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Quantum Phase Space Representations and Their Negativities Ninnat Dangniam

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University of New Mexico UNM Digital Repository Physics & Astronomy ETDs Electronic Theses and Dissertations Spring 4-13-2018 Quantum Phase Space Representations and Their Negativities Ninnat Dangniam Follow this and additional works at: https://digitalrepository.unm.edu/phyc_etds Part of the Astrophysics and Astronomy Commons, and the Physics Commons Recommended Citation Dangniam, Ninnat. "Quantum Phase Space Representations and Their eN gativities." (2018). https://digitalrepository.unm.edu/ phyc_etds/180 This Dissertation is brought to you for free and open access by the Electronic Theses and Dissertations at UNM Digital Repository. It has been accepted for inclusion in Physics & Astronomy ETDs by an authorized administrator of UNM Digital Repository. For more information, please contact [email protected]. Ninnat Dangniam Candidate Physics and Astronomy Department This dissertation is approved, and it is acceptable in quality and form for publication: Approved by the Dissertation Committee: Carlton M. Caves , Chairperson Akimasa Miyake Daniel Finley Christopher Ferrie Cristopher Moore Quantum Phase Space Representations and Their Negativities by Ninnat Dangniam B.S., Physics, University of Oregon, 2011 B.S., Mathematics, University of Oregon, 2011 DISSERTATION Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Physics The University of New Mexico Albuquerque, New Mexico May 2018 iii c 2018, Ninnat Dangniam iv Acknowledgments Before anything else, I would like to thank my parents, who not only believe in me, but also support whatever life choice I pursue, both emotionally and financially, something that not all parents can do. I would like to express my deepest gratitude to the three C's: my advisor Carlton Caves, Christopher Ferrie, and Christopher Jackson, without whom this dissertation would not have been possible. It has been a valuable experience to be exposed to contrasting styles of mentoring, with Carl's high standards and emphasis on under- standing explicit examples instead of getting caught up in generalities|a \pedestrian approach" as he humbly puts it|on one hand, and Chris Ferrie's more whimsical, intuitive style on the other hand. Although it has only been a year since I started collaborating with Chris Jackson, I always enjoy his presence and contagious passion on all things mathematical. He also thoroughly read and gave a detailed feedback on every technical chapter of the dissertation. I would also like to thank the mem- bers of my dissertation committee: Carl, Chris Ferrie, Akimasa Miyake, Cristopher Moore, and Daniel Finley for taking time out of their busy schedules to read this dissertation. My time at UNM has been one of the best times in my life, and everyone here is to be thanked for that. I have learned much from the faculty and post-docs in CQuIC (in addition to the three C's), especially Ivan Deutsch, Andrew Landahl, Akimasa Miyake, Cristopher Moore, Howard Barnum, Robin Blume-Kohout, Joshua Combes, and Rafael Alexander. It is also my pleasure to interact with other UNM faculty members, in particular, David Dunlap and Rouzbeh Allahverdi. My spe- cial thanks also go to fellow graduate students Jonathan Gross for just being an all-around brilliant groupmate, Jacob Miller for all the interesting random conver- sations, and Adrian Chapman and Qi Xiaodong for being officemates and surviving together. I thank Jonathan, Qi, Travis Scholten, and Anupam Mitra for organizing the outstanding computing workshop, and last but not least, the CQuIC Program Specialist, Gloria Cordova, for putting up with us and making our lives here easier. I am indebted to Stephen Bartlett, Andrew Doherty, Steve Flammia, and every- one at the Quantum Science Research Group at the University of Sydney for the hospitality during my two visits. v Quantum Phase Space Representations and Their Negativities by Ninnat Dangniam B.S., Physics, University of Oregon, 2011 B.S., Mathematics, University of Oregon, 2011 Ph.D., Physics, University of New Mexico, 2018 Abstract A classical simulation scheme of quantum computation given a restricted set of states and measurements may be|occasionally, but only occasionally|interpreted naturally as a statistical simulation of positive quasi-probability distributions on a phase space. In this dissertation, we explore phase space representations for finite- dimensional quantum systems and their negativities beyond the usual analogues of the Wigner function. The first line of study focuses on a characterization tool for valid quasi-probability distributions of (possibly mixed) quantum states. A quantum generalization of Bochner's theorem from classical probability theory simultaneously characterizes both the set