Quantum Brachistochrone for Mixed States

We present a general formalism based on the variational principle for finding the time-optimal quantum evolution of mixed states governed by a master equation, when the Hamiltonian and the Lindblad operators are subject to certain constraints. The problem reduces to solving first a fundamental equation (the {\it quantum brachistochrone}) for the Hamiltonian, which can be written down once the constraints are specified, and then solving the constraints and the master equation for the Lindblad and the density operators. As an application of our formalism, we study a simple one-qubit model where the optimal Lindblad operators control decoherence and can be simulated by a tunable coupling with an ancillary qubit. It is found that the evolution through mixed states can be more efficient than the unitary evolution between given pure states. We also discuss the mixed state evolution as a finite time unitary evolution of the system plus an environment followed by a single measurement. For the simplest choice of the constraints, the optimal duration time for the evolution is an exponentially decreasing function of the environment's degrees of freedom.Comment: 8 pages, 3 figure

Brachistochrone of Entanglement for Spin Chains

We analytically investigate the role of entanglement in time-optimal state evolution as an appli- cation of the quantum brachistochrone, a general method for obtaining the optimal time-dependent Hamiltonian for reaching a target quantum state. As a model, we treat two qubits indirectly cou- pled through an intermediate qubit that is directly controllable, which represents a typical situation in quantum information processing. We find the time-optimal unitary evolution law and quantify residual entanglement by the two-tangle between the indirectly coupled qubits, for all possible sets of initial pure quantum states of a tripartite system. The integrals of the motion of the brachistochrone are determined by fixing the minimal time at which the residual entanglement is maximized. Entan- glement plays a role for W and GHZ initial quantum states, and for the bi-separable initial state in which the indirectly coupled qubits have a nonzero value of the 2-tangle.Comment: 9 pages, 4 figure

Square Root Actions, Metric Signature, and the Path-Integral of Quantum Gravity

We consider quantization of the Baierlein-Sharp-Wheeler form of the gravitational action, in which the lapse function is determined from the Hamiltonian constraint. This action has a square root form, analogous to the actions of the relativistic particle and Nambu string. We argue that path-integral quantization of the gravitational action should be based on a path integrand $\exp[ \sqrt{i} S ]$ rather than the familiar Feynman expression $\exp[ i S ]$, and that unitarity requires integration over manifolds of both Euclidean and Lorentzian signature. We discuss the relation of this path integral to our previous considerations regarding the problem of time, and extend our approach to include fermions.Comment: 32 pages, latex. The revision is a more general treatment of the regulator. Local constraints are now derived from a requirement of regulator independenc

A fully-discrete scheme for systems of nonlinear Fokker-Planck-Kolmogorov equations

We consider a system of Fokker-Planck-Kolmogorov (FPK) equations, where the dependence of the coefficients is nonlinear and nonlocal in time with respect to the unknowns. We extend the numerical scheme proposed and studied recently by the authors for a single FPK equation of this type. We analyse the convergence of the scheme and we study its applicability in two examples. The first one concerns a population model involving two interacting species and the second one concerns two populations Mean Field Games

Time-optimal CNOT between indirectly coupled qubits in a linear Ising chain

We give analytical solutions for the time-optimal synthesis of entangling gates between indirectly coupled qubits 1 and 3 in a linear spin chain of three qubits subject to an Ising Hamiltonian interaction with equal coupling $J$ plus a local magnetic field acting on the intermediate qubit. The energy available is fixed, but we relax the standard assumption of instantaneous unitary operations acting on single qubits. The time required for performing an entangling gate which is equivalent, modulo local unitary operations, to the $\mathrm{CNOT}(1, 3)$ between the indirectly coupled qubits 1 and 3 is $T=\sqrt{3/2} J^{-1}$, i.e. faster than a previous estimate based on a similar Hamiltonian and the assumption of local unitaries with zero time cost. Furthermore, performing a simple Walsh-Hadamard rotation in the Hlibert space of qubit 3 shows that the time-optimal synthesis of the $\mathrm{CNOT}^{\pm}(1, 3)$ (which acts as the identity when the control qubit 1 is in the state $\ket{0}$, while if the control qubit is in the state $\ket{1}$ the target qubit 3 is flipped as $\ket{\pm}\rightarrow \ket{\mp}$) also requires the same time $T$.Comment: 9 pages; minor modification

Why is Spacetime Lorentzian?

