Topological quantum computation employs two-dimensional quasiparticles called anyons [10, 5]. The generally accepted mathematical basis for the theory of anyons is the framework of modular tensor categories. That framework, as presented in  or  or  involves a substantial amount of category theory and is, as a result, considered rather difficult to understand. For example, Trebst et al. [14, page 385] write “In general terms, we can describe anyons by a mathematical framework called tensor category theory. Here we will not delve into this difficult mathematical subject .” Similarly, Bonderson [4, page 13] writes “In mathematical terminology, anyon models are known as unitary braided tensor categories, but we will avoid descending too far into the abstract depths of category theory .”
Is the complexity of the current framework necessary? We do not think so. Our opinion is based on the following.
Our own experience, admittedly modest. In , after describing modular tensor categories, we presented a simplification based on Yoneda’s Lemma. Then we exhibited some computations for the particular case of Fibonacci anyons, including the computation of the braiding operators that are central to proposed uses of anyons for quantum computation. It turned out that a good part of the axiomatics of tensor categories was not needed for those computations.
Only isomorphisms, rather than category theory’s more general morphisms, are used in the anyon theory.
Physicists tend to avoid category theory.
The computations in  can be based on a much simpler framework, which looks less like category theory and more like familiar algebra. We introduce that framework here.
The main idea of the proposed framework is to work with ordinary algebraic structures, like rings, amplified with a notion of witnesses for equations. The rules of equational logic are then accompanied by constructions of new witnesses from old. For example, where equational logic says that, from an equation , one can infer , our framework will say that, from a witness for , one can construct a witness for . The other rules of equational logic are treated similarly; see Section 2 for details. We call this new framework witness algebra.
Traditional equational logic can be viewed as the special case of witness algebra where all witnesses are trivial and therefore do not need to be mentioned. In our work with anyons, the witnesses for the associativity and commutativity of multiplication will be highly nontrivial. Other witnesses, for example those for the associativity and commutativity of addition, will not be entirely trivial but will amount to minor bookkeeping information. See Section 4 for details.
Category-Theoretic Remark 1.
Readers who care only about the witness algebra framework and not the traditional category-theoretic one can skip this and subsequent category-theoretic remarks. These will serve to connect our framework to the traditional one but will not be essential for the development of witness algebra itself.
Our work, having emerged from category theory, can be translated back into category-theoretic terminology. The result of that translation would be a theory of groupoids (categories in which all morphisms are isomorphisms) enhanced with additional structure. For example, in our braid semirings (defined in Section 3), the enhancement would be two monoidal structures, one of which (corresponding to addition) is symmetric while the other (corresponding to multiplication) is braided, plus a distributive law connecting the two, plus suitable coherence axioms.
Somewhat ironically, easing the category-theoretical complexity of the anyon theory required some category-theoretical investigation. The associativity, commutativity and distributivity isomorphisms should satisfy appropriate coherence conditions. What are these conditions? Much of the work has been done in the literature but there was a gap which we filled in the companion paper . The situation is explained in detail in Section 3.
Gödel suggested in unpublished work [6, Section V] that modal logic could profitably be amplified by introducing justifications. This idea was independently rediscovered and extensively studied by Artemov; see for example . The role of justifications in modal logic is similar to the role of witnesses in our witness algebra. But, instead of modal logic, we work with elementary algebra. Another difference is that Artemov’s justifications can be nested; that is, “ is a justification for ” is itself a formula that can admit justifications. We use witnesses only for equations, and “ is a witness for ” is not itself an equation, so it cannot be witnessed.
2. Witnessed Equational Logic
To establish terminology, we recall some definitions from universal algebra. A signature is a collection of operation symbols, each having a specified number of argument places (arity). An algebra of signature consists of a set , the universe of , together with interpretations of the symbols in on ; an -ary symbol is interpreted as an -ary operation. The interpretations of nullary operation symbols are viewed as elements of ; that is, nullary operation symbols are individual constants.
It is often convenient to use the same symbol for an algebra and its universe; in such cases it should be clear from the context whether the symbol denotes the algebra or its universe.
2.2. Witness Frames
We begin the discussion of witness algebras with the special case where the signature is empty. In this case, algebras of signature are merely sets with no additional structure. Nevertheless, the key concept of witnessing appears even in this context and is easier to explain without the additional baggage of a nontrivial signature. So we begin with this special case; we use the name “witness frame” for witness algebras for the empty signature.111Since algebras for the empty signature are sets, it would seem reasonable to call witness algebras for the empty signature “witness sets”. Unfortunately, that sounds too much like a set of witnesses. We have chosen the terminology “witness frame” to suggest that these are basic systems to which additional algebraic structure (nonempty signatures) could be attached.
The idea here is that we have a set , each of whose elements might be viewed in a variety of ways. Witnesses will indicate whether and how two views represent the same element . As a fairly typical example (similar to what we shall later use for anyon computations), the elements of
might be some vector spaces, and a view of a vector space might be that space equipped with a particular basis. A witness should then indicate how two vector spaces with specified bases (i.e., two views) are actually the same vector space (though the bases might be quite different). Such a witness could be an invertible matrix transforming the one basis into the other, i.e., expressing each vector of the latter basis as a linear combination of the former basis.
Thus, a witness frame really involves two sets: the set and the set of views of elements of . To formalize these notions, it is convenient to take the views as the basic entities; elements of can then be identified with equivalence classes under the equivalence relation of “some witness shows that the two views represent the same element of .” (Of course, the formal definition will need to ensure, among other things, that this is an equivalence relation.)
This discussion leads to the somewhat unusual situation that the entities of primary interest and importance, the elements of , are not the ones taken to be basic in the formalization. We shall emphasize the importance of the equivalence classes, the elements of , by the following terminology and notation.
