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In physics, an open quantum system is a quantum-mechanical system which interacts ("couples") with an external quantum system, typically referred to as the environment. In reality, no quantum system is completely isolated from its surroundings (as in closed systems). This leads to some interesting quantum characteristics, namely the loss of the original information characterizing the system, also known as quantum dissipation. Due to increasing complexity caused by the openness of the system to the environment, approximation techniques have been developed to make calculations more feasible, starting in the late 20th century. As time has passed, these approximation techniques have become increasingly precise to results found by experimentation. Open quantum systems and their treatments have been explored namely in fields such as quantum optics, quantum measurement theory, quantum statistical mechanics, quantum information science, quantum thermodynamics, quantum cosmology, and quantum biology.

Quantum system and environment
No quantum system can be completely isolated from its environment. As a direct result, a quantum system can never be in a pure state. A pure state is unitary equivalent to a zero temperature ground state forbidden by the third law of thermodynamics. This brings us to the differences between a closed system and an open system: a closed system can be in a pure state and only treatment of the system itself is necessary, since complete neglect of the external environment is allowed. Hence, a complete description of a quantum system requires the inclusion of the environment. The outcome of this process of embedding is that we obtain the state of the whole universe as described by a single wavefunction $$\Psi$$.

However, even if the combined system is a pure state and can be represented by a wavefunction $$ \Psi $$, a subsystem in general cannot be described by one wavefunction because there is a basis of possible states that the wavefunction can take on. This observation motivated the formalism of density matrices or density operators introduced by John von Neumann in 1927 and independently, but less systematically by Lev Landau in 1927 and Felix Bloch in 1946. A density matrix ρ has matrix elements generated by the probabilities of all possible states. This formalism obviously allows for more complicated dynamics than simply considering one wavefunction. In general, the state of a subsystem is described by the density operator $$ \rho $$ and an observable by the scalar product $$ (\rho \cdot {\bf A}) = Tr\{ \rho {\bf A} \} $$. There is no way to know if the combined system is pure based on the knowledge of the observables of the subsystem. In particular, if the combined system has quantum entanglement, the system state is not a pure state.

In addition, because these open quantum systems are so complex, certain characteristics can be taken into account that can reduce both the complexity of the calculations and the runtime needed to make them happen. A couple of examples are listed below:


 * Markovianity - Often characterized as the "memorylessness" of a system, it refers to the consideration that any future conditions of the system in question depend only on the current condition of the system and no past conditions. We call systems that do also consider conditions of the past as "non-Markovian."
 * Ergodicity - The ability of a system as a whole to be characterized by a random sample. It refers to how entangled states within a system are, such that if a measurement was taken, one could assume that this one sample could generally characterize the entire system.

Some examples of actual approximations are:


 * Born Approximation - Interaction between the system and the bath is assumed to be weak, such that the condition of the bath does not change. Alternatively, this allows perturbative treatment of the system-bath interaction term and does not change the information of the actual system.
 * Markov Approximation - The memory effects between the system and the bath are neglected, i.e:

$$\hat{\rho_S}=(t-\tau) \approx \hat{\rho_S}(t)$$

where $$t$$represents the time of the current condition of the system and $$\tau$$represents the time of the past conditions of the system.


 * Secular Approximation - Considering large time steps between the maximum time of consideration and the minimum time of consideration allows averaging of the time scale to zero.



Open quantum system dynamics
The theory of open quantum systems seeks an economical treatment of the dynamics of observables that can be associated with the system. Typical observables are energy and quantum coherence, while loss of energy to the environment is known as quantum dissipation and loss of coherence is known as quantum decoherence. This decoherence effect decays the very fragile coherence of the internal system. Now, this surrounding environment is typically very large, making exact solutions impossible to calculate and unable to be treated perturbatively. The reduction problem is difficult; therefore, a diversity of approaches have been pursued. A typical objective is to derive a reduced description of the open system where the system's dynamics are considered explicitly and the bath is described implicitly.

In any case of treatment, the main assumption is that the entire world is a large closed system, and therefore, time evolution is governed by a unitary transformation generated by a global Hamiltonian. For the combined system bath scenario, the global Hamiltonian can be decomposed into:
 * $$ H=H_S+H_B+H_{SB} $$

where $$H_S$$ is the system's Hamiltonian, $$H_B $$ is the bath Hamiltonian, and $$H_{SB}$$ is the system-bath interaction. The state of the system is obtained from a partial trace over the combined system and bath: $$\rho_S (t) =Tr_B (\rho_{SB} (t)) $$. As a real world application, the bath (resevoir) can function as heating or cooling devices causing a gradient with the internal system, which could be seen as a nanomachine that exchanges energy and matter with its surroundings.

Since open quantum systems generally cannot be described by a unitary operator, for a dynamic description, first-order differential equations (known as "master equations") detailing how probabilities associated with measurement outcomes evolve in time are used. This description is necessary because quantum systems are generally dominated by stochastic processes (statistically random) such that we must be able to characterize system dynamics with a probabilistic description. A few examples of master equations and techniques of treatment are discussed below.

