1) The document discusses pre-selection, post-selection, and measurements in quantum mechanics, highlighting several paradoxes that can arise. 2) It shows that while these "logical PPS paradoxes" do not necessarily imply theories must be contextual, there is a related proof of contextuality for each paradox when pre- and post-selection projectors are not orthogonal. 3) The author conjectures that the existence of logical PPS paradoxes, combined with an "uncertainty principle" constraint, would imply contextuality and discusses open questions about applications in quantum information theory.

1. Pre- and Post-Selection Paradoxes, Measurement-Disturbance and Contextuality in Quantum Mechanics M. S. Leifer (joint work with R. W. Spekkens) 10th November 2004 Coming soon to an arXiv/quant-ph near you!

11. 2.3) The “Three-Box Paradox” | 1 i | 2 i | 3 i Pre-selection: |  pre i = | 1 i + | 2 i + | 3 i Post-selection: |  post i = | 1 i + | 2 i - | 3 i Intermediate measurements: M: { P M,1 = | 1 ih 1 |, P M,2 = | 2 i h 2 | + | 3 ih 3 | } N: { P N,1 = | 2 ih 2 |, P N,2 = | 1 ih 1 | + | 3 ih 3 | } p ( X M = 1 | A pre , A post , M) = p ( X N = 1 | A pre , A post , N) = 1 Reason: h  post | P M,2 |  pre i = h  post | P N,2 |  pre i = 0 Counterfactual interpretation: David Blaine is in both boxes at the same time!

21. 4.2) Failure of the Product Rule Q - P - R + Projectors onto Eigenvalues R + = P + Q - + P - Q + R + = P + Q + + P - Q - X ­ Z Q - Q + I ­ Z P - P + X ­ I -1 +1 Observable