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arXiv:quant-ph/0402135 · Physical Review A

Spherical-code key-distribution protocols for qubits

Joseph M. Renes

Published Nov 16, 2004• 60 citations
arXivHTMLPDF

Abstract

Recently spherical codes were introduced as potentially more capable ensembles for quantum key distribution. Here we develop specific key-creation protocols for the two qubit-based spherical codes, the trine and tetrahedron, and analyze them in the context of a suitably tailored intercept/resend attack, both in standard form, and in a ``gentler'' version whose back action on the quantum state is weaker. When compared to the standard unbiased basis protocols, Bennett-Brassard 1984 (BB84) and six-state, two distinct advantages are found. First, they offer improved tolerance of eavesdropping, the trine besting its counterpart BB84 and the tetrahedron the six-state protocol. Second, the key error rate may be computed from the sift rate of the protocol itself, removing the need to sacrifice key bits for this purpose. This simplifies the protocol and improves the overall key rate.

DOI
10.1103/physreva.70.052314

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Recently spherical codes were introduced as potentially more capable ensembles for quantum key distribution. Here we develop specific key-creation protocols for the two qubit-based spherical codes, the trine and tetrahedron, and analyze them in the context of a suitably tailored intercept/resend attack, both in standard form, and in a ``gentler'' version whose back action on the quantum state is weaker. When compared to the standard unbiased basis protocols, Bennett-Brassard 1984 (BB84) and six-state, two distinct advantages are found. First, they offer improved tolerance of eavesdropping, the trine besting its counterpart BB84 and the tetrahedron the six-state protocol. Second, the key error rate may be computed from the sift rate of the protocol itself, removing the need to sacrifice key bits for this purpose. This simplifies the protocol and improves the overall key rate.
Quantum key distribution (QKD) protocols are cryptographic techniques with security based only on the laws of quantum mechanics. Two prominent QKD schemes are the Bennett-Brassard 1984 and Bennett 1992 protocols that use four and two quantum states, respectively. In 2000, Phoenix et al. proposed a new family of three-state protocols that offers advantages over the previous schemes. Until now, an error rate threshold for security of the symmetric trine spherical code QKD protocol has been shown only for the trivial intercept-resend eavesdropping strategy. In this Letter, we prove the unconditional security of the trine spherical code QKD protocol, demonstrating its security up to a bit error rate of 9.81%. We also discuss how this proof applies to a version of the trine spherical code QKD protocol where the error rate is evaluated from the number of inconclusive events.
Quantum key distribution protocols based on equiangular spherical codes are introduced and their behavior under the intercept/resend attack investigated. Such protocols offer a greater range of secure noise tolerance and speed options than protocols based on their cousins, the mutually-unbiased bases, while also enabling the determination of the channel noise rate without the need to sacrifice key bits. For fixed number of signal states in a given dimension, the spherical code protocols offer Alice and Bob more noise tolerance at the price of slower key generation rates.
Mutually unbiased bases have been extensively studied in the literature and are simple and effective in quantum key distribution protocols, but they are not optimal. Here equiangular spherical codes are introduced as a more efficient and robust resource for key distribution. Such codes are sets of states that are as evenly spaced throughout the vector space as possible. In the case the two parties use qubits and face the intercept/resend eavesdropping strategy, they can make use of three equally-spaced states, called a \emph{trine}, to outperform the original four-state BB84 protocol in both speed and reliability. This points toward the optimality of spherical codes in arbitrary dimensions.
We introduce a new quantum key distribution protocol that uses d-level quantum systems to encode an alphabet with c letters. It has the property that the error rate introduced by an intercept-and-resend attack tends to one as the numbers c and d increase. In dimension d=2, when the legitimate parties use a complete set of three mutually unbiased bases, the protocol achieves a quantum bit error rate of 57.1%. This represents a significant improvement over the 25% quantum bit error rate achieved in the BB84 protocol or 33% in the six-state protocol.
Quantum key distribution (QKD) allows two spatially separated parties to securely generate a cryptographic key. The first QKD protocol, published by C. H. Bennett and G. Brassard in 1984 (BB84), describes how this is achieved by transmitting individual qubits and exchanging classical authenticated information. Any attempt to eavesdrop on the protocol introduces errors detectable by the legitimate parties. This Licentiate Thesis studies the recently introduced EEQKD protocol which builds on BB84. In EEQKD, the qubits sent individually in BB84 are entangled and thus not directly available to an eavesdropper who is, in this protocol, provided only one-by-one access to the transmission. The maximal information an eavesdropper can gain using a straightforward intercept-resend (IR) attack, is obtained for a given error rate. The secure key generation rate of EEQKD is estimated in practical scenarios including qubit loss and quantum channel noise. In addition, an exquisite vulnerability is exposed: For a particular setting of qubit entanglement, paradoxically proving most useful in the face of an IR attack, EEQKD reduces to BB84. Part of this research has been published in O. Ahonen et al., Physical Review A 78, 032314 (2008).
Mutually unbiased bases have been extensively studied in the literature and are simple and effective in quantum key distribution protocols, but they are not optimal. Here equiangular spherical codes are introduced as a more efficient and robust resource for key distribution. Such codes are sets of states that are as evenly spaced throughout the vector space as possible. In the case of qubits, the two parties can make use of three equally-spaced states, called a trine, to outperform the original four-state BB84 protocol in both speed and reliability. This points toward the optimality of spherical codes in arbitrary dimensions.