Counting channel uses of the lossy bosonic channel or definition of channel uses

Question

The PLOB-bound ("Fundamental Limits of Repeaterless Quantum Communications") gives an asymptotic upper bound on the secret-key rate per used lossy bosonic channel. However, I'm not sure how to count the number of used channels in a physical implementation.

For example, let's assume that I use the BB84 protocol encoded in the polarization degrees of freedom of a single photon pulse. One could think that one uses two lossy optical channels per sent photon due to the two polarizations, but this assumes that the photon would have a single frequency contradicting the spatial localization of the photon pulse. However, if I assume a localized photon pulse, then the photon does not have a defined frequency anymore, but has a specific linewidth. Therefore, I would possibly use an uncountable number of frequencies and therefore also channels.

Have I misunderstood anything or is this bound not useful in realistic implementations of QKD?

Answer 1

I think your misunderstanding is in the wording of the secret key rate; The PLOB-Bound offers an asymptotic, ultimate-upper bound on the secret key rate per use of a lossy bosonic channel.

This bound on the secret key rate is computed as a regularisation, where one considers the infinite limit of $ n \rightarrow \infty$ transmissions across the channel in order to create an infinitely long secret key. Specifically, the regularised relative entropy of entanglement of a teleportation resource state $\hat{\rho}$, \begin{equation} \mathcal{C}(\mathcal{E}) \leq E_R^\infty(\hat{\rho}) := \lim_{n\to\infty} {1\over n} E_R(\hat{\rho}^{\otimes n}). \end{equation} Each transmission used within the QKD protocol is therefore considered a use of the channel. This considers only one quantum channel - one does not introduce a new channel for each degree of freedom that the quantum information being transmitted may have.

The generality of the PLOB-Bound and its versatility to a variety of channels using Discrete Variable or Continuous Variable quantum information, makes it extremely useful as an optimal upper bound to strive for in realistic implementations of QKD. If you would like more insight into the techniques used in this paper (channel simulation, teleportation stretching, etc.), check out this paper.

Furthermore, more recent work using these techniques attempts to determine more expedient bounds on the secret key rate for CVQKD. Here they investigate sub-optimal, but finite energy resource states within the teleportation stretching protocols, which converge to the optimal bound in the limit of infinite energy resource states.