If scattering amplitudes are ordinary complex numbers (not quaternions) there is a universal algebraic relationship between the six coherent cross sections of any three scatterers (taken singly and pairwise). A violation of this relationship would indicate either that scattering amplitudes are quaternions, or that the superposition principle fails. Some possible experimental tests involve neutron interferometry, K_S-meson regeneration, and low energy proton-proton scattering.
We consider a quaternionic quantum formalism for the description of quantum states and quantum dynamics. We prove that generalized quantum measurements on physical systems in quaternionic quantum theory can be simulated by usual quantum measurements with positive operator valued measures on complex Hilbert spaces. Furthermore, we prove that quaternionic quantum channels can be simulated by completely positive trace preserving maps on complex matrices. These novel results map all quaternionic quantum processes to algorithms in usual quantum information theory.
Solutions of quaternionic quantum mechanics (QQM) are difficult to grasp, even in simple physical situations. In this article, we provide simple and understandable free particle quaternionic solutions, that can be easily compared to complex quantum mechanics (CQM). As an application, we study the scattering of quaternionic particles through a scalar step potential. We also provide a general solution method for the quaternionic Schrödinger equation, which can be applied to more sophisticated and physically interesting models.
We characterize the quasianti-Hermitian quaternionic operators in QQM by means of their spectra; moreover, we state a necessary and sufficient condition for a set of quasianti-Hermitian quaternionic operators to be anti-Hermitian with respect to a uniquely defined positive scalar product in a infinite dimensional (right) quaternionic Hilbert space. According to such results we obtain two alternative descriptions of a quantum optical physical system, in the realm of quaternionic quantum mechanics, while no alternative can exist in complex quantum mechanics, and we discuss some differences betwe…
If $Ψ$ is a quaternionic wave function, then $iΨ\neq Ψi$. Thus, there are two versions of the quaternionic Schrödinger equation (QSE). In this article, we present the second possibility for solving the QSE, following on from a previous article. After developing the general methodology, we present the quaternionic free particle solution and the scattering of the quaternionic particle through a scalar barrier.
In this article we obtained the harmonic oscillator solution for quaternionic quantum mechanics ($\mathbbm{H}$QM) in the real Hilbert space, both in the analytic method and in the algebraic method. The quaternionic solutions have many additional possibilities if compared to complex quantum mechanics ($\mathbbm{C}$QM), and thus there are many possible applications to these results in future research.
We consider non-local properties of quanternionic quantum mechanics, in which the complex numbers are replaced by the quaternions as the underlying algebra. Specifically, we show that it is possible to construct a non-local box. This allows one to rule out quaternionic quantum mechanics using assumptions about communication complexity or information causality.
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.
The breakdown of Ehrenfest's theorem imposes serious limitations on quaternionic quantum mechanics (QQM). In order to determine the conditions in which the theorem is valid, we examined the conservation of the probability density, the expectation value and the classical limit for a non-anti-hermitian formulation of QQM. The results also indicated that the non-anti-hermitian quaternionic theory is related to non-hermitian quantum mechanics, and thus the physical problems described with both of the theories should be related.
We study the elastic scattering of quantum particles based on a real Hilbert space approach to quaternionic quantum mechanics ($\mathbbm H$QM) and derive expression for the wave function, the phase shifts, as well as the optical theorem for the case of a hard sphere scattering potential. The strong agreement between these new quaternionic results and the corresponding results in complex quantum mechanics reinforce the validity of the $\mathbbm H$QM generalization of ordinary complex quantum mechanics ($\mathbbm C$QM).