of valid Wigner functions and the subset of positive ones. We ex- tend this theorem to discrete phase spaces based on projective representations of abelian groups, including the discrete Wigner functions of Gross (J. Math. Phys. 47, 122107 (2006)) and Gibbons et al. (Phys. Rev. A 70, 062101 (2004)). vi The rest of the dissertation is then dedicated to phase space representations co- variant with respect to a general Lie group. This means, in particular, that operators that lie in the same orbit of the group have the same amount of negativity in their quasi-probabilities|a natural requirement if one would like to interpret negativity as non-classicality, and the group action as a classical dynamics. We show that the construction due to Brif and Mann (Phys. Rev. A 59, 971 (1999)) of group covariant phase space representations satisfying the so-called Stratonovich-Weyl cor- respondence is essentially unique, given the group and an input state in the relevant representation. As an application, we construct quasi-probability representations on a compact phase space of fermionic Gaussian states tailored to a classically simulat- able problem of fermionic linear optics and find that, in the first non-trivial case of four fermionic modes, the quasi-probabilities of the Gaussian states exhibit a con- siderable amount of negativity. vii Contents List of Figuresx 1 Introduction1 2 Representation theory 13 2.1 Overview.................................. 13 2.2 Fundamental notions........................... 14 2.3 Real, complex, and quaternionic representations............ 21 2.4 Harmonic analysis on homogeneous spaces............... 22 2.4.1 Regular representations and the group algebra......... 24 2.4.2 Induced representations..................... 25 2.4.3 Fourier transform......................... 28 2.5 Representations of Lie groups and Lie algebras............. 30 2.5.1 Lie groups and Lie algebras................... 31 2.5.2 Matrix Lie groups......................... 34 2.5.3 Representations of semisimple Lie algebras........... 37 2.5.4 Roots and weights........................ 40 Contents viii 2.5.5 Realizations of classical Lie algebras by creation and annihila- tion operators........................... 45 2.6 Harmonic analysis on multiplicity-free spaces.............. 49 2.6.1 Symmetric spaces and KAK decomposition.......... 49 2.6.2 Spherical functions........................ 51 3 Quasi-Probability Representations 54 3.1 The operational definition........................ 54 3.2 Frame formalism............................. 57 3.2.1 Inevitability of negativity.................... 59 3.3 Group covariance............................. 61 3.4 Stratonovich-Weyl correspondence.................... 61 3.4.1 Construction and uniqueness of SW frames........... 62 3.4.2 G-coherent-state quasi-probability functions.......... 69 4 Commutative Phase Spaces 71 4.1 Overview.................................. 71 4.2 The Wigner function........................... 72 4.3 The discrete Wigner function...................... 74 4.4 The quantum Bochner's theorem.................... 77 4.5 Discrete Quantum Bochner's theorem.................. 79 4.5.1 Projective unitary frames.................... 81 4.5.2 Characterizing projective unitary frames............ 83 Contents ix 4.5.3 Examples and non-examples................... 85 5 Fermionic Phase Spaces 88 5.1 Overview.................................. 88 5.2 Fermionic linear optics.......................... 89 5.2.1 Jordan-Wigner transformation and matchgates......... 92 5.2.2 Fermionic Gaussian states.................... 95 5.2.3 Classical simulatability and approaches to universality.... 99 5.3 Fermionic phase spaces.......................... 102 5.4 The spinor representations........................ 103 5.5 Four modes................................ 107 5.5.1 Matchgate-circuit parametrization of four-mode Gaussian states107 5.5.2 Integration measure........................ 112 5.5.3 U(4)-singlets and spherical functions.............. 114 5.5.4 Gaussian quasi-probability functions are negative....... 117 6 Summary and Open Problems 119 Appendix A Proof of the Discrete Hudson's Theorem 125 Appendix B Measure on Homogeneous K¨ahlerManifolds 129 References 133 x List of Figures 1.1 In a quantum world, a cat can be dead or alive; knowing that an electron here is in the \spin up" state instantaneously collapses an electron on the other side of the universe to the \spin down" state. Prompted by this description of the quantum world, an intelligent seven-year-old might say: \What's the big deal? The cat is either alive or dead. We just couldn't tell until we look at it. Also, the pair of electrons is no different from a pair of gloves. If I know that one of the gloves is left handed, the other one must be right handed." And the kid would be right! Classical
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