Random matrix theory is used to represent generic loss of coherence of a fixed central system coupled to a quantum-chaotic environment, represented by a random matrix ensemble, via random interactions. We study the average density matrix arising from the ensemble induced, in contrast to previous studies where the average values of purity, concurrence, and entropy were considered; we further discuss when one or the other approach is relevant. The two approaches agree in the limit of large environments. Analytic results for the average density matrix and its purity are presented in linear response approximation. The two-qubit system is analysed, mainly numerically, in more detail.
The aim of this letter is to analyze the effect on decoherence of the randomness of the coupling coefficients involved in the interaction Hamiltonian. By studying the spin-bath model with computer simulations, we show that such randomness greatly improves the "efficiency" of decoherence and, then, its physical meaning deserves to be considered.
We present a new method for the solution of the behavior of an ensemble of qubits in a random time-dependent external field. The forward evolution in time is governed by a transfer matrix. The elements of this matrix determine the various decoherence times. The method provides an exact solution in cases where the noise is piecewise constant in time. We show that it applies, for example, to a realistic model of decoherence of electron spins in semiconductors. Results are obtained for the non-perturbative regimes of the models, and we see a transition from weak relaxation to overdamped behavior …
In order to analyze the effect of chaos or order on the rate of decoherence in a subsystem, we aim to distinguish effects of the two types of dynamics by choosing initial states as random product states from two factor spaces representing two subsystems. We introduce a random matrix model that permits to vary the coupling strength between the subsystems. The case of strong coupling is analyzed in detail, and we find no significant differences except for very low-dimensional spaces.
This course aims to introduce the student to random matrix models for decoherence and fidelity decay. They present a very powerful alternate approach, that emphasizes the disordered character of many environments and uncontrollable perturbations/couplings. The inherent integrability of such models makes analytic studies possible. We limit our considerations to linear response treatment, as high fidelity and small decoherence are the backbone of quantum information processes. For fidelity decay, where experimental results are available, a comparison with experiments shows excellent agreement wi…
We analyze the Stern-Gerlach experiment in phase space with the help of the matrix Wigner function, which includes the spin degree of freedom. Such analysis allows for an intuitive visualization of the quantum dynamics of the apparatus. We include the interaction with the environment, as described by the Caldeira-Leggett model. The diagonal terms of the matrix provide us with information about the two components of the state, that arise from interaction with the magnetic field gradient. In particular, from the marginals of these components, we obtain an analytical formula for the position and …
Dissipation and decoherence, and the evolution from pure to mixed states in quantum physics are handled through master equations for the density matrix. By embedding elements of this matrix in a higher-dimensional Liouville-Bloch equation, the methods of unitary integration are adapted to solve for the density matrix as a function of time, including the non-unitary effects of dissipation and decoherence. The input requires only solutions of classical, initial value time-dependent equations. Results are illustrated for a damped driven two-level system.
The decoherence matrix studied by Gudder and Sorkin (2011) can be considered as a map from the set of all the pairs of $n$-length paths to complex numbers, which is induced by the discrete-time quantum walk. The decoherence matrix is one of the decoherence functionals which present their historical quantum measure theory. In this paper, we compute the von Neumann entropy of the decoherence matrix. To do so, we use the result that the eigensystem of the decoherence matrix can be expressed by a corresponding correlated random walk.
Quantum random walk in a two-dimensional lattice with randomly distributed traps is investigated. Distributions of quantum walkers are evaluated dynamically for the cases of Hadamard, Fourier, and Grover coins, and quantum to classical transition is examined as a function of the density of the traps. It is shown that traps act as a serious and additional source of quantum decoherence. Furthermore, non-trivial temporal dependence of the standard deviation of the probability distribution of the walker is found when the trapping imperfections are introduced.