We revisit the formulation of quantum mechanics over the quaternions and investigate the dynamical structure within this framework. Similar to standard complex quantum mechanics, time evolution is then mediated by a unitary operator which can be written as the exponential of the generator of time shifts. By imposing physical assumptions on the correspondence between the energy observable and the generator of time shifts, we prove that quaternionic quantum theory admits a time evolution only for systems with a quaternionic dimension of at most two. Applying the same strategy to standard complex…
It has been recently pointed out by Caves, Fuchs, and Rungta that real quantum mechanics (that is, quantum mechanics defined over real vector spaces provides an interesting foil theory whose study may shed some light on just which particular aspects of quantum entanglement are unique to standard quantum theory, and which ones are more generic over other physical theories endowed with this phenomenon. Following this work, we discuss some entanglement properties of two-rebits systems, making a comparison with the basic properties of two-qubits systems, i.e., the ones described by standard comple…
In his recent article [arXiv:1604.04950], Adler questions the usefulness of the bound found in our experimental search for genuine effects of hyper-complex quantum mechanics [arXiv:1602.01624]. Our experiment was performed using a black-box (instrumentalist) approach to generalized probabilistic theories; therefore, it does not assume a priori any particular underlying mechanism. From that point of view our experimental results do indeed place meaningful bounds on possible effects of "post-quantum theories", including quaternionic quantum mechanics. In his article, Adler compares our experimen…
The density operators obtained by taking partial traces do not represent proper mixtures of the subsystems of a compound physical system, but improper mixtures, since the coefficients in the convex sums expressing them never bear the ignorance interpretation. As a consequence, assigning states to these subsystems is problematical in standard quantum mechanics (subentity problem). Basing on the proposal provided in the SR interpretation of quantum mechanics, where improper mixtures are considered as true nonpure states conceptually distinct from proper mixtures, we show here that proper and imp…
In looking for quaternionic violations of quantum mechanics, we discuss the delay time for pure quaternionic potentials. The study shows in which energy region it is possible to amplify the difference between quaternionic and complex quantum mechanics.
We investigate the lateral displacement of electronic waves when partially and totally reflected by a quaternionic potential. Following the analogy between Quantum Mechanics and Optics we introduce a refractive index for the complex and the pure quaternionic case. For incidence greater than the critical one (total reflection) the quaternionic potentials amplify the lateral displacement found in the complex case. For incidence below the critical angle, we find, in the quaternionic case, an additional shift. The analytical formula found for the quaternionic lateral displacement gives the possibi…
We define the quaternionic quantum walk on a finite graph and investigate its properties. This walk can be considered as a natural quaternionic extension of the Grover walk on a graph. We explain the way to obtain all the right eigenvalues of a quaternionic matrix and a notable property derived from the unitarity condition for the quaternionic quantum walk. Our main results determine all the right eigenvalues of the quaternionic quantum walk by using complex eigenvalues of the quaternionic weighted matrix which is easily derivable from the walk. Since our derivation is owing to a quaternionic …
Recently, the quaternionic quantum walk was formulated by the first author as a generalization of discrete-time quantum walks. We treat the right eigenvalue problem of quaternionic matrices to analysis the spectra of its transition matrix. The way to obtain all the right eigenvalues of a quaternionic matrix is given. From the unitary condition on the transition matrix of the quaternionic quantum walk, we deduce some properties about it. Our main results, Theorem 5.3, determine all the right eigenvalues of a quaternionic quantum walk by use of those of the corresponding weighted matrix. In addi…
Quantum mechanics with quaternionic mass is considered. The momentum eigen-value equation with quaternionic mass yields the Klein-Gordon equation with a mass consisting of longitudinal and traverse masses. The scalar field total mass is found to be a sum of these masses. This field appears to be connected with two subfields conserving linear momentum. It is found that a particle with real mass satisfies the quantum Telegraph equation, whereas that one with quaternionic mass satisfies the Klein-Gordon equation. A quantum force acting on the particle is found to be proportional to its velocity. …
By using the analogy between optics and quantum mechanics, we obtain the Snell law for the planar motion of quantum particles in the presence of quaternionic potentials.
A formulation of quaternionic quantum mechanics ($\mathbb{H}$QM) is presented in terms of a real Hilbert space. Using a physically motivated scalar product, we prove the spectral theorem and obtain a novel quaternionic Fourier series. After a brief discussion on unitary operators in this formalism, we conclude that this quantum theory is indeed consistent, and can be a valuable tool in the search for new physics.
We solve Klein-Gordon equation (KGE) in the framework of the real Hilbert space approach to quaternionic quantum mechanics ($\mathbbm H$QM). The presented solution is the simplest ever obtained for quaternionic quantum theories, and the closest to the complex solution. The scattering of a quaternionic charged scalar particle from an electric field is also obtained. A remarkable feature of quaternionic scalar particles is the existence of massive light cone particles.
