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Introduction to Decoherence Theory
Abstract
This is an introduction to the theory of decoherence with an emphasis on its microscopic origins and on a dynamic description. The text corresponds to a chapter soon to be published in: A. Buchleitner, C. Viviescas, and M. Tiersch (Eds.), Entanglement and Decoherence. Foundations and Modern Trends, Lecture Notes in Physics, Vol 768, Springer, Berlin (2009)
... Furthermore, the transport equation (72) is equivalent to the linear Boltzmann equation for the Wigner function of the quantum particle [25,32,33], as shown in Appendix G. This reveals an interesting connection with decoherence theory of a quantum particle in a gas of moving atoms [68][69][70][71][72]. In the Foldy-Lax model, where the scatterers are fixed and the disorder is quenched, the randomness of the scatterer positions leads to decoherence, which is characterized by a finite coherence length [71][72][73][74] ...
... The former name comes from the structure of the diagrams shown in Fig. 14(c). The validity of this approximation is difficult to justify rigorously, but it seems relevant to describe the wave propagation in the weak scattering regime (69) in the absence of localization [25,47,83]. In fact, the ladder diagrams in Fig. 14(c) account for the incoherent propagation of the wave, while the cross diagrams in the end of Fig. 14(b) account for the coherent effects, such as the coherent backscattering [7,23,25,27,63,[83][84][85][86]. ...
... Finally, the coherence length can be explicitly obtained from definition (74) using the far-field behavior (75) of the density matrix. In the weak scattering regime (69), the fast oscillations of the sine function in Eq. (75) can be asymptotically approached by the average value sin 2 x = 1 2 , and the integrals involved in Eq. (74) read ...
In a previous paper [Phys. Rev. A 105, 042205 (2022)], the distribution of resonance poles in the complex plane of the wavenumber k associated to the multiple scattering of a quantum particle in a random point field was numerically discovered. This distribution presented two distinctive structures: a set of peaks at small k when the wavelength is larger than the interscatterer distance, and a band almost parallel to the real axis at larger k. In this paper, a detailed theoretical study based on wave transport theory is proposed to explain the origin of these structures and to predict their distribution in the complex k plane. First, it is shown that the peaks at small k can be understood using an effective wave equation for the average wavefunction over the disorder. Then, that the band at large k can be described by the Bethe-Salpeter equation for the square modulus of the wavefunction, which is derived from the diagrammatic method. This study is supported by careful comparisons with numerical simulations. The largest simulations revealed the presence of filaments alike quantum scars for the wavefunction in the bulk of the disordered medium.
... In system-environment evolutions which lead to pure dephasing of the system, pure state entanglement has a particularly meaningful interpretation. Entanglement, which in this case is directly linked to system decoherence, describes the amount information about the state of the system which can be extracted from the environment [8,9]. It has been recently shown for mixed states, that without such information transfer, pure dephasing is not accompanied by entanglement generation [10,11]. ...
... The measure has a straightforward physical interpretation in contrast to all other measures of mixed state entanglement, directly linking the amount of information about the qubit state which is contained in the environment to the amount of QEE. This means that the pure state interpretation of entanglement for the class of interactions studied [8,9] can be directly extended to mixed states, even though the link between entanglement and decoherence cannot [10][11][12]. ...
... meaning that the state (3) is separable if and only if the state of the environment conditional on one of the two pointer states of the qubit is the same as the state of the environment conditional on the other pointer state at time t, otherwise the qubit is entangled with its environment. Separability by no means excludes qubit decoherence which is proportional to TrR 01 (t) (the only exception is the situation when the state of the environment is also initially pure; then decoherence without entanglement is impossible [8,9]) and there are ample examples for realistic qubits undergoing decoherence both accompanied by QEE [18,19] and not accompanied by QEE [12,[20][21][22][23][24][25]. There are even more examples of systems which undergo this type of decoherence, which have never been classified in terms of their entangling or separable nature [26][27][28][29][30][31][32][33][34]. ...
We propose a qubit-environment entanglement measure which is tailored for evolutions that lead to pure dephasing of the qubit which are abundant in solid state scenarios. The measure can be calculated directly form the density matrix, but in fact does not require the knowlegde of the full density matrix, and it is enough to know the initial qubit state and the states of the environment conditional on qubit pointer states. In contrast to all other measures of mixed state entanglement, the measure a straightforward physical interpretation directly linking the amount of information about the qubit state which is contained in the environment to the amount of qubit-environmnent entanglement. This allows for a direct extension of the pure state interpretation of entanglement generated during pure dephasing to mixed states, even though pure-state conclusions about qubit decoherence are not transferable.
... Environmental decoherence describes the unwanted interaction between a qubit and its surroundings. These environmental effects can significantly impact the efficiency of manipulating quantum information [59,60]. To reflect our understanding of the environmental influences, it is crucial to have an accurate mathematical description of the errors [61]. ...
... In noisy environments, how quantum information changes over time can be described using mathematical tools called quantum channels. In fact, a quantum channel denoted by ϕ, is a mathematical function that transforms a pure state into a mixed state due to the influence of its surroundings [59,62]. In general, a quantum channel is fully described by the Kraus representation as follows ...
Quantum metrology leverages quantum effects such as squeezing, entanglement, and other quantum correlations to boost precision in parameter estimation by saturating quantum Cramer Rao bound, which can be achieved by optimizing quantum Fisher information or Wigner-Yanase skew information. This work provides analytical expressions for quantum Fisher and skew information in a general three-qubit X-state and examines their evolution under phase damping, depolarization, and phase-flip decoherence channels. To illustrate the validity of our method, we investigate their dynamics for a three-qubit Greenberger-Horne-Zeilinger (GHZ) state subjected to various memoryless decoherence channels. Closed-form expressions for QFI and SQI are derived for each channel. By comparing these metrics with the entanglement measure of concurrence, we demonstrate the impact of decoherence on measurement precision for quantum metrology. Our results indicate that phase damping and phase-flip channels generally allow for better parameter estimation compared to depolarization. This study provides insights into the optimal selection of noise channels for enhancing precision in quantum metrological tasks involving multi-qubit entangled states.
... From the aforementioned observations, we infer that the qubits in the D-Wave machine are also affected by the amplitude fluctuation of σ z . The decay rate of the first excited state typically increases with the transition matrix |⟨g|σ (z) j |e⟩| [44] where |g⟩ and |e⟩ denote the ground and first excited state of the Hamiltonian, respectively. We numerically show that the transition matrix of the four qubits in the transverse-field Ising model is much smaller than that of a single qubit in the transverse and longitudinal field Hamiltonian (See Fig. 4). ...
... In order to elucidate the mechanism of its small incoherent decay rate, we employ perturbation theory and obtain the explicit form of the ground state when the transverse magnetic field is small. Thus, we can obtain the transition matrix, which is useful for quantifying the robustness of the first excited state against decoherence [44]. ...
Quantum annealing has been demonstrated with superconducting qubits. Such a quantum annealer has been used to solve combinational optimization problems. Moreover, it serves as a quantum simulator for investigating the properties of the quantum many-body systems.
However, the coherence properties of actual devices provided by D-Wave Quantum Inc. have not been explored sufficiently. Here, we measure the energy relaxation of the excited state in quantum annealing with the D-Wave device. Specifically, we investigate the incoherent decay rate of the first excited states of a fully connected Ising model with a transverse field. We find that the decay rate of the excited states of the model is orders of magnitude smaller than that of the excited state of a single qubit, and we qualitatively explain this phenomenon by using theoretical methods. Since our numerical simulations show that the first excited state during QA for the model is entangled, our experimental results indicate that the long-lived entangled state can be generated during QA with the D-Wave machine.
... From the observations mentioned above, we infer that the qubits in the D-Wave machine are also affected by the amplitude fluctuation of σ z . The energy relaxation rate of the first excited state typically increases as transition matrix | g|σ (z) j |e | increases [37] where |g and |e denote the ground and first excited state of the Hamiltonian, respectively. We numerically show that the transition matrix of the four qubits in the transverse-field Ising model is much smaller than that of a single qubit in the transverse and longitudinal fields Hamiltonian (See Fig. 4). ...
... In order to elucidate the mechanism of its long energy relaxation time, we adopt the perturbation theory and obtain the explicit form of the ground state when the transverse magnetic field is small. This allows us to obtain the transition matrix, which is useful to quantify the robustness of the first excited state against decoherence [37]. ...
Quantum annealing has been demonstrated with superconducting qubits. Such a quantum annealer has been used to solve combinational optimization problems and is also useful as a quantum simulator to investigate the properties of the quantum many-body systems. However, the coherence properties of actual devices provided by D-Wave Quantum Inc. are not sufficiently explored. Here, we propose and demonstrate a method to measure the coherence time of the excited state in quantum annealing with the D-Wave device. More specifically, we investigate the energy relaxation time of the first excited states of a fully connected Ising model with a transverse field. We find that the energy relaxation time of the excited states of the model is orders of magnitude longer than that of the excited state of a single qubit, and we qualitatively explain this phenomenon by using a theoretical model. The reported technique provides new possibilities to explore the decoherence properties of quantum many-body systems with the D-Wave machine.
... Although δp could be negative, in turns out that p − is always non-negative. To see this, we simply insert the expressions (15) and (16) into the above formula of p − , we have ...
... This is why there is no resonance effect in this case. The commutator ofĤ int at different times is not an operator, but just a c-number [15] ...
We consider quantum decoherence and Landauer’s principle in qubit-cavity quantum field theory (QFT) interaction, treating the qubit as the system and cavity QFT as the environment. In particular, we investigate the changes that occur in the system with a pure initial state and environment during the decoherence process, with or without energy dissipation, and compare the results with the case in which the initial state of the system is a mixed state and thus decoherence is absent. When we choose an interaction Hamiltonian such that the energy and coherence of the system change simultaneously, the population change of the system and the energy change are the same when the initial state is mixed. However, the decoherence terms increase the von Neumann entropy of the system. In this case the energy change and decoherence of the system are not independent physical processes. The decoherence process maintains unitarity. On the other hand, if the interaction Hamiltonian does not change the energy of the system, there is only the decoherence effect. The environment will be a distribution in the basis of the displaced number state and always increases the energy. Landauer’s principle is satisfied in both cases.
... The term γ φ D[n]ρ introduces a dephasing effect [20]. In the most narrow sense [21], the effect of dephasing is the decay of the off-diagonal elements ofρ expressed in the energy eigenbasis of the system. These elements are called coherences. ...
... and p(φ; 0) = δ(φ), (E. 21) it is seen that (E.14) is satisfied. The solution to (E. 19) with (E.21) is given by [39] p(φ; t) = 1 ...
