Popescu-Rohrlich box fraction of nonobjective information and distinguishing quantum theory

It is demonstrated that identifying information-theoretic limitations of quantum Bell nonlocality alone cannot completely distinguish quantum theory from generalized nonsignaling theories. To this end, an information-theoretic concept of certifying nonobjective information by the Popescu-Rohrlich box fraction is employed. Furthermore, in the aforementioned demonstration, a partial answer to the question of what distinguishes quantum theory from generalized nonsignaling theories emerges beyond the one provided by the principle of information causality alone. This is accomplished by demonstrating that postquantum models identified by the information causality are isolated by the emergence of the Popescu-Rohrlich box fraction of nonobjective information in Bell-local boxes of a nonsignaling model, over the other nonsignaling models.

💡 Research Summary

The paper tackles the longstanding problem of distinguishing quantum theory from the broader class of generalized nonsignaling theories (GNSTs). While Bell nonlocality demonstrates that quantum mechanics violates local realism, it does not, by itself, separate quantum correlations from all possible nonsignaling correlations. Prior work introduced the principle of information causality (IC) as an information‑theoretic constraint that rules out many post‑quantum correlations, yet IC fails to exclude all nonsignaling boxes that respect the Tsirelson bound. Consequently, a complete operational principle that singles out the quantum set remains elusive.

To address this gap, the author introduces the notion of “non‑objective information,” defined as the negation of “objective information” (the latter being a repeatable, nondisturbing test that extracts classical data from a state, as formalized by Perinotti). In the framework of causal operational probabilistic theories, objective information characterizes null‑discord states; non‑objective information therefore corresponds to non‑null discord, a generic signature of nonclassicality beyond entanglement.

The paper then focuses on “dimensionally restricted nonlocality” (DRNL). In the simplest bipartite scenario (two binary inputs and outputs per party), a Bell‑local box can be written as a convex mixture of deterministic boxes. If the hidden variable λ that mediates the correlations has a dimension bounded by the number of measurement outcomes (here dλ ≤ 2), the box is said to exhibit DRNL. A nonlinear determinant witness N_L, built from the covariance matrix of the correlators ⟨A_x B_y⟩, detects DRNL: N_L > 0 iff the box cannot be simulated with a classical λ of unrestricted dimension. The author proves Proposition 1: whenever DRNL (N_L > 0) arises from a state ρ in a causal probabilistic theory, the state must contain non‑objective information (i.e., non‑null discord). This links a purely operational nonlocality test to an information‑theoretic property of the underlying state.

The central quantitative tool is the “Popescu‑Rohrlich (PR) box fraction,” denoted p_PR. PR boxes are the eight extremal nonlocal vertices of the nonsignaling polytope. Any nonsignaling box P can be decomposed as
 P = p_PR P_PR + (1 − p_PR) P_L^{Γ=0},
where P_L^{Γ=0} is a Bell‑local box with a vanishing value of a nonlinear measure Γ introduced in the appendix. For such a decomposition, Γ(P) = 4 p_PR, so Γ > 0 directly quantifies the PR‑box component. The PR‑box fraction therefore serves as a measure of the amount of non‑objective information present in the box: a nonzero p_PR implies DRNL (N_L > 0) and hence non‑null discord.

Armed with these definitions, the author examines two families of GNST models. The first family consists of “post‑quantum‑only” models in which every nonlocal correlation is a mixture of a PR box and a local box that attains the classical CHSH value B = 2:
 P = c_0 P_PR + (1 − c_0) P_L.
Lemma 1 shows that, within this family, the PR‑box fraction is equivalent to Bell‑nonlocality itself; thus any nonzero p_PR automatically signals the presence of non‑objective information and distinguishes these models from quantum theory.

The second family comprises the quantum set. Quantum correlations respect the Tsirelson bound (CHSH ≤ 2√2) and, crucially, have p_PR = 0 (or arbitrarily small when approximated by noisy PR boxes). While they satisfy information causality, they do not exhibit DRNL with a PR‑box component. Consequently, the simultaneous satisfaction of IC and the absence of a PR‑box fraction uniquely characterizes the quantum set among the examined GNSTs.

The paper’s main contribution is therefore twofold: (i) it establishes a rigorous link between DRNL and non‑objective information via the PR‑box fraction, and (ii) it demonstrates that the combination of information causality and a vanishing PR‑box fraction provides a stronger, more discriminating principle than IC alone. This result narrows the gap toward a complete operational axiom that isolates quantum theory, and it also clarifies the resource‑theoretic role of PR‑box components in tasks such as secure key distribution.

In summary, by introducing the PR‑box fraction of non‑objective information and showing how it complements information causality, the author offers a novel, information‑theoretic criterion that separates quantum correlations from broader nonsignaling correlations, advancing our understanding of the foundational limits of quantum nonlocality.