Compositional Quantum Logic
Quantum logic aims to capture essential quantum mechanical structure in order-theoretic terms. The Achilles’ heel of quantum logic is the absence of a canonical description of composite systems, given descriptions of their components. We introduce a framework in which order-theoretic structure comes with a primitive composition operation. The order is extracted from a generalisation of C*-algebra that applies to arbitrary dagger symmetric monoidal categories, which also provide the composition operation. In fact, our construction is entirely compositional, without any additional assumptions on limits or enrichment. Interpreted in the category of finite-dimensional Hilbert spaces, it yields the projection lattices of arbitrary finite-dimensional C*-algebras. Interestingly, there are models that falsify standardly assumed correspondences, most notably the correspondence between noncommutativity of the algebra and nondistributivity of the order.
In 1936, Birkhoff and von Neumann questioned whether the full Hilbert space structure was needed to capture the essence of quantum mechanics . They argued that the order-theoretic structure of the closed subspaces of state space, or equivalently, of the projections of the operator algebra of observables, may already tell the entire story. To be more precise, we need to consider an order together with an order-reversing involution on it, a so-called orthocomplementation, which can also be cast as an orthogonality relation. Support along those lines comes from Gleason’s theorem , which characterises the Born rule in terms of order-theoretic structure. In turn, via Wigner’s theorem , this fixes unitarity of the dynamics.
These developments prompted Mackey to formulate his programme for the mathematical foundations of quantum mechanics: the reconstruction of Hilbert space from operationally meaningful axioms on an order-theoretic structure . In 1964, Piron “almost” completed that programme for the infinite-dimensional case . Full completion was achieved much more recently, by Solèr in 1995 .
Birkhoff and von Neumann coined the term ‘quantum logic’, in light of the developments in algebraic logic which were also subject to an order-theoretic paradigm. In particular they observed that the distributive law for meets and joins, which is key to the deduction theorem in classical logic, fails to hold for the lattice of closed subspaces for a Hilbert space .
This failure of distributivity and hence the absence of a deduction theorem resulted in rejection of the quantum ‘logic’ idea by a majority of logicians. However, while the name quantum logic was retained, many of its researchers also rejected the direct link to logic, and simply saw quantum logic as the study of the order-theoretic structure associated to quantum phenomena, as well as other structural paradigms that were proposed thereafter .
The quantum logic paradigm.
In the Mackey-Piron-Solèr reconstruction, the elements of the partially ordered set become the projections on the resulting Hilbert space, that is, the self-adjoint idempotents of the algebra of operators on the Hilbert space:
Conversely, the ordering can be recovered from the composition structure on these projections:
and the orthogonality relation can be recovered from it, too:
In fact, the reconstruction does not produce Hilbert space, but Hilbert space with superselection rules. That is, depending on the particular nature of the ordering that we start with, it either produces quantum theory or classical theory, or combinations thereof.
The presence of “quantumness” is famously heralded in order-theoretic terms by the failure of the distributive law, giving rise to the following comparison.
This translates as follows to the level of operator algebra.
Thus, the combination yields the following slogan.
This is indeed the case for the projection lattices of arbitrary von Neumann algebras: the projection lattice is distributive if and only if the algebra is commutative , and has been a guiding thought within the quantum structures research community.
Categorical quantum mechanics.
More recently, drawing on modern developments in logic and computer science, and mainly a branch called type-theory, Abramsky and Coecke introduced a radically different approach to quantum structures that has gained prominence, which takes compositional structure as the starting point . Proof-of-concept was provided by the fact that many quantum information protocols which crucially rely on the description of compound quantum systems could be very succinctly derived at a high level of abstraction.
In what is now known as categorical quantum mechanics, composition of systems is treated as a primitive connective, typically as a so-called dagger symmetric monoidal category. Additional axioms may then be imposed on such categories to capture the particular nature of quantum compoundness. In other words, a set of equations that axiomatise the Hilbert space tensor products is generalised to a broad range of theories. Importantly, at no point is an underlying vector-space like structure assumed.
