Mixed vs Pure States
A pure state carries maximal quantum information; a mixed state represents classical uncertainty or entanglement with an inaccessible system.
Two kinds of states
A pure state is a single state vector |psi>, the most complete description a quantum system can have. A mixed state is a statistical ensemble of pure states, described by a density matrix that is not a rank-one projector. The distinction is central to modelling real, noisy hardware.
Where mixed states come from
- Classical uncertainty: the preparation randomly yields |0> or |1> with equal probability
- Entanglement: a subsystem of an entangled pure state is mixed when viewed alone
- Decoherence: interaction with an environment turns a pure superposition into a mixture
A key contrast
Consider two states that both give 50/50 outcomes when measured in the computational basis. The pure state (|0>+|1>)/sqrt(2) gives a definite result if measured in the {|+>,|->} basis. The mixed state 'half |0>, half |1>' gives 50/50 in every basis. They are physically different despite matching in one basis — off-diagonal coherences distinguish them.
Detecting the difference
Compute the purity Tr(rho^2). It equals 1 for a pure state and is strictly less than 1 for a mixed state, bottoming out at 1/d for the maximally mixed state. On the Bloch ball, pure states sit on the surface and mixed states in the interior, with the fully mixed state at the centre.
Why it matters
Quantum algorithms are designed on pure states, but hardware drifts toward mixed states through decoherence. The whole discipline of error correction and fault tolerance exists to keep the working state as pure as possible. Fidelity metrics quantify how close a noisy mixed state is to the intended pure target.