A notion of optimal packings of subspaces with mixed-rank and solutions

Abstract

We resolve a longstanding open problem by reformulating the Grassmannian fusion frames to the case of mixed dimensions and show that this satisfies the proper properties for the problem. In order to compare elements of mixed dimension, we use a classical embedding to send all fusion frame elements to points on a higher dimensional Euclidean sphere, where they are given "equal footing". Over the embedded images -a compact subset in the higher dimensional embedded sphere -we define optimality in terms of the corresponding restricted coding problem. We then construct infinite families of solutions to the problem by using maximal sets of mutually unbiased bases and block designs. Finally, we show that using Hadamard 3-designs in this construction leads to infinite examples of maximal orthoplectic fusion frames of constant-rank. Moreover, any such fusion frames constructed by this method must come from Hadamard 3-designs.

Quantum state tomography as a numerical optimization problem

“…. , m} in an optimal, maximally spread fashion has been studied for both F = R and F = C [30][31][32][33][34][35][36][37][38][39][40]. Typically, optimality here refers to maximizing the minimum chordal distance d 2 c (P j , P i ) = l − tr(P † j P i ) where P j is the projector on the subspace U j , i.e.…”

“…We describe the realization of such a system-in NV centers in diamond. We reveal the relation between our optimization problem of finding the optimal QST measurement set and packing problems in Grassmannian manifolds, which have been studied in great detail [30][31][32][33][34][35][36][37][38][39][40] and are relevant for many fields, such as wireless communication, coding theory, and machine learning [39,[41][42][43][44]. As we are able to approximate the optimal measurement scheme of the qubit-qutrit system, we solve a greater problem, namely we find a close approximation to an optimal Grassmannian packing of half-dimensional subspaces in Hilbert space of dimension six.…”

Quantum state tomography as a numerical optimization problem

“…. , m} in an optimal, maximally spread fashion has been studied for both F = R and F = C [28][29][30][31][32][33][34][35][36][37][38]. Typically, optimality here refers to maximizing the minimum chordal distance d 2 c (P j , P i ) = l − Tr(P † j P i ) where P j is the projector on the subspace U j , i.e., min i =j d 2 c (P j , P i ) shall be maximal.…”

“…We describe the realization of such a system -in NV centers in diamond. We reveal the relation between our optimization problem of finding the optimal QST measurement set and packing problems in Grassmannian manifolds, which have been studied in great detail [28][29][30][31][32][33][34][35][36][37][38] and are relevant for many fields, such as wireless communication, coding theory, and machine learning [37,[39][40][41][42]. As we are able to approximate the optimal measurement scheme of the qubit-qutrit system, we solve a greater problem, namely we find an optimal Grassmannian packing of halfdimensional subspaces in Hilbert space of dimension six.…”