Representation Of Quantum States Research Articles

Eigenstate extraction with neural-network tomography

We discuss quantum state tomography via a stepwise reconstruction of the eigenstates of the mixed states produced in experiments. Our method is tailored to the experimentally relevant class of nearly pure states or simple mixed states, which exhibit dominant eigenstates and thus lend themselves to low-rank approximations. The developed scheme is applicable to any pure-state tomography method, promoting it to mixed-state tomography. Here, we demonstrate it with machine learning-inspired pure-state tomography based on neural-network representations of quantum states. The latter have been shown to efficiently approximate generic classes of complex (pure) states of large quantum systems. We test our method by applying it to experimental data from trapped ion experiments with four to eight qubits.

Probing criticality in quantum spin chains with neural networks

The numerical emulation of quantum systems often requires an exponential number of degrees of freedom which translates to a computational bottleneck. Methods of machine learning have been used in adjacent fields for effective feature extraction and dimensionality reduction of high-dimensional datasets. Recent studies have revealed that neural networks are further suitable for the determination of macroscopic phases of matter and associated phase transitions as well as efficient quantum state representation. In this work, we address quantum phase transitions in quantum spin chains, namely the transverse field Ising chain and the anisotropic XY chain, and show that even neural networks with no hidden layers can be effectively trained to distinguish between magnetically ordered and disordered phases. Our neural network acts to predict the corresponding crossovers finite-size systems undergo. Our results extend to a wide class of interacting quantum many-body systems and illustrate the wide applicability of neural networks to many-body quantum physics.

SU(2) Symmetry of Qubit States and Heisenberg–Weyl Symmetry of Systems with Continuous Variables in the Probability Representation of Quantum Mechanics

In view of the probabilistic quantizer–dequantizer operators introduced, the qubit states (spin-1/2 particle states, two-level atom states) realizing the irreducible representation of the S U ( 2 ) symmetry group are identified with probability distributions (including the conditional ones) of classical-like dichotomic random variables. The dichotomic random variables are spin-1/2 particle projections m = ± 1 / 2 onto three perpendicular directions in the space. The invertible maps of qubit density operators onto fair probability distributions are constructed. In the suggested probability representation of quantum states, the Schrödinger and von Neumann equations for the state vectors and density operators are presented in explicit forms of the linear classical-like kinetic equations for the probability distributions of random variables. The star-product and quantizer–dequantizer formalisms are used to study the qubit properties; such formalisms are discussed for photon tomographic probability distribution and its correspondence to the Heisenberg–Weyl symmetry properties.

Conventional Quantum Statistics with a Probability Distribution Describing Quantum System States

The review of a new probability representation of quantum states is presented, where the states are described by conventional probability distribution functions. The invertible map of the probability distribution onto density operators in the Hilbert space is found using the introduced operators called a quantizer–dequantizer, which specify the invertible map of operators of quantum observables onto functions and a product of the operators onto an associative product (star product) of the functions. Examples of a quantum oscillator and a spin-1/2 particle are considered. The kinetic equations for probabilities, specifying the evolution of the states of a quantum system, which are equivalent to Schrodinger and von Neumann equations, are derived explicitly.

Overcoming the Tradeoff Between Accuracy and Compactness in Decision Diagrams for Quantum Computation

Quantum computation promises to solve many hard or infeasible problems substantially faster than classical solutions. The involvement of big players like Google, IBM, Intel, Rigetti, or Microsoft furthermore led to a momentum which increases the demand for automated design methods for quantum computations. In this context, decision diagrams for quantum computation provide a major pillar as they allow to efficiently represent quantum states and quantum operations which, otherwise, have to be described in terms of exponentially large state vectors and unitary matrices. However, current decision diagrams for the quantum domain suffer from a tradeoff between accuracy and compactness, since: 1) small errors that are inevitably introduced by the limited precision of floating-point arithmetic can harm the compactness (i.e., the size of the decision diagram) significantly and 2) overcompensating these errors (to increase compactness) may lead to an information loss and introduces numerical instabilities. In this article, we describe and evaluate the effects of this tradeoff which clearly motivates the need for a solution that is perfectly accurate and compact at the same time. More precisely, we show that the tradeoff indeed weakens current design automation approaches for quantum computation (possibly leading to corrupted results or infeasible run-times). To overcome this, we propose an alternative approach that utilizes an algebraic representation of the occurring complex and irrational numbers and outline how this can be incorporated in a decision diagram which is suited for quantum computation. Evaluations show that-at the cost of an overhead which is moderate in many cases-the proposed algebraic solution indeed overcomes the tradeoff between accuracy and compactness that is present in current numerical solutions.

