Multidimensional Hilbert Space Research Articles

Relativistic dilation and contraction of the probabilities of quantum states of light at angular incidence

In this work we propose a formulation for the treatment of quantized states of light in photon-matter interaction. It is built in the multidimensional Hilbert space to describe the orbital angular momentum states, in such that probabilities naturally arise in a relativistic setting. Our findings indicate that the classical-relativistic variability of the probabilities can describe the quantized states of light, with the state variable being the variation of the photon angular momentum. It was verified that the probability to find a specific quantized state of light at angular incidence can be treated by relativistic probabilities. Interestingly, our results indicate that, when both states have equal odds, there is a synchronization between source and observer. We found that the relativistic effect in the perspective of the source referential is able to change the odds of an event occurring, dilating and contracting the probabilities of finding a quantized state of light at angular incidence.

Quantum-probabilistic SVD: complex-valued factorization of matrix data

The paper reports a method for compressed representation of matrix data on the principles of quantum theory. The method is formalized as complex-valued matrix factorization based on standard singular value decomposition. The developed approach establishes a bridge between standard methods of semantic data analysis and quantum models of cognition and decision. According to the quantum theory, real-valued observable quantities are generated by wavefunctions being complex-valued vectors in multidimensional Hilbert-space. Wavefunctions are defined as superpositions of basis vectors encoding composition of semantic factors. Basis vectors are found by singular value decomposition of the initial data matrix transformed to a real-valued amplitude form. Phase-dependent superposition amplitudes are found to optimize approximation of the source data. The resulting model represents the observed real-valued data as generated from a small number of basis wavefunctions superposed with complex-valued coefficients. The method is tested for random matrices of sizes from 3 × 3 to 12 × 12 and dimensionality of latent Hilbert-space from 2 to 4. The best approximation is achieved by encoding latent factors in normalized complex-valued amplitude vectors interpreted as wavefunctions generating the data. In terms of approximation fitness, the developed method surpasses standard truncated SVD of the same dimensionality. The mean advantage over the considered range of parameters is 22 %. The method permits cognitive interpretation in accord with the existing quantum models of cognition and decision. The method can be integrated in the algorithms of semantic data analysis including natural language processing. In these tasks, the obtained improvement of approximation translates to the increased precision of similarity measures, principal component analysis, advantage in classification, and document ranking methods. Integration with quantum models of cognition and decision is expected to boost methods of artificial intelligence and machine learning improving imitation of natural thinking.

Multimode superconducting circuits for realizing strongly coupled multiqubit processor units

Inter-qubit coupling and qubit connectivity in a processor are crucial for achieving high fidelity multi-qubit gates and efficient implementation of quantum algorithms. Typical superconducting processors employ relatively weak transverse inter-qubit coupling which are activated via frequency tuning or microwave drives. Here, we propose a class of multi-mode superconducting circuits which realize multiple transmon qubits with all-to-all longitudinal coupling. These "artificial molecules" directly implement a multi-dimensional Hilbert space that can be easily manipulated due to the always-on longitudinal coupling. We describe the basic technique to analyze such circuits, compute the relevant properties and discuss how to optimize them to create efficient small-scale quantum processors with universal programmability.

Study and Analysis of Multidimensional Hilbert Space Filling Curve and Its Applications - A Survey

Study and Analysis of Multidimensional Hilbert Space Filling Curve and Its Applications - A Survey

Integrated Optics Modules Based Proposal for Quantum Information Processing, Teleportation, QKD, and Quantum Error Correction Employing Photon Angular Momentum

To address key challenges for both quantum communication and quantum computing applications in a simultaneous manner, we propose to employ the photon angular momentum approach by invoking the well-known fact that photons carry both the spin angular momentum (SAM) and the orbital angular momentum (OAM). SAM is associated with polarization, while OAM is associated with azimuthal phase dependence of the complex electric field. Given that OAM eigenstates are mutually orthogonal, in principle, an arbitrary number of bits per single photon can be transmitted. The ability to generate/analyze states with different photon angular momentum, by using either holographic or interferometric methods, allows the realization of quantum states in multidimensional Hilbert space. Because OAM states provide an infinite basis state, while SAM states are 2-D only, the OAM can also be used to increase the security for quantum key distribution (QKD) applications and improve computational power for quantum computing applications. The goal of this paper is to describe photon angular momentum based deterministic universal quantum qudit gates, namely, {generalized-X, generalized-Z, generalized-CNOT} qudit gates, and different quantum modules of importance for various applications, including (fault-tolerant) quantum computing, teleportation, QKD, and quantum error correction. For instance, the basic quantum modules for quantum teleportation applications include the generalized-Bell-state generation module and the QFT-module. The basic quantum module for quantum error correction and fault-tolerant computing is the nonbinary syndrome calculator module. The basic module for entanglement assisted QKD is either the generalized-Bell-state generation module or the Weyloperator-module. The possibility of implementing all these modules in integrated optics is discussed as well. Finally, we provide security analysis of entanglement assisted multidimensional QKD protocols, employing the proposed qudit modules, by taking into account the imperfect generation of OAM modes.

