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SIAM Journal on Discrete Mathematics

A Characterization of Projective Unitary Equivalence of Finite Frames and Applications

Tuan-Yow Chien, Shayne F. D. Waldron

Published Jan 1, 2016• 29 citations
Publisher

Abstract

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.

DOI
10.1137/15m1042140

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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.
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 (Gram 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 is based on an algorithm to recover the sequence of vectors (up to projective unitary equivalence) from a small subset of these projective invariants. We consider the implications for the characterisation of SICs, MUBs and harmonic frames up to projective unitary equivalence. We also extend our results to projective similarity of vectors.
The principle of complementarity lies at the heart of quantum mechanics. In finite dimensional quantum systems this principle is captured by pairs of observables which are given by mutually unbiased bases (MUBs). Two orthonormal bases B and C of Cd are mutually unbiased if |<b|c>|2 = 1/d holds for all vectors b ∈ B and c ∈ C. This implies that whenever we are given a vector from one of these bases and perform a measurement with respect to any other of the bases, then there is no information gained from this measurement. A basic question about MUBs is how many of them can be found in a given dimension d. While constructions of maximal sets of d+1 such bases are known for system of prime power dimension d, it is unknown whether this bound can be achieved for any non-prime power dimension. We review the known constructions of MUBs and demonstrate that maximal sets of MUBs come with a rich combinatorial structure by showing that they actually are the same objects as the complex projective 2-designs with angle set {0,1/d}. Furthermore, we address the problem of constructing positive operator-valued measures (POVMs) in finite dimension d consisting of d2 operators of rank one which have an inner product equal to uniform or very 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. We also give a simple proof of the fact that symmetric informationally complete POVMs are complex projective 2-designs with angle set {1/(d+1)}. Moreover, we show that MUBs and SIC-POVMs form uniform tight frames in Cd.
Abstract In quantum mechanics, mutually unbiased bases (MUBs) represent orthonormal bases (ONBs) that are as ‘far apart’ as possible, and their classification reveals rich underlying geometric structure. Given a complex inner product space, we construct the space of its ONBs as a discrete quotient of the complete flag manifold. We introduce a metric on this space, which corresponds to the ‘MUBness’ distance. This allows us to describe equivalence between sets of MUBs in terms of the geometry of this space. The subspace of bases that are unbiased with respect to the standard basis decomposes into orbits under a certain group action, and this decomposition corresponds to the classification of complex Hadamard matrices. More generally, we consider a list of k MUBs, that one wishes to extend. The candidates are points in the subspace comprising all bases which are unbiased with respect to the entire list. This space also decomposes into orbits under a group action, and we prove that points in distinct orbits yield inequivalent MUB lists. Thus, we generalize the relation between complex Hadamard matrices and MUBs. As an application, we identify new symmetries that reduce the parameter space of MUB triples in dimension 4 by a factor of 4.
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.
Bargmann invariants, a class of unitary-invariant quantities arising from the overlaps of quantum state vectors, provide a profound and unifying framework for understanding the relative geometry of the projective Hilbert space. This survey offers a comprehensive overview of their theoretical characterization and practical applications, with particular emphasis on recent progress in determining the full structure of their admissible set. The core of this review demonstrates how these invariants serve as a powerful tool for characterizing the intrinsic geometry of the space of quantum states, leading to applications in determining local unitary equivalence and constructing a complete set of polynomial invariants for mixed states. On the operational side, we review the cycle-test quantum circuit for the direct estimation of Bargmann invariants without full state tomography, and demonstrate their utility in witnessing quantum imaginary, discriminating local unitary equivalence, and detecting entanglement via partial-transpose moments---with explicit complete invariant sets provided for two-qubit systems. By connecting fundamental geometric classification with experimentally feasible estimation protocols,this survey establishes Bargmann invariants as indispensable probes of the relational, noncommutative, and geometric structure of quantum states, and identifies key open problems for multipartite high-dimensional systems and for quantum resource theories.
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.