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arXiv:quant-ph/0302078 · Physical Review A

Security of quantum key distributions with entangled qudits

Thomas Durt, Dagomir Kaszlikowski, Jing‐Ling Chen, L. C. Kwek

Published Mar 19, 2004• 124 citations
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Abstract

We consider a generalization of Ekert's entanglement-based quantum cryptographic protocol where qubits are replaced by $N$- or $d$-dimensional systems (qudits). In order to study its robustness against optimal incoherent attacks, we derive the information gained by a potential eavesdropper during a cloning-based individual attack. In doing so, we generalize Cerf's formalism for cloning machines and establish the form of the most general cloning machine that respects all the symmetries of the problem. We obtain an upper bound on the error rate that guarantees the confidentiality of qudit generalizations of the Ekert's protocol for qubits.

DOI
10.1103/physreva.69.032313

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We consider a generalization of Ekert's entanglement-based quantum cryptographic protocol where qubits are replaced by NNN- or ddd-dimensional systems (qudits). In order to study its robustness against optimal incoherent attacks, we derive the information gained by a potential eavesdropper during a cloning-based individual attack. In doing so, we generalize Cerf's formalism for cloning machines and establish the form of the most general cloning machine that respects all the symmetries of the problem. We obtain an upper bound on the error rate that guarantees the confidentiality of qudit generalizations of the Ekert's protocol for qubits.
The study of quantum cryptography and quantum entanglement have traditionally been based on two-level quantum systems (qubits). In this paper, we consider a generalization of Ekert's entanglement-based quantum cryptographic protocol where qubits are replaced by three-level systems (qutrits). In order to investigate the security against the optimal individual attack, we derive the information gained by a potential eavesdropper applying a cloning-based attack. We exhibit the explicit form of this cloner, which is distinct from the previously known cloners, and conclude that the protocol is more robust than those based on entangled qubits as well as unentangled qutrits.
We consider two quantum cryptographic schemes relying on encoding the key into qudits, i.e., quantum states in a d-dimensional Hilbert space. The first cryptosystem uses two mutually unbiased bases (thereby extending the BB84 scheme), while the second exploits all d+1 available such bases (extending the six-state protocol for qubits). We derive the information gained by a potential eavesdropper applying a cloning-based individual attack, along with an upper bound on the error rate that ensures unconditional security against coherent attacks.
We investigate the security bounds of quantum-cryptographic protocols using ddd-level systems. In particular, we focus on schemes that use two mutually unbiased bases, thus extending the Bennett-Brassard 1984 quantum-key-distribution scheme to higher dimensions. Under the assumption of general coherent attacks, we derive an analytic expression for the ultimate upper security bound of such quantum-cryptography schemes. This bound is well below the predictions of optimal cloning machines. The possibility of extraction of a secret key beyond entanglement distillation is discussed. In the case of qutrits we argue that any eavesdropping strategy is equivalent to a symmetric one. For higher dimensions such an equivalence is generally no longer valid.
We propose an extension of quantum key distribution based on encoding the key into quNits, i.e. quantum states in an N -dimensional Hilbert space. In our protocol, we adopt asymmetric measurement scheme resulting in an efficiency higher than previous protocols and a generalized Bell inequality [Phys. Rev. Lett. 92 (2004) 130404] is employed to detect the presence of an eavesdropper Eve. We also derive the information gained by a potential eavesdropper Eve applying a cloning-based attack and the maximal error rate which measures the robustness of the protocol. The result shows that the security of our scheme increases with the dimension N .
We analyze the security of an Ekert type bipartite key distribution protocol coming under cloning attacks by an untrusted observer. Precisely, an eavesdropper is assumed to intercept and apply an asymmetric, phase-variant cloner on one or both qubits which are transmitted over public channel from an authenticated source to the trusted parties. We partially characterize two-qubit states from the perspective of designing a QKD protocol that can generate a distillable secret key among the trusted users even under such cloning attacks. We rely upon Devetak Winter framework for computing key rate so as to identify states, which when used in the protocol, secret-key can be distilled. Utility of an entangled state, for designing a successful QKD protocol, changes from single-qubit to double-qubits cloning attacks. Interestingly, we can distinguish some useful families of two-qubit entangled states when only one of the transmitted qubits is cloned by a malicious third party.
The Ekert quantum key distribution protocol uses pairs of entangled qubits and performs checks based on a Bell inequality to detect eavesdropping. The N-DEB protocol uses instead pairs of entangled qudits to achieve better noise resistance than the Ekert protocol. It performs checks based on the Bell CGLMP inequality for qudits. In this paper, we present the generalization for qudits of our protocol h3DEB (for qutrits). This protocol also uses pairs of entangled qudits, but achieves even better noise resistance than N-DEB and is showed to be secure against the same family of cloning attacks than N-DEB. This gain of performance is obtained by using another inequality called here hCHSH-ddd. For each party, the hCHSH-ddd inequality involves 2d2d2d observables. We explain how the parties can measure these observables and thus are able to check the violation of hCHSH-ddd. In the presence of noise, this violation allows the parties to ensure the secrecy of the key because it guarantees the absence of a local Trojan horse attack. The advantage of our proposed scheme is that it results in an increased resistance to noise while remaining secure against individual attacks.