Analysis of the logical foundations of quantum mechanics indicates the possibility of constructing a theory using quaternionic Hilbert spaces. Whether this mathematical structure reflects reality is a matter for experiment to decide. We review the only direct search for quaternionic quantum mechanics yet carried out and outline a recent proposal by the present authors to look for quaternionic effects in correlated multi-particle systems. We set out how such experiments might distinguish between the several quaternionic models proposed in the literature.
We present some striking global consequences of a model quaternionic quantum field theory which is locally complex. We show how making the quaternionic structure a dynamical quantity naturally leads to the prediction of cosmic strings and non-baryonic hot dark matter candidates.
This study examines Quaternion Dirac solutions for an infinite square well. The quaternion result does not recover the complex result within a particular limit. This raises the possibility that quaternionic quantum mechanics may not be understood as a correction to complex quantum mechanics, but it may also be a structure that can be used to study phenomena that cannot be described through the framework of complex quantum mechanics.
Assuming the standard axioms for quaternionic quantum theory and a spatially localized scattering interaction, the $S$-matrix in quaternionic quantum theory is complex valued, not quaternionic. Using the standard connections between the $S$-matrix, the forward scattering amplitude for electromagnetic wave scattering, and the index of refraction, we show that the index of refraction is necessarily complex, not quaternionic. This implies that the recent optical experiment of Procopio et al. based on the Peres proposal does not test for hyper-complex or quaternionic quantum effects arising within…
In this paper, we analyze, by using a matrix approach, the dynamics of a non-relativistic particle in presence of a quaternionic potential barrier. The matrix method used to solve the quaternionic Schrodinger equation allows to obtain a closed formula for the transmission coefficient. Up to now, in quaternionic quantum mechanics, almost every discussion on the dynamics of non-relativistic particle was motived by or evolved from numerical studies. A closed formula for the transmission coefficient stimulates an analysis of qualitative differences between complex and quaternionic quantum mechanic…
The spectral singularity have been extensively studied over the last one and half decade for different non-Hermitian potentials in non-Hermitian quantum mechanics. The nature of spectral singularities have not been studied for the case of quaternionic potential. In the present work we perform an analytical study for the scattering from a quaternionic point interaction represented by delta function. New features of spectral singularities are observed which are different than the case of complex (non-quaternionic) point interaction. Most notable difference is the occurrence of spectral singulari…
The discrete-time quantum walk (QW) is determined by a unitary matrix whose component is complex number. Konno (2015) extended the QW to a walk whose component is quaternion.We call this model quaternionic quantum walk (QQW). The probability distribution of a class of QQWs is the same as that of the QW. On the other hand, a numerical simulation suggests that the probability distribution of a QQW is different from the QW. In this paper, we clarify the difference between the QQW and the QW by weak limit theorems for a class of QQWs.
Scattering discussion due to Double Dirac Equation in Quaternionic version of relativistic quantum mechanics has been studied in this paper in details. In such a quantum mechanics Dirac equation in presence vector and scalar potential has been considered. Then a Quaternionic double Dirac delta potential comes to our considered system which causes to scatter the particles. Scattering states of the particles have been derived as well as reflected and transmission coefficients are calculated.
We define a quaternionic extension of the Szegedy walk on a graph and study its right spectral properties. The condition for the transition matrix of the quaternionic Szegedy walk on a graph to be quaternionic unitary is given. In order to derive the spectral mapping theorem for the quaternionic Szegedy walk, we derive a quaternionic extension of the determinant expression of the second weighted zeta function of a graph. Our main results determine explicitly all the right eigenvalues of the quaternionic Szegedy walk by using complex right eigenvalues of the corresponding doubly weighted matrix…
A quaternionic version of Quantum Mechanics is constructed using the Schwinger's formulation based on measurements and a Variational Principle. Commutation relations and evolution equations are provided, and the results are compared with other formulations.
We consider the competition between decoherence processes and an iterated quantum purification protocol. We show that this competition can be modelled by a nonlinear map onto the quaternion space. This nonlinear map has complicated behaviours, inducing a fractal border between the area of the quantum states dominated by the effects of the purification and the area of the quantum states dominated by the effects of the decoherence. The states on the border are unstable. The embedding in a 3D space of this border is like a quaternionic Julia set or a Mandelbulb with a fractal inner structure.
The reduction paradigm of quantum interferometry is reanalyzed. In contrast to widespread opinion it is shown to be amenable to straightforward mathematical treatment within ``every-users'' simple-minded single particle quantum mechanics (without reduction postulate or the like), exploiting only its probabilistic content.
