Schmidt Decomposition Research Articles

Many-body entanglement in fermion systems

We introduce a general bipartite-like representation and Schmidt decomposition of an arbitrary pure state of $N$ indistinguishable fermions, based on states of $M<N$ and $(N-M)$ fermions. It is directly connected with the reduced $M$- and $(N-M)$-body density matrices (DMs), which have the same spectrum in such states. The concept of $M$-body entanglement emerges naturally in this scenario, generalizing that of one-body entanglement. Rigorous majorization relations satisfied by the normalized $M$-body DM are then derived, which imply that the associated entropy will not increase, on average, under a class of operations which have these DMs as post-measurement states. Moreover, such entropy is an upper bound to the average bipartite entanglement entropy generated by a class of operations which map the original state to a bipartite state of $M$ and $N-M$ effectively distinguishable fermions. Analytic evaluation of the spectrum of $M$-body DMs in some strongly correlated fermionic states is also provided.

A Review of Model Order Reduction Methods for Large-Scale Structure Systems

The large-scale structure systems in engineering are complex, high dimensional, and variety of physical mechanism couplings; it will be difficult to analyze the dynamic behaviors of complex systems quickly and optimize system parameters. Model order reduction (MOR) is an efficient way to address those problems and widely applied in the engineering areas. This paper focuses on the model order reduction of high-dimensional complex systems and reviews basic theories, well-posedness, and limitations of common methods of the model order reduction using the following methods: center manifold, Lyapunov–Schmidt (L-S), Galerkin, modal synthesis, and proper orthogonal decomposition (POD) methods. The POD is a powerful and effective model order reduction method, which aims at obtaining the most important components of a high-dimensional complex system by using a few proper orthogonal modes, and it is widely studied and applied by a large number of researchers in the past few decades. In this paper, the POD method is introduced in detail and the main characteristics and the existing problems of this method are also discussed. POD is classified into two categories in terms of the sampling and the parameter robustness, and the research progresses in the recent years are presented to the domestic researchers for the study and application. Finally, the outlooks of model order reduction of high-dimensional complex systems are provided for future work.

Quantum Information in Neural Systems

Identifying the physiological processes in the central nervous system that underlie our conscious experiences has been at the forefront of cognitive neuroscience. While the principles of classical physics were long found to be unaccommodating for a causally effective consciousness, the inherent indeterminism of quantum physics, together with its characteristic dichotomy between quantum states and quantum observables, provides a fertile ground for the physical modeling of consciousness. Here, we utilize the Schrödinger equation, together with the Planck–Einstein relation between energy and frequency, in order to determine the appropriate quantum dynamical timescale of conscious processes. Furthermore, with the help of a simple two-qubit toy model we illustrate the importance of non-zero interaction Hamiltonian for the generation of quantum entanglement and manifestation of observable correlations between different measurement outcomes. Employing a quantitative measure of entanglement based on Schmidt decomposition, we show that quantum evolution governed only by internal Hamiltonians for the individual quantum subsystems preserves quantum coherence of separable initial quantum states, but eliminates the possibility of any interaction and quantum entanglement. The presence of non-zero interaction Hamiltonian, however, allows for decoherence of the individual quantum subsystems along with their mutual interaction and quantum entanglement. The presented results show that quantum coherence of individual subsystems cannot be used for cognitive binding because it is a physical mechanism that leads to separability and non-interaction. In contrast, quantum interactions with their associated decoherence of individual subsystems are instrumental for dynamical changes in the quantum entanglement of the composite quantum state vector and manifested correlations of different observable outcomes. Thus, fast decoherence timescales could assist cognitive binding through quantum entanglement across extensive neural networks in the brain cortex.

