Conjugates, Filters and Quantum Mechanics

The Jordan structure of finite-dimensional quantum theory is derived, in a conspicuously easy way, from a few simple postulates concerning abstract probabilistic models (each defined by a set of basic measurements and a convex set of states). The key assumption is that each system A can be paired with an isomorphic $\textit{conjugate}$ system, $\overline{A}$, by means of a non-signaling bipartite state $\eta_A$ perfectly and uniformly correlating each basic measurement on A with its counterpart on $\overline{A}$. In the case of a quantum-mechanical system associated with a complex Hilbert space $\textbf{H}$, the conjugate system is that associated with the conjugate Hilbert space $\overline{\textbf{H}}$, and $\eta_A$ corresponds to the standard maximally entangled EPR state on $\textbf{H} \otimes \overline{\textbf{H}}$. A second ingredient is the notion of a $\textit{reversible filter}$, that is, a probabilistically reversible process that independently attenuates the sensitivity of detectors associated with a measurement. In addition to offering more flexibility than most existing reconstructions of finite-dimensional quantum theory, the approach taken here has the advantage of not relying on any form of the"no restriction"hypothesis. That is, it is not assumed that arbitrary effects are physically measurable, nor that arbitrary families of physically measurable effects summing to the unit effect, represent physically accessible observables. An appendix shows how a version of Hardy's"subpace axiom"can replace several assumptions native to this paper, although at the cost of disallowing superselection rules.


I. INTRODUCTION AND BACKGROUND
This paper derives the Jordan structure of finitedimensional quantum theory from a very lean set of postulates, and in a conspicuously easy way -easy, at any rate, if one knows the Koecher-Vinberg Theorem, which relates euclidean Jordan algebras to homogeneous, selfdual cones 1 . This brings us within hailing distance of orthodox finite-dimensional quantum theory: every euclidean Jordan algebra is a direct sum of self-adjoint parts of real, complex or quaternionic matrix algebras, spin factors, and the exceptional jordan algebra of self-adjoint 3 × 3 matrices over the octonions [10].
In fact, I present two slightly different results, both resting on the idea of a conjugate system. Here is a sketch. As is well known, any mixed state on a quantummechanical system is the marginal of a pure bipartite state on a composite of two copies of that system. This latter state perfectly correlates some pair of basic observables on these two copies. Using not two copies of the same system but rather, a system and its conjugate system (associated with the conjugate of the first system's Hilbert space), the maximally mixed state arises as the marginal of a bipartite state -essentially, the maximally entangled state -perfectly correlating every observable with its conjugate.
These correlational features can be abstracted. A finite-dimensional probabilistic model A is characterized by a set of basic measurements and a finite-dimensional convex set of states. From this data, one can construct, in a canonical way, a pair of ordered real vector space V(A) and E(A) ≤ V(A) * , the former generated by A's states, and the latter by its basic measurement outcomes. Suppose that all basic measurements of A have a common number n of outcomes. Define a conjugate for A to be a model A, plus a fixed isomorphism γ A : A ≃ A and a fixed bipartite, non-signaling state η A between A and A, such that, for every basic measurement outcome x of A, η A (x, γ A (x)) = 1/n. Assume now that (i) A has a conjugate system, and, further, that (ii) every state of A dilates to a non-signaling bipartite state between A and A that perfectly correlates some pair of basic measurements. Further, suppose (iii) every basic measurement outcome has probability one in a unique state, and (iv) every non-singular state (every state strictly positive on basic measurement outcomes) can be prepared, up to normalization, from the maximally mixed state by means of a reversible process.
Theorem 1 Subject to conditions (i)-(iii), E(A) is self-dual, and isomorphic to V(A). Subject to condition (iv), V(A) is homogeneous. Thus, subject to conditions (i)-(iv), E(A) is homogeneous and self-dual, and hence, by the Koecher-Vinberg Theorem, has the structure of a euclidean Jordan algebra.
In the presence of a conjugate, a slightly stronger preparability hypothesis than (iv) yields both the homogeneity and the self-duality of E(A) + , without the need for assumptions (ii) and (iii) above. A filter is a process Φ (that is, a positive mapping V(A) → V(A)) that independently attenuates the reliability of each outcome of some basic measurement, in the sense that for each outcome x, there is a constant t x ∈ (0, 1] with Φ(α)(x) = t x α(x) for all states α. Suppose A has a conjugate, A: starting in the canonical state η A , we can apply Φ to the first component of the composite system AA to obtain a new bipartite state (Φ ⊗ 1 A )(η A ) (where 1 A is the identity transformation on A's state space). We can also apply the counterpart of Φ to A, obtaining (1 ⊗ Φ)(η A ). Call Φ symmetric iff these states are the same.
Theorem 2: Let A have a conjugate system A. If every interior state of A arises from the maximally mixed state by a symmetric reversible filter, then A satisfies (ii) and (iii); hence, E(A) + is homogeneous and self-dual.
The proofs of both of these results are quite short and straightforward. Several of the ideas in this paper were earlier explored, and somewhat similar results derived, in [17] and [18]. However, the approach taken here is much simpler and more direct.

