Symmetry, Compact Closure and Dagger Compactness for Categories of Convex Operational Models

Abstract

In the categorical approach to the foundations of quantum theory, one begins with a symmetric monoidal category, the objects of which represent physical systems, and the morphisms of which represent physical processes. Usually, this category is taken to be at least compact closed, and more often, dagger compact, enforcing a certain self-duality, whereby preparation processes (roughly, states) are interconvertible with processes of registration (roughly, measurement outcomes). This is in contrast to the more concrete “operational” approach, in which the states and measurement outcomes associated with a physical system are represented in terms of what we here call a convex operational model: a certain dual pair of ordered linear spaces–generally, not isomorphic to one another. On the other hand, state spaces for which there is such an isomorphism, which we term weakly self-dual, play an important role in reconstructions of various quantum-information theoretic protocols, including teleportation and ensemble steering. In this paper, we characterize compact closure of symmetric monoidal categories of convex operational models in two ways: as a statement about the existence of teleportation protocols, and as the principle that every process allowed by that theory can be realized as an instance of a remote evaluation protocol—hence, as a form of classical probabilistic conditioning. In a large class of cases, which includes both the classical and quantum cases, the relevant compact closed categories are degenerate, in the weak sense that every object is its own dual. We characterize the dagger-compactness of such a category (with respect to the natural adjoint) in terms of the existence, for each system, of a symmetric bipartite state, the associated conditioning map of which is an isomorphism.

1 Categorical Semantics and Quantum Foundations

One natural way to formalize a physical theory is as some kind of category, ${\mathcal C}$, the objects of which are the systems, and the morphisms of which are the processes, contemplated by that theory. In order to provide some apparatus for representing compound systems, it is natural to assume further that ${\mathcal C}$ is a symmetric monoidal category. In the categorical semantics for quantum theory pioneered by Abramsky and Coecke [1], Selinger [38, 39], and others (e.g., [2, 3, 7, 17]), it is further assumed that ${\mathcal C}$ is at least compact closed, and more usually, dagger compact. This last condition enforces a certain self-duality, in that there is a bijection between the states of a system $A \in {\mathcal C}$, represented by elements of ${\mathcal C}(I,A)$, and and the measurement-outcomes associated with that system, represented by elements of ${\mathcal C}(A,I)$. The motivating example here is the category FDHilb of finite-dimensional complex Hilbert spaces and unitary mappings—that is, the category of finite-dimensional “closed” quantum systems and unitary processes. Many of the information-processing features of finite-dimensional quantum systems occur in any dagger-compact category, notably, conclusive (that is, post-selected) teleportation and entanglement-swapping protocols. On the other hand, if our interest in a categorical reformulation of quantum theory is mainly foundational, rather than strictly one of systematization, these strong structural assumptions need further justification, or at any rate, further motivation.

There is an older tradition, stemming from Mackey’s work on the foundations of quantum mechanics [35], in which an individual physical (or, more generally, probabilistic) system is represented by a set of states, a set of observables or measurements, and an assignment of probabilities to measurement outcomes, conditional upon the state. From this basic idea, one is led to a representation of systems by pairs of ordered real vector spaces—the convex operational models of our title—and of physical processes, by certain positive linear mappings between such spaces. The motivating example is the category of Hermitian parts of $C^{\ast }$ algebras and completely positive mappings.

This “convex operational” approach, in contrast to the categorical one, is conservative of classical probabilistic concepts, but liberal as to how systems may be combined and transformed, so long as this probabilistic content is respected. In particular, there is no standing assumption of monoidality; rather, systems are combined using any of a variety of “non-signaling” products. Nor is there, in general, any hint of the kind of self-duality mentioned above—indeed, the natural dual object for a convex operational model is not itself an operational model. Nevertheless, here again various familiar “quantum” phenomena—such as no-cloning and no-broadcasting theorems, information-disturbance tradeoffs, teleportation and entanglement swapping protocols, and ensemble steering—emerge naturally and in some generality [8, 9, 12, 15]. A key idea here is that of a remote evaluation protocol [9] (of which teleportation is a special case), which reduces certain kinds of dynamical processes to purely classical conditioning.

It is obviously of interest to see how far such convex operational theories can be treated formally, that is, as categories, and more especially, as symmetric monoidal categories; equally, one would like to know how much of the special structure assumed in the categorical approach can be given an operational motivation. Some first steps toward addressing these issues are taken in [13, 14]. Here, we aim to make further progress, albeit along a somewhat narrower front. We focus on symmetric monoidal categories of convex operational models—what we propose to call probabilistic theories. We show that such a theory admits a compact closed structure if and only if every system allowed by the theory can be teleported (conclusively, though not necessarily with probability 1) through a copy of itself—or, equivalently, if and only if every process contemplated by the theory can be represented as a remote evaluation protocol. We then specialize further, to consider weakly self-dual theories, in which for every system A there is a bipartite state $\gamma _{A}$ on $A \otimes A$ corresponding to an isomorphism between A and its dual, and an effect corresponding to its inverse. (Such state spaces figure heavily in earlier treatments of teleportation protocols [9] and ensemble steering [12] in general probabilistic theories.) We show that if the state implementing weak self-duality can be chosen to be symmetric for every A, then a weakly self-dual monoidal probabilistic theory is not merely compact closed, but dagger compact.

Organization and Notation

Sections 2 and 3 provide quick reviews of the category-theoretic and the convex frameworks, respectively, mainly following [1] for the former and [8, 9, 12–14] for the latter. Section 4 makes precise what we mean by a monoidal probabilistic theory, as a symmetric monoidal category of convex operational models, and establishes that all such theories have the property of allowing remote evaluation [9]; when the state spaces involved are weakly self-dual, teleportation arises as a special case. Section 5 contains the results on categories of weakly self-dual state spaces described above. Section 6 discusses some of the further ramifications of these results.

We assume that the reader is familiar with basic category-theoretic ideas and notation, as well as with the probabilistic machinery of quantum theory. We write ${\mathcal C}, {\mathcal D}$ etc. for categories, $A \in {\mathcal C}$, to indicate that A is an object of ${\mathcal C}$, and ${\mathcal C}(A,B)$ for the set of morphisms between objects $A, B \in {\mathcal C}$. Except as noted, all vector spaces considered here will be finite-dimensional and real. We write $\mbox {\bf Vec}_{\mathbb {R}}$ for the category of finite-dimensional real vector spaces and linear maps. The dual space of a vector space A is denoted by $A^{\ast }$. An ordered vector space is a real vector space V equipped with a regular—that is, closed, convex, pointed, generating—cone $V_{+}$, and ordered by the relation $x \leq y \Leftrightarrow y - x \in A_{+}$. A linear mapping $\phi : V \rightarrow W$ between ordered linear spaces V and W is positive if $\phi (V_{+}) \subseteq W_{+}$. We write ${\cal L}_{+}(A,B)$ for the cone of positive linear mappings from A to B. The special case in which $B = {\mathbb R}$, the positive linear functionals on A, is the dual cone of $A_+$, denoted $A_+^*$. The category of ordered linear spaces and positive linear maps we denote by Ordlin. Finally, we make the standing assumption that, except where otherwise indicated, all vector spaces considered here are finite dimensional.

2 The Category-Theoretic Perspective

A monoidal category [36] is a category ${\mathcal C}$ equipped with a bifunctor¹ $\otimes : {\mathcal C} \times {\mathcal C} \rightarrow {\mathcal C}$, a distinguished unit object I, and natural associativity and left and right unit deletion isomorphisms,

$$\begin{array}{rll} \alpha_{A,B,C} &:& A \otimes (B \otimes C) \cong (A \otimes B) \otimes C, \\ \lambda_{A} : I \otimes A &\cong& A, \qquad\quad \rho_{A} : A \otimes I \cong A, \end{array} $$

subject to some coherence conditions (for which, see [36]). If these isomorphisms are identities, the category is called strict monoidal. Every monoidal category is equivalent to a strict one, so setting $A\otimes I = A$ etc, is harmless and we will do this throughout.

