Source
Barnum, Gaebler & Wilce (2009; Found. Phys. 2013), LIT-081, read in full in NOTE-082.
What was actually shown
Setting. Finite-dimensional ordered linear spaces with a regular cone and an order unit (Def. 2.1). Every effect in [0, u_A] is a measurement outcome (no restriction). Composites lie between the minimal and maximal tensor products, and local tomography is a working assumption (p. 7).
Steering as order theory. A bipartite state ω is a positive map ω̂ : A* → B. It steers its B-marginal when every ensemble of that marginal is the image of an observable on A (Def. 5.1). Theorem 5.4: ω steers its marginal iff ω̂ restricted to [0, u_A] is a strong quotient map onto [0, ω_B], every chain lifting to a chain. Both directions are proved.
From steering to geometry.
- Cor. 5.5: if ω steers an interior marginal and ω̂ is injective, ω̂ is an order-isomorphism. When dim A = dim B, the steering states are exactly the isomorphism states.
- Thm 4.1: a cone is homogeneous iff every normalised interior state is the marginal of an isomorphism state with one fixed ancilla isomorphic to A*. Cor. 4.2: for irreducible A, homogeneous and weakly self-dual iff every interior state is the marginal of an isomorphism state in A ⊗max A.
- Prop. 5.8: universal self-steering implies every irreducible state space is homogeneous and weakly self-dual. With the extra assumption that direct summands of a theory’s state spaces are in the theory, this extends to all state spaces (p. 15).
What could have come out otherwise. A non-homogeneous cone could have admitted self-steering states for every interior state. The proof rules it out: matching dimensions force the steering map to be bijective.
What this does not say
- That steering forces quantum theory. The paper’s §6 argues that self-steering plus strong self-duality, local tomography and a qubit would give finite-dimensional C*-algebraic quantum theory, via Koecher–Vinberg and Hanche-Olsen. Strong self-duality is a hypothesis there (“If one could motivate the stronger assumption”, p. 16), and the operational match to Hanche-Olsen’s conditions is deferred to work “elsewhere” (NOTE-082, C7, weak). The weak-to-strong gap is the paper’s own stated limitation.
- That uniform universal steering already gives homogeneity. Prop. 5.7 claims it in one line. Cor. 5.5 covers only an injective steering map, and a steering system of larger dimension need not give one (NOTE-082, corrections). Only the self-steering case, Prop. 5.8, is secured.
- That including all steering states yields a weakly self-dual composite. For two qubits the steering product equals the maximal tensor product, which is not weakly self-dual (Fact B.3).
- Anything about ψ-ontology. NOTE-082 records that the paper has no ontological-model content; its link to PBR is only a shared ancestor in Schrödinger’s 1935–36 papers.
- Real-Hilbert-space quantum theory. It violates local tomography and so falls outside the framework (fn. 5).
Connections
- The record’s general-probabilistic-theory vocabulary is LIT-073, whose “steering” is a different, channel notion (NOTE-082). The GPT structure theorem for noncontextuality is THEORY-016.
- PBR (THEORY-605) shares the Schrödinger lineage and nothing technical.