Quantum Mechanics of the Symmetric Rotor

Following our preceding work on spherical and linear rotors [C. Stopera and J. A. Morales, J. Chem. Phys. 152, 134112 (2020)], we reformulate an earlier rotational coherent state (CS) set to obtain a temporally stable (TS) CS set for symmetric rotors. Being TS, the new CSs remain within its own set during dynamics by evolving exclusively through their parameters. The TS CS set is now appropriate to reconstruct quantum rotational properties from classical-mechanics simulations of chemical reactions. Following literature precedents, we enforce temporal stability by incorporating action-angle-related phase factors into two parameters of the original CS set. Proofs and final expressions of the symmetric-rotor CS turn out more intricate than those of its spherical-rotor counterpart. We demonstrate and examine the key properties of the new CS set: continuity, resolution of unity, temporal stability, action identity, minimum uncertainty relationships, and quasi-classical behavior. Regarding the last property, we demonstrate that the body-fixed z-component of the CS angular momentum average evolves exactly as its classical counterpart, and that the x- and y-components display an astonishing analogy with their classical counterparts in terms of functional form, precession angular velocities, amplitudes, and phases. We elucidate some of these properties via computer simulations of a rotating benzene molecule represented with the CS set. We discuss the utilization of this CS set to reconstruct quantum rotational properties of symmetric-rotor molecules from classical-mechanics simulations. The new CS set is appropriate to establish quantum-classical connections for rotational properties in chemical dynamics, statistical mechanics, spectroscopy, nuclear physics, and quantum computing.Following our preceding work on spherical and linear rotors [C. Stopera and J. A. Morales, J. Chem. Phys. 152, 134112 (2020)], we reformulate an earlier rotational coherent state (CS) set to obtain a temporally stable (TS) CS set for symmetric rotors. Being TS, the new CSs remain within its own set during dynamics by evolving exclusively through their parameters. The TS CS set is now appropriate to reconstruct quantum rotational properties from classical-mechanics simulations of chemical reactions. Following literature precedents, we enforce temporal stability by incorporating action-angle-related phase factors into two parameters of the original CS set. Proofs and final expressions of the symmetric-rotor CS turn out more intricate than those of its spherical-rotor counterpart. We demonstrate and examine the key properties of the new CS set: continuity, resolution of unity, temporal stability, action identity, minimum uncertainty relationships, and quasi-classical behavior. Regarding the last property, we demonstrate that the body-fixed z-component of the CS angular momentum average evolves exactly as its classical counterpart, and that the x- and y-components display an astonishing analogy with their classical counterparts in terms of functional form, precession angular velocities, amplitudes, and phases. We elucidate some of these properties via computer simulations of a rotating benzene molecule represented with the CS set. We discuss the utilization of this CS set to reconstruct quantum rotational properties of symmetric-rotor molecules from classical-mechanics simulations. The new CS set is appropriate to establish quantum-classical connections for rotational properties in chemical dynamics, statistical mechanics, spectroscopy, nuclear physics, and quantum computing.

The Journal of Chemical PhysicsMay 16, 20220 citations

Joint Definability, Context–Instantiated Geometry, and the Inapplicability of Fine’s Theorem

Bryan Sanctuary

We extend our description of the electron’s quaternion spin (Q–spin) as a geometric rotor. This follows from incorporating both the symmetric polarization and the antisymmetric coherence into the theory. Polarization is observed only through the local process we call contextual instantiation. For an EPR pair the observed correlation is reconstructed statistically from the coincidence populations and follows from the scalar invariant of the quaternion rotor, yielding the observed correlation. Numerical simulations reproduce this correlation using only local detector events, and show the same data retain complementary polarization and coherence structure producing a lemniscate rotor portrait. Because spin observables are not jointly definable on a single Kolmogorov probability space, the CHSH quadruple cannot be formed and Bell’s theorem is therefore not applicable. The results provide a unified geometric description of spin, measurement, and Bell correlations, contributing to the foundations of quantum mechanics.We extend our description of the electron’s quaternion spin (Q–spin) as a geometric rotor. This follows from incorporating both the symmetric polarization and the antisymmetric coherence into the theory. Polarization is observed only through the local process we call contextual instantiation. For an EPR pair the observed correlation is reconstructed statistically from the coincidence populations and follows from the scalar invariant of the quaternion rotor, yielding the observed correlation. Numerical simulations reproduce this correlation using only local detector events, and show the same data retain complementary polarization and coherence structure producing a lemniscate rotor portrait. Because spin observables are not jointly definable on a single Kolmogorov probability space, the CHSH quadruple cannot be formed and Bell’s theorem is therefore not applicable. The results provide a unified geometric description of spin, measurement, and Bell correlations, contributing to the foundations of quantum mechanics.

