Related papers: Mutually Unbiased Bases and Orthogonal Latin Squar…
We show that in a complex d-dimensional vector space, one can find O(d) bases whose elements form a 2-design. Such vector sets generalize the notion of a maximal collection of mutually unbiased bases (MUBs). MUBs have manifold applications…
Mutually unbiased bases of a Hilbert space can be constructed by partitioning a unitary error basis. We consider this construction when the unitary error basis is a nice error basis. We show that the number of resulting mutually unbiased…
Constructing four six-dimensional mutually unbiased bases (MUBs) is an open problem in quantum physics and measurement. We investigate the existence of four MUBs including the identity, and a complex Hadamard matrix (CHM) of Schmidt rank…
In this work, the concept of mutually unbiased frames is introduced as the most general notion of unbiasedness for sets composed by linearly independent and normalized vectors. It encompasses the already existing notions of unbiasedness for…
Mutually unbiased bases (MUBs), which are such that the inner product between two vectors in different orthogonal bases is a constant equal to 1/sqrt{d), with d the dimension of the finite Hilbert space, are becoming more and more studied…
Every Latin square has three attributes that can be even or odd, but any two of these attributes determines the third. Hence the parity of a Latin square has an information content of 2 bits. We extend the definition of parity from Latin…
We show that if a set of four mutually unbiased bases (MUBs) in $\mathbb{C}^6$ exists and contains the identity, then any other basis in the set contains at most two product states and at the same time has Schmidt rank at least three. Here…
We investigate MacNeish's conjecture (known to be false in general) in the setting of what we call "transitive" Mutually Orthogonal Latin Squares (MOLS). When we restrict our attention to "simply transitive" MOLS, we find that the…
We show that any partial Latin square of order $n$ can be embedded in a Latin square of order at most $16n^2$ which has at least $2n$ mutually orthogonal mates. We also show that for any $t\geq 2$, a pair of orthogonal partial Latin squares…
A long-standing open problem asks if there can exist 7 mutually unbiased bases (MUBs) in $\mathbb{C}^6$, or, more generally, $d + 1$ MUBs in $\mathbb{C}^d$ for any $d$ that is not a prime power. The recent work of Kolountzakis, Matolcsi,…
Finding four six-dimensional mutually unbiased bases (MUBs) containing the identity matrix is a long-standing open problem in quantum information. We show that if they exist, then the $H_2$-reducible matrix in the four MUBs has exactly nine…
Mutually unbiased bases (MUB) have many applications in quantum information processing and quantum cryptography. Several complex MUB's in $\mathbb{C}^d$ for some dimension $d$ and with larger size have been constructed. On the other hand,…
There are fairly large families of unitarily inequivalent complete sets of N+1 mutually unbiased bases (MUBs) in C^N for various prime powers N. The number of such sets is not bounded above by any polynomial as a function of N. While it is…
We exhibit an infinite family of {\it triplets} of mutually unbiased bases (MUBs) in dimension 6. These triplets involve the Fourier family of Hadamard matrices, $F(a,b)$. However, in the main result of the paper we also prove that for any…
We exhibit an operational connection between mutually unbiased bases and symmetric infomationally complete positive operator-valued measures. Assuming that the latter exists, we show that there is a strong link between these two structures…
We describe a particular class of pairs of quantum observables which are extremal in the convex set of all pairs of compatible quantum observables. The pairs in this class are constructed as uniformly noisy versions of two mutually unbiased…
We use combinatorial and Fourier analytic arguments to prove various non-existence results on systems of real and complex unbiased Hadamard matrices. In particular, we prove that a complete system of complex mutually unbiased Hadamard…
The paper gives a short introduction to mutually unbiased bases and the Welch bounds and demonstrates that the latter is a good technical tool to explore the former. In particular, a criterion for a system of vectors to satisfy the Welch…
We construct two mutually unbiased bases by maximally entangled states (MUMEB$s$) in $\mathbb{C}^{2}\otimes \mathbb{C}^{3}$. This is the first example of MUMEB$s$ in $\mathbb{C}^{d}\otimes \mathbb{C}^{d'}$ when $d\nmid d'$, namely $d'$ is…
A Galois unitary is a generalization of the notion of anti-unitary operators. They act only on those vectors in Hilbert space whose entries belong to some chosen number field. For Mutually Unbiased Bases the relevant number field is a…