Compound Systems and Entanglement

Postulate Given two quantum systems with state spaces given independently by Hilbert spaces and , as a compound (joint) system their overall state space is , the tensor product of the two Hilbert spaces.

Remark

As a result, state spaces of quantum systems grow large very rapidly: a collection of n qubits will have a state space isomorphic to , requiring complex numbers to specify its state vector exactly. In contrast, a classical system consisting of bits can have its state specified by a single binary number of length .

Def Product State and Entangled State For a compound system with state space , a product state is a state of the form with and . An entangled state is a state not of this form.

Remark

The definition of product and entangled state also generalizes to systems with more than two components. When using Dirac notation, if and are chosen states, we will often write for their product state .

Def Bell State The Bell basis for a pair of qubits with state space is the orthonormal basis given by the following states: The state is often called ‘the Bell state’, and is very prominent in quantum information. The Bell states are maximally entangled, meaning that they induce an extremely strong correlation between the two systems involved.