A short history of frames and quantum designs

Abstract

In this survey, we relate frame theory and quantum information theory, focusing on quantum 2-designs. These are arrangements of weighted subspaces which are in a specific sense optimal for quantum state tomography. After a brief introduction, we discuss the role of POVMs in quantum theory, developing the importance of quantum 2-designs. In the final section, we collect many if not most known examples of quantum-2 designs to date.

Trace optimality of SIC POVMs

“…1 In the complex projective space CP d−1 -made of all unit vectors in C d up to phase -a SIC POVM corresponds to a regular d 2 -simplex of maximal size, a so called tight simplex. As one would expect, such highly symmetrical objects are distinguished by various optimality properties [21,27,14,4,10]. Our goal is to put on record the following general characterization.…”

The Solution to the Frame Quantum Detection Problem

“…Proof. We will show that (1) implies (2). Let T, S be Hilbert Schmidt selfadjoint operators such that T x k , x k = Sx k , x k , for all k.…”

“…In this section we will solve the finite dimensional injectivity problem and the state estimation problem for both the real and complex cases. These problems were originally solved by Scott [17] (See also [2]) where the solutions are called informationally complete quantum measurements. We will have to redo this here since we need much more information about the solutions and need proofs in a format that will easily generalize to infinite dimensions.…”

“…(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 a frame to solve the state estimation problem for all measurements in ℓ1 and show that there is no injective frame for which the state estimation problem is solvable for all measurements in ℓ2.…”

“…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.…”