Pure j-Positive States Without Exactness: An explicit system–ancilla counterexample for canonical purification

Jinmin Yi and Ruizhi Liu

PF:2026.000013 · Operator Algebras

Pure $j$-Positive States Without Exactness\\ \large An explicit system--ancilla counterexample for canonical purification

Primary · Analysis & Differential Equations / math.OA — Operator Algebras · Secondary · Mathematical Physics / math.MP — Mathematical Physics

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Jinmin Yi and Ruizhi Liu · Perimeter Institute

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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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Lemma 6.1

PF:2026.000013/v1/LEM-6.1

For the action of $K$ on $G\times X$ defined by k\cdot(s,x)=(ksk^{-1},kx), the only finite orbit is the singleton $\{(e,0)\}$.

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Section codes

PF:2026.000013/v1/SEC-11. Historical context and the question

PF:2026.000013/v1/SEC-22. Conventions: conjugate algebra, $j$-positivity, purity, and exactness

PF:2026.000013/v1/SEC-33. Canonical purification in standard form

PF:2026.000013/v1/SEC-44. Main theorem

PF:2026.000013/v1/SEC-55. The finitary symmetric-group construction

PF:2026.000013/v1/SEC-66. Purity

PF:2026.000013/v1/SEC-77. A projection witnessing failure of exactness

PF:2026.000013/v1/SEC-88. The conditional-expectation form of the example

PF:2026.000013/v1/SEC-99. Realization on the two-sided qubit chain

PF:2026.000013/v1/SEC-1010. Consequences and scope

PF:2026.000013/v1/SEC-1111. Historical remarks

PF:2026.000013/v1/SEC-1212. The regular-commutant identity for $G$

PF:2026.000013/v1/SEC-1313. Conjugate-algebra and opposite-algebra conventions

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Definition 2.1: $j$-positivity
PF:2026.000013/v1/DEF-2.1
A state $\varphi$ on $\bar\A\otimes_{\min}\A$ is $j$-positive if \varphi(\bar a\otimes a)\ge0\qquad(a\in\A). Equivalently, $B_\varphi(a,b)=\varphi(\bar a\otimes b)$ is a positive-semidefinite sesquilinear form.
Definition 2.2: Exactness / system--ancilla Haag duality
PF:2026.000013/v1/DEF-2.2
The doubled state is \emph{exact} when N=M'. Here $M$ is the von Neumann algebra of the original system and $N$ is the von Neumann algebra of the purification ancilla. Thus exactness is the statement that the ancilla supplies the full commutant of the system in the doubled GNS representation.
Definition 2.3: Purity
PF:2026.000013/v1/DEF-2.3
The state $\varphi$ is pure if its GNS representation is irreducible: (M\vee N)'=\C\one, \qquad\text{equivalently}\qquad M\vee N=B(\Hh).
Theorem 4.1: Pure $j$-positive non-exact state
PF:2026.000013/v1/THM-4.1
There exists a unital $C^*$-algebra $\A$ and a state $\varphi$ on $\bar\A\otimes_{\min}\A$ such that: \item $\varphi$ is pure; \item $\varphi$ is $j$-positive; \item the two marginals coincide and are tracial; \item an antiunitary involution $J$ fixes the GNS vector and implements $j$; \item both the system and ancilla von Neumann algebras are isomorphic to the separable hyperfinite $II_1$ factor; \item exactness fails strictly: $N\subsetneq M'$. One may take either \[ \A=C^*\!\left(\Symfin(\Nzero)\right) \qquad\text{or}\qquad \A=\A_2=\bigotimes_{n=1}^{\infty}M_2(\C). \] For the second choice, place the ancilla copy on the left half-chain and the original-system copy on the right half-chain. This gives a pur…
Corollary 4.2
PF:2026.000013/v1/COR-4.2
By Theorem 4.1, purity, $j$-positivity, and a fixed factor marginal do not determine Woronowicz's canonical purification. Exactness, equivalently system--ancilla Haag duality, is an independent hypothesis.
Lemma 6.1
PF:2026.000013/v1/LEM-6.1
For the action of $K$ on $G\times X$ defined by k\cdot(s,x)=(ksk^{-1},kx), the only finite orbit is the singleton $\{(e,0)\}$.
Proposition 6.2
PF:2026.000013/v1/PROP-6.2
The state $\varphi$ is pure.
Proposition 9.1
PF:2026.000013/v1/PROP-9.1
The state $\psi$ is pure and $j$-positive, both marginals equal $\tau_2$, and its system and ancilla von Neumann algebras are the same $M$ and $N$ as in the group construction. Consequently $N\subsetneq M'$.