We expand on the idea that spacetime signature should be treated as a dynamical degree of freedom in quantum field theory. It has been argued that the probability distribution for signature, induced by massless free fields, is peaked at the Lorentzian value uniquely in D=4 dimensions. This argument is reviewed, and certain consistency constraints on the generalized signature (i.e. the tangent space metric \eta_{ab}(x)=\mbox{diag}[e^{i\theta(x)},1,1,1]) are derived. It is shown that only one dynamical "Wick angle" $\theta(x)$ can be introduced in the generalized signature, and the magnitude of fluctuations away from Lorentzian signature $\delta \theta = \pi - \theta$ is estimated to be of order $(l_P/R)^3$, where $l_P$ is the Planck length, and $R$ is the length scale of the Universe. For massless fields, the case of D=2 dimensions and the case of supersymmetry are degenerate, in the sense that no signature is preferred. Mass effects lift this degeneracy, and we show that a dynamical origin of Lorentzian signature is also possible for (broken) supersymmetry theories in D=6 dimensions, in addition to the more general non-supersymmetric case in D=4 dimensions.Comment: 26 pages, plain LaTeX, NBI-HE-93-3

Structural properties and thermoelectric performance of the double-filled skutterudite (Sm,Gd)y(FexNi1-x)4Sb12

The structural and thermoelectric properties of the filled skutterudite (Sm,Gd)y(FexNi1-x)4 Sb12 were investigated and critically compared to the ones in the Sm-containing system with the aim of unravelling the effect of double filling on filling fraction and thermal conductivity. Several samples (x = 0.50-0.90 and y = 0.15-0.48) were prepared by melting-sintering, and two of them were densified by spark plasma sintering in order to study their thermoelectric features. The crystallographic study enables the recognition of the role of the filler size in ruling the filling fraction and the compositional location of the p/n crossover: It has been found that the former lowers and the latter moves toward lower x values with the reduction of the filler ionic size, as a consequence of the progressively weaker interaction of the filler with the Sb12 cavity. The analysis of thermoelectric properties indicates that, despite the Sm3+/Gd3+ small mass difference, the contemporary presence of these ions in the 2a site significantly affects the thermal conductivity of both p- and n-compositions. This occurs by reducing its value with respect to the Sm-filled compound at each temperature considered, and making the overall thermoelectric performance of the system comparable to several multi-filled (Fe, Ni)-based skutterudites described in the literature

Dynamical Determination of the Metric Signature in Spacetime of Nontrivial Topology

The formalism of Greensite for treating the spacetime signature as a dynamical degree of freedom induced by quantum fields is considered for spacetimes with nontrivial topology of the kind ${\bf R}^{D-1} \times {\bf T}^1$, for varying $D$. It is shown that a dynamical origin for the Lorentzian signature is possible in the five-dimensional space ${\bf R}^4 \times {\bf T}^1$ with small torus radius (periodic boundary conditions), as well as in four-dimensional space with trivial topology. Hence, the possibility exists that the early universe might have been of the Kaluza-Klein type, \ie multidimensional and of Lorentzian signature.Comment: 10 pages, LaTeX file, 4 figure

On the Non-renormalization of the AdS Radius

We show that the relation between the 't Hooft coupling and the radius of AdS is not renormalized at one-loop in the sigma model perturbation theory. We prove this by computing the quantum effective action for the superstring on AdS_5 x S^5 and showing that it does not receive any finite alpha' corrections. We also show that the central charge of the interacting worldsheet conformal field theory vanishes at one-loop.Comment: 13 pages, harvmac. v2: refs added, version to be published on JHE