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Physical Review A

Quantum phase for an arbitrary system with finite-dimensional Hilbert space

D. Arsenović, Nikola Burić, Dragomir M Davidović, Slobodan Prvanović

Published Apr 26, 2012• 2 citations
Publisher

Abstract

A representation of the phase observable in terms of a positive-operator-valued measure for an arbitrary quantum system with a finite Hilbert space is consistently defined. The phase for systems with rational relations between the energy eigenvalue differences is treated explicitly and the phase in the case of the irrational relations is obtained as a well-defined limit of the rational approximations.

DOI
10.1103/physreva.85.044103

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Comment on “Quantum phase for an arbitrary system with finite-dimensional Hilbert space”

Michael J. W. Hall, David Thomas Pegg

A construction of covariant quantum phase observables, for Hamiltonians with a finite number of energy eigenvalues, has been recently given by D. Arsenovi\ifmmode \acute{c}\else \'{c}\fi{} et al. [Phys. Rev. A 85, 044103 (2012)]. For Hamiltonians generating periodic evolution, we show that this construction is just a simple rescaling of the known canonical ``time'' or ``age'' observable, with the period $T$ rescaled to $2\ensuremath{\pi}$. Further, for Hamiltonians generating quasiperiodic evolution, we note that the construction leads to a phase observable having several undesirable features, including (i) having a trivially uniform probability density for any state of the system, (ii) not reducing to the periodic case in an appropriate limit, and (iii) not having any clear generalization to an infinite energy spectrum. In contrast, we note that a covariant time observable has been previously defined for such Hamiltonians, which avoids these features. We also show how this ``quasiperiodic'' time observable can be represented as the well-defined limit of a sequence of periodic time observables.

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A representation of the phase observable in terms of a positive-operator-valued measure for an arbitrary quantum system with a finite Hilbert space is consistently defined. The phase for systems with rational relations between the energy eigenvalue differences is treated explicitly and the phase in the case of the irrational relations is obtained as a well-defined limit of the rational approximations.
A construction of covariant quantum phase observables, for Hamiltonians with a finite number of energy eigenvalues, has been recently given by D. Arsenovi\ifmmode \acute{c}\else \'{c}\fi{} et al. [Phys. Rev. A 85, 044103 (2012)]. For Hamiltonians generating periodic evolution, we show that this construction is just a simple rescaling of the known canonical ``time'' or ``age'' observable, with the period TTT rescaled to $2\ensuremath{\pi}$. Further, for Hamiltonians generating quasiperiodic evolution, we note that the construction leads to a phase observable having several undesirable features, including (i) having a trivially uniform probability density for any state of the system, (ii) not reducing to the periodic case in an appropriate limit, and (iii) not having any clear generalization to an infinite energy spectrum. In contrast, we note that a covariant time observable has been previously defined for such Hamiltonians, which avoids these features. We also show how this ``quasiperiodic'' time observable can be represented as the well-defined limit of a sequence of periodic time observables.
The quantum observables used in the case of quantum systems with finite-dimensional Hilbert space are defined either algebraically in terms of an orthonormal basis and discrete Fourier transformation or by using a continuous system of coherent states. We present an alternative approach to these important quantum systems based on the finite frame quantization. Finite systems of coherent states, usually called finite tight frames, can be defined in a natural way in the case of finite quantum systems. Novel examples of such tight frames are presented. The quantum observables used in our approach are obtained by starting from certain classical observables described by functions defined on the discrete phase space corresponding to the system. They are obtained by using a finite frame and a Klauder–Berezin–Toeplitz-type quantization. Semi-classical aspects of tight frames are studied through lower symbols of basic classical observables.
Phase-space representations can be used to reveal resources for quantum computational advantage. In odd-dimensional qudit systems, states and operations admitting non-negative quasi-probability representations are efficiently classically simulable, and analogous conclusions hold for bosonic systems, usually represented in quadrature phase space. However, when quantum information is encoded in bosonic systems, the link between physical phase-space representations and computational resources becomes nontrivial: large negativities may arise in the physical space even when the encoded model remains efficiently simulable. We establish a general framework connecting physical phase spaces of bosonic systems to the effective computational phase spaces associated with arbitrary encodings. We show that bosonic Hilbert spaces with finitely many modes can always be exactly encoded on a quantum computer, without truncation or approximations, and that for any encoding there exists an adapted physical phase-space representation in which all provably classically simulable states and protocols are non-negative. Conversely, negativity can always persist in physical representations for arbitrary encodings. Our results formalize the operational meaning of phase-space negativity in bosonic quantum information and its relation with computational resources.
Phase-space representation of quantum state vectors is obtained within the framework of the relative-state formulation. For this purpose, the Hilbert space of a quantum system is enlarged by introducing an auxiliary quantum system. Relative-position state and relative-momentum state are defined in the extended Hilbert space of the composite quantum system and expressions of basic operators such as canonical position and momentum operators, acting on these states, are obtained. Phase-space functions which represent a state vector of the relevant quantum system are obtained in terms of the relative-position states and the relative-momentum states. The absolute-square of the phase-space function represents the probability distribution of the phase-space variables. Time-evolution of a quantum system is investigated in terms of the phase-space functions. The relations to the phase-space representations formulated by the other methods are obtained.
In the Hilbert space formulation of quantum mechanics, ideal measurements of physical variables are discussed using the spectral theory of Hermitian operators and the corresponding projector-valued measures (PVMs). However, more general types of measurements require the treatment in terms of positive-operator-valued measures (POVMs). In the Hamiltonian formulation of quantum mechanics, canonical coordinates are related to PVM. In this paper the results of an analysis of various aspects of applications of POVMs in the Hamiltonian formulation are reported. Several properties of state parameters and quantum observables given by POVMs or represented in an overcomplete basis, including the general Hamiltonian treatment of the Neumark extension, are presented. An analysis of the phase operator, given by the corresponding POVMs, in the Hilbert space and the Hamiltonian frameworks is also given.