In systems with infinitely many degrees of freedom, fundamental results from quantum information theory can fail. An important example is the uniqueness of purifications: Even when two subsystems, described by commuting von Neumann algebras $A$ and $B$ , are tomographically complete, purifications of a state on $A$ need not be related by unitaries in $B$ . It was recently shown that this occurs precisely when Haag duality fails, i.e., when the commutant $B'$ is strictly larger than $A$ . This raises the question of which fundamental entanglement properties survive in such a setting. We show that, for a pure global state, the ability to steer any ensemble decomposition of the marginal state on $A$ by measurements on $B$ is equivalent to the existence of a state-preserving conditional expectation from $B'$ onto $A$ . This establishes a direct connection between quantum steering and subfactor theory. The key observation is that steering is equivalent to the existence of extensions of ensemble decompositions from $A$ to $B'$ . Working with general Jordan algebras, we prove that unital positive maps have state-preserving left inverses if and only if ensemble decompositions can be lifted. For the inclusion $A\hookrightarrow B'$ , a left inverse is precisely a conditional expectation, yielding the characterization above.In systems with infinitely many degrees of freedom, fundamental results from quantum information theory can fail. An important example is the uniqueness of purifications: Even when two subsystems, described by commuting von Neumann algebras $A$ and $B$, are tomographically complete, purifications of a state on $A$ need not be related by unitaries in $B$. It was recently shown that this occurs precisely when Haag duality fails, i.e., when the commutant $B'$ is strictly larger than $A$. This raises the question of which fundamental entanglement properties survive in such a setting. We show that, for a pure global state, the ability to steer any ensemble decomposition of the marginal state on $A$ by measurements on $B$ is equivalent to the existence of a state-preserving conditional expectation from $B'$ onto $A$. This establishes a direct connection between quantum steering and subfactor theory. The key observation is that steering is equivalent to the existence of extensions of ensemble decompositions from $A$ to $B'$. Working with general Jordan algebras, we prove that unital positive maps have state-preserving left inverses if and only if ensemble decompositions can be lifted. For the inclusion $A\hookrightarrow B'$, a left inverse is precisely a conditional expectation, yielding the characterization above.

quant-pharXiv:2608.10783Aug 12, 20260 citations

Quantum Sufficiency for Self-Adjoint Statistical Models via Likelihood-Type Operators on Real $*$ -Subalgebras and Real Jordan AlgebrasQuantum Sufficiency for Self-Adjoint Statistical Models via Likelihood-Type Operators on Real $*$-Subalgebras and Real Jordan Algebras

Koichi Yamagata

We develop a theory of quantum sufficiency on real *-subalgebras and real Jordan algebras for models consisting of general self-adjoint operators, including derivatives of states. Square-root likelihood ratios and symmetric logarithmic derivatives serve as self-adjoint likelihood-type objects, allowing ordinary quantum statistical models and local models to be treated within a unified framework that admits degenerate reference states. We introduce sufficient real unital positive maps and show that their Hermitian fixed points form a real Jordan algebra admitting a sufficient Jordan conditional expectation. The real and complex *-subalgebras generated by this algebra admit sufficient real and complex completely positive conditional expectations, respectively. We characterize the minimal sufficient real *-subalgebra by the likelihood-ratio set and $ρ$ -modular invariance. The minimal sufficient real Jordan algebra is generated by the likelihood-ratio set together with the operator obtained by normalizing every irreducible diagonal block of the reference state, relative to the minimal sufficient real *-subalgebra, to trace one. We also obtain Koashi-Imoto type decompositions for real *-subalgebras and real Jordan algebras admitting sufficient conditional expectations, explicitly accounting for inequivalent irreducible Jordan representations. These results separate the likelihood-ratio aspect of sufficiency from its noncommutative modular aspect and identify real Jordan structure as a natural framework for quantum statistical sufficiency.We develop a theory of quantum sufficiency on real *-subalgebras and real Jordan algebras for models consisting of general self-adjoint operators, including derivatives of states. Square-root likelihood ratios and symmetric logarithmic derivatives serve as self-adjoint likelihood-type objects, allowing ordinary quantum statistical models and local models to be treated within a unified framework that admits degenerate reference states. We introduce sufficient real unital positive maps and show that their Hermitian fixed points form a real Jordan algebra admitting a sufficient Jordan conditional expectation. The real and complex *-subalgebras generated by this algebra admit sufficient real and complex completely positive conditional expectations, respectively. We characterize the minimal sufficient real *-subalgebra by the likelihood-ratio set and $ρ$-modular invariance. The minimal sufficient real Jordan algebra is generated by the likelihood-ratio set together with the operator obtained by normalizing every irreducible diagonal block of the reference state, relative to the minimal sufficient real *-subalgebra, to trace one. We also obtain Koashi-Imoto type decompositions for real *-subalgebras and real Jordan algebras admitting sufficient conditional expectations, explicitly accounting for inequivalent irreducible Jordan representations. These results separate the likelihood-ratio aspect of sufficiency from its noncommutative modular aspect and identify real Jordan structure as a natural framework for quantum statistical sufficiency.

quant-pharXiv:2604.23292Apr 28, 20260 citations

Conditional expectation and commutativity in von Neumann algebras

Maciej Mączyński

International Journal of Theoretical PhysicsJun 1, 19773 citations

Conditional expectations on von Neumann algebras and the causal independence of quantized fields

K. Yu. Dadashyan, Sergey S. Horujy

Theoretical and Mathematical PhysicsFeb 1, 19810 citations

A

and

B

, are tomographically complete, purifications of a state on

A

need not be related by unitaries in

B

. It was recently shown that this occurs precisely when Haag duality fails, i.e., when the commutant

B'

is strictly larger than

A

. This raises the question of which fundamental entanglement properties survive in such a setting. We show that, for a pure global state, the ability to steer any ensemble decomposition of the marginal state on

A

by measurements on

B

is equivalent to the existence of a state-preserving conditional expectation from

B'

onto

A

. This establishes a direct connection between quantum steering and subfactor theory. The key observation is that steering is equivalent to the existence of extensions of ensemble decompositions from

A

B'

. Working with general Jordan algebras, we prove that unital positive maps have state-preserving left inverses if and only if ensemble decompositions can be lifted. For the inclusion

A\hookrightarrow B'

, a left inverse is precisely a conditional expectation, yielding the characterization above.

Quantum Sufficiency for Self-Adjoint Statistical Models via Likelihood-Type Operators on Real

*

-Subalgebras and Real Jordan Algebras

ρ

-modular invariance. The minimal sufficient real Jordan algebra is generated by the likelihood-ratio set together with the operator obtained by normalizing every irreducible diagonal block of the reference state, relative to the minimal sufficient real *-subalgebra, to trace one. We also obtain Koashi-Imoto type decompositions for real *-subalgebras and real Jordan algebras admitting sufficient conditional expectations, explicitly accounting for inequivalent irreducible Jordan representations. These results separate the likelihood-ratio aspect of sufficiency from its noncommutative modular aspect and identify real Jordan structure as a natural framework for quantum statistical sufficiency.