Device-independent secret sharing and a stronger form of Bell nonlocality

M. G. M. Moreno, Samuraí Brito, Ranieri V. Nery, and Rafael Chaves

Phys. Rev. A 101, 052339 – Published 27 May 2020

Abstract

Bell nonlocality, the fact that local hidden-variable models cannot reproduce the correlations obtained by measurements on entangled states, is a cornerstone in our modern understanding of quantum theory. Apart from its fundamental implications, nonlocality is also at the core of device-independent quantum information processing, the successful implementation of which is achieved without precise knowledge of the physical apparatus. Here we show that a stronger form of Bell nonlocality, for which even some nonlocal hidden-variable models cannot reproduce the quantum predictions, allows us to circumvent possible attacks in the implementation of secret sharing, a paradigmatic communication protocol in which a secret split amid many possibly untrusted parts can be decoded only if they collaborate among themselves.

Received 30 September 2019
Accepted 6 May 2020

DOI:https://doi.org/10.1103/PhysRevA.101.052339

Physics Subject Headings (PhySH)

Nonlocality Quantum cryptography Quantum entanglement Quantum networks

Quantum Information, Science & Technology

Authors & Affiliations

M. G. M. Moreno ¹, Samuraí Brito ¹, Ranieri V. Nery¹, and Rafael Chaves ^1,2

¹International Institute of Physics, Federal University of Rio Grande do Norte, 59070-405 Natal, Brazil
²School of Science and Technology, Federal University of Rio Grande do Norte, 59078-970 Natal, Brazil

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Vol. 101, Iss. 5 — May 2020

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Images

Figure 1
DAG representation of causal structures. In classical causal models the common source of correlations is described by a random variable $Λ$ , while in a quantum description we have a density operator $ρ$ describing a (potentially entangled) multipartite quantum state. (a) Tripartite scenario with an external eavesdropper that can share correlations with Alice, Bob, and Charlie but has no direct access to their inputs or outputs. (b) If Alice is untrusted, she can correlate her input with the source $Λ$ (represented by an arrow from $X$ to $Λ$ ) and try to guess the output $c$ via an extra outcome $e$ (unknown to Bob and Charlie). Notice that the same set of correlations can be obtained when the direction of the arrow between $X$ and $Λ$ is reversed [34]. (c) Causal structure where the input of Alice is broadcast to Bob and Charlie and can generate the same NS correlations as the DAG in Fig. 1. (d) DAG characterizing the correlations obtainable by an untrusted Bob.
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Figure 2
Guessing probability as a function of Svetlichny's inequality violation. The upper bound for Eve's guessing probability $P_{guess} : = p (e = c | x^{*} y^{*} z^{*})$ for a generic quantum behavior is shown by the solid curve. This bound is obtained from the level $2 + ABC + ABE + BCE + ABCE$ of the NPA hierarchy [49]. Even with the approximation, it can be seen that for the maximal quantum violation $4 \sqrt{2}$ of the inequality, perfect security for Charlie's bit can be obtained as the guessing probability approaches 0.5. The dashed curve represents the optimal NS guessing probability.
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Figure 3
Guessing probability as a function of the quantum state visibility. We consider the parties to share a state $ρ_{v} = v | GHZ 〉 〈 GHZ | + (1 - v) 1 / 8$ and perform the optimal measurements maximizing the violation of (6). The straight blue curve gives the guessing probability $p (e = c | x^{*} y^{*} z^{*})$ considering the level $2 + ABC + ABE + BCE + ABCE$ in the semidefinite approximation of the quantum set (8), while the dashed straight line considers the NS set (7) (see the Appendix for more details). The dot-dashed red curve shows the second level of approximation to the quantum set without the extra terms; it can be seen that this approximation to the set of quantum distributions is very crude for the scenario considered here but already reveals the onset of security for Charlie's bit for high values of the visibility.
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Figure 4
DAG representation of another way of picturing an untrusted part; here the lack of ”free will” becomes more evident.
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Figure 5
Guessing probability with the secret-sharing condition imposed. Upper bound for Eve's guessing probability for a generic quantum behavior as function of the violation of inequality (6) with the additional constraint that $a \oplus b \oplus c = 0$ . It should be noted that, in contrast to the case with no secret sharing (Fig. 2 of the main text), now the optimization becomes infeasible for $S \approx 5.29$ , before the quantum bound $4 \sqrt{2}$ . Nonetheless, it is still possible to certify some randomness on Charlie's bit as the guessing probability approaches $52 %$ for the values of violation shown here.
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