Evolution of a Qubit Density Matrix in a Probabilistic Representation of Quantum States

A system of equations is obtained for a generalized qubit density matrix that depends on two moments in time. It determines the evolution of the quantum state of a qubit and transforms into the von Neumann equation for a qubit density matrix at coinciding times. The states of a harmonic oscillator in a probabilistic representation of quantum states are considered. Kinetic equations for the probabilities of projections of a spin +1/2 onto three perpendicular directions in space that determine the state of a qubit are obtained.

Monogamy of correlations and entropy inequalities in the Bloch picture

We investigate monogamy of correlations and entropy inequalities in the Bloch representation. Here, both can be understood as direct relations between different correlation tensor elements and thus appear intimately related. To that end we introduce the split Bloch basis, that is particularly useful for representing quantum states with low dimensional support and thus amenable to purification arguments. Furthermore, we find dimension dependent entropy inequalities for the Tsallis 2-entropy. In particular, we present an analogue of the strong subadditivity and a quadratic entropy inequality. These relations are shown to be stronger than subadditivity for finite dimensional cases.

Quantum Aspects of High Dimensional Conceptual Space: a Model for Achieving Consciousness

Cognitive frameworks best represent cognitive information and knowledge. Several classical probability theory (CPT)-based cognitive frameworks were proposed in the literature. Recently, CPT has failed in explaining certain cognitive processes while quantum theories were successful in explaining the same. In this work, we integrate two cognitive frameworks namely conceptual spaces and 3-way formal concept analysis (3WFCA) to propose a high dimensional conceptual space (HDCS). Our new insights into the analysis of this proposal reveal its quantum characteristics. Among different cognitive processes that can be modelled, our interest is on phenomenal consciousness. Accordingly, we have proposed a formal method to achieve consciousness. We have also proposed an algorithm for the conceptual scaling of a cognitive scenario. HDCS represents a cognitive state using quality dimensions, attributes and their relations. Subsequently, the proposed model makes novel use of agent-environment interaction paradigm for guiding the interaction of HDCS with conceptually scaled cognitive scenario. Cognitive-state representation in HDCS is analogous to quantum state representation in a N-qubit system. Cognitive states are learnt through the parallel accumulation of evidences against all the attributes of multiple dimensions. The interaction between the HDCS and the scenario facilitates the identification and removal of uncertainties. The identified uncertainty coupled with its resolution time resembles Heisenberg-like uncertainty. The analogous of quantum two-slit self-interference would be comparison of evidence accumulation in HDCS with the preceding version of itself. Consequently, this research reveals that modelling consciousness requires a high dimensional space rather than set theoretic structures and support from quantum theories.

Detector-Agnostic Phase-Space Distributions.

The representation of quantum states via phase-space functions constitutes an intuitive technique to characterize light. However, the reconstruction of such distributions is challenging as it demands specific types of detectors and detailed models thereof to account for their particular properties and imperfections. To overcome these obstacles, we derive and implement a measurement scheme that enables a reconstruction of phase-space distributions for arbitrary states whose functionality does not depend on the knowledge of the detectors, thus defining the notion of detector-agnostic phase-space distributions. Our theory presents a generalization of well-known phase-space quasiprobability distributions, such as the Wigner function. We implement our measurement protocol, using state-of-the-art transition-edge sensors without performing a detector characterization. Based on our approach, we reveal the characteristic features of heralded single- and two-photon states in phase space and certify their nonclassicality with high statistical significance.

Deep Learning of Matrix Product States

Machine learning and especially deep learning architectures provide a fresh perspective on the study of many body physics phenomena. In this paper, we employ Restricted Boltzmann machines (RBM) to represent quantum many-body states and find connections that can be made useful to quantum many-body physics research, ultimately leading to a better understanding of the fundamental nature of entanglement entropy in quantum physics. In this work, we establish the conditions for translating RBMs into Matrix Product States (MPS), showing that deep learning algorithms can be exploited as a powerful tool for an efficient representation of quantum states. We present an algorithm for mapping an RBM into an MPS, with a specific proof for Ising model. We discuss the upper entropy bound and entanglement properties resulting from such a connection, together with the consequences of our results in a broader context.

