text stringlengths 128 2.05k |
|---|
Because the effects are required to respect the monoidal structure of [MATH] , the CPM category is itself a symmetric monoidal category with tensor product [MATH] extended as follows to arbitrary morphisms [MATH] and [MATH] |
[EQUATION] This makes [MATH] an environment structure for [MATH] . Furthermore, the specific choice [MATH] satisfies the following additional requirement, which makes [MATH] a dagger-compact category: |
[EQUATION] The condition above is in fact equivalent to requiring closure of the family under conjugation: [EQUATION] In this work, we will keep the additional requirement above explicit, and not include it as part of the definition of environment structure |
2.3 A symmetry perspective Previous literature on the CPM construction has focussed mostly on the connection with positive operators and completely positive maps |
in presence of compact closed structure. States [MATH] in [MATH] correspond to positive morphisms in [MATH] [EQUATION] where [MATH] is some morphism and [MATH] is some object. Morphisms in [MATH] then correspond to super-operators in [MATH] , sending positive morphisms to positive morphisms: |
[EQUATION] where [MATH] is some morphism and [MATH] is some object. In this work, we will instead take a “symmetry” perspective on the CPM construction. To begin with, we observe that the following defines a group homomorphism [MATH] from the finite abelian group [MATH] to the group of monoidal automorphisms [MATH] (i.... |
[EQUATION] where [MATH] is the identity functor and [MATH] is the conjugation functor. The doubling functor can then be re-cast as follows, in terms of the group homomorphism [MATH] |
[EQUATION] The actual choice of ordering for the tensor product is essentially irrelevant, as any two choices will lead to functors which are naturally isomorphic via conjugation by a permutation of the objects. |
Out of all these natural permutation isomorphisms, we will in particular be interested in the ones corresponding to the regular action of [MATH] on the indices, i.e the natural transformations [MATH] for all [MATH] |
[EQUATION] Using them, we can see that the autofunctors [MATH] on [MATH] are all naturally isomorphic, via conjugation by the permutations [MATH] , to the identity functor: |
[EQUATION] This means that the morphisms in [MATH] are essentially invariant under the [MATH] -action given by the autofunctors [MATH] |
In order to extend this invariance to the morphisms in [MATH] , we need to make the following assumption on the discarding maps: |
[EQUATION] Note that this assumption is different from the additional assumption [MATH] , but is equally satisfied by the traditional choice [MATH] . In terms of the autofunctors and natural transformation above, this means that: |
[EQUATION] As a consequence, we get that the autofunctors [MATH] on [MATH] are all naturally isomorphic, via conjugation by the permutations [MATH] , to the identity functor (using the fact that the generic morphism in [MATH] takes the form [MATH] for some [MATH] in [MATH] ): |
[EQUATION] Restricted to the monoid [MATH] of scalars for [MATH] , the autofunctors [MATH] define a group homomorphism from [MATH] to the group of monoid automorphisms of [MATH] . The invariance argument above shows that the scalars of [MATH] and [MATH] always fall within the sub-set of elements of [MATH] which are lef... |
as a dagger symmetric monoidal category, so that the conjugation functor is automatically linear: this means that [MATH] is naturally a commutative semiring [MATH] with involution , and the action of [MATH] coincides with the action of the involution. If [MATH] is a field and conjugation is non-trivial, we can define [... |
[EQUATION] For example, in the case of [MATH] we have [MATH] and is complex conjugation, so that [MATH] is the quadratic extension [MATH] . If [MATH] is any test made of product effects and [MATH] is a product state on [MATH] which is normalised |
, then the above yields the Born rule for the ( [MATH] -valued) probabilities of test outcomes: [EQUATION] In the [MATH] case of [MATH] we recover the familiar form [MATH] |
The higher-order CPM construction 3.1 The folded category Consider a symmetric monoidal category [MATH] . Let [MATH] be a finite abelian group, and [MATH] be a group homomorphism from [MATH] to the group of monoidal automorphisms [MATH] . The following definition generalises the construction of the doubled category fro... |
Definition 1 The [MATH] -folding functor is the endofunctor [MATH] on [MATH] defined as follows: [EQUATION] From now on, we will require that [MATH] be chosen in such a way that the folding functor [MATH] is injective on objects, i.e. that [MATH] implies [MATH] for all [MATH] . The following is then well-defined. |
