Quantum Brachistochrone Problem and the Geometry of the State Space in Pseudo-Hermitian Quantum Mechanics

Abstract

A non-Hermitian operator with a real spectrum and a complete set of eigenvectors may serve as the Hamiltonian operator for a unitary quantum system provided that one makes an appropriate choice for the defining inner product of the physical Hilbert state. We study the consequences of such a choice for the representation of states in terms of projection operators and the geometry of the state space. This allows for a careful treatment of the quantum Brachistochrone problem and shows that it is indeed impossible to achieve faster unitary evolutions using PT -symmetric or other non-Hermitian Hamiltonians than those given by Hermitian Hamiltonians.Pacs numbers: 03.67.Lx, 02.30.Yy, Since the publication of the pioneering work of Bender and Boettcher [1] on non-Hermitian PT -symmetric Hamiltonian operators, there have appeared numerous research articles exploring the mathematical properties of such operators and their possible physical applications. Recently, it has been suggested that one can obtain arbitrarily fast quantum evolutions using a class of such Hamiltonians [2]. If true, this will have drastic consequences in quantum computation, because for example it removes the bound on the time-optimal unitary NOT operations [3] that is obtained within the framework of conventional (Hermitian) quantum mechanics [4,5,6,7]. As pointed out in [8], this seems to contradict the equivalence of the quantum theory based on such non-Hermitian Hamiltonians and the Hermitian quantum mechanics [9,10]. In this article, we offer a comprehensive treatment of this problem that is based on a detailed study of the projective space PH phys of physical states. In particular, we obtain the explicit form of the natural metric tensor on PH phys and unravel the subtleties of the quantum Brachistochrone problem for a general unitary quantum system that is defined by a non-Hermitian Hamiltonian.In general if a linear (possibly non-Hermitian) operator has a complete set of eigenvectors and a real spectrum, then it can serve as the Hamiltonian operator for a unitary quantum system provided that the physical Hilbert space of the system is defined using an appropriate inner product [11,12,13]. This leads to a quantum theory that turns out to be equivalent to the conventional quantum mechanics [9,10]. In other words, this theory, that we refer to as Pseudo-Hermitian Quantum Mechanics [14], is an alternative representation of the conventional quantum mechanics. The key ingredient of this representation is that the inner product of the physical Hilbert space H phys is determined by the Hamiltonian operator of the system. This has led to the discovery of an intriguing structural similarity between quantum mechanics and general theory of relativity [15]. It has also found applications in dealing with the Hilbert-space problem in quantum cosmology [16], the old problem of constructing a unitary first-quantized quantum theory of Klein-Gordon fields [17], bound state scattering [18], and ghosts in certain quantum field theories [19].In [2] the authors co...

Optimal Time Evolution for Hermitian and Non-Hermitian Hamiltonians

“…The arbitrarily short alternative complex pathway from an up state to a down state, as illustrated by this thought experiment, is reminiscent of the short alternative distance between two widely separated space-time points as measured through a wormhole in general relativity. This comparison is of course controversial, and it has subsequently motivated much research and it has generated a lively debate in the literature [2,15,33,35,36,49,52,53,56]. We emphasise that the entire package of flipping the spin is not realised by a unitary operation.…”

Equivalência entre a mecânica quântica e a mecânica quântica PT simétrica

“…Sem perda de generalidade pode-se descrever todo sistema quântico com matrizes hermitianas. Uma demonstração matematicamente formal desta afirmação pode ser encontrada em [17]. Nesse artigo concluímos que o efeito de transição entre estados quânticos de maneira mais rápida que mecânica quântica usual apenas se trata de uma interpretação equivocada do comportamento dos estados.…”

Complex PT -symmetric nonlinear Schrödinger equation and Burgers equation

“…The notion of pseudo-Hermiticity has also been introduced [21][22][23][24] and it has been shown that every Hamiltonian with a real spectrum is pseudo-Hermitian [25]. 2 More types of pseudo-Hermitian Hamiltonians have also been studied (see [27][28][29][30][31][32][33][34][35][36][37] and references therein). 2 A Hamiltonian H is said to be pseudo-Hermitian if its adjoint H † satisfies H † = η + Hη −1 + , where η + is a positive-definite (metric) operator.…”