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Quantum master equation

The quantum master equation is a fundamental mathematical framework in quantum mechanics that describes the time evolution of the density operator for open quantum systems, accounting for both coherent dynamics driven by the system's Hamiltonian and irreversible dissipative processes arising from interactions with an external environment under the Markovian approximation.^[1] This equation, most commonly expressed in the Lindblad form, ensures that the evolution preserves the complete positivity and trace of the density operator, thereby maintaining the physical interpretability of probabilities and quantum coherence.^[1] It takes the general form \dot{\rho}(t) = -i[H, \rho(t)] + \sum_k \Gamma_k (L_k \rho(t) L_k^\dagger - \frac{1}{2} \{L_k^\dagger L_k, \rho(t)\}), where H is the system Hamiltonian, \Gamma_k are dissipation rates, and L_k are Lindblad operators representing specific environmental coupling channels.^[1] The quantum master equation emerged in the mid-1970s as a rigorous characterization of quantum dynamical semigroups, independently derived by Gorini, Kossakowski, Sudarshan, and Lindblad to provide the most general form of Markovian generators for completely positive trace-preserving maps on the space of density operators.^[2]^[3] Prior approaches, such as the Redfield equation, offered perturbative descriptions but could violate positivity for strong dissipation, whereas the Lindblad structure guarantees mathematical consistency with quantum axioms even in the presence of noise.^[1] This formalism has since become indispensable for modeling realistic quantum systems, where isolation from the environment is impossible, and it underpins derivations from microscopic system-bath interactions via techniques like the Born-Markov or secular approximations.^[4] Key properties of the Lindblad master equation include its invariance under unitary transformations of the jump operators and its ability to capture decoherence and relaxation in controlled settings, with the purity \operatorname{Tr}[\rho^2] monotonically decreasing under Hermitian Lindbladians, reflecting the second law of thermodynamics in quantum terms.^[1] Extensions beyond strict Markovianity, such as time-dependent or non-local forms, address correlated environments, but the standard version remains the cornerstone for analytical and numerical simulations.^[4] In applications, the quantum master equation is pivotal in quantum optics for simulating cavity QED and laser dynamics, where it models photon loss and atomic spontaneous emission.^[5] In quantum information science, it facilitates the study of error correction, entanglement generation, and noise mitigation in qubits, enabling predictions of fidelity in gate operations and quantum memories.^[6] Further, it informs condensed matter physics by describing transport in mesoscopic systems and quantum biology through decoherence in photosynthetic complexes, highlighting its broad impact across disciplines.^[1] Numerical tools, including quantum trajectory methods, solve these equations efficiently for complex many-body scenarios.^[4]

Overview

Definition and motivation

A quantum master equation is a differential equation governing the time evolution of the density operator \rho(t) of a quantum system that interacts with an external environment, thereby incorporating both reversible coherent dynamics and irreversible processes such as dissipation and decoherence. This formalism provides a reduced description of the system's state by tracing out the environmental degrees of freedom, allowing for the modeling of non-unitary evolution that is essential in realistic settings where perfect isolation is unattainable.^[7] The primary motivation for developing quantum master equations arises from the inherent openness of quantum systems in nature, where interactions with surrounding baths—such as phonons in solids or electromagnetic fields—lead to energy exchange, loss of quantum coherence, and approach to thermal equilibrium. These environmental couplings introduce effects that cannot be captured by the unitary Schrödinger equation alone, necessitating a probabilistic framework to describe phenomena like relaxation and noise in fields ranging from quantum optics to quantum computing. Without such equations, predictions for experimental outcomes in open systems would remain incomplete, as they fail to account for the irreversible entropy production inherent to system-environment entanglement.^[7]^[8] Historically, the quantum master equation traces its roots to early efforts in the late 1920s to extend classical master equations to quantum statistics, with Wolfgang Pauli deriving the first quantum master equation in 1928, known as the Pauli master equation, which incorporated transition probabilities under weak coupling assumptions.^[9] Its development accelerated in the 1960s within quantum optics, where researchers like Melvin Lax introduced the quantum regression theorem to handle correlation functions in Markovian approximations, enabling the analysis of light-matter interactions in lasers and resonators. This period saw the integration of density operator methods, originally formalized by von Neumann in the 1930s, into a coherent theory for open systems, laying the groundwork for modern treatments of decoherence and dissipation.^[8] The basic structure of a quantum master equation separates coherent and incoherent contributions, typically written as

