Fermat's Library | q-2019-07-08-158 annotated/explained version.

Conjugates, Filters and Quantum Mechanics

Alexander Wilce

Department of Mathematics and Computer Science, Susquehanna University

July 2, 2019

The Jordan structure of ﬁnite-dimensional quantum theory is derived, in a conspic-

uously 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 conjugate system,

A, by means of a non-signaling bipartite state η

perfectly and uniformly correlating

each basic measurement on A with its counterpart on A. In the case of a quantum-

mechanical system associated with a complex Hilbert space H, the conjugate system

is that associated with the conjugate Hilbert space H, and η

corresponds to the stan-

dard maximally entangled EPR state on H ⊗ H. A second ingredient is the notion

of a 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 quan-

tum 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.)

1 Introduction and Overview

A number of recent papers, notably [11, 14, 20, 25, 28], have succeeded in deriving the mathemat-

ical apparatus of ﬁnite-dimensional quantum mechanics (henceforth: QM) from various packages

of broadly operational, probabilistic, or information-theoretic assumptions. These assumptions

are, however, rather strong, and the derivations themselves are not trivial. This paper aims at a

slightly broader target, and ﬁnds it much easier to hit.

Speciﬁcally, the Jordan structure of ﬁnite-dimensional quantum theory is derived, in a con-

spicuously easy way, from a few simple principles. This still brings us within hailing distance

of standard QM, owing to the classiﬁcation theorem for ﬁnite-dimensional formally real Jordan

algebras as direct sums of real, complex and quaternionic quantum systems, spin factors (“bits”

of arbitrary dimension), and the exceptional Jordan algebra [22]. In contrast, all of the cited

reconstructions make use of strong axioms that rule out real and quaternionic systems, and even

complex quantum systems with superselection rules, more or less by ﬁat. Since there are good

arguments for taking real and quaternionic quantum systems seriously (see [4] for a forceful ar-

gument in this direction), it is of interest to have an axiomatic scheme that accommodates them.

Alexander Wilce: wilce@susqu.edu

Accepted in Quantum 2019-06-24, click title to verify 1

arXiv:1206.2897v8 [quant-ph] 1 Jul 2019

I shall have more to say on this point below.

Correlation in quantum mechanics The approach taken here begins with a simple and well-

known observation about ﬁnite-dimensional quantum systems. Let H be an n-dimensional com-

plex Hilbert space

, representing a ﬁnite-dimenional quantum system. Recall that the conjugate

Hilbert space,

H, is the same abelian group, but endowed with the scalar multiplication (c, x) 7→ cx

(where the scalar multiplication on the right is that in H, and c is the complex conjugate of c ∈ C),

and with inner product (x, y) 7→ hy, xi. It is customary to write x for the vector x ∈ H, regarded

as a vector in H, so that cx = cx, or, equivalently, cx = c x. The inner product on H is then

given by hx, yi = hy, xi = hx, yi.

Suppose now that W is any density operator on H, with spectral decomposition W =

x∈E

for some orthonormal basis E, where p

is the rank-one projection associated with a

unit vector x ∈ E. Then W is the marginal, or reduced state, of the pure bipartite state

x∈E

1/2

x ⊗ x ∈ H ⊗ H (1)

The fact that mixed quantum-mechanical states arise in this way as marginals of pure states on

larger systems is the starting point for the reconstruction of QM in [11]. Here, we focus instead

on the correlational features of Ψ

. A straightforward calculation shows that if a, b are any two

operators on H, then

h(a ⊗ b)Ψ

, Ψ

i = Tr(W

1/2

b).

In particular, if a and b commute with W , then we have

h(a ⊗ b)Ψ

, Ψ

i = Tr(W ab). (2)

It follows that the state Ψ

perfectly correlates any projection-valued observable that commutes

with W , with its counterpart on H: if a and b are mutually orthogonal projections, both com-

muting with W , then the joint probability of observing a and b is hΨ

, a ⊗ bi = Tr(Wab) = 0,

while the joint probability of a and a is hΨ

, a ⊗ ai = Tr(W a). Where a = p

is the rank-one

projection associated with a unit vector x, this means that the conditional state of the conjugate

system, given a measurement result x on the system corresponding to H, is the “collapsed” state

corresponding x. In eﬀect, the entangled state Ψ

allows the conjugate system to retain a record

of the measurement result on the ﬁrst system — even though no signal need have passed between

the two.

A striking special case arises where W =

1, the maximally mixed state: in this case, Ψ

the “EPR” state

Ψ =

√

x∈E

x ⊗ x,

the expansion being independent of the choice of the orthonormal basis E. As every observable

commutes with W , Ψ perfectly, and uniformly, correlates every observable on H with its coun-

terpart on H. Thus, if we imagine that the system corresponding to H is controlled by Alice and

that corresponding to H, by Bob, then if Alice and Bob happen to make the same measurement,

A word on notation: I follow the mathematicians’ convention that a complex inner product h , i is conjugate-

linear in the second, rather than the ﬁrst argument. Thus, in terms of Dirac notation, hx, yi = hy|xi.

One can think of H as the space of bras hx| corresponding to the kets |xi ∈ H, but I prefer to avoid this

representation, since I want to stress the idea that H represents a quantum system in its own right. Thus, using

Dirac notation we might write |xi = hx|.

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they are bound to obtain the same result, with uniform probability 1/n. Notice, also, that by (2)

we have

h(a ⊗ b)Ψ, Ψi =

Tr(ab)

for all observables a and b, so the state Ψ in some sense explains the normalized trace inner product.

Correlation in General Probabilistic Theories These correlational features make sense in

a much more general setting. As explained in more detail below, a probabilistic model is char-

acterized by a set of basic measurements or experiments, and a convex set of states, with each

state α assigning a probability α(x) to every outcome x of every basic measurement. Given two

such models A and B, a bipartite state ω on A and B is an assignment of joint probabilities

ω(x, y) to all outcomes x and y of basic A- and B-measurements, respectively, having well deﬁned

conditional and marginal (reduced) probability weights corresponding to states of A and B.

We now impose some restrictions on the probabilistic models under consideration. First, we

require all state spaces to be ﬁnite-dimensional (we are, after all, only attempting to recover

ﬁnite-dimensional QM). Secondly, we require that models be uniform, in the sense that

(i) all basic measurements have a common, ﬁnite number of outcomes, n, called the rank of A;

and

(ii) there exists a maximally mixed state, ρ, deﬁned by ρ(x) = 1/n for all basic measurement

outcomes x

These conditions are satisﬁed by ﬁnite-dimensional quantum-mechanical models, including those

involving superselection sectors, provided that we restrict attention to maximal observables, i.e.,

those consisting of rank-one projections. More generally, condition (i) is reasonable if we think

of basic measurements as maximally informative, so that each has the largest possible number of

outcomes, and cannot be further reﬁned. Given condition (i), the maximally-mixed state is well-

deﬁned mathematically, so in (ii), we are only requiring that it count as a physically accessible

state.

The following is a direct translation of the correlational features discussed above for quantum-

mechanical systems, into the language of probabilistic models.

Deﬁnition 1. A conjugate of a (uniform) probabilistic model A is a model A, together with an

isomorphism γ taking each basic measurement outcome x of A to an outcome x := γ(x) of A,

such that

(a) Every state α of A is the marginal of some state ω on A and

A (in general, depending

on α) correlating some basic measurement E of A with its counterpart on A so that

for all x ∈ E,

ω(x, x) = α(x)

so that ω(x, y) = α(x)δ

x,y

(b) The maximally mixed state ρ arises as the marginal of a bipartite state η

uniformly

correlating every basic measurement with its counterpart, in the sense that

(x, x) =

for all basic measurement outcomes x, where n is the rank of A.

Evidently, in the quantum-mechanical case, where α corresponds to a density operator W ,

the state Ψ

supplies the correlating state ω, while the bipartite state η corresponds to the EPR

state Ψ

= Ψ.

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Mathematically, the existence of a conjugate system has aﬃnities with the puriﬁcation postu-

late of [11], though we do not require the correlating bipartite state ω above to be pure. Physically,

a conjugate system A allows for the formation of records of the outcomes of measurements on A in

causally separated systems, exactly as in the quantum case. Condition (a) above simply requires

that, for every state α, there be at least one basic measurement on A that can be thus recorded

and later “read oﬀ” by performing the corresponding measurement on

A. Condition (b) requires

that, where A is in the maximally mixed state, it be possible to record every basic measurement

in this way.

From correlation to Jordan algebras Remarkably little is required, beyond the existence of a

conjugate, to secure a representation of A in terms of a formally real Jordan algebra. This depends

on a classic mathematical result, the Koecher-Vinberg Theorem [18]. A ﬁnite-dimensional ordered

vector space E with positive cone E

is self-dual if it carries an inner product such that a ∈ E

iﬀ ha, bi ≥ 0 for all b ∈ E

. If the group of invertible linear mappings E → E carrying E

onto

itself acts transitively on the interior of the cone E

, then E is said to be homogeneous. The

Koecher-Vinberg Theorem asserts that if E is both homogeneous and self-dual, it can be endowed

with a formally real Jordan structure for which E

= {a

|a ∈ E}.

Any probabilistic model A gives rise in a natural way to two ordered vector spaces: a space

V(A), generated by A’s states, and a space E(A) ≤ V(A)

∗

generated by evaluation functionals

x : α 7→ α(x) associated with basic measurement outcomes x. Since we are assuming that the state

space is ﬁnite dimensional, both of the spaces E(A) and V(A) are also ﬁnite dimensional, and it

is easy to see that they have the same dimension. If we can show that E(A) is homogeneous and

self-dual, then the Koecher-Vinberg Theorem will provide a formally real (equivalently, euclidean)

Jordan structure on E(A) for which the cone of squares coincides with E(A)

Call a probabilistic model A sharp if, for every basic measurement outcome x, there is a unique

state δ

with δ

(x) = 1. Physically, this is a way of saying that basic measurements are maximally

informative: if we can predict the outcome with certainty, we know the system’s state exactly.

Theorem 1. Suppose A is sharp and has a conjugate. Then the state η

gives rise to a self-

dualizing inner product on E(A), with respect to which E(A) and V(A) are isomorphic as ordered

vector spaces.

It follows that if V(A) is homogeneous, so is E(A), whence, by the Koecher-Vinberg Theorem,

the latter carries a formally real Jordan structure. But the homogeneity of V(A) has a direct

physical interpretation: it says that for every non-singular state — that is, every state α with

α(x) > 0 for every basic measurement outcome — there exists a probabilistically reversible process

T — deﬁned below, but, roughly, one that can be reversed by another process with non-zero

probability — such that T (ρ) = rα where r ∈ [0, 1]. In other words, every non-singular state

can be prepared, up to normalization, by applying a probabilistically reversible process to the

maximally mixed state.

In fact, it is enough to assume less. By a ﬁlter for a basic measurement with outcome-set E, I

mean a process Φ — that is, a positive linear mapping Φ : V(A) → V(A) — that independently

attenuates the reliability of each outcome x ∈ E, so that for every state α, Φ(α)(x) = t

α(x) for

some constant t

(independent of α). If we think of basic measurement outcomes as detectors,

the existence of such ﬁlters, with arbitrary coeﬃcients, is plausible; in standard QM, not only do

they exist but, if t

> 0 for every x ∈ E, then Φ can be chosen to be probabilistically reversible.

Where a probabilistic model A shares this feature, I will say that A has arbitrary reversible ﬁlters.

Corollary 1. Suppose that A is sharp, has a conjugate, and has arbitrary reversible ﬁlters. Then

E(A) is homogeneous and self-dual.

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If we adopt a stronger assumption about ﬁlters, we can weaken the requirement that A have

a conjugate, and eliminate entirely the hypothesis that A is sharp. Let us say that A has a weak

conjugate if the maximally mixed state ρ is the marginal of a uniformly correlating state η

on A

and A, as in condition (b) in Deﬁnition 1, but not assuming that every state is the marginal of a

correlating state, i.e., not assuming condition (a). That is, we require only that there exist a joint

state on two copies of A — an analogue of the EPR state — in which, if the same measurement is

performed on each copy, the results are guaranteed to be the same, but are otherwise completely

random.

By applying the ﬁlter Φ to the system A, and then computing the canonical bipartite state

, we obtain a new bipartite state. Equally, we could begin by applying the counterpart of Φ to

the conjugate system, obtaining another bipartite state. If these two bipartite states are in fact

the same, we say that Φ is symmetric.

Corollary 2. Let A have a weak conjugate. If every non-singular state of A can be prepared, up

to normalization, from the maximally mixed state by a symmetric, reversible ﬁlter, then E(A) is

homogeneous and self-dual.

