# From diagrams to quantum theory

This is a Perspective on "Reconstructing quantum theory from diagrammatic postulates" by John H. Selby, Carlo Maria Scandolo, and Bob Coecke, published in Quantum 5, 445 (2021).

By John van de Wetering (Radboud University Nijmegen and Oxford University).

Quantum mechanics was developed over a number of decades, starting with ad-hoc solutions to specific problems before expanding into the full theories of Heisenberg’s matrix mechanics and Schrödinger’s wave mechanics. That these were just two sides of the same coin was realised by von Neumann who with his seminal book Mathematische Grundlagen der Quantenmechanik [1] wrote down what we now consider to be the mathematics of quantum theory: Hilbert spaces, complex numbers, self-adjoint operators, unitary dynamics, the Born rule, tensor products. This tremendous achievement immediately raised some deep questions though. Why does nature seem to run on Hilbert space? And why a complex one, and not, say, a real one or even a quaternionic one? Why the Born rule and not some other way to get probabilities out of vectors in a Hilbert space?

A general way to resolve these questions can be found by looking at the other major theory of physics: relativity. Its development was pretty much the antithesis to that of quantum mechanics. Instead of starting from ad-hoc solutions and progressing towards a general theory, Einstein started from a small set of physical principles, from which he saw which mathematics were needed (albeit with considerable effort). The physical evidence for this theory generally only came later. One can then wonder what it would have been like to have a quantum Einstein’ that also started from some set of physical principles and from that derived the mathematics and consequences of quantum theory. What would those principles be?

This question has been asked and (partially) answered by many people over the last hundred years. The early work in this area mostly focused on trying to find generalisations of quantum theory. Here it was felt that the seeming arbitrariness of complex Hilbert spaces must mean that there is some more general theory lying behind it. Seminal work in this area was the quantum logic approach of Birkhoff and von Neumann [2], where they focused on the order theoretic structure of the sharp observables of a quantum system, and the Jordan algebra approach of Jordan, von Neumann and Wigner, where they tried generalising the algebra of observables of a quantum system [3]. While both approaches found some space for a more general theory, one was mostly lead back to the theory of Hilbert spaces. Some other early results that showed that the mathematical content of quantum theory was perhaps less arbitrary than initially thought were Stone’s theorem on one-parameter unitary groups [4], which tells you that once you know your dynamics are unitary that then the Schrödinger equation is pretty much the only way in which you can continuously evolve a system; and Gleason’s theorem [5] that sheds light on the nature of the Born rule.

The need for a complete axiomatic reconstruction of quantum theory from a set of principles was outlined by Mackey in 1957 [6]. He wished to do this using the quantum logic approach (for an overview of work in this area of quantum logic, see [7]). Modern work in this field, which is commonly referred to as trying to reconstruct quantum theory, focuses on a more operational approach, where the act of state preparation, transformation and measurement are taken as fundamental. This approach considers probabilities and convex state spaces as given, and builds from there. While such approaches were considered earlier [8], they really took off after Hardy’s influential 2001 preprint [9]. This framework was formalised under the moniker of generalised probabilistic theories [10]. In the 20 years since Hardy’s preprint, many different reconstructions have appeared, each using a different set of principles (a selection in no particular order: [11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27]). This reveals a rather surprising perspective on quantum theory: there might not be a single right’ set of physical principles that pinpoints the mathematics of quantum theory. Instead there are many different perspectives, each of which leads to the same destination. For instance, some of these reconstructions are based on the information processing properties of physical systems [16,11,19,21,22], others on properties of entanglement [17,14], properties of pure processes [18,24], properties of sequential measurements [23,25] or on any other of a multitude of considerations [9,12,15,20].

The present paper offers another such perspective on quantum theory, one where states, processes and effects can all be treated on the same footing and all of quantum theory can be understood through the use of diagrams. This paper, Reconstructing quantum theory from diagrammatic postulates, by Selby, Scandolo and Coecke [28], is a result of the work done in categorical quantum mechanics over the last two decades.

