Orbits of mutually unbiased bases

Blanchfield, Kate

doi:10.1088/1751-8113/47/13/135303

Cited by 13 publications

(26 citation statements)

References 41 publications

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“…SIC vectors do saturate the inequality for all prime dimensions, and being MUS they have the additional property that all the simplices we see in the d + 1 projections are of the same size [18,24]. Alltop vectors also saturate the inequality as one can see from a formula given by Khaterinejad [31], and they have the additional property that d out of d + 1 projected simplices share the same size and the same orientation, as follows from the fact that every Alltop vector is left invariant by a unitary operator that cycles through d of the bases in the stabilizer MUB [9]. A glance at Fig.…”

Section: Quantitative Measures and An Inequalitymentioning

confidence: 84%

States that are far from being stabilizer states

Andersson¹,

Bengtsson²,

Blanchfield³

et al. 2015

J. Phys. A: Math. Theor.

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Abstract:Stabilizer states are eigenvectors of maximal commuting sets of operators in a finite Heisenberg group. States that are far from being stabilizer states include magic states in quantum computation, MUB-balanced states, and SIC vectors. In prime dimensions the latter two fall under the umbrella of Minimum Uncertainty States (MUS) in the sense of Wootters and Sussman. We study the correlation between two ways in which the notion of "far from being a stabilizer state" can be quantified, and give detailed results for low dimensions. In dimension 7 we identify the MUB-balanced states as being antipodal to the SIC vectors within the set of MUS, in a sense that we make definite. In dimension 4 we show that the states that come closest to being MUS with respect to all the six stabilizer MUBs are the fiducial vectors for Alltop MUBs.

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Section: Quantitative Measures and An Inequalitymentioning

confidence: 84%

States that are far from being stabilizer states

Andersson¹,

Bengtsson²,

Blanchfield³

et al. 2015

J. Phys. A: Math. Theor.

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“…(14) reveals that the unitary equivalence between Ivanović and Alltop MUB constructions [27] is actually due to a unitary from the third level of the Clifford hierarchy. In addition, Eq.…”

Section: Mutually Unbiased Basesmentioning

confidence: 96%

“…We are interested in those with non-degenerate spectra. The Clifford group contains exactly p(p + 1)(p − 1) cyclic subgroups of such elements [14]. They relate to the Alltop MUBs, introduced in the next section.…”

Section: The Clifford Hierarchymentioning

confidence: 99%

“…Such states can be obtained by acting on the stabilizer states with elements of the third level of the Clifford hierarchy, which consists of those unitaries that map the Weyl-Heisenberg group into the Clifford group [12]. When acting on the stabilizer states, a unitary at this third level turns the usual MUB into another MUB, which is itself an orbit of the Weyl-Heisenberg group with one extra basis appended [13,14]. These are the magic states that we will discuss, under the name of Alltop vectors.…”

Section: Introductionmentioning

confidence: 99%

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Order 3 symmetry in the Clifford hierarchy

Bengtsson¹,

Blanchfield²,

Campbell³

et al. 2014

J. Phys. A: Math. Theor.

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We investigate the action of the first three levels of the Clifford hierarchy on sets of mutually unbiased bases comprising the Ivanovic MUB and the Alltop MUBs. Vectors in the Alltop MUBs exhibit additional symmetries when the dimension is a prime number equal to 1 modulo 3 and thus the set of all Alltop vectors splits into three Clifford orbits. These vectors form configurations with so-called Zauner subspaces, eigenspaces of order 3 elements of the Clifford group highly relevant to the SIC problem. We identify Alltop vectors as the magic states that appear in the context of fault-tolerant universal quantum computing, wherein the appearance of distinct Clifford orbits implies a surprising inequivalence between some magic states.

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“…Clearly, the min a,b |V c c[a, b]| is going to be the largest if V c c is constant with the exception of a point (0, 0). This holds if {π(a, b)c} is an equiangular frame, namely, if c is a scalar multiple of a fiducial vector [19], [20], [21], [22], [23]. Assuming without loss of generality that…”

Section: B Finite-dimensional Gabor Framesmentioning

confidence: 99%

Estimation of Overspread Scattering Functions

Pfander

Zheltov²

2015

IEEE Trans. Signal Process.

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In many radar scenarios, the radar target or the medium is assumed to possess randomly varying parts. The properties of a target are described by a random process known as the spreading function. Its second order statistics under the WSSUS assumption are given by the scattering function. Recent developments in operator sampling theory suggest novel channel sounding procedures that allow for the determination of the spreading function given complete statistical knowledge of the operator echo from a single sounding by a weighted pulse train. We construct and analyze a novel estimator for the scattering function based on these findings. Our results apply whenever the scattering function is supported on a compact subset of the time-frequency plane. We do not make any restrictions either on the geometry of this support set, or on its area. Our estimator can be seen as a generalization of an averaged periodogram estimator for the case of a non-rectangular geometry of the support set of the scattering function

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Orbits of mutually unbiased bases

Cited by 13 publications

References 41 publications

States that are far from being stabilizer states

States that are far from being stabilizer states

Order 3 symmetry in the Clifford hierarchy

Estimation of Overspread Scattering Functions

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