# Density matrix

of a state $\rho$ defined on the algebra $\mathfrak A ( {\mathcal H})$ of bounded linear operators acting on a Hilbert space ${\mathcal H}$

The positive nuclear operator $\widetilde \rho \in \mathfrak A ( {\mathcal H})$ such that

$$\tag{1 } \rho ( A) = \mathop{\rm tr} A \widetilde \rho ,\ \ A \in \mathfrak A ( {\mathcal H}),$$

where $\mathop{\rm tr} \widetilde \rho = 1$. Conversely, any state $\rho$, i.e. any linear positive $( \rho ( A ^ {*} A) \geq 0)$ normalized $( \rho ( E) = 1)$ functional on $\mathfrak A( {\mathcal H})$, can be represented in the form (1), i.e. it has a density matrix $\widetilde \rho$, which is moreover unique.

The concept of a density matrix arose in statistical physics in defining a Gibbs quantum state. Let a quantum system occupying a finite volume $V$ in $\mathbf R ^ {3}$ be described by the vectors of a certain Hilbert space ${\mathcal H} _ {V}$, by the Hamiltonian $H _ {V} ^ {0}$ and, possibly, by some set of mutually commuting "first integrals" $H _ {V} ^ {1} \dots H _ {V} ^ {k}$, $k = 1, 2 ,\dots$. A Gibbs state for such a system is a state on $\mathfrak A( {\mathcal H} _ {V} )$ defined by the density matrix

$$\tag{2 } \widetilde \rho = Z ^ {-} 1 \mathop{\rm exp} \{ - \beta ( H _ {V} ^ {0} + \mu _ {1} H _ {V} ^ {1} + \dots + \mu _ {k} H _ {V} ^ {k} ) \} ,$$

where $Z$ is a normalizing factor and $\beta > 0$, $\mu _ {1} \dots \mu _ {k}$ are real parameters.

In addition to the density matrix (2), the state of a system in quantum statistical physics may be defined by means of the so-called reduced density matrix. In the simplest case of a system of identical particles (bosons or fermions) described by the vectors of a Fock [Fok] space ${\mathcal H} _ {V}$, the reduced density matrix $\widehat \rho$ of a state $\rho$ is the set of (in general, generalized) functions

$$\tag{3 } \widehat \rho = \{ \widehat \rho _ {m,n} ( x _ {1} \dots x _ {m} , y _ {1} \dots y _ {n} ) :\ x _ {i} \in V , y _ {j} \in V ,$$

$${} i = 1 \dots m,\ j = 1 \dots n; \ m , n = 0, 1 ,\dots \} ,$$

where

$$\widehat \rho _ {m,n} ( x _ {1} \dots x _ {m} , y _ {1} \dots y _ {n} ) = \ \rho \left ( \prod _ { i= } 1 ^ { m } a( x _ {i} ) \prod _ { j= } 1 ^ { n } a ^ {*} ( y _ {j} ) \right ) ,$$

and where $a( x), a ^ {*} ( y)$, $x , y \in \mathbf R ^ {3}$, are the creation operators and annihilation operators, respectively, acting in ${\mathcal H} _ {V}$. If the creation and annihilation operators in $\mathfrak A( {\mathcal H} _ {V} )$ are replaced by some other system of generators $\{ {a _ \lambda } : {\lambda \in {\mathcal L} } \}$( ${\mathcal L}$ is a certain set of indices), then the reduced density matrix $\widehat \rho$ for a state $\rho$ is defined by analogy with (3) as the set of values of $\rho$ on all possible monomials of the form

$$a _ {\lambda _ {1} } \dots a _ {\lambda _ {n} } ,\ \ \lambda _ {i} \in {\mathcal L} ,\ i = 1 \dots n, n = 1, 2 ,\dots .$$

The reduced density matrix is particularly convenient for determining the limiting Gibbs state in a $C ^ {*}$- algebra $\mathfrak A _ \infty$ of so-called quasi-local observables: $\mathfrak A _ \infty = {\cup _ {V \in \mathbf R ^ {2} } \mathfrak A( {\mathcal H} _ {V} ) } bar$( the bar denotes closure in the uniform topology).

#### References

 [1] L.D. Landau, E.M. Lifshitz, "Statistical physics" , Pergamon (1980) (Translated from Russian) [2] D. Ruelle, "Statistical mechanics: rigorous results" , Benjamin (1974)

The statement that any state $\rho$ has a representation (1) has been proved for finite-dimensional ${\mathcal H}$ only.

The functions defined by (3) are the quantum analogues of distribution functions.

How to Cite This Entry:
Density matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Density_matrix&oldid=46627
This article was adapted from an original article by R.A. Minlos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article