\frac{d\rho}{dt} = -i[H, \rho] + \mathcal{D}(\rho),

where H is the effective Hamiltonian driving unitary evolution through the commutator [H, \rho] = H\rho - \rho H, and \mathcal{D}(\rho) denotes the dissipator term that models the environment-induced incoherent effects, such as jump processes or diffusive broadening. This form ensures the equation preserves the trace and positivity of \rho, reflecting physical probabilities.^[7]

Relation to closed-system dynamics

In closed quantum systems, the dynamics of the density operator \rho are governed by the von Neumann equation,

\frac{d\rho}{dt} = -i [H, \rho],

where H is the Hamiltonian of the system and \hbar = 1. This equation describes unitary evolution, which is reversible and preserves the trace of \rho (ensuring total probability conservation) as well as the purity of the state, meaning pure states remain pure and the von Neumann entropy S(\rho) = -\operatorname{Tr}(\rho \ln \rho) stays constant.^[10]^[11] When the system couples to an external environment, the dynamics transition to an open quantum system description via the quantum master equation, which incorporates non-unitary terms arising from tracing over the environmental degrees of freedom. These additional terms lead to trace-preserving but non-unitary maps, allowing for irreversible processes while maintaining probability conservation.^[10]^[11] Key differences emerge in the physical consequences: unlike the reversible evolution of closed systems, master equations permit entropy increase, decoherence (loss of quantum superpositions), and relaxation to steady states, reflecting the irreversible influence of the environment. For instance, a pure state |\psi\rangle in a closed system evolves unitarily to another pure state, but under the master equation, environmental interactions can transform it into a mixed state, as seen in examples like the damping of Schrödinger cat states where coherence is lost.^[10]^[11]

Theoretical foundations

Density operator formalism

The density operator, also known as the density matrix, provides a general framework for describing the state of a quantum system when it is not fully known or when it represents a statistical mixture of pure states, extending beyond the limitations of wave function descriptions. Introduced by John von Neumann, it formalizes the probabilistic interpretation of quantum mechanics for ensembles. For an ensemble of pure states {|\psi_i\rangle} with probabilities {p_i} satisfying \sum_i p_i = 1 and p_i \geq 0, the density operator is defined as

