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Mathematische Nachrichten

Conditional Local States and Decomposition of Locally Normal States

Volkmar Liebscher

Published Jan 1, 1993• 0 citations
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Abstract

Abstract For infinite boson systems in the framework of [5, 9] it is proven that all locally normal states possess conditional local states. This result is used to find a tail‐field‐measurable decomposition of these states and to characterize their position distributions in completion to [6] as ‐Point processes. As an example mixed coherent states are investigated.

DOI
10.1002/mana.19931640109

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Abstract For infinite boson systems in the framework of [5, 9] it is proven that all locally normal states possess conditional local states. This result is used to find a tail‐field‐measurable decomposition of these states and to characterize their position distributions in completion to [6] as ‐Point processes. As an example mixed coherent states are investigated.
Specifying their (normally ordered) characteristic functions we determine all states of the boson C *-Weyl algebra which satisfy Glauber's coherence condition and are not realizable as density operators in Fock space. The pure ones are shown to be just the eigenstates of the annihilation operators in their GNS-representations (in contrast to the Fock case) and are characterized in many equivalent manners. The central decomposition of an arbitrary coherent state has the macroscopic phase variable as parameter and is supported by the pure coherent states, which is in fact the only way for a maximal decomposition. The set of all coherent states with the same absolute factorizing function is proven to be a Bauer simplex. The appearence of a classical coherent field part is studied in detail in the GNS-representations and shown to correspond to an enlargement of the set of one boson states by just one additional mode.
Based on a model for general beam splittings we search for states of boson systems which are invariant under the combination of the evolution given by the splitting procedure and some inherent evolution. It turns out that for finite systems only trivial invariant normal states may appear. However, for locally normal states on a related quasilocal algebra representing states of infinite boson systems, one can find examples of nontrivial invariant states. We consider as example a beam splitting combined with a contraction compensating the loss of intensity caused by the splitting process. In general, we observe interesting connections between the splitting procedure and certain thinning operations in classical probability theory. Several applications to physics seem to be natural since these beam splitting models are used to describe measuring procedures on electromagentic fields.
We analyze the first order coherent, states (and various sub-classes) over an arbitrary C*-Weyl algebra. Their normally ordered characteristic functions are expressed in terms of an infinite, positive-definitite matrix, which exhibits an additional positive-definiteness condition if the states are classical. In the latter case the matrix elements are identified as moments of probability measures on C. By going over to measures on bC, the Bohr compactification of C, there arises a Bauer simplex of generalized classical coherent states
Based on certain conditional expectations leading to conditional local states we provide a Gibbs Theory analogous to [20,21][20, 21][20,21] .As example, we look at specifications corresponding to coherent states and the ideal Bose gas.
It is shown that a finite number of conditions are not sufficient to determine the locality of transformations between two probability distributions of pure states as well as the locality of transformations between two $d\ifmmode\times\else\texttimes\fi{}d$ mixed states with $d\ensuremath{\geqslant}4$. As an example, an infinite, but minimal, set of necessary and sufficient conditions for the existence of a local procedure that converts one probability distribution of two pure pair of qubits into another one is found.