High-dimensional Hilbert Space Research Articles

Optimizing the signal-to-noise ratio of biphoton distribution measurements

Single-photon-sensitive cameras can now be used as massively parallel coincidence counters for entangled photon pairs. This enables measurement of biphoton joint probability distributions with orders-of-magnitude greater dimensionality and faster acquisition speeds than traditional raster scanning of point detectors; to date, however, there has been no general formula available to optimize data collection. Here we analyze the dependence of such measurements on count rate, detector noise properties, and threshold levels. We derive expressions for the biphoton joint probability distribution and its signal-to-noise ratio (SNR), valid beyond the low-count regime up to detector saturation. The analysis gives operating parameters for global optimum SNR that may be specified prior to measurement. We find excellent agreement with experimental measurements within the range of validity, and discuss discrepancies with the theoretical model for high thresholds. This work enables optimized measurement of the biphoton joint probability distribution in high-dimensional joint Hilbert spaces.

A large-alphabet three-party quantum key distribution protocol based on orbital and spin angular momenta hybrid entanglement

The orthogonality of the orbital angular momentum (OAM) eigenstates enables a single photon carry an arbitrary number of bits. Moreover, additional degrees of freedom (DOFs) of OAM can span a high-dimensional Hilbert space, which could greatly increase information capacity and security. Moreover, the use of the spin angular momentum---OAM hybrid entangled state can increase Shannon dimensionality, because photons can be hybrid entangled in multiple DOFs. Based on these observations, we develop a hybrid entanglement quantum key distribution (QKD) protocol to achieve three-party quantum key distribution without classical message exchanges. In our proposed protocol, a communicating party uses a spatial light modulator (SLM) and a specific phase hologram to modulate photons' OAM state. Similarly, the other communicating parties use their SLMs and the fixed different phase holograms to modulate the OAM entangled photon pairs, producing the shared key among the parties Alice, Bob and Charlie without classical message exchanges. More importantly, when the same operation is repeated for every party, our protocol could be extended to a multiple-party QKD protocol.

Demultiplexing of photonic temporal modes by a linear system

Temporally and spatially overlapping but field-orthogonal photonic temporal modes (TMs) that intrinsically span a high-dimensional Hilbert space are recently suggested as a promising means of encoding information on photons. Presently, the realization of photonic TM technology, particularly to retrieve the information it carries, i.e., demultiplexing of photonic TMs, is mostly dependent on nonlinear medium and frequency conversion. Meanwhile, its miniaturization, simplification, and optimization remain the focus of research. In this paper, we propose a scheme of TM demultiplexing using linear systems consisting of resonators with linear couplings. Specifically, we examine a unidirectional array of identical resonators with short environment correlations. For both situations with and without tunable couplers, propagation formulas are derived to demonstrate photonic TM demultiplexing capabilities. The proposed scheme, being entirely feasible with current technologies, might find potential applications in quantum information processing.

Nonlinear Low-Rank Matrix Completion for Human Motion Recovery.

Human motion capture data has been widely used in many areas, but it involves a complex capture process and the captured data inevitably contains missing data due to the occlusions caused by the actor's body or clothing. Motion recovery, which aims to recover the underlying complete motion sequence from its degraded observation, still remains as a challenging task due to the nonlinear structure and kinematics property embedded in motion data. Low-rank matrix completion based methods have shown promising performance in short-time-missing motion recovery problems. However, low-rank matrix completion, which is designed for linear data, lacks the theoretic guarantee when applied to the recovery of nonlinear motion data. To overcome this drawback, we propose a tailored nonlinear matrix completion model for human motion recovery. Within the model, we first learn a combined low-rank kernel via multiple kernel learning. By exploiting the learned kernel, we embed the motion data into a high dimensional Hilbert space where motion data is of desirable low-rank and we then use the low-rank matrix completion to recover motions. In addition, we add two kinematic constraints to the proposed model to preserve the kinematics property of human motion. Extensive experiment results and comparisons with five other state-of-the-art methods demonstrate the advantage of the proposed method.