\hat{\rho} = \sum_i p_i |\psi_i\rangle\langle\psi_i|. $$ In the context of open quantum systems interacting with an environment, the system's density operator is obtained by tracing over the environmental [degrees of freedom](/page/Degrees_of_freedom) from the total pure state density operator: \hat{\rho}_S = \mathrm{Tr}_E(|\Psi\rangle\langle\Psi|), where |\Psi\rangle describes the combined system-environment state.[](https://global.oup.com/academic/product/the-theory-of-open-quantum-systems-9780199213900) This construction captures the reduced description of the system, accounting for environmental influences without explicit knowledge of the environment's state.[](https://global.oup.com/academic/product/the-theory-of-open-quantum-systems-9780199213900) The density operator possesses key mathematical properties that ensure its physical interpretability: it is Hermitian (\hat{\rho}^\dagger = \hat{\rho}), positive semi-definite (eigenvalues \geq 0), and normalized such that \mathrm{Tr}(\hat{\rho}) = 1. These properties arise directly from the probabilistic weights and the outer product structure, guaranteeing that probabilities derived from it are non-negative and sum to unity. The trace of \hat{\rho}^2, known as the purity, equals 1 for pure states (where \hat{\rho} = |\psi\rangle\langle\psi|) and is strictly less than 1 for mixed states, quantifying the degree of quantum coherence or classical uncertainty in the description.[](https://arxiv.org/pdf/2303.08738) Expectation values of observables \hat{A} are computed as \langle \hat{A} \rangle = \mathrm{Tr}(\hat{\rho} \hat{A}), which generalizes the pure-state formula \langle\psi|\hat{A}|\psi\rangle and holds for any Hermitian operator \hat{A}. This trace form leverages the cyclic property of the trace operation, making it basis-independent and suitable for computations in any representation.[](https://arxiv.org/pdf/2303.08738) The time evolution of \hat{\rho} is governed by linear superoperators acting on the operator space, preserving these properties under unitary or dissipative dynamics, though the specific form depends on the system's context.[](https://global.oup.com/academic/product/the-theory-of-open-quantum-systems-9780199213900) For pure states, the density operator reduces to the projector |\psi\rangle\langle\psi|, recovering the standard [wave function](/page/Wave_function) formalism where all information is complete. However, its true utility emerges in scenarios involving mixed states or subsystems, such as open quantum systems, where the wave function alone cannot adequately represent the partial loss of [coherence](/page/Coherence) due to environmental [coupling](/page/Coupling).[](https://global.oup.com/academic/product/the-theory-of-open-quantum-systems-9780199213900) This formalism thus serves as the foundational tool for analyzing statistical ensembles and reduced dynamics in [quantum theory](/page/Quantum_theory).[](https://arxiv.org/pdf/2303.08738) ### Open quantum systems In the framework of open quantum systems, the total quantum system is conceptually partitioned into a subsystem of interest, referred to as the [system](/page/System) with [Hilbert space](/page/Hilbert_space) $\mathcal{H}_S$, and an external environment or [bath](/page/Bath) with [Hilbert space](/page/Hilbert_space) $\mathcal{H}_E$. The overall [Hilbert space](/page/Hilbert_space) is then the [tensor product](/page/Tensor_product) $\mathcal{H} = \mathcal{H}_S \otimes \mathcal{H}_E$, and the total [Hamiltonian](/page/Hamiltonian) governing the unitary evolution is $H = H_S + H_E + H_\mathrm{int}$, where $H_S$ acts on the [system](/page/System), $H_E$ on the environment, and $H_\mathrm{int}$ describes their interaction.[](https://arxiv.org/pdf/1104.5242.pdf)[](https://global.oup.com/academic/product/the-theory-of-open-quantum-systems-9780199213900) Solving the full [Schrödinger equation](/page/Schrödinger_equation) for the composite system is generally intractable, particularly because the environment often comprises a large number of [degrees of freedom](/page/Degrees_of_freedom), leading to rapid growth in [computational complexity](/page/Computational_complexity) and entanglement.[](https://arxiv.org/pdf/1104.5242.pdf) To describe the dynamics of the system alone, one seeks a reduced description that traces out the environmental [degrees of freedom](/page/Degrees_of_freedom), typically via the system's density operator $\rho_S = \mathrm{Tr}_E(\rho)$, where $\rho$ is the total density operator.[](https://global.oup.com/academic/product/the-theory-of-open-quantum-systems-9780199213900) This reduction relies on key assumptions to make the dynamics tractable: the system is weakly coupled to the environment, justifying the [Born approximation](/page/Born_approximation) where the system's state has negligible back-action on the environment, and the dynamics are Markovian, meaning environmental correlations decay much faster than the system's characteristic timescales.[](https://pubs.aip.org/aip/jcp/article/33/5/1338/206419/Ensemble-Method-in-the-Theory-of-Irreversibility)[](https://arxiv.org/pdf/1104.5242.pdf) These assumptions give rise to irreversible [evolution](/page/Evolution) in the reduced [system](/page/System) [description](/page/Description), manifesting as decoherence—the progressive loss of off-diagonal elements in $\rho_S$ that erodes quantum superpositions—and [dissipation](/page/Dissipation), involving unidirectional energy flow from the [system](/page/System) to the [environment](/page/Environment).[](https://global.oup.com/academic/product/the-theory-of-open-quantum-systems-9780199213900)[](https://arxiv.org/pdf/1104.5242.pdf) ## Derivation ### Microscopic approach The microscopic approach to deriving the quantum master equation begins with the [total](/page/Total) [Hamiltonian](/page/Hamiltonian) of the combined [system](/page/System) and [environment](/page/Environment), partitioned as $ H = H_S + H_E + H_I $, where $ H_S $ governs the [isolated system](/page/Isolated_system), $ H_E $ the [environment](/page/Environment), and $ H_I $ their [interaction](/page/Interaction). The [dynamics](/page/Dynamics) of the [total](/page/Total) state $ |\Psi(t)\rangle $ follow the [von Neumann](/page/Von_Neumann) equation $ i\hbar \frac{d}{dt} |\Psi(t)\rangle = H |\Psi(t)\rangle $, and the reduced [system](/page/System) density operator is obtained via [partial trace](/page/Partial_trace), $ \rho_S(t) = \mathrm{Tr}_E [ |\Psi(t)\rangle \langle \Psi(t)| ] $.[](https://doi.org/10.1093/acprof:oso/9780199213900.001.0001) To derive an equation for $ \rho_S(t) $, perturbative methods are employed under the weak-coupling limit, transforming to the [interaction picture](/page/Interaction_picture) where the unperturbed evolution separates system and environment. The [Born approximation](/page/Born_approximation) assumes weak interaction such that the total density operator factorizes approximately as $ \rho(t) \approx \rho_S(t) \otimes \rho_E $, with $ \rho_E $ the stationary environment state, valid for short correlation times relative to system evolution.[](https://doi.org/10.1093/acprof:oso/9780199213900.001.0001) Further, the Markov approximation neglects memory effects by assuming environment correlations decay rapidly, allowing integration over past times with a delta-function-like form for the bath autocorrelation functions, yielding a time-local equation for $ \rho_S $. This is justified when the environment relaxation time is much shorter than the system timescales.[](https://doi.org/10.1093/acprof:oso/9780199213900.001.0001) Finally, the secular approximation eliminates rapidly oscillating terms in the [interaction picture](/page/Interaction_picture), retaining only resonant contributions for long-time dynamics where system frequencies are well-separated from bath fluctuations. The resulting [master equation](/page/Master_equation) takes a form with a unitary [Hamiltonian](/page/Hamiltonian) term and a dissipator resembling the Lindblad [structure](/page/Structure), though specific operator details arise from the interaction form and are discussed elsewhere.[](https://doi.org/10.1093/acprof:oso/9780199213900.001.0001) ### Phenomenological approach The phenomenological approach constructs the quantum master equation by extending the von Neumann equation for closed quantum systems with heuristically motivated dissipative terms that capture environmental interactions without deriving them from a full microscopic model.[](https://journals.aps.org/prxquantum/abstract/10.1103/PRXQuantum.5.020202) The von Neumann equation governs the unitary evolution of the density operator $\rho$ as

