論文

We establish a deformation framework for highly symmetric solutions to the Einstein equations. In this framework, four-dimensional metrics are constructed from three-dimensional eta-Einstein metrics admitting a deformation locally determined by a single function. Under this deformation, the resulting spacetime solves the Einstein equations with a string-cloud source. Within this framework, a wide range of symmetric spacetimes can be treated in a unified manner. These include FLRW, Kantowski-Sachs, and LRS Bianchi cosmological models [including Taub-NUT-(A)dS solutions], as well as Reissner-Nordstrom-(A)dS black holes admitting spherical, planar, or hyperbolic symmetry. In the cosmological setting, the deformation leaves the evolution equations for the scale factors unchanged, and hence the expansion history coincides with that of the corresponding undeformed models. For the deformed Reissner-Nordstrom-(A)dS black holes, the structure of Killing horizons is insensitive to the deformation.

The equation of motion of an extended object in spacetime reduces to an ordinary differential equation in the presence of symmetry. By properly defining the symmetry with the notion of cohomogeneity, we discuss the method for classifying all these extended objects. We carry out the classification for the strings in the five-dimensional anti-de Sitter space by the effective use of the local isomorphism between SO(4,2) and SU(2,2). In the case where the string is described by the Nambu-Goto action, we present a general method for solving the trajectory. We then apply the method to one of the classification cases, where the spacetime naturally obtains a Hopf-like bundle structure, and find a solution. The geometry of the solution is analyzed and found to be a timelike, helicoidlike surface.

We study the integrability of the equations of motion for the Nambu-Goto strings with a cohomogeneity-one symmetry in Minkowski spacetime. A cohomogeneity-one string has a world surface which is tangent to a Killing vector field. By virtue of the Killing vector, the equations of motion reduce to the geodesic equation in the orbit space. Cohomogeneity-one strings are classified into seven classes (types I to VII). We investigate the integrability of the geodesic equations for all the classes and find that the geodesic equations are integrable. For types I to VI, the integrability comes from the existence of Killing vectors on the orbit space which are the projections of Killing vectors on Minkowski spacetime. For type VII, the integrability is related to a projected Killing vector and a nontrivial Killing tensor on the orbit space. We also find that the geodesic equations of all types are exactly solvable, and show the solutions. © 2010 IOP Publishing Ltd.

A gauge-invariant quantum theory of the Friedmann-Robertson-Walker (FRW) universe with dust is studied in terms of the Ashtekar variables. We use the reduced phase space quantization which has following advantages: (i) fundamental variables are all gauge invariant, (ii) there exists a physical time evolution of gauge-invariant quantities, so that the problem of time is absent, and (iii) the reduced phase space can be quantized in the same manner as in ordinary quantum mechanics. In the FRW model, the dynamical components of the Ashtekar variables are given by a single quantity p and its conjugate momentum, where p is related to the scale factor a as a alpha root vertical bar p vertical bar and its sign gives the orientation of triads. We solve a scattering problem in terms of ingoing and outgoing energy eigenstates. We show that the incident wave is reflected in rate 1/4 and transmitted in rate 3/4 at the classical singularity p = 0. Analyzing the dynamics of a wave packet, we show that the classical initial singularity is replaced by a big bounce in quantum theory. A possible interpretation of the result is that the wave function of the universe has been in a superposition of states representing right-handed and left-handed systems before the big bounce.

We discuss constants of motion of a particle under an external field in a curved spacetime, taking into account the Hamiltonian constraint, which arises from the reparametrization invariance of the particle orbit. As the necessary and sufficient condition for the existence of a constant of motion, we obtain a set of equations with a hierarchical structure, which is understood as a generalization of the Killing tensor equation. It is also a generalization of the conventional argument in that it includes the case when the conservation condition holds only on the constraint surface in the phase space. In that case, it is shown that the constant of motion is associated with a conformal Killing tensor. We apply the hierarchical equations and find constants of motion in the case of a charged particle in an electromagnetic field in black hole spacetimes. We also demonstrate that gravitational and electromagnetic fields exist in which a charged particle has a constant of motion associated with a conformal Killing tensor.

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 a symmetric 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 CNOT(1, 3) between the indirectly coupled qubits 1 and 3 is T = root 3/2J(-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 Hilbert space of qubit 3 shows that the time-optimal synthesis of the CNOT(+/-)(1, 3) (which acts as the identity when the control qubit 1 is in the state vertical bar 0 >, while if the control qubit is in the state vertical bar 1 >, the target qubit 3 is flipped as vertical bar +/-> -> vertical bar -/+>) also requires the same time T.

The crease set of an event horizon or a Cauchy horizon is an important object which determines the qualitative properties of the horizon. In particular, it determines the possible topologies of the spatial sections of the horizon. By Fermat's principle in geometric optics, we relate the crease set and the Maxwell set of a smooth function in the context of singularity theory. We thereby give a classification of generic topological structures of the Maxwell sets and the generic topologies of the spatial section of the horizon.

We analyze the amplification by the Aharonov-Albert-Vaidman weak quantum measurement on a Sagnac interferometer [Dixon et al., Phys. Rev. Lett. 102, 173601 (2009)] up to all orders of the coupling strength between the measured system and the measuring device. The amplifier transforms a small tilt of a mirror into a large transverse displacement of the laser beam. The conventional analysis has shown that the measured value is proportional to the weak value, so that the amplification can be made arbitrarily large in the cost of decreasing output laser intensity. It is shown that the measured displacement and the amplification factor are in fact not proportional to the weak value and rather vanish in the limit of infinitesimal output intensity. We derive the optimal overlap of the pre- and postselected states with which the amplification become maximum. We also show that the nonlinear effects begin to arise in the performed experiments so that any improvements in the experiment, typically with an amplification greater than 100, should require the nonlinear theory in translating the observed value to the original displacement.

