The purpose of this paper is to study the equivalence relation on unitary bases defined by R. F. Werner [{\it J. Phys. A: Math. Gen.} {\bf 34} (2001) 7081], relate it to local operations on maximally entangled vectors bases, find an invariant for equivalence classes in terms of certain commuting systems, and, relate it to mutually unbiased bases and Hadamard matrices. Illustrations are given in the context of latin squares and projective representations as well. Applications to quantum tomography are indicated.
The purpose of this paper is to study the equivalence relation on unitary bases defined by R. F. Werner [{\it J. Phys. A: Math. Gen.} {\bf 34} (2001) 7081], relate it to local operations on maximally entangled vectors bases, find an invariant for equivalence classes in terms of certain commuting systems, and, relate it to mutually unbiased bases and Hadamard matrices. Illustrations are given in the context of latin squares and projective representations as well. Applications to quantum tomography are indicated.
We analyze some special properties of a system of two qubits, and in particular of the so-called Bell basis for this system, and discuss the possibility of extending these properties to higher dimensional systems. We give a general construction for orthonormal bases of maximally entangled vectors, which works in any dimension, and is based on Latin squares and complex Hadamard matrices. However, for none of these bases the special properties of the operation of complex conjugation in Bell basis hold, namely that maximally entangled vectors have up-to-a-phase real coefficients and that factorizable unitaries have real matrix elements.
In this paper, we study the concrete construction of mutually unbiased maximally entangled bases in bipartite systems C d C kd(k ∈ Z+). We first analyze and simplify the sufficient and necessary conditions of two maximally entangled bases to be mutually unbiased in C d C kd, then we use matrix forms to illustrate these conditions in C2 C2 k and C3 C3 k and generalize them in C d C kd. Thus we find that the sufficient and necessary conditions of two maximally entangled bases to be mutually unbiased in C d C kd are translating to the conditions of transit matrices T kd between two orthonormal bases in C kd satisfy, that is, T kd can be divided into k2 submatrices of d × d satisfying same equations. So the problem of constructing mutually unbiased maximally entangle bases in C d C kd(k ∈ Z+) are changing to the choice of transit matrices in C kd. According the above equations of transit matrices in C kd, we first construct two transit matrices in C2, C3 and C4, then we find that using any unitary matrix with nonzero entities in C k to tensor product the above chosen transit matrices in C d(d = 2, 3, 4) from left, we can easily get the transit matrices in C kd. Hence the choice of transit matrices in C kd is changing to the choice of unitary matrices with nonzero entities in C k. Until now, the problem of constructing mutually unbiased maximally entangled bases in C d C kd is really simplified to the choice of unitary matrices in C k.Since the unitary matrices with nonzero entities in C k are always exist, we can confirm that mutually unbiased maximally entangled bases in C d C kd(d = 2, 3, 4; k ∈ Z+) always exist. Based on the two transit matrices we construct in each space in C kd(d = 2, 3, 4; k = 1, 2, 3), we want to describe the concrete construction of two mutually unbiased maximally entangled bases in C d C kd(d = 2, 3, 4; k = 1, 2, 3). As examples, we present a pair of mutually unbiased maximally entangled bases in C2 C6 and C3 C6, whose dimensions are not prime powers.
We introduce orthogonal quantum Latin squares, which restrict to traditional orthogonal Latin squares, and investigate their application in quantum information science. We use quantum Latin squares to build maximally entangled bases, and show how a pair of mutually unbiased maximally entangled bases can be constructed in square dimension from orthogonal quantum Latin squares. We also compare our construction to an existing construction due to Beth and Wocjan and show that ours is strictly more general.
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.
Many applications of finite tight frames (e.g., the use of SICs and mutually unbiased bases (MUBs) in quantum information theory and harmonic frames for the analysis of signals subject to erasures) depend only on the vectors up to projective unitary equivalence. It is well known that two finite sequences of vectors in inner product spaces are unitarily equivalent if and only if their respective inner products (Gramian matrices) are equal. Here we present a corresponding result for the projective unitary equivalence of two sequences of vectors (lines) in inner product spaces, i.e., that a finite number of (Bargmann) projective (unitary) invariants are equal. This result is equivalent to finding a rank-one matrix completion of a certain matrix. We give an algorithm to recover the sequence of vectors (up to projective unitary equivalence) from a small subset of these projective invariants and apply it to SICs, MUBs, and harmonic frames. We also extend our results to the projective similarity of vectors.