Exciton propagation via quantum walks based on non-Hermitian coin flip operations

We examine the coherent propagation of the one-dimensional Frenkel exciton (correlated electron-hole pair system) based on a model of a quantum walker in multi-dimensional Hilbert space. The walk is governed by a non-Hermitian coin flip operation coupled to a generalized shift process. The dissipative coin flip operation is associated with amplitude leakages at occupied sites, typical of processes which occur when an exciton is transferred along dimer sites in photosynthetic protein complexes. We analyze the characteristics probability distribution of the one-dimensional quantum walk for various system parameters, and examine the complex interplay between non-Markovian signatures and amplitude leakages within the Hilbert position subspace. The visibility of topological defects such as exceptional points, and non-Markovian signatures via quantum tomography based spectroscopic measurements are discussed.

Open quantum dynamics of single-photon optomechanical devices

We study the quantum dynamics of a Michelson interferometer with Fabry-Perot cavity arms and one movable end mirror, and driven by a single photon---an optomechanical device previously studied by Marshall et al. as a device that searches for gravity decoherence. We obtain an exact analytical solution for the system's quantum mechanical equations of motion, including details about the exchange of the single photon between the cavity mode and the external continuum. The resulting time evolution of the interferometer's fringe visibility displays interesting new features when the incoming photon's frequency uncertainty is narrower or comparable to the cavity's line width---only in the limiting case of much broader-band photon does the result return to that of Marshall et al., but in this case the photon is not very likely to enter the cavity and interact with the mirror, making the experiment less efficient and more susceptible to imperfections. In addition, we show that in the strong-coupling regime, by engineering the incoming photon's wave function, it is possible to prepare the movable mirror into an arbitrary quantum state of a multidimensional Hilbert space.

Modal Characterization using Principal Component Analysis: application to Bessel, higher-order Gaussian beams and their superposition

The modal characterization of various families of beams is a topic of current interest. We recently reported a new method for the simultaneous determination of both the azimuthal and radial mode indices for light fields possessing orbital angular momentum. The method is based upon probing the far-field diffraction pattern from a random aperture and using the recorded data as a ‘training set'. We then transform the observed data into uncorrelated variables using the principal component analysis (PCA) algorithm. Here, we show the generic nature of this approach for the simultaneous determination of the modal parameters of Hermite-Gaussian and Bessel beams. This reinforces the widespread applicability of this method for applications including information processing, spectroscopy and manipulation. Additionally, preliminary results demonstrate reliable decomposition of superpositions of Laguerre-Gaussians, yielding the intensities and relative phases of each constituent mode. Thus, this approach represents a powerful method for characterizing the optical multi-dimensional Hilbert space.

High spatial entanglement via chirped quasi-phase-matched optical parametric down-conversion

By making use of the spatial shape of paired photons, parametric down-conversion allows the generation of two-photon entanglement in a multidimensional Hilbert space. How much entanglement can be generated in this way? In principle, the infinite-dimensional nature of the spatial degree of freedom renders unbounded the amount of entanglement available. However, in practice, the specific configuration used, namely its geometry, the length of the nonlinear crystal and the size of the pump beam, can severely limit the value that could be achieved. Here we show that the use of quasi-phase-matching engineering allows to increase the amount of entanglement generated, reaching values of tens of ebits of entropy of entanglement under different conditions. Our work thus opens a way to fulfill the promise of generating massive spatial entanglement under a diverse variety of circumstances, some more favorable for its experimental implementation.

