A factorization condition which must be satisfied by the first
n correlation functions for the electromagnetic field operators has been used to define nth-order coherence. The first-order coherence condition has been shown to imply maximum fringe contrast in interference patterns. In the present paper we investigate the mathematical consequences of assuming the condition for maximum fringe contrast. By considering the correlation functions as scalar products and formulating rigorous inequalities for them we are able to show that the assumed condition in turn implies factorization of the first-order correlation function. By extending the same methods we are able to show that all of the higher order correlation functions factorize into forms similar to those required for full coherence, but differing from them through the inclusion of a sequence of constant numerical factors. These coefficients are shown to furnish a convenient description of the higher-order coherence properties of the field. Their values are presented for some typical examples. We derive a number of inequalities satisfied by the coefficients for the case of fields which possess positive-definite weight functions in the
P representation. Some inequalities obeyed by the correlation functions for such fields are derived as well.
Using the quantum phase space technique and the information-theory like the Wehrl entropy, the Husimi function and the Wehrl entropy of the quantum pure states and the corresponding mixed states in thermo field dynamics are studied. It is found that the Husimi function and the Wehrl entropy of the thermal coherent state agree with that of the corresponding mixed states. And the Wehrl entropy of thermal coherent state is not related with the displacement factor. Therefore for a quantum system, the quantum fluctuations of the observable quantities and corresponding uncertainty relation are also not related with the displacement factor in the thermal coherent state.