Elements of an uncountable set (of the cardinality of the continuum) require a countable amount of information to describe.
Perhaps this goes too far, but this is why one typically prefers separable Hilbert spaces in QM (and functional analysis in general). Admittedly non-separable Hilbert spaces (which lack even a countable basis) are somewhat rare in practice.
Perhaps this goes too far, but this is why one typically prefers separable Hilbert spaces in QM (and functional analysis in general). Admittedly non-separable Hilbert spaces (which lack even a countable basis) are somewhat rare in practice.