RAPID: Measuring Deformation of Biological Tissues from MR Images Through the Riemannian Pseudo Kernel

Due to the nonlinear deformation of nonrigid and nonuniform tissues, it is challenging to accurately measure the displacements of feature points distributed on the inner parts, boundaries, and separatrices of tissue layers. To address this challenge, we propose a feature point matching technique called RAPID to measure MR 2D slice deformation of nonuniform and nonrigid biological tissues. We propose to use the covariance of several neighboring point statistics computed around a keypoint, as the keypoint descriptor. Inspired by the kernel methods, we advocate adopting a Riemannian pseudo kernel to map SPD matrices to a high dimensional Hilbert space, where the Euclidean geometry applies. We compare our RAPID with two existing schemes (i.e., SIFT and SURF). Our experimental results show that our RAPID is superior to SIFT and SURF, because the benefits offered by RAPID are two-fold. First, our RAPID increases the number of matched data points. Second, RAPID substantially improves the key-point matching accuracy of SIFT and SURF.

Cross Euclidean-to-Riemannian Metric Learning with Application to Face Recognition from Video.

Riemannian manifolds have been widely employed for video representations in visual classification tasks including video-based face recognition. The success mainly derives from learning a discriminant Riemannian metric which encodes the non-linear geometry of the underlying Riemannian manifolds. In this paper, we propose a novel metric learning framework to learn a distance metric across a Euclidean space and a Riemannian manifold to fuse average appearance and pattern variation of faces within one video. The proposed metric learning framework can handle three typical tasks of video-based face recognition: Video-to-Still, Still-to-Video and Video-to-Video settings. To accomplish this new framework, by exploiting typical Riemannian geometries for kernel embedding, we map the source Euclidean space and Riemannian manifold into a common Euclidean subspace, each through a corresponding high-dimensional Reproducing Kernel Hilbert Space (RKHS). With this mapping, the problem of learning a cross-view metric between the two source heterogeneous spaces can be converted to learning a single-view Euclidean distance metric in the target common Euclidean space. By learning information on heterogeneous data with the shared label, the discriminant metric in the common space improves face recognition from videos. Extensive experiments on four challenging video face databases demonstrate that the proposed framework has a clear advantage over the state-of-the-art methods in the three classical video-based face recognition scenarios.

Majorana Representation and Mean Field Approach for Interacting-Boson System☆☆Supported by the National Natural Science Foundation of China under Grant Nos. 11405008, 11175044, and the Plan for Scientific and Technological Development of Jilin Province under Grant No. 20160520173JH

The Majorana representation, which represents a quantum state by stars on the Bloch sphere, provides us an intuitive tool to study the quantum evolution in high dimensional Hilbert space. In this work, we investigate the second quantized model and the mean-field model for the interacting-boson system in the Majorana representation. It is shown that the motions of states in the two models are same in the linear case. Furthermore, the contribution of the nonlinear interaction to the star motions in the second quantized model can be expressed by a single star part which is equal to the nonlinear part of the equation for the star in mean-field model under large boson number limit and an extra part caused by the correlation between stars. These differences and relations can not only be reflected by the population differences between the two boson modes in the two models, but also lie with the differences between the continuous changes of the second quantized evolution with the nonlinear interacting strength and the critical behavior of the mean-field evolution which related to the self-trapping effect. The reason of the difference between the two models is also discussed by an effective Hamiltonian.

Realization of quantum permutation algorithm in high dimensional Hilbert space**Project supported by the Fundamental Research Funds for the Central Universities and the National Natural Science Foundation of China (Grant Nos. 11374008, 11374238, 11374239, and 11534008).