Motivated by quantum experiments with nanomechanical systems, the evolution of a Kerr oscillator with focus on creation of states with a negative Wigner function is investigated. Using the phase space formalism, results are presented that demonstrate an asymptotic behavior in the large squeezing regime for the negativity of a squeezed vacuum state under unitary evolution. The analysis and model are extended to squeezed vacuum states of open systems, adding the decoherence effects of damping and dephasing. To increase experimental relevance, the regime of strong damping is considered. These effects are investigated, yielding similar asymptotic results for the behavior of these effects in the large squeezing regime. Combining these results, it is shown that a weak nonlinearity as compared to damping may be improved by increasing the squeezing of the initial state. It is also shown that this may be done without exacerbating the effects of dephasing.
... Environmentally induced dephasing of superpositions of pointer states of a controlled quantum system is commonly associated with creation of system-environment entanglement, or at least the presence of the latter is deemed to be necessary in order to call this process quantum decoherence [1][2][3]. However, as has been pointed out in literature, this association holds only when the initial states of both the qubit and the environment are pure [1][2][3][4]. ...
... Environmentally induced dephasing of superpositions of pointer states of a controlled quantum system is commonly associated with creation of system-environment entanglement, or at least the presence of the latter is deemed to be necessary in order to call this process quantum decoherence [1][2][3]. However, as has been pointed out in literature, this association holds only when the initial states of both the qubit and the environment are pure [1][2][3][4]. In the more general, and much more realistic, case of mixed environmental states, dephasing of the system does not have to be accompanied by establishment of system-environment entanglement, and intuitions concerning distinguishing between "quantum decoherence" and "dephasing due to classical environmental noise" (understood here strictly as leading to no systemenvironment entanglement) that are built in works focusing on pure-state vs "classical" environments become unreliable [5][6][7][8][9][10][11][12][13]. ...
We analyze the relationship between qubit-environment entanglement that can be created during the pure dephasing of the qubit and the effectiveness of the spin-echo protocol. We focus here on mixed states of the environment. We show that whereas the echo protocol can obviously counteract classical environmental noise, it can also undo dephasing associated with qubit-environment entanglement, and there is no obvious difference in its efficiency in these two cases. Additionally, we show that qubit-environment entanglement can be generated at the end of the echo protocol even when it is absent at the time of application of the local operation on the qubit (the π pulse). We prove that this can occur only at isolated points in time, after fine-tuning of the echo protocol duration. Finally, we discuss the conditions under which the observation of specific features of the echo signal can serve as a witness of the entangling nature of the joint qubit-environment evolution.
... In system-environment evolutions which lead to pure dephasing (PD) of the system, pure-state entanglement has a particularly meaningful interpretation. Entanglement, which in this case is directly linked to system decoherence, describes the amount of information about the system state which can be extracted from the environment [8][9][10]. It has been recently shown for mixed states that without such information transfer, PD is not accompanied by entanglement generation [11,12]. ...
... If there are no correlations (classical or quantum) in the initial state of the environment,R(0) = kR k (0), then the density matrices of the environment conditional on the qubit pointer state also retain this form,R ii (t ) = kR k ii (t ). In this case F (R 00 (t ),R 11 (t )) = k F R k 00 (t ),R k 11 (t ) (10) and much smaller matrices have to be diagonalized to find the value of the PD entanglement measure. This feature does not simplify the complexity of calculating any of the other measures of mixed-state entanglement. ...
We propose a qubit-environment entanglement measure which is tailored for evolutions that lead to pure dephasing of the qubit, such as are abundant in solid-state scenarios. The measure can be calculated directly form the density matrix without minimization of any kind. In fact, it does not require knowledge of the full density matrix and it is enough to know the initial qubit state and the states of the environment conditional on qubit pointer states. This yields a computational advantage over standard entanglement measures, which becomes large when there are no correlations between environmental components in the conditional environmental states. In contrast to all other measures of mixed-state entanglement, the measure has a straightforward physical interpretation directly linking the amount of information about the qubit state which is contained in the environment to the amount of qubit-environment entanglement. This allows for a direct extension of the pure-state interpretation of entanglement generated during pure dephasing to mixed states, even though pure-state conclusions about qubit decoherence are not transferable.
... Many attempts have been made and will likely continue, but the possibility of a more fundamental type of uncertainty should not be dismissed. physics, and in particular underpins the Lindblad equation, which provides a differential description of such Markovian open-system dynamics [30][31][32]. ...
... In the present context, there is an approximately well-defined quantum conditional probability p i n ;t n |i n−1 ;t n−1 for the system to be in the actual ontic state i n at time t n given that its actual ontic state was i n−1 at time t n−1 . Given a sequence of such actual ontic states at particular coarse-grained times i n ;t n M n=0 , the probability associated with this overall coarse-grained ontic trajectory is This description of coarse-grained ontic trajectories is in keeping with the literature on quantum trajectories associated with the Lindblad equation and other related types of open-system dynamics [32,33]. ...
Any realist interpretation of quantum theory must grapple with the measurement problem and the status of state-vector collapse. In a no-collapse approach, measurement is typically modeled as a dynamical process involving decoherence. We describe how the minimal modal interpretation closes a gap in this dynamical description, leading to a complete and consistent resolution to the measurement problem and an effective form of state collapse. Our interpretation also provides insight into the indivisible nature of measurement—the fact that you can’t stop a measurement part-way through and uncover the underlying ‘ontic’ dynamics of the system in question. Having discussed the hidden dynamics of a system’s ontic state during measurement, we turn to more general forms of open-system dynamics and explore the extent to which the details of the underlying ontic behavior of a system can be described. We construct a space of ontic trajectories and describe obstructions to defining a probability measure on this space.
... Environmentally induced dephasing of superpositions of pointer states of a controlled quantum system is commonly associated with creation of system-environment entanglement, or at least the presence of the latter is deemed to be necessary in order to call this process quantum decoherence [1][2][3]. However, as has been pointed out in literature, this association holds only when the initial states of both the qubit and the environment are pure [1][2][3][4]. ...
... Environmentally induced dephasing of superpositions of pointer states of a controlled quantum system is commonly associated with creation of system-environment entanglement, or at least the presence of the latter is deemed to be necessary in order to call this process quantum decoherence [1][2][3]. However, as has been pointed out in literature, this association holds only when the initial states of both the qubit and the environment are pure [1][2][3][4]. In the more general, and much more realistic case of mixed environmental states, dephasing of the system does not have to be accompanied by establishment of system-environment entanglement, and intuitions concerning distinguishing between "quantum decoherence" and "dephasing due to classical environmental noise" (understood here strictly as leading to no systemenvironment entanglement) that are built in works focusing on pure-state vs "classical" environments become unreliable [5][6][7][8][9][10][11][12][13]. ...
We analyze the relationship between qubit-environment entanglement that can be created during the pure dephasing of the qubit initialized in a superposition of its pointer states, and the effectiveness of the spin echo protocol. Commonly encountered intuitions connecting the amount of decoherence with the amount of qubit-environment entanglement - suggesting that large echo signal corresponds to undoing of a large amount of entanglement - hold only for pure initial states of the environment, which is obviously a rarely encountered case, and we focus here on mixed states of the environment. We show that while the echo protocol can obviously counteract classical environmental noise (but it does not have to, if the noise is not mostly of low-frequency character), it can also undo dephasing associated with qubit-environment entanglement, and there is no obvious difference in its efficiency in these two cases. Additionally, we show that qubit-environment entanglement can be generated at the end of the echo protocol even when it is absent at the time of application of the local operation on the qubit (the pulse). We prove that this can occur only at isolated points in time, after fine-tuning of the echo protocol duration. Finally, we discuss the conditions under which the observation of specific features of the echo signal can serve as a witness of the entangling nature of the joint qubit-environment evolution.
... For the choice of the pointer basis to be definite, the interaction of the system with its observed environments must be of pure dephasing type [16,20,21] as well. Such an evolution does not disturb central system occupations and has a particularly simple form, even when the observed environments are taken into account. ...
... Let us now consider a qubit interacting with an environment of dimension d. Imagine that there exists a time t for which the d qubit states |ψ n (t) divide into two groups I and II such that: i) all of the states within each group are identical up to a global phase factor: |ψ n I/II (t) = e iΘn I/II |ψ I/II (21) for all n I ∈ Group I and n II ∈ Group II; ii) the states in the two groups are orthogonal with respect to each other ψ n I |ψ n II = |a 0 | 2 + |a 1 | 2 e i(φn I (t)−φn II (t)) = 0. (22) In this case the state (12) takes the form ...
We study separable system-environment evolutions of pure dephasing type in the context of objectivity. We find that it can lead to the natural emergence of spectrum broadcast structures (SBSs), but at discrete time instances. Contrary to the standard way of obtaining SBS states, which requires entanglement, reaching such states here does not require decoherence (no unobserved environments are necessary). Yet the biggest difference is the basis with respect to which the SBS states are formed. Here it is not the pointer basis of the system given by the interaction with the environment, but their equal superposition basis, the so-called mutually unbiased basis. The price to pay is a simple SBS structure with just one environment and its momentary character.
... In the context of Eq. (3), dephasing dampens the off-diagonal density matrix elements nn (k, t ), n = n towards zero. This damping is traditionally modeled by phenomenological dephasing rates, γ n,n (k) [57,[59][60][61][62], which here we take as independent of k and the band indices: γ nn (k) := γ = 1/T 2 with n = n and T 2 is the dephasing time. The typical values for T 2 used in the literature range from 1 to 100 fs [9,10,16,37,58,59,[61][62][63][64]. ...
Experimental indications have been reported suggesting that laser pulses shining on materials with relativistic dispersion can produce currents that survive long after the illumination has died out. Such residual currents, i.e., remnant currents, have applications in petahertz logical gates. The remnant currents' strength strongly depends on the pulse shape. We develop an analytical formula that allows one to optimize the pulse shape for remnant current production; we predict remnant currents exceeding the values observed so far by up to five orders of magnitude. This can be achieved by using single-cycle pulses instead of the previously employed multicycle pulses. In fact, remnant currents can be almost as strong as the peak current under irradiation.
Published by the American Physical Society 2024
... For systems with finite Hilbert spaces, the eigenoperator method is appropriate [5]. For scattering processes, however, the collisional method (or repeated interaction scheme) is often more suitable [21,22]. We focus on two relevant approaches: i ) The first approach considers the complete evolution for a short time interval, ∆t, expanding the time evolution operator and tracing over the environment [23]. ...
We develop a collisional framework for neutrino propagation within open quantum systems, termed the \emph{Collisional Approach for Open Neutrino Systems} (CAONS). A Born-Markov equation is derived, linking decoherence, dissipation, decay rates, and scattering cross sections. Perturbation theory is not required and the resultant master equation is applied to visible and invisible neutrino decays and propagation through a stationary medium. Comparing with previous studies on neutrino decoherence, we show that current bounds on decoherence parameters can significantly tighten constraints on neutrino couplings with dark-matter. Lastly, we establish connections between CAONS and non-Hermitian Hamiltonian approaches.
