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N-partners Quantum Communication and Bell's inequalities

Valerio Scarani, Nicolas Gisin

Published Jan 23, 2001• 1 citation
OpenAlex

Abstract

We present strong evidences for the existence of a link between the security of quantum communication protocols involving N partners and the violation of some Bell's inequalities. The link allows to establish optimal eavesdropping strategies for all N; in particular, we discuss eavesdropping on quantum secret sharing (N=3). The arguments are independent of the local hidden variable debate.

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We present strong evidences for the existence of a link between the security of quantum communication protocols involving N partners and the violation of some Bell's inequalities. The link allows to establish optimal eavesdropping strategies for all N; in particular, we discuss eavesdropping on quantum secret sharing (N=3). The arguments are independent of the local hidden variable debate.
Quantum secret-sharing protocols involving N partners are key distribution protocols in which Alice encodes her key into $N\ensuremath{-}1$ qubits, in such a way that all the other partners must cooperate in order to retrieve the key. On these protocols, several eavesdropping scenarios are possible: some partners may want to reconstruct the key without the help of the other ones, and consequently collaborate with an Eve that eavesdrops on the other partners' channels. For each of these scenarios, we give the optimal individual attack that Eve can perform. A link with Bell's inequality is demonstrated analytically for half of the scenarios, and is conjectured on the basis of numerical estimates for the other scenarios: the authorized partners have a higher information on the key than the unauthorized ones if and only if they can violate a Bell's inequality.
In secret sharing protocols, a secret is to be distributed among several partners such that leaving out any number of them, the rest do not have the complete information. Strong multiqubit correlations in the state by which secret sharing is carried out had been proposed as a criterion for security of such protocols against individual attacks by an eavesdropper. However we show that states with weak multiqubit correlations can also be used for secure secret sharing. That our state has weak multiqubit correlations is shown from the perspective of violation of local realism, and also by showing that its higher-order correlations are described by lower ones. We then present a unified criterion for security of secret sharing in terms of violation of local realism, which works when the secret sharing state is the Greenberger-Horne-Zeilinger state (with strong multiqubit correlations), as well as states of a different class (with weak multiqubit correlations).
We show that rejected-data protocols for quantum cryptography are secure against generalized intercept/resend strategies of an eavesdropper provided that the legitimate users of the communication channel use at least three bases. We discuss the connection between this result and the recently-developed protocol based on violations of a suitably-constructed Bell-type inequality for single particles. We also give a new estimate of the probability that an eavesdropper remains undetected under the original protocol and thereby show that the optimal strategies available to an eavesdropper are further limited to those which randomize the errors between the sub-ensembles of data. This result also has implications for the way in which the legitimate users of the channel choose their test data.
Quantum correlations which violate a Bell inequality are presumed to power better-than-classical protocols for solving communication complexity problems (CCPs). How general is this statement? We show that violations of correlation-type Bell inequalities allow advantages in CCPs, when communication protocols are tailored to emulate the Bell no-signaling constraint (by not communicating measurement settings). Abandonment of this restriction on classical models allows us to disprove the main result of, inter alia, \cite{BZ02}; we show that quantum correlations obtained from these communication strategies assisted by a small quantum violation of the CGLMP Bell inequalities do not imply advantages in any CCP in the input/output scenario considered in the reference. More generally, we show that there exists quantum correlations, with nontrivial local marginal probabilities, which violate the I3322 Bell inequality, but do not enable a quantum advantange in any CCP, regardless of the communication strategy employed in the quantum protocol, for a scenario with a fixed number of inputs and outputs