\frac{d\rho}{dt} = -i [H, \rho],

where $H$ is the system [Hamiltonian](/page/Hamiltonian) and $\hbar = 1$. To account for [dissipation](/page/Dissipation), one adds a [superoperator](/page/Superoperator) $\mathcal{D}[\rho]$ in the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) form,

\mathcal{D}[\rho] = \sum_k \gamma_k \left( L_k \rho L_k^\dagger - \frac{1}{2} { L_k^\dagger L_k, \rho } \right),

with positive rates $\gamma_k$ and jump operators $L_k$ chosen to represent specific decoherence channels; this structure ensures the evolution preserves trace, positivity, and complete positivity of $\rho$. The physical motivation for these terms arises from interpreting the jump operators $L_k$ as corresponding to elementary irreversible processes, such as energy relaxation or [dephasing](/page/Dephasing) induced by the [environment](/page/Environment). For instance, in a two-level atom undergoing [spontaneous emission](/page/Spontaneous_emission), the operator $L = \sqrt{\gamma} \sigma_-$, where $\sigma_-$ is the lowering operator and $\gamma$ is the decay rate, models the transition from excited to [ground state](/page/Ground_state) via [photon](/page/Photon) emission.[](https://journals.aps.org/prxquantum/abstract/10.1103/PRXQuantum.5.020202) This approach allows practitioners to incorporate empirically known relaxation rates directly into the equation, facilitating practical simulations in fields like [quantum optics](/page/Quantum_optics). A key advantage of the phenomenological method is its [simplicity](/page/Simplicity): it enables modeling of open-system [dynamics](/page/Dynamics) using only the desired [dissipation](/page/Dissipation) rates, bypassing the need for detailed [knowledge](/page/Knowledge) of the environment's [spectral](/page/Spectral) properties or [coupling](/page/Coupling) strengths.[](https://journals.aps.org/prxquantum/abstract/10.1103/PRXQuantum.5.020202) However, it requires careful selection of the $L_k$ to guarantee complete positivity and physical consistency; improper choices can lead to unphysical evolutions, such as negative probabilities, underscoring its [heuristic](/page/Heuristic) nature compared to rigorous microscopic derivations. As an illustrative example, consider a damped harmonic oscillator at zero temperature, with Hamiltonian $H = \omega a^\dagger a$ (where $a$ is the annihilation operator and $\omega$ is the frequency) and damping rate $\gamma$. The phenomenological master equation is

