First of all, it is important to note that the new regulation is coming in two separate pieces: the (draft)_implementing_regulation is the main document, which is in turn referenced by the (draft)_delegated_regulation, which specifically targets the design of UAS hardware that may be used within the ‘Open’ category (more below). It also attempts to standardise a number of aspects relevant to third-country operation.
Both documents have already passed the public consultation phase, meaning that experts and lobbyists from diverse interest groups have been able to give extensive feedback. The next step is the EU Commission’s approval, expected for February 28th, 2019. Let’s see what’s in store.
Zoning is a key principle of EU UAS regulations and played an important part in securing the required cross-national support due to their granting of considerable regulatory freedom to individual member states.
Each member state are free to determine an arbitrary number of geographical flight zones. Each of these zones may come with a number of operational restrictions that can be applied for various reasons. For example, UAS operations may be restricted in environmental zones. Safety and/or security considerations might also prohibit UAS operation entirely, e.g. above crowded city centres, gliding fields, international airports, embassies and other government sites, and zones of military interest such as border zones and test sites.
In accordance with EU guidelines, member states will have to ensure that zoning maps are freely downloadable in digital format and kept up-to-date.
EU cross-border operations are automatically permitted as long as zones on both sides are unrestricted. Otherwise, authorisation has to be sought with the respective National Aviation Authorities (NAAs).
All UAS activity will be classified into three operational categories. If you are a hobbyist, then you will most likely fall into the least restricted, or ‘open’, category. This category is mostly categorised by drone weight, as well as whether operation is above, close to or far away from people. Independently of weight, all drones in the ‘open’ category may not more than 120m above ground and can only be operated in unrestricted flight zones (see below).
Understandably, the EU commission has put effort into preventing toy drones from being overregulated. Hence, drones < 250g do not have to be registered, can be operated by anyone and may even fly above crowds. The same applies to all drones < 900g, as long as the operator is age 16 or above and has completed a brief online drone course.
Any drone > 900g will have to be registered.
For drones >= 900g but < 4kg, operation is not permitted within ca. 30m of people (see Draft Delegated Annex). Note that this may effectively prevent the use of drones >900g in urban areas. Even in non-urban areas, operators are required to pass both an online course and a theoretical exam at a certified test centre.
Still heavier drones < 25kg may only fly at least ca. 150m from people, effectively preventing use of drones >4kg within urban areas. Operation will require taking an online training course.
CE markings are obligatory for all drone hardware within the ‘Open’ category, and these will place different technical capability requirements for drones operating within different subcategories.
Note that even though EU directives only require insurance for drones > 20kg, national laws usually demand taking insurance for all ‘Open’ drone operations.
Operation under increased risk falls within the ‘specified’ category. This requires obtaining authorisation by your National Aviation Authority (NAA) based on a Specific Operation Risk Assessment (SORA) ****.
To simplify the task of creating a SORA, prefabricated risk assessments covering a range of standard operational scenarios will be made available. If your drone operations fall within such a standard scenario, creating a SORA might be as easy as filling in a simple form**.
If your operations do not fall under a standard scenario, a full SORA needs to be carried out. This first requires identification of two specific metrics summarising both Ground Risk Class and Air Risk Class respectively.
These risk-related numbers can be improved on through the use of strategic mitigation measures. For example, flying at night might circumvent traffic-related zone restrictions.
The resulting final residual Ground Risk Class and residual Air Risk Class are then combined into a single Specific Assurance and Integrity Level (SAIL) number using a simple lookup-table.
Note: If the resulting SAIL is larger than 7, it is understood that the amount of risk indicated cannot be effectively handled by ‘Specified’ operation and approval within the ‘Certified’ category has to be sought.
Otherwise, Operational Safety Objectives (OSOs) apply progressively depending on the SAIL number. These are specific requirements related technical features (such as communication abilities), fail safe features, human error and adverse weather conditions. If the relevant OSOs are met, authorisation is normally granted***.
Operators that regularly conduct operations within the ‘Specified’ category can attain a Light UAS Operator Certificate (LUC). A LUC grants certain privileges that simplify the process of gaining authorisation for ‘Specified’ operations that do not fall within standard scenarios.
One risk aspect that is difficult to control are accidental changes in either or both ground and air risk – for example, if a drone accidentally enters a restricted zone. In practice, this particular risk is usually managed by the use of (physical) buffer zones. However, feasibility often demand that such buffers are insufficiently large to accommodate for a full range escape. Additional regulation may address this issue in the future.