Definition 2 The [MATH] -folded category [MATH] is the image of the [MATH] -folding functor. The folding functor is also a functor [MATH] which is bijective on objects. |
Lemma 3 The [MATH] -folded category is a symmetric monoidal category [MATH] , with tensor product [MATH] defined as follows: [EQUATION] |
The folding functor [MATH] is a monoidal functor under this choice of monoidal structure. The choice of ordering for the tensor product is essentially irrelevant—as was the case for the ordinary doubling construction—since all possible choices lead to folding functors which are naturally isomorphic via conjugation by p... |
[EQUATION] For example, for [MATH] we would have the following natural isomorphisms: [EQUATION] By using the natural isomorphisms [MATH] , we can see that the monoidal autofunctors [MATH] on [MATH] are all naturally isomorphic, via conjugation by the permutations [MATH] , to the identity functor: |
[EQUATION] This means that the morphisms in [MATH] are essentially invariant under the [MATH] -action given by the monoidal autofunctors [MATH] . If [MATH] is the commutative monoid of scalars for [MATH] , then [MATH] restricts to an action of [MATH] on [MATH] by monoid isomorphisms, and we can consider the sub-monoid ... |
3.2 The higher-order CPM construction In the previous section, we have seen that an environment structure for the CPM category can be defined by choosing a family of effects—the discarding maps—which respect the monoidal structure of the doubled category. Here we will be interested in the more general context where we ... |
Before moving on to do so, recall that the tensor product of maps in the CPM category involved a permutation in order to obtain a domain in the correct form. For the traditional second-order case, this only involved a swap. To deal succinctly with the more general case, we define the following natural isomorphism [MATH... |
[EQUATION] For example, for [MATH] we would have the following isomorphism: [EQUATION] In particular, the symmetric monoidal structure of the [MATH] -folded category can be expressed in terms of [MATH] . If [MATH] and [MATH] are morphisms in [MATH] , then we have that: |
[EQUATION] Definition 4 A multi-environment structure for [MATH] is a family [MATH] of sets [MATH] of effects on [MATH] in [MATH] which satisfies the following three conditions: |
(i) for all [MATH] and all [MATH] we have that [MATH] (ii) we have that [MATH] (iii) for all [MATH] and all [MATH] we have that [MATH] |
In particular, an environment structure for [MATH] is a multi-environment structure where each set [MATH] contains exactly one element, which we denote by [MATH] |
The multi-environment structures for a fixed [MATH] can be partially ordered by object-wise subset inclusion. The partial order is in fact a lattice, with meet given by object-wise set intersection and join given by suitable closure of object-wise set union. |
Definition 5 Given a multi-environment structure [MATH] , the [MATH] -CPM category , which we denote by [MATH] , is defined to be the smallest sub-category of [MATH] which contains [MATH] as well as all maps in the following form: |
[EQUATION] for all pairs of objects [MATH] and all effects [MATH] in the multi-environment structure. Lemma 6 We can extend the tensor product [MATH] of [MATH] as follows to turn [MATH] into a symmetric monoidal category, having [MATH] as a monoidal subcategory: |
[EQUATION] where [MATH] and [MATH] are generic morphisms in [MATH] We refer to the SMC constructed above as a higher-order CPM construction . By analogy to the traditional CPM construction, it is easy to see that the morphisms of [MATH] can always be put into the following normal form—by sliding the multi-environment e... |
[EQUATION] As a consequence of condition (iii) for the multi-environment structure [MATH] , we get that the autofunctors [MATH] on [MATH] are all naturally isomorphic, via conjugation by the permutations [MATH] , to the identity functor: |
[EQUATION] 3.3 Functoriality of the higher-order CPM construction In order to understand the functorial and iterative properties of the higher-order CPM construction, we introduce the notion of a “universe of symmetric monoidal structures”. |
Definition 7 Let [MATH] be the category of (suitably small) symmetric monoidal categories and monoidal functors between them. By a SMC-universe we mean a category [MATH] equipped with a faithful functor [MATH] with image [MATH] forming a sub-category of [MATH] . We refer to [MATH] as the underlying SMC functor |
SMC-universes are essentially categories where the objects are symmetric monoidal categories and the morphisms are monoidal functors between them, where the categories can be thought to have been equipped with some additional information, and the morphisms between them restricted based upon that information. Some inter... |