\frac{d\rho}{dt} = -i [\omega a^\dagger a, \rho] + \gamma \left( a \rho a^\dagger - \frac{1}{2} { a^\dagger a, \rho } \right),

where the single jump operator $L = \sqrt{\gamma} a$ describes energy loss to the [environment](/page/Environment) through successive [photon](/page/Photon) emissions, leading to thermalization toward the [vacuum](/page/Vacuum) state.[](https://arxiv.org/abs/quant-ph/0609144) This form is justified by microscopic treatments under weak coupling assumptions but is readily applied phenomenologically for qualitative analysis of oscillator decoherence.[](https://journals.aps.org/prxquantum/abstract/10.1103/PRXQuantum.5.020202) ## Standard forms ### Lindblad equation The Lindblad equation, also referred to as the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) equation, provides the most general Markovian [master equation](/page/Master_equation) for the [time evolution](/page/Time_evolution) of the density operator $\rho$ in open quantum systems, ensuring that the [dynamics](/page/Dynamics) remain physically valid under the assumptions of a Markovian [environment](/page/Environment).[](https://arxiv.org/abs/1906.04478) This form arises as the generator of a completely positive trace-preserving (CPTP) dynamical [semigroup](/page/Semigroup) and is widely used to model dissipative processes in [quantum mechanics](/page/Quantum_mechanics).[](https://doi.org/10.1007/BF01608499) The equation is expressed as

\frac{d\rho}{dt} = -i [H, \rho] + \sum_k \Gamma_k \left( L_k \rho L_k^\dagger - \frac{1}{2} { L_k^\dagger L_k, \rho } \right),

where $H$ denotes the effective [Hamiltonian](/page/Hamiltonian) incorporating coherent evolution and possible Lamb shifts from the [environment](/page/Environment), $\Gamma_k$ are the dissipation rates, and the $L_k$ are the Lindblad operators that capture the system's [interaction](/page/Interaction) with the [bath](/page/Bath) through specific jump processes.[](https://doi.org/10.1007/BF01608499) The dissipative term involves the action of these operators, which can represent phenomena such as relaxation or decoherence, while the anticommutator ensures the overall structure aligns with physical constraints.[](https://arxiv.org/abs/1906.04478) Key properties of this equation include complete positivity, which guarantees that the map preserves the positivity of [density](/page/Density) operators even when extended to larger systems via tensor products with the [identity](/page/Identity), and [trace](/page/Trace) preservation, as the time [derivative](/page/Derivative) of the [trace](/page/Trace) vanishes: $\frac{d}{dt} \operatorname{[Tr](/page/.tr)}(\rho) = 0$.[](https://arxiv.org/abs/1906.04478) These features ensure that the evolved $\rho(t)$ remains a valid [density](/page/Density) [operator](/page/Operator)—Hermitian, [positive semidefinite](/page/Positive_semidefinite), and normalized—for any initial state, thereby maintaining the physical interpretability of probabilities and [expectation](/page/Expectation) values in Markovian [dynamics](/page/Dynamics).[](https://doi.org/10.1007/BF01608499) The Gorini–Kossakowski–Sudarshan–Lindblad theorem establishes that this form is unique: any generator of a CPTP [semigroup](/page/Semigroup) on the space of bounded operators must take this structure, providing a rigorous foundation for modeling irreversible quantum processes.[](https://doi.org/10.1007/BF01608499) Steady states of the Lindblad equation are [stationary](/page/Stationary) solutions satisfying $\frac{d\rho}{dt} = 0$, which typically yield unique mixed or [thermal](/page/Thermal) states depending on the [Hamiltonian](/page/Hamiltonian) and Lindblad operators, representing long-time equilibria where dissipation balances any driving.[](https://arxiv.org/abs/1906.04478)[](https://ocw.mit.edu/courses/22-51-quantum-theory-of-radiation-interactions-fall-2012/b24652d7f4f647f05174797b957bd3bf_MIT22_51F12_Ch8.pdf) A representative example is the pure dephasing of a [qubit](/page/Qubit), modeled by a single Lindblad operator $L = \sqrt{\gamma} \sigma_z$, where $\sigma_z$ is the Pauli Z operator and $\gamma$ is the [dephasing](/page/Dephasing) rate; this leads to [exponential decay](/page/Exponential_decay) of the off-diagonal [coherence](/page/Coherence) terms in $\rho$ as $e^{-\gamma t}$ while leaving the diagonal populations unchanged.[](https://ocw.mit.edu/courses/22-51-quantum-theory-of-radiation-interactions-fall-2012/b24652d7f4f647f05174797b957bd3bf_MIT22_51F12_Ch8.pdf) ### Redfield equation The [Redfield equation](/page/Redfield_equation) represents a second-order perturbative [master equation](/page/Master_equation) for the dynamics of open quantum systems, derived from the microscopic interaction between a [system](/page/System) and its [environment](/page/Environment) without invoking the [secular approximation](/page/Secular_approximation).[](https://doi.org/10.1093/acprof:oso/9780199213900.001.0001) It arises in the weak-coupling limit, where the system-bath interaction [Hamiltonian](/page/Hamiltonian) is treated perturbatively to second order, assuming the bath correlation time is short compared to the system evolution but allowing for the retention of off-diagonal terms in the [interaction picture](/page/Interaction_picture). This approach captures intermediate regimes where non-Markovian effects are weak, providing a bridge between fully Markovian descriptions and more exact non-perturbative methods.[](https://arxiv.org/abs/2108.03128) In the [Schrödinger picture](/page/Schrödinger_picture) and in the energy eigenbasis of the system [Hamiltonian](/page/Hamiltonian), the [Redfield equation](/page/Redfield_equation) takes the form

