A parameterization of the density operator, a coherence vector representation, which uses a basis of orthogonal, traceless, Hermitian matrices is discussed. Using this parameterization we find the region of permissible vectors which represent a density operator. The inequalities which specify the region are shown to involve the Casimir invariants of the group. In particular cases, this allows the determination of degeneracies in the spectrum of the operator. The identification of the Casimir invariants also provides a method of constructing quantities which are invariant under local unitary operations. Several examples are given which illustrate the constraints provided by the positivity requirements and the utility of the coherence vector parameterization.

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Three groups of people ranging in age from 64 to 88 years performed tasks of word generation, paired-associate recall, and free and cued recall. The groups differed in socioeconomic status, verbal intelligence, and apparent levels of daily activity. A fourth group, consisting of young undergraduates, was also tested. Results showed that whereas there were age-related differences in some tests, these age differences were strongly modulated by characteristics of the participants and characteristics of the tasks. The findings are discussed in a contextualist framework.

In this paper we give an explicit parametrization for all two qubit density matrices. This is important for calculations involving entanglement and many other types of quantum information processing. To accomplish this we present a generalized Euler angle parametrization for SU (4) and all possible two qubit density matrices. The important group-theoretical properties of such a description are then manifest. We thus obtain the correct Haar (Hurwitz) measure and volume element for SU (4) which follows from this parametrization. In addition, we study the role of this parametrization in the Peres-Horodecki criteria for separability and its corresponding usefulness in calculating entangled two qubit states as represented through the parametrization. *

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We propose a polynomial-time algorithm for simulation of the class of pairing Hamiltonians, e.g., the BCS Hamiltonian, on an NMR quantum computer. The algorithm adiabatically finds the lowlying spectrum in the vicinity of the gap between ground and first excited states, and provides a test of the applicability of the BCS Hamiltonian to mesoscopic superconducting systems, such as ultra-small metallic grains.PACS numbers: 03.67. Lx,74.20.Fg The potential of quantum computers (QCs) to provide exponential speed-up in the simulation of quantum physics problems was originally conjectured by Feynman [1], confirmed by Lloyd [2], and later studied theoretically by a number of authors, e.g., [3,4,5,6,7]. NMR-QC experiments performing quantum physics simulations were reported in [8]. Current QC technology is limited to fewer than 10 qubits and the testing of simple algorithms [9]. QCs of the next generation, with 10-100 qubits, have the potential to solve hard problems in quantum many-body theory. We show here how this observation can be applied to the problem of simulating the class of pairing Hamiltonians with general, i.e., arbitrary long-range interactions. The pairing Hamiltonians are of wide interest in condensed matter and nuclear physics [10]. An important example of a pairing Hamiltonian is the BCS model of low-T c superconductivity. We provide an algorithm for testing the validity of the general BCS Hamiltonians of finite particle-number systems, pertinent to nuclear systems and mesoscopic condensed-phase systems, such as ultra-small metallic grains [11,12,13,14]. These grains provide a fertile testing ground for the BCS ansatz for the ground state wave function. The BCS wave function is a superposition of different Fermion numbers and is expected to be exact in the thermodynamic limit [15]. In contrast, in ultra-small metallic grains the number of states N within the Debye frequency cutoff from the Fermi energy is only ∼ 100. A similar estimate holds for the number of states within a few major shells for medium or heavy nuclei. In systems with finite particle number the BCS ansatz is doubtful, and at the same time exact numerical diagonalization of the general BCS Hamiltonian is impractical beyond a few tens of electron pairs [12]. Various approximations have been proposed [16], but it would clearly be desirable to have an exact numerical solution for the problem. In [5,6] efficient QC algorithms were presented for simulating a many-body fermionic system. While the BCS Hamiltonian describes a system of interacting fermions, it does so at the level of an effective field theory. This can be expressed in terms of an interacting spin system [15], or parafermions [17]. Therefore the fermionic simulation algorithms [5] are not directly applicable. Further, while a number of authors have recently considered simulation of one Hamiltonian in terms of another [7], the connection of these phenomenological Hamiltonians to those of manybody condensed matter and nuclear physics is not a priori clear. Here we clarify the co...

Decoherence-induced leakage errors can couple a physical or encoded qubit to other levels, thus potentially damaging the qubit. They can therefore be very detrimental in quantum computation and require special attention. Here we present a general method for removing such errors by using simple decoupling and recoupling pulse sequences. The proposed gates are experimentally accessible in a variety of promising quantum computing proposals.PACS numbers: 03.67.Lx,03.65.Yz The unit of quantum information is the qubit: an idealized two-level system consisting of a pair of orthonormal quantum states. However, this idealization neglects other levels which are typically present and can mix with those defining the qubit. Such mixing, the prevention of which is the subject of this work, is known as "leakage". Leakage may be the result of the application of logical operations, or induced by system-bath coupling. In the former case, a rather general solution was proposed in [1]. Here we are interested in decoherence-induced leakage. E.g., in the ion-trap QC proposal the two-level approximation may break down and spontaneous transitions may leak population out of those levels that represent the qubit in the ion [2]. This is part of a more general problem: quantum computation (QC) depends on reliable components and a high degree of isolation from a noisy environment. When these conditions are satisfied, it is known that it is possible to stabilize a quantum computer using an encoding of a "logical qubit" into several physical qubits. Methods which profitably exploit such an encoding are, e.g., (closed-loop) quantum error correcting codes (QECC) [3,4] and (open-loop) decoherencefree subspaces or subsystems (DFS) [5,6,7,8]. The logical qubits of these codes can also undergo leakage errors, which are particularly serious: by mixing states from within the code and outside the code space, leakage completely invalidates the encoding. A simple procedure to detect and correct leakage, which can be incorporated into a fault-tolerant QECC circuit, was given in [3]. This scheme is, however, not necessarily compatible with all encodings [9]. Here we present a universal, open-loop solution to leakage elimination, which makes use of fast and strong "bang-bang" (BB) pulses [10,11]. We first give a general scheme for protecting qubits (whether encoded or physical) from leakage errors using an efficient pulse sequence. Then we illustrate the general result with examples taken from a variety of promising QC proposals. Particularly important is the fact that our scheme is experimentally feasible in these examples, in the sense that we only make use of the naturally available interactions.Universal leakage elimination operator.-Here we give a general, existential argument for eliminating all leakage errors on encoded or physical qubits. Suppose that n twolevel systems (e.g., electron spins in quantum dots [12]) are used to encode one logical qubit, or that a N -level Hilbert space H N supports a two-dimensional physical qubit subspace (e.g., h...