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
A
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
A
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
1
, 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.
2
Suppose now that W is any density operator on H, with spectral decomposition W =
P
x∈E
λ
x
p
x
for some orthonormal basis E, where p
x
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
Ψ
W
:=
X
x∈E
λ
1/2
x
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
W
. A straightforward calculation shows that if a, b are any two
operators on H, then
h(a ⊗ b)Ψ
W
, Ψ
W
i = Tr(W
1/2
aW
1/2
b).
In particular, if a and b commute with W , then we have
h(a ⊗ b)Ψ
W
, Ψ
W
i = Tr(W ab). (2)
It follows that the state Ψ
W
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
W
, a ⊗ bi = Tr(Wab) = 0,
while the joint probability of a and a is hΨ
W
, a ⊗ ai = Tr(W a). Where a = p
x
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
W
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
n
1, the maximally mixed state: in this case, Ψ
W
is
the “EPR” state
Ψ =
1
√
n
X
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,
1
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.
2
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|.
Accepted in Quantum 2019-06-24, click title to verify 2
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 =
1
n
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
A
uniformly
correlating every basic measurement with its counterpart, in the sense that
η
A
(x, x) =
1
n
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
W
supplies the correlating state ω, while the bipartite state η corresponds to the EPR
state Ψ
1
n
1
= 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
2
|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
b
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
x
with δ
x
(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
A
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
x
α(x) for
some constant t
x
(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
x
> 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
A
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
η
A
, 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
(c) Systems have arbitrary reversible filters.
Alternatively, and more compactly:
(b
0
) Systems have weak conjugates;
(c
0
) 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
i
} of minimal idempotents satisfying e
i
·
e
j
= 0 for i 6= j,
and
P
i
e
i
= 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.)
3
Secondly, it rests on fewer and (arguably) simpler assumptions. Certainly, the second package
of asumptions (that is, (b
0
) and (c
0
) 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
4
; 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.
3
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].
4
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
2
= 1 or J
2
= −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
2
= −1 or J
2
= 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.
5
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
X
of real-valued functions on
a set X is ordered by the cone R
X
+
of functions taking non-negative values. Another example is
the space L
sa
(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
E
∗
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
P
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
1
, ..., p
n
are non-negative real
numbers summing to 1, and α
1
, ..., α
n
are states in Ω(A), then the function p
1
α
1
+ · · · + p
n
α
n
also
5
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].
Accepted in Quantum 2019-06-24, click title to verify 7
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
α
x, xi for some
density operator W
α
, and all unit vectors x ∈ X(H). Real and quaternionic quantum models,
corresponding to real or quaternionic Hilbert spaces, are defined in the same way.
Remark: Not every physically accessible observable on a finite-dimensional quantum system is
represented by an orthonormal basis. Rather, the general observable corresponds to a positive-
operator-valued 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
A
: V(A) → R, called the unit effect of A, given
by u
A
(α) =
P
x∈E
α(x), where E is any test in M(A). Note that if α ∈ V(A)
+
, then u
A
(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
A
(β) > 0. Thus, any non-zero β ∈ V(A)
+
can be normalized to yield a state
e
β = u
A
(β)
−1
β ∈ Ω(A). A positive linear functional f ∈ V(A)
∗
+
with f ≤ u
A
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
b
x : V(A) → R, given by
b
x(α) = α(x) for all α ∈ V(A), and u
A
=
P
x∈E
b
x.
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),
b
x(α) = Tr(αp
x
) where p
x
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
b
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
b
x associated with
outcomes x ∈ X(A), ordered by the cone E(A)
+
of finite linear combinations
P
i
t
i
b
x
i
, x
i
∈ X(A),
with coefficients t
i
≥ 0.
Since we are assuming that V(A) and (hence) V(A)
∗
are finite-dimensional, the set of func-
tionals
b
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.
6
Such
a mapping extends uniquely to a positive linear mapping T : V(A) → V(A) with T (α)(u
A
) ≤ 1
for all α ∈ Ω(A).
Definition 3. A process T : V(A) → V(A) is probabilistically reversible — hereafter, just p-
reversible
7
— 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
A
) ≤ u
A
. T is lossless iff T
∗
(u
A
) = u
A
. 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
E
and E(A) ' (R
E
)
∗
, with
the standard inner product on R
E
providing the order-isomorphism. If H is a finite-dimensional
6
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.
7
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
sa
(H)
of self-adjoint operators on H, ordered by the cone of positive operators. For any x ∈ X(H),
the evaluation functional
b
x ∈ V(A) is then given by W 7→ hW x, xi = Tr(W P
x
). It follows that
E(A(H)) ' L
sa
(H)
∗
' L
sa
(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
E
and the trace inner product on L
sa
(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
2
·
b) = a
2
·
(a
·
b)
for all a, b ∈ J (with a
2
= 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
P
k
i=1
a
2
i
= 0 implies a
i
= 0 for all i.
