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Civil War Book Review

USING WEIGHTS FOR THE DESCRIPTION OF STATES OF BOSON SYSTEMS

Volkmar Liebscher

Published Jan 1, 2009• 1 citation
Publisher

Abstract

For a locally normal state of a boson system we construct, based on the Skorokhod integral and Malliavin derivative respectively their compound versions, weights showing similar properties as the reduced and the compound Campbell measures of classical point process theory.As application we provide several characterisations of coherent states and states with property (γ h ), generalising a respective notion from point process theory and appearing in coherence considerations [37].

DOI
10.31390/cosa.3.2.02

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For a locally normal state of a boson system we construct, based on the Skorokhod integral and Malliavin derivative respectively their compound versions, weights showing similar properties as the reduced and the compound Campbell measures of classical point process theory.As application we provide several characterisations of coherent states and states with property (γ h ), generalising a respective notion from point process theory and appearing in coherence considerations [37].
We construct a new family of boson coherent states using a specially designed function which is a solution of a functional equation dε(q,x)/dx=ε(q,qx) with 0⩽q⩽1 and ε(q,0)=1. We use this function in place of the usual exponential to generate new coherent states |q,z〉 from the vacuum, which are normalized and continuous in their label z. These states allow the resolution of unity, and a corresponding weight function is furnished by the exact solution of the associated Stieltjes moment problem. They also permit exact evaluation of matrix elements of an arbitrary polynomial given as a normally-ordered function of boson operators. We exemplify this by showing that the photon number statistics for these states is sub-Poissonian. For any q<1 the states |q,z〉 are squeezed; we obtain and discuss their signal to quantum noise ratio. The function ε(q,x) allows a natural generation of multiboson coherent states of arbitrary multiplicity, which is impossible for the usual coherent states. For q=1 all the above results reduce to those for conventional coherent states. Finally, we establish a link with q-deformed bosons.
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
As an analogy of best separable approximation (BSA) in the framework of entanglement theory, here we concentrate on the notion of best incoherent approximation, with application to characterizing and quantifying quantum coherence. From both analytical and numerical perspectives, we have demonstrated that the weight-based coherence measure displays some unusual properties, in sharp contrast to other popular coherence quantifiers. First, by deriving a closed formula for single-qubit states, we have shown the exact value of weight-based coherence measure does not depend solely on the off-diagonal element but relies on the geometry of a given qubit state, thus exhibiting a rich structure even in this simplest case. Second, albeit the fact that almost all popular coherence measures only assign a maximal value to pure maximally coherent states, we have demonstrated the existence of mixed maximally coherent states (MMCS) with respect to this coherence measure and discussed the characteristic feature of MMCS in high-dimensional Hilbert spaces. Especially, we present several important families of MMCS by gaining insights from the numerical simulations. Moreover, it is pointed out that some considerations in this work can be generalized to general convex resource theories and a numerical method of improving the computational efficiency for finding the BSA is also discussed.
Abstract We employ reduced density matrices to investigate the localization of relativistic particles within the framework of (1+1)(1+1)(1+1) ( 1 + 1 ) -dimensional quantum field theory. In the case of bosons, we prove that scalar particles cannot be localized within any compact region, although they can be made arbitrarily close to localized states. For massless fermions, by exploiting properties of the reduced density matrices of coherent states, we show that the fermionic single-particle state which, under constructive bosonization, corresponds to a bosonic coherent state is strictly localized at a single spatial point; however, the energy of this state is infinite. Moreover, since the notion of spatial localization based on reduced density matrices is consistent with relativistic causality, even a particle with infinite energy that is localized at a single spatial point will neither spread outside the light cone nor propagate superluminally. Indeed, the explicit expression for the localized fermionic state derived in our work shows that a massless fermion can preserve its sharply localized profile while propagating at the speed of light, in full agreement with causality.
Twirled and swirled worlds describe generalised probabilistic theories (GPTs) restricted by symmetry-imposed superselection rules. We extend these constructions to a fully compositional setting for finite-dimensional GPTs and compact groups, treating states, effects, and multipartite transformations uniformly as processes. We distinguish system actions from process-space actions. We define actions on a theory as a whole by requiring the corresponding process-space actions to respect sequential and parallel composition. Averaging physical theory actions defines twirling, while averaging nonphysical theory actions defines swirling. We show that invariant subtheories might not satisfy process tomography, and establish conditions under which they do, involving invariant dynamically faithful states or internal classical reference frames. For locally tomographic parent theories, we prove that every theory symmetry admits a representation by linear system actions and that the invariant subtheory satisfies process tomography. An invariant multipartite process space fails local tomography if and only if some bipartition has sides sharing a non-trivial irreducible representation. We characterise when ancillary systems are necessary, which representations they must carry, and when entanglement is required for ancilla-free process tomography. We develop a framework for unifying twirled subtheories and extend incoherent simulations to compact groups. Finally, we recover bilocal classical theory as a twirled subtheory of classical theory and real quantum theory as a swirled subtheory of complex quantum theory, deriving their tomographic properties from their representation structure.