Information Causality Characterizes the Set of Quantum Correlations in the Simplest Bell Scenario

The paper in detail

The scenario is the simplest Bell experiment. Alice and Bob receive bits \(x,y\in\{0,1\}\) and output bits \(a,b\in\{0,1\}\). Writing outcomes as signs \((-1)^a,(-1)^b\), the behaviour is fixed by marginals and correlators: \[\mathbb P(a,b|x,y)=\tfrac14\bigl(1+(-1)^a\langle a_x\rangle+(-1)^b\langle b_y\rangle+(-1)^{a\oplus b}\langle a_xb_y\rangle\bigr).\] The paper works with the renormalized correlators \[D_{x,y}=\frac{\langle a_xb_y\rangle-\langle a_x\rangle\langle b_y\rangle}{\sqrt{(1-\langle a_x\rangle^2)(1-\langle b_y\rangle^2)}},\] which remove the part of each correlation that the marginals already explain.

Which values of the four \(D_{x,y}\) can quantum mechanics produce? The answer is the Tsirelson–Landau–Masanes (TLM) criterion \[\bigl|\arcsin D_{0,0}+\arcsin D_{0,1}+\arcsin D_{1,0}-\arcsin D_{1,1}\bigr|\le\pi,\] together with its relabellings, which move the minus sign to each of the other correlators. With vanishing marginals this is exactly the quantum boundary. In general, the quantum, macroscopically local and almost-quantum sets all have this same projection onto the four \(D_{x,y}\). Can a principle about communication, rather than about Hilbert spaces, single out this boundary?

The principle is information causality: a message through a channel of capacity \(\mathcal C\) cannot give Bob more than \(\mathcal C\) bits of information about Alice’s data, whatever no-signalling correlations they share. The authors use a generalized form, \[I(x_0;m^{\prime},b\,|\,y=0)+I(x_1;m^{\prime},b\,|\,y=1,x_0)\le I(m;m^{\prime})\le\mathcal C.\] Alice holds two bits \(x_0,x_1\), and Bob tries to learn \(x_y\). Alice measures \(x=x_0\oplus x_1\) on her half of the correlation and sends a noisy encoding \(m\) of \((x_0,x_1,a)\). Bob receives the channel output \(m^{\prime}\) and also uses his own outcome \(b\).

Theorem 1 states that generalized information causality implies the TLM criterion. In this scenario it therefore reproduces the quantum boundary of the correlator projection. Theorem 2 states that the principle is strictly stronger than macroscopic locality. The point with \(\langle a_0\rangle=\langle a_1\rangle=\tfrac35\), \(\langle b_0\rangle=\tfrac12\), \(\langle b_1\rangle=0\) and suitable correlators satisfies TLM. Yet with a perfect one-bit channel and uncorrelated inputs, the left-hand side equals \(\tfrac14\log_25+\tfrac3{10}\log_23\approx1.056>1\). It remains open whether information causality picks out exactly the quantum set in the full eight-parameter space of behaviours.

Transferable methods

1. Keep the full measurement record

   Information causality was first stated using a guess \(g\) for \(x_y\). The authors instead score the full record \((m^{\prime},b)\) that Bob holds, and they condition the second term on \(x_0\) because the inputs may be correlated.

   Why does keeping the record help? Any guess is a function \(g=f(m^{\prime},b)\), so the data-processing inequality gives \[I(x_i;g\,|\,y=i)\le I(x_i;m^{\prime},b\,|\,y=i).\] The generalized left-hand side is therefore never smaller than the original, while the right-hand side \(I(m;m^{\prime})\le\mathcal C\) is unchanged. A larger left-hand side under the same budget rules out more correlations.

   Why condition on \(x_0\)? Alice’s bits are drawn with \(\mathbb P(x_0,x_1)=\tfrac14\bigl(1+(-1)^{x_0\oplus x_1}\varepsilon\bigr)\), so each bit is uniform, but the pair is correlated when \(\varepsilon\ne0\). Part of what Bob could learn about \(x_1\) is then already in \(x_0\). The template is the chain rule \(I(x_0,x_1;M)=I(x_0;M)+I(x_1;M\,|\,x_0)\): the conditioned term counts only new information. The correlation \(\varepsilon\) becomes a free parameter, optimized in the next technique.

   How to reuse it

   When bounding a protocol, apply the data-processing inequality as late as possible. Every coarse-graining (decoding to a bit, discarding side information) can only lower the information being scored, so postponing it gives the strongest bound the principle allows. Correlated inputs, scored with conditional terms, are a further parameter worth optimizing.

