general discussion of the phenomenon of BEC, under conditions much broader than those realized in the equilibrium noninteracting system introduced in Chapter 1. A definition of BEC in a general (nonequilibrium, noninteracting) Bose system is given in terms of the eigenvalues of the single-particle density matrix; alternative definitions are also discussed. The important concepts of order parameter and superfluid velocity are introduced. The question is raised: why should BEC occur, and when does it (not)? For Fermi systems, Cooper pairing is defined in terms of the eigenvalues of the two-particle density matrix; again, alternative definitions are briefly discussed, and possible reasons for the occurrence of the phenomenon reviewed. The chapter concludes with an overview of the consequences of BEC/Cooper pairing, and with a discussion of some unusual cases in which the BEC is “fragmented”. Two appendices review the second-quantization formalism and the properties of number and phase operators.