Characterization of two-qubit perfect entanglers

Here we consider perfect entanglers from another perspective. It is shown that there are some {\em special} perfect entanglers which can maximally entangle a {\em full} product basis. We have explicitly constructed a one-parameter family of such entanglers together with the proper product basis that they maximally entangle. This special family of perfect entanglers contains some well-known operators such as {\textsc{cnot}} and {\textsc{dcnot}}, but {\em not} {\small{\sqrt{\rm{\textsc{swap}}}}}. In addition, it is shown that all perfect entanglers with entangling power equal to the maximal value, 2/9, are also special perfect entanglers. It is proved that the one-parameter family is the only possible set of special perfect entanglers. Also we provide an analytic way to implement any arbitrary two-qubit gate, given a proper special perfect entangler supplemented with single-qubit gates. Such these gates are shown to provide a minimum universal gate construction in that just two of them are necessary and sufficient in implementation of a generic two-qubit gate.Comment: 6 pages, 1 eps figur

The Maslov Gerbe

Let Lag(E) be the grassmannian of lagrangian subspaces of a complex symplectic vector space E. We construct a Maslov class which generates the second integral cohomology of Lag(E), and we show that its mod 2 reduction is the characteristic class of a flat gerbe with structure group Z_2. We explain the relation of this gerbe to the well-known flat Maslov line bundle with structure group Z_4 over the real lagrangian grassmannian, whose characteristic class is the mod 4 reduction of the real Maslov class.Comment: 8 page

Muon capture on light nuclei

This work investigates the muon capture reactions 2H(\mu^-,\nu_\mu)nn and 3He(\mu^-,\nu_\mu)3H and the contribution to their total capture rates arising from the axial two-body currents obtained imposing the partially-conserved-axial-current (PCAC) hypothesis. The initial and final A=2 and 3 nuclear wave functions are obtained from the Argonne v_{18} two-nucleon potential, in combination with the Urbana IX three-nucleon potential in the case of A=3. The weak current consists of vector and axial components derived in chiral effective field theory. The low-energy constant entering the vector (axial) component is determined by reproducting the isovector combination of the trinucleon magnetic moment (Gamow-Teller matrix element of tritium beta-decay). The total capture rates are 393.1(8) s^{-1} for A=2 and 1488(9) s^{-1} for A=3, where the uncertainties arise from the adopted fitting procedure.Comment: 6 pages, submitted to Few-Body Sys

Asymptotically Optimal Quantum Circuits for d-level Systems

As a qubit is a two-level quantum system whose state space is spanned by |0>, |1>, so a qudit is a d-level quantum system whose state space is spanned by |0>,...,|d-1>. Quantum computation has stimulated much recent interest in algorithms factoring unitary evolutions of an n-qubit state space into component two-particle unitary evolutions. In the absence of symmetry, Shende, Markov and Bullock use Sard's theorem to prove that at least C 4^n two-qubit unitary evolutions are required, while Vartiainen, Moettoenen, and Salomaa (VMS) use the QR matrix factorization and Gray codes in an optimal order construction involving two-particle evolutions. In this work, we note that Sard's theorem demands C d^{2n} two-qudit unitary evolutions to construct a generic (symmetry-less) n-qudit evolution. However, the VMS result applied to virtual-qubits only recovers optimal order in the case that d is a power of two. We further construct a QR decomposition for d-multi-level quantum logics, proving a sharp asymptotic of Theta(d^{2n}) two-qudit gates and thus closing the complexity question for all d-level systems (d finite.) Gray codes are not required, and the optimal Theta(d^{2n}) asymptotic also applies to gate libraries where two-qudit interactions are restricted by a choice of certain architectures.Comment: 18 pages, 5 figures (very detailed.) MatLab files for factoring qudit unitary into gates in MATLAB directory of source arxiv format. v2: minor change

Yang-Mills theory for bundle gerbes

Given a bundle gerbe with connection on an oriented Riemannian manifold of dimension at least equal to 3, we formulate and study the associated Yang-Mills equations. When the Riemannian manifold is compact and oriented, we prove the existence of instanton solutions to the equations and also determine the moduli space of instantons, thus giving a complete analysis in this case. We also discuss duality in this context.Comment: Latex2e, 7 pages, some typos corrected, to appear in J. Phys. A: Math. and Ge

BRST, anti-BRST and their geometry

We continue the comparison between the field theoretical and geometrical approaches to the gauge field theories of various types, by deriving their Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST trasformation properties and comparing them with the geometrical properties of the bundles and gerbes. In particular, we provide the geometrical interpretation of the so--called Curci-Ferrari conditions that are invoked for the absolute anticommutativity of the BRST and anti-BRST symmetry transformations in the context of non-Abelian 1-form gauge theories as well as Abelian gauge theory that incorporates a 2-form gauge field. We also carry out the explicit construction of the 3-form gauge fields and compare it with the geometry of 2--gerbes.Comment: A comment added. To appear in Jour. Phys. A: Mathemaical and Theoretica

Non-Abelian Tensor Multiplet Equations from Twistor Space

We establish a Penrose-Ward transform yielding a bijection between holomorphic principal 2-bundles over a twistor space and non-Abelian self-dual tensor fields on six-dimensional flat space-time. Extending the twistor space to supertwistor space, we derive sets of manifestly N=(1,0) and N=(2,0) supersymmetric non-Abelian constraint equations containing the tensor multiplet. We also demonstrate how this construction leads to constraint equations for non-Abelian supersymmetric self-dual strings.Comment: v3: 23 pages, revised version published in Commun. Math. Phy