Quantum Variance Research Articles

Subsystem eigenstate thermalization hypothesis for translation invariant systems.

The eigenstate thermalization hypothesis for translation invariant quantum spin systems has been proved recently by using random matrices. In this paper, we study the subsystem version of the eigenstate thermalization hypothesis for translation invariant quantum systems without referring to random matrices. We first find a relation between the quantum variance and the Belavkin-Staszewski relative entropy. Then, by showing the small upper bounds on the quantum variance and the Belavkin-Staszewski relative entropy, we prove the subsystem eigenstate thermalization hypothesis for translation invariant quantum systems with an algebraic speed of convergence in an elementary way. The proof holds for most of the translation invariant quantum lattice models with exponential or algebraic decays of correlations.

Quantum Reversible Audio Steganalysis Using Quantum Schmidt Decomposition and Quantum Support Vector Machine

With the development of various methods of quantum steganography, the discovery of these secret communications has become a necessity. This paper presents the first quantum steganalysis approach that uses statistical features of the quantum frequency domain to detect the steganographic methods in quantum communication networks. In this method, by proposing different quantum circuits, statistical features such as quantum mean, quantum square mean root, quantum variance, and quantum standard deviation are extracted to increase the accuracy of distinguishing between stego and clean signals. In the proposed method, the Schmidt coefficient vector extracted from the Quantum Haar Wavelet Transform (QHWT) is used to calculate the statistical features of the audio signal so that the main and dominant coefficients below the frequency sub-bands of the audio signal can be identified and selected. The proposed method uses a Quantum Support Vector Machine (QSVM) to classify the extracted feature vectors. The higher accuracy of the proposed frequency-based steganalysis method indicates its higher efficiency in detecting stego and clean signals than the quantum steganalysis methods based on the time domain.

Uncertainty and Quantum Variance at the light of Quantum Information Geometry

Uncertainty and Quantum Variance at the light of Quantum Information Geometry

Quantum variance for dihedral Maass forms

We establish an asymptotic formula for the weighted quantum variance of dihedral Maass forms on Γ 0 ( D ) ∖ H \Gamma _0(D) \backslash \mathbb H in the large eigenvalue limit, for certain fixed D D . As predicted in the physics literature, the resulting quadratic form is related to the classical variance of the geodesic flow on Γ 0 ( D ) ∖ H \Gamma _0(D) \backslash \mathbb H , but also includes factors that are sensitive to underlying arithmetic of the number field Q ( D ) \mathbb Q(\sqrt {D}) .

Quantum fluctuations and semiclassicality in an inflaton-driven evolution

A semiclassical description of quantum systems is applied to probe the dynamics of the cosmological model of an inflationary universe with quadratic inflaton potential, described in a quantum framework of geometrodynamics. The systematic analysis, focusing in particular on the inflationary and post-inflationary epochs, revealed several surprising and counterintuitive features: (i) during inflation the universe rapidly spreads out in volume which leads to significant relative variance by the end of inflation; (ii) despite that, the quantum evolution can still be described to high accuracy by semiclassical methods; (iii) moreover, in the post-inflationary epoch, as the order of included quantum corrections increases, the quantum trajectory approaches the classical one and the description involving second-order corrections only is actually the least accurate there. The consequence of the latter is that the effects of the quantum variances are washed out by the higher-order quantum corrections.

Thermal Critical Dynamics from Equilibrium Quantum Fluctuations.

We show that quantum fluctuations display a singularity at thermal critical points, involving the dynamical z exponent. Quantum fluctuations, captured by the quantum variance [Frérot etal., Phys. Rev. B 94, 075121 (2016)PRBMDO2469-995010.1103/PhysRevB.94.075121], can be expressed via purely static quantities; this in turn allows us to extract the z exponent related to the intrinsic Hamiltonian dynamics via equilibrium unbiased numerical calculations, without invoking any effective classical model for the critical dynamics. These findings illustrate that, unlike classical systems, in quantum systems static and dynamic properties remain inextricably linked even at finite-temperature transitions, provided that one focuses on static quantities that do not bear any classical analog-namely, on quantum fluctuations.

