More Constructions of Approximately Mutually Unbiased Bases

Abstract

Let $m$ be a positive integer and $p$ a prime number. We prove the orthogonality of some character sums over the finite field $\mathbb{F}_{p^{m}}$ or over a subset of a finite field and use this to construct some new approximately mutually unbiased bases of dimension $p^{m}$ over the complex number field $\mathbb{C}$, especially with $p=2$.

Spectral distribution of random matrices from Mutually Unbiased Bases

“…where W γ 1 ,γ 2 is defined in (9). If v γ 1 ∩γ 2 = 0, then the same statement holds with an extra factor of n in the numerator of the error term.…”

Constructing Mutually Unbiased Bases from Quantum Latin Squares

“…In addition we prove that our construction can produce bases that are unobtainable by existing methods [18,20]. We also introduce the concept of mutually weak orthogonal quantum Latin squares (MOQLS) which generalise mutually orthogonal Latin squares (MOLS), about which a significant body of research exists in connection with quantum information, and particularly pertaining to the connection between MOLS and MUBs [5,10,14]. Mutually unbiased bases are of fundamental importance to quantum information, as they capture the physical notion of complementary observables, quantities that cannot be simultaneously measured.…”

Spectral distribution of random matrices from mutually unbiased bases

“…for almost all dimensions n, and their construction can be extended to all dimensions n by assuming certain conjectures about the gap between consecutive primes. Some other variants of AMUBs have been studied in [10,20,28]. It can be seen that Theorem 1.1 also holds true for AMUBs.…”