The views, which are formally the basic entities, will be called the raw elements of the witness frame, and the symbol will be used to mean that two of them are the same view, not merely representatives of the same element of . The equivalence relation of “some witness says they’re the same” will be denoted by , and the equivalence classes, the elements of , will be called the true elements of the witness frame. In the vector space example above, would mean that the views and are the same vector space with the same basis, whereas would mean that they are the same vector space with possibly different bases.
Our use of to denote an equivalence relation coarser than what one might consider complete equality, , is unusual but not unprecedented. For example, in combinatorial group theory, when discussing groups and their presentations, one sometimes writes to mean that the words and represent the same group element, while means that they are identical as words.
We emphasize that our unusual use of the equality symbol applies only to raw elements. In other contexts, the equality symbol retains its customary meaning. In particular, if and are witnesses, then means that they are the same witness.
After these explanations of the underlying intention, we are ready for the definition of witness frames.
A witness frame consists of a set of raw elements and, for all elements , pairwise disjoint (possibly empty) sets of witnesses for equality between and . If , we write and we say that witnesses that . We call a witness if it is in one of the sets . The system of raw elements and witnesses is required to have the following structure.
For each , there is a specified witness .
For each witness , there is a specified .
For each pair of witnesses and , there is a specified witness .
These specifications are subject to the following axioms.
If then .
If then and .
If , , and ,
If we ignore the witnesses and pay attention only to the equations, then requirements (W1), (W2), and (W3) correspond to the usual axioms and rules of equational logic (in the case of the empty signature), saying that equality is reflexive, symmetric, and transitive. Thus, in witness frames these three requirements ensure that the relation introduced in the following definition is an equivalence relation.
In any witness frame, with the notation of the preceding definition, we define, for ,
The elements of are called raw elements of the witness frame, and equality between them is symbolized by . The equivalence classes with respect to the relation just defined are called true elements of the witness frame.
We have built into the definition of witness frames that the sets for the various pairs are disjoint. In other words, a witness witnesses only a single equation. This convention is very convenient for theoretical purposes. It implies in particular that the inverse of any witness is uniquely defined and that the composition of any two witnesses can be defined in at most one way. In some concrete situations, on the other hand, it is tempting to re-use the same witness for several equations. Indeed, this temptation arose in our vector space example above; one and the same matrix can serve as the transformation matrix between many pairs of bases. Fortunately, there is an easy solution for this problem, namely to “mark” each witness with the equation that we want it to witness. That is, if the same is in both and , we replace it by as an element of , and we replace it by as an element of . More generally, we adopt the following convention, intended to give us the best of both worlds — authorization to re-use witnesses when describing concrete examples while maintaining disjointness of the sets of witnesses for official purposes.
If a witness frame is described in a way that allows the sets to overlap, then it is to be understood that the actual, official witnesses for an equation are not the described ’s but rather the marked versions .
Our definition of witness frames includes requirements (W4), (W5), and (W6), which say that certain combinations of witnesses are equal. In each case, the required equality makes sense because the two sides are witnesses of the same equation. The intention behind these requirements is that the specifications in (W1), (W2), and (W3) should not be made randomly or arbitrarily but in some coherent way. For example, the witness in (W3) should not be just any witness for but rather one that combines, in a sensible way, the information in the witnesses and (and the transitive law of equality).
Category-Theoretic Remark 6.
Category theorists will recognize witness frames as just a notational variant of the familiar notion of groupoid, a category in which all morphisms are isomorphisms. Our raw elements are the objects of the groupoid, our witnesses are the morphisms, and amounts to . Our (W1), (W3), (W4), and (W6) amount to the definition of a category, while (W2) and (W5) provide the inverses that make all the morphisms isomorphisms. The true elements are the connected components of the groupoid.
Notice that we compose witnesses in what is often called diagrammatic order. That is, the composition of with is written as , not as or as .
The following lemma records some basic properties of the operations on witnesses that are involved in witness frames.
The operation on witnesses admits cancellation. That is, if and are defined and equal, then . Similarly if .
Inversion is involutive. That is, all witnesses satisfy .
For part (1), assume , where and so that the operation is defined for these witnesses. Then compute
The proof under the hypothesis is symmetrical.
For part (2), notice that, if then , and use part (1) to cancel . ∎
The definition of witness frames requires that the raw elements, and therefore also the true elements, constitute a set rather than a proper class. Everything we do, however, would work just as well if we used classes instead. So, for example,we could deal with a witness frame whose true elements are all of the vector spaces, not just some. Except for this remark, we shall ignore the set-class distinction in this paper.
2.3. Witness Algebras
In the preceding subsection, we dealt with the case of the empty signature, where algebras are just sets. In keeping with that special case, although our witness frames had considerable structure, as described in (W1)–(W6), the true elements formed just a set with no additional structure. In the present subsection, we deal with the case of general signatures . A witness -algebra will have enough additional structure to make the true elements into a -algebra in the usual sense.
Let be a signature. A witness -algebra is a witness frame (with notation as above) together with actions of the operation symbols from on raw elements and on witnesses, as follows. Let be -ary. Then:
is an -ary operation on the raw elements.
If for then is defined and
If and for then
To avoid a proliferation of notation, we use for all three of a symbol in , its action as an operation on , and its action on witnesses (as long as the context prevents ambiguity).
When the arity is zero, a 0-ary operation on is, of course, a function from the one-element set into . It is convenient (and customary in algebra and logic) to identify such a function with its unique value. Thus, 0-ary operations amount to constants.
If we pay attention only to witnessed equality and not to the specific witnesses, then clause (2) in Definition 9 says that the algebraic laws of equality concerning function symbols are obeyed: we can substitute equals for equals. This clause and the clauses in the earlier definition of witness frames give all the usual laws of equational logic.
