Post-selective attack with multi-mode projection onto Fock subspace

Abstract

In this work we present a comprehensive analysis of a post-selective attack on quantum key distribution protocols employing phase-encoded linearly independent coherent states (or similar alternatives). The attack relies on multimode projection onto a Fock subspace and enables probabilistic extraction of information by an eavesdropper. We derive analytical expressions for the information accessible to the adversary and show that it depends only on three protocol parameters: the mean photon number of the signal states, the phase separation in the information basis, and the expected optical loss of the quantum channel. Several optical realizations of phase-encoded quantum key distribution protocols are analyzed to illustrate the applicability of the results. Possible countermeasures against the proposed attack are also discussed.

Figures (3)

• Figure 1: The principal scheme of the proposed attack, as it is described in Section \ref{['sec-attackdescript']}. A sender transmits a coherent state $|\alpha e^{i\phi_j}\rangle$, that is split by unitary transformation (basically a beamsplitter) into $|\sqrt{\eta_1}\alpha e^{i\phi_j}\rangle$ and $|\sqrt{\eta_2}\alpha e^{i\phi_j}\rangle$. The former one is augmented by $|\sqrt{\eta_1}\alpha \rangle$ (made from a reference beam) to form $|\psi(\sqrt{\eta_1}\alpha,\phi_j)\rangle$ as in Eq. \ref{['addingmode']}; the latter one is transferred further to a receiver. Then the projection $Q^{(n)}$ as in Eq. \ref{['project']} is applied to $|\psi(\sqrt{\eta_1}\alpha,\phi_j)\rangle$ to form the reduced state $|\xi^{(n)}(\sqrt{\eta_1}\alpha,\phi_j)\rangle$ as in Eq. \ref{['reduced']}. If the outcome of the projection yields $n=0$, then the transferred to a receiver state is blocked (triggered by a control, denoted by a dashed arrow); otherwise ($n\ge1$), it travels further to a receiver, and the reduced state is stored in a quantum memory (QM)
• Figure 2: Comparison of the information obtained by performing the proposed attack, denoted by $I$ and expressed in Eq. \ref{['info']} with the one obtained by performing a general collective attack, i.e., the Holevo bound, denoted by $\chi$ and expressed in Eq. \ref{['holevo']}, assuming $\Delta=\pi/2$. Three regions are shown: 1) the top region (light gray) defined by $\eta_L\ge1-(1-\frac{|\alpha|^2}{2})\chi$, where $I<\chi$, 2) the middle region (gray) defined by $\frac{|\alpha|^2}{2}\le\eta_L\le1-(1-\frac{|\alpha|^2}{2})\chi$, where $I>\chi$, and 3) the bottom region (dark gray) defined by $\eta_L\le\frac{|\alpha|^2}{2}$, where $\chi<I=1$
• Figure 3: Comparison of the information obtained by performing the proposed attack, denoted by $I$ and expressed in Eq. \ref{['info']} with the one obtained by performing a general collective attack, i.e., the Holevo bound, denoted by $\chi$ and expressed in Eq. \ref{['holevo']}, for several values of $\Delta$. The gray region denotes $I>\chi$, solid black line is described by $I=\chi$, black dashed line is $\eta_L=\frac{|\alpha|^2}{2}$