2. Expand around zero capacity

   The channel biases its binary output by a small parameter \(e_c\): \(\mathbb P(m^{\prime}|m)=\tfrac12+\tfrac12(-1)^{m^{\prime}}e_cf_m\), with \(f_{x_0,x_1,a}=(-1)^{a\oplus x_0}\kappa_{x_0\oplus x_1}/\sqrt{1-\langle a_{x_0\oplus x_1}\rangle^2}\) and free real \(\kappa_0,\kappa_1\). As \(e_c\to0\) both sides of the inequality vanish, and the authors compare their leading terms.

   Every probability in the problem has the form \(P_e(z)=\xi(z)+e\,\eta(z)\) with \(\sum_z\eta(z)=0\). Mutual information is an average of relative entropies, and relative entropy is second order near equality: \[D(P_e\Vert\xi)=\frac{e^2}{2\ln2}\sum_z\frac{\eta(z)^2}{\xi(z)}+O(e^3).\] This is the content of the limit \(\lim_{e\to0}\partial_e^2(\xi+e\eta)\log(\xi+e\eta)=\eta^2/\xi\) used in the paper. Dividing the information-causality inequality by \(e_c^2\) and letting \(e_c\to0\) turns entropies into a quadratic \(\chi^2\)-type expression in \(\kappa_0,\kappa_1\).

   With \(p=\tfrac{1+\varepsilon}2\) and \(q=\tfrac{1-\varepsilon}2\), the leading-order condition (the paper’s Eq. 35) reads \(\kappa^{\mathsf T}M\kappa\le0\) for \(\kappa=(\kappa_0,\kappa_1)\), where \[M=\begin{pmatrix}-p\bigl(p(1-D_{0,0}^2)+q(1-D_{0,1}^2)\bigr)&pq\,\Delta\\ pq\,\Delta&-q\bigl(q(1-D_{1,0}^2)+p(1-D_{1,1}^2)\bigr)\end{pmatrix},\] with \(\Delta=D_{0,0}D_{1,0}-D_{0,1}D_{1,1}\). Alice may choose any \(\kappa\), so the principle demands \(-M\succeq0\). The diagonal entries are non-positive automatically; the remaining condition is \(\det M\ge0\), i.e. \[\Delta^2\le\bigl(u(1-D_{0,0}^2)+1-D_{0,1}^2\bigr)\bigl(u^{-1}(1-D_{1,0}^2)+1-D_{1,1}^2\bigr),\qquad u=\tfrac{1+\varepsilon}{1-\varepsilon}.\]

   The principle must hold for every source, so the strongest constraint uses the \(u>0\) that minimizes the right-hand side. With \(\alpha=1-D_{0,0}^2\), \(\beta=1-D_{0,1}^2\), \(\gamma=1-D_{1,0}^2\) and \(\delta=1-D_{1,1}^2\), the right-hand side is \(\alpha\gamma+\beta\delta+u\alpha\delta+\beta\gamma/u\). By the AM–GM inequality it is minimized at \(u=\sqrt{\beta\gamma/(\alpha\delta)}\), where it equals \((\sqrt{\alpha\gamma}+\sqrt{\beta\delta})^2\). Hence \[\bigl|D_{0,0}D_{1,0}-D_{0,1}D_{1,1}\bigr|\le\sqrt{(1-D_{0,0}^2)(1-D_{1,0}^2)}+\sqrt{(1-D_{0,1}^2)(1-D_{1,1}^2)}.\] This is the algebraic form of TLM. It is equivalent to the arcsine inequality together with its relabellings (we checked the equivalence numerically on random points). Check: at the Tsirelson point \(D_{0,0}=D_{0,1}=D_{1,0}=-D_{1,1}=1/\sqrt2\) both sides equal 1. The PR box (\(D_{1,1}=-1\), the others 1) gives \(2\le0\), which fails.

   How to reuse it

   This pipeline applies to any entropic inequality that is hard to optimize directly. (i) Perturb around a point where both sides vanish. (ii) Expand to second order, where relative entropy becomes the \(\chi^2\) form \(\tfrac12\sum\eta^2/\xi\). (iii) If free parameters enter linearly in \(\eta\), the condition becomes positivity of a matrix, which can be checked through its minors. (iv) Optimize the remaining parameters in closed form. The cost is that only the leading order is captured. The paper’s Theorem 2 shows that finite-capacity protocols impose constraints this limit misses.

Source: Information causality principle; Results, Theorem 1 and Appendix A. Based on the linked paper version. Derivations marked as editorial, and suggested uses in other problems, are editorial guidance rather than results of the paper.

Context & limitations

Why it matters. Connects limits on communication to limits on nonlocal correlations.

Limitations. The stated characterization concerns the simplest bipartite scenario; it is not a characterization of all Bell scenarios.

Description based on the full text of the linked arXiv version, read 2026-10-19; equations follow the paper’s notation.