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arXiv:1810.05654 · Journal of the Optical Society of America B

Entropic uncertainty relations and the measurement range problem, with consequences for high-dimensional quantum key distribution

J. Eli Bourassa, Hoi‐Kwong Lo

Published Jan 31, 2019• 7 citations
arXivHTMLPDF

Abstract

The measurement range problem, where one cannot determine the data outside the range of the detector, limits the characterization of entanglement in high-dimensional quantum systems when employing, among other tools from information theory, the entropic uncertainty relations. Practically, the measurement range problem weakens the security of entanglement-based large-alphabet quantum key distribution (QKD) employing degrees of freedom including time-frequency or electric field quadrature. We present a modified entropic uncertainty relation that circumvents the measurement range problem under certain conditions and apply it to well-known QKD protocols. For time-frequency QKD, although our bound is an improvement, we find that high channel loss poses a problem for its feasibility. In homodyne-based continuous variable QKD, we find our bound provides a quantitative way to monitor for saturation attacks.

DOI
10.1364/josab.36.000b65

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The measurement range problem, where one cannot determine the data outside the range of the detector, limits the characterization of entanglement in high-dimensional quantum systems when employing, among other tools from information theory, the entropic uncertainty relations. Practically, the measurement range problem weakens the security of entanglement-based large-alphabet quantum key distribution (QKD) employing degrees of freedom including time-frequency or electric field quadrature. We present a modified entropic uncertainty relation that circumvents the measurement range problem under certain conditions and apply it to well-known QKD protocols. For time-frequency QKD, although our bound is an improvement, we find that high channel loss poses a problem for its feasibility. In homodyne-based continuous variable QKD, we find our bound provides a quantitative way to monitor for saturation attacks.
We derive a new entropic quantum uncertainty relation involving min-entropy. The relation is tight and can be applied in various quantum-cryptographic settings. Protocols for quantum 1-out-of-2 Oblivious Transfer and quantum Bit Commitment are presented and the uncertainty relation is used to prove the security of these protocols in the bounded quantum-storage model according to new strong security definitions. As another application, we consider the realistic setting of Quantum Key Distribution (QKD) against quantum-memory-bounded eavesdroppers. The uncertainty relation allows to prove the security of QKD protocols in this setting while tolerating considerably higher error rates compared to the standard model with unbounded adversaries. For instance, for the six-state protocol with one-way communication, a bit-flip error rate of up to 17% can be tolerated (compared to 13% in the standard model). Our uncertainty relation also yields a lower bound on the min-entropy key uncertainty against known-plaintext attacks when quantum ciphers are composed. Previously, the key uncertainty of these ciphers was only known with respect to Shannon entropy.
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It has been shown recently that the framework of quantum sampling, as introduced by Bouman and Fehr, can lead to new entropic uncertainty relations highly applicable to finite-key cryptographic analyses. Here we revisit these so-called sampling-based entropic uncertainty relations, deriving newer, more powerful, relations and applying them to source-independent quantum random number generators and high-dimensional quantum key distribution protocols. Along the way, we prove several interesting results in the asymptotic case for our entropic uncertainty relations. These sampling-based approaches to entropic uncertainty, and their application to quantum cryptography, hold great potential for deriving proofs of security for quantum cryptographic systems, and the approaches we use here may be applicable to an even wider range of scenarios.
Entropic uncertainty relations are underpinning to compute the quantitative security bound in quantum cryptographic applications, such as quantum random number generation (QRNG) and quantum key distribution (QKD). All security proofs derive a relation between the information accessible to the legitimate group and the maximum knowledge that an adversary may have gained, Eve, which exploits entropic uncertainty relations to lower bound Eve's uncertainty about the raw key generated by one party, Alice. The standard entropic uncertainty relations is to utilize the smooth min- and max-entropies to show these cryptographic applications' security by computing the overlap of two incompatible measurements or positive-operator valued measures (POVMs). This paper draws one case of the POVM-versioned standard entropic uncertainty relation yielding the trivial bound since the maximum overlap in POVMs always produces the trivial value, "one." So, it fails to tie the smooth min-entropy to show the security of the quantum cryptographic application.
It has been shown recently that the framework of quantum sampling, as introduced by Bouman and Fehr, can lead to new entropic uncertainty relations highly applicable to finite-key cryptographic analyses. Here we revisit these so-called sampling-based entropic uncertainty relations, deriving newer, more powerful, relations and applying them to source-independent quantum random number generators and high-dimensional quantum key distribution protocols. Along the way, we prove several interesting results in the asymptotic case for our entropic uncertainty relations. These sampling-based approaches to entropic uncertainty, and their application to quantum cryptography, hold great potential for deriving proofs of security for quantum cryptographic systems, and the approaches we use here may be applicable to an even wider range of scenarios.
By applying the framework of entropic uncertainty relation (EUR) and the Quantum Leftover Hash Lemma (QLHL), we introduce a security-proof method for variable-length side-channel-secure (SCS) quantum key distribution (QKD) against coherent attacks. This method reframes composable security as a statistical fluctuation problem of phase errors, enabling direct proofs against coherent attacks through observables and virtual observables. It yields tight key rates for the SCS protocol and reduces pulse requirements by over two orders of magnitude compared to prior works that employ the post-selection technique. {Utilizing the characteristic of the SCS protocol that it has no untagged-bit bit-flip errors, we prove that the secure key length for the SCS protocol can be determined after Cascade error correction, using its actual parity disclosures. We also explain how the same argument extends to protocols with compatible bit-error-pattern and phase-error measurements.} We further identify sufficient conditions under which the final key length may be determined after error correction in a broader class of QKD protocols. Under the framework of EUR and QLHL, we clarify the applicability of several commonly used concentration bounds to variable-length QKD and the appropriate manner of their implementation. This work enhances the practical value of the SCS protocol and clarifies the security justification of key-rate formulas used in practical variable-length QKD implementations.