A symmetric monoidal category (SMC) is a monoidal category further equipped with a natural family of symmetry isomorphisms,

$$\sigma_{A,B} :A \otimes B \cong B \otimes A, $$

again, subject to some coherence conditions (for which, again, see [36]). Unlike the other isomorphisms, these symmetry isomorphisms cannot generally be made strict.

Examples of SMCs include commutative monoids (as one-object categories), the category of sets and mappings (with $A \otimes B = A \times B$), and—of particular relevance for us—the category of (say, finite-dimensional) vector spaces over a field K and K-linear maps, with $A \otimes B$ the usual tensor product. Another source of examples comes from logic: one can regard the set of sentences of a logical calculus as a category, with proofs, composed by concatenation, as morphisms. In this context, one can take conjunction, $\wedge$, as a monoidal product.

Much more broadly, if somewhat less precisely, if one views the objects of a category ${\mathcal C}$ as “systems” (of whatever sort), and morphisms as “processes” between systems, then a natural interpretation of the product in a symmetric monoidal category is as a kind of accretive composition: $A \otimes B$ is the system that consists of the two systems A and B sitting, as it were, side by side, without any special interaction; $f \otimes g$ represents the processes $f : A \rightarrow X$ and $g : A \rightarrow Y$ acting in parallel. Taking this point of view, it is helpful to regard processes of the form $I \rightarrow A$, where I is the monoidal unit in ${\mathcal C}$, as states, associated with ways of preparing the system A. Similarly, we regard processes of the form $a : A \rightarrow I$ as “effects”, or measurement-outcomes. We shall henceforth adhere to the convention of denoting states by lower-case Greek letters $\alpha , \beta , ...$ and effects, lower-case Roman letters $a, b, ...$.

In any monoidal category, one can regard endomorphisms $s \in {\mathcal C}(I,I)$ as “scalars” acting on elements of ${\mathcal C}(A,B)$ by $sx = s \otimes x$. In every monoidal category, ${\mathcal C}(I,I)$ is a commutative monoid, even if ${\mathcal C}$ is not symmetric. When ${\mathcal C}(I,I)$ is isomorphic to a particular monoid S, we shall say that ${\mathcal C}$ is a symmetric monoidal category over S.

2.1 Compact Closed Categories

A dual for an object A of a symmetric monoidal category ${\mathcal C}$ is an object B and two morphisms, the unit, $\eta : I \rightarrow B \otimes A$ (not to be confused with the tensor unit I) and the co-unit, $\epsilon : A \otimes B \rightarrow I$, such that

$$\begin{array}{*{20}{l}} {A\xrightarrow{{{1_{A}} \otimes \eta }}A \otimes B \otimes A\xrightarrow{{\epsilon \otimes {1_{A}}}}A = {1_{A}}} \hfill \\ {B\xrightarrow{{\eta \otimes {1_{B}}}}B \otimes A \otimes B\xrightarrow{{{1_{B}} \otimes \epsilon }}B = {1_{B}}.} \hfill \\ \end{array}$$

(1)

Duals are unique up to a canonical isomorphism. Indeed, if $(B_{1},\eta _{1},\epsilon _{1})$ and $(B_{2},\eta _{2},\epsilon _{2})$ are duals for A, then $\phi := (\ensuremath {1_{B_{2}}} \otimes \epsilon _{1}) \circ (\eta _{2} \otimes \ensuremath {1_{B_{1}}}) : B_{1} \rightarrow B_{2}$ has inverse $\phi ^{-1} = ( \ensuremath {1_{B_{1}}}\otimes \epsilon _{2}) \circ (\eta _{1} \otimes \ensuremath {1_{B_{2}}})$; moreover, $\eta _{2} = (\phi \otimes \ensuremath {1_{A}}) \circ \eta _{1}$. Some, for example the authors of [17], use the term compact structure to refer to what we are calling a dual, i.e., a particular choice of $(A',\eta _{A},\epsilon _{A})$ for a given object $A \in {\mathcal C}$ or, when applied to a category, a particular choice of dual for each object.

A symmetric monoidal category ${\mathcal C}$ is compact closed ² if for every object A in the category, there is a dual, $(A',\eta _{A},\epsilon _{A})$, where $A'$ is an object of the category.³ As thus defined, compact closedness is a property of the SMC ${\mathcal C}$, not an additional structure: it requires the existence of at least one dual for each object, but not the explicit specification of a distinguished one. The alternative definition of compact closed category, which differs only in requiring a choice of duals be specified [32], is perhaps more common. Owing to the uniqueness up to isomorphism mentioned above, the various possible choices of duals are largely—but not entirely—equivalent. A compact structure is said to be degenerate iff $A' = A$; a compact closed category ${\mathcal C}$ with a distinguished compact structure is said to be degenerate if every object’s compact structure is degenerate. This does depend on an explicit choice of duals, and thus imposes some non-trivial structure⁴ beyond compact closure. This is the setting that will most interest us below.

Remark 1

If $(A',\eta _{A},\epsilon _{A})$ and $(B',\eta _{B}, \epsilon _{B})$ are duals for objects $A, B \in {\mathcal C}$, then we can construct a canonical dual $(A' \otimes B', \eta _{AB}, \epsilon _{AB})$ for $A \otimes B$ by setting $\eta _{AB} = \tau \circ (\eta _{A} \otimes \eta _{B})$ and $\epsilon _{AB} = (\eta _{A} \otimes \eta _{B}) \circ \tau ^{-1}$, where

$$\tau = 1_{A'} \otimes \sigma_{AB'} \otimes 1_{B} : (A' \otimes A) \otimes (B' \otimes B) \simeq (A' \otimes B') \otimes (A \otimes B).$$

Since all duals are isomorphic we are free to assume that $A\otimes B$ has this particular dual.

In any compact closed category, an assignment $A \mapsto A'$ extends to a contravariant functor $(-)' : {\mathcal C}^{op} \rightarrow {\mathcal C}$, called the adjoint, taking morphisms $\phi : A \rightarrow B$ to $\phi ' : B' \rightarrow A'$ defined by:

(2)

The functor ′ is nearly involutive, in that there are natural isomorphisms $w_{A}: A'' \rightarrow A$. To say that ′ is involutive is just to say that $A''=A$ and $\phi = \phi ''$; note that this does not imply that $w_{A} = \ensuremath {1_{A}}$.

Remark 2

In the classic treatment of coherence for compact closed categories in [32], one has that $\sigma \circ \eta _{A} = (1_{A} \otimes w_{A}) \circ \eta _{A'}$; a similar condition holds for $\epsilon$ ([32], eq. (6.4)ff.). In the case of a degenerate category, this implies that

$$ \sigma \circ \eta_{A} = (1_{A} \otimes w_{A}) \circ \eta_{A}. $$

(3)

It is easy to show that if the units—or, equivalently, co-units—are symmetric, in the sense that, for every object $A \in {\mathcal C}$, $\eta _{A} = \sigma _{A,A} \circ \eta _{A}$ or, equivalently, $\epsilon _{A} = \epsilon _{A} \circ \sigma _{A,A}$, then the the functor ′ is involutive. However the converse does not necessarily hold, unless $w_{A} = \ensuremath {1_{A}}$. In general, it is not clear what coherence requirements are appropriate for the degenerate categories we consider, nor whether the functors involved are always strict. Therefore, in Section 5 we will establish explicitly that the involutiveness of the adjoint is equivalent to the symmetry of the unit and co-unit for the compact closed categories of convex operational models considered in this paper.

Remark 3

For an arbitrary degenerate compact closed category, there is no guarantee that $\eta _{A}$ will be symmetric. (We thank Peter Selinger (P. Selinger, 2010, personal communication. See also [40]) for supplying a nice example involving a category of plane tangles). Thus, it is a non-trivial constraint on such a category that the canonical adjoint be an involution. This will be important below.

