Separable state

In quantum mechanics, separable quantum states are states without quantum entanglement.

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

Let ${\displaystyle H_{1}}$ and ${\displaystyle H_{2}}$ be quantum mechanical state spaces, that is, finite-dimensional Hilbert spaces with basis states ${\displaystyle \{|{a_{i}}\rangle \}_{i=1}^{n}}$ and ${\displaystyle \{|{b_{j}}\rangle \}_{j=1}^{m}}$, respectively. By a postulate of quantum mechanics, the state space of the composite system is given by the tensor product

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