In the definition of witness algebras, clause (1) makes the raw elements into a -algebra, which we call the raw algebra. Clause (2) makes the witnessed equality relation , which we already know is an equivalence relation, into a congruence relation. As a result, the quotient, the set of true elements, becomes a -algebra as well. We call it the true algebra (or the true -algebra) of the witness algebra.
The purpose of clause (3) is, as with some of the clauses in the definition of witness frames earlier, to require that the witnesses in clause (2) should not be just arbitrarily chosen witnesses for the relevant equations but should be chosen in some reasonable way based on .
Any -algebra can be converted trivially into a witness algebra whose raw and true algebras are both the given . Just take all the sets to be singletons. (We could even choose to take just a single, trivial witness, say 0, to witness exactly those equalities where and are the same raw element. Convention 5 would make this choice “legal” by replacing the single witness 0 with different witnesses for different equations.) In this way, we can regard ordinary algebras as a special case of witness algebras.
More generally, consider a -algebra , a congruence relation on it, and the quotient algebra . Then, using only one witness (until Convention 5 turns it into many witnesses), we can produce a witness algebra whose raw algebra is and whose true algebra is . It suffices to define to hold if and only if .
As already mentioned in connection with some of the clauses in our definitions, our witnesses behave similarly to deductions in equational logic. To emphasize the similarity, we can use a notation where a witness is displayed above the equation that it witnesses, resembling a deduction of that equation. Then we have, for all ,
corresponding to the axiom of equational logic. For any
we can depict as
so that it looks like followed by an application of the inference rule “from infer .” Similarly, for any
we can depict as
so that it looks like and followed by an application of the inference rule “from and infer .” Finally, for an -ary function symbol , we can depict as
The similarity between witnesses and deductions suggests the following variation on the theme of Example 11.
Suppose is the -algebra presented by some generators and relations, and suppose is the quotient obtained by imposing some additional relations. Then we can almost obtain a witness algebra by taking as the raw algebra, taking the witnesses for any equation to be the formal deductions of this equation from the relations defining , and taking witnesses , and to be compositions of deductions as depicted above. The reason for “almost” in the preceding sentence is that the axioms for witness algebras require certain identifications between deductions. For example, two consecutive uses of symmetry as in
should have no effect (see part (2) of Lemma 7); this deduction should be identified with
A special case of this example occurs when is the algebra in a certain variety presented by certain generators and relations, and is presented by the same generators and relations in a smaller variety, given by more identities. For example, might be the group with a certain presentation while is the abelian group with the same presentation (the abelianization of ). The witnesses would then be deductions (modulo identifications required by the axioms) from the commutative law.
Category-Theoretic Remark 13.
Since witness algebras amount to groupoids, our definition of witness -algebras makes the operations act on both the objects (raw elements) and morphisms (witnesses). These operations constitute functors from powers of the groupoid to itself. In general, the definition of “functor” also requires preservation of identity morphisms, so it would seem that we need to require
In the case of groupoids, though, this preservation of identity elements follows from compatibility with composition. This last obsrevation will be useful even apart from the category-theoretic point of view, so we formulate it as the following proposition.
If is -ary then . In particular, if is a nullary operation symbol, then the corresponding witness is .
To simplify the notation, we assume for now that is binary, and we write and instead of and . Note that both and witness the equation . Furthermore, according to requirement (3) in the definition of witness algebras, we have
The desired conclusion now follows by cancellation (part (1) of Lemma 7).
The argument for generalizes easily to larger arities and to . For , the argument still works and in fact becomes simpler, but it requires some notational caution, as follows. If is 0-ary then, remembering that 0-ary operations on a set amount to elements, we have a raw element , and we also have a witness (also called if no confusion results, but here confusion would result) for the equation . Then we have, from the definition of witness algebra,
and cancellation (part (1) of Lemma 7) gives us . ∎
3. Braid Semirings
In this section, we introduce the particular algebraic theory underlying the application of witness algebra to anyons. The basic idea is quite simple, but various complications arise in the details.
We shall arrange our witness algebras so that the associated true algebras are commutative semirings with unit. Here “semiring” is defined like “ring” except that additive inverses are not required to exist. Some authors use “rig” to mean “semiring” (the idea being that removing the letter “n” from “ring” corresponds to removing negatives from the definition). Because multiplication will be commutative in our semirings, we adopt the following convention for brevity.
“Rig” means commutative semiring with unit.
Thus, the signature for our algebras consists of two binary operations and and two constants and . A rig is an algebra for this signature in which
both and are associative and commutative,
and are identity elements for and , respectively, and
distributes over .
Distributivity here means not only that multiplication distributes over sums of two elements222We begin here to use capital letters for elements of our raw algebras, for two reasons. First, in the application to anyons, these raw elements will be tuples of vector spaces, for which capital letters are more natural than lower-case letters. Second, we shall need to import some diagrams from . That paper was written from the point of view of category theory, so raw elements were objects of categories, denoted, as usual, by capital letters., (which implies distribution over any larger number of summands) but also distribution over zero summands, i.e., . (This equation would be redundant in the presence of additive inverses, but it needs to be assumed in the axiomatization of rigs.)
The raw algebras admit an even simpler description: They are algebras for the same signature but subject to no equations. This means that, for the true algebras to satisfy the equations that define rigs, we need to introduce witnesses for all those equations. That is, our witness algebras must have (at least) the following witnesses specified as part of the structure, for all raw elements :
We shall refer to these witnesses as the rig witnesses.
So far, our description of the desired witness algebras can be easily summarized: Algebras for the signature with enough specified witnesses to make the true algebra a rig. More is needed, though, for this structure to make good sense and (more importantly) to be useful for anyon theory. We need to specify, or at least constrain, how the rig witnesses listed here interact with each other and with other witnesses that might be present. Let us begin with two examples before presenting the general situation.
