aa r X i v : . [ m a t h . OA ] O c t QUANTUM SETS
ANDRE KORNELL
Department of MathematicsUniversity of California, Davis
Abstract.
A quantum set is defined to be simply a set of nonzero finite-dimensional Hilbertspaces. Together with binary relations, essentially the quantum relations of Weaver, quan-tum sets form a dagger compact category. Functions between quantum sets are certainbinary relations that can be characterized in terms of this dagger compact structure, andthe resulting category of quantum sets and functions generalizes the category of ordinarysets and functions in the manner of noncommutative mathematics. In particular, this cat-egory is dual to a subcategory of von Neumann algebras. The basic properties of quantumsets are presented thoroughly, with the noncommutative dictionary in mind, and with aneye to convenient application. As a motivating example, a notion of quantum graph coloringis derived within this framework, and it is shown to be equivalent to the notion that appearsin the quantum information theory literature. introduction This paper concerns the quantum generalization of sets, in the sense of noncommutativegeometry. It does not concern the quantization of sets as a primitive mathematical notion,nor the quantization of the familiar cumulative hierarchy described by Zermelo-Fraenkel settheory, as in [34] [29]. Rather, it concerns the quantization of sets as trivial structures, that is,as structures without structure. The category of quantum sets and functions described heregeneralizes the category of ordinary sets and functions in the same sense that the oppositeof the category of unital C*-algebras and unital ∗ -homomorphisms generalizes the categoryof ordinary compact Hausdorff spaces and continuous maps [16].Ordinary sets can be identified with discrete topological spaces, with atomic measurespaces, and indeed, with many elementary examples from familiar classes of structures. Theidentification of sets with discrete topological spaces is particularly significant to noncom-mutative mathematics, because the quantum generalization of locally compact Hausdorfftopological spaces is well established; it is the class of C*-algebras. Likewise, the identifica-tion of sets with atomic measure spaces is also particularly significant, because von Neumannalgebras are a quantum generalization of suitably well-behaved measure spaces [13, 343B].To simplify the discussion, we consider measure spaces up to equivalence of measure, andup to subsets of measure zero, so that indeed, atomic measure spaces correspond exactly to E-mail address : [email protected] . ets. Thus, up to isomorphism, we expect quantum sets to be in canonical bijective corre-spondence with a class of von Neumann algebras, whose commutative members are exactlythe von Neumann algebras of ordinary atomic measure spaces. Likewise for C*-algebras andordinary discrete topological spaces.The class of atomic, or synonymously, fully atomic von Neumann algebras is the obviouscandidate for a canonical quantum generalization of atomic measure spaces, but it is not thequantum generalization used in this paper. Recall that a von Neumann algebra is said to beatomic if and only if every nonzero projection is above a minimal nonzero projection. It iswell known that a von Neumann algebra is atomic if and only if it is an ℓ ∞ -direct sum of type Ifactors [3, IV.2.2.1]. As a quantum generalization of atomic measure spaces, this class of vonNeumann algebras suffers from a number of related problems, each of which takes some timeto expound. One such problem is that an ultraweakly closed unital ∗ -subalgebra of an atomicvon Neumann algebra need not itself be atomic, whereas the quotient of an atomic measurespace must be atomic. In the appropriate variant of Gelfand duality, ultraweakly closed unital ∗ -subalgebras correspond to quotient measure spaces. The existence of nondiagonalizableoperators is another such problem (theorem 5.4). The quantum generalization of atomicmeasure spaces that is used in this paper definitionally requires that every ultraweakly closed ∗ -subalgebra be atomic. Such a von Neumann algebra is an ℓ ∞ -direct sum of finite type Ifactors, i.e., of simple matrix algebras.Essentially the same class of operator algebras arises as the quantum generalization ofdiscrete topological spaces in the theory of locally compact quantum groups. The C*-algebrasare the established quantum generalization of locally compact Hausdorff spaces, and locallycompact quantum groups are defined to be C*-algebras with additional structure, whichspecifies a group operation on the Gelfand spectrum of the C*-algebra whenever the C*-algebra is commutative [24]. Pontryagin duality is a fundamental duality among abelian locally compact groups, and research into compact quantum groups has been substantiallyguided by the goal of generalizing Pontryagin duality to include nonabelian groups. In thecontext of this generalized duality, the dual of a nonabelian locally compact group is a locallycompact quantum group that is abelian in a suitable sense. It is an elementary fact thatthe Pontryagin dual of an abelian compact group is discrete, and vice versa, and this factgeneralizes to the noncommutative setting, naturally yielding a notion of discrete quantumgroup, and therefore a natural notion of discrete quantum space [11]. Consequently, a discretequantum space is defined to be a c -direct sum of simple matrix algebras.The same picture of quantum sets emerges in both settings. The ℓ ∞ -direct sum generalizesthe disjoint union of well-behaved measure spaces, and the c -direct sum generalizes thedisjoint union of locally compact Hausdorff spaces. Thus, a quantum set apparently consistsof atoms, each possessing a positive integer internal dimension. We define a quantum set tobe a structure that is fully determined by an ordinary set At( X ) of nonzero finite-dimensionalcomplex Hilbert spaces. The atomic quantum measure space constructed from X is the ℓ ∞ -direct sum L X ∈ At( X ) L ( X ), and the discrete quantum topological space constructed from X is the c -direct sum L X ∈ At( X ) L ( X ), where L ( X ) denotes the algebra of all linear operatorson a finite-dimensional Hilbert space X , which are automatically bounded. We refrain fromidentifying X with At( X ) to avoid ambiguity when we generalize familiar notions fromordinary sets to quantum sets, and to protect the intuition that ordinary sets are a subclass f quantum sets, rather than vice versa. Each ordinary set S can be regarded as a quantumset ‘ S by replacing each of its elements with a one-dimensional atom.One of the broader ambitions of this article is to provide a cohesive foundation for non-commutative discrete mathematics, and establishing a uniform system of terminology is asignificant part of that goal. Noncommutative mathematics considers many notions thatgeneralize familiar notions to the noncommutative setting, and they all need names. Thereis a sound and firmly established convention that these names should refer to the notionsbeing generalized, as an aid to intuition and to memory. The modifier “quantum” also ap-pears naturally in this context as a way of distinguishing the generalization from the originalnotion, but often its appearance does more harm than good.As an example, consider the generalization of the notion “is equinumerous to”. We brieflysuppose that the appropriate generalization of this notion to quantum sets is to equate X and Y iff P X ∈ At( X ) (dim( X )) = P Y ∈ At( Y ) (dim( Y )) . In fact, this supposition is well-founded,but because this article is focused on the categorical properties of quantum sets, we cannotgive it a worthy defense here, beyond observing that this invariant behaves appropriately forinjective and surjective morphisms in the category qSet . However, this proposed general-ization illustrates the desirability of two conventions in notation and terminology.If the equation in the above paragraph holds, shall we say that X is equinumerous to Y ,or that X is quantum equinumerous to Y ? All else being equal, it is preferable to limit theproliferation of the word “quantum” in text and speech already saturated with the word.In fact, there is another reason to prefer the unqualified term “equinumerous”: it implicitlycarries the assurance that two ordinary sets are equinumerous in this generalized sense ifand only if they are equinumerous in the familiar sense. It is not immediately clear from theterminology that ‘ N is not “quantum equinumerous” to ‘ R . After all, there are nonisomorphicsimple graphs that are quantum isomorphic. Once we have chosen to use “equinumerous”instead of “quantum equinumerous”, it becomes absolutely necessary to distinguish eachquantum set X from the ordinary set At( X ) of its atoms. Indeed, quantum sets X and Y can fail to be equinumerous even if At( X ) and At( Y ) are equinumerous, and vice versa. Convention.
Use the modifier “quantum” when generalizing a class of structures, e.g.,quantum sets, quantum groups, quantum graphs, etc. Do not use the modifier “quantum”when generalizing a notion that agrees with the ordinary notion on ordinary structures,e.g., equinumerous quantum sets, a homomorphism between quantum groups, a connectedquantum graph, etc.We will also use the modifier “quantum” for variants of ordinary notions that allow somevariable to range over a class of quantum structures. For example, the quantum chromaticnumber of an ordinary graph is the least number of colors for which there exists a quantumfamily of graph colorings (definition 1.1).1.1. connection to quantum information theory.
Together with [27], our category ofquantum sets occupies a point of contact between noncommutative mathematics and quan-tum information theory. One branch of research in quantum information theory considerscooperative games where the two players must coordinate their responses to a referee withoutcommunicating with one another. If the two players are allowed to make measurements onquantum systems that are entangled with each other, then they may have a winning strategyeven if no winning strategy exists without this aid. This phenomenon is termed “quantum seudotelepathy”; its investigation has been surveyed by Brassard, Broadbent, and Tapp[4]. They essentially credit Heywood and Redhead with discovering the first such game [19],obtained from the proof of the Kochen-Specker theorem [20].We highlight one line of research leading from quantum pseudotelepathy to our quantumsets. It was discovered that one example in quantum communication complexity can beunderstood as quantum pseudotelepathy for a graph coloring game [5, theorem 4] [15] [14][9, 4.1]. Such a game is played with a fixed simple graph and a fixed set of colors, whichmay or may not suffice to properly color the graph. There are two cooperating players,traditionally named Alice and Bob, and a referee. The referee separates the players, andthen queries each player about a vertex on the graph. Both players must respond with acolor, and neither player hears the question for, or the answer from, the other player. If thevertices are equal, the players win iff they respond with the same color; if the vertices areadjacent, the players win iff they respond with different colors; if the vertices are neitherequal nor adjacent, the players win automatically. These victory conditions are chosen sothat a deterministic classical winning strategy for the two players is the same thing as aproper graph coloring. However, if the two players possess quantum systems entangled witheach other, then a winning strategy may be available even when no proper graph coloringexists. To implement such a strategy, each player performs a measurement on their quantumsystem before responding to the referee.An ordinary graph coloring is a function from vertices to colors, and when a quantumstrategy is available, then one such strategy can be described by a family of functions indexedby a quantum set (proposition 1.1). The first step toward this connection is the resultof Cameron, Montanaro, Newman, Severini, and Winter [6, proposition 1] that when aquantum winning strategy is available, there must be a quantum winning strategy thatuses projective measurement. In our framework, each projective measurement correspondsto a function from a quantum set describing the quantum system, to the classical set ofpossible outcomes, so in retrospect, this proposition might be viewed as the first appearanceof quantum sets in this context. From the perspective of noncommutative mathematics, thisresult is the quantum analog of the proposition that when a probabilistic classical winningstrategy exists, so does a deterministic such strategy.The next step toward this connection between pseudotelepathy and our quantum setsis taken in the paper of Manˇcinska and Roberson on quantum homomorphisms betweenordinary graphs [26]. A proper graph coloring can be characterized as a graph homomorphismto the complete simple graph of colors, and the graph coloring game can be generalized to agame for which the deterministic classical strategies are exactly the graph homomorphismsfrom one given simple graph to another. Manˇcinska and Roberson begin their paper bygeneralizing the proposition of Cameron et al. to quantum homomorphism games. In doingso, they all but define a quantum homomorphism to be a quantum winning strategy that usesprojective measurement [26, corollary 2.2]. Such a definition is made explicitly by Abramsky,Barbosa, de Silva, and Zapata [1, theorem 7]. They define a quantum homomorphism offinite relational structures in terms of projective measurements; their definition is nontriviallycompatible [1, theorem 16] with the definition implicit in [26, corollary 2.2].The work of Musto, Reutter, and Verdon [27] [28] cements the analogy. They introduce acategory of quantum sets and quantum functions, and define quantum homomorphisms to befunctions preserving adjacency [27, definition 5.4]. Their quantum sets and their quantum unctions are not strictly speaking the same as our quantum sets and functions, but thenotions are very closely related. Their quantum sets are essentially our finite quantum sets,and our functions are essentially their one-dimensional quantum functions. Formally, ourcategory of finite quantum sets and functions is equivalent to their category of quantum setsand one-dimensional quantum functions.In fact, even the quantum functions of dimension other than one appear naturally in ourframework. A quantum function is essentially a quantum family of functions indexed by anatomic quantum set, which determines the dimension of that quantum function. We describethis bijective correspondence in detail toward the end of this introductory section, afterdemonstrating the relevance of our framework to the graph coloring game in subsection 1.3.1.2. summary of contents. We begin the development by defining a category qRel ofquantum sets and their binary relations, which is a dagger compact category with biproducts,or synonymously, a strongly compact closed category with biproducts [2]. The category
Rel of ordinary sets and ordinary binary relations can be canonically included into qRel as a fullsubcategory. We notate this inclusion S ‘ S . The usual partial order on binary relationsgeneralizes to binary relations between quantum sets.We define a subcategory qSet of qRel , by directly applying the usual characterization offunctions as binary relations satisfying a pair of conditions expressed in terms of the daggercompact structure on Rel . There is a contravariant equivalence between the category ofquantum sets and functions, and the category of hereditarily atomic von Neumann alge-bras and unital normal ∗ -homorphisms; we define a von Neumann algebra to be hereditarilyatomic just in case every von Neumann subalgebra is atomic. This contravariant equiva-lence generalizes the familiar equivalence between ordinary sets and commutative atomicvon Neumann algebras, which is defined by S ℓ ∞ ( S ).Our quantum sets can be characterized as sets of nonzero finite-dimensional Hilbert spaces,or equivalently, as hereditarily atomic von Neumann algebras. Analogies between sets andHilbert spaces, or equivalently, type I factors, are well established in the literature [17] [31][38]. The distinguishing feature of our definition is essentially that we allow infinite ℓ ∞ -directsums of factors, but restrict to finite type I factors. Our immediate purpose is to obtaina category of quantum sets and binary relations that is dagger compact, and a category ofquantum sets and functions that is complete, cocomplete, and closed.Researchers working with quantum groups have settled on the same notion of discreteness.Specifically, the discrete quantum groups, those locally compact quantum groups that aredual to compact quantum groups, can also be characterized as locally compact quantumgroups whose C*-algebras are c -direct sums of matrix C*-algebras [30] [12] [35]. A recentpaper investigates the action of compact quantum groups on “discrete quantum spaces”[11], essentially the quantum sets of the present paper. I thank Piotr So ltan for bringingthis exciting connection to my attention.Our definition of binary relations is essentially that of Weaver [37]; it arose from his workwith Kuperberg on quantum metrics [23], itself motivated in part by quantum informationtheory. It is straightforward that the category qRel of quantum sets and binary relationsis equivalent to the category of hereditarily atomic von Neumann algebras and quantumrelations in Weaver’s sense. The category of all von Neumann algebras and quantum relationsis sadly not dagger compact. e denote the monoidal structure on qRel and qSet by −×− , and its monoidal unit by .A monoidal product X ×X of two quantum sets is generally not their product in the categorytheory sense, but this monoidal structure does retain a number of important properties ofthe usual Cartesian product of ordinary sets. First, ‘ S × ‘ S is naturally isomorphic to‘( S × S ), as functors in ordinary sets S and S . Second, this monoidal structure has aterminal monoidal unit in qSet , which allows us to define canonical projection morphisms P : X × X → X and P : X × X → X . Third, the monoidal product X × X satisfiesthe uniqueness property of the category-theoretic product.Explicitly, the symmetric monoidal category ( qSet , × , )(1) is finitely complete,(2) is finitely cocomplete,(3) is closed,(4) has a terminal monoidal unit ,(5) has, for each pair of morphisms ( F i : Y → X i | i ∈ { , } ), at most one morphismmaking the following diagram commute, YX X × X X F F ! P P (6) and has, for every monic Z X , a unique “classical” function from X to thecoproduct ⊎ making the following diagram into a pullback square: Z X ⊎ J ! The function J : ֒ → ⊎ is the inclusion of into the second summand of the coproduct.In any symmetric monoidal category with terminal unit, we define a morphism F : X → Y to be “classical” just in case there is a morphism
X → Y × X whose first component is F and whose second component is the identity on X .The monoidal structure × may be viewed as formalizing the compatibility of observables.If a quantum system is described by some quantum set X , then its observables correspond tofunctions X → ‘ R . Two observables F and F are then compatible in the familiar physicalsense if and only if there is a function X → ‘ R × ‘ R whose first and second components are F and F respectively. A classical observable is then an observable that is compatible withevery observable. More generally, a classical function between quantum sets is essentially adeterministic quantum operation that is compatible with every observable. Such functionsare characterized by proposition 10.8.Each topos may be regarded as a symmetric monoidal category by equipping it with itscategory-theoretic product. Adjusting the definition of topoi in this way, we find that asymmetric monoidal category satisfying properties (1) – (6) above is a topos if and only if itsmonoidal product coincides with its category-theoretic product, in other words, if and only ifit is Cartesian monoidal. This observation suggests two tantalizing notions: first, that thereis a direct quantum generalization of topos theory, and second, that monoidal categories eneralize Cartesian categories like quantum objects generalize their classical counterpartsin noncommutative mathematics.Thus, the category of quantum sets and functions retains a number of familiar properties ofthe category of ordinary sets and functions. However, the incompatibility of morphisms thatis formalized by the discrepancy between the monoidal product × , and the category-theoreticproduct ∗ , is both essential to the quantum setting, and a significant departure from thebehavior of ordinary sets. Furthermore, the terminal object is not a generating object, andthis fact complicates the direct comparison of our category to standard category-theoreticaccounts of set theory, such as Lawvere’s elementary theory of the category of sets [25].However, we mention that the axiom of choice fails in its standard categorical formulations,because state preparation is a nondeterministic quantum operation, i.e., it is not describedby a function. Thus, the measurement of a qubit is a surjective function H → ‘ { , } thathas no right inverse ‘ { , } → H . Here, H is a quantum set which consists of a singletwo-dimensional atom, formalizing the qubit.The opposite of the category of von Neumann algebras and unital normal ∗ -homomorphismswith the spatial tensor product is also a symmetric monoidal category satisfying (1) – (6)[22], and it likewise arises from the category of von Neumann algebras and quantum rela-tions in Weaver’s sense [21, unpublished]. The arguments are essentially the same; theiradaptibility to quantum sets rests on the fact that ℓ ∞ -direct sums of finite type I factorsform a hereditary class of von Neumann algebras, in other words, that the von Neumannsubalgebras of any von Neumann algebra in this class are themselves in this class. So ltan’squantum families of maps are an even earlier instance of the same construction of exponentialoperator algebras [33].Effectus theory [7] offers another generalization of topoi to the quantum setting. It is amuch broader generalization than the one suggested by properties (1) – (6), and in partic-ular, it is directly applicable to categories formalizing probabilistic quantum operations bycompletely positive maps. The morphisms of qSet are deterministic in the sense of [22], butit is possible to reason indirectly about probabilistic processes by constructing a quantumset of subprobability distributions F ( X ) for each quantum set X [40, 4.3.4], following thesame approach as for the categories of von Neumann algebras in [39]. Furthermore, I expectthat quantum sets form an adequate model of the quantum lambda calculus, as in [8].In noncommutative geometry, operator algebras are viewed as algebras of complex-valuedfunctions on a quantum object. Our definition of quantum sets incarnates this intuition. Aquantum set X is a concrete object distinct from and simpler than the corresponding operatoralgebra ℓ ( X ), and furthermore, up to equivalence of categories, this operator algebra literallyconsists of functions from the quantum set X to a canonical quantum set C , which has thestructure of a ring, internal to the category qSet . Formally, we show that the functors ℓ ( − ) and Fun( − ; C ) are naturally isomorphic, as contravariant functors to the category of ∗ -algebras over C .In the appendix, we also show that the functors Proj( ℓ ( − )), Fun( − ; ‘ { , } ), and Rel( − ; )are naturally isomorphic. The binary relations from a quantum set X to the terminal quan-tum set correspond to the predicates on X , which we naively expect to be quantum setsin the their own right. We exhibit a technical modification to the definition of quantum setsthat incorporates this intuition, without undermining preceding arguments. n some places, the presentation is more explicit and verbose than strictly necessary. I haveincluded calculation and discussion where a citation or a terse remark might do. A minimalistapproach would be at odds with the secondary, expository goals of this article. The purposeof this article is to provide a convenient foundation for noncommutative discrete mathe-matics, a growing area that includes both researchers in noncommutative mathematics, andresearchers in other academic disciplines. For researchers in the latter category, this articlemight serve as an introduction to noncommutative mathematics, for better or for worse.1.3. quantum families of graph colorings. We now demonstrate the connection to quan-tum pseudotelepathy with an example. The purpose of this example is not only to show thatthe notion of quantum graph coloring implicit in [6] can be formalized in our framework,but furthermore to show that it can be motivated within noncommutative mathematics, en-tirely apart from the graph coloring game. This example applies several defined notions, soI suggest that the interested reader look over the example to get a general sense of what isdone, and then come back to it once they are more familiar with the notions being applied.Let G be an ordinary graph, and T an ordinary set, intuitively of colors. A proper graphcoloring of G is a function f : G → T satisfying the familiar condition that the values of f on any two adjacent vertices must be distinct. Similarly, we define a family of proper graphcolorings indexed by a set X to be a function f : G × X → T such that f ( g , x ) = f ( g , x )for all pairs of adjacent vertices g and g , and all indices x . Equivalently, we can ask thatthe inverse image of each color for the function f ( · , x ) be an independent subset of G , foreach index x , or that the inverse images of each color for the functions f ( g , · ) and f ( g , · )be disjoint subsets of X , for each adjacent pair of vertices g and g .Thus, we want to define a quantum family of proper graph colorings, indexed by a quantumset X , to be a function F : ‘ G × X → ‘ T such that the inverse images of each color underthe functions F ( g , · ) and F ( g , · ) are disjoint, for each adjacent pair of vertices g and g .We explain what this should mean. The quantum set ‘ G is the quantum set canonicallyidentified with the ordinary set G ; it has a one-dimensional atom for each element of G (section 2). The notation ‘ G × X refers to the Cartesian product of quantum sets (definition2.2). The morphism F should be a function (definition 4.1, section 3). The notation F ( g , · )refers to the function F g = F ◦ (‘cst g × I X ) : X → ‘ T (definition 3.2), where cst g is just theordinary constant function with value g ; likewise for the notation F ( g , · ).For each color t ∈ T , let us write { t } T for that singleton considered as a predicate on T ,that is, as a subset of T . It is canonically identified with a predicate ‘ { t } T on the quantumset ‘ T (definition A.1). Taking inverse images, we obtain two predicates F ⋆g (‘ { t } T ) and F ⋆g (‘ { t } T ) (definition A.2). We ask that these two predicates be disjoint (definition A.1) . Definition 1.1.
