From SICs and MUBs to Eddington

Bengtsson, Ingemar

doi:10.1088/1742-6596/254/1/012007

Cited by 24 publications

(21 citation statements)

References 29 publications

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“…Theorems 1, 3, and 4 also establish a one-toone correspondence between SICs and NQPRs of quantum mechanics with minimal negativity, namely, perfect QPRs. Based on the study of SICs [19][20][21][22][23], we can construct perfect QPRs at least in dimensions 2 to 16,19,24,28,31,35,37,43,48, and with extremely high precision up to dimension 67 (the number of known solutions is increasing continually). There is every reason to believe that such representations exist for any Hilbert space of finite dimension.…”

Section: Theorem 4 the Channel Negativity Of Any Nqprmentioning

confidence: 99%

Quasiprobability Representations of Quantum Mechanics with Minimal Negativity

Zhu

2016

Phys. Rev. Lett.

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Quasiprobability representations, such as the Wigner function, play an important role in various research areas. The inevitable appearance of negativity in such representations is often regarded as a signature of nonclassicality, which has profound implications for quantum computation. However, little is known about the minimal negativity that is necessary in general quasiprobability representations. Here we focus on a natural class of quasiprobability representations that is distinguished by simplicity and economy. We introduce three measures of negativity concerning the representations of quantum states, unitary transformations, and quantum channels, respectively. Quite surprisingly, all three measures lead to the same representations with minimal negativity, which are in one-to-one correspondence with the elusive symmetric informationally complete measurements. In addition, most representations with minimal negativity are automatically covariant with respect to the Heisenberg-Weyl groups. Furthermore, our study reveals an interesting tradeoff between negativity and symmetry in quasiprobability representations. [2][3][4][5][6][7], which are very useful in many research areas, such as quantum optics, quantum tomography, and quantum computing. The inevitable appearance of negativity in such representations is often regarded as a signature of nonclassicality, which is closely related to other nonclassical features [5,6,[8][9][10]. In particular, it was shown by Spekkens that negativity and contextuality are equivalent notions of nonclassicality [9]. Moreover, negativity has been recognized as a resource in quantum computation and has attracted increasing attention in the quantum information community [11][12][13][14][15][16][17]. To be specific, in the paradigm of magic state quantum computation [18], negativity in the Wigner function is necessary for computational speedup and is directly related to the efficiency of classical simulation [13,14,17].To fully understand the distinction between quantum theory and classical theory in terms of negativity, one should not only consider the representation of quantum states, but also representations of measurements and transformations. In addition, it is crucial to consider the whole class of QPRs instead of a particular choice. This advice is also helpful for understanding the power of quantum information processing as well as the connection between negativity and contextuality [5, 9, 10]. However, little is known about the degree of negativity in any QPR other than the Wigner function and its discrete analogs.In this paper we initiate a systematic study of the degree of negativity in general QPRs. Our focus is an important family of QPRs in finite dimensions that is distinguished by simplicity and economy. It includes most examples proposed in the literature, such as many discrete Wigner functions [2,3,5,6]. In these representations, information is encoded without redundancy; the analogy to classical probability theory is preserved as far as possible except for the app...

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Section: Theorem 4 the Channel Negativity Of Any Nqprmentioning

confidence: 99%

Quasiprobability Representations of Quantum Mechanics with Minimal Negativity

Zhu

2016

Phys. Rev. Lett.

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“…The symmetric group S 3 contains the permutation matrices I, X and X 2 of the Pauli group, where X = Taking the eigensystem of the latter matrices, it is not difficult check that there exists two types of qutrit magic states of the form (0, 1, ±1) ≡ 1 √ 2 (|1 ± |2 ). Then, taking the action of the nine qutrit Pauli matrices, one arrives at the well known Hesse SIC [10,11,12].…”

Section: Permutation Gates Magic States and Informationally Completementioning

confidence: 99%

Magic informationally complete POVMs with permutations

2017

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Eigenstates of permutation gates are either stabilizer states (for gates in the Pauli group) or magic states, thus allowing universal quantum computation (Planat, Rukhsan-Ul-Haq 2017 Adv. Math. Phys. 2017, 5287862 (doi:10.1155/2017/528786210.1155/2017/5287862)). We show in this paper that a subset of such magic states, when acting on the generalized Pauli group, define (asymmetric) informationally complete POVMs. Such informationally complete POVMs, investigated in dimensions 2–12, exhibit simple finite geometries in their projector products and, for dimensions 4 and 8 and 9, relate to two-qubit, three-qubit and two-qutrit contextuality.

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“…Hughston [20] has shown that the SIC vectors of the single SIC for t = 0 can be obtained from the inflection points of a family of cubic elliptic curves on the complex projective plane known as the Hesse pencil (see also Bengtsson [21].) There are 8 SICs with parameter value t = π 9 that are unitarily equivalent to the single SIC for t = 0, and we call these 9 SICs the Hesse SICs.…”

Section: Properties Of Weyl-heisenberg Qutrit Sicsmentioning

confidence: 99%

Exploring the geometry of qutrit state space using symmetric informationally complete probabilities

Tabia

Appleby

2013

Phys. Rev. A

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We examine the geometric structure of qutrit state space by identifying the outcome probabilities of symmetric informationally complete (SIC) measurements with quantum states. We categorize the infinitely many qutrit SICs into 8 SIC-families corresponding to independent orbits of the extended Clifford group. Every SIC can be uniquely identified from a set of geometric invariants that we use to establish several properties of the convex body of qutrits, which include a simple formula describing its extreme points, an expression for the rotation between the probability vectors for distinct qutrit SICs, and a polar equation for its boundary states.

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From SICs and MUBs to Eddington

Cited by 24 publications

References 29 publications

Quasiprobability Representations of Quantum Mechanics with Minimal Negativity

Quasiprobability Representations of Quantum Mechanics with Minimal Negativity

Magic informationally complete POVMs with permutations

Exploring the geometry of qutrit state space using symmetric informationally complete probabilities

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