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We study various measures of classicality of the states of open quantumsystems subject to decoherence. Classical states are expected to be stable inspite of decoherence, and are thought to leave conspicuous imprints on theenvironment. Here these expected features of environment-induced superselection(einselection) are quantified using four different criteria: predictabilitysieve (which selects states that produce least entropy), purification time(which looks for states that are the easiest to find out from the imprint theyleave on the environment), efficiency threshold (which finds states that can bededuced from measurements on a smallest fraction of the environment), andpurity loss time (that looks for states for which it takes the longest to losea set fraction of their initial purity). We show that when pointer states --the most predictable states of an open quantum system selected by thepredictability sieve -- are well defined, all four criteria agree that they areindeed the most classical states. We illustrate this with two examples: anunderdamped harmonic oscillator, for which coherent states are unanimouslychosen by all criteria, and a free particle undergoing quantum Brownian motion,for which most criteria select almost identical Gaussian states (although, inthis case, predictability sieve does not select well defined pointer states.)Comment: 10 pages, 13 figure

Predictability sieve, pointer states, and the classicality of quantum trajectories