Security of Quantum Key Distribution Usingd-Level Systems

Abstract

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 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.

Robustness of a quantum key distribution with two and three mutually unbiased bases

“…(7). Using the normalization of the ancilla states and taking into account the expression (4), we can prove that …”

“…The cases with two and four bases are studied analytically in [7], while in [10,11] the case with three mutually unbiased bases is studied numerically. The results obtained both with the most general unitary eavesdropping strategy and with the optimal quantum cloning machine are exactly the same.…”

“…In these sections we find the optimal eavesdropping strategy and we compare the results with those ones obtained by an optimal quantum cloning machine. The generalization to four-dimensional systems and the comparison with the corresponding results derived from cloning attacks [7,10,11] is discussed in Sec. III, in a protocol with only two unbiased bases.…”

“…After the BB84 protocol, suggested originally by Bennett and Brassard in 1984 [1] and based on the transmission of single qubits, many generalized quantum key distribution protocols appeared in the literature: the six-state protocol for qubits [2,3], the generalization to qutrits in [4] and to ququarts in [5], and then subsequent generalizations to arbitrary dimensions [6,7]. Several aspects of the security of these protocols have already been analyzed [2,3,6,7,8]. Here we limit our attention to quantum key distribution protocols based on the transmission of single particles, and do not consider entanglement based schemes [9].…”

Mutually Unbiasedness between Maximally Entangled Bases and Unextendible Maximally Entangled Systems in ℂ 2 ⊗ ℂ 2 k $\mathbb {C}^{2}\otimes \mathbb {C}^{2^{k}}$

“…Then two mutually unbiased pairs of a maximally entangled basis and an unextendible maximally entangled system are constructed; lastly, explicit constructions are obtained for mutually unbiased MEB and UMES in C 2 ⊗C 4 and C 2 ⊗ C 8 , respectively. [5,6], mean king problem [7], quantum teleportation and superdense coding [8][9][10], and in quantifying wave-particle duality in multipath interferometers [4]. Two orthogonal bases…”