Bifurcation of Gap Solitons in Coupled Mode Equations in d Dimensions

We consider a system of first order coupled mode equations in {mathbb {R}}^d describing the envelopes of wavepackets in nonlinear periodic media. Under the assumptions of a spectral gap and a generic assumption on the dispersion relation at the spectral edge, we prove the bifurcation of standing gap solitons of the coupled mode equations from the zero solution. The proof is based on a Lyapunov–Schmidt decomposition in Fourier variables and a nested Banach fixed point argument. The reduced bifurcation equation is a perturbed stationary nonlinear Schrödinger equation. The existence of solitary waves follows in a symmetric subspace thanks to a spectral stability result. A numerical example of gap solitons in {mathbb {R}}^2 is provided.

Expressibility of comb tensor network states (CTNS) for the P-cluster and the FeMo-cofactor of nitrogenase

Polynuclear transition metal complexes such as the P-cluster and the FeMo-cofactor of nitrogenase with eight transition metal centers represent a great challenge for current electronic structure methods. In this work, we initiated the use of comb tensor network states (CTNS), whose underlying topology has a one-dimensional backbone and several one-dimensional branches, as a many-body wavefunction ansatz to tackle these challenging systems. As an important first step, we explored the expressive power of CTNS with different underlying topologies. To this end, we presented an algorithm to express a configuration interaction (CI) wavefunction into CTNS based on the Schmidt decomposition. The algorithm was illustrated for representing approximate CI wavefunctions obtained from selected CI calculations for the P-cluster and the FeMo-cofactor into CTNS with three chemically meaningful comb structures, which successively group orbitals belonging to the same atom into branches. The conventional matrix product states (MPS) representation was obtained as a special case. We also discussed the insights gained from such decompositions, which shed some light on the future developments of efficient numerical tools for polynuclear transition metal complexes.

Bootstrap embedding with an unrestricted mean-field bath.

A suite of quantum embedding methods have recently been developed where the Schmidt decomposition is applied to the full system wavefunction to derive basis states that preserve the entanglement between the fragment and the bath. The quality of these methods can depend heavily on the quality of the initial full system wavefunction. Most of these methods, including bootstrap embedding (BE) [M. Welborn et al; J. Chem. Phys. 145, 074102 (2016)], start from a spin-restricted mean-field wavefunction [call this restricted BE (RBE)]. Given that spin-unrestricted wavefunctions can capture a significant amount of strong correlation at the mean-field level, we suspect that starting from a spin-unrestricted mean-field wavefunction will improve these embedding methods for strongly correlated systems. In this work, BE is generalized to an unrestricted Hartree-Fock bath [call this unrestricted BE (UBE)], and UBE is applied to model hydrogen ring systems. UBE's improved versatility over RBE is utilized to calculate high spin symmetry states that were previously unattainable with RBE. Ionization potentials, electron affinities, and spin-splittings are computed using UBE with accuracy on par with spin-unrestricted coupled cluster singles and doubles. Even for cases where RBE is viable, UBE converges more reliably. We discuss the limitations or weaknesses of each calculation and how improvements to RBE and density matrix embedding theory these past few years can also improve UBE.

Efficient compression of the environment of an open quantum system

In order to simulate open quantum systems, many approaches (such as Hamiltonian-based solvers in dynamical mean-field theory) aim for a reproduction of a desired environment spectral density in terms of a discrete set of bath states, mimicking the open system as a larger closed problem. Existing strategies to find a compressed representation of the environment for this purpose can be numerically demanding, or lack the compactness and systematic improvability required for an accurate description of the system propagator. We propose a method in which bath orbitals are constructed explicitly by an algebraic construction based on the Schmidt-decomposition of response wave functions, efficiently and systematically compressing the description of the full environment. These resulting bath orbitals are designed to directly reproduce the system Green's function, not hybridization, which allows for consideration of the relevant system energy scales to optimally model. This results in an accurate and efficient truncation of the environment, with applications in a wide range of numerical simulations of open quantum systems.

