Common eigenvalues for two states or two bases of same state?

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nomadreid
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Two questions:
If you have two states which have at least one common eigenvalue, then are the two states distinguishable?
If you have one state but measure it with two different bases, can one conclude anything if the two measurements have a common eigenvalue?
Thanks
 
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For the first question, think about the H atom. The discrete energy spectrum is degenerate. 2 states with the same energy are distinguishable, since the angular momentum can have different measurable values.
 
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For the second question, I think it's important to specify what is being measured. For example, if you are measuring the intrinsic angular momentum of a particle in two different bases and you obtain the same eigenvalue in both bases, one basis doesn't imply anything about the state of the particle in the other since you can't prepare a particle with a definite value of angular momentum in both bases. There is probably a conclusion that can be drawn about total angular momentum though.

It depends.
 
Thanks, dextercioby and wotanub
I am still trying (without success) to understand the argument in http://arxiv.org/abs/1111.3328 that a state which has a common value between the supports of two different measures of that state cannot be real.
 
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dextercioby, sorry for the delay in posing this question based on your answer:
For the first question, think about the H atom. The discrete energy spectrum is degenerate. 2 states with the same energy are distinguishable, since the angular momentum can have different measurable values.
Sounds reasonable, but I am trying to jive this with the following quote (source given in my previous post):
"...if two probability distributions μL(x,p) and μL'(x,p) have overlapping support, i.e., if there is some region Δ of phase space where both distributions are non-zero, then the labels L and L' cannot refer to a physical property of the system."
whereby he defines a "label" L a collection of probability distributions {μL(λ)}, whereby λ is a state.
Just before that the authors give an example which imply that two energy states must have disjoint support to be distinguishable; your example gives two energy states which do not have disjoint support yet are distinguishable.
I am, to put it mildly, confused.