Related papers: Hjelmslev Geometry of Mutually Unbiased Bases
For a system of N qubits, spanning a Hilbert space of dimension d=2^N, it is known that there exists d+1 mutually unbiased bases. Different construction algorithms exist, and it is remarkable that different methods lead to sets of bases…
A complex Hilbert space of dimension six supports at least three but not more than seven mutually unbiased bases. Two computer-aided analytical methods to tighten these bounds are reviewed, based on a discretization of parameter space and…
In this paper, we consider the problem of Mutually Unbiased Bases in prime dimension $d$. It is known to provide exactly $d+1$ mutually unbiased bases. We revisit this problem using a class of circulant $d \times d$ matrices. The…
In this thesis we study symmetric structures in Hilbert spaces known as symmetric informationally complete positive operator-valued measures (SIC-POVMs), mutually unbiased bases (MUBs), and MUB-balanced states. Our tools include symmetries…
This paper deals with bases in a finite-dimensional Hilbert space. Such a space can be realized as a subspace of the representation space of SU(2) corresponding to an irreducible representation of SU(2). The representation theory of SU(2)…
We propose a unifying phase-space approach to the construction of mutually unbiased bases for a two-qubit system. It is based on an explicit classification of the geometrical structures compatible with the notion of unbiasedness. These…
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 study mutually unbiased maximally entangled bases (MUMEB's) in bipartite system $\mathbb{C}^d\otimes\mathbb{C}^d (d \geq 3)$. We generalize the method to construct MUMEB's given in [16], by using any commutative ring $R$ with $d$…
We develop a strong connection between maximally commuting bases of orthogonal unitary matrices and mutually unbiased bases. A necessary condition of the existence of mutually unbiased bases for any finite dimension is obtained. Then a…
We consider the notion of unitary transformations forming bases for subspaces of $M(d,\mathbb{C})$ such that the square of Hilbert-Schmidt inner product of matrices from the differing bases is a constant. Moving from the qubit case,…
We initiate a systematic study of triplets of mutually unbiased bases (MUBs). We show that in $\mathbb{C}^d$ each MUB-triplet is characterized by a $d\times d\times d$ object that we call a Hadamard cube. We describe the basic properties of…
We study sets of pure states in a Hilbert space of dimension d which are mutually unbiased (MU), that is, the squares of the moduli of their scalar products are equal to zero, one, or 1/d. These sets will be called a MU constellation, and…
We consider the primitive quaternionic reflection groups of type P for H^2 that are obtained from Blichfeldt's collineation groups for C^4.These are seen to be intimately related to the maximal set of five quaternionic mutually unbiased…
The paper concerns Weyl-Heisenberg covariant SIC-POVMs (symmetric informationally complete positive operator valued measures) and full sets of MUBs (mutually unbiased bases) in prime dimension. When represented as vectors in generalized…
A complete set of mutually unbiased bases in a Hilbert space of dimension $d$ defines a set of $d+1$ orthogonal measurements. Relative to such a set, we define a "MUB-balanced state" to be a pure state for which the list of probabilities of…
A set of mutually unbiased bases (MUBs) is said to be unextendible if there does not exist another basis that is unbiased with respect to the given set. Here, we prove the existence of smaller sets of MUBs in prime-squared dimensions…
Mutually unbiased bases that can be cyclically generated by a single unitary operator are of special interest, since they can be readily implemented in practice. We show that, for a system of qubits, finding such a generator can be cast as…
In a quantum system having a finite number $N$ of orthogonal states, two orthonormal bases $\{a_i\}$ and $\{b_j\}$ are called mutually unbiased if all inner products $<a_i|b_j>$ have the same modulus $1/\sqrt{N}$. This concept appears in…
Mutually unbiased bases generalize the X, Y and Z qubit bases. They possess numerous applications in quantum information science. It is well-known that in prime power dimensions N=p^m (with p prime and m a positive integer) there exists a…
Let q be a power of 2. We show by representation theory that there exists a q x q unitary matrix of multiplicative order q+1 whose powers generate q+1 pairwise mutually unbiased base in C^q. When q is a power of an odd prime, there is a q x…