The proofs of these results are all quite short and straightforward. In summary, we recover

euclidean Jordan algebras from either of two distinct, but related, sets of assumptions about

physical systems represented by uniform, ﬁnite-dimensional probabilistic models:

(a) systems are sharp;

(b) Systems have conjugates; and

Alternatively, and more compactly:

) Systems have weak conjugates;

) All non-singular states can be prepared by reversible symmetric ﬁlters

Any euclidean Jordan algebra gives rise to a probabilistic model in which basic measurements

correspond to Jordan frames, i.e., sets {e

} of minimal idempotents satisfying e

= 0 for i 6= j,

and

= 1, where 1 is the Jordan unit. (In standard QM, these would correspond to maximally

ﬁne-grained projective measurements.) In Appendix A, it is shown that any such model satisﬁes

all of the assumptions above and, conversely, if A satisﬁes either package of assumptions, the

Jordan product on E(A) can be chosen so that the set of basic measurements is precisely the

set of Jordan frames. Thus, these two sets of assumptions are in fact equivalent, and exactly

characterize this class of euclidean Jordan-algebraic probabilistic models.

There is actually a third possibility, in which condition (b) in the deﬁnition of a conjugate is

replaced by a rather weak symmetry assumption and a version of Hardy’s subspace axiom [20].

Again, the resulting package of assumptions is satisﬁed by, and hence, characterizes, Jordan mod-

els. The details are spelled out in Appendix B.

Other reconstructions of QM The approach of this paper oﬀers some signiﬁcant advantages

over the reconstructions of quantum mechanics cited earlier. First, it is simply easier, in the sense

that our results are obtained with less mathematical eﬀort. (This, notwithstanding the length

Accepted in Quantum 2019-06-24, click title to verify 5

of this paper, which owes to the inclusion of many details intended to make the paper easier to

follow.)

Secondly, it rests on fewer and (arguably) simpler assumptions. Certainly, the second package

of asumptions (that is, (b

) and (c

) above) is smaller than anything found in earlier reconstructions

of QM. Other reconstructions tend to impose strong constraints on subsystems, in eﬀect assuming

that every face of the state space corresponds to the state space of a “sub-system”, satisfying the

remaining axioms. Nothing like this is needed here (though, as mentioned above, if one ﬁnds a such

a “subspace axiom” compelling, it can be put to good use in the present approach; see Appendix

B for details). A related assumption, also used in several of the cited papers, is that all systems

having the same “information capacity” — the maximal number of states sharply distinguishable

by single-shot measurement — are isomorphic. The present approach entirely avoids such an

assumption. It also does without the assumption, commonly called the no-restriction hypothesis

[21], used in [25] for bits, that all mathematically possible eﬀects — that is, aﬃne functionals

assigning probabilities to states — correspond to physically accessible measurement results. More

recently, the interesting paper [7] derives the same Jordan-algebraic structure arrived at here, but

in a diﬀerent way. In addition to a strong symmetry postulate, this paper assumes a weak form of

the no restriction hypothesis, namely, that all ﬁnite sets of “allowed” eﬀects that sum to the unit

eﬀect (the eﬀect identically 1 on all states) correspond to accessible measurements, along with a

kind of spectral decomposition for states. Here, we manage without any form of no-restriction

assumption, and a spectral decomposition for states is derived, rather than postulated.

Finally, all of the earlier reconstructions of QM cited above assume some form of local to-

mography. This is the doctrine that the state of a bipartite system is determined by the joint

probabilities it assigns to outcomes of measurements on the two component systems. This prin-

ciple has a certain intuitive appeal; moreover, it is well known, and easy to see on dimensional

grounds, that among ﬁnite-dimensional real, complex and quaternionic quantum mechanics, only

in the complex version are composites locally tomographic.

More generally [19, 8], the only probabilistic theory in which systems correspond to Jordan

models and composite systems are locally tomographic, and which includes at least one system

having the structure of a qubit, is ﬁnite-dimensional complex quantum mechanics. Thus, if one in-

sists on local tomography, it can be added to the list of assumptions discussed above, and leads to

standard, complex QM (with superselection rules). One should perhaps not rush to embrace local

tomography as a universal principle, however. The very fact that it excludes real and quaternionic

quantum theory suggests that it is too strong. There are natural ways of representing complex

Hilbert spaces in terms of real or quaternionic ones, and vice versa

; moreover, these representa-

tions have physical meaning, in that bosonic or fermionic (complex) quantum systems can very

naturally be modelled in terms of the corresponding real or, respectively, quaternionic Hilbert

spaces. Again, see, e.g., [4] a cogent development of this line of thought. In any case, 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.

And also notwithstanding my appeal to the Koecher-Vinberg Theorem, as this is itself a very accessible result.

See [18] for a not terribly taxing proof. A number of the other reconstructions mentioned here also depend on

nontrivial mathematical results, e.g., the classiﬁcation of transitive actions of compact groups on spheres is used in

[25].

A real or quaternionic Hilbert space can be regarded as a complex Hilbert space equipped with a designated anti-

unitary operator J satisfying, respectively, J

= 1 or J

= −1; conversely, a complex Hilbert space is essentially

equivalent to a real or quaternionic one equipped with, respectively, an orthogonal or a simplectic operator J

satisfying J

= −1 or J

= 1.

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Organization The balance of this paper is arranged as follows. Section 2 provides general

background on probabilistic models, ordered vector spaces, Jordan algebas and so on, making

more precise many of the technical terms used above. This material will be familiar to many, but

probably not to all, readers. Section 3 contains the proof of Theorem 1; Corollaries 1 and 2 are

proved in Section 4. Section 5 collects some ﬁnal thoughts, inlcuding a few further remarks on

how the approach of this paper compares to the reconstructions of QM cited above. Appendix A

contains additional information on probabilistic models associated with euclidean Jordan algebras,

and Appendix B shows how a version of the “subspace axiom”, plus a symmetry assumption, can

replace some of the assumptions native to this paper.

Several of the ideas developed here were earlier explored, and somewhat similar results derived,

in [30] and [31], but the approach taken here is much simpler and more direct, and seems to go a

good deal farther.

2 Background

The mathematical framework for this paper is that of “generalized probabilistic theories” [10], in

the idiom of [30, 9], which I now quickly review.

In a few places, set oﬀ in numbered deﬁnitions,

my usage diﬀers slightly from that of these last-cited works. See [18, 2] for more information on

ordered vector spaces and Jordan algebras.

Ordered vector spaces An ordered vector space is a real vector space E equipped with a closed,

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 non-negative

scalars, given by a ≤ b iﬀ b − a ∈ E

. As an illustration, the space R

of real-valued functions on

a set X is ordered by the cone R

of functions taking non-negative values. Another example is

the space L

(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 is positive iﬀ T (E

) ⊆ F

. If

T is bijective and T

−1

is also positive, then T is an order isomorphism. If E and F are ﬁnite

dimensional with dim(E) = dim(F ), T is an order isomorphism iﬀ T (E

) = F

. The dual space

∗

of a ﬁnite-dimensional ordered linear space carries a natural ordering, deﬁned by the dual

cone, E

∗

, consisting of positive linear functionals f ∈ E

∗

Probabilistic Models As discussed above, a probabilistic model is characterized by a set M(A)

of basic measurements or tests, and a set Ω(A) of states. It is convenient to identify each test

with its outcome-set, so that M(A) is simply a collection of non-empty sets (a test space, in the

language of [9]). Let X(A) stand for the union of this collection; that is, X(A) is the space of

all outcomes of all basic measurements. States are understood as assignments of probabilities to

measurement-outcomes, that is, as functions α : X(A) → [0, 1] such that

x∈E

α(x) = 1 for all

tests E ∈ M(A) (but not all such functions necessarily correspond to states). As mentioned above,

a state α ∈ Ω(A) is non-singular iﬀ α(x) > 0 for all x ∈ X(A). To reﬂect the possibility of forming

statistical mixtures, I also assume that Ω(A) is convex, that is, if p

, ..., p

are non-negative real

numbers summing to 1, and α

, ..., α

are states in Ω(A), then the function p

+ · · · + p

also

This is a variant of the standard “convex-operational” framework developed in the 1960s and 1970s by Ludwig,

Davies and Lewis and others (e.g., [24, 15, 16]), specialized to ﬁnite dimensions, and with additional structure

deriving from work of Foulis and Randall [17].

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belongs to Ω(A). Finally, I assume that Ω(A) is closed under pointwise limits, whence, compact

as a subset of [0, 1]

X(A)

in the product topology.

By way of illustration, in the simplest 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) = hW

x, xi for some

density operator W

, 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 measure. 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 privileged (or perhaps, simply convenient)

catalogue of such measurements, suﬃciently large to determine the system’s states.

The spaces V(A) and E(A). Any probabilistic model A gives rise in a canonical way to an

ordered vector space V(A). This 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; that is, β ∈ V(A)

iﬀ β = tα

for some state α ∈ Ω(A) and some real constant t ≥ 0. An element of V(A)

of the form tα

with α ∈ Ω(A) and t ≤ 1 is said to be sub-normalized. One can show that the interior of V(A)

consists exactly of multiples of non-singular states. The dimension of a model A is the dimension

of V(A). As mentioned in the introduction, it is assumed in this paper that all probabilistic models

are ﬁnite-dimensional.

There is a canonical positive linear functional u

: V(A) → R, called the unit eﬀect of A, given

by u

(α) =

x∈E

α(x), where E is any test in M(A). Note that if α ∈ V(A)

, then u

(α) = 1

precisely when α ∈ Ω(A), and that every non-zero element α ∈ V(A)

has the form β = tα for

a unique t = u

(β) > 0. Thus, any non-zero β ∈ V(A)

can be normalized to yield a state

β = u

(β)

−1

β ∈ Ω(A). A positive linear functional f ∈ V(A)

∗

with f ≤ u

is usually called

an eﬀect on A, and can be thought of as representing an “in-principle” measurement outcome,

with probability 0 ≤ f(α) ≤ 1 in state α. Every outcome x ∈ X(A) corresponds to an eﬀect

x : V(A) → R, given by

x(α) = α(x) for all α ∈ V(A), and u

x∈E

If A = A(H) is the quantum mechanical model associated with a ﬁnite-dimensional Hilbert

space H, as discussed above, then, identifying Ω(H) with the convex set of density operators on

H, V(A) can be identiﬁed with the ordered vector space L(H) of self-adjoint operators on H,

with its usual cone of positive operators. We can then also identify V(H)

∗

with L(H), using the

trace inner product. That is, if a ∈ L(H), we can deﬁne a positive linear functional a ∈ V(A)

∗

by setting a(α) = Tr(aα) for all α ∈ V(A) = L(H), and all such functionals arise uniquely from

elements of L(H). In this setting, measurement-outcomes are unit vectors of H, and, for each

x ∈ X(H), and for each density operator α on V(A),

x(α) = Tr(αp

) where p

is the rank-one

projection associated with x. More generally, eﬀects correspond to positive operators between 0

and 1.

The spectral theorem for self-adjoint operators tells us that every eﬀect in V(A(H)) = L(H)

is a positive linear combination of functionals

x corresponding to measurement outcomes. This

will not be the case for probabilistic models in general. It is therefore useful to deﬁne a smaller

cone, as follows:

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Deﬁnition 2. The space E(A) is the span, in V(A)

∗

, of the set of eﬀects

x associated with

outcomes x ∈ X(A), ordered by the cone E(A)

of ﬁnite linear combinations

, x

∈ X(A),

with coeﬃcients t

≥ 0.

Since we are assuming that V(A) and (hence) V(A)

∗

are ﬁnite-dimensional, the set of func-

tionals

x in fact spans V(A)

∗

(since they separate points of V(A).) Thus, E(A) and V(A)

∗

are

identical as vector spaces. However, their cones are generally quite diﬀerent. If a ∈ E(A)

, then

a(α) ≥ 0 for all α ∈ V(A)

, so the cone E(A)

is contained in the dual cone V(A)

∗

, but the

inclusion is usually proper. Thus, E(A) and V(A) are generally distinct as ordered vector spaces.

Remark: The space E(A) will be a useful technical tool in what follows, but should not necessarily

be regarded as anything more than that. In particular, it is not assumed that all physically mean-

ingful eﬀects reside in E(A)

, nor that every eﬀect in E(A)

is physically meaningful. In fact, it

will not be necessary to take any position at all on which eﬀects, other than those associated with

measurement outcomes, are physically signiﬁcant. Thus, as mentioned in the introduction, we

avoid the so-called no-restriction hypothesis [21], namely, the asumption that all eﬀects in V(A)

∗

are physically accessible. This assumption is often made in the literature, sometimes explicitly

(e.g., [25]), sometimes not.

