Critical error rate of quantum-key-distribution protocols versus the size and dimensionality of the quantum alphabet

A quantum-information analysis of how the size and dimensionality of the quantum alphabet affect the critical error rate of the quantum-key-distribution (QKD) protocols is given on an example of two QKD protocols--the six-state and ∞-state (i.e., a protocol with continuous alphabet) ones. In the case of a two-dimensional Hilbert space, it is shown that, under certain assumptions, increasing the number of letters in the quantum alphabet up to infinity slightly increases the critical error rate. Increasing additionally the dimensionality of the Hilbert space leads to a further increase in the critical error rate