UAS operation falls within the ‘certified’ category if the risk is roughly comparable to or exceeding the risks associated with manned aviation.
UAS operation falls within the ‘Certified’ category if the risk is roughly comparable to or exceeding the risks associated with manned aviation. Unlike the first two operations categories, EU regulation for ‘Certified’ operation is as of yet largely unspecified and will likely be adopted in multiple stages.
In any case, ‘Certified’ operation is expected to require adequate certification of both UAS hardware and control software and, unless operation is entirely autonomous, will require the drone pilot to be adequately licensed.
Currently, authorisation for ‘Certified’ operation needs to be sought in line with National Aviation Authority (NAA) regulation. In addition, the European Union Aviation Safety Authority (EASA) accepts applications in its present remit. Note that certain hardware systems may require independent approval from additional authorities (e.g. Datalink, Detect and Avoid systems etc).
The current EU stance on drone flight beyond visual line of sight, i.e. the drone operator does not stay in visual contact with the drone, is that the neither communications nor control technology necessary for such automation are mature enough yet to be regulated. This is reflected also in SORA methodology, which puts tight constraints on B-VLOS. B-VLOS operations are therefore not going to be enabled by EU regulation in the near future.
The EU Commission is expected to approve the two EU UAS guideline drafts on February 28th. This could mean that regulation could be adopted as early as June 2019. This would mean that regulation may come into force from June 2020.
However, it is expected that the enforcement of some regulation may be delayed up until June 2021. For example, member states might require ample time in order to decide on matters related to Zoning areas and making sure public access to relevant information is routinely warranted under EU guidelines.
Market regulation is expected to be subject to a three year transition period until June 2022 to allow manufacturers, such as e.g. DJI, to adapt to the new regulatory environment. This means that during the transition period, older drones may still be operated even if they do not conform with new EU UAS regulations.
From a high-level perspective, current US regulation may be thought of entailing similar operations categories as the upcoming EU UAS regulation drafts. Roughly speaking, the main difference is that US equivalents of both ‘open’ and ‘certified’ categories are increasingly protruding into the ‘specified’ sector equivalent.
As for B-VLOS, the US currently offers standardised training and testing procedures to achieve the status of Remote Pilot in Command. With this certification, which currently has no EU equivalent, interested parties may conduct limited B-VLOS operations. Recent B-VLOS permissions granted by the US Department of Transportation to a select number of commercial enterprises, however, indicate a trend of progressive relaxation of B-VLOS requirements for business purposes, such as delivery or surveillance.
In the future, it is conceivable that future regulation on both sides of the Atlantic related to B-VLOS might ultimately draw from experience gathered during past military operations. For illustration, in Iraq in 2008, US military drone operators maintained (largely) safe inter-operation with manned operations by using separate designated runways for UAS under B-VLOS and personally conducting routine ATC radio communications.
Many important aspects of UAS operation are understood to fall outside the scope of EU UAS guidelines. These include:
Overall, it seems as if the new set of EU UAS guidelines will achieve the EU Commission’s goal of creating a set of coherent regulations crucial to UAS operations. As they cannot be easily restricted by individual member states, it is to be expected that the new guidelines will have an enabling effect.
However, significant gaps related to the ‘Certified’ category and B-VLOS operation still need to be closed and some minor issues in the remaining two categories still need to be addressed. This is of particular importance in order to keep up with fast-paced advances in US regulation.
*unless the UK undergoes a Hard Brexit, it will very certainly strive to stay part of the single European sky airspace – even if exiting with a deal. Hence EU UAS regulation will most likely apply to UK drone operators despite of Brexit. In case of a Hard Brexit,
** Note that there is currently still disagreement among member states as to whether NAAs are required to individually assess applications or whether authorisation is granted by default. While the former is favoured by the UK, other countries such as Denmark highlight the impracticability of this approach and suggest to rely on occasional routine checks instead.
*** Note that the guidelines do not explicitly require risk assessments to be conducted under SORA. However, no other risk assessment methodology has so far been approved.
**** Official SORA v2.0 documentation had not been publicly released by the time of writing
I (Christian Schroeder de Witt) am Technical and Outreach Officer at Oxford Drone Society. Disclaimer: I am not a lawyer, and any information given below may be wrong or incomplete. Please double-check for yourself (sources provided)! If you spot any errors, please contact me.