Quantum algorithms provide a more efficient way to solve computational tasks than classical algorithms. We experimentally realize quantum permutation algorithm using light’s orbital angular momentum degree of freedom. By exploiting the spatial mode of photons, our scheme provides a more elegant way to understand the principle of quantum permutation algorithm and shows that the high dimension characteristic of light’s orbital angular momentum may be useful in quantum algorithms. Our scheme can be extended to higher dimension by introducing more spatial modes and it paves the way to trace the source of quantum speedup.

Guest Editorial – Special Issue on Quantum Circuits

Guest Editorial – Special Issue on Quantum Circuits

Advanced Relativity: Multidimensionality of Consciousness and Mind, Origin of Life, PSI Phenomena

Evolution of life has its informational basis in higher-dimensional Hilbert spaces which also represent the origin of consciousness and mind. The idea of neurology that mind has its origin in neuronal activity of the brain, which is three-dimensional, is here improved with the model that the mind states exist in higher-dimensional Hilbert spaces. Mind is multidimensional and has the ability of PSI phenomena.

Noise suppression by quantum control before and after the noise

We discuss the possibility of protecting the state of a quantum system that goes through noise, by measurements/operations before and after the noise process. The aim is to seek for the optimal protocol that makes the input and output states as close as possible and clarify the role of the measurements therein. We consider two cases, one can perform quantum measurements/operations (i) only after the noise process and (ii) both before and after that. We prove in the two-dimensional Hilbert space that, in the case (i), the noise suppression is essentially impossible for all types of noise and, in the case (ii), the optimal protocol for the depolarizing noise is either the "do nothing" protocol or the "discriminate & reprepare" protocol. These protocols are not "truly quantum" and can be considered as classical. They involve no measurement or only use the measurement outcomes. These results describe the fundamental limitations in quantum mechanics from the viewpoint of control theory. Finally, we conjecture that a statement similar to the case (ii) holds for higher-dimensional Hilbert spaces and present some numerical evidence.

Discriminant Analysis on Riemannian Manifold of Gaussian Distributions for Face Recognition With Image Sets.

To address the problem of face recognition with image sets, we aim to capture the underlying data distribution in each set and thus facilitate more robust classification. To this end, we represent image set as the Gaussian mixture model (GMM) comprising a number of Gaussian components with prior probabilities and seek to discriminate Gaussian components from different classes. Since in the light of information geometry, the Gaussians lie on a specific Riemannian manifold, this paper presents a method named discriminant analysis on Riemannian manifold of Gaussian distributions (DARG). We investigate several distance metrics between Gaussians and accordingly two discriminative learning frameworks are presented to meet the geometric and statistical characteristics of the specific manifold. The first framework derives a series of provably positive definite probabilistic kernels to embed the manifold to a high-dimensional Hilbert space, where conventional discriminant analysis methods developed in Euclidean space can be applied, and a weighted Kernel discriminant analysis is devised which learns discriminative representation of the Gaussian components in GMMs with their prior probabilities as sample weights. Alternatively, the other framework extends the classical graph embedding method to the manifold by utilizing the distance metrics between Gaussians to construct the adjacency graph, and hence the original manifold is embedded to a lower-dimensional and discriminative target manifold with the geometric structure preserved and the interclass separability maximized. The proposed method is evaluated by face identification and verification tasks on four most challenging and largest databases, YouTube Celebrities, COX, YouTube Face DB, and Point-and-Shoot Challenge, to demonstrate its superiority over the state-of-the-art.To address the problem of face recognition with image sets, we aim to capture the underlying data distribution in each set and thus facilitate more robust classification. To this end, we represent image set as the Gaussian mixture model (GMM) comprising a number of Gaussian components with prior probabilities and seek to discriminate Gaussian components from different classes. Since in the light of information geometry, the Gaussians lie on a specific Riemannian manifold, this paper presents a method named discriminant analysis on Riemannian manifold of Gaussian distributions (DARG). We investigate several distance metrics between Gaussians and accordingly two discriminative learning frameworks are presented to meet the geometric and statistical characteristics of the specific manifold. The first framework derives a series of provably positive definite probabilistic kernels to embed the manifold to a high-dimensional Hilbert space, where conventional discriminant analysis methods developed in Euclidean space can be applied, and a weighted Kernel discriminant analysis is devised which learns discriminative representation of the Gaussian components in GMMs with their prior probabilities as sample weights. Alternatively, the other framework extends the classical graph embedding method to the manifold by utilizing the distance metrics between Gaussians to construct the adjacency graph, and hence the original manifold is embedded to a lower-dimensional and discriminative target manifold with the geometric structure preserved and the interclass separability maximized. The proposed method is evaluated by face identification and verification tasks on four most challenging and largest databases, YouTube Celebrities, COX, YouTube Face DB, and Point-and-Shoot Challenge, to demonstrate its superiority over the state-of-the-art.