Machine learning approach to dynamical properties of quantum many-body systems

Variational representations of quantum states abound and have successfully been used to guess ground-state properties of quantum many-body systems. Some are based on partial physical insight (Jastrow, Gutzwiller projected, and fractional quantum Hall states, for instance), and others operate as a black box that may contain information about the underlying structure of entanglement and correlations (tensor networks, neural networks) and offer the advantage of a large set of variational parameters that can be efficiently optimized. However, using variational approaches to study excited states and, in particular, calculating the excitation spectrum, remains a challenge. We present a variational method to calculate the dynamical properties and spectral functions of quantum many-body systems in the frequency domain, where the Green's function of the problem is encoded in the form of a restricted Boltzmann machine (RBM). We introduce a natural gradient descent approach to solve linear systems of equations and use Monte Carlo to obtain the dynamical correlation function. In addition, we propose a strategy to regularize the results that improves the accuracy dramatically. As an illustration, we study the dynamical spin structure factor of the one dimensional $J_1-J_2$ Heisenberg model. The method is general and can be extended to other variational forms.

Time-dependent density matrix renormalization group quantum dynamics for realistic chemical systems.

Electronic and/or vibronic coherence has been found by recent ultrafast spectroscopy experiments in many chemical, biological, and material systems. This indicates that there are strong and complicated interactions between electronic states and vibration modes in realistic chemical systems. Therefore, simulations of quantum dynamics with a large number of electronic and vibrational degrees of freedom are highly desirable. Due to the efficient compression and localized representation of quantum states in the matrix-product state (MPS) formulation, time-evolution methods based on the MPS framework, which we summarily refer to as tDMRG (time-dependent density-matrix renormalization group) methods, are considered to be promising candidates to study the quantum dynamics of realistic chemical systems. In this work, we benchmark the performances of four different tDMRG methods, including global Taylor, global Krylov, and local one-site and two-site time-dependent variational principles (1TDVP and 2TDVP), with a comparison to multiconfiguration time-dependent Hartree and experimental results. Two typical chemical systems of internal conversion and singlet fission are investigated: one containing strong and high-order local and nonlocal electron-vibration couplings and the other exhibiting a continuous phonon bath. The comparison shows that the tDMRG methods (particularly, the 2TDVP method) can describe the full quantum dynamics in large chemical systems accurately and efficiently. Several key parameters in the tDMRG calculation including the truncation error threshold, time interval, and ordering of local sites were also investigated to strike the balance between efficiency and accuracy of results.

Superposition Principle and Born’s Rule in the Probability Representation of Quantum States

The basic notion of physical system states is different in classical statistical mechanics and in quantum mechanics. In classical mechanics, the particle system state is determined by its position and momentum; in the case of fluctuations, due to the motion in environment, it is determined by the probability density in the particle phase space. In quantum mechanics, the particle state is determined either by the wave function (state vector in the Hilbert space) or by the density operator. Recently, the tomographic-probability representation of quantum states was proposed, where the quantum system states were identified with fair probability distributions (tomograms). In view of the probability-distribution formalism of quantum mechanics, we formulate the superposition principle of wave functions as interference of qubit states expressed in terms of the nonlinear addition rule for the probabilities identified with the states. Additionally, we formulate the probability given by Born’s rule in terms of symplectic tomographic probability distribution determining the photon states.

Quantum Neural Network States: A Brief Review of Methods and Applications

AbstractOne of the main challenges of quantum many‐body physics is the exponential growth in the dimensionality of the Hilbert space with system size. This growth makes solving the Schrödinger equation of the system extremely difficult. Nonetheless, many physical systems have a simplified internal structure that typically makes the parameters needed to characterize their ground states exponentially smaller. Many numerical methods then become available to capture the physics of the system. Among modern numerical techniques, neural networks, which show great power in approximating functions and extracting features of big data, are now attracting much interest. In this work, the progress in using artificial neural networks to build quantum many‐body states is reviewed. The Boltzmann machine representation is taken as a prototypical example to illustrate various aspects of the neural network states. The classical neural networks are also briefly reviewed, and it is illustrated how to use neural networks to represent quantum states and density operators. Some physical properties of the neural network states are discussed. For applications, the progress in many‐body calculations based on neural network states, the neural network state approach to tomography, and the classical simulation of quantum computing based on Boltzmann machine states are briefly reviewed.

