Akio Hosoya

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.

Strange weak values

We develop a formal theory of the weak values with emphasis on the consistency conditions and a probabilistic interpretation in the counter-factual processes. We present the condition for the choice of the post-selected state to give a negative weak value of a given projection operator and strange values of an observable in general. The general framework is applied to Hardys paradox and the spin-1/2 system to explicitly address the issues of counter-factuality and strange weak values. The counter-factual arguments which characterize the paradox specify the pre-selected state, and a complete set of the post-selected states clarifies how the strange weak values emerge.

Weak values with decoherence

The weak value of an observable is experimentally accessible by weak measurements as theoretically analyzed by Aharonov et al and recently experimentally demonstrated. We introduce a W operator associated with the weak values and give a general framework of quantum operations to the W operator in parallel with the Kraus representation of the completely positive map for the density operator. The decoherence effect is also investigated in terms of the weak measurement by a shift of a probe wavefunction of continuous variable. As an application, we demonstrate how the geometric phase is affected by the bit flip noise.

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.

Optimal probe wave function of weak-value amplification

The weak measurement proposed by Aharonov and his colleagues extracts information of a physical quantity of the system by the post selection as the shifts of the argument of the probe wavefunction. The shift is called the weak value and is larger for the post-selected state more orthogonal to the initial state. Recently, the signal amplification by the weak measurement has been extensively studied. In the present work, we explicitly obtain the optimal probe wavefunction and the amplification factor for a given weak value, which is calculated from the experimental setup. It is shown that the amplification factor has no upper bound in contrast to the Gaussian probe wavefunction and that the amplified signal is sharp.

Relative information entropy and Weyl curvature of the inhomogeneous Universe

Penrose conjectured a connection between entropy and Weyl curvature of the Universe. This is plausible, as the almost homogeneous and isotropic Universe at the onset of structure formation has negligible Weyl curvature, which then grows (relative to the Ricci curvature) due to the formation of large-scale structure, and thus reminds us of the second law of thermodynamics. We study two scalar measures to quantify the deviations from a homogeneous and isotropic space-time: the relative information entropy and a Weyl tensor invariant, and show their relation to the averaging problem. We calculate these two quantities up to second order in standard cosmological perturbation theory and find that they are correlated and can be linked via the kinematical backreaction of a spatially averaged universe model.

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.

Quantum computers and unstructured search: finding and counting items with an arbitrarily entangled initial state

Abstract Grovers quantum algorithm for an unstructured search problem and the COUNT algorithm by Brassard et al. are generalized to the case when the initial state is arbitrarily and maximally entangled. This ansatz might be relevant with quantum subroutines, when the computational qubits and the environment are coupled, and in general when the control over the quantum system is partial.

Gravitational Collapse in Painlevé-Gullstrand Coordinates

We construct an exact solution for the spherical gravitational collapse in asingle coordinate patch. To describe the dynamics of collapse, we use ageneralized form of the Painleve-Gullstrand coordinates in the Schwarzschildspacetime. The time coordinate of the form is the proper time of a free-fallingobserver so that we can describe the collapsing star not only outside but alsoinside the event horizon in a single coordinate patch. We show the bothsolutions corresponding to the gravitational collapse from infinity and from afinite radius.