Coherence-breaking channels and coherence sudden death

Quantum noise is ubiquitous to quantum systems as they incessantly interact with their surroundings and results in degrading useful resources such as coherence for single quantum systems and quantum correlations for multipartite systems. Given the importance of these resources in various quantum information processing protocols, it is of utmost importance to characterize how deteriorating is a particular noise scenario (quantum channel) in reference to a certain resource? Here we develop a theory of coherence breaking channels for single quantum systems. Any quantum channel on a single quantum system will be called a coherence breaking channel if it is an incoherent channel and maps any state to an incoherent state. We explicitly and exhaustively characterize these coherence breaking channels. Moreover, we define the coherence breaking indices for incoherent quantum channels and present various examples to elucidate this concept. We further introduce the concept of coherence sudden death under noisy evolutions and make an explicit connection of the phenomenon of coherence sudden death with the coherence breaking channels and the coherence breaking indices together with various suggestive examples. Furthermore, for higher dimensional Hilbert spaces, we establish the typicality of the dynamics of coherence under any incoherent quantum channel exploiting the concentration of measure phenomenon.

Berry phase and quantum entanglement in Majorana's stellar representation

By presenting the evolution of a quantum state with the trajectories of the Majorana stars on the Bloch sphere, the Majorana's stellar provides an intuitive geometric picture to study a quantum system with high-dimensional Hilbert space. We study the Berry phase and quantum entanglement by distributions and motions of these stars on the Bloch sphere. It is shown that both of these unique characters of quantum state can be perfectly represented by the Majorana stars. The former is expressed by the solid angles of Majorana star loops and the distance between stars. For the latter, the distances between stars are also found to be a tool for measuring and classifying the multiparticle entanglement of a symmetric multiqubit pure state. To demonstrate our theory, we study a typical spin model which is equivalent to an interacting boson model or an interacting multiqubit system. The self-trapping phenomenon within is also discussed via the Majorana stars.

Non-linear dynamic modeling of glucose in type 1 diabetes with kernel adaptive filters.

We propose a non-linear recursive solution to the problem of short-term prediction of glucose in type 1 diabetes. The Fixed Budget Quantized Kernel Least Mean Square (QKLMS-FB) algorithm is employed to construct a univariate model of subcutaneous glucose concentration, which: (i) handles nonlinearities by transforming the input space into a high-dimensional Reproducing Kernel Hilbert Space and, (ii) finds a sparse solution by retaining a representative subset of the training input vectors. The dataset comes from the continuous multi-day recordings of 15 type 1 patients in free-living conditions. QKLMS-FB produces an average root mean squared error of 18.66±3.19 mg/dl for a prediction horizon of 30 min with 82.04% of hypoglycemic readings and 93.30% of hyperglycemic ones being classified as clinically accurate or with benign errors. The effect of the prediction horizon is more evident in the hypoglycemic range.

Construction of high-dimensional universal quantum logic gates using a Λ system coupled with a whispering-gallery-mode microresonator.

