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arXiv:2107.06411 · Physical Review Applied

Upper Bounds on Device-Independent Quantum Key Distribution Rates in Static and Dynamic Scenarios

Eneet Kaur, Karol Horodecki, Siddhartha Das

Published Nov 10, 2022• 13 citations
arXivHTMLPDF

Abstract

In this work, we develop upper bounds on key rates for device-independent quantum key distribution (DI-QKD) protocols and devices. We study the reduced $\mathrm{cc}$-squashed entanglement and show that it is a convex functional. As a result, we show that the convex hull of the currently known bounds is a tighter upper bound on the device-independent key rates of the standard Clauser-Horne-Shimony-Holt (CHSH)-based protocol. We further provide tighter bounds for DI-QKD key rates achievable by any protocol applied to the CHSH-based device. This bound is based on reduced relative entropy of entanglement optimized over decompositions into local and nonlocal parts. In the dynamical scenario of quantum channels, we obtain upper bounds for device-independent private capacity for the CHSH-based protocols. We show that the device-independent private capacity for the CHSH-based protocols on depolarizing and erasure channels is limited by the secret key capacity of dephasing channels.

DOI
10.1103/physrevapplied.18.054033

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In this work, we develop upper bounds on key rates for device-independent quantum key distribution (DI-QKD) protocols and devices. We study the reduced cc\mathrm{cc}cc-squashed entanglement and show that it is a convex functional. As a result, we show that the convex hull of the currently known bounds is a tighter upper bound on the device-independent key rates of the standard Clauser-Horne-Shimony-Holt (CHSH)-based protocol. We further provide tighter bounds for DI-QKD key rates achievable by any protocol applied to the CHSH-based device. This bound is based on reduced relative entropy of entanglement optimized over decompositions into local and nonlocal parts. In the dynamical scenario of quantum channels, we obtain upper bounds for device-independent private capacity for the CHSH-based protocols. We show that the device-independent private capacity for the CHSH-based protocols on depolarizing and erasure channels is limited by the secret key capacity of dephasing channels.
We provide several general upper bounds on the rate of a key secure against a quantum adversary in the device-independent conference key agreement (DI-CKA) scenario. They include bounds by reduced entanglement measures and those based on multipartite secrecy monotones such as a multipartite squashed entanglement-based measure, which we refer to as reduced c-squashed entanglement. We compare the latter bound with the known lower bound for the protocol of conference key distillation based on the parity Clauser-Horne-Shimony-Holt game. We also show that the gap between the DI-CKA rate and the device-dependent rate is inherited from the bipartite gap between device-independent and device-dependent key rates, giving examples that exhibit the strict gap.
Device-independent quantum key distribution (DIQKD) is one of the most challenging tasks in quantum cryptography. The protocols and their security are based on the existence of Bell inequalities and the ability to violate them by measuring entangled states. We study the entanglement needed for DIQKD protocols in two different ways. Our first contribution is the derivation ofupper boundson the key rates of CHSH-based DIQKD protocols in terms of the violation of the inequality; this sets an upper limit on the possible DI key extraction rate from states with a given violation. Our upper bound improves on the previously known bound of Kaur et al. Our second contribution is the initiation of the study of the role ofbound entangledstates in DIQKD. We present a revised Peres conjecture stating that such states cannot be used as a resource for DIQKD. We give a first piece of evidence for the conjecture by showing that the bound entangled state found by Vertesi and Brunner, even though it can certify DI randomness, cannot be used to produce a key using protocols analogous to the well-studied CHSH-based DIQKD protocol.
We initiate a systematic study to provide upper bounds on device-independent key, secure against a non-signaling adversary (NSDI), distilled by a wide class of operations, currently used in both quantum and non-signaling device-independent protocols. These operations consist of a direct measurements on the devices followed by Local Operations and Public Communication (MDLOPC). We employ the idea of squashing on the secrecy monotones, which provide upper bounds on the key rate in secret key agreement (SKA) scenario, and show that secrecy monotones are the upper bounds on NSDI key. As an important instance, an upper bound on NSDI key rate called squashed non-locality, has been constructed. It exhibits several important properties, including convexity, monotonicity, additivity on tensor products, and asymptotic continuity. Using this bound, we identify numerically a domain of two binary inputs and two binary outputs non-local devices for which the non-locality is zero, and therefore one can not distil key from them via MDLOPC operations. These are mixtures of Popescu-Rohrlich (PR) and anti-PR box with the weight of PR box less than 80%80\%80%. This example confirms the intuition that non-locality need not imply secrecy in the non-signaling scenario. The approach is general, describing how to construct other tighter yet possibly less computable upper bounds. Our technique for obtaining upper bounds is based on the non-signaling analog of quantum purification: the complete extension, which yields equivalent security conditions as previously known in the literature.
Abstract In this paper, we introduce intrinsic non-locality and quantum intrinsic non-locality as quantifiers for Bell non-locality, and we prove that they satisfy certain desirable properties such as faithfulness, convexity, and monotonicity under local operations and shared randomness. We then prove that intrinsic non-locality is an upper bound on the secret-key-agreement capacity of any device-independent protocol conducted using a device characterized by a correlation p, while quantum intrinsic non-locality is an upper bound on the same capacity for a correlation arising from an underlying quantum model. We also prove that intrinsic steerability is faithful, and it is an upper bound on the secret-key-agreement capacity of any one-sided-device-independent protocol conducted using a device characterized by an assemblage ρ ˆ . Finally, we prove that quantum intrinsic non-locality is bounded from above by intrinsic steerability.
In this paper, we introduce intrinsic non-locality and quantum intrinsic non-locality as quantifiers for Bell non-locality, and we prove that they satisfy certain desirable properties such as faithfulness, convexity, and monotonicity under local operations and shared randomness. We then prove that intrinsic non-locality is an upper bound on the secret-key-agreement capacity of any device-independent protocol conducted using a device characterized by a correlation p, while quantum intrinsic non-locality is an upper bound on the same capacity for a correlation arising from an underlying quantum model. We also prove that intrinsic steerability is faithful, and it is an upper bound on the secret-key-agreement capacity of any one-sided-device-independent protocol conducted using a device characterized by an assemblage ρˇ. Finally, we prove that quantum intrinsic non-locality is bounded from above by intrinsic steerability.
Abstract Device-independent quantum key distribution (DI-QKD) can guarantee the security even with untrusted devices. Unfortunately, conventional DI-QKD protocols can tolerate low noises only and require high detection efficiencies to achieve positive key rates. To improve the noise robustness, one promising solution is implementing the two-way classical postprocessing, which has the advantage of reducing the bit errors. In this paper, we study the DI-QKD with B-step two-way classical postprocessing under individual attacks. We adopt the tool of convex combination attack, i.e. an optimal individual attack, to upper bound the Devetak-Winter key rate. We show that, by using the B-step procedure, our protocol can tolerate detection efficiencies as low as 81.0 % and depolarising noise of 0.799, which is better than the thresholds for the protocol with one-way error correction. This result can serve as the lower bounds on the critical noise and detection efficiency for the scenarios under general attacks. Our work justifies the advantage of two-way classical postprocessing for DI-QKD, thus offering a step towards its applications.