A EJA J is also an ordered vector space with positive cone J
+
= {a
2
|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
sa
(H) of self-adjoint operators on a finite-dimensional real,
complex or quaternionic Hilbert space H, with a · b =
1
2
(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
n
:= R × R
n
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
2
. 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
1
, ..., e
n
} 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 =
P
x∈E
t
x
x
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,
b
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
(x,y)∈E×F
ω(x, y) = 1.
Such a weight is said to be non-signaling if, in addition, the marginal weights
ω
1
(x) :=
X
y∈F
ω(x, y) and ω
2
(y) :=
X
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
ω
1|y
(x) =
ω(x, y)
ω
2
(y)
and ω
2|x
(y) :=
ω(x, y)
ω
1
(x)
,
when ω
2
(y) and ω
1
(x) are non-zero. This gives us the following bipartite law of total probability
[17]
ω
2
=
X
x∈E
ω
1
(x)ω
2|x
and ω
1
=
X
y∈F
ω
2
(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
1
and ω
2
) 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
b
ω : X(A) → V(B) and
b
ω
∗
: X(B) → V(A) by
b
ω(x)(y) = ω(x, y) =
b
ω
∗
(y)(x).
Note that
b
ω(x) = ω
1
(x)ω
2|x
for every x ∈ X(A), i.e.,
b
ω(x) can be understood as the un-normalized
conditional state of B given the outcome x on A, and similarly for
b
ω
∗
(y).
The conditioning map
b
ω extends uniquely to a positive linear mapping E(A) → V(B), which
I also denote by
b
ω, such that
b
ω(
b
x) =
b
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(
b
ω
∗
(y))
Accepted in Quantum 2019-06-24, click title to verify 11
for all y ∈ X(B). If f =
b
x, we have T (
b
x) = ω
1
(x)ω
2|x
∈ V(B)
+
. Thus, the range of T lies in
V(B).) In the same way,
b
ω
∗
defines a positive linear mapping
b
ω
∗
: E(B) → V(A). Notice that
b
ω need not take V(A)
∗
+
into V(B)
+
. This is the principal reason for working with E(A) rather
than V(A)
∗
. If
b
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 =
1
√
n
P
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
sa
(H). Since it
is precisely this inner product that makes L
sa
(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(
b
x) =
d
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
A
, η
A
) consisting of a probabilistic model A, an isomorphism γ
A
: A ' A, and a bipartite
state η
A
on A and A such that
(a) η
A
(x, γ
A
(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
A
(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
A
(x)) = 1/n for all x ∈ X(A),
the bipartite state η
t
defined by η
t
(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
t
)/2. We therefore can, and do, assume in what follows
that the chosen correlating state η
A
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
y
where δ
y
is the
unique state in which y has probability 1. Thus, η(x, y) = δ
y
(x). In this case, therefore, η = η
t
,
i.e., η is automatically symmetric.
8
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
A
(x, y) =
|hΨ, x ⊗ yi|
2
. 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
W
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
A
to identify A with its con-
jugate. Certainly one can define a bipartite state η
0
A
(x, y) = η
A
(x, γ
A
(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
η
0
(x, y) := |h(1 ⊗ J)
−1
Ψ, x ⊗ yi|
2
= |hΨ, x ⊗ Jyi|
2
gives a state on A(H ⊗ H), correlating along
the anti-unitary isomorphism γ
0
(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
δ
x
∈ Ω(A) with δ
x
(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
(a, γ
A
(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
(a, · ) for a ∈ E(A).
9
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
b
x ∈ E(A), and write x for γ
A
(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
x
.
Lemma 1. For every α ∈ V(A)
+
, there exists a test E such that
α =
X
x∈E
α(x)δ
x
(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
1
where ω correlates some test E ∈ M(A) with E ∈ M(A) along the bijection
8
It follows that, where δ
x
and δ
y
are the unique states making x and y certain, we have δ
y
(x) = δ
x
(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
x
(y) is a valid non-signaling probabilty weight on A and A.
9
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 =
P
x∈E
ω
2
(x)ω
1|x
. Since ω
is correlating we have ω
1|x
(x) = 1. Thus, by sharpness, ω
1|x
= δ
x
. Hence, α =
P
x∈E
ω
2
(x)δ
x
. It
follows that ω
2
(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
A
is an isomorphism state.
Proof: We need to show that
c
η
A
: E(A) → V(A) is an order-isomorphism. Since E(A) and V(A)
have the same dimension, it is enough to show that
c
η
A
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 =
P
x∈E
α(x)δ
x
. Since η
A
(x, x) = 1/n, we have
c
η
A
(x) =
1
n
δ
x
for every
x ∈ X(A). Hence,
c
η
A
(
P
x∈E
nα(x)x) = α.
Lemma 3. Every a ∈ E(A) has a representation a =
P
x∈E
t
x
x for some test E ∈ M(A) and
some coefficients t
x
.