The discrete-time quantum walk (QW) has been extensively and intensively investigated for the last decade, whose coin operator is defined by a unitary matrix. We extend the QW to a walk determined by a unitary matrix whose component is quaternion. We call this model quaternionic quantum walk (QQW) and present some properties. This paper is the first step for the study on QQWs.
The three and five-dimensional convex sets of two-level complex and quaternionic quantum systems are studied in the Bayesian thermostatistical framework introduced by Lavenda. Associated with a given parameterization of each such set is a quantum Fisher (Helstrom) information matrix. The square root of its determinant (adopting an ansatz of Harold Jeffreys) provides a reparameterization-invariant prior measure over the set. Both such measures can be properly normalized and their univariate marginal probability distributions (which serve as structure functions) obtained. Gibbs (posterior) proba…
Optical quantum interferometry represents the oldest example of quantum metrology and it is at the source of quantum technologies. The original squeezed state scheme is now a significant element of the last version of gravitational wave detectors and various additional uses have been proposed. Further quantum enhanced schemes, from SU(1,1) interferometer to twin beam correlation interferometry, have also reached the stage of proof of principle experiments enlarging the field of experimental quantum interferometry and paving the way to several further applications ranging from Planck scale sign…
Non-classical states of light find applications in enhancing the performance of optical interferometric experiments, with notable example of gravitational wave-detectors. Still, the presence of decoherence hinders significantly the performance of quantum-enhanced protocols. In this review, we summarize the developments of quantum metrology with particular focus on optical interferometry and derive fundamental bounds on achievable quantum-enhanced precision in optical interferometry taking into account the most relevant decoherence processes including: phase diffusion, losses and imperfect inte…
We discuss the role of an external phase reference in quantum interferometry. We point out inconsistencies in the literature with regard to the use of the quantum Fisher information (QFI) in phase estimation interferometric schemes. We discuss the interferometric schemes with and without an external phase reference and show a proper way to use QFI in both situations.
We present an axiomatic modification of quaternionic quantum mechanics with a possible-worlds semantics capable of predicting essential "nonquantum" features of an observable universe model - the dimensionality and topology of spacetime, the existence, the signature and a specific form of a metric on it, and certain naturally distinguished directions (vistas) in spacetime unrelated to its metric properties.
Beginning with the quaternionic generalization of the quantum wave equation, we construct a simple model of relativistic quantum electrodynamics for massive dyons. A new quaternionic form of unified relativistic wave equation consisting of vector and scalar functions is obtained, and also satisfy the quaternionic momentum eigenvalue equation. Keeping in mind the importance of quantum field theory, we investigate the relativistic quantum structure of electromagnetic wave propagation of dyons. The present quantum theory of electromagnetism leads to generalized Lorentz gauge conditions for the el…
The patterns of fringes produced by an interferometer have long been important testbeds for our best contemporary theories of physics. Historically, interference has been used to contrast quantum mechanics to classical physics, but recently experiments have been performed that test quantum theory against even more exotic alternatives. A physically motivated family of theories are those where the state space of a two-level system is given by a sphere of arbitrary dimension. This includes classical bits, and real, complex and quaternionic quantum theory. In this paper, we consider relativity of …
The reduction paradigm of quantum interferometry and the objectivation problem in quantum measurements are reanalyzed. Both are shown to be amenable to straightforward mathematical treatment within "every-users" simple-minded quantum mechanics without reduction postulate etc., using only its probabilistic content.
We analyse the effect of decoherence and noise on quantum Fourier transform interferometry, in which a boson sampling photonic network is used to measure optical phase gradients. This novel type of metrology is shown to be robust against phase decoherence. One can also measure gradients using lower order correlations without substantial degradation. Our results involve estimation of up to a 100 x 100 matrix permanent.
Quaternionic quantum Hamiltonians describing nonrelativistic spin particles require the ambient physical space to have five dimensions. The quantum dynamics of a spin-1/2 particle system characterised by a generic such Hamiltonian is worked out in detail. It is shown that there exists, within the structure of quaternionic quantum mechanics, a canonical reduction to three spatial dimensions upon which standard quantum theory is retrieved. In this dimensional reduction, three of the five dynamical variables are shown to oscillate around a cylinder, thus behaving in a quasi one-dimensional manner…
This chapter reviews recent experiments on matter wave interferometry with large molecules. Starting from an elementary introduction to matter wave physics we discuss far-field diffraction and near-field interferometry with thermally excited many-body systems. We describe the constraints imposed by decoherence and dephasing effects, and present an outlook to the future challenges in macromolecule and cluster interferometry.