Correlations of quantum curvature and variance of Chern numbers

We analyse the correlation function of the quantum curvature in complex quantum systems, using a random matrix model to provide an exemplar of a universal correlation function. We show that the correlation function diverges as the inverse of the distance at small separations. We also define and analyse a correlation function of mixed states, showing that it is finite but singular at small separations. A scaling hypothesis on a universal form for both types of correlations is supported by Monte-Carlo simulations. We relate the correlation function of the curvature to the variance of Chern integers which can describe quantised Hall conductance.

Quantum motion, coherent states and geometric phase of a generalized damped pendulum

In this work, we analyze the quantum dynamics of a generalized pendulum with a time-varying mass increasing exponentially and constant gravitation. By using Lewis–Riesenfeld invariant approach and Fock states, we solve the time-dependent Schrödinger equation for this system and write its solutions in terms of solutions of the Milne–Pinney equation. We also construct coherent states for the quantized pendulum and use both Fock and coherent states to investigate some important physical proprieties of the quantized pendulum such as eigenvalues of the angular displacement and momentum, their quantum variances as well as the respective uncertainty principle. Finally, we derive the geometric, dynamical and Berry phases for the time-dependent generalized pendulum.

Entanglement and Entropy Squeezing for Moving Two Two-Level Atoms Interaction with a Radiation Field

In this paper, we analyzed squeezing in the information entropy, quantum state fidelity, and qubit-qubit entanglement in a time-dependent system. The proposed model consists of two qubits that interact with a two-mode electromagnetic field under the dissipation effect. An analytical solution is calculated by considering the constants for the equations of motion. The effect of the general form of the time-dependent for qubit-field coupling and the dissipation term on the temporal behavior of the qubit-qubit entanglement, quantum state fidelity, entropy, and variance squeezing are examined. It is shown that the intervals of entanglement caused more squeezing for the case of considering the time-dependent parameters. Additionally, the entanglement between the qubits became more substantial for the case of time dependence. Fidelity and negativity rapidly reached the minimum values by increasing the effect of the dissipation parameter. Moreover, the amount of variance squeezing and the amplitude of the oscillations decreased considerably when the time dependence increased, but the fluctuations increased substantially. We show the relation between entropy and variance squeezing in the presence and absence of the dissipation parameter during the interaction period. This result enables new parameters to control the degree of entanglement and squeezing, especially in quantum communication.

Classical field theory limit of many-body quantum Gibbs states in 2D and 3D

We provide a rigorous derivation of nonlinear Gibbs measures in two and three space dimensions, starting from many-body quantum systems in thermal equilibrium. More precisely, we prove that the grand-canonical Gibbs state of a large bosonic quantum system converges to the Gibbs measure of a nonlinear Schrodinger-type classical field theory, in terms of partition functions and reduced density matrices. The Gibbs measure thus describes the behavior of the infinite Bose gas at criticality, that is, close to the phase transition to a Bose–Einstein condensate. The Gibbs measure is concentrated on singular distributions and has to be appropriately renormalized, while the quantum system is well defined without any renormalization. By tuning a single real parameter (the chemical potential), we obtain a counter-term for the diverging repulsive interactions which provides the desired Wick renormalization of the limit classical theory. The proof relies on a new estimate on the entropy relative to quasi-free states and a novel method to control quantum variances.

The spectral decomposition of |theta |^2

Let theta be an elementary theta function, such as the classical Jacobi theta function. We establish a spectral decomposition and surprisingly strong asymptotic formulas for langle |theta |^2, varphi rangle as varphi traverses a sequence of Hecke-translates of a nice enough fixed function. The subtlety is that typically |theta |^2 notin L^2. Applications to the subconvexity, quantum variance and 4-norm problems are indicated.