This article is heavily informed by an excellent presentation held by Klavs Anderson, Chief Flight Inspector for the Danish Transport, Construction and Housing Authority and contributor to the JARUS guidelines on Specific Operations Risk Assessment (SORA).
]]>Definition 1. [1] The category FdHilb consists of a symmetric monoidal category (SMC) with finite-dimensional complex Hilbert spaces as objects and linear transformations as arrows. Arrow composition is provided by ordinary matrix multiplication. The monoidal structure is provided by the ordinary matrix tensor product $\otimes$.
Remember how we complained right at the beginning that one reason for Dirac notation being clumsy because it allows for meaningless global phases? Well, FdHilb currently has exactly the same issue. Let’s fix this!
Definition 2. [2] The category $\mathbf{FdHilb}_{wp}$ has the same objects and arrows as FdHilb, however linear maps are subject to the equivalence condition $f\equiv g$ iff there exists $\theta\in\mathbb{R}$ such that $f=e^{i\theta}g$.
So what have we achieved so far? Let’s pick some simple object within $\mathbf{FdHilb}_{wp}$ – for example, the well-known Qubit of type $\mathbb{C}^2$. But what can we do with this? The answer is: Not that much. We can juxtapose multiple Qubits into composite systems, and define composite linear transformations to get from one composite system to another. All of this, we can of course do either formally, using equations, or graphically, using the graphical notation introduced for SMCs.
So how can we get to do more exciting things? Let’s look into the nature of Hilbert spaces and augment $\mathbf{FdHilb}_{wp}$ with appropriate inner structure, starting with general properties and then gradually moving on to finer structure.
What again distinguishes a Hilbert space from an Euclidean space? We faintly remember from our undergrad times that Hilbert spaces admit a scalar product, that, unlike in Euclidean spaces, has to be positive-definite complex. This additional constraint gives rise to the concept of a dual space, which we propose to capture as follows:
Definition 3. [3] A compact closed category is a symmetric monoidal category (SMC) where each object $A$ is assigned a dual object $A^*$ together with a unit map $\eta_A:I\rightarrow A^*\otimes A$ and a counit map $\epsilon_A:A\otimes A^*\rightarrow I$, such that \(\\lambda^{-1}\_A\\circ(\\epsilon\_A\\otimes A)\\circ\\alpha^{-1}\_{A,A^\*,A}\\circ(A\\otimes\\eta\_A)\\circ\\rho\_A=id\_{A}\) and \(\\rho^{-1}\_A\\circ(A^\*\\otimes\\epsilon\_A)\\circ\\alpha\_{A^\*,A,A^\*}\\circ(\\eta\_A\\otimes A^\*)\\circ\\lambda\_A=id\_{A^\*}\)
The consistency conditions on $\eta_A$ and $\epsilon_A$ maybe looking slightly outlandish, but if you have a closer look (or draw them as a little diagram), you will realize that they are just constraining $\eta$ and $\epsilon$ to be appropriate primal/dual space preparators and destructors.
What other properties make Hilbert spaces special? After having introduced structure reflecting positive-definite complex scalar products, we now need to impose an important property on morphisms: unitarity.
Definition 4. [4] A $\mathbf{\dagger}$-symmetric monoidal category ($\dagger$-SMC) is a symmetric monoidal category equipped with an identity-on-objects contravariant endofunctor $(-)^\dagger: \mathbf{C}^{op}\rightarrow\mathbf{C}$, which assigns to each morphism $f:A\rightarrow B$ and adjoint morphism $f^\dagger:B\rightarrow A$, which coherently preserves the monoidal structure, i.e.: $(f\circ g)^\dagger=g^\dagger\circ f^\dagger, \ (f\otimes g)^\dagger=f^\dagger\otimes g^\dagger,\ 1^\dagger_A=1_A$ and $f^{\dagger\dagger}=f$. Further, for the natural isomorphisms $\lambda, \rho,\alpha$ and $\sigma$ of the symmetric monoidal structure, the adjoint and the inverse coincide.
Combining both unitarity for morphism and a positive-definite complex scalar products:
Definition 13. [5] A $\mathbf{\dagger}$-compact closed category is a $\mathbf{\dagger}$-symmetric monoidal category that is also, and such that the following diagram commutes:
$\dagger$-compact closed categories equally admit a diagrammatic calculus:
And, to represent primal and dual spaces and the associated scalar product (we enclose the equivalent Dirac bra-ket notation):
Congratulations, now we have captured the fundamental properties of Hilbert spaces.
