Density matrix and reduced states

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 $ρ$ .^[1]

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.^[2]

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.^[3]

System–environment model

The total Hamiltonian is typically written as

$\hat{H} = {\hat{H}}_{S} + {\hat{H}}_{E} + {\hat{H}}_{int},$

where:

${\hat{H}}_{S}$ describes the system,
${\hat{H}}_{E}$ describes the environment,
${\hat{H}}_{int}$ 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.

References

↑ Breuer, Heinz-Peter; Petruccione, Francesco (2002). The Theory of Open Quantum Systems. Oxford University Press.
↑ Zurek, Wojciech H. (2003). Decoherence and the Transition from Quantum to Classical. Springer.
↑ Breuer, Heinz-Peter; Petruccione, Francesco (2002). The Theory of Open Quantum Systems. Oxford University Press.

Author: Harold Foppele

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[1] Breuer, Heinz-Peter; Petruccione, Francesco (2002). The Theory of Open Quantum Systems. Oxford University Press.

[2] Zurek, Wojciech H. (2003). Decoherence and the Transition from Quantum to Classical. Springer.

[3] Breuer, Heinz-Peter; Petruccione, Francesco (2002). The Theory of Open Quantum Systems. Oxford University Press.

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Physics:Quantum Open quantum systems

Density matrix and reduced states

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

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