2.2 Daggers

A dagger category [1, 39] is a category ${\mathcal C}$ together with an involutive functor $(-)^{\dagger } : {\mathcal C}^{op} \rightarrow {\mathcal C}$ that acts as the identity on objects. That is, $A^{\dagger } = A$ for all $A \in {\mathcal C}$, and, if $f \in {\mathcal C}(A,B)$, then $f^{\dagger } \in {\mathcal C}(B,A)$, with

$$(g \circ f)^{\dagger} = f^{\dagger} \circ g^{\dagger} \quad \text{and} \quad f^{\dagger \dagger} = f $$

for all $f \in {\mathcal C}(A,B)$ and $g \in {\mathcal C}(B,C)$.⁵ We say that f is unitary iff $f^{\dagger } = f^{-1}$. A dagger-monoidal category is a symmetric monoidal category with a dagger such that (i) all the canonical isomorphisms defining the symmetric monoidal structure are unitary, and (ii)

$$(f \otimes g)^{\dagger} = f^{\dagger} \otimes g^{\dagger} $$

for all morphisms f and g in ${\mathcal C}$. Finally, a dagger-monoidal category ${\mathcal C}$ is dagger compact if it is compact closed and

$$\eta_{A} = \sigma_{A,A'} \circ \epsilon_{A}^{\dagger} $$

for every A, i.e.:

commutes. In the case of a degenerate compact closed category, the canonical adjoint ′ functions as a dagger if it is involutive. However, as remarked above, this is a nontrivial condition.

In the work of Abramsky and Coecke [1], a symmetric monoidal category is interpreted as a physical theory, in which a morphism $\alpha : I \rightarrow A$ is interpreted as a state on the system A, and a morphism $b = \beta ^{\dag } : A \rightarrow I$ is understood as the registration of an effect (e.g., a measurement outcome) on A. The scalar $\beta ^{\dag } \circ \alpha : I \rightarrow I$ is understood, somewhat figuratively in the abstract setting, as the “probability” that the given effect will occur when the given state obtains.⁶ This raises the obvious question of how to implement the compelling idea that probabilities should be identified with real numbers in the interval $[0,1]$ without passing through Hilbert space. One way to do this is simply to posit a mapping $p : {\mathcal C}(I,I) \rightarrow [0,1]$, whereby the scalars of ${\mathcal C}$ can be interpreted probabilistically. Another is to examine the symmetric monoidal possibilities in cases in which the category consists, ab initio, of concretely described probabilistic models of a reasonably simple and general sort. In this paper, we concentrate on this second strategy. As a first step, in the next section we describe the kinds of concrete probabilistic models we have in mind.

3 Convex Operational Models and their Duals

The more traditional approach to modeling probabilistic physical theories [16, 35] begins by associating to each individual physical (or other probabilistic) system a triple $(X,\Sigma ,p)$—sometimes called a Mackey triple—where $\Sigma$ is a set of possible states, X is a set of possible measurement-outcomes, and $p : X \times \Sigma \rightarrow [0,1]$ assigns to each pair $(x,s)$ the probability, $p(x,s)$, that x will occur, if measured, when the system’s state is s.

This minimal apparatus can be “linearized” in a natural way. The probability function p gives us a mapping $\Sigma \rightarrow [0,1]^{X}$, namely $s \mapsto p( \cdot , s)$. We can plausibly identify each state $s \in \Sigma$ with its image under this mapping (thus identifying states if they cannot be distinguished statistically by the outcomes in X). Having done so, let $\Omega$ denote the point-wise closed, hence compact, convex hull of $\Sigma \subseteq [0,1]^{X}$, representing the set of possible probabilistic mixtures of states in $\Sigma$. Every measurement outcome $x \in X$ can be represented by the affine functional $a_{x} : \Omega \rightarrow [0,1]$, given by $a_{x}(\alpha ) = \alpha (x)$ for all $\alpha \in \Omega$. More broadly, we can regard any affine functional $a : \Omega \rightarrow [0,1]$ as representing a mathematically possible measurement outcome, having probability $a(\alpha )$ in state $\alpha \in \Omega$. Such functionals are called effects in the literature.

In general, the (mixed) state space $\Omega$ we have just constructed will have infinite affine dimension. Accordingly, for the next few paragraphs, we suspend our standing finite-dimensionality assumption. Now, any compact convex set $\Omega$ can be embedded, in a canonical way, as a base for the positive cone $V_{+}(\Omega )$ of a regularly ordered linear $V(\Omega )$ [4]. This means that every $\rho \in V_{+}(\Omega )$ has the form $\rho = t \alpha$ for a unique scalar $t \geq 0$ and a unique vector $\alpha \in \Omega$ (hence, $\Omega$ spans $V(\Omega )$). This space $V(\Omega )$ is complete in a natural norm, the base norm, the unit ball of which is given by the closed convex hull of $\Omega \cup -\Omega$. Moreover, $V(\Omega )$ has the following universal property: every bounded affine mapping $L : \Omega \rightarrow \textbf {M}$, where $\textbf {M}$ is any real Banach space, extends uniquely to a bounded linear mapping $L : V(\Omega ) \rightarrow \textbf {M}$.

In particular, every affine functional on $\Omega$—in particular, every effect—extends uniquely to a linear functional in $V(\Omega )^{\ast }$. In particular, there is a unique unit functional $u_{\Omega } \in V(\Omega )^{\ast }$ such that $u_{\Omega }(\alpha ) = 1$ for $\alpha \in \Omega$, and $\Omega = u_{\Omega }^{-1}(1) \cap V_{+}(\Omega )$. Thus, effects correspond to positive functionals $a \in V(\Omega )^{\ast }$ with $0 \leq a \leq u_{\Omega }$.

One often regards any effect $a \in V(\Omega )^{\ast }$ as a bona fide measurement outcome. This is the point of view, e.g., of [8, 9]. However, we may sometimes wish to privilege certain effects as “physically accessible”. This suggests the following more general formulation:

Definition 4

A convex operational model (COM) is a triple $(A,A^{\#},u_{A})$ where

• (i) A is a complete base-normed space with (strictly positive) unit functional $u_{A}$, and

• (ii) $A^{\#}$ is a weak-$\ast$ dense subspace of $A^{\ast }$, ordered by a chosen regular cone $A^{\#}_{+} \subseteq A^{\ast }_{+}$ containing $u_{A}$.

An effect on A is a functional $a \in A^{\#}_{+}$ with $a \leq u_{A}$.

Henceforth, where no ambiguity seems likely, we write A for the triple $(A,A^{\#},u_{A})$. Also, we now revert, for the balance of this paper, to our standing assumption that all COMs are finite-dimensional. In this case, the weak-$\ast$ density assumption above simply says that $A^{\#} = A^{\ast }$ as vector spaces. Even in this situation, however, the chosen cone $A^{\#}_{+}$ will generally be smaller than the dual cone $A^{\ast }_{+}$, so the positive cone $(A^{\#})_{+}^{\ast }$ will in general be larger than $A_{+}$. It is useful to regard normalized elements of the former cone as mathematically consistent probability assignments on the effects in $A^{\#}$, from which the model singles out those in $A_{+}$ as physically possible. In the special case in which $A^{\#}_+ = A^{\ast }_+$—as, e.g., in the case of quantum systems—we shall say that the COM A is saturated.

Example 5

Let E be a finite set, thought of as the outcome set for a discrete classical experiment. Take $A = {\mathbb {R}}^{E}$, with $A_{+}$ the cone of non-negative functions on E, and let $u_{A}(f) = \sum _{x \in E} f(x)$. Then $\Omega = u^{-1}(1)$ is simply the set of probability weights on E. Geometrically, this last is a simplex. In finite dimensions, every simplex has this form. Accordingly, we say a COM is classical iff its normalized state space is a simplex.

Example 6

Let H be a finite-dimensional complex Hilbert space, and let $A = {\cal L}_{h}(\textbf {H})$, the space of Hermitian operators $a : \textbf {H} \rightarrow \textbf {H}$, with the usual positive cone, i.e, $A_{+}$ consists of all Hermitian operators of the form $a^{\dagger } a$. Let $u_{A}(a) = \text {Tr}(a)$. Then $\Omega _{A}$ is the convex set of density operators on H, i.e., the usual space of mixed quantum states.

Example 7

Let $(X,\Sigma ,p)$ be any Mackey triple. Construct the state-space $\Omega$ and the associated ordered Banach space $V(\Omega )$ as described above. Letting $A^{\#}_{+}$ be the cone in $V^{\ast }(\Omega )$ generated by the evaluation functionals $a_{x}$, $x \in X$, we have a convex operational model. This will be finite-dimensional iff the span of (the image of) $\Sigma$ in ${\Bbb R}^{X}$ is finite dimensional.