The following important theorem allows custom yanking and bending of wires in $\dagger$-compact categories:
Theorem 2. [5, 6, 7] An equation in the symbolic language of a $\dagger$-compact category follows from the axioms of $\dagger$-compact categories if and only if it holds up to isotopy in the graphical language.
The symbolic language of $\dagger$-compact categories allows for insightful representations of quantum phenomena, such as quantum teleportation and entanglement swapping. We will not discuss these examples at this stage, but we will get back to them later.
Taking a step back, we realize that we have not yet considered a quantum mechanical process of fundamental importance: measurement.
[1] Coecke, B., and Paquette, E. O. Categories for the practising physicist. arXiv e-print 0905.3010, May 2009.
[2] Duncan, R., and Perdrix, S. Graph states and the necessity of Euler decomposition. arXiv e-print 0902.0500, Feb. 2009.
[3] Selinger, P. Dagger compact closed categories and completely positive maps. Electronic Notes in Theoretical Computer Science 170 (2007) 139-163.
[4] Coecke, B., and Duncan, R. Interacting quantum observables: Categorical algebra and diagrammatics. arXiv e-print 0906.4725, June 2009. New J. Phys. 13 (2011) 043016
[5] G. M. Kelly and M. L. Laplaza (1980) Coherence for compact closed categories. Journal of Pure and Applied Algebra 19, 193–213.
[6] P. Selinger (2005) Dagger compact closed categories and completely positive maps. Electronic Notes in Theoretical Computer Science 170, 139–163. www.mathstat.dal.ca/∼selinger/papers.html#dagger
[7] P. Selinger (2010) Autonomous categories in which $A\simeq A^*$. Extended abstract. In: Proceedings of the 7th International Workshop on Quantum Physics and Logic, May 29-30, Oxford. www.mscs.dal.ca/∼selinger/papers.html#halftwist
[8] Abramsky, S., and Coecke, B. A categorical semantics of quantum
protocols. arXiv e-print quant-ph/0402130, Feb. 2004. Proceedings of the 19th IEEE conference on Logic in Computer Science (LiCS’04). IEEE Computer Science Press (2004).
The properties of these compositions vary across different physical theories: Maxwell’s equations of classical electrodynamics, for example, respect the superposition principle of charges and currents, which is captured by scalar and vector addition (which are an instantiation of the direct sum $\oplus$ known from abstract algebra). Similarly, classical mechanics obeys a principle of superposition of forces, again based on the direct sum.
In contrast, quantum mechanics uses the direct sum as an internal composition for the superposition of quantum states within a given system. The external composition operation used to tie different quantum mechanical systems together is, however, the tensor product $\otimes$.
One might thus assume that the major difference between classical mechanics and quantum mechanics is their different compositional nature. However, it turns out that the respective categorification of classical mechanics, namely ClassMech, gives rise to a notion of compositionality that is more tensor product-like than direct sum-like. It has thus been postulated that classical mechanics might admit some information-theoretic properties that are traditionally only ascribed to quantum mechanics, such as e.g. a no-cloning theorem (albeit in a different sense). Unfortunately, the precise nature of the compositionality of ClassMech is not yet sufficiently well understood and therefore it is currently not trivial to compare the compositional properties of quantum mechanics and classical mechanics directly [2].
In any case, we conclude that compositionality seems a fundamental characteristic of any physical theory [1]. We will now see that symmetric monoidal categories offer just the right structure to capture compositionality at an abstract level.
Definition 1. A monoidal category $(\mathbf{C},\otimes,I)$ is a category $\mathbf{C}$ equipped with a bifunctor $-\otimes -:\mathbf{C}\times\mathbf{C}\rightarrow\mathbf{C}$, a distinguished unit object $I$, natural unit isomorphisms $\lambda_A:A\simeq I\otimes A$ and $\rho_A:A\simeq A\otimes I$, and a natural associativity isomorphism $\alpha_{A,B,C}: A\otimes(B\otimes C)\simeq (A\otimes B)\otimes C$, which are subject to certain coherence equations that we omit here for brevity.
A monoidal category naturally seems like a good start in our quest to identify a proto-physical categorical structure because compositionality can be expressed in two different ways: Either through composition of morphisms (which lends itself to a form of causal compositionality), or, alternatively, using the natural associativity isomorphism (which lends itself naturally to composition of subsystems).