Suppose we have witnesses and . Then the fact that and commute under addition has, in addition to the witness above, another witness that works via the commutativity of and , namely . If we think of and as giving us alternative ways to view and , then this second witness for is just an alternative way to view the original witness . So it is reasonable to require that these two witnesses be equal333Recall that equality of witnesses means genuine identity, not existence of some meta-witness.. Equivalently,
This requirement can be summarized as “The witnesses respect witnessed equalities.” In Section 3.1, we will impose analogous requirements on all of the other rig witnesses. The justifications for these requirements are the same as for here.
Category-Theoretic Remark 17.
In category theory, the requirement in Example 16 is expressed by saying that is a natural transformation. The analogous requirements for the other rig witnesses will say that all of them are natural transformations.
We have two witnesses for , namely and . It seems reasonable to require that they coincide.
The requirements in Example 16 and Example 18 are qualitatively different. In the former, we were concerned with how a rig witness () interacts with arbitrary other witnesses ( and ); in the latter, we are concerned with how rig witnesses interact with each other. In the former, there is an obvious generalization from the considered there to all the other rig witnesses. In the latter, there is no obvious generalization, and indeed there is considerable freedom in choosing what requirements of this sort should be imposed.
The rest of this section is devoted to presenting the requirements that we impose on the witnessess in our braid rigs (also called braid semirings). We present these requirements in three parts, subsections 3.1, 3.3, and 3.4, with an intermediate subsection 3.2 explaining how some of the requirements were chosen.
The requirements in subsection 3.1 are analogous to what we saw in Example 16. This subsection is simply the evident generalization of the example from to all rig witnesses. Subsections 3.3 and 3.4 present requirements analogous to that in Example 18. As indicated above, we have some freedom in choosing requirements of this sort. Subsection 3.2 discusses how we chose to exercise that freedom in making the decisions in the next two subsections.
Much of the material in subsections 3.2, 3.3, and 3.4 has already appeared, up to notational and terminological differences, in our earlier paper . For the reader’s convenience, we import it here, with the necessary modifications, rather than contenting ourselves with a list of the changes.
Braid rigs will be defined as witness algebras for the rig signature , equipped with all the rig witnesses listed above and satisfying all the requirements imposed in the following subsections.
3.1. Rig Witnesses Respect Witnessed Equalities
Suppose we have witnesses
Then we require that all rig witnesses involving any of and the corresponding rig witnesses involving match via the given . In detail, these requirements are as follows.
Notice that the fourth of these equations is what we had in Example 16. All ten of the equations have the same general form: The first factor on the left and the second on the right are from one of our ten types of rig witnesses, with unprimed subscripts on the left and primed on the right. The other factors are built from (some of) . How they are built matches the specifications of the rig witness. (In the last of these equations, in strict analogy to the previous ones, the left side would have been ; we have performed the trivial simplification to .)
Category-Theoretic Remark 19.
In the language of category theory, these ten equations merely say that the rig witnesses constitute ten natural transformations.
3.2. Coherence Conditions
We turn next to identities, known as coherence conditions, between various compositions of rig witnesses. We have seen one coherence condition, , in Example 18, but there is no evident way to determine all of the coherence conditions that should be imposed on our rig witnesses. In fact, as we shall see, the choice of coherence conditions is influenced by the intended application. For example, since addition alone and multiplication alone are subject to the same requirements in the definition of rig (associativity, unit, and commutativity), it would seem natural to impose the same coherence conditions on the purely additive and purely multiplicative rig witnesses. That approach, however, turns out to be completely inappropriate for the study of non-abelian anyons. (See the discussion of symmetry versus braiding later in the present subsection.) Accordingly, we devote this subsection to explaining how we selected suitable coherence conditions to be satisfied by braid rigs; the conditions themselves will be presented in the next two subsections.
There are two mathematical constraints on our selection of coherence conditions, plus a practical consideration that also influenced our choices. The first and most important mathematical constraint is that our coherence conditions should be satisfied in the witness algebras arising in anyon models, the witness algebras that we propose as a simplification of modular tensor categories. Axioms that fail in the intended examples are useless. So we must not make our coherence conditions too strong.
The second mathematical constraint is that our coherence conditions should not be too weak; they should entail all the information needed in our computations of specific examples. For example, our coherence conditions should support the computations, as in , of the associativity and braiding matrices for Fibonacci anyons.
For practical purposes, we stay close to the coherence conditions already available in the literature for structures resembling some of our rig witnesses. Let us briefly summarize the relevant literature.
If we consider either addition by itself or multiplication by itself, then the definition of rigs requires that we have a commutative monoid. An analogous structure has been studied in category theory, namely symmetric monoidal categories, and suitable coherence conditions were found by Mac Lane  and simplified by Kelly . Here is an example to clarify what “suitable” means.
In ordinary (not witness) algebra, the associative law for addition, , implies that one can safely omit parentheses in sums of any number of terms. For example, one can deduce , and similarly for more summands and for other arrangements of the parentheses. In fact, for the specific case of , two deductions are available. One goes via , and the other goes via and .
In witness algebra, the corresponding facts are as follows. Given witnesses as above, for all raw elements , we can construct, by composing them, witnesses for and similarly for more summands and for other arrangements of the parentheses. In fact, for the specific case of , two such compositions are available, corresponding to the two deductions in ordinary algebra. A typical coherence condition would require that these two compositions coincide.
