Purity and the Trace of rho-Squared

Purity, Tr(rho-squared), measures how pure a quantum state is, ranging from 1 for pure states down to 1 over d for maximally mixed states.

One number for pure versus mixed

Purity is a scalar that says how close a state is to being pure. It is defined as gamma = Tr(rho^2), the trace of the squared density matrix. It gives a single, easily computed test to distinguish pure from mixed states.

The range

• Pure state: Tr(rho^2) = 1 (maximum)
• Mixed state: Tr(rho^2) < 1
• Maximally mixed state in dimension d: Tr(rho^2) = 1/d (minimum)

For a qubit (d = 2) purity runs from 1 down to 1/2. A pure state has a single eigenvalue of 1; the maximally mixed state I/2 has two eigenvalues of 1/2, giving Tr(rho^2) = 1/4 + 1/4 = 1/2.

Why it works

Write rho in its eigenbasis with eigenvalues lambda_i. Then Tr(rho^2) = sum_i lambda_i^2. Since the eigenvalues are non-negative and sum to one, this quantity is maximised when one eigenvalue is 1 (pure) and minimised when they are all equal (maximally mixed). It measures how concentrated the eigenvalue distribution is.

Relation to the Bloch vector

For a single qubit, purity ties directly to the Bloch vector length: Tr(rho^2) = (1 + |r|^2)/2. Pure states sit on the surface with |r| = 1; mixed states lie inside; the origin is maximally mixed. Purity is a radial coordinate of the Bloch ball.

Practical use

Purity is a cheaper diagnostic than full state tomography and is used to gauge how much decoherence a state has suffered. A dropping purity over time signals that noise is turning coherent superpositions into classical mixtures, the process error correction is built to resist. It complements fidelity, which compares against a specific target.