Nonlocality as Resource

Bell nonlocality is an operational resource when parties cannot communicate during a task and cannot reproduce the required correlations using shared randomness alone. Resource language turns a violation of a model into a quantified advantage. It leaves the interpretive question of how nature realizes those correlations separate.

The local baseline

For inputs $x,y$ and outputs $a,b$, a local behavior has the form

$$p(a,b\mid x,y)= \int d\lambda\,\mu(\lambda) p(a\mid x,\lambda)p(b\mid y,\lambda),$$

with $\mu$ independent of the freely selected inputs. The shared variable can contain arbitrarily much classical information. Stochastic local responses are already included.

Shared randomness, local relabeling, and local processing preserve this set. Local wirings of several independent boxes also preserve locality when they introduce no interparty communication and no selection loophole. Different nonlocality resource theories specify different admissible operations, so conversion claims must name their rules.

A game converts a witness into a payoff

In the CHSH game, uniform bits $x,y$ are supplied to the parties, who win when $a\oplus b=xy$. With the usual correlator convention,

$$p_{\mathrm{win}}=\frac12+\frac{S}{8}.$$

Local strategies have $S\leq2$, giving $p_{\mathrm{win}}\leq3/4$. A quantum strategy using an entangled pair reaches $S=2\sqrt2$, giving $\cos^2(\pi/8)$. A PR box wins with certainty but is outside the quantum set.

This advantage is rigorously defined without assigning trajectories or hidden properties to individual particles. The full inequality derivation appears in CHSH Inequality.

Noise, activation, and comparison

For a behavior mixed with uniform white noise,

$$p_v=v\,p_{\mathrm{ideal}}+(1-v)p_{\mathrm{white}},$$

the CHSH correlator scales as $S_v=vS_{\mathrm{ideal}}$. For the maximal quantum strategy, violation requires $v>1/\sqrt2$. Falling below that threshold proves failure of this witness, not necessarily locality under every possible experiment or transformation.

Likewise, a higher value of one Bell expression does not guarantee superiority for every task. Randomness, key generation, and communication complexity have different performance measures and assumptions. Quantum Information develops the cryptographic protocol rather than treating a Bell violation as a complete security proof.

Exercises

Why must the communication restriction specify when messages are forbidden?

Solution

Shared randomness may be distributed before the inputs arrive without spoiling the local baseline. A message carrying Alice’s input during a CHSH trial can instead let Bob choose a response that wins perfectly. The timing of communication changes the feasible behavior set and therefore the claimed resource.

References

• Brunner, Nicolas, Daniel Cavalcanti, Stefano Pironio, Valerio Scarani, and Stephanie Wehner, “Bell nonlocality,” Reviews of Modern Physics 86, 419–478 (2014). Review.
• Clauser, John F., Michael A. Horne, Abner Shimony, and Richard A. Holt, “Proposed experiment to test local hidden-variable theories,” Physical Review Letters 23, 880–884 (1969). Paper.