Proof: If a ∈ E(A)
+
, then by Lemma 1,
c
η
A
(a) =
P
x∈E
t
x
δ
x
for some E ∈ M(A) and coefficients
t
x
≥ 0. By Lemma 2,
b
η
A
is an order-isomorphism. Applying
b
η
−1
A
to the expansion above gives
a =
P
x∈E
t
x
x. For an arbitrary a ∈ E(A), we have a = a
1
− a
2
with a
1
, a
2
∈ E(A)
+
. Choose
N ≥ 0 with a
2
≤ Nu. Thus, b := a + Nu
A
= a
1
+ (Nu
A
− a
2
) ≥ 0. So b :=
P
x∈E
t
x
x for some
E ∈ M(A) and thus
a = b − Nu
A
=
X
x∈E
t
x
x − N(
X
x∈E
x) =
X
x∈E
(t
x
− N )x.
Lemma 4. The function ha, bi := η
A
(a, γ
A
(b)) is an inner product on E(A).
Proof: η
A
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 =
P
x∈E
t
x
x for some test E and coeffcients
t
x
. Now
ha, ai =
*
X
x∈E
t
x
x,
X
y∈E
t
y
y
+
=
X
x,y∈E×E
t
x
t
y
hx, yi =
X
x,y∈E×E
t
x
t
y
η
A
(x, y) =
1
n
X
x∈E
t
x
2
≥ 0.
This is zero only when all coefficients t
x
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 =
P
x∈E
t
x
x for some test E. Thus, for all y ∈ E we have ha, yi = t
y
≥ 0, whence,
a ∈ E(A)
+
.
Remarks There are several directions in which we can usefully modify the assumptions of Theorem
1.
(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
b
η
A
(E
+
),
from which it follows that
b
η is a linear isomorphism, and hence (as the vector spaces involved are
finite-dimensional) an homeomorphism. Thus,
b
η
A
(E
+
) is closed, and so, contains the closure of
V
◦
+
, i.e., V
+
. In other words,
b
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
A
, 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
A
: 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
A
. 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
sa
(H
A
). 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
x for every x ∈ E, where 0 ≤ t
x
≤ 1. This gives us a completely
positive linear mapping Φ : E(A) → E(A), namely Φ(a) = V aV . If t
x
> 0 for every x ∈ E, Φ
has a completely positive inverse Φ
−1
(a) = V
−1
aV
−1
. For each x ∈ E, the corresponding effect
b
x ∈ E(A) ' L
sa
(H) is the rank-one projection operator p
x
. It is easy to check that V p
x
V = t
x
p
x
,
i.e., that Φ(
b
x) = t
x
b
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
x
≤ 1 with
Φ(α)(x) = t
x
α(x)
for all all states α ∈ Ω(A). Equivalently, Φ
∗
(x) = t
x
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
x
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 =
P
x∈E
α(x)δ
x
. 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 =
P
x∈E
1
n
δ
x
. Then
Φ(ρ) =
1
n
P
x∈E
α(x)δ
x
=
1
n
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
A
into a new sub-normalized bipartite state ω, given by ω(x, y) =
η
A
(Φ
∗
x, y) for all x ∈ X(A), y ∈ X(B). Noticing that Φ
∗
(x) = t
x
x for every x ∈ E, we see that
ω correlates E with E: if x, y ∈ E with x 6= y, we have
ω(x, y) = η
A
(t
x
x, y) = t
x
η
A
(x, y) = 0.
In other words, ω is correlating. It follows that the normalized bipartite state
e
ω :=
1
ω(u
A
, u
A
)
ω
is likewise correlating. Since ω
1
= 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
e
α :=
1
u
A
(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 ' 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
A
iff η
A
(Φ
∗
(x), y) = η
A
(x, Φ
∗
(y)) for all x, y ∈ X(A), i.e., iff η
A
◦ (Φ
∗
⊗ id
A
) =
η
A
◦ (id
A
⊗ 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
(a, γ
A
(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
A
◦ (Φ
∗
⊗ id
A
) = η
A
◦ (id
A
⊗ Φ
∗
).
Accepted in Quantum 2019-06-24, click title to verify 16
For each outcome x ∈ X(A), let δ
x
denote the conditional state (η
A
)
1|x
. Then for all x ∈ E, and
all outcomes y ∈ X, we have
ω
1|x
(y) =
η
A
(Φ
∗
(y), x)
η
A
(Φ
∗
(u
A
), x)
=
η
A
(y, Φ
∗
(x))
η
A
(u
A
, Φ
∗
(x))
=
η
A
(y, t
x
x)
η
A
(u
A
, t
x
x)
=
η
A
(y, x)
η
A
(u
A
, x)
= (η
A
)
1|x
(y) = δ
x
(y).
It follows that ω
1|x
= δ
x
. It is easy to check that ω
1
= Φ((η
A
)
1
) = Φ(ρ) = α; also, by the law of
total probability (3), ω
1
=
P
x∈E
ω
2
(x)ω
1|x
=
P
x∈E
t
x
δ
x
, where t
x
= ω
2
(x). Thus, every state in
Ω(A) is a convex combination of the states δ
x
, and the cone generated by these states coincides
with V(A)
+
. It follows that
b
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
x
= 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
A
, 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
(c) A is a Jordan model.
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.
10
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 :=
b
µ ◦
b
ω
−1
is a reversible transformation on V(A)
with
b
µ = φ ◦
b
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].
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
B.
10
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.
11
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.
12
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<