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Numerical security framework for quantum key distribution with bypass channels

Lewis Wooltorton, Twesh Upadhyaya, Mohsen Razavi

Abstract

Satellite based quantum key distribution (QKD) aims to establish secure key exchange over long distances despite significant technological challenges. To alleviate some of these challenges, Ghalaii et al. [PRX Quantum 4, 040320 (2023)] proposed that any airborne eavesdropper up to a certain size can be detected by classical monitoring techniques, limiting the transmission efficiencies of any undetected Eve. This creates a new QKD scenario in which some of the transmitted signal from Alice to Bob bypasses Eve entirely. In this manuscript, we develop a general framework for computing key rates in this "bypass" scenario for discrete variable protocols. We first numerically support a conjecture that the performance of BB84 with single photons does not improve under bypass constraints, and go on to find new regimes that do. Specifically, we find improvements when the receiver's detectors have an efficiency mismatch and when BB84 is implemented using weak coherent pulses under certain squashing assumptions. Technically, our framework is realized by including marginal constraints on the source to account for bypass effects, combined with existing numerical approaches for minimizing the key rate and squashing and dimension reduction techniques to handle photonic states of unbounded dimension.

Numerical security framework for quantum key distribution with bypass channels

Abstract

Satellite based quantum key distribution (QKD) aims to establish secure key exchange over long distances despite significant technological challenges. To alleviate some of these challenges, Ghalaii et al. [PRX Quantum 4, 040320 (2023)] proposed that any airborne eavesdropper up to a certain size can be detected by classical monitoring techniques, limiting the transmission efficiencies of any undetected Eve. This creates a new QKD scenario in which some of the transmitted signal from Alice to Bob bypasses Eve entirely. In this manuscript, we develop a general framework for computing key rates in this "bypass" scenario for discrete variable protocols. We first numerically support a conjecture that the performance of BB84 with single photons does not improve under bypass constraints, and go on to find new regimes that do. Specifically, we find improvements when the receiver's detectors have an efficiency mismatch and when BB84 is implemented using weak coherent pulses under certain squashing assumptions. Technically, our framework is realized by including marginal constraints on the source to account for bypass effects, combined with existing numerical approaches for minimizing the key rate and squashing and dimension reduction techniques to handle photonic states of unbounded dimension.
Paper Structure (23 sections, 4 theorems, 112 equations, 8 figures, 1 table)

This paper contains 23 sections, 4 theorems, 112 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. QKD with a bypass channel
  3. Security analysis
  4. Key rate formula
  5. Problem constraints
  6. Reliable lower bounds
  7. Dimension Reduction
  8. Application to BB84 with single photons
  9. Single photon bypass model
  10. Single photon with mismatched detector efficiencies
  11. Heuristic upper bounds
  12. Lower bounds
  13. Application to BB84 with weak coherent pulses
  14. WCP bypass model
  15. Applying dimension reduction
  16. ...and 8 more sections

Key Result

Lemma 1

Let $(\ket{\psi^{\mathrm{init}}},U_{1},U_{BE},U_{2}, M_{A},N_{B} )$ be any quantum strategy in the bypass scenario that gives rise to the post-selected state $\tilde{\rho}$ in eq:pms. Then the following holds: where $V_{x}: \mathcal{H}_{A}\otimes \mathcal{H}_{BF} \to \mathcal{H}_{\tilde{A}}\otimes \mathcal{H}_{A}\otimes \mathcal{H}_{BF}$, where $\mathcal{H}_{\tilde{A}} \cong \mathbb{C}^{2}$, is

Figures (8)

  • Figure 1: The general bypass channel model. Each round, Alice prepares an entangled state $\ket{\psi}$ and measures one part, whilst sending the other to Bob along spatial mode $B$. First, the signal is interfered with the vacuum at a beam splitter with transmissivity $\eta_{AE}$, which models the lossy channel between Alice and Eve, who is described by an unknown quantum channel $\mathcal{E}$. After Eve's attack, the signal is interfered with another mode $F'$ at a beam splitter with transmittance $\eta_{EB}$ modeling a lossy channel between Eve and Bob, before reaching Bob's collection device, modeled by a channel $\mathcal{E}_{T}$. Any signal reflected at the first beam splitter travels along the bypass mode $F$, and undergoes an unknown quantum channel $\mathcal{E}_{F}$, before
  • Figure 2: The bypass channel model considered in this work. This model is a special case of \ref{['fig:bypass_gen']}, where the bypass channel is lossless (captured by choosing $\mathcal{E}_{F}$ as the identity channel), there is no loss between Eve and Bob (choosing $\eta_{EB} = 1$) and we model Bob's collection device $\mathcal{E}_{T}$ as a beam splitter with transmitivity $\eta_{T}$. Eve's attack is viewed as the action of a quantum channel $\mathcal{E}$ on mode $B$, which is related to the unitary description in the text via its Stinespring dilation $\mathcal{E}(\rho_{B}) = \tr_{E}[U_{BE}(\rho_{B} \otimes \ketbra{0}{0}_{E})U_{BE}^{\dagger}]$.
  • Figure 3: Secret key rate of single photon BB84 in the presence of bypass channels. $\eta_{AE}$ denotes the fraction of signal sent from Alice and received by Eve, which can be bounded via monitoring techniques. "Normal QKD" indicates the rate in the absence of bypass channels, and "bypass model lower bound" refers to that derived in ghalaii2023satellitebased; here both coincide. "Bypass model this work" denotes the results of our numerical calculation; we observe an invariant rate as we vary the bypass parameter $\eta_{T}$, which we hence set to $1$ in this plot. Simulation parameters are given in \ref{['tab:num']}
  • Figure 4: Secret key rate of single photon BB84 with detector efficiency mismatch in the presence of bypass channels. $\eta_{1}$ is the efficiency of Bob's first detector, while the other has unity efficiency. We set the bypass parameter $\eta_{T}=\eta_{AE}$; hence the curves serve as heuristic upper bounds on the key rate. In (a), we set $q=0.1$ in \ref{['eq:depol']} to calculate the observed statistics, and vary $\eta_{AE}$ at different detector efficiencies. In (b), we take different values of $\eta_{AE}$, and vary the noise $q$ at a single detector efficiency.
  • Figure 5: Secret key rate of single photon BB84 with detector efficiency mismatch, in the presence of bypass channels. $\eta_{1}$ is the efficiency of Bob's first detector. The efficiency of Bob's second detector is fixed at $\eta_{2}=1$. In (a) for any fixed value of $\eta_{AE}$ and $eta_{1}$ the rate is minimized over feasible values of $\eta_{T}$. In (b), we show the nature of this minimization at $\eta_{1}=0.09$. The rates are compared to a normal QKD key rate derived from our numerical framework when $\eta_{AE} = \eta_{T} = 1$, highlighting improvements from the bypass channel. The noise parameter is set at $q=0.1$ for both plots.
  • ...and 3 more figures

Theorems & Definitions (9)

  • Lemma 1
  • Remark 1
  • Theorem 1: Coles12 Theorem 1
  • proof
  • proof
  • Lemma 2
  • proof
  • Corollary 1
  • proof