Diagrams and GPTs for Quantum Gravity

This is a Perspective on "Any consistent coupling between classical gravity and quantum matter is fundamentally irreversible" by Thomas D. Galley, Flaminia Giacomini, and John H. Selby, published in Quantum 7, 1142 (2023).

By Andrea Di Biagio (Institute for Quantum Optics and Quantum Information (IQOQI) Vienna, Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna and Basic Research Community for Physics e.V., Mariannenstraße 89, Leipzig, Germany).

What’s new?

In their new article [1], Thomas Galley, Flaminia Giacomini, and John Selby, formalise and prove, within the framework of Generalised Probabilistic Theories (GPT), that if quantum matter is allowed to affect the gravitational field, then either the gravitational field isn’t classical or the interaction is irreversible. Additionally, in the presence of superselection sectors, only the classical information is allowed to flow into the classical system.

This result may not surprise some. This is how one would formulate such a hypothesis over a beer: Say that a variable’s evolution depends on the value of another variable, and suppose that this variable is in a quantum superposition. What is the classical variable supposed to do?

    • It does not budge (since the quantum variable does not have a value)
    • It also goes into a superposition (then it is not a classical variable)
    • Collapses the quantum variable, then moves (irreversible)
    • Breaks quantum mechanics in some other way.

In fact, several arguments to this effect are already present in the literature. Some notable examples include Eppley and Hannah’s 1977 paper [2], which argued that the interaction of classical gravitational waves with quantum matter would allow for the violation of the Heisenberg uncertainty principle (thus rendering matter not-that-quantum); Penrose’s argument for why classical spacetime would induce the collapse of a superposition of a massive quantum system [3, chapter 30]; and Feynman’s argument for why detecting the gravitational deflection of an object by another mass that is verifiably in a superposition of positions would be evidence that the gravitational field itself was in a quantum superposition [4, chapter 23].

Indeed, several models of classical gravity sourced by quantum matter seem to foster this hypothesis. Some have the irreversibility baked in, be it in the form of an objective collapse mechanism in the Diósi-Penrose gravitational collapse model [5,6], or a stochastic differential equation for the non-classical matter and the classical gravitational field in Jonathan Oppenheim’s “postquantum” model [7]. In theories with a classical field with deterministic evolution, such as Schrödinger-Newton [8] and semiclassical gravity [9,10], the non-linearity of the dynamics allows one to distinguish non-orthonormal vectors in the matter Hilbert space, effectively classicising it [11].

So, what is the point of Galley, Giacomini, and Selby’s article? In short: it provides a no-go theorem—with a clear and concise proof—within an established meta-theoretic framework, thus establishing the result once and for all for a whole swathe of possible theories, be they classical, quantum, or more exotic. Therefore, this is a milestone result in the field of quantum gravity, restricting what kind of theories one should look for.

Diagrams strike again

One peculiarity of this work, especially seen from the quantum gravity community’s perspective, is the use of GPTs and diagrammatic calculus for the proof, and in the next sections I will briefly comment on these notions. Before that, I would like to point out that this is not the first time the trio used these tools to formalise and generalise an argument about quantum gravity.

In 2017, Sougato Bose and collaborators [12] as well as Marletto and Vedral [13] argued that if two masses in a superposition of locations could become entangled as a result of the gravitational interaction alone (gravity-mediated entanglement, or GME), this would be evidence for the non-classical nature of the gravitational interaction. The argument in [12] was based on the well-known result that in quantum mechanics, local operations and classical communication (LOCC) does not allow one to increase entanglement between two systems, while the argument in [13] is based on the meta-theoretic framework of constructor theory (the full argument is in [14]). Galley, Giacomini, and Selby [15] later provided another theory-independent argument, this time in GPTs, with a straightforward diagrammatic proof.