... Classicality emerges when the system losses the superpositions and turns into an incoherent mixture through decoherence.[1, 2, 10,11]. This can be directly seen through the three-time Leggett-Garg inequality (LGI) [12][13][14], where the authors considered measuring a dichotomous (±1 valued) observablê Q at three different instances t 1 , t 2 and t 3 , as the system evolves in time. ...
Quantum theory contravenes classical macrorealism by allowing a system to be in a superposition of two or more physically distinct states, producing physical consequences radically different from that of classical physics. We show that a system, upon subjecting to transform under superposition of unitary operators, exhibits enhanced non-macrorealistic feature - as quantified by violation of the Leggett-Garg inequality (LGI) beyond the temporal Tsirelson bound. Moreover, this superposition of unitaries also provides robustness against decoherence by allowing the system to violate LGI and thereby retain its non-macrorealistic behavior for a strikingly longer duration. Using an NMR register, we experimentally demonstrate the superposition of unitaries with the help of an ancillary qubit and verify these theoretical predictions.
... In the literature, decoherence has been studied quite rigorously using the scattering model [29,30,32,53,54], and has also been studied from a field theory approach [35,36]. The standard practice for air molecule scattering, photon scattering, and blackbody emission and absorption is to solve the decoherence rate in the limit where the environmental particle's wavelength is big or small compared to the size of the spatial superposition, see [32,55,56]. ...
We provide a solution for decoherence in spatial superpositions due to scattering/collision with air molecules. This result reproduces the short- and long-wavelength limits known in the literature. We compare the decoherence rate with several existing interpolations in the literature and evaluate the decoherence rate and experimental parameters when creating macroscopic quantum spatial superpositions (i.e., micron-size spheres). Finally, we consider the decoherence rate's time dependence while creating and closing the spatial superposition in an interferometer setup.
... However, the physical realization of entangled states for practical purposes is often hampered by decoherence, which arises from the interaction of a quantum system with the surrounding environment [5][6][7][8]. Decoherence and noise destroy entanglement, and are a significant obstacle to maintaining the quantum coherence required for effective quantum information processing [9][10][11][12][13][14][15]. Understanding and mitigating the effects of decoherence is thus crucial for the advancement of quantum technologies [16][17][18][19][20]. ...
We study how stronger noise can enhance the entanglement in inhomogeneously monitored quantum systems. We consider a free fermions model composed of two coupled chains - a system chain and an ancilla chain, each subject to its own different noise - and explore the dynamics of entanglement within the system chain under different noise intensities. Our results demonstrate that, contrary to the detrimental effects typically associated with noise, certain regimes of noise on the ancilla can significantly enhance entanglement within the system. Numerical simulations demonstrate the robustness of such entanglement enhancement across various system sizes and noise parameters. This enhancement is found to be highly dependent on the hopping strength in the ancilla, suggesting that the interplay between unitary dynamics and noise can be tuned to optimize the entanglement of a quantum system.
... In the quantum theory, we see that in the statistical approach we enter the regime of open quantum systems, decoherence, entanglement, partial tracing and non-equilibrium statistical mechanics [108] which leads to master equations of Lindbladt type for the "statistical operator", i.e., the density matrix that replaces Z 0 . In the effective approach of the quantum theory one tries to solve the exact Schrödinger or Heisenberg equations following the above idea of the classical theory of solving first the equations for the X, Y sector and after that for the Q, P sector. ...
In a seminal work, Hawking showed that natural states for free quantum matter fields on classical spacetimes that solve the spherically symmetric vacuum Einstein equations are KMS states of non-vanishing temperature. Although Hawking’s calculation does not include the backreaction of matter on geometry, it is more than plausible that the corresponding Hawking radiation leads to black hole evaporation which is, in principle, observable. Obviously, an improvement of Hawking’s calculation including backreaction is a problem of quantum gravity. Since no commonly accepted quantum field theory of general relativity is available yet, it has been difficult to reliably derive the backreaction effect. An obvious approach is to use the black hole perturbation theory of a Schwarzschild black hole of fixed mass and to quantize those perturbations. However, it is not clear how to reconcile perturbation theory with gauge invariance beyond linear perturbations. In recent work, we proposed a new approach to this problem that applies when the physical situation has an approximate symmetry, such as homogeneity (cosmology), spherical symmetry (Schwarzschild), or axial symmetry (Kerr). The idea, which is surprisingly feasible, is to first construct the non-perturbative physical (reduced) Hamiltonian of the reduced phase space of fully gauge invariant observables and only then apply perturbation theory directly in terms of observables. The task to construct observables is then disentangled from perturbation theory, thus allowing to unambiguously develop perturbation theory to arbitrary orders. In this first paper of the series we outline and showcase this approach for spherical symmetry and second order in the perturbations for Einstein–Klein–Gordon–Maxwell theory. Details and generalizations to other matter and symmetry and higher orders will appear in subsequent companion papers.
... In the current pursuit of large-scale quantum computation, two major obstacles are encountered: hardware noise [1][2][3][4] and scalability limitations [5][6][7]. Errorcorrected [8][9][10] modular quantum computers [11,12] offer a promising route to overcome both these challenges. ...
Connecting multiple smaller qubit modules by generating high-fidelity entangled states is a promising path for scaling quantum computing hardware. The performance of such a modular quantum computer is highly dependent on the quality and rate of entanglement generation. However, the optimal architectures and entanglement generation schemes are not yet established. Focusing on modular quantum computers with solid-state quantum hardware, we investigate a distributed surface code's error-correcting threshold and logical failure rate. We consider both emission-based and scattering-based entanglement generation schemes for the measurement of non-local stabilizers. Through quantum optical modeling, we link the performance of the quantum error correction code to the parameters of the underlying physical hardware and identify the necessary parameter regime for fault-tolerant modular quantum computation. In addition, we compare modular architectures with one or two data qubits per module. We find that the performance of the code depends significantly on the choice of entanglement generation scheme, while the two modular architectures have similar error-correcting thresholds. For some schemes, thresholds nearing the thresholds of non-distributed implementations () appear feasible with future parameters.
... Interaction of qubits with their environments that fluctuate in an uncontrolled manner leads to decoherence [51] of their quantum states. In this process, the entanglement, which requires existence of a coherent superposition of at least two product states of the two qubits, is also destroyed [37]. ...
We analyze in detail a procedure of entangling of two singlet–triplet ( S – T 0 ) qubits operated in a regime when energy associated with the magnetic field gradient, Δ B z , is an order of magnitude smaller than the exchange energy, J , between singlet and triplet states (Shulman et al. in Science 336:202, 2012). We have studied theoretically a single S – T 0 qubit in free induction decay and spin echo experiments. We have obtained analytical expressions for the time dependence of components of its Bloch vector for quasistatic fluctuations of Δ B z and quasistatic or dynamical 1 / f β -type fluctuations of J . We have then considered the impact of fluctuations of these parameters on the efficiency of the entangling procedure which uses an Ising-type coupling between two S – T 0 qubits. In particular, we have obtained an analytical expression for evolution of two qubits affected by 1 / f β -type fluctuations of J . This expression indicates the maximal level of entanglement that can be generated by performing the entangling procedure. Our results deliver also an evidence that in the above-mentioned experiment S – T 0 qubits were affected by uncorrelated 1 / f β charge noises.
... Moreover, it is straightforward to argue that an electron, for instance, which has interacted with an apparatus or, more generally, with its environment will have an extremely well localized wave function. Such localization processes are well understood in the context of decoherence theory (see, e.g., [23] and references therein). ...
This work aims to shed some light on the meaning of the positive energy assumption in relativistic quantum theory and its relation to questions of localization of quantum systems. It is shown that the positive energy property of solutions of relativistic wave equations (such as the Dirac equation) is very fragile with respect to state transformations beyond free time evolution. Paying attention to the connection between negative energy Dirac wave functions and pair creation processes in second quantization, this analysis leads to a better understanding of a class of problems known as the localization problem of relativistic quantum theory (associated for instance with famous results of Newton and Wigner, Reeh and Schlieder, Hegerfeldt or Malament). Finally, this analysis is reflected from the perspective of a Bohmian quantum field theory.
... A very similar procedure has been performed at the level of the time-evolution operator in [111] for the spin-boson model. This can be carried over almost immediately by recalling the adjoint action of these operators on the density operator and the fact that the S α (t) can be ordered arbitrarily inside the time-ordering T (S) ← . ...
We consider the coupling of a scalar field to linearised gravity and derive a relativistic gravitationally induced decoherence model using Ashtekar variables. The model is formulated at the gauge invariant level using suitable geometrical clocks in the relational formalism, broadening existing gauge invariant formulations of decoherence models. For the construction of the Dirac observables we extend the known observable map by a kind of dual map where the role of clocks and constraints is interchanged. We also discuss a second choice of geometrical clocks existing in the ADM literature. Then we apply a reduced phase space quantisation on Fock space and derive the final master equation choosing a Gibbs state for the gravitational environment and using the projection operator technique. The resulting master equation is not automatically of Lindblad type, a starting point sometimes assumed for phenomenological models, but still involves a residual time dependence at the level of the effective operators in the master equation due to the form of the correlation functions that we express in terms of thermal Wightman functions. Furthermore, we discuss why in the model analysed here the application of a second Markov approximation in order to obtain a set of time independent effective system operators is less straightforward than in some of the quantum mechanical models.
... This is the story of decoherence. 1 Di®erent approaches have been used to study quantum open systems including time-dependent Hamiltonians, nonlinear Schr€ odinger equations, master equations for the reduced density matrix describing the system of interest, the functional integral approach and stochastic dynamics in Hilbert space. [2][3][4][5] Practically, a simple e®ective description is given when coupling to the environment is taken into account through the inclusion of dissipative and stochastic terms in the dynamical equations describing the system. 2 Physical systems subjected to analytical potentials can be approximated by a harmonic oscillator near the local minima of the potential energy function. ...
Some aspects of quantum damped harmonic oscillator (DHO) obeying a Markovian master equation are considered in the absence of thermal noise. The continuity equation is derived and Bohmian trajectories are constructed. As a solution of the master equation, we take a single coherent state and compute analytically the relative entropy of coherence, [Formula: see text], in the energy, position and momentum bases. Although [Formula: see text] is constant in both the position and the momentum bases, it is a decreasing function of time in the energy basis becoming zero at long times, revealing its role as the preferred basis. Then, quantum coherence is computed for a superposition of two coherent states, a cat state, and also a superposition of two cat states in the energy basis as a function of separation, in the complex plane, between the two superposed states. It is seen that the quantum coherence increases with this separation. Furthermore, quantum coherence of superposition is compared to that of decomposed states in the superposition. Finally, by considering a system of two noninteracting DHOs, the effect of quantum statistics is studied on the coherence of reduced single-particle states, the joint detection probability and the mean square separation of particles. Our computations show that the single-particle coherence for antisymmetric states is always less than that of symmetric ones. Furthermore, boson anti-bunching and fermion bunching is seen in this open system. This behavior of bosons is the matter-wave analogue of photon anti-bunching seen in a modified Hanbury Brown–Twiss (HBT) interferometer.
