Minimal Informationally Complete quantum measurements, or MICs, illuminate the structure of quantum theory and how it departs from the classical. Central to this capacity is their role as tomographically complete measurements with the fewest possible number of outcomes for a given finite dimension. Despite their advantages, little is known about them. We establish general properties of MICs, explore constructions of several classes of them, and make some developments to the theory of MIC Gram matrices. These Gram matrices turn out to be a rich subject of inquiry, relating linear algebra, number theory and probability. Among our results are some equivalent conditions for unbiased MICs, a characterization of rank-1 MICs through the Hadamard product, several ways in which immediate properties of MICs capture the abandonment of classical phase space intuitions, and a numerical study of MIC Gram matrix spectra. We also present, to our knowledge, the first example of an unbiased rank-1 MIC which is not group covariant. This work provides further context to the discovery that the symmetric informationally complete quantum measurements (SICs) are in many ways optimal among MICs. In a deep sense, the ideal measurements of quantum physics are not orthogonal bases.
Minimal Informationally Complete quantum measurements, or MICs, illuminate the structure of quantum theory and how it departs from the classical. Central to this capacity is their role as tomographically complete measurements with the fewest possible number of outcomes for a given finite dimension. Despite their advantages, little is known about them. We establish general properties of MICs, explore constructions of several classes of them, and make some developments to the theory of MIC Gram matrices. These Gram matrices turn out to be a rich subject of inquiry, relating linear algebra, number theory and probability. Among our results are some equivalent conditions for unbiased MICs, a characterization of rank-1 MICs through the Hadamard product, several ways in which immediate properties of MICs capture the abandonment of classical phase space intuitions, and a numerical study of MIC Gram matrix spectra. We also present, to our knowledge, the first example of an unbiased rank-1 MIC which is not group covariant. This work provides further context to the discovery that the symmetric informationally complete quantum measurements (SICs) are in many ways optimal among MICs. In a deep sense, the ideal measurements of quantum physics are not orthogonal bases.
Generalized symmetric informationally complete (SIC) measurements are SIC measurements that are not necessarily rank 1. They are interesting originally because of their connection with rank-1 SICs. Here we reveal several merits of generalized SICs in connection with quantum state tomography and Lie algebra that are interesting in their own right. These properties uniquely characterize generalized SICs among minimal informationally complete (IC) measurements although, on the face of it, they bear little resemblance to the original definition. In particular, we show that in quantum state tomography generalized SICs are optimal among minimal IC measurements with given average purity of measurement outcomes. Besides its significance to the current study, this result may help us to understand tomographic efficiencies of minimal IC measurements under the influence of noise. When minimal IC measurements are taken as bases for the Lie algebra of the unitary group, generalized SICs are uniquely characterized by the antisymmetry of the associated structure constants.
Wigner functions provide a way to do quantum physics using quasiprobabilities, that is, ``probability'' distributions that can go negative. Informationally complete positive operator-valued measures, a much younger subject than phase-space formulations of quantum mechanics, are less familiar but provide wholly probabilistic representations of quantum theory. In this paper, we show that the Born rule links these two classes of structure and discuss the art of interconverting between them. In particular, we demonstrate that the operator bases corresponding to minimal discrete Wigner functions (Wigner bases) are orthogonalizations of minimal informationally complete measurements (MICs). By not imposing a particular discrete phase-space structure at the outset, we push Wigner functions to their limits in a suitably quantified sense, revealing an extremality condition satisfied by the symmetric informationally complete measurements (SICs). This contributes another member to the set of optimality criteria that SICs are known to satisfy. Finally, we speculate that astute choices of MICs from the orthogonalization preimages of Wigner bases may in general give quantum measurements that are conceptually illuminating for the quasiprobability representations defined by those Wigner bases.
