Security of Quantum Key Distribution Using<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi mathvariant="italic">d</mml:mi></mml:math>-Level Systems

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.