Quantum entanglement is the failure of a joint quantum state to be expressible as a convex combination of product states. For
finite-dimensional Hilbert spaces 
and 
, a density matrix 
is separable when it can be written
 |
with 
,
,
and density matrices 
and 
. A state that is not separable is entangled. A pure
state is separable precisely when its defining unit vector
is a tensor product.
For example, 
is an entangled pure state of two quantum bits. The unentanglement
promise in QMA(2) and the approximation problem in the no-disentanglers
conjecture depend on this distinction between separable states and arbitrary
joint states.
