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

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?

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 $$\hat{\rho}$$, $$$$\mathcal{C}(\mathcal{E}) \leq E_R^\infty(\hat{\rho}) := \lim_{n\to\infty} {1\over n} E_R(\hat{\rho}^{\otimes n}).$$$$ 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.