A quantum family of graph colorings indexed by a quantum set X is afunction F : ‘ G × X → ‘ T such that the predicates F ⋆g (‘ { t } T ) and F ⋆g (‘ { t } T ) are disjoint foreach color t ∈ T , whenever g , g ∈ G are adjacent vertices. Proposition 1.2.
The following are equivalent:(1) There is a winning strategy for the graph coloring game using quantum entanglement.(2) There exists a quantum family of graph colorings indexed by a quantum set
X 6 = ‘ ∅ .(3) There exists a quantum family of graph colorings indexed by a quantum set H , with At( H ) = { H } for some nonzero finite-dimensional Hilbert space H . his proposition refers to the graph coloring game investigated in [6], also described insubsection 1.1. The quantum family of graph colorings in (3) is essentially a function ofdimension dim( H ) from G to T in the sense of [27]. The empty quantum set ‘ ∅ has no atomsat all (definition 2.2). Proof.
Let H be a nonzero finite-dimensional Hilbert space, and let H be the quantum setwhose only atom is H . The quantum set ‘ G × H is a coproduct of copies of H , one foreach element of G (section 8). The functions ‘cst g × I H : H → ‘ G × H are the injections forthis coproduct. Thus, a function F : ‘ G × H → ‘ T is uniquely determined by its restrictions F g : H → ‘ T , for g ∈ G .Under the duality between quantum sets and hereditarily atomic von Neumann alge-bras, functions F g : H → ‘ T correspond to unital normal ∗ -homomorphisms F ⋆g : ℓ ∞ ( T ) → L ( H ) (definition 5.2, theorem 7.6). The expression L ( H ) denotes the von Neumann al-gebra of all linear operators on H , which are essentially matrices. The unital normal ∗ -homorphism F ⋆g : ℓ ∞ ( T ) → L ( H ) is uniquely determined by the projections p gt = F ⋆g ( δ t ),for t ∈ T , where δ t ∈ ℓ ∞ ( T ) is the function that takes value 1 at t , and vanishes otherwise.Thus, we have a bijection between functions F : ‘ G × H → ‘ T and families of projections( p gt ∈ L ( H ) | g ∈ G, t ∈ T ) satisfying P t ∈ T p gt = 1 H for each g ∈ G .Let g and g be adjacent vertices. The condition that the predicate F ⋆g (‘ { t } T ) be dis-joint from the predicate F ⋆g (‘ { t } T ) is equivalent to the condition that the projection F ⋆g ( δ t )be orthogonal to the projection F ⋆g ( δ t ) (theorem A.8). Therefore, we have a bijectionbetween quantum families of graph colorings indexed by H , and families of projections( p gt ∈ L ( H ) | g ∈ G, t ∈ T ) satisfying P t ∈ T p gt = 1 H for each vertex g , and p g t · p g t = 0,for each adjacent pair of vertices g and g , and each color t . Appealing to [6, section II], weestablish (1) ⇔ (3).The implication (3) ⇒ (2) is trivial. We assume (2), that there exists a quantum familyof graph colorings F : ‘ G × X → ‘ T for some nonempty quantum set X . Let H be any atomof X , and let H be the quantum set whose only atom is H . Let J : H ֒ → X be the inclusionfunction (proposition 10.1). We claim that the function ˜ F = F ◦ (‘id G × J ) : ‘ G × H → T isa quantum family of graph colorings. We compute that for all vertices g ,˜ F g = F ◦ (‘id G × J ) ◦ (‘cst g ◦ I H ) = F ◦ (‘cst g × I X ) ◦ J = F g ◦ J. Let g and g be adjacent vertices, and let t be a color. By assumption, F ⋆g (‘ { t } T ) and F ⋆g (‘ { t } T ) are disjoint. The inverse images of disjoint predicates are disjoint, so ˜ F ⋆g (‘ { t } T ) = J ⋆ ( F ⋆g (‘ { t } T )) and ˜ F ⋆g (‘ { t } T ) = J ⋆ ( ˜ F ⋆g (‘ { t } T )) are disjoint (theorem A.8). Therefore, ˜ F isa quantum family of graph colorings, indexed by H . (cid:3) We have described a class of quantum strategies as being equivalently quantum familiesof functions. The quantum functions of Musto, Reutter, and Verdon [27, definition 3.11]are likewise equivalently quantum families of functions. Their category QSet [27, definition3.18] is weakly equivalent as a 1-category to the category C whose objects are finite quantumsets, and whose nonzero morphisms are families of functions indexed by atomic quantum sets.More precisely, an object of C is a finite quantum set in our sense, and a morphism of C from X to Y is a function X × H → Y , with H empty or atomic. Composition is defined inthe expected way. The weak equivalence from C to QSet is closely related to the equivalencein theorem 7.6, but it is covariant. Each object X in C is taken to the finite-dimensional on Neumann algebra ℓ ∞ ( X ), which is also uniquely a special symmetric dagger Frobeniusalgebra [36, theorem 4.6], i.e., a quantum set in the sense of Musto, Reutter, and Verdon.Each morphism F : X ×H → Y from X to Y in C is taken to the linear map obtained from thecoalgebra homomorphism ( F ⋆ ) † by putting it through the following natural homomorphisms: L ( ℓ ∞ ( X ) ⊗ L ( H ) , ℓ ∞ ( Y )) ∼ = L ( ℓ ∞ ( X ) ⊗ H ⊗ H ∗ , ℓ ∞ ( Y )) ∼ = L ( ℓ ∞ ( X ) ⊗ H, H ⊗ ℓ ∞ ( X ))Here, H is the unique atom of H , if H is nonempty, and otherwise H = 0. This is a functionin the sense of Musto, Reutter, and Verdon [27, definition 3.11]; the dimension of H is thedimension of the function. We do not prove the claimed equivalence of categories, which isbarely outside the scope of this article.1.4. notation and terminology. For the benefit of researchers working in physics andcomputer science, the development is initially framed in terms of Hilbert spaces, rather thanoperator algebras. Our binary relations correspond to the quantum relations of Weaver [37],but this Hilbert space framing avoids operator topologies. The Hermitian adjoint is rendered † , and the symbol ∗ is reserved for the Banach space adjoint. Nevertheless, we retain the stockterm “ ∗ -homomorphism”, to mean an algebra homomorphism that preserves the Hermitianadjoint † . We write H ≤ K when H is a subspace of K . We write L ( H, K ) for the spaceof all bounded linear operators from H to K . If V ≤ L ( H , H ) and W ≤ L ( H , H ), wewrite W · V for the span of the set { wv | v ∈ V, w ∈ W } . We also use the dot as a visualseparator in products. Homomorphisms are not assumed to be unital, but representationsare assumed to be nondegenerate. Von Neumann algebras are assumed to be concrete andto contain the identity operator. A von Neumann subalgebra of A is an ultraweakly closed ∗ -subalgebra that need not contain the identity operator, but the Hilbert space shrinkscorrespondingly. An ortholattice is a bounded lattice, not necessarily distributive, equippedwith an orthocomplementation operation. For quantum sets X and Y , we write Fun( X ; Y )for the set of functions from X to Y , we write Par( X ; Y ) for the set of partial functions from X to Y , and we write Rel( X ; Y ) for the set of binary relations from X to Y .For reference, I suggest Operator Algebras [3] and
Categorical Quantum Mechanics [2]; theterm “strongly compact closed” is used synonymously with “dagger compact”.1.5. acknowledgements.
The appearance of [27] spurred me to finally put pen to paper;I thank David Reutter and Dominic Verdon for useful discussion. I thank David Robersonfor his detailed comments about the literature in quantum pseudotelepathy, which is new tome. I thank Neil Ross and Peter Selinger for organizing QPL 2018, an enriching experiencethat motivated the discussion of quantum pseudotelepathy in this introductory section. Ithank an anonymous referee for advocating the inclusion of “fissions” in sections 6 and 7.Finally, I am deeply grateful to Bert Lindenhovius and Michael Mislove for their confidencein this approach to quantization, and for many hours of enriching discussion. ontents
1. introduction 11.1. connection to quantum information theory 31.2. summary of contents 51.3. quantum families of graph colorings 81.4. notation and terminology 101.5. acknowledgements 102. quantum sets 113. binary relations between quantum sets 134. functions between quantum sets 165. hereditarily atomic von Neumann algebras 186. three perspectives on a function 197. functoriality 248. completeness and cocompleteness 289. quantum function sets 3210. subobjects of a quantum set 3411. operators as functions on a quantum set 37References 40Appendix A. predicates on quantum sets 41Appendix B. the corange of a partial function 45Appendix C. material quantum sets 462. quantum sets
Definition 2.1.
A quantum set X is completely determined by a set At( X ) of nonzerofinite-dimensional Hilbert spaces, called the atoms of X .Formally X = At( X ), so each quantum set is also an ordinary set, i.e., a set in the ordinarysense. Of course, in the standard formalization of mathematics in set theory, the same istrue of each real number. Formalized as a Dedekind cut, the real number π is equal to theset Q ∩ ( −∞ , π ), but it is good mathematical practice to draw a distinction between thetwo objects. We similarly draw a distinction between X and At( X ). We write X ∝ X tomean that X is an atom of X , that is, as an abbreviation for X ∈ At( X ). The symbol ∝ isintended to suggest the word “atom”.Thus, the atoms of X are not the elements of X , except on a purely formal level. Theintuition is that the genuine elements of X may be mathematically fictional, like the pointsof a quantum compact Hausdorff space. They may be inextricably mixed together, havingno individual identity. The atoms of X merely correspond to those subsets of X that areindecomposable with respect to the union operation, which we now define. Definition 2.2. (1) A quantum set X is empty iff At( X ) = ∅ .(2) A quantum set X is finite iff At( X ) is finite.(3) A quantum set X is a subset of a quantum set Y , written X ⊆ Y , iff At( X ) ⊆ At( Y ).
4) The union
X ∪ Y of quantum sets X and Y is defined byAt( X ∪ Y ) = At( X ) ∪ At( Y ) . (5) The Cartesian product X × Y of quantum sets X and Y is defined byAt( X × Y ) = { X ⊗ Y | X ∝ X and Y ∝ Y } . Each of the five notions just defined are generalizations of the corresponding notions forordinary sets, in the following precise sense. We identify each ordinary set S with a quantumset ‘ S , which has a one-dimensional atom for each element of S . Formally, ‘ S is defined byAt(‘ S ) = { C s | s ∈ S } , where C s = ℓ ( { s } ), with the understanding that C s = C s ′ whenever s and s ′ are distinct elements of S .For all ordinary sets S and T , we now have that ‘ S is empty if and only if S is empty inthe ordinary sense, that ‘ S is finite if and only if S is finite in the ordinary sense, and that‘ S is a subset of ‘ T if and only if S is a subset of T in the ordinary sense. Furthermore,for all ordinary sets S and T , we have that ‘ S ∪ ‘ T = ‘( S ∪ T ). However, the equation‘ S × ‘ T = ‘( S × T ) is only true up to isomorphism. The equality becomes strict if thedagger compact category FdHilb of finite-dimensional Hilbert spaces and linear operatorsis tailored appropriately (appendix C).We say that quantum sets X and Y are isomorphic iff there exists an ordinary bijectionAt( X ) → At( Y ) that preserves the dimensions of atoms. This condition is equivalent to theexistence of an invertible morphism from X to Y in the category qSet of quantum sets andfunctions (proposition 4.4).This pattern of generalization does strongly suggest a notion of membership for quantumsets. It is certainly natural to say that an object x is an element of a quantum set X iff ‘ { x } ⊆ X , or equivalently, iff C x ∈ At( X ). One-dimensional atoms not of the form C x also intuitively correspond to individual elements, albeit not to specific objects in themathematical universe. As with the generalized Cartesian product, this flaw can be repaired(appendix C), but it is inconsequential to the structural approach taken in this article.Thus, we define a quantum set X to be a singleton iff it has exactly one atom, and thatatom is one-dimensional. More inclusively, we define a quantum set X to be atomic iff itsimply has exactly one atom. The two notions coincide for ordinary sets, considered asquantum sets in the manner described above. We use the established term for the formernotion, because it formalizes the intuition that X has exactly one element, and the latternotion does not. For example, any atomic quantum set that is not a singleton admits asurjection onto ‘ { , } (proposition 8.1).Our overall approach is to use established terms in an unqualified way only when thatterm retains its familiar meaning whenever all the relevant quantum sets are of the form‘ S . The notation is chosen to blur the distinction between S and ‘ S , and to emphasize thedistinction between X and At( X ). Nevertheless, since we will occasionally wish to define aquantum set by specifying its atoms, we introduce the following notation. Definition 2.3.
Let M be an ordinary set of finite-dimensional Hilbert spaces. We write Q M for the unique quantum set such that At( Q M ) = { H ∈ M | dim( H ) = 0 } .The definition of quantum sets (definition 2.1) might more naturally be given to includezero-dimensional Hilbert spaces. Chris Heunen observed that this yields equivalent cate-gories qRel and qSet ; adding zero-dimensional atoms to a quantum set does not affect its somorphism class. We exclude zero-dimensional atoms so that the isomorphism of quantumsets in qSet is equivalent to their isomorphism in the naive sense. Nevertheless, sometimesthe most natural definition of a given quantum set includes Hilbert spaces that may be zero-dimensional, and in those cases, the convention in definition 2.3 is useful and appropriate.3. binary relations between quantum sets We write L ( H, K ) for the set of bounded linear operators from H to K . Definition 3.1.
A binary relation from a quantum set X to a quantum set Y is a functionthat assigns to each pair of atoms, X of X and Y of Y , a subspace R ( X, Y ) ≤ L ( X, Y ) oflinear operators. We write Rel( X ; Y ) for the set of all binary relations from X to Y .For ordinary sets S and T , the binary relations between ‘ S and ‘ T are in canonical bi-jective correspondence with ordinary binary relations between S and T , because for allone-dimensional Hilbert spaces X and Y , the vector space L ( X, Y ) is itself one-dimensional.If r is an ordinary binary relation from S to T , we write ‘ r for the corresponding binaryrelation from ‘ S to ‘ T . Definition 3.2.
Let X , Y , and Z be quantum sets. If R is a binary relation from X to Y ,and S is a binary relation from Y to Z , then their composition is the relation from X to Z defined by ( S ◦ R )( X, Z ) = span { sr | ∃ Y ∝ Y : s ∈ S ( Y, Z ) and r ∈ R ( X, Y ) } . Quantum sets and their binary relations form a category qRel . The identity relation I X on a quantum set X is defined as follows: I X ( X, X ) is spanned by the identity operator on X , and I X ( X, X ′ ) = 0 if X = X ′ .Identifying each ordinary set S with its quantum set counterpart ‘ S , the category Rel ofordinary sets and ordinary binary relations is a full subcategory of qRel . The functor S ‘ S is an equivalence of categories from Rel to the full subcategory of quantum sets whose atomsare one-dimensional. Furthermore, this is an equivalence of dagger monoidal categories forthe canonical dagger monoidal structure on qRel , which we proceed to describe.
Definition 3.3.
The Cartesian product of binary relations R and R is the binary relation R × R from X × X to Y × Y defined by( R × R )( X ⊗ X , Y ⊗ Y ) = span { r ⊗ r | r ∈ R ( X , Y ) , r ∈ R ( X , Y ) } , where R is a binary relation from a quantum set X to a quantum set Y , and R is a binaryrelation from a quantum set X to a quantum set Y .As we will see, the Cartesian product is a symmetric monoidal structure on qRel withmonoidal unit = Q{ C } . Definition 3.4.