In contrast to quantum logic, this approach led to an abstract language with high expressive power, that enabled one to address concretely posed problems in the area of quantum computing (see e.g. ) and quantum foundations (see e.g. ), and that has even led to interesting connections between quantum structures and the structure of natural language .
One of the key insights of this approach is the fact that many notions that are primitive in Hilbert space theory, and hence quantum theory, can actually be recovered in compositional terms. For example, given the pure operations of a theory, one can define mixed operations in purely compositional terms, which together give rise to a new dagger symmetric monoidal category . We will refer to this construction, as (Selinger’s) CPM–construction. While this construction applies to arbitrary dagger symmetric monoidal categories (as shown in ), Selinger also assumed compactness , something that we will also do in this paper. These structures are called dagger compact categories.
Another example, also crucial to this paper, is the fact that orthonormal bases can be expressed purely in terms of certain so-called dagger Frobenius algebras, which only rely on dagger symmetric monoidal structure . In turn, these dagger Frobenius algebras enable one to define derived concepts such as stochastic maps. All of this still occurs within the language of dagger symmetric monoidal categories . We will refer to this construction as the Stoch–construction. Similarly, finite-dimensional C*-algebras can also be realised as certain dagger Frobenius algebras, internal to the dagger compact category of finite-dimensional Hilbert spaces and linear maps, the tensor product, and the linear algebraic adjoint .
Recently , the authors proposed a construction, called the CP*–construction, that generalises this correspondence to certain dagger Frobenius algebras in arbitrary dagger compact caterories. At the same time, this construction unifies the CPM–construction and the Stoch-construction, starting from a given dagger compact category. The resulting structure is an abstract approach to classical-quantum interaction, with Selinger’s CPM–fragment playing the role of the “purely quantum”, and the abstract stochastic maps fragment playing the role of the “purely classical”.
Overview of this paper.
In this paper, we take this framework of “generalised C*-algebras” as a starting point, and investigate the structure of the dagger idempotents. We will refer to these as in short as projections too, since these dagger idempotents provide the abstract counterpart to projections of concrete C*-algebras.
We show that, just as in the concrete case, one always obtains a partially ordered set with an orthogonality relation. However, equation breaks down in general. More specifically, in the dagger compact category of sets and relations with the Cartesian product as tensor and the relational converse as the dagger, there are commutative algebras with nondistributive projection lattices.
As mentioned above, the upshot of our approach is that it resolves a problem that rendered quantum logic useless for modern purposes: providing an order structure representing compound systems at an abstract level, given the ones describing the component systems. Since we start with a category with monoidal structure, of course composition for objects is built in from the start, and it canonically lifts to algebras thereon. Let us emphasise that our framework relies solely on dagger categorical and compositional structure: the (sequential) composition of morphisms, and the (parallel) tensor product of morphisms. This is a key improvement over previous work  that combines order-theoretic and compositional structure.
Diagrams will be read from bottom to top. Wires represent the objects of the category, while boxes or dots or any other entity with incoming and outcoming wires – possibly none – represents a morphism, and their type is determined by the respective number of incoming and outgoing wires. The directions of arrows on wires represent duals of the compact structure.
Our main objects of study are symmetric Frobenius algebras, defined as follows. Let us emphasise that this is a larger class of Frobenius algebra than just the commutative ones, which previous works on categorical quantum mechanics have mainly considered.
Symbolically, we denote the multiplication of two points , that is,
as . Also, since the multiplication fixes its unit, and the dagger fixes the comultiplication given the multiplication, we will usually represent our algebras as .
We write for the category of finite-dimensional Hilbert spaces and linear maps, with the tensor product as the monoidal structure, and linear adjoint as the dagger.
