Information Causality

Information causality limits how much information a receiver can obtain about independent sender data using a bounded classical message and previously shared resources. It strengthens no-signaling by allowing a nonzero communication budget. Its best-known application constrains certain postquantum correlations; it is not a complete characterization of every quantum correlation.

Helpful background. No-signaling distinguishes marginal independence from Bell locality.

The random-access task

Alice receives independent unbiased bits $A_0,\ldots,A_{N-1}$. Bob receives an independent index $b$ and tries to guess $A_b$ using a guess $\beta$. They may share a resource prepared independently of the bits and index, and Alice may send at most $m$ classical bits. Information causality requires

$$\sum_{k=0}^{N-1} I(A_k:\beta\mid b=k)\leq m.$$

Mutual information is measured in bits. The expression sums performance across alternative index choices; Bob does not output all the bits in one run. Independence of the source bits and of the index is part of the task.

For $m=0$, a violation would give Bob information about freely supplied remote data without communication. For $m>0$, the principle constrains how shared correlations supplement the message.

A one-bit PR-box protocol

Let a PR box produce outputs $\alpha,\gamma$ on inputs $x,y$ satisfying $\alpha\oplus\gamma=xy$, with uniform local marginals. Alice sets $x=A_0\oplus A_1$ and sends $c=A_0\oplus\alpha$. Bob inputs $y=b$ and outputs $\beta=c\oplus\gamma$. Then

$$\beta=A_0\oplus b(A_0\oplus A_1)=A_b.$$

Both alternative guesses are perfect. Their mutual informations sum to $2$, although only one classical bit was sent. The box itself remains no-signaling; the contradiction is with the stronger communication principle.

The isotropic threshold

For an isotropic noisy box, let the required parity hold with probability $(1+E)/2$, with $0\leq E\leq1$. A standard concatenated protocol using independent copies of the box for $N=2^n$ bits has success probability $(1+E^n)/2$ for each selected bit. Information causality consequently requires

$$2^n\left[1-h_2\!\left(\frac{1+E^n}{2}\right)\right]\leq1,$$

where $h_2$ is binary entropy. For small bias, $1-h_2((1+\epsilon)/2)=\epsilon^2/(2\ln2)+O(\epsilon^4)$. When $2E^2>1$, the left side eventually exceeds one. Thus this family requires $E\leq1/\sqrt2$, corresponding to CHSH $S=4E\leq2\sqrt2$.

This calculation concerns the specified family and protocol. Recovering a familiar boundary in one slice does not prove that information causality singles out the full quantum set in arbitrary scenarios.

Exercises

Why does the PR protocol not let Bob learn both bits after one run?

Solution

Bob chooses one box input and obtains one output. The counterfactual performance for the other input is assessed in another run; the protocol supplies no second independent outcome for the same box. Information causality deliberately constrains this random-access ability, not only the size of a simultaneously decoded string.

References

• Pawłowski, Marcin, Tomasz Paterek, Dagomir Kaszlikowski, Valerio Scarani, Andreas Winter, and Marek Żukowski, “Information causality as a physical principle,” Nature 461, 1101–1104 (2009). Paper.
• Brunner, Nicolas, Daniel Cavalcanti, Stefano Pironio, Valerio Scarani, and Stephanie Wehner, “Bell nonlocality,” Reviews of Modern Physics 86, 419–478 (2014). Review.