Hubert de Guise

Quantum encodings in spin systems and harmonic oscillators

We show that higher-dimensional versions of qubits, or qudits, can be encoded into spin systems and into harmonic oscillators, yielding important advantages for quantum computation. Whereas qubit-based quantum computationis adequate for analyses of quantum vs classical computation, in practice qubits are often realized in higher-dimensional systems by truncating all but two levels, thereby reducing the size of the precious Hilbert space. We develop natural qudit gates for universal quantum computation, and exploit the entire accessible Hilbert space. Mathematically, we give representations of the generalized Pauli group for qudits in coupled spin systems and harmonic oscillators, and include analyses of the qubit and the infinite-dimensional limits.

Multicomplementary operators via finite Fourier transform

A complete set of d + 1 mutually unbiased bases exists in a Hilbert space of dimension d, whenever d is a power of a prime. We discuss a simple construction of d + 1 disjoint classes (each one having d − 1 commuting operators) such that the corresponding eigenstates form sets of unbiased bases. Such a construction works properly for prime dimension. We investigate an alternative construction in which the real numbers that label the classes are replaced by a finite field having d elements. One of these classes is diagonal, and can be mapped to cyclic operators by means of the finite Fourier transform, which allows one to understand complementarity in a similar way as for the position–momentum pair in standard quantum mechanics. The relevant examples of two and three qubits and two qutrits are discussed in detail.

Generalized Multiphoton Quantum Interference

Non-classical interference of photons lies at the heart of optical quantum information processing. This effect is exploited in universal quantum gates as well as in purpose-built quantum computers that solve the BosonSampling problem. Although non-classical interference is often associated with perfectly indistinguishable photons this only represents the degenerate case, hard to achieve under realistic experimental conditions. Here we exploit tunable distinguishability to reveal the full spectrum of multi-photon non-classical interference. This we investigate in theory and experiment by controlling the delay times of three photons injected into an integrated interferometric network. We derive the entire coincidence landscape and identify transition matrix immanants as ideally suited functions to describe the generalized case of input photons with arbitrary distinguishability. We introduce a compact description by utilizing a natural basis which decouples the input state from the interferometric network, thereby providing a useful tool for even larger photon numbers.

Geometric Phase of Three-Level Systems in Interferometry

We present the first scheme for producing and measuring an Abelian geometric phase shift in a three-level system where states are invariant under a non-Abelian group. In contrast to existing experiments and proposals for experiments, based on U(1)-invariant states, our scheme geodesically evolves U(2)-invariant states in a four-dimensional SU(3)/U(2) space and is physically realized via a three-channel optical interferometer.

Quantum phases of a qutrit

We consider various approaches to treat the phases of a qutrit. Although it is possible to represent qutrits in a convenient geometrical manner by resorting to a generalization of the Poincare sphere, we argue that the appropriate way of dealing with this problem is through phase operators associated with the algebra su(3). The rather unusual properties of these phases are caused by the small dimension of the system and are explored in detail. We also examine the positive operator-valued measures that can describe the qutrit phase properties.

Some nonunitary, indecomposable representations of the Euclidean algebra \mathfrak {e}(3)

The Euclidean group E(3) is the noncompact, semidirect product group . It is the Lie group of orientation-preserving isometries of three-dimensional Euclidean space. The Euclidean algebra is the complexification of the Lie algebra of E(3). We construct three distinct families of finite-dimensional, nonunitary representations of and show that each representation is indecomposable. The representations of the first family are explicitly realized as subspaces of the polynomial ring with the action of given by differential operators. The other families are constructed via duals and tensor products of the representations within the first family. We describe subrepresentations, quotients and duals of these indecomposable representations.

Some finite dimensional indecomposable representations of E(2)

We describe the construction of some finite dimensional nonunitary representations of E(2), the Lie group of Euclidean transformations in the plane. Some properties of these representations are also discussed, with emphasis on indecomposable representations.

Coincidence Landscapes for Three-Channel Linear Optical Networks

We use permutation-group methods plus SU(3) group-theoretic methods to determine the action of a three-channel passive optical interferometer on controllably delayed single-photon pulse inputs to each channel. Permutation-group techniques allow us to relate directly expressions for rates and, in particular, investigate symmetries in the coincidence landscape. These techniques extend the traditional Hong-Ou-Mandel effect analysis for two-channel interferometry to valleys and plateaus in three-channel interferometry. Our group-theoretic approach is intuitively appealing because the calculus of Wigner

VECTOR PHASE MEASUREMENT IN MULTIPATH QUANTUM INTERFEROMETRY

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Dynamical symmetry reduction and discrete tomography of a Ξ atom

functions partially accounts for permutational symmetries and directly reveals the connections among