3.1 Processes as Positive Mappings

Definition 8

A morphism of COMs from $(A,A^{\#},u_{A})$ to $(B,B^{\#},u_{B})$ is a positive linear map $\phi : A \rightarrow B$ such that the usual linear adjoint map $\phi ^* : B^{*} \rightarrow A^{*}$ is positive with respect to the designated cones $A^{\#}_{+}$ and $B^{\#}_{+}$.

The set of morphisms of COMs from A to B is clearly a sub-cone of ${\cal L}_{+}(A,B)$. It is clear that the composition of two mappings of COMs is again a mapping of COMs, so that COMs form a concrete category.

Definition 9

Let A and B be COMs. A process from A to B is a morphism $\phi : A \rightarrow B$ such that, for every state $\alpha \in \Omega _{A}$, $u_{B}(\phi (\alpha )) \leq 1$, or, equivalently, if $\phi ^{\ast }(u_{B}) \leq u_{A}$.

If $\phi : A \rightarrow B$ is a process, we can regard $u_{B}(\phi (\alpha ))$ as the probability that the process represented by $\phi$ occurs. If we regard ${\mathbb {R}}$ as an COM with $u_{\mathbb {R}}$ the identity mapping on ${\mathbb {R}}$, this is consistent with our understanding of $a(\alpha )$ as the probability of the effect $a : A \rightarrow {\mathbb {R}}$ occurring. Notice that a positive linear map $\phi : {\mathbb {R}} \rightarrow A$ is a process if and only if $\phi (1)$ is a sub-normalized state, while a positive functional $f: A \rightarrow {\mathbb {R}}$ is a process if and only if $f \in A^{\#}_{+}$ and $f \leq u_{A}-$in other words, if and only if f is an effect. Finally, since $\Omega _{A}$ is compact, $u_{A}(\phi (\alpha ))$ attains a maximum value, say M on $\Omega _{A}$. $M^{-1}\phi$ is a process, so every morphism of COMs is a positive multiple of a process.

3.2 Bipartite States and Composite Systems

Given two separate systems, represented by COMs A and B, we should expect that any state of the composite system AB will induce a joint probability assignment $p(a,b)$ on pairs of effects $a \in A^{\#}$, $b \in B^{\#}$. If the two systems can be prepared independently, we should also suppose that, for any two states $\alpha \in A$ and $\beta \in B$, the product state $\alpha \otimes \beta$, given by $(\alpha \otimes \beta )(a,b) = \alpha (a) \beta (b)$, will be a legitimate joint state. Finally, if the two systems do not interact, the choice of measurement made on A ought not to influence the statistics of measurement outcomes on B, and vice versa. This latter “no-signaling” condition is equivalent [44] to the condition that the joint probability assignment p extends to a bilinear form on $A^{\#} \times B^{\#}$, normalized so that $p(u_{A}, u_{B}) = 1$. Abstractly, then, one makes the following definition.

Definition 10

A (normalized, non-signaling) bipartite state between convex operational models A and B is a bilinear form $\omega : A^{\#} \times B^{\#} \rightarrow {\Bbb R}$ that is positive, in the sense that $\omega (a,b) \geq 0$ for all effects $a \in A^{\#}$ and $b \in B^{\#}$, and normalized (satisfies $\omega (u_{A},u_{B}) = 1$).

Implicit in this definition is the assumption, lately called local tomography [18], that a joint state is determined by the joint probabilities it assigns to measurement outcomes associated with the local systems A and B. As has been pointed out by many authors, e.g. [6, 15, 33], this condition is violated in both real and quaternionic quantum theory, and can therefore be made to serve as an axiom separating standard complex QM from these. A more general notion of non-signaling bipartite state would merely associate, rather than identify, each such state with a positive bilinear form on $A^{\#} \times B^{\#}$. See the remarks following Definition 11 for more on this.

It is clear that any product $\omega = \alpha \otimes \beta$ of normalized states $\alpha \in A$ and $\beta \in B$ defines a non-signaling state; hence, so do convex combinations of product states. Non-signaling states arising in this way, as mixtures of product states, are said to be separable or unentangled. An entangled non-signaling state is one that is not a convex combination of product states. Many of the basic properties of entangled quantum states actually hold for entangled states in this much more general setting [8, 33].

The space $\mathfrak {B}(A^{\#},B^{\#})$ of all bilinear forms on $A^{\#} \times B^{\#}$, ordered by the cone of all positive bilinear forms, is the maximal tensor product, $A \otimes _{max} B$, of A and B. This notation is reasonable, since (in finite dimensions), $\mathfrak {B}(A^{\#},B^{\#})$ is one model of the tensor product $(A^{\#})^{\ast } \otimes (B^{\#})^{\ast }$—thus, of the vector-space tensor product $A \otimes B$.⁷ Ordering $A \otimes B$ instead by the generally much smaller cone of unentangled states, that is, the cone generated by the product states, gives the minimal tensor product, $A \otimes _{min} B$. It is important to note that these coincide only when A or B is classical [8]. If A and B are quantum state spaces, then the cone of bipartite density matrices for the composite system lies properly between the maximal and minimal cones. This indicates the need for something more general:

Definition 11

A (locally tomographic) composite of COMs A and B is a convex operational model $(AB, (AB)^{\#},u_{AB})$, such that $AB \subseteq \mathfrak {B}(A^{\#},B^{\#})$, with $u_{AB} = u_{A} \otimes u_{B}$, $\alpha \otimes \beta \in (AB)_{+}$ for all $\alpha \in A_{+}$, $\beta \in B_{+}$, and $a \otimes b \in (AB)^{\#}_{+}$ for all $a \in A^{\#}_{+}, b \in B^{\#}_{+}$.

It is worth stressing that there are perfectly reasonable theories that are not locally tomographic. Indeed, one of these is quantum mechanics over the real, rather than complex, scalars. We might more generally define a composite in the wide sense of COMs A and B to be a COM $(AB,(AB)^{\#},u_{AB})$, together with (i) a positive linear embedding (injection) $i : A \otimes _{min} B \rightarrow AB$, and (ii) a positive map $r : AB \rightarrow A \otimes _{max} B$, surjective as a linear map, such that for all $a \in A^{\#}, b \in B^{\#}$,

$$r(i(\alpha \otimes \beta))(a,b) = a(\alpha) b(\beta),$$

i.e., $r \circ i$ is the canonical embedding of $A \otimes _{min} B$ in $A \otimes _{max} B$. However, we shall make no use of this extra generality here. Accordingly, we assume henceforth that all composites are locally tomographic, as per Definition 11 above.

3.3 Conditioning and Remote Evaluation

A bipartite state $\omega$ on A and B gives rise to a positive linear mapping $\hat {\omega } : A^{\#}\rightarrow B$ with

$$b(\hat{\omega}(a)) = \omega(a,b) $$

for all $a \in A^{\#}$ and $b \in B^{\#}$; dually, a bipartite effect $f \in (AB)^{\#}$ gives rise to a linear map $\hat {f}: A \rightarrow B^{\#}$, given by $\hat {f}(\alpha )(\beta ) = f(\alpha \otimes \beta )$, and subject to $f(\alpha ) \leq u_{B}$ for all $\alpha \in \Omega _{A}$.

The marginals of $\omega$ are given by $\omega _{B} = \hat {\omega }(u_{A})$ and $\omega _{A} = \hat {\omega }^{\ast }(u_{B})$. Note that $\omega$ is normalized iff $u_{B}(\hat {\omega }(u_{A})) = 1$. We can define the conditional states of A and B given (respectively) effects $b \in [0,u_{B}]$ and $a \in [0,u_{A}]$ by

$$\omega_{A|b} := \frac{\hat{\omega}^*(b)}{\omega_{B}(b)} \ \ \text{and} \ \ \omega_{B|a} := \frac{\hat{\omega}(a)}{\omega_{A}(a)} $$

provided the marginal probabilities $\omega _{B}(b)$ and $\omega _{A}(a)$ are non-zero. Accordingly, we refer to $\hat {\omega }(a)$ as the un-normalized conditional state. Notice that the linear adjoint, $\hat {\omega }^{*} : B^{\#} \rightarrow A$, of $\hat {\omega }$ represents the same state, but evaluated in the opposite order: $\hat {\omega }^{*}(b)(a) = \hat {\omega }(a)(b) = \omega (a,b)$.

