Some of the strangeness of quantum mechanics was nicely isolated in Bell's Theorem, proposed in 1964 by John Bell, which seems to indicate there's a fundamental nonlocal nature to reality - according to quantum theory.

The simplest and most straightforward exposition of quantum weirdness I've found is in the article Quantum Mysteries for Anyone reprinted in "Boojums All the Way Through" by David Mermim.

can be downloaded as an ebook. Has a lot of quantum theory mixed in with very well-written biography by a physicist and long-time contributor to the New Yorker magazine.
For the more mathematically minded:

Quantum Theory, Concepts and Methods by Asher Peres, was the most helpful single reference for me.

The Feynman Lectures on Physics: Volume III is a classic.

Quantum Computation and Quantum Information by Isaac L. Chuang, Michael A. Nielsen is a clear exposition of those topics.
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