Abstract
Alignment is a geometric relation between pairs of Weyl–Heisenberg SICs, one in dimension d and another in dimension , manifesting a well-founded conjecture about a number-theoretical connection between the SICs. In this paper, we prove that if d is even, the SIC in dimension of an aligned pair can be partitioned into (d − 2)2 tight d2-frames of rank and, alternatively, into d2 tight (d − 2)2-frames of rank . The corresponding result for odd d is already known, but the proof for odd d relies on results which are not available for even d. We develop methods that allow us to overcome this issue. In addition, we provide a relatively detailed study of parity operators in the Clifford group, emphasizing differences in the theory of parity operators in even and odd dimensions and discussing consequences due to such differences. In a final section, we study implications of alignment for the symmetry of the SIC.
Highlights
An informationally complete POVM is one that can be used to reconstruct any quantum state, pure or mixed
Since an n-dimensional state is given by an n × n unit-trace Hermitian matrix, and, by n2 − 1 real parameters, a minimal informationally complete POVM has to consist of n2 unit rank elements, giving n2 − 1 independent measurement results
We have proven that the property of alignment of WH-SICs in even dimensions of the form d(d − 2) implies that the SICs can be partitioned into sets of equiangular tight frames, in two different ways
Summary
An informationally complete POVM is one that can be used to reconstruct any quantum state, pure or mixed. SICs are exceptional among informationally complete POVMs in the sense that the information overlap of the measurement results is minimal, making them optimal candidates for state tomography [2] These remarkable tomographic properties reflect that the SIC elements constitute an equiangular tight frame of maximally many vectors. Zauner conjectured that in all finite dimensions at least one SIC exists that is covariant under the discrete Weyl-Heisenberg group, and he further conjectured that at least one such SIC has an order 3 unitary symmetry These conjectures have been guiding the search for SIC-POVMs ever since. Chinese remaindering can be applied in odd dimensions of the form we are interested in, since for odd d the factors d and d − 2 are relatively prime, and has been used to prove the existence of embedded tight frames in the SIC in the larger dimension of an aligned pair [7].
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