A formalism for multiport devices is developed in which the creation and annihilation operators at the output are related with those at the input through a finite Fourier transform. By considering two specific inputs of number states and coherent states, we show how these devices can be used to produce multimode Schrödinger cat states and can be combined to create multipath interferometers. We introduce uncertainty relations for the distributions of particles in the input modes and the correlations between the output modes. These allow us to determine how the phase resolution of the output state scales with the number of ports and particles. This scaling is of fundamental interest for schemes that seek to use multipath interferometers for enhanced measurement precision.

Various inequalities (Boole inequality, Chung–Erdös inequality, Frechet inequality) for Kolmogorov (classical) probabilities are considered. Quantum counterparts of these inequalities are introduced, which have an extra ‘quantum correction’ term, and which hold for all quantum states. When certain sufficient conditions are satisfied, the quantum correction term is zero, and the classical version of these inequalities holds for all states. But in general, the classical version of these inequalities is violated by some of the quantum states. For example in bipartite systems, classical Boole inequalities hold for all rank one (factorizable) states, and are violated by some rank two (entangled) states. A logical approach to CHSH inequalities (which are related to the Frechet inequalities), is studied in this context. It is shown that CHSH inequalities hold for all rank one (factorizable) states, and are violated by some rank two (entangled) states. The reduction of the rank of a pure state by a quantum measurement with both orthogonal and coherent projectors, is studied. Bounds for the average rank reduction are given.

A quantum statistical formalism which accommodates both positive and negative temperatures is studied. The Dirac contour representation is used to extend the harmonic oscillator Hilbert space into a larger space which is suitable for the description of quantum systems at both positive and negative temperatures. The analytic continuation of various physical quantities into the negative temperature region is examined in detail. The formalism can be useful for the description of systems which are excited to higher states, and which decay into the lower states.