Symmetric Informationally-Complete Quantum States as Analogues to Orthonormal Bases and Minimum-Uncertainty States

Appleby, D. M.; Dang, Hoan Bui; Fuchs, Christopher A.

doi:10.48550/arxiv.0707.2071

Cited by 29 publications

(45 citation statements)

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“…the set {N j } is called a symmetric informationally complete POVM [26]. Weyl-Heisenberg (WH) covariant SIC-sets of states in prime dimensions are examined in reference [41]. WH SIC-sets, whenever they exist, consist solely of minimum uncertainty states with respect to Rényi's 2-entropy for a complete set of MUBs [41].…”

Section: A Mutually Unbiased Bases and Symmetric Informationally Comp...mentioning

confidence: 99%

“…Weyl-Heisenberg (WH) covariant SIC-sets of states in prime dimensions are examined in reference [41]. WH SIC-sets, whenever they exist, consist solely of minimum uncertainty states with respect to Rényi's 2-entropy for a complete set of MUBs [41]. Klappenecker et al [42] discussed approximate versions of SIC-POVMs, where a small deviation from uniformity of the inner products is allowed.…”

Section: A Mutually Unbiased Bases and Symmetric Informationally Comp...mentioning

confidence: 99%

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Uncertainty relations for MUBs and SIC-POVMs in terms of generalized entropies

2013

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We formulate uncertainty relations for mutually unbiased bases and symmetric informationally complete measurements in terms of the Rényi and Tsallis entropies. For arbitrary number of mutually unbiased bases in a finite-dimensional Hilbert space, we give a family of Tsallis α-entropic bounds for α ∈ (0; 2]. Relations in a model of detection inefficiences are obtained. In terms of Rényi's entropies, lower bounds are given for α ∈ [2; ∞). State-dependent and state-independent forms of such bounds are both given. Uncertainty relations in terms of the min-entropy are separately considered. We also obtain lower bounds in term of the so-called symmetrized entropies. The presented results for mutually unbiased bases are extensions of some bounds previously derived in the literature. We further formulate new properties of symmetric informationally complete measurements in a finite-dimensional Hilbert space. For a given state and any SIC-POVM, the index of coincidence of generated probability distribution is exactly calculated. Short notes are made on potential use of this result in entanglement detection. Further, we obtain state-dependent entropic uncertainty relations for a single SIC-POVM. Entropic bounds are derived in terms of the Rényi α-entropies for α ∈ [2; ∞) and the Tsallis α-entropies for α ∈ (0; 2]. In the Tsallis formulation, a case of detection inefficiences is briefly mentioned. For a pair of symmetric informationally complete measurements, we also obtain an entropic bound of Maassen-Uffink type.

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Section: A Mutually Unbiased Bases and Symmetric Informationally Comp...mentioning

confidence: 99%

Section: A Mutually Unbiased Bases and Symmetric Informationally Comp...mentioning

confidence: 99%

Uncertainty relations for MUBs and SIC-POVMs in terms of generalized entropies

2013

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“…Interestingly, unlike the case in dimension two, it is not always possible to map three vectors to another three vectors in the same SIC POVM with unitary or antiunitary operations. In addition, in quantum state tomography with a SIC POVM in dimension three, the set of occurring probabilities is not permutation invariant, according to [16].…”

Section: Sic Povms In Three-dimensional Hilbert Spacementioning

confidence: 99%

“…The trace of the triple product tr(ρ 1 ρ 2 ρ 3 ) is invariant under unitary transformation. This invariant has been applied by Appleby et al [16] to studying the set of occurring probabilities of measurement outcomes of SIC POVMs. According to (1), |tr(ρ 1 ρ 2 ρ 3 )| = 1 8 for d = 3, so the relevant invariant is the phase of the trace, φ ′ = arg[tr(ρ 1 ρ 2 ρ 3 )], with −π ≤ φ ′ < π.…”

Section: Infinitely Many Inequivalent Sic Povmsmentioning

confidence: 99%

SIC POVMs and Clifford groups in prime dimensions

Zhu¹

2010

J. Phys. A: Math. Theor.

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Abstract. We show that in prime dimensions not equal to three, each group covariant symmetric informationally complete positive operator valued measure (SIC POVM) is covariant with respect to a unique Heisenberg-Weyl (HW) group. Moreover, the symmetry group of the SIC POVM is a subgroup of the Clifford group. Hence, two SIC POVMs covariant with respect to the HW group are unitarily or antiunitarily equivalent if and only if they are on the same orbit of the extended Clifford group. In dimension three, each group covariant SIC POVM may be covariant with respect to three or nine HW groups, and the symmetry group of the SIC POVM is a subgroup of at least one of the Clifford groups of these HW groups respectively. There may exist two or three orbits of equivalent SIC POVMs for each group covariant SIC POVM, depending on the order of its symmetry group. We then establish a complete equivalence relation among group covariant SIC POVMs in dimension three, and classify inequivalent ones according to the geometric phases associated with fiducial vectors. Finally, we uncover additional SIC POVMs by regrouping of the fiducial vectors from different SIC POVMs which may or may not be on the same orbit of the extended Clifford group.

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“…Concerning measurement schemes, mutually unbiased bases (MUBs) [41][42][43][44][45] and symmetric informationally complete (SIC) POVMs [46][47][48][49][50][51][52][53], which stand for very efficient von Neumann measurements and POVMs [54], respectively, are two focuses in the current research community. Great efforts have been directed to solving their existence problem and understanding their structure [41-44, 46-48, 50-52]; since they are closely related to the physics in finite-dimensional Hilbert spaces [45,49,53].…”

Section: Introductionmentioning

confidence: 99%

Connections of geometric measure of entanglement of pure symmetric states to quantum state estimation

2011

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We study the geometric measure of entanglement (GM) of pure symmetric states related to rank 1 positiveoperator-valued measures (POVMs) and establish a general connection with quantum state estimation theory, especially the maximum likelihood principle. Based on this connection, we provide a method for computing the GM of these states and demonstrate its additivity property under certain conditions. In particular, we prove the additivity of the GM of pure symmetric multiqubit states whose Majorana points under Majorana representation are distributed within a half sphere, including all pure symmetric three-qubit states. We then introduce a family of symmetric states that are generated from mutually unbiased bases and derive an analytical formula for their GM. These states include Dicke states as special cases, which have already been realized in experiments. We also derive the GM of symmetric states generated from symmetric informationally complete POVMs (SIC POVMs) and use it to characterize all inequivalent SIC POVMs in three-dimensional Hilbert space that are covariant with respect to the Heisenberg-Weyl group. Finally, we describe an experimental scheme for creating the symmetric multiqubit states studied in this article and a possible scheme for measuring the permanence of the related Gram matrix.

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Symmetric Informationally-Complete Quantum States as Analogues to Orthonormal Bases and Minimum-Uncertainty States

Cited by 29 publications

References 1 publication

Uncertainty relations for MUBs and SIC-POVMs in terms of generalized entropies

Uncertainty relations for MUBs and SIC-POVMs in terms of generalized entropies

SIC POVMs and Clifford groups in prime dimensions

Connections of geometric measure of entanglement of pure symmetric states to quantum state estimation

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