Information causality beyond the random access code model

1. Problem Statement

The paper addresses the fundamental challenge of characterizing the set of quantum correlations ($\mathcal{Q}$) within the broader set of no-signaling correlations. While the No-Signaling (NS) principle is necessary, it is insufficient to define quantum mechanics, as it allows for "super-quantum" correlations (e.g., PR-boxes) that violate Tsirelson's bound.

Information Causality (IC) was proposed as a physical principle to recover the quantum boundary. However, the standard definition of IC relies on a specific task: the Random Access Code (RAC). In the RAC scenario, Alice sends a string of $N$ bits, and Bob must retrieve a specific bit $A_i$ based on an input $t$. The principle states that the mutual information Bob can gain about the specific requested bit cannot exceed the communication capacity $k$.

The Limitation: The authors argue that the RAC-based definition is too restrictive because it enforces a specific "bit-to-bit" correspondence between Alice's input string and Bob's retrieval task. This structure may fail to capture the true limits of "potential information" in all scenarios, leading to gaps where the IC boundary does not perfectly coincide with the quantum boundary (specifically in certain slices of the no-signaling polytope).

2. Methodology

The authors propose a redefinition of Information Causality that decouples the principle from the specific RAC task, focusing instead on the global structure of information and the concept of redundancy.

A. New Quantifier: Redundant Information

Instead of summing mutual informations conditioned on specific retrieval indices (as in RAC), the new quantifier, $IC_{red}$, considers the total information Alice's variable $A$ holds with respect to Bob's multiple outputs $B_1, \dots, B_M$, while subtracting redundant information.

The formula for $M=2$ (the simplest Bell scenario) is:
$$IC_{red} = I(A; B_1) + I(A; B_2) - I_r(A; B_1, B_2)$$
Where $I_r$ is the redundant information. The authors utilize the measure proposed by Harder, Salge, and Polani to calculate $I_r$:

1. Convex Hull Construction: For each value of $B_i$, a probability distribution $p_{A|b_i}$ is constructed. The convex hull $C_i$ of these distributions is defined.
2. Projection: The distribution $p_{A|b_1}$ is projected onto the convex hull $C_2$ using Kullback-Leibler (KL) divergence minimization to find the "closest" element.
3. Projected Information: The information gained by this projection is calculated.
4. Redundancy: $I_r$ is defined as the minimum of the projected information in both directions ($B_1 \to B_2$ and $B_2 \to B_1$).

B. Protocol and Scenario

• Scenario: The 2-2-2 Bell scenario (two parties, two inputs, two outputs).
• Protocol: Alice inputs $x = a_1 \oplus a_2$ into a no-signaling box. She sends a bit $m$ through a noisy binary symmetric channel with capacity $k$ to Bob. Bob inputs $y$ to retrieve $B_i$.
• Test Cases: The authors test three families of no-signaling boxes (convex combinations of PR-boxes, local deterministic boxes, and white noise) previously studied by Allcock et al. [15].
• Numerical Approach:
  1. Boundary Tightening: They calculate the maximum $IC_{red}$ for various no-signaling correlations to see if they violate the bound $IC_{red} \leq k$.
  2. Quantum Verification: They generate 2-2-2 quantum correlations using extremal points (parameterized by entangled states $|\phi_\theta\rangle$ and measurement angles) and random mixtures. They verify that all generated quantum correlations satisfy $IC_{red} \leq k$.
  3. Quasi-convexity Check: They test if $IC_{red}$ is a quasi-convex function (i.e., mixing resources does not improve performance), a necessary physical property for any valid information principle.

3. Key Contributions

1. Decoupling from RAC: The paper successfully reformulates Information Causality without relying on the specific winning condition of the Random Access Code, allowing for a more general assessment of "potential information."
2. Introduction of Redundancy: By explicitly subtracting redundant information ($I_r$), the new quantifier avoids overcounting information that is present in multiple outputs of Bob, providing a tighter bound.
3. Tightening the IC Boundary: The authors provide the first improvement on the IC boundary for the 2-2-2 scenario since the original 2009 study.
4. Numerical Evidence of Quasi-convexity: The study provides strong numerical evidence that the new $IC_{red}$ function is quasi-convex on the set of correlations, supporting its physical validity.

4. Results

The authors compared the original RAC-based IC ($IC_{RAC}$) with the new Redundant Information-based IC ($IC_{red}$) across three specific slices of the no-signaling polytope:

• Case 1 ($R_{NS} = P^{010}_{NL}$): Both the original and new definitions recover the quantum boundary exactly (within numerical precision).
• Case 2 ($R_{NS} = P^{110}_{NL}$):
  • Original IC: Failed to recover the quantum boundary; it only detected violations for boxes that violated the Tsirelson bound, leaving a significant gap between the IC limit and the quantum limit.
  • New IC ($IC_{red}$): Successfully recovered the exact quantum boundary. The new definition correctly identified that the potential information in these correlations does not exceed the channel capacity, aligning perfectly with quantum mechanics.
• Case 3 ($R_{NS} = P^{0000}_{L}$ - Local Deterministic): Both definitions yield the same boundary, which still has a gap with the true quantum boundary. The authors suggest this gap may be real or due to the current limitations of the redundancy measure (e.g., it might need to depend on joint distributions $p_{B_1 B_2}$ rather than just marginals).

Verification on Quantum Correlations:

• Extensive numerical sampling of 2-2-2 extremal quantum correlations and their random mixtures ($M=2$ to $50$) showed zero violations of the $IC_{red} \leq k$ bound.
• The distribution of $IC_{red}$ values for mixed states shifted toward zero as the number of mixed components increased, confirming the quasi-convexity property.

5. Significance

• Refining Quantum Principles: This work demonstrates that the "Random Access Code" was an artifact of the original formulation that obscured the true power of Information Causality. By removing this constraint, the principle becomes a more robust candidate for defining the quantum set.
• Closing the Gap: The ability of $IC_{red}$ to recover the quantum boundary in the $P^{110}_{NL}$ case (where the old method failed) is a major step toward proving that Information Causality might be the sole principle needed to single out quantum correlations from super-quantum ones.
• Future Directions: The paper highlights that while the 2-2-2 scenario is well-handled, the definition of redundant information for $M > 2$ indicators remains an open problem in classical information theory. Solving this is crucial for extending these results to multipartite scenarios.

In conclusion, Yu and Scarani provide a rigorous, redundancy-aware reformulation of Information Causality that offers a tighter, more accurate bound on quantum correlations, bringing the principle closer to fully characterizing the quantum world.