This paper describes the dynamics of a quantum two-level system (qubit) under the influence of an environment modeled by an ensemble of random matrices. In distinction to earlier work, we consider here separable couplings and focus on a regime where the decoherence time is of the same order of magnitude than the environmental Heisenberg time. We derive an analytical expression in the linear response approximation, and study its accuracy by comparison with numerical simulations. We discuss a series of unusual properties, such as purity oscillations, strong signatures of spectral correlations (i…
Decoherence is believed to deteriorate the ability of a purification scheme that is based on the idea of driving a system to a pure state by repeatedly measuring another system in interaction with the former and hinder for a pure state to be extracted asymptotically. Nevertheless, we find a way out of this difficulty by deriving an analytic expression of the reduced density matrix for a two-qubit system immersed in a bath. It is shown that we can still extract a pure state if the environment brings about only dephasing effects. In addition, for a dissipative environment, there is a possibility…
In this paper we use the Fano representation of two-qubit states from which we can identify a correlation matrix containing the information about the classical and quantum correlations present in the bipartite quantum state. To illustrate the use of this matrix, we analyze the behavior of the correlations under non-dissipative decoherence in two-qubit states with maximally mixed marginals. From the behavior of the elements of the correlation matrix before and after making measurements on one of the subsystems, we identify the classical and quantum correlations present in the Bell-diagonal stat…
It is commonly stated that decoherence in open quantum systems is due to growing entanglement with an environment. In practice, however, surprisingly often decoherence may equally well be described by random unitary dynamics without invoking a quantum environment at all. For a single qubit, for instance, pure decoherence (or phase damping) is always of random unitary type. Here, we construct a simple example of true quantum decoherence of two qubits: we present a feasible phase damping channel of which we show that it cannot be understood in terms of random unitary dynamics. We give a very int…
The hypothesis of the random flow of time is considered. To do this, the concepts of microscopic random time and macroscopic mean time, as well as random modular time are introduced. The possibilities of experimental verification of the hypothesis of the random flow of time are discussed. The possibility of interpreting the time-like fifth coordinate (introduced by V.A. Fock) as absolute time is considered and the question of the reality of absolute time is discussed. The absolute time, which manifests itself as the proper time of the particles, is assumed to be random.
We study the behavior of Quantum Darwinism (Zurek, [8]) within the iterative, random unitary operations qubit-model of pure decoherence (Novotny et al, [6]). We conclude that Quantum Darwinism, which describes the quantum mechanical evolution of an open system S from the point of view of its environment E, is not a generic phenomenon, but depends on the specific form of input states and on the type of S-E-interactions. Furthermore, we show that within the random unitary model the concept of Quantum Darwinism enables one to explicitly construct and specify artificial input states of environment…
We put forward a new, versatile and highly-scalable experimental setup for the realization of discrete two-dimensional quantum random walks with a single-qubit coin and tunable degree of decoherence. The proposed scheme makes use of a small number of simple optical components arranged in a multi-path Mach-Zehnder-like configuration, where a weak coherent state with average photon number of 1 is injected. Environmental effects (decoherence) are generated by a spatial light modulator, which introduces pure dephasing in the transverse spatial plane, perpendicular to the direction of propagation o…
Dissipation and decoherence, and the evolution from pure to mixed states in quantum physics are handled through master equations for the density matrix. Master equations such as the Lindblad equation preserve the trace of this matrix. Viewing them as first-order time-dependent operator equations for the elements of the density matrix, a unitary integration procedure can be adapted to solve for these matrix elements. A simple model for decoherence preserves the hermiticity of the density matrix. A single, classical Riccati equation is the only one requiring numerical handling to obtain a full s…
We examine an exactly solvable model of decoherence - a spin-system interacting with a collection of environment spins. We show that in this model (introduced some time ago to illustrate environment-induced superselection) generic assumptions about the coupling strengths lead to a universal (Gaussian) suppression of coherence between pointer states. We explore the regimes of validity of these results and discuss their relation to the spectral features of the environment and to the Loschmidt echo (or fidelity). Finally, we comment on the observation of such time dependence in spin echo experime…
An "almost diagonal" reduced density matrix (in coordinate representation) is usually a result of environment induced decoherence and is considered the sign of classical behavior. We point out that the proton of a ground state hydrogen atom can possess such a density matrix. We demonstrate on this example that an "almost diagonal" reduced density matrix may derive from an interaction with a low number of degrees of freedom (in our case with a single electron) which play the role of the environment. We also show that decoherence effects in our example can only be observed if the interaction wit…
We propose a selfconsistent quantum mechanical approach to study the dynamics of a two-level system subject to random time evolution. This randomness gives rise to competing effects between dissipative and non-dissipative decoherence with a consequent slow down of the atomic decay rate.
