The Solution to the Frame Quantum Detection Problem

TL;DR

This paper provides a comprehensive solution to the frame quantum detection problem, including classifications, constructions, and density results for both finite and infinite dimensional cases, addressing injectivity and state estimation.

Contribution

It offers complete classifications and methods for solving the frame quantum detection problem in finite and infinite dimensions, including density and approximation results.

Findings

Frames solving the injectivity problem are open and dense in finite dimensions.

Parseval frames solving injectivity are dense among Parseval frames.

In infinite dimensions, frames solving injectivity are neither open nor dense.

Abstract

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 $\ell_1$ and show that there is no injective frame for which the state estimation problem is solvable for all measurements in $\ell_2$. (4) When the state estimation problem does not have an exact solution, we give the best approximation to a solution.