... The Lindblad master equation 31,32,[47][48][49][50][51] is the most general Markovian description of a system density matrix ρ S (t) interacting with the environment. It forms a dynamical semigroup of reduced density matrices that are trace-preserving (norm-preserving) and completely positive for population. ...
We derive the L-MFE method to incorporate Lindblad jump operator dynamics into the mean-field Ehrenfest (MFE) approach. We map the density matrix evolution of Lindblad dynamics onto pure state coefficients using trajectory averages. We use simple assumptions to construct the L-MFE method that satisfies this exact mapping. This establishes a method that exactly reproduces Lindblad decay dynamics using a wavefunction description, with deterministic changes of the magnitudes of the quantum expansion coefficients, while only adding on a stochastic phase. We further demonstrate that when including nuclei in the Ehrenfest dynamics, the L-MFE method gives semi-quantitatively accurate results, with the accuracy limited by the accuracy of the approximations present in the semiclassical MFE approach. This work provides a general framework to incorporate Lindblad dynamics into semiclassical or mixed quantum-classical simulations.
... The Markov dynamics of states evolving in noisy environments are modeled by the maps between the spaces of operators known as quantum channels. Indeed, a quantum channel is a map Φ that acts on the density operators; Φ : ρ −→ Φ (ρ), and incorporates the evolution of a pure quantum state to a mixed quantum state [69,70]. Mathematically, the description of the quantum channel can be completely characterized by the Kraus representation as ...
In statistical estimation theory, it has been shown previously that the Wigner-Yanase skew information is bounded by the quantum Fisher information associated with the phase parameter. Besides, the quantum Cramér-Rao inequality is expressed in terms of skew information. Since these two fundamental quantities are based on the concept of quantum uncertainty, we derive here their analytical formulas for arbitrary two qubit X-states using the same analytical procedures. A comparison of these two informational quantifiers for two quasi-Werner states composed of two bipartite superposed coherent states is examined. Moreover, we investigated the decoherence effects on such quantities generated by the phase damping, depolarization and amplitude damping channels. We showed that decoherence strongly influences the quantum criteria during the evolution and these quantities exhibit similar dynamic behaviors. This current work is characterized by the fact that these two concepts play the same role and capture similar properties in quantum estimation protocols.
... The latter representation of the Lindblad generator W has been, to our knowledge, first reported in Ref. [8], in discussing the emission statistics of two and three-level atomic systems. References [6,7,78] provide an extensive discussion of the subject. In Equation (5) , has a direct effect on the environment. ...
We consider Markovian open quantum systems subject to stochastic resetting, which means that the dissipative time evolution is reset at randomly distributed times to the initial state. We show that the ensuing dynamics is non-Markovian and has the form of a generalized Lindblad equation. Interestingly, the statistics of quantum-jumps can be exactly derived. This is achieved by combining techniques from the thermodynamics of quantum-jump trajectories with the renewal structure of the resetting dynamics. We consider as an application of our analysis a driven two-level and an intermittent three-level system. Our findings show that stochastic resetting may be exploited as a tool to tailor the statistics of the quantum-jump trajectories and the dynamical phases of open quantum systems.
... The Markov dynamics of states evolving in noisy environments are modeled by the maps between the spaces of operators known as quantum channels. Indeed, a quantum channel is a map Φ that acts on the density operators; Φ : ρ −→ Φ (ρ), and incorporates the evolution of a pure quantum state to a mixed quantum state [64,65]. Mathematically, the description of the quantum channel can be completely characterized by the Kraus representation as ...
In statistical estimation theory, it has been shown previously that the Wigner-Yanase skew information is bounded by the quantum Fisher information associated with the phase parameter. Besides, the quantum Cram\'er-Rao inequality is expressed in terms of skew information. Since these two fundamental quantities are based on the concept of quantum uncertainty, we derive here their analytical formulas for arbitrary two-qubit X-states using the same analytical procedures. A comparison of these two informational quantifiers for two quasi-Werner states composed of two bipartite superposed coherent states is examined. Moreover, we investigated the decoherence effects on such quantities generated by the phase damping, depolarization and amplitude damping channels. We showed that decoherence strongly influences the initial quantum criteria and these quantities exhibit similar dynamic behaviors. This current work is characterized by the fact that these two concepts play the same role and capture similar properties in quantum estimation protocols.
... The Lindblad master equation 31,32,[49][50][51][52][53] is the most general Markovian description of a system density matrixρ S (t) interacting with the environment. It forms a dynamical semigroup of reduced density matrices that are trace-preserving (norm-preserving) and completely positive for population. ...
We derive the -MFE method to incorporate Lindblad jump operator dynamics into the mean-field Ehrenfest (MFE) approach. We map the density matrix evolution of Lindblad dynamics onto pure state coefficients using trajectory averages. We use simple assumptions to construct the -MFE method that satisfies this exact mapping. This establishes a method that uses independent trajectories which exactly reproduces Lindblad decay dynamics using a wavefunction description, with deterministic changes of the magnitudes of the quantum expansion coefficients, while only adding on a stochastic phase. We further demonstrate that when including nuclei in the Ehrenfest dynamics, the -MFE method gives semi-quantitatively accurate results, with the accuracy limited by the accuracy of the approximations present in the semiclassical MFE approach. This work provides a general framework to incorporate Lindblad dynamics into semiclassical or mixed quantum-classical simulations.
... A very similar procedure has been performed at the level of the time-evolution operator in [107] for the spin-boson model. This can be carried over almost immediately by recalling the adjoint action of these operators on the density operator and the fact that the S α (t) can be ordered arbitrarily inside the time-ordering T (S) ← . ...
We consider the coupling of a scalar field to linearised gravity and derive a relativistic gravitationally induced decoherence model using Ashtekar variables. The model is formulated at the gauge invariant level using suitable geometrical clocks in the relational formalism, broadening existing gauge invariant formulations of decoherence models. For the construction of the Dirac observables we extend the known observable map by a kind of dual map where the role of clocks and constraints is interchanged. We also discuss a second choice of geometrical clocks existing in the ADM literature. Then we apply a reduced phase space quantisation on Fock space and derive the final master equation choosing a Gibbs state for the gravitational environment and using the projection operator technique. The resulting master equation is not automatically of Lindblad type, a starting point sometimes assumed for phenomenological models, but still involves a residual time dependence at the level of the effective operators in the master equation due to the form of the correlation functions that we express in terms of thermal Wightman functions. Furthermore, we discuss why in the model analysed here the application of a second Markov approximation in order to obtain a set of time independent effective system operators is less straightforward than in some of the quantum mechanical models.
... We are interested in how fast this object decoheres and begins to behave classically. After the pioneering work of Joos and Zeh (in [2]), many authors have studied this problem, e.g., [18][19][20][21][22]. Take the calculation of Hornberger as example. ...
While the issues of dissipation, fluctuations, noise and decoherence in open quantum systems (with autocratic divide) analyzed via Langevin dynamics are familiar subjects, the treatment of corresponding issues in closed quantum systems is more subtle, as witnessed by Boltzmann’s explanation of dissipation in a macroscopic system made up of many equal constituents (a democratic system). How to extract useful physical information about a closed democratic system with no obvious ways to distinguish one constituent from another, nor the existence of conservation laws governing certain special kinds of variables, e.g., the hydrodynamic variables—this is the question we raise in this essay. Taking the inspirations from Boltzmann and Langevin, we study (a) how a hierarchical order introduced to a closed democratic system—defined either by substance or by representation, and (b) how hierarchical coarse-graining, executed in a specific order, can facilitate our understanding in how macro-behaviors arise from micro-dynamics. We give two examples in: (a) the derivation of correlation noises in the Bogolyubov-Born-Green-Kirkwood-Yvonne hierarchy and the use of a Boltzmann-Langevin equation to study the decoherence of the lower order correlations; and (b) the derivation of quantum fluctuation forces by ordered coarse-grainings of the relevant variables in the medium, the quantum field and the internal degrees of freedom of an atom.
... The manifestation of coherence phenomena is a hallmark of quantum mechanics, differentiating it from classical phenomenon due to the property of superposition in quantum reality due to the linearity of the Schroedinger equation [1][2][3][4]. ...
... The formal solution of the Lindblad equation in the form of Eqs. (2)-(4) can be written as [6,7,62] ...
We consider Markovian open quantum systems subject to stochastic resetting, which means that the dissipative time evolution is reset at randomly distributed times to the initial state. We show that the ensuing dynamics is non-Markovian and has the form of a generalized Lindblad equation. Interestingly, the statistics of quantum-jumps can be exactly derived. This is achieved by combining techniques from the thermodynamics of quantum-jump trajectories with the renewal structure of the resetting dynamics. We consider as an application of our analysis a driven two-level and an intermittent three-level system. Our findings show that stochastic resetting may be exploited as a tool to tailor the statistics of the quantum-jump trajectories and the dynamical phases of open quantum systems.
... F ollowing early pioneering studies [1][2][3][4] , the investigation of the quantum-to-classical transition via the mechanism of decoherence has become a very active area of research, both experimentally and theoretically, playing an increasingly central role in the research area on the foundations of quantum mechanics (QM) and the appearance of a classical world at the macroscopic scale, as one may gather, e.g., from the many excellent existing reviews on the subject (i.e., see for instance refs. [5][6][7][8][9][10][11][12][13] and references therein). In broad terms, decoherence appears to be due to the inevitable interaction and the ensuing creation of entanglement between a given quantum system and the environment in which it is embedded. ...
Schemes of gravitationally induced decoherence are being actively investigated as possible mechanisms for the quantum-to-classical transition. Here, we introduce a decoherence process due to quantum gravity effects. We assume a foamy quantum spacetime with a fluctuating minimal length coinciding on average with the Planck scale. Considering deformed canonical commutation relations with a fluctuating deformation parameter, we derive a Lindblad master equation that yields localization in energy space and decoherence times consistent with the currently available observational evidence. Compared to other schemes of gravitational decoherence, we find that the decoherence rate predicted by our model is extremal, being minimal in the deep quantum regime below the Planck scale and maximal in the mesoscopic regime beyond it. We discuss possible experimental tests of our model based on cavity optomechanics setups with ultracold massive molecular oscillators and we provide preliminary estimates on the values of the physical parameters needed for actual laboratory implementations.