A. General Probabilistic Theories
The general framework for this paper is that of [6,17], which I'll now quickly review. In a few places (set off in numbered definitions), my usage differs slightly from that of the cited works. See [2,10] for further information on ordered vector spaces and on Jordan algebras.
Probabilistic Models A probabilistic model is characterized by a set M(A) of basic measurements or tests, and a set Ω(A) of states. Identifying each test with its outcome-sest, M(A) is simply a collection of non-empty sets (a test space, in the language of [6]). I'll write X(A) for the union of this collection, i.e., the space of all outcomes of all basic measurements. I understand a state to be an assignment of probabilities to measurementoutcomes, that is, a function α : X(A) → [0, 1] such that x∈E α(x) = 1 for all tests E ∈ M(A). To reflect the possibility of forming statistical mixtures, I also assume that Ω(A) is convex. 2 By the dimension of a model A, I mean the affine dimension of Ω(A). In the simplest finite-dimensional classical model, M(A) consists of a single, finite test, and Ω(A) is the simplex of all probability weights on that test. Of more immediate interest to us is the quantum model A(H) = (M(H), Ω(H)) associated with a complex Hilbert space H. The test space M(H) is the set of orthonormal bases of H; thus, the outcome-space X(H) is the set of unit vectors of H. The state space Ω(H) consists of the quadratic forms associated with density operators on H, so that a state α ∈ Ω(H) has the form α(x) = W α x, x for some density operator W α , and all unit vectors x ∈ X(H). Real and quaternionic quantum models, corresponding to real or quaternionic Hilbert spaces, are defined in the same way.
Remark: Not every physically accessible observable on a finite-dimensional quantum system is represented by an orthonormal basis. Rather, the general observable corresponds to a positive-operator-valued measurement. Similarly, for an arbitrary probabilistic model A, the test space M(A) may, but need not, represent a complete catalogue of all possible measurements one might make on the system represented by A: rather, it is some convenient catalogue of such measurements, sufficiently large to determine the system's states. 2 It is usually also assumed that Ω(A) is closed (hence, compact) in the product topology on R X . This assumption isn't needed here, however.
The spaces E(A) and V(A) An ordered vector space is a real vector space E equipped with a convex cone E + with E + ∩ −E + = {0} and E = E + − E + -that is, E is spanned by E + . The cone induces a partial order, invariant under translation and multiplication by nonnegative scalars, given by a ≤ b iff b − a ∈ E + . An example is the space L h (H) of hermitian operators on a real, complex or quaternionic Hilbert space, ordered by the cone of positive semi-definite operators. A linear mapping T : E → F between ordered vector spaces E and F is positive iff T (E + ) ⊆ F + . The dual cone, E * + ⊆ E * , consists of positive linear functionals f ∈ E * .
Any probabilistic model A gives rise in a canonical way to a pair ordered vector spaces E(A) and V(A). The latter is simply the span of the state space Ω(A) in the space R X(A) , ordered by the cone V(A) + of non-negative multiples of states. Every outcome x ∈ X(A) corresponds to a linear evaluation functional The space E(A) is the span of these functionals in V(A) * , ordered by the cone E(A) + consisting of all finite linear combinations i t i x i having non-negative coefficients t i . Note that a ∈ E(A) + implies a(α) ≥ 0 for all α ∈ V(A) + . The converse is generally false. Note, too, that there is a distinguished vector u A ∈ E(A) + given by u A = x∈E x; this is independent of the choice of E ∈ M(A). Any state α ∈ Ω(A) satisfies u A (α) = 1; again, the converse is generally false. 3 Processes A physical process on a system A is naturally represented by an affine (convex-linear) mapping T : Ω(A) → V(A) such that, for every α ∈ Ω(A), T (α) = pβ for some β ∈ Ω(A) and some constant 0 ≤ p ≤ 1 (depending on α), which we can regard as the probability that the process occurs, given that the initial state is α. Such a mapping extends uniquely to a positive linear mapping T : V(A) → V(A) with T (α)(u A ) ≤ 1 for all α ∈ Ω(A). I will say that a process T : V(A) → V(B) prepares a state α of A from another state, β, if α is a multiple of T (β), i.e., T (β) coincides with α up to normalization.
A process T : V(A) → V(B) has a dual action on V (A) * , given by T * (f ) = f • T for all f ∈ V(A) * , with T * (u) ≤ u. In our finite-dimensional setting, we can identify V(A) * with E(A) as vector spaces, but not, generally, as ordered vector spaces. While Φ * will preserve the dual cone V(A) * + , it is not required that T * preserve the cone E(A) ≤ V(A) * . This reflects the idea that not every physically accessible measurement need appear among the tests in M(A), as discussed above.
A process T : V(A) → V(A) reversible iff there is another process, S, such that, for every state α, there exists a constant p ∈ (0, 1] with S(T (α)) = pα. In other words, S allows us to recover α from T (α), up to normalization. It is not hard to see that p must be independent of α, so that S = pT −1 . In particular, T is an order-isomorphism of V(A).
Self-Duality and Jordan Algebras For both classical and quantum models, the ordered spaces E(A) and V(A) are isomorphic. In the former case, where M(A) consists of a single test E and Ω(A) is the simplex of all probability weights on E, we have V(A) ≃ R E and E(A) ≃ (R E ) * . The standard inner product on R E provides an order-isomorphism between these spaces, that is, a linear bijection taking the positive cone of one space onto that of the other. If H is a finite-dimensional real or complex Hilbert space, we have an affine isomorphism between the state space of Ω(H) and the set of density operators on H, allowing us to identify V(A(H)) with L h (H). For any x ∈ X(H), the evaluation functional , with the latter isomorphism implemented by the tracial inner product a, b = Tr(ab). More generally, any inner product , on an ordered vector space E defines a positive linear mapping E → E * . If this is an order-isomorphism, one says that E is selfdual with respect to this inner product. This is equivalent to the condition a ∈ E + iff a, b ≥ 0 for all b ∈ E + . In this language, the standard inner product on R E and the tracial inner product on L h (H) are self-dualizing, where E is a finite set and H, a finite-dimensional Hilbert space.
In fact, any euclidean Jordan algebra, ordered by its cone of squares, is self-dual with respect to its canonical tracial inner product. Another property shared by all euclidean Jordan algebras is homogeneity: the group of order-automorphisms of E acts transitively on the interior of the positive cone E + . The Koecher-Vinberg Theorem [10] states that, conversely, any self-dual, homogenous order-unit space E can be equipped with the structure of a euclidean Jordan algebra. This structure is unique, up to the choice of an element of the interior of the cone E + to serve as a unit for the Jordan algebra.

