Latest revision as of 11:33, 22 May 2026

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Open systems in quantum mechanics, an open quantum system is a system that interacts with its surrounding environment. 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. In quantum mechanics, an open quantum system is a system that interacts with its surrounding environment. The density operator provides a general description of quantum states, including both pure states and statistical mixtures: For a system composed of a subsystem S and environment E, the total state lives in

Density matrix

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

$ρ = \sum_{i} p_{i} | ψ_{i} ⟩ ⟨ ψ_{i} |,$

where $p_{i}$ are probabilities.

It satisfies:

$ρ \geq 0$ (positive)
$T r (ρ) = 1$ (normalized)
$ρ^{†} = ρ$ (Hermitian)

The expectation value of an observable $\hat{A}$ is

$⟨ \hat{A} ⟩ = T r (ρ \hat{A}) .$

Reduced density matrix

For a system composed of a subsystem $S$ and environment $E$ , the total state lives in

$ℋ_{S} \otimes ℋ_{E} .$

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

$ρ_{S} = {T r}_{E} (ρ_{S E}) .$

This operation removes environmental degrees of freedom.

Mixed states and entanglement

Even if the combined system $ρ_{S E}$ is in a pure state, the reduced state $ρ_{S}$ 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.^[1]

Basic idea

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

$| ψ ⟩_{S} \otimes | E_{0} ⟩ \to \sum_{i} c_{i} | i ⟩_{S} \otimes | E_{i} ⟩ .$

The environment effectively "records" information about the system.

Loss of coherence

The reduced density matrix of the system becomes

$ρ_{S} = {T r}_{E} (ρ_{S E}) .$

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

$ρ_{i j} \to 0 (i \neq j) .$

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.^[2]

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 {{Short description|Quantum Collection topic on Quantum Open systems}}
 {{Quantum book backlink|Open quantum systems}}
+{{Quantum article nav|previous=Physics:Quantum optics|previous label=Optics|next=Physics:Quantum Master equation|next label=Master equation}}
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-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>
+'''Open systems''' in quantum mechanics, an open quantum system is a system that interacts with its surrounding environment. 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. In quantum mechanics, an open quantum system is a system that interacts with its surrounding environment. The density operator provides a general description of quantum states, including both pure states and statistical mixtures: For a system composed of a subsystem S and environment E, the total state lives in
-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.
 </div>
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 * is essential in quantum information and thermodynamics.
-=Decoherence=
+== 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>
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 It is a central concept in understanding real-world quantum systems.
-=Environment coupling=
+== 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>
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 {{Author|Harold Foppele}}
-{{Sourceattribution|Quantum Open quantum systems|1}}
+{{Sourceattribution|Physics:Quantum Open systems|1}}

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Physics:Quantum Open systems: Difference between revisions

Latest revision as of 11:33, 22 May 2026

Density matrix

Reduced density matrix

Mixed states and entanglement

Physical significance

Decoherence

Basic idea

Loss of coherence

Pointer states

Relation to measurement

Physical significance

Environment coupling

System–environment model

Weak coupling

Markovian approximation

Non-Markovian dynamics

Physical significance

See also

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