Why Isn’t Nature More Quantum?
Quantum physics draws a boundary. Information theory approaches the same limit from a completely different direction.
Alice and Bob are playing a simple guessing game: QSeaBattle. Alice sees a board of n bits and can send Bob one bit, while Bob is asked about a randomly chosen position. Their job is simple: Bob has to guess whether the bit at the requested position is 0 or 1.
We discussed this game in previous posts and ended with a surprisingly simple result. The classical strategy became more and more complicated as we increased the number of bits, but quantum mechanics did the opposite: It replaced the many-faced geometry representing all optimal classical strategies with a sphere. For two bits we had a circle, for three a sphere, and in general a hypersphere,
c₁² + c₂² + ⋯ + cₙ² = 1.
Here cᵢ represents the advantage for index i: 2Pᵢ − 1, with Pᵢ Bob’s probability of answering correctly when asked for bit i.
The spherical boundary comes from physics. Behind it sits the mathematical structure of quantum mechanics: Hilbert space, operators representing observables, and the Born rule that turns quantum amplitudes into probabilities. Almost exactly a century ago, the probabilistic interpretation of the new quantum mechanics established the rules from which this quantum limit ultimately follows. Famously, Born’s 1926 scattering paper initially stated the probability interpretation incorrectly and corrected it in a proof: it was the square of the amplitude that mattered.
So quantum mechanics tells us where Nature stops. But that is not quite the same as understanding why it stops there.
Let us ignore quantum mechanics for a moment and imagine that, in this game, Bob can correctly guess whichever of Alice’s two bits he is asked for, with 100% certainty. In our geometry, such a resource would take us all the way to the corner c₁ = c₂ = cᵢ = 1.
That sounds as if we must be violating some fundamental principle. Perhaps it would require instantaneous communication, violating Einstein’s prohibition on faster-than-light communication? It does not. It is possible to construct hypothetical correlations that give Bob this extraordinary ability while still preventing Alice from sending him any information faster than light [1].
So we have a strange situation when we ask why Alice and Bob cannot have a perfect win-rate. Geometry allows it. Causality allows it. Yet Nature apparently does not. What, then, is wrong with Bob being able to know whichever bit he chooses?
Perhaps the problem is not causality, but information. Alice sent only one bit. Should Bob really be able to use that one bit, together with his shared resource, to recover any of the independent bits perfectly? This suggests a different way of looking at the quantum boundary. Instead of starting from quantum physics and asking what correlations it permits, can we start from information and recover the same restriction?
Can an information-theoretical principle explain why Nature stops at the quantum boundary?
The Information Boundary
Now evaluate the same game from an information perspective. For each possible question, Bob’s advantage cᵢ corresponds to a success probability.
Pᵢ = ½(1 + cᵢ).
From that probability, we can calculate how much information Bob’s answer carries about Alice’s bit. For a uniformly random bit, this is
Iᵢ = 1 − h₂(Pᵢ),
where h₂ is the binary entropy.
Alice sends Bob only one classical bit. In their 2009 paper “Information Causality as a Physical Principle,” Pawłowski and colleagues proposed that this should impose a deeper restriction: In whatever way Alice and Bob exploit their shared resources, the total information available through Bob’s possible retrieval choices should not exceed the information Alice actually communicates [2]. For our game,
Σᵢ Iᵢ ≤ 1.
They called this proposed physical principle Information Causality.
Information Causality adds the information associated with all Bob’s possible retrieval choices. Importantly, Bob does not actually make all those choices: in any one run he chooses one question and obtains one answer (in fact, once he uses his quantum resource for one measurement, he can’t go back and ask what he’d have learned from another). We will return to what it means to add the information associated with these alternative possibilities.
The proposal is stronger than merely using the inequality as a convenient mathematical bound. The suggestion is that Nature itself respects this information budget. So let us take the principle seriously and see what boundary it draws.
Figure 1 combines the two perspectives in a single picture. The circle is the boundary supplied by quantum physics. The rounded shape outside it is the boundary supplied by the information principle.
They are not the same.
For two bits, Information Causality is more permissive than quantum mechanics. Along the diagonal there is visible room between the quantum point and the point where the information budget is exhausted. So if Information Causality is going to tell us something about why Nature selects the quantum limit, the answer is not simply that its finite-game boundary reproduces the quantum circle. Perhaps two bits are simply too few.
As the Game Gets Bigger
With three bits we get one last chance to see the complete geometry before it escapes into higher dimensions.
Figure 2 shows the full three-dimensional advantage space. Figure 3 shows two cross-sections through it. The sphere becomes a circle in each cut, while the information boundary produces a different, somewhat larger curve. Just as for two bits, information theory still leaves room outside the quantum boundary.
For four bits, the full advantage space is already four-dimensional, so we can no longer draw it. But the cross-sections give us a way forward. We can take equivalent two-dimensional cuts through higher-dimensional shapes and watch what happens as the game grows.
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There is one particularly useful direction: the symmetric direction (1,1,…,1), where Bob has the same advantage for every possible question. On the quantum sphere, if every coordinate has the same value c, then
n c² = 1,
so the quantum boundary lies at
c = 1/√n.
As the game grows, the quantum advantage available for each question therefore shrinks: 1/√2 for two bits, 1/2 for four, 1/√8 for eight, and so on.
