[Paper Review] Complementarity between signalling and local indeterminacy in quantum nonlocal correlations

S. Aravinda, R. Srikanth|arXiv (Cornell University)|Sep 17, 2013

Statistical Mechanics and Entropy4 citations

TL;DR

This paper establishes a quantitative complementarity relation between signaling (S) and local indeterminacy (I) in nonlocal correlations, proving that S + 2I ≥ 1 for any resource simulating singlet-state statistics. It confirms a conjecture by Hall (2010) and Kar et al. (2011), demonstrating that nonlocality arises from a trade-off between signaling and indeterminacy, enabling device-independent randomness certification even without assuming no-signaling or quantum mechanics.

ABSTRACT

The correlations that violate the CHSH inequality are known to have complementary contributions from signaling and local indeterminacy. This complementarity is shown to represent a strengthening of Bell's theorem, and can be used to certify randomness in a device-independent way, assuming neither the validity of quantum mechanics nor even no-signaling. We obtain general nonlocal resources that can simulate the statistics of the singlet state, encompassing existing results. We prove a conjecture due to Hall (2010) and Kar et al. (2011) on the complementarity for such resources.

Motivation & Objective

• To formalize and quantify the complementarity between signaling and local indeterminacy in nonlocal correlations.
• To prove the conjecture by Hall (2010) and Kar et al. (2011) that S + 2I ≥ 1 for resources simulating singlet-state statistics.
• To establish a device-independent certification of randomness using only the complementarity relation, without assuming quantum mechanics or no-signaling.
• To unify existing models of nonlocality by showing that local bias (I < 1/2) fails to simulate singlet statistics due to violation of the complementarity bound.

Proposed method

• Defining signaling (S) as the maximum variation in outcome probability due to input changes, and indeterminacy (I) as the maximum min{P(o|xy), 1−P(o|xy)} over settings.
• Deriving the entropic versions HS and HI using mutual information and Shannon entropy to quantify signal and indeterminacy in information-theoretic terms.
• Constructing a general nonlocal resource as a convex combination of deterministic boxes, including signaling and non-signaling components.
• Using pre-shared randomness and directional vectors to simulate singlet-state correlations via a protocol involving sgn(·) functions and XOR operations.
• Proving that any simulation of singlet statistics must satisfy S + 2I ≥ 1 by setting communication cost C = 1 and deriving the inequality from the general resource form.
• Establishing the entropic complementarity HS + HI ≥ 1 by optimizing over outcome probabilities and using entropy bounds.

Experimental results

Research questions

• RQ1 What is the quantitative relationship between signaling and local indeterminacy in nonlocal correlations that simulate singlet-state statistics?
• RQ2 Does the conjecture S + 2I ≥ 1 hold for all nonlocal resources simulating the singlet state, as proposed by Hall (2010)?
• RQ3 Can randomness be certified in a device-independent way using only the complementarity between signaling and indeterminacy, without assuming no-signaling or quantum mechanics?
• RQ4 Why do local-bias models (I < 1/2) fail to reproduce singlet statistics, and how does complementarity explain this failure?
• RQ5 Is the entropic version of the complementarity relation, HS + HI ≥ 1, valid for nonlocal resources simulating the singlet state?

Key findings

• The paper proves that any nonlocal resource simulating singlet-state statistics must satisfy the inequality S + 2I ≥ 1, confirming a conjecture by Hall (2010).
• The entropic version of the complementarity relation, HS + HI ≥ 1, is established, verifying a conjecture by Kar et al. (2011) using information-theoretic measures of signal and indeterminacy.
• The bound S + 2I ≥ 1 explains why local-bias models (I < 1/2) cannot reproduce singlet statistics, as they violate the inequality even when S = 0.
• When S = 0 (no-signaling), the bound implies I = 1/2, showing that unbiased marginals are necessary for singlet simulation, consistent with Branciard et al. (2008).
• The complementarity relation provides a device-independent method to certify 1 bit of randomness from nonlocal correlations, even in the presence of signaling.
• The result strengthens Bell’s theorem by showing that nonlocality arises from a fundamental trade-off between signaling and indeterminacy, not just nonlocality alone.