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Open Physics

Separability criteria via wavelet transform on homogenous spaces and projective representations

Mahdi Karamati, Mohammad Reza Rezapour

Published Jul 23, 2009• 0 citations
Publisher

Abstract

Abstract The intimate connection between the Banach space wavelet reconstruction method for each unitary representation of a given group and homogenous space, and the quantum entanglement description using group theory were both studied in our previous articles. Here, we present a universal description of quantum entanglement using group theory and non-commutative characteristic functions for homogenous space and projective representation of compact groups on Banach spaces for some well known examples, such as: Moyal representation for a spin; Dihedral and Permutation groups.

DOI
10.2478/s11534-009-0103-z

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Abstract The intimate connection between the Banach space wavelet reconstruction method for each unitary representation of a given group and homogenous space, and the quantum entanglement description using group theory were both studied in our previous articles. Here, we present a universal description of quantum entanglement using group theory and non-commutative characteristic functions for homogenous space and projective representation of compact groups on Banach spaces for some well known examples, such as: Moyal representation for a spin; Dihedral and Permutation groups.
The intimate connection between the Banach space wavelet reconstruction method for each unitary representation of a given group and some of well-known quantum tomographies, such as tomography of rotation group, spinor tomography and tomography of unitary group, is established. Also both the atomic decomposition and Banach frame nature of these quantum tomographic examples are revealed in detail. Finally, we consider separability criteria for any state with group theoretical wavelet transform on Banach spaces.
We present a universal description of quantum entanglement using group theory and noncommutative characteristic functions. It leads to reformulations of the separability problem, which allows us to generalize the latter, thus connecting the theory of entanglement and harmonic analysis. As an example, we translate and analyze the positivity of partial transpose criterion and a simple criterion for pure states into the group-theoretical language. We also show that when applied to finite groups, our formalism embeds the separability problem in a given dimension into a higher dimensional but highly symmetric one. Finally, our formalism reveals a connection between the very existence of entanglement and group noncommutativity.
Using the notions of frame transform and of square integrable projective representation of a locally compact group G , we introduce a class of isometries (tight frame transforms) from the space of Hilbert–Schmidt operators in the carrier Hilbert space of the representation into the space of square integrable functions on the direct product group G × G . These transforms have remarkable properties. In particular, their ranges are reproducing kernel Hilbert spaces endowed with a suitable 'star product' which mimics, at the level of functions, the original product of operators. A 'phase space formulation' of quantum mechanics relying on the frame transforms introduced in the present paper, and the link of these maps with both the Wigner transform and the wavelet transform are discussed.