Purification of Mixed States

Any mixed state can be viewed as part of a larger pure state, a construction that underlies channels, entropy, and open-system reasoning.

Every mixture hides a pure whole

Given a mixed state rho_A, there always exists a pure state |psi_AB> on an enlarged system such that Tr_B(|psi_AB> is a purification of rho_A. Concretely, diagonalize rho_A = sum_i p_i |i> = sum_i sqrt(p_i) |i>_A |i>_B. Tracing out B recovers rho_A exactly.

The reference system

The extra system B is a mathematical reference; it need not be physical. Its minimal dimension equals the rank of rho_A. Purification formalizes the idea that classical uncertainty can always be reinterpreted as entanglement with an inaccessible partner. This is the church of the larger Hilbert space: dynamics that look noisy and irreversible on A are unitary and reversible on AB.

import numpy as np
p = np.array([0.7, 0.3])
rho = np.diag(p)
psi = np.zeros(4)
for i in range(2):
    psi[i*2 + i] = np.sqrt(p[i])   # sum sqrt(p_i)|ii>
m = psi.reshape(2,2)
print(np.round(m@m.conj().T,3))     # recovers rho

Freedom in the purification

Purifications are not unique. Any two purifications of the same rho_A differ only by a unitary (or isometry) acting on the reference B. This freedom, sometimes called the unitary equivalence of purifications, is the engine behind many proofs, including Uhlmann's theorem for fidelity and the Stinespring dilation of channels.

Why it matters

Purification lets one prove statements about mixed states and noisy channels by lifting to the pure, unitary setting where tools are sharper. It gives the operational meaning of a mixed state as a subsystem, underlies thermofield-double constructions in physics, and is the conceptual basis for treating decoherence as entanglement leaking into the environment rather than genuine randomness.