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University of Waterloo

1. Dang, Hoan. Studies of symmetries that give special quantum states the "right to exist".

Degree: 2015, University of Waterloo

URL: http://hdl.handle.net/10012/9306

In this thesis we study symmetric structures in Hilbert spaces known as symmetric
informationally complete positive operator-valued measures (SIC-POVMs), mutually unbiased bases (MUBs), and MUB-balanced states. Our tools include symmetries such
as the Weyl-Heisenberg (WH) group symmetry, Clifford unitaries, Zauner symmetry, and
Galois-unitaries (g-unitaries). In the study of SIC-POVMs, we found their geometric
significance as the ``most orthogonal'' bases on the cone of non-negative operators. While investigating SICs, we discovered a linear dependency property of the orbit of an arbitrary vector with the Zauner symmetry under the WH group. In dimension d = 3, the linear dependency structures arising from certain special SIC states are identified with the Hesse configuration known from the study of elliptic curves in mathematics. We provide an analytical explanation for linear dependencies in every dimension, and a numerical analysis based on exhaustive numerical searches in dimensions d = 4 to 9. We also study the relations among normal vectors of the hyperplanes spanned by the linearly dependent sets, and found 2-dimensional SICs embedded in the Hilbert space of dimension d = 6, and 3-dimensional SICs for d = 9. A full explanation is given for the case d = 6. Another study in the thesis focuses on the roles of g-unitaries in the theory of mutually unbiased bases. G-unitaries are, in general, non-linear operators defined to generalize the notion of anti-unitaries. Due to Wigner's theorem, their action has to be restricted to a smaller region of the Hilbert space, which consists of vectors whose components belong to a specific number field. G-unitaries are relevant to MUBs when this number field is the cyclotomic field. In this case, we found that g-unitaries simply permuted the bases in the standard set of MUBs in odd prime-power dimensions. With their action further restricted only to MUB vectors, g-unitaries can be represented by rotations in the Bloch space, just as ordinary unitary operators can. We identify g-unitaries that cycle through all d + 1 bases in prime power dimensions d = p^n where n is odd (the problem in even prime power dimensions has been solved using ordinary unitaries). Each of these MUB-cycling g-unitaries always leaves one state in the Hilbert space invariant. We provide a method for calculating these eigenvectors. Furthermore, we prove that when d = 3 mod 4, they are MUB-balanced states in the sense of Wootters & Sussman and Amburg et al.

Subjects/Keywords: quantum physics; quantum information; symmetry; SIC-POVM; MUB; Galois unitary; g-unitary; Weyl-Heisenberg group; Clifford group; MUB-balanced; MUB-cycling

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Dang, H. (2015). Studies of symmetries that give special quantum states the "right to exist". (Thesis). University of Waterloo. Retrieved from http://hdl.handle.net/10012/9306

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Dang, Hoan. “Studies of symmetries that give special quantum states the "right to exist".” 2015. Thesis, University of Waterloo. Accessed August 05, 2020. http://hdl.handle.net/10012/9306.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Dang, Hoan. “Studies of symmetries that give special quantum states the "right to exist".” 2015. Web. 05 Aug 2020.

Vancouver:

Dang H. Studies of symmetries that give special quantum states the "right to exist". [Internet] [Thesis]. University of Waterloo; 2015. [cited 2020 Aug 05]. Available from: http://hdl.handle.net/10012/9306.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Dang H. Studies of symmetries that give special quantum states the "right to exist". [Thesis]. University of Waterloo; 2015. Available from: http://hdl.handle.net/10012/9306

Not specified: Masters Thesis or Doctoral Dissertation

2. Kopp, Gene. Indefinite Theta Functions and Zeta Functions.

Degree: PhD, Mathematics, 2017, University of Michigan

URL: http://hdl.handle.net/2027.42/140957

We define an indefinite theta function in dimension g and index 1 whose modular parameter transforms by a symplectic group, generalizing a construction of Sander Zwegers used in the theory of mock modular forms. We introduce the indefinite zeta function, defined from the indefinite theta function using a Mellin transform, and prove its analytic continuation and functional equation. We express certain zeta functions attached to ray ideal classes of real quadratic fields as indefinite zeta functions (up to gamma factors). A Kronecker limit formula for the indefinite zeta function – and by corollary, for real quadratic fields – is obtained at s=1. Finally, we discuss two applications related to Hilbert's 12th problem: numerical computation of Stark units in the rank 1 real quadratic case, and computation of fiducial vectors of Heisenberg SIC-POVMs.
*Advisors/Committee Members: Lagarias, Jeffrey C (committee member), Doering, Charles R (committee member), Koch, Sarah Colleen (committee member), Prasanna, Kartik (committee member), Snowden, Andrew (committee member), Zieve, Michael E (committee member).*

Subjects/Keywords: number theory; indefinite theta function; zeta function; real quadratic field; Kronecker limit formula; SIC-POVM; Mathematics; Science

…Flammia, McConnell and Yard [5, 6].
An *SIC*-*POVM* is a set of d2 “equiangular complex… …one *SIC*-*POVM*
in every dimension up to d = 121 [37, 36]. The case d = 4 is… …described in detail is
[8]. An overview of the *SIC*-*POVM* problem is provided by the… …preprint [17].
1.5.1
*SIC*-*POVM* example
The numerical example for the Stark conjecture… …*SIC*-*POVM* according
to conjectures of Appleby et. al. [6], which are verified in…

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Kopp, G. (2017). Indefinite Theta Functions and Zeta Functions. (Doctoral Dissertation). University of Michigan. Retrieved from http://hdl.handle.net/2027.42/140957

Chicago Manual of Style (16^{th} Edition):

Kopp, Gene. “Indefinite Theta Functions and Zeta Functions.” 2017. Doctoral Dissertation, University of Michigan. Accessed August 05, 2020. http://hdl.handle.net/2027.42/140957.

MLA Handbook (7^{th} Edition):

Kopp, Gene. “Indefinite Theta Functions and Zeta Functions.” 2017. Web. 05 Aug 2020.

Vancouver:

Kopp G. Indefinite Theta Functions and Zeta Functions. [Internet] [Doctoral dissertation]. University of Michigan; 2017. [cited 2020 Aug 05]. Available from: http://hdl.handle.net/2027.42/140957.

Council of Science Editors:

Kopp G. Indefinite Theta Functions and Zeta Functions. [Doctoral Dissertation]. University of Michigan; 2017. Available from: http://hdl.handle.net/2027.42/140957