Recall that is a compact category , that is, we can coherently pick a compact structure on each object as follows. If is a Hilbert space and is its conjugate space, the triple:
is a compact structure which can be shown to be independent of the choice of basis–see  for more details. We depict the maps and respectively as:
and compactness means that they satisfy:
Each symmetric dagger Fobenius algebra also canonically induces a ‘self-dual’ compact structure. The cups and caps of this compact structure are given by:
and one easily verifies that it follows from the axioms of a symmetric Frobenius algebra that the required ‘yanking’ conditions hold:
A projection of a C*-algebra is a *-idempotent. In this section we will recast this definition in light of Theorem ?, that is, we will identify what these projections are when a C*-algebra is presented as a symmetric dagger Frobenius algebra in , as in .
We claim that the projections of a C*-algebra arise as points satisfying:
where the symmetric dagger Frobenius algebra is the one induced by Theorem ?. Note that the first condition is simply idempotence of -multiplication of points, and the second one is self-conjugateness with respect to the compact structure induced by the symmetric dagger Frobenius algebra. Symbolically, we denote this conjugate of as .
A C*-algebra is realised as a symmetric dagger Frobenius algebra as follows. Each finite dimensional C*-algebra decomposes as a direct sum of matrix algebras. These can then be represented as endomorphism monoids in , which are triples of the following form: Diagrammatically, for an endomorphism monoid the multiplication and its unit respectively are: The elements of the matrix algebra are then represented by underlying points: By compactness, each point of type is of this form. By Theorem ? we know that all symmetric dagger Frobenius algebras in arise in this manner.
We can now verify the above stated claim on how the projections of a C*-algebra arise in this representation. For these points the conditions of equation respectively become:
that is, using again compactness, , i.e. idempotence and self-adjointness.
We can now generalise the definition of projection to points to arbitrary symmetric dagger Frobenius algebras in any dagger symmetric monoidal category.
The next section studies the structure of these generalised projections of abstract C*-algebras.
Before that, we compare abstract projections to copyable points. These played a key role for commutative abstract C*-algebras, because they correspond to the elements of an orthonormal basis that determines the algebra . However, as we will now see, in the noncommutative case, there simply do not exist enough copyable points (whereas the projections do have interesting structure, as the next section shows). Recall that a point is copyable when the following equation is satisfied.
The middle equation follows from symmetry of .
Let us examine what this implies for the example of in above. Equivalently, we may speak about -by- matrices, so that becomes actual matrix multiplication. Because it is well known that the central elements of matrix algebras are precisely the scalars, any copyable point is simply a scalar by the previous lemma. But substituting back into shows that the only scalar satisfying this equation is (unless ). That is, no noncommutative symmetric dagger Frobenius algebra in can have nontrivial copyable points. This explains why we prefer to work with (abstract) projections.
4Quantum logics for abstract C*-algebras
We will assume that an algebra always has a zero projection.
First, note the following stardard equation for Frobenius algebras: Then, the result follows from associativity:
(i) We have
where the middle equation follows from Lemma ?. (ii) If then, by self-conjugateness of projections and (i), .
The order is defined as . Reflexivity follows by the idempotence of projections. If and then by Lemma ? (ii) we have , so the order is anti-symmetric. If and then , so the order is transitive.
Orthogonality is defined as . Symmetry follows by Lemma ? (ii) and anti-reflexivity above 0 by idempotence of projections. If , and then where we twice relied on Lemma ? (ii).
Given a symmetric dagger Frobenius algebra , we will denote the partial order and orthogonality of the previous theorem as . The following two examples correspond to the “pure classical” and the “pure quantum” in the “concrete” case of .
Unfold the definitions of Theorem ?.
In general, every commutative monoid of idempotents is a meet-semilattice with respect to the order , and if it is furthermore finite, then it is even a (complete) lattice. As shown in , in this case the notion of an idempotent can be generalised to arbitrary types . Considered together for all algebras, this always yields a cartesian bicategory of relations in the sense of Carboni-Walters . The conclusion we draw from the previous proposition is the following: considering noncommutative algebras obstructs the construction of the categorical operation of composition.
