Separable vs Entangled States
Separable states factor into independent parts or classical mixtures of products; entangled states cannot, and are the useful quantum resource.
The dividing line
A multi-qubit state is separable if it can be built from independent single-qubit states, possibly combined with classical randomness. If it cannot be so built, it is entangled. This distinction separates states that classical resources can mimic from those that need genuine quantum correlation.
Pure-state case
A pure state is separable exactly when it factors as a tensor product: |psi_A> tensor |psi_B>. For example |01> = |0> tensor |1> is separable. The Bell state (|00>+|11>)/sqrt(2) admits no such factorisation, so it is entangled. For pure states, checking separability is straightforward via the Schmidt decomposition — a Schmidt rank of one means separable.
Mixed-state case
For mixed states the definition is broader: rho is separable if it can be written as a probabilistic mixture sum_i p_i (rho_A^i tensor rho_B^i) of product states. This includes classically correlated states, which are not entangled even though the parts are correlated. Deciding separability for general mixed states is computationally hard.
Detecting entanglement
- Schmidt rank > 1 for pure states means entangled
- Nonzero entanglement entropy of a reduced state means entangled (pure case)
- The PPT (positive partial transpose) criterion detects entanglement for small systems
- Entanglement witnesses are observables whose value flags entanglement
Why the distinction matters
Separable states offer no quantum computational advantage — a computation that stays separable can be tracked classically qubit by qubit. Entanglement is a necessary resource for quantum speedup and for protocols like teleportation. Knowing whether a state has crossed from separable to entangled is knowing whether it has any quantum power to offer.