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Repair Quantum Collection B backlink template

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{{Quantum book backlink|Open quantum systems}}
In quantum mechanics, an '''open quantum system''' is a system that interacts with its surrounding environment. Such systems cannot be fully described by a single wavefunction; instead, they are described using a '''density operator''' <math>\rho</math>.<ref>{{cite book |last=Breuer |first=Heinz-Peter |last2=Petruccione |first2=Francesco |title=The Theory of Open Quantum Systems |publisher=Oxford University Press |year=2002}}</ref>
This interaction leads to phenomena such as decoherence and dissipation, which cause the loss of quantum coherence and energy into the environment. As a result, the system’s dynamics are typically governed by master equations that account for both unitary evolution and environmental effects.

[[File:Open_quantum_systems_yellow.jpg|thumb|400px|Open quantum systems: interaction with the environment, decoherence, dissipation, and quantum noise.]]

=== Density matrix ===
The density operator provides a general description of quantum states, including both pure states and statistical mixtures:

<math>
\rho = \sum_i p_i |\psi_i\rangle \langle \psi_i|,
</math>

where <math>p_i</math> are probabilities.

It satisfies:

* <math>\rho \ge 0</math> (positive)
* <math>\mathrm{Tr}(\rho) = 1</math> (normalized)
* <math>\rho^\dagger = \rho</math> (Hermitian)

The expectation value of an observable <math>\hat{A}</math> is

<math>
\langle \hat{A} \rangle = \mathrm{Tr}(\rho \hat{A}).
</math>

=== Reduced density matrix ===

For a system composed of a subsystem <math>S</math> and environment <math>E</math>, the total state lives in

<math>
\mathcal{H}_S \otimes \mathcal{H}_E.
</math>

The state of the subsystem alone is obtained by taking the '''partial trace''' over the environment:

<math>
\rho_S = \mathrm{Tr}_E(\rho_{SE}).
</math>

This operation removes environmental degrees of freedom.

=== Mixed states and entanglement ===

Even if the combined system <math>\rho_{SE}</math> is in a pure state, the reduced state <math>\rho_S</math> is generally mixed. This reflects entanglement between the system and its environment.

=== Physical significance ===

The density matrix formalism:

* allows description of open systems,
* captures statistical mixtures and decoherence,
* is essential in quantum information and thermodynamics.

=Decoherence=

'''Decoherence''' is the process by which a quantum system loses its coherent superposition due to interaction with its environment. It provides a mechanism for the emergence of classical behavior from quantum systems.<ref>{{cite book |last=Zurek |first=Wojciech H. |title=Decoherence and the Transition from Quantum to Classical |publisher=Springer |year=2003}}</ref>

=== Basic idea ===

When a quantum system interacts with its environment, the combined system becomes entangled:

<math>
|\psi\rangle_S \otimes |E_0\rangle \rightarrow \sum_i c_i |i\rangle_S \otimes |E_i\rangle.
</math>

The environment effectively "records" information about the system.

=== Loss of coherence ===

The reduced density matrix of the system becomes

<math>
\rho_S = \mathrm{Tr}_E(\rho_{SE}).
</math>

Off-diagonal elements (coherences) in the density matrix decay over time:

<math>
\rho_{ij} \rightarrow 0 \quad (i \ne j).
</math>

This suppresses interference effects.

=== Pointer states ===

Certain states, called '''pointer states''', remain stable under environmental interaction. These states form the preferred basis in which classical behavior emerges.

=== Relation to measurement ===

Decoherence explains why quantum superpositions are not observed at macroscopic scales. It does not by itself select a single outcome, but it explains the apparent collapse of the wavefunction in practical terms.

=== Physical significance ===

Decoherence:

* explains the quantum-to-classical transition,
* limits coherence in quantum systems,
* is a major challenge in quantum computing and information processing.

It is a central concept in understanding real-world quantum systems.

=Environment coupling=

In an open quantum system, the system of interest interacts with an external '''environment''' (or bath). This interaction is responsible for decoherence, dissipation, and noise.<ref>{{cite book |last=Breuer |first=Heinz-Peter |last2=Petruccione |first2=Francesco |title=The Theory of Open Quantum Systems |publisher=Oxford University Press |year=2002}}</ref>

=== System–environment model ===

The total Hamiltonian is typically written as

<math>
\hat{H} = \hat{H}_S + \hat{H}_E + \hat{H}_{\text{int}},
</math>

where:

* <math>\hat{H}_S</math> describes the system,
* <math>\hat{H}_E</math> describes the environment,
* <math>\hat{H}_{\text{int}}</math> represents the interaction.

=== Weak coupling ===

In many cases, the interaction between system and environment is weak. This allows approximate descriptions where:

* the environment acts as a reservoir,
* the system evolves with small perturbations.

This regime is often treated using perturbation theory.

=== Markovian approximation ===

If the environment has no memory (fast relaxation), the dynamics are called '''Markovian'''. In this case:

* the system evolution depends only on its current state,
* memory effects can be neglected.

This approximation leads to simple evolution equations.

=== Non-Markovian dynamics ===

If the environment retains memory, the system exhibits '''non-Markovian''' behavior:

* information can flow back from environment to system,
* coherence can partially recover,
* dynamics become more complex.

=== Physical significance ===

Environment coupling:

* explains why real quantum systems are never perfectly isolated,
* determines decoherence rates,
* is central to quantum technologies and noise control.

=See also=
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

=References=
{{reflist|3}}

{{Author|Harold Foppele}}

{{Sourceattribution|Quantum Open quantum systems|1}}

Physics:Quantum Open systems - Revision history

imported>WikiHarold: Repair Quantum Collection B backlink template

imported>WikiHarold: Repair Quantum Collection B backlink template