Two entanglement measures

In a previous work we introduced an entanglement number e(ψ) for a vector state ψ. This number relied on the Schmidt decomposition of ψ which may be difficult to compute. We now present a method for finding e(ψ) that is simple and efficient. We show that e(ψ) is continuous in the vector norm. We next extend e(ψ) to an entanglement number e(ρ) for a general mixed state ρ and show that ρ is separable (not entangled) if and only if e(ρ) = 0. We next define a related quantity es (ρ) ≥ e(ρ) which we call the spectral entanglement number. We argue that es (ρ) is easier to compute than e(ρ) and that the physical motivation for es (ρ) is superior to that of e(ρ). It is shown that es (ρ) is continuous in the operator norm. Although e(ρ) is a convex function, we show that es (ρ) need not be convex. An open problem is to characterize the states ρ such that e(ρ) = es (ρ). Many illustrative examples are presented.

Construction of many-body-localized models where all the eigenstates are matrix-product-states

The inverse problem of ‘eigenstates-to-Hamiltonian’ is considered for an open chain of N quantum spins in the context of many-body-localization. We first construct the simplest basis of the Hilbert space made of 2N orthonormal matrix-product-states (MPS), that will thus automatically satisfy the entanglement area-law. We then analyze the corresponding N local integrals of motions (LIOMs) that can be considered as the local building blocks of these 2N MPS, in order to construct the parent Hamiltonians that have these 2N MPS as eigenstates. Finally we study the matrix-product-operator form of the diagonal ensemble density matrix that allows to compute long-time-averaged observables of the unitary dynamics. Explicit results are given for the memory of local observables and for the entanglement properties in operator-space, via the generalized notion of Schmidt decomposition for density matrices describing mixed states.

Local matrix product operators: Canonical form, compression, and control theory

We present a new method for compressing matrix product operators (MPOs) which represent sums of local terms, such as Hamiltonians. Just as with area law states, such local operators may be fully specified with a small amount of information per site. Standard matrix product state (MPS) tools are ill-suited to this case, due to extensive Schmidt values that coexist with intensive ones, and Jordan blocks in the transfer matrix. We ameliorate these issues by introducing an "almost Schmidt decomposition" that respects locality. Our method is "$\varepsilon$-close" to the accuracy of MPS-based methods for finite MPOs, and extends seamlessly to the thermodynamic limit, where MPS techniques are inapplicable. In the framework of control theory, our method generalizes Kung's algorithm for model order reduction. Several examples are provided, including an all-MPO version of the operator recursion method (Lanczos algorithm) directly in the thermodynamic limit. All results are accompanied by practical algorithms, well-suited for the large MPOs that arise in DMRG for long-range or quasi-2D models.

A constitutive framework for finite viscoelasticity and damage based on the Gram–Schmidt decomposition

A novel thermodynamic framework for the continuum mechanical response of nonlinear solids is described. Large deformations, nonlinear hyperelasticity, viscoelasticity, and property changes due to evolution of damage in the material are all encompassed by the general theory. The deformation gradient is decomposed in Gram–Schmidt fashion into the product of an orthogonal matrix and an upper triangular matrix, where the latter can be populated by six independent strain attributes. Strain attributes, in turn, are used as fundamental independent variables in the thermodynamic potentials, rather than the usual scalar invariants of deformation tensors as invoked in more conventional approaches. A complementary set of internal variables also enters the thermodynamic potentials to enable history and rate dependence, i.e., viscoelasticity, and irreversible stiffness degradation, i.e., damage. Governing equations and thermodynamic restrictions imposed by the entropy production inequality are derived. Mechanical, thermodynamic, and kinetic relations are presented for material symmetries that reduce to cubic or isotropic thermoelasticity in the small strain limit, restricted to isotropic damage. Representative models and example problems demonstrate utility and flexibility of this theory for depicting nonlinear hyperelasticity, viscoelasticity, and/or damage from cracks or voids, with physically measurable parameters.

Asymptotic security analysis of teleportation-based quantum cryptography

We prove that the teleportation-based quantum cryptography protocol presented in Gordon and Rigolin (Opt Commun 283:184, 2010), which is built using only orthogonal states encoding the classical bits that are teleported from Alice to Bob, is asymptotically secure against all types of individual and collective attacks. We then investigate modifications to that protocol leading to greater secret-key rates and to security against coherent attacks. In other words, we show an unconditional secure quantum key distribution protocol that does not need non-orthogonal quantum states to encode the bits of the secret key sent from Alice to Bob. We also revisit the security proof of the BB84 protocol by exploring the non-uniqueness of the Schmidt decomposition of its entanglement-based representation. This allows us to arrive at a secure transmission of the key for a slightly greater quantum bit error rate (quantum communication channel’s noise) when compared to its standard security analysis.

