About quantum measurement problem, proper or improper mixture? Generally a quantum measurement is regarded as resulting in a definite outcome due to "state collapse" and the post-measurement state is described as a proper mixture with the ignorance of the measurement result, i.e. a mixture of pure states with certain probabilities. At least the linear QM does not find this idea inconvenient.
But the decoherence "solution" of the measurement problem does not answer the "collapse" problem and in fact the joint system of the object and the apparatus is entangled with the environment and therefore is an improper mixture.
For nonlinear QM (for example with closed timelike curves), whether the post-measurment state is a proper or improper mixture does matter.
Is there evidence to illustrate that after measuring subsystem A of an EPR pair AB in state |00>+|11>, subsystem B should be described by a proper mixture or an improper mixture? Since the current experimental results on this issue is still in the field of linear QM, so I am wondering if there is a definite answer for this.   
 A: Just to clear things out:


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*A mixed state $\rho$ can be represented as a proper mixture when it is realized as an ensemble of identical systems, each prepared in a pure state. In one familiar example the individual pure states are the eigenstates of $\rho$ and occur with frequencies given by the eigenvalues of $\rho$. But since a mixed state usually has other distinct representations as a superposition of non-orthogonal pure states, the ensemble realization is anything but unique. In all these different realizations the statistics of any single-system observable as measured on the entire ensemble reproduces the statistics described by $\rho$.  

*Single systems individually prepared in a mixed state described by $\rho$ provide an improper mixture representation of $\rho$. This is because an ensemble of such systems is basically a collection of clones, each in an identical mixed state, as opposed to the proper mixture of systems in different pure states. 
The issue of whether it is possible to distinguish between proper and improper mixtures is independent of the occurrence or not of "collapse" in quantum measurements, and is nevertheless a fundamental one. 


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*If proper and improper mixtures are indistinguishable by any physically realizable measurements then the time evolutions of the corresponding ensemble statistics are also  indistinguishable. A well-known argument then shows that the co-existence of quantum dynamics and quantum entanglement with a finite upper limit for the speed of propagation of all interactions (aka relativity, no-signaling clause) is possible if and only if quantum dynamics is linear. That is, the evolution of any quantum system is given by a linear map on the cone of the system's quantum states $\rho$. The current framework of quantum dynamics is considered fundamentally linear and there are experiments that claim to have verified linearity to a high degree of precision. In this sense such experiments may be considered as (indirect?) tests of the indistinguishability of proper and improper ensembles. 

*If it were possible to distinguish proper and improper mixtures in any way, the necessary linearity of quantum dynamics could be brought into question, which in turn would raise a host of other fundamental issues. Interestingly enough, the question of whether there can exist meaningful and self-consistent nonlinear extensions of quantum theory was raised in the early '90 by Steven Weinberg and a variety of studies have concluded that the answer is likely negative, usually on grounds that the various non-linear aspects proposed would necessarily imply the possibility of by-passing the finite speed-of-propagation limit. The no-signaling argument mentioned before was actually the climax of that debate in the early 2000's.  
To answer the question: As far as quantum measurements as we currently know and apply them are concerned, proper mixtures are indistinguishable from improper mixtures. In other words, no set of measurements involving standard observables can distinguish between a proper mixture ensemble and an improper mixture ensemble described by the same density matrix $\rho$. This is in fact the basis of quantum measurements on mixed states, whether collapse occurs or not. Unfortunately, to the extent that I am aware, attempts to probe the proper vs. improper mixture problem by more sophisticated and dedicated experimental approaches have not been reported so far. Maybe somebody else can back this up or correct me otherwise. But then it would always be nice to see such a probe ;)