The archetypal quantum interferometry experiment yields an interference pattern that results from the indistinguishability of two spatiotemporal paths available to a photon or to a pair of entangled photons. A fundamental challenge in quantum interferometry is to perform such experiments with a higher number of paths, and over large distances. In particular, the distribution of such highly entangled states in long-haul optical fibers is one of the core concepts behind quantum information networks. We demonstrate that using indistinguishable frequency paths instead of spatiotemporal ones allows…
We discuss modern developments in quantum optics with organic molecules, clusters and nanoparticles -- in particular recent realizations of near-field matter-wave interferometry. A unified theoretical description in phase space allows us to describe quantum interferometry in position space and in the time domain on an equal footing. In order to establish matter-wave interferometers as a universal tool, which can accept and address a variety of nanoparticles, we elaborate on new quantum optical elements, such as diffraction gratings made of matter and light, as well as their absorptive and disp…
We characterize operationally meaningful quantum gains in a paradigmatic model of lossless multiple-phase interferometry and stress insufficiency of the analysis based solely on the concept of quantum Fisher information. We show that the advantage of the optimal simultaneous estimation scheme amounts to a constant factor improvement when compared with schemes where each phase is estimated separately -- contrary to a widely cited results claiming a better precision scaling in terms of the number of phases involved.
This paper descrives the first experimental evidence of antimatter-wave interference, a process at the heart of Quantum Physics and its interpretation. For the case of ordinary matter particles, interference phenomena have been observed in a variety of cases, ranging to electrons up to complex molecules. Here I present the first demonstration of single-positrons quantum interference.
A novel single-photon Mach-Zehnder interferometer terminated at two different frequencies realizes the nonlinear frequency conversion of optical quantum superposition states. The information-preserving character of the relevant unitary transformation has been experimentally demonstrated for input qubits and ebits. Besides its own intrinsic fundamental interest, the new scheme will find important applications in modern quantum information technology.
Nonlinear interferometers that replace beamsplitters in Mach-Zehnder interferometers with nonlinear amplifiers for quantum-enhanced phase measurements have drawn increasing interest in recent years, but practical quantum sensors based on nonlinear interferometry remain an outstanding challenge. Here, we demonstrate the first practical application of nonlinear interferometry by measuring the displacement of an atomic force microscope microcantilever with quantum noise reduction of up to 3~dB below the standard quantum limit, corresponding to a quantum-enhanced measurement of beam displacement o…
Quantum phenomena such as entanglement can improve fundamental limits on the sensitivity of a measurement probe. In optical interferometry, a probe consisting of $N$ entangled photons provides up to a $\sqrt{N}$ enhancement in phase sensitivity compared to a classical probe of the same energy. Here, we employ high-gain parametric down-conversion sources and photon-number-resolving detectors to perform interferometry with heralded quantum probes of sizes up to $N=8$ (i.e. measuring up to 16-photon coincidences). Our probes are created by injecting heralded photon-number states into an interfero…
We address the use of entanglement to improve the precision of generalized quantum interferometry, i.e. of binary measurements aimed to determine whether or not a perturbation has been applied by a given device. For the most relevant operations in quantum optics, we evaluate the optimal detection strategy and the ultimate bounds to the minimum detectable perturbation. Our results indicate that entanglement-assisted strategies improve the discrimination in comparison with conventional schemes. A concrete setup to approach performances of the optimal strategies is also suggested.
Quantum non-locality based on the well-known Bell inequality is of kinematic nature. A different type of quantum non-locality, the non-locality of the quantum equation of motion, is recently put forward with connection to the Aharonov-Bohm effect [Nature Phys. 6, 151 (2010)]. Evolution of the displacement operator provides an example to manifest such dynamic quantum non-locality. We propose an experiment using single-atom interferometry to test such dynamic quantum non-locality. We show how to measure evolution of the displacement operator with clod atoms in a spin-dependent optical lattice po…