Now compare that known quantum boundary with the information boundary obtained from the one-bit budget. The information boundary converges to a shape that touches the quantum boundary at the (1,0,0,…) axis and bulges out in the uniform (1,1,1,…) direction. The two shapes do not converge: for every n, Information Causality still allows some points beyond the quantum sphere.
Why do larger games become sensitive? Along the symmetric direction, both boundaries sit at a radius that barely moves as n grows: the quantum radius stays exactly 1, the information radius approaches about √(2 ln 2) ≈ 1.177. So in absolute terms the two stay roughly the same distance apart, however large the game gets. But that fixed gap is shared out over more and more bits as n grows — so the amount of extra advantage allowed per bit keeps shrinking. A correlation only slightly stronger than the quantum one can look harmless when there are just two bits to spread it over, yet when that same slight excess is repeatedly used to construct larger games, its advantage eventually carries us across the information boundary. In the familiar language of analysis: for every ε > 0, there exists an n for which information theory objects if we take the strength of the correlation to exceed the highest quantum value by ε.
This is the key result of Information Causality. Pawłowski and colleagues showed that even an arbitrarily small excess in the underlying correlation beyond the quantum limit eventually violates the one-bit information budget [2]. The information boundary never literally becomes the finite-n quantum sphere, yet the strongest correlation that survives this test for arbitrarily large games is precisely the quantum one.
Go even slightly beyond the quantum limit, and eventually there is a guessing game large enough for information theory to object.
Why Isn’t Nature More Quantum?
We have arrived at the same limiting correlation strength by two very different roads.
The first road began with physics. The mathematical structure developed in the first years of quantum mechanics became the modern framework of Hilbert spaces, operators, and probabilities. Within that framework, quantum mechanics determines how far Alice and Bob can go.
The second road did not assume quantum mechanics. We started with a guessing game, one communicated bit and a proposed principle about information. At any finite game size, that principle is actually more permissive than quantum mechanics. Yet when Pawłowski and colleagues ask which correlations can survive the information test for arbitrarily large games, the same quantum limit reappears.
That coincidence is the remarkable part.
Perhaps Information Causality is telling us why Nature is not more quantum: correlations stronger than the quantum limit would eventually allow Alice and Bob to violate a fundamental information budget.
But does that close the story? Not completely.
The first thing to wonder about is the information sum itself. Bob never retrieves all n bits. He is asked for one bit, chooses the corresponding measurement, and obtains one answer. In the quantum case, that choice matters: once he has used his shared quantum resource, he cannot go back and ask what he would have learned had he chosen another measurement instead.
Information Causality nevertheless adds the mutual information associated with all these choices. In a sense, it puts all the things Bob could have learned into one information budget, even though he never can learn them simultaneously. Should we interpret this sum as the amount of information Bob actually has — or as a measure of what he could have learned under mutually exclusive choices?
There is also something peculiar about this limit. In advantage space, the quantum boundary remains a hypersphere of radius one. Along the symmetric direction, the information boundary approaches radius √(2 ln 2) (≈1.177, as we saw). When n grows, the advantage as well as the difference is distributed over more and more indices, making the value for each index approach zero. We distinguish quantum mechanics from a slightly stronger theory in a curious limit: Bob faces more and more possible questions, while his chance of correctly answering any particular one gets closer and closer to a random guess.
None of this makes Information Causality less remarkable. One route begins with the physical and mathematical structure of quantum mechanics. The other begins with a proposed principle about information. Yet both single out the same limiting correlation strength.
The question is what we should make of that coincidence.
Information Causality is not the only attempt to understand why Nature stops where it does. Communication-complexity arguments [3,4] and principles such as Macroscopic Locality [5] constrain the post-quantum world from different directions.
Perhaps information really is the fundamental principle hiding behind the quantum boundary. Or perhaps Information Causality is seeing the shadow of something deeper — a principle of geometry, consistency, or symmetry that we have not yet identified.
So, after starting with a simple guessing game, we have arrived at a question that is still very much alive: Why isn’t Nature more quantum?
Does Information Causality explain the quantum boundary — or merely find the same limit from another direction?
I am curious about your view on this question. Please leave your comments and subscribe to receive future posts. Thank you for reading.
References
[1] S. Popescu and D. Rohrlich, “Quantum Nonlocality as an Axiom,” Foundations of Physics 24, 379–385 (1994). https://doi.org/10.1007/BF02058098
[2] M. Pawłowski, T. Paterek, D. Kaszlikowski, V. Scarani, A. Winter and M. Żukowski, “Information Causality as a Physical Principle,” Nature 461, 1101–1104 (2009). https://doi.org/10.1038/nature08400
[3] W. van Dam, “Implausible Consequences of Superstrong Nonlocality,” Natural Computing 12, 9–12 (2013); originally arXiv:quant-ph/0501159 (2005).
[4] G. Brassard, H. Buhrman, N. Linden, A. A. Méthot, A. Tapp and F. Unger, “Limit on Nonlocality in Any World in Which Communication Complexity Is Not Trivial,” Physical Review Letters 96, 250401 (2006). https://doi.org/10.1103/PhysRevLett.96.250401
[5] M. Navascués and H. Wunderlich, “A Glance Beyond the Quantum Model,” Proceedings of the Royal Society A 466, 881–890 (2010). https://doi.org/10.1098/rspa.2009.0453