Definition 1: Let us say that a model
If the model A is both homogeneous and self-dual, then E(A) is homogeneous and self-dual, and hence, E(A) + is the cone of squares of a euclidean Jordan algebra. Of these two properties, homogeneity is the easier to interpret in physical terms: it amounts to assumption (iv) in the introduction, namely, that every state in the interior of V(A) + can be obtained, up to normalization, from some particular such state by means of a reversible process. Self-duality is less easily motivated, but will emerge from the other assumptions sketched in the introduction.
Composite Systems A composite of two probabilistic models A and B is a model AB, together with a mapping X(A) × X(B) → X(AB) allowing us to interpret a pair (x, y) of outcomes belonging to the two systems separately as a single product outcome xy ∈ X(AB), in such a way that, for any tests E ∈ M(A) and F ∈ M(B), the set EF = {xy|x ∈ E, y ∈ F } is a test in AB. It follows that any state ω ∈ Ω(AB) pulls back to a functionwhich I'll also denote by ω -on X(A) × X(B), given by ω(x, y) = ω(xy), satisfying x∈E,y∈F ω(xy) = 1 for all tests E ∈ M(A), F ∈ M(B). One understands ω(x, y) as the joint probability of the outcomes x and y in the bipartite state ω.
Remark: This definition is weaker than that used in, e.g., [6], where it is also assumed that, for any pair of states α ∈ Ω(A) and β ∈ Ω(B), there exists a unique state α ⊗ β on AB such that (α ⊗ β)(x, y) = α(x)β(y) for all outcomes x ∈ X(A), y ∈ X(B). This assumption is not needed in what follows.

Non-Signaling Composites A state ω on a composite
AB is non-signaling iff the marginal states ω 1 (x) = y∈F and ω 2 (y) = x∈E ω(xy) are well-defined, i.e., independent of the choice of tests E and F . One can then also define conditional states ω 2|x (y) := ω(x, y)/ω 1 (x) (with, say, ω 2|x identically zero if ω 1 (x) = 0), and similarly for ω 1|y . This gives us the following bipartite version of the law of total probability [11]: for any choice of E ∈ M(A) or F ∈ M(B). Classical and quantum bipartite states are clearly non-signaling.