If we use associativity to rearrange parentheses in sums with more than four summands, there will, in general, be many deductions for a single equation in ordinary algebra, and therefore many witnesses, composites of witnesses of type , for the same equation in witness algebra. We would like all these witnesses for the same equation to coincide. So, a priori, we would impose infinitely many coherence conditions, with more and more variables. Fortunately, Mac Lane showed in  that the coherence condition for associativity with four summands implies all the other coherence conditions for associativity. Moreover, he found a small number of coherence conditions for associativity, unit, and commutativity that imply that, whenever two reasonable compositions of these witnesses witness the same equation, they coincide. (“Reasonable” requires careful formulation, to avoid, for example, expecting the special case of commutativity to coincide with .) We shall adopt Mac Lane’s coherence conditions, as simplified by Kelly , for the additive structure of our braid semirings, i.e., for the rig witnesses of the forms .
Although it seems natural and simple to treat multiplication the same way, we shall not adopt these same conditions for the multiplicative structure. The reason lies in the behavior of anyons that we intend to model. Here is a rough explanation of the situation; a more detailed (and thus more accurate) explanation can be found in [2, 13]. Think of a product as representing an anyon (or anyon system) of type located next to one of type . The commutativity witness represents interchanging the locations of and . Anyons inhabit a two-dimensional space, and so there are two different ways to move from, say, the left of to the right of : could pass in front of or behind it. If we (arbitrarily) take to represent the transposition that moves in front of , then represents the transposition moving behind , and we do not want these to always coincide. In other words, we do not want the so-called symmetry condition
to hold in general. Indeed, the left side of the symmetry condition represents moving all the way around , back to its original location. The non-triviality of such braiding operations is the key to the usefulness of anyons in quantum computation. But the symmetry condition is among the coherence conditions of Mac Lane and Kelly. Our multiplicative structure should therefore be subject only to some weaker system of coherence conditions, allowing non-trivial braiding.
Fortunately, Joyal and Street [7, 8] have given a system of coherence conditions that accomplishes exactly what we need. Their notion of “braided monoidal category” is like “symmetric monoidal category” except that the symmetry condition is omitted (and another condition, deducible using symmetry but not otherwise, is added). The coherence conditions for braided monoidal categories are included among the axioms for modular tensor categories; see [2, 13]. We shall adopt the Joyal–Street coherence conditions for the multiplicative structure of our braid semirings.
Beyond the additive and multiplicative structures, whose coherence conditions we borrow from Mac Lane, Kelly, Joyal, and Street, we also have the distributivity witnesses which connect the additive and multiplicative structures.
The available literature concerning coherence conditions for distributivity is the paper  of Laplaza. He introduces and justifies a system of such coherence conditions for the situation where both the additive and multiplicative structures are symmetric monoidal structures. In our situation, however, only the additive structure is symmetric; the multiplicative structure is merely braided. As a result, we must modify Laplaza’s coherence conditions to work properly with braided multiplication. We have carried out this modification in  and proved some theorems there that indicate its appropriateness. In the present paper, we shall only record the coherence conditions that we found and some remarks about them, referring to  for details.
Rather than writing these conditions as equations, we shall exhibit them as diagrams, in accordance with the following conventions. Each diagram will be a directed graph, with vertices labeled by raw elements and with directed edges labeled by witnesses. If labels an edge from a vertex to a vertex then . Consider a path in the underlying undirected graph (obtained by forgetting the orientations of the edges); note that the edges in such a path may be directed forward or backward along the path. Associate to this path the witness for obtained, by composing (by ), in order along the path, the labels of those edges that are directed forward along the path and the inverses of the labels of the edges directed backward along the path. In our diagrams, the underlying undirected graph will always be just a cycle, so for each pair of vertices , there will be exactly two paths from to , associated with two witnesses for . The diagram is to be interpreted as the equation saying that these two witnesses are equal. It is easy to check that, if we had chosen two other vertices and instead of and , then the resulting equation between two witnesses for would be equivalent to the equation described here between witnesses for . (The proof uses clauses (W4), (W5), and (W6) of the definition of witness frames.)
For another equivalent way to interpret the diagram, consider any vertex in the cycle and consider a “path” that goes from , once around the cycle (in either direction), ending back at . (We put “path” in quotation marks because, strictly speaking, a path should not have a repeated vertex.) Associated with this “path” is a witness for . The equations that interpret the diagram as above are equivalent not only to each other but also to the equation .
3.3. Coherence for Addition and Multiplication
Our coherence conditions for the additive structure, , are Kelly’s simplification  of Mac Lane’s coherence conditions  for symmetric monoidal categories. In the diagram form explained in Convention 21, they are the following Figures 1–4, in whose captions we have given names for the conditions. (The first figure, the pentagon, is the previously discussed case of two witnesses for moving the parentheses, in a sum of four terms, from the extreme left to the extreme right.)
Our coherence conditions for the multiplicative structure, , are those given by Joyal and Street [7, 8] for braided monoidal categories, namely the following Figures 5–8. Note that they differ from the additive ones by the absence of symmetry and the presence of a second hexagon condition. This second hexagon condition is like the first but with every replaced with and the direction of the associated edge reversed. In view of our Convention 21 about reading diagrams as equations, this replacement is equivalent to replacing every with . Thus, the two hexagon conditions are equivalent in the presence of symmetry, so we needed only one of them in the additive situation. But when symmetry is unavailable, the two hexagon conditions must both be assumed.
In the names of the mutiplicative hexagon conditions, “in front of” and “behind” refer to the way two anyons are interchanged by the commutativity witnesses . This corresponds to the customary picture of braided commutativity in terms of geometric braids (the same picture that gave the name “braided” to this weakening of symmetry).
Category-Theoretic Remark 22.
The content of this subsection is that our groupoid is equipped with two monoidal structures, a symmetric one written with and a braided one written with . These two structures will be connected by distributivity, whose coherence conditions constitute the next subsection.