High-efficient quantum secret sharing based on the Chinese remainder theorem via the orbital angular momentum entanglement analysis

We investigate a novel quantum secret sharing (QSS) based on the Chinese remainder theory (CRT) in multi-dimensional Hilbert space with the orbital angular momentum (OAM) entanglement analysis. The secret is divided and then allotted to two or more participants who prepare pairs of photons in the OAM-entanglement states. The initial secret can be restored jointly by legal participants via the OAM-entanglement analysis on the corresponding photons. Its security is guaranteed from the OAM entanglement of photons that are established through the spin angular momentum (SAM) entanglement analysis performed on the generated SAM-based OAM hybrid entanglement photons. It provides an alternative technique for the QSS while producing the OAM entanglement photons in the combined multi-dimensional OAM Hilbert space, where the CRT is conducted properly for sharing the conventional secret among legal participants.

Entanglement swapping with pure orbital angular momentum Bell-state analysis

We investigate a framework of an orbital angular momentum (OAM) entanglement swapping in a multi-dimensional Hilbert space with the spin angular momentum (SAM)-based orbital angular momentum (OAM) Bell-sate analysis. By the implementations of entanglement swapping with the SAM and OAM Bell-state measurements subsequently, the OAM entanglement states (qudits) are generated and then transferred between photons in multi-dimensional Hilbert space in a point-to-point fashion. In the proposed scheme, two pairs of the SAM-based OAM hybrid entanglement photons are deployed to conduct the successive SAM and OAM Bell-state measurements. It provides an alternative technique to transfer pure OAM Bell-states in qudits, which illustrates a possible experimental approach for devising a full repeater in a complex quantum computation network where entanglement swapping serves as a critical constituent.

Orbital angular momentum correlations of entangled paired photons

The generation of paired photons entangled in the spatial degree of freedom, i.e., in orbital angular momentum, offers a convenient physical resource to investigate the nature of entanglement in a multidimensional Hilbert space with controllable dimensionality. The two main physical processes that generate pairs of photons which show correlations in orbital angular momentum are (a) spontaneous parametric down-conversion (SPDC), and (b) Raman transitions induced in atomic ensembles. One question naturally arises: what kinds of correlations exist between the orbital angular momentum of the generated photons? The answer might be different if we consider the whole quantum state of the generated photons, i.e., all possible directions where the pairs of photons can be emitted, or if we consider only a small section of the full set of directions.

Physical realization of coupled Hilbert-space mirrors for quantum-state engineering

Manipulation of superpositions of discrete quantum states has a mathematical counterpart in the motion of a unit-length statevector in an N-dimensional Hilbert space. Any such statevector motion can be regarded as a succession of two-dimensional rotations. But the desired statevector change can also be treated as a succession of reflections, the generalization of Householder transformations. In multidimensional Hilbert space such reflection sequences offer more efficient procedures for statevector manipulation than do sequences of rotations. We here show how such reflections can be designed for a system with two degenerate levels—a generalization of the traditional two-state atom—that allows the construction of propagators for angular momentum states. We use the Morris–Shore transformation to express the propagator in terms of Morris–Shore basis states and Cayley–Klein parameters, which allows us to connect properties of laser pulses to Hilbert-space motion. Under suitable conditions on the couplings and the common detuning, the propagators within each set of degenerate states represent products of generalized Householder reflections, with orthogonal vectors. We propose physical realizations of this novel geometrical object with resonant, near-resonant and far-off-resonant laser pulses. We give several examples of implementations in real atoms or molecules.

On the calculation of absolute spectral densities

A new method of calculating the absolute spectral density of a Hamiltonian operator is derived and discussed. The spectral density is expressed as the solution of an integral equation in which the kernel is a renormalized one-sided energy correlation function of the full microcanonical density operator and a microcanonical density operator for a reference Hamiltonian. The integral operator associated with this equation transforms a known spectral density function for the reference Hamiltonian into the spectral density of the full Hamiltonian. The integral equation, by virtue of its formulation in energy space, is inherently one-dimensional and offers no storage difficulties, and the elements of its kernel may be computed by applying the Lanczos algorithm to randomly selected eigenfunctions of the reference Hamiltonian. This spectral density correlation method offers a number of advantages over variational methods. In particular, it has the potential for overcoming the hitherto largely insurmountable problem of tracing over a multidimensional Hilbert space in order to compute the spectral density of a nonseparable molecular Hamiltonian.