Distinguishability of countable quantum states and von Neumann lattice

TL;DR

This paper establishes conditions for unambiguous distinguishability of countably infinite pure quantum states, linking it to generalizations of linear independence, and applies the theory to von Neumann lattices in phase space.

Contribution

It provides a necessary and sufficient condition for distinguishability of infinite pure states, extending linear independence concepts, and analyzes the von Neumann lattice case.

Findings

Distinguishability requires the lattice area to exceed Planck's constant.

Identifies subtle behaviors at the distinguishability threshold.

Connects measurement theory with the fundamental quantum of area in phase space.

Abstract

Condition for distinguishability of countably infinite number of pure states by a single measurement is given. Distinguishability is to be understood as possibility of an unambiguous measurement. For finite number of states, it is known that the necessary and sufficient condition of distinguishability is that the states are linearly independent. For infinite number of states, several natural classes of distinguishability can be defined. We give a necessary and sufficient condition for a system of pure states to be distinguishable. It turns out that each level of distinguishability naturally corresponds to one of the generalizations of linear independence to families of infinite vectors. As an important example, we apply the general theory to von Neumann's lattice, a subsystem of coherent states which corresponds to a lattice in the classical phase space. We prove that the condition for distinguishability is that the area of the fundamental region of the lattice is greater than the Planck constant, and also find subtle behavior on the threshold. These facts reveal the measurement theoretical meaning of the Planck constant and give a justification for the interpretation that it is the smallest unit of area in the phase space. The cases of uncountably many states and of mixed states are also discussed.