Processes A physical process on a system represented by a probabilistic model A is naturally

represented by an aﬃne (that is, 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

) ≤ 1

for all α ∈ Ω(A).

Deﬁnition 3. A process T : V(A) → V(A) is probabilistically reversible — hereafter, just p-

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-automorphism

of V(A).

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

. T is lossless iﬀ T

∗

) = 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 T

∗

will preserve the dual cone V(A)

∗

, it is not required, a priori, 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.

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

and E(A) ' (R

)

∗

, with

the standard inner product on R

providing the order-isomorphism. If H is a ﬁnite-dimensional

To be clear, we are not suggesting that all such positive mappings on V(A) represent physically allowable

processes. Indeed, in QM, only completely positive mappings are physically allowable.

It should be noted that my usage is slightly nonstandard here: ordinarily, the adjective reversible is reserved

for processes that are probabilistically reversible, in the above sense, with probability one.

Accepted in Quantum 2019-06-24, click title to verify 9

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 the space L

(H)

of self-adjoint operators on H, ordered by the cone of positive operators. For any x ∈ X(H),

the evaluation functional

x ∈ V(A) is then given by W 7→ hW x, xi = Tr(W P

). It follows that

E(A(H)) ' L

(H)

∗

' L

(H), with the latter isomorphism implemented by the trace inner

product.

More generally, call an inner product h , i on an ordered vector space E positive iﬀ ha, bi ≥ 0

for all a, b ∈ E

. We then have a positive linear mapping E → E

∗

, namely a 7→ ha, ·i. If this

is an order-isomorphism, one says that E is self-dual with respect to this inner product. This is

equivalent to the condition a ∈ E

iﬀ ha, bi ≥ 0 for all b ∈ E

. In this language, the standard

inner product on R

and the trace inner product on L

(H) are self-dualizing, for any ﬁnite set

E and ﬁnite-dimensional Hilbert space H.

In fact, any euclidean Jordan algebra, ordered by its cone of squares, is self-dual with respect

to its canonical inner product. Recall here that a Jordan algebra is a real commutative (but not

necessarily associative) unital algebra (J ,

) satisfying the Jordan identity

a · (a

b) = a

for all a, b ∈ J (with a

= a

a). A euclidean Jordan algebra (EJA) is a ﬁnite-dimensional Jordan

algebra J equipped with an inner product h , i such that ha

b, ci = hb, a

ci for all a, b, c ∈ J . This

is equivalent to the condition that J be formally real, i.e, that

i=1

= 0 implies a

= 0 for all i.

A EJA J is also an ordered vector space with positive cone J

= {a

|a ∈ E}, and it can be shown

that this cone is self-dual with respect to the given inner product [18]. Examples of euclidean

Jordan algebras include the space L

(H) of self-adjoint operators on a ﬁnite-dimensional real,

complex or quaternionic Hilbert space H, with a · b =

(ab + ba), and with ha|bi = Tr(ab).

The exceptional Jordan algebra of self-adjoint hermitian matrices over the Octonions is also a

euclidean Jordan algebra. Finally, one obtains a euclidean Jordan algebra, called a spin factor, by

deﬁning on V

:= R × R

a product (t, x) · (s, y) = (ts + hx, yi, ty + sx). This essentially exhausts

the possibilities: according to the Jordan-von Neumann-Wigner classiﬁcation theorem [22], every

euclidean Jordan algebra is a direct sum of euclidean Jordan algebras of these ﬁve types.

We can associate a probabilistic model to an EJA J in the following way. An idempotent in

J is an element e = e

. An idempotent is minimal, or primitive, iﬀ for any idempotent f ≤ e,

f = 0 or f = e. Two idempotents e, f are Jordan-orthogonal iﬀ e

f = 0. A maximal pairwise

Jordan-orthogonal set {e

, ..., e

} of primitive idempotents summing to the Jordan unit is called

a Jordan frame.

Deﬁnition 4. The Jordan model A(J ) = (X(J ), M(J ), Ω(J )) corresponding to a euclidean

Jordan algebra J has X(J ) the set of primitive idempotents, M(J ), the set of Jordan frames of

J , and Ω(J ) the set of states of the form α(x) := ha, xi where a ∈ E(A)

satisﬁes ha, 1i = 1. The

spectral theorem for EJAs ([18], Theorem III.1.2) expresses every a ∈ J in the form a =

x∈E

where E is a Jordan frame. Therefore, J = E(A), and the model A(J ) is self-dual.

Besides self-duality, all euclidean Jordan algebras share a property called homogeneity: the

group of order-automorphisms of J acts transitively on the interior of the positive cone J

The Koecher-Vinberg Theorem [18] states that, conversely, any ﬁnite-dimensional homogeneous,

self-dual, homogenous ordered vector space J can be equipped with the structure of a euclidean

Jordan algebra.

Deﬁnition 5. A probabilistic model A is homogeneous iﬀ V(A) is homogeneous, and self-dual iﬀ

E(A) carries an inner product with respect to which it is self-dual and E(A)

' V(A)

, in the

Accepted in Quantum 2019-06-24, click title to verify 10

sense that a ∈ E(A)

iﬀ α(x) := ha,

xi deﬁnes an element of V(A)

, and every element of V(A)

arises in this way.

If A is both homogeneous and self-dual — henceforth, HSD — then E(A) is also homogeneous

and self-dual, and thus, by the Koecher-Vinberg Theorem, can be made into a Jordan algebra.

In Appendix A, it is shown that this can be done in such a way that A is actually isomorphic to

the Jordan model corresponding to E(A).

Bipartite States and Conditioning A joint probability weight on a pair of models A and B is

a mapping ω : X(A) × X(B) → R such that, for all E ∈ M(A) and F ∈ M(B),

(x,y)∈E×F

ω(x, y) = 1.

Such a weight is said to be non-signaling if, in addition, the marginal weights

(x) :=

y∈F

ω(x, y) and ω

(y) :=

x∈E

ω(x, y)

are well-deﬁned, i.e., independent of the choice of tests F ∈ M(B) and E ∈ M(A), respectively.

The idea is that such a state precludes the sending of signals between A and B based solely on

the choice of what test to perform.

If ω is non-signalling, then given outcomes y ∈ X(B) and x ∈ X(A), we can deﬁne conditional

probability weights ω

1|y

and ω

2|x

on A and B, respectively, by setting

1|y

(x) =

ω(x, y)

(y)

and ω

2|x

(y) :=

ω(x, y)

(x)

when ω

(y) and ω

(x) are non-zero. This gives us the following bipartite law of total probability

[17]

x∈E

(x)ω

2|x

and ω

y∈F

(y)ω

1|y

(3)

which will be exploited below.

Deﬁnition 6. Let ω be a non-signaling joint probability weight on A and B. If all conditional

weights ω

1|x

and ω

2|y

(and hence, the marginals ω

and ω

) of ω belong to Ω(A) and Ω(B),

respectively, then we say that ω is a bipartite state on the models A and B.

If H and K are real or complex Hilbert spaces, every density operator W on H ⊗ K gives rise

to a bipartite state on A(H) and A(K), given by ω(x, y) = hW x ⊗ y, x ⊗ yi.

The conditioning map If ω is a bipartite state on A and B, deﬁne the associated conditioning

maps

ω : X(A) → V(B) and

∗

: X(B) → V(A) by

ω(x)(y) = ω(x, y) =

∗

(y)(x).

Note that

ω(x) = ω

(x)ω

2|x

for every x ∈ X(A), i.e.,

ω(x) can be understood as the un-normalized

conditional state of B given the outcome x on A, and similarly for

∗

(y).

The conditioning map

ω extends uniquely to a positive linear mapping E(A) → V(B), which

I also denote by

ω, such that

ω(

x) =

ω(x) for all outcomes x ∈ X(A). (To see this, consider

the positive linear mapping T : V(A)

∗

→ R

X(B)

deﬁned, for f ∈ V(A)

∗

, by T (f )(y) = f(

∗

(y))

Accepted in Quantum 2019-06-24, click title to verify 11

for all y ∈ X(B). If f =

x, we have T (

x) = ω

(x)ω

2|x

∈ V(B)

. Thus, the range of T lies in

V(B).) In the same way,

∗

deﬁnes a positive linear mapping

∗

: E(B) → V(A). Notice that

ω need not take V(A)

∗

into V(B)

. This is the principal reason for working with E(A) rather

than V(A)

∗

. If

ω : E(A) → V(A) is an order isomorphism, ω is said to be an isomorphism state [5].

Composite Systems As the language here suggests, one wants to view (some) bipartite states

as elements of the state space of a composite model. Broadly, a composite of two probabilistic

models A and B, is a model AB equipped with a mapping X(A) × X(B) → V(AB)

∗

, taking

every pair of outcomes x ∈ X(A) and y ∈ X(B), to a product eﬀect xy, such that every state

ω ∈ Ω(AB) pulls back to a bipartite state ω(x, y) := ω(xy) on A and B. While nothing in the

mathematical development to follow depends on the choice of such a composite model, questions

of interpretation may hinge on such a choice.

3 Conjugate Systems

Let H be an n-dimensional complex Hilbert space and H is its conjugate space. As discussed

in the introduction, the maximally engangled “EPR” state, deﬁned by Ψ =

√

x∈E

x ⊗ x,

where E is any orthonormal basis for H, establishes a perfect, uniform correlation betweeen

every projection-valued observable on the system associated with H, and its counterpart on H.

Moreover, Ψ eﬀectively deﬁnes the normalized trace inner product on E(H) = L

(H). Since it

is precisely this inner product that makes L

(H) self-dual, one might guess that the existence of

a uniformly correlating bipartite state is implicated in self-duality more generally.

As a ﬁrst step, we need to generalize the relationship between the models A(H) and A(H). In

order to do this, as mentioned in the introduction, we need ﬁrst to impose the minor restriction

that, henceforth, all models are uniform, meaning that all tests have a common cardinality n, and

that the maximally mixed state ρ, given by ρ(x) = 1/n for all x ∈ X(A), belongs to Ω(A). An iso-

morphism between two models A and B is a bijection φ : X(A) → X(B) such that φ(E) ∈ M(B)

iﬀ E ∈ M(A), and β ◦ φ ∈ Ω(A) iﬀ β ∈ Ω(B). It is straightforward that such a mapping gives

rise to an order isomorphism — which I’ll also denote by φ — from E(A) to E(B), deﬁned by

φ(

x) =

φ(x) for all x ∈ X(A). The following reprises, and makes more precise, the deﬁnition of a

conjugate system (Deﬁnition 1).

Deﬁnition 1 (bis) Let A be a uniform probabilistic model of rank n. A conjugate for A is a triple

(A, γ

, η

) consisting of a probabilistic model A, an isomorphism γ

: A ' A, and a bipartite

state η

on A and A such that

(a) η

(x, γ

(x)) = 1/n for every x ∈ X(A).

(b) Every state α ∈ Ω(A) is the marginal of some bipartite state ω on A and A that correlates

some test E ∈ M(A) with its counterpart on A, so that ω(x, γ(x)) = α(x) for every x ∈ E.

As remarked earlier, the marginals of the perfectly correlating state η in part (a) are the maxi-

mally mixed states ρ on A and ρ on A and A, respectively. Where no ambiguity is likely, I write

x for γ

(x). If A satisﬁes (a), but not necessarily (b), I will call it a weak conjugate for A.

Given any bipartite state on A and A satisfying (a) i.e., η(x, γ

(x)) = 1/n for all x ∈ X(A),

the bipartite state η

deﬁned by η

(x, γ(y)) = η(y, γ(x)) is also perfectly uniformly correlating,

Accepted in Quantum 2019-06-24, click title to verify 12

whence, so is the symmetic state (η + η

)/2. We therefore can, and do, assume in what follows

that the chosen correlating state η

is always symmetric. If A is sharp, it is easy to show that

η is uniquely determined by condition (a) of Deﬁnition 1: since η(x, y) = 0 for outcomes y 6= x

belonging to a common test, and η(y, y) = 1/n, we have η

1|y

(y) = 1, i.e,. η

1|y

= δ

where δ

is the

unique state in which y has probability 1. Thus, η(x, y) = δ

(x). In this case, therefore, η = η

i.e., η is automatically symmetric.