Definition 2:
A non-signaling composite of A and B, I mean a composite AB such that every state ω of AB is non-signaling, and the conditional states ω 1|y and ω 2|x are valid states of A and B, respectively, for all outcomes y ∈ X(A) and x ∈ X(A).
As an example, if H and K are real or complex Hilbert spaces, there is a natural mapping It is straightforward to check that this makes A(H ⊗ K) a non-signaling composite of A(H) and A(K), in the above sense.
It follows from (1) that if ω ∈ Ω(AB) is non-signaling, the marginal states ω 1 and ω 2 also belong to Ω(A) and Ω(B), respectively. It is not hard to show that a state ω on a non-signaling composite AB gives rise to a bilinear form ω : E(A)×E(B) → R, uniquely defined by ω( x, y) = ω(xy) for all outcomes x ∈ X(A), y ∈ X(B), and hence also to a positive linear conditioning map given by ω(a)(b) = ω(a, b). Note that for any x ∈ X(A), ω(x) = ω 1 (x)ω 2|x , whence the terminology. If ω is an order-isomorphism, then ω is called an isomorphism state.
Probabilistic Theories As a rule, one wants to think of physical theories, not as loosely structured classes, but rather, as categories of systems [1,5]. In what follows, a probabistic theory is understood to be a category, of probabilistic models, with morphisms corresponding to processes -that is, if A and B are models belonging to the theory, then morphism A → B are positive linear mappings T : V(A) → V(B), subject to the condition that T (α)(u B ) ≤ 1 for all α ∈ Ω(A), as discussed above. A monoidal probabilistic theory is one closed under some definite operation A, B → AB of forming non-signaling composites, and subject also to the further condition that, for all models A 1 , A 2 , B 1 and B 2 ∈ C and all pro- for all x ∈ X(A 1 ) and y ∈ X(A 2 ). The category of quantum models and completely positive mappings is a monoidal probabilistic theory in this sense, as is the category of real quantum models and completely positive mappings. It will simplify the discussion to assume, in the balance of this paper, that we are working in some monoidal non-signaling probabilistic theory, so that we can take advantage of (2).

II. CORRELATIONAL PROPERTIES OF QUANTUM COMPOSITES
In this section, H, K are finite-dimensional real or complex Hilbert spaces (with inner products conjugatelinear in the second argument, in the complex case). The space of linear operators on H is L(H); the space of Hermitian operators, L h (H). As discussed above, if A(H) is the corresponding quantum probabilistic model (with X(H) the set of unit vectors of H, M(H), the set of orthonormal bases of H, and Ω(H), the set of states associated with density operators on H), then . If x ∈ H, write x for the same vector in H, so that, for any scalar c, cx = cx, and cx = c x. Note that inner product on H is given by x, y = x, y = y, x . If T ∈ L(H), write T for the operator T x = T x. While T → T is conjugate linear as a mapping from L(H) to L(H), it is linear as a mapping between the real vector spaces E(H) = L h (H) and E(H) = L h (H). Indeed, T → T defines an order-isomorphism between these spaces.
For any vectors x, y ∈ H, let x ⊙ y denote the rankone operator on H given by (x ⊙ y)z = z, y x. (In Dirac notation, this is |x y|.) If x is a unit vector, then x⊙x = P x , the orthogonal projection operator associated with x.
The mapping x, y → x⊙y is sesquilinear, that is, linear in its first, and conjugate linear in its second, argument; it therefore extends uniquely to a linear mapping H⊗H → L(H). It is easy to check that this is inective, and hence, on dimensional grounds, an isomorphism.
Suppose now that α is a state on A(H), represented by a density operator W on H, so that α(x) = W x, x for all unit vectors x ∈ X(H). Let W have spectral resolution where E is an orthonormal basis for H (so that λ x = α(x) for all x ∈ E). Functional calculus gives us W 1/2 = x∈E λ 1/2 x x⊙x. Using the isomorphism L(H) ≃ H⊗H, This is a unit vector, and so, in turn, defines a pure bipartite state ω on the composite quantum system for all y, z ∈ X(H). The marginal, or reduced, state ω 1 on first component system assigns to an effect a a probability In other words, ω 1 = α. A similar computation gives us ω 2 = α, the state on A(H) corresponding to W . Indeed, the state ω is symmetric, in the sense that ω(x, y) = ω(y, x) for all x, y ∈ X(H). Notice that, by (3), we have | x ⊗ x, v W | 2 = α(x) for every x ∈ E. Evidently, the pure state ω corresponding to v W sets up a perfect correlation between E ∈ M(H) and the corresponding test E = {x|x ∈ E} ∈ M(H). This works for any basis E diagonalizing W , i.e., ω correlates every such basis with the corresponding basis E for H. Indeed, if U is a unitary operator on H with U W = W U , then ω(U x, U y) = ω(x, y) for all x, y ∈ X(H).

