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Concept# Separable state

Summary

In quantum mechanics, separable states are quantum states belonging to a composite space that can be factored into individual states belonging to separate subspaces. A state is said to be entangled if it is not separable. In general, determining if a state is separable is not straightforward and the problem is classed as NP-hard.
Consider first composite states with two degrees of freedom, referred to as bipartite states. By a postulate of quantum mechanics these can be described as vectors in the tensor product space . In this discussion we will focus on the case of the Hilbert spaces and being finite-dimensional.
Let and be orthonormal bases for and , respectively. A basis for is then , or in more compact notation . From the very definition of the tensor product, any vector of norm 1, i.e. a pure state of the composite system, can be written as
where is a constant.
If can be written as a simple tensor, that is, in the form with a pure state in the i-th space, it is said to be a product state, and, in particular, separable. Otherwise it is called entangled. Note that, even though the notions of product and separable states coincide for pure states, they do not in the more general case of mixed states.
Pure states are entangled if and only if their partial states are not pure. To see this, write the Schmidt decomposition of as
where are positive real numbers, is the Schmidt rank of , and and are sets of orthonormal states in and , respectively.
The state is entangled if and only if . At the same time, the partial state has the form
It follows that is pure --- that is, is projection with unit-rank --- if and only if , which is equivalent to being separable.
Physically, this means that it is not possible to assign a definite (pure) state to the subsystems, which instead ought to be described as statistical ensembles of pure states, that is, as density matrices. A pure state is thus entangled if and only if the von Neumann entropy of the partial state is nonzero.
Formally, the embedding of a product of states into the product space is given by the Segre embedding.

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