... It has recently been shown that for evolutions which lead to pure dephasing of a qubit or even a larger system [24][25][26][27][28], there exists a straightforward signature, which entanglement leaves on the state of the environment [24,25] (after the qubit/system state is traced out). This is, on the one hand, the reason why it is important to know if qubit-environment entanglement (QEE) is generated, since only decoherence with generation of QEE is accompanied by the information about the qubit state leaking out into the environment [29], similarly as in the case of a pure environment [30,31]. On the other hand, it also serves as the basis for the possibility of direct measurement of QEE. ...
We propose a scheme for the detection of qubit–environment entanglement which requires only operations and measurements on the qubit, all within reach of current experimental state-of-the-art. The scheme works for any type of interaction which leads to pure dephasing of the qubit as long as the initial qubit state is pure. The scheme is direct in the sense that it allows the detection of entanglement present in the system at time τ after the initialization of the qubit in a superposition state. It requires a measurement on the qubit at time τ and a comparison of the post-measurement evolution to the evolution obtained by a modified scheme. It becomes particularly simple when one of the qubit states is neutral with respect to the environment, such as in case of the most common choice of the NV center spin qubit or for excitonic charge qubits, when the environment is initially at thermal equilibrium. In this case, the post-measurement evolution needs to be compared only to the standard decoherence which is obtained without any qubit manipulation after the preparation of the initial state.
... Decoherence refers to the gradual decay of a system's ability to show quantum interference, and is caused by its irreversible interaction with an environment [97]. Decoherence can explain the emergence of classicality in open quantum systems [98], and it poses a fundamental challenge to quantum superposition tests because perfect isolation is impossible. ...
Quantum theory is incredibly successful, explaining the microscopic world with great accuracy, from the behaviour of subatomic particles to chemical reactions to solid-state electronics. There is not a single experimental finding challenging its predictions, and ever more quantum phenomena are exploited in technology, including interferometric sensing and quantum cryptography. In order to explore novel applications and test the validity of quantum physics at the macroscale researchers strive to prepare ever heavier and bigger objects in quantum superpositions. Experiments with levitated microscale particles are about to push this quest into uncharted waters.
... We are interested in how fast this object decoheres and begins to behave classically. After the pioneering work of Joos and Zeh (in [2]), many authors have studied this problem, e.g., [18,19,20,21,22]. Take the calculation of Hornberger as example. ...
While the issues of dissipation, fluctuations, noise and decoherence in open quantum systems (with autocratic divide) analyzed via Langevin dynamics are familiar subjects, the treatment of corresponding issues in closed quantum systems is more subtle, as witnessed by Boltzmann's explanation of dissipation in a macroscopic system made up of many equal constituents (a democratic system). How to extract useful physical information about a closed democratic system with no obvious ways to distinguish one constituent from another, nor the existence of conservation laws governing certain special kinds of variables, e.g., the hydrodynamic variables -- this is the question we raise in this essay. Taking the inspirations from Boltzmann and Langevin, we study a) how a hierarchical order introduced to a closed democratic system -- defined either by substance or by representation, and b) how hierarchical coarse-graining, executed in a specific order, can facilitate our understanding in how macro-behaviors arise from micro-dynamics. We give two examples in: a) the derivation of correlation noises in the BBGKY hierarchy and how using a Boltzmann-Langevin equation one can study the decoherence of the lower order correlations; and b) the derivation of quantum fluctuation forces by ordered coarse-grainings of the relevant variables in the medium, the quantum field and the internal degrees of freedom of an atom.
... This is very closely related to the rate of scattering of particles off a single target Γ sc = n v σ and is similar to decoherence of a single point particle whose center of mass is in a superposition; see refs. [41][42][43][44][45][46][47][48][49][50][51]. However the difference is that we are considering the rate of decoherence of a superposition of two different states for the DM target. ...
Decoherence describes the tendency of quantum sub-systems to dynamically lose their quantum character. This happens when the quantum sub-system of interest interacts and becomes entangled with an environment that is traced out. For ordinary macroscopic systems, electromagnetic and other interactions cause rapid decoherence. However, dark matter (DM) may have the unique possibility of exhibiting naturally prolonged macroscopic quantum properties due to its weak coupling to its environment, particularly if it only interacts gravitationally. In this work, we compute the rate of decoherence for light DM in the galaxy, where a local density has its mass, size, and location in a quantum superposition. The decoherence is via the gravitational interaction of the DM overdensity with its environment, provided by ordinary matter. We focus on relatively robust configurations: DM perturbations that involve an overdensity followed by an underdensity, with no monopole, such that it is only observable at relatively close distances. We use non-relativistic scattering theory with a Newtonian potential generated by the overdensity to determine how a probe particle scatters off of it and thereby becomes entangled. As an application, we consider light scalar DM, including axions. In the galactic halo, we use diffuse hydrogen as the environment, while near the earth, we use air as the environment. For an overdensity whose size is the typical DM de Broglie wavelength, we find that the decoherence rate in the halo is higher than the present Hubble rate for DM masses ma ≲ 5 × 10⁻⁷ eV and in earth based experiments it is higher than the classical field coherence rate for ma ≲ 10⁻⁶ eV . When spreading of the states occurs, the rates can become much faster, as we quantify. Also, we establish that DM BECs decohere very rapidly and so are very well described by classical field theory.
... It is clear that decoherence does not require a measurement. Decoherence is now commonly described in terms of the movement of information, using terms such as "information deposited in the environment" and "environment as witness" [16], "information transfer from the system to the environment" [17], etc. Transfer of which-path information to the environment is certainly sufficient to cause decoherence, but is it a necessary condition? ...
Quantum computation is often limited by environmentally-induced decoherence. We examine the loss of coherence for a two-branch quantum interference device in the presence of multiple witnesses, representing an idealized environment. Interference oscillations are visible in the output as the magnetic flux through the branches is varied. Quantum double-dot witnesses are field-coupled and symmetrically attached to each branch. The global system—device and witnesses—undergoes unitary time evolution with no increase in entropy. Witness states entangle with the device state, but for these blind witnesses, which-path information is not able to be transferred to the quantum state of witnesses—they cannot “see” or make a record of which branch is traversed. The system which-path information leaves no imprint on the environment. Yet, the presence of a multiplicity of witnesses rapidly quenches quantum interference.
... Dissipation and decoherence are effects that are induced on a quantum system by its environment [1,2]. These effects may be seen as a detrimental factor that is to be reduced and/or undone, and this viewpoint is taken in the majority of quantum computing schemes, as well as in other tasks in quantum information processing and storage [3][4][5]. ...
We study the dynamics of a Lipkin-Meshkov-Glick model in the presence of Markovian dissipation, with a focus on late-time dynamics and the approach to thermal equilibrium. Making use of a vectorized bosonic representation of the corresponding Lindblad master equation, we use degenerate perturbation theory in the weak-dissipation limit to analytically obtain the eigenvalues and eigenvectors of the Liouvillian superoperator, which in turn give access to closed-form analytical expressions for the time evolution of the density operator and observables. Our approach is valid for large systems, but takes into account leading-order finite-size corrections to the infinite-system result. As an application, we show that the dissipative Lipkin-Meshkov-Glick model equilibrates by passing through a continuum of thermal states with damped oscillations superimposed, until finally reaching an equilibrium state with a temperature that in general differs from the bath temperature. We discuss limitations of our analytic techniques by comparing to exact numerical results.
... Due to the fact that solving directly the dynamic equation of the total atom-cavity-reservoir is so dif cult, we often need to make some approximations, such as rotating wave approximation, Born approximation, Markov approximation and let ρ SE (0) = ρ S (0) ⊗ ρ E (0) though some association information between the atom and its environment will be lost [41][42][43][44]. If the total system has only one initial excitation and the reservoir is at zero temperature, we then obtain the following master equation for the atom-cavity system in the dressed-state basis {|α 1,+ , |α 1,− , |α 0 } [45] ...
In this work, we obtain an analytical representation of the density operator of an atom in a dissipative cavity when the reservoir is at zero temperature and the total number of excitations is N = 1. We also investigated the quantum speed limit time (QSLT) of the atom and the non-Markovianity in the dynamics process. The results show that the QSLT and the non-Markovianity can be effectively manipulated by the atom–cavity coupling and the reservoir parameters. Both of the atom–cavity coupling and the detuning can induce a sudden transition from Markovian to non-Markovian dynamics, and this transition is the main physical reason of the quantum speed-up process. The critical value of the sudden transition from no speed-up to speed-up the atom–cavity coupling and the reservoir parameters. The corresponding physical explanation is also provided for our results.
... It has recently been shown that for evolutions which lead to pure dephasing of a qubit or even a larger system [24][25][26][27][28], there exists a straihtforward signature, which entanglement leaves on the state of the environment [24,25] (after the qubit/system state is traced out). This is, on one hand, the reason why it is important to know if qubit-environment entanglement (QEE) is generated, since only decoherence with generation of QEE is accompanied by the information about the qubit state leaking out into the environment [29], similarly as in the case of a pure environment [30,31]. On the other hand, it also serves as the basis for the possibility of direct measurement of QEE. ...
We propose a scheme for the detection of qubit-environment entanglement at time which requires only operations and measurements on the qubit, all within reach of current experimental state-of-the-art. The scheme works for any type of interaction which leads to pure dephasing of the qubit as long as the initial qubit state is pure. It becomes particularly simple when one of the qubit states is neutral with respect to the environment, such as in case of the most common choice of the NV center spin qubit or for excitonic charge qubits, when the environment is initially at thermal equilibrium.
This work aims to shed some light on the meaning of the positive energy assumption in relativistic quantum theory and its relation to questions of localization of quantum systems. It is shown that the positive energy property of solutions of relativistic wave equations (such as the Dirac equation) is very fragile with respect to state transformations beyond free time evolution. Paying attention to the connection between negative energy Dirac wave functions and pair creation processes in second quantization, this analysis leads to a better understanding of a class of problems known as the localization problem of relativistic quantum theory (associated for instance with famous results of Newton & Wigner, Reeh & Schlieder, Hegerfeldt or Malament). Finally, this analysis is reflected from the perspective of a Bohmian quantum field theory.
Several phenomenological approaches to quantum gravity predict the existence of a minimal measurable length and/or a maximum measurable momentum near the Planck scale. When embedded into the framework of quantum mechanics, such constraints induce a modification of the canonical commutation relations and thus a generalization of the Heisenberg uncertainty relations, commonly referred to as generalized uncertainty principle (GUP). Different models of quantum gravity imply different forms of the GUP. For instance, in the framework of string theory the GUP is quadratic in the momentum operator, while in the context of doubly special relativity it includes an additional linear dependence. Among the possible physical consequences, it was recently shown that the quadratic GUP induces a universal decoherence mechanism, provided one assumes a foamy structure of quantum spacetime close to the Planck length. Along this line, in the present work we investigate the gravitational decoherence associated to the linear-quadratic GUP and we compare it with the one associated to the quadratic GUP. We find that, despite their similarities, the two generalizations of the Heisenberg uncertainty principle yield decoherence times that are completely uncorrelated and significantly distinct. Motivated by this result, we introduce a theoretical and experimental scheme based on cavity optomechanics to measure the different time evolution of nonlocal quantum correlations corresponding to the two aforementioned decoherence mechanisms. We find that the deviation between the two predictions occurs on time scales that are macroscopic and thus potentially amenable to experimental verification. This scenario provides a possible setting to discriminate between different forms of the GUP and therefore different models of quantum gravity.