Quantum singular value decomposer

We present a variational quantum circuit that produces the Singular Value Decomposition of a bipartite pure state. The proposed circuit, that we name Quantum Singular Value Decomposer or QSVD, is made of two unitaries respectively acting on each part of the system. The key idea of the algorithm is to train this circuit so that the final state displays exact output coincidence from both subsystems for every measurement in the computational basis. Such circuit preserves entanglement between the parties and acts as a diagonalizer that delivers the eigenvalues of the Schmidt decomposition. Our algorithm only requires measurements in one single setting, in striking contrast to the $3^n$ settings required by state tomography. Furthermore, the adjoints of the unitaries making the circuit are used to create the eigenvectors of the decomposition up to a global phase. Some further applications of QSVD are readily obtained. The proposed QSVD circuit allows to construct a SWAP between the two parties of the system without the need of any quantum gate communicating them. We also show that a circuit made with QSVD and CNOTs acts as an encoder of information of the original state onto one of its parties. This idea can be reversed and used to create random states with a precise entanglement structure.

Spectral shaping of the biphoton state from multiplexed thermal atomic ensembles

We theoretically investigate the spectral property of a biphoton state from multiplexed thermal atomic ensembles. This biphoton state originates from the cascade emissions, which can be generated by two weak pump fields under four-wave mixing condition. Under this condition, a signal photon from the upper transition, chosen in a telecommunication bandwidth, can be generated along with a correlated idler photon from the lower infrared transition. We can spectrally shape the biphoton state by multiplexing the atomic ensembles with frequency-shifted emissions, where the entropy of entanglement can be analyzed via Schmidt decompositions. We find that this spectral entanglement increases when more vapor cells are multiplexed with correlated or anti-correlated signal and idler fields. The eigenvalues in Schmidt bases approach degenerate under this multiplexing scheme, and corresponding Schmidt numbers can be larger than the number of the multiplexed vapor cells, showing the enlarged entropy of entanglement and excess correlated modes in continuous frequency spaces. We also investigate the lowest entropy of entanglement allowed in the multiplexing scheme, which is preferential for generating a pure single photon source. This shows the potentiality to spectrally shape the biphoton source, where high-capacity spectral modes can be applied in long-distance quantum communication and multimode quantum information processing.

A constitutive model for lung mechanics and injury applicable to static, dynamic, and shock loading

A continuum model for lung parenchyma is constructed. The model describes the thermomechanical response over a range of loading rates—from static to dynamic to shock waves—and a range of stress states, including isotropic expansion, triaxial extension, simple shear, and plane wave compression. Nonlinear elasticity, viscoelasticity, and damage are included, with the latter associated with changes of biological function as well as mechanical stiffness. A Gram–Schmidt decomposition of the deformation gradient leads to strain attributes that enter the thermodynamic potentials as state variables. A free energy function is designed for loading at low to moderate rates and tensile pressures, whereby the tissue response, with surface tension, is preeminent. An internal energy function is designed for wave propagation analysis, including shock waves, whereby compressibility of the air inside the alveoli is addressed via a composite stiffness based on a closed-cell assumption. The model accurately represents the response to triaxial loading, pressure relaxation, and dynamic torsion with relatively few parameters. Longitudinal wave speeds are reasonable for ranges of internal airway pressure and transpulmonary pressure. Airway pressure strongly affects the response to plane wave compression. Criteria for local injury and damage progression depend on a normalized energy density and its gradient, where the latter is paramount for impact problems involving fast pressure rises. Results suggest that local damage associated with edema is induced at load intensities much lower than those that would cause significant stiffness changes due to rupture, major tearing, or local collapse.

