Liouville Representation of Quantum Mechanics with Application to Relaxation Processes

Eduard Prugovečki

Masashi Ban

A.S. Davydov

Kieran Flatt, Stephen Mark Barnett, Sarah Croke

Probabilities enter quantum mechanics via Born's rule, the uniqueness of which was proven by Gleason. Busch subsequently relaxed the assumptions of this proof, expanding its domain of applicability in the process. Extending this work to sequential measurement processes is the aim of this paper. Given only a simple set of postulates, a probability measure is derived utilising the concept of Liouville space and the most general permissible quantum channel arises in the same manner. Super-Liouville space is constructed and a Bayesian interpretation of this object is provided. An important application of the new method is demonstrated, providing an axiomatic derivation of important results of the BB84 protocol in quantum cryptography.

U. Fano

The theoretical results of Baranger, Kolb, and Griem on pressure broadening are rederived by a more compact and flexible procedure directly applicable to other relaxation processes. Pressure broadening is worked out to first order in the pressure, including previously disregarded corrections. The procedure adapts the concepts and techniques of scattering theory to the Liouville representation of density matrices. Its key quantity is a frequency-dependent relaxation operator $〈{M}_{c}(\ensuremath{\omega})〉$ introduced by Zwanzig.