Definition 3:
A non-signaling state ω on a composite AB is correlating iff there exists some pair of tests E ∈ M(A) and F ∈ M(B) and a bijection f : E → F such that ω(x, y) = 0 for (x, y) ∈ E × F with y = f (x) -equivalently, x∈E ω(x, f (x)) = 1.
Using this language, we can summarize the foregoing discussion as follows: every state α on a quantum model A(H) is the marginal of a pure, correlating state on A (H ⊗ H), invariant under the group of unitaries of the form U ⊗ U with U stabilizing α. 4 In the special case where α is the maximally mixed state, i.e., where W = 1/n, where n = dim(H)) and 1 is the identity operator on H, we the corresponding vector from (2) is v W = 1 n x∈E x ⊗ x =: Ψ, the maximally entangled stateSince W = 1/n is diagonalized by every orthonormal basis, the expansion above is valid for all E ∈ M(H). Let's denote the corresponding non-signaling state on The following is well known, and straightforward to verify: Lemma 1: Let W be any density operator on H, and let ω be the pure state on A(H ⊗ H) corresponding to v W , as given by (2) and (3) above. Then the conditioning map corresponding to ω is given by ω(a) = W 1/2 aW 1/2 for all a ∈ E(H).
Corollary 1: Let W and ω be as in Lemma 1. If W is non-singular, then ω is an isomorphism state.
Remark: Quaternionic Hilbert spaces present special problems, owing to the difficulty of defining a tensor product over a non-commutative division ring. However [3], one can view a quaternionic Hilbet space as a pair (H, J) where H is a complex Hilbert space and J is an anti-unitary operator on H satisfying J 2 = −1. Likewise, a real Hilbert spaces can be identified with complex Hilbert spaces equipped with anti-unitary operator J satisfying J 2 = 1. Given two quaternionic Hilbert spaces (H 1 , J 1 ), a natural candidate for the tensor product of two pairs (H i , J i ) with J i anti-unitary and satisfying J 2 = −1, is (H 1 ⊗ H 2 , J 1 ⊗ J 2 ). Since J 2 i = −1, we have (J 1 ⊗ J 2 ) 2 = 1, i.e., the tensor product should be thought of as a real Hilbert space. Understanding composites of quaternionic quantum systems in this way, with a little care one can show that QM over H still enjoys the correlational features just disucussed. In other words, these features are equally consistent with real, complex and quaternionic quantum mechanics.

III. CONJUGATE SYSTEMS
We've seen that the composite quantum system A(H ⊗ H) has some very strong correlational properties. As I'll now show, one can derive the Jordan structure of finite-dimensional QM from these properties, with a minimum of fuss. As a first step, we need to generalize the relationship between A(H) and A(H). Throughout this section, let A is a model of uniform rank n, meaning that all tests have cardinality n. By an isomorphism between two models A and B, I mean a bijection φ :

Definition 4:
where γ A : A ≃ A is an isomorphism and η A is a non-signaling state on (some composite of) A and A such that (a) η(x, γ A (y)) = η(y, γ A (x)) and (b) η A (x, γ A (x)) = 1/n for every x, y ∈ X(A). 5 In the context of finite-dimensional quantum mechanics, where A = A(H) for a Hilbert space H, we can take A = A(H), with γ A (x) = x; for the state η A , we can use the standard maximally entangled state on H ⊗ H. Thus, we can think of η A for an arbitrary probabilistic model A, as a generalized maximally entangled state.
Remark: One might wonder whether one can use the isomorphism γ A to simply identify A with its conjugate. Certainly we can use γ A to pull the correlator η A on AA back to a positive bilinear form on A (a)). However, whether this corresponds to a legitimate bipartite state on a legitimate composite AA of A with itself, depends on the particular probabilistic theory at hand. For example, if A = A(H) is the quantum model associated with a Hilbert space H, and A is the model associated with H in the usual way, then the state η A pulls back along the isomorphism a → a to a bilinear form on L h (H)×L h (H) -but one associated with the non-completely positive partial-transpose operation, which corresponds to no state on A(H ⊗ H). Thus, the notion of a conjugate system is best understood as applying to an entire probabilistic theory, rather than to a single probabilistic model.

Four Axioms
We can now restate the four conditions discussed in the introduction in more precise terms:

Axiom 4 (State Preparation):
Every state α with α(x) > 0 for all outcomes x, can be prepared by a reversible process from the maximally mixed state ρ(x) ≡ 1/n. Axioms 1-3 are trivially satisfied in discrete classical probability theory. As observed in Section 2 they are also satisfied in finite-dimensional quantum theory. Axiom 4 is equivalent to the homogeneity of the cone V(A) + , and hence, also satisfied by classical and quantum probability theory.
We are now ready to prove Theorem 1. The main order of business is to show that a model satisfying Axioms 1, 2 and 3 is self-dual. The proof is not difficult. I'll break it up into a sequence of even easier lemmas. In the interest of readability, in what follows I conflate outcomes x ∈ X(A) with the corresponding effects x ∈ E(A), and write x for γ A (x).

