Quantum Case Research Articles

Thermodynamics of a single mesoscopic phononic mode

In recent decades, the laws of thermodynamics have been pushed down to smaller and smaller scales, within the theoretical field of stochastic thermodynamics and state-of-the-art experiments performed on microfabricated mesoscopic systems. These measurements concern thermal properties of electrons, photons, and mesoscopic mechanical objects. Here we report on the measurements of thermal fluctuations of a single mechanical mode in equilibrium with a heat reservoir. The device under study is a nanomechanical beam with a first flexural mode resonating at $3.8\phantom{\rule{0.16em}{0ex}}\mathrm{MHz}$, cooled down to temperatures in the range from 100 to $400\phantom{\rule{0.16em}{0ex}}\mathrm{mK}$. The technique is constructed around a microwave optomechanical setup using a cryogenic high electron mobility transistor, and is based on two parametric amplifications implemented in series: an in-built optomechanical ``blue detuned'' pumping plus a traveling wave parametric amplifier stage. We demonstrate our ability to resolve energy fluctuations of the mechanical mode in real time up to the fastest relevant speed given by the mechanical relaxation rate. The energy probability distribution is then exponential, matching the expected Boltzmann distribution. The variance of fluctuations is found to be ${({k}_{B}T)}^{2}$ with no free parameters. Our microwave detection floor is about three times the standard quantum limit at $6\phantom{\rule{0.16em}{0ex}}\mathrm{GHz}$; the resolution of our fastest acquisition tracks reached about $100\phantom{\rule{0.16em}{0ex}}\mathrm{phonons}$, and is directly related to the rather poor optomechanical coupling of the device (${g}_{0}/2\ensuremath{\pi}\ensuremath{\approx}0.5\phantom{\rule{0.16em}{0ex}}\mathrm{Hz}$). This result is deeply in the classical regime, but shall be extended to the quantum case in the future with systems presenting a much larger ${g}_{0}$ (up to $2\ensuremath{\pi}\ifmmode\times\else\texttimes\fi{}250\phantom{\rule{0.16em}{0ex}}\mathrm{Hz}$), potentially reaching the resolution of a single mechanical quantum. We believe that it will open an experimental field, phonon-based quantum stochastic thermodynamics, with fundamental implications for quantum heat transport and macroscopic mechanical quantum coherence.

Shortcut to synchronization in classical and quantum systems

Synchronization is a major concept in nonlinear physics. In a large number of systems, it is observed at long times for a sinusoidal excitation. In this paper, we design a transiently non-sinusoidal driving to reach the synchronization regime more quickly. We exemplify an inverse engineering method to solve this issue on the classical Van der Pol oscillator. This approach cannot be directly transposed to the quantum case as the system is no longer point-like in phase space. We explain how to adapt our method by an iterative procedure to account for the finite-size quantum distribution in phase space. We show that the resulting driving yields a density matrix close to the synchronized one according to the trace distance. Our method provides an example of fast control of a nonlinear quantum system, and raises the question of the quantum speed limit concept in the presence of nonlinearities.

Objective trajectories in hybrid classical-quantum dynamics

Consistent dynamics which couples classical and quantum degrees of freedom exists, provided it is stochastic. This dynamics is linear in the hybrid state, completely positive and trace preserving. One application of this is to study the back-reaction of quantum fields on space-time which does not suffer from the pathologies of the semi-classical equations. Here we introduce several toy models in which to study hybrid classical-quantum evolution, including a qubit coupled to a particle in a potential, and a quantum harmonic oscillator coupled to a classical one. We present an unravelling approach to calculate the dynamics, and provide code to numerically simulate it. Unlike the purely quantum case, the trajectories (or histories) of this unravelling can be unique, conditioned on the classical degrees of freedom for discrete realisations of the dynamics, when different jumps in the classical degrees of freedom are accompanied by the action of unique operators on the quantum system. As a result, the “measurement postulate'' of quantum theory is not needed; quantum systems become classical because they interact with a fundamentally classical field.

