Given an initial quantum state |ψI and a final quantum state |ψF in a Hilbert space, there exist Hamiltonians H under which |ψI evolves into |ψF . Consider the following quantum brachistochrone problem: Subject to the constraint that the difference between the largest and smallest eigenvalues of H is held fixed, which H achieves this transformation in the least time τ ? For Hermitian Hamiltonians τ has a nonzero lower bound. However, among non-Hermitian PT -symmetric Hamiltonians satisfying the same energy constraint, τ can be made arbitrarily small without violating the time-energy uncertainty principle. This is because for such Hamiltonians the path from |ψI to |ψF can be made short. The mechanism described here is similar to that in general relativity in which the distance between two space-time points can be made small if they are connected by a wormhole. This result may have applications in quantum computing.PACS numbers: 11.30. Er, 03.65.Ca, 03.65.Xp Suppose that one wishes to transform unitarily a state |ψ I in a Hilbert space to a different state |ψ F by means of a Hamiltonian H. In Hermitian quantum mechanics, such a transformation requires a nonzero amount of time, provided that the difference between the largest and the smallest eigenvalues of H is held fixed. However, if we extend quantum mechanics into the complex domain while keeping the energy eigenvalues real, then under the same energy constraint it is possible to achieve such a transformation in an arbitrarily short time. In this paper we demonstrate this by means of simple examples.The paper is organized as follows: We first review why in Hermitian quantum mechanics there is an unavoidable lower bound τ on the time required to transform one state into another. In particular, we consider the minimum time required to flip unitarily a spin-up state into a spindown state. We then summarize briefly how Hermitian quantum mechanics can be extended into the complex domain while retaining the reality of the energy eigenvalues, the unitarity of time evolution, and the probabilistic interpretation. In this complex framework we show how a spin-up state can be transformed arbitrarily quickly to a spin-down state by a simple non-Hermitian Hamiltonian. Then we discuss the transformation between pairs of states by more general complex non-Hermitian Hamiltonians. We make some comments regarding possible experimental consequences of these ideas.In Hermitian quantum mechanics how does one achieve the transformation |ψ I → |ψ F = e −iHt/ |ψ I in the shortest time t = τ ? Since τ is the minimum of all possible evolution times t, the Hamiltonian associated with τ is the "quantum brachistochrone" [1]. Finding the optimal evolution time requires only the solution to a much simpler problem, namely, finding the optimal evolution time for the 2 × 2 matrix Hamiltonians acting in the twodimensional subspace spanned by |ψ I and |ψ F [2].To solve the Hermitian version of the two-dimensional quantum brachistochrone problem one can choose the basis so that the initial and ...

A cyclic thermodynamic heat engine runs most efficiently if it is reversible. Carnot constructed such a reversible heat engine by combining adiabatic and isothermal processes for a system containing an ideal gas. Here, we present an example of a cyclic engine based on a single quantum-mechanical particle confined to a potential well. The efficiency of this engine is shown to equal the Carnot efficiency because quantum dynamics is reversible. The quantum heat engine has a cycle consisting of adiabatic and isothermal quantum processes that are close analogues of the corresponding classical processes.

t is possible to extract work from a quantum-mechanical system whose dynamics is governed by a time-dependent cyclic Hamiltonian. An energy bath is required to operate such a quantum engine in place of the heat bath used to run a conventional classical thermodynamic heat engine. The effect of the energy bath is to maintain the expectation value of the system Hamiltonian during an isoenergetic process. It is shown that the existence of such a bath leads to equilibrium quantum states that maximise the von Neumann entropy. Quantum analogues of certain thermodynamic relations are obtained that allow one to define the temperature of the energy bath.

Given a sequence of numbers {an}, it is always possible to find a set of Feynman rules that reproduce that sequence. For the special case of the partitions of the integers, the appropriate Feynman rules give rise to graphs that represent the partitions in a clear pictorial fashion. These Feynman rules can be used to generate the Bell numbers B(n) and the Stirling numbers S(n,k) that are associated with the partitions of the integers.

For a given ensemble of $N$ independent and identically prepared particles, we calculate the binary decision costs of different strategies for measurement of polarised spin 1/2 particles. The result proves that, for any given values of the prior probabilities and any number of constituent particles, the cost for a combined measurement is always less than or equal to that for any combination of separate measurements upon sub-ensembles. The Bayes cost, which is that associated with the optimal strategy (i.e., a combined measurement) is obtained in a simple closed form.Comment: 11 pages, uses RevTe