However, monoidal categories are still too general to describe a physical theory: For one, is there really any reason for why $A$, $A\otimes I$, $I\otimes A$ etc are distinguishable objects? If we regard $I$ as an ’empty subsystem’, i.e. $I$ is indistinguishable from nothing, then any object juxtaposed with $I$ should be indistinguishable from the object itself. This means that juxtaposition with $I$ should be reduced to a purely semantic meaning that is solely required in order to ensure consistency of juxtaposition operations. The following constraint on monoidal categories achieves this task:
Definition 2. [3] A monoidal category is called strict when $\lambda, \rho, \alpha$ are all identities, i.e. the objects made isomorphic are in fact equal to each other. Every monoidal category is equivalent to a strict monoidal category.
For another, it really should not matter we regard system $A$ to be juxtaposed with system $B$, or system $B$ juxtaposed with system $A$: For physical systems, both operations must refer to the same composite system. It may seem as if this issue might be solved by simply requiring $A\otimes B=B\otimes A$. It can be easily seen, however, would violate types: Imagine we transformed from system $A\otimes B$ to system $D\otimes E$ by using process $f$ to get from $A$ to $D$ and process $g$ to get from $B$ to $E$. We can write this as a composite process \(A\\otimes B\\stackrel{f\\otimes g}{\\rightarrow}\\rightarrow D\\otimes E\)
If now $A\otimes B=B\otimes A$, then we cannot distinguish anymore which process was applied to which system: \(A\\otimes B=B\\otimes A\\stackrel{f\\otimes g}{\\rightarrow}D\\otimes E=E\\otimes D\). In order to be able to apply composite morphisms to composite systems we therefore need to impose a slightly more subtle constraint:
Definition 8. A symmetric monoidal category (SMC) is a monoidal category equipped with a natural symmetry isomorphism $\sigma_{A,B}:A\otimes B\simeq B\otimes A$ such that $\sigma^{-1}_{A,B}=\sigma_{B,A}$, and again subject to some coherence conditions which we omit.
It has been conjectured that SMCs are the natural categorical description of any physical theory, i.e. the smallest common denominator [3].
A very neat feature of SMCs is that they admit a graphical calculus, which can be used to reason in a way that is as powerful as conventional formal reasoning:
Theorem 1. [3] A formal equational statement holds for SMCs if and only if it holds up to isomorphism in the graphical calculus. The graphical calculus is given by:
With SMCs, we now have succeeded in defining a structure that promises to capture any physical theory. What is more, we have also seen that this structure admits an elegant graphical calculus.
The obvious next question is: How do we adapt SMCs to a specific physical theory? The obvious answer is that we first have to identify objects and morphisms appropriately. We will see though that we can leverage SMCs even more by defining inner structure.
[1] Hardy, L. On the theory of composition in physics. arXiv e-print 1303.1537, Mar. 2013.
[2] Fenyes, A. Limitations on cloning in classical mechanics. arXiv e-print 1010.6103, Oct. 2010.
[3] Coecke, B., and Paquette, E. O. Categories for the practising physicist. arXiv e-print 0905.3010, May 2009
]]>This is not entirely a new idea: physicists use Noether’s theorem to understand conservation laws by studying the actions that respect them. They also use Feynman diagrams, which provide exactly the same types of process abstraction [1].
In short, a category consists of objects and morphisms between them. Morphisms may be composed by the associative operation $\circ$, as long as types are preserved. For example, morphisms $f : A \rightarrow B$ and $g : B \rightarrow C$ may be composed to form a morphism $g \circ f : A \rightarrow C$. Furthermore, each object $A$ gives rise to an identity morphism $1_A$. But let’s better use the necessary mathematical rigour:
Definition 1. [3] A category $C$ consists of:
a family $ | C | $ of objects, |
a set of morphisms $C(a, b)$ for each pair of objects $a, b \in | C | $ |
for any object $a\in | C | $, there exists a morphism $1_a\in C(a,a)$ called identity, such that for any $f\in C(a,b)$ we have $f=f\circ 1_a=1_b\circ f$ (existence of identity) |
Categories were first introduced by Samuel Eisenberg and Saunders MacLane as a byproduct of their research on natural transformations (which we will encounter later)
As we will see over time, arrows and morphisms within categories may themselves admit rich inter-relationships, which, for notational simplicity, are usually expressed as commutative diagrams:
Definition 1b. [3] For any category $\mathcal{C}$, if one forgets that the arrows in $\mathcal{C}$ may be composed into new arrows and additionally, if one discards of all identity arrows, then one ends up with a category just defining a graph. Such graphs may be used in order to efficiently notate inter-relationships between the remaining arrows and morphisms as so-called commutative diagrams.
