Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Relative Phase and Time States

Masataka Ohta

Published Jun 1, 2019• 0 citations
OpenAlex

Abstract

It is pointed out that, while Suskind and Glogower claim “The phase operator for an oscillator is shown not to exist.”, they actually showed absolute phase operator, which is physically meaningless, does not exist. It is also pointed out that, they showed countably infinitely many trigonometric relative (relative between two oscillators) phase states, which form complete basis and are, in a sense, best possible quantum “phase states”, exist. In addition, continuously infinitely many phase states are constructed at classical, that is, infinitely many quanta, limit. Similarly, relative time states and operator are constructed with two oscillators with different angular velocities.

Related papers

Quantum mechanical phase and time operator

Leonard Susskind, Jonathan Glogower

The phase operator for an oscillator is shown not to exist. It is replaced by a pair of non-commuting sin and cos operators which can be used to define uncertainty relations for phase and number. The relation between phase and angle operators is carefully discussed. The possibility of using a phase variable as a quantum clock is demonstrated and the states for which the clock is most accurate are constructed.

Physics Physique Fizika•Jul 1, 1964•812 citations

Eigenstates of linear combinations of phase operators

C V Sukumar

The eigenstates of linear combinations of the Susskind and Glogower phase operators [Physics 1, 49 (1964)] for the harmonic oscillator are constructed. It is shown that such eigenstates are squeezed states.

arXiv:1809.03844•Physical Review A•Feb 1, 1993•3 citations

Dual eigenvectors of the Susskind-Glogower phase operators for a two-mode field

Xuefen Xu

We find that there exist dual eigenvectors |(?),q〉_* of the phase state-vectors |(?),q〉of the two-mode Susskind-Glogower phase operator e~(iφ).By contour integration we show that |(?),q〉_* is the common eigenvector of e~(-iφ)and the photon number difference operator.The properties of |(?),q〉_* are studied,it turns out that |(?),q〉_* and |(?),q〉can compose a completeness relation in contour integration form.A phase state representation for two-mode field is thus presented.

Chinese Journal of Quantum Electronics•Jan 1, 2006•0 citations

Quantum phase and quantum phase operators: some physics and some history

Michael Martin Nieto

After reviewing the role of phase in quantum mechanics, I discuss, with the aid of a number of unpublished documents, the development of quantum phase operators in the 1960's. Interwoven in the discussion are the critical physics questions of the field: are there (unique) quantum phase operators and are there quantum systems which can determine their nature? I conclude with a critique of recent proposals which have shed new light on the problem.

arXiv:hep-th/9304036•Physica Scripta•Jan 1, 1993•61 citations

Covariant Path Integrals and Observation Regularity on Physical--Observation State Spaces

Xianwei Meng

The physical--observation dual-axis (PODA) structure treats physical and observation configurations as distinct arguments of joint dynamics. We study the analytic conditions under which this structure admits rigorous path integrals and finite complex measures for sequential quantum histories. On compact smooth joint geometry, a specified covariant Hamiltonian has a Feynman--Kac representation. A uniform complex-time product estimate permits simultaneous removal of Abel damping and time slicing at arbitrary relative speeds. For fixed finite bounded-readout protocols, class operators converge strongly and the trace-class branch-pair operators converge in trace norm. Finite instruments with complete Kraus families also yield trace-norm convergence of their record states. Energy-form regularity gives quantitative error bounds. A completed cylinder-test space carries a consistent, strongly positive history functional. The stronger finite-measure question is controlled by covariant transport: calibrated rank-one readouts on a finite-dimensional fibre satisfy the sharp estimate \( \|\mu_\gamma\|_{TV}\leq e^{2c_\gamma} \), where \( c_\gamma \) is the off-diagonal covariant variation. In the specified common-action class, conserved energy and smooth, globally labelled rank-one readouts imply this regularity when the kinetic geometry is complete and the effective potential is bounded below, or when the chosen energy confines the trajectory to a compact set. In infinite dimension, finitely many readout sectors allow unbounded internal energies and bounded transitions. The construction extends to countable outcomes when the operator of block norms is bounded on \( \ell^2 \); nonatomic readouts admit a double-Dyson construction when an absolute integral kernel is bounded on \( L^2 \). In these Dyson constructions, pure preparations with rank-one discrete readouts or scalar continuous fibres have exact variation \( \|e^{T\mathsf B}v\|_2^2 \). Whole-sector and spatial-domain cutoffs converge in variation, whereas nonatomic readout quantization converges weakly and remains separated in variation norm. Mixed curvature survives in the covariant constructions, and invariant quantum embeddings recover physical-sector histories. Unrestricted basis switching, a bounded-Hamiltonian countable example and free-particle position histories establish the limits of finite-measure existence; exact physical recovery transfers the last obstruction to the joint theory. Analytic two-axis models exhibit the resulting dynamics and verify the distinct convergence statements. A proposed two-mode trapped-ion realization links the magnetic oscillator's transport phase to independent spectral and driven-response tests. Exact finite-time Gaussian evolution shows that nearly complete wave-packet return can coexist with a substantial phase correction. The protocol therefore infers the coupling from finite-time interference and predicts the other observables without refitting, with calibration and uncertainty requirements explicit. These are model predictions, not measured data. Each construction is tied to its stated geometric, dynamical and readout hypotheses.

