Pure State Density Matrix Research Articles

A Quantum-Like Tensor Compression Sentence Representation Based on Constraint Functions for Semantics Analysis

To emphasize the semantic impact of local semantic and grammatical information among adjacent words in the input text, we establish a constraint functions-based quantum-like tensor compression sentence representation model by integrating the concept of extending the pure state-based density matrix to the mixed-state projection operator in quantum mechanics. The provided model highlights the semantic significance of mixed word associations in the input text, simultaneously reducing the reliance on information derived solely from dictionary statistics. We combine the correlation coefficient with the attention mechanism to establish the correlation coefficient between words. The quantum-like sentence representation based on pure state density matrix is extended to the projection operator of mixed states. Combining the acquisition of maximum in convex optimization, a constraint functions-based quantum-like text representation pruning model is established to reduce redundant information caused by dimensional expansion of tensor operations. The experimental results on SICK-2014, STS-benchmark, and STS-companion show that the provided model is more effective than the mainstream models in mining semantic information, especially more sensitive to the negative semantics of sentences.

Classical Chaos Described by a Density Matrix

In this paper, a reference to the semiclassical model, in which quantum degrees of freedom interact with classical ones, is considered. The classical limit of a maximum-entropy density matrix that describes the temporal evolution of such a system is analyzed. Here, it is analytically shown that, in the classical limit, it is possible to reproduce classical results. An example is classical chaos. This is done by means a pure-state density matrix, a rather unexpected result. It is shown that this is possible only if the quantum part of the system is in a special class of states.

The quantum cocktail party problem

The cocktail party problem refers to the famous selective attention problem of how to find out the signal of each individual sources from signals of a number of detectors. In the classical cocktail party problem, the signal of each source is a sequence of data such as the voice from a speaker, and each detector detects signal as a linear combination of all sources. This problem can be solved by a unsupervised machine learning algorithm known as the independent component analysis. In this work we propose a quantum analog of the cocktail party problem. Here each source is a density matrix of a pure state and each detector detects a density matrix as a linear combination of all pure state density matrix. The quantum cocktail party problem is to recover the pure state density matrix from a number observed mixed state density matrices. We propose the physical realization of this problem, and how to solve this problem through either classical Newton's optimization method or by mapping the problem to the ground state of an Ising type of spin Hamiltonian.

Towards a formal definition of static and dynamic electronic correlations.

Some of the most spectacular failures of density-functional and Hartree-Fock theories are related to an incorrect description of the so-called static electron correlation. Motivated by recent progress in the N-representability problem of the one-body density matrix for pure states, we propose a method to quantify the static contribution to the electronic correlation. By studying several molecular systems we show that our proposal correlates well with our intuition of static and dynamic electron correlation. Our results bring out the paramount importance of the occupancy of the highest occupied natural spin-orbital in such quantification.

On Classical and Quantum Logical Entropy

The notion of a partition on a set is mathematically dual to the notion of a subset of a set, so there is a logic of partitions dual to Boole's logic of subsets (Boolean logic is usually mis-specified as propositional logic). The notion of an element of a subset has as its dual the notion of a distinction of a partition (a pair of elements in different blocks). Boole developed finite logical probability as the normalized counting measure on elements of subsets so there is a dual concept of logical entropy which is the normalized counting measure on distinctions of partitions. Thus the logical notion of information is a measure of distinctions. Classical logical entropy naturally extends to the notion of quantum logical entropy which provides a more natural and informative alternative to the usual Von Neumann entropy in quantum information theory. The quantum logical entropy of a post-measurement density matrix has the simple interpretation as the probability that two independent measurements of the same state using the same observable will have different results. The main result of the paper is that the increase in quantum logical entropy due to a projective measurement of a pure state is the sum of the absolute squares of the off-diagonal entries (coherences) of the pure state density matrix that are zeroed (decohered) by the measurement, i.e., the measure of the distinctions (decoherences) created by the measurement.

Gross–Pitaevskii Equation for the Density Matrix in the Position Representation

We consider the generalized pure-state density matrix, which depends on different time moments, and obtain the evolution equation for this density matrix for the case where the density matrix corresponds to solutions of the Gross–Pitaevskii equation.

