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 computation is 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.PACS numbers: 03.67. Lx, Quantum computation may be able to perform certain tasks more efficiently than a classical computer; for example, Shor's algorithm [1] for factoring prime numbers on a quantum computer is exponentially faster than any known algorithm on a classical computer. The standard model of a quantum computer involves coupling together two-level quantum systems (qubits) such that the Hilbert space of the system grows exponentially in the number of qubits.A major obstacle to universal quantum computing is the limit on the number of coupled qubits that can be achieved in a physical system [2]. The use of ddimensional, or qudit, quantum computing enables a much more compact and efficient information encoding than for qubit computing. Qudit quantum information processing employs fewer coupled quantum systems: a considerable advantage for the experimental realization of quantum computing. The harmonic oscillator is a system that naturally provides qudits as quanta in its energy spectrum. Qubits are obtained by restricting the dynamics to just two of these quanta, namely the vacuum state |0 and the first excited state |1 ; e.g., photons in cavity QED [3] and interferometry [4]. However, the control of entanglement in larger Hilbert spaces is now feasible (e.g., orbital angular momentum states of photons [5]). Our aim is to show that the restriction to two-dimensional Hilbert spaces is not necessary and that higher-dimensional Hilbert spaces are an advantage, particularly when the number of achievable coupled systems is limited and entanglement between systems with larger Hilbert spaces is physically possible.A quantum computer also requires gates, realized as the unitary evolution under some Hamiltonian. For qubits, a universal set of gates is given by arbitrary SU(2) rotations of a single qubit along with some nonlinear coupling transformation between adjacent qubits generated by a two-qubit Hamiltonian [6]. For qudit quantum computation, the issue of creating a universal set of gates is more involved. In particular, it is not possible to treat coupled qudits as a collection of qubits, because (typically) one does not have access to "pairwise" Hamiltonians between two arbitrary levels of coupled qudits. For example, in a system of coupled oscillators realize...

Nonclassical interference of photons lies at the heart of optical quantum information processing. Here, we exploit tunable distinguishability to reveal the full spectrum of multiphoton nonclassical interference. We investigate this 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 that decouples the input state from the interferometric network, thereby providing a useful tool for even larger photon numbers.

Bases for SU(3) irreps are constructed on a space of three-particle tensor products of two-dimensional harmonic oscillator wave functions. The Weyl group is represented as the symmetric group of permutations of the particle coordinates of these space. Wigner functions for SU(3) are expressed as products of SU(2) Wigner functions and matrix elements of Weyl transformations. The constructions make explicit use of dual reductive pairs which are shown to be particularly relevant to problems in optics and quantum interferometry.Comment: : RevTex file, 11 pages with 2 figure

A complete set of d + 1 mutually unbiased bases exists in a Hilbert spaces 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.

Entanglement, or quantum inseparability, is a crucial resource in quantum information applications, and therefore the experimental generation of separated yet entangled systems is of paramount importance. Experimental demonstrations of inseparability with light are not uncommon, but such demonstrations in physically well-separated massive systems, such as distinct gases of atoms, are new and present significant challenges and opportunities. Rigorous theoretical criteria are needed for demonstrating that given data are sufficient to confirm entanglement. Such criteria for experimental data have been derived for the case of continuous-variable systems obeying the Heisenberg-Weyl (position- momentum) commutator. To address the question of experimental verification more generally, we develop a sufficiency criterion for arbitrary states of two arbitrary systems. When applied to the recent study by Julsgaard, Kozhekin, and Polzik [Nature 413, 400 - 403 (2001)] of spin-state entanglement of two separate, macroscopic samples of atoms, our new criterion confirms the presence of spin entanglement.Comment: 11 pages, 1 figur

We develop a framework for solving the action of a three-channel passive optical interferometer on single-photon pulse inputs to each channel using SU(3) group-theoretic methods, which can be readily generalized to higher-order photon-coincidence experiments. We show that features of the coincidence plots versus relative time delays of photons yield information about permanents, immanants, and determinants of the interferometer SU(3) matrix.

We demonstrate a method for general linear optical networks that allows one to factorize any SU(n) matrix in terms of two SU(n − 1) blocks coupled by an SU(2) entangling beam splitter. The process can be recursively continued in a straightforward way, ending in a tidy arrangement of SU(2) transformations. The method hinges only on a linear relationship between input and output states, and can thus be applied to a variety of scenarios, such as microwaves, acoustics, and quantum fields.

We propose an operational form for the kernel of a mapping between an operator acting in a Hilbert space of a quantum system with SU(n) symmetry group and its symbol in the corresponding classical phase space. For symmetric irreps of SU(n), this mapping is bijective. We briefly discuss complications that will occur in the general case.