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Density matrix is an operator used in quantum mechanics to describe the state of a quantum system. It provides a unified formalism for both pure states, represented by state vectors, and mixed states, represented by statistical ensembles of state vectors.^[1]^[2] The density matrix is important in quantum statistical mechanics, quantum measurement theory, and the theory of open quantum systems, where a subsystem is generally not described by a single wavefunction.^[3]

Definition

A quantum state may be represented by a density matrix $ρ$ , which is a linear operator acting on the system's Hilbert space. For a pure state $| ψ ⟩$ , the density matrix is

$ρ = | ψ ⟩ ⟨ ψ | .$ ^[4]

More generally, for an ensemble of states ${| ψ_{i} ⟩}$ occurring with probabilities $p_{i}$ , the density matrix is

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

with

$p_{i} \geq 0, \sum_{i} p_{i} = 1 .$ ^[5]

Thus the density matrix extends the usual state-vector formalism to cases where there is classical uncertainty about which pure state has been prepared.

Properties

A density matrix $ρ$ satisfies the following conditions:^[6]^[7]

Hermiticity
$ρ = ρ^{†}$
Unit trace
$T r (ρ) = 1$
Positive semidefiniteness
$ρ \geq 0$

These conditions are not only necessary but also sufficient: any operator satisfying them is a valid density matrix.^[8]

For a pure state, the density matrix is idempotent:

$ρ^{2} = ρ .$

Equivalently,

$T r (ρ^{2}) = 1 .$

For a mixed state,

$T r (ρ^{2}) < 1 .$ ^[9]

The quantity $T r (ρ^{2})$ is called the purity of the state.

Matrix representation

If ${| n ⟩}$ is an orthonormal basis, the density matrix may be written in components as

$ρ_{m n} = ⟨ m | ρ | n ⟩ .$

For a two-level system with

$| ψ ⟩ = α | 0 ⟩ + β | 1 ⟩,$

the corresponding pure-state density matrix is

$ρ = (\begin{matrix} | α |^{2} & α β^{*} \\ α^{*} β & | β |^{2} \end{matrix}) .$ ^[10]

The diagonal elements represent populations in the chosen basis, while the off-diagonal elements represent quantum coherences.^[11]

Expectation values

The expectation value of an observable represented by an operator $A$ is given by

$⟨ A ⟩ = T r (ρ A) .$ ^[12]^[13]

This formula applies to both pure and mixed states, which is one reason the density matrix formalism is so useful.

Reduced density matrix

For a composite system with Hilbert space $ℋ_{A} \otimes ℋ_{B}$ , the state of subsystem $A$ is described by the reduced density matrix

$ρ_{A} = {T r}_{B} (ρ_{A B}),$

where ${T r}_{B}$ denotes the partial trace over subsystem $B$ .^[14]^[15]

Even if the total system is in a pure state, the reduced density matrix of a subsystem may be mixed. This feature is central to the study of quantum entanglement, decoherence, and open-system dynamics.^[16]

Time evolution

For a closed quantum system, the density matrix evolves according to the von Neumann equation

$i ℏ \frac{d ρ}{d t} = [H, ρ],$

where $H$ is the Hamiltonian operator.^[17]^[18]

This is the density-matrix analogue of the Schrödinger equation. For open systems interacting with an environment, the evolution is more general and is often described by a Lindbladian or other quantum master equations.^[19]^[20]

Physical significance

The density matrix formalism is indispensable when:

the preparation procedure produces an ensemble rather than a definite pure state;
only a subsystem of a larger entangled system is considered;
decoherence suppresses phase relations in a preferred basis;
thermal equilibrium states are studied in quantum statistical mechanics.^[21]^[22]

In these contexts, the density matrix provides a more general and physically realistic description than a single wavefunction.

Table of contents (176 articles)

Index

Core theory

Foundations
Conceptual and interpretations
Mathematical structure and systems
Atomic and spectroscopy
Wavefunctions and modes
Quantum dynamics and evolution
Measurement and information
Quantum information and computing

Matter (by scale)

Matter (by scale)

Applications and extensions

Quantum optics and experiments
Open quantum systems
Quantum field theory
Statistical mechanics and kinetic theory
Condensed matter and solid-state physics
Plasma and fusion physics
Timeline
Advanced and frontier topics

Methods and tools

Methods and tools

Full contents

1. Foundations (11)↑

2. Conceptual and interpretations (13)↑

3. Mathematical structure and systems (13)↑

4. Atomic and spectroscopy (14)↑

Physics:Quantum Atomic structure and spectroscopy
Physics:Quantum Hydrogen atom
Physics:Quantum number
Physics:Quantum Multi-electron atoms
Physics:Quantum Fine structure
Physics:Quantum Hyperfine structure
Physics:Quantum Isotopic shift
Physics:Quantum defect
Physics:Quantum Zeeman effect
Physics:Quantum Stark effect
Physics:Quantum Spectral lines and series
Physics:Quantum Selection rules
Physics:Quantum Fermi's golden rule
Physics:Quantum beats