the category [MATH] of symmetric monoidal categories and monoidal functors (equipped with the identity [MATH] ); the category [MATH] of dagger symmetric monoidal categories and dagger monoidal functors (equipped with the sub-category inclusion [MATH] into [MATH] ); |
the category [MATH] of dagger-compact categories with chosen duals and dagger monoidal functors preserving chosen duals (equipped again with the sub-category inclusion into [MATH] ). |
The reason to introduce the additional layer of abstraction given by the underlying SMC functor, rather than simply considering sub-categories of [MATH] as in the examples above, can be summarised as follows: in order to characterise the higher-order CPM construction as a functor, we need to equip symmetric monoidal ca... |
Definition 8 By a morphism of SMC-universes [MATH] we mean a functor [MATH] together with a natural transformation [MATH] . We denote the category of SMC-universes and morphisms between them by [MATH] |
One interesting example of morphism between SMC universes is given by the traditional CPM construction, which we can write as [MATH] in the following way: |
the functor [MATH] is [MATH] , sending a dagger-compact category [MATH] to [MATH] and a dagger monoidal functor [MATH] to its restriction [MATH] (recalling that [MATH] is defined as a sub-category of [MATH] in this work); |
the natural transformation [MATH] is given by the doubling functor [MATH] , where we have used the fact that [MATH] is a sub-category of [MATH] , so that we have [MATH] and [MATH] |
Definition 9 Let [MATH] be a SMC-universe. Then the category [MATH] is defined as follows. (i) The objects of [MATH] are in the form [MATH] , where; |
[MATH] is an object in [MATH] [MATH] is a group homomorphism from a finite abelian group [MATH] to the automorphisms [MATH] [MATH] is a multi-environment structure for [MATH] |
the [MATH] -CPM category [MATH] is an object of [MATH] the [MATH] -folding functor [MATH] is a morphism of [MATH] the [MATH] -folding functor [MATH] is injective on objects; |
(ii) The morphisms [MATH] in [MATH] , where [MATH] and [MATH] are both actions for the same finite abelian group [MATH] , are the morphisms [MATH] in [MATH] which satisfy the following: |
[MATH] is [MATH] -equivariant , in the sense that for all [MATH] we have: [EQUATION] [MATH] respects the multi-environment structure , in the sense that for all [MATH] we have: |
[EQUATION] If [MATH] and [MATH] are not actions for the same finite abelian group [MATH] , then there are no morphisms between [MATH] and [MATH] |
Lemma 10 There is a faithful and surjective functor [MATH] which sends [MATH] to [MATH] and is the identity on morphisms. The category [MATH] is an SMC-universe with underlying SMC functor [MATH] defined by [MATH] |
The definition of the SMC-universe [MATH] allows us to detail the functorial properties of the higher-order CPM construction in full generality. |
Lemma 11 Let [MATH] be a sub-category of [MATH] , seen as a SMC-universe where [MATH] is the sub-category inclusion (so that [MATH] ). Then the higher-order CPM construction can be used to define a morphism of SMC-universes [MATH] as follows: |
the functor [MATH] is the one sending an object [MATH] of [MATH] to the object [MATH] of [MATH] , and acting as the identity on morphisms; |
the natural transformation [MATH] is given by the [MATH] -folding functor [MATH] In particular, the result above shows that the higher-order CPM construction is functorial over monoidal functors which are [MATH] -equivariant and respect the multi-environment structure. |
3.4 The higher-order CPM construction as an Eilenberg-Moore algebra The traditional CPM construction can be iterated, but the combined result of multiple iterations is not itself a CPM construction: this is because traditional CPM construction is defined to be second-order (i.e. it corresponds to a [MATH] symmetry), wh... |
If we have two actions [MATH] and [MATH] which commute, i.e. which satisfy [MATH] for all [MATH] , then we can combine them into a new action [MATH] as follows: |
[EQUATION] This way, [MATH] can be taken to define a commutative monoid operation on the actions of finite abelian groups over a fixed object [MATH] of a fixed SMC-universe [MATH] , where the unit is the trivial action [MATH] given by the identity automorphism. |
If [MATH] is a multi-environment structure for [MATH] , then we can define a multi-environment structure [MATH] for [MATH] as [MATH] . Similarly, if [MATH] is a multi-environment structure for [MATH] then we can define a multi-environment structure [MATH] for [MATH] as [MATH] . Given both [MATH] and [MATH] , we use the... |
[EQUATION] We also define the trivial multi-environment structure [MATH] for [MATH] as follows: [EQUATION] This way, [MATH] can be taken to define a commutative monoid operation on multi-environment structures, compatible with the commutative monoid operation previously defined on the underlying actions. |