Lemma 3: η A is an isomorphism state.
Proof: We need to show that η A : E(A) → V(A) is a linear isomorphism with a positive inverse. It is enough to show that η A maps the positive cone of E(A) onto that of V(A). Since x → x is an isomorphism between A and A, we can apply Lemma 2 to A: if α ∈ V + (A), we have α = x∈E α(x)δ x . Since η A (x, x) = 1/n, we have η A (x) = 1 n δ x for every x ∈ X(A). Hence, η A ( x∈E nα(x)x) = α.
Lemma 4: Every a ∈ E(A) has a representation a = x∈E t x x for some test E ∈ M(A) and some coefficients t x .
Proof:If a ∈ E(A) + , then by Lemma 2, η A (a) = x∈F t x δ x for some E ∈ M(A). By Lemma 3, η A is an order-isomorphism. Applying η −1 A to this expansion gives the desired result. Now for an arbitrary a ∈ E(A) + , we can find some N such that a ≤ N u. If a = a 1 − a 2 with a 1 , a 2 ∈ E(A) + , we can find N ≥ 0 with a 2 ≤ N u. Thus, b := a + N u = a 1 + (N u − a 2 ) ≥ 0.
Proof: By assumption, it's symmetric. It's also bilinear, because η A is non-signaling and γ A is linear. We need to show that , is positive semi-definite. Let a ∈ E(A). From Lemma 4, we have a = x∈E t x x for some test E and some coeffcients t x . Now This is zero only where all coefficients t x are zero, i.e., only for a = 0.

Lemma 2 tells us that E(A) ≃ V(A), so it remains only to show that the inner product
, For the reverse inclusion, suppose a, b ≥ 0 for all b ∈ E(A) + . Then a, y ≥ 0 for all y ∈ X. By Lemma 4, a = x∈E t x x for some test E. Thus, for all y ∈ E we have a, y = t y ≥ 0, whence, a ∈ E(A) + .
Axiom 4 now guarantees then V(A) and hence, E(A), is homogeneous. Indeed, as discussed above, a reversible process is an order-automorphism φ : V(A) → V(A); to say that this prepares α, up to normalization, from ρ, is simply to say that α = tφ(ρ) for some appropriate constant t > 0 (namely, t = u A (φ(ρ)) −1 ). Since tφ is again an order-automorphism, it follows that the group of order-automorphisms of V(A) act transitively on the interior of V(A) + , i.e., V(A) is homogeneous.
We now have the advertised result: Subject to axioms 1-4, E(A) is homogeneous and self-dual, whence, by the Koecher-Vinberg Theorem, can be equipped with a euclidean Jordan structure making E(A) + the cone of squares. Indeed, this Jordan structure is unique, subject to the condition that u A act as the unit for this Jordan structure. (One can further show that the outcome-set X(A) is precisely the set of primitive idempotents in E(A) with respect to this Jordan structure, and that basic measurements in M(A) correspond to Jordan frames, i.e., maximal pairwise orthogonal sets of idempotents. See Lemma 5 in [7].) Remarks (1) Note that the only point at which Axioms 2 and 3 were invoked was in the proof of Lemma 2. Thus, any other assumptions leading to the representation (4) could be used instead. Moreover, since η A is a linear isomorphism, and hence (in finite dimensions) an homeomorphism, it is enough to obtain this representation for states in the interior of Ω. From this we have, as in the proof of Lemma 3, that the interior of the cone V + is in the range of η A , from which it follows that η is a linear isomorphism, and hence, that K : is an open sub-cone of E + , spanning the latter. Moreover, every point in the interior of K has a corresponding "spectral" decomposition of the form x∈E t x x with t x ≥ 0, where E is some test in M(A). Arguing exactly as in the proof of Lemma 4, we can extend this to arbitrary points a ∈ E by decomposing a as a 1 − a 2 , where a 1 , a 2 ∈ K. The rest of the proof of Theorem 1 then proceeds just as before.
(2) Since axioms 1 and 2 have such a similar character, it is natural to look for a single principle that encompasses them both. Suppose G is a group acting transitively on the outcome-space X(A) of the model A, and leaving the state-space Ω(A) invariant. If G is compact, there will exist an invariant state, ρ, obtained by group averging; by the transitivity of G on outcomes, this state must be constant, i.e., ρ is the maximally mixed state ρ(x) ≡ 1/n. Now consider the following variant of Axiom 2 (here G α denotes the stabilizer in G of the state α): Axiom 2 ′ There exists a system A and an isomorphism γ A : A ≃ A, such that every state α is the marginal, ω 1 , of some correlating bipartite state ω on AB with ω(gx, γ A (gy)) = ω(x, γ A (y)) for every g ∈ G α .
As observed in Section 2, this is satisfied by finitedimensinal quantum models. Applied to the maximally mixed state ρ, this produces a correlator η A turning A into a conjugate in the sense of Definition 3. Thus, we have Corollary 2: Let A carry a compact, transitive-onoutcomes group action, as described above, and satisfy Axioms 2 ′ and 3. Then E(A) is self-dual.
(3) Given Axioms 1-3, any condition guaranteeing the homogeneity of V(A) will also secure that of E(A). As observed in [4], homogeneity follows from the requirement that every interior state of A be the marginal of an isomorphism state on a composite of two isomorphic copies of A. Thus, in place of Axiom 4, we could simply strengthen Axiom 2 to Axiom 2 ′′ Every interior state of A is the marginal of an isomorphism state on AA, correlating some pair of tests.
Corollary 3: For any model A satisfying Axioms 1, 2 ′′ and 3, E(A) is homogeneous and self-dual.
For irreducible systems, isomorphism states are pure [4], so Axiom 2 ′′ is related to the purification postulate of [8]. The latter asserts that every system A has a "conjugate system" (in their usage) B, such that every state on A arises as the marginal of a pure state of AB, unique up to symmetries of B.

