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Non-restrictive state reduction and analytical bounds in a multipartite device-independent scenario

Federico Grasselli, Gláucia Murta, Hermann Kampermann, Dagmar Bruß

Published Apr 29, 2020• 0 citations
OpenAlex

Abstract

Multiparty quantum cryptography based on distributed entanglement will find its natural application in the upcoming quantum networks. The security of many multipartite device-independent (DI) protocols, such as DI conference key agreement, relies on bounding the von Neumann entropy of the parties' outcomes conditioned on the eavesdropper's information, given the violation of a multipartite Bell inequality. We consider three parties testing the Mermin-Ardehali-Belinskii-Klyshko (MABK) inequality and certify the privacy of their outcomes by bounding the conditional entropy of a single party's outcome and the joint conditional entropy of two parties' outcomes. From the former bound, we show that genuine multipartite entanglement is necessary to certify the privacy of a party's outcome, while the latter significantly improve previous results. We obtain the entropy bounds thanks to two general results of independent interest. The first one drastically simplifies the quantum setup of an $N$-partite Bell scenario. The second one provides an upper bound on the violation of the MABK inequality by an arbitrary $N$-qubit state, as a function of the state's parameters.

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Multiparty quantum cryptography based on distributed entanglement will find its natural application in the upcoming quantum networks. The security of many multipartite device-independent (DI) protocols, such as DI conference key agreement, relies on bounding the von Neumann entropy of the parties' outcomes conditioned on the eavesdropper's information, given the violation of a multipartite Bell inequality. We consider three parties testing the Mermin-Ardehali-Belinskii-Klyshko (MABK) inequality and certify the privacy of their outcomes by bounding the conditional entropy of a single party's outcome and the joint conditional entropy of two parties' outcomes. From the former bound, we show that genuine multipartite entanglement is necessary to certify the privacy of a party's outcome, while the latter significantly improve previous results. We obtain the entropy bounds thanks to two general results of independent interest. The first one drastically simplifies the quantum setup of an NNN-partite Bell scenario. The second one provides an upper bound on the violation of the MABK inequality by an arbitrary NNN-qubit state, as a function of the state's parameters.
Multiparty quantum cryptography based on distributed entanglement will find its natural application in the upcoming quantum networks.The security of many multipartite device-independent (DI) protocols, such as DI conference-key agreement, relies on bounding the von Neumann entropy of the parties' outcomes conditioned on the eavesdropper's information, given the violation of a multipartite Bell inequality.We consider three parties testing the Mermin-Ardehali-Belinskii-Klyshko (MABK) inequality and certify the privacy of their outcomes by bounding the conditional entropy of a single party's outcome and the joint conditional entropy of two parties' outcomes.From the former bound, we show that genuine multipartite entanglement is necessary to certify the privacy of a party's outcome, while the latter significantly improves previous results.We obtain the entropy bounds thanks to two general results of independent interest.The first one drastically simplifies the quantum setup of an N -partite Bell scenario.The second one provides an upper bound on the violation of the MABK inequality by an arbitrary N -qubit state, as a function of the state's parameters.
Device-independent (DI) protocols, such as DI conference key agreement (DICKA) and DI randomness expansion (DIRE), certify private randomness by observing nonlocal correlations when two or more parties test a Bell inequality. While most DI protocols are restricted to bipartite Bell tests, harnessing multipartite nonlocal correlations may lead to better performance. Here, we consider tripartite DICKA and DIRE protocols based on testing multipartite Bell inequalities, specifically: the Mermin-Ardehali-Belinskii-Klyshko (MABK) inequality, and the Holz and the Parity-CHSH inequalities introduced in the context of DICKA protocols. We evaluate the asymptotic performance of the DICKA (DIRE) protocols in terms of their conference key rate (net randomness generation rate), by deriving lower bounds on the conditional von Neumann entropy of one party's outcome and two parties' outcomes. For the Holz inequality, we prove a tight analytical lower bound on the one-outcome entropy and conjecture a tight lower bound on the two-outcome entropy. We additionally re-derive the analytical one-outcome entropy bound for the MABK inequality with a much simpler method and obtain a numerical lower bound on the two-outcome entropy for the Parity-CHSH inequality. Our simulations show that DICKA and DIRE protocols employing tripartite Bell inequalities can significantly outperform their bipartite counterparts. Moreover, we establish that genuine multipartite entanglement is not a precondition for multipartite DIRE while its necessity for DICKA remains an open question.