Lemma 12

(Remote Evaluation 1) Let A,B and C be convex operational models. For any bipartite effect on $f \in (AB)^{\ast }$ and any bipartite state $\omega \in BC$ , and for any state $\alpha \in A$,

$$(f \otimes -)(\alpha \otimes \omega) = \hat{\omega}(\hat{f}(\alpha)).$$

Proof

It is straightforward that this holds where f and $\omega$ are a product effect and a product state, respectively. Since these generate $(AB)^{\ast }$ and AB, the result follows. □

Operationally, this says that one can implement the transformation $\hat {\omega }\,\circ \hat {f}$ by preparing the tripartite system ABC in state $\alpha \otimes \omega$, where $\alpha \in A$ is the “input” state to be processed, and then making a measurement on AB, of which f is a possible outcome: the un-normalized conditional state of C, given the effect f on AB, is exactly $\hat {\omega }(\hat {f}(\alpha ))$. Thus, the process $\phi := \hat {\omega }\, \circ \hat {f} : A \rightarrow C$ becomes a special case of conditioning. In [9], we have called this protocol remote evaluation. Note that conclusive, or post-selected, teleportation arises as the special case of remote evaluation in which, up to some specified isomorphism, $C \simeq A$ and $\hat {\omega } \circ \hat {f} \simeq \ensuremath {1_{A}}$. We shall return to this point below.

4 Categories of Convex Operational Models

We now wish to chart some connections between the two approaches outlined above. In the first place, we will bring some category-theoretic order to the concepts developed in the preceding section.

Since morphisms of COMs compose, we can define a category Com of all convex operational models and COM morphisms. As described in Section 3.1, the hom-sets ${\bf Com}(A,B)$ are themselves cones. Let I denote the COM $\mathbb R$ with its standard cone and order unit, i.e. $I = ({\mathbb R},{\mathbb R}, 1)$.

Definition 13

A category of COMs is a subcategory ${\mathcal C}$ of Com such that: (i) ${\mathcal C}(A,B)$ is a (regular) sub-cone of the cone ${\bf Com}_{+}(A,B)$; (ii) ${\mathcal C}$ contains the distinguished COM I; (iii) ${\mathcal C}(I,A) \simeq A$; and (iv) ${\mathcal C}(A,I) \simeq A^{\#}$. We call a such a category finite-dimensional if all state spaces $A \in {\mathcal C}$ are finite dimensional.

A more general definition would require only that ${\mathcal C}(A,B)$ be some set of processes, in the sense of Definition 9, between A and B. However, we should like to be able to construct random mixtures of processes, so ${\mathcal C}(A,B)$ should at least be convex. Allowing for the taking of limits as a reasonable idealization, it is plausible to take ${\mathcal C}(A,B)$ also to be closed. Finally, one should require that, if $\phi : A \rightarrow B$ is a physically valid process, then so is $t\phi$ for any $t \in [0,1]$–this reflecting the possibility of attenuating a process (as, for instance, by some filter that admits only a fraction t of incident systems, but otherwise leaves systems unchanged, or by in some other way conditioning its occurrence on an event assigned a probability less than 1). This much given, the physically meaningful processes between two systems should generate a closed, convex, pointed cone of positive mappings, as per Definition 13.

In [8, 9, 12], a probabilistic theory is defined, rather loosely, to be any class of COMs (or “probabilistic models”) that is equipped with some device, or devices, for forming composite systems. Tightening this up considerably, we make the following definition.

Definition 14

A monoidal category of COMs is a category of COMs equipped with a monoidal structure, such that (i) the monoidal unit is the COM I; (ii) for every $A, B \in {\mathcal C}$, $A \otimes B$ is a non-signaling composite in the sense of Definition 11.

While Definition 14 does not require it, in the rest of the paper we will assume that all monoidal categories of COMs are symmetric monoidal, and (in accordance with our standing assumtion), finite dimensional.

As an example, the category FDCom of all finite-dimensional convex operational models and positive mappings can be made into a monoidal category in two ways, using either the maximal or the minimal tensor product. Another example is the “box-world” considered, e.g., in [22, 41]: here, state spaces are constructed by forming maximal tensor products of basic systems, the normalized state spaces of which are two-dimensional squares. Another example is afforded by the category of quantum-mechanical systems, represented as the self-adjoint parts of complex matrix algebras. Here, the appropriate monoidal product of two systems A and B is what is sometimes referred to as the “spatial” tensor product, obtained by forming tensor products of the Hilbert spaces on which the A and B act, and taking the self-adjoint operators on this space.

4.1 Remote Evaluation Again

We now reformulate the conditioning maps and remote evaluation protocol discussed above in purely categorical terms. In fact, both make sense in any symmetric monoidal category ${\mathcal C}$. Suppose, then, that $\omega : I \rightarrow B \otimes A$ is a “bipartite state”, i.e, a state of the composite system $B \otimes A$. Then there is a canonical mapping $\hat {\omega } : {\mathcal C}(B,I) \rightarrow {\mathcal C}(I,A)$ given by

(4)

Dually, if $f \in {\mathcal C}(A \otimes B, I)$, there is a natural mapping $\hat {f} : {\mathcal C}(I,A) \rightarrow {\mathcal C}(B,I)$ given by

(5)

Note that if ${\mathcal C}$ is already a category of COMs, then $\hat {\omega }$ and $\hat {f}$ are exactly the maps discussed in the last section. This has a simple but important corollary, namely, that these maps are indeed morphisms of COMs. Another consequence is that any monoidal category of COMs is closed under conditioning – that is, if $\omega$ is a normalized bipartite state of such a theory, belonging, say, to a composite system AB, then for every effect a on A and b on B, the composite states $\omega _{B|a}$ and $\omega _{A|b}$ are indeed states of A and B, respectively (as opposed to merely being elements of $(A^{\#})^{\ast }$ and $(B^{\#})^{\ast }$).⁸

The remote evaluation protocol of Lemma 12 also has a purely category-theoretic formulation:

Lemma 15

(Remote Evaluation 2) Let $\omega : I \rightarrow B \otimes C$ and $f : A \otimes B \rightarrow I$ in ${\mathcal C}$. Then

$$ \hat{\omega}(\hat{f}(\alpha)) \ = \ (f \otimes \ensuremath{1_{C}})\circ (\ensuremath{1_{A}} \otimes \omega) \circ \alpha = (f \otimes \ensuremath{1_{C}}) \circ (\alpha \otimes \omega) $$

(6)

for all $\alpha \in {\mathcal C}(I,A)$. Dually, for every $\beta \in {\mathcal C}(I,B)$ , we have

$$ \hat{\omega}^{\ast}(\hat{f}^{\ast}(\beta)) \ = \ (\ensuremath{1_{A}} \otimes f) \circ (\omega \otimes \beta). $$

(7)

Proof

We prove (6), the proof of (7) being similar. Tensoring the diagram (5) with C (on the right) gives the right-hand triangle in the diagram below. Applying (4) to compute $\hat {\omega }(\hat {f}(\alpha ))$ gives the lower triangle. The square commutes by the bifunctoriality of the tensor.

Chasing around the diagram gives the desired result. □

Suppose that, in the preceding lemma, $\omega (1) \in A \otimes B$ is a normalized state, and $f : B \otimes C \rightarrow I$ is an effect, i.e, $0 \leq f \leq u_{BC}$. Then, in operational terms, the Lemma says that the mapping $\hat {\omega } \circ \hat {f}$ is represented, within the category ${\mathcal C}$, by the composite morphism $(f \otimes \ensuremath {1_{C}}) \circ (\ensuremath {1_{A}} \otimes \omega )$.⁹ In other words: preparing BC in joint state $\omega$, and then measuring AB and obtaining f , guarantees that the “un-normalized conditional state” of C is $\hat {\omega }(\hat {f}(\alpha ))$, where $\alpha$ is the state of A.

