Experimental evidence for a non-globally trace-preserving POVM
Raymond JensenVisiting Assistant Professor, Mathematics
Northern State University, Aberdeen SD [email protected]
www.slideshare.net/rwjst4
Single photon
• Mach Zehnder interferometer Rotation transform—to basis of detectors (see Ruhla, Physics of Chance, pp. 162ff., Oxford 1995)
Measurement results, according to “orthodox” quantum theory• Cohen-Tannoudji, et al p.291 (Wiley, 1977)
Measurement results, according to “orthodox” quantum theory
• Similarly,
• IOW NO interference pattern.• “Orthodox” QM does NOT agree with the experimental results!
On the other hand, if we do a modification to the “orthodox” formula
• Sum inside the norm instead of outside• This is NOT “orthodox” QM!
On the other hand, if we do a modification to the “orthodox” formula
• Matches experimental results (like the single photon mode did)• This is NOT “orthodox” QM!
Difference between the two methods: associated operators• “Orthodox” quantum theory: von Neumann projector
• where
Difference between the two methods: associated operators• “Orthodox” quantum theory: von Neumann projector
• where • New operator “POVM”:
• where
Properties of
• does NOT obey completeness (“trace preservation”) throughout the entire Hilbert space but only on a subset .• on • This is why is NOT “orthodox” QM. • e.g. gives (super-normalized). Does such a state give rise to interference effects?
Properties of
• a. on U• b. (self adjoint, or Hermitian)• c. (Positive semidefinite)• Therefore is a “POVM” on U.• d. NOT idempotent:
GHZ “Bell” experiment
• If Bob measures “which way” () then BCHSH inequality is satisfied for Alice.• If Bob destroys “which way”
information (), then the BCHSH inequality is violated for Alice—same statistics as in Clauser, Aspect experiments.
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