Consider a quantum system prepared in state ψ, a unit vector in a d -dimensional Hilbert space. Let b 1 , ..., b d be an orthonormal basis and suppose that, with some probability 0 < p < 1, ψ 'collapses', i.e., gets replaced by b k (possibly times a phase factor) with Born's probability |〈 b k |ψ〉| 2 . The question we investigate is: how well can any quantum experiment on the system determine afterwards whether a collapse has occurred? The answer depends on how much is known about the initial vector ψ. We provide a number of different results addressing several variants of the question. In each case, no experiment can provide more than rather limited probabilistic information. In case ψ is drawn randomly with uniform distribution over the unit sphere in Hilbert space, no experiment performs better than a blind guess without measurement; that is, no experiment provides any useful information.
Consider a quantum system prepared in state ψ, a unit vector in a d -dimensional Hilbert space. Let b 1 , ..., b d be an orthonormal basis and suppose that, with some probability 0 < p < 1, ψ 'collapses', i.e., gets replaced by b k (possibly times a phase factor) with Born's probability |〈 b k |ψ〉| 2 . The question we investigate is: how well can any quantum experiment on the system determine afterwards whether a collapse has occurred? The answer depends on how much is known about the initial vector ψ. We provide a number of different results addressing several variants of the question. In each case, no experiment can provide more than rather limited probabilistic information. In case ψ is drawn randomly with uniform distribution over the unit sphere in Hilbert space, no experiment performs better than a blind guess without measurement; that is, no experiment provides any useful information.
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 b1, …, 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 bk|ψ2. 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 b1, …, bd as known. For certain values of p, we show that the set of ψs for which any experiment E 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 E such that the set of ψs for which E is more reliable than blind guessing has measure greater than half the sphere, though with a conjectured maximum of 64% of the sphere.
GAP measures (also known as Scrooge measures) are a natural class of probability distributions on the unit sphere of a Hilbert space that come up in quantum statistical mechanics; for each density matrix ρ there is a unique measure GAPρ. We describe and prove a property of these measures that was not recognized so far: If a wave function Ψ is GAPρ distributed and a collapse occurs, then the collapsed wave function Ψ′ is again GAP distributed (relative to the appropriate ρ′). This fact applies to collapses due to a quantum measurement carried out by an observer, as well as to spontaneous collapse theories such as CSL or GRW. More precisely, it is the conditional distribution of Ψ′, given the measurement outcome (respectively, the noise in CSL or the collapse history in GRW), that is GAPρ′.
We present a set of exact system solutions to a model we developed to study wave function collapse in the quantum spin measurement process. Specifically, we calculated the wave function evolution for a simple harmonic oscillator of spin \frac{1}{2}, with its magnetic moment in interaction with a magnetic field, coupled to an environment that is a bath of harmonic oscillators. The system's time evolution is described by the direct product of two independent Hilbert spaces: one that is defined by an effective Hamiltonian, which represents a damped simple harmonic oscillator with its potential well divided into two, based on the spin and the other that represents the effect of the bath, i.e., the Brownian motion. The initial states of this set of wave functions form an orthonormal basis, defined as the eigenstates of the system. If the system is initially in one of these states, the final result is predetermined, i.e., the measurement is deterministic. If the bath is initially in the ground state,and the wave function is initially a wave packet at the origin, it collapses into one of the two potential wells depending on the initial spin. If the initial spin is a vector in the Bloch sphere not parallel to the magnetic field, the final distribution among the two potential wells is given by the Born rule applied to the initial spin state with the well-known ground state width. Hence, the result is also predetermined. We discuss its implications to the Bell theorem[1]. We end with a summary of the implications for the understanding of the statistical interpretation of quantum mechanics.
-dimensional Hilbert space, suppose a pure state |ψ⟩ is subjected to a complete orthogonal measurement. The measurement effectively maps |ψ⟩ to a point (p1,…,pd) in the appropriate probability simplex. It is a known fact-which depends crucially on the complex nature of the system's Hilbert space-that if |ψ⟩ is distributed uniformly over the unit sphere, then the resulting ordered set (p1,…,pd) is distributed uniformly over the probability simplex; that is, the resulting measure on the simplex is proportional to dp1⋯dpd-1. In this paper we ask whether there is some foundational significance to this uniform measure. In particular, we ask whether it is the optimal measure for the transmission of information from a preparation to a measurement in some suitably defined scenario. We identify a scenario in which this is indeed the case, but our results suggest that an underlying real-Hilbert-space structure would be needed to realize the optimization in a natural way.
Three facts related to the collapse of quantum mechanical wavefunctions following measurements or observations are presented and explained. They are (1) in all stochastic theories, both classical and quantum mechanical, probability densities collapse following observations; (2) the laws of probability determine the probability density just after an observation in terms of that just before the observation and the result of the observation; (3) the wavefunction just after an observation can be found from that just before the observation by using the result of the observation, the laws of probability, and the assumption that immediate repetition of an observation yields the same result and does not change the wavefunction except for an arbitrary phase factor.
The main result of the paper can be summarized as follows. The number of distinguishable quantum states in a 2-dimensional Hilbert space is proportional to the square root of the number of identical copies of each state measured and to the total length of the angle intervals occupied by the state vectors. Surprisingly, it does not depend on the position of the arcs comprising the range on the unit circle. These results can be generalized for the N-dimensional Hilbert space of states of a quantum system.As in the 2-dimensional case, the unit sphere can be reduced to the non-negative orthant of the unit sphere in the real N-dimensional Euclidean space.It turns out [6] that the number of distinguishable states depends only on the area Ω of the domain on the unit sphere from which the states can be chosen, but does not depend on the shape and position of this domain. The optimal distribution is uniform over the domain in angular (polar) coordinates, and the number of distinguishable states is W(n,Ω)=Ω(cn)2N−1 where c is a constant.