Remark 16

An important point here is that any process that factors as $\hat {\omega } \circ \hat {f}$ can be simulated by a remote evaluation protocol, using what amounts to classical conditioning–in particular, without need to invoke any mysterious “collapse” of the state, or for that matter, any other physical dynamics at all.

4.2 Teleportation, Conditional Dynamics and Compact Closure

Suppose that, in the remote evaluation protocol of Lemma 15, we have $C = A$. Suppose further that the mapping $\hat {\omega } : B^{\#} \rightarrow A$ has a right inverse — that is, suppose there exists a positive linear map $\hat {r}:A \to B^\#$ such that $\hat {\omega } \circ \hat {r} = \ensuremath {1_{A}}$. Then we can re-scale $\hat {r}$ to obtain an effect f on $A \otimes B$ by

$$f(\alpha , \beta ) = c \hat{r}(\alpha)(\beta), $$

for a small enough positive constant c. Upon obtaining the result f in a measurement on $A \otimes B$ when the composite system is in state $\alpha \otimes \omega$, the un-normalized conditional state of C is:

$$\hat{\omega}(\hat{f}(\alpha)) = c \alpha. $$

The normalized conditional state will be exactly $\alpha$. This is what is meant, in quantum-information theory, by a conclusive, correction-free teleportation protocol. Adopting this language, we will say that it is possible to teleport system A through system B if and only if there exists such a pair $\hat {\omega }, \hat {r}$.

If $\hat {\omega }$ is in fact an isomorphism $A^\# \cong B$, then $\hat {r} = \hat {\omega }^{-1}$, and system B can also be teleported through system A. When this is the case, Lemma 15 tells us that $\omega : I \rightarrow B \otimes A$ and $f : A \otimes B \rightarrow I$ with $\hat {f} = \hat {\omega }^{-1}$, provide respectively a unit and co-unit making $(B,\omega ,f)$ a dual for A. Thus, a compact closed category of COMs is exactly one in which every system A is paired with a second system $B = A'$, in such a way that each system can be teleported through the other.

Proposition 17

Let ${\mathcal C}$ be a monoidal category of COMs. The following are equivalent.

• (a) ${\mathcal C}$ is compact closed.

• (b) Every $A \in {\mathcal C}$ can be teleported through some $B \in {\mathcal C}$ , which in turn can be teleported through A.

• (c) Every morphism in ${\mathcal C}$ has the form $\hat {\omega } \circ \hat {f}$ for some bipartite state $\omega$ and bipartite effect f.

Proof

The equivalence of (a) and (b) is clear from the preceding discussion. To see that these are in turn equivalent to (c), suppose first that (a) holds, and let $(A', \eta _{A}, e_{A})$ be the dual for A. Suppose that $\phi : A\to B$ is a morphism in ${\mathcal C}$, and define $\omega _{\phi } = (\ensuremath {1_{A'}}\otimes \phi ) \circ \eta _{A}.$ By Remote Evaluation (Lemma 15), we have

$$\hat{\omega_{\phi}}(\hat{e_{A}}(\alpha)) \ = \ (e_{A} \otimes \ensuremath{1_{B}})\circ (\ensuremath{1_{A}} \otimes \omega_{\phi}) \circ \alpha $$

for every $\alpha \in {\mathcal C}(I,A)$. Since ${\mathcal C}$ is compact closed, the following diagram commutes:

and hence $\hat {\omega _{\phi }}(\hat {e_{A}}(\alpha )) = \phi (\alpha )$. Since ${\mathcal C}(I,A) \cong A$ we have $\phi = \hat {\omega _{\phi }} \circ \hat {e_{A}}$ as required. Conversely, if (c) holds, then for each A, the identity mapping $\ensuremath {1_{A}}$ factors as $\hat {\omega }_{A} \circ \hat {f}_{A}$ for some $\omega _{A} \in B \otimes A$ and some $f \in A \otimes B$. It follows that $\hat {\omega }_{A} = \hat {f}_{A}^{-1}$, so this gives us a compact closed structure. □

5 Weakly Self-Dual Theories

In a compact closed category ${\mathcal C}$, the internal adjoint $' : {\mathcal C} \rightarrow {\mathcal C}$ described in Section 3.1 establishes an isomorphism ${\mathcal C} \simeq {\mathcal C}^{op}$. In particular, for every object A in the category, understood as a “physical system”, there is a distinguished isomorphism between the system’s state space ${\mathcal C}(I,A)$ and the space ${\mathcal C}(A,I)$ of effects.

In contrast, a convex operational model A is not generally isomorphic to its dual. Indeed, there is a type issue: A has, by definition, a distinguished unit functional $u_{A} \in A^{\#}$; in order for $A^{\#}$ to be treated as a COM, one would need to privilege a state $\alpha _{o} \in A$ to serve as an order unit on $A^{\#}$. Only in special cases is there a natural way of doing so.¹⁰ Beyond this, there is the more fundamental problem that, geometrically, the cones $A_{+}$ and $A^{\#}_{+}$ are generally not isomorphic. This said, those convex operational models that are order-isomorphic to their duals are of considerable interest – not only because both classical and quantum systems exhibit this sort of self-duality, but because it appears to be a strong constraint, in some measure characteristic of these theories.

5.1 Weak vs Strong Self-Duality

A finite-dimensional ordered vector space A (or its cone, $A_{+}$) is said to be self-dual iff there exists an inner product–that is, a positive-definite, hence also symmetric and non-degenerate, bilinear form $\langle \ , \ \rangle$ – on A such that

$$A_{+} = A^{+} := \{ a \in A | \langle a, x \rangle \geq 0 \ \forall x \in A_{+}\}. $$

In this case, we have $A \simeq A^{\ast }$, as ordered spaces, via the canonical isomorphism $a \mapsto \langle a , . \rangle$. A celebrated theorem of Koecher and Vinberg [21, 28, 42] states that if $A_{+}$ is both self-dual and homogeneous, meaning that the group of order-automorphisms of A acts transitively on the interior of $A_{+}$, then A is isomorphic, as an ordered space, to a formally real Jordan algebra ordered by its cone of squares. The Jordan-von Neumann-Wigner [26] classification of such algebras then puts us within hailing distance of quantum mechanics.

Definition 18

A COM $(A,A^{\#},u_{A})$ is weakly self-dual (WSD) iff there exists an order-isomorphism $\phi : A \simeq A^{\#}$. We shall say that A is symmetrically self-dual iff $\phi$ can be so chosen that $\phi (\alpha )(\beta ) = \phi (\beta )(\alpha )$ for all $\alpha , \beta \in A$.

Note that, for a given linear map $\phi : A \rightarrow A^{\#}$, the bilinear form $\langle \alpha , \beta \rangle := \phi (\alpha )(\beta )$ is non-degenerate iff $\phi$ is a linear isomorphism, and symmetric iff $\phi = \phi ^{\ast }$. If we don’t require saturation, any finite-dimensional ordered linear space A can serve as the state space for a weakly self-dual COM, simply by setting $(A^{\#})_{+} = \phi (A_+)$ for some nonsingular positive linear mapping $A \rightarrow A^{\ast }$, and taking any point in the interior of $(A^{\#})_+$ for $u_{A}$. However in the saturated case, weak self-duality is a real constraint on the geometry of the state cone, although strong self-duality is an even stronger one.

Note that, for a given linear map $\phi : A \rightarrow A^{\#}$, the bilinear form $\langle \alpha , \beta \rangle := \phi (\alpha )(\beta )$ is non-degenerate iff $\phi$ is a linear isomorphism, and symmetric iff $\phi = \phi ^{\ast }$. Thus, A will be self-dual, in the classical sense described above, iff $\langle \ , \ \rangle$ is positive-definite, and $A^{\#} = A^{\ast }$, i.e, A is saturated. To emphasize the distinction, we shall henceforth refer to this situation as strong self-duality.