We exploit the properties of chain mapping transformations of bosonic environments to identify a finite collection of modes able to capture the characteristic features, or fingerprint, of the environment. Moreover we show that the countable infinity of residual bath modes can be replaced by a universal Markovian closure, namely, a small collection of damped modes undergoing a Lindblad-type dynamics whose parametrization is independent of the spectral density under consideration. We show that the Markovian closure provides a quadratic speedup with respect to standard chain mapping techniques and makes the memory requirement independent of the simulation time, while preserving all the information on the fingerprint modes. We illustrate the application of the Markovian closure to the computation of linear spectra but also to nonlinear spectral response, a relevant experimentally accessible many body coherence witness for which efficient numerically exact calculations in realistic environments are currently lacking.
We exploit the properties of chain mapping transformations of bosonic environments to identify a finite collection of modes able to capture the characteristic features, or fingerprint, of the environment. Moreover we show that the countable infinity of residual bath modes can be replaced by a universal Markovian closure, namely a small collection of damped modes undergoing a Lindblad-type dynamics whose parametrization is independent of the spectral density under consideration. We show that the Markovian closure provides a quadratic speed-up with respect to standard chain mapping techniques and makes the memory requirement independent of the simulation time, while preserving all the information on the fingerprint modes. We illustrate the application of the Markovian closure to the computation of linear spectra but also to non-linear spectral response, a relevant experimentally accessible many body coherence witness for which efficient numerically exact calculations in realistic environments are currently lacking.
Complex molecules are intriguing objects at the interface between quantum and classical phenomena. Compared to the electrons, neutrons, or atoms studied in earlier matter-wave experiments, they feature a much more complicated internal structure, but can still behave as quantum objects in their center-of-mass motion. Molecules may involve a large number of vibrational modes and highly excited rotational states, they can emit thermal photons, electrons, or even atoms, and they exhibit large cross sections for collisional interactions with residual background gases. This makes them ideal candidates for decoherence experiments which we review in this contribution.KeywordsMatter-wave decoherenceMolecule interferometryQuantum-to-classical transition
We propose a two-frequency driving scheme in dynamic atomic force microscopy that maximizes the interaction time between tip and sample. Using a stochastic description of the cantilever dynamics, we predict large classical squeezing and a small amount of skewness of the tip's phase-space probability distribution. Strong position squeezing will require close contact between tip and surface, while momentum squeezing would also be possible in the van der Waals region of the tip-surface force. Employing a generalized Caldeira-Leggett model, we predict that surface-dependent dissipative forces may be the dominant source of quantum effects and propose a procedure to isolate quantum effects from thermal fluctuations.
Quantum theory is incredibly successful, explaining the microscopic world with great accuracy, from the behaviour of subatomic particles to chemical reactions to solid-state electronics. There is not a single experimental finding challenging its predictions, and ever more quantum phenomena are exploited in technology, including interferometric sensing and quantum cryptography. In order to explore novel applications and test the validity of quantum physics at the macroscale, researchers strive to prepare ever heavier and bigger objects in quantum superpositions. Experiments with levitated microscale particles are about to push this quest into uncharted waters.
The study of quantum dynamics featuring memory effects has always been a topic of interest within the theory of open quantum system. The latter is concerned with providing useful conceptual and theoretical tools for the description of the reduced dynamics of a system interacting with an external environment. Definitions of non-Markovian processes have been introduced trying to capture the notion of memory effect by studying features of the quantum dynamical map providing the evolution of the system states, or changes in the distinguishability of the system states themselves. We introduce basic notions in the framework of open quantum systems. We stress in particular analogies and differences with models used for introducing modifications of quantum mechanics which should help in dealing with the measurement problem. We further discuss recent developments in the treatment of non-Markovian processes and their role in considering more general modifications of quantum mechanics.
The consideration of measuring instruments as macroscopic bodies leads to neglect of the microscopic processes that occur during measurements. This disregard is not justified in general cases. As an example of measurements using microscopic instruments, the scattering of a photon by an electron with electron interference at two slits (Compton effect) was used. The amount of information that can be obtained in such a process is inversely proportional to the wavelength of the incident photon. At large photon wavelengths (soft measurements), the pure state of the electron can be disrupted by an arbitrarily small extent; accordingly, the amount of information extracted in such an experiment is also arbitrarily small. It is shown that the energy price for a bit obtained in such a measurement tends toward a constant value for increasing the photon wavelength. Microscopic instruments can be used in situations where energy costs for measurements are important.
We present a wave-function approach to the study of the evolution of a small system when it is coupled to a large reservoir. Fluctuations and dissipation originate in this approach from quantum jumps that occur randomly during the time evolution of the system. This approach can be applied to a wide class of relaxation operators in the Markovian regime, and it is equivalent to the standard master-equation approach. For systems with a number of states N much larger than unity this Monte Carlo wave-function approach can be less expensive in terms of calculation time than the master-equation treatment. Indeed, a wave function involves only N components, whereas a density matrix is described by N² terms. We evaluate the gain in computing time that may be expected from such a formalism, and we discuss its applicability to several examples, with particular emphasis on a quantum description of laser cooling.
This book treats the central physical concepts and mathematical techniques used to investigate the dynamics of open quantum systems. To provide a self-contained presentation, the text begins with a survey of classical probability theory and with an introduction to the foundations of quantum mechanics, with particular emphasis on its statistical interpretation and on the formulation of generalized measurement theory through quantum operations and effects. The fundamentals of density matrix theory, quantum Markov processes, and completely positive dynamical semigroups are developed. The most important master equations used in quantum optics and condensed matter theory are derived and applied to the study of many examples. Special attention is paid to the Markovian and non-Markovian theory of environment induced decoherence, its role in the dynamical description of the measurement process, and to the experimental observation of decohering electromagnetic field states. The book includes the modern formulation of open quantum systems in terms of stochastic processes in Hilbert space. Stochastic wave function methods and Monte Carlo algorithms are designed and applied to important examples from quantum optics and atomic physics. The fundamentals of the treatment of non-Markovian quantum processes in open systems are developed on the basis of various mathematical techniques, such as projection superoperator methods and influence functional techniques. In addition, the book expounds the relativistic theory of quantum measurements and the density matrix theory of relativistic quantum electrodynamics.
First, we summarize the argument against deterministic nonlinear Schrodinger equations. We recall that any such equation activates quantum nonlocality in the sense that that information could be signalled in a finite time over arbitrarily large distances. Next we introduce a deterministic nonlinear Schrodinger equation. We justify it by showing that it is closest, in a precise sense, to the master equations for mixed states used to describe the evolution of open quantum systems. We also illustrate some interesting properties of this equation. Finally, we show that this equation can avoid the signalling problem if one adds noise to it in a precise way. Cases of both discrete and continuous noise are introduced explicitly and related to the density operator evolution. The relevance for the classical limit of the obtained stochastic equations is illustrated on a classically chaotic kicked anharmonic oscillator.
This is the first part of what will be a two-part review of distribution functions in physics. Here we deal with fundamentals and the second part will deal with applications. We discuss in detail the properties of the distribution function defined earlier by one of us (EPW) and we derive some new results. Next, we treat various other distribution functions. Among the latter we emphasize the so-called P distribution, as well as the generalized P distribution, because of their importance in quantum optics.
After they have interacted, quantum particles generally behave as a single nonseparable entangled system. The concept of entanglement plays an essential role in quantum physics. We have performed entanglement experiments with Rydberg atoms and microwave photons in a cavity and tested quantum mechanics in situations of increasing complexity. Entanglement resulted either from a resonant exchange of energy between atoms and the cavity field or from dispersive energy shifts affecting atoms and photons when they were not resonant. With two entangled particles (two atoms or one atom and a photon), we have realized new versions of the Einstein-Podolsky-Rosen situation. The detection of one particle projected the other, at a distance, in a correlated state. This process could be viewed as an elementary measurement, one particle being a âmeterâ measuring the other. We have performed a âquantum nondemolitionâ measurement of a single photon, which we detected repeatedly without destroying it. Entanglement is also essential to understand decoherence, the process accounting for the classical appearance of the macroscopic world. A mesoscopic superposition of states (âSchrödinger catâ) gets rapidly entangled with its environment, losing its quantum coherence. We have prepared a Schrödinger cat made of a few photons and studied the dynamics of its decoherence, in an experiment which constitutes a glimpse at the quantum/classical boundary. We have also investigated entanglement as a resource for the processing of quantum information. By using quantum two-state systems (qubits) instead of classical bits of information, one can perform logical operations exploiting quantum interferences and taking advantage of the properties of entanglement. Manipulating as qubits atoms and photons in a cavity, we have operated a quantum gate and applied it to the generation of a complex three-particle entangled state. We finally discuss the perspectives opened by these experiments for further fundamental studies.
We present a panoramic view on various attempts to "solve" the problems of quantum measurement and macro-objectivation, i.e., of the transition from a probabilistic quantum mechanic microscopic world to a deterministic classical macroscopic world. After a general introduction we describe in some detail both schemes that require some change or extension of the formalism (as hidden variable models and spontaneous collapse models) and those that do not introduce a real collapse of the wave function (as many worlds models, decoherence and quantum histories approaches, informational and modal interpretation, relational quantum mechanics, etc.). A large bibliography is provided for the readers who want to examine closely these studies.
We study the loss of spatial coherence in the extended wave function of fullerenes due to collisions with background gases. From the gradual suppression of quantum interference with increasing gas pressure we are able to support quantitatively both the predictions of decoherence theory and our picture of the interaction process. We thus explore the practical limits of matter wave interferometry at finite gas pressures and estimate the required experimental vacuum conditions for interferometry with even larger objects.
Emergent quantum technologies have led to increasing interest in decoherence--the processes that limit the appearance of quantum effects and turn them into classical phenomena. One important cause of decoherence is the interaction of a quantum system with its environment, which 'entangles' the two and distributes the quantum coherence over so many degrees of freedom as to render it unobservable. Decoherence theory has been complemented by experiments using matter waves coupled to external photons or molecules, and by investigations using coherent photon states, trapped ions and electron interferometers. Large molecules are particularly suitable for the investigation of the quantum-classical transition because they can store much energy in numerous internal degrees of freedom; the internal energy can be converted into thermal radiation and thus induce decoherence. Here we report matter wave interferometer experiments in which C70 molecules lose their quantum behaviour by thermal emission of radiation. We find good quantitative agreement between our experimental observations and microscopic decoherence theory. Decoherence by emission of thermal radiation is a general mechanism that should be relevant to all macroscopic bodies.
