Least Maximum Entropy and Minimum Uncertainty Coherent States

Abstract

A density matrix form of the Heisenberg uncertainty relation is obtained by means of the principle of maximum entropy (PME) subject to appropriate constraints. The least maximum entropy of zero is attained for the Heisenberg equality in concordance with the known property that the equality holds for appropriate pure states and their unitary equivalents. The case when the constraints involve expectations of a quadratic form in position and momentum operators is considered in detail. A few unitary transformations of importance, namely those corresponding to rotation in phase space, squeezing, and time evolution which leave the Heisenberg equality intact are discussed. The relation of these to the appropriately defined “correlated minimum uncertainty coherent states” are also discussed.