| 1. |
- Appleby, D. M., et al.
(författare)
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GALOIS UNITARIES, MUTUALLY UNBIASED BASES, AND MUB-BALANCED STATES
- 2015
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Ingår i: Quantum information & computation. - 1533-7146. ; 15:15-16, s. 1261-1294
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Tidskriftsartikel (refereegranskat)abstract
- A Galois unitary is a generalization of the notion of anti-unitary operators. They act only on those vectors in Hilbert space whose entries belong to some chosen number field. For Mutually Unbiased Bases the relevant number field is a cyclotomic field. By including Galois unitaries we are able to remove a mismatch between the finite projective group acting on the bases on the one hand, and the set of those permutations of the bases that can be implemented as transformations in Hilbert space on the other hand. In particular we show that there exist transformations that cycle through all the bases in all dimensions d - p(n) where p is an odd prime and the exponent n is odd. (For even primes unitary MUB-cyclers exist.) These transformations have eigenvectors, which are MUB-balanced states (i.e. rotationally symmetric states in the original terminology of Wootters and Sussman) if and only if d = 3 modulo 4. We conjecture that this construction yields all such states in odd prime power dimension.
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| 2. |
- Appleby, D.M., et al.
(författare)
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Systems of Imprimitivity for the Clifford Group
- 2014
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Ingår i: Quantum information & computation. - : Rinton Press. - 1533-7146. ; 14:3-4, s. 339-360
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Tidskriftsartikel (refereegranskat)abstract
- It is known that if the dimension is a perfect square the Clifford group can be represented by monomial matrices. Another way of expressing this result is to say that when the dimension is a perfect square the standard representation of the Clifford group has a system of imprimitivity consisting of one dimensional subspaces. We generalize this result to the case of an arbitrary dimension. Let k be the square-free part of the dimension. Then we show that the standard representation of the Clifford group has a system of imprimitivity consisting of k-dimensional subspaces. To illustrate the use of this result we apply it to the calculation of SIC-POVMs (symmetric informationally complete positive operator valued measures), constructing exact solutions in dimensions 8 (hand-calculation) as well as 12 and 28 (machine-calculation).
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| 3. |
- Appleby, D. M., et al.
(författare)
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The monomial representations of the Clifford group
- 2012
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Ingår i: Quantum information & computation. - : Rinton Press, Inc.. - 1533-7146. ; 12:5-6, s. 404-431
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Tidskriftsartikel (refereegranskat)abstract
- We show that the Clifford group-the normaliser of the Weyl-Heisenberg group-can be represented by monomial phase-permutation matrices if and only if the dimension is a square number. This simplifies expressions for SIC vectors, and has other applications to SICs and to Mutually Unbiased Bases. Exact solutions for SICs in dimension 16 are presented for the first time.
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| 4. |
- Dang, Hoan Bui, et al.
(författare)
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Linear Dependencies in Weyl-Heisenberg Orbits
- 2013
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Ingår i: Quantum Information Processing. - : Springer Science and Business Media LLC. - 1570-0755 .- 1573-1332. ; 12:11, s. 3449-3475
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Tidskriftsartikel (refereegranskat)abstract
- 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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