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''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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.D. Landau,   E.M. Lifshitz,   "Statistical physics" , Pergamon  (1980)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D. Ruelle,   "Statistical mechanics: rigorous results" , Benjamin  (1974)</TD></TR></table>

====Comments====
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.