The thesis advanced in all these arguments is that a local interaction with a classical system cannot mediate the creation of entanglement. One of the upshots of the theory-independent proofs is the clarification of the locality concept involved in these no-go theorems. It becomes clear that the relevant notion of locality is not the relativistic concept related to spacetime. Instead, it is a form of locality based on the concept of subsystems within the GPT framework. Gravity being local, here, means that a) there is a subsystem assigned to gravity and that b) this system acts as a mediator, meaning that the evolution $E$ of the three system can be written as sequential product of evolutions $E_i$ such that no $E_i$ takes both masses as input. It remains to be seen whether this notion of locality is relevant in field theories; see [16] for a discussion.

GPTs

Generalised Probabilistic Theories are a well-established framework in the study of quantum foundations and quantum information theory, but are not as well known to the wider community. While the articles [15,1] cover the essentials to understand the proofs, more comprehensive introductions can be found in [17,18,19], for example.

The core idea behind the GPT approach is that the least a scientific theory can do is give probabilistic predictions about what can happen in the lab. A theory can do much more, of course: it can give predictions for phenomena outside the lab, it can offer useful pictures both for intuition and computation, and may provide an ontology for the physics. However, arguably, it would not be a good scientific theory if it could not provide testable predictions for things you can do in the laboratory. GPT is a meta-theoretic framework that studies what different theories can predict within laboratory setups.

The main elements of a GPT are states, transformations, and effects. States are to be thought of as equivalence classes of experimental preparations for a given system, while effects should be thought of as different kinds of measurements that can be performed on it. Probabilities are obtained by composing states with effects. Transformations relate states to other states and as such contain information about the structure of the state space and characterise the operations that can be done on the system between measurements.

GPTs have been used to study which theories could exist beyond quantum theory. One of the first insights that came out of studying GPTs was that things that we thought were counterintuitive about quantum mechanics are actually common properties of GPTs [20,19]. For example, only classical probability theory features non-disturbing measurements, and the property that a mixed state can be written as a mixture of maximum-information states in a unique way. Indeed, one of the great successes of GPT has been its use in understanding precisely what is unique and special about quantum mechanics, in the space of possible GPTs [18,21,22,23,17], the aim of the programme of reconstructing quantum mechanics from operational axioms (or diagrammatic axioms, in the case of [24]). GPTs are also used in designing theory-independent cryptographic protocols.

The scope of GPTs, though, is not unlimited, as they cannot be applied, at the moment, in situations with indefinite causal structure [25], or with superobservers such as extended Wigner’s Friend scenarios [26,27]. Another serious limitation, as the authors of [1] point out, is that the relationship between a specific theory and its GPT is not always straightforward. This is the case, as mentioned, for Schrödinger-Newton, where the ostensibly quantum matter gets mapped to a classical GPT. In other cases, the GPT is unknown, like for the Reginatto-Hall classical-quantum hybrid model [28].

It also is worth mentioning that GPTs are but one of several partially overlapping meta-theoretic frameworks, including Operational Probabilistic Theories (OPTs) [29,30], process theories [31], the positive formalism [32] and constructor theory [14,33], which are used for similar purposes by different sub-communities within quantum foundations and quantum information theory.

Diagrams

An exciting aspect of GPTs is the existence of an expressive, faithful, and complete diagrammatic calculus for them. The use of diagrams to work with linear maps has been pioneered by Roger Penrose [3, chapter 12] who used it to keep track of abstract indices in tensorial equations, but this kind of diagrams has found a killer app with GPTs, OPTs, and process theories.

The basic elements of a GPT diagram are boxes with wires used to represent linear maps. The wires are associated with domain and codomain of the maps, just like abstract indices. Joining wires means sequential composition ($\circ$), and parallel wires represent parallel composition ($\otimes$).
An example of why this is useful: Let $f_1,f_2:V\rightarrow V$ and $g_1,g_2:W\rightarrow W$ be four linear maps. Then one can prove the following, seemingly nontrivial equation
\begin{equation}
(f_2\circ f_1)\otimes(g_2\circ g_1)=(f_2\otimes g_2)\circ(f_1\otimes g_2).
\end{equation}
On the other hand, if one were to build the diagrams corresponding to the left-hand side and to the right-hand side, one obtains the single diagram:

