There has been great interest in finding sets of $m$ mutually unbiased bases which are compatible with a given space $\mathbb{C}^d$, specially in physics due to their interesting applications in quantum information theory. Several general results have been obtained so far, but surprising results may occur for definite $(m,d)$-values. One such case that has remained an open question (the simplest case) is the one regarding the existence of $m=4$ mutually orthogonal bases for $d=6$. In the present work we introduce a new approach to the problem by translating it into an optimization procedure for a given pair $(m,d)$.
New approach to finding the maximum number of mutually unbiased bases in C6
There has been great interest in finding sets of m mutually unbiased bases which are compatible with a given space Cd, specially in physics due to their interesting applications in quantum information theory. Several general results have been obtained so far, but surprising results may occur for definite (m,d)-values. One such case that has remained an open question (the simplest case) is the one regarding the existence of m=4 mutually orthogonal bases for d=6. In the present work we introduce a new approach to the problem by translating it into an optimization procedure for a given pair (m,d).
There has been great interest in finding sets of m mutually unbiased bases which are compatible with a given space C, specially in physics due to their interesting applications in quantum information theory. Several general results have been obtained so far, but surprising results may occur for definite (m, d)-values. One such case that has remained an open question (the simplest case) is the one regarding the existence of m = 4 mutually orthogonal bases for d = 6. In the present work we introduce a new approach to the problem by translating it into an optimization procedure for a given pair (m, d).
The study of Mutually Unbiased Bases continues to be developed vigorously, and presents several challenges in the Quantum Information Theory. Two orthonormal bases in Cd,BandB′ are said mutually unbiased if ∀b∈B,b′∈B′ the scalar product b⋅b′ has modulus d−1/2. In particular this property has been introduced in order to allow an optimization of the measurement-driven quantum evolution process of any state ψ∈Cd when measured in the mutually unbiased bases B_jofCd. At present it is an open problem to find the maximal umber of mutually Unbiased Bases when d is not a power of a prime number. \noindent In this article, we revisit the problem of finding Mutually Unbiased Bases (MUB's) in any dimension d. The method is very elementary, using the simple unitary matrices introduced by Schwinger in 1960, together with their diagonalizations. The Vandermonde matrix based on the d-th roots of unity plays a major role. This allows us to show the existence of a set of 3 MUB's in any dimension, to give conditions for existence of more than 3 MUB's for d even or odd number, and to recover the known result of existence of d+1 MUB's for d a prime number. Furthermore the construction of these MUB's is very explicit. As a by-product, we recover results about Gauss Sums, known in number theory, but which have apparently not been previously derived from MUB properties.
In this note I point out that (1) pairs of real mutually unbiased bases (i.e., orthonormal bases of R^d) can only exist in dimensions 2 or d where d is a multiple of 4 and that (2) triples of real mutually unbiased bases can only exist for dimensions d that are also squares. For the case d=2^{2m} the maximal number of real MUBs is given by (d+2)/2 (this follows from known results on extremal euclidean line-sets). In the case d=s^2 where s is an even number that is not a power of 2, we have the lower bound N_MOLS(s)+2, where N_MOLS(s) denotes the maximal number of mutually orthogonal Latin square of order s, provided that there exists a Hadamard matrix of size s. It is not known how good this bound is. Moreover, I observe that the question of deciding if there always two real MUBs is equivalent to the Hadamard conjecture.
A set of k orthonormal bases of Cd is called mutually unbiased if ∣⟨e,f⟩∣2=1/d whenever e and f are basis vectors in distinct bases. A natural question is for which pairs (d,k) there exist~k mutually unbiased bases in dimension d. The (well-known) upper bound k≤d+1 is attained when~d is a power of a prime. For all other dimensions it is an open problem whether the bound can be attained. Navascués, Pironio, and Acín showed how to reformulate the existence question in terms of the existence of a certain C∗-algebra. This naturally leads to a noncommutative polynomial optimization problem and an associated hierarchy of semidefinite programs. The problem has a symmetry coming from the wreath product of Sd and Sk. We exploit this symmetry (analytically) to reduce the size of the semidefinite programs making them (numerically) tractable. A key step is a novel explicit decomposition of the Sd≀Sk-module C([d]×[k])t into irreducible modules. We present numerical results for small d,k and low levels of the hierarchy. In particular, we obtain sum-of-squares proofs for the (well-known) fact that there do not exist d+2 mutually unbiased bases in dimensions~d=2,3,4,5,6,7,8. Moreover, our numerical results indicate that a sum-of-squares refutation, in the above-mentioned framework, of the existence of more than 3 MUBs in dimension 6 requires polynomials of total degree at least~12.
Finding all the mutually unbiased bases in various dimensions is a problem of fundamental interest in quantum information theory and pure mathematics. The general problem formulated in finite-dimensional Hilbert spaces is open. In the categorical approach to quantum mechanics one can find examples of categories which behave ``like'' the category of finite-dimensional Hilbert spaces in various ways but are subtly different. One such category is the category of sets and relations, Rel. One can formulate the concept of mutually unbiased bases here as well. In this note we classify all the mutually unbiased bases in this category by relating it to a standard question in combinatorics.
Mutually unbiased bases for quantum degrees of freedom are central to all theoretical investigations and practical exploitations of complementary properties. Much is known about mutually unbiased bases, but there are also a fair number of important questions that have not been answered in full as yet. In particular, one can find maximal sets of N + 1 mutually unbiased bases in Hilbert spaces of prime-power dimension N = p M , with p prime and M a positive integer, and there is a continuum of mutually unbiased bases for a continuous degree of freedom, such as motion along a line. But not a single example of a maximal set is known if the dimension is another composite number (N = 6, 10, 12,…). In this review, we present a unified approach in which the basis states are labeled by numbers 0, 1, 2, …, N - 1 that are both elements of a Galois field and ordinary integers. This dual nature permits a compact systematic construction of maximal sets of mutually unbiased bases when they are known to exist but throws no light on the open existence problem in other cases. We show how to use the thus constructed mutually unbiased bases in quantum-informatics applications, including dense coding, teleportation, entanglement swapping, covariant cloning, and state tomography, all of which rely on an explicit set of maximally entangled states (generalizations of the familiar two–q-bit Bell states) that are related to the mutually unbiased bases. There is a link to the mathematics of finite affine planes. We also exploit the one-to-one correspondence between unbiased bases and the complex Hadamard matrices that turn the bases into each other. The ultimate hope, not yet fulfilled, is that open questions about mutually unbiased bases can be related to open questions about Hadamard matrices or affine planes, in particular the notorious existence problem for dimensions that are not a power of a prime. The Hadamard-matrix approach is instrumental in the very recent advance, surveyed here, of our understanding of the N = 6 situation. All evidence indicates that a maximal set of seven mutually unbiased bases does not exist — one can find no more than three pairwise unbiased bases — although there is currently no clear-cut demonstration of the case.