Supercritical Poincaré–Andronov–Hopf Bifurcation in a Mean-Field Quantum Laser Equation

We deal with the dynamical system properties of a Gorini–Kossakowski–Sudarshan–Lindblad equation with mean-field Hamiltonian that models a simple laser by applying a mean-field approximation to a quantum system describing a single-mode optical cavity and a set of two-level atoms, each coupled to a reservoir. We prove that the mean-field quantum master equation has a unique regular stationary solution. In case a relevant parameter $$C_\mathfrak {b} $$ , i.e., the cavity cooperative parameter, is less than 1, we prove that any regular solution converges exponentially fast to the equilibrium, and so the regular stationary state is a globally asymptotically stable equilibrium solution. We obtain that a locally exponential stable limit cycle is born at the regular stationary state as $$C_\mathfrak {b} $$ passes through the critical value 1. Then, the mean-field laser equation has a Poincare–Andronov–Hopf bifurcation at $$C_\mathfrak {b} =1 $$ of supercritical-like type. Namely, we derive rigorously, at the level of density matrices—for the first time—the transition from a global attractor quantum state, where the light is not emitted, to a locally stable set of coherent quantum states producing coherent light. Moreover, we establish the local exponential stability of the limit cycle in case a relevant parameter is between the first and second laser thresholds appearing in the semiclassical laser theory. Thus, we get that the coherent laser light persists over time under this condition. In order to prove the exponential convergence of the quantum state, as the time goes to $$+ \infty $$ , we develop a new technique for proving the exponential convergence in open quantum systems that is based on a new variation of constant formula, which is obtained by combining probabilistic techniques with classical arguments from the semigroup theory. Furthermore, applying our main results we find the long-time behavior of the von Neumann entropy, the photon number statistics, and the quantum variance of the quadratures.

Variational Principles in Quantum Monte Carlo: The Troubled Story of Variance Minimization.

We investigate the use of different variational principles in quantum Monte Carlo, namely, energy and variance minimization, prompted by the interest in the robust and accurate estimation of electronic excited states. For two prototypical, challenging molecules, we readily reach the accuracy of the best available reference excitation energies using energy minimization in a state-specific or state-average fashion for states of different or equal symmetry, respectively. On the other hand, in variance minimization, where the use of suitable functionals is expected to target specific states regardless of the symmetry, we encounter severe problems for a variety of wave functions: as the variance converges, the energy drifts away from that of the selected state. This unexpected behavior is sometimes observed even when the target is the ground state and generally prevents the robust estimation of total and excitation energies. We analyze this problem using a very simple wave function and infer that the optimization finds little or no barrier to escape from a local minimum or local plateau, eventually converging to a lower-variance state instead of the target state. For the increasingly complex systems becoming in reach of quantum Monte Carlo simulations, variance minimization with current functionals appears to be an impractical route.

A sufficient entanglement criterion based on quantum fisher information and variance

We derive criterion in the form of inequality based on quantum Fisher information and quantum variance to detect multipartite entanglement. It can be regarded as complementary of the well-established positive partial transpose criterion in the sense that it can also detect bound entangled states. The inequality is motivated by Y Akbari-Kourbolagh et al [Phys. Rev A. 99, 012304 (2019)], which introduced a multipartite entanglement criterion based on quantum Fisher information. Our criterion is experimentally measurable for detecting any N-qudit pure state mixed with white noisy. We take several examples to illustrate that our criterion has good performance for detecting certain entangled states.

Equidistribution of toral eigenfunctions along hypersurfaces

We prove a new quantum variance estimate for toral eigenfunctions. As an application, we show that, given any orthonormal basis of toral eigenfunctions and any smooth embedded hypersurface with nonvanishing principal curvatures, there exists a density one subsequence of eigenfunctions that equidistribute along the hypersurface. This is an analogue of the Quantum Ergodic Restriction theorems in the case of the flat torus, which in particular verifies the Bourgain–Rudnick’s conjecture on L^2 -restriction estimates for a density one subsequence of eigenfunctions in any dimension. Using our quantum variance estimates, we also obtain equidistribution of eigenfunctions against measures whose supports have Fourier dimension larger than d − 2 . In the end, we also describe a few quantitative results specific to dimension 2.

