Phase error estimation for passive detection setups with imperfections and memory effects

TL;DR

A generic framework to bound the phase error rate for quantum key distribution protocols using passive detection setups with imperfections and memory effects is developed, which can be used in proof techniques based on the entropic uncertainty relation or phase error correction, to prove security in the finite-size regime against coherent attacks.

Abstract

We develop a generic framework to bound the phase error rate for quantum key distribution protocols using passive detection setups with imperfections and memory effects. This framework can be used in proof techniques based on the entropic uncertainty relation or phase error correction, to prove security in the finite-size regime against coherent attacks. Our framework can incorporate on-the-fly announcements of click/no-click outcomes on Bob's side. In the case of imperfections without memory effects, it can be combined with proofs addressing source imperfections in a modular manner. We apply our framework to compute key rates for the decoy-state BB84 protocol, when the beam splitting ratio, the detection efficiency, and dark counts of the detectors are only known to be within some ranges. We also compute key rates in the presence of memory effects in the detectors. In this case, our results allow for protocols to be run at higher repetition rates, resulting in a significant improvement in the secure key generation rate.

Figures (11)

• Figure 1: Passive detection setups on Bob's side. The beam splitter has a splitting ratio $s$. Each of the four detectors has a different efficiency $\eta_i$ and dark count rate $d_i$. The parameters $s$, $\eta_i$, and $d_i$ are only partially characterized: specifically, $\eta_i \in [\eta^l_i, \eta^u_i]$, $d_i \in [d^l_i, d^u_i]$, and the beam splitting ratio, denoted $s$, lies within the interval $[s^* - \theta, s^* + \theta]$. Loss can be modeled as a beam splitter sitting in front of a perfect threshold detector. Dark counts can be modeled as a post-processing map applied to the outcomes obtained without dark counts.
• Figure 2: Proof structure: The tilde notation ($\tilde{n}$) on values or random variables such as $\tilde{n}_{{(X,1)}}$ indicates that these quantities are not directly accessible to Alice and Bob in the actual protocol.
• Figure 3: Applying \ref{['lemma:twostep']} to break measurements into a multi-step process, for the rounds where Bob receives a photon number $m$ such that $1\leq m <M$.
• Figure 4: The phase error rate estimation protocol described in \ref{['subsubsec:fullphaseprotocol']}: The colors indicate which POVM is used to complete the measurement given the outcome of the previous step. A tilde $\tilde{n}$ denotes variables that cannot be directly observed. The superscript $(0, \text{M}, >\text{M})$ indicates that the POVM element belongs to the corresponding photon number subspace.
• Figure 5: The filtering process. In each round, Bob performs either the filter or the full measurement depending on the click patterns observed in the previous $l_c$ rounds, to determine whether the current round results in a click or no-click. After $n$ rounds, the final shared state, conditioned on the filtering outcomes, is denoted by ${\rho_{A^{n_{(\text{keep,click})}}B^{n_{(\text{keep,click})}}E^n}}_{|\Omega(n_{(\text{keep,click})})}$.

Theorems & Definitions (54)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Lemma 1
• Remark 9
• Lemma 2
• Remark 10
• Remark 11
• Remark 12