The wisdom in the qubit

This is a Perspective on "Improving social welfare in non-cooperative games with different types of quantum resources" by Alastair A. Abbott, Mehdi Mhalla, and Pierre Pocreau, published in Quantum 8, 1376 (2024).

By Giannicola Scarpa (Escuela Técnica Superior de Ingeniería de Sistemas Informáticos, Universidad Politécnica de Madrid, 28031 Madrid, Spain).

Published:	2024-09-05, volume 8, page 82
Doi:	https://doi.org/10.22331/qv-2024-09-05-82
Citation:	Quantum Views 8, 82 (2024)

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Prologue

Inverness, 11th century. Macbeth sits in front of his quantum computer, his hands shaking, a drop of sweat falling on his forehead. He types.

$\textbf{Macbeth:}$ “What shall I do?”

$\textbf{Computer:}$ “Pursuing murder will lead to guilt and downfall.”

$\textbf{Macbeth:}$ “I’m destined for greatness! The witches told me so.”

$\textbf{Computer:}$ “They give classical advice. They tell you what you want to hear. I, instead, use entangled advice between you and King Duncan. He might be already strengthening his defences.”

$\textbf{Macbeth:}$ “Doesn’t this expose me?”

$\textbf{Computer:}$ “Quantum advice is no-signalling, he cannot tell what your intentions are. He trusts you. Choose honour over ambition, it will secure your legacy.”

$\textbf{Macbeth:}$ (pausing) “Perhaps you are right. I will not act tonight.”

The Scottish nobleman walks out, enjoying the calm evening, happy with his rudimentary quantum communication device, wondering what a universal quantum computer will do in 20 years…

A different kind of resource

Quantum computers are not just horsepower. I like to think of quantum computing as something different, a new twist which sometimes allows us to solve problems in an unexpected way. There is a long road ahead before a fully-fledged quantum internet will be operational [1], but in the meantime we have the privilege and honour to be among the first to imagine new applications. Prepare-and-measure scenarios and entanglement generation will be among the first technologies to be developed and promise to give us great advances in security. But not only. One of the less well-known applications of these early quantum technologies is decision making. In particular, one can employ shared quantum states as correlation devices in game theory.

Many people use a game-theoretical approach when they first study Bell inequalities through the famous CHSH game (as explained, for example, in [2], adapted from the original inequality [3]). This intuitive and successful example shows that players who share a quantum state can outperform players who use shared randomness for a collaborative task. From this example, one might think that quantum resources can create $\textit{stronger correlations}$, but the notion of strength depends on the application. In fact, I simply like to think that sharing entanglement can create $\textit{more correlations}$. The fact that they make it possible to optimise the CHSH collaborative task is a fortunate design choice.
This is interesting, because game theory models a lot more than collaboration. Very often it models situations of conflicting interests, where players have different goals and the outcome of a game depends on joint decisions. And even more interestingly, correlation devices and advice play a prominent role in the field.

Wisdom on demand

As a simple example, consider a game where two drivers need to cross an intersection. They arrive at the same time and they need to decide who passes first. A traffic light solves this problem by a mere suggestion, never enforced to the players. The traffic light simply displays a red light for one player and a green light for the other. If a player gets a red light, knowing that the other player has a green light, they will follow the advice to avoid a crash. Of course, life is not always that simple and there are more complicated advice models, where players actively communicate with a mediator, who analyses the information of all the players and then gives each of them advice [4]. The main concept, however, remains the same: rational players willingly follow the advice when it is designed to be convenient. There is a main disadvantage to this: the mediator’s advice given to one player could reveal important information about another player. In a real-life scenario this raises privacy concerns and in the theoretical works it gives rise to potentially infinite structures to represent what players know and to find out if they will follow the advice.

People have wondered if one could use quantum-information resources as correlation devices in games of conflicting interests for decades. The earliest example I found is [5], and another notable example is [6]. My colleagues and I compiled these and other related results from the quantum-information and game-theory literature and proposed a general mathematical framework to integrate the quantum-information formalism and the game-theoretical one [7]. We concluded that quantum resources can implement more equilibria than classical correlations alone (like in the traffic-light example), and fewer equilibria than an informed mediator can enable. Here, the keyword is $\textit{equilibrium}$. We knew similar results for the number of correlations, but in game theory we want these correlations to give rise to an equilibrium, which can be interpreted as predictable behaviour of the players.