If $\phi : A \simeq A^{\#}$ is an order-isomorphism implementing A’s self-duality, then $\phi ^{-1} : A^{\#} \simeq A$ defines an un-normalized bipartite non-signaling state $\gamma$ in $A \otimes _{max} A$ with $\phi ^{-1} = \hat {\gamma }$ —that is, $\gamma (a,b) = \phi ^{-1}(a)(b)$. Following [12], we shall call such a state an isomorphism state. It is shown in [12] that such a state is necessarily pure in $A \otimes _{max} B$.¹¹ In this language, A is WSD iff $A \otimes _{max} B$ contains an isomorphism state.

Example 19

Let A be the convex operational model of a basic quantum-mechanical system, i.e., the space of self-adjoint operators associated with the system’s Hilbert space H. The standard maximally entangled state on $A \otimes A$ is the pure state associated with the unit vector

$$\Psi = \frac{1}{\sqrt{d}}\sum_{i} x_{i} \otimes x_{i} $$

where $\{x_{1},...,x_{n}\}$ is an orthonormal basis for H. Using this, one has a mapping

$$R : T \mapsto R_{T} := (T \otimes 1) P_{\Psi} $$

taking operators $T: \textbf {H} \rightarrow \textbf {H}$ to operators ${\cal B}(\textbf {H} \otimes \textbf {H})$. It is a basic result, due to Choi and, independently, Jamiolkowski, that this is a linear isomorphism, taking the cone of completely positive maps on ${\cal B}(\textbf {H})$ onto the cone of positive operators on $\textbf {H} \otimes \textbf {H}$. Note that $R^{-1}$ maps [31] $\rho$ to $T_{\rho }$ where the latter is given by

$$ \langle x | T_{\rho} (\sigma) y\rangle = d \mbox{Tr}(\rho ((| y \rangle\langle x |) \otimes \sigma^{T})) $$

where $| y \rangle \langle x |$ is the operator $z \mapsto \langle z, x \rangle y$, and the transpose is defined relative to the chosen orthonormal basis. This gives us a state $\gamma \in A \otimes A$ with $\hat {\gamma } : A^{\ast } \simeq A$, namely,

$$\hat{\gamma}(a)(b) = \mbox{Tr}(P_{\Psi}(a \otimes b)) = \mbox{Tr}(P_{\Psi}(a \otimes 1) (1 \otimes b)). $$

5.2 Categories of Self-dual COMs

Let ${\mathcal C}$ be a monoidal category of COMs, as described in Section 4. There is a distinction between requiring that a state space $A \in {\mathcal C}$ be weakly self-dual, which implies only that there exist an order-isomorphism $A^{\#} \simeq A$ —an isomorphism in Ordlin —and requiring that this correspond to an (un-normalized) state $\gamma \in (AA)_+$, hence, to an element of ${\mathcal C}(I,A \otimes A)$. We now focus on categories in which this latter condition holds for every system.

Definition 20

A symmetric monoidal category ${\mathcal C}$ of COMs is weakly self-dual (WSD) iff for every $A \in {\mathcal C}$, there exists a pair $(\gamma _{A}, f_{A})$ consisting of a bipartite state $\gamma _{A} \in A \otimes A$ and a positive functional $f_{A} \in (A\otimes A)^{\#} = {\mathcal C}(A \otimes A,I)$ (a multiple of an effect) such that (i) $\gamma _{A}$ is an isomorphism state, and (ii) $\hat {f}_{A} = \hat {\gamma }_{A}^{-1}$. If $\gamma _{A}$ can be chosen to be symmetric for every $A \in {\mathcal C}$, we shall say that ${\mathcal C}$ is symmetrically self-dual (SSD).

Note that this is stronger than merely requiring every COM $A \in {\mathcal C}$ to be weakly self-dual. Equivalently, we may say that category ${\mathcal C}$ of COMs is WSD iff every system can be equipped with a designated conclusive, correction-free teleportation protocol $\gamma_{A} \in A \otimes A, f_{A} \in (A \otimes A)^{\#}$, whereby A can be teleported “through itself”. Thus, by Proposition 17, we have:

Theorem 21

A monoidal category ${\mathcal C}$ of convex operational models is weakly self-dual iff it is compact closed, and can be equipped with a compact structure such that $A' = A$ for all objects $A \in {\mathcal C}$.

Recall that any morphism $\phi$ in a compact closed category has a categorial adjoint, $\phi '$. In the context of a WSD category of COMs, this has a useful interpretation in terms of the linear adjoint, $\phi ^{\ast }$:

Lemma 22

Let ${\mathcal C}$ be any WSD category of ASPs, regarded as compact closed as above. Let $\phi : A \rightarrow B$. Then the canonical adjoint mapping $\phi ' : B \rightarrow A$ is given by

$$\phi' = (\hat{f}_{B} \circ \phi \circ \hat{\gamma}_{A})^* = \hat{\gamma}_{A}^* \circ \phi^* \circ \hat{f}_{B}^* $$

Proof

We must show that, for any $\beta \in B$ – that is, any $\beta \in {\mathcal C}(I,B)$ – we have $\phi '(\beta ) := \phi ^{*} \circ \beta = \hat {\gamma }_{A}^*(\phi ^{*}(\hat {f}_{B}^*(\beta )))$, where $\phi ^* : B^{\#} \rightarrow A^{\#}$ is the linear adjoint. Let $\omega := (\ensuremath {1_{A}} \otimes \phi ) \circ \gamma _{A} : I \rightarrow A \otimes B$. Then we have

$$\begin{array}{rll} \phi'(\beta) & = & \phi' \circ \beta \\ & = & (\ensuremath{1_{A}} \otimes f_{B}) \circ (\ensuremath{1_{A}} \otimes \phi \otimes \ensuremath{1_{B}}) \circ (\gamma_{A} \otimes \ensuremath{1_{B}}) \circ \beta\\ & = & (\ensuremath{1_{A}} \otimes f_{B}) \circ (\ensuremath{1_{A}} \otimes \phi \otimes \beta) \circ \gamma_{A} \\ & = & (\ensuremath{1_{A}} \otimes (f_{B} \circ (\ensuremath{1_{B}}\otimes \beta))) \circ ((\ensuremath{1_{A}} \otimes \phi) \circ \gamma_{A}) \\ & = & (\ensuremath{1_{A}} \otimes \hat{f}^* (\beta)) \circ \omega )\\ & = & \hat{\omega}^{*}(\hat{f}^{*}_{B}(\beta)). \end{array} $$

Now, for any $b : B \rightarrow I$, we have

$$\begin{array}{rll} \hat{\omega}^{*}(b) & = & \widehat{(\sigma_{B,B} \circ \omega)}(b)\\ & = & (\ensuremath{1_{A}} \otimes b) \circ \omega \\ & = & (\ensuremath{1_{A}} \otimes b) \circ (\ensuremath{1_{A}} \otimes \phi) \circ \gamma_{A}\\ & = & (\ensuremath{1_{A}} \otimes (b \circ \phi)) \circ \gamma_{A} \\ & = & (\ensuremath{1_{A}} \otimes \phi^{*}(b)) \circ \gamma_{A}\\ & = & \hat{\gamma}^{*}_{A}(\phi^{*}(b)). \end{array} $$

With $b = \hat {f}^{*}_{B}(\beta )$, this gives the desired result. □

Corollary 23

For all $A \in {\mathcal C}$, $f_{A}' = \sigma _{A,A} \circ \gamma _{A}$.

Proof

Note first that $f^{*}_{A}(1) = f_{A} \in (A \otimes A)^{\#}$. Thus, the preceding Lemma gives us

$$ f_{A}'(1) = (\hat{\gamma}_{A \otimes A}^{*} \circ f_{A}^{*} \circ \hat{f}^{*}_{I})(1) $$

Since $f_{I} = f_{I}^{*} = \ensuremath {1_{I}}$, we have

$$ f_{A}'(1) = (\hat{\gamma}_{A}^{*} \otimes \hat{\gamma}_{A}^{*})(f_{A}). $$

Thus, for every $a, b \in A^{\#}$, we have

$$\begin{array}{rll} f_{A}'(1)(a,b) & = & (\hat{\gamma}_{A}^{*} \otimes \hat{\gamma}_{A}^{*})(f_{A})(a,b) \\ & = & f_{A}(\hat{\gamma}_{A}(a),\hat{\gamma}_{A}(b)) \\ & = & \hat{f}_{A}^{*}(\hat{\gamma}_{A}(b))(\hat{\gamma}^{*}_{A}(a)) \\ & = & b (\hat{\gamma}_{A}^{*}(a)) \\ & = & \gamma_{A}(b,a) \\ & = & (\sigma \circ \gamma_{A})(a,b). \end{array} $$

□

Theorem 24

Let ${\mathcal C}$ be a symmetrically self-dual monoidal category of COMs. Then ${\mathcal C}$ is dagger compact with $\dagger$ given by the canonical adjoint $' : {\mathcal C}^{op} \rightarrow {\mathcal C}$.

