A. R. P. Rau

Quantum discord for qubit–qudit systems

We present two formulae to calculate quantum discord, a kind of quantum correlation, between a qubit and a second party of arbitrary dimension d. The first formula is the original entropic definition and the second is a recently proposed geometric distance measure which leads to an analytical formulation. The tracing over the qubit in the entropic calculation is reduced to a very simple prescription. And, when the d-dimensional system is a so-calledX-state, the density matrix having non-zero elements only along the diagonal and anti-diagonal, the entropic calculation can also be carried out analytically. Such states of the full bipartite qubit–qudit system may be named ‘extended X-states’, whose density matrix is built of four block matrices, each visually appearing as an X. The optimization involved in the entropic calculation is generally over two parameters, reducing to one for many cases, and avoided altogether for an overwhelmingly large set of density matrices as our numerical investigations demonstrate. In the case of N = 2, extended X-states encompass the entire 15-dimensional parameter space, that is, they represent the full qubit–qubit system.Quantum discord, a kind of quantum correlation, is defined as the difference between quantum mutual information and classical correlation in a bipartite system. It has been discussed so far for small systems with only a few independent parameters. We extend here to a much broader class of states when the second party is of arbitrary dimension d, so long as the first, measured, party is a qubit. We present two formulæ to calculate quantum discord, the first relating to the original entropic definition and the second to a recently proposed geometric distance measure which leads to an analytical formulation. The tracing over the qubit in the entropic calculation is reduced to a very simple prescription. And, when the d-dimensional system is a so-called X state, the density matrix having non-zero elements only along the diagonal and anti-diagonal so as to appear visually like the letter X, the entropic calculation can be carried out analytically. Such states of the full bipartite qubit-qudit system may be named “extended X states”, whose density matrix is built of four block matrices, each visually appearing as an X. The optimization involved in the entropic calculation is generally over two parameters, reducing to one for many cases, and avoided altogether for an overwhelmingly large set of density matrices as our numerical investigations demonstrate. Our results also apply to states of a N-qubit system, where “extended X states” consist of (2 − 1) states, larger in number than the (2 − 1) of X states of N qubits. While these are still smaller than the total number (2 − 1) of states of N qubits, the number of parameters involved is nevertheless large. In the case of N = 2, they encompass the entire 15-dimensional parameter space, that is, the extended X states for N = 2 represent the full qubit-qubit system.

Useful alternative to the multipole expansion of 1 / r potentials

Few-body problems involving Coulomb or gravitational interactions between pairs of particles, whether in classical or quantum physics, are generally handled through a standard multipole expansion of the two-body potentials. We discuss an alternative based on a compact, cylindrical Green’s function expansion that should have wide applicability throughout physics. Two-electron “direct” and “exchange” integrals in many-electron quantum systems are evaluated to illustrate the procedure which is more compact than the standard one using Wigner coefficients and Slater integrals.

Threshold energy and angular distributions in multiple ionization

The Wannier theory for double escape near threshold (1965) makes detailed predictions for the energy and angular distributions of the outgoing electrons. Not only the dependence on the excess energy above threshold but even the numerical coefficient in front that governs these distributions can be derived as a function of the charge Z on the residual core. These are of interest with the advent of recent accurate experiments and computer calculations. Such coefficients are presented and it is pointed out that there is a dramatic disappearance of some correlations for Z>or=3. Recent computer calculations for Z=2 can, perhaps, be extended to verify this result. Similar non-monotonic dependences on Z are to be expected for triple and other multiple escape.

Geometric phase for SU(N) through fiber bundles and unitary integration

Geometric phases are important in quantum physics and now central to fault tolerant quantum computation. For spin-1/2 and SU(2), the Bloch sphere

Variational principles, variational identities, and supervariational principles for wavefunctions

S^2

Four-level and two-qubit systems, subalgebras, and unitary integration

, together with a U(1) phase, provides a complete SU(2) description. We generalize to

The quantum defect: Early history and recent developments

N

Time-dependent treatment of a general three-level system

-level systems and SU(

GeneralizedXstates ofNqubits and their symmetries

N

Effect of an electric field on autoionising states of H

) in terms of a