5Composing quantum logics
Given two symmetric dagger Frobenius algebras we can define their tensor as follows.
It is easily seen to inherit the entire algebraic structure. So we can define a compositional structure on the corresponding partial orders with orthogonality as follows.
By a bi-order map we mean a function of two variables that preserves the order in each argument separately when the other one is fixed (cf. bilinearity of the tensor product).
If then .
6Commutativity versus distributivity
Having abstracted projection lattices to the setting of arbitrary dagger symmetric monoidal categories, we can now consider other models than Hilbert spaces.
We will be interested in the category of sets and relations, where the monoidal structure is taken to be Cartesian product, and the dagger is given by relational converse. This setting will provide a counterexample to equation . Here, symmetric dagger Frobenius algebras were identified by Pavlovic (in the commutative case) and Heunen–Contreras–Cattaneo (in general) in  and , respectively. They are in 1-to-1 correspondence with small groupoids. As it turns out, even in the commutative case, groupoids may yield nondistributive projection lattices.
This follows directly from .
It immediately follows that in , like in , the abstract projection lattice is a complete lattice, even though we are not dealing with finite sets.
The collection of subgroupoids is closed under arbitrary intersections.
In fact, for our counterexample to equation , it suffices to consider groups (i.e. single-object groupoids). In this case abstract projections correspond to subgroups, and it is known precisely under which conditions the lattice of subgroupoids is distributive, thanks to the following classical theorem due to Ore. A group is locally cyclic when any finite subset of its elements generates a cyclic group.
Perhaps the simplest example of an abelian group that is not locally cyclic is . It has three nontrivial subgroups, namely:
But evidently distributivity breaks down: .
By Theorem ?, we know that the converse (distributive commutative) holds for groups, but what about for arbitrary groupoids. Consider the groupoid with two objects and the only non-identity arrows and . The lattice of subgroupoids has the following Hasse diagram: which is indeed distributive, but . Thus we have proven the following corollary.
Let us finish by remarking on the copyable points in . As in , they differ from the projections. But unlike in , where there are only trivial copyable points, copyable points in are more interesting, for similar reasons as the above corollary.
A point of in corresponds to a subset . Copyability now means precisely that
Hence if , and has , then also . Because is a groupoid, this means that is precisely (the set of morphisms of a) connected component of .
From the point of view of traditional quantum logic, a number of questions arise, in particular about which categorical structure yields which order structure:
when is the orthogonality relation an orthocomplementation?
when do we obtain an orthoposet?
when do we obtain an orthomodular poset?
when is the partial order a (complete) lattice?
when is this lattice Boolean, modular or orthomodular?
Conversely, what does the lattice structure say about the category? An important first step is the characterisation of dagger Frobenius algebras in more example categories besides and .
There is a clear intuition of the comultiplication of the algebra being a “logical broadcasting operation” in the sense of . A more general question then arises on the general operational significance of the partial ordering and orthogonality relation constructed in this paper.
One of the more recent compelling results which emerged from quantum logic is the Faure-Moore-Piron theorem  on the reconstruction of dynamics from the lattice structure together with the its operational interpretation. A key ingredient is the reliance on Galois adjoints. Does this construction have a counterpart within our framework, and its (to still be understood) operational significance?
- See also the survey , which provides a comprehensive overview of the entire reconstruction, drawing from the fundamental theorem of projective geometry. Reconstructions of quantum theory have recently seen a great revival . In contrast to the Piron-Solèr theorem, this more recent work is mainly restricted to the finite-dimensional case, and focuses on operational axioms concerning how (multiple) quantum and classical systems interact.
- There is an earlier unification of the CPM–construction and the Stoch-construction , into which our construction faithfully embeds, see . However, this construction does not support the interpretation of “generalised C*-algebras” .
- The construction in  needs the rather strong extra assumption of dagger biproducts, while the construction in  requires the weaker assumption of dagger kernels. The intersection of both constructions can be made to work, provided one additionally assumes a weak form of additive enrichment .
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