We propose two semi-device-independent approaches that are able to quantify unknown multipartite quantum entanglement experimentally, where the only information that has to be known beforehand is quantum dimension, and the concept that plays a key role is nondegenerate Bell inequalities. Specifically, using the nondegeneracy of multipartite Bell inequalities, we obtain useful information on the purity of target quantum state. Combined with an estimate of the maximal overlap between the target state and pure product states, and a continuous property of the geometric measure of entanglement, we shall prove the information on purity allows us to give a lower bound for this entanglement measure. In addition, we show that a different combination of the above results also converts to a lower bound for the relative entropy of entanglement. As a demonstration, we apply our approach on 5-partite qubit systems with the Mermin-Ardehali-Belinskii-Klyshko (MABK) inequality, and show that useful lower bounds for the geometric measure of entanglement can be obtained if the Bell expression value is larger than 3.60, and those for the relative entropy of entanglement can be given if the Bell expression value is larger than 3.80, where the Tsirelson bound is 4.
We present a security analysis of conference key agreement (CKA) in the most adversarial model of device independence (DI). Our protocol can be implemented by any experimental setup that is capable of performing Bell tests [specifically, the Mermin-Ardehali-Belinskii-Klyshko (MABK) inequality], and security can in principle be obtained for any violation of the MABK inequality that detects genuine multipartite entanglement among the NNN parties involved in the protocol. As our main tool, we derive a direct physical connection between the NNN-partite MABK inequality and the Clauser-Horne-Shimony-Holt (CHSH) inequality, showing that certain violations of the MABK inequality correspond to a violation of the CHSH inequality between one of the parties and the other $N\ensuremath{-}1$. We compare the asymptotic key rate for device-independent conference key agreement (DICKA) to the case where the parties use $N\ensuremath{-}1$ device-independent quantum key distribution protocols in order to generate a common key. We show that for some regime of noise the DICKA protocol leads to better rates.
Abstract Position-based cryptography allows a party to prove its geographic location by responding to challenges from verifiers. Classical position verification is impossible against entangled adversaries. Quantum position verification overcomes this but requires trusted measurement devices at the verifi cation locations. We introduce Device-Independent Position Verification using the Mermin-Ardehali Belinskii-Klyshko four-party Bell inequality. Our protocol uses four verifiers at the corners of a square and a single prover. The MABK inequality has classical bound BMABK ≤ 4, while quantum mechan ics achieves BMABK = 4√2 ≈ 5.657 using the four-qubit Greenberger-Horne-Zeilinger state. We prove security against general quantum adversaries using relativistic no-go theorems and the entropy accumulation theorem. Using realistic experimental parameters (four-photon GHZ state fidelity F = 0.85, detector efficiency η = 0.65), we predict a Bell violation BMABK = 4.25, exceeding the classical bound by 0.25. The protocol tolerates up to 15% noise and achieves verification distances up to 30 km with current technology. This work provides the first practical path to location verification on untrusted hardware.
We present a security analysis of a multiparty quantum key distribution (QKD) based on Mermin-Ardehali-Belinskiĭ-Klyshko (MABK) type multipartite nonlocality. Our analysis provides an asymptotic secret key rate of the multiparty QKD under the restriction that successive trials are completely independent. In our analysis, we construct the relation between a secret key rate of the multiparty QKD and an expectation value of MABK operator. We show that side channel attacks, which can threaten the information theoretic (IT) security analysis of multiparty QKD, are prevented in our analysis. We compare secret key rates obtained by using the IT analysis, the existing fully device-independent analysis, and our analysis. It is shown that efficiency of the multiparty QKD can be improved with increasing number of authorized parties in our analysis as well.