Циклические фробениусовы алгебры

In this paper, we introduce the notion of cyclic Frobenius algebras (CF-algebras). Canonical structures of CF-algebras exist on associative and Poisson algebras. It turns out that the modern theory of integrable systems yields non-trivial examples of CF-algebras. In the theory of the KdV hierarchy there is a structure of CF-algebra which leads to explicit expressions for the first integrals of the N-th Novikov hierarchy, suitable for classical, non-Abelian and quantum cases.

Entanglement and coherence in the Bernstein-Vazirani algorithm

Quantum algorithms can outperform their classical counterparts in various tasks, the most prominent example being Shor's algorithm for efficient prime factorization on a quantum computer. It is clear that one of the reasons for the speedup is the superposition principle of quantum mechanics, which allows a quantum processor to be in a superposition of different states at the same time. While such a superposition can lead to entanglement across different qubits of the processors, there also exist quantum algorithms that outperform classical ones using superpositions of individual qubits without entangling them. As an example, the Bernstein-Vazirani algorithm allows one to determine a bit string encoded into an oracle. While the classical version of the algorithm requires multiple calls of the oracle to learn the bit string, a single query of the oracle is enough in the quantum case. In this article, we analyze in detail the quantum resources in the Bernstein-Vazirani algorithm. For this, we introduce and study its probabilistic version, where the goal is to guess the bit string after a single call of the oracle. We show that in the absence of entanglement, the performance of the algorithm is directly related to the amount of quantum coherence in the initial state. We further demonstrate that a large amount of entanglement in the initial state prevents the algorithm from achieving optimal performance. We also apply our methods to quantum computation with mixed states, proving that pseudopure states achieve optimal performance for a given purity in the Bernstein-Vazirani algorithm. We further investigate quantum resources in the one clean qubit model, showing that the model can exhibit speedup over any known classical algorithm even with an arbitrarily little amount of multipartite entanglement, general quantum correlations, and coherence.

Bound on energy dependence of chaos

We conjecture a chaos energy bound, an upper bound on the energy dependence of the Lyapunov exponent for any classical/quantum Hamiltonian mechanics and field theories. The conjecture states that the Lyapunov exponent $\ensuremath{\lambda}(E)$ grows no faster than linearly in the total energy $E$ in the high energy limit. In other words, the exponent $c$ in $\ensuremath{\lambda}(E)\ensuremath{\propto}{E}^{c}(E\ensuremath{\rightarrow}\ensuremath{\infty})$ satisfies $c\ensuremath{\le}1$. This chaos energy bound stems from thermodynamic consistency of out-of-time-order correlators and applies to any classical/quantum system with finite $N/\mathrm{large}$ $N$ ($N$ is the number of degrees of freedom) under plausible physical conditions on the Hamiltonians. To the best of our knowledge the chaos energy bound is satisfied by any classically chaotic Hamiltonian system known, and is consistent with the cerebrated chaos bound by Maldacena, Shenker, and Stanford, which is for quantum cases at large $N$. We provide arguments supporting the conjecture for generic classically chaotic billiards and multiparticle systems. The existence of the chaos energy bound may put a fundamental constraint on physical systems and the Universe.

High-order corrections to the radiation-free dynamics of an electron in the strongly radiation-dominated regime

A system of reduced equations is proposed for electron motion in the strongly radiation-dominated regime for an arbitrary electromagnetic field configuration. The approach developed here is used to analyze various scenarios of electron dynamics in this regime: motion in rotating electric and magnetic fields and longitudinal acceleration in a plane wave and in a plasma wakefield. The results obtained show that this approach is able to describe features of electron dynamics that are essential in certain scenarios, but cannot be captured in the framework of the original radiation-free approximation [Samsonov et al., Phys. Rev. A 98, 053858 (2018) and A. Gonoskov and M. Marklund, Phys. Plasmas 25, 093109 (2018)]. The results are verified by numerical integration of the nonreduced equations of motion with account taken of radiation reaction in both semiclassical and fully quantum cases.