Proof

By Theorem 21, ${\mathcal C}$ is degenerate compact closed, with a compact structure on $A \in {\mathcal C}$ given by $(A, \gamma _{A}, f_{A})$, where $\gamma _{A}$ is symmetric. Define $(\cdot )^{\dag } : {\mathcal C}^{\text {op}} \to {\mathcal C}$ by $(\cdot )^{\dag } = (\cdot )'$; then $(\cdot )^{\dag }$ is a monoidal functor which is the identity on objects. Since $\sigma _{A,B}' = \sigma _{A,B}^{-1}$ in any compact closed category, ${\mathcal C}$ is dagger-monoidal. That $f_{A}^{\dag } = \sigma _{A,A} \circ \gamma _{A}$ is immediate from Corollary 23. □

Lemma 25

For all A, let $\tau _{A} : A \rightarrow A$ be the order-isomorphism given by

$$\tau_{A} := \hat{\gamma}_{A} \circ \hat{f}^{*}_{A}. $$

Then, for all $\phi \in {\mathcal C}(A,B)$ , we have

$$\phi'' = \tau_{B}^{-1} \circ \phi \circ \tau_{A}. $$

Proof

Notice first that $\tau _{B}^{-1} = (\hat {\gamma }_{B} \circ \hat {f}^{*}_{B})^{-1} = \hat {\gamma }_{B}^{*} \circ \hat {f}_{B}$. By Lemma 22, we have

$$\begin{array}{rll} \ \ \phi'' = (\hat{f}_{A} \circ \phi' \circ \hat{\gamma}_{B})^{*} & = & (\hat{f}_{A} \circ (\hat{f}_{B} \circ \phi \circ \hat{\gamma}_{A})^{*} \circ \hat{\gamma}_{B})^{*} \\ & = & \hat{\gamma}_{B}^{*} \circ (\hat{f}_{B} \circ \phi \circ \hat{\gamma}_{A})^{**} \circ \hat{f}_{A}^{*} \\ & = & (\hat{\gamma}_{B}^{*} \circ \hat{f}_{B}) \circ \phi \circ (\hat{\gamma}_{A} \circ \hat{f}_{A}^{*})\\ & = & \tau_{B}^{-1} \circ \phi \circ \tau_{A} \ \ \end{array} $$

□

Corollary 26

Let $\phi : A \rightarrow B$ with $\phi '' = \phi$. Then $\phi \circ \tau _{A} = \tau _{B} \circ \phi$.

Theorem 27

For any object A in a weakly self-dual category ${\mathcal C}$ of convex operational models, the following are equivalent: (i) $\phi '' = \phi$ for all $\phi \in {\mathcal C}(A,A)$ , (ii) $\tau _{A} = \ensuremath {1_{A}}$ , and (iii) $f_{A}$ and $\gamma _{A}$ are symmetric as a bilinear forms.

Proof

(i) implies (ii): From (i), and the fact that the morphisms in ${\mathcal C}(A,B)$ are a basis for ${\cal L}(A,B)$, $(\cdot )''$ is the identity map. Let $E_{i}, F_{j}$ be bases of the spaces ${\cal L}(A,A), {\cal L}(B,B)$ of linear maps on vector spaces A,B respectively. Then the maps $X \mapsto F_{j} X E_{i}$, where $X \in {\cal L}(A,B)$, are a basis for the space of linear maps from ${\cal L}(A,B)$ to itself. Using this fact, we can expand the map $(\cdot )'': \phi \mapsto \tau _{B}^{-1} \circ \phi \circ \tau _{A}$, which is a map from ${\cal L}(A,B)$ to itself, in a basis $M_{ij}: \phi \mapsto F_{j} X E_{i}$ where $E_{0}= 1_{B}, F_{0}=1_{A}$. By the uniqueness of expansions in bases and the fact that $(\cdot )''$ is the identity map, we get $\tau _{A} = 1_{A}, \tau _{B} = 1_{B}$. (ii) implies (iii): By (ii) we have $\tau _{A} := \hat {\gamma }_{A} \circ \hat {f}^*_{A} = 1_{A}$, so $\hat {f}^*_{A} = \hat {\gamma }_{A}^{-1} = \hat {f}_{A}$. Since $f(a,b) \equiv \hat {f}(a)(b) \equiv \hat {f}^*(b)(a)$, $\hat {f}^*_{A} = \hat {f}_{A}$ is equivalent to symmetry of $f_{A}$. Symmetry of $\gamma _{A}$ then follows from the fact that $\gamma _{A} = f_{A}^{-1}$.(iii) implies (i): If $f_{A}$ (hence, also $\gamma _{A}$) are symmetric, then we have $f_{A}^{\ast } = f_{A}$, whence, $\tau _{A} = \gamma _{A} \circ \hat {f}_{A} = 1_{A}$; thus, by Lemma 25, $\phi '' = \phi$ for all $\phi \in {\mathcal C}(A,A)$. □

Applying Theorem 24, we now have the

Corollary 28

A WSD monoidal category of COMs is dagger compact with respect to the canonical adjoint $' : {\mathcal C}^{op} \rightarrow {\mathcal C}$ , if and only if it is symmetrically self-dual.

Theorem 27 tells us that if ${\mathcal C}$ is a saturated weakly self-dual theory in which $'$ is an involution, then the interior of $A_+$ is a domain of positivity in the sense of Koecher [30]. If we further suppose that every irreducible state space in ${\mathcal C}$ is homogeneous, meaning that $G(A_+)$ acts transitively on the interior of $A_{+}$ for every $A \in {\mathcal C}$ (a condition one can motivate physically in several ways, e.g., [12, 45]), then we are close to requiring that every state space in ${\mathcal C}$ be a formally real (also called Euclidean) Jordan algebra [28, 42]. This line of thought will be pursued in a sequel to this paper.

6 Conclusion

We began with the observation that compact closure, and still more, dagger compactness, represent strong constraints on a physical theory, thought of as a symmetric monoidal category of “systems” and “processes”. Our results cast some light on the operational (or, if one prefers, the physical) content of these assumptions in the concrete—but still very general—context of probabilistic (or convex operational) theories. In particular, we have seen that, in probabilistic theories qua categories of COMs, compact closure amounts to the condition that all processes–that is, all dynamics–can be induced by the kind of conditioning that occurs in a teleportation-like protocol. Indeed, in such a theory, a process between systems A and B amounts to a choice of bipartite state on $A \otimes B$. Finally, we have established that for weakly self-dual theories symmetric weak self-duality implies the existence of a dagger compatible with the compact closed structure. As in the special case of quantum mechanics, a dagger amounts to reversing the order of conditioning.

Several directions for further study suggest themselves. It would be interesting to identify necessary and sufficient conditions for the COM representations discussed in Section 4.1 to yield finite dimensional models–and, equally, one would like to know how far the other results of Sections 4 and 5 extend to infinite-dimensional systems. Our definition of category of weakly self-dual state spaces assumes the existence of a state that induces, by conditioning, an isomorphism from the state cone to the effect cone, and an effect inducing its inverse; but as we noted, it would be interesting to investigate conditions under which this follows just from weak self-duality of the objects. As mentioned at the end of Section 5, the consequences of homogeneity of the state-spaces should also be explored.

Perhaps the most urgent task, though, is to identify operational and category-theoretic conditions equivalent to the strong self-duality of a probabilistic theory.¹²