Mutually unbiased maximally entangled bases from difference matrices

Based on maximally entangled states, we explore the constructions of mutually unbiased bases in bipartite quantum systems. We present a new way to construct mutually unbiased bases by difference matrices in the theory of combinatorial designs. In particular, we establish $q$ mutually unbiased bases with $q-1$ maximally entangled bases and one product basis in $\mathbb{C}^q\otimes \mathbb{C}^q$ for arbitrary prime power $q$. In addition, we construct maximally entangled bases for dimension of composite numbers of non-prime power, such as five maximally entangled bases in $\mathbb{C}^{12}\otimes \mathbb{C}^{12}$ and $\mathbb{C}^{21}\otimes\mathbb{C}^{21}$, which improve the known lower bounds for $d=3m$, with $(3,m)=1$ in $\mathbb{C}^{d}\otimes \mathbb{C}^{d}$. Furthermore, we construct $p+1$ mutually unbiased bases with $p$ maximally entangled bases and one product basis in $\mathbb{C}^p\otimes \mathbb{C}^{p^2}$ for arbitrary prime number $p$.Comment: 24 page

Novel Constructions of Mutually Unbiased Tripartite Absolutely Maximally Entangled Bases

We develop a new technique to construct mutually unbiased tripartite absolutely maximally entangled bases. We first explore the tripartite absolutely maximally entangled bases and mutually unbiased bases in $\mathbb{C}^{d} \otimes \mathbb{C}^{d} \otimes \mathbb{C}^{d}$ based on mutually orthogonal Latin squares. Then we generalize the approach to the case of $\mathbb{C}^{d_{1}} \otimes \mathbb{C}^{d_{2}} \otimes \mathbb{C}^{d_{1}d_{2}}$ by mutually weak orthogonal Latin squares. The concise direct constructions of mutually unbiased tripartite absolutely maximally entangled bases are remarkably presented with generality. Detailed examples in $\mathbb{C}^{3} \otimes \mathbb{C}^{3} \otimes \mathbb{C}^{3},$ $\mathbb{C}^{2} \otimes \mathbb{C}^{2} \otimes \mathbb{C}^{4}$ and $\mathbb{C}^{2} \otimes \mathbb{C}^{5} \otimes \mathbb{C}^{10}$ are provided to illustrate the advantages of our approach

Entanglement in mutually unbiased bases

One of the essential features of quantum mechanics is that most pairs of observables cannot be measured simultaneously. This phenomenon is most strongly manifested when observables are related to mutually unbiased bases. In this paper, we shed some light on the connection between mutually unbiased bases and another essential feature of quantum mechanics, quantum entanglement. It is shown that a complete set of mutually unbiased bases of a bipartite system contains a fixed amount of entanglement, independently of the choice of the set. This has implications for entanglement distribution among the states of a complete set. In prime-squared dimensions we present an explicit experiment-friendly construction of a complete set with a particularly simple entanglement distribution. Finally, we describe basic properties of mutually unbiased bases composed only of product states. The constructions are illustrated with explicit examples in low dimensions. We believe that properties of entanglement in mutually unbiased bases might be one of the ingredients to be taken into account to settle the question of the existence of complete sets. We also expect that they will be relevant to applications of bases in the experimental realization of quantum protocols in higher-dimensional Hilbert spaces.Comment: 13 pages + appendices. Published versio

On Approximately Symmetric Informationally Complete Positive Operator-Valued Measures and Related Systems of Quantum States

We address the problem of constructing positive operator-valued measures (POVMs) in finite dimension $n$ consisting of $n^2$ operators of rank one which have an inner product close to uniform. This is motivated by the related question of constructing symmetric informationally complete POVMs (SIC-POVMs) for which the inner products are perfectly uniform. However, SIC-POVMs are notoriously hard to construct and despite some success of constructing them numerically, there is no analytic construction known. We present two constructions of approximate versions of SIC-POVMs, where a small deviation from uniformity of the inner products is allowed. The first construction is based on selecting vectors from a maximal collection of mutually unbiased bases and works whenever the dimension of the system is a prime power. The second construction is based on perturbing the matrix elements of a subset of mutually unbiased bases. Moreover, we construct vector systems in \C^n which are almost orthogonal and which might turn out to be useful for quantum computation. Our constructions are based on results of analytic number theory.Comment: 29 pages, LaTe

Symplectic spreads, planar functions and mutually unbiased bases

In this paper we give explicit descriptions of complete sets of mutually unbiased bases (MUBs) and orthogonal decompositions of special Lie algebras $sl_n(\mathbb{C})$ obtained from commutative and symplectic semifields, and from some other non-semifield symplectic spreads. Relations between various constructions are also studied. We show that the automorphism group of a complete set of MUBs is isomorphic to the automorphism group of the corresponding orthogonal decomposition of the Lie algebra $sl_n(\mathbb{C})$. In the case of symplectic spreads this automorphism group is determined by the automorphism group of the spread. By using the new notion of pseudo-planar functions over fields of characteristic two we give new explicit constructions of complete sets of MUBs.Comment: 20 page

Bipartite entangled stabilizer mutually unbiased bases as maximum cliques of Cayley graphs

We examine the existence and structure of particular sets of mutually unbiased bases (MUBs) in bipartite qudit systems. In contrast to well-known power-of-prime MUB constructions, we restrict ourselves to using maximally entangled stabilizer states as MUB vectors. Consequently, these bipartite entangled stabilizer MUBs (BES MUBs) provide no local information, but are sufficient and minimal for decomposing a wide variety of interesting operators including (mixtures of) Jamiolkowski states, entanglement witnesses and more. The problem of finding such BES MUBs can be mapped, in a natural way, to that of finding maximum cliques in a family of Cayley graphs. Some relationships with known power-of-prime MUB constructions are discussed, and observables for BES MUBs are given explicitly in terms of Pauli operators.Comment: 8 pages, 1 figur