We derive analytical upper bounds for the entanglement of generalized Greenberger-Horne-Zeilinger states coupled to locally depolarizing and dephasing environments, and for local thermal baths of arbitrary temperature. These bounds apply for any convex quantifier of entanglement, and exponential entanglement decay with the number of constituent particles is found. The bounds are tight for depolarizing and dephasing channels. We also show that randomly generated initial states tend to violate these bounds, and that this discrepancy grows with the number of particles.
The decoherence of a fast quantum particle in a gas is studied by applying the Kramers-Moyal expansion to the quantum master equation for the reduced density matrix of the particle. This expansion leads to a general form of the Caldeira-Leggett master equation accounting for the angular variation of the differential cross section. The equation describes the decoherence in both the longitudinal and transverse directions with respect to the particle motion. It is shown that, when the differential cross section is concentrated in the forward direction, transverse decoherence dominates. The cohere…
We present a first-principles derivation of spatial atomic-sublevel decoherence near dielectric and metallic surfaces. The theory is based on the electromagnetic-field quantization in absorbing dielectric media. We derive an expression for the time-variation of the off-diagonal matrix element of the atomic density matrix for arbitrarily shaped substrates. For planar multilayered substrates we find that for small lateral separations of the atom's possible positions the spatial coherence decreases quadratically with the separation and inversely to the squared atom-surface distance.
Using the path integral representation of the density matrix propagator of quantum Brownian motion, we derive its asymptotic form for times greater than the localization time, $ (\hbar / γk T )^{\half}$, where $γ$ is the dissipation and $T$ the temperature of the thermal environment. The localization time is typically greater than the decoherence time, but much shorter than the relaxation time, $γ^{-1}$. We use this result to show that the reduced density operator rapidly evolves into a state which is approximately diagonal in a set of generalized coherent states. We thus reproduce, using a co…
In this work, two experimentally feasible methods of decoherence engineering-one based on the application of stochastic classical kicks and the other based on temporally randomized pulse sequences are combined. A different coupling interaction is proposed, which leads to amplitude damping as compared to existing methods which model phase damping, utilizing the $zz$ coupling interaction. The decoherence process on combining the stochastic kick method and the randomized pulse sequence method and the effectiveness of dynamical decoupling under these coupling interactions are analyzed. Finally, a …
For a discrete time quantum walk (QW) on the $N$-cycle, allowing for decoherence on the coin, we derive a number of new results, including an explicit formula for the position probability distribution. For a QW of this type, we show that the mixing behavior tends, in the long-run, to a uniform distribution, regardless of the initial state of the system and irrespective of the parity of the number of nodes $N$. These results confirm the findings of previous authors who arrived at similar conclusions through extensive numerical simulations. In particular, we infer that the mixing time $\bar{M(ε)…
Exact solutions of the Caldeira-Leggett Master equation for the reduced density matrix for a free particle and for a harmonic oscillator system coupled to a heat bath of oscillators are obtained for arbitrary initial conditions. The solutions prove that the Fourier transform of the density matrix at time t with respect to (x + x')/2, where x and x' are the initial and final coordinates, factorizes exactly into a part depending linearly on the initial density matrix and a part independent of it. The theorem yields the exact initial state dependence of the density operator at time t and its even…
We consider how randomness can be made to play a useful role in quantum information processing - in particular, for decoherence control and the implementation of quantum algorithms. For a two-level system in which the decoherence channel is non-dissipative, we show that decoherence suppression is possible if memory is present in the channel. Random switching between two potentially harmful noise sources can then provide a source of stochastic control. Such random switching can also be used in an advantageous way for the implementation of quantum algorithms.
If a quantum experiment includes random processes, then the results of repeated measurements can appear consistent with irreversible decoherence even if the system's evolution prior to measurement was reversible and unitary. Two thought experiments are constructed as examples.
