Bulletin of the American Physical Society
APS March Meeting 2011
Volume 56, Number 1
Monday–Friday, March 21–25, 2011; Dallas, Texas
Session J29: Focus Session: Quantum Information for Quantum Foundations - Structures in Hilbert Space |
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Sponsoring Units: GQI Chair: Asa Ericsson, Institute Mittag-Leffler Room: C148 |
Tuesday, March 22, 2011 11:15AM - 11:27AM |
J29.00001: Proofs of the Kochen-Specker theorem based on two qubits Mordecai Waegell, P.K. Aravind The observables for a pair of qubits yield a system of 60 rays and 105 bases in a complex Hilbert space of four dimensions that contains over a hundred million parity proofs of the Kochen-Specker theorem. An overview of these proofs is given and they are compared with those in other 4-d systems, such as the 600-cell. The significance of the results is discussed. [Preview Abstract] |
Tuesday, March 22, 2011 11:27AM - 11:39AM |
J29.00002: Proofs of the Kochen-Specker theorem based on the 600-cell P.K. Aravind, Mordecai Waegell, Norman Megill, Mladen Pavicic It is shown that the system of 60 rays and 75 bases derived from the vertices of a 600-cell (a regular polytope in four dimensions) contains over a hundred million parity proofs of the Kochen-Specker theorem. An overview of the proofs is given, some examples of them are presented and the significance of the results is discussed. [Preview Abstract] |
Tuesday, March 22, 2011 11:39AM - 11:51AM |
J29.00003: MUB Entanglement Patterns by Transformations in Phase Space Jay Lawrence All possible MUB entanglement patterns for systems of N prime-state particles are obtained from standard ones by unitary transformations in the Hilbert space, thus preserving the relationships between the generalized Pauli operators, the phase point operators, and the MUB projectors. The transformations are described geometrically in discrete phase space. Illustrative examples show the invariance of the total entanglement content and the connection of entanglement with Galois fields. Different field representations for the same dimension may produce inequivalent MUB sets. This work provides alternative constructions and generalizes previous work on qubit systems [1,2]. \\[4pt] [1] J L Romero, G Bjork A B Klimov, and L L Sanchez-Soto, Phys. Rev. A {\bf 72}, 062310 (2005).\\[0pt] [2] A B Klimov, J L Romero, G Bjork, and L L Sanchez-Soto, Ann. Phys. {\bf 324}, 53 (2009). [Preview Abstract] |
Tuesday, March 22, 2011 11:51AM - 12:03PM |
J29.00004: Entanglement in Mutually Unbiased Bases Marcin Wiesniak, Tomasz Paterek, Anton Zeilinger Higher-dimensional Hilbert spaces are still not fully explored. One issue concerns mutually unbiased bases (MUBs). For primes [1] and their powers (e.g. [2]), full sets of MUBs are known. The question of existence of all MUBs in composite dimensions is still open. We show that for all full sets of MUBs of a given dimension a certain entanglement measure of the bases is constant. This fact could be an argument either for or against the existence of full sets of MUBs in some dimensions and tells us that almost all MUBs are maximally entangled for high-dimensional composite systems, whereas this is not the case for prime dimensions. We present a new construction of MUBs in squared prime dimensions. We use only one entangling operation, which simplifies possible experiments. The construction gives only product states and maximally entangled states. \\[4pt] [1] I. D. Ivanovi\'c, J. Phys. A 14, 3241 (1981). \\[0pt] [2] W. K. Wootters and B. D. Fields, Ann. Phys. (N.Y.) 191, 363 (1989). [Preview Abstract] |
Tuesday, March 22, 2011 12:03PM - 12:15PM |
J29.00005: Qutrits under a microscope Gelo Noel Tabia Gleason's theorem states that the set of quantum states is complete, in the sense that density operators specify the unique probability measure definable on the lattice of Hilbert space of projection operators according to the Born Rule. Particularly, Gleason showed that the theorem holds in all finite dimensions if and only if it holds in dimension 3. This suggests that the essential features defining the probability structure of quantum theory can already be found in 3-dimensional quantum systems. Hence, we establish key geometric properties of qutrit state space as they are expressed in terms of symmetric, informationally-complete (SIC) measurements. We provide a variety of important results, which include an elegant formula for describing pure qutrits, affine plane symmetries and the Hesse configuration in qutrit SICs derived from algebraic structure constants for $\mathrm{GL}(3,C)$, and a comparison of the SIC and generalized Bloch representations by analyzing plane cross-sections of qutrit state space. In addition, we present a new way of implementing SIC-POVMs using multi-port devices built from waveguide-based micro-optics, in particular, by proposing experimental circuits for qubits and qutrits. [Preview Abstract] |