Theorem 12 The map [MATH] can be extended to an endofunctor of [MATH] by defining its action on morphisms [MATH] of [MATH] as follows: |
the functor [MATH] is given by: [EQUATION] where [MATH] is the group homomorphism [MATH] , and we define the multi-environment structure [MATH] |
the natural transformation [MATH] is given by [MATH] The endofunctor [MATH] is a monad with the following multiplication [MATH] and unit [MATH] |
the functor [MATH] for the multiplication [MATH] is given by: [EQUATION] the functor [MATH] for the unit [MATH] is given by: [EQUATION] |
the natural transformations for both the multiplication [MATH] and the unit [MATH] are identity functors: [EQUATION] If [MATH] is a sub-category of [MATH] , then [MATH] is an Eilenberg-Moore algebra for the monad [MATH] |
Examples 4.1 Iterated CPM construction The simplest example of higher-order CPM construction is given by iterating the traditional second-order CPM construction on a dagger-compact category [MATH] . At the first level, this means choosing the following monoidal [MATH] action [MATH] on [MATH] |
[EQUATION] together with the environment structure [MATH] given by the caps. The [MATH] -th iteration of the second-order construction is captured by the higher-order construction with [MATH] action [MATH] and associated multi-environment structure [MATH] . Explicitly, the group action [MATH] takes the following form: |
[EQUATION] Explicitly, the effects in the multi-environment structure [MATH] are [MATH] and all [MATH] [EQUATION] In particular, the double-dilation construction of Zwart and Coecke |
arises as the fourth-order CPM construction with [MATH] group action and effects in the multi-environment structure [EQUATION] A handy way of visualising the [MATH] symmetry of the iterated CPM construction theory is to imagine the objects in the tensor product [MATH] to be arranged on the vertices of an [MATH] -dimens... |
[EQUATION] 4.2 Categories of free finite-dimensional modules Iteration of the traditional CPM construction only yields higher-order examples corresponding [MATH] conjugating symmetries. In order to construct more interesting examples, we focus on a family of dagger-compact categories with much richer structure, namely ... |
, these categories have been shown to capture a number of well-studied toy models of quantum theory, including real quantum theory, hyperbolic quantum theory, modal quantum theories and the category fRel of finite sets and relations. |
We define [MATH] to have natural numbers [MATH] as objects, and the [MATH] -by- [MATH] [MATH] -valued matrices as morphisms [MATH] . The category has tensor product given by the Kronecker product of matrices, [MATH] -linear structure given by the [MATH] -linear structure of matrices and each object [MATH] comes with a ... |
A fairly standard way of making higher-order CPM constructions on [MATH] -Mat is to consider a homomorphism [MATH] from some finite abelian group [MATH] into the semiring automorphisms of [MATH] . The action [MATH] can then be defined as the identity [MATH] on objects and as follows on morphisms: |
[EQUATION] In particular, picking [MATH] and [MATH] will yield a [MATH] -folded category which is isomorphic to the one obtained from the traditional second-order CPM construction. |
Because all objects [MATH] can be uniquely decomposed (up to permutation) as a product of primes, a multi-environment structure [MATH] can be defined by taking sets [MATH] of effects for all primes [MATH] , and then closing them under tensor product [MATH] . As a special case, an environment structure generalising the ... |
[EQUATION] In the [MATH] case of double-dilation, we might consider replacing the [MATH] effects with the [MATH] effects defined above, and doing so results in the double-mixing construction. |
If [MATH] is any vector on [MATH] , then we define its norm [MATH] to be the following higher-order generalisation of the quadratic trace [MATH] for pure states in quantum theory: |
[EQUATION] Normalisation of a pure state [MATH] in the higher-order CPM category [MATH] has nothing to do with inner products, and is instead the same as having coordinates with norms [MATH] adding to 1. |
By using the traces [MATH] , it is not hard to show that the scalars in [MATH] are exactly the closure under addition of the subset [MATH] |
[EQUATION] As a consequence, the scalars of [MATH] form a sub-semiring [MATH] of [MATH] . The same trick can be used to show that arbitrary morphisms are closed under addition, so that [MATH] is enriched in [MATH] -modules. |
We now show that a categorical [MATH] -probabilistic theory can be constructed inside the Karoubi envelope for [MATH] : this allows one to study the natural interface between “quantum” systems in [MATH] and “classical” systems with a notion of non-determinism defined by the semiring [MATH] . This includes natural defin... |