Quantum effects on the propagation of surface magnetoplasmon polaritons in a graphene-plasmonic structure

We investigate the properties of surface magnetoplasmon polaritons (SMPPs) in a graphene-plasmonic structure which is constructed as a graphene film sandwiched with two semi-infinite dielectrics under a perpendicular configuration. By solving Maxwell equations and quantum magneto-hydrodynamic equations with considering the quantum statistical and quantum diffraction effects, we deduce the dispersion relation of graphene SMPPs (GSMPPs) in detail. We show how the graphene electron density, the external magnetic field, and the dielectric constant, affect the features of the dispersion of GSMPPs in both classical and quantum cases. We find that the quantum effects (QEs) significantly alter the properties of GSMPPs, which are entirely different from those in a classical model. We find that the propagation speed of classical GSMPPs has small increases while the propagation speed of quantum GSMPPs has fast and sharp increases along with the increases in graphene electron density. We further find that the propagation speed decreases gradually by increasing the applied magnetic field in both classical and quantum GSMPPs. Moreover, we also find that the propagation speed of classical GSMPPs has fast decreases tending to zero at large wavenumber while the propagation speed of quantum GSMPPs has slow decreases tending to infinity with increasing the dielectric constant. Our findings elucidate that QEs play a crucial role in the properties of GSMPPs and their response to different parameters.

Spatiotemporal dynamics of classical and quantum density profiles in low-dimensional spin systems

We provide a detailed comparison between the dynamics of high-temperature spatiotemporal correlation functions in quantum and classical spin models. In the quantum case, our large-scale numerics are based on the concept of quantum typicality, which exploits the fact that random pure quantum states can faithfully approximate ensemble averages, allowing the simulation of spin-$1/2$ systems with up to $40$ lattice sites. Due to the exponentially growing Hilbert space, we find that for such system sizes even a single random state is sufficient to yield results with extremely low noise that is negligible for most practical purposes. In contrast, a classical analog of typicality is missing. In particular, we demonstrate that, in order to obtain data with a similar level of noise in the classical case, extensive averaging over classical trajectories is required, no matter how large the system size. Focusing on (quasi-)one-dimensional spin chains and ladders, we find a remarkably good agreement between quantum and classical dynamics. This applies not only to cases where both the quantum and classical model are nonintegrable, but also to cases where the quantum spin-$1/2$ model is integrable and the corresponding classical $s\to\infty$ model is not. Our analysis is based on the comparison of space-time profiles of the spin and energy correlation functions, where the agreement is found to hold not only in the bulk but also in the tails of the resulting density distribution. The mean-squared displacement of the density profiles reflects the nature of emerging hydrodynamics and is found to exhibit similar scaling for quantum and classical models.

Assemblages and steering in general probabilistic theories

We study steering in the framework of general probabilistic theories. We show that for dichotomic assemblages, steering can be characterized in terms of a certain tensor cross norm, which is also related to a steering degree given by steering robustness. Another contribution is the observation that steering in general probabilistic theories (GPTs) can be conveniently treated using Choquet theory for probability measures on the state space. In particular, we find a variational expression for universal steering degree for dichotomic assemblages and obtain conditions characterizing unsteerable states analogous to some conditions recently found for the quantum case. The setting also enables us to rather easily extend the results to infinite dimensions and arbitrary numbers of measurements with arbitrary outcomes.

Non-Separability of Physical Systems as a Foundation of Consciousness.

