### Abstract

We are concerned with the problem of detecting with high probability whether a wave function has collapsed or not, in the following framework: A quantum system with a d-dimensional Hilbert space is initially in state ψ; with probability 0 < p < 1, the state collapses relative to the orthonormal basis b
1, …, bd
. That is, the final state ψ′ is random, it is ψ with probability 1 − p and bk
(up to a phase) with p times Born’s probability
. Now an experiment on the system in state ψ′ is desired that provides information about whether or not a collapse has occurred. Elsewhere [C. W. Cowan and R. Tumulka, J. Phys. A: Math. Theor. 47, 195303 (2014)], we identify and discuss the optimal experiment in case that ψ is either known or random with a known probability distribution. Here, we present results about the case that no a priori information about ψ is available, while we regard p and b
1, …, bd
as known. For certain values of p, we show that the set of ψs for which any experiment
is more reliable than blind guessing is at most half the unit sphere; thus, in this regime, any experiment is of questionable use, if any at all. Remarkably, however, there are other values of p and experiments
such that the set of ψs for which
is more reliable than blind guessing has measure greater than half the sphere, though with a conjectured maximum of 64% of the sphere.

Received 20 February 2014
Accepted 09 August 2015
Published online 26 August 2015

Acknowledgments:
Both authors are supported in part by NSF Grant No. SES-0957568. R.T. is supported in part by Grant No. 37433 from the John Templeton Foundation and by the Trustees Research Fellowship Program at Rutgers, the State University of New Jersey.

Article outline:

I. INTRODUCTION
II. IF *ψ* IS KNOWN OR RANDOM WITH KNOWN DISTRIBUTION
III. NO PRIOR INFORMATION ABOUT *ψ*
IV. PROOFS
A. The spectrum of *A*_{p}
(*E*)
B. Computing and bounding Λ_{
p
}(*E*)

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