For example, given $f:a\rightarrow b$, $g:b\rightarrow c$, $k:a\rightarrow c$ and $l:c\rightarrow d$ and, additionally, $g\circ f=l\circ k$ then we may represent this equivalently as the easily digestable commutative diagram:
As categories become more complex, commutative diagrams will prove to be extremely useful notational tools.
But what kind of mathematical entities may serve as objects and morphisms? In principle, anything goes – as long as Definition 1 is fulfilled. For example, many familiar mathematical structures from physics and computer science may be categorified:
Definition 2. [2] A monoid is a triple $(m, \bullet, 1_\bullet)$ with $m$ being a set and $\bullet$ is an associative multiplication operation with identity $1_\bullet\in m$.
In other words, a monoid is a single object that serves as both target and source of all arrows and, together with a binary operation, determines the category $\textbf{C(m)}$ of a monoid $\mathbf{m}$.
Monoids are very useful structural building blocks. In fact, it can be shown that every one-object category is a monoid. A particularly interesting type of monoid is one where each arrow admits an inverse: Such monoids exactly correspond to the widely-used concept of a group.
Moving to many-object categories, the category of finite sets and relations (FRel) consists of finite sets as object and relations between sets as arrows. We will see later how FRel is intimately related to fundamental arithmetic operations such as addition and multiplication.
Another particularly useful many-object category is FdVect, which is an important step toward a category over Hilbert spaces that we will encounter later:
Definition 3. [2] The category $\mathbf{FdVect}_\mathbb{K}$ admits the following:
Category Theory is so abstract that it is sometimes referred to as “generalized abstract nonsense”. However, we will see that it indeed underpins extremely powerful structure!
At first it may seem as if the concept of a category is so abstract and versatile that virtually anything seems to correspond to a category – thus rendering its concept arbitrary, and ultimately, useless.
This is, however, not the case. For simplicity, we here only give a brief example of a structure that does not admit a straight-forward categorification:
Consider a mathematical entity labelled Empire that admits both objects and morphisms. Take its objects to be planar maps marking the territories of simply-connected countries. Let the existence of a morphism $f:a\rightarrow b$ signify that country $a$ borders on country $b$. Now postulate that Empire is in fact a category. Can you spot the problem?
Exactly! Unless all countries border on each other, Empire does not define a category, as the attribute “bordering on” is intransitive. This makes arrow composition ill-defined. At this point, let this example suffice to illustrate that the concept of a category is abstract, but not arbitrary.
As stated above, the inventors of categories, Eisenberg and MacLane, were not primarily interested in them as such. Rather, categories first arose as a necessary building block in order to define higher-level structures dubbed natural transformations. It is very insightful and I strongly recommend reading their original paper [3], which gives a great introduction to natural transformations from within the realms of linear algebra. Let us now see ourselves what the whole fuss is about. First, however, we need to define some more lower-level structure.
Functors are structure-preserving maps between categories:
Definition 4. [3] For categories $\mathcal{C}$ and $\mathcal{D}$, a functor $F: \mathcal{C}\rightarrow\mathcal{D}$ is a pair of functions $(F_0, F_1)$ for which
The category Cat has small categories (i.e. objects and arrows constitute sets [3]) as objects and functors as morphisms. If we restrict Cat to one-object categories, then we end up in a category that is isomorphic to Mon, whose arrows are monoid homomorphisms.
Definition 5. [3] A homomorphism is a function $f:s\rightarrow t$ with the property that if $x\alpha y$ in $s$ then $f(x)\beta f(y)$ in $t$, where $(s,\alpha)$ and $(t,\beta)$ are sets with relations on them. Monoid homomorphisms additionally preserve the identity element [4].
Note the analogy of Definition 5 to monoid categories $\textbf{C(m)}$: Cat is merely living one abstraction level further up.