3.4. Coherence for Distributivity
Our coherence requirements for the distributivity witnesses and are given by Figures 9–18, taken from our paper . We refer to this paper for motivation and additional information about these conditions, in particular their connections with Laplaza’s coherence conditions for the case where both and are symmetric.
We shall sometimes use the usual conventions from algebra that means and that, for example, means , not .
It will be convenient to have the following notation for the deviation from symmetry.
Thus, symmetry amounts to the requirement that . In the general braided situation, . Pictorially, if we imagine as interchanging with by moving in front of , then moves all the way around back to its initial position, first passing in front of and then returning behind .
We now present our coherence conditions for distributivity, along with some remarks intended to make them easier to understand.
We have required witnesses for the distributive law but not for the analogous law . This is reasonable, since we have commutativity and can therefore deduce either of these distributive laws from the other. In terms of witnesses, we have
In fact, we have a second, equally good witness for the same equation:
In terms of the braiding picture of products, the first of these witnesses moves behind , , and , and the second witness moves in front of these other factors. There is no reason to prefer one of these witnesses to the other, so we shall impose a coherence condition saying that they are equal. Rather than writing out that equality, we simplify it a bit by “clearing fractions”, i.e., by multiplying both sides by factors to cancel the inverses that occur in our two witnesses. Once that is done, each side of the desired coherence condition involves a composition of two witnesses, a composition that fits our definition of above. Thus, the desired coherence condition takes the simple form of the left diagram in Figure 9.
The right diagram in Figure 9 is essentially the analog of the left for a sum of no summands in place of the sum of two summands. The precise analog would result from the left diagram by changing the vertex labels on the left to and on the right to , changing the horizontal arrows to , changing the left vertical arrow , and changing the right vertical arrow to . (To see that this last is correct, use Proposition 14 with being the 0-ary operation 0.) Multiplying by the inverse of , we simplify the desired equality to , which is depicted on the right side of Figure 9.
It is worthwhile to keep in mind the two witnesses (equal by the left part of Figure 9) described above for . They will occur twice in Figure 17, and that rather large figure becomes easier to understand if one realizes that, in the two places indicated by dashed lines444Red in the pdf version of the paper, what looks like a composition of three witnesses can be understood as just a witness for distributivity from the right rather than the left.
Notice also that we have an analogous pair of witnesses (equal by the right part of Figure 9) with no summands rather than two,
The next three coherence conditions, Figures 10 through 12, say that distribution respects additive manipulations — commutativity, associativity, and unit properties. That is, given where is a sum, it doesn’t matter whether we perform additive manipulations within and then apply distributivity or first apply distributivity and then perform the corresponding manipulations on the resulting sum.
In these and subsequent figures, we indicate in the caption the corresponding condition in .
Next are four coherence conditions saying that, when distributing a product of several factors across a sum, it doesn’t matter whether one distributes the whole product at once or the individual factors one after the other. The case of a product of two factors distributing across a sum of two summands is the obvious one; it implies (in the presence of the other coherence conditions) the cases with more factors or summands. It is, however, also necessary to cover the cases where the number of factors or the number of summands is zero. So we get the four coherence conditions in Figures 13 through 16. In our names for the conditions, the numbers 2 or 0 refer first to the number of factors and second to the number of summands.
The remaining coherence conditions for distributivity, in Figures 17 and 18, concern a product of two sums, like . Distributivity lets us expand this as a sum of four products, but there is a choice whether to apply distributivity first from the left, obtaining , or from the right, obtaining . One coherence condition (Figure 17) says that both choices produce the same final result, up to associativity and commutativity of addition. (Unfortunately, the associativity and commutativity make the diagram rather large. It gets even larger because a single witness for distributivity from the right looks like a witness for distributivity from the left flanked by two commutativity witnesses.) In addition, there are analogous but far simpler coherence conditions for the case where one or both of the factors is the sum of no terms rather than of two. Our labels for these conditions include numbers 2 or 0 indicating the number of summands in each factor.
This completes our list of coherence conditions and allows us to define braid rigs.
A braid rig (or braid semiring) is a witness algebra for the signature together with chosen witnesses of the forms specified above and subject to the coherence conditions in subsections 3.1, 3.3, and 3.4.
We emphasize that the specifications of the chosen witnesses in braid rigs make the associated true algebras into rigs.
4. Unitary Fusion Semirings
In this section, we describe the particular braid rigs that are used in anyon models. We begin by describing the true rigs, as this description will be rather straightforward. Afterward, we shall describe the raw elements and witnesses.
4.1. True Algebra
The true rigs associated to our unitary fusion rigs wlll be rigs in which
there is a finite set of additive generators,
each element is a finite sum of generators in a unique way (up to order and parentheses), and
one of the generators is the multiplicative unit element .
The first two requirements in this list say that, as far as the additive structure of the rig is concerned, it is the free commutative monoid on the finite set of generators. Note that the finite sums in requirement (2) include the empty sum . In connection with requirement (3), we adopt the convention that the generators are numbered so that .
We also adopt the standard convention that , for a natural number and a rig element , means the sum of copies of .
Apart from and our intention to produce a rig, no requirements are imposed here on the multiplicative structure. Of course, the distributive law for rigs and requirement (2) together imply that the multiplicative structure is completely determined by the products of the generators. Thus, in the true rig associated to any unitary fusion rig, we shall have a system of equations of the form
where the are natural numbers. These equations, which in anyon theory are usually called fusion rules, suffice to determine the whole true rig. They define the products of the generators, and we extend the definition to arbitrary elements, i.e., sums of generators, by distributivity.
