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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Comparative characteristics of quantum key distribution protocols with alphabets corresponding to the regular polyhedrons on the Bloch sphere

Сыч Денис Васильевич, B. A. Grishanin, Victor N. Zadkov

Published Jun 1, 2005• 1 citation
Publisher

Abstract

Possibilities of improving characteristics of quantum key distribution (QKD) protocols via variation of character set in quantum alphabets are investigated. QKD protocols with discrete alphabets letters of which form regular polyhedrons on the Bloch sphere (tetrahedron octahedron cube icosahedron and dodecahedron which have 4, 6, 8, 12, and 20 vertexes) and QKD protocol with continuous alphabet which corresponds to the limiting case of a polyhedron with infinitive number of vertexes are considered. Stability of such QKD protocols to the interceptresend and optimal eavesdropping strategies at the individual attacks is studied in detail. It is shown that in case of optimal eavesdropping strategy after safety bases reconciliation critical error rate of the QKD protocol with continuous alphabet surpasses all other protocols. Without basis reconciliation the highest critical error rate have the protocol with tetrahedron-type alphabet.

DOI
10.1117/12.620514

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Possibilities of improving characteristics of quantum key distribution (QKD) protocols via variation of character set in quantum alphabets are investigated. QKD protocols with discrete alphabets letters of which form regular polyhedrons on the Bloch sphere (tetrahedron octahedron cube icosahedron and dodecahedron which have 4, 6, 8, 12, and 20 vertexes) and QKD protocol with continuous alphabet which corresponds to the limiting case of a polyhedron with infinitive number of vertexes are considered. Stability of such QKD protocols to the interceptresend and optimal eavesdropping strategies at the individual attacks is studied in detail. It is shown that in case of optimal eavesdropping strategy after safety bases reconciliation critical error rate of the QKD protocol with continuous alphabet surpasses all other protocols. Without basis reconciliation the highest critical error rate have the protocol with tetrahedron-type alphabet.
Possibilities of improving critical error rate of quantum key distribution (QKD) protocols for different strategies of eavesdropping are investigated. QKD-protocols with discrete alphabets letters of which form regular polyhedrons on the Bloch sphere (tetrahedron octahedron cube icosahedron and dodecahedron which have 4, 6, 8, 12 and 20 vertexes respectively) and QKD-protocol with continuous alphabet which corresponds to the limiting case of a polyhedron with infinitive number of vortexes are considered. Stability of such QKD-protocols to the noise in a quantum channel which is due to the Eve's interference that apply either intercept-receipt or optimal eavesdropping strategy at the individual attacks is studied in detail. It is shown that in case of optimal eavesdropping strategy after bases reconciliation the QKD-protocol with continuous alphabet surpasses all other protocols in terms of noise-resistance. Without basis reconciliation the highest critical error rate have the protocol with tetrahedron-type alphabet.
The problem of increasing the critical error rate of quantum-cryptography protocols by varying a set of letters in a quantum alphabet for space of a fixed dimensionality is studied. Quantum alphabets forming regular polyhedra on the Bloch sphere and the continual alphabet equally including all the quantum states are considered. It is shown that, in the absence of basis reconciliation, a protocol with the tetrahedral alphabet has the highest critical error rate among the protocols considered, while after the basis reconciliation, a protocol with the continual alphabet possesses the highest critical error rate.
A novel quantum key distribution protocol (QKD) based on all unselected states of a quantum sys- tem that set the alphabet with a continuous set of letters is proposed. Employing all states of the Hilbert space leads to a maximum quantum uncertainty of transmitted states, and, therefore, an eavesdropper receives the minimum amount of information. For the case of a two-dimensional Hilbert space, our protocol allows secure transmission at an error rate better than that for the BB84-protocol and is comparable with the characteristics of the best known QKD protocols. However, with increasing the dimensionality of the Hilbert space, the critical error rate for our protocol increases and, in the limit of infinite-dimensional space, the protocol becomes non- threshold.
We present the scheme of compatible quantum information analysis of the quantum key distribution (QKD) protocols, which give answers to the following questions: is it possible to improve the quantum bit error rate (QBER) of the 6-state protocol by employing more states, up to infinity, and can we essentially improve the QBER if the multidimensional Hilbert space with dimensionality more than 3 is used? Also, a novel quantum key distribution (QKD) protocol, based on all unselected states of a quantum system, which set the alphabet with continuous set of letters, is proposed. Employing all states of the Hilbert space leads to the maximal quantum uncertainty of transmitted states and therefore an eavesdropper receives the minimal amount of information. For the case of two-dimensional Hilbert space, our protocol allows secure transmission at the error rate higher than that one for the BB84-protocol and comparable with the characteristics of the best known QKD-protocols. However, with increasing the dimensionality of the Hilbert space the critical error rate for our protocol increases and in the limit of infinite-dimensional space the protocol becomes non-threshold.
A quantum-information analysis of how the size and dimensionality of the quantum alphabet affect the critical error rate of the quantum-key-distribution (QKD) protocols is given on an example of two QKD protocols---the six-state and \ensuremath{\infty}-state (i.e., a protocol with continuous alphabet) ones. In the case of a two-dimensional Hilbert space, it is shown that, under certain assumptions, increasing the number of letters in the quantum alphabet up to infinity slightly increases the critical error rate. Increasing additionally the dimensionality of the Hilbert space leads to a further increase in the critical error rate.
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.