Purification Overview

Purification represents a mixed state as part of a larger pure state. It is a mathematical construction for tracking possible correlations, not a claim that a particular auxiliary system was present in the experiment. Purification develops the equivalence theorems and applications.

Helpful background. Reduced density matrices explains tracing out a factor, and spectral decomposition supplies the eigenvalues used below.

Building a Purifying State

A purification of $\rho_A$ is a normalized vector $|\Psi\rangle_{AR}$ such that

$$\operatorname{Tr}_R|\Psi\rangle\langle\Psi|=\rho_A.$$

In finite dimensions, write the nonzero spectral terms as $\rho_A=\sum_{k=1}^{r}p_k|k\rangle\langle k|$. Choose an orthonormal set of $r$ reference kets and form

$$|\Psi\rangle_{AR}=\sum_{k=1}^{r}\sqrt{p_k}\,|k\rangle_A|k\rangle_R.$$

Its squared norm is $\sum_kp_k=1$. Tracing over $R$ kills the $k\neq\ell$ terms because the reference kets are orthogonal, leaving $\rho_A$. The minimum reference dimension is the rank $r$; a smaller reference cannot supply enough Schmidt terms.

Nonuniqueness

For $\rho_A=\operatorname{diag}(3/4,1/4)$, one choice is

$$|\Psi\rangle=\frac{\sqrt3}{2}|00\rangle+\frac12|11\rangle.$$

Applying any unitary to $R$ produces another purification with the same $A$ reduction. For example, flipping the reference basis gives $\sqrt3|01\rangle/2+|10\rangle/2$. The reference’s labels have changed, while Alice’s local statistics have not.

More generally, purifications are related by an isometry on the occupied reference subspaces; if the reference spaces have the same finite dimension, this relation extends to a unitary. No particular purification singles out the actual preparation ensemble.

Quick Check

Why would $\sqrt3|00\rangle/2+|10\rangle/2$ fail to purify $\operatorname{diag}(3/4,1/4)$?

Solution

Both terms carry the same reference ket $|0\rangle_R$. The state factors as $(\sqrt3|0\rangle+|1\rangle)_A|0\rangle_R/2$, so tracing out $R$ leaves a pure state with nonzero off-diagonal coherence. Distinct orthogonal reference kets are what remove those cross terms in the desired construction.

References

• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th anniversary ed., Cambridge University Press, 2010, sec. 2.5.
• J. Watrous, The Theory of Quantum Information, Cambridge University Press, 2018, ch. 2. Author’s manuscript.