So, when working with diagrams, one need not worry about distributing sequential and parallel composition, it happens naturally. Similarly, formulas involving several operations on different systems appear more readable. Consider for example the quantum operation
\begin{equation}
(\mathbb{I}_A\otimes\langle\phi|)V_{AB}(U_A\otimes\mathbb{I}_A),
\end{equation}
versus the corresponding diagram:

To perform derivations with diagrams, besides thinking of the boxes as being able to slide along the wires, one can use previously established equalities between diagrams to substitute one part of a diagram with another. The diagrammatic calculus, with specific diagrammatic manipulation rules, has been shown to be faithful and complete, in the sense that an equality established via diagrammatic reasoning is true if and only if it can be established using standard formulas. This result is one of the great successes of category theory applied to physics; see [31] for historical notes on this development.

The proofs in [15,1] are prime examples of such derivations. For example, one of the key steps in [1] is to establish that the existence of a reversible interaction with a classical system implies the existence of a non-disturbance measurement. A classical system $C$ is one that allows for the existence of non-disturbing measurements, which are maps $m:C\rightarrow C\otimes C’$, where $C’$ is another classical system, that satisfy

where the “earth” symbol represents the discard operation, which models the effect of ignoring, losing, or destroying a system—the equivalent of the partial trace in quantum mechanics. A line with no box represents the identity, thus, the previous formula means that discarding the register system $C’$ after doing the measurement $m$ is like not doing anything to the system $C$ at all. The existence of a reversible interaction $R$ of a possibly non-classical system $S$ and the classical system $C$ can be used to define a non-disturbing measurement for $S$:

where $p$ is some normalised state for the classical system. Here is the diagrammatic derivation to show that this is indeed a non-disturbing measurement for $S$:

where we used, in order, the definition of $\tilde{m}$, the defining property of a non-disturbing measurement for $m$, the fact that $R$ is reversible, and the fact that $p$ is normalised, which means that discarding it returns the empty diagram: the number $1$.

I invite the reader to have a look at the rest of the proof [1] and compare it, at glance, with the equivalent algebraic proof (both are in the appendix). Without knowing the meaning of the symbols, the diagrammatic proofs certainly speak a little more clearly.

While they are not a silver bullet—inevitably, part of the computations and proofs have to be done in the standard formalism—they are a useful tool, and it is likely that they are here to stay. Coecke and Kissinger [31] make a strong sell of diagrammatic reasoning as a high-level language for quantum theory and other process theories, while providing a thorough (if lengthy!) introduction. Personally, I have taken a lot of value from diagrammatic reasoning. As a student, it drastically accelerated my understanding of entanglement and quantum algorithms. As a researcher, I use diagrams in an exploratory way, as a guide to reasoning, and to make sure my algebraic formulas make sense. They can be designed and typeset in using the free to use package TikZiT.

Closing words

The work by Galley, Giacomini, and Selby is a significant contribution to the field of quantum gravity. By formalising and proving a long-held idea within the meta-theoretic GPT framework, it provides solid ground for future research. It is also an example for a practical application of diagrammatic reasoning.

Part of the impact of their work will depend on the extent to which it is possible to find the correct GPT for proposed concrete models, as it would widen its range of applicability. Conversely, knowing the GPT corresponding to one’s model allows one to extract qualitative predictions from it by applying the theorems. For example, while it might be hard to conclude whether one’s theory allows for the creation of gravity-mediated entanglement, it can be demonstrated that the corresponding GPT features a classical mediator. Then, by using the theorem in [15], one learns that no entanglement is created. Thus, this mapping between model and GPT is an interesting future direction of research supported by these articles.

By applying tools and insights from quantum foundations and quantum information theory to quantum gravity, this work underscores the value of interdisciplinary approaches. It is a testament to the potential of interdisciplinary projects like QISS, ISRQI, and It from qubit aiming to bridge the gap between the communities working in thee different fields.

► BibTeX data

► References

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