For each finite-dimensional Hilbert space H , write H ∗ = L ( H, C ) for thedual Hilbert space. For each linear operator v ∈ L ( H, K ), write v ∗ ∈ L ( K ∗ , H ∗ ) for theBanach space adjoint of v , defined by v ∗ ( ϕ ) = ϕ ◦ v . For each subspace V ≤ L ( H, K ),write V ∗ = { v ∗ | v ∈ V } ≤ L ( K ∗ , H ∗ ). The dual of a quantum set X is the quantum set X ∗ = Q{ X ∗ | X ∝ X } . The dual of a binary relation R from X to Y is the binary relation R ∗ from Y ∗ to X ∗ defined by R ∗ ( Y ∗ , X ∗ ) = R ( X, Y ) ∗ . s in any compact closed category, each morphism set Rel( X ; Y ) is naturally isomorphicto Rel( ; Y × X ∗ ) as a functor in X and Y . Definition 3.5.
Let H and K be finite-dimensional Hilbert spaces. For each linear operator v ∈ L ( H, K ), write v † ∈ L ( K, H ) for the Hermitian adjoint of v , defined by h v † k | h i = h k | vh i . For each subspace V ≤ L ( H, K ), write V † = { v † | v ∈ V } ≤ L ( K, H ). The adjointof a binary relation R from X to Y is the binary relation R † from Y to X defined by R † ( Y, X ) = R ( X, Y ) † .The category FdHilb of finite-dimensional Hilbert spaces and linear operators is thecanonical example of a dagger compact category; formally, (
FdHilb , ⊗ , C , † ) is a daggercompact category. Similarly, ( Rel , × , {∗} , † ) is a dagger compact category, where R † denotesthe transpose of an ordinary binary relation R . Our category qRel might be viewed as amutual generalization of Rel and
FdHilb : An ordinary binary relation can be pictured as amatrix of truth values. Similarly a binary relation between quantum sets can be pictured asa matrix of morphisms of the category whose objects are nonzero finite-dimensional Hilbertspaces and whose morphisms are subspaces of operators from one Hilbert space to another.
Definition 3.6.
We define the category qAtRel : The objects are nonzero finite-dimensionalHilbert spaces. The morphisms from an object X to an object Y are subspaces of L ( X, Y ),that is, vector spaces of linear operators from X to Y .This definition might more naturally be given to include zero-dimensional Hilbert spaces,and likewise the definition of quantum sets (definition 2.1). Lemma 3.7 and theorem 3.8 aretrue if zero-dimensional Hilbert spaces are included. Lemma 3.7.
The category qAtRel is dagger compact if it is equipped with(1) the monoidal product defined on morphisms by V ⊗ W = span { v ⊗ w | v ∈ V, w ∈ W } ,(2) the monoidal unit C , and(3) the involution defined by V † = { v † | v ∈ V} .Proof. The full proof is a tedious perusal of the defining properties of a dagger compactcategory. The most involved step is likely the first one: showing that the monoidal productis a bifunctor, which amounts to the observation that all the products involved are linear.The braidings, associators, and unitors of
FdHilb are simply replaced with their spans, andwe must check that these spans form natural transformations, quite like we check that themonoidal product is a bifunctor. The triangle and pentagon identies lift to qAtRel in astraightforward way. Similarly, the units and counits for dual Hilbert spaces are replacedwith their spans, and their identities lift to qAtRel . All but one of the desired conditionson the involution † are equations, which hold trivially. The remaining condition expressesthe compatibility of the involution with duality, which lifts to qAtRel , just as before: X ⊗ X ∗ C X ∗ ⊗ X σ X,X ∗ η X ǫ † X X ⊗ X ∗ C X ∗ ⊗ X C · σ X,X ∗ C · η X C · ǫ † X (cid:3) xplicitly, the unit η , the counit ǫ , and the braiding σ in the above diagrams are definedby η X (1) = dim( X ) X i =1 x i ⊗ ˆ x i ǫ X (ˆ x ⊗ x ′ ) = ˆ x ( x ′ ) σ X,X ∗ ( x ′ ⊗ ˆ x ) = ˆ x ⊗ x ′ , where { x , . . . , x dim( X ) } is any orthonormal basis of X , and { ˆ x , . . . , ˆ x dim( X ) } is the corre-sponding orthonormal basis of X ∗ . Typically, the unit is defined to be a map to X ∗ ⊗ X andthe counit is defined to be a map from X ⊗ X ∗ ; we choose the opposite convention to bettervisualize the correspondence between operators in L ( X, Y ) and vectors in Y ⊗ X ∗ , becauseoperators are applied on the left.In the category qRel , the unit H and the counit E are defined by H X ( C , X ⊗ X ∗ ) = C · η X and E X ( X ∗ ⊗ X, C ) = C · ǫ X , for all quantum sets X and all atoms X ∝ X , with all othercomponents vanishing. Theorem 3.8.
The structure ( qRel , × , , † ) is a dagger compact category.Proof. The full proof follows the same pattern as the proof of lemma 3.7, with most of thework going toward establishing the functoriality of the monoidal product. However, ratherthan working with vector spaces of morphisms, as we do in lemma 3.7, we work with sets ofmorphisms, appealing to lemma 3.7 for the properties of these morphisms.The objects of qRel are sets of objects of qAtRel , and the morphisms of qRel are es-sentially matrices of morphisms of qAtRel , with composition defined analogously to matrixmultiplication. In this context, the sum of a family of morphisms ( V j | j ∈ J ) from a Hilbertspace X to a Hilbert space Z is their naive algebraic sum P j V j = span S j V j . This algebraicsum commutes with composition; this is why the composition of binary relations is associa-tive. This algebraic sum also commutes with the tensor product, so the Cartesian productof quantum sets is a functor. The naturality of the braidings, associators, and unitors, andmore generally the commutativity of the other relevant diagrams can be checked atomwise,because in each expression for a composition of binary relations, all but one summand V j will be 0. (cid:3) The coproduct of two quantum sets X and Y in the category qRel is their disjoint union X ⊎ Y , which is defined up to isomorphism by At(
X ⊎ Y ) = At( X ) ⊎ At( Y ). Assuming thatAt( X ) and At( Y ) are disjoint, the inclusions J : X ֒ → X ⊎ Y and K : Y ֒ → X ⊎ Y are thendefined by J ( X, X ) = C · X for X ∝ X , and K ( Y, Y ) = C · Y for Y ∝ Y , with the othercomponents vanishing. If R is a quantum relation from X to a quantum set Z , and S isa quantum relation from Y to Z , we can define a binary relation [ R, S ] from
X ⊎ Y to Z ,by [ R, S ]( X, Z ) = R ( X, Z ) and [
R, S ]( Y, Z ) = S ( Y, Z ), for X ∝ X , Y ∝ Y , and Z ∝ Z . It isstraightforward to check that [ R, S ] ◦ J = R , that [ R, S ] ◦ K = S , and that together thesetwo equations uniquely determine [ R, S ].It follows by general arguments that qRel has biproducts and is enriched over commutativemonoids [2, 3.2, 5.2]. One may check that the monoidal product of this enrichment is thedisjunction ∨ , defined below. In fact, it is easy to see directly from definition 3.1 thatcomposition respects infinitary disjunction. Altogether, the set of binary relations from aquantum set X to a quantum set Y carries the structure of a complete orthomodular lattice,but composition does not generally respect the rest of this structure. efinition 3.9. Let X and Y be quantum sets. The set Rel( X ; Y ) of binary relations from X to Y is canonically a complete orthomodular lattice, with operations defined atomwise.Explicitly, for R, S ∈ Rel( X ; Y ), we define:(1) R ∨ S = ( R ( X, Y ) ∨ S ( X, Y ) | X ∝ X , Y ∝ Y )(2) R ∧ S = ( R ( X, Y ) ∧ S ( X, Y ) | X ∝ X , Y ∝ Y )(3) ¬ R = ( R ( X, Y ) ⊥ | X ∝ X , Y ∝ Y )We extend (1) and (2) to arbitrary families in Rel( X ; Y ), in the obvious way. We also write:(4) R ≤ S ⇔ ∀ X ∝ X : ∀ Y ∝ Y : R ( X, Y ) ≤ S ( X, Y )(5) R ⊥ S ⇔ ∀ X ∝ X : ∀ Y ∝ Y : R ( X, Y ) ⊥ S ( X, Y )4. functions between quantum sets
Definition 4.1.
A function from a quantum set X to a quantum set Y is a binary relation F from X to Y such that F † ◦ F ≥ I X and F ◦ F † ≤ I Y . We write Fun( X ; Y ) for the set ofall functions from X to Y .The identity binary relation I X on a quantum set X satisfies I †X = I X and I X ◦ I X = I X ,so it is a function. Furthermore, if F is a function from X to a quantum set Y , and G is afunction from Y to a quantum set Z , then G ◦ F is also a function; we display one of thetwo relevant computations:( G ◦ F ) † ◦ ( G ◦ F ) = F † ◦ G † ◦ G ◦ F ≥ F † ◦ I Y ◦ F = F † ◦ F ≥ I X Thus, quantum sets and functions form a subcategory qSet of qRel . Proposition 4.2.
Let F be a function from a quantum set X to a quantum set Y . If F isinvertible in qRel , then F − = F † .Proof. F † = F † ◦ F ◦ F − ≥ I X ◦ F − = F − F † = F − ◦ F ◦ F † ≤ F − ◦ I Y = F − (cid:3) Definition 4.3.
Let R be a binary relation from a quantum set X to a quantum set Y . Wesay that R is(1) injective iff R † ◦ R ≤ I X , and(2) surjective iff R ◦ R † ≥ I Y .As a consequence of proposition 4.2, an invertible function must be both injective andsurjective. Conversely, a function F : X → Y that is both injective and surjective satisfies F † ◦ F = I X and F ◦ F † = I Y , so it is invertible. Evidently, if F is invertible in qRel , thenits inverse F − = F † is also a function. Proposition 4.4.
Let F be an invertible function from a quantum set X to a quantum set Y . Then there exists an ordinary invertible function f from At( X ) to At( Y ) , and a familyof unitaries ( u X : X → f ( X ) | X ∝ X ) such that F ( X, f ( X )) = C · u X for all X ∝ X , with F ( X, Y ) = 0 whenever Y = f ( X ) . Conversely, every relation from X to Y that is of thisform is an invertible function. roof. As we have seen, the invertibility of a quantum function is equivalent to the systemof equations F † ◦ F = I X and F ◦ F † = I Y . Equivalently, _ Y ∝ Y F ( X, Y ) † · F ( X ′ , Y ) = I X ( X ′ , X ) and _ X ∝ X F ( X, Y ) · F ( X, Y ′ ) † = I Y ( Y ′ , Y ) , for all X, X ′ ∝ X , and for all Y, Y ′ ∝ Y , respectively. These equations imply that for all X ∝ X and Y ∝ Y , we have F ( X, Y ) † · F ( X, Y ) ≤ C · X and F ( X, Y ) · F ( X, Y ) † ≤ C · Y , so if F ( X, Y ) is nonzero, then it is spanned by a single unitary operator.Fix X ∝ X . At least one of the spaces F ( X, Y ) must be nonzero. Furthermore, if F ( X, Y )and F ( X, Y ′ ) are both nonzero, then both are spanned by a unitary, and we have I Y ( Y ′ , Y ) ≥ F ( X, Y ) · F ( X, Y ′ ) † = 0, so Y ′ = Y . Thus, there is a unique atom f ( X ) ∝ Y such that F ( X, f ( X )) = 0. Choose a unitary u X that spans F ( X, f ( X )). As we vary X , we obtain afunction f : At( X ) → At( Y ), with each space F ( X, f ( X )) spanned by a unitary u X , and allother spaces F ( X, Y ) vanishing.Assume that there exists an atom Y ∝ Y that is not in the range of f . For this atom,we have F ( X, Y ) = 0 for all X ∝ X , so we obtain a contradiction: 1 Y ∈ I Y ( Y , Y ) = 0.Therefore, f is a bijection.It is easy to see that any function from X to Y of the given form is invertible, simply byverifying the two equations given at the beginning of this proof. In each join, at most oneof the terms is nonzero, and if there is such a term, then it is spanned by the product of aunitary operator with its adjoint. (cid:3) Both injectivity and surjectivity have natural dual notions.
Definition 4.5.
Let R be a binary relation from a quantum set X to a quantum set Y . Wesay that R is(1) coinjective iff R ◦ R † ≤ I Y , and(2) cosurjective iff R † ◦ R ≥ I X .Thus, a function is a binary relation that is both coinjective and cosurjective. Definition 4.6.
A partial function from a quantum set X to a quantum set Y is a coinjectivebinary relation from X to Y .Quantum sets and partial functions form a subcategory qPar of qRel , and qSet is asubcategory of qPar . Proposition 4.7.
Let F be a partial function from a quantum set X to a quantum set Y .If F is invertible in qRel , then F − = F † , so F is a function.Proof. The relation F ◦ F † is an invertible subrelation of I Y ; therefore F ◦ F † = I Y . Similarly,we can prove that F † ◦ F = I X , if we can show that F † ◦ F ≤ I X . We can: F † ◦ F = F − ◦ F ◦ F † ◦ F ≤ F − ◦ I Y ◦ F = I X (cid:3) Not every invertible relation is a function. Let a ∈ L ( C , C ) be an invertible matrix thatisn’t a scalar multiple of a unitary matrix. The relation R from Q{ C } to Q{ C } defined by R ( C , C ) = C · a is evidently invertible, but not coinjective. . hereditarily atomic von Neumann algebras Definition 5.1.
Let X be a quantum set. Define: ℓ ( X ) = Y X ∝ X L ( X )This has the structure of a ∗ -algebra over C , equipped with the product topology.For any ordinary set S , the ∗ -algebra ℓ (‘ S ) is canonically isomorphic to ℓ ( S ) = C S . We willlater show that the self-adjoint elements of ℓ ( X ) are in canonical bijective correspondencewith functions from X to ‘ R (proposition 11.2). Similarly, the normal elements of ℓ ( X ) arein canonical bijective correspondence with functions from X to ‘ C . Arbitrary elements of ℓ ( X ) are in bijective correspondence with functions from X to a canonical quantum set C (definition 11.7), which intuitively consists of complex numbers whose real and imaginaryparts need not be simultaneously observable. Definition 5.2. ℓ ∞ ( X ) = (cid:26) a ∈ ℓ ( X ) (cid:12)(cid:12)(cid:12)(cid:12) sup X ∝ X k a ( X ) k ∞ < ∞ (cid:27) c ( X ) = n a ∈ ℓ ( X ) (cid:12)(cid:12)(cid:12) lim X →∞ k a ( X ) k ∞ = 0 o The limit above is in the sense of one-point compactification; in other words, for each ǫ >
0, we should have k a ( X ) k ∞ < ǫ for all but finitely many X ∝ X . The ∗ -algebras ℓ ∞ ( X )and c ( X ) are canonically represented on the ℓ -direct sum of the Hilbert spaces in At( X ),isometrically for the operator norm, and the norm k a k = sup X ∝ X k a ( X ) k ∞ . Represented in this way, c ( X ) is a concrete C*-algebra, and ℓ ∞ ( X ) is a von Neumann alge-bra. In the context of noncommutative mathematics, the C*-algebra c ( X ) is the operatoralgebra associated to X considered as a discrete quantum topological space, and the vonNeumann algebra ℓ ∞ ( X ) is the operator algebra associated to X considered as an atomicquantum measure space. We might venture to say that ℓ ( X ) is the operator algebra associ-ated to X considered as a quantum set. Definition 5.3.
A von Neumann algebra A is hereditarily atomic just in case every vonNeumann subalgebra of A is atomic.Recall that a von Neumann algebra is said to be atomic, or sometimes fully atomic, if everynonzero projection is above a minimal projection. Equivalently, a von Neumann algebra isatomic if and only if every projection is the sum of some family of pairwise orthogonalminimal projections. Proposition 5.4.
Let A be a von Neumann algebra. The following are equivalent:(1) A is hereditarily atomic(2) A is isomorphic to ℓ ∞ ( X ) for some quantum set X (3) every self-adjoint operator a in A is diagonalizable e call a self-adjoint operator a diagonalizable just in case there is a family of pairwiseorthogonal projections ( p α | α ∈ R ) such that a = P α αp α , with convergence in the ultraweaktopology. Observe that if a is diagonalizable, then the family ( p α ) is unique, and eachprojection p α is a spectral projection of a . As a consequence, a is diagonalizable in A if andonly if it is diagonalizable in any given von Neumann subalgebra B of A that contains a . Proof. (1) ⇒ (2). Let A be a hereditarily atomic von Neumann algebra. The center of A is atomic, so A is an ℓ ∞ -direct sum of factors. Every factor that is not finite type I has avon Neumann subalgebra isomorphic to L ∞ ([0 , , dt ), which is not atomic, so A must be adirect sum of finite type I factors. Choosing an irreducible representation for each factor, weobtain a quantum set X such that ℓ ∞ ( X ) ∼ = A .(2) ⇒ (3). Let A be isomorphic to ℓ ∞ ( X ), and let a ∈ A be self adjoint. Without loss ofgenerality, we assume A = ℓ ∞ ( X ). Each self-adjoint operator a ( X ) can be diagonalized in L ( X ) by the spectral theorem for self-adjoint matrices. Altogether, we have a diagonalizationof a ; a bounded net converges ultraweakly in an ℓ ∞ -direct sum of von Neumann algebras ifand only if it coverges ultraweakly in each summand.(3) ⇒ (1). Assume that every self-adjoint operator in A is diagonalizable. Let B bea von Neumann subalgebra of A , and let p be a nonzero projection in B . Choose a max-imal abelian von Neumann subalgebra C of the von Neumann algebra pBp . If C has avon Neumann subalgebra isomorphic to L ∞ ([0 , , dt ), then C contains a nondiagonalizableself-adjoint operator, contradicting our assumption on A ; indeed, the identity function is anondiagonalizable element of L ∞ ([0 , , dt ), and diagonalizability in a von Neumann subalge-bra is equivalent to diagonalizability in the larger von Neumann algebra, as we observed justabove the proof. Therefore, C is atomic, and in particular, it contains a minimum projection q . The projection q is also a minimal projection in pBp , because C is maximal abelian. So, q is a minimum projection in B that is below p . We conclude that B is atomic, and moregenerally, that A is hereditarily atomic. (cid:3) Definition 5.5.
We write M ∗ for the category of hereditarily atomic von Neumann algebrasand normal ∗ -homomorphisms. We write M ∗ for the category of hereditarily atomic vonNeumann algebras and unital normal ∗ -homomorphisms.The letter “M” is intended to suggest matrix algebras.6. three perspectives on a function We now show that each partial function F from a quantum set X to a quantum set Y induces a ∗ -homomorphism F ⋆ from ℓ ( Y ) to ℓ ( X ). The immediate significance of this con-struction is that it yields a contravariant equivalence of categories, from the category qSet of quantum sets and functions, to the category M ∗ of hereditarily atomic von Neumannalgebras and unital normal ∗ -homomorphisms (theorem 7.6), placing our objects and mor-phisms within the usual framework of noncommutative mathematics, and enabling the useof operator-algebraic techniques.If X and Y are quantum sets whose atoms are all one-dimensional, then a function F from X to Y is effectively an ordinary function between ordinary sets, and F ⋆ is effectively justprecomposition by F . The unital ∗ -homomorphism F ⋆ can be viewed as precomposition by F in the general case too (theorem 11.9). et X and Y be arbitrary quantum sets. Let F be an arbitrary partial function from X to Y , and consider a pair of atoms X ∝ X and Y ∝ Y . By definition of partial function, F ◦ F † ≤ I Y , so F ( X, Y ) · F ( X, Y ) † ≤ C · Y . We now recall a result classifying operatorspaces of this kind [32, 4.2.7]. Lemma 6.1.
