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Physical Review A

Critical error rate of quantum-key-distribution protocols versus the size and dimensionality of the quantum alphabet

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

Published Nov 30, 2004• 28 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.70.052331

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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.
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.
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.
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.
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.
Quantum key distribution(QKD) could be the most significant application of quantum information theory. In nearly four decades, although substantial QKD protocols are developed, the BB84 protocol and its variants are still the most researched ones. It is well-known that the secure bound of qubit error rate(QBER) of BB84 protocol is about 11%\%% while it can be increased to 12.6%\%% by six-state protocol. It would not be surprising that employing more basis could increase the bound. However, what is the optimal one? In this paper, investigations of asymptotically optimal QKD protocols are proposed. Precisely, We present an abstraction of prepare-measure QKD protocols and investigate two special cases which are optimal among all protocols coding by the same states. Our analyses demonstrate that the asymptotically optimal QBER bounds coding by orthogonal qubits are about 22.17%\%% for both general and memoryless attacks while the bounds coding by non-orthogonal states in two mutually unbiased basis are about 25.30%\%% for general attacks and 27.00%\%% for memoryless attacks. The significant improvement indicates the huge space for previous protocols to improve their security while the optimality of our protocols demonstrates the ultimate potential of their security.