The objective of this paper is the linear reconstruction of a vector, up to a unimodular constant, when all phase information is lost, meaning only the magnitudes of frame coefficients are known. Reconstruction algorithms of this type are relevant for several areas of signal communications, including wireless and fiber-optical transmissions. The algorithms discussed here rely on suitable rank-one operator valued frames defined on finite-dimensional real or complex Hilbert spaces. Examples of such operator-valued frames are the rank-one Hermitian operators associated with vectors from maximal sets of equiangular lines or maximal sets of mutually unbiased bases. A more general type of examples is obtained by a tensor product construction. We also study erasures and show that in addition to loss of phase, a maximal set of mutually unbiased bases can correct for erased frame coefficients as long as no more than one erasure occurs among the coefficients belonging to each basis, and at least one basis remains without erasures.
The objective of this paper is the linear reconstruction of a vector, up to a unimodular constant, when all phase information is lost, meaning only the magnitudes of frame coefficients are known. Reconstruction algorithms of this type are relevant for several areas of signal communications, including wireless and fiber-optical transmissions. The algorithms discussed here rely on suitable rank-one operator valued frames defined on finite-dimensional real or complex Hilbert spaces. Examples of such operator-valued frames are the rank-one Hermitian operators associated with vectors from maximal sets of equiangular lines or maximal sets of mutually unbiased bases. A more general type of examples is obtained by a tensor product construction. We also study erasures and show that in addition to loss of phase, a maximal set of mutually unbiased bases can correct for erased frame coefficients as long as no more than one erasure occurs among the coefficients belonging to each basis, and at least one basis remains without erasures.
In this work, the concept of mutually unbiased frames is introduced as the most general notion of unbiasedness for sets composed by linearly independent and normalized vectors. It encompasses the already existing notions of unbiasedness for orthonormal bases, regular simplices, equiangular tight frames, positive operator valued measure, and also includes symmetric informationally complete quantum measurements. After introducing the tool, its power is shown by finding the following results about the last mentioned class of constellations: (i) real fiducial states do not exist in any even dimension, and (ii) unknown d -dimensional fiducial states are parameterized, a priori, with roughly 3d/2 real variables only, without loss of generality. Furthermore, multi-parametric families of pure quantum states having minimum uncertainty with regard to several choices of d+1 orthonormal bases are shown, in every dimension d . These last families contain all existing fiducial states in every finite dimension, and the bases include maximal sets of d+1 mutually unbiased bases, when d is a prime number.
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.
Finite frames have many applications in quantum informatics, communications, coding theory, and other fields. In this article we investigate their use in quantum tomography. The family of vectors comprising a frame can be used to define a protocol of quantum tomography. To do this one estimates the probabilities of projective measurements on quantum states corresponding to the frame vectors. The state under investigation can then be reconstructed by one of several quantum tomography techniques. Since quantum tomography performance depends greatly on the choice of protocol, the problem of finding an optimal protocol is of considerable interest. WeexamineafamilyofunitnormframesconsistingofnumericalsolutionstoFeketepackingproblemincomplex space. Notable members of this family are Mutually Unbiased Bases (MUBs), which were computed in all investigated prime power dimensions, as well as Symmetrical Informationally Complete Positive Operator Valued Measures(SIC-POVM) in all dimensions. This family contains other uniform tight frames, in which the number of vectors lies between that of full sets of Mutually Unbiased Bases and Symmetrical Informationally Complete POVMs, and which all share the same internal structure. We detail some properties of these frames and their performance as quantum tomography protocols.
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.
Two orthonormal bases in the d-dimensional Hilbert space are said to be unbiased if the square modulus of the inner product of any vector of one basis with any vector of the other equals 1 d. The presence of a modulus in the problem of finding a set of mutually unbiased bases constitutes a source of complications from the numerical point of view. Therefore, we may ask the question: Is it possible to get rid of the modulus? After a short review of various constructions of mutually unbiased bases in Cd, we show how to transform the problem of finding d + 1 mutually unbiased bases in the d-dimensional space Cd (with a modulus for the inner product) into the one of finding d(d+1) vectors in the d2-dimensional space Cd2 (without a modulus for the inner product). The transformation from Cd to Cd2 corresponds to the passage from equiangular lines to equiangular vectors. The transformation formulas are discussed in the case where d is a prime number.
Mutually unbiased bases (MUBs), which are such that the inner product between two vectors in different orthogonal bases is a constant equal to 1/sqrt{d), with d the dimension of the finite Hilbert space, are becoming more and more studied for applications such as quantum tomography and cryptography, and in relation to entangled states and to the Heisenberg-Weil group of quantum optics. Complete sets of MUBs of cardinality d+1 have been derived for prime power dimensions d=p^m using the tools of abstract algebra. Presumably, for non prime dimensions the cardinality is much less. Here we reinterpret MUBs as quantum phase states, i.e. as eigenvectors of Hermitean phase operators generalizing those introduced by Pegg & Barnett in 1989. We relate MUB states to additive characters of Galois fields (in odd characteristic p) and to Galois rings (in characteristic 2). Quantum Fourier transforms of the components in vectors of the bases define a more general class of MUBs with multiplicative characters and additive ones altogether. We investigate the complementary properties of the above phase operator with respect to the number operator. We also study the phase probability distribution and variance for general pure quantum electromagnetic states and find them to be related to the Gauss sums, which are sums over all elements of the field (or of the ring) of the product of multiplicative and additive characters. Finally, we relate the concepts of mutual unbiasedness and maximal entanglement. This allows to use well studied algebraic concepts as efficient tools in the study of entanglement and its information aspects