If A(H) is the quantum probabilistic model associated with a ﬁnite-dimensional Hilbert space,

then the EPR state Ψ turns A(H) := A(H) into a conjugate in this sense, with η

(x, y) =

|hΨ, x ⊗ yi|

. In fact, as pointed out in the introduction, A(H) is a conjugate for A(H), since

every density operator is the marginal of a state Ψ

correlating an eigenbasis for W with its

conjugate. All of this works equally well for real quantum systems, taking H = H. With a little

care, it can be shown to work for quaternionic systems as well.

Remark: One might wonder whether one can use the isomorphism γ

to identify A with its con-

jugate. Certainly one can deﬁne a bipartite state η

(x, y) = η

(x, γ

(y)). However, whether

this corresponds to a legitimate state of any physically reasonable composite AA of A with it-

self, depends on the particular probabilistic theory at hand. For example, if A = A(H) is the

quantum model associated with a complex Hilbert space H, then (a, b) 7→ Tr(ab) corresponds

to no state on A(H ⊗ H). On the other hand, any choice of an anti-unitary operator J acting

on H yields a unitary isomorphism J : H → H, given by J(x) = J(x) for all x ∈ H. Deﬁning

(x, y) := |h(1 ⊗ J)

−1

Ψ, x ⊗ yi|

= |hΨ, x ⊗ Jyi|

gives a state on A(H ⊗ H), correlating along

the anti-unitary isomorphism γ

(x) := J

−1

x. Thus, whether we choose to treat A(H) as its own

conjugate, or as distinct from its conjugate, is to an extent a matter of convention.

Conjugates and Self-Duality We are now ready to prove Theorem 1, which, for convenience,

I restate. Recall that a model A is sharp iﬀ, for every outcome x ∈ X(A), there is a unique state

∈ Ω(A) with δ

(x) = 1. Both classical and quantum models are sharp.

Theorem 1 (bis) Let A be sharp and have a conjugate A. Then ha, bi := η

(a, γ

(b)) is a

self-dualizing inner product on E(A), and induces an order-isomorphism E(A) → V(A) given by

a 7→ η

(a, · ) for a ∈ E(A).

The proof is not diﬃcult. It will be convenient to break it up into a sequence of even easier

lemmas. In the interest of readability, below I conﬂate x ∈ X(A) with the corresponding eﬀect

x ∈ E(A), and write x for γ

(x). Until further notice, the hypotheses of Theorem 1 are in force.

The ﬁrst step is to obtain a kind of weak “spectral” decomposition for states in Ω(A) in terms of

the states δ

Lemma 1. For every α ∈ V(A)

, there exists a test E such that

α =

x∈E

α(x)δ

(4)

Proof: We can assume that α is a normalized state, i.e, that α ∈ Ω(A). Since A is a conjugate

for A, α = ω

where ω correlates some test E ∈ M(A) with E ∈ M(A) along the bijection

It follows that, where δ

and δ

are the unique states making x and y certain, we have δ

(x) = δ

(y) for all

x, y ∈ X(A). Indeed, this last condition could substitute for condition (a) in the deﬁnition of η, as it implies that

η(x, y) := δ

(y) is a valid non-signaling probabilty weight on A and A.

Here, I am identifying V(A), as a vector space, with E(A)

∗

Accepted in Quantum 2019-06-24, click title to verify 13

x 7→ x. By the law of total probability (1) for non-signaling states, α =

x∈E

(x)ω

1|x

. Since ω

is correlating we have ω

1|x

(x) = 1. Thus, by sharpness, ω

1|x

= δ

. Hence, α =

x∈E

(x)δ

. It

follows that ω

(x) = α(x) for every x ∈ E, giving us (4). 

We will refer to the decomposition in equation (4) as a spectral decomposition for α.

Lemma 2. η

is an isomorphism state.

Proof: We need to show that

: E(A) → V(A) is an order-isomorphism. Since E(A) and V(A)

have the same dimension, it is enough to show that

maps the positive cone of E(A) onto that

of V(A). Since x 7→ x is an isomorphism between A and A, we can apply Lemma 1 to A: if

α ∈ V

(A), we have α =

x∈E

α(x)δ

. Since η

(x, x) = 1/n, we have

(x) =

for every

x ∈ X(A). Hence,

(

x∈E

nα(x)x) = α. 

Lemma 3. Every a ∈ E(A) has a representation a =

x∈E

x for some test E ∈ M(A) and

some coeﬃcients t

Proof: If a ∈ E(A)

, then by Lemma 1,

(a) =

x∈E

for some E ∈ M(A) and coeﬃcients

≥ 0. By Lemma 2,

is an order-isomorphism. Applying

−1

to the expansion above gives

a =

x∈E

x. For an arbitrary a ∈ E(A), we have a = a

− a

with a

, a

∈ E(A)

. Choose

N ≥ 0 with a

≤ Nu. Thus, b := a + Nu

= a

+ (Nu

− a

) ≥ 0. So b :=

x∈E

x for some

E ∈ M(A) and thus

a = b − Nu

x∈E

x − N(

x∈E

x) =

x∈E

− N )x. 

Lemma 4. The function ha, bi := η

(a, γ

(b)) is an inner product on E(A).

Proof: η

is bilinear and, by assumption, symmetric. We need to show that h , i is positive

deﬁnite. Let a ∈ E(A). From Lemma 3, we have a =

x∈E

x for some test E and coeﬀcients

. Now

ha, ai =

x∈E

y∈E

x,y∈E×E

hx, yi =

x,y∈E×E

(x, y) =

x∈E

≥ 0.

This is zero only when all coeﬃcients t

are zero, i.e., only for a = 0. 

Proof of Theorem 1, concluded: Lemma 2 tells us that E(A) ' V(A), so it remains only to show

that the inner product h , i is self-dualizing. Clearly ha, bi = η(a, b) ≥ 0 for all a, b ∈ E(A)

Suppose a ∈ E(A) is such that ha, bi ≥ 0 for all b ∈ E(A)

. Then ha, yi ≥ 0 for all y ∈ X. By

Lemma 3, a =

x∈E

x for some test E. Thus, for all y ∈ E we have ha, yi = t

≥ 0, whence,

a ∈ E(A)

. 

Remarks There are several directions in which we can usefully modify the assumptions of Theorem

(1) In the proof of Theorem 1, the only point at which we needed to assume that A satisﬁes

condition (b) in the deﬁnition of a conjugate was in order to obtain the spectral decomposition

— equation (4) — of Lemma 1. Thus, if we are willing simply to assume such decompositions

are available, as in [7], then a weak conjugate suﬃces. Alternatively, any postulate or postulates

leading to such decompositions can replace condition (b). For instance, certain versions of the

symmetry and “subspace” axioms used in [20, 14, 25] imply a spectral decomposition. This is

Accepted in Quantum 2019-06-24, click title to verify 14

spelled out in Appendix B. Another approach to obtaining such a decomposition can be found in

a recent paper of G. Chiribella and C. M. Scandolo [12].

(2) In fact, it is even enough if (4) holds for states in the interior of Ω(A). From this we have,

as in the proof of Lemma 2, that the interior, V

◦

, of the cone V(A)

is contained in

from which it follows that

η is a linear isomorphism, and hence (as the vector spaces involved are

ﬁnite-dimensional) an homeomorphism. Thus,

) is closed, and so, contains the closure of

◦

, i.e., V

. In other words,

η is an order-isomorphism. The proofs of Lemmas 3 and 4, and the

rest of the proof of Theorem 1, then proceed just as before.

(3) The deﬁnition of a conjugate for a probabilistic model A requires the existence of the

uniformly, universally correlating state η

, and that arbitrary states of A arise as marginals of

bipartite states on A and A correlating some test E ∈ M(A) with its conjugate twin. One might

wonder whether there is some reasonably simple postulate that will imply both of these condi-

tions. Suppose that 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 averaging; by the transitivity of G on outcomes, this state must be constant. It

follows that all tests have the same ﬁnite size size, say n, and that ρ is the maximally mixed state

ρ(x) ≡ 1/n. That is, the model is uniform. Now let γ

: x 7→ x be an isomorphism between A and

a model A. Suppose that every state α ∈ Ω(A) is the marginal of a correlating state ω ∈ Ω(AA)

such that ω(gx, gy) = ω(x, y) for all g ∈ G with α ◦g = α. It is easily checked that this is satisﬁed

by ﬁnite-dimensional quantum models. Applied to the maximally mixed state ρ, this produces a

perfectly, uniformly correlating state η

. Thus, A is a conjugate in the sense of Deﬁnition 3.

4 Filters

We have just seen that if A is sharp and has a conjugate, then E(A) is self-dual, and isomorphic

to V(A). Suppose now that every non-singular state of A can be prepared, up to normalization,

from the maximally mixed state ρ(x) ≡ 1/n by some reversible process. This guarantees that

V(A), and hence, E(A), is homogeneous, so that, by the Koecher-Vinberg Theorem, E(A) carries

a euclidean Jordan structure making E(A)

the cone of squares.

In fact, we can say something more interesting. 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. This can always be done through

post-processing, using a classical ﬁlter. In QM, it can also be accomplished by subjecting the

system to a suitable process prior to measurement. To see this, let A be a ﬁnite-dimensional

quantum system, with corresponding Hilbert space H, and identify E(A) with L

). If E is

an orthonormal basis representing a basic measurement on this system, deﬁne a positive operator

V : H → H by setting V x = t

1/2

x for every x ∈ E, where 0 ≤ t

≤ 1. This gives us a completely

positive linear mapping Φ : E(A) → E(A), namely Φ(a) = V aV . If t

> 0 for every x ∈ E, Φ

has a completely positive inverse Φ

−1

(a) = V

−1

. For each x ∈ E, the corresponding eﬀect

x ∈ E(A) ' L

(H) is the rank-one projection operator p

. It is easy to check that V p

V = t

i.e., that Φ(

x) = t

x for every x ∈ E.

Deﬁnition 7. A ﬁlter for a test E of a probabilistic model A is a positive linear mapping

Φ : V(A) → V(A) such that, for every outcome x ∈ E, there exists a coeﬃcient 0 ≤ t

≤ 1 with

Φ(α)(x) = t

α(x)

for all all states α ∈ Ω(A). Equivalently, Φ

∗

(x) = t

x for every x ∈ E.

Accepted in Quantum 2019-06-24, click title to verify 15

As noted above, in QM, not only do ﬁlters with arbitrary coeﬃcients exist for every test, but

they can be implemented p-reversibly, so long as the coeﬃcients t

are all non-zero. I will say

that a general probabilistic model with this feature has arbitrary reversible ﬁlters.

Corollary 1 (bis) Suppose that A is sharp and has a conjugate A. If A has arbitrary reversible

ﬁlters, then E(A) is homogeneous and self-dual.

Proof: A is self-dual by Theorem 1. Let α be a normalized state in the interior of V(A)

By Lemma 1, α has a spectral decomposition α =

x∈E

α(x)δ

. Let Φ be a ﬁlter for E with

coeﬃcients α(x). Since α is non-singular, α(x) > 0 for all x ∈ E, so Φ can be chosen to be

reversible. Now expand the maximally mixed state ρ, with ρ(x) ≡ 1/n, as ρ =

x∈E

. Then

Φ(ρ) =

x∈E

α(x)δ

α. Thus, any non-singular state can be prepared, up to normalization,

by a reversible ﬁlter, and it follows that V(A) is homogeneous. In view of Theorem 1, E(A) is

self-dual, and E(A) ' V(A), whence, also homogeneous. 

State preparation by reversible ﬁlters Suppose now that A has only a weak conjugate A,

and that Φ is a ﬁlter for a test E ∈ M(A). By applying Φ to one of the two systems A and A,

we can convert the correlator η

into a new sub-normalized bipartite state ω, given by ω(x, y) =

(Φ

∗

x, y) for all x ∈ X(A), y ∈ X(B). Noticing that Φ

∗

(x) = t

x for every x ∈ E, we see that

ω correlates E with E: if x, y ∈ E with x 6= y, we have

ω(x, y) = η

x, y) = t

(x, y) = 0.

In other words, ω is correlating. It follows that the normalized bipartite state

ω :=

ω(u

, u

)

is likewise correlating. Since ω

= Φ(ρ), it follows that any state preparable from ρ by a ﬁlter —

that is, any state of the form α =

]

Φ(ρ), where Φ is a ﬁlter and

α :=

(α)

α — is the marginal

of a correlating state, and hence enjoys a spectral decomposition as in Equation (4). Thus, if

every state is so preparable, the weak conjugate A is actually a conjugate. So, in the presence of

sharpness, we can replace the assumption that the conjugate is strong, by the requirement that

every state be preparable by a ﬁlter. In fact, by strengthening this preparability assumption, it

is even possible to omit the hypothesis that A is sharp.