A fully quantum treatment of Einstein's Brownian motion is given, stressing in particular the role played by the two original requirements of translational invariance and connection between dynamics of the Brownian particle and atomic nature of the medium. The former leads to a clearcut relationship with a generator of translation-covariant quantum-dynamical semi-groups recently characterized by Holevo, the latter to a formulation of the fluctuation-dissipation theorem in terms of the dynamic structure factor, a two-point correlation function introduced in seminal work by Van Hove, directly related to density fluctuations in the medium and therefore to its atomistic, discrete nature. A microphysical expression for the generally temperature dependent friction coefficient is given in terms of the dynamic structure factor and of the interaction potential describing the single collisions. A comparison with the Caldeira-Leggett model is drawn, especially in view of the requirement of translational invariance, further characterizing general structures of reduced dynamics arising in the presence of symmetry under translations.
Although coupling to a super-Ohmic bosonic reservoir leads only to partial dephasing on short time scales, exponential decay of coherence appears in the Markovian limit (for long times) if anharmonicity of the reservoir is taken into account. This effect not only qualitatively changes the decoherence scenario but also leads to localization processes in which superpositions of spatially separated states dephase with a rate that depends on the distance between the localized states. As an example of the latter process, we study the decay of coherence of an electron state delocalized over two semiconductor quantum dots due to anharmonicity of phonon modes.
We analyse dissipation in quantum computation and its destructive impact on
efficiency of quantum algorithms. Using a general model of decoherence, we
study the time evolution of a quantum register of arbitrary length coupled with
an environment of arbitrary coherence length. We discuss relations between
decoherence and computational complexity and show that the quantum
factorization algorithm must be modified in order to be regarded as efficient
and realistic.
Dissipation, the irreversible loss of energy and coherence, from a microsystem is the result of coupling to a much larger macrosystem (or reservoir) that is so large that one has no chance of keeping track of all of its degrees of freedom. The microsystem evolution is then described by tracing over the reservoir states, which results in an irreversible decay as excitation leaks out of the initially excited microsystems into the outer reservoir environment. Earlier treatments of this dissipation used density matrices to describe an ensemble of microsystems, either in the Schr\"odinger picture with master equations, or in the Heisenberg picture with Langevin equations. The development of experimental techniques to study single quantum systems (for example, single trapped ions, or cavity-radiation-field modes) has stimulated the construction of theoretical methods to describe individual realizations conditioned on a particular observation record of the decay channel. These methods, variously described as quantum-jump, Monte Carlo wave function, and quantum-trajectory methods, are the subject of this review article. We discuss their derivation, apply them to a number of current problems in quantum optics, and relate them to ensemble descriptions.
A method is described for calculating the heat generated in a quantum computer due to loss of quantum phase information. Amazingly enough, this heat generation can take place at zero temperature. and may explain why it is impossible to extract energy from vacuum fluctuations. Implications for optical computers and quantum cosmology are also briefly discussed.
There are fundamental limits to the accuracy with which one can determine the state of a quantum system. I give an overview of the main approaches to quantum state discrimination. Several strategies exist. In quantum hypothesis testing, a quantum system is prepared in a member of a known, finite set of states, and the aim is to guess which one with the minimum probability of error. Error free discrimination is also sometimes possible, if we allow for the possibility of obtaining inconclusive results. If no prior information about the state is provided, then it is impractical to try to determine it exactly, and it must be estimated instead. In addition to reviewing these various strategies, I describe connections between state discrimination, the manipulation of quantum entanglement, and quantum cloning. Recent experimental work is also discussed. Comment: Contemporary Physics 2000, in press
We consider spatially separated qubits coupled to a thermal bosonic field that
causes pure dephasing. Our focus is on the entanglement of two Bell states which
for vanishing separation are known as robust and fragile entangled states. The
reduced two-qubit dynamics is solved exactly and explicitly. Our results allow
us to gain information about the robustness of two-qubit decoherence-free
subspaces with respect to physical parameters such as temperature, qubit-bath
coupling strength and spatial separation of the qubits. Moreover, we clarify the
relation between single-qubit coherence and two-qubit entanglement and identify
parameter regimes in which the terms robust and fragile are no longer
appropriate.
This book describes the phenomena that arise from the interaction between quantum systems and their environment. Since the first edition appeared in 1996, the concepts of decoherence have become firmly established experimentally and are now widely used in the literature. Its major consequences are the emergence of "classicality", superselection rules, the border line between microscopic and macroscopic behavior, the emergence of classical spacetime, and the appearance of quantum jumps.
Most of the new developments in this rapidly evolving field are discussed in this second edition: chaos theory, quantum information, neuroscience, primordial fluctuations in cosmology, black holes and string theory, experimental tests, and interpretational issues. While the major part of the book is concerned with environmental decoherence derived from a universal Schrödinger equation, later chapters address complementary or competing approaches, such as consistent histories, open system dynamics, algebraic methods, and collapse models.
In quantum mechanics, two states |ψ
1〉 and |ψ
2〉 of a system may be superposed to give a new state |ψ〉 = (|ψ
1〉 + |ψ
2〉)/√2. Sometimes, due to environmental influences, such a superposition is not dynamically robust and decays into a mixture ρ = 1/2 (|ψ
1〉〈ψ
1|+|ψ
2〉〈ψ
2|). These lectures are concerned with the nature of robust states and the time scales involved during the transition from the coherent superposition to the mixture, i.e the time scale of decoherence.
This chapter presents the main ideas and methods of quantum optics. It shows how mesoscopic states could be prepared and studied using the conceptually simple methods of quantum optics. It analyzes simple models of decoherence and describes the experiments which illustrate the fundamental connexion between environment induced decoherence, entanglement and complementarity. It describe beam splitters and particle interference effects followed by Schrödinger cats in Cavity Quantum Electrodynamics experiments. The chapter also discusses mesoscopic state superpositions of atoms in Bose Einstein condensates. Schrödinger cat states in quantum or atomic optics are characterized by their extreme fragility and sensitivity to decoherence, which occurs at a rate essentially proportional to the number of particles in the system. This puts severe limits to the size of these cat states. They have to be built within a finite time, to let the processes responsible for the preparation of the superposition to take place. This preparation time must be shorter than the decoherence time of the final cat state and this sets, in practice, an absolute limit to the number of particles in the system. In Cavity Quantum Electrodynamics (CQED), the maximum number of photons involved in cat states could not exceed a few hundred, even if the technology of cavities were considerably improved.
Major revision of the previous article published on the same website: 'The Role of Decoherence in Quantum Mechanics', in E. N. Zalta (ed.), The Stanford Encyclopedia of Philosophy (Winter 2003 Edition), http://plato.stanford.edu/archives/win2003/entries/qm-decoherence/ [under the editorial responsibility of J. Norton] (25pp.).
The course presents some basic concepts and rigorous results of a new scientiflc discipline { quantum information theory. It starts with a thorough reconsideration of the mathematical foundations of quantum theory from the modern statistical/information theoretic viewpoint, requiring only a minor prior knowledge of standard quantum mechanics. The concepts of quantum state, observable and (irreversible) dynamics of a quantum system are carefully discussed and appropriately generalized. A central notion of quantum communication channel is explained with a number of examples. The mathematical framework of the lectures is operator (matrix) theory in flnite dimensional Hilbert space, so that knowledge of a basic linear algebra, along with basic probability, would be a su‐cient technical prerequisite.
Dynamical processes in macroscopic systems are often approximately described by kinetic and hydrodynamic equations. One of the central problems in nonequilibrium statistical mechanics is to understand the approximate validity of these equations starting from a microscopic model. We discuss a variety of classical as well as quantum-mechanical models for which kinetic equations can be derived rigorously. The probabilistic nature of the problem is emphasized: The approximation of the microscopic dynamics by either a kinetic or a hydrodynamic equation can be understood as the approximation of a non-Markovian stochastic process by a Markovian process.
The form of the interaction Hamiltonian between the apparatus and its environment is sufficient to determine which observable of the measured quantum system can be considered "recorded" by the apparatus. The basis that contains this record—the pointer basis of the apparatus—consists of the eigenvectors of the operator which commutes with the apparatus-environment interaction Hamiltonian. Thus the environment can be said to perform a nondemolition measurement of an observable diagonal in the pointer basis.
This book describes the phenomena that arise from the interaction
between quantum systems with their environment. The emerging
irreversible dynamics of local systems explains the classical behaviour
of macroscopic objects. The emergence of superselection rules, observed
particle aspects of quantum fields, the occurrence of quantum jumps, and
the emergence of classical spacetime from quantum gravity are also
discussed. This approach, which is based on the assumed universality of
quantum mechanics, is compared and contrasted with others, such as
consistent histories, open-system dynamics, and explicit collapse
mechanisms.
It is shown how S-matrix theory and the concept of continuous quantum measurements can be combined to yield Markovian master equations which describe the environmental interaction non-perturbatively. The method is then applied to obtain the master equation for the effects of a gas on the internal dynamics of an immobile complex quantum system, such as a trapped molecule, in terms of the exact multi-channel scattering amplitudes.
It is shown that a systematic Markovian approximation yields a given new term to the known master equation, ensuring conservation of positivity for arbitrary initial conditions and for all times.
Contents 1 Introduction and overview 2 Quantum measurements 2.1 Bit-by-bit measurement and quantum entanglement 2.2 Interactions and the information transfer in quantum measurements 2.3 Monitoring by the environment and decoherence 2.4 One-bit environment for a bit-by-bit measurement 2.5 Decoherence of a single (qu)bit 2.6 Decoherence, einselection, and controlled shifts 3 Dynamics of quantum open systems: Master equations 3.1 Master equation: Perturbative evaluation 3.2 Example 1: Perturbative master equation in quantum Brownian motion 3.3 Example 2: Perturbative master equation for a two-level system coupled to a bosonic heat bath 3.4 Example 3: Perturbative master equation for a particle interacting with a quantum field 3.5 Exact master equation for quantum Brownian motion 4 Einselection in quantum Brownian motion 4.1 Decoherence of a superposition of two coherent states 4.2 Predictability sieve and preferred states for QBM 4.3 Energy eigenstates can also be selected by the environment! 5 Deconstructing decoherence: Landscape beyond the standard models 5.1 Saturation of the decoherence rate at large distances 5.2 Decoherence at zero temperature 5.3 Preexisting correlations between the system and the environment 6 Decoherence and chaos 6.1 Quantum predictability horizon: How the correspondence is lost 6.2 Exponential instability vs. decoherence 6.3 The arrow of time: A price of classicality? 6.4 Decoherence, einselection, and the entropy production. 7 How to fight against decoherence: Quantum error correcting codes 7.1 How to protect a classical bit 7.2 How to protect a quantum bit 7.3 Stabilizer quantum error-correcting codes 8 Discussion
The dependence of macroscopic systems upon their environment is studied under the assumption that quantum theory is universally valid. In particular scattering of photons and molecules turns out to be essential even in intergalactic space in restricting the observable properties by locally destroying the corresponding phase relations. The remaining coherence determines the ‘classical’ properties of the macroscopic systems. In this way local classical properties have their origin in the nonlocal character of quantum states.