Techniques of treatment

 * A formal construction of a local equation of motion with a Markovian property is an alternative to a reduced derivation. The theory is based on an axiomatic approach. The basic starting point is a completely positive map. Again, the assumption is that the initial system-environment state is uncorrelated ($$ \rho(0)=\rho_S \otimes \rho_B $$) and the combined dynamics is generated by a unitary operator. Such a map falls under the category of Kraus operator. The most general type of a time-homogeneous master equation with Markovian property describing non-unitary evolution of the density matrix ρ that is trace-preserving and completely positive for any initial condition is the Gorini–Kossakowski–Sudarshan–Lindblad equation (GKSL) equation:


 * $$\dot\rho_S=-{i\over\hbar}[H_S,\rho_S]+{\cal L}_D(\rho_S) $$

where $$ H_S$$ is a (Hermitian) Hamiltonian part and $${\cal L}_D$$:


 * $${\cal L}_D(\rho_S)=\sum_n \left(V_n\rho_S V_n^\dagger-\frac{1}{2}\left(\rho_S V_n^\dagger V_n + V_n^\dagger V_n\rho_S\right)\right)$$

is the dissipative part described implicitly through system operators $$ V_n $$ represents the influence of the bath on the system. The Markov property imposes that the system and bath are uncorrelated at all times: $$ \rho_{SB}=\rho_S \otimes \rho_B $$. The GKSL equation is unidirectional and leads any initial state $$ \rho_S$$ to a steady state solution which is an invariant of the equation of motion $$ \dot \rho_S(t \rightarrow \infty ) = 0 $$. The family of maps generated by the GKSL equation forms a quantum dynamical semigroup. In some fields such as quantum optics, the term Lindblad superoperator is often used to express the quantum master equation for a dissipative system. E.B. Davis, derived the GKSL with Markovian property master equations using perturbation theory and additional approximations, such as the rotating wave or secular, thus fixing the flaws of the Redfield equation. Davis construction is consistent with the Kubo-Martin-Schwinger stability criterion for thermal equilibrium (i.e. the KMS state) . An alternative approach to fix the Redfield has been proposed by J. Thingna, J.-S. Wang, and P. Hänggi that allows for system-bath interaction to play a role in equilibrium differing from the KMS state.


 * When the interaction between the system and the environment can be considered weak, a time-dependent perturbation theory seems appropriate, where the density matrix of the internal system itself evolves with time but the density matrix of the bath is kept constant. The typical assumption is that the system and bath are initially uncorrelated: $$ \rho(0)=\rho_S \otimes \rho_B $$. The idea was originated by Bloch and followed by Redfield, creating what is now known as the Redfield equation. The Redfield equation is a Markovian master equation that describes the time evolution of the density matrix $$ \rho $$. Its main drawback is that it does not conserve the positivity of the density operator, a crucial property.
 * Nakajima–Zwanzig equation: Employing projection operator techniques, the derivation highlights that the problem of the reduced dynamics is non-local in time. It is the most general non-Markovian master equation, written as:
 * $$\partial_t{\rho }_\mathrm{S}=\mathcal{P}{\cal L}{{\rho}_\mathrm{S}}+\int_{0}^{t}{dt'\mathcal{K}({t}'){{\rho }_\mathrm{S}}(t-{t}')}.$$

The effect of the bath is hidden in the memory kernel $$ \kappa (\tau)$$. Additional assumptions of a fast bath are required to lead to a time local equation: $$ \partial_t \rho_S = {\cal L } \rho_S $$.


 * Analogue of classical dissipation theory: Developed by R. Kubo and Y. Tanimura, this approach is connected to Hierarchical equations of motion, which embed the density operator in a larger space of auxiliary operators such that a time local equation is obtained for the whole set and their memory is contained in the auxiliary operators.


 * Caldeira–Leggett or Harmonic Bath Model: In 1981, A. Caldeira and A. J. Leggett proposed a simplifying assumption in which the bath is decomposed to normal modes represented as harmonic oscillators linearly coupled to the system. As a result, the influence of the bath can be summarized by the bath spectral function. To proceed and obtain explicit solutions, typically the path integral formulation description of quantum mechanics is employed.

The harmonic normal-mode bath leads to a physically consistent picture of quantum dissipation. Nevertheless, its ergodic properties are too weak. The dynamics does not generate wide scale quantum entanglement between the bath modes.


 * An alternative bath model is a spin bath. At low temperature and weak system-bath coupling these two bath models are equivalent. But for higher excitations, the spin bath has strong ergodic properties. Once the system is coupled significant entanglement is generated between all modes. A spin bath can simulate a harmonic bath but the opposite is not true.

An example of a natural system coupled to a spin bath is an nitrogen-vacancy (NV) center in diamond. The color center is the system and the bath consists of 13C impurities which interact with the system via magnetic dipole-dipole interaction.

An increasingly relevant context for open quantum systems is that of quantum computers. Quantum noise makes for