The reconstruction of unknown quantum states via minimal informationally complete measurements (MICs) is a cornerstone of quantum tomography. Although the statistical properties of these measurements are well-understood, their geometric structure has remained elusive. In this work, we establish a correspondence between the class of minimal s-tight informationally complete measurements, encompassing, among others, tight IC and morphophoric measurements, and the classical geometry of orthocentric simplices. In particular, we prove a three-way equivalence: a MIC is s-tight if and only if its measurement vectors, upon suitable rescaling, form the vertices of an acute orthocentric simplex with the orthocentre at the origin, and such simplices are precisely the homothetically self-dual ones. This geometric manifestation of operational ''tightness'' provides a bridge between the physical world and Euclidean geometry. Furthermore, the s-tight class is fully characterised by its measurement directions: the angles between them must be obtuse and satisfy a cross-ratio condition. We determine the space of admissible direction configurations: modulo rotations, every such configuration is encoded by a single probability vector, the ''skeleton'' of the measurement, together with an orientation class, so that the moduli space of s-tight MIC directions is Δd+1∘×{±1}. Conversely, every acute orthocentric simplex with the orthocentre at the origin can be anchored in the state space, generating a class of minimal s-tight IC measurements that contains exactly one tight IC measurement up to overall rescaling.
Abstract We discuss symmetric quantum measurements and the associated covariant observables modelled, respectively, as instruments and positive-operator-valued measures. The emphasis of this work are the optimality properties of the measurements, namely, extremality, informational completeness, and the rank-1 property which contrast the complementary class of (rank-1) projection-valued measures. The first half of this work concentrates solely on finite-outcome measurements symmetric w.r.t. finite groups where we derive exhaustive characterizations for the pointwise Kraus-operators of covariant instruments and necessary and sufficient extremality conditions using these Kraus-operators. We motivate the use of covariance methods by showing that observables covariant with respect to symmetric groups contain a family of representatives from both of the complementary optimality classes of observables and show that even a slight deviation from a rank-1 projection-valued measure can yield an extreme informationally complete rank-1 observable. The latter half of this work derives similar results for continuous measurements in (possibly) infinite dimensions. As an example we study covariant phase space instruments, their structure, and extremality properties.
This thesis will focus on deriving informationally-optimal quantum state tomographic measurements on single-qubit systems, with the definition of informationally optimal to be defined as those measurements which maximize the average information gain. The informationally-optimal measurements that will be covered include projective measurements (formally building off of the work of [1] and putting it on firm, information-theoretic foundations), and more general types of quantum measurements called directional (also known as rank-one) positive operator-valued measure (POVM) measurements, and, finally, adaptive directional POVM measurements. For projective measurements, we build on the work of Wootters and Fields ([1]) and show, via analytical methods and maximizing the average information gained from the measurements, that orthogonal projective measurements are optimal. We then go on to define what we call directional POVMs, which are POVMs whose elements are given by weighted projection operators. We consider these measurements in the context of static quantum state tomography (QST), wherein we use the same choice of measurement throughout our analysis/experiment. By examining these directional POVMs acting on single qubit systems, we show that the most informationally optimal of these types of measurements are informationally complete directional POVMs that are both minimal, informationally complete (MIC) and symmetric, informationally complete (SIC) POVMs, simultaneously. Finally, we deal with the case of adaptive directional POVM measurements.
The unpredictability of quantum physics gives rise to intrinsic randomness. In an adversarial scenario, any additional degrees of freedom must be attributed to an eavesdropper with correlations to the measurement set-up. The true randomness is then quantified by the probability that she correctly guesses the measurement outcomes, optimised over all possible strategies. Extremal measurements are appealing here, since they do not allow information to leak to such an eavesdropper. Beyond projective measurements, however, a simple question remains open: how much intrinsic randomness can be generated by a given extremal measurement? In a step towards solving it, we characterise the randomness generated by any unbiased extremal rank-one measurement acting on any state, solving the problem explicitly in dimension two. Four-outcome qubit measurements of this type are tomographic, so these results hold for fully source-device-dependent randomness too. The tetrahedral symmetric informationally complete (SIC) measurement, we find, has the least intrinsic randomness within this class. We also present the skewed SIC family of measurements, and use them to partially solve an open problem: we prove that 2logd bits of randomness, the maximal amount, can be generated device-dependently (or source-device-independently) in any dimension in which there exists a SIC measurement.