We investigate the patterns in distributions of localizable entanglement over a pair of qubits for random multi-qubit pure states. We observe that the mean of localizable entanglement increases gradually with increasing the number of qubits of random pure states while the standard deviation of the distribution decreases. The effects on the distributions, when the random pure multi-qubit states are subjected to local as well as global noisy channels, are also investigated. Unlike the noiseless scenario, the average value of the localizable entanglement remains almost constant with the increase …
We relate disentanglement and decoherence rates in a pair of three-level atoms subjected to multi-local and collective pure dephasing noise acting in a preferred basis. The bipartite entanglement decay rate, as bounded from above by the negativity, is found to be greater than or equal to the dephasing-decoherence rates characterized by the decay of off-diagonal elements in the corresponding full density matrix describing the system or the reduced density matrix describing either qutrit, extending previous results for qubit pairs subject to such noise.
We discuss a few mathematical aspects of random dynamical decoupling, a key tool procedure in quantum information theory. In particular, we place it in the context of discrete stochastic processes, limit theorems and CPT semigroups on matrix algebras. We obtain precise analytical expressions for expectation and variance of the density matrix and fidelity over time in the continuum-time limit depending on the system Lindbladian, which then lead to rough short-time estimates depending only on certain coupling strengths. We prove that dynamical decoupling does not work in the case of intrinsic (i…
We calculate the two-qubit disentanglement due to classical random telegraph noise using the quasi-Hamiltonian method. This allows us to obtain analytical results even for strong coupling and mixed noise, important when the qubits have tunable working point. We determine when entanglement sudden death and revival occur as functions of qubit working point, noise coupling strength and initial state entanglement. For extended Werner states, we show that the concurrence is related to the difference of two functions: one is related to dephasing and the other longitudinal relaxation. A physical inte…
We find an exact analytical solution of the reduced density matrix from the Feynman-Vernon influence functional theory for a wave packet influenced by an environment containing a few discrete modes. We obtain two intrinsic energy scales relating to the time scales of the system and the environment. Different relationship between these two scales alters the overall form of the solution of the system. We also introduce a decoherence measure for a single wave packet which is defined as the ratio of Schrödinger uncertainty over the delocalization extension of the wave packet and characterizes the …
We propose easy implementable protocols for robust quantum key distribution with the collective dephasing channel or collective rotating channel. In these protocols, Bob only takes passive photon detection to measure the polarization qubits in the random bases. The source for the protocol with collective rotating channel is made by type 2 spontaneous parametric down conversion with random unitary rotation and phase shifter, no quantum disentangler is required. A simple proof for unconditionally security is shown.
We study decoherence of two non-interacting qubits. The environment and its interaction with the qubits are modelled by random matrices. Decoherence, measured in terms of purity, is calculated in linear response approximation. Monte Carlo simulations illustrate the validity of this approximation and of its extension by exponentiation. The results up to this point are also used to study one qubit decoherence. Purity decay of entangled and product states are qualitatively similar though for the latter case it is slower. Numerical studies for a Bell pair as initial state reveal a one to one corre…
It is discussed the decoherence problems in ensemble large-scale solid state NMR quantum computer based on the array of P donor atoms having nuclear spin I = 1/2. It is considered here, as main mechanisms of decoherence for low temperature (< 0.1 K), the adiabatic processes of random modulation of qubit resonance frequency determined by secular part of nuclear spin interaction with electron spin of the basic atoms, with impurity paramagnetic atoms and also with nuclear spins of impurity diamagnetic atoms. It was made estimations of allowed concentrations of magnetic impurities and of spin temp…
The coupled-channels density-matrix technique for nuclear reaction dynamics, which is based on the Liouville-von Neumann equation with Lindblad dissipative terms, is developed with the inclusion of full angular momentum couplings. It allows a quantitative study of the role and importance of quantum decoherence in nuclear scattering. Formulae of asymptotic observables that can reveal effects of quantum decoherence are given. A method for extracting energy-resolved scattering information from the time-dependent density matrix is introduced. As an example, model calculations are carried out for t…
We analyze the quantum process tomography (QPT) in the presence of decoherence, focusing on distinguishing local and non-local decoherence mechanisms for a two-partite system from experimental QPT data. In particular, we consider the $\sqrt{\rm iSWAP}$ gate realized with superconducting phase qubits and calculate the QPT matrix $χ$ in the presence of several local and non-local decoherence processes. We determine specific patterns of these decoherence processes, which can be used for a fast identification of the main decoherence mechanisms from an experimental $χ$-matrix.