Let X and Y be finite-dimensional Hilbert spaces. There are canonical bijectivecorrespondences between(1) linear spaces V of operators from X to Y , satisfying V · V † ≤ C · Y ,(2) coisometries w : X → Y ⊗ H , with H a Hilbert space up to unitary equivalence, and(3) ∗ -homomorphisms ρ : L ( Y ) → L ( X ) . Thus, we identify two coisometries w : X → Y ⊗ H and w ′ : X → Y ⊗ H ′ iff there is aunitary operator u : H → H ′ such that w ′ = (1 ⊗ u ) w . Proof.
We describe the three constructions. Let V be any linear space of operators from X to Y , satisfying V · V † ≤ C · Y . We have an inner product on V defined by vv ′† = ( v ′ | v ) · Y .Choose an orthonormal basis ( v , . . . , v n ) for this inner product. Let ( e , . . . , e n ) be thestandard basis of C n , and define w : X → Y ⊗ C n by w ( x ) = P ni =1 v i ( x ) ⊗ e i , for all x ∈ X .It is easy to check that w † ( y ⊗ e i ) = v † i ( y ), for all y ∈ Y . A short computation shows that w is a coisometry: h w † ( y ⊗ e i ) | w † ( y ′ ⊗ e j ) i = h v † i y | v † j y ′ i = h y | v i v † j y ′ i = h y | y ′ i · δ ij = h y ⊗ e i | y ′ ⊗ e j i Here, δ ij is the Kronecker delta symbol. Adjusting that computation slightly, we find thatif w ′ is obtained using another choice of orthonormal basis, then h w † ( y ⊗ e i ) | w ′† ( y ′ ⊗ e j ) i = h y | y ′ i u ij for some unitary matrix ( u ij ), so ww ′† = (1 ⊗ u ) for some unitary operator u on C n .Let w : X → Y ⊗ H be any coisometry. Define ρ : L ( Y ) → L ( X ) by ρ ( b ) = w † ( b ⊗ w .It is easy to check that ρ is a ∗ -homomorphism, and that ρ is the same if H is replaced bya unitarily equivalent Hilbert space.Let ρ : L ( Y ) → L ( X ) be any ∗ -homomorphism. Define V to be the space of linear op-erators v from X to Y such that bv = vρ ( b ) for all b ∈ L ( Y ). It follows immediately that V · V † ⊆ L ( Y ) ′ = C · Y .Fix V , a linear space of operators from X to Y satisfying V · V † ≤ C ·
1. Performing thefirst two constructions, we obtain a ∗ -homomorphism ρ defined by ρ ( b ) = P ni =1 v † i bv i , where( v , . . . , v n ) is some orthonormal basis for V . It is easy to check that each basis element v j satisfies v j ρ ( b ) = bv j , so V is a subspace of the linear space ˜ V constructed from ρ via thethird construction. Conversely, each operator ˜ v in ˜ V is in V because˜ v = 1 · ˜ v = ˜ v · ρ (1) = n X i =0 ˜ vv † i v i = n X i =0 ( v i | ˜ v ) v i . Therefore ˜ V = V .Thus, the three constructions compose to the identity. In particular, the third constructionis surjective. The second construction is also surjective, by the representation theory of L ( Y ).The first construction is surjective, because the construction w span { (1 ⊗ ˆ e i ) w | ≤ i ≤ n } ,where ˆ e i = h e i | · i , is easily seen to be a right inverse. It follows immediately that all threeconstructions are bijective. (cid:3) emma 6.1 suggests three different ways of looking at a function between quantum sets:as a relation, as a family of isometries, and as a ∗ -homomorphism. We use the term “fission”for the second of these three notions, because a function from a quantum set X to a quantumset Y splits each atom of X into finitely many atoms of Y , and the corresponding family ofisometries makes this splitting explicit. Definition 6.2.
Let X and Y be quantum sets. A partial fission f from X to Y is a family offinite-dimensional Hilbert spaces ( H YX | X ∝ X , Y ∝ Y ), together with a family of coisometries( f YX : X → Y ⊗ H YX | X ∝ X , Y ∝ Y ) such that ( f Y X ) · ( f Y X ) † = 0 whenever Y = Y . Weidentify partial fissions f and f ′ from X to Y whenever there is a family of unitary operators( u YX : H YX → H ′ YX | X ∝ X , Y ∝ Y ) such that f ′ YX = (1 ⊗ u YX ) f YX for all X ∝ X and Y ∝ Y .The orthogonality condition implies that for each X ∝ X , we have dim( H YX ) = 0 for all butfinitely many Y ∝ Y . Intuitively, the partial fission splits each atom X ∝ X into atoms of Y ,with each atom Y ∝ Y occurring dim( H YX ) times. It is possible to formulate a definition thatis even closer to this intuition, but it would obscure that, for example, the many orthogonaldecompositions of a two-dimensional Hilbert space X into subspaces isomorphic to C areequivalent in the sense that they describe the same function from Q{ X } to Q{ C } . In general,the support projections of the coisometries ( f YX | Y ∝ Y ) need not sum to the identity on X ;a part of the atom X may be simply lost. For each X ∝ X , we only havedim( X ) ≥ X Y ∝ Y dim( Y ) · dim( H YX ) . Theorem 6.3.
Let X and Y be quantum sets. There are canonical bijective correspondencesbetween(1) partial functions F from X to Y ,(2) partial fissions f from X to Y , and(3) ∗ -homomorphisms φ : ℓ ( Y ) → ℓ ( X ) , continuous for the product topologies.Explicitly, the bijections are given by the following constructions: • Let F be a partial function from X to Y . For each X ∝ X , and each Y ∝ Y , choose anorthonormal basis B YX of F ( X, Y ) for the inner product defined by ( v | v ′ )1 Y = v ′ · v † .Equip F ( X, Y ) † with the corresponding inner product, defined by ( h | h ′ )1 Y = h † · h ′ .Then the family ( f YX : X → Y ⊗ F ( X, Y ) † | X ∝ X , Y ∝ Y ) defined by f YX ( x ) = X v ∈ B YX v ( x ) ⊗ v † is a partial fission from X to Y that does not depend on the choice of bases. • Let f be a partial fission from X to Y . Then the map φ : ℓ ( Y ) → ℓ ( X ) defined by φ ( b )( X ) = X Y ∝ Y ( f YX ) † ( b ( Y ) ⊗ f YX ) is a continuous ∗ -homomorphism. • Let φ : ℓ ( Y ) → ℓ ( X ) be a continuous ∗ -homomorphism. Then the binary relation F from X to Y defined by F ( X, Y ) = { v ∈ L ( X, Y ) | b ( Y ) · v = v · φ ( b )( X ) for all b ∈ ℓ ( Y ) } is a partial function from X to Y . urthermore, the three constructions compose the identity in each cyclical order.Proof. Lemma 6.1 yields canonical bijective correspondences between(1) families of subspaces ( F ( X, Y ) ≤ L ( X, Y ) | X ∝ X , Y ∝ Y ) satisfying F ( X, Y ) · F ( X, Y ) † ≤ C · Y ,(2) families of coisometries ( f YX : X → Y ⊗ H YX | X ∝ X , Y ∝ Y ), up to unitary equivalenceof the coefficient Hilbert spaces H YX , and(3) families of ∗ -homomorphisms ( φ YX : L ( Y ) → L ( X ) | X ∝ X , Y ∝ Y ).We have three equivalent orthogonality conditions under these correspondences:(1) F ( X, Y ) · F ( X, Y ) † = 0 whenever Y = Y (2) ( f Y X ) · ( f Y X ) † = 0 whenever Y = Y (3) φ Y X ( L ( Y )) · φ Y X ( L ( Y )) = 0 whenever Y = Y The implications (1) ⇒ (2) and (2) ⇒ (3) are immediate. Therefore, assume (3). For all v ∈ F ( X, Y ) and v ∈ F ( X, Y ), we find that v · v † = 1 · v · v † · v · φ Y X (1) · φ Y X (1) · v † = 0,whenever Y = Y . Thus, (3) ⇒ (1).The families of subspaces satisfying condition (1) are exactly the partial functions from X to Y , essentially by definition. The families of coisometries satisfying condition (2) are exactlythe partial fissions from X to Y , by definition. The families of ∗ -homomorphisms satisfyingcondition (3) correspond bijectively to the continuous ∗ -homomorphisms φ : ℓ ( Y ) → ℓ ( X )via the equation φ ( b )( X ) = P Y ∝ Y φ YX ( b ( Y )) for X ∝ X and Y ∝ Y , by lemma 6.4 below. (cid:3) Lemma 6.4.
Let X and Y be quantum sets. For each X ∝ X , write π X : ℓ ( X ) → L ( X ) forthe projection map, and for each Y ∝ Y , write ι Y : L ( Y ) → ℓ ( Y ) for the inclusion map.For each continuous ∗ -homomorphism φ : ℓ ( Y ) → ℓ ( X ) , the family of ∗ -homomorphisms ( φ YX = π X ◦ φ ◦ ι Y | X ∝ X , Y ∝ Y ) satisfies φ Y X ( L ( Y )) · φ Y X ( L ( Y )) = 0 for all distinct Y , Y ∝ Y . Conversely, for each family of ∗ -homomorphisms ( φ YX : L ( Y ) → L ( X ) | X ∝ X , Y ∝ Y ) satisfying φ Y X ( L ( Y )) · φ Y X ( L ( Y )) = 0 for all distinct Y , Y ∝ Y , the equation φ ( b )( X ) = X Y ∝ Y φ YX ( b ( Y )) defines a continuous ∗ -homomorphism φ : ℓ ( Y ) → ℓ ( X ) . The two constructions are inverses.Furthermore, the same proposition is true for normal ∗ -homomorphisms ℓ ∞ ( Y ) → ℓ ∞ ( X ) ,in place of continuous ∗ -homomorphisms ℓ ( Y ) → ℓ ( X ) . As a consequence, each continuous ∗ -homomorphism ℓ ( Y ) → ℓ ( X ) restricts to a normal ∗ -homomorphism ℓ ∞ ( Y ) → ℓ ∞ ( X ) ,and this defines a bijection between the two sets of morphisms. As elsewhere, we use the product topologies on ℓ ( X ) and ℓ ( Y ). Proof.
Let φ : ℓ ( Y ) → ℓ ( X ) be any continuous ∗ -homomorphism, and let Y , Y ∝ Y be dis-tinct. By definition of the canonical inclusion maps ι Y and ι Y , we certainly have that ι Y ( L ( Y )) · ι Y ( L ( Y )) = 0. Therefore, the family ( φ YX = π X ◦ φ ◦ ι Y | X ∝ X , Y ∝ Y ) satisfies φ Y X ( L ( Y )) · φ Y X ( L ( Y )) = 0, for all distinct Y , Y ∝ Y . Furthermore, for all b ∈ ℓ ( Y ) and all X ∝ X , X Y ∝ Y φ YX ( b ( Y )) = X Y ∝ Y π X ( φ ( ι Y ( b ( Y )))) = π X φ X Y ∝ Y ι Y ( b ( Y )) !! = π X ( φ ( b )) = φ ( b )( X ) . e have used the assumed continuity of φ , and the tautological continuity of π X .Now, let ( φ YX : L ( Y ) → L ( X ) | X ∝ X , Y ∝ Y ) be any family of ∗ -homomorphisms satis-fying this property, and fix X ∝ X . Since X is finite-dimensional, φ YX ( L ( Y )) = 0 for allbut finitely many Y ∝ Y . It follows that for all b ∈ ℓ ( Y ), the sum is P Y ∝ Y φ YX ( b ( Y )) istrivially convergent, and furthermore, this expression defines a continuous ∗ -homomorphism φ X : ℓ ( Y ) → L ( X ). Therefore, the ∗ -homomorphism φ : ℓ ( Y ) → ℓ ( X ) defined by φ ( b )( X ) = φ X ( b ) is also continuous, by definition of the product topology. Furthermore, for all X ∝ X , Y ∝ Y , and m ∈ L ( Y ), π X ( φ ( ι Y ( m )) = φ ( ι Y ( m ))( X ) = φ X ( ι Y ( m )) = X Y ′ ∝ Y φ Y ′ X ( ι Y ( m )( Y ′ )) = φ YX ( m ) . Therefore, the two constructions are inverses.The argument works just the same with normal ∗ -homomorphisms ℓ ∞ ( Y ) → ℓ ∞ ( X ),in place of continuous ∗ -homomorphisms ℓ ( Y ) → ℓ ( X ), with a minor adjustment in thejustifications. In the forward direction, we observe that the sum P Y ∝ Y ι Y ( b ( Y )) convergesto b ultraweakly, and that φ is ultraweakly continuous because it is normal. In the backwarddirection, we observe that ℓ ∞ ( Y ) is the product of the family ( L ( Y ) | Y ∝ Y ) in the categoryof von Neumann algebras and normal ∗ -homomorphisms. (cid:3) Lemma 6.5.
Let X and Y be quantum sets, and let F be a partial function from X to Y .Let f be the corresponding partial fission, and let φ be the corresponding ∗ -homomorphism,in the sense of theorem 6.3. Then, the following are equivalent(1) F † ◦ F ≥ I (2) For all X ∝ X , we have P Y ∝ Y f YX † f YX = 1 .(3) φ (1) = 1A partial fission satisfying condition (2) is thus reasonably called simply a fission. Proof.
Condition (2) is easily seen to be equivalent to condition (3): φ (1) = 1 ⇔ ∀ X ∝ X : φ (1)( X ) = 1 X ⇔ ∀ X ∝ X : X Y ∝ Y f Y † X (1 ⊗ f YX = 1 ⇔ ∀ X ∝ X : X Y ∝ Y f Y † X f YX = 1To demonstrate the equivalence between condition (1) and condition (2), choose bases( B YX ) for the spaces ( F ( X, Y )). Condition (1) is equivalent to P Y ∝ Y F ( X, Y ) † · F ( X, Y ) ∋ X for all X ∝ X , and condition (2) is equivalent to the equation P Y ∝ Y P v ∈ B YX v † v = 1 X for all X ∝ X . Thus, condition (2) clearly implies condition (1).So, assume condition (1), and fix X ∝ X . The definition of a function implies that( F ( X, Y ) † · F ( X, Y ) | Y ∝ Y ) is a family of pairwise-orthogonal ∗ -subalgebras of L ( X ). Thus,the identity operator 1 X can be written as a sum of orthogonal projections P Y ∝ Y p Y , with p Y the identity of F ( X, Y ) † · F ( X, Y ). Fix Y ∝ Y , and for convenience, enumerate theorthonormal basis B YX = { v , v , . . . , v n } . Since p Y ∈ F ( X, Y ) † · F ( X, Y ), we can write p Y = P ni,j =1 α ij v † i v j for some doubly indexed family of complex numbers ( α ij | ≤ i, j ≤ d ),and the orthonormality of the basis easily implies that α ij is the Dirac delta symbol. Thus, Y = P di =1 v † i v i = P v ∈ B YX v † v . We vary Y ∝ Y to conclude that 1 X = P Y ∝ Y P v ∈ B YX v † v , asdesired. (cid:3) functoriality We now define the composition of partial fissions, to prove that the bijective correspon-dences in theorem 6.3 are functorial.
Definition 7.1.
Let X , Y , and Z be quantum sets. Let f be a partial fission from X to Y ,and let g be a partial fission from Y to Z . Explicitly, f has components f YX : X → Y ⊗ H YX ,and g has components g ZY : Y → Z ⊗ K ZY . For all X ∝ X and Z ∝ Z , the Hilbert space L ZX = L Y ∝ Y K ZY ⊗ H YX is finite-dimensional (definition 6.2). The composition g ◦ f is thepartial fission from X to Z whose components ( g ◦ f ) ZX : X → Z ⊗ L ZX are defined by( g ◦ f ) ZX = X Y ∝ Y ( g ZY ⊗ · f YX . Each individual term ( g ZY ⊗ · f YX is a coisometry from X to Z ⊗ K ZY ⊗ H YX , so the sum hasfinitely many nonzero terms and is an operator from X to L Y ∝ Y Z ⊗ K ZY ⊗ H YX = Z ⊗ L ZX .For fixed X ∝ X , it is easy to see that [( g ZY ⊗ · f YX ] · [( g Z ′ Y ′ ⊗ · f Y ′ X ] † = 0 unless Y = Y ′ and Z = Z ′ . This implies that ( g ◦ f ) ZX is a coisometry for each Z ∝ Z , and furthermore that[( g ◦ f ) ZX ] · [( g ◦ f ) ZX ] † = 0 unless Z = Z ′ . We vary X to conclude that g ◦ f is a coisometry. Proposition 7.2.
