nan Arvind

The real symplectic groups in quantum mechanics and optics

We present a utilitarian review of the family of matrix groups Sp(2n, ℛ), in a form suited to various applications both in optics and quantum mechanics. We contrast these groups and their geometry with the much more familiar Euclidean and unitary geometries. Both the properties of finite group elements and of the Lie algebra are studied, and special attention is paid to the so-called unitary metaplectic representation of Sp(2n, ℛ). Global decomposition theorems, interesting subgroups and their generators are described. Turning ton-mode quantum systems, we define and study their variance matrices in general states, the implications of the Heisenberg uncertainty principles, and develop a U(n)-invariant squeezing criterion. The particular properties of Wigner distributions and Gaussian pure state wavefunctions under Sp(2n, ℛ) action are delineated.

Implementing quantum-logic operations, pseudopure states, and the Deutsch-Jozsa algorithm using noncommuting selective pulses in NMR

We demonstrate experimentally the usefulness of selective pulses in NMR to perform quantum computation.Three different techniques based on selective pulse excitations have been proposed to prepare a spin system in a pseudopure state. We describe the design of ‘‘portmanteau’’ gates using the selective manipulation of level populations. A selective pulse implementation of the Deutsch-Jozsa algorithm for two and three-qubit quantum computers is demonstrated.

Implementation of a Deutsch-like quantum algorithm utilizing entanglement at the two-qubit level on an NMR quantum-information processor

We describe the NMR implementation of a recently proposed quantum algorithm involving quantum entanglement at the level of two qubits. The algorithm solves a generalization of the Deutsch problem, and distinguishes between even and odd functions using fewer function calls than is possible classically. The manipulation of entangled states of the two qubits is essential here, unlike the Deutsch-Jozsa algorithm and Grover’s search algorithm for two bits.

Bargmann invariants and off-diagonal geometric phases for multilevel quantum systems: A unitary-group approach

We investigate the geometric phases and the Bargmann invariants associated with multilevel quantum systems. In particular, we show that a full set of ‘‘gauge-invariant’’ objects for an n-level system consists of n geometric phases and 1/2 (n-1)(n-2) algebraically independent four-vertex Bargmann invariants. In the process of establishing this result, we develop a canonical form for U(n) matrices that is useful in its own right. We show that the recently discovered ‘‘off-diagonal’’ geometric phases [N. Manini and F. Pistolesi, Phys. Rev. Lett. 8, 3067 (2000)] can be completely analyzed in terms of the basic building blocks developed in this work. This result liberates the off-diagonal phases from the assumption of adiabaticity used in arriving at them.

Experimental construction of generic three-qubit states and their reconstruction from two-party reduced states on an NMR quantum information processor

We experimentally explore the state space of three qubits on an NMR quantum information processor. We construct a scheme to experimentally realize a canonical form for general three-qubit states up to single-qubit unitaries. This form involves a non-trivial combination of GHZ and W-type maximally entangled states of three qubits. The general circuit that we have constructed for the generic state reduces to those for GHZ and W states as special cases. The experimental construction of a generic state is carried out for a nontrivial set of parameters and the good fidelity of preparation is confirmed by complete state tomography. The GHZ and W-states are constructed as special cases of the general experimental scheme. Further, we experimentally demonstrate a curious fact about three-qubit states, where for almost all pure states, the two-qubit reduced states can be used to reconstruct the full three-qubit state. For the case of a generic state and for the W-state, we demonstrate this method of reconstruction by comparing it with the directly tomographed three-qubit state.

Quantum entanglement and quantum computational algorithms

The existence of entangled quantum states gives extra power to quantum computers over their classical counterparts. Quantum entanglement shows up qualitatively at the level of two qubits. We demonstrate that the one- and the two-bit Deutsch-Jozsa algorithm does not require entanglement and can be mapped onto a classical optical scheme. It is only for three and more input bits that the DJ algorithm requires the implementation of entangling transformations and in these cases it is impossible to implement this algorithm classically.

Gaussian-Wigner distributions and hierarchies of nonclassical states in quantum optics: The single-mode case

A recently introduced hierarchy of states of a single-mode quantized radiation field is examined for the case of centered Gaussian-Wigner distributions. It is found that the onset of squeezing among such states signals the transition to the strongly nonclassical regime. Interesting consequences for the photon-number distribution, and explicit representations for them, are presented. The effects of nonideal detection are also carefully analyzed.

Quantum entanglement in the NMR implementation of the Deutsch-Jozsa algorithm

A scheme to execute an n-bit Deutsch-Jozsa (DJ) algorithm using n qubits has been implemented for up to three qubits on an NMR quantum computer. For the one- and the two-bit Deutsch problem, the qubits do not get entangled, and the NMR implementation is achieved without using spin-spin interactions. It is for the three-bit case, that the manipulation of entangled states becomes essential. The interactions through scalar J-couplings in NMR spin systems have been exploited to implement entangling transformations required for the three bit DJ algorithm.

Bargmann invariants and geometric phases: A generalized connection

We develop the broadest possible generalization of the well known connection between quantum-mechanical Bargmann invariants and geometric phases. The key concept is that of null phase curves in quantum-mechanical ray and Hilbert spaces. Examples of such curves are developed. Our generalization is shown to be essential for properly understanding geometric phase results in the cases of coherent states and of Gaussian states. Differential geometric aspects of null phase curves are also briefly explored.

Experimental construction of aWsuperposition state and its equivalence to the Greenberger-Horne-Zeilinger state under local filtration

We experimentally construct a novel three-qubit entangled W-superposition (