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Condition for distinguishability of countably infinite number of pure statesby a single measurement is given. Distinguishability is to be understood aspossibility of an unambiguous measurement. For finite number of states, it isknown that the necessary and sufficient condition of distinguishability is thatthe states are linearly independent. For infinite number of states, severalnatural classes of distinguishability can be defined. We give a necessary andsufficient condition for a system of pure states to be distinguishable. Itturns out that each level of distinguishability naturally corresponds to one ofthe generalizations of linear independence to families of infinite vectors. Asan important example, we apply the general theory to von Neumann's lattice, asubsystem of coherent states which corresponds to a lattice in the classicalphase space. We prove that the condition for distinguishability is that thearea of the fundamental region of the lattice is greater than the Planckconstant, and also find subtle behavior on the threshold. These facts revealthe measurement theoretical meaning of the Planck constant and give ajustification for the interpretation that it is the smallest unit of area inthe phase space. The cases of uncountably many states and of mixed states arealso discussed.

Distinguishability of countable quantum states and von Neumann lattice