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''of a state <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d0310901.png" /> defined on the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d0310902.png" /> of bounded linear operators acting on a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d0310903.png" />''
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The positive [[Nuclear operator|nuclear operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d0310904.png" /> such that
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{{TEX|auto}}
 +
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d0310905.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
''of a state  $  \rho $
 +
defined on the algebra  $  \mathfrak A ( {\mathcal H}) $
 +
of bounded linear operators acting on a Hilbert space  $  {\mathcal H} $''
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d0310906.png" />. Conversely, any state <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d0310907.png" />, i.e. any linear positive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d0310908.png" /> normalized <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d0310909.png" /> functional on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d03109010.png" />, can be represented in the form (1), i.e. it has a density matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d03109011.png" />, which is moreover unique.
+
The positive [[Nuclear operator|nuclear operator]]  $  \widetilde \rho  \in \mathfrak A ( {\mathcal H}) $
 +
such that
  
The concept of a density matrix arose in statistical physics in defining a Gibbs quantum state. Let a quantum system occupying a finite volume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d03109012.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d03109013.png" /> be described by the vectors of a certain Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d03109014.png" />, by the Hamiltonian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d03109015.png" /> and, possibly, by some set of mutually commuting "first integrals"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d03109016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d03109017.png" />. A Gibbs state for such a system is a state on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d03109018.png" /> defined by the density matrix
+
$$ \tag{1 }
 +
\rho ( A)  =   \mathop{\rm tr} A \widetilde \rho  ,\ \
 +
A \in \mathfrak A ( {\mathcal H}),
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d03109019.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
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.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d03109020.png" /> is a normalizing factor and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d03109021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d03109022.png" /> are real parameters.
+
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
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d03109023.png" />, the reduced density matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d03109024.png" /> of a state <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d03109025.png" /> is the set of (in general, generalized) functions
+
$$ \tag{2 }
 +
\widetilde \rho    =  Z  ^ {-} 1  \mathop{\rm exp} \{ - \beta ( H _ {V}  ^ {0} + \mu _ {1} H _ {V}  ^ {1}
 +
+ \dots + \mu _ {k} H _ {V}  ^ {k} ) \} ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d03109026.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
where  $  Z $
 +
is a normalizing factor and  $  \beta > 0 $,
 +
$  \mu _ {1} \dots \mu _ {k} $
 +
are real parameters.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d03109027.png" /></td> </tr></table>
+
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
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d03109028.png" /></td> </tr></table>
+
$$
 +
\widehat \rho  _ {m,n} ( x _ {1} \dots x _ {m} , y _ {1} \dots y _ {n} )  = \
 +
\rho \left ( \prod _ { i= } 1 ^ { m }  a( x _ {i} ) \prod _ { j= } 1 ^ { n }  a  ^ {*} ( y _ {j} ) \right ) ,
 +
$$
  
and where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d03109029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d03109030.png" />, are the creation operators and annihilation operators, respectively, acting in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d03109031.png" />. If the creation and annihilation operators in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d03109032.png" /> are replaced by some other system of generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d03109033.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d03109034.png" /> is a certain set of indices), then the reduced density matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d03109035.png" /> for a state <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d03109036.png" /> is defined by analogy with (3) as the set of values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d03109037.png" /> on all possible monomials of the form
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d03109038.png" /></td> </tr></table>
+
$$
 +
a _ {\lambda _ {1}  } \dots a _ {\lambda _ {n}  } ,\ \
 +
\lambda _ {i} \in {\mathcal L} ,\
 +
i = 1 \dots n, n = 1, 2 ,\dots .
 +
$$
  
The reduced density matrix is particularly convenient for determining the limiting Gibbs state in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d03109039.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d03109040.png" /> of so-called quasi-local observables: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d03109041.png" /> (the bar denotes closure in the uniform topology).
+
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====
 
====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>
 
<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====
 
====Comments====
The statement that any state <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d03109042.png" /> has a representation (1) has been proved for finite-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031090/d03109043.png" /> only.
+
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.
 
The functions defined by (3) are the quantum analogues of distribution functions.

Revision as of 17:32, 5 June 2020


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.

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