The effect of the interaction depends essentially on whether it permanently ‘measures’ discrete or continuous quantities. For discrete variables (here exemplified by two-state systems) the classical properties are given by the measurement basis. The continuous case, studied for translational degrees of freedom, leads to a competition between destruction of coherence by the interaction and dispersion of the wave packet by the internal dynamics. A non-phenomenological Boltzmann-type master equation is derived for the density matrix of the center of mass. Its solutions show that the much-discussed dispersion hardly ever shows up even for small dust particles or large molecules. Instead the coherence length decreases towards the thermal de Broglie wave length of the object, whereas the incoherent spread increases. The Ehrenfest theorems are shown nevertheless to remain valid for recoil-free interactions. Some consequences of these investigations for the quantum theory of measurement are pointed out.
It is proved that the reduced dynamics of anN-level system coupled to a free quantum gas converges to a quantum dynamical semigroup in the low density limit. The proof uses a perturbation series of the quantum BBGKY-hierarchy, and the analysis of this series is based on scattering theory. The limiting semigroup contains the full scattering cross section, but it does not depend on the statistics of the reservoir. The dynamics of the semigroup is discussed.
The goal of non-equilibrium statistical mechanics is to explain the macroscopic behavior of matter from the dynamics of its microscopic constituents, i.e. atoms or molecules. Because of the large number of particles involved, this is a rather ambitious program. Therefore, as an intermediate step, one tries to write down an approximately valid dynamics as governed by a kinetic equation. Examples are the Boltzmann equation for a dilute gas, the hydrodynamic equations for an “aged” fluid, etc., There is a lot known about the interrelationship between the microscopic and kinetic description of a physical system and it would be rather hopeless to press all these results into one lecture. Therefore I would like to do three things.
The report presents an exhaustive review of the recent attempt to overcome the difficulties that standard quantum mechanics meets in accounting for the measurement (or macro-objectification) problem, an attempt based on the consideration of nonlinear and stochastic modifications of the Schrödinger equation. The proposed new dynamics is characterized by the feature of not contradicting any known fact about microsystems and of accounting, on the basis of a unique, universal dynamical principle, for wavepacket reduction and for the classical behavior of macroscopic systems. We recall the motivations for the new approach and we briefly review the other proposals to circumvent the above mentioned difficulties which appeared in the literature. In this way we make clear the conceptual and historical context characterizing the new approach. After having reviewed the mathematical techniques (stochastic differential calculus) which are essential for the rigorous and precise formulation of the new dynamics, we discuss in great detail its implications and we stress its relevant conceptual achievements. The new proposal requires also to work out an appropriate interpretation; a procedure which leads us to a reconsideration of many important issues about the conceptual status of theories based on a genuinely Hilbert space description of natural processes. Attention is also paid to many problems which are naturally raised by the dynamical reduction program. In particular we discuss the possibility and the problems one meets in trying to develop an analogous formalism for the relativistic case. Finally we discuss the experimental implications of the new dynamics for various physical processes which should allow, in principle, to test it against quantum mechanics. The review covers the work which has been done in the last 15 years by various scientists and the lively debate which has accompanied the elaboration of the new proposal.
We apply the influence-functional method of Feynman and Vernon to the study of Brownian motion at arbitrary temperature. By choosing a specific model for the dissipative interaction of the system of interest with its environment, we are able to evaluate the influence functional in closed form and express it in terms of a few parameters such as the phenomenological viscosity coefficient. We show that in the limit h→0 the results obtained from the influence functional formalism reduce to the classical Fokker-Planck equation. In the case of a simple harmonic oscillator with arbitrarily strong damping and at arbitrary temperature, we obtain an explicit expression for the time evolution of the complete density matrix ϱ(x, x′, t) when the system starts in a particular kind of pure state. We compare our results with those of other approaches to the problem of dissipation in quantum mechanics.
Bibliogr. na konci kapitol
The purpose of measurements is the determination of properties of the physical system under investigation. In this sense the general conception of measurement is that of an unambiguous comparison: the object system S, prepared in a state T, is brought into a suitable contact — a measurement coupling — with another, independently prepared system, the measuring apparatus from which the result related to the measured observable E is determined by reading the value of the pointer observable. It is the goal of the quantum theory of measurement to investigate whether measuring processes, being physical processes, are the subject of quantum mechanics. This question, ultimately, is the question of the universality of quantum mechanics (see Chapter I).
The effect of the environment on a quantum system is studied on an exactly solvable model: a harmonic oscillator interacting with a one-dimensional massless scalar field. We show that in an open quantum system, dissipation can cause decorrelation on a time scale significantly shorter than the relaxation time which characterizes the approach of the system to thermodynamic equilibrium. In particular, we demonstrate that the density matrix decays rapidly toward a mixture of approximate eigenstates'' of the pointer observable,'' which commutes with the system-environment interaction Hamiltonian. This observable can be regarded as continuously, if inaccurately, monitored by the scalar field environment. Both because in a harmonic oscillator the state of the system rotates in the phase space and because the effective environment measurement'' is weak, the system, on the short collision'' time scale (1/{Gamma}), maintains a coherence in this pointer observable on time scales of order ({gamma}/{Omega}ln({Gamma}/{Omega})){sup 1/2} and on longer time scales settles into a mixture of coherent states with a dispersion approximately consistent with the vacuum state. The master equation satisfied by the exact solution differs from the other master equations derived both for the high-temperature limit and for {ital T}=0. We discuss these differences and study the transition region between the high- and low-temperature regimes. We also consider the behavior of the system in the short-time transient'' regime. For {ital T}=0, we find that, in the long-time limit, the system behaves as if it were subject to 1/{ital f} noise.'' The generality of our model is considered and its predictions are compared with previous treatments of related problems. Some of the possible applications of the results to experimentally realizable situations are outlined.
The rate at which pure initial states deteriorate into mixtures is computed for a harmonic oscillator interacting with an environment in thermal equilibrium. The decoherence process resulting from this interaction selects a set of states characterized by maximal stability (or minimal loss of predictive power) which can be quantified by the rate of increase in either linear or statistical entropy. In the weak coupling limit, coherent states are shown to produce the least entropy, thus becoming the natural counterparts of classical points in phase space.
The equation for the quantum motion of a Brownian particle in a gaseous environment is derived by means of S-matrix theory. This quantum version of the linear Boltzmann equation accounts nonperturbatively for the quantum effects of the scattering dynamics and describes decoherence and dissipation in a unified framework. As a completely positive master equation it incorporates both the known equation for an infinitely massive Brownian particle and the classical linear Boltzmann equation as limiting cases.
We show how the effective equation of motion for a distinguished quantum particle in an ideal gas environment can be obtained by means of the monitoring approach introduced in [EPL 77, 50007 (2007)]. The resulting Lindblad master equation accounts for the quantum effects of the scattering dynamics in a non-perturbative fashion and it describes decoherence and dissipation in a unified framework. It incorporates various established equations as limiting cases and reduces to the classical linear Boltzmann equation once the state is diagonal in momentum. Comment: published version, 20 pages, 2 figures; now in fancy two-column format
Environment-induced decoherence and superselection have been a subject of intensive research over the past two decades, yet their implications for the foundational problems of quantum mechanics, most notably the quantum measurement problem, have remained a matter of great controversy. This paper is intended to clarify key features of the decoherence program, including its more recent results, and to investigate their application and consequences in the context of the main interpretive approaches of quantum mechanics.
We re-derive the quantum master equation for the decoherence of a massive Brownian particle due to collisions with the lighter particles from a thermal environment. Our careful treatment avoids the occurrence of squares of Dirac delta functions. It leads to a decoherence rate which is smaller by a factor of 2 pi compared to previous findings. This result, which is in agreement with recent experiments, is confirmed by both a physical analysis of the problem and by a perturbative calculation in the weak coupling limit. Comment: 33 pages, 4 figures
Decoherence is caused by the interaction with the environment. Environment monitors certain observables of the system, destroying interference between the pointer states corresponding to their eigenvalues. This leads to environment-induced superselection or einselection, a quantum process associated with selective loss of information. Einselected pointer states are stable. They can retain correlations with the rest of the Universe in spite of the environment. Einselection enforces classicality by imposing an effective ban on the vast majority of the Hilbert space, eliminating especially the flagrantly non-local "Schr\"odinger cat" states. Classical structure of phase space emerges from the quantum Hilbert space in the appropriate macroscopic limit: Combination of einselection with dynamics leads to the idealizations of a point and of a classical trajectory. In measurements, einselection replaces quantum entanglement between the apparatus and the measured system with the classical correlation. Comment: Final version of the review, with brutally compressed figures. Apart from the changes introduced in the editorial process the text is identical with that in the Rev. Mod. Phys. July issue. Also available from http://www.vjquantuminfo.org
We study dynamics of quantum open systems, paying special attention to those aspects of their evolution which are relevant to the transition from quantum to classical. We begin with a discussion of the conditional dynamics of simple systems. The resulting models are straightforward but suffice to illustrate basic physical ideas behind quantum measurements and decoherence. To discuss decoherence and environment-induced superselection einselection in a more general setting, we sketch perturbative as well as exact derivations of several master equations valid for various systems. Using these equations we study einselection employing the general strategy of the predictability sieve. Assumptions that are usually made in the discussion of decoherence are critically reexamined along with the ``standard lore'' to which they lead. Restoration of quantum-classical correspondence in systems that are classically chaotic is discussed. The dynamical second law -it is shown- can be traced to the same phenomena that allow for the restoration of the correspondence principle in decohering chaotic systems (where it is otherwise lost on a very short time-scale). Quantum error correction is discussed as an example of an anti-decoherence strategy. Implications of decoherence and einselection for the interpretation of quantum theory are briefly pointed out.
Classical properties of an open quantum system emerge through its interaction with other degrees of freedom (decoherence). We treat the case where this interaction produces a Markovian master equation for the system. We derive the corresponding distinguished local basis (pointer basis) by three methods. The first demands that the pointer states mimic as close as possible the local non-unitary evolution. The second demands that the local entropy production be minimal. The third imposes robustness on the inherent quantum and emerging classical uncertainties. All three methods lead to localized Gaussian pointer states, their formation and diffusion being governed by well-defined quantum Langevin equations.