Now let’s fix two categories $\mathcal{C}$, $\mathcal{D}$ and look at the family of functors $\mathcal{F}$ between them. One may form a new category with those functors as objects, and the arrows between these objects being so-called natural transformations:
Definition 6. [3] Consider two categories $\mathcal{C}$, $\mathcal{D}$ and the family of functors $\mathcal{F}$ between them. For $\mathcal{F}$, $\mathcal{G}\in\mathcal{F}$, a natural transformation $\eta: F \rightarrow G$ associates to each object $X$ in $\mathcal{C}$ a morphism $\eta_X: F(X) \rightarrow G(X)$ between objects of $\mathcal{D}$, such that for every morphism $f: X\rightarrow Y$ the following commutative diagram holds:
It is of course possible to continue by defining morphisms between natural transformations and so on. This actually leads to interesting insights: Fundamental algebraic operations, like subtraction and addition, may be seen as having resulted from an infinite categorification of set-theoretical operations like unions and disjoints [1]. However, the knowledge of functors and natural transformations suffices for many practical applications of category theory, including the purposes of Categorical Quantum Mechanics.
How can a physical theory be cast into the language of category theory? There are many possibilities. An intuitive one is to construct a category Phys, whose objects are physical states and morphisms are processes between such states. This is an example of an abstract category which can be concretized in order to reflect the intrinsics of a specific physical theory:
For example, quantum mechanics implies objects to be Hilbert spaces and morphisms to be linear maps. Classical mechanics suggests a category ClassMech with Poisson manifolds as objects and Poisson maps as morphisms [23].
However, there is more to a physical theory than just a basic categorical structure – as we shall see next.
[1] Baez, J. C., and Dolan, J. From finite sets to Feynman diagrams. Featured in ’Mathematics Unlimited – 2001 and Beyond’, vol. 1, eds. Bjoern Engquist and Wilfried Schmid, Springer, Berlin, 2001, pp. 29-50.
[2] Coecke, B., and Paquette, E. O. Categories for the practising physicist. arXiv e-print 0905.3010, May 2009.
[3] Samuel Eilenberg and Saunders MacLane, General theory of natural equivalences, Trans. Amer. Math. Soc. 58 (1945), 231-294. PDF here.
[4] Barr, M., and Wells, C. Category Theory for Computing Science.
Prentice Hall International Series in Computer Science, 1990.
For one, the use of Hilbert spaces results in structural overkill: Every physical system \(\Psi\) corresponds to an infinite number of unit vector representations \(e^{i\theta}|\Psi\rangle,\ \theta\in\mathbb{R}\). For another, the true clumsiness of the Dirac-von Neumann approach lies in its failure to adequately capture the compositionality of the theory.
To illustrate our claims, consider the following example which is taken from a quantum secret sharing protocol [1, p.99]:
It needs to be shown that this expression is proportional to \(|0\rangle+e^{i\alpha}|1\rangle\). This can be done brute-force, using the matrix representations of the individual gates. The above circuit involves the manipulation of four distinct physical \(2^n\) qubits, which are the smallest informational unit in the realm of quantum mechanics. Matrix size scales as \(2^n\), meaning that \(4\) qubits result in $16\times 16$ matrices, with roughly $n^3=4096$ scalar multiplications performed per matrix multiplication.
Performing these calculations densely on a classical computer won’t work beyond roughly a dozen qubits. Exploiting the problem’s underlying sparsity is usually difficult due to its non-trivial structure. Many recent important quantum computation models, such as measurement-based quantum computation [2], depend on thousands or millions of entangled qubits.
But even if tractable, brute-force calculations yield little physical insight. If one wishes to understand the physical operation of the described system better, one may resort to circuit representation:
How does one arrive at this circuit diagram? Each qubit corresponds to a distinct line. The tensor product $\otimes$ is represented by juxtaposition. Unitary evolution, corresponding to $\circ$, is represented by sequential composition. Two-qubit gates, such as CNot, are depicted by connecting the lines of the respective qubits, which intuitively indicates that there is some information flow between control and gate. The power of this pictorial representation lies in abstracting away the well-known interchange identity \((g\\circ f)\\otimes(g’\\circ f’) = (g\\otimes g’)\\circ(f\\otimes f’)\) with $f,f’,g,g’$ being linear maps, which allows to trace individual qubits through the system. Provided that only standard gates are being employed and the network isn’t too dense, this method can be used to avoid the complexity of matrix manipulations.
