Continuous frames in tensor product Hilbert spaces, localization operators and density operators

TL;DR

This paper explores the properties of continuous frames in tensor product Hilbert spaces, their dual systems, and applications to quantum density operators, advancing the theoretical understanding of frame theory in physics.

Contribution

It provides a comprehensive characterization of dual systems for continuous frames in tensor product spaces and links these to quantum density operators.

Findings

Preservation of frame properties under tensor products.

Full characterization of dual systems for continuous frames.

Application to quantum density operators and partial trace multipliers.

Abstract

Continuous frames and tensor products are important topics in theoretical physics. This paper combines those concepts. We derive fundamental properties of continuous frames for tensor product of Hilbert spaces. This includes, for example, the consistency property, i.e. preservation of the frame property under the tensor product, and the description of the canonical dual tensors by those on the Hilbert space level. We show the full characterization of all dual systems for a given continuous frame, a result interesting by itself, and apply this to dual tensor frames. Furthermore, we discuss the existence on non-simple tensor product (dual) frames. Continuous frame multipliers and their Schatten class properties are considered in the context of tensor products. In particular, we give sufficient conditions for obtaining partial trace multipliers of the same form, which is illustrated with examples related to short-time Fourier transform and wavelet localization operators. As an application, we offer an interpretation of a class of tensor product continuous frame multipliers as density operators for bipartite quantum states, and show how their structure can be restricted to the corresponding partial traces.