Compatibility conditions from a Gram–Schmidt decomposition of deformation gradient in two dimensions

Planar deformations are analyzed in the context of a Gram-Schmidt decomposition of deformation gradient into a rotation and upper triangular stretch, where the latter consists of up to three possibly nonzero terms. Necessary—and sufficient under suitable restrictions—conditions for integrability of deformation gradient and stretch (geometrically, vanishing torsion and curvature of a certain affine connection) reduce to a system of three partial differential equations (PDEs). This system appears more convenient than corresponding compatibility conditions from polar decompositions, especially for motions involving simple shear. Admissible families of triangular stretch corresponding to simple shear, pure shear, dilation, and uniaxial strain are analyzed. Also considered are simultaneous simple shear and axial stretch, for which a non-trivial solution with non-vanishing rotation gradient is constructed.

Joint Schmidt-type decomposition for two bipartite pure quantum states

It is well known that the Schmidt decomposition exists for all pure states of a two-party quantum system. We demonstrate that there are two ways to obtain an analogous decomposition for arbitrary rank-1 operators acting on states of a bipartite finite-dimensional Hilbert space. These methods amount to joint Schmidt-type decompositions of two pure states where the two sets of coefficients and local bases depend on the properties of either state, however, at the expense of the local bases not all being orthonormal and in one case the complex-valuedness of the coefficients. With these results we derive several generally valid purity-type formulae for one-party reductions of rank-1 operators, and we point out relevant relations between the Schmidt decomposition and the Bloch representation of bipartite pure states.

Physical observable for one-dimensional the spin-1 Blume–Capel model

We carry out an extensive study, within the infinite matrix product state algorithm, one-dimensional spin-1 Blume–Capel model on the spin-chain and subject to a changeable external magnetic field. The phase transition points are obtained by investigating the behavior of physical observable and as well as exacting the critical exponent from the scaling relationship. The final results are achieved for the physical observable-quantum entanglement entropy, local order parameter and its scaling: the scaling relationship between quantum entanglement entropy and truncation dimension, the scaling of Schmidt decomposition coefficient λmax and truncation dimension, the critical exponent of local order parameter near the phase transition point is extracted with the scaling behavior for the model with exchange coupling γ=0.6. These techniques are an easy way to exact the critical exponent for quantum phase transitions.

Explicit properties of the simplest inhomogeneous matrix-product-state including the Riemann metric of the MPS manifold

We consider the simplest inhomogeneous matrix-product-state for an open chain of N quantum spins that involves only two angles per site and two angles per bond with the following direct physical meanings. The two angles associated to the site k are the two Bloch angles that parametrize the two orthonormal eigenvectors of the reduced density matrix of the spin k alone. The two angles associated to the bond parametrize the entanglement properties of the Schmidt decomposition across the bond . Explicit results are given for the reduced density matrix of two consecutive sites that is needed to evaluate the energy of two-body Hamiltonians, and for the reduced density matrix of two sites at distance r that is needed to evaluate the spin–spin correlations at distance r. The global structure of the MPS manifold as parametrized by these angles is then characterized by its explicit Riemann metric. Finally, the generalizations to any tree-like structure without loops and to the chain with periodic boundary conditions are discussed.

SU(4) description of bilayer skyrmion-antiskyrmion pairs

The antiferromagnetic coupling and entanglement between skyrmion lattices are treated in magnetic bilayer systems. We first formulate the problem of large bilayer skyrmions using the -theory. We have considered bilayer skyrmions under the presence of Dzyaloshinskii-Moriya (DMI) and Zeeman interactions confined in a two-dimensional chiral magnet such as Fe0.5Co0.5Si. We parametrize bilayer skyrmions using the SU(4) representation, and represent each skyrmion and antiskyrmion using the Schmidt decomposition. The reduced density matrices for skyrmion and antiskyrmion have been calculated. The conditions for maximal, partial entanglement and separable bilayer skyrmions are presented. Our results can be used for generating entanglement in systems with a large number of spins.