However, questions remain: Firstly, can the map \(\\text{Dirac-von Neumann term}\\mapsto \\text{circuit diagram}\) be expressed in a mathematically rigorous fashion? Secondly, there are some obvious rewrites that can be directly performed on the circuit diagram, for example, $\text{CNot}_{12} \circ \text{CNot}_{12} := \text{Id}_{12}$ corresponding to:
Are there further opportunities for graphical manipulation? Apart from being intrinsically suited to human abilities, this could open new avenues for efficient automatized manipulation beyond physically-unaware sparse-matrix methods.
In their seminal paper [3], Abramsky and Coecke developed a rigorous mathematical framework for translating Dirac-von Neumann terms into circuit diagrams based on category theory. The study of interactions between complementary observables lead to a simple, yet powerful, set of graphical rewrite rules, dubbed the ZX-calculus [4].
Returning to the above example, the ZX-calculus allows to convert the circuit diagram to the following form:
A short series of rewrites transforms this diagram to the expected form:
Categorical quantum mechanics doesn’t stop there. The mathematical framework is sufficiently abstract to accommodate a wide range of graphical calculi. Instead of interactions between complementary qubits, one can study the interactions of three-partite entangled states. This results in the graphical GHZ/W-calculus [5]. The ZX-calculus diagram from above may now be represented as follows:
A different approach may be taken to handle so-called mixed states, which correspond to classical superpositions of quantum states: Selinger’s CPM construction allows the graphical representation of density matrices and completely positive maps using the ZX-calculus [6].
For example, suppose one would like to apply a conditional $X$-gate to the above example state with probability $\frac{1}{2}$, leaving the state intact otherwise. In von Neumann-formalism, the resulting density matrix is obtained by treating both scenarios individually. In Selinger’s CPM construction, this corresponds to the following graphical manipulation:
, where the resultant diagram corresponds to the maximally mixed state $|0\rangle\langle 0| + |1\rangle\langle 1|$. This may not immediately seem like a clear advantage. Measurement correlations of and conditional gates on $n$-qubit entangled states, however, in the worst case result in $2^n$ distinct scenarios to be considered. The CPM construction comes with clear and simple high-level rules that help to prune unnecessary complexity, whereas such strategies may be hard to come by using the Dirac-von Neumann notation.
Categorical Quantum Mechanics gives rise to more than just graphical calculi. It is a thoroughly principled framework that sheds light on the underlying compositionality of the physical world.
Yet again, Categorical quantum mechanics offers much more than just a rigorous framework for graphical calculi. It provides an abstraction away from Hilbert spaces based on information-theoretic axioms. Many characteristic features of quantum information processing, such as the no-cloning theorem [7], quantum teleportation [8], entanglement swapping [9], quantum steering [10] and gate teleportation [11], are trivially encompassed [12]. The formalism also seems ideal for the study of multi-partite entanglement[13].
One way to better understand the oddness of quantum mechanics is to compare and contrast quantum phenomena within a large space of generalised theories. Categorical quantum mechanics helped to clarify issues around one particularly popular such theory, namely Spekken’s toy model [14]. This in turn may have deeper interpretational consequences [15].
Can categorical quantum mechanics help coming up with a theory of quantum gravity information processing?
But why is it so important to thoroughly understand the information-theoretic properties of quantum mechanics? Information is always physical, and conversely, every physical theory gives rise to an associated information processing theory. It is clear that a successful theory of quantum gravity will need to have both quantum mechanics and general relativity as limiting cases. It follows that a theory of quantum gravity information processing needs to, at least in the limit, respect both a general relativistic information processing theory and quantum information processing. This latter issue has received a lot of attention recently: Tentative theories of quantum gravity information processing have been suggested [16], and the interaction between motion under spatial curvature and quantum entanglement has been studied experimentally [17].
At the same time, there is an ongoing debate questioning the role of string theory in physics [18][19][20]. Undoubtedly, more than 30 years of research in string theory have yielded plenty of exciting mathematics [21], however, whether or not string theory will ever result in a useful formulation of physical reality seems as of yet undecidable. Even if string theory one day turns out to offer a concise description of physical reality, alternative routes may deepen our understanding of it.
We need more research into Categorical Quantum Mechanics!
Despite the unfortunate case that infinite dimensional Hilbert spaces have so far resisted full categorical treatment[31], the framework of Categorical Quantum Mechanics does hold some promise to yield insight into more fundamental questions. A particularly interesting feature is that its finite dimensional formalism seems to be well-suited to relativistic extensions [15].
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