Tatsuhiko Koike

Time-optimal quantum evolution

We present a general framework for finding the time-optimal evolution and the optimal Hamiltonian for a quantum system with a given set of initial and final states. Our formulation is based on the variational principle and is analogous to that for the brachistochrone in classical mechanics. We reduce the problem to a formal equation for the Hamiltonian which depends on certain constraint functions specifying the range of available Hamiltonians. For some simple examples of the constraints, we explicitly find the optimal solutions.

Time Optimal Unitary Operations

Department of Physics, Keio University, Yokohama, Japan(Dated: January 15, 2007)Extending our previous work on time optimal quantum state evolution [A. Carlini, A. Hosoya, T.Koike and Y. Okudaira, Phys. Rev. Lett. 96, 060503 (2006)], we formulate a variational principle forfinding the time optimal realization of a target unitary operation, when the available Hamiltoniansare subject to certain constraints dictated either by experimental or by theoretical conditions. Sincethe time optimal unitary evolutions do not depend on the input quantum state this is of more directrelevance to quantum computation. We explicitly illustrate our method by considering the case ofa two-qubit system self-interacting via an anisotropic Heisenberg Hamiltonian and by deriving thetime optimal unitary evolution for three examples of target quantum gates, namely the swap ofqubits, the quantum Fourier transform and the entangler gate. We also briefly discuss the case inwhich certain unitary operations take negligible time.

Limits on amplification by Aharonov-Albert-Vaidman weak measurement

We analyze the amplification by the Aharonov-Albert-Vaidman weak quantum measurement on a Sagnac interferometer [P. B. 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 post-selected 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.

Time optimal quantum evolution of 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 may be reduced to solving first a fundamental equation, which can be written down once the constraints are specified, for the Hamiltonian 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 can be simulated by a tunable coupling with an ancillary qubit.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 environments degrees of freedom.

Dynamics of compact homogeneous universes

A complete description of dynamics of compact locally homogeneous universes is given, which, in particular, includes explicit calculations of Teichmuller deformations and careful counting of dynamical degrees of freedom. We regard each of the universes as a simply connected four-dimensional space–time with identifications by the action of a discrete subgroup of the isometry group. We then reduce the identifications defined by the space–time isometries to ones in a homogeneous section, and find a condition that such spatial identifications must satisfy. This is essential for explicit construction of compact homogeneous universes. Some examples are demonstrated for Bianchi II, VI0, VII0, and I universal covers.

Upper bound for entropy in asymptotically de Sitter space-time

We investigate the nature of asymptotically de Sitter space-times containing a black hole. We show that if the matter fields satisfy the dominant energy condition and cosmic censorship holds in the considered space-time, the area of the cosmological event horizon for an observer approaching a future timelike infinity does not decrease; i.e., the second law is satisfied. We also show under the same conditions that the total area of the black hole and the cosmological event horizon, a quarter of which is the total Bekenstein-Hawking entropy, is less than 12\ensuremath{\pi}/\ensuremath{\Lambda}, where \ensuremath{\Lambda} is the cosmological constant. The physical implications are also discussed.

Constants of motion for constrained Hamiltonian systems: A particle around a charged rotating black hole

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.

Hamiltonian structures for compact homogeneous universes

Hamiltonian structures for spatially compact locally homogeneous vacuum universes are investigated, provided that the set of dynamical variables contains the Teichmuller parameters, parameterizing the purely global geometry. One of the key ingredients of our arguments is a suitable mathematical expression for quotient manifolds, where the universal cover metric carries all the degrees of freedom of geometrical variations, i.e., the covering group is fixed. We discuss general problems concerned with the use of this expression in the context of general relativity, and demonstrate the reduction of the Hamiltonians for some examples. For our models, all the dynamical degrees of freedom in Hamiltonian view are unambiguously interpretable as geometrical deformations, in contrast to the conventional open models.

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 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 , 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 |0, while if the control qubit is in the state |1, the target qubit 3 is flipped as | ? ? |) also requires the same time T.

Exactly solvable strings in Minkowski spacetime

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.