SIC-POVMS and MUBS: Geometrical Relationships in Prime Dimension

Appleby, D. M.

doi:10.1063/1.3109944

Cited by 44 publications

(45 citation statements)

References 43 publications

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“…We now have a different line of argument singling out these subspaces for attention, and we suggest that this hints at a deeper connection between MUBs and SICs. Indeed, one weak link is already known [22,23]. In dimension p = 3 there is a very direct link, effectively discovered by Hesse in a different language [24], and elaborated on since [25].…”

Section: Introductionmentioning

confidence: 99%

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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Section: Introductionmentioning

confidence: 99%

Order 3 symmetry in the Clifford hierarchy

Bengtsson¹,

Blanchfield²,

Campbell³

et al. 2014

J. Phys. A: Math. Theor.

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“…Moreover, the vectors being sought are then always fully determined by a complete set of mutually unbiased bases, in a geometrically natural way. Although it is considerably harder to see (one will have to read several papers in order to patch a proof together [6][7][8][9]), this statement holds also when d = 3. Numerically, SICs have been found in every dimension where they have been looked for (this includes all dimensions d ≤ 121, and a few more [10,11]), but no existence proof has been found, and beyond three dimensions it is very hard to see an underlying pattern in the solutions.…”

Section: Introducing the Weyl-heisenberg Groupmentioning

confidence: 99%

The Number Behind the Simplest SIC–POVM

Bengtsson

2017

Found Phys

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“…To see this we use the analysis in [33]. Let t j = Tr(G j ) and let Q j (respectively N j ) be the set of quadratic residues (respectively non-residues) in Z qj .…”

Section: Lemmamentioning

confidence: 99%

Linear dependencies in Weyl–Heisenberg orbits

Dang

Blanchfield

Bengtsson

et al. 2013

Quantum Inf Process

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Five years ago, Lane Hughston showed that some of the symmetric informationally complete positive operator valued measures (SICs) in dimension 3 coincide with the Hesse configuration (a structure well known to algebraic geometers, which arises from the torsion points of a certain elliptic curve). This connection with elliptic curves is signalled by the presence of linear dependencies among the SIC vectors. Here we look for analogous connections between SICs and algebraic geometry by performing computer searches for linear dependencies in higher dimensional SICs. We prove that linear dependencies will always emerge in Weyl-Heisenberg orbits when the fiducial vector lies in a certain subspace of an order 3 unitary matrix. This includes SICs when the dimension is divisible by 3 or equal to 8 mod 9. We examine the linear dependencies in dimension 6 in detail and show that smaller dimensional SICs are contained within this structure, potentially impacting the SIC existence problem. We extend our results to look for linear dependencies in orbits when the fiducial vector lies in an eigenspace of other elements of the Clifford group that are not order 3. Finally, we align our work with recent studies on representations of the Clifford group.

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SIC-POVMS and MUBS: Geometrical Relationships in Prime Dimension

Cited by 44 publications

References 43 publications

Order 3 symmetry in the Clifford hierarchy

Order 3 symmetry in the Clifford hierarchy

The Number Behind the Simplest SIC–POVM

Linear dependencies in Weyl–Heisenberg orbits

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