University of Minnesota
School of Physics & Astronomy

Condensed Matter Seminar

Wednesday, April 2nd 2008
12:20 pm:
Speaker: Raymond F. Bishop, The University of Manchester
Subject: Thermal and More General Mixed Coherent States for Open Systems and Quantum Information Theory II

A complete formulation of quantum information theory must simultaneously take into account at least three important concepts or principles, namely: (a) quantum entanglement, (b) quantum coherence versus decoherence (i.e., in the presence of dissipation), and (c) the quantum-classical limit (or quantum-classical interface). We discuss how the mixed coherent states that we have introduced can provide a valuable tool in implementing these ideas.

The (over-)complete set of pure Glauber (harmonic oscillator) coherent states is a very useful basis for many purposes. The P-representation then provides a diagonal expansion of an arbitrary operator in ? in terms of projection operators onto the coherent states. We discuss the extension of these results to the analogous mixed states introduced by us, which describe comparable displaced harmonic oscillator systems in thermodynamic equilibrium at nonzero temperatures T. These thermal coherent states provide a very useful "random" (or "thermal" or "noisy") basis in H, since the corresponding statistical density operator provides a probability measure on H. We prove a resolution of the identity for these states and use it to generalise the usual pure (T = 0) coherent state formalism to the mixed (T ne 0) case. An important and unexpected result is that our temperature-dependent P- and Q-representations are the analytic continuations to negative temperatures of each other.

The above formalism for thermal coherent states is further generalised to a broader class of so-called negative-binomial mixed states, in terms of which a resolution of the identity operator in H is again constructed. The thermal coherent states are just the limiting case k = ½ of this larger class, characterised by a new parameter k. The negative-binomial distribution is itself intimately related to the discrete series of SU(1,1) representations. Indeed, the Hilbert space of a two-mode harmonic oscillator can be expressed as a direct sum of an infinite number of subspaces each of which is related to a particular representation in the discrete series of SU(1,1) representations. We have shown previously how such states can be useful for processes involving general fluctuation-dissipation phenomena. We consider the pure SU(1,1) coherent states discussed by Perelomov in the two-mode harmonic oscillator Hilbert space, and show that the partial trace with respect to one of the two modes leads, rather miraculously, to our negative-binomial mixed states. This observation is then used to show how the formalism of thermofield dynamics may be generalised to a correspondingly much broader negative-binomial field dynamics, which we expect to have many uses for open systems.

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