Security bound of two-basis quantum-key-distribution protocols using qudits

We investigate the security bounds of quantum-cryptographic protocols using $d$-level systems. In particular, we focus on schemes that use two mutually unbiased bases, thus extending the Bennett-Brassard 1984 quantum-key-distribution scheme to higher dimensions. Under the assumption of general coherent attacks, we derive an analytic expression for the ultimate upper security bound of such quantum-cryptography schemes. This bound is well below the predictions of optimal cloning machines. The possibility of extraction of a secret key beyond entanglement distillation is discussed. In the case of qutrits we argue that any eavesdropping strategy is equivalent to a symmetric one. For higher dimensions such an equivalence is generally no longer valid.

Figure 1

(Color online) $2d$-state QKD protocols: The threshold disturbance for entanglement distillation as a function of dimension. The triangles refer to Eq. (35) and arbitrary coherent attacks whereas the circles correspond to Eq. (36) and optimal cloning machines.

Figure 2

(Color online) $2d$-state QKD protocols: The regions of the independent parameters $D$ and $x-y$ for which the qudit-pair state ${\tilde{\rho }}_{AB}^{\left(j_1\right)}$ is (a) NPPT and distillable; (b) PPT; (c) NPPT but the reduction criterion is satisfied. From the top to the bottom, the “distillability maps” correspond to $d=5$, 4, and 3, respectively. The non-negativity of $x$, $y$, and $u$ (straight lines) defines the region of parameters where the protocols operate while the distillability condition (29) separates distillable from nondistillable states. The threshold disturbances for entanglement distillation are indicated by black dots. The triangles correspond to the separable state (33). Note the different scales of the horizontal axis.