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Poincare Journal of Analysis and Applications

INJECTIVITY PROBLEM VIA g-FRAMES

ASGHAR RAHIMI, ABDOLLAH GHASEMZADEH, AND BAYAZ DARABY

Published May 12, 2024• 1 citation
Publisher

Abstract

The main issues in quantum detection theory are the injectivity and state estimation problems, which recently linked and partially studied via frame theory on Hilbert spaces.Inspired by some recent researches and works on the quantum detection problem by (discrete) frames, continuous frames and fusion frames, in this note we examine the quantum detection problem for generalized frames (g-frames).In the theory of g-frames, instead of using the frame element itself, we use a family of bounded operators.We characterize injectivity problem by g-frames and with the help of these characterizations, we give some equivalent conditions for the injectivity problems.We answer the injectivity problems both for trace and Hilbert-Schmidt classes of operators.

DOI
10.46753/pjaa.2023.v010i03.008

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The main issues in quantum detection theory are the injectivity and state estimation problems, which recently linked and partially studied via frame theory on Hilbert spaces.Inspired by some recent researches and works on the quantum detection problem by (discrete) frames, continuous frames and fusion frames, in this note we examine the quantum detection problem for generalized frames (g-frames).In the theory of g-frames, instead of using the frame element itself, we use a family of bounded operators.We characterize injectivity problem by g-frames and with the help of these characterizations, we give some equivalent conditions for the injectivity problems.We answer the injectivity problems both for trace and Hilbert-Schmidt classes of operators.
A quantum injective frame is a frame whose measurements for density operators can be used as a distinguishing feature in a quantum system, and the frame quantum detection problem demands a characterization of all such frames. Very recently, the quantum detection problem for continuous as well as discrete frames in both finite and infinite dimensional Hilbert spaces received significant attention. The quantum detection problem pertaining to the characterization of informationally complete positive operator-valued measures (POVM) can be split into two cases: The quantum injectivity or state separability problem and the rang analysis or quantum state estimation problem. Building upon this notion, this note is aimed at the quantum detection problem for fusion frames. The injectivity of a family of vectors and a family of closed subspaces is characterized in terms of some operator equations in Hilbert–Schmidt and trace classes.
Some recent references have considered quantum detection problems in discrete and continuous frames. This paper examines the quantum detection problem with g-frame in finite dimensions. We will discuss two issues related to the problem at hand: the quantum injectivity problem and the solvability of the quantum state estimation problem. This work addresses the quantum detection problem in a broader context and provides multiple descriptions of quantum injective g-frames. Specifically, we also offer a quantitative stability outcome for g-frames.
A quantum injective frame is a frame that can be used to distinguish quantum systems by measuring density operators. The quantum detection problem aims to characterize all such frames. Re-cently, there has been significant attention on the problem of quantum detection in both finite and infinite dimensional Hilbert spaces, for both continuous and discrete frames. This paper aims to present various characterizations for injective continuous g-frames. The work solves the quantum detection problem in a more general context and provides characterizations for a broader class of injective continuous g-frames.
We will give a complete solution to the frame quantum detection problem. We will solve both cases of the problem: the quantum injectivity problem and quantum state estimation problem. We will answer the problem in both the real and complex cases and in both the finite dimensional and infinite dimensional cases. Finite Dimensional Case: (1) We give two complete classifications of the sets of vectors which solve the injectivity problem - for both the real and complex cases. We also give methods for constructing them. (2) We show that the frames which solve the injectivity problem are open and dense in the family of all frames. (3) We show that the Parseval frames which give injectivity are dense in the Parseval frames. (4) We classify all frames for which the state estimation problem is solvable, and when it is not solvable, we give the best approximation to a solution. Infinite Dimensional Case: (1) We give a classification of all frames which solve the injectivity problem and give methods for constructing solutions. (2) We show that the frames solving the injectivity problem are neither open nor dense in all frames. (3) We give necessary and sufficient conditions for the state estimation problem to always have solutions for measurements in ℓ1\ell_1ℓ1​ and show that there is no injective frame for which the state estimation problem is solvable for all measurements in ℓ2\ell_2ℓ2​. (4) When the state estimation problem does not have an exact solution, we give the best approximation to a solution.
We continue the study of Phase Retrieval in the Quantum Information setting in the style of \cite{liu2023phase}, in a manner which is more general in two primary ways: (a) Instead of only pure states we consider also states (and positive semidefinite operators) of bounded rank, as is done in Quantum Tomography; (b) We consider general super operators instead of only quantum channels. We show that in order to have phase retrieval with respect to positive semidefinite operators of bounded rank, it suffices to discriminate between perfectly distinguishable pairs (i.e. those with orthogonal supports). From this we prove that phase retrievability is always Lipschitz stable with respect to the trace norm, and then we use the Fuchs-van de Graaf inequalities to deduce stability with respect to the Bures--Wasserstein distance (thus generalizing the known stability results for phase retrieval in Frame Theory). For Hermitian-preserving super operators taking values in block-diagonal matrices, we characterize phase retrievability in terms of conditions inspired by the classical complement property from Frame Theory.
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.