Equiangular lines, mutually unbiased bases, and spin models

Abstract

We use difference sets to construct interesting sets of lines in complex space. Using (v,k,1)-difference sets, we obtain k^2-k+1 equiangular lines in C^k when k-1 is a prime power. Using semiregular relative difference sets with parameters (k,n,k,l) we construct sets of n+1 mutually unbiased bases in C^k. We show how to construct these difference sets from commutative semifields and that several known maximal sets of mutually unbiased bases can be obtained in this way, resolving a conjecture about the monomiality of maximal sets. We also relate mutually unbiased bases to spin models.Comment: 23 pages; no figures. Minor correction as pointed out in arxiv.org:1104.337

Graph-state formalism for mutually unbiased bases

“…Here, the resulting states may no longer be graph-states, but belong to the class of LME states [25]. This gives a new perspective on other constructions such as the one by Alltop [12], which for p ≥ 5 was shown to be equivalent to the construction by Wootters and Fields up to a permutation of the vector components [52]. Note that for LME states, such a permutation can always be rephrased in terms of general phase gates [25].…”

A Criterion for Attaining the Welch Bounds with Applications for Mutually Unbiased Bases

“…In fact, a more general result [16] is known: the existence of a relative (n, k, n, λ)-difference set implies the existence of k MUHs in C n .…”

Close Encounters with Boolean Functions of Three Different Kinds

“…Such objects are called equiangular lines and, in the context of quantum tomography, are known as SIC-POVMs. Whilst a number of SIC-POVM sequences have been found over unwieldy alphabets, only one 2 × 2 × 2 SIC-POVM array has been found (the Hoggar lines), and this is not over the alphabet {1, −1} [16]. Moreover, [16] has shown that SIC-POVM arrays in C n 2 do not exist for n > 3.…”

“…Whilst a number of SIC-POVM sequences have been found over unwieldy alphabets, only one 2 × 2 × 2 SIC-POVM array has been found (the Hoggar lines), and this is not over the alphabet {1, −1} [16]. Moreover, [16] has shown that SIC-POVM arrays in C n 2 do not exist for n > 3. When viewing the n-variable Boolean function, a, as a quantum state of n qubits, as described by pure-state vector |A >= 2 −n/2 A, then the action of the HW group on |A > identifies the qubit bit-flip, phase-flip, and combined phase-flip then bit-flip errors on |A > (the action of unitaries X, Z, and XZ), respectively.…”

“…But, in the next section, we identify perfect pairs of type-I, II, and III functions which are also constructible, whereas the far stricter HW criteria does not appear to allow such pairs. However, recent activity [18,16] has identified N -element sequences and (one) array which realise the value of |H s,t (a)| 2 = 1 N +1 everywhere, which is the theoretical min-max. Such objects are called equiangular lines and, in the context of quantum tomography, are known as SIC-POVMs.…”