Antisymmetric Subspace Research Articles

Uncertainty, Monogamy, and Locking of Quantum Correlations

Squashed entanglement and entanglement of purification are quantum-mechanical correlation measures and are defined as certain minimizations of entropic quantities. In this paper, we present the first nontrivial calculations of both quantities. Our results lead to the conclusion that both measures can drop by an arbitrary amount when only a single qubit of a local system is lost. This property is known as locking and has previously been observed for other correlation measures such as accessible information, entanglement cost, and logarithmic negativity. In the case of squashed entanglement, the results are obtained using an inequality that can be understood as a quantum channel analogue of well-known entropic uncertainty relations. This inequality may prove a useful tool in quantum information theory. The regularized entanglement of purification is known to equal the entanglement needed to prepare many copies of a quantum state by local operations and a sublinear amount of communication. Here, monogamy of quantum entanglement (i.e., the impossibility of a system being maximally entangled with two others at the same time) leads to an exact calculation for all quantum states that are supported either on the symmetric or on the antisymmetric subspace of a d/spl times/d-dimensional system.

Atomic spectral methods for molecular electronic structure calculations

Theoretical methods are reported for ab initio calculations of the adiabatic (Born-Oppenheimer) electronic wave functions and potential energy surfaces of molecules and other atomic aggregates. An outer product of complete sets of atomic eigenstates familiar from perturbation-theoretical treatments of long-range interactions is employed as a representational basis without prior enforcement of aggregate wave function antisymmetry. The nature and attributes of this atomic spectral-product basis are indicated, completeness proofs for representation of antisymmetric states provided, convergence of Schrodinger eigenstates in the basis established, and strategies for computational implemention of the theory described. A diabaticlike Hamiltonian matrix representative is obtained, which is additive in atomic-energy and pairwise-atomic interaction-energy matrices, providing a basis for molecular calculations in terms of the (Coulombic) interactions of the atomic constituents. The spectral-product basis is shown to contain the totally antisymmetric irreducible representation of the symmetric group of aggregate electron coordinate permutations once and only once, but to also span other (non-Pauli) symmetric group representations known to contain unphysical discrete states and associated continua in which the physically significant Schrodinger eigenstates are generally embedded. These unphysical representations are avoided by isolating the physical block of the Hamiltonian matrix with a unitary transformation obtained from the metric matrix of the explicitly antisymmetrized spectral-product basis. A formal proof of convergence is given in the limit of spectral closure to wave functions and energy surfaces obtained employing conventional prior antisymmetrization, but determined without repeated calculations of Hamiltonian matrix elements as integrals over explicitly antisymmetric aggregate basis states. Computational implementations of the theory employ efficient recursive methods which avoid explicit construction the metric matrix and do not require storage of the full Hamiltonian matrix to isolate the antisymmetric subspace of the spectral-product representation. Calculations of the lowest-lying singlet and triplet electronic states of the covalent electron pair bond (H(2)) illustrate the various theorems devised and demonstrate the degree of convergence achieved to values obtained employing conventional prior antisymmetrization. Concluding remarks place the atomic spectral-product development in the context of currently employed approaches for ab initio construction of adiabatic electronic eigenfunctions and potential energy surfaces, provide comparisons with earlier related approaches, and indicate prospects for more general applications of the method.

Simple optical measurement of the overlap and fidelity of quantum states

We present the experimental results of a direct measurement of the overlap of both pure and mixed polarization states of photons. The fidelity and purity of mixed states were also measured. The experimental apparatus exploits the fact that a beam splitter can distinguish the singlet Bell state from the other Bell states, i.e., it realizes projections into the symmetric and antisymmetric subspaces of photons' Hilbert space.

Localization of chaos in the discrete nonlinear Schrödinger equation.

We partition the perturbation phase space in the three-element discrete nonlinear Schr\odinger equation into symmetric and antisymmetric subspaces. We then show that chaotic motion in the neighborhood of symmetric trajectories is confined to the antisymmetric space. Chaos occurs in the system at arbitrarily low levels of nonlinearity, in agreement with previous calculations. We call this phenomenon ``microchaos.''

On the Existence of Solutions to Time-Dependent Hartree-Fock Equations

where Xj=(x], xj, xj)ei2, Aj= ^ ( d / 3 x j ) 2 and Q(x V(x) are real functions such that V(x) = V(—x). If the system obeys the Fermi statistics, it is natural to treat (1.1) in the anti-symmetric subspace of L(R). Taking note of this anti-symmetry and using the variational principle, Dirac ([3], [4]) has derived the following time-dependent version of Hartree-Fock equation in order to obtain an approximate solution of (1.1) :