Purification Principle

The purification principle says that every mixed state can be realized as the marginal of a pure state of a larger system, and that purifications are essentially unique under reversible transformations on the purifying system. As a reconstruction principle, it constrains how local uncertainty and global pure descriptions fit together. The standard Hilbert-space construction is developed in Purification.

Existence and essential uniqueness

For a finite-dimensional quantum state $\rho_A$, purification supplies a unit vector $\lvert\Psi\rangle_{AB}$ with

$$\operatorname{Tr}_B \lvert\Psi\rangle\langle\Psi\rvert=\rho_A.$$

If two purifications use the same sufficiently large purifying Hilbert space, they are related by a unitary on that space. With different ancillary dimensions, the precise relation uses an isometry on the relevant support, or enlargement to a common environment.

An operational formulation replaces partial trace by discarding and unitaries by reversible transformations. Both clauses matter. Existence alone permits many arbitrary extensions; essential uniqueness expresses that the extra freedom is confined to the inaccessible purifying system.

Why a classical register is insufficient

Take the qubit state

$$\rho_A=p\lvert0\rangle\langle0\rvert +(1-p)\lvert1\rangle\langle1\rvert, \qquad0<p<1.$$

The coherent state $\sqrt p\,\lvert00\rangle+\sqrt{1-p}\,\lvert11\rangle$ purifies it. A classical record instead gives the joint state

$$p\lvert00\rangle\langle00\rvert +(1-p)\lvert11\rangle\langle11\rvert.$$

This extension has the same marginal but remains mixed. The distinction can be detected by joint measurements even though it is invisible on $A$ alone.

In ordinary classical probability, a pure joint distribution is concentrated on one pair of sample points. Its marginal is also concentrated on one point. Therefore a nontrivial classical mixture has no pure classical purification. Keeping a label of the preparation does not evade this argument.

What the principle explains and does not

Within appropriate operational axioms, purification ties irreversible local processes to reversible interactions with discarded environments. This connects state uncertainty with the structure behind quantum channels and dilations. Chiribella, D’Ariano, and Perinotti develop these consequences without initially assuming Hilbert space.

Purification does not imply that a particular laboratory mixed state actually arose from a physically accessible pure environment. It is a representation or physical-realizability principle within the theory. Nor does an enlarged pure state explain why a particular outcome is experienced; that remains part of the measurement problem.

Exercises

For the coherent and classically correlated extensions above, calculate the expectation of $X_A\otimes X_B$.

Solution

The coherent extension gives $2\sqrt{p(1-p)}$, while the classical mixture gives zero. The observable connects $\lvert00\rangle$ with $\lvert11\rangle$ and hence detects the joint off-diagonal terms. At $p=0$ or $1$, the extensions coincide and the distinction disappears.

References

• Chiribella, Giulio, Giacomo Mauro D’Ariano, and Paolo Perinotti, “Probabilistic theories with purification,” Physical Review A 81, 062348 (2010). Paper.
• Nielsen, Michael A., and Isaac L. Chuang, Quantum Computation and Quantum Information, 10th anniversary edition, Cambridge University Press, 2010 — purification and channel dilations. Publisher.