The isomorphism γ

: A ' A extends to an order-automorphism V(A) ' V(A), given by

α 7→ α, with α(x) = α(x) for all x ∈ X(A). Hence, a positive linear mapping Φ : V(A) → V(A)

has a counterpart Φ : V(A) → V(A), given by Φ(α) = Φ(α). Let us say that Φ is symmetric

with respect to η

iﬀ η

(Φ

∗

(x), y) = η

(x, Φ

∗

(y)) for all x, y ∈ X(A), i.e., iﬀ η

◦ (Φ

∗

⊗ id

) =

◦ (id

⊗ Φ

∗

Lemma 5 Let A have a weak conjugate A. Suppose that every state of A is preparable by a

symmetric ﬁlter. Then ha, bi := η

(a, γ

(b)) is a self-dualizing inner product on E(A).

Proof: Let α = Φ(ρ), where Φ is a symmetric ﬁlter for some test E. Consider the bipartite state

ω := η

◦ (Φ

∗

⊗ id

) = η

◦ (id

⊗ Φ

∗

Accepted in Quantum 2019-06-24, click title to verify 16

For each outcome x ∈ X(A), let δ

denote the conditional state (η

)

1|x

. Then for all x ∈ E, and

all outcomes y ∈ X, we have

1|x

(y) =

(Φ

∗

(y), x)

(Φ

∗

), x)

(y, Φ

∗

(x))

, Φ

∗

(x))

(y, t

, t

(y, x)

, x)

= (η

)

1|x

(y) = δ

(y).

It follows that ω

1|x

= δ

. It is easy to check that ω

= Φ((η

)

) = Φ(ρ) = α; also, by the law of

total probability (3), ω

x∈E

(x)ω

1|x

x∈E

, where t

= ω

(x). Thus, every state in

Ω(A) is a convex combination of the states δ

, and the cone generated by these states coincides

with V(A)

. It follows that

η maps E(A)

onto V(A)

, as in the proof of Lemma 2. The proof

that ha, bi := η(a, b) deﬁnes an inner product on E(A) now proceeds as in the proof of Lemmas 3

and 4. 

In fact, we can do a bit better:

Corollary 2 (bis) Let A have a weak conjugate, and suppose that every interior state is prepara-

ble by a reversible symmetric ﬁlter. Then A is homogeneous and self-dual.

Proof: The preparability assumption clearly makes V(A) homogeneous. The proof of Lemma 5

shows that all states in the interior of Ω can be decomposed as in equation (4) with respect to

the states δ

= η

1|x

. As noted in Remark (2) following the proof of Theorem 1, this is enough to

secure the self-duality of E(A), and its isomorphism with V(A). 

It follows from the KV theorem that, for any model A satisfying the hypotheses of either

Corollary 1 or Corollary 2, E(A) carries a Jordan product compatible with the inner product

arising from η

, i.e, E(A) is a euclidean Jordan algebra. In fact, one can prove more: the unit

eﬀect u coincides with the Jordan unit, and M(A) is precisely the set of Jordan frames. In other

words, E(A) is a Jordan moel. The proof is given in Appendix A, where it is also shown that any

Jordan model satisﬁes the hypotheses of both corollaries. Thus, these two sets of hypotheses are

equivalent, and exactly characterize the class of Jordan models. To summarize:

Theorem 2 For a ﬁnite-dimensional, uniform probabilistic model A, the following statements are

equivalent:

(a) A is sharp, has a conjugate, and has arbitrary reversible ﬁlters

(b) A has a weak conjugate, and all non-singular states can be prepared by reversible

symmetric ﬁlters

It should be stressed that all of the assumptions going into (a) and (b) are what [7] calls single-

system postulates, at least to the extent that the existence of a conjugate (or weak conjugate) is

a property of a single system. In any event, these assumptions, whether seen as pertaining to a

single system A or to the pair (A, A), are quite diﬀerent in ﬂavor from local tomography or the

subspace axiom, which place constraints on an entire theory’s worth of probabilistic models.

Accepted in Quantum 2019-06-24, click title to verify 17

5 Conclusion

We’ve seen that either of two related packages of assumptions — given in (a) and (b) of Theorem

2 — lead in a very simple way the homogeneity and self-duality of the space E(A) associated with

a probabilistic model A, and hence, by the Koecher-Vinberg Theorem, to A’s having a euclidean

Jordan structure. While this is not the only route one can take to deriving this structure (see, e.g,

[27] and [31] for approaches stressing symmetry principles), it does seem especially straightforward.

As discussed in the introduction, several other recent papers (e.g, [20, 28, 14, 25, 11]) have

derived standard ﬁnite-dimensional quantum mechanics, over C, from operational or information-

theoretic axioms. Besides the fact that the mathematical development here is quicker and easier,

the axiomatic basis is considerably diﬀerent, and arguably leaner, making no appeal to the struc-

ture of subsystems, or to the isomorphism of systems with the same information-carrying capacity,

or to local tomography. The last two points are particularly important: by avoiding local tomog-

raphy, we allow for real and quaternionic quantum systems; by not insisting that physical systems

having the same information capacity be isomorphic, we allow for quantum theory with supers-

election rules, and for physical theories in which real and quaternionic systems can coexist. Of

course, the door has been opened a bit wider than this: our postulates are also compatible with

spin factors and with the exceptional Jordan algebra.

Of the reconstructions cited above, the one having the strongest aﬃnity with the approach of

this paper is that of [11], the key postulate of which is that every state dilates to — that is, arise

as the marginal of — a pure state on a larger, composite system, unique up to symmetries of the

ancillary system. Condition (a) in the deﬁnition of a conjugate, requiring that every state dilate

to a correlating state, has a somewhat similar character, albeit with the emphasis on the dilated

state’s correlational properties, rather than its purity. To make the connection more explicit,

suppose we require that every non-singular state α on A dilate to a correlating isomorphism state

ω (which is the case, in the presence of our other assumptions). If µ is another isomorphism

state with the same marginal state α, then φ :=

µ ◦

−1

is a reversible transformation on V(A)

with

µ = φ ◦

ω, i.e., µ(a, b) = ω(a, φ(b)) for all a, b ∈ E(A). Now, as shown in [5], if V(A)

is irreducible as an ordered vector space, isomorphism states are pure. Thus, in the irreducible

case, we have a version of the puriﬁcation postulate for non-singular states. In view of these

connections, it seems plausible that the approach taken here might be adapted to considerably

simplify the mathematical development in [11].

An assumption that is common to nearly all of the cited earlier reconstructions is some version

of Hardy’s subspace postulate, which requires (roughly speaking) that the result of constraining

a physical system to the set of states making a particular measurement-outcome impossible, also

count as a physical system. This very powerful assumption, while not needed in the development

above, can readily be adapted to the framework of this paper, and can to a large extent replace our

assumptions above about the existence of reversible ﬁlters. The details can be found in Appendix

It can be shown [6] that the exceptional Jordan algebra can be ruled out on the grounds that one can form no

satisfactory composite of two euclidean Jordan algebras if either has an exceptional direct summand. Whether the

spin factors can also be discarded, or whether they have some physical role to play, remains an open question.

Going in the other direction, in [12], the authors derive a version of part (a) of our deﬁnition of a conjugate, that

is, the existence of a dilation perfectly correlating two tests, from axioms similar to those of [11]. More recently, in

the conext of a compact closed category of processes, the authors of [29] introduce a stronger, “symmetric” version

of the puriﬁcation postulate, and show that when combined with suitable versions of sharpness and the existence of

a “classical interface”, this implies that all states can be prepared from the maximally mixed state by a reversible

process, allowing them to prove that their analogue of the cone V(A)

is homogeneous and self-dual.

Accepted in Quantum 2019-06-24, click title to verify 18

It is worth remarking that the subspace axiom applies, not to an indidual probabilistic model

but to a class of probabilistic models, that is, to an entire probabilistic theory. (In the language of

[7], it is not a single-system postulate.) As a rule, one wants to think of a physical theory, not as a

loosely structured class, but as a category of systems, with morphisms corresponding to processes.

To allow for composite systems, it is natural to take this to be a symmetric monoidal category

[1]. This brings us to the interesting question of whether one can construct symmetric monoidal

categories of probabilistic models, in which (say) the hypotheses of Corollary 1 are satisﬁed by

all systems. This is indeed possible for special Jordan algebras (those not having the exceptional

Jordan algebra as a direct summand). Restricting attention to Jordan models corresponding to

direct sums of real, complex and quaternionic matrix algebras, one can even arrange for this cat-

egory to be compact closed [5]. This implies that many standard quantum-information theoretic

protocols, notably conclusive teleportation and entanglement-swapping, are still available in this

non-locally tomographic setting.

Acknowledgements I wish to thank Giulio Chiribella, Chris Heunen, Matt Leifer and Markus

M¨uller for helpful comments on earlier drafts of this paper. This work was supported in part by

a grant (FQXi-RFP3-1348) from the FQXi foundation.

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A Jordan Models

Let J be a euclidean Jordan algebra. As discussed earlier, this is associated with a probabilistic

model A(J ) = (X(J ), M(J ), Ω(J )), where X(J ) is the set of primitive idempotents (that is,

minimal projections) in J , M(J ) is the set of Jordan frames (maximal pairwise orthogonal sets

of minimal projections), and Ω(J ) is the set of states on (X(J ), M(J )) arising from states on J ,

that is, restrictions to X(J ) of positive, normalized linear functionals on J . Using the self-duality

of J , it’s easy to show that E(A) ' J ' V(A).

In this appendix, it is shown that any probabilistic model satisfying the conditions of Corollary

1 is actually a Jordan model of this form, and, conversely, that any such Jordan model satisﬁes

the hypotheses of Corollary 2 (which imply those of Corollary 1).

Accepted in Quantum 2019-06-24, click title to verify 21

A.1 Direct Sums and central projections

At several points, we will need to use some basic facts about direct sum decompositions of Jordan

algebras. The direct sum of Jordan algebras J

, ..., J

is the algebraic direct sum J = J

⊕· · ·⊕J

of the vector spaces J

, consisting of n-tuples (a

, ..., a

) with a

∈ J

, with the Jordan product

deﬁned by (a

)

) = (a

), where

is the Jordan product on the i-th summand. Identifying

a ∈ J

with (a

) where a

= 0 for j 6= i and a

= a for j = i, we can treat each J

as a subalgebra

of J , and write (a

) ∈ J as

i=1

. With this understood, we have a

b = 0 if a ∈ E

and

b ∈ E

with i 6= j. The unit is evidently 1 =

i=1

where 1

is the unit in J

. Note that the

canonical projection map π

: J → J

is a Jordan homomorphism. Thus, e ∈ E is an idempotent

iﬀ e

:= π(e) is an idempotent in J

. As e =

, it follows that e is a primitive idempotent iﬀ

e = e

∈ E

for some i = 1, ..., n. In other words, every primitive idempotent of J lives in one of

the summands J

If each J

is a euclidean Jordan algebra with inner product h , i

, then we endow J =

with the usual inner product, that is, ha, bi =

, b

. where a

= π

(a) and b

= π

(b). Thus,

the summands J

are orthogonal to one another as subspaces of J . A euclidean Jordan algebra

is simple iﬀ it is not isomorphic to a direct sum of non-trivial Jordan algebras. Every EJA is

isomorphic to a direct sum of simple EJAs in an essentially unique way. It will be helpful brieﬂy

to review how this decomposition works. Elements a and b in an EJA J operator commute

with one another iﬀ a

x) = b

x) for every x ∈ J , i.e., iﬀ the operators L

and L

of left

Jordan-multiplication by a and b commute. An element of J is central iﬀ it operator-commutes

with all elements of J . If J =

, then each of the units 1

∈ J

is central. Note that

these elements are also idempotents, or projections. A Jordan ideal of an EJA J is a subspace

of the form cJ where c is a central projection, which is then unique. In this case, the mapping

x 7→ c

x is a Jordan homomorphism from J onto cJ ; indeed, x 7→ (c

x, c

x) provides a canonical

isomorphism J ' cJ ⊕ c

J . More generally, recall that idempotents e, f in a Jordan algebra are

Jordan-orthogonal iﬀ e

f = 0. If {c

} is a maximal pairwise Jordan-orthogonal set of central

idempotents, then it is straightforward to show that

= 1 and, hence, that J '

J via

the mapping φ : a 7→ (c

a). Moreover, each of the summands J

:= c

J is simple. For if c

= p + q

where p and q are central projections in c

J , then p, q are central in J , and are Jordan-orthogonal

to every c

with j 6= i, so {c

}

i6=j

∪ {p, q} is a larger pairwise Jordan-orthogonal set of central

projections, a contradiction.