Superhorizon entanglement entropy from particle decay in inflation

In inflationary cosmology all particle states decay as a consequence of the lack of kinematic thresholds. The decay of an initial single particle state yields an \emph{entangled quantum state of the product particles}. We generalize and extend a manifestly unitary field theoretical method to obtain the time evolution of the quantum state. We consider the decay of a light scalar field with mass $M\ll H$ with a cubic coupling in de Sitter space-time. Radiative corrections feature an infrared enhancement manifest as poles in $\Delta=M^2/3H^2$ and we obtain the quantum state in an expansion in $\Delta$. To leading order in $\Delta$ the pure state density matrix describing the decay of a particle with sub-horizon wavevector is dominated by the emission of superhorizon quanta, describing \emph{entanglement between superhorizon and subhorizon fluctuations and correlations across the horizon}. Tracing over the superhorizon degrees of freedom yields a mixed state density matrix from which we obtain the entanglement entropy. Asymptotically this entropy grows with the \emph{physical} volume as a consequence of more modes of the decay products crossing the Hubble radius. A generalization to localized wave packets is provided. The cascade decay of single particle states into many particle states is discussed. We conjecture on \emph{possible} impact of these results on non-gaussianity and on the ``low multipole anomalies'' of the CMB.

Entanglement entropy in particle decay

The decay of a parent particle into two or more daughter particles results in an entangled quantum state as a consequence of conservation laws in the decay process. Recent experiments at Belle and BaBar take advantage of quantum entanglement and the correlations in the time evolution of the product particles to study CP and T violations. If one (or more) of the product particles are not observed, their degrees of freedom are traced out of the pure state density matrix resulting from the decay, leading to a mixed state density matrix and an entanglement entropy. This entropy is a measure of the loss of information present in the original quantum correlations of the entangled state. We use the Wigner-Weisskopf method to construct an approximation to this state that evolves in time in a {\em manifestly unitary} way. We then obtain the entanglement entropy from the reduced density matrix of one of the daughter particles obtained by tracing out the unobserved states, and follow its time evolution. We find that it grows over a time scale determined by the lifetime of the parent particle to a maximum, which when the width of the parent particle is narrow, describes the phase space distribution of maximally entangled Bell-like states. The method is generalized to the case in which the parent particle is described by a wave packet localized in space. Possible experimental avenues to measure the entanglement entropy in the decay of mesons at rest are discussed.

The energy, orbitals and electric properties of the ozone molecule with ensemble density functional theory

For a molecule with a complicated electronic structure, the solution of the Kohn–Sham equations of density functional theory (DFT) may require the use of an ensemble density matrix rather than a pure-state density matrix. In this paper, we first outline how an ensemble density matrix may be parameterised in a form, which ensures that all occupation numbers are between zero and two and that the total number of electrons is constant. A previously developed method for the optimisation of both orbitals and occupation numbers is summarised, and a time-independent linear response method for ensemble densities is developed. The optimisation and linear response methods are applied to the energy, dipole moment and static polarisability of the ground state of ozone at various bond angles. When comparing the results of pure-state and ensemble DFT calculations, it is found that the latter gives better agreement with wave-function methods and in particular provides smooth and differentiable potential and property curves.

New two-mode intermediate momentum-coordinate representation with quantum entanglement and its application

We construct four linear composite operators for a two-particle system and give common eigenvectors of those operators. The technique of integration within an ordered product (IWOP) of operators is employed to prove that those common eigenvectors are complete and orthonormal. Therefore, a new two-mode intermediate momentum-coordinate representation which involves quantum entanglement for a two-particle system is proposed and applied to some two-body dynamic problems. Moreover, the pure-state density matrix |ξ1, ξ2〉C,D C,D〈ξ1, ξ2| is a Radon transform of Wigner operator.

