Faster than Hermitian Quantum Mechanics

Abstract

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 ...

Optimal Time Evolution for Hermitian and Non-Hermitian Hamiltonians

“…We emphasise that the entire package of flipping the spin is not realised by a unitary operation. This follows from the fact that P T -symmetric quantum theory is unitary, and as such it respects the Fleming bound (17) applicable to all unitary theories [14]. The point is that the 'black box' scheme described above actually consists of three regimes: (i) the preparation of a spin-up state in the Hermitian setup, (ii) the fast unitary motion to flip the spin using a P T -symmetric Hamiltonian, and (iii) the recovery of a spindown state in the Hermitian setup.…”

“…[14] to realise this effect in a laboratory. The setup is as follows: we use a Stern-Gerlach filter to create a beam of spin-up electrons.…”

Equivalência entre a mecânica quântica e a mecânica quântica PT simétrica

“…A hamiltoniana estudada por [11] se apresenta como capaz de obter um comportamento anômalo com respeito a transição de estados, segundo os autores,é possível obter uma transição de maneira mais rápida do que no caso da mecânica quântica usual.…”

“…Para entender melhor este fenômeno, vamos detalhar os raciocínios desenvolvidos em [11]; afim de relacionar a mecânica quântica usual com a PT simétrica, vamos calcular o tempo de transição entre os estados nas duas situações. O cálculo do tempo mínimo de transição entre estadosé o famoso problema da braquistócrona quântica.…”

“…Além disso, vamos analisar a Braquistócrona Quântica sob o ponto de vista da mecânica quântica usual e da PT simétrica [9][10][11]. Através dessa análise, podemos ser induzidos a acreditar em comportamentos anômalos, que geram transições infinitamente rápidas entre estados quânticos [11], vamos entender esse fenômeno e identificar que essa transição rápida se dá devido a uma interpretação equivocada dos estados quânticos. …”