Categorical Quantum Mechanics (CQM) was initiated by Abramsky and Coecke in 2004 [29]. It takes an abstract view of quantum theory where the fundamental building block is the process, which is commonly presented diagrammatically as a box with some input and output wires. Crucially, these processes can be composed both vertically, representing one process happening after one another, and horizontally, such that the processes happen in parallel (in terms of linear maps, these two types of composition correspond to the standard composition of linear maps, respectively the tensor product of linear maps). That such a two-dimensional representation is necessary to understand quantum theory should come as no surprise to anyone who has worked with quantum circuits, where you also have two axes of composition corresponding to the time direction and the space direction. Where CQM goes beyond the types of diagrams familiar from quantum circuits is that it assumes the existence of cups and caps. These are special types of processes that allow one to bend wires every which way you want, giving complete freedom to wire diagrams: making loops, connecting inputs of processes together, etc. (the technical name for this is that CQM deals with compact closed categories). In perhaps more familiar terms, the cup is the maximally entangled Bell state, and the cap the Bell effect. By putting cups on all the inputs of a process, we transform it into a state, which gives us the familiar Choi-Jamioªkowski isomorphism between states and processes. It is the cups and caps and the resulting process-state duality that allows one to treat states, effects, and general processes on the same footing.

Another one of the structures that is crucial to CQM is the dagger. The dagger allows one to vertically flip diagrams, swapping inputs and outputs in the process. Intuitively, the dagger allows you to flip the direction of time of the diagram. For instance, if we take the dagger of a state, then we get the effect that tests for that state. It is these ideas borrowed from CQM-diagrams, cups and caps, and a dagger-that form the backbone of the present paper. The other assumptions needed to reconstruct quantum theory are inspired by the work done on other reconstructions, specifically that of Chiribella, D’Ariano and Perinotti [16], an influential reconstruction that is also sometimes referred to as the Pavia reconstruction, after the location of the institution of the authors.

Like most other modern reconstructions, the present paper presupposes the framework of generalised probabilistic theories (GPTs): probabilities are given by real numbers; we can take probabilistic combinations of processes; and we assume that all state spaces are finite-dimensional. The paper does present a novel take on this though. Instead of viewing this classical interface’ as something external, the authors include it as part of their diagrams by allowing two different types of inputs and outputs: classical ones and general ones. For instance, while in a GPT a measurement is usually viewed as a list of effects, in this work a measurement is a process that has a quantum input and a classical output, neatly capturing what it actually means to do a measurement.

A principle that was assumed in the Pavia reconstruction [16] is that each state can be purified. That is, each mixed state arises as a pure state on some composite system where we do not have access to part of the system, commonly referred to as the environment’. In quantum theory we can purify not just states, but general processes, where such a purification is known as Stinespring dilation [30]. Interestingly, states on classical systems can generally not be purified (this is because a marginal of a classical pure state is also a pure state, so that we can never get a mixed state out of discarding an environment system). This led the authors of the present paper to introduce a different type of purification that they call symmetric purification. It turns out that any quantum, classical, or quantum-classical process has a symmetric purification. This is a valuable observation and might itself prove to be useful in a broader context, for instance allowing one to more easily give proofs of statements that apply equally well in the classical as in the quantum setting.

A final principle that the paper assumes, which it shares in common with many other reconstructions, is that the processes allow local tomography. This principle says that we can fully characterise a process acting on multiple systems by testing it with local processes (i.e. that are not entangled across the subsystems). Local tomography has so far proven itself to be one of the few ways in which we can distinguish quantum theory from real quantum theory, where the complex numbers are replaced by real numbers [31], and as such it is often used in the final steps of a reconstruction proof to distinguish these closely related possible theories.

The main result of the paper can then be summarised as follows: Quantum theory is the most general theory where

• we can compose processes both sequentially and in parallel,
• we can use cups and caps to switch inputs and outputs,
• we have a dagger to vertically flip diagrams and transform
states into tests,
• we have a classical interface to interact with the theory,
• this interface is finite-dimensional and locally tomographic,
• and where every process has a symmetric purification.

Here by quantum theory’ we mean that each system is represented by a (finite-dimensional) C*-algebra. A possibility that is allowed by these principles is that every system is classical, corresponding to the commutative C*-algebras. Hence, this reconstruction does not pinpoint’ anything inherently quantum, but instead establishes the similarities between the classical and the quantum. The authors identify some additional assumptions that can break this symmetry, so that the purely quantum systems can be identified. These assumptions can be summarised as `information gain always causes disturbance’ [32].

While categorical quantum mechanics started in a fairly abstract world where the goal was to say as much as possible about quantum theory with the fewest number of assumptions, with this reconstruction the authors have made a bridge between this abstract world and the concrete nature of quantum mechanics. It shows that once we have the basic CQM structure—cups, caps and a dagger—the concrete quantum world is not very far away.

### ► References

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