The effects of decoherence for quantum system coupled with a bosonic field are investigated. An application of the stochastic golden rule shows that in the stochastic limit the dynamics of such a system is described by a quantum stochastic differential equation. The corresponding master equation describes convergence of a system to equilibrium. In particular it predicts exponential damping for off-diagonal matrix elements of the system density matrix, moreover these elements for a generic system will decay at least as \exp(-tN{kT\over\hbar}), where N is a number of particles in the system. As …
The probability operator is derived from first principles for an equilibrium quantum system. It is also shown that the superposition states collapse into a mixture of states giving the conventional von Neumann trace form for the quantum average. The mechanism for the collapse is found to be quite general: it results from the conservation law for a conserved, exchangeable variable (such as energy) and the entanglement of the total system wave function that necessarily follows. The relevance of the present results to the einselection mechanism for decoherence, to the quantum measurement problem,…
In this work the problem of characterizing matrix material structure from embedded electron spin decoherence is studied both theoretically and experimentally. Theoretical calculation using nuclear spin bath model and cluster correlation expansion method shows that the positions of decoherence time scale extremums among single crystal orientations of the matrix material coincide with those of the nearest neighbour proton dipolar couplings. This finding is confirmed by single crystal pulsed EPR experiment performed on $γ$-irradiated malonic acid (MA). Electron spin decoherence decay profile in p…
In order to analyze the effect of chaos or order on the rate of decoherence in a subsystem we aim to distinguish effects of the two types of dynamics from those depending on the choice of the wave packet. To isolate the former we introduce a random matrix model that permits to vary the coupling strength between the subsystems. The case of strong coupling is analyzed in detail, and we find at intermediate times a weak effect of spectral correlations that is reminiscent of the correlation hole.
We present a simple theoretical description of two recent experiments where damping of Rabi oscillations, which cannot be attributed to dissipative decoherence, has been observed. This is obtained considering the evolution time or the Hamiltonian as random variables and then averaging the usual unitary evolution on a properly defined, model-independent, probability distribution.
We study the non-Markovian dynamics of a qubit made up of a two-level atom interacting with an electromagnetic field (EMF) initially at finite temperature. Unlike most earlier studies where the bath is assumed to be fixed, we study the coherent evolution of the combined qubit-EMF system, thus allowing for the back-action from the bath on the qubit and the qubit on the bath in a self-consistent manner. In this way we can see the development of quantum correlations and entanglement between the system and its environment, and how that affects the decoherence and relaxation of the system. We find …
We study time evolution of entanglement between two qubits, which are part of a larger system, after starting from a random initial product state. We show that, due to randomness in the initial product state, entanglement is present only between directly coupled qubits and only for short times. Time dependence of the entanglement appears essentially independent of the specific hamiltonian used for time evolution and is well reproduced by a parameter-free two-body random matrix model.
We study the behavior of an open quantum system, with an $N$--dimensional space of states, whose density matrix evolves according to a non--unitary map defined in two steps: A unitary step, where the system evolves with an evolution operator obtained by quantizing a classically chaotic map (baker's and Harper's map are the two examples we consider). A non--unitary step where the evolution operator for the density matrix mimics the effect of diffusion in the semiclassical (large $N$) limit. The process of decoherence and the transition from quantum to classical behavior are analyzed in detail b…
The decoherence of a quantum system $S$ coupled to a quantum environment $E$ is considered. For states chosen uniformly at random from the unit hypersphere in the Hilbert space of the closed system $S+E$ we derive a scaling relationship for the sum of the off-diagonal elements of the reduced density matrix of $S$ as a function of the size $D_E$ of the Hilbert space of $E$. This sum decreases as $1/\sqrt{D_E}$ as long as $D_E\gg 1$. This scaling prediction is tested by performing large-scale simulations which solve the time-dependent Schr{ö}dinger equation for a ring of spin-1/2 particles, four…
The behaviors of one-dimensional quantum random walks are strikingly different from those of classical ones. However, when decoherence is involved, the limiting distributions take on many classical features over time. In this paper, we study the decoherence on both position and ``coin'' spaces of the particle. We propose a new analytical approach to investigate these phenomena and obtain the generating functions which encode all the features of these walks. Specifically, from these generating functions, we find exact analytic expressions of several moments for the time and noise dependence of …
We look at two possible routes to classical behavior for the discrete quantum random walk on the line: decoherence in the quantum ``coin'' which drives the walk, or the use of higher-dimensional coins to dilute the effects of interference. We use the position variance as an indicator of classical behavior, and find analytical expressions for this in the long-time limit; we see that the multicoin walk retains the ``quantum'' quadratic growth of the variance except in the limit of a new coin for every step, while the walk with decoherence exhibits ``classical'' linear growth of the variance even…
Two non-interacting qubits are coupled to an environment. Both coupling and environment are represented by random matrix ensembles. The initial state of the pair is a Bell state, though we also consider arbitrary pure states. Decoherence of the pair is evaluated analytically in terms of purity; Monte Carlo calculations confirm these results and also yield the concurrence of the pair. Entanglement within the pair accelerates decoherence. Numerics display the relation between concurrence and purity known for Werner states, allowing us to give a formula for concurrence decay.