High-dimensional quantum system provides a higher capacity of quantum channel, which exhibits potential applications in quantum information processing. However, high-dimensional universal quantum logic gates is difficult to achieve directly with only high-dimensional interaction between two quantum systems and requires a large number of two-dimensional gates to build even a small high-dimensional quantum circuits. In this paper, we propose a scheme to implement a general controlled-flip (CF) gate where the high-dimensional single photon serve as the target qudit and stationary qubits work as the control logic qudit, by employing a three-level Λ-type system coupled with a whispering-gallery-mode microresonator. In our scheme, the required number of interaction times between the photon and solid state system reduce greatly compared with the traditional method which decomposes the high-dimensional Hilbert space into 2-dimensional quantum space, and it is on a shorter temporal scale for the experimental realization. Moreover, we discuss the performance and feasibility of our hybrid CF gate, concluding that it can be easily extended to a 2n-dimensional case and it is feasible with current technology.

Nonlinear Landau-Zener tunneling in Majorana’s stellar representation

By representing the evolution of a quantum state with the trajectories of the stars on a Bloch sphere, the Majorana’s stellar representation provides an intuitive way to understand quantum motion in a high dimensional projective Hilbert space. In this work we show that the Majorana’s representation offers a very interesting and intuitive way to understand the nonlinear Landau-Zener tunneling. In particular, the breakdown of adiabaticity in this tunneling phenomenon can be understood as some of the stars never reaching the south pole. We also establish a connection between the Majorana stars in the second quantized model and the single star in the mean field model by using the reduced density matrix.

Foetal ECG extraction by support vector regression

A new method to extract the foetal electrocardiogram (ECG) using an abdominal ECG signal and a thoracic ECG signal of the mother is presented. The nonlinear mapping of the maternal ECG component from thoracic signal to an abdominal signal is approximated by the support vector regression (SVR) method. The foetal ECG signal is then extracted by subtracting the mapped thoracic signal from an abdominal signal. In an SVR algorithm, the nonlinear regression problem in an original signal space is transformed into a linear regression problem in high-dimensional Hilbert space by kernel technology. It avoids the ‘overfitting’ problem and could obtain good results for the small sample training set. The validity of the proposed method is validated by the real ECG signals, and compared with the existing method.

Observation of Four-Photon Orbital Angular Momentum Entanglement.

We demonstrate genuine multipartite quantum entanglement of four photons in their orbital angular momentum degrees of freedom, where a high-dimensional discrete Hilbert space is attached to each photon. This can encode more quantum information compared to the qubit case, but it is a long-standing problem to entangle more than two such photons. In our experiment we use pulsed spontaneous parametric down-conversion to produce the photon quadruplets, which allows us to detect about one four-photon event per second. By means of quantum state reconstruction and a suitable witness operator we find that the photon quadruplets form a genuine multipartite entangled symmetric Dicke state. This opens a new tool for addressing foundational questions in quantum mechanics, and for exploration of novel high-dimensional multiparty quantum information applications such as secret sharing.

Photonic EPR State from Quadratic Waveguide Array with Alternating Positive and Negative Couplings**Supported by the State Key Program for Basic Research in China under Grant No. 2012CB921802, the National Natural Science Foundations of China under Grant Nos. 91321312, 11321063 and 11422438

We propose the generation of photonic EPR state from quadratic waveguide array. Both the propagation constant and the nonlinearity in the array are designed to possess a periodical modulation along the propagation direction. This ensures that the photon pairs can be generated efficiently through the quasi-phase-matching spontaneous parametric down conversion by holding the spatial EPR entanglement in the fashion of correlated position and anticorrelated momentum. The Schmidt number which denotes the degree of EPR entanglement is calculated and it can approach a high value when the number of illuminated waveguide channels and the length of the waveguide array are properly chosen. These results suggest the quadratic waveguide array as a compact platform for engineering photonic quantum states in a high-dimensional Hilbert space.