The bijective correspondences of theorem 6.3 are functorial. Explicitly:Let X , Y , and Z be quantum sets. Let F be a partial function from X to Y with correspondingpartial fission f and homomorphism φ . Similarly, let G be a partial function from Y to Z with corresponding partial fission g and homomorphism ψ . It follows that the partial function G ◦ F has corresponding partial fission g ◦ f , and corresponding homomorphism φ ◦ ψ .Proof. By definition of the corresponding homomorphisms φ and ψ , given in the statementon theorem 6.3, we have that for all c ∈ ℓ ∞ ( Z ) and all X ∝ X , φ ( ψ ( c ))( X ) = X Y ∝ Y X Z ∝ Z f Y † X ( g Z † Y ( c ( Z ) ⊗ g ZY ⊗ f YX = X Y ∝ Y X Z ∝ Z f Y † X ( g Z † Y ⊗ c ( Z ) ⊗ ⊗ g ZY ⊗ f YX . The second tensor factor in c ( Z ) ⊗ ⊗ K ZY , and the third is identity on H YX .On the other hand, the homomorphism θ corresponding to g ◦ f is defined by θ ( c )( X ) = X Z ∝ Z ( g ◦ f ) Z † X ( c ( z ) ⊗ g ◦ f ) Zx = X Z ∝ Z X Y ∝ Y ( g ZY ⊗ f YX ! † · ( c ( Z ) ⊗ · X Y ′ ∝ Y ( g ZY ′ ⊗ f Y ′ X ! = X Z ∝ Z X Y ∝ Y X Y ′ ∝ Y f Y † X ( g Z † Y ⊗ c ( Z ) ⊗ g ZY ′ ⊗ f Y ′ X . The second tensor factor in c ( Z ) ⊗ L ZX = L Y ∝ Y K ZY ⊗ H YX . Terms with Y = Y ′ do not contribute because the ranges of ( g ZY ⊗ f YX and ( g ZY ′ ⊗ f Y ′ X are Z ⊗ K ZY ⊗ H YX and ⊗ K ZY ′ ⊗ H Y ′ X respectively, and each is invariant for c ( Z ) ⊗
1. On each summand K ZY ⊗ H YX ,the identity on L ZX can also be written as the tensor product of the identity on K ZY and thethe identity on H YX , which appears in our expression for φ ( ψ ( c )). Thus, we find that thetwo expressions are equal; in other words, φ ◦ ψ = θ . We conclude that the correspondencebetween partial fissions and normal ∗ -homomorphisms is functorial, contravariantly.We now argue that the first correspondence in theorem 6.3 is also functorial. Fix X ∝ X and Z ∝ Z . For each Y ∝ Y , the composition map G ( Y, Z ) ⊗ F ( X, Y ) → ( G ◦ F )( X, Z ) isan isometry, as a direct consequence of the definition of the inner products on these threespaces. Furthermore, for distinct
Y, Y ′ ∝ Y , the ranges of the corresponding compositionmaps are orthogonal in ( G ◦ F )( X, Z ), because F ( X, Y ) · F ( X, Y ′ ) † = 0, as an immediateconsequence of the definition of partial function. Together, the ranges of the compositionmaps G ( Y, Z ) ⊗ F ( X, Y ) → ( G ◦ F )( X, Z ), for Y ∝ Y , span ( G ◦ F )( X, Z ), so they forman orthogonal decomposition of this Hilbert space. In other words, these composition mapstogether form a unitary operator u ZX from L Y ∝ Y G ( Y, Z ) ⊗ F ( X, Y ) to ( G ◦ F )( X, Z ).For each Y ∝ Y , choose an orthonormal basis B YX for F ( X, Y ), and an orthonormal basis C ZY for G ( Y, Z ). For all x ∈ X , we have( g ◦ f ) ZX ( x ) = X Y ∝ Y ( g ZY ⊗ f YX ( x ))= X Y ∝ Y X v ∈ B YX ( g ZY ⊗ v ( x ) ⊗ v † )= X Y ∝ Y X v ∈ B YX X w ∈ C ZY w ( v ( x )) ⊗ w † ⊗ v † The set { w ⊗ v | Y ∝ Y , w ∈ C ZY , v ∈ B YX } is an orthonormal basis of L Y ∝ Y G ( Y, Z ) ⊗ F ( X, Y ), so the set { u ZX ( w ⊗ v ) | Y ∝ Y , w ∈ C ZY , v ∈ B YX ) } is an orthonormal basis of( G ◦ F )( X, Z ). By definition of u ZX , we have that u ZX ( w ⊗ v ) = wv , so the partial fission h corresponding to G ◦ F satisfies h ZX ( x ) = X Y ∝ Y X v ∈ B YX X w ∈ C ZY w ( v ( x )) ⊗ ( wv ) † . Allowing X and Z to vary, we identify the partial fission h and the partial fission g ◦ f according to definition 6.2. Explicitly, we can use the family of unitaries defined by u ZX = † ◦ u ZX ◦ ( † ⊗ † ). (cid:3) Definition 7.3.
For i ∈ { , } , let X i and Y i be quantum sets, and let f i : X i → Y i be apartial fission, with components ( f i ) Y i X i : X i → Y i ⊗ ( H i ) Y i X i , for all X i ∝ X i and Y i ∝ Y i . Up tothe obvious permutation of tensor factors, the tensor product partial fission f ⊗ f is definedsimply by taking the tensor product in each component. Explicitly,( f ⊗ f ) Y ⊗ Y X ⊗ X = (1 ⊗ σ ⊗ · (( f ) Y X ⊗ ( f ) Y X ) : X ⊗ X → Y ⊗ Y ⊗ ( H ) Y X ⊗ ( H ) Y X , where σ is the braiding of the tensor product. Lemma 7.4.
For each i ∈ { , } , let X i and Y i be quantum sets, and let F i be a partial func-tion from X i to Y i , with corresponding partial fission f i , and corresponding homomorphism φ i . It follows that the partial function F × F has corresponding partial fission f ⊗ f , andcorresponding homomorphism φ ⊗ φ . n this context, the continuous homomorphism φ ⊗ φ : ℓ ( Y × Y ) → ℓ ( X × X ) isdefined by ( φ ⊗ φ ) Y ⊗ Y X ⊗ X = ( φ ) Y X ⊗ ( φ ) Y X : L ( Y ⊗ Y ) → L ( X ⊗ X ) (lemma 6.4). Forall b ∈ ℓ ( Y ) and b ∈ ℓ ( Y ), writing b ⊗ b for the element of ℓ ( Y × Y ) defined by( b ⊗ b )( Y ⊗ Y ) = b ( Y ) ⊗ b ( Y ), we find that( φ ⊗ φ )( b ⊗ b )( X ⊗ X ) = X Y ⊗ Y ∝ Y ×Y ( φ ⊗ φ ) Y ⊗ Y X ⊗ X (( b ⊗ b )( Y ⊗ Y ))= X Y ∝ Y X Y ∝ Y ( φ ) Y X ( b ( Y )) ⊗ ( φ ) Y X ( b ( Y ))= φ ( b )( X ) ⊗ φ ( b )( X ) . Proof of lemma 7.4.
For all b ∈ ℓ ( Y ) and b ∈ ℓ ( Y ), and for all X ∝ X and X ∝ X ,( φ ⊗ φ )( b ⊗ b )( X ⊗ X ) = φ ( b )( X ) ⊗ φ ( b )( X )= X Y ∝ Y ( f ) Y † X ( b ( Y ) ⊗ f ) Y X ! ⊗ X Y ∝ Y ( f ) Y † X ( b ( Y ) ⊗ f ) Y X ! = X Y ∝ Y X Y ∝ Y (( f ) Y X ⊗ ( f ) Y X ) † ( b ( Y ) ⊗ ⊗ b ( Y ) ⊗ f ) Y X ⊗ ( f ) Y X )= X Y ∝ Y X Y ∝ Y (( f ) Y X ⊗ ( f ) Y X ) † (1 ⊗ σ ⊗ † · (( b ⊗ b )( Y ⊗ Y ) ⊗ ⊗ · (1 ⊗ σ ⊗ f ) Y X ⊗ ( f ) Y X ) . The span of { b ⊗ b | b ∈ ℓ ( Y ) , b ∈ ℓ ( Y ) } is clearly dense in ℓ ( Y × Y ), since thisalgebra is the closure of the span of its factors. Thus, φ ⊗ φ is equal to the homomorphismcorresponding to f ⊗ f .Fix X ∝ X , X ∝ X , Y ∝ Y , and Y ∝ Y . Choose orthonormal bases ( B ) Y X for ( B ) Y X for F ( X , Y ) and F ( X , Y ) respectively. The set { v ⊗ v | v ∈ ( B ) Y X , v ∈ ( B ) Y X } is anorthonormal basis for F ( X , Y ) ⊗ F ( X , Y ). For all vectors x ⊗ x in an arbitrary atom X ⊗ X ∝ X × X ,( f ⊗ f ) Y ⊗ Y X ⊗ X ( x ⊗ x ) = (1 ⊗ σ ⊗ f ) Y X ( x ) ⊗ ( f ) Y X ( x ))= (1 ⊗ σ ⊗ X v ∈ ( B ) Y X X v ∈ ( B ) Y X v ( x ) ⊗ v † ⊗ v ( x ) ⊗ v † = X v ∈ ( B ) Y X X v ∈ ( B ) Y X v ( x ) ⊗ v ( x ) ⊗ v † ⊗ v † = X v ∈ ( B ) Y X X v ∈ ( B ) Y X ( v ⊗ v )( x ⊗ x ) ⊗ ( v ⊗ v ) † . Vectors of the form x ⊗ x span X ⊗ X , so we can vary X , X , Y , and Y to concludethat f ⊗ f is the partial fission corresponding to F ⊗ F . (cid:3) Definition 7.5.
Let X and Y be quantum sets. Let F : X → Y be a partial function. Write F ⋆ : ℓ ( Y ) → ℓ ( X ) for the continuous ∗ -homomorphism φ that corresponds to F in the senseof theorem 6.3. e will also write F ⋆ for the restriction of F ⋆ : ℓ ( Y ) → ℓ ( X ) to a normal ∗ -homomorphism ℓ ∞ ( Y ) → ℓ ∞ ( X ), which exists by lemma 6.4. Theorem 7.6.
There is a contravariant monoidal equivalence from qPar to M ∗ , whichrestricts to a contravariant monoidal equivalence from qSet to M ∗ . It takes each quantumset X to the von Neumann algebra ℓ ∞ ( X ) , and it takes each function F : X → Y to thenormal ∗ -homomorphism F ⋆ : ℓ ∞ ( Y ) → ℓ ∞ ( X ) . This a monoidal functor in the strong sense,for the Cartesian product of quantum sets and the spatial tensor product of von Neumannalgebras. This is an equivalence of categories in the weak sense that it is a full and faithfulfunctor with the property that every hereditarily atomic von Neumann algebra is isomorphicto something in its image. The monoidal structure on qPar is just the restriction of the monoidal structure on qRel ,defined in section 3: The monoidal product of two quantum sets X and Y is their generalizedCartesian product X × Y = Q{ X ⊗ Y | X ∝ X , Y ∝ Y } , and the monoidal unit is the quantumset = Q{ C } . The monoidal product of partial functions F : X → Y and F : X → Y is defined by ( F × F )( X ⊗ X , Y ⊗ Y ) = F ( X , Y ) ⊗ F ( X , Y ), for X ∝ X , X ∝ X , Y ∝ Y , and Y ∝ Y .The monoidal product of two von Neumann algebras A ⊆ L ( H ) and B ⊆ L ( K ) is theirspatial tensor product A ⊗ B = span { a ⊗ b ∈ L ( H ⊗ K ) | a ∈ A, b ∈ B } , and the monoidalunit is the one-dimensional von Neumann algebra L ( C ). The monoidal product of normal ∗ -homomorphisms φ : A → B and φ : A → B is defined by ( φ ⊗ φ )( a ⊗ a ) = φ ( a ) ⊗ φ ( a ), for all a ∈ A and a ∈ A . For general von Neumann algebras, theexistence of such a normal ∗ -homomorphism φ ⊗ φ follows from the special case of injective φ and φ , and the special case of surjective φ and φ .The spatial tensor product of two hereditarily atomic von Neumann algebras is equal totheir categorical tensor product, because each is an ℓ ∞ -direct sum of type I factors [18]. Inother words, if A and A are hereditarily atomic von Neumann algebras, and B is any vonNeumann algebra, then a pair of normal ∗ -homomorphisms φ : A → B and φ : A → B will factor through A ⊗ A if and only if every operator in the image of φ commutes withevery operator in the image of φ . Proof of theorem 7.6.
Theorem 6.3 and proposition 7.2 give us a contravariant equivalencefrom the category of quantum sets and partial functions, to a category of topological ∗ -algebras and continuous ∗ -homomorphisms. The contravariant equivalence maps each func-tion F : X → Y to the continuous ∗ -homomorphism φ : ℓ ( Y ) → ℓ ( X ), which restricts toa normal ∗ -homomorphism ℓ ∞ ( Y ) → ℓ ∞ ( X ) by lemma 6.4. This defines a contravariantequivalence qPar → M ∗ by proposition 5.4 and lemma 6.4, which restricts to a con-travariant equivalence qSet → M ∗ by lemma 6.5. Both equivalences are monoidal be-cause the von Neumann algebra isomorphism ℓ ∞ ( X ) ⊗ ℓ ∞ ( Y ) → ℓ ∞ ( X × Y ) defined by a ⊗ b ( X ⊗ Y a ( X ) ⊗ b ( Y )) is natural by lemma 7.4. (cid:3) Proposition 7.7.
Let X and Y be quantum sets. Let F : X → Y be a partial function. Foreach X ∝ X and each Y ∝ Y , the vector space L ( X, Y ) is canonically a Hilbert space withthe inner product h w | w ′ i = Tr( w † w ′ ) . If ( B YX | X ∝ X , Y ∝ Y ) is any family of orthonormalbases for the Hilbert spaces ( F ( X, Y ) | X ∝ X , Y ∝ Y ) for these inner products, then for all ∈ ℓ ( Y ) and all X ∝ X , we have F ⋆ ( b )( X ) = X Y ∝ Y X w ∈ B YX dim( Y ) · w † b ( Y ) w. For fixed b and X , only finitely many of the terms in the above sum are nonzero.Proof. Fix b ∈ ℓ ( Y ) and X ∝ X . Choose orthonormal bases ( B YX | Y ∝ Y ) for the Hilbertspaces ( F ( X, Y ) | Y ∝ Y ) equipped with the inner products defined by h w | w ′ i = Tr( w † w ′ ).Referring to the statement of theorem 6.3, we find that for all w, w ′ ∈ F ( X, Y ), h w | w ′ i = Tr( w † w ′ ) = Tr( w ′ w † ) = Tr(( w | w ′ ) · Y ) = ( w | w ′ ) · dim( Y ) . It follows that for each Y ∝ Y , the set ˆ B YX = { w · p dim( Y ) | w ∈ B YX } is an orthonormalbasis for the inner product ( · | · ). We now calculate that for all x, x ′ ∈ X , h x | φ ( b )( X ) x ′ i = X Y ∝ Y h f YX x | ( b ( Y ) ⊗ f YX x ′ i = X Y ∝ Y X v ∈ ˆ B YX X v ′ ∈ ˆ B YX h v ( x ) ⊗ v † | b ( Y ) v ′ ( x ′ ) ⊗ v ′† i = X Y ∝ Y X v ∈ ˆ B YX X v ′ ∈ ˆ B YX h v ( x ) | b ( Y ) v ′ ( x ′ ) i · ( v † | v ′† )= X Y ∝ Y X v ∈ ˆ B YX h v ( x ) | b ( Y ) v ( x ′ ) i = X Y ∝ Y X v ∈ ˆ B YX h x | v † b ( Y ) vx ′ i . Therefore, φ ( b )( X ) = X Y ∝ Y X v ∈ ˆ B YX v † b ( Y ) v = X Y ∝ Y X w ∈ B YX dim( Y ) · w † b ( Y ) w. (cid:3) completeness and cocompleteness In this section, we establish the basic properties of the category qSet of quantum setsand functions, by leveraging its contravariant duality with the category M ∗ of hereditarilyatomic von Neumann algebras and unital normal ∗ -homomorphisms (theorem 7.6). Thearguments follow those of [22]; their applicability relies on the trivial fact that any vonNeumann subalgebra of a hereditarily atomic von Neumann algebra is itself hereditarilyatomic. As a consequence, any quotient of a hereditarily atomic von Neumann algebra byan ultraweakly closed ∗ -ideal is also hereditarily atomic. Proposition 8.1.
Let F be a function from a quantum set X to a quantum set Y . Thefollowing are equivalent:(1) F is surjective(2) F is epic in qSet (3) F ⋆ is monic in M ∗ (4) F ⋆ is injective as a map ℓ ∞ ( Y ) → ℓ ∞ ( X ) Proof. (1) ⇒ (2): Assume that F is surjective. Then, F is both surjective and coinjectiveas a binary relation, so F ◦ F † = I Y . For all functions G and G from Y to a quantum set Z , if G ◦ F = G ◦ F , then G ◦ F ◦ F † = G ◦ F ◦ F † , so G = G . Thus, F is epic. he equivalence (2) ⇔ (3) is immediate from the duality between qSet and M ∗ .(3) ⇒ (4): Assume F ⋆ is not injective. Then, its kernel is equal to (1 − p ) · ℓ ∞ ( Y ) forsome central projection p = 1. For i ∈ { , } , define φ i : C ⊕ C ⊕ C → ℓ ∞ ( Y ) to be theunital normal ∗ -homomorphism that takes the i -th minimal projection in C ⊕ C ⊕ C to 1 − p and takes the third minimal projection to p . Immediately, for i ∈ { , } , the homomorphism F ⋆ ◦ φ i : C ⊕ C ⊕ C → ℓ ∞ ( X ) takes the third minimal projection to the identity, and it iscompletely determined by this property. Thus, F ⋆ is not monic.(4) ⇒ (1): Assume F is not surjective. This assumption is equivalent to the strictinequality F ◦ F † < I Y , so there is some atom Y ∝ Y such that ( F ◦ F † )( Y , Y ) = 0. Thedefinition of composition then implies that F ( X, Y ) = 0 for all atoms X ∝ X . Let q ∈ ℓ ∞ ( Y )be the projection defined by q ( Y ) = 1 Y , and q ( Y ) = 0 for Y = Y . The characterization of F ⋆ given in proposition 7.7 immediately implies that F ⋆ ( q ) = 0, so F ⋆ is not injective. (cid:3) Definition 8.2.
Let Y be a quantum set, and let X be a subset of Y . We define theinclusion function J YX : X ֒ → Y by J YX ( X, Y ) = C · X if X = Y , with J YX ( X, Y ) vanishingotherwise.
Lemma 8.3.
Write J = J YX . The unital normal ∗ -homomorphism J ⋆ : ℓ ∞ ( Y ) → ℓ ∞ ( X ) isdefined by J ⋆ ( b )( X ) = b ( X ) , for X ∝ X . It is surjective, and its kernel is (1 − p ) · ℓ ∞ ( Y ) ,where p is the central projection defined by p ( Y ) = 1 Y for Y ∝ X , with p ( Y ) = 0 otherwise.Proof. For all X ∝ X , the Hilbert space F ( X, X ) is one-dimensional. The norm of theidentity operator 1 X as a vector in this Hilbert space is p dim( X ), so { X / p dim( X ) } is an orthonormal basis for F ( X, X ). For X = Y , the Hilbert space F ( X, Y ) is zero-dimensional. Applying proposition 7.7, we find that for all b ∈ ℓ ∞ ( Y ) and all X ∝ X , wehave J ⋆ ( b )( X ) = 1 † X b ( X )1 X = b ( X ) as claimed. It follows immediately that J ⋆ is surjective.We also see that an operator b is in the kernel of J ⋆ if and only if b ( Y ) = 0 for all Y ∝ X . Inother words, b is in the kernel of J ⋆ if and only if p · b = 0. (cid:3) Proposition 8.4.
Let F be a function from X to Y . The following are equivalent:(1) F is injective(2) F is monic in qSet (3) F ⋆ is epic in M ∗ (4) F ⋆ is surjective as a map ℓ ∞ ( Y ) → ℓ ∞ ( X ) Proof.