We also found out something surprising: with a measure of social welfare in place (a simple one: the sum of all the players’ utilities) one can find out that there exist some games where the equilibrium that maximises social welfare can be implemented with quantum resources, without the need for a mediator, but not with classical correlations. That is, for some particular games, one can implement a socially optimal equilibrium either with an informed mediator (which might cause privacy concerns) or by instructing players to correlate autonomously by using their quantum computers and a trusted uninformed source of entangled states: $\textit{quantum advice}$.

Less is more

Enter the surprising work of Abbott, Mhalla, and Pocreau [8]. As part of one of their main results they introduce $\textit{quantum correlated advice}$, a seemingly innocent change to the setting of the game: what if players do not have quantum computers, but use a third-party service such as one in the cloud? Can this service be a meaningful intermediary, similar to a less informed version of a mediator?

Under this change, the set of achievable correlations is the same as in the standard quantum model discussed above, where players use their own quantum computers. Indeed, one can ask the services to implement any desired operation. So where’s the innovation? Again, the keyword is $\textit{equilibrium}$. The service providers can limit the number of inputs they receive from the players, this way constraining the quantum operations that players can perform. This trick reduces the number of possible cheating strategies of the players: instead of allowing any possible operation, it forces the players to stick to the agreed usage of the quantum resource. By doing so, this system can reach more equilibria. Indeed, cheating strategies can be seen as ways out of an equilibrium. In other words, quantum-correlated advice denies players full control of quantum information, thus limiting the possible cheating strategies and, by doing so, gives rise to more equilibria. It can implement all the correlations that standard quantum advice can, but in a more controlled way. The featured image of [8] clearly illustrates this change with respect to the standard quantum model of [7].

But are these additional equilibria of any use? Not always. However, importantly, the authors find that there are some games for which the socially optimal equilibrium is implementable with quantum-correlated advice, but not with standard quantum advice. That is, for some particular games, one can implement a socially optimal equilibrium either with an informed mediator (which might cause privacy concerns) or by instructing each player to correlate through their own informed service provider who in turn uses quantum information and promises to stick to the agreed allowed usage. Notably, one cannot implement the socially optimal equilibrium if a player has direct access to the quantum resource.

I find the work very well motivated, because this is exactly how most quantum resources will be accessed in the near future. It is possible nowadays to access quantum computers remotely, and in the future companies will probably provide access to the quantum internet. This new model still reasonably preserves privacy, because players’ inputs are locally processed by their own chosen provider. For comparison, in the original quantum model players are still required to trust their hardware, while in the informed-mediator scenario every player involved must trust the same entity. Therefore, we can say that quantum-correlated advice sits sandwiched between the two previous settings also privacy-wise.

Can the service providers collude with the players to allow them to cheat? Certainly, but one can imagine a system where providers would be audited and trusted, as it happens today with certification authorities in cybersecurity.

Conclusions

While quantum information can make Macbeth a boring play, it for sure can entertain me in new ways: Abbott and colleagues surprised me by exhibiting yet another rich and complex phenomenon in game theory. The presence of conflicting interests and different social-welfare measures creates a counter-intuitive result: capping players’ access to resources by adding intermediaries can improve the overall welfare while never enforcing choices for the players, who ultimately choose their actions autonomously and are motivated solely by maximising their utility functions given what they know.

I look forward to seeing this research evolve in two main directions: more applied examples of how quantum resources can help with real situations, and more theoretical studies of these beautiful classes of equilibria and correlations. There is much more wisdom in the qubit, waiting to be discovered.

Acknowledgements

We acknowledge financial support from PID2020-113523GB-I00 funded by MCIN/AEI/10.13039/501100011033 and by ‘ERDF A way of making Europe’. We thank Patricia Contreras-Tejada for useful discussions.

► BibTeX data

@article{Scarpa2024wisdominqubit, doi = {10.22331/qv-2024-09-05-82}, url = {https://doi.org/10.22331/qv-2024-09-05-82}, title = {The wisdom in the qubit}, author = {Scarpa, Giannicola}, journal = {{Quantum Views}}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {8}, pages = {82}, month = sep, year = {2024} }

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This View is published in Quantum Views under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.