A.2 The unit eﬀect as the Jordan unit

Suppose A is a model satisfying the hypotheses of Corollary 1. In particular, then, A is HSD,

so E := E(A) has a Jordan structure. We wish to show that the unit eﬀect u ∈ E(A) is (or

can be taken to be) the Jordan unit, and that each outcome x ∈ X(A) — or, more exactly, the

corresponding eﬀect

x — is a primitive idempotent.

In fact, we will ultimately establish more,

namely, that X(A) corresponds exactly to the set of primitive idempotents, and M(A), to the

set of Jordan frames, of E(A).

In what follows, normalize the inner product on E so that hx, xi = 1 for every x ∈ X(A).

(This is possible, since the inner product arising from a correlator assigns every outcome the same

norm.) Note that we also have hu, xi = 1, and hu, ui = n, the rank of A. Every state α ∈ V(A)

corresponds to a unique a ∈ E(A)

with α(x) = ha, xi. In particular, ha, ui = 1. Conversely,

This can actually be established very directly by appealing to a slightly stronger version of the Koecher-Vinberg

Theorem, as in [8]. The proof given here is more self-contained and elementary.

Accepted in Quantum 2019-06-24, click title to verify 22

every a ∈ E(A)

with ha, ui = 1 corresponds to a state in this way, since E(A) ' V(A) as ordered

spaces.

The KV theorem produces a Jordan structure on E(A) in which the Jordan unit, 1, is ﬁxed by

every order-automorphism that is also an orthogonal transformation relative to the inner product.

That is, writing Aut(E) for the group of order-automorphisms and Aut(E)

for the stabilizer of 1

therein, Aut(E) ∩ O(E) ⊆ Aut(E)

. Moreover, the stabilizer of 1 in the connected component G

of Aut(E) is then exactly K := G ∩ SO(E) [18]. In particular, every order-automorphism in the

connected component of the identity ﬁxing 1 is an orthogonal transformation with respect to h , i.

In the interest of notational simplicity, from this point on I will identify outcomes x ∈ X(A)

with the corresponding evaluation functionals

x ∈ E(A), treating each test E ∈ M(A) as a set of

eﬀects.

A.1 Lemma: For each x ∈ X(A), there exists a primitive idempotent e

and a scalar t

> 0

such that x = t

, and every primitive idempotent corresponds to an outcome in this way.

Proof: Since A is sharp and hx, xi = hx, ui = 1, we have δ

= hx| for every x ∈ X(A); that is,

(a) = hx, ai for all a ∈ E(A). In particular, as δ

is a pure state, x is ray-extremal for every

x ∈ X(A). Hence, x = t

for some primitive idempotent e

. Conversely, since X(A) generates

E(A), every ray-extremal element of E(A)

must be a multiple of some x ∈ X(A). In particular,

then, each primitive idempotent is a multiple of an outcome, and vice versa. 

Notice that if x and y are distinct outcomes belonging to a common test, we have he

, e

i =

hx, yi = 0. Hence, 1 = hx, xi = t

, so t

= 1/ke

k for every x.

A.2 Theorem: The Jordan product on E(A) can be so chosen that u

is the Jordan unit, 1 and

every x ∈ X(A) is a primitive idempotent.

Proof: We consider in turn the case in which E(A) is simple and the general case in which E(A)

is a direct sum of simple ideals. Throughout, we write u for u

Case 1: E(A) is simple. By [18] Corollary IV.2.7, the group K := G ∩ SO(E), where G

is the connected component of the identity in Aut(E), acts transitively on the set of primitive

idempontents. Since K consists of orthogonal transformations, all primitive idempotents have

the same norm. It follows that, for E simple, kek ≡ c > 0 for all primitive idempotents, whence,

that we have t

≡ t = 1/c for all x. That is, if E(A) is simple, x = te

for every x ∈ X(A).

Now redeﬁne the Jordan product on E by setting a ◦ b := t

−1

b. It is easy to check that this

gives a Jordan product deﬁning the same positive cone, with unit 1

:= t1. Also note that the

new Jordan product continues to ineract with the given innner product in the desired way, i.e.,

ha ◦ b, ci = hb, a ◦ ci for all a, b, c ∈ E. We also have, for each outcome x ∈ X(A),

x ◦ x = t

−1

(te

) = t(e

) = te

= x,

so x is an idempotent with respect to this new Jordan product; moreover, since the positive cone

is unchanged, this is still ray-extremal, hence, a primitive idempotent. We now have

hx, 1

i = hx

, 1

i = hx, xi = 1.

By the spectral decomposition for states (equation (4)), we have 1

x∈E

x for some test

Accepted in Quantum 2019-06-24, click title to verify 23

E ∈ M(A) and coeﬃcients s

. But for every x ∈ E,

y∈F

hx, yi = hx, 1

i = 1,

so 1

x∈E

x = u

Case 2: E(A) a direct sum of simple ideals. Let E =

i=1

where each E

is a simple Jordan

algebra. By Lemma A.1, we still have a correspondence x 7→ e

between outcomes x ∈ X(A) and

primitive idempotents e

∈ E, with x = t

for some t

> 0. As remarked earlier, each primitive

idempotent lies in a unique summand E

whence, the same is true for outcomes. The argument

given above shows that, for all x ∈ E

, we have t

= t

for a constant t

> 0 depending only on

the summand. Adjusting the Jordan product on each summand as in Case 1, we obtain a new

Jordan product on E given by

) ◦ (b

) =

−1

rendering each x ∈ X(A) ∩ E

— and hence, each outcome x ∈ X(A) — a primitive idem-

potent. By the same argument as in Case 1, we have hx, 1i = 1 for all x ∈ X. Expanding

1 =

x∈E

x for some test E ∈ M(A), we then have (again, just as in the irreducible case) that

1 = hx, 1i =

y∈E

ix, yi = s

, whence, s

= 1 for all x ∈ E, and hence, 1 = u

. 

From now on, we treat u

as the Jordan unit of E(A) without further comment, and revert

to writing u, rather than 1, for the unit in an abstract Jordan algebra. It will be convenient at

this point to adopt the notation x ⊥ y to indicate that two outcomes x, y ∈ X(A) are distinct

and belong to a common test.

A.3 Corollary: Let x, y ∈ X(A). Then x ⊥ y ⇒ x

y = 0.

Proof: By [2], Prop. 2.18, if p, q are projections in an EJA, then p ≤ u − q implies p

q = 0. By

Proposition A.2, x and y are projections. If x ⊥ y, then x + y ≤ u, so x ≤ u − y. .

Idempotents p, q in a Jordan algebra are Jordan-orthogonal iﬀ p

q = 0. Recall that a Jordan

frame in an EJA E is a set E of non-zero Jordan-orthogonal primitive idempotents summing to

the unit. It follows from Proposition A.2 and Corollary A.3 that every test in M(A) deﬁnes a

Jordan frame in E(A). We now show that, conversely, every Jordan frame in E(A) belongs to

M(A).

A.4 Theorem: Let A be any probabilistic model satifsying the hypotheses of Corollary 1 or of

Corollary 2, and let E(A) have the Jordan structure in which u

is the Jordan unit, as per Lemma

A.2. Then every Jordan frame of E(A) corresponds to a test in M(A). Hence, A is a Jordan

model.

Proof: Theorem A.2 and Corollary A.3 tell us that the set X(A) of outcomes of A is exactly the

set of primitive idempotents (still continuing to identify outcomes with the corresponding eﬀects),

and that every test E ∈ M(A) is a Jordan frame of E(A). The Spectral Theorem for euclidean

Jordan algebras ([18] Theorem III.1.1) tells us that if J is any EJA with unit u and a ∈ J , then

there exists a unique family of mutually Jordan-orthogonal non-zero idempotents e

and distinct

coeﬃcients s

, such that a =

. Suppose the idempotents e

are primitive, and that a can

Accepted in Quantum 2019-06-24, click title to verify 24

also be expanded as

for some pairwise Jordan-orthogonal non-zero idempotents f

and

coeﬃcients t

(not a priori distinct). Then by collecting those terms with common coeﬃcients, we

can also write a =

where p

= k}. Since the f

are pairwise Jordan-orthogonal

idempotents, it is easy to check that p

is also an idempotent, with p

= 0 for k 6= k

. It

follows from the uniqueness of the spectral expansion that, for each k, there is some i with k = s

and p

= e

. Since e

is primitive, this means that the sum p

: t

= k} has a unique

term, call it f

, with f

= e

. It follows that the coeﬃcients t

were after all distinct, and the

projections f

coincide with the projections e

. In other words, if a can be expanded with dinstinct

coeﬃcients relative to some family of mutually orthogonal primitive idempotents, it can have no

other such expansion in terms of mutually orthogonal idempotents, with any coeﬃcients. Now

let F = {e

, ..., e

} be a Jordan frame of E(A). Choosing distinct coeﬃcients s

, i = 1, ..., n, let

a =

. By our spectral decomposition for HSD models (equation (4)), there exists a test

E ∈ M(A) and coeﬃcients t

, x ∈ E such that a =

x∈E

x. Since E is a Jordan frame, the

uniqueness result above tells us that E = F . Thus, every Jordan frame of E(A) is a test in M(A),

as advertised. 

A.3 Conjugates and Filters for Jordan Models

As discussed in Section 5, the question of what it means, physically, to say that a given system

has a conjugate really depends on the probabilistic theory — and the notion of composite system

— with which one is concerned. But if we are content to interpret this idea very broadly, we can

understand the composite AA as the “maximal” tensor product A ⊗

max

A [9], the states of which

are simply the positive, normalized bilinear forms on E(A) × E(A). In particular, if A = A(J )

is the model associated with a euclidean Jordan algebra J , the canonical inner product on J ,

normalized so that hu, ui = 1, supplies exactly the needed bilinear form. Thus, every Jordan

model can be regarded as its own weak conjugate. (At least, this is so if we construe “A has

a weak conjugate” to mean only that there exists a perfectly correlating positive, bilinear form

on E(A) — mathematically, a weaker condition than that A have a conjugate in any particular

probabilistic theory.)

We now show that Jordan models support arbitrary ﬁlters, and that every state of such a

model can be prepared by a symmetric ﬁlter. For any element a of a euclidean Jordan algebra J ,

the operator U

: J → J deﬁned by

(x) := 2a · (a · x) − a

· x

is positive ([2] Theorem 1.25). If J is special, i.e., if J consists of self-adjoint operators on a real,

complex or quaternionic Hilbert space H, with a ·b =

(ab+ba), one can check that U

(x) = axa.

Below, we shall see that if a =

x∈E

x, where E is a Jordan frame, then U

is a ﬁlter with

coeﬃcients t

Let J be an EJA. An element a ∈ J is invertible iﬀ there exists an element b of the associative

sub-algebra generated by a and u, such that a

b = u. This element, which is clearly unique, is

then the inverse of a, denoted by a

−1

. The following collects some facts about invertibility that

will be needed in a number of places below.

A.5 Proposition: The following are equivalent:

(a) a is invertible;

(b) There exists some b ∈ J with a

b = u and a

b = a;

is invertible. In this case, U

−1

= U

−1

Accepted in Quantum 2019-06-24, click title to verify 25

Proof: That (a) implies (b) is clear. That (b) implies (a) is a consequence of the fact (the Shirsov-

Cohn Theorem; see [2], Proposition 1.14) that the Jordan algebra generated by two elements and

u is special, plus the fact that (a) and (b) are equivalent in special Jordan algebras. See [2] Lemma

1.16 and Proposition 1.17 for details. The equivalence of (a) and (c) is Lemma 1.23 in [2]. 

We shall also need the following elementary observation:

A.6 Lemma: If a has a spectral decomposition a =

x∈E

x where E is a Jordan frame and

6= 0 for all x ∈ E, then a is invertible. In particular, a is invertible if lies in the interior of J

Proof: By spectral theory, a =

x∈E

x where E is a Jordan frame. If t

6= 0 for all x, let

b =

x∈E

−1

x = f (a) where f (x) = x

−1

, so that b ∈ C(a, u), the Jordan subalgebra generated

by a and u. Observe that, by the Jordan-orthogonality of elements of a Jordan frame,

b =

x,y∈E

−1

y =

−1

x =

x∈E

x = u.

Thus a is invertible with inverse b. 

A.7 Lemma: Let J be any EJA. Every state of the model A(J ) is preparable by symmetric ﬁlter,

and every non-singular state, by a reversible symmetric ﬁlter.

