We discuss correspondence between the predictions of quantum theories for rotation angle formulated in infinite and finite dimensional Hilbert spaces, taking as example, the calculation of matrix elements of phase-angular momentum commutator. A new derivation of the matrix elements is presented in infinite space, making use of a unitary transformation that maps from the state space of periodic functions to non-periodic functions, over which the spectrum of angular momentum operator is in general, fractional. The approach can be applied to finite dimensional Hilbert space also, for which identical matrix elements are obtained.
In this paper we construct the commutators of the Fedosov * (a generalization of the Moyal star product) on the phase space of S2. It is shown that this product obeys the standard angular momentum commutation relations in ordinary nonrelativistic quantum mechanics. The purpose of this paper is three-fold. One is to find an exact, non-perturbative solution to the commutators of the Fedosov *. The other is to verify that these commutation relations correspond to angular momentum commutation relations as in ordinary nonrelativistic quantum mechanics. The last is to show a more general computation…
We discuss conditional expectations that can be used as generalizations of the partial trace for quantum systems with an infinite-dimensional Hilbert space of states.
A quaternionic commutator bracket for position and momentum shows that the quaternionic wave function, \emph{viz.} $\widetildeψ=(\frac{i}{c}\,ψ_0\,,\vecψ)$, represents a state of a particle with orbital angular momentum, $L=3\,\hbar$, resulting from the internal structure of the particle. This angular momentum can be attributed to spin of the particle. The vector $\vecψ$, points along the direction of $\vec{L}$. When a charged particle is placed in an electromagnetic fields the interaction energy reveals that the magnetic moments interact with the electric and magnetic fields giving rise to te…
Uncertainty relations between two general non-commuting self-adjoint operators are derived in a Krein space. All of these relations involve a Krein space induced fundamental symmetry operator, $J$, while some of these generalized relations involve an anti-commutator, a commutator, and various other nonlinear functions of the two operators in question. As a consequence there exist classes of non-self-adjoint operators on Hilbert spaces such that the non-vanishing of their commutator implies an uncertainty relation. All relations include the classical Heisenberg uncertainty principle as formulat…
We elaborate an approach to quantum fluctuations of angular momentum based on the diagonalization of the covariance matrix in two versions: real symmetric and complex Hermitian. At difference with previous approaches this is SU(2) invariant and avoids any difficulty caused by nontrivial commutators. Meaningful uncertainty relations are derived which are nontrivial even for vanishing mean angular momentum. We apply this approach to some relevant states.
In this paper we find a simple rule to reproduce the algebra of quantum observables using only the commutators and operators which appear in the Koopman-von Neumann (KvN) formulation of classical mechanics. The usual Hilbert space of quantum mechanics becomes embedded in the KvN Hilbert space: in particular it turns out to be the subspace on which the quantum positions Q and momenta P act irreducibly.
The practically useful criteria of separable states $ρ=\sum_{k}w_{k}ρ_{k}$ in $d=2\times2$ are discussed. The equality $G({\bf a},{\bf b})= 4[\langle ψ|P({\bf a})\otimes P({\bf b})|ψ\rangle-\langle ψ|P({\bf a})\otimes{\bf 1}|ψ\rangle\langle ψ|{\bf 1}\otimes P({\bf b})|ψ\rangle]=0$ for any two projection operators $P({\bf a})$ and $P({\bf b})$ provides a necessary and sufficient separability criterion in the case of a separable pure state $ρ=|ψ\rangle\langleψ|$. We propose the separability criteria of mixed states, which are given by ${\rm Tr}ρ\{{\bf a}\cdot {\bf σ}\otimes {\bf b}\cdot {\bf σ}\…
We test experimentally the quantum ``paradox'' proposed by Lucien Hardy in 1993 [Phys. Rev. Lett. 71, 1665 (1993)] by using single photons instead of photon pairs. This is achieved by addressing two compatible degrees of freedom of the same particle, namely its spin angular momentum, determined by the photon polarization, and its orbital angular momentum, a property related to the optical transverse mode. Because our experiment involves a single particle, we cannot use locality to logically enforce non-contextuality, which must therefore be assumed based only on the observables' compatibility.…
Novel analysis of finite dimensional Hilbert space is outlined. The approach bypasses general, inherent, difficulties present in handling angular variables in finite dimensional problems: The finite dimensional, d, Hilbert space operators are underpinned with finite geometry which provide intuitive perspective to the physical operators. The analysis emphasizes a central role for projectors of mutual unbiased bases (MUB) states, extending thereby their use in finite dimensional quantum mechanics studies. Interrelation among the Hilbert space operators revealed via their (finite) dual affine pla…
We examine the implementation of an arbitrary U(4) gate consisting of CNOT gates and single qubit unitary gates for the Hilbert space of photon spin polarization and two states of photon orbital angular momentum. Our scheme improves over a recently proposed one that uses q-plates because the fidelity is limited only by losses thus in principle it could be used to achieve a perfect transformation.
