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arXiv:1903.12154 · Physical Review A

Limitations on a device-independent key secure against a nonsignaling adversary via squashed nonlocality

Marek Winczewski, Tamoghna Das, Karol Horodecki

Published Nov 29, 2022• 6 citations
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Abstract

We initiate a systematic study to provide upper bounds on device-independent keys, secure against a nonsignaling adversary (NSDI). We employ the idea of ``squashing'' on the secrecy monotones and show that squashed secrecy monotones are the upper bounds on the NSDI key. Our technique for obtaining upper bounds is based on the nonsignaling analog of quantum purification: the complete extension. As an important instance of an upper bound, we construct a measure of nonlocality called ``squashed nonlocality.'' Using this bound, we identify numerically a certain domain of two binary inputs and two binary outputs of nonlocal devices for which the squashed nonlocality is zero. Therefore one can not distill a secure key from these nonlocal devices via a considered (standard) class of operations. Showing a connection of our approach to the one in New J. Phys. 8, 126 (2006), we provide, to our knowledge, the tightest known upper bound in the (3,2,2,2) scenario. Moreover, we formulate a security condition, equivalent to known ones, for the considered class of protocols. To achieve this, we introduce a nonsignaling norm that constitutes an analogy to the trace norm used in the security condition of the quantum key distribution.

DOI
10.1103/physreva.106.052612

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We initiate a systematic study to provide upper bounds on device-independent keys, secure against a nonsignaling adversary (NSDI). We employ the idea of squashing'' on the secrecy monotones and show that squashed secrecy monotones are the upper bounds on the NSDI key. Our technique for obtaining upper bounds is based on the nonsignaling analog of quantum purification: the complete extension. As an important instance of an upper bound, we construct a measure of nonlocality called squashed nonlocality.'' Using this bound, we identify numerically a certain domain of two binary inputs and two binary outputs of nonlocal devices for which the squashed nonlocality is zero. Therefore one can not distill a secure key from these nonlocal devices via a considered (standard) class of operations. Showing a connection of our approach to the one in New J. Phys. 8, 126 (2006), we provide, to our knowledge, the tightest known upper bound in the (3,2,2,2) scenario. Moreover, we formulate a security condition, equivalent to known ones, for the considered class of protocols. To achieve this, we introduce a nonsignaling norm that constitutes an analogy to the trace norm used in the security condition of the quantum key distribution.
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.
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.
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.
The aim of this thesis is to understand the fundamental limitations on secret key distillation in various settings of quantum key distribution. We first consider quantum steering, which is a resource for one-sided device-independent quantum key distribution. We introduce a conditional mutual information based quantifier for quantum steering, which we call intrinsic steerability. Next, we consider quantum non-locality, which is a resource for device-independent quantum key distribution. In this context, we introduce a quantifier, intrinsic non-locality, which is a monotone in the resource theory of Bell non-locality. Both these quantities are inspired by intrinsic information and squashed entanglement and are based on conditional mutual information. The idea behind these quantifiers is to suppress the correlations that can be explained by a local hidden variable or by an inaccessible quantum system, thus quantifying the remaining intrinsic correlations. We then prove various properties of these two monotones, which includes the following: monotonicity under free operations, additivity under tensor product of objects, convexity, and faithfulness, among others. Next, we prove that intrinsic steerability is an upper bound on the secret-key-agreement capacity of an assemblage, and intrinsic non-locality is an upper bound on the secret-key-agreement capacity of a quantum probability distribution. Thus we prove that these quantities are upper bounds on the achievable key rates in one-sided device-independent and device-independent quantum key distribution protocols. We also calculate these bounds for certain honest devices. The study of these upper bounds is instrumental in understanding the limitations of protocols that can be designed for various settings. These upper bounds inform us that, even if one considers the best possible protocol, there is no possibility of exceeding the upper bounds on key rates without a quantum repeater. The upper bounds introduced in this thesis are an important step for initiating this line of research in one-sided device-independent and in device-independent quantum key distribution.
Device-independent quantum key distribution is a secure quantum cryptographic paradigm that allows two honest users to establish a secret key, while putting minimal trust in their devices. Most of the existing protocols have the following structure: first, a bipartite nonlocal quantum state is distributed between the honest users, who perform local projective measurements to establish nonlocal correlations. Then, they announce the implemented measurements and extract a secure key by postprocessing their measurement outcomes. We show that no protocol of this form allows for establishing a secret key when implemented on any correlation obtained by measuring local projective measurements on certain entangled nonlocal states, namely, on a range of entangled two-qubit Werner states. To prove this result, we introduce a technique for upper bounding the asymptotic key rate of device-independent quantum key distribution protocols, based on a simple eavesdropping attack. Our results imply that either different reconciliation techniques are needed for device-independent quantum key distribution in the large-noise regime, or Bell nonlocality is not sufficient for this task.
Device-independent quantum key distribution allows for proving the security of a shared cryptographic key between two distant parties with potentially untrusted devices. The security proof is based on the measurement outcome statistics (correlation) of a Bell experiment, and security is guaranteed by the laws of quantum theory. While it is known that the observed correlation must be Bell nonlocal in order to prove security, recent results show that Bell nonlocality is in general not sufficient for standard device-independent quantum key distribution. In this work, we show that conversely, there is no lower bound on the amount of nonlocality that is sufficient for device-independent quantum key distribution. Even more so, we show that from certain correlations that exhibit arbitrarily small nonlocality, one can still extract unbounded device-independent key rates. Therefore, a quantitative relation between device-independent key rates and Bell nonlocality cannot be drawn in general. Our main technique comprises a rigorous connection between self-testing and device-independent quantum key distribution, applied to a recently discovered family of Bell inequalities with arbitrarily many measurement outcomes.