Separable State - Separable Pure States

Separable Pure States

For simplicity, the following assumes all relevant state spaces are finite dimensional. First, consider separability for pure states.

Let and be quantum mechanical state spaces, that is, finite dimensional Hilbert spaces with basis states and, respectively. By a postulate of quantum mechanics, the state space of the composite system is given by the tensor product

with base states, 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

$|\psi\rangle = \Sigma_{i,j} c_{i,j} | a_i \rangle \otimes | b_j \rangle =\Sigma_{i,j} c_{i,j} | a_i b_j \rangle$

If a pure state can be written in the form where is a pure state of the i-th subsystem, it is said to be separable. Otherwise it is called entangled. Formally, the embedding of a product of states into the product space is given by the Segre embedding. That is, a quantum-mechanical pure state is separable if and only if it is in the image of the Segre embedding.

A standard example of an (un-normalized) entangled state is

$|\psi\rangle = \begin{bmatrix} 1 \\ 0 \\ 0 \\ 1 \end{bmatrix} \in H \otimes H$

where H is the Hilbert space of dimension 2. We see that when a system is in an entangled pure state, it is not possible to assign states to its subsystems. This will be true, in the appropriate sense, for the mixed state case as well.

The above discussion can be extended to the case of when the state space is infinite dimensional with virtually nothing changed.

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