We provide exact analytical solutions for the problem of time-optimal transfer of coherence from one spin-polarization to a threefold coherence in a trilinear Ising chain with a fixed energy available and subject to local controls with a non-negligible time cost. The time of transfer is optimal and consistent with a previous numerical result obtained assuming instantaneous local controls.

We analytically determine the minimal time and the optimal control laws required for the realization, up to an assigned fidelity and with a fixed energy available, of entangling quantum gates (CNOT) between indirectly coupled qubits of a trilinear Ising chain. The control is coherent and open loop, and it is represented by a local and continuous magnetic field acting on the intermediate qubit. The time cost of this local quantum operation is not restricted to be zero. When the matching with the target gate is perfect (fidelity equal to 1), we provide exact solutions for the case of equal Ising coupling. For the more general case when some error is tolerated (fidelity smaller than 1), we give perturbative solutions for unequal couplings. Comparison with previous numerical solutions for the minimal time to generate the same gates with the same Ising Hamiltonian but with instantaneous local controls shows that the latter are not time optimal. © 2013 IOP Publishing Ltd.

We study the dynamics of the Nambu-Goto membranes with cohomogeneity one symmetry, i.e., the membranes whose trajectories are foliated by homogeneous surfaces. It is shown that the equation of motion reduces to a geodesic equation on a certain manifold, which is constructed from the original spacetime and Killing vector fields thereon. A general method is presented for classifying the symmetry of cohomogeneity one membranes in a given spacetime. The classification is completely carried out in Minkowski spacetime. We analyze one of the obtained classes in depth and derive an exact solution.

The condition for distinguishability of a countably infinite number of pure states by a single measurement is given. Distinguishability is to be understood as the possibility of an unambiguous measurement. For a finite number of states, it is known that the necessary and sufficient condition of distinguishability is that the states are linearly independent. For an infinite number of states, several natural classes of distinguishability can be defined. We give a necessary and sufficient condition for a system of pure states to be distinguishable. It turns out that each level of distinguishability naturally corresponds to one of the generalizations of linear independence to families of infinite vectors. As an important example, we apply the general theory to von Neumann's lattice, a subsystem of coherent states which corresponds to a lattice in the classical phase space. We prove that the condition for distinguishability is that the area of the fundamental region of the lattice is greater than the Planck constant, and also find subtle behavior on the threshold. These facts reveal the measurement theoretical meaning of the Planck constant and give a justification for the interpretation that it is the smallest unit of area in the phase space. The cases of uncountably many states and of mixed states are also discussed.

We analytically investigate the role of entanglement in time-optimal state evolution as an application 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 coupled 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. Entanglement plays a role for W and Greenberger-Horne-Zeilinger (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.

We discuss the possibility of protecting the state of a quantum system that goes through noise by measurements and operations before and after the noise process. The aim is to seek the optimal protocol that makes the input and output states as close as possible and to clarify the role of the measurements therein. We consider two cases: one can perform quantum measurements and operations (i) only after the noise process and (ii) both before and after. We prove in a two-dimensional Hilbert space that, in case (i), the noise suppression is essentially impossible for all types of noise and, in case (ii), the optimal protocol for the depolarizing noise is either the "do nothing" protocol or the "discriminate and reprepare" protocol. These protocols are not "truly quantum" and can be considered as classical. They involve no measurement or only use the measurement outcomes. These results describe the fundamental limitations in quantum mechanics from the viewpoint of control theory. Finally, we conjecture that a statement similar to case (ii) holds for higher-dimensional Hilbert spaces and present some numerical evidence.

We discuss protection of a quantum state that goes through a noise process by measurements and operations before and after the noise process. In our previous work, we showed the nonexistence of "truly quantum" protocols that protect an unknown qubit state against depolarizing noise better than "classical" ones. Toward identifying the class of noise processes that is optimally suppressed by such a "classical" protocol, we extend our previous result in two directions. First, we show that the statement is also true in any finite-dimensional Hilbert spaces, which was previously conjectured; the optimal protocol is either the do nothing protocol or the discriminate and reprepare protocol, depending on the strength of the noise. Second, in the case of a single qubit, we show that essentially the same conclusion holds for any unital noise. Thus, the noise must be nonunital for a control protocol beyond "classical" ones to exist.

© 2018 authors. Published by the American Physical Society. Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. We discuss the disturbance by measurements which unambiguously discriminate between given candidate states. We prove that such an optimal measurement necessarily changes distinguishable states indistinguishable when the inconclusive outcome is obtained. The result was previously shown by Chefles [Phys. Lett. A 239, 339 (1998)PYLAAG0375-960110.1016/S0375-9601(98)00064-4] under restrictions on the class of quantum measurements and on the definition of optimality. Our theorems remove these restrictions and are also applicable to infinitely many candidate states. Combining with our previous results, one can obtain concrete mathematical conditions for the resulting states. The method may have a wide variety of applications in contexts other than state discrimination.

We study the classical dynamics of the Nambu-Goto strings with a null symmetry in curved spacetimes admitting a null Killing vector field. The Nambu-Goto equation is reduced to first-order ordinary differential equations and is always integrable in contrast to the case of non-null symmetries where integrability requires additional spacetime symmetries. It is found that in the case of null symmetry, an almost contact structure associated with the metric dual 1-form η of the null Killing vector field emerges naturally. This structure determines the allowed class of string worldsheets in such a way that the tangent vector fields of the worldsheet lie in ker dη. In the special case that the almost contact structure becomes a contact structure, its Reeb vector field completely characterizes the worldsheet. We apply our formulation to the strings in the pp-waves, the Einstein static universe and the Gödel universe. We also study their worldsheet geometry in detail.