Sporadic SICs and the Normed Division Algebras

Stacey, Blake C.

doi:10.1007/s10701-017-0087-2

Cited by 26 publications

(35 citation statements)

References 69 publications

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“…where P (D|H) is a matrix of conditional probabilities. A SIC [16][17][18][19][20][21][22][23][24][25] is a MIC for which all the H i are rank-1 and SICs have yet to be proven to exist in all finite dimensions d, but they are widely believed to [24] and have even been experimentally demonstrated in some low dimensions [26][27][28][29]. The SIC projectors associated with a SIC are the pure states ρ i = dH i .…”

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confidence: 99%

Symmetric informationally complete measurements identify the irreducible difference between classical and quantum systems

DeBrota

Fuchs

Stacey

2020

Phys. Rev. Research

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We describe a general procedure for associating a minimal informationally complete quantum measurement (or MIC) with a purely probabilistic representation of the Born Rule. Such representations provide a way to understand the Born Rule as a consistency condition between probabilities assigned to the outcomes of one experiment in terms of the probabilities assigned to the outcomes of other experiments. In this setting, the difference between quantum and classical physics is the way their physical assumptions augment bare probability theory: Classical physics corresponds to a trivial augmentation-one just applies the Law of Total Probability (LTP) between the scenarioswhile quantum theory makes use of the Born Rule expressed in one or another of the forms of our general procedure. To mark the irreducible difference between quantum and classical, one should seek the representations that minimize the disparity between the expressions. We prove that the representation of the Born Rule obtained from a symmetric informationally complete measurement (or SIC) minimizes this distinction in at least two senses-the first to do with unitarily invariant distance measures between the rules, and the second to do with available volume in a reference probability simplex (roughly speaking a new kind of uncertainty principle). Both of these arise from a useful result in majorization theory. This work complements recent studies in quantum computation where the deviation of the Born Rule from the LTP is measured in terms of negativity of Wigner functions. arXiv:1805.08721v3 [quant-ph] 30 Apr 2020

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confidence: 99%

Symmetric informationally complete measurements identify the irreducible difference between classical and quantum systems

DeBrota

Fuchs

Stacey

2020

Phys. Rev. Research

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“…We have chosen the Hesse SIC (which is derived from the Hesse configuration in design theory) due to its special symmetry properties, which are unique among the infinite family of qutrit SIC-POVMs that exist [44]. However, other SIC-POVMs such as the Norrell states would also be interesting to analyse [45,46].…”

Section: B21 Upper Bounds U M Dmentioning

confidence: 99%

Linking entanglement detection and state tomography via quantum 2-designs

2019

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We present an experimentally feasible and efficient method for detecting entangled states with measurements that extend naturally to a tomographically complete set. Our detection criterion for bipartite systems with equal dimensions is based on measurements from subsets of a quantum 2-design, e.g. mutually unbiased bases or symmetric informationally complete states, and has several advantages over standard entanglement witnesses. First, as more detectors in the measurement are applied, there is a higher chance of witnessing a larger set of entangled states, in such a way that the measurement setting converges to a complete setup for quantum state tomography. Secondly, our method is twice as effective as standard witnesses in the sense that both upper and lower bounds can be derived. Thirdly, the scheme can be readily applied to measurement-device-independent scenarios.For quantum information applications it is often more interesting to learn if multipartite quantum states are entangled than to identify quantum states themselves [1,2]. This is in fact what direct detection of entanglement executes, which utilizes an entanglement witness that works with individual measurements followed be postprocessing of the outcomes [3], to provide an experimentally feasible approach for this task [4]. Entanglement detection under less assumptions, for instance, when detectors are not trusted [5][6][7] or dimensions are unknown [8], is of practical significance for cryptographic applications.For the practical usefulness of entanglement detection, it is worth exploring the experimental resources. If a priori information about a quantum state is given, a set of entanglement witnesses may be constructed accordingly and exploited for entanglement detection. With no a priori information multiple entanglement witnesses may be required. One possible method is quantum state tomography which determines a ddimensional quantum state with O(d 2 ) measurements. Then, theoretical tools such as positive maps [9], e.g. partial transpose, or numerical tests involving semidefinite programming [10] can be applied. For entanglement witnesses, however, little is known about the minimal measurements for their realization. In fact, it may happen that repeating experiments for multiple witnesses may be less cost effective than state tomography [11], and quite possible that no useful information is obtained, neither for entanglement detection nor for quantum state identification. This raises questions on the usefulness of entanglement witnesses, in particular when a priori information about a particular state is not available.A useful experimental setup for entanglement detection may distinguish the largest collection of entangled states with as few measurements as possible. It is noteworthy that a tomographically complete measurement can ultimately identify a quantum state so that theoretical tools may completely determine whether it is entangled or separable. From a practical point of view, it would be therefore highly desirable that measurements for ...

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“…(The exception is generated, when d = 8, by another group. Like so many exceptional structures it is related to octonions [15], and here we will assume that it can be left aside as a curiousity. )…”

Section: Introducing the Weyl-heisenberg Groupmentioning

confidence: 99%