相关论文: Constructions of Mutually Unbiased Bases
Construction of a large class of Mutually Unbiased Bases (MUBs) for non-prime power composite dimensions ($d = k\times s$) is a long standing open problem, which leads to different construction methods for the class Approximate MUBs (AMUBs)…
A new construction of complex Hadamard matrices of composite order d=pq, with primes p,q, is presented which is based on pairs of mutually unbiased bases containing only product states. For product dimensions d < 100, we illustrate the…
The notion of unbiased orthogonal designs is introduced as a generalization among unbiased Hadamard matrices, unbiased weighing matrices and quasi-unbiased weighing matrices. We provide upper bounds and several constructions for mutually…
There exists a set $A$ of positive integers such that the number of representations of a large positive integer $m$ as a sum of two elements of $A$ grows with a lower bound of order $\log m$, but for which there is no subset $D$ of $A$…
We consider questions posed in a recent paper of Mandayam, Bandyopadhyay, Grassl and Wootters [10] on the nature of "unextendible mutually unbiased bases." We describe a conceptual framework to study these questions, using a connection…
Mutually unbiased bases (MUBs) have been used in several cryptographic and communications applications. There has been much speculation regarding connections between MUBs and finite geometries. Most of which has focused on a connection with…
Mutually Unbiased Bases (MUBs) constitute a fundamental geometric structure in quantum theory, known for providing an optimal measurement scheme for quantum state tomography. In prime and prime-power dimensions, analytical constructions of…
We present a detailed computational and algebraic study of Mutually Unbiased Bases (MUBs) in finite-dimensional Hilbert spaces, with a particular focus on dimensions 2, 3, 4, and the challenging case of 6. Starting from the Hadamard-phase…
Bases of finite-dimensional Hilbert spaces (in dimension d) of relevance for quantum information and quantum computation are constructed from angular momentum theory and su(2) Lie algebraic methods. We report on a formula for deriving in…
For a field $F$ and an integer $d\geq 1$, we consider the universal associative $F$-algebra $A$ generated by two sets of $d+1$ mutually orthogonal idempotents. We display four bases for the $F$-vector space $A$ that we find attractive. We…
In quantum information, complementarity of quantum mechanical observables plays a key role. If a system resides in an eigenstate of an observable, the probability distribution for the values of a complementary observable is flat. The…
We study entanglement witness and present a construction of entanglement witnesses in terms of the mutually unbiased measurements (MUMs). These witnesses include the entanglement witnesses constructed from mutually unbiased bases (MUBs) as…
Quantum systems with variables in ${\mathbb Z}(d)$ are considered, and three different structures are studied. The first is weak mutually unbiased bases, for which the absolute value of the overlap of any two vectors in two different bases…
We derive a framework for quantifying entanglement in multipartite and high dimensional systems using only correlations in two unbiased bases. We furthermore develop such bounds in cases where the second basis is not characterized beyond…
We show that, for every set of $n$ points in the $d$-dimensional unit cube, there is an empty axis-parallel box of volume at least $\Omega(d/n)$ as $n\to\infty$ and $d$ is fixed. In the opposite direction, we give a construction without an…
In a previous paper, one of us pointed out that the anomalous dimension matrices for all physical processes that have been calculated to date are complex symmetric, if stated in an orthonormal basis. In this paper we prove this fact and…
An old problem of Moser asks: how large of a union-free subfamily does every family of m sets have? A family of sets is called union-free if there are no three distinct sets in the family such that the union of two of the sets is equal to…
The unextendible entangled basis with any arbitrarily given Schmidt number $k$ (UEBk) in $\mathbb{C}^{d_1}\otimes\mathbb{C}^{d_2}$ is proposed in [Phys. Rev. A 90 (2014) 054303], $1<k\leq \min\{d_1,d_2\}$, which is a set of orthonormal…
The dimension of a linear space is the maximum positive integer $d$ such that any $d$ of its points generate a proper subspace. For a set $K$ of integers at least two, recall that a pairwise balanced design PBD$(v,K)$ is a linear space on…
I introduce a new notion, that extends the mutually unbiased bases (MUB) conditons to more than two bases. These, I call the nUB conditions, and the corresponding bases $n$-fold unbiased. They naturally appear while optimizing generic…