Separable state

Separable quantum states are those without Quantum entanglement.

Separability for Pure States

We will first consider separability for pure states. If we have two particles, whose Hilbert spaces have base states $\{|{a_{i}}\rangle \}_{i=1}^{n}$ and $\{|{b_{i}}\rangle \}_{i=1}^{n}$, then their compound system will have the base states $\{|{a_{i}}\rangle \bigotimes |{b_{j}}\rangle \}$, or in more compact notation $\{|a_{i}b_{j}\rangle \}$. These base states satisfy completeness and orthogonality for the compound system as they represent every combination of the possible states of the two individual systems. Of course these are only one of an infinitely many set of valid base states. A valid combined state of a system is can be written as any linear superposition of these base states

$|\psi \rangle =\Sigma _{i,j}c_{i,j}|a_{i}b_{j}\rangle$

Most such compound states cannot be represented as the tensor product of individual states of the individual particles. If the system cannot be represented in such a way then the particles cannot be considered separately but only as part of the larger system. Such particles are known as being entangled as they are inextricably linked and an action on one will affect the other. If the system can be represented as the tensor product of individual states it is known as being separable and is not entangled at all.

Separability for Mixed States

A more general definition of separability is that a separable state can be created from any other state using local actions and classical communication and an entangled state cannot. Operationally, a state $\rho$ is separable if there exist $p_{k}\geq 0$, $\{|{a_{k}}\rangle \}$ $\in$ $H_{A}$ and $\{\rangle {b_{k}}\}$ $\in$ $H_{B}$ for which a density operator can be expressed as

$\rho =\sum _{k}p_{k}\rho _{A}^{k}\otimes \rho _{B}^{k}$

where

$\sum _{k}p_{k}=1$

If such a decomposition does not exist then $\rho$ is entangled.

Separability Criterion

A separability criterion is a criterion a state must satisfy in order to be separable. One of the most important is the Peres-Horodecki criterion.