IV. FILTERS
To this point, I've been leaning heavily on the assumption of sharpness (Axiom 3). It would surely be preferable to define δ x to be the conditional state η 1|x , for each x ∈ X(A), and to prove that this is the unique state making x certain. In fact, this can be done if we replace Axiom 2 with a slightly stronger, but very plausible, axiom concerning the existence of certain processes called filters [17]. In many kinds of laboratory experiments, the distinct outcomes of an experiment correspond to physical detectors, the efficiency of which can independently be attenuated, if desired, by the experimenter.
In fact, this can be done reversibly. Let A be a finite-dimensional quantum system, with corresponding Hilbert space H, and identify E(A) with L h (H A ). If E is an orthonormal basis representing a basic measurement on this system, define a positive self-adjoint operator V : H → H by setting V x = t 1/2 x x for every x ∈ E, where 0 < t x ≤ 1. This gives us a completely positive linear mapping φ : E(A) → E(A), namely φ(a) = V aV . This has a completely positive inverse φ −1 (a) = V −1 aV −1 , hence, is an order automorphism. For each x ∈ E, the corresponding effect x ∈ E(A) ≃ L h (H) is the projection operator P x . It is easy to check that V P x V = t x P x , i.e., that φ( x) = t x x for every x ∈ E. Let us say that Φ prepares a state α if α equals φ(ρ) up to normalization, where where ρ is the maximally mixed state ρ(x) ≡ 1/n. If every interior state is preparable by a reversible filter, then V(A) + is homogeneous. Suppose that A has a conjugate system, A, and that φ is a filter for a test E ∈ M(A). By applying φ to one wing of the composite system AA, we can convert the correlator η A into a new non-signaling, sub-normalized joint state ω, given by ω(x, y) = η A (φ * x, y) for all x ∈ X(A), y ∈ X(B). Noticing that Φ * (x) = t x x for every x ∈ E, we see that ω correlates E with E: if x, y ∈ E with x = y, we have ω(x, y) = η A (t x x, y) = t x η A (x, y) = 0. Since ω 1 = ρ • φ, it follows that any state preparable from ρ by a filter, is the marginal of a correlating state. So, in the presence of sharpness we can replace Axiom 2 by the axiom that every state be preparable by a filter. In fact, we can do a bit better.
The isomorphism γ A : A ≃ A extends to an orderautomorphism V(A) ≃ V(A), given by α → α, with α(x) = α(x) for all x ∈ X(A). Hence, a positive linear mapping Φ : Lemma 6: Let A have a conjugate A. Suppose that every state of A is preparable by a symmetric filter. Then a, b := η A (a, γ A (b)) is a self-dualizing inner product on E(A).
Proof: Let α = Φ(ρ), where Φ is a symmetric filter on some test E. Consider the bipartite state For each outcome x ∈ X(A), let δ x denote the conditional state (η A ) 1|x . For all x ∈ E, we have Now ω 1 = Φ((η A ) 1 ) = Φ(ρ) = α, and also, by the law of total probability (1), . Thus, every state on A is a convex combination of the states δ x . Hence, the cone generated by these states coincides with V(A) + . We now have that η maps E(A) + onto V(A) + , as in the proof of Lemma 3. The proof that a, b := η(a, b) defines an inner product on E(A) now proceeds as in the proof of Lemmas 4 and 5.
This suggests another axiom, combining Axiom 1 with a strengthened form of Axiom 4: Axiom 4b A has a conjugate system, and every interior state is preparable by a reversible symmetric filter.
This clearly implies the homogeneity of V(A). In fact, it is strong enough to allow us to do without Axioms 2 and 3. Indeed, as noted in Remark (1) following the proof of Theorem 1, it is sufficient to obtain the decomposition (4) for points in the interior of V + . By Lemma 6, for any system satisfying Axiom 5, all states in the interior of Ω can be decomposed as in equation (4); as noted in Remark (1) following the proof of Theorem 1, this is enough to secure the self-duality of E(A), and its isomorphism with V(A). This proves Theorem 2.