\frac{d\rho}{dt} = -i [H, \rho] + \sum_{\alpha, \beta} R_{\alpha \beta} \left( A_\alpha \rho A_\beta^\dagger - \frac{1}{2} { A_\beta^\dagger A_\alpha, \rho } \right),

where $H$ is the system [Hamiltonian](/page/Hamiltonian), $A_\alpha$ are the system operators in the eigenbasis, and $R_{\alpha \beta}$ is the Redfield relaxation tensor that encodes the bath correlation functions and Fourier transforms at Bohr frequencies $\omega_{\alpha} - \omega_{\beta}$.[](https://doi.org/10.1093/acprof:oso/9780199213900.001.0001)[](https://qutip.readthedocs.io/en/latest/guide/dynamics/dynamics-bloch-redfield.html) This formulation originates from the original work on relaxation in spin systems, later generalized to arbitrary open quantum systems. A key limitation of the Redfield equation is its potential violation of positivity for the density operator $\rho$, particularly in strong-coupling scenarios or over long evolution times, where negative eigenvalues can emerge due to the non-secular terms. To address this, modifications such as the Bloch-Redfield approach introduce secular approximations or regularization techniques to restore positivity while preserving physical accuracy. In the secular approximation, which neglects rapidly oscillating cross-terms, the Redfield equation reduces to the Lindblad form, with relaxation rates given by $\Gamma(\omega) = \int_{-\infty}^{\infty} C(\tau) e^{i \omega \tau} d\tau$, where $C(\tau)$ is the bath correlation function. This limiting case ensures complete positivity and trace preservation, aligning with the general Gorini-Kossakowski-Lindblad-Sudarshan structure.[](https://doi.org/10.1093/acprof:oso/9780199213900.001.0001) The Redfield equation finds application in intermediate coupling regimes, such as spin-phonon relaxation in solid-state systems like quantum dots or superconducting qubits, where it accurately models decoherence without full Markovian assumptions. ## Applications ### Quantum optics In quantum optics, master equations provide a fundamental framework for describing the dissipative dynamics of light-matter interactions, particularly in systems like optical cavities and lasers. The approach originated in the 1960s with the work of Marlan O. Scully and Willis E. Lamb Jr., who developed quantum master equations to analyze optical coherence and laser operation, marking a shift from semiclassical theories to fully quantum treatments of [photon statistics](/page/Photon_statistics) and field fluctuations.[](https://link.aps.org/doi/10.1103/PhysRev.159.208) A foundational example is the damped cavity mode, where a single electromagnetic field mode couples to a dissipative reservoir, such as through mirror losses. The dynamics are captured by a Lindblad master equation of the form

\dot{\rho} = \frac{\kappa}{2} \left( 2 a \rho a^\dagger - a^\dagger a \rho - \rho a^\dagger a \right),

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Influence of local channels on Bell states of two qubits | Phys. Rev. A
Nov 22, 2005 · We will analyze in detail physical processes such as decoherence, decay, quantum homogenization, etc., in terms of the concurrence-versus-purity ...
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[PDF] arXiv:quant-ph/0508106v1 15 Aug 2005
Aug 15, 2005 · We will analyze in detail physical processes such as decoherence, decay, quantum homog- enization, etc. in terms of the concurrence vs. purity.
[31]
Nonperturbative quantum dynamical decoupling | Phys. Rev. A
Aug 27, 2013 · This paper presents a nonperturbative dynamical control approach based on the exact stochastic Schrödinger equation.
[32]
[PDF] Simulating Non-Markovian Quantum Dynamics on NISQ Computers ...
Feb 27, 2025 · Our approach enables the implementation of arbitrary quantum master equations on noisy intermediate-scale quantum (NISQ) computers.
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Fidelity-enhanced variational quantum optimal control | Phys. Rev. A
May 28, 2025 · In this study, we propose a method for creating robust pulses based on the stochastic Schrödinger equation. This equation describes individual ...<|separator|>