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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.

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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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Definition

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