Requiring that a Hamiltonian be Hermitian is overly restrictive. A consistent physical theory of quantum mechanics can be built on a complex Hamiltonian that is not Hermitian but satisfies the less restrictive and more physical condition of space-time reflection symmetry (PT symmetry). One might expect a non-Hermitian Hamiltonian to lead to a violation of unitarity. However, if PT symmetry is not spontaneously broken, it is possible to construct a previously unnoticed symmetry C of the Hamiltonian. Using C, an inner product whose associated norm is positive definite can be constructed. The procedure is general and works for any PT-symmetric Hamiltonian. Observables exhibit CPT symmetry, and the dynamics is governed by unitary time evolution. This work is not in conflict with conventional quantum mechanics but is rather a complex generalization of it.

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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 ...

The Hermiticity condition in quantum mechanics required for the characterisation of (a) physical observables and (b) generators of unitary motions can be relaxed into a wider class of operators whose eigenvalues are real and whose eigenstates are complete. In this case, the orthogonality of eigenstates is replaced by the notion of biorthogonality that defines the relation between the Hilbert space of states and its dual space. The resulting quantum theory, which might appropriately be called 'biorthogonal quantum mechanics', is developed here in some detail in the case for which the Hilbert space dimensionality is finite. Specifically, characterisations of probability assignment rules, observable properties, pure and mixed states, spin particles, measurements, combined systems and entanglements, perturbations, and dynamical aspects of the theory are developed. The paper concludes with a brief discussion on infinite-dimensional systems.

The manifold of pure quantum states can be regarded as a complex projective space endowed with the unitary-invariant Riemannian geometry of Fubini and Study. According to the principles of geometric quantum mechanics, the detailed physical characteristics of a given quantum system can be represented by specific geometrical features that are selected and preferentially identified in this complex manifold. In particular, any specific feature of projective geometry gives rise to a physically realisable characteristic in quantum mechanics. Here we construct a number of examples of such geometrical features as they arise in the state spaces for spin-1 2 , spin-1, and spin-3 2 systems, and for pairs of spin-1 2 systems. A study is undertaken on the geometry of entangled states, and a natural measure is assigned to the degree of entanglement of a given state for a general multi-particle system. The properties of this measure are analysed in detail for the entangled states of a pair of spin-1 2 particles, thus enabling us to determine the structure of the space of maximally entangled states. With the specification of a quantum Hamiltonian, the resulting Schrödinger trajectories induce an isometry of the Fubini-Study manifold. For a generic quantum evolution, the corresponding Killing trajectory is quasiergodic on a toroidal subspace of the energy surface. When the dynamical trajectory is lifted orthogonally to Hilbert space, it induces a geometric phase shift on the wave function. The uncertainty of an observable in a given state is the length of the gradient vector of the level surface of the expectation of the observable in that state, a fact that allows us to calculate higher order corrections to the Heisenberg relations. A general mixed state is determined by a probability density function on the state space, for which the associated first moment is the density matrix. The advantage of the idea of a general state is in its applicability in various attempts to go beyond the standard quantum theory, some of which admit a natural phase-space characterisation.