A density tensor hierarchy for open system dynamics: retrieving the noise

We develop a density tensor hierarchy for open system dynamics that recovers information about fluctuations (or ‘noise’) lost in passing to the reduced density matrix. For the case of fluctuations arising from a classical probability distribution, the hierarchy is formed from expectations of products of pure state density matrix elements and can be compactly summarized by a simple generating function. For the case of quantum fluctuations arising when a quantum system interacts with a quantum environment in an overall pure state, the corresponding hierarchy is defined as the environmental trace of products of system matrix elements of the full density matrix. Whereas all members of the classical noise hierarchy are system observables, only the lowest member of the quantum noise hierarchy is directly experimentally measurable. The unit trace and idempotence properties of the pure state density matrix imply descent relations for the tensor hierarchies, that relate the order n tensor, under contraction of appropriate pairs of tensor indices, to the order n − 1 tensor. As examples to illustrate the classical probability distribution formalism, we consider a spatially isotropic ensemble of spin-1/2 pure states, a quantum system evolving by an Itô stochastic Schrödinger equation and a quantum system evolving by a jump process Schrödinger equation. As examples to illustrate the corresponding trace formalism in the quantum fluctuation case, we consider the tensor hierarchies for collisional Brownian motion of an infinite mass Brownian particle and for the weak coupling Born–Markov master equation. In different specializations, the latter gives the hierarchies generalizing the quantum optical master equation and the Caldeira–Leggett master equation. As a further application of the density tensor, we contrast stochastic Schrödinger equations that reduce and that do not reduce the state vector, and discuss why a quantum system coupled to a quantum environment behaves like the latter. The descent relations for our various examples are checked in a series of appendices.

Entangled Husimi operator as a pure state density matrix of two-mode squeezed coherent state

We introduce the entangled Husimi operator Δ h ( σ , γ , κ ) which is endowed with definite physical meaning, and find that there corresponds a special two-mode squeezed coherent state | σ , γ 〉 κ representation such that Δ h ( σ , γ , κ ) = | σ , γ 〉 κ 〈 κ σ , γ | , a pure state density matrix! The Weyl ordered form and the normally ordered form of Δ h ( q , p , κ ) are derived that provides us with an operator version to examine various properties of the entangled Husimi distribution, which is concise and neat.

Quantum Mechanical Nature in Liquid NMR Quantum Computing**The project supported by National Fundamental Research Program under Grant No. 001CB309308, and National Natural Science Foundation of China under Grant No. 60073009, the Hang Tian Science Fund, and the Excellent Young University Teachers' Fund of the Education Ministry of China

The quantum nature of bulk ensemble NMR quantum computing — the center of recent heated debate, is addressed. Concepts of the mixed state and entanglement are examined, and the data in a two-qubit liquid NMR quantum computation are analyzed. The main points in this paper are: i) Density matrix describes the “state” of an average particle in an ensemble. It does not describe the state of an individual particle in an ensemble; ii) Entanglement is a property of the wave function of a microscopic particle (such as a molecule in a liquid NMR sample), and separability of the density matrix cannot be used to measure the entanglement of mixed ensemble; iii) The state evolution in bulk-ensemble NMR quantum computation is quantum-mechanical; iv) The coefficient before the effective pure state density matrix, є, is a measure of the simultaneity of the molecules in an ensemble. It reflects the intensity of the NMR signal and has no significance in quantifying the entanglement in the bulk ensemble NMR system. The decomposition of the density matrix into product states is only an indication that the ensemble can be prepared by an ensemble with the particles unentangled. We conclude that effective-pure-state NMR quantum computation is genuine, not just classical simulations.

Structure and properties of Hughston’s stochastic extension of the Schrödinger equation

Hughston has recently proposed a stochastic extension of the Schrödinger equation, expressed as a stochastic differential equation on projective Hilbert space. We derive new projective Hilbert space identities, which we use to give a general proof that Hughston’s equation leads to state vector collapse to energy eigenstates, with collapse probabilities given by the quantum mechanical probabilities computed from the initial state. We discuss the relation of Hughston’s equation to earlier work on norm-preserving stochastic equations, and show that Hughston’s equation can be written as a manifestly unitary stochastic evolution equation for the pure state density matrix. We discuss the behavior of systems constructed as direct products of independent subsystems, and briefly address the question of whether an energy-based approach, such as Hughston’s, suffices to give an objective interpretation of the measurement process in quantum mechanics.

Density matrix of the Universe.

The Hartle-Hawking wave function for the Universe gives a pure-state density matrix that may be expressed as a path integral over compact Euclidean four-geometries and matter fields which have two disjoint pieces, one of which is bounded by the first argument (a three-geometry with its matter configuration) of the density matrix and the other of which is bounded by the second. Here an alternate proposal is suggested, that the density matrix of the Universe is given by a path integral over all compact Euclidean four-geometries bounded by both three-geometries. Connected four-geometries joining the two three-geometries would be a new contribution to the density matrix which would not factorize, so the entire density matrix proposed here would represent an impure mixed state of the Universe.

Covariant polarization density matrix for pure states of particles with integral spin

A closed expression is obtained for the covariant polarization density matrix of pure states of particles with integral spin. The application of the results to the decay f0 →γ is considered.