Quantum Variance for Eisenstein Series

Abstract In this paper, we prove an asymptotic formula for the quantum variance for Eisenstein series on $\operatorname{PSL}_2(\mathbb{Z})\backslash \mathbb{H}$. The resulting quadratic form is compared with the classical variance and the quantum variance for cusp forms. They coincide after inserting certain subtle arithmetic factors, including the central values of certain L-functions.

Reconstructing the quantum critical fan of strongly correlated systems using quantum correlations

Albeit occurring at zero temperature, quantum critical phenomena have a huge impact on the finite-temperature phase diagram of strongly correlated systems, giving experimental access to their observation. Indeed, the existence of a gapless, zero-temperature quantum critical point induces the existence of an extended region in parameter space—the quantum critical fan (QCF)—characterized by power-law temperature dependences of all observables. Identifying experimentally the QCF and its crossovers to other regimes (renormalized classical, quantum disordered) remains nonetheless challenging. Focusing on paradigmatic models of quantum phase transitions, here we show that quantum correlations—captured by the quantum variance of the order parameter—exhibit the temperature scaling associated with the QCF over a parameter region much broader than that revealed by ordinary correlations. The link existing between the quantum variance and the dynamical susceptibility paves the way to an experimental reconstruction of the QCF using spectroscopic techniques.

Estimates of the quantum Fisher information in the S=1 antiferromagnetic Heisenberg spin chain with uniaxial anisotropy

The quantum Fisher information is of considerable interest not only for quantum metrology but also because it is a useful entanglement measure for finite temperature mixed states. In particular, it estimates the degree to which multipartite entanglement is present. Recent results have related the quantum Fisher information to experimentally measurable probes. While in principle possible, a direct evaluation of the quantum Fisher information at finite temperatures is technically challenging and here we show that a simple estimate can be obtained for materials where the single mode approximation is valid. We focus on the $S=1$ anti-ferromagnetic Heisenberg model with uniaxial anisotropy. Quantum Monte Carlo thechniques are used to determine low temperature correlations from which the quantum Fisher information can be estimated within the single mode approximation. The quantum Fisher information is compared to the quantum variance for the staggered magnetization operators in the transverse direction and inequalities between the quantum Fisher information, the quantum variance and the full variance are discussed. Both the quantum and full variance as well as the quantum Fisher information are examined at finite temperatures above the isotropic point and at the quantum critical point for the Haldane-N\'{e}el transtion. A finite size scaling study of the quantum Fisher information is performed at the quantum critical point and used to confirm the Ising nature of the Haldane-N\'{e}el transition.

The Quantum variance of the Modular Surface

The Quantum variance of the Modular Surface

Many-body multivaluedness of particle-current variance in closed and open cold-atom systems

The quantum variance of an observable is a fundamental quantity in quantum mechanics, and the variance provides additional information other than the average itself. By examining the relation between the particle-current variance $(\delta J)^2$ and the average current $J$ in both closed and open interacting fermionic systems, we show the emergence of a multi-valued Lissajous curve between $\delta J$ and $J$ due to interactions. As a closed system we considered the persistent current in a benzene-like lattice enclosing an effective magnetic flux and solved it by exact diagonalization. For the open system, the steady-state current flowing through a few lattice sites coupled to two particle reservoirs was investigated using a Lindblad equation. In both cases, interactions open a loop and change the topology of the corresponding $\delta J$-$J$ Lissajous curve, showing that this effect is model-independent. We finally discuss how the predicted phenomena can be observed in ultracold atoms, thus offering an alternative way of probing the dynamics of many-body systems.