Quantum ReportsSep 19, 20260 citations

Symmetry and quantum mechanics

William Martin McClain

Springer eBooksJan 1, 20090 citations

Quantum Mechanics in Terms of Symmetric Measurements

Christopher A. Fuchs

Bulletin of the American Physical SocietyMar 13, 20060 citations

Multimode rotation-symmetric bosonic codes from homological rotor codes

Yijia Xu, Yixu Wang, Victor V. Albert

We develop quantum information processing primitives for the planar rotor, the state space of a particle on a circle. The $n$ -rotor Clifford group, $\text{U}{(1)}^{n(n+1)/2}\ensuremath{\rtimes}{\text{GL}}_{n}(\mathbb{Z})$, is represented by continuous $\text{U}(1)$ gates generated by polynomials quadratic in angular momenta, as well as discrete ${\text{GL}}_{n}(\mathbb{Z})$ gates generated by momentum sign-flip and sum gates. Our understanding of this group allows us to establish connections between homological rotor error-correcting codes [Vuillot, Ciani, and Terhal, Commun. Math. Phys. 405, 53 (2024)] and oscillator quantum codes, including Gottesman-Kitaev-Preskill codes and rotation-symmetric bosonic codes. Inspired by homological rotor codes, we provide a systematic construction of multimode rotation-symmetric bosonic codes by making a parallel between oscillator Fock states and rotor states with fixed non-negative angular momentum. This family of homological number-phase codes protects against dephasing and changes in occupation number. Encoding and decoding circuits for these codes can be derived from the corresponding rotor Clifford operations. As a result of independent interest, we show how to nondestructively measure the oscillator phase using conditional occupation-number addition and postselection. We also outline several rotor and oscillator varieties of the Gottesman-Kitaev-Preskill-stabilizer codes [Phys. Rev. Lett. 125, 080503 (2020).].We develop quantum information processing primitives for the planar rotor, the state space of a particle on a circle. The $n$-rotor Clifford group, $\text{U}{(1)}^{n(n+1)/2}\ensuremath{\rtimes}{\text{GL}}_{n}(\mathbb{Z})$, is represented by continuous $\text{U}(1)$ gates generated by polynomials quadratic in angular momenta, as well as discrete ${\text{GL}}_{n}(\mathbb{Z})$ gates generated by momentum sign-flip and sum gates. Our understanding of this group allows us to establish connections between homological rotor error-correcting codes [Vuillot, Ciani, and Terhal, Commun. Math. Phys. 405, 53 (2024)] and oscillator quantum codes, including Gottesman-Kitaev-Preskill codes and rotation-symmetric bosonic codes. Inspired by homological rotor codes, we provide a systematic construction of multimode rotation-symmetric bosonic codes by making a parallel between oscillator Fock states and rotor states with fixed non-negative angular momentum. This family of homological number-phase codes protects against dephasing and changes in occupation number. Encoding and decoding circuits for these codes can be derived from the corresponding rotor Clifford operations. As a result of independent interest, we show how to nondestructively measure the oscillator phase using conditional occupation-number addition and postselection. We also outline several rotor and oscillator varieties of the Gottesman-Kitaev-Preskill-stabilizer codes [Phys. Rev. Lett. 125, 080503 (2020).].

arXiv:2311.07679Physical Review AAug 1, 20247 citations

We develop quantum information processing primitives for the planar rotor, the state space of a particle on a circle. The

n

-rotor Clifford group, $\text{U}{(1)}^{n(n+1)/2}\ensuremath{\rtimes}{\text{GL}}_{n}(\mathbb{Z})$, is represented by continuous

\text{U}(1)

gates generated by polynomials quadratic in angular momenta, as well as discrete

{\text{GL}}_{n}(\mathbb{Z})

gates generated by momentum sign-flip and sum gates. Our understanding of this group allows us to establish connections between homological rotor error-correcting codes [Vuillot, Ciani, and Terhal, Commun. Math. Phys. 405, 53 (2024)] and oscillator quantum codes, including Gottesman-Kitaev-Preskill codes and rotation-symmetric bosonic codes. Inspired by homological rotor codes, we provide a systematic construction of multimode rotation-symmetric bosonic codes by making a parallel between oscillator Fock states and rotor states with fixed non-negative angular momentum. This family of homological number-phase codes protects against dephasing and changes in occupation number. Encoding and decoding circuits for these codes can be derived from the corresponding rotor Clifford operations. As a result of independent interest, we show how to nondestructively measure the oscillator phase using conditional occupation-number addition and postselection. We also outline several rotor and oscillator varieties of the Gottesman-Kitaev-Preskill-stabilizer codes [Phys. Rev. Lett. 125, 080503 (2020).].