# What can we learn from trivial measurements?

This is a Perspective on "No-free-information principle in general probabilistic theories" by Teiko Heinosaari, Leevi Leppäjärvi, and Martin Plávala, published in Quantum 3, 157 (2019).

By Marius Krumm (Institute for Quantum Optics and Quantum Information (Vienna), Faculty of Physics, University of Vienna).

The discovery of quantum mechanics radically changed our understanding of the world. Today, trying to reveal the mysteries of quantum mechanics has become one of the most fruitful fields of modern science, both for applications and foundational understanding. A key strategy is to identify non-classical aspects of physical systems and investigate their consequences. However, when doing so, one easily misses that non-classicality does not necessarily imply that the system is actually behaving according to quantum mechanics. Indeed, there do exist important reasons to consider the possibility that quantum theory might not be the ultimate picture of the world. Among them, the most prominent one is the problem of combining quantum theory and general relativity (see e.g. [1]). Other recent results expand the Wigner’s friend thought-experiment [2] and show that the universal validity of quantum theory seems to be in contradiction to the classical perception of observers and consistency of the physical world [3,4].

However, if not classical or quantum physics then what else might be there? As an answer to this question the framework of General/Operational Probabilistic Theories (GPT/OPT) was invented [5,6,7]. In this framework one takes the point of view of an experimenter who is combining black boxes – devices with input settings and pointers that are only described by their input-output-statistics. The assumptions of the framework are very weak: Access to randomness and the possibility of the usual statistical analysis of lab experiments on the input-output-statistics. Therefore the framework manages to accommodate a large variety of physical theories. To single out one of them one explicitly uses physical or informational postulates about the inner-working of the devices or their statistics. This led to many derivations of quantum theory from physical or information-theoretical principles without relying on purely mathematical concepts such as Hilbert spaces, path integrals or Hermitian operators, e.g. [5,7,8,9]. Another line of research is to construct new theories that might eventually replace quantum theory – examples include theories with higher-order interference [10] or modifications of quantum theory for quantum gravity [1].

As an example for applications, this framework can be used as an approach to (semi-) device-independent-cryptography: Instead of assuming full quantum theory and a particular physical implementation of the devices, one treats the devices as black boxes and just adds a minimal list of physical assumptions to guarantee safety. For example, it turns out that the no-cloning principle is very generic among Operational Probabilistic Theories. Therefore, secure-money-schemes relying on the no-cloning principle can be extended to a wide class of these theories without needing most of the structure of quantum theory [11].

Investigations in the framework already lead to many surprises. In particular, features that were believed to be specifically quantum, e.g. entanglement, Bell-non-locality or the no-broadcasting property, can be found in many OPTs and are therefore only of limited use to figure out what makes quantum theory special among them (see e.g. [6]). Such surprises motivate us to re-evaluate our basic intuitions in OPTs, even those that refer to trivial objects.

The work by Teiko Heinosaari, Leevi Leppäjärvi, and Martin Plávala considers one such seemingly trivial object: Trivial measurements, i.e. measurements that are not able to reveal any information about the measured system. They start from a simple question: What measurements are trivial, i.e. useless? The deep mathematical analysis that the authors perform shows that it can be surprisingly difficult to answer this question.

To approach the question of trivial measurements from an operational point of view, the authors consider three principles about information gain in measurements.

One such principle is the no-free-information principle: Every observable that is compatible with all the other observables must be trivial. Or in other words: Measuring observables that reveal properties of the system prevents some other observables from being measured.

Another principle is the no-information-without-disturbance principle. This principle says that every measurement revealing new properties must lead to a disturbance of the state, i.e. useful measurements are always invasive.

The third principle is the no-broadcasting-principle: It is impossible to create perfect local copies of systems.

All these principles formalize the intuition that one cannot acquire information without unwanted side-effects, however it is not clear whether they are equivalent. The authors argue that the no-free-information principle implies the no-information-without-disturbance principle. Their argument is that if one could measure a non-trivial observable without disturbance, that non-trivial observable would be compatible with every other observable. Furthermore, the authors argue that the no-information-without-disturbance principle implies the no-broadcasting principle. They say if this was not true, one could use the broadcasting process to measure non-trivial observables on the copies.

Motivated by these principles, the authors define three classes of measurements. The first class is the set of measurements whose outcome is independent of the input state. Their outcome is generated by ignoring the physical system and simply using a random number generator. The second class of measurements is the set of measurements that can be implemented without disturbance. Here, without disturbance means that if one forgets the outcome, the state after the measurement is the same as before. The third class of measurements is given by observables that are compatible with every other observable. That means that for every other observable, there exists a fine-grained observable that reveals both of the observables.

The authors formalize these classes of measurements and show that the first is contained in the second, and the second is contained in the third, just as expected by their intuitive arguments. Furthermore, the authors mathematically analyse the properties of observables that can be measured without disturbance and of those observables that are compatible with all the others, providing several useful mathematical results. Some of these results explore the connection between observables compatible with all the others on one hand and simulation irreducible observables on the other hand.

A key insight of the paper is that the considered classes of measurements can be different. Therefore, the aforementioned principles are not necessarily equivalent. The authors provide a simple example of a theory in which the observables that can be measured without disturbance and the observables that are compatible with all others do not form the same set. But the authors also provide several theorems that can be used to determine whether the no-free-information principle and the no-information-without-disturbance principle hold in a theory, and find important connections to simulation-irreducible observables.

As an important class of examples, the authors investigate the polygon state spaces. They discover that the no-information-without-disturbance principle is true in all non-classical of these state spaces, while the validity of the no-free-information principle depends on whether the number of corners is even or odd. In particular, for the odd polygon state spaces, the authors also investigate the connection between the noise of an observable and an observable being compatible with every other observable.

So, in conclusion, what did we learn from the trivial measurements? Perhaps the most important lesson is that we should be careful about declaring things as trivial. When looking beyond classical and quantum theory, trivial objects can be difficult to recognize as such. While the authors analysed the statement that useful measurements come with a cost, one should wonder whether there exist other examples – examples of questions that restrict the structure of the physical world while only talking about trivialities at first glance. The authors have demonstrated that the answers to such questions can critically depend on their formalization.

On a more specific note, the authors have provided an approach to define and analyse different notions of trivial measurements, together with information-theoretical principles that connect these notions. There are several opportunities to expand this line of research: A fruitful approach might be to investigate the power of these principles for reconstructions of quantum theory or even new theories. One previous example that shows that such a direction can indeed be very fruitful is given by [12]: The authors of [12] analyse state spaces in which measurements without information-gain can be implemented without disturbance. They find that all such theories that are discrete (i.e. a polytope) must already be classical.

Another direction might be to quantify the violation of the principles or investigate the robustness of the results to noise: How is the validity of the principles affected if we assume that perfect measurements do not exist and allow the compatibility requirements to only hold within some accuracy bounds?

Furthermore, the authors found technical results relating the validity of the principles, simulation irreducible observables and noise – perhaps an interesting physical insight connecting these three seemingly different things is waiting to be found.

### ► References

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