We follow the pattern of the previous proof.(1) ⇒ (2): Assume that F is injective. Then, F is both injective and cosurjective as abinary relation, so F † ◦ F = I X . For all functions G and G from a quantum set Z to X , if F ◦ G = F ◦ G , then F † ◦ F ◦ G = F † ◦ F ◦ G , so G = G . Thus, F is monic.The equivalence (2) ⇔ (3) is immediate from the duality between qSet and M ∗ .(3) ⇒ (4): Assume F ⋆ is not surjective. Then, F ⋆ ( ℓ ∞ ( Y )) is a proper ultraweaklyclosed ∗ -subalgebra of ℓ ∞ ( X ) that contains the identity of ℓ ∞ ( X ), and it is possible tofind an automorphism of ℓ ∞ ( X ) that fixes the operators in F ⋆ ( ℓ ∞ ( Y )) (lemma 8.5). Thisdemonstrates that F ⋆ is not epic.(4) ⇒ (1): Assume that F ⋆ is surjective. The kernel of F ⋆ is equal to (1 − p ) · ℓ ∞ ( Y ) forsome central projection p . The central projections of ℓ ∞ ( Y ) are clearly in bijective correspon-dence with the subsets of Y , with p corresponding to the subset Z = Q{ Y ∝ Y | p ( Y ) = 1 Y } .Let J = J YZ : Z ֒ → Y be the corresponding inclusion map (definition 8.2). By lemma 8.3, he functions F ⋆ : ℓ ∞ ( Y ) → ℓ ∞ ( X ), and J ⋆ : ℓ ∞ ( Y ) → ℓ ∞ ( Z ) have the same kernel, so theyfactor through each other via an isomorphism of Neumann algebras. Dually, the functions F : X → Y and J : Z → Y must also factor through each other via an isomorphism in qSet . Thus, it is sufficient to show that J : Z → Y is injective (proposition 4.2). This isstraightforward. (cid:3)
Lemma 8.5.
Let A be an atomic von Neumann algebra, and let B be a proper von Neumannsubalgebra of A that contains the unit of A . There is a nontrivial automorphism φ of A thatfixes each operator in B .Proof. Being an atomic von Neumann algebra, A is the ℓ ∞ -direct sum of type I factors: A = L i A i . The inclusion homomorphism ρ : B ֒ → A decomposes into unital normal ∗ -homomorphisms ρ i : B → A i , which we may regard as representations of B .If any such representation ρ i is reducible, then the bicommutant theorem yields a non-trivial unitary operator u i ∈ A i that commutes with every operator in ρ i ( B ). The unitaryoperator u ∈ A that is u i in the direct summand A i and the identity in every other directsummand, is then in the commutant of B , but not in the center of A , so conjugation by u isthe desired automorphism.If all the representations ρ i : B → A i are irreducible, then some pair of them ρ i and ρ i must be unitarily equivalent, as otherwise the minimal central projections of A would all bein B , implying that B = A . The automorphism of A that exchanges the summands A i and A i according to this unitary equivalence is then a nontrivial automorphism of A that fixesthe elements of B . (cid:3) Proposition 8.6.
The category M ∗ of hereditarily atomic von Neumann algebras and unitalnormal ∗ -homomorphisms is complete, so the category qSet of quantum sets and functionsis cocomplete.Proof. See propositions 5.1 and 5.3 in [22], which show that the category W ∗ of von Neu-mann algebras and unital normal ∗ -homomorphisms has all products and all equalizers. The ℓ ∞ -direct sum of any family of hereditarily atomic von Neumann algebras is clearly itselfhereditarily atomic, so the subcategory of hereditarily atomic von Neumann algebras is closedunder products. The equalizer of two unital normal ∗ -homomorphism B ⇒ A is an ultra-weakly closed ∗ -subalgebra of B , and is therefore hereditarily atomic when B if hereditarilyatomic, so the subcategory of hereditarily atomic von Neumann algebras is also closed underequalizers. (cid:3) Proposition 8.7.
The category M ∗ of hereditarily atomic von Neumann algebras and unitalnormal ∗ -homomorphisms is cocomplete, so the category qSet of quantum sets and functionsis complete.Proof. See propositions 5.5 and 5.7 in [22]. The subcategory of hereditarily atomic vonNeumann algebras is not closed under coproducts: Davis showed that the von Neumannalgebra of bounded operators on a separable infinite-dimensional Hilbert space is generatedby three projections [10], so the coproduct of three copies of C in the category of all vonNeumann algebras is not hereditarily atomic. Thus, we will need to modify the proof of thecited proposition 5.5 in [22], rather than appeal to the proposition itself.For any family ( A j ) of von Neumann algebras, we define a joint representation of ( A j ) tobe a family unital normal ∗ -homomorphisms ( ρ j : A j → L ( H )), for some Hilbert space H . e construct the universal hereditarily atomic joint representation ( π j ) of ( A j ) by taking adirect sum of all finite-dimensional joint representations. This representation is hereditarilyatomic in the sense that the von Neumann algebra A = ( S j π j ( A j )) ′′ generated by the setof all represented operators is hereditarily atomic, being a subalgebra of an ℓ ∞ -direct sumof finite-dimensional von Neumann algebras.The universal hereditarily atomic joint representation of ( A j ) is indeed universal amongits hereditarily atomic joint representations, in the sense that every hereditarily atomic jointrepresentation ( ρ j ) of ( A j ) factors uniquely through A via a unital normal ∗ -homomorphism:Since, by assumption, the von Neumann algebra ( S j ρ j ( A j )) ′′ is hereditarily atomic, we mayassume without loss of generality that this von Neumann algebra is an irreducibly representedfinite type I factor. In this case, the joint representation ( ρ j ) is unitarily equivalent to one ofthe summands of the universal hereditarily atomic joint representation, giving us the desiredhomomorphism as conjugation by an isometry. This normal ∗ -homomorphism is unique,because its values on the generators of A are determined by the joint representation ( ρ j ).If ( A j ) is an indexed family of hereditarily atomic von Neumann algebras, then the algebra A is their coproduct. Indeed, any cocone on the family ( A j ) in M ∗ to some hereditarilyatomic von Neumann algebra B is a hereditarily atomic joint representation of ( A j ), andtherefore factors uniquely through A , as we have shown. The inclusions of the coproduct areexactly the representations π j .The category of hereditarily atomic von Neumann algebras also has coequalizers, be-cause it is closed under coequalizers as a subcategory of the category of von Neumannalgebras and unital normal ∗ -homomorphisms. Indeed, the coequalizer of two unital normal ∗ -homomorphisms is a quotient of the codomain von Neumann algebra, which is hereditarilyatomic if the codomain is itself hereditarily atomic. Thus, the category M ∗ is cocomplete. (cid:3) In an effort to emphasize the duality between our two categories, we will use the symbols ⊕ , ⊛ , and ⊗ for the product, the coproduct, and the monoidal product on the category M ∗ ,and we will use the symbols ⊎ , ∗ , and × for the coproduct, the product, and the monoidalproduct on the category qSet . We will refer to the former three operations as the directsum, the free product, and the tensor product of hereditarily atomic von Neumann algebras;we will refer to the latter three operations as the disjoint union, the total product, and theCartesian product of quantum sets. Readers with a background in type theory may chooseto perfect this notational correspondence by using + for the disjoint union of quantum sets.We comment briefly on the exact definitions of these six operations. The direct sum L j A j of an indexed family ( A j ) of hereditarily atomic von Neumann algebras is obtained in thestandard way, by first taking the ℓ -direct sum of their Hilbert spaces. Similarly, the tensorproduct A ⊗ A of hereditarily atomic von Neumann algebras A and A is obtained by firsttaking the tensor product of the Hilbert spaces; this is the “spatial” tensor product, thoughby a result of Guichardet [18, proposition 8.6], it coincides with the “categorical” tensorproduct. The free product ⊛ j A j is obtained as in the proof of proposition 8.7, with theuniversal hereditarily atomic joint representation constructed as the direct sum of irreduciblerepresentations on Hilbert spaces of the form C n , so such irreducible representations forma set. The disjoint union U j X j of an indexed family ( X j ) of quantum sets is the union S j X j × ‘ { j } , and the Cartesian product X × X is exactly as we defined it in section 2. Thetotal product of an indexed family of quantum sets is only defined up to natural isomorphism. . quantum function sets Theorem 9.1.
The category qSet of quantum sets and functions, equipped with the monoidalproduct × , is a closed symmetric monoidal category.Proof. See theorem 9.1 in [22]. Our construction here follows the same basic pattern as ourconstruction of the total product in the proof of proposition 8.5: we work in the category M ∗ of hereditarily atomic von Neumann algebras and unital normal ∗ -homomorphisms, and weconstruct the desired universal object by taking a direct sum of finite-dimensional represen-tations of the appropriate kind. Our task is to prove the following claim: for all hereditarilyatomic von Neumann algebras A and B , there is a hereditarily atomic von Neumann algebra B ⊛ A and unital normal ∗ -homomorphism ε : B → B ⊛ A ⊗ A that is universal among unitalnormal ∗ -homomorphisms φ : B → C ⊗ A , for C a hereditarily atomic von Neumann algebra,in the sense that there is a unique unital normal ∗ -homomorphism π : B ⊛ A → C such that( π ⊗ A ) ◦ ε = φ : B B ⊛ A ⊗ AC ⊗ A εφ π ⊗ The shortest proof of this claim is essentially that of theorem 9.1 in [22]. We take a slightlylonger route, which I hope the reader will find more intuitive.We first prove the above claim just in the special case that A is a matrix algebra, i.e., A = L ( C d ). In this case, we construct ε as the direct sum of all representations of B on aHilbert space of the form C n ⊗ C d , for n a nonnegative integer. Indexing the direct summands,we have a representation of B on the Hilbert space L i ( C n i ⊗ C d ) ∼ = ( L i C n i ) ⊗ C d . We nowdefine the von Neumann algebra B ⊛ L ( C d ) to be the smallest von Neumann algebra on theHilbert space L i C n i such that ε ( B ) ⊆ B ⊛ L ( C d ) ⊗ L ( C d ). Certainly, ε ( B ) ⊆ L i L ( C n i ⊗ C d ) = L i ( L ( C n i ) ⊗ L ( C d )) ∼ = ( L i L ( C n i )) ⊗ L ( C d ), so the von Neumann algebra B ⊛ L ( C d ) is asubalgebra of the hereditarily atomic von Neumann algebra L i L ( C n i ); it is therefore itselfhereditarily atomic. It is not immediately apparent that B ⊛ L ( C d ) is well-defined, in otherwords, that there is a smallest such von Neumann algebra. We may construct it by takinggenerators of the form ( ⊗ Tr)((1 ⊗ a ) ε ( b )), for a ∈ L ( C d ) and b ∈ B .Let φ : B → C ⊗ L ( C d ) be any unital normal ∗ -homomorphism, with C a hereditarilyatomic von Neumann algebra. Since C is hereditarily atomic, we may assume without lossof generality that C = L j L ( C m j ), for some indexed family of positive integers ( m j ). Wethen have that φ is a map from B to L j L ( C m j ⊗ C d ); in particular, we may view it as adirect sum of representations of B on Hilbert spaces of the form C m ⊗ C d . A unital normal ∗ -homomorphism π : B ⊛ L ( C d ) → C making the diagram commute can now be constructed bymatching representations. It is unique because its values are determined on the generatorsof B ⊛ L ( C d ) : π (( ⊗ Tr)((1 ⊗ a ) ε ( b ))) = ( π ⊗ Tr)((1 ⊗ a ) ε ( b )) = ( ⊗ Tr)((1 ⊗ a )( π ⊗ )( ε ( b ))) = (1 ⊗ Tr)((1 ⊗ a ) φ ( b ))We have proved the desired claim in the special case that A is a matrix algebra. Translatingthis conclusion to the symmetric monoidal category qSet of quantum sets and functionsequipped with the Cartesian product, we find that the functor ( − × X ) has a right adjointwhenever X is atomic, i.e., whenever it has exactly one atom. In any symmetric monoidal ategory, this implies that arbitrary coproducts of such objects X enjoy the same property.Since, every quantum set is the coproduct of a family of atomic quantum sets, we havesucceded in showing that qSet is a closed symmetric monoidal category. (cid:3) Definition 9.2.
Let X be a quantum set. We write ( − ) X for the right adjoint of the functor( − ) × X , where both functors are from the category qSet to itself. For any quantum set Y , we call Y X the quantum function set from X to Y .Closed symmetric monoidal categories are common, and their basic properties are widelyknown. We review some of this basic theory in the context of our discussion of quantum sets.The expression Y X defines a functor, contravariant in X , and covariant in Y . Left adjointspreserve colimits and right adjoints preserve limits; thus, the functor ( − ) × X preservescolimits, and the functor ( − ) X preserves limits. In particular ( Y ∗ Y ) X ∼ = Y X ∗ Y X ; thesame principle holds for total products of infinite families. The adjunction between thefunctors ( − ) × X and ( − ) X can be internalized: we have an isomorphism ( Z Y ) X ∼ = Z Y×X ,natural in all variables. We also have a natural isomorphism Y X + X ∼ = Y X ∗ Y X ; the sameprinciple holds for disjoint unions of infinite families. In particular, if a quantum set X isdecomposed as the union of its atomic subsets, X = S X ∝ X Q{ X } , then Y X ∼ = ∗ X ∝ X Y Q{ X } . The computation that establishes this natural isomorphism implicitly appears in the laststep of our proof of theorem 9.1.Each quantum set X has a maximum classical subset which consists of just its one-dimensional atoms, and which is naturally isomorphic to ‘Fun( ; X ). In particular, theclassical subset of Y X is naturally isomorphic to ‘Fun( ; Y X ) ∼ = ‘Fun( X ; Y ). Identifying Set with a subcategory of qSet via the functor S ‘ S , we might say that the classical subsetof Y X consists of functions from X to Y . Proposition 9.3.
The category qSet of quantum sets and functions is not Cartesian closed.
Due to a clash of terminology, this proposition refers to the absence of right adjointsfor functors of the form ( − ) ∗ X , not for functors of the form ( − ) × X . The product ∗ is Cartesian in the category-theoretic sense: it has the same universal property in thecategory qSet as the ordinary Cartesian product has in the category Set . However, it is theproduct × that generalizes the ordinary Cartesian product in a manner appropriate to thenoncommutative dictionary. To begin with, the product × is a quantum generalization ofthe ordinary Cartesian product in the sense that ‘ S × ‘ T is naturally isomorphic to ‘( S × T )for ordinary sets S and T , and ∗ is not a quantum generalization of the ordinary Cartesianproduct in this sense. The product × is also effectively the established generalization of theordinary Cartesian product in noncommutative geometry. For example, for any quantum set X , a quantum group comultiplication on the von Neumann algebra ℓ ∞ ( X ) is a unital normal ∗ -homomorphism ℓ ∞ ( X ) → ℓ ∞ ( X ) ⊗ ℓ ∞ ( X ), so dually a quantum group multiplication on X is a function X × X → X , not a function
X ∗ X → X . We now also see that evaluation,the counit of the adjunction in definition 9.2, is a function
X × Y X → Y .To draw a conceptual distinction between the two products, we compare the Cartesianproduct X ×X to the total product
X ∗X , for a fixed quantum set X . The Cartesian product X × X intuitively consists of pairs of elements of X : an element of X in one hand, and an lement of X in the other. By contrast, the total product X ∗ X ∼ = X ‘ { , } intuitively consistsof functions to X , from the quantum set ‘ { , } . Classically, for each ordinary set S , wehave a canonical bijective correspondence between functions { , } → S and ordered pairsfrom S ; we evaluate each function on input 1 to obtain the first entry of the ordered pair,and then we evaluate it on input 2 to obtain the second entry of the ordered pair. We haveno such correspondence in the quantum setting because we cannot duplicate the function toevaluate it twice; there is no duplication morphism X ‘ { , } → X ‘ { , } × X ‘ { , } . Proof of proposition 9.3.
In any Cartesian closed category, the category-theoretic productpreserves colimits in each variable. The category-theoretic product in qSet is the totalproduct, and it does not preserve colimits: the quantum set ( ⊎ ) ∗ ( ⊎ ) has uncountablymany atoms because there are uncountably many inequivalent irreducible joint representa-tions of a pair of von Neumann algebras, each isomorphic to C , on any two-dimensionalHilbert space, but the quantum set [ ∗ ( ⊎ )] ⊎ [ ∗ ( ⊎ )] has only four atoms, becausethe terminal object is also the unit for the total product ∗ . (cid:3) subobjects of a quantum set Let X be a quantum set. Recall from category theory that the subobjects of X aredefined via the category of all monomorphisms into X , i.e., of all injective functions into X .A morphism in this category from a monomorphism Z X to a monomorphism Z X is a function Z → Z making the triangle commute. If such a function Z → Z exists, itis unique, so the category of monomorphisms into X is a preorder. The subobjects of X arethe equivalence classes of this preorder. Proposition 10.1.
Let X be a quantum set. The map taking each subset of X to theequivalence class of its inclusion into X is an isomorphism of partial orders.Proof. The statement of proposition 8.4, together with the proof of (4) ⇒ (1), shows thatevery monomorphism into X is isomorphic to the inclusion function of some subset Y ⊆ X into X . Distinct inclusion functions induce homomorphisms with distinct kernels (lemma8.3), so no two distinct inclusions can be isomorphic. Thus, we have a bijection between thesubsets of X and the subobjects of X in qSet .If Z ⊆ Y ⊆ X , then we may easily check that J XY ◦ J YZ = J XZ , so our bijection is monotone(definition 8.2). Conversely, assume that Y and Z are subsets of X , and that there is afunction F : Z → Y such that J XY ◦ F = J XZ . Let Z ∝ Z ⊆ X . By definition of composition,we have that _ Y ∝ Y J XY ( Y, Z ) · F ( Z, Y ) = J XZ ( Z, Z ) . The space J XZ ( Z, Z ) is nonempty, so for some Y ∝ Y , we also have that J XY ( Y, Z ) is nonempty,which implies that Z = Y ∝ Y . Therefore, Z ⊆ Y . We conclude that our bijection is anisomorphism of partial orders. (cid:3)
Thus, modulo formalities, the subobjects of a quantum set X are exactly its subsets. Atopos, by definition, must have a subobject classifier Ω, an object admitting a bijection,natural in X , between the subobjects of X , and the morphisms from X to Ω. Assume forthe sake of contradiction that the category qSet has a subobject classifier Ω, and consider itsuniversal property in the dual category M ∗ : the central projections of any hereditarily atomic on Neumann algebra A must be in bijection with the unital normal ∗ -homomorphisms from ℓ ∞ (Ω) to A . In particular, there must be exactly two unital normal ∗ -homomorphisms from ℓ ∞ (Ω) to L ( C n ) for any positive integer n . It follows immediately that Ω does not haveatoms of dimension larger than 1, and that it must in fact have exactly two atoms; in otherwords, we conclude Ω ∼ = ⊎ . But there are uncountably many functions from X = Q{ C } to ⊎ , contradicting that Q{ C } has only two subobjects. Thus, we have demonstratedthe following: Proposition 10.2.
The category qSet of quantum sets and functions does not have a sub-object classifier.
However, there is a bijection between the subobjects of a quantum set X and the “classical”functions from X to ⊎ . The rest of this section is dedicating to defining and motivatingthis term.We begin by examining the Cartesian product monoidal structure on qSet . The unit of theCartesian product is also the terminal object, so we have projection functions P : X × X →X and P : X × X → X defined by P = U ◦ ( I X × ! ) and P = U ◦ ( ! × I X ), where !denotes the unique map to the terminal object, and U denotes the right or left unitor, asappropriate.The salient feature of the Cartesian product monoidal structure we have defined on qRel is that it coincides with the Cartesian product on ordinary sets. If we think of functions F and F from a fixed quantum set X to ordinary sets ‘ S and ‘ S , respectively, as generalizedobservables, then the compatibility of the two observables should mean the existence of afunction from X to the Cartesian product ‘ S × S that combines the two observables. Thisleads to the following definition. Definition 10.3.