Decoherence challenges in Nanoscience: A Quantum Phase Space perspective
Faculty of Sciences, University of Antananarivo（塔那那利佛大学理学院）；Institut National des Sciences et Techniques Nucléaires（国家核科学与技术研究所）；TWAS Madagascar Chapter, Malagasy Academy（马尔加什科学院 TWAS 马达加斯加分会）
We study a natural notion of decoherence on quantum random walks over the hypercube. We prove that in this model there is a decoherence threshold beneath which the essential properties of the hypercubic quantum walk, such as linear mixing times, are preserved. Beyond the threshold, we prove that the walks behave like their classical counterparts.
We investigate the time evolution of the chirality reduced density matrix for a discrete-time quantum walk on a one-dimensional lattice, which is obtained by tracing out the spatial degree of freedom. We analyze the standard case, without decoherence, and the situation where decoherence appears in the form of broken links in the lattice. By examining the trace distance for possible pairs of initial states as a function of time, we conclude that the evolution of the reduced density matrix is non-Markovian, in the sense defined in [H. P. Breuer, E. M. Laine, and J. Piilo, Phys. Rev. Lett. 103, 2…
We solve the time-dependent Schrodinger equation for the combination of a spin system interacting with a spin bath environment. In particular, we focus on the time development of the reduced density matrix of the spin system. Under normal circumstances we show that the environment drives the reduced density matrix to a fully decoherent state, and furthermore the diagonal elements of the reduced density matrix approach those expected for the system in the canonical ensemble. We show one exception to the normal case is if the spin system cannot exchange energy with the spin bath. Our demonstrati…
A common misconception is that decoherence gives the eigenstates that we observe to be fairly definite about a subsystem (e.g., approximate eigenstates of position) as the elements of the Schmidt basis in which the density matrix of the subsystem is diagonal. Here I show that in simple examples of linear systems with gaussian states, the Schmidt basis states have as much mean uncertainty about position as the full density matrix with its combination of different possibilities.
Decoherence is the process by which quantum systems interact and become correlated with their external environments; quantum trajectories are a powerful technique by which decohering systems can be resolved into stochastic evolutions, conditioned on different possible ``measurements'' of the environment. By calling on recently-developed tools from quantum information theory, we can analyze simplified models of decoherence, explicitly quantifying the flow of information and randomness between the system, the environment, and potential observers.
We study the time evolution of a qubit linearly coupled with a quantum environment under a sequence of short pi pulses. Our attention is focused on the case where qubit-environment interactions induce the decoherence with population decay. We assume that the environment consists of a set of bosonic excitations. The time evolution of the reduced density matrix for the qubit is calculated in the presence of periodic short pi pulses. We confirm that the decoherence is suppressed if the pulse interval is shorter than the correlation time for qubit-environment interactions.
We illustrate through numerical results a number of features of environment-induced decoherence under a broad class of apparatus-environment interactions in quantum measurements wherein the reduced system-apparatus density matrix evolves towards the nearest separable state and, in addition, there occurs a mixing in relevant groups of apparatus microstates (see below). The resulting final state is unique and correctly embodies the measurement statistics even in the absence of environment-induced superselection because of energy differences between these groups of states. The partial transpose r…