V. CONCLUSION
We've seen that any of several related sets of assumptions, e.g., Axioms 1-4, or Axioms 2',3 and 4, pr Axioms 1, 2' and 3, or the two-part Axiom 4b, lead in a very simple way the homogeneity and self-duality of the cone E(A) + associated with a probabilistic model A. Hence, by the Koecher-Vinberg Theorem, the space E(A) carries a canonical Jordan struture. While this is not the only route one can take to deriving this structure (see, e.g, [15] and [18] for approaches stressing symmetry principles), it does seem especially straightforward.
Among Jordan-algebraic probabilistic theories, finitedimensional quantum mechanics over C can be singled out as follows. A non-signaling composite system AB locally tomographic iff every state ω ∈ AB is uniquely determined by the joint probability function ω(x, y) that it induces. It is well known, and easy to see on dimensional grounds, that among finite-dimensional real, complex and quaternionic quantum mechanics, only the complex version is locally tomographic. Call a probabilistic theory monoidal iff it is a symmetric monoidal category, in which the monoidal product is a non-signaling composite in the sense of Definition 2 above. By exploiting a result of Hanche-Olsen [12], one can show [6] that a Jordan-algebraic theory in which all composites are locally tomographic, and which contains at least one system having the structure of a qubit, must be a direct sum of finite-dimensional complex matrix algebras -that is, finite-dimensional complex QM with superselection rules. One should perhaps not rush to embrace local tomography as a universal principle, however. Indeed, the very fact that it excludes real and quaternionic quantum theory suggests that it is too strong: see [3] for some cogent reasons not to exclude these cases.
Several other recent papers (e.g, [8,9,13,14,16]) have derived standard finite-dimensional quantum mechanics, over C, from operational axioms. Besides the fact that the mathematical development here is much quicker and easier (modulo invocation of the KV theorem), the axiomatic basis is different, and arguably leaner. The papers cited in the introduction tend to impose strong constraints on "subspaces", along the lines of assuming that every face of the state space corresponds to the state space of a system satisfying the remaining axioms. A related assumption, also used in several of the cited papers, is that all systems characterized by the same "information-carrying capacity" are isomorphic. The present approach entirely avoids such assumptions. I also avoid the assumption, used in [14,18] that every element of V(A) * corresponds to a physically accessible measurement result. Finally, all of the cited papers assume some form of local tomography. In view of the comments above, it seems valuable to be able to delineate clearly what does and what does not depend on this assumption (particularly if we are interested in the possibilities for a "post-quantum" theory).
This brings us to the interesting question of whether all euclidean Jordan algebras actually satisfy the axioms discussed here. Let X denote the set of primitive (that is, minimal) idempotents in E, let M denote the set of Jordan frames (i.e., maximal sets of pairwise orthogonal idempotents), and let Ω denote the set of states on E (i.e., positive linear functionals α ∈ E * with α(u) = 1, where u is the unit element of E). Then A = (M, Ω) is a probabilistic model with E(A) = E. In particular, A is self-dual and sharp. Taking η A (a, b) = 1 r Tr(ab), where r is the rank of E, we have a perfectly correlating, nonsignaling bipartite state. Using the spectral theorem for Jordan algebras, plus the quadratic representation [10], one can show that every state of A can be prepared by a filter. Hence, every state is the marginal of a correlating bipartite state, as discussed in Section 4.
What isn't obvious is how to interpret the state η just defined. In orthodox QM, in fact, it isn't a state at all: rather, one needs to invoke the complex conjugate Hilbert space. What meaning one should attach to η A must depend on the choice of the conjugate system A, and on that of the composite system AA --and this, in turn, depends, not on the individual model A, but on the entire probabilistic theory at hand. So the question remains: can we embed arbitrary euclidean Jordan algebras in a probabilistic theory in which axioms 1, 2' and 3 are satisfied? Even assuming that this is possible mathematically, there is still a question of how to interpret the conjugate system A. In quantum theory, one can regard the conjugate Hilbert space H as representing a time-reversed version of the system represented by H. Whether some such interpretation can be maintained more generally remains to be addressed. Alternatively, one might view the existence of a conjugate as a way of formulating von Neumann's "projection postulate": if we take A to represent the system at one moment, and A, the system "immediately afterwards", then η A represents a state in which we expect that, whatever measurement is made, and whatever result is secured, if that same measurement were immediately repeated, the result would be the same. In any case, the meaning of the conjugate, and of the correlator η A , must certainly depend on the entire probabilistic theory. These matters will be taken up elsewhere.