Let F : X → Y and F : X → Y be functions. We say that F and F are compatible just in case there is a function F from X to Y × Y such that P ◦ F = F and P ◦ F = F , where P and P are the projection functions just defined.Viewing this definition in the category of hereditarily atomic von Neumann algebras, F and F are compatible just in case there is a unital normal ∗ -homomorphism φ from thetensor product ℓ ∞ ( Y ) ⊗ ℓ ∞ ( Y ) to ℓ ∞ ( X ) such that φ ( b ⊗ b ) = F ⋆ ( b ) · F ⋆ ( b ) for all b ∈ ℓ ∞ ( Y ) and b ∈ ℓ ∞ ( Y ). In particular, the images of F ⋆ and F ⋆ must commute. Thisnecessary condition is also sufficient, because the categorical tensor product coincides withthe spatial tensor product for hereditarily atomic von Neumann algebras [18, proposition8.6]. Thus, we have established the following: Lemma 10.4.
Let F and F be functions, as in definition 10.3 above. The functions F and F are compatible if and only if every element in the image of F ⋆ commutes with everyelement in the image of F ⋆ . We use the term “classical” for functions that do not consume their arguments:
Definition 10.5.
A function out of a quantum set X is classical just in case it is compatiblewith the identity function I X . A quantum set X is said to be classical just in case the identityfunction I X is classical. Proposition 10.6.
A quantum set X is classical if and only if each atom of X is one-dimensional. roof. By lemma 10.4, the quantum set X is classical if and only if ℓ ∞ ( X ) is commutative,or equivalently, if and only if the matrix algebra L ( X ) is commutative for each X ∝ X . (cid:3) Lemma 10.7.
Let F be a function from a quantum set X to a quantum set Y . The followingare equivalent:(1) F is classical(2) F is compatible with every function out of X (3) F factors through the canonical function Q : X ։ ‘At( X ) X ‘At( X ) Y FQ The canonical surjection Q : X → ‘At( X ) is intuitively the quotient map that contractseach atom to be one-dimensional; it is defined by Q ( X, C X ) = L ( X, C X ) for X ∝ X , with theother components vanishing. Proof.
By lemma 10.4, the function F is classical if and only if the image of F ⋆ is in thecenter of ℓ ∞ ( X ). The latter condition is equivalent to F factoring through the inclusion ofthe center into ℓ ∞ ( X ), or equivalently, through Q ⋆ . Thus, (1) is equivalent to (3).If F is compatible with I X , then it is also compatible with G ◦ I X = G , for any function G out of X , so (1) implies (2). The converse is trivial. (cid:3) Proposition 10.8.
Let X and Y be quantum sets. For every classical function F from X to Y , there is a unique ordinary function f from At( X ) to Fun( ; Y ) such that F = J ◦ ‘ f ◦ Q ,where Q is the canonical surjection from X to ‘At( X ) , and J is the canonical injection from ‘Fun( ; Y ) to Y . X Y ‘At( X ) ‘Fun( ; Y ) FQ ‘ f J Up to canonical natural isomorphism, the functor ‘Fun( ; − ) takes each quantum set Y to its maximum classical subset, and the injection J is its inclusion function. Proof.
In light of lemma 10.7, to establish the existence of f , it is sufficient to show that everyfunction from a classical quantum set ‘ T to the quantum set Y factors through the inclusionfunction of the maximum classical subset of Y . Reasoning in the category of hereditarilyatomic von Neumann algebras, we simply observe that any unital normal ∗ -homomorphisminto a commutative von Neumann algebra must vanish on each noncommutative factor, andtherefore must factor through the quotient by these noncommutative factors. The uniquenessof f follows by propositions 8.1 and 8.4. (cid:3) Proposition 10.9.
For every injective function J : Z X there is a unique classicalfunction F : X → ⊎ such that the following diagram is a pullback square: Z X ⊎ J T ! F he function Z → at the top of the diagram is the unique function from Z to the terminalobject . The function T : ⊎ is the injection taking the singleton to the secondsummand of ⊎ .Proof. Equivalently, we are to show that for every surjective unital normal ∗ -homomorphism π from a hereditarily atomic von Neumann algebra A to a hereditarily atomic von Neumannalgebra C , there is a unique unital normal ∗ -homomophism φ : C → A with central imagethat makes the following diagram a pushforward square: C C A C π φ ! A unital normal ∗ -homorphism φ : C → A with central image is simply a way of writingthe unit of A as a sum of two central projections p and p . A cocone on the diagram C A C φ is essentially just a unital normal ∗ -homomorphism from A to a hereditarily atomic vonNeumann algebra B that takes the second central projection p in A to the identity of B , sothe colimit of this diagram is the quotient map A ։ p A .Up to isomorphism of codomains, every surjective unital normal ∗ -homomorphism π : A → C is of this form, so there does exist a unital normal ∗ -homomorphism φ : C → A withcentral image making the first diagram of the proof into a pushforward square. This φ isunique, because it is completely determined by the projection p , and two quotient maps A ։ p A and A ։ p ′ A are distinct up to isomorphism of codomains whenever p and p ′ are distinct central projections. (cid:3) We have shown that the category qSet behaves in many ways like a topos, as discussedin section 1.2. 11. operators as functions on a quantum set
Definition 11.1.
Let X be a quantum set. An observable on X is a function from X to ‘ R .We begin this section by demonstrating that the observables on X are in bijective cor-respondence with the self-adjoint operators in ℓ ( X ). We later show that under this corre-spondence, the sum of two self-adjoint operators arises in the expected way from a function‘ R ∗ ‘ R → ‘ R .Write Herm( A ) for the vector space of self-adjoint elements of a von Neumann algebra A . Proposition 11.2.
The contravariant functors
Fun( − ; ‘ R ) and Herm( ℓ ( − )) , from the cat-egory of quantum sets and functions to the category of sets and functions, are naturallyisomorphic. The natural isomorphism takes each function F : X → ‘ R to the self-adjointoperator F ⋆ ( r ) , where r is the element of ℓ (‘ R ) defined by r ( C α ) = α . roof. For each function F : X → Y we consider the following diagram:Fun( Y ; ‘ R ) Hom ( ℓ ∞ (‘ R ) , ℓ ∞ ( Y )) Herm( ℓ ( Y ))Fun( X ; ‘ R ) Hom ( ℓ ∞ (‘ R ) , ℓ ∞ ( X )) Herm( ℓ ( X )) ∼ = ◦ F F ⋆ ◦ ∼ = F ⋆ ∼ = ∼ = The commutative square on the left comes from the contravariant equivalence of qSet and M ∗ . The isomorphism on the top right is defined to take each ψ ∈ Hom ( ℓ ∞ (‘ R ) , ℓ ∞ ( Y )) tothe self-adjoint operator P α ∈ R α · ψ ( e α ) in ℓ ( Y ), where for each real number α , we write e α for the corresponding minimal projection in ℓ ∞ (‘ R ). The isomorphism on the bottom rightis defined likewise.The square on the right commutes because for each unital normal ∗ -homomorphism ψ from ℓ ∞ ( R ) to ℓ ∞ ( Y ), F ⋆ X α ∈ R α · ψ ( e α ) ! = X α ∈ R α · F ⋆ ( ψ ( e α )) = X α ∈ R α · ( F ⋆ ◦ ψ )( e α ) . The interchange of F ⋆ with summation is justified because F ⋆ is continuous as a function from ℓ ( Y ) to ℓ ( X ). Therefore, we have a natural isomorphism between the functors Fun( − ; ‘ R )and Herm( ℓ ( − )).Setting Y = ‘ R , and tracking the identity function I ∈ Fun( Y ; ‘ R ) through the diagram,we find that our natural isomorphism takes F = I ◦ F ∈ Fun( X ; ‘ R ) to F ⋆ ( P α ∈ R αe α ) = F ⋆ ( r ). (cid:3) Definition 11.3.
Write Q and Q for the two projection functions ‘ R ∗ ‘ R → ‘ R . Define + : ‘ R ∗ ‘ R → ‘ R to be the unique function such that + ⋆ ( r ) = Q ⋆ ( r ) + Q ⋆ ( r ), where r is theelement of ℓ (‘ R ) defined by r ( C α ) = α . Definition 11.4.
Let F and F be observables on a quantum set X . Their sum F + F isdefined to be the observable + ◦ h F , F i , where h F , F i is the unique function from X to‘ R ∗ ‘ R defined by Q ◦ h F , F i = F and Q ◦ h F , F i = F . Proposition 11.5.
Under the natural isomorphism of proposition 11.2, the sum of observ-ables on X corresponds to the sum of self-adjoint operators in ℓ ( X ) .Proof. This is immediate from the definition:( F + F ) ⋆ ( r ) = h F , F i ⋆ ( + ⋆ ( r )) = h F , F i ⋆ ( Q ⋆ ( r ) + Q ⋆ ( r ))= h F , F i ⋆ ( Q ⋆ ( r )) + h F , F i ⋆ ( Q ⋆ ( r ))= F ⋆ ( r ) + F ⋆ ( r ) (cid:3) Thus, we may think of the observables on a quantum set X as being functions into ‘ R ,and we may think of their additive structure as coming from the additive structure on ‘ R ,just as we do classically. The structure (‘ R , + , ‘0) is like a quantum group, but its groupproduct is defined on the total product ‘ R ∗ ‘ R , as opposed to the Cartesian product ‘ R × ‘ R .The group product of a quantum group is in effect defined on just the Cartesian product ofthat quantum group with itself.The product of two self-adjoint operators is generally not self-adjoint, but it is possibleto extend the ordinary product operation of the real numbers from the Cartesian product R × ‘ R to the total product ‘ R ∗ ‘ R , by working with the Jordan product. However, we willinstead turn to the ∗ -algebra ℓ ( X ), and work with the product there. Lemma 11.6.
The contravariant functors
Fun( − ; ‘ R ∗ ‘ R ) and ℓ ( − ) , from the category ofquantum sets and functions to the category of sets and functions, are naturally isomorphic.The natural isomorphism takes each function F : X → ‘ R ∗ ‘ R to the self-adjoint operator F ⋆ ( s ) , where s is the element of ℓ (‘ R × ‘ R ) defined by Q ⋆ ( r ) + iQ ⋆ ( r ) , and r is the elementof ℓ (‘ R ) defined by r ( C α ) = α .Proof. We compose natural isomorphisms:Fun( X ; ‘ R ∗ ‘ R ) ∼ = Fun( X ; ‘ R ) × Fun( X ; ‘ R ) ∼ = Herm( ℓ ( X )) × Herm( ℓ ( X )) ∼ = ℓ ( X )The first isomorphism is the universal property of the total product; the second isomorphismis from proposition 11.2; and the third isomorphism is the bijection taking each pair of self-adjoint operators ( a , a ) to the operator a + ia . We track a function F from a quantumset X to ‘ R ∗ ‘ R through this chain of bijections: F ( Q ◦ F, Q ◦ F ) (( Q ◦ F ) ⋆ ( r ) , ( Q ◦ F ) ⋆ ( r )) ( Q ◦ F ) ⋆ ( r ) + i ( Q ◦ F ) ⋆ ( r )( Q ◦ F ) ⋆ ( r ) + i ( Q ◦ F ) ⋆ ( r ) = F ⋆ ( Q ⋆ ( r )) + iF ⋆ ( Q ⋆ ( r )) = F ⋆ ( Q ⋆ ( r ) + iQ ⋆ ( r )) = F ⋆ ( s ) (cid:3) Definition 11.7.
Define C = ‘ R ∗ ‘ R . Define s ∈ ℓ ( C ) as in lemma 11.6. Define(1) a function + : C ∗ C → C by + ⋆ ( s ) = Q ⋆ ( s ) + Q ⋆ ( s ),(2) a function · : C ∗ C → C by · ⋆ ( s ) = Q ⋆ ( s ) Q ⋆ ( s ), and(3) a function : C → C by ⋆ ( s ) = s † .Define the inclusion function ‘ C ֒ → C to be the canonical injective function h P , P i : ‘ R × ‘ R ‘ R ∗ ‘ R . Definition 11.8.
Let X be a quantum set, and let F and F be functions from X to C .Define(1) F + F = + ◦ h F , F i ,(2) F · F = · ◦ h F , F i , and(3) F = ◦ F .For each complex number α , define Cst α to be the function from X to C given by the followingcomposition: X ‘ C C ! α Theorem 11.9.
The contravariant functors
Fun( − ; C ) and ℓ ( − ) , from the category ofquantum sets and functions, to the category of unital ∗ -algebras over C and unital ∗ -homo-morphisms, are naturally isomorphic. The natural isomorphism takes each function F : X →C to the operator F ⋆ ( s ) .Proof. It remains only to show that the natural isomorphism of lemma 11.6 respects thestructure of definition 11.8.( F + F ) ⋆ ( s ) = h F , F i ⋆ ( + ⋆ ( s )) = h F , F i ⋆ ( Q ⋆ ( s ) + Q ⋆ ( s ))= h F , F i ⋆ ( Q ⋆ ( s )) + h F , F i ⋆ ( Q ⋆ ( s ))= F ⋆ ( s ) + F ⋆ ( s ) F · F ) ⋆ ( s ) = h F , F i ⋆ ( · ⋆ ( s )) = h F , F i ⋆ ( Q ⋆ ( s ) · Q ⋆ ( s ))= h F , F i ⋆ ( Q ⋆ ( s )) · h F , F i ⋆ ( Q ⋆ ( s ))= F ⋆ ( s ) · F ⋆ ( s ) F ⋆ ( s ) = F ⋆ ( ⋆ ( s )) = F ⋆ ( s † ) = F ⋆ ( s ) † Cst ⋆α ( s ) = ! ⋆ ( α ⋆ ( h P , P i ⋆ ( Q ⋆ ( r ) + iQ ⋆ ( r )))= ! ⋆ ( α ⋆ ( P ⋆ ( r ) + iP ⋆ ( r ))= ! ⋆ ( α ⋆ ( P ⋆ ( r )) + iα ⋆ ( P ⋆ ( r )))= ! ⋆ (Re( α ) + i Im( α ))= ! ⋆ ( α ) = α · (cid:3) References [1] S. Abramsky, R. S. Barbosa, N. de Silva, and O. Zapata,
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Appendix A. predicates on quantum sets An ordinary predicate p on an ordinary set S is essentially just a subset of S , but itssuperset S is part of its data. We generalize this notion to the quantum setting. Definition A.1.
A predicate on a quantum set X is a function that assigns to each atom X ∝ X a subspace P ( X ) ≤ X .If p is an ordinary predicate on an ordinary set S , we define the predicate ‘ p on thequantum set ‘ S in the expected way: ‘ p ( C s ) is equal to C s if s ∈ p , and it vanishes otherwise.We write Pred( X ) for the set of predicates on a quantum set X . It is a complete ortho-modular lattice, with the partial order and the orthocomplemention defined atomwise (cf.definition 3.9). When P ≤ P ⊥ , we say that that P and P are disjoint. In this appendix,we extend Pred to a contravariant functor from the category qSet of quantum sets andfunctions to the category of ortholattices and monotone functions. We go on to exhibit afew naturally isomorphic functors. Definition A.2.
Let R be a binary relation from a quantum set X to a quantum set Y .Define the ordinary function R ⋆ from Pred( X ) to Pred( Y ) by R ⋆ ( P )( Y ) = span { rx | X ∝ X , x ∈ P ( X ) , r ∈ R ( X, Y ) } . he function R ⋆ is clearly monotone. If R and S are composable binary relations, then( S ◦ R ) ⋆ = S ⋆ ◦ R ⋆ , for essentially the same reason that the composition of binary relationsis associative. Thus, we have a covariant functor from the category of quantum sets andbinary relations to the category of partially ordered sets and monotone functions. We callthis functor the direct image functor.We define the inverse image functor by composing the direct image functor with the adjointfunctor, writing R ⋆ = ( R † ) ⋆ . This is a contravariant functor from the category of quantumsets and binary relations to the category of partially ordered sets and monotone functions.The functor Pred mentioned at the beginning of this section is just the restriction of thisinverse image functor to qSet . Overloading notation, we will also write Pred for the inverseimage functor itself. Lemma A.3.
The inverse image functor
Pred( − ) from the category qRel of quantum setsand binary relations to the category of ortholattices and monotone functions is naturallyisomorphic to the functor Rel( − ; ) . The natural isomorphism takes each predicate P on aquantum set X to the binary relation R defined by R ( X, C ) = {h x | · i | x ∈ P ( X ) } .Proof. On the level of Hilbert spaces, this natural isomorphism is just the canonical antiuni-tary from a Hilbert space X to its dual X ∗ = L ( H, C ); it is natural because h a † y | · i = h y | a ·i for all y ∈ Y , for each a ∈ L ( X, Y ). The reasoning lifts to binary relations, as usual. (cid:3)
Any binary relation R from a quantum set X to is a partial function, since the condition R ◦ R † ≤ I is satisfied automatically. Thus, we can equivalently say that Pred( − ) isnaturally isomorphic to Par( − ; ).Under the contravariant equivalence between qPar and M ∗ , partial functions from X to correspond to normal ∗ -homomorphisms from C to ℓ ∞ ( X ). A normal ∗ -homomorphism φ from C to any von Neumann algebra A is uniquely determined by the projection φ (1), so wehave a bijective correspondence between the predicates on X and the projections in ℓ ∞ ( X ).To state this correspondence as a natural isomorphism of functors, we write Proj( − ) for thefunctor that takes each von Neumann algebra A to its partially ordered set of projections,and that simply restricts each normal ∗ -homomorphism out of A to this set. Lemma A.4.
The functors
Par( − ; ) and Proj( ℓ ∞ ( − )) are naturally isomorphic as con-travariant functors from the category of quantum sets and partial functions to the categoryof ortholattices and monotone functions. For any quantum set X , the natural isomorphismtakes each partial function R in Par( X ; ) to R ⋆ (1) , where R ⋆ is the normal ∗ -homomorphismcorresponding to R .Proof. It is immediate from proposition 7.7 that the assignment R R ⋆ (1) is monotone.Its inverse is also monotone, as we can see from the expression for R in theorem 6.3 as theintertwining space of R ⋆ : R ( X, C ) = { v ∈ L ( X, C ) | αv = vR ⋆ ( α )( X ) for all α ∈ C } = { v ∈ L ( X, C ) | v = vR ⋆ (1)( X ) } Finally, the functoriality of ( − ) ⋆ implies immediately that we have a natural transformation:for any partial function F from a quantum set X to a quantum set Y , and any partial function R from Y to , we have F ⋆ ( R ⋆ (1)) = ( R ◦ F ) ⋆ (1). (cid:3) As a functor to the category
Set of sets and functions, Par( − ; ) is naturally isomorphicto Fun( − ; ⊎ ); this is apparent in M ∗ . In general, the morphism set Fun( X ; Y ), for rbitrary quantum sets X and Y , has no canonical order structure, so the components ofthis natural isomorphism are bijections, rather than order isomorphisms. However, thesebijections do become order isomorphisms if each morphism set Fun( X ; ⊎ ) is given apartial order structure from ⊎ , in the manner of definition 11.8. Definition A.5.
