Entanglement-enhanced quantum key distribution

We present and analyze a quantum key distribution protocol based on sending entangled $N$-qubit states instead of single-qubit ones as in the trail-blazing scheme by Bennett and Brassard 1984 (BB84). Since the qubits are sent and acknowledged individually, an eavesdropper is limited to accessing them one by one. In an intercept-resend attack, this fundamental restriction allows one to make the eavesdropper’s information on the transmitted key vanish if even one of the qubits is not intercepted. The implied upper bound $1∕\left(2N\right)$ for this information is further shown not to be the lowest, as the information can be reduced to less than 30% of that in the BB84 scheme in the case $N=2$. In general, the protocol is at least as secure as BB84.

Figure 1

(Color online) Quantum circuit for the proposed protocol with the intercept-resend attack. The circuit is repeated until a large enough number of bits has been transmitted. The required classical communication is not shown. Semicircles represent projective measurements. Gate $U_N$ is an $N$-qubit entangler announced in public. Gate $R(\beta ,\gamma )$ rotates a qubit by the angles $\gamma$ and $\beta$ with respect to the $z$ and $y$ axes, respectively. Each $a_i$, $e_i$, and $b_i$ is a binary variable representing a bit in Alice’s, Eve’s, and Bob’s raw key, respectively.

Figure 2

(Color online) Eve’s information per bit on Alice’s sifted key as a function of the observed QBER for the BB84 protocol with cloning and intercept-resend (IR) attacks (dashed lines), and for our protocol using $U_2=C(\pi ∕32,3\pi ∕8,\pi ∕32)$ with the corresponding optimal IR attack (solid line). The arrow shows the effect of engaging $U_2$ while keeping the fraction of intercepted qubits $\xi =0.8$ constant.

Figure 3

(Color online) Eve’s mutual information on Alice’s sifted key as a function of the induced QBER for different gates $U_2=C\left(\mathbf{c}\right)$. Eve uses the IR attack and measures in the ${\sigma }_z$ eigenbasis. The red dots (blue crosses) correspond to Eve measuring both (only one) of the two entangled qubits, in which case Eve’s maximal mutual information is between 0.5 and 0.125 (0.25 and 0).

Figure 4

(Color online) Eve’s mutual information on Alice’s sifted key as a function of the induced QBER sampled over all possible measurement bases for Eve. The entangling gate is fixed to $U_2=C\left({\mathbf{c}}^{*}\right)$, where ${\mathbf{c}}^{*}=(\pi ∕32,3\pi ∕8,\pi ∕32)$. The red dots (blue crosses) correspond to Eve measuring both (only one) of the two entangled qubits, in which case Eve’s maximal mutual information is between 0 and 0.2237 (0 and 0.0284).

Figure 5

(Color online) Relative key rate as a function of the yet unknown factor $\delta$ by which the scheme changes the QBER. The solid line represents our protocol with $U_2=C\left({\mathbf{c}}^{*}\right)$. The dashed blue line shows the rate for a maximally improving gate, i.e., one that hides all information sent via the quantum channel $(s=0)$. The dotted red line shows the rate for the BB84 scheme. The QBER in the BB84 protocol is fixed to 6%.