Proof: Let α be any state on J . By self-duality, there exists a unique w ∈ J

with α(b) = hw, bi

for all b ∈ J . The spectral theorem gives us a Jordan frame E and a decomposition w =

x∈E

Let a =

x∈E

1/2

x ∈ J

: then we have U

(u) = w and U

(x) = t

x for every x ∈ E, so U

is a

ﬁlter. Since left multiplication by a is self-adjoint with respect to the inner product on J , so is

, whence, U

is a symmetric ﬁlter. Finally, if w lies in the interior of J

, then the coeﬃcients

, and hence also t

1/2

, are all strictly positive. Thus, a is invertible by Lemma A.5, and thus U

is invertible with inverse U

−1

, again a positive operator, by Proposition A.5. 

As noted in the discussion preceding Lemma 5, every state preparable by a symmetric re-

versible ﬁlter is the marginal of a correlating state. Hence, every Jordan model is its own (strong)

conjugate. Since such a model is evidently sharp, the conditions of Corollary 1 are satisﬁed. This

proves Theorem 2: every probabilistic model satisfying the hypotheses of Corollary 1 or Corollary

2 is a Jordan model, and every Jordan model satisﬁes the hypotheses of both Corollaries.

B Symmetry and Subspace Axioms

In most other recent reconstructions of QM [20, 14, 25, 11], one encounters some version of a

subspace axiom. Informally, the idea is that if we constrain the states of a given system so as to

render a certain measurement result impossible, we obtain what amounts the state space of a new

system in its own right, to which any other axioms must then apply. In practice, the subsets of the

state space arising in this way are (some of the) faces of the larger state space

. An assumption

of this sort was made by Hardy [20], and, following his lead, is also central in reconstructions by

Dakic and Brukner [14], Masanes and Mueller [25] and Chiribella, D’Ariano and Perinotti [11],

although the precise formulation varies somewhat from one set of authors to another.

Recall that a face of a convex set K is a convex set F ⊆ K such that for all α, β ∈ K and all t ∈ (0, 1),

tα + (1 − t)β ∈ F implies α, β ∈ F .

Accepted in Quantum 2019-06-24, click title to verify 26

In this Appendix it is established that, for uniform probabilistic models, a certain version of

this subspace axiom, plus a very reasonable compactness assumption, enforce a spectral decom-

position of states (equation (4) in Section 3). Thus, these assumptions can replace condition (b)

in the deﬁnition of a conjugate, at least as far as Theorem 1 is concerned. Moreover, a stronger

version of the subspace axiom, which seems about equally well motivated, implies the existence

of arbitrary reversible ﬁlters. Combined with a rather weak and operationally natural symmetry

assumption, this in turn yields the homogeneity of the space V(A). In the presence of weak

conjugates, it follows that V(A) is both homogeneous and self-dual.

Probabilistic Theories Before proceeding, it will be important to clarify the term “probabilistic

theory”, up to this point used rather freely and informally. In the very broadest sense, a proba-

bilistic theory is simply a class of probabilistic models, together with some designated processes

that are singled out for study. But allowing for the composition of processes, and assuming that

identity operators count as processes, it is very natural to assume that a probabilistic theory is a

category of probabilistic models and processes. This is the point of view taken in the “categorical

quantum mechanics” programme of Abramsky, Coecke and others [1]. It is also usual in this

context also to assume that this category has a (symmetric) monoidal structure, allowing for the

formation of composite systems. However, for my purposes, it will be enough to require only that

we are given a class C of probabilistic models and, for each model A ∈ C, a preferred monoid

Proc(A) of allowed processes, i.e., positive linear mappings T : V(A) → V(A) with u

(T (α)) ≤ 1

for all α ∈ Ω(A).

We continue to identify outcomes with the corresponding eﬀects, regarding X(A) as a subset

of E(A) and M(A), as a collection of subsets of E(A). We can then deﬁne a symmetry of A to be

a dual process g = φ

∗

where φ ∈ Proc(A) is invertible, such that gM(A) = M(A). I will write

G(A) for the set of all symmetries of A, noting that this is a group. As above, we use the notation

x ⊥ y to indicate that x and y are distinct outcomes belonging to a common test. Notice that

this does not (yet) imply that x and y are orthogonal with respect to any inner product on E(A).

B.1 The Subspace Axiom and Spectrality

Masanes and Mueller’s Framework In the reconstruction due Masanes and Mueller [25],

one begins by specifying the state-space of a physical system, taken to be a ﬁnite-dimensional

compact convex set Ω. Measurement results are associated with eﬀects, i.e., aﬃne functionals

a : Ω → R with 0 ≤ a(α) ≤ 1 for all α ∈ Ω. Of course, a(α) is understood as the probability of

the result associated with a being obtained, when the system’s state is α. Masanes and Mueller

do not assume that all eﬀects correspond to physically realizable measurement results, but do

seem, tacitly, to deﬁne a measurement to be any list a

, ..., a

of allowed eﬀects that sum to the

unit eﬀect u (where u(α) = 1 for all α ∈ Ω(A)). Further assumptions, explicit in [26], are that

the set of allowed eﬀects is topologically closed, convex, and closed under a 7→ u − a. Masanes

and Mueller call states α

, ..., α

∈ Ω(A) perfectly distinguishable iﬀ there exist aﬃne functionals

: Ω → R with 0 ≤ a

(α) ≤ 1, representing measurement results, such that a

+ · · · + a

= u,

where u is the unit eﬀect (u(α) = 1 for all α ∈ Ω) and a

(α

) = δ

i,j

for all i, j = 1, ..., n. The

information capacity of the system is the maximum size of such a perfectly distinguishable set

of states. A complete measurement is a measurement that perfectly distinguishes a maximum

We can, of course, regard this as a degenerate category in which there are no morphisms between diﬀerent

objects. My intention, however, is simply to leave it open which mappings between diﬀerent systems count as

processes, as nothing will depend on this. This is another respect in which the approach taken here is “single-

system”.

Accepted in Quantum 2019-06-24, click title to verify 27

number of states. In the actual development of their results, they eﬀectively take M(A) to be

the set of complete measurements in this sense.

Masanes and Mueller’s Subspace Axiom Here is how Masanes and Mueller explain their

version of the subspace axiom in [26]:

Postulate 2 (Equivalence of subspaces). Let S

and S

N−1

be systems with ca-

pacities N and N − 1, respectively. If E

, ..., E

is a complete measurement on S

then the set of states ω ∈ S

with E

(ω) = 0 is equivalent to S

N−1

The notion of equivalence needs some discussion. Postulate 2 states the equivalence

of S

N−1

and

N−1

:= {ω ∈ S

(ω) = 0}.

Denote the real linear space which contains S

by V

; deﬁne V

N−1

analogously, and

set V

N−1

:= Span(S

N−1

). Equivalence means ﬁrst of all that there is an invertible

linear map L : V

N−1

→ V

N−1

such that L(S

N−1

) = S

N−1

. But it also means that

transformations ... on one of them can be implemented on the other... To be more

speciﬁc, deﬁne G

N−1

as the set of transformations in S

that preserve S

N−1

...

N−1

:= {T ∈ G

| T S

N−1

= S

N−1

The set of reversible transformations G

N−1

is deﬁned as the restriction of all these

transformations to S

N−1

...

A Reformulation Returning to our own formalism, suppose that x ∈ X(A), and let

:= { α ∈ Ω(A) | α(x) = 0 }.

Note that this is a face of Ω(A), corresponding exactly to the restricted state space contemplated

in Masanes and Mueller’s subspace axiom. If we wish to treat this as the state space of a model

in our sense, we must associate a test space with it. The simplest option is to deﬁne, for x ∈ X,

M(A)

= { E \ {x} | E ∈ M(A) with x ∈ E}.

Notice that the outcome-space of M(A)

is the set of outcomes in X(A) that are distinguishable

from x; that is, the union of the tests in M(A)

is the set { y ∈ X(A) | y ⊥ x }. In the

special case of a quantum model, say A = A(H), this is the right deﬁnition, since if K = x

⊥

the subspace of H orthogonal to a unit vector x, then any frame F ∈ M(K) extends to a frame

E = F ∪ {x} of H, so that M(K) = M(H)

Any state α ∈ F

deﬁnes a probability weight α

on M(A)

, simply by restricting α to

outcomes in

M(A)

, i.e., to outcomes y ∈ X(A) with y ⊥ x. The mapping α 7→ α

is ob-

viously aﬃne, but in general is not injective. For a simple example, consider the “square bit”:

M(A) = {{x, x

}, {y, y

}}, with Ω(A) the set of all probability weights thereon. Then Ω(A) is

isomorphic to the unit square in R

under the mapping α 7→ (α(x), α(y)). The face F

can be

identiﬁed with the right-hand face of the square, i.e., F

= {(0, t)|0 ≤ t ≤ 1}. On the other

hand, M(A)

is the trivial test space {{x

}}, which has only a single probabilty weight. We

will therefore need to assume that M(A)

is large enough to separate points of F

, i.e., that if

α, β ∈ F

are distinct, then there exists some y ⊥ x with α(y) 6= β(y). This makes α 7→ α

injection, so that we can identify F

with a set of probability weights on M(A)

Accepted in Quantum 2019-06-24, click title to verify 28

With this in mind, the following now seems to capture the spirit of Masanes and Mueller’s

axiom:

Deﬁnition B.1 Say that a probabilistic theory C has the subspace property iﬀ, for every A ∈ C

and every x ∈ X(A),

(i) M(A)

separates points of F

(ii) The model A

:= (M(A)

, F

) belongs to C, and

(iii) every symmetry in G(A

) extends to a symmetry g ∈ G(A) with gx = x.

As an example, let A = A(H) be the quantum model associated with a Hilbert space H. If

x ∈ X(H) is a unit vector, then F

is the set of density operators W such that Tr(W x) = 0,

or, equivalently, such that P W P = W where P is the orthogonal projection onto the sub-

space K

= x

⊥

orthogonal to x. In this case M(A)

is the set of orthonormal bases of K,

so M(A)

= M(K), and A(H)

= A(K

Lemma B.2 Let C be a probabilistic theory with the subspace property, in which every model is

uniform. Then every model in C is sharp.

Proof: By induction on rank. Clearly, all models of rank 1 are sharp. Assume the result holds for

all models of rank n or lower. Suppose A has rank n + 1 (with n > 0), and let x ∈ X(A). Since

n + 1 > 1, we can ﬁnd some y ⊥ x. Since A

∈ C has rank n, A

is sharp. Therefore, there exists

a unique δ

∈ F

— and hence, a unique δ

∈ Ω(A) — with δ

(x) = 1. 

Remark: This is the only use we make of Condition (i) in Deﬁnition B.1. If one is content to

assume that every model in C is sharp, this condition can be dispensed with.

When dealing with ﬁnite-dimensional probabilistic models, one usually assumes that the state-

space Ω(A) is compact. It is equally natural to suppose that X(A) is compact in the topology

inherited from E(A) — equivalently, in, the coarsest topology making every state α : X(A) → [0, 1]

continuous. This is certainly the case for quantum models, where X(A) is the unit sphere in

a ﬁnite-dimensional Hilbert space, and can be shown to hold for the test space of “complete

measurements” considered (implicitly) in [25].

Let us say that a probabilistic model A is spectral if it is sharp, and every states α ∈ Ω(A)

has a spectral decomposition α =

x∈E

α(x)δ

for some E ∈ M(A).

Proposition B.3 Let C be a probabilistic theory with the subspace property, in which all models

are uniform and have compact outcome-spaces. Then every model in A is spectral.

Proof: By induction on the rank of A ∈ C. Spectrality is trivial for models of rank 1 (which have

only a single state). Assume the result holds for all A ∈ C having rank < n = rank(A). Let

α ∈ Ω(A). Since X(A) is compact and α is continuous on X(A), α takes its minimum value, m,

0 ≤ m < 1, at some point x

∈ X(A). Thus, β := α − mδ

is non-negative on X(A), hence,

belongs to V(A)

. Now, u

(β) = 1 −m, so β

:= (1 − m)

−1

β ∈ Ω(A), and α = (1 − m)β

+ mδ

Since β

) = 0, β

belongs to the face F

:= {α ∈ Ω(A)|α(x

) = 0}. Because C has the subspace

property, A

belongs to C. Since A

has rank n − 1, our inductive hypothesis implies that

y∈F

(y)δ

for some F ∈ M(A

) = M(A)

, i.e., for some F = E \ {x

}, x

∈ E ∈ M(A).

Accepted in Quantum 2019-06-24, click title to verify 29

But now

α = (1 − m)β

+ mδ

y∈F

(1 − m)β(y)δ

+ mδ

which gives a spectral decomposition for α. 

