Experimental Evidence of a non-Globally Trace-Preserving Positive Operator Valued Measure

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

The experimentGrangier, Roger, Aspect, Europhysics Lett. 1(4) p.173 (1986)

Results

Single photon

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√2¿

Single photon

• Mach Zehnder interferometer Rotation transform—to basis of detectors (see Ruhla, Physics of Chance, pp. 162ff., Oxford 1995)

Single photon

• Mach Zehnder interferometer Rotation transform—to basis of detectors

Single photon

• Mach Zehnder interferometer Rotation transform—to basis of detectors

Match with experimental data

But the system has two entangled photons

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√2¿

But the system has two entangled photons

But the system has two entangled photons

Measurement results, according to “orthodox” quantum theory• Cohen-Tannoudji, et al p.291 (Wiley, 1977)

Measurement results, according to “orthodox” quantum theory

Measurement results, according to “orthodox” quantum theory

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

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

• OTOH Completeness holds for Bell states, GHZ state

Properties of

• a. on U• b. (self adjoint, or Hermitian)• c. (Positive semidefinite)• Therefore is a “POVM” on U.• d. NOT idempotent:

QM is “complete”

But not all operators are complete on the entire space

GHZ “Bell” experiment

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.

GHZ “Bell” experiment