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)

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.