The coefficients in the fusion rules, called fusion coefficients, are subject to several constraints, because the multiplication operation is required to be associative and commutative with as the unit element. Specifically, to ensure that , we must have
where is the Kronecker delta. To ensure that , we must have
To ensure the associativity equation , we must have
(The two sides of this equation are simply the coefficients of in the two sides of the associativity equation.) It is well known that the associativity, commutativity, and unit laws, which we have ensured for the generators by means of these constraints on the fusion coefficients, imply the corresponding laws for all elements, because products of arbitrary elements were defined from the products of generators via distributivity.
To summarize this description of the true rigs associated to unitary fusion rigs: Such a true rig is completely specified by a positive integer and a system of fusion coefficients subject to the constraints (1), (2), and (3). The additive structure is a free monoid on , and the multiplicative structure is given by the fusion rules and distributivity.
4.2. Witness Frame
We now turn from the true rigs to the full structure of unitary fusion rigs, i.e., to the raw elements and witnesses. Of course, these must be defined in a way that produces true algebras of the sort described above.
For the rest of this section, we work with a fixed set of fusion rules, and we use the notation as above for the fusion coefficients. In particular, we have a fixed value for , the number of generators other than .
The raw elements are, by definition, all of the -tuples of finite-dimensional complex Hilbert spaces
each with a specified orthonormal basis, such that the elements of each are the formal linear combinations (over ) of basis elements. When we speak of basis elements, we always mean elements of the specified bases.
The witnesses for the equality of two raw elements, , are all of the -tuples of Hilbert-space isomorphisms (i.e., unitary transformations) between the corresponding components of and ,
It is not required that the unitary transformations respect the specified bases. When they do, i.e., when each is induced by a bijection between the specified bases of and of , we call a basic witness.
The reflexivity witnesses are just -tuples of identity maps. Composition and inversion of witnesses are done componentwise.
Raw elements are added by forming the componentwise direct sum of the Hilbert spaces. In more detail, the component of has as its specified basis the disjoint union of the specified bases of the components and of and , respectively.
Before proceeding further, we need to add some details about the additive structure just defined. There are several ways to formally define direct sums of vector spaces. The different ways produce isomorphic results, so it usually doesn’t matter which way one chooses. In our situation, though, our witnesses are themselves (tuples of) isomorphisms, and when we need to manipulate these witnesses, it will not do to say that things are well-defined up to isomorphism. We therefore describe a few ways to formalize direct sums and, afterward, indicate a notation that will be convenient for the rest of our work.
(1) The most common construction of the direct sum of two vector spaces and
is the set of ordered pairswith and . Addition and scalar multiplication are defined componentwise.
Theoretically, this works well, but it becomes awkward in some situations that we shall have to consider. Notice that, in direct sums of three vector spaces we shall have elements of the form in and elements of the form in . Mathematicians frequently ignore the distinction, because of the obvious isomorphism, but in our situation, the obvious isomorphism is part of the structure we are defining, so it cannot simply be swept under the rug. Of course, with more than three summands, we would have an even greater proliferation of parenthesis patterns, making it more difficult to see the structures involved.
One could also introduce sums of the form (without parentheses) as a vector space of ordered triples , and, depending on one’s set-theoretic conventions, such a triple might or might not be considered the same as one (but not both) of and . Similarly for direct sums of more vector spaces. When we consider multiplication of our raw elements, we shall need to deal with direct sums of many vector spaces at a time, with no natural parenthesization of the summands, nor even a really natural ordering. Representing elements of the direct sum by tuples (or by tuples of tuples of …) becomes increasingly arbitrary and awkward.
(2) Another way to construct direct sums like is to begin with the specified bases for and for and to take the disjoint union of these bases as a basis for ; the other elements of are then formal linear combinations of these basis elements.
An immediate difficulty here concerns the notion of disjoint union: What if the two bases are not disjoint? (As an extreme example, we might have with the same specified basis.) Fortunately, there is an easy solution, namely to tag the elements of our bases, so that the disjoint union consists of elements and with and in the bases of and of , respectively. The tag notation can be extended to non-basis vectors from and . Given a vector , the corresponding vector in is obtained by expanding as a linear combination of basis vectors and replacing each of those basis vectors by . We call the resulting vector . Similarly, if , the corresponding vector in is called .
This approach works well for a direct sum of many spaces, even if these are given as an indexed family without any particular ordering. We can take the basis elements of to be tagged elements of the specified bases of the ’s, i.e., ordered pairs with and . This observation will be useful because such naturally indexed but not naturally ordered direct sums will occur in our discussion of multiplication of raw elements. Incidentally, note that, if we imposed some arbitrary ordering on the indices and then used the traditional approach (1) above, then what we have written as here would be an -tuple with one component equal to and all the other components equal to 0; the location of the would encode (in effect, is written in unary notation).
(3) There is a way to attach tags to vectors, as in (2), without the need for specified bases. In (2), we did this as syntactic sugar, writing , when is a (possibly non-basis) vector and is a tag, for something constructed out of tagged basis vectors. Without resorting to syntactic sugar, we can achieve the same goal as follows. Select some 1-dimensional Hibert spaces (copies of ), one space for each tag that we might want to use, and let be a fixed unit vector in (the copy in that space of ). Then define the direct sum to consist of formal sums of vectors from and . In general, consists of formal sums of vectors from the spaces . Because is one-dimensional and spanned by , every vector in has the form for a unique . A fairly common simplification of the notation for tensor products would write as just , which brings us back to almost the same notation as in (2).
(4) Having introduced witness algebra as a generalization of universal algebra, we mention another viewpoint that we hope will appeal to universal algebraists. In each of the preceding three approaches, a vector from a summand appears in as with additional information that indicates the value of . In (1), the additional information is the location of the component amid many other components (and parentheses) in a tuple (of tuples …); in (2) it is the tag in ; and in (3) it is the tag in . We can view any such arrangement of tags as an operation (in the sense of universal algebra) applied to . Nesting of tags becomes composition of operations. This more abstract point of view allows considerably more freedom in tagging. We have not yet had need for this freedom, but it may prove useful in further studies.
