Normal covariant quantization maps

Abstract

We consider questions related to quantizing complex valued functions defined on a locally compact topological group. In the case of bounded functions, we generalize R. Werner's approach to prove the characterization of the associated normal covariant quantization maps.

Universal joint-measurement uncertainty relation for error bars

“…This fundamental fact has been proven and extensively studied by several authors using different techniques [9][10][11][12]. The marginal observables of G m are of the form (9), with the probability measures…”

Covariant observables on a nonunimodular group

“…On the other hand, they can be used in quantization. Namely, in the context of a locally compact, second countable topological group G with a Haar measure λ, each positive normal covariant map Γ : L ∞ (G, λ) → L(H), with L(H) the set of bounded operators on some separable Hilbert space H, is eligible to represent a quantization procedure [11,13]. The maps of this kind correspond to covariant positive operator measures via the association B(G) B → Γ (χ B ) ∈ L(H), where B(G) is the Borel σ -algebra of G and χ B the characteristic function of the set B.…”

“…The canonical examples of covariant observables are constructed, e.g., in [4], and there are (at least) two completely different ways to obtain their characterization: a group theoretical approach [3], and a direct approach [9,11,13] based on the theory of integration with respect to vector measures. The latter approach was presented by Werner in [13] in the context where G = R 2n , and it was generalized to the case of a unimodular group in [11]. The assumption of unimodularity was quite essential in the proof, and there is no trivial way of getting rid of it.…”

“…As in [13] and [11], the proof relies on the fact that the Banach space of trace class operators on a separable Hilbert space has the Radon-Nikodým property. The essential difference is that Lemma 3.1 of [13] and its generalization [11,Lemma 2] no longer hold if G is not unimodular. That result must be replaced by a weaker version, which is a consequence of the classical work of Duflo and Moore [6] concerning square integrable representations.…”

Measurement Uncertainty Relations for Position and Momentum: Relative Entropy Formulation

“…Although Q and P are not jointly measurable, the following well-known result says that there are plenty of covariant phase-space observables [4] (Theorem 4.8.3), [63,64]. In (43) below, we use the parity operator Π on H, which is such that…”