Tuesday, March 22, 2011 12:15PM - 12:27PM |
J29.00006: A Linear Dependency Structure Arising from Weyl-Heisenberg Symmetry Hoan Bui Dang, Marcus Appleby, Ingemar Bengtsson, Kate Blanchfield, Asa Ericsson, Christopher Fuchs, Matthew Graydon, Gelo Tabia The Weyl-Heisenberg (WH) group was used by Hermann Weyl to construct finite-dimensional quantum mechanics in the earliest days of the theory and, through its ubiquitous use in quantum information theory, is even more important today. While investigating properties of symmetric informationally-complete (SIC) measurements, we found a linear dependency structure in a class of Weyl-Heisenberg covariant sets when certain conditions on the dimensionality of the Hilbert space are met. This result reveals more structure in WH symmetry than previously noted and helps us gain a better understanding of quantum state space. For example in the Quantum Bayesian framework of Fuchs and collaborators, the number of zeros of a quantum state in a SIC representation is directly related to this linear dependency. [Preview Abstract] |
Tuesday, March 22, 2011 12:27PM - 12:39PM |
J29.00007: Regrouping phenomena of SIC POVMs covariant with respect to the Heisenberg--Weyl group Huangjun Zhu Symmetric informationally complete positive operator valued measures (SIC POVMs) covariant with respect to the Heisenberg--Weyl (HW) group form disjoint orbits under the action of the normalizer of the HW group---the (extended) Clifford group. Additional SIC POVMs can be obtained by a suitable regrouping of the fiducial vectors on certain orbits, for example, in Hilbert spaces of dimension three, four, eight and twelve. To understand these SIC POVM regrouping phenomena, we need to go beyond the Clifford group and consider a larger group, in particular the normalizer of the Clifford group. We prove that, when the dimension of the Hilbert space is not a multiple of four, the HW group is a characteristic subgroup of the Clifford group, and the normalizer of the Clifford group is itself; when the dimension is a multiple of four, there are exactly two normal subgroups in the Clifford group that are isomorphic to the HW group, which are conjugated to each other in the normalizer of the Clifford group. Based on this observation, we provide a unified framework for understanding the regrouping phenomena mentioned above and those potential candidates. [Preview Abstract] |
Tuesday, March 22, 2011 12:39PM - 12:51PM |
J29.00008: Quantum Computational Geodesic Derivative Howard Brandt In recent developments in the differential geometry of quantum computation, the quantum evolution is described in terms of the special unitary group of n-qubit unitary operators with unit determinant. The group manifold is taken to be Riemannian. In the present work the geodesic derivative is clarified. This is applicable to investigations of conjugate points and the global characteristics of geodesic paths in the group manifold, and the determination of optimal quantum circuits for carrying out a quantum computation. [Preview Abstract] |
Tuesday, March 22, 2011 12:51PM - 1:03PM |
J29.00009: Affine Maps of the Polarization Vector for Quantum Systems of Arbitrary Dimension Mark Byrd, C. Allen Bishop, Yong-Cheng Ou The operator-sum decomposition (OS) of a mapping from one density matrix to another has many applications in quantum information science. To this mapping there corresponds an affine map which provides a geometric description of the density matrix in terms of the polarization vector representation. This has been thoroughly explored for qubits since the components of the polarization vector are measurable quantities (corresponding to expectation values of Hermitian operators) and also because it enables the description of map domains geometrically. Here we extend the OS-affine map correspondence to qudits, briefly discuss general properties of the map, the form for particular important cases, and provide several explicit results for qutrit maps. We use the affine map and a singular-value-like decomposition, to find positivity constraints that provide a symmetry for small polarization vector magnitudes (states which are closer to the maximally mixed state) which is broken as the polarization vector increases in magnitude (a state becomes more pure). The dependence of this symmetry on the magnitude of the polarization vector implies the polar decomposition of the map can not be used as it can for the qubit case. However, it still leads us to a connection between positivity and purity for general d-state systems. [Preview Abstract] |