For the purposes of this paper, we shall use the tag notation as in (2), including the use of tags with non-basis vectors. It will do no harm if the reader views as syntactic sugar for the of (3), nor will it do harm if the reader views the tagging operation as in (4).
In view of the definition of the addition operation on raw witnesses, it is clear that the raw element consisting of zero-dimensional Hilbert spaces serves as the additive identity element.
Addition of witnesses is defined componentwise in the obvious way. That is, if and , then has as its component the isomorphism given by
This completes our description of how addition works in our unitary fusion rig. Notice that this structure depends on our fixed fusion rules only through , the number of non-1 generators. The fusion coefficients will affect only the multiplicative structure.
Notice also the following property of witness addition, which will be generalized later and will be useful in the verification of several of the requirements for witness algebras.
Tag Invariance (preliminary form): When the sum of witnesses acts on a vector, it leaves the tags unchanged and merely applies the summand witnesses in the unique reasonable way.
Let us check that addition as defined here for raw elements produces the desired additive structure in the true algebra. Two raw elements are equal in the true algebra if and only if they are componentwise isomorphic, which means just that corresponding components have the same dimension. The elements of the true algebra thus correspond bijectively to -tuples of dimensions, natural numbers , and thus to the formal sums
that we want as elements of the true algebra. This correspondence can be formalized by defining, for each , the raw element to have (with specified basis ) in component and zero in all other components. Then every raw element is equal (i.e., componentwise isomorphic) to a unique sum of these ’s. The equivalence classes in the true algebra of the ’s serve as the additive generators of the true algebra, and addition of raw elements corresponds to the formal addition used in our earlier description of the true rig.
Addition essentially works on (specified) bases; the Hilbert space structure (linear structure and inner product) just comes along for the ride. By this we mean two things. First, the specified bases in are built purely set-theoretically (no linear combinations involved) from the specified bases in and . Second, if and happen to be basic witnesses (recall that this means they respect specified bases), then is also basic. Everything we have done so far would continue to work if raw elements were -tuples of finite sets rather than Hilbert spaces.
The product of raw elements and has in its component the direct sum of copies of the tensor product for all and .
Quite generally, when forming the tensor product of two Hilbert spaces and with specified bases, we let the specified basis of consist of ordered pairs where and range over the specified bases of and of , respectively.
In our present situation, when forming a direct sum of many such tensor products, we must adjoin tags to identify the various components of the sum. In accordance with Convention 32, we write a typical basis element of the component of in the form
where and are basis elements from the components of and of , and where . Here the tags and serve to identify the tensor product in which is a basis element, and the last tag serves to tell which of the copies of this tensor product our basis element is in. It will often be convenient in calculations (though not technically required) to add a subscript indicating which component of this basis element is in; thus, instead of , we may write .
Recall that we defined to be the raw element consisting of in component and in all other components. Let us calculate the product of two of these raw elements, say . For any , the component of this product, as described above, is the direct sum of numerous tensor products, but, because of the many 0 components in and , many of these tensor products will be 0. Indeed, the only non-zero summands occur when and , and those summands are . So the direct sum of these tensor products will be the direct sum of one-dimensional spaces. Since this happens for every , we see that the product has the same dimensions, in all components, as . It follows that the true elements represented by the raw elements satisfy the given fusion rules. Thus, our definition of mutiplication of raw elements produces the correct true algebra.
In particular, we can define the element 1 of our unitary fusion rig to be , and this serves as a multiplicative identity element in the true algebra.
We define the multiplication of witnesses so that the principle of Tag Invariance applies to them, just as for addition. That is, if and then is the -tuple whose component sends to
We repeat the principle of Tag Invariance, now in its final form, including both addition and multiplication.
Tag Invariance: When the sum or product of witnesses acts on a vector, it leaves the tags unchanged and merely applies the summand and factor witnesses in the unique reasonable way.
This completes the definition of a witness algebra. To make it into a braid semiring, we must specify the associativity, commutativity, and unit witnesses for both addition and multiplication; specify the distributivity witnesses; verify that these rig witnesses respect witnessed equalities (Section 3.1); and verify the coherence conditions (Sections 3.3 and 3.4).
4.5. Additive Rig Witnesses
In this subsection, we specify witnesses for the associative, commutative, and identity laws of addition.
For the associative law, , let us consider what happens in one component, say the , and let us omit, for brevity, the subscripts .
We begin by observing that the standard basis vectors for have three possible forms. Basis elements from look like or , and they provide basis elements and in . In addition, provides basis vectors . Similarly, we see that the standard basis vectors for are of the three forms , , and . Now we can define the associativity isomorphism in the obvious way:
Restoring the subscripts that we omitted for brevity earlier, we should write this as . It is the component of the associativity witness for .
Before proceeding to other witnesses for addition, we point out an important property, which will also hold for many — but not all — of the rig witnesses to be introduced later. It concerns what happens to tags (the 0’s and 1’s in our present situation) and the generic elements of specified bases (the in our present situation).
Tag Manipulation: All rig witnesses, with the exception of the multiplicative associativity and commutativity witnesses and , merely manipulate tags, leaving vectors from the given Hilbert spaces unchanged. These manipulations of the tags do not depend on the particular vectors from the given Hilbert spaces.
The exceptional witnesses and are what makes unitary fusion rigs interesting and useful for quantum computation.
The two principles of Tag Invariance (for sums of witnesses) and Tag Manipulation (for ) together imply that respects witnessed equality in the sense explained in Section 3.1. That is, if , , and then