We present two novel solutions of real Hilbert state quaternionic quantum mechanics ($\mathbb H$QM). Firstly, we observe that the angular momentum operator admits two different classes of physically non-equivalent free particles. As a second result, we study the Larmor precession to observe that it has a quaternionic solution where a novel phenomenological interpretation is possible, as well as a different of spin is possible, and these results may encourage experimental and theoretical investigations of the quaternionic theory.
We report the first quantum entanglement experiment in South Africa. The spatial modes of the entangled photon pair are investigated with their potential for high-dimensional entanglement. The generation, measurement and characterisation of the entangled states are examined in detail, where we show high-dimensional entanglement in a Hilbert space of dimension 25. We highlight the experimental challenges contained within each step and provide practical techniques for future experiments in the quantum regime.
We propose a scheme enabling the universal coupling of angular momentum of $N$ remote noninteracting qubits using linear optical tools only. Our system consists of $N$ single-photon emitters in a $Λ$-configuration that are entangled among their long-lived ground-state qubits through suitably designed measurements of the emitted photons. In this manner, we present an experimentally feasible algorithm that is able to generate any of the $2^N$ symmetric and nonsymmetric total angular momentum eigenstates spanning the Hilbert space of the $N$-qubit compound.
We present a new implementation of the BB84 quantum key distribution protocol that employs a $d$-dimensional Hilbert space spanned by spatial modes of the propagating beam that have a definite value of orbital angular momentum. Each photon carries $\log d$ bits of information, increasing the key generation rate of the protocol. The states used in the transmission part of the protocol are invariant under rotations about the propagation direction, making this implementation independent of the alignment between the reference frames of the sender and receiver. The protocol still works when these r…
In Dirac's canonical quantization theory on systems with second-class constraints, the commutators between the position, momentum and Hamiltonian form a set of algebraic relations that are fundamental in construction of both the quantum momentum and the Hamiltonian. For a free particle on a two-dimensional sphere or a spherical top, results show that the well-known canonical momentum p_{θ} breaks one of the relations, while three components of the momentum expressed in the three-dimensional Cartesian system of axes as p_{i} (i=1,2,3) are satisfactory all around. This momentum is not only geome…
Quantum state tomography is an important tool for quantum communication, computation, metrology, and simulation. Efficient quantum state tomography on a high dimensional quantum system is still a challenging problem. Here, we propose a novel quantum state tomography method, auxiliary Hilbert space tomography, to avoid pre-rotations before measurement in a quantum state tomography experiment. Our method requires projective measurements in a higher dimensional space that contains the subspace that includes the prepared states. We experimentally demonstrate this method with orbital angular moment…
Covariant integral quantisation using coherent states for semidirect product groups is studied and applied to the motion of a particle on the circle. In the present case the group is the Euclidean group E$(2)$. We implement the quantisation of the basic classical observables, particularly the $2π$-periodic discontinuous angle function and the angular momentum, and compute their corresponding lower symbols. An important part of our study is devoted to the angle operator given by our procedure, its spectrum and lower symbol, its commutator with the quantum angular momentum, and the resulting Hei…
The complex Hilbert space of standard quantum mechanics may be treated as a real Hilbert space. The pure states of the complex theory become mixed states in the real formulation. It is then possible to generalize standard quantum mechanics, keeping the same set of physical states, but admitting more general observables. The standard time reversal operator involves complex conjugation, in this sense it goes beyond the complex theory and may serve as an example to motivate the generalization. Another example is unconventional canonical quantization such that the harmonic oscillator of angular fr…
Satellite-based quantum communications enable a bright future for global-scale information security. However, the spin orbital momentum of light, currently used in many mainstream quantum communication systems, only allows for quantum encoding in a two-dimensional Hilbert space. The orbital angular momentum (OAM) of light, on the other hand, enables quantum encoding in higher-dimensional Hilbert spaces, opening up new opportunities for high-capacity quantum communications. Due to its turbulence-induced decoherence effects, however, the atmospheric channel may limit the practical usage of OAM. …
The Hilbert hotel is an old mathematical paradox about sets of infinite numbers. This paradox deals with the accommodation of a new guest in a hotel with an infinite number of occupied rooms. Over the past decade, there have been many attempts to implement these ideas in photonic systems. In addition to the fundamental interest that this paradox has attracted, this research is motivated by the implications that the Hilbert hotel has for quantum communication and sensing. In this work, we experimentally demonstrate the fractional vortex Hilbert's hotel scheme proposed by G. Gbur [Optica 3, 222-…
We present a quantum mechanical analysis of the orbital angular momentum of a class of recently discovered elliptically-symmetric stable light fields --- the so-called Ince-Gauss modes. We study, in a fully quantum formalism, how the orbital angular momentum of these beams varies with their ellipticity and discover several compelling features, including: non-monotonic behavior, stable beams with real continuous (non-integer) orbital angular momenta, and orthogonal modes with the same orbital angular momenta. We explore, and explain in detail, the reasons for this behavior. These features may h…