Remarks: Subspace axioms are usually paired with a conceptually distinct requirement that all

systems of a given information capacity are isomorphic. This immediately rules out any non-

classical theory involving superselection rules, or in which there are more than one kind of “bit”.

As we do not impose such an isomorphism requirement, we avoid this restriction.

B.2 Subspaces plus symmetry

In addition to one or another version of subspace postulate, [20, 14, 25, 11] also assume that every

model A to carries a preferred group G(A) of symmetries, understood as acting on the state space,

which is consistent with our choice of G(A) in the previous section. The ﬁrst three of the cited

papers assume a symmetry postulate requiring that G(A) act transitively on pure states, i.e., on

the extreme points of Ω(A). This is also a consequence of the “puriﬁcation postulate” used in [11].

Deﬁnition B.4 A probabilistic model A is symmetric iﬀ G(A) acts transitively on X(A). If A is

sharp and all pure states are of the form {δ

|x ∈ X(A)}, this implies G(A) also acts transitively

on pure states.

As an example, let A = A(H) be the quantum model associated with a Hilbert space H. As

discussed earlier, this means that M(A) is the set of orthonormal bases of H, X(A) is (therefore)

the unit sphere of H, and Ω(A) is the set of density operators on H. If we take G(A) to be the

group of unitary operators on H, then G(A) certainly acts transitively on X(A).

Notice that, in this example, G(A) acts transitively also on M(A). In [30] and elsewhere, a

uniform test space with a preferred group of symmetries G(A) is said to be fully symmetric if, for

every pair of tests E, F ∈ M(A) and every bijection f : E → F , there exists an element g ∈ G(A)

such that gx = f(x) for every x ∈ E. If A has rank n, we can deﬁne an ordered test to be an

n-tuple (x

, ..., x

) ∈ X

where {x

, ..., x

} ∈ M(A). Then full symmetry is the condition that

G(A) act transitively on ordered tests, where g(x

, ..., x

) = (gx

, ..., gx

Lemma B.5 Suppose that C has the subspace property. If every A ∈ C is uniform and symmetric,

then every A ∈ C is fully symmetric.

Proof: Every symmetric model of rank 1 is trivially fully symmetric. Suppose every model in C

having rank ≤ n is fully symmetric, and let A ∈ C be a model of rank n + 1. Let (x

, ..., x

)

and (y

, ...y

) be any ordered tests of A. By symmetry, we can ﬁnd some g

∈ G(A) such that

= y

. Now (y

, ..., y

) and (g

, ...., g

) are both ordered tests of A

. By our induction

hypothesis, A

is fully symmetric, so there exists some g

∈ G(A

) with y

= g

for every

i = 1, ...., n. By the subspace property, g

extends to a symmetry g ∈ G(A) with gy

= y

and

gz = g

z for every z ⊥ y

— in particular, gg

= g

= y

for i = 1, ...., n. Hence, gg

takes

, ...., x

) to (y

, ..., y

). 

Accepted in Quantum 2019-06-24, click title to verify 30

B.3 A Strengthened Subspace Postulate

Since we are considering physical processes φ ∈ Proc(A) that need not be normalization-preserving

(in particular, need not be associated with symmetries in G(A)), the following stronger version

of the subspace property seems quite as well-motivated a the weaker version discussed above.

Deﬁnition B.6 Let C be a probabilistic theory in which all systems are sharp. Say that C has

the strong subspace property (SSP) iﬀ, for every system A ∈ C and every test E ∈ M(A), if x ∈ E,

then (i) A

∈ C and (ii) any (reversible) process φ ∈ Proc(A

) lifts to a (reversible) process in

Proc(A) ﬁxing δ

Note that the signiﬁcant diﬀerence here from the subspace property of Deﬁnition B.1 is the

requirement that all processes on A

, and not only symmetries, lift to processes on A, which can

be taken to be reversible if the original processes are.

Lemma B.7 If C satisﬁes the SSP, every model in C has arbitrary reversible ﬁlters.

Proof: A system of rank 1 automatically has arbitrary ﬁlters: the only test has a single outcome,

x, and there is just one state, δ

. The mapping δ

7→ t

deﬁnes a p-reversible positive mapping

on V(A) ' R.

Now suppose, for purposes of induction, that every system of rank < n has arbitrary reversible

ﬁlters. Let E be a test, and let φ

: V(E \ {x}) → V(E \ {x}) be a ﬁlter with φ(δ

) = t

for

all y ∈ E \ {x}. Extend this to φ : V(A) → V(A) ﬁxing δ

. Now let z ∈ E \ {x}. Repeating the

argument, ﬁnd ψ a ﬁlter on A with ψ(δ

) = δ

for all y ∈ E \ {x} and ψ(δ

) = t

with t

< 1.

(In other words, for z ∈ E \ {y}, we have t

= 1 if z 6= x and t

the given value.) Composing, we

have a positive mapping with

(ψ ◦ φ)(δ

) = ψ(t

) = t

ψ(δ

) = t

for z 6= x, and

(ψ ◦ φ)(δ

) = ψ(δ

) = t

Thus, we have arbitrary p-reversible ﬁlters. 

In [30], it is shown that the existence of arbitrary reversible ﬁlters plus full symmetry implies

the homogeneity of the state space. So we have

Proposition B.8 Let C satisfy the SSP. If every A ∈ C is symmetric, then for every A ∈ C,

V(A) is homogeneous.

Proof: By Lemma B.7 above, we have arbitrary reversible ﬁlters; by Lemma B.5 in previous sub-

section, every A ∈ C is fully symmetric. 

The proofs of Theorem 1 and Corollary 1 imply that, in the presence of a spectral decom-

postion for states, sharpness, the existence of a weak conjugate, and the existence of arbitrary

reversible ﬁlters are enough to secure the self-duality of E(A). Thus, Propositions B.3 and B.8

gives us a third route to Jordan models, to put alongside Corollaries 1 and 2 in the main body of

this paper:

Accepted in Quantum 2019-06-24, click title to verify 31

Corollary B.9 Let C satisfy the SSP. If every model A in C has a compact outcome-space X(A),

is symmetric and has a weak conjugate, then A is homogeneous and self-dual, and hence, a Jordan

model.

B.4 Jordan models and the SSP

The preceding results provide an alternative route from four operationally meaningful axioms

— sharpness, symmetry, the SSP and the existence of weak conjugates — to euclidean Jordan

algebras. The question remains whether all EJAs arise in this way. In fact, the full symmetry

enforced by Lemma B.5 signiﬁcantly constrains the possibilities.

B.10 Lemma: Let A = A(J ) be the Jordan model associated with a Jordan algebra J . If A is

fully symmetric, then J is either simple, or a direct sum of one-dimensional Jordan algebras.

Proof: Suppose J =

i=1

where J

, ..., J

are simple euclidean Jordan algebras, with n ≥ 2.

Let E

be a Jordan frame for J

; then E =

i=1

is a Jordan frame for J . Let x

∈ E

and

∈ E

. By full symmetry, there exists a symmetry g ∈ G(A) such that gx

= x

, gx

= x

and gy = y for all y ∈ E \ {x

, x

}. Now let p

be the central projection associated with the i-th

summand, so that p

and p

are the central covers of (the smallest central projections above) x

and x

, respectively. Hence, g(p

) is the central cover of gx

= x

, i.e, g(p

) = p

, and similarly

= p

. But now if y ∈ E

is any point other than x

, we have y = gy ≤ gp

= p

, which

are impossible, as y is orthogonal to p

. So E

= {x

}, and J

is one-dimensional. Since J

arbitrary, all summands are one-dimensional. 

Thus, a probabilistic theory satisfying the hypotheses of Corollary B.9 will comprise only

simple Jordan models and classical systems. In particular, such a theory cannot accommodate

superselection rules. This is also true, however, of earlier reconstructions based on versions of the

subspace axiom that assume all systems of the same information capacity to be isomorphic. (If

the theory is monoidal, roughly meaning that it allows for the formation of composite systems —

and meaning more precisely that the category in question has a symmetric monoidal structure,

satisfying some reasonable constraints on how this interacts with the theory’s probabilisic appa-

ratus; see [9] for details — then there are further constraints. These matters are discussed in [5].

For a discussion of how monoidal “process theories” in the sense of [1, 13] can be represented as

probabilistic theories in the sense of this paper, see [32].)

We now show that the assumptions discussed in this Appendix imply no further constraints

on Jordan models. To this end, it will be enough to exhibit a probabilistic theory C containing all

simple Jordan models, and assigning to each such model A a monoid Proc(A) of allowed processes

in such a way that A is symmetric and C enjoys the SSP.

Proposition B.11: Let C be the probabilistic theory consisting of all full, simple euclidean Jordan

models, with processes on a given model A(J ) consisting of composites of maps of the form U

a ∈ J

. Then every model in C is symmetric, and satisﬁes the SSP.

The structure group, Γ(J ), of a euclidean Jordan algebra J consists of all non-singular positive mappings

φ : J → J such that U

(a) = φU

∗

for every a ∈ J . A theorem of Koecher [23] shows that the connected

component of the identity in Γ(J ) consists exactly of composites of mappings of the form U

. In other words, in

this theory, G(A) is precisely the connected identity component of Γ(J ).

Accepted in Quantum 2019-06-24, click title to verify 32

Proof: Let A = A(J ) = (M(J ), Ω(J )) be the Jordan model associated with an EJA J . Thus,

M(J ) is the set of Jordan frames, and X(J ) the set of primitive idempotents, of J . If J is simple,

symmetry (indeed, full symmetry) of A under G(A) follows from ([18], IV.2.5)

It follows easily

that G(A) acts transitively on Jordan frames even if J is not simple (since each such frame is the

disjoint union of frames chosen from each simple summand). Let p ∈ P (J ). Then ([2], Prop. 1.38,

Lemma 1.39, Prop. 1.43, Prop. 2.32 and remarks preceding latter), J

:= U

(J ) is a hereditary

Jordan subalgebra, meaning that if 0 ≤ a ≤ b ∈ J

, then a ∈ J

as well. The unit of J

is p. It

is important to note here that (J

)

= J

∩ J

(that is, an element of J

is positive in J

iﬀ it

is positive in J ). This follows from spectral theory; see [2] Prop. 1.22 and subsequent discussion.

It follows that if a, b ∈ J

, a ≤ b in J

iﬀ a ≤ b as elements of J . One can also show ([2], Lemma

1.45) that if a ∈ J

and b ∈ J

, then a

b = 0.

Now let e be a primitive idempotent in J that happens to lie in J

: then e is an idempotent

in J

as well. If f is another nonzero idempotent in J

with f ≤ e, then f is still idempotent

in J and, by remarks above, f ≤ e in J , whence, as e is minimal, e = f. Thus, e is primitive in

. Conversely, let e be primitive idempotent in J

. Then e is still idempotent in J . If f is an

idempotent of J with 0 < f ≤ e in J , then f ∈ J

, since the latter is hereditary. As e is primitive

in J

, f = e. Thus, e is primitive in J .

This shows that X(J

) = J

∩ X(J ). It follows that M(J

) is the set of all sets of primitive

idempotents of J summing to p. Now let p = u − x where x is a primitive idempotent of J . Then

F ∈ M(J

) iﬀ F ∪ {x} ∈ M(J ). In other words, M(J

) = M(A)

in the notation of Deﬁnition

B.1.

It remains to show that any (p-reversible) process in Proc(A

) extends to a (reversible) process

in Proc(A) leaving x ﬁxed. It suﬃces to show this holds for processes of the form U

, with a ∈ J

Since x

b = b

x = 0 for all b ∈ J

, a straightforward calculation then shows that U

a+x

b = U

for all b ∈ J

, and that U

a+x

x = x. Hence, U

a+x

is the desired extension of U

. Note that if U

invertible, then a is invertible in J

. By Lemma A.4 (b), there exists some b ∈ J

with a

b = p

and a

b = a. Since x

b = 0 for all b ∈ J

, we have

(a + x)

(b + x) = a

b + x

= a

b + x = p + x = u,

and

(a + x)

(b + x) = (a

+ x)

(b + x) = a

b + x

= a + x.

Invoking Lemma A.4 (b) again, this implies that b + x is the inverse of a + x in J , whence, by

Lemma A.4 (c), U

a+x

is invertible. 

Or, more exactly, from the proof thereof. For any two Jordan-orthogonal primitive idempotents x, y there exists

a symmetry — an element s ∈ A with s

= u — such that U

(x) = y ([18], IV.2.4). This, plus an induction, allows

one to construct an automorphism of the form U

taking one Jordan frame to another.

Accepted in Quantum 2019-06-24, click title to verify 33

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