The Schmidt decomposition expresses a vector 
in the tensor product of
two finite-dimensional Hilbert spaces 
and 
as
 |
where the 
form an orthonormal set in 
, the 
form an orthonormal set
in 
,
and the Schmidt coefficients 
are positive. The coefficients are uniquely determined
up to order, and 
is called the Schmidt rank.
After bases are chosen, the Schmidt decomposition is the singular value decomposition of the
coefficient matrix of 
.
In quantum information, a normalized pure bipartite state is unentangled exactly when its Schmidt rank is 1 (Nielsen and Chuang 2000).
