Finite frames constructed by solving Fekete problem and accuracy of quantum tomography protocols based on them

Abstract

Finite frames have many applications in quantum informatics, communications, coding theory, and other fields. In this article we investigate their use in quantum tomography. The family of vectors comprising a frame can be used to define a protocol of quantum tomography. To do this one estimates the probabilities of projective measurements on quantum states corresponding to the frame vectors. The state under investigation can then be reconstructed by one of several quantum tomography techniques. Since quantum tomography performance depends greatly on the choice of protocol, the problem of finding an optimal protocol is of considerable interest. WeexamineafamilyofunitnormframesconsistingofnumericalsolutionstoFeketepackingproblemincomplex space. Notable members of this family are Mutually Unbiased Bases (MUBs), which were computed in all investigated prime power dimensions, as well as Symmetrical Informationally Complete Positive Operator Valued Measures(SIC-POVM) in all dimensions. This family contains other uniform tight frames, in which the number of vectors lies between that of full sets of Mutually Unbiased Bases and Symmetrical Informationally Complete POVMs, and which all share the same internal structure. We detail some properties of these frames and their performance as quantum tomography protocols.