Pure $j$-Positive States Without Exactness\\ \large An explicit system--ancilla counterexample for canonical purification
Uploaded by: Ruizhi Liu
Generated by: Jinmin Yi and Ruizhi Liu
Woronowicz's purification of a factor state on a $C^*$-algebra is characterized by a doubled extension that is pure, $j$-positive, and exact. Exactness is the system--ancilla commutant equality: in the GNS representation of the doubled state, the von Neumann algebra generated by the ancilla copy equals the commutant of the original-system algebra. In his 1972 paper Woronowicz explicitly observed that ``the existence of pure non exact states is not clear.'' We give a concrete construction showing that exactness is independent of purity and $j$-positivity, even after the marginal state is fixed.
The first model uses the finitary symmetric group $G=\operatorname{Sym}_{\mathrm{fin}}(\mathbb N_0)$ and the point stabilizer $K=\{g:g(0)=0\}$. On $\ell^2(G)\otimes\ell^2(\mathbb N_0)$ we construct commuting left and twisted-right representations whose vector state satisfies
\[
\varphi(\bar u_g\otimes u_h)=\mathbf 1_{\{g=h\in K\}}.
\]
The resulting state is pure and $j$-positive, and both marginals are the canonical trace. The projection $\mathbf 1\otimes\ket 0\!\bra 0$ lies in the commutant of the system algebra but not in the ancilla algebra. Thus exactness fails strictly. The same state admits the conditional-expectation formula
\[
\varphi(\bar a\otimes b)=\tau\!\left(E(a)^*E(b)\right),
\qquad E:R\to Q\subsetneq R,
\]
where $R$ is the hyperfinite $II_1$ factor and $Q$ is a proper subfactor. By the uniqueness of the separable hyperfinite $II_1$ factor, the construction restricts to two copies of the qubit UHF algebra. Placing the system copy and the ancilla copy on opposite half-chains realizes the doubled state as a pure state on the two-sided qubit chain. Both copy marginals are the product trace, while the system--ancilla commutant equality fails.
This establishes that exactness is an independent part of Woronowicz's characterization of canonical purification. The qubit-chain realization is spatially inhomogeneous, and its $j$-positive form is degenerate; translation-invariant and strictly $j$-positive variants are separate questions.
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