5. Wavefunctions and modes (9)↑

Physics:Quantum Wavefunction
Physics:Quantum Superposition principle
Physics:Quantum Eigenstates and eigenvalues
Physics:Quantum Boundary conditions and quantization
Physics:Quantum Standing waves and modes
Physics:Quantum Normal modes and field quantization
Physics:Number of independent spatial modes in a spherical volume
Physics:Quantum Density of states
Physics:Quantum carpet

6. Quantum dynamics and evolution (17)↑

7. Measurement and information (9)↑

8. Quantum information and computing (10)↑

9. Quantum optics and experiments (5)↑

Physics:Quantum Nonlinear King plot anomaly in calcium isotope spectroscopy
Physics:Quantum optics beam splitter experiments
Physics:Quantum Ultra fast lasers
Physics:Quantum Experimental quantum physics
Physics:Quantum optics

Template:Quantum optics operators

10. Open quantum systems (9)↑

11. Quantum field theory (19)↑

12. Statistical mechanics and kinetic theory (9)↑

13. Condensed matter and solid-state physics (13)↑

14. Plasma and fusion physics (8)↑

Physics:Quantum Fusion reactions and Lawson criterion
Physics:Quantum Plasma (fusion context)
Physics:Quantum Magnetic confinement fusion
Physics:Quantum Inertial confinement fusion
Physics:Quantum Plasma instabilities and turbulence
Physics:Quantum Tokamak core plasma
Physics:Quantum Tokamak edge physics and recycling asymmetries
Physics:Quantum Stellarator

15. Timeline (8)↑

16. Advanced and frontier topics (9)↑

Author: Harold Foppele

References

↑ John von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955.
↑ R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
↑ H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press, 2002.
↑ J. J. Sakurai and Jim Napolitano, Modern Quantum Mechanics, 2nd ed., Addison-Wesley, 2011.
↑ Michael A. Nielsen and Isaac L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2000.
↑ Nielsen and Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2000.
↑ Breuer and Petruccione, The Theory of Open Quantum Systems, Oxford University Press, 2002.
↑ Karl Blum, Density Matrix Theory and Applications, 3rd ed., Springer, 2012.
↑ Sakurai and Napolitano, Modern Quantum Mechanics, 2nd ed., Addison-Wesley, 2011.
↑ Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
↑ Breuer and Petruccione, The Theory of Open Quantum Systems, Oxford University Press, 2002.
↑ von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955.
↑ Sakurai and Napolitano, Modern Quantum Mechanics, 2nd ed., Addison-Wesley, 2011.
↑ Nielsen and Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2000.
↑ Breuer and Petruccione, The Theory of Open Quantum Systems, Oxford University Press, 2002.
↑ Wojciech H. Zurek, "Decoherence, einselection, and the quantum origins of the classical," Reviews of Modern Physics 75, 715-775 (2003).
↑ von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955.
↑ Blum, Density Matrix Theory and Applications, 3rd ed., Springer, 2012.
↑ G. Lindblad, "On the generators of quantum dynamical semigroups," Communications in Mathematical Physics 48, 119-130 (1976).
↑ Breuer and Petruccione, The Theory of Open Quantum Systems, Oxford University Press, 2002.
↑ Blum, Density Matrix Theory and Applications, 3rd ed., Springer, 2012.
↑ Zurek, "Decoherence, einselection, and the quantum origins of the classical," Reviews of Modern Physics 75, 715-775 (2003).

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[1] John von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955.

[2] R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.

[3] H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press, 2002.

[4] J. J. Sakurai and Jim Napolitano, Modern Quantum Mechanics, 2nd ed., Addison-Wesley, 2011.

[5] Michael A. Nielsen and Isaac L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2000.

[6] Nielsen and Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2000.

[7] Breuer and Petruccione, The Theory of Open Quantum Systems, Oxford University Press, 2002.

[8] Karl Blum, Density Matrix Theory and Applications, 3rd ed., Springer, 2012.

[9] Sakurai and Napolitano, Modern Quantum Mechanics, 2nd ed., Addison-Wesley, 2011.

[10] Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.

[11] Breuer and Petruccione, The Theory of Open Quantum Systems, Oxford University Press, 2002.

[12] von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955.

[13] Sakurai and Napolitano, Modern Quantum Mechanics, 2nd ed., Addison-Wesley, 2011.

[14] Nielsen and Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2000.

[15] Breuer and Petruccione, The Theory of Open Quantum Systems, Oxford University Press, 2002.

[16] Wojciech H. Zurek, "Decoherence, einselection, and the quantum origins of the classical," Reviews of Modern Physics 75, 715-775 (2003).

[17] von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955.

[18] Blum, Density Matrix Theory and Applications, 3rd ed., Springer, 2012.

[19] G. Lindblad, "On the generators of quantum dynamical semigroups," Communications in Mathematical Physics 48, 119-130 (1976).

[20] Breuer and Petruccione, The Theory of Open Quantum Systems, Oxford University Press, 2002.

[21] Blum, Density Matrix Theory and Applications, 3rd ed., Springer, 2012.

[22] Zurek, "Decoherence, einselection, and the quantum origins of the classical," Reviews of Modern Physics 75, 715-775 (2003).

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Physics:Quantum Density matrix

Definition

Properties

Matrix representation

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Reduced density matrix

Time evolution

Physical significance

See also

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