•Oct 10, 2026•0 citations

Qubit State Represented by Pendulum Oscillations

Masataka Ohta

Dirac’s misconception that classical and quantum superpositions are different because absolute phase is quantum physically meaningless is, even today, seemingly blindly believed by many physicists. However, there is no such difference because, time translation symmetry means absolute time and, thus, absolute phase are classical physically meaningless, too. As a qubit can be a polarization division multiplexed (PDM) quadrature amplitude modulated (QAM) symbol of light, its state has direct correspondence with polarization state of classical light. Even more intuitively, the state may be represented by two-dimensional mechanical oscillations of a pendulum, which enables intuitive understanding of not-really-quantum superpositioned states of a qubit even by elementary school pupils.

IEICE Technical Report; IEICE Tech. Rep.•Oct 1, 2018•0 citations
It is pointed out that, while Suskind and Glogower claim “The phase operator for an oscillator is shown not to exist.”, they actually showed absolute phase operator, which is physically meaningless, does not exist. It is also pointed out that, they showed countably infinitely many trigonometric relative (relative between two oscillators) phase states, which form complete basis and are, in a sense, best possible quantum “phase states”, exist. In addition, continuously infinitely many phase states are constructed at classical, that is, infinitely many quanta, limit. Similarly, relative time states and operator are constructed with two oscillators with different angular velocities.
The physical--observation dual-axis (PODA) structure treats physical and observation configurations as distinct arguments of joint dynamics. We study the analytic conditions under which this structure admits rigorous path integrals and finite complex measures for sequential quantum histories. On compact smooth joint geometry, a specified covariant Hamiltonian has a Feynman--Kac representation. A uniform complex-time product estimate permits simultaneous removal of Abel damping and time slicing at arbitrary relative speeds. For fixed finite bounded-readout protocols, class operators converge strongly and the trace-class branch-pair operators converge in trace norm. Finite instruments with complete Kraus families also yield trace-norm convergence of their record states. Energy-form regularity gives quantitative error bounds. A completed cylinder-test space carries a consistent, strongly positive history functional. The stronger finite-measure question is controlled by covariant transport: calibrated rank-one readouts on a finite-dimensional fibre satisfy the sharp estimate ∥μγ∥TV≤e2cγ \|\mu_\gamma\|_{TV}\leq e^{2c_\gamma} ∥μγ​∥TV​≤e2cγ​, where cγ c_\gamma cγ​ is the off-diagonal covariant variation. In the specified common-action class, conserved energy and smooth, globally labelled rank-one readouts imply this regularity when the kinetic geometry is complete and the effective potential is bounded below, or when the chosen energy confines the trajectory to a compact set. In infinite dimension, finitely many readout sectors allow unbounded internal energies and bounded transitions. The construction extends to countable outcomes when the operator of block norms is bounded on ℓ2 \ell^2 ℓ2; nonatomic readouts admit a double-Dyson construction when an absolute integral kernel is bounded on L2 L^2 L2. In these Dyson constructions, pure preparations with rank-one discrete readouts or scalar continuous fibres have exact variation ∥eTBv∥22 \|e^{T\mathsf B}v\|_2^2 ∥eTBv∥22​. Whole-sector and spatial-domain cutoffs converge in variation, whereas nonatomic readout quantization converges weakly and remains separated in variation norm. Mixed curvature survives in the covariant constructions, and invariant quantum embeddings recover physical-sector histories. Unrestricted basis switching, a bounded-Hamiltonian countable example and free-particle position histories establish the limits of finite-measure existence; exact physical recovery transfers the last obstruction to the joint theory. Analytic two-axis models exhibit the resulting dynamics and verify the distinct convergence statements. A proposed two-mode trapped-ion realization links the magnetic oscillator's transport phase to independent spectral and driven-response tests. Exact finite-time Gaussian evolution shows that nearly complete wave-packet return can coexist with a substantial phase correction. The protocol therefore infers the coupling from finite-time interference and predicts the other observables without refitting, with calibration and uncertainty requirements explicit. These are model predictions, not measured data. Each construction is tied to its stated geometric, dynamical and readout hypotheses.
Dirac’s misconception that classical and quantum superpositions are different because absolute phase is quantum physically meaningless is, even today, seemingly blindly believed by many physicists. However, there is no such difference because, time translation symmetry means absolute time and, thus, absolute phase are classical physically meaningless, too. As a qubit can be a polarization division multiplexed (PDM) quadrature amplitude modulated (QAM) symbol of light, its state has direct correspondence with polarization state of classical light. Even more intuitively, the state may be represented by two-dimensional mechanical oscillations of a pendulum, which enables intuitive understanding of not-really-quantum superpositioned states of a qubit even by elementary school pupils.