. When [MATH] , the construction presented here reduces to the one originally detailed in On every object [MATH] of [MATH] , we can consider the copy map [MATH] for the special commutative [MATH] -Frobenius algebra [MATH] associated with the standard orthonormal basis. Combining these maps with the environment structur... |
[EQUATION] Lemma 13 The full subcategory of the Karoubi envelope for [MATH] spanned by objects in the form [MATH] is isomorphic to [MATH] , i.e. it behaves as the category of [MATH] -probabilistic classical systems. As a consequence, the full sub-SMC of the Karoubi envelope spanned by objects in the form [MATH] —the “q... |
In the categorical [MATH] -probabilistic theory defined above, a generic normalised quantum-to-classical process [MATH] —the generalisation of a POVM, if you will—is defined by a classically-indexed family [MATH] of effects [MATH] in [MATH] such that [MATH] . In the sharp case where [MATH] for some orthonormal family [... |
[EQUATION] The non-quadratic nature of the Born rule suggests that categorical [MATH] -probabilistic theories obtained using the higher-order CPM construction might display higher-order interference phenomena, and this indeed turns out to be the case: in recent work by |
, a variation on the double-mixing construction was used to construct a probabilistic theory of “density hypercubes”, displaying interference of order up to four and possessing hyper-decoherence maps. |
Conclusions and future work We have shown that the CPM construction can be generalised from the traditional [MATH] conjugating symmetry to arbitrary finite abelian group symmetries, in a completely functorial way. We have provided a categorical description of the closure of our higher-order CPM constructions under iter... |
We have constructed a broad family of semiring-based examples generalising the traditional second-order ones, and we have proved that they can all be studied using the operational framework of categorical probabilistic theories. As shown by recent work on higher-order interference and hyper-decoherence, these new examp... |
Finally, this work defines generalised CPM constructions in such a way that they are embedded into the original SMC, for mathematical ease of definitions and proofs. This leads to the technical requirement that the folding functor be injective on objects, which could be avoided by adopting a construction analogous to |
. Such a modification is conceptually simple but technically convoluted, and is left to future work. Appendix A Proofs Lemma 3 The [MATH] -folded category is a symmetric monoidal category [MATH] , with tensor product [MATH] defined as follows: |
[EQUATION] The folding functor [MATH] is a monoidal functor under this choice of monoidal structure. Proof. The proof is entirely straightforward. |
Lemma 6 We can extend the tensor product [MATH] of [MATH] as follows to turn [MATH] into a symmetric monoidal category, having [MATH] as a monoidal subcategory: |
[EQUATION] where [MATH] and [MATH] are generic morphisms in [MATH] Proof. note that [MATH] is defined by composing all maps from [MATH] , where [MATH] is already well-defined, with all maps in the following form, for all effects [MATH] in the multi-environment structure: |
[EQUATION] Note that by the way we have extended the definition of [MATH] from [MATH] to [MATH] , the maps above can be equivalently written in the following form: |
[EQUATION] By picking a suitable composite system for [MATH] and using the symmetry isomorphisms from [MATH] , we can place the effect [MATH] on any output of type [MATH] of any map in [MATH] . It is therefore enough to show that things work out in the case of [MATH] , for any choice of [MATH] and [MATH] . But this fol... |
[EQUATION] Remark One may wonder why we didn’t simply define [MATH] as [MATH] [MATH] and the effects in [MATH] . The reasons for this is technical: the product [MATH] is not defined on the entirety of [MATH] , so such a definition would be mathematically imprecise. The reader should however feel free to reason about [M... |
Remark One may also wonder why we had to consider maps in the form [MATH] , rather than going directly for [MATH] . This is because [MATH] had to be defined—for the reasons explained in the previous Remark—as a sub- category of [MATH] spanned by certain maps, and not as a sub- SMC of [MATH] . This means that the existe... |
Proof. The proof is entirely straightforward. Lemma 11 Let [MATH] be a sub-category of [MATH] , seen as a SMC-universe where [MATH] is the sub-category inclusion (so that [MATH] ). Then the higher-order CPM construction can be used to define a morphism of SMC-universes [MATH] as follows: |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.