Practical security analysis of two-way quantum-key-distribution protocols based on nonorthogonal states

Within the broad research scenario of quantum secure communication, two-way quantum key distribution (TWQKD) is a relatively new proposal for sharing secret keys that is not yet fully explored. We analyze the security of TWQKD schemes that use qubits prepared in nonorthogonal states to transmit the key. Investigating protocols that employ an arbitrary number of bases for the channel preparation, we show, in particular, that the security of the LM05 protocol cannot be improved by the use of more than two preparation bases. We also provide an alternative proof of unconditional security for a deterministic TWQKD protocol recently proposed in Beaudry et al., Phys. Rev. A 88, 062302 (2013). In addition, we introduce a deterministic protocol named “TWQKD six-state” and compute an analytical lower bound (which can be tightened) for the maximum amount of information that an eavesdropper could extract in this case. An interesting advantage of our approach to the security analysis of TWQKD is the great simplicity and transparency of the derivations.

Figure 1

Schematic illustration of the quantum part of the general TWQKD protocol, with Alice at the left side, Bob at the right side, and an eventual eavesdropper (Eve) at the center (the classical communication channel is omitted). The states $\left\{\right|{\psi }_i\rangle {\}}_i$ and the unitary operations ${\left\{U_i^A\right\}}_i$ are chosen probabilistically by Bob and Alice, respectively. Bob performs the decoding, measuring the qubit in the preparation basis, and, for CM, Alice measures the received qubit in any of the possible preparation bases (randomly chosen). Alice also chooses randomly between EM and CM with probabilities $p_e$ and $p_c$, respectively. See the general protocol description in the main text for more details.

Figure 2

Schematic illustration of the quantum part of the TWQKD six-state protocol (the classical communication channel is omitted). Alice (on the left side) and Bob (on the right side) perform each of their possible choices with equal probability, i.e., basis direction $\widehat{k}$ for the state preparation, Alice's encoding operation (in EM), and Alice's measurement basis (in CM).

Figure 3

Comparison of the Alice-Bob mutual information $I(A:B)$ (blue solid line) and Eve's information $\left(I_E\right)$ for different protocols, as a function of the forward noise $\left(Q_f\right)$. It is assumed that the overall noise (Q) and the forward noise coincide, which is the case if the QFC and the QBC are correlated (see, e.g., in Ref. [19], some comments in this respect). $I_E$ for the generalized LM05 protocol is plotted as a (dark-green) dotted line. The upper bound to $I_E$ for the $\mathrm{LM}05{}^{\prime }$ protocol is represented by a (dark-yellow) dashed line. The lower bound to $I_E$ for the TWQKD six-state is drawn as the (dark-red) dash-dotted line.