Write B = { , } . Define t to be the element of ℓ (‘ B ) satisfying t ( C α ) = α for both values of α ∈ B . Define(1) a function ∨ : ‘ B ∗ ‘ B → ‘ B by ∨ ⋆ ( t ) = Q ⋆ ( t ) ∨ Q ⋆ ( t ),(2) a function ∧ : ‘ B ∗ ‘ B → ‘ B by ∧ ⋆ ( t ) = Q ⋆ ( t ) ∧ Q ⋆ ( t ), and(3) a function ∼ : ‘ B → ‘ B by ∼ ⋆ ( t ) = 1 − t . Definition A.6.
Let X be a quantum set, and let F and F be functions from X to ‘ B .Define(1) F ∨ F = ∨ ◦ h F , F i ,(2) F ∧ F = ∧ ◦ h F , F i , and(3) ∼ F = ∼ ◦ F .Define F ≤ F just in case F ∨ F = F . Lemma A.7.
The functors
Fun( − ; ‘ B ) and Proj( ℓ ∞ ( − )) are naturally isomorphic as con-travariant functors from the category of quantum sets and functions to the category of par-tially ordered sets and monotone functions. For any quantum set X , the natural isomorphismtakes each function F in Fun( X ; ‘ B ) to F ⋆ ( t ) .Proof. For any quantum set X , the assignment F F ⋆ ( t ) is a bijection from Fun( X ; ‘ B ) toProj( ℓ ∞ ( X )) thanks to the contravariant equivalence of qSet and M ∗ described in theorem7.6. This assignment is immediately seen to be natural in X , just as in the proof of propo-sition A.4. To show that this assignment is an order isomorphism, we demonstrate that itpreserves ∨ and ∧ , just as in the proof of theorem 11.9:( F ∨ F ) ⋆ ( t ) = h F , F i ⋆ ( ∨ ⋆ ( t )) = h F , F i ⋆ ( Q ⋆ ( t ) ∨ Q ⋆ ( t ))= h F , F i ⋆ ( Q ⋆ ( t )) ∨ h F , F i ⋆ ( Q ⋆ ( t ))= F ⋆ ( t ) ∨ F ⋆ ( t )( F ∧ F ) ⋆ ( t ) = h F , F i ⋆ ( ∧ ⋆ ( t )) = h F , F i ⋆ ( Q ⋆ ( t ) ∧ Q ⋆ ( t ))= h F , F i ⋆ ( Q ⋆ ( t )) ∧ h F , F i ⋆ ( Q ⋆ ( t ))= F ⋆ ( t ) ∧ F ⋆ ( t )Any unital normal ∗ -homomorphism preserves the meets and joins of projections, becausein any von Neumann algebra the meet of two projections p and q is equal to the ultraweaklimit of the sequence (( pq ) n | n ∈ N ). (cid:3) In fact, for each quantum set X , the partial order Fun( X ; ‘ B ) is a complete orthomodularlattice with orthocomplementation F
7→ ∼ F , because it is order isomorphic to the completeorthomodular lattice Proj( ℓ ∞ ( X )), and the order isomorphism takes ∼ to the orthocomple-mentation: ( ∼ F ) ⋆ ( t ) = F ⋆ ( ∼ ⋆ ( t )) = F ⋆ (1 − t ) = 1 − F ⋆ ( t ) heorem A.8. The functors
Pred( − ) , Rel( − ; ) , Proj( ℓ ∞ ( − )) , and Fun( − ; ‘ B ) are nat-urally isomorphic as contravariant functors from the category of quantum sets and functionsto the category of ortholattices and ortholattice morphisms. For us, an ortholattice morphism from one ortholattice to another is a monotone functionthat preserves meets, joins, and orthocomplements, as well as the top and bottom elements.
Proof.
By lemmas A.3, A.4, and A.7, the four given functors are naturally isomorphic asfunctions into the category of ortholattices and monotone functions. The components ofthe three given natural isomorphisms are easily seen to be ortholattice isomorphisms: Theypreserve meets and joins, as well as the top and bottom elements simply by virtue of beingorder isomorphisms. Each component of the natural isomorphism in lemma A.3 preservesorthocomplementation because the map x
7→ h x | · i is antiunitary. Each component of thenatural isomorphism in lemma A.4 preserves orthocomplementation because the subspaces { v ∈ L ( X, C ) | v = vp } and { v ∈ L ( X, C ) | v = v (1 − p ) } are orthocomplements for everyHilbert space X , and every projection p in L ( X ). We have already observed that eachcomponent of the natural isomorphism in lemma A.7 preserves orthocomplementation.Let R be a function from a quantum set X to a quantum set Y , and let F be any elementof Fun( Y ; ‘ B ). Immediately, ( ∼ F ) ◦ R = ∼ ◦ F ◦ R = ∼ ( F ◦ R ). Therefore, Fun( − ; ‘ B ) is afunctor into the category of ortholattices and ortholattice morphisms. Bootstrapping alongour three natural isomorphisms, we conclude that the other three functors are also into thecategory of ortholattices and ortholattice morphisms. (cid:3) Pred( X ) Rel( X ; )Proj( ℓ ∞ ( X )) Fun( X ; ‘ B ) ∼ = A. A. ∼ = A. ∼ = For reference, we describe all twelve natural isomorphisms of theorem A.8. To simplifyexpressions, we will suppress the canonical isomorphisms ‘ B ∼ = ⊎ , C ∼ = C , and C ∼ = C . P R ( X, C ) = P ( X ) ⊥ p ( X ) = proj P ( X ) F ( X, C ) = P ( X ) P ( X ) = R ( X, C ) ⊥ Rp = R ⋆ (1) [ R † , ¬ R † ] † P ( X ) = p ( X ) · X R ( X, C ) = { v | vp = v } p F ( X, C ) = { v | vp = 0 } P ( X ) = F ( X, C ) J † ◦ Fp = F ⋆ ( t ) F We use the notation ( · ) for the polar of a subspace: For each X ∝ X , the Hilbert space X ∗ = L ( X, C ) is the Hilbert space dual of X . For H ≤ X and K ≤ X ∗ we write: H = { v ∈ X ∗ | v ( x ) = 0 for all x ∈ H } K = { x ∈ X | v ( x ) = 0 for all v ∈ K } ⊥ = { x ∈ X | h x ′ | x i = 0 for all x ′ ∈ H } K ⊥ = { v ∈ X ∗ | vv ′† = 0 for all v ′ ∈ K } For any function F from X to ‘ B , and each X ∝ X , we have F ( X, C ) = F ( X, C ) ⊥ , sinceunder the identifications C ∼ = C and C ∼ = C , inner products from the subspaces F ( X, C )and F ( X, C ) of L ( X, C ) vanish, while outer products contain the identity 1 L ( X, C ) . Thenotation [ R , R ] refers to the universal property of the disjoint union as the coproduct of qRel , and J is the inclusion of the first summand ‘ { } = Q{ C } into ‘ B . Appendix B. the corange of a partial function Definition B.1.
Let G be a partial function from a quantum set X to a quantum set Y .The corange of G is the predicate G ⋆ ( T Y ) on X , where T Y is the maximum predicate on Y .Intuitively, the corange of G is its domain of definition, but I prefer to reserve the term“domain” for the source object X of G . Following the natural isomorphisms of propositionsA.3 and A.4, the corange of G corresponds to the projection G ⋆ (1), as both T Y ∈ Pred( Y )and 1 ∈ Proj( ℓ ∞ ( Y )) are the top elements of these two partial orders. Thus, G is a functionif and only if its corange is T X (theorem 7.6). Proposition B.2.
Let G : X → Y be a partial function, and let F : Y → Z be a function.Then, the corange of F ◦ G is equal to the corange of G .Proof. ( F ◦ G ) ⋆ (1) = G ⋆ ( F ⋆ (1)) = G ⋆ (1) (cid:3) Thus, for each predicate P on a quantum set X , we have a category of partial functions outof X with corange P , whose morphisms are postcomposition by functions. Examining thiscategory from M ∗ , we see that it has an initial object, corresponding to the homomorphism φ : C → ℓ ∞ ( X ) mapping 1 to p , and it has a terminal object, corresponding to the inclusion φ : p · ℓ ∞ ( X ) · p ֒ → ℓ ∞ ( X ). Definition B.3.
Let P be a predicate on a quantum set X . Define R P be the binary relationfrom X to corresponding to P under the natural isomorphism of proposition A.3. Proposition B.4.
Let G be a partial function from a quantum set X to a quantum set Y with corange P . Then, R P factors uniquely through G via a function. X Y G R P ! Proof.
In the light of the contravariant equivalence between qSet and M ∗ , it is enoughto show that R ⋆P (1) = G ⋆ (1). The projection R ⋆P (1) corresponds to P via the naturalisomorphisms of propositions A.3 and A.4, by the latter proposition, and as we have alreadyobserved, G ⋆ (1) corresponds to the corange of G . (cid:3) Definition B.5.
Let P be a predicate on a quantum set X . Assume that P ( X ) = P ( X ′ )for distinct atoms X, X ′ ∝ X . Define the quantum set P by P = Q{ P ( X ) | X ∝ X } . For each X ∝ X , let u X ∈ L ( P ( X ) , X ) be the inclusion isometry. Define the binary relation K P from X to P by K P ( X, P ( X )) = C · u † X for X ∝ X , with the other components of K P vanishing. ecause each inclusion operator u X is an isometry, K P ◦ K † P = I P , so K P is a surjectivepartial function from X to P . Intuitively, K † P is the inclusion of P into X , but in general P is not a subset of X , and K † P is not a function. Our assumption on the predicate P ensuresthat the atoms of P are in one-to-one correspondence with those atoms of X on which P isnonzero. In general, distinct atoms of X may have equal subspaces. Proposition B.6.
Same assumptions as of definition B.5. Let G be a partial function froma quantum set X to a quantum set Y with corange P . Then, G factors uniquely through K P via a function. X P Y K P G ! Proof.
For all b ∈ ℓ ∞ ( P ), and all X ∝ X , we have that K ⋆P ( b )( X ) = u X b ( P ( X )) u † X (proposi-tion 7.7). Since each operator u X is an isometery, K ⋆P is injective, and furthermore, the imageof K ⋆P is exactly p · ℓ ∞ ( X ) · p , where p is the projection defined by p ( X ) = u X u † X = proj P ( X ) for X ∝ X . Thus, K ⋆P factors through the inclusion p · ℓ ∞ ( X ) · p ֒ → ℓ ∞ ( X ) via an isomorphismof von Neumann algebras. The projection p corresponds to the predicate P under the iso-morphism Proj( ℓ ∞ ( X )) ∼ = Pred( X ) described in appendix A, so G ⋆ (1) = p . The propositionnow follows by the contravariant equivalence of theorem 7.6, from the universal property ofthe inclusion p · ℓ ∞ ( X ) · p ֒ → ℓ ∞ ( X ) expressed in the following commutative diagram: ℓ ∞ ( X ) p · ℓ ∞ ( X ) · p ℓ ∞ ( Y ) G ⋆ ! (cid:3) Appendix C. material quantum sets The definition of quantum sets given in section 2 is intuitively correct up to a weak equiv-alence of categories, but it is nonetheless a bit sloppy. A couple of the resulting blemishesare mentioned in that section. First, the one-dimensional atoms of a quantum set neednot correspond to actual elements, though they intuitively should. Second, the equation‘ S × ‘ T = ‘( S × T ) for ordinary sets S and T is only true modulo natural isomorphism,though intuitively it should be exactly true.We also have an odd caveat in definition B.5 and proposition B.6. This caveat is neces-sary because distinct atoms of a quantum set may nevertheless have a nonzero subspace incommon. This unpleasant phenomenon necessitates our definition of predicates as functionsassigning subspaces to atoms, rather than as quantum sets in their own right. It could havebeen avoided by simply requiring that distinct atoms of a quantum set be disjoint in defini-tion 2.1. This requirement would have no effect on the development of our theory, apart fromcompelling some discussion of the set-theoretic details of various constructions. I judged thediscussion of these technicalities to be harmful to the expository goals of the article.Our distinction between quantum sets and predicates is appropriate to the structural, i.e.,category-theoretic approach we have taken. In this section, we provide alternative definitionsthat express a more material conception of quantum sets. Roughly speaking, we identify uantum sets with predicates on a universe of uratoms. Formally, we change the definitionof quantum sets in a way that preserves the soundness of the arguments in this article. Weomit the proofs, which are tedious and straightforward.The most significant change is replacing the dagger compact category FdHilb of finite-dimensional Hilbert spaces and linear operators with a weakly equivalent dagger compactcategory
FdHilb ′ , which we now define. Definition C.1.
On objects,
FdHilb ′ has the following structure:(1) An object of FdHilb ′ is a subspace of ℓ ( M ), where M is any finite set.(2) The tensor product of two such objects X ≤ ℓ ( M ) and Y ≤ ℓ ( N ) is defined by X ⊗ Y = span { ( m, n ) x ( m ) · y ( n ) | m ∈ M, n ∈ N } ≤ ℓ ( M × N ) . (3) Each object is its own canonical dual, i.e., ℓ ( M ) ∗ = ℓ ( M ) by definition.As usual, we assume that ℓ ( M ) and ℓ ( N ) are disjoint as sets whenever M and N are dis-tinct, as a consequence of our formalization of mathematics in set theory. Each object of theform ℓ ( M ) is essentially a finite-dimensional Hilbert space, equipped with an orthonormalbasis. Consequently, our morphisms are essentially matrices. Definition C.2.
On morphisms,
FdHilb ′ has the following structure:(1) A morphism of FdHilb ′ from X ≤ ℓ ( M ) to Y ≤ ℓ ( N ) is an element of Y ⊗ X .(2) If a is a morphism from X ≤ ℓ ( M ) to Y ≤ ℓ ( N ), and b is a morphism from Y ≤ ℓ ( N ) to Z ≤ ℓ ( P ), then their composition is given by( b ◦ a )( p, m ) = X n ∈ N b ( p, n ) · a ( n, m ) . (3) If a i is a morphism from X i ≤ ℓ ( M i ) to Y i ≤ ℓ ( N i ), for i ∈ { , } , then the tensorproduct of a and a is defined by( a ⊗ b )(( n , n ) , ( m , m )) = a ( n , m ) · a ( n , m ) . (4) If a is a morphism from X ≤ ℓ ( M ) to Y ≤ ℓ ( N ) then we define its duals by a † ( m, n ) = a ( n, m ) and a ∗ ( m, n ) = a ( n, m ) . Proposition C.3.
The category
FdHilb ′ is dagger compact, and it is weakly equivalent tothe dagger compact category FdHilb . The equivalence is identity on objects. The equivalencetakes each morphism a from ℓ ( M ) to ℓ ( N ) to the linear operator defined by f n X m ∈ M a ( n, m ) · f ( m ) ! . Definition C.4 (in place of definition 2.1) . A quantum set X is completely determined bya set At( X ) of nonzero Hilbert spaces in FdHilb ′ that are pairwise disjoint.Note that X ≤ ℓ ( M ) and Y ≤ ℓ ( N ) are disjoint if and only if M = N , because X consists of functions with domain M , and Y consists of functions with domain N . Thus,quantum sets are now in bijective correspondence with predicates of small support on a“quantum class” U , where At( U ) = { ℓ ( M ) | M is a finite set } . roposition C.5. For each quantum set X , and each Hilbert space H , define X ( H ) to bethe unique atom of X that is a subspace of H , if such an atom exists, and define X ( H ) tobe the zero subspace of H , if such an atom does not exist. This defines a bijection between quantum sets and those predicates P on U in the senseof definition A.1 that have small support in the sense that P ( H ) = 0 for all but set-many H ∈ At( U ). Thus, like ordinary sets, quantum sets don’t quite form an ortholattice. Definition C.6.
Let X and Y be quantum sets. Define Y ≤ X iff Y ( H ) ≤ X ( H ) for all H ∈ At( U ). Furthermore, define X ∨ Y and X ∧ Y , and X \ Y if X ≥ Y , by(1) (
X ∨ Y )( H ) = X ( H ) ∨ Y ( H ),(2) ( X ∧ Y )( H ) = X ( H ) ∧ Y ( H ), and(3) ( X \ Y )( H ) = X ( H ) ∧ Y ( H ) ⊥ ,for all H ∈ At( U ). Definition C.7 (in place of definition A.1) . For each quantum set X , define Pred ′ ( X ) = {P ≤ X } .The construction P 7→ ( X
7→ P ( X )) defines an isomorphism of ortholattices from thepredicates on X as defined above, to the predicates on X as in definition A.1. We mayimplement a similar identification for binary relations. Definition C.8 (in place of definition 3.1) . For each quantum set X and each quantum set Y , define Rel ′ ( X ; Y ) = {R ≤ Y × X } .The construction R 7→ (( X, Y )
7→ R ( Y ⊗ X )) defines an isomorphism of ortholatticesfrom the binary relations from X to Y as defined above, to the binary relations from X to Y as in definition 3.1, provided that L ( X, Y ) is identified with the space of morphisms from X to Y in the category FdHilb ′ . Proceeding as in section 3, we obtain a category qRel ′ , eachof whose morphisms is also one of its objects. Echoing the formalization of mathematicsin set theory, everything is a quantum set: each function, each predicate, and each binaryrelation. With the exception of our definition of union, we may proceed with these quantumsets and binary relations as before.The trouble with definition 2.2(4) of the union of two quantum sets is that it is no longerguaranteed to produce a quantum set. Indeed it is neither fully at home in the materialapproach taken in this appendix, nor in the structural approach taken in the rest of thearticle. Definition 2.2(4) produces a quantum set in the sense of this appendix if and onlyif both X and Y are subsets of X ∨ Y , and in that case
X ∪ Y = X ∨ Y . It is thereforereasonable to use that notation in that case. However, apart from definition of 2.2(4), whichserves only an expository purpose, the rest of the arguments remain sound.
Theorem C.9.
Every result in all preceding sections and appendices is correct using defi-nition C.4 in place of definition 2.1, definition C.7 in place of definition A.1, definition C.8in place of definition 3.1, and
FdHilb ′ in place of FdHilb .Proof.
Every argument remains valid just as it is. (cid:3)
In addition to turning our morphisms into quantum sets, this definition also tightens thecorrespondence between ordinary sets and classical quantum sets. roposition C.10. The functor ‘( · ) : Rel → qRel ′ is an isomorphism of dagger compactcategories from Rel onto the full subcategory of qRel ′ consisting of classical quantum sets. In other words, the construction S ‘ S is a one-to-one correspondence between ordinarysets and quantum sets whose atoms are all one-dimensional, and it exactly preserves alldagger compact structure. In particular, for all ordinary sets S and T , we have ‘ S × ‘ T =‘( S × T ). For concreteness, we may take the monoidal unit of Set to be the set 1 = {∅} , sothe monoidal unit of qSet is = ‘1 = Q{ C ∅ } = Q{ ℓ ( {∅} ) } . Thus, where we speak of theHilbert space C , we mean C ∅ ..