Does information causality single out the set of quantum correlations?

Information causality bounds Bob's knowledge by Alice's message

Information causality is a physical principle stated in a random-access-code scenario. Alice holds a database of n independent random dits, Bob is asked to guess one chosen entry, and the two parties may share non-local resources. IC requires that the total information Bob gains about the individual entries does not exceed the information Alice communicates to him, even when those shared resources are used.

The paper works with a formulation in which the sum over entries of the mutual information between each entry and Bob's corresponding guess is bounded by the information the message carries about Alice's database, inputs, and outcomes. This version extends the noisy-channel formulation to an arbitrary number of copies and subsumes the original definition. A set of non-local boxes satisfies IC with database size n when any number of independent boxes from that set obey the inequality for all protocols; IC over all n is written with an infinity subscript.

The stakes are set by what IC was meant to do. Classical and quantum theory satisfy the requirement, while other generalized probabilistic theories such as boxworld do not. Among proposed information-theoretic principles, IC stood out because it had not yet been shown to be satisfied by some non-quantum non-local correlations. The paper closes that gap.

Ghostworld GPTs supply the finite-length counterexamples

For any finite database length n at least 2, the authors exhibit a bipartite set of correlations that satisfies IC for all compatible protocols and that contains post-quantum correlations in the Bell scenario with n+1 inputs and 2 outcomes per party. The construction treats the ghostworld sets of correlations as bipartite state spaces of a family of generalized probabilistic theories indexed by n, with effect spaces inherited from boxworld.

Working at the level of a GPT rather than a fixed box size lets the authors prove properties of the theory independently of system size, and therefore for all input and output counts simultaneously. The proof of IC for these GPT correlations is entropic, using a conditional measurement entropy defined as an infimum over fine-grained measurements, and it is offered as an alternative to the standard GPT proof of IC.

The authors also report that in the Bell scenario with n+1 inputs and 2 outcomes per party, the constructed sets of correlations are neither a superset nor a subset of the quantum correlations. That comparison matters because it rules out the simpler hope that the counterexamples merely sit above the quantum set in a nested way.

One fixed scenario covers every database size

The finite-length result leaves open whether a single Bell scenario can host a set satisfying IC for all n at once. To get there, the authors build a smaller set of correlations, unrelated to GPTs, in the setting with three binary measurements per party. They first prove IC for the restricted case where Alice and Bob have access to only one copy of their shared non-local resource, which settles the formulation of IC that does not use concatenation.

They then extend the argument to an unbounded number of copies, which yields the general result. The proof techniques come from contraction coefficients of divergences and the associated strong data-processing inequalities, applied to non-local correlations. A fixed-input contraction coefficient bounds how much the Kullback-Leibler divergence between two distributions can shrink under a channel, and a coefficient strictly below one gives a strong data-processing inequality that can bound the mutual-information terms appearing in IC.

The resulting set contains all local behaviors and some post-quantum ones, and IC holds for every adaptive protocol that combines independent resources from this set. The authors note that the set is not closed under local relabellings, but the conclusion also holds for its closure under finite local wirings and shared randomness.

Three binary inputs per party are enough to break the principle

The headline construction lives in the Bell scenario with 3 inputs and 2 outputs per party. The authors exhibit a set of correlations in that scenario that satisfies IC for all database sizes n and that includes post-quantum points. The specific behavior involved satisfies the inequality for a range of a parameter, and it lies outside the quantum commuting set.

The consequence is stated sharply: all known formulations of information causality fail to fully characterize the bipartite set of quantum correlations, already in the 3-input, 2-outcome Bell scenario. The failure is not confined to a large or exotic scenario, and it does not require an unbounded number of inputs or outcomes.

The paper also connects the counterexample to other proposed principles. Using the Navascues-Pironio-Acin hierarchy, the authors observe that the relevant behavior satisfies macroscopic locality but lies outside the almost quantum set. Their results therefore imply that there exist behaviors outside the almost quantum set that satisfy information causality.

The almost quantum comparison sharpens the failure

Almost quantum correlations form a strict superset of quantum correlations that was previously shown to satisfy several other proposed physical principles. The paper's counterexamples sit outside that set while still satisfying IC, so IC is not merely failing to exclude some correlations that other principles already exclude. It is admitting correlations that a broader-than-quantum candidate set rejects.

This cuts in a specific direction. If IC were a principle that captured quantum correlations, one might expect its satisfying set to sit inside or around the almost quantum set. The construction shows that the IC-satisfying set reaches beyond it, which is a stronger statement than a single post-quantum point would provide.

The authors leave open whether all almost quantum correlations satisfy information causality. That question is distinct from the one they answer, and the paper does not claim a complete characterization of the relationship between the two sets.

The entropic proof works without a full GPT entropy

A methodological contribution is that IC can be established for GPTs, and for the corresponding sets of correlations, without defining an entropy that satisfies all the properties used in the usual GPT proof of IC. The authors show this is possible from just the construction of bipartite state and effect spaces.

The conditional measurement entropy they use does not reduce to the conditional von Neumann entropy in the quantum case. That is a deliberate feature of the approach rather than an oversight: the proof needs a quantity that behaves well for the ghostworld theories, not one that matches the quantum entropy.

The authors flag the exploration of other entropy measures for GPTs as an interesting direction for future research. The entropic proof of IC for GPTs is presented as potentially of independent interest beyond the specific counterexamples.

What the result does not settle

The paper answers the question in the negative for the formulations of IC known to the authors. It does not claim that no strengthening of IC could ever recover the bipartite quantum set, and it explicitly raises the question whether IC can be further strengthened or whether an alternative principle is needed.

The results concern bipartite correlations in Bell scenarios. The constructions are stated for specific input and output counts, and the general claim is about the bipartite quantum set, not about multipartite settings or about correlations outside the Bell scenario framework.

The authors also note that extending one reformulation of IC beyond two decoding procedures remains an open problem. That limitation belongs to the broader IC literature the paper engages with, and it bounds how far the present counterexamples can be read as covering every conceivable variant.

The priority is the fixed three-input scenario

The most decision-relevant result is not the finite-length construction but the fixed scenario: a set of correlations in the 3-input, 2-outcome Bell scenario that satisfies IC for all database sizes n and still contains post-quantum points. That is the smallest setting the paper needs to defeat every known formulation at once, and it is where a reader should focus when judging how strong the failure is.

The comparison to keep in view is between the IC-satisfying set and the almost quantum set. The counterexamples lie outside almost quantum, so IC is admitting correlations that a strict superset of quantum correlations already excludes. For anyone weighing IC as a candidate principle for quantum correlations, that placement is the operative threshold.

The finite-length ghostworld results and the fixed-scenario construction answer different questions: the first shows that for each finite n there is a set with post-quantum correlations in an n+1-input scenario, while the second shows that a single 3-input scenario already suffices for all n. The second is the stronger statement about the principle's reach, and it is the one that closes the question the paper set out to answer.

Sources used in this answer