Physics:Quantum Density matrix: Difference between revisions

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{{Short description|Quantum Collection topic on Quantum Density matrix}}
{{Short description|Quantum Collection topic on Quantum Density matrix}}


{{Quantum book backlink|Mathematical structure and systems}}
{{Quantum book backlink|Mathematical structure and systems}}
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'''Density matrix''' is an operator used in [[Physics:quantum mechanics|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.<ref>John von Neumann, ''Mathematical Foundations of Quantum Mechanics'', Princeton University Press, 1955.</ref><ref>R. Shankar, ''Principles of Quantum Mechanics'', 2nd ed., Springer, 1994.</ref> The density matrix is important in quantum statistical mechanics, quantum measurement theory, and the theory of [[Physics:open quantum system|open quantum systems]], where a subsystem is generally not described by a single wavefunction.<ref>H.-P. Breuer and F. Petruccione, ''The Theory of Open Quantum Systems'', Oxford University Press, 2002.</ref>
'''Density matrix''' is a Book I topic in the Quantum Collection. 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. 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. 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.
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== Definition ==
== Definition ==


A quantum state may be represented by a density matrix <math>\rho</math>, which is a linear operator acting on the system's [[Hilbert space]]. For a pure state <math>|\psi\rangle</math>, the density matrix is
A quantum state may be represented by a density matrix <math>\rho</math>, which is a linear operator acting on the system's Hilbert space. For a pure state <math>|\psi\rangle</math>, the density matrix is


<math>\rho = |\psi\rangle\langle\psi|.</math><ref>J. J. Sakurai and Jim Napolitano, ''Modern Quantum Mechanics'', 2nd ed., Addison-Wesley, 2011.</ref>
<math>\rho = |\psi\rangle\langle\psi|.</math><ref>J. J. Sakurai and Jim Napolitano, ''Modern Quantum Mechanics'', 2nd ed., Addison-Wesley, 2011.</ref>
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== Time evolution ==
== Time evolution ==


For a closed quantum system, the density matrix evolves according to the [[Superoperator|von Neumann equation]]
For a closed quantum system, the density matrix evolves according to the von Neumann equation


<math>i\hbar \frac{d\rho}{dt} = [H,\rho],</math>
<math>i\hbar \frac{d\rho}{dt} = [H,\rho],</math>
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where <math>H</math> is the Hamiltonian operator.<ref>von Neumann, ''Mathematical Foundations of Quantum Mechanics'', Princeton University Press, 1955.</ref><ref>Blum, ''Density Matrix Theory and Applications'', 3rd ed., Springer, 2012.</ref>
where <math>H</math> is the Hamiltonian operator.<ref>von Neumann, ''Mathematical Foundations of Quantum Mechanics'', Princeton University Press, 1955.</ref><ref>Blum, ''Density Matrix Theory and Applications'', 3rd ed., Springer, 2012.</ref>


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 [[Physics:Lindbladian|Lindbladian]] or other quantum master equations.<ref>G. Lindblad, "On the generators of quantum dynamical semigroups," ''Communications in Mathematical Physics'' 48, 119-130 (1976).</ref><ref>Breuer and Petruccione, ''The Theory of Open Quantum Systems'', Oxford University Press, 2002.</ref>
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.<ref>G. Lindblad, "On the generators of quantum dynamical semigroups," ''Communications in Mathematical Physics'' 48, 119-130 (1976).</ref><ref>Breuer and Petruccione, ''The Theory of Open Quantum Systems'', Oxford University Press, 2002.</ref>


== Physical significance ==
== Physical significance ==
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==See also==
==See also==
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}
* [[Partial trace]]
* partial trace
* [[Physics:Coherence|Quantum coherence]]
* Quantum coherence
* [[Physics:Open quantum system|Open quantum system]]
* Open quantum system
* [[Quantum entanglement]]
* quantum entanglement


{{Author|Harold Foppele}}
{{Author|Harold Foppele}}

Revision as of 08:12, 20 May 2026



← Back to Mathematical structure and systems

Density matrix is a Book I topic in the Quantum Collection. 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. 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. 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.

Quantum Density matrix.

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|.[1]

More generally, for an ensemble of states {|ψi⟩} occurring with probabilities pi, the density matrix is

ρ=∑ipi|ψi⟩⟨ψi|,

with

pi≥0,∑ipi=1.[2]

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:[3][4]

  1. Hermiticity
    ρ=ρ†
  2. Unit trace
    Tr(ρ)=1
  3. Positive semidefiniteness
    ρ≥0

These conditions are not only necessary but also sufficient: any operator satisfying them is a valid density matrix.[5]

For a pure state, the density matrix is idempotent:

ρ2=ρ.

Equivalently,

Tr(ρ2)=1.

For a mixed state,

Tr(ρ2)<1.[6]

The quantity Tr(ρ2) is called the purity of the state.

Matrix representation

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

ρmn=⟨m|ρ|n⟩.

For a two-level system with

|ψ⟩=α|0⟩+β|1⟩,

the corresponding pure-state density matrix is

ρ=(|α|2αβ*α*β|β|2).[7]

The diagonal elements represent populations in the chosen basis, while the off-diagonal elements represent quantum coherences.[8]

Expectation values

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

⟨A⟩=Tr(ρA).[9][10]

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 ℋA⊗ℋB, the state of subsystem A is described by the reduced density matrix

ρA=TrB(ρAB),

where TrB denotes the partial trace over subsystem B.[11][12]

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.[13]

Time evolution

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

iℏdρdt=[H,ρ],

where H is the Hamiltonian operator.[14][15]

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.[16][17]

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.[18][19]

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

See also

Table of contents (198 articles)

Index

Full contents

9. Quantum optics and experiments (5) Back to index
Experimental quantum physics: qubits, dilution refrigerators, quantum communication, and laboratory systems.
Experimental quantum physics: qubits, dilution refrigerators, quantum communication, and laboratory systems.
14. Plasma and fusion physics (8) Back to index
Conceptual illustration of plasma physics in a fusion context, showing magnetically confined ionized gas in a tokamak and the collective behavior governed by electromagnetic fields and transport processes.
Conceptual illustration of plasma physics in a fusion context, showing magnetically confined ionized gas in a tokamak and the collective behavior governed by electromagnetic fields and transport processes.
  • partial trace
  • Quantum coherence
  • Open quantum system
  • quantum entanglement


Author: Harold Foppele


References

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

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