A hypothesis is presented that non-separability of degrees of freedom is the fundamental property underlying consciousness in physical systems. The amount of consciousness in a system is determined by the extent of non-separability and the number of degrees of freedom involved. Non-interacting and feedforward systems have zero consciousness, whereas most systems of interacting particles appear to have low non-separability and consciousness. By contrast, brain circuits exhibit high complexity and weak but tightly coordinated interactions, which appear to support high non-separability and therefore high amount of consciousness. The hypothesis applies to both classical and quantum cases, and we highlight the formalism employing the Wigner function (which in the classical limit becomes the Liouville density function) as a potentially fruitful framework for characterizing non-separability and, thus, the amount of consciousness in a system. The hypothesis appears to be consistent with both the Integrated Information Theory and the Orchestrated Objective Reduction Theory and may help reconcile the two. It offers a natural explanation for the physical properties underlying the amount of consciousness and points to methods of estimating the amount of non-separability as promising ways of characterizing the amount of consciousness.

Experimental validation of the Kibble-Zurek mechanism on a digital quantum computer

The Kibble-Zurek mechanism (KZM) captures the essential physics of nonequilibrium quantum phase transitions with symmetry breaking. KZM predicts a universal scaling power law for the defect density which is fully determined by the system’s critical exponents at equilibrium and the quenching rate. We experimentally tested the KZM for the simplest quantum case, a single qubit under the Landau-Zener evolution, on an open access IBM quantum computer (IBM-Q). We find that for this simple one-qubit model, experimental data validates the central KZM assumption of the adiabatic-impulse approximation for a well isolated qubit. Furthermore, we report on extensive IBM-Q experiments on individual qubits embedded in different circuit environments and topologies, separately elucidating the role of crosstalk between qubits and the increasing decoherence effects associated with the quantum circuit depth on the KZM predictions. Our results strongly suggest that increasing circuit depth acts as a decoherence source, producing a rapid deviation of experimental data from theoretical unitary predictions.

Proportionality of gravitational and electromagnetic radiation by an electron in an intense plane wave

Accelerated charges emit both electromagnetic and gravitational radiation. Classically, it was found that the electromagnetic energy spectrum radiated by an electron in a monochromatic plane wave is proportional to the corresponding gravitational one. Quantum mechanically, it was shown that the amplitudes of graviton photoproduction and Compton scattering are proportional to each other at tree level. Here, by combining strong-field QED and quantum gravity, we demonstrate that the amplitude of nonlinear graviton photoproduction in an arbitrary plane wave is proportional to the corresponding amplitude of nonlinear Compton scattering. Also, introducing classical amplitudes we prove that the proportionality relies on the semiclassical nature of the electron's motion in a plane wave and on energy-momentum conservation laws, leading to the same proportionality constant in the classical and quantum case. These results deepen the intertwine between gravity and electromagnetism into both a nonlinear and a quantum level.

Scaling Theory of Wave Confinement in Classical and Quantum Periodic Systems.

Functional defects in periodic media confine waves-acoustic, electromagnetic, electronic, spin, etc.-in various dimensions, depending on the structure of the defect. While defects are usually modeled by a superlattice with a typical band-structure representation of energy levels, determining the confinement associated with a given band is highly nontrivial and no analytical method is known to date. Therefore, we propose a rigorous method to classify the dimensionality of wave confinement. Starting from the confinement energy and the mode volume, we use finite-size scaling to find that ratios of these quantities raised to certain powers yield the confinement dimensionality of each band. Our classification has negligible additional computational costs compared to a band structure calculation and is valid for any type of wave, both quantum and classical, and in any dimension. In the quantum regime, we illustrate our method on electronic confinement in 2D hexagonal boron nitride (BN) with a nitrogen vacancy, in agreement with previous results. In the classical case, we study a three-dimensional photonic band gap cavity superlattice, where we identify novel acceptorlike behavior. We briefly discuss the generalization to quasiperiodic lattices.

