Tata Lectures on Theta III

Mumford, David; Nori, Madhav; Norman, Peter

doi:10.1007/978-0-8176-4579-3

Cited by 127 publications

(125 citation statements)

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“…Note that U ( 0 2 1 2 ) corresponds to the Zauner matrix in Equation (19) of Section 2. The SIC fiducial defined in Equation (25) is then precisely the intersection point of the 2-dimensional subspaces H 1 from each of these four symplectic canonical order 3 unitaries.…”

Section: The Weyl-heisenberg and Clifford Groupsmentioning

confidence: 99%

“…The short orbit arises because it contains only linearly dependent sets invariant under the subgroup {1, X 2 Z 4 , X 4 Z 2 }. This subgroup commutes with the Zauner unitary defined in Equations (13) and (19), which leaves the fiducial SIC vector invariant. Twenty two of these WH orbits contain sets that are invariant under the Zauner matrix (or sets invariant under a WH conjugate D p U Z D † p of the Zauner matrix); the other 6 are invariant under the action of U M .…”

Section: Numerical Linear Dependenciesmentioning

confidence: 99%

“…The WH group appears naturally in the theory of elliptic curves [19], suggesting this might be an advantageous framework for the study of SICs. Furthermore, there is a very close connection between elliptic curves and linear dependencies among SIC vectors in dimension 3, first noted by Lane Hughston [20].…”

Section: Introductionmentioning

confidence: 99%

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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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Section: The Weyl-heisenberg and Clifford Groupsmentioning

confidence: 99%

Section: Numerical Linear Dependenciesmentioning

confidence: 99%

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Linear dependencies in Weyl–Heisenberg orbits

Dang

Blanchfield

Bengtsson

et al. 2013

Quantum Inf Process

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“…where ξ g is an eighth root of unity depending on the group element g [1]. Now, we like to find a compatible function on the orbifold in which the complex structure is preserved, g · T = T .…”

Section: Orbifolding and Classical Theta Functionmentioning

confidence: 99%

“…Classical theta functions [1] can be regarded as state functions on classical tori, and have played an important role in the string loop calculation [2,3]. Recently, Manin [4,5,6] introduced the concept of quantum theta function as a quantum counterpart of classical theta function.…”

Section: Introductionmentioning

confidence: 99%

Symmetry of quantum torus with crossed product algebra

Chang-Young

Kim

2006

Journal of Mathematical Physics

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In this paper, we study the symmetry of quantum torus with the concept of crossed product algebra. As a classical counterpart, we consider the orbifold of classical torus with complex structure and investigate the transformation property of classical theta function. An invariant function under the group action is constructed as a variant of the classical theta function. Then our main issue, the crossed product algebra representation of quantum torus with complex structure under the symplectic group is analyzed as a quantum version of orbifolding. We perform this analysis with Manin's so-called model II quantum theta function approach. The symplectic group Sp(2n, Z) satisfies the consistency condition of crossed product algebra representation. However, only a subgroup of Sp(2n, Z) satisfies the consistency condition for orbifolding of quantum torus.

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2000

The Fourier‐Analytic Proof of Quadratic Reciprocity

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Tata Lectures on Theta III

Cited by 127 publications

References 0 publications

Linear dependencies in Weyl–Heisenberg orbits

Linear dependencies in Weyl–Heisenberg orbits

Symmetry of quantum torus with crossed product algebra

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