Tuesday, March 22, 2011 1:03PM - 1:15PM |
J29.00010: Pseudo-unitary freedom in the operator-sum representation Yong Cheng Ou, Mark S. Byrd A general dynamical map can be written in an operator-sum representation (OSR) by using a spectral decomposition, which needs not be completely positive. The OSR is not unique; there is freedom to choose a different set of operators in the OSR, yet still obtain the same map. We will show that, whereas the freedom for completely positive maps is unitary, the freedom for not completely positive maps is pseudo-unitary. [Preview Abstract] |
Tuesday, March 22, 2011 1:15PM - 1:27PM |
J29.00011: Long-range spin-coupled interactions: a \emph{Gedankenexperiment} on the nature of spin Ian Durham What is intrinsic spin? It is at the heart of the quantum information revolution and yet it defies many of the efforts to better understand it, even to the point of pushing particle physics beyond the Standard Model. Long assumed to require the relativistic theory of Dirac, in 1967 L\'{e}vy-Lablond demonstrated that this was not the case: it is not necessarily a relativistic effect. In this article, we apply the L\'{e}vy-Lablond model to a simple \emph{Gedankenexperiment} that suggests the existence of a quasi-fundamental long-range spin-coupled interaction. Calculations of the eigenfunctions of a test particle and the coupling constant of the force gives insight into the behavior of the potential that gives rise to this interaction. For large separation distances the potential looks like a simple potential well while for very small separation distances it exhibits a more complex nature. This, in turn, sheds additional light on the nature of intrinsic spin and a suggests a path for future research. [Preview Abstract] |
Tuesday, March 22, 2011 1:27PM - 1:39PM |
J29.00012: Quantum simulation of time-dependent Hamiltonians and the convenient illusion of Hilbert space Rolando Somma, David Poulin, Angie Qarry, Frank Verstraete We consider the manifold of all quantum many-body states that can be generated by arbitrary time-dependent local Hamiltonians in a time that scales polynomially in the system size, and show that it occupies an exponentially small volume in Hilbert space. This implies that the overwhelming majority of states in Hilbert space are not physical as they can only be produced after an exponentially long time. We establish this fact by making use of a time-dependent generalization of the Suzuki-Trotter expansion, followed by a counting argument. This also demonstrates that a computational model based on arbitrarily rapidly changing Hamiltonians is no more powerful than the standard quantum circuit model. [Preview Abstract] |
Tuesday, March 22, 2011 1:39PM - 1:51PM |
J29.00013: Mathematical Constraint on Realistic Theories James Franson We consider realistic theories in which some physical property f(r,t) is assumed to exist regardless of whether or not we measure it. It is shown that the value of f(r,t) at position r and time t is completely determined by its value at all other locations r$'$ and earlier times t$' <$ t provided that f(r,t) has continuous second partial derivatives [1]. Mathematical functions of this kind are sufficiently general to describe many situations of physical interest. These results are based on a mathematical identity that is similar in some respects to Cauchy's integral theorem and it can be viewed as a generalization of Green's third identity. The physical implications of weak determinism of this kind will be discussed and it will be contrasted with the properties of quantum systems. \\[4pt] [1] J.D. Franson, arXiv: 1007.1941. [Preview Abstract] |
Tuesday, March 22, 2011 1:51PM - 2:03PM |
J29.00014: Construction of optimal witness for unknown two-qubit entanglement S.-S.B. Lee, H.S. Park, H. Kim, S.-K. Choi, H.-S. Sim Whether entanglement in a state can be detected, distilled, and quantified without full state reconstruction is a fundamental open problem. We demonstrate a new scheme encompassing these three tasks for arbitrary two-qubit entanglement, by constructing the optimal entanglement witness for polarization-entangled mixed-state photon pairs without full state reconstruction. With better efficiency than quantum state tomography, the entanglement is maximally distilled by newly developed tunable polarization filters, and quantified by the expectation value of the witness, which equals the concurrence. This scheme is extendible to multiqubit Greenberger-Horne-Zeilinger entanglement. This work is to appear in Physical Review Letters. [Preview Abstract] |
Tuesday, March 22, 2011 2:03PM - 2:15PM |
J29.00015: ABSTRACT WITHDRAWN |
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