Probability Representation of Quantum States as a Renaissance of Hidden Variables— God Plays Coins

We develop an approach where the quantum system states and quantum observables are described as in classical statistical mechanics -- the states are identified with probability distributions and observables, with random variables. An example of the spin-1/2 state is considered. We show that the triada of Malevich's squares can be used to illustrate the qubit state. We formulate the superposition principle of quantum states in terms of probabilities determining the quantum states. New formulas for nonlinear addition rules of probabilities providing the probabilities associated with the interference of quantum states are obtained. The evolution equation for quantum states is given in the form of a kinetic equation for the probability distribution identified with the state.

Classical (Local and Contextual) Probability Model for Bohm-Bell Type Experiments: No-Signaling as Independence of Random Variables.

We start with a review on classical probability representations of quantum states and observables. We show that the correlations of the observables involved in the Bohm–Bell type experiments can be expressed as correlations of classical random variables. The main part of the paper is devoted to the conditional probability model with conditioning on the selection of the pairs of experimental settings. From the viewpoint of quantum foundations, this is a local contextual hidden-variables model. Following the recent works of Dzhafarov and collaborators, we apply our conditional probability approach to characterize (no-)signaling. Consideration of the Bohm–Bell experimental scheme in the presence of signaling is important for applications outside quantum mechanics, e.g., in psychology and social science. The main message of this paper (rooted to Ballentine) is that quantum probabilities and more generally probabilities related to the Bohm–Bell type experiments (not only in physics, but also in psychology, sociology, game theory, economics, and finances) can be classically represented as conditional probabilities.

Wave-kinetic approach to the Schrödinger–Newton equation

We discuss the Schrödinger–Newton (SN) equation in the context of cold atom physics. For that purpose, we establish the wave-kinetic equation equivalent to the SN equation, which stays valid in different physical scenarios relevant to cold atoms. They include: (1) the usual scenario of matter confined in a self-gravitating field, (2) atomic molasses, confined and cooled by laser beams in a magneto-optical trap (MOT), (3) Bose–Einstein condensates, with or without long range dipolar interactions, and (4) electron states in a quantum plasma. We show that these different systems can be described by a formally identical equation, and they also manifest similar elementary excitations. The wave-kinetic equation is obtained by following the well-known Wigner–Moyal procedure, allowing the representation of quantum states in a classical phase-space. It is particularly well suited to discuss kinetic properties associated with Landau damping, as shown. We also consider generalisation of the SN equation onto the relativistic domain, and its impact on the proposed wave-kinetic description.

Quasistates and quasiprobabilities

The quasiprobability representation of quantum states addresses two main concerns, the identification of nonclassical features and the decomposition of the density operator. While the former aspect is a main focus of current research, the latter decomposition property has been studied less frequently. In this contribution, we introduce a method for the generalized expansion of quantum states in terms of so-called quasistates. In contrast to the quasiprobability decomposition through nonclassical distributions and pure-state operators, our technique results in classical probabilities and nonpositive semidefinite operators, defining the notion of quasistates, that carry the information about the nonclassical characteristics of the system. Therefore, our method presents a complementary approach to prominent quasiprobability representations. We explore the usefulness of our technique with several examples, including applications for quantum state reconstruction and the representation of nonclassical light. In addition, using our framework, we also demonstrate that inseparable quantum correlations can be described in terms of classical joint probabilities and tensor-product quasistates for an unambiguous identification of quantum entanglement.

Quantum suprematism picture of Triada of Malevich’s squares for spin states and the parametric oscillator evolution in the probability representation of quantum mechanics

Review of tomographic probability representation of quantum states is presented both for oscillator systems with continious variables and spin–systems with discrete variables. New entropy–information inequalities are obtained for Franck–Condon factors. Density matrices of qudit states are expressed in terms of probabilities of artificial qubits as well as the quantum suprematism approach to geometry of these states using the triadas of Malevich squares is developed. Examples of qubits, qutrits and ququarts are considered.

Quantum non-Gaussianity and quantification of nonclassicality

The algebraic quantification of nonclassicality, which naturally arises from the quantum superposition principle, is related to properties of regular nonclassicality quasiprobabilities. The latter are obtained by non-Gaussian filtering of the Glauber-Sudarshan $P$~function. They yield lower bounds for the degree of nonclassicality. We also derive bounds for convex combinations of Gaussian states for certifying quantum non-Gaussianity directly from the experimentally accessible nonclassicality quasiprobabilities. Other quantum-state representations, such as $s$-parametrized quasiprobabilities, insufficiently indicate or even fail to directly uncover detailed information on the properties of quantum states. As an example, our approach is applied to multi-photon-added squeezed vacuum states.