Spontaneous Collapse Theories and Temporal Primitivism about Time’s Direction

Two views on the direction of time can be distinguished—primitivism and non-primitivism. According to the former, time’s direction is an in-built, fundamental property of the physical world. According to the latter, time’s direction is a derivative property of a fundamentally directionless reality. In the literature, non-primitivism has been widely supported since most (if not all) our fundamental dynamical laws are time-reversal invariant. In this paper, I offer a way out to the primitivist. I argue that we do have good grounds to support a primitive direction of time in the quantum realm. The rationale depends on exploiting the metaphysical and dynamical underdetermination of quantum theories to make a case in favor of primitivism. In particular, primitivism can be grounded in spontaneous collapse theories (e.g., GRW and CSL). The specific sense in which these theories capture a primitive direction of time is that, when the ontology of the theory is seriously taken into account, it does not remain invariant under time reversal. In taking GRW with a matter-density field (GRWm), I will argue that primitivism about the direction of time can be defended in the quantum case.

Aspects of entropy in classical and in quantum physics

Entropy has played an essential role in the history of physics. Its mathematical definition and applications have changed over time till today. In this paper, we first review the historical evolution of these various points of view, from the thermodynamic definition to information entropy from Shannon in classical physics, up to the modern concept of Neumann’s quantum entropy. As a specific example, we consider entanglement entropy and compare the phase space approach in classical physics to the Hilbert space approach in quantum physics in simple model systems. We derive a general expression for the entanglement entropy of fermions and bosons in arbitrary partitions of Hilbert space, valid beyond the thermodynamic limit. Next, we compare thermodynamic heat engines with quantum heat engines. Finally, we proceed to the more general concept of quantum (computational) complexity and argue, using the concept of entanglement entropy, that the Heisenberg time in classically chaotic systems coincides with the time when maximal complexity is reached in the quantum case for systems with all–all interactions.

Simple circuit simulations of classical and quantum Turing machines

We construct reversible Boolean circuits efficiently simulating reversible Turing machines. Both the circuits and the simulation proof are rather simple. Then we give a fairly straightforward generalization of the circuits and the simulation proof to the quantum case.

Quantum fluctuations in electrical multiport linear systems

We present an extension of the classical Nyquist-Thevenin theorem for multiport classical electrical networks by Twiss to the quantum case. Conversely, we extend the quantum fluctuation-dissipation result for one port electrical systems to the multiport case, both reciprocal and nonreciprocal. Our results are extended to lossy systems by depicting resistive components as continuous limits of purely lossless lumped-element networks. Simple circuit examples are analyzed, including a linear system lacking a direct impedance representation.

Computing Eigenvalues of Diagonalizable Matrices on a Quantum Computer

Computing eigenvalues of matrices is ubiquitous in numerical linear algebra problems. Currently, fast quantum algorithms for estimating eigenvalues of Hermitian and unitary matrices are known. However, the general case is far from fully understood in the quantum case. Based on a quantum algorithm for solving linear ordinary differential equations, we show how to estimate the eigenvalues of diagonalizable matrices that only have real eigenvalues. The output is a superposition of the eigenpairs, and the overall complexity is polylog in the dimension for sparse matrices. Under an assumption, we extend the algorithm to diagonalizable matrices with complex eigenvalues.

Quantum and classical annealing in a continuous space with multiple local minima

The protocol of quantum annealing is applied to an optimization problem with a one-dimensional continuous degree of freedom, a variant of the problem proposed by Shinomoto and Kabashima. The energy landscape has a number of local minima, and the classical approach of simulated annealing is predicted to have a logarithmically slow convergence to the global minimum. We show by extensive numerical analyses that quantum annealing yields a power-law convergence, thus an exponential improvement over simulated annealing. The power is larger, and thus the convergence is faster, than a prediction by an existing phenomenological theory for this problem. Performance of simulated annealing is shown to be enhanced by introducing quasiglobal searches across energy barriers, leading to a power-law convergence but with a smaller power than in the quantum case and thus a slower convergence classically even with quasiglobal search processes. We also reveal how diabatic quantum dynamics, quantum tunneling in particular, steers the systems toward the global minimum by a meticulous choice of annealing schedule. This latter result explicitly contrasts the role of tunneling in quantum annealing against the classical counterpart of stochastic optimization by simulated annealing.