Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · quant-ph/0510234

Quantization of the canonically conjugate pair angle and orbital angular momentum

H. A. Kastrup
View original source ↗Read source document ↗Export metadata ↓

Abstract

The question how to quantize a classical system where an angle phi is one of the basic canonical variables has been controversial since the early days of quantum mechanics. The problem is that the angle is a multivalued or discontinuous variable on the corresponding phase space. The remedy is to replace phi by the smooth periodic functions cos phi and sin phi. In the case of the canonical pair (phi,l),l: orbital angular momentum (OAM), the phase space S_(phi,l) ={phi in R mod 2pi, l in R} has the global structure S^1 x R of a cylinder on which the Poisson brackets of the 3 functions cos phi, sin phi and l obey the Lie algebra of the euclidean group E(2) in the plane. This property provides the basis for the quantization of the system in terms of irreducible unitary representations of the group E(2) or of its covering groups. A crucial point is that - due to the fact that the subgroup SO(2) = S^1 is multiply connected - these representations allow for fractional OAM l = n + c, c in [0,1). Such c not 0 have already been observed in cases like the Aharonov-Bohm and the fractional quantum Hall effects and they correspond to the quasi-momenta of Bloch waves in ideal crystals. The proposal of the present paper is to look for fractional OAM in connection with the quantum optics of Laguerre-Gaussian laser modes in external magnetic fields. The quantum theory of the phase space S_(phi,l) in terms of unitary representations of E(2) allows for two types of "coherent" states the properties of which are discussed in detail: Non-holomorphic minimal uncertainty states and holomorphic ones associated with Bargmann-Segal Hilbert spaces.

Explore related subjects

quant-phmath-phmath.MPphysics.optics

Keep this discovery

BibTeXRIS

H. A. Kastrup. 2006-03-29. Quantization of the canonically conjugate pair angle and orbital angular momentum. https://doi.org/10.1103/physreva.73.052104

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Probing the Error-Mitigation Threshold with Matrix Product States

Quantum error mitigation relies on accurate noise characterization, but mismatches between the actual and characterized noise can be amplified and drive a sharp threshold between successful and failed mitigation. In random circuits, this threshold maps onto a random-field Ising transition, but previous exact numerics were limited to small one-dimensional and all-to-all systems, leaving explicit two-dimensional architectures unresolved. We develop a fixed-bond-dimension matrix-product-state method for the replicated transfer dynamics that extends threshold calculations beyond exact propagation while retaining the finite-size signatures of the transition. At system sizes beyond previous exact studies, we recover the predicted absence of a threshold for quenched disorder in 1D, obtain a sharper annealed all-to-all critical point, and resolve architecture-dependent finite-depth thresholds in 2D square and heavy-hex circuits. These results establish replicated tensor-network dynamics as a practical tool for probing error-mitigation thresholds in large and higher-dimensional noisy circuits.

quant-ph↗
arXiv2026

Low-cost algorithm-to-execution framework for surface-code quantum computing

The execution of useful quantum algorithms on fault-tolerant processors requires more than a mapping from logical gates to encoded operations: the spatial organization, non-Clifford resource supply, and execution schedule must also be determined while keeping physical overhead within practical limits. Although the theoretical hierarchy from logical circuits to fault-tolerant operations is well established, these implementation choices are often specified and optimized separately. Here we develop a low-cost algorithm-to-execution framework for surface-code quantum computing. From hierarchical algorithm descriptions, it constructs dependency-preserving logical schedules and an executable workload capturing logical interactions, operation parallelism, and time-resolved non-Clifford demand, thereby linking logical computation to surface-code organization, resource-state preparation, and fault-tolerant execution in a traceable workflow. We apply the framework to twenty benchmark circuits across seven algorithm families and a hierarchically composed application-scale elliptic-curve discrete-logarithm workload. Physical costs vary substantially even for circuits with similar logical resource counts. Under our direct-rotation calibration, non-Clifford implementation selection reduces space-time volume by up to 241.5 times versus an all-synthesis baseline for the QAOA amplitude-amplification workload. Circuit-specific surface-code layouts reduce routed-latency estimates for all twenty benchmarks; thirteen also reduce space-time volume because communication savings outweigh added spatial overhead. These results show that low-cost fault-tolerant execution depends on computation scheduling and organization, not aggregate logical resource counts alone.

quant-ph↗
arXiv2026

Sample-optimal learning of stabilizer states

It is well-known that learning a pure $n$-qubit stabilizer state $|\psi\rangle$ both requires, and can be accomplished with, access to a number of copies of $|\psi\rangle$ linear in $n$. However, the precise constant coefficient of this scaling does not appear to have been determined. Here we prove that $L_\delta(n)$, the smallest number of copies from which a quantum procedure can identify any stabilizer state with failure probability at most $0<\delta<1/8$, satisfies $n+\lceil\log_2(1/\delta)\rceil-3\leq L_\delta(n)\leq n+\left\lceil\log_2(1/\delta)\right\rceil+4$. We present a polynomial-time quantum learning algorithm that saturates this bound, achieving a constant factor improvement in sample-complexity over previously known approaches. As an immediate corollary, we obtain via the Choi-Jamiolkowski isomorphism an algorithm for learning an unknown $n$-qubit Clifford unitary from $2n+\left\lceil\log_2(1/\delta)\right\rceil+4$ queries, the $n$-dependence of which we show to be optimal. Our proof technique, which involves Fourier analysis on the abelian group $\mathbb{Z}_4^n \times \mathbb{F}_2^{n(n-1)/2}$, seems to be qualitatively different to previous approaches to stabilizer state learning, and may be of some independent interest; in particular, it admits natural generalisations to further problems in quantum learning theory.

quant-ph↗