A variation of the Bell experiment

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Discussion Overview

The discussion revolves around a variation of the Bell experiment, focusing on the transformations applied to qubits in a Bell state and the implications for measurement and state collapse. Participants explore the mathematical outcomes of different transformations and their interpretations in the context of quantum mechanics.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • One participant presents a mathematical derivation showing that applying an interferometer transformation to the first qubit after preparing a Bell state results in equal probabilities for the qubit states.
  • Another participant questions the validity of the transformation applied to the second qubit, suggesting that it is not reversible and that a measurement must occur to collapse the wave function, leading to a mixed state.
  • A third participant discusses the concept of measurement in quantum mechanics, noting that information leakage about the state can be considered a form of measurement, and raises concerns about the implications of path information in interferometer setups.
  • A later reply emphasizes that modern interpretations often consider decoherence as a point at which an observation has occurred, adding complexity to the discussion of measurement and state collapse.

Areas of Agreement / Disagreement

Participants express differing views on the reversibility of the transformations and the nature of measurement in quantum mechanics. There is no consensus on the assumptions made regarding state collapse and the implications of the transformations applied.

Contextual Notes

The discussion highlights limitations in assumptions about measurement and state preparation, particularly regarding the transition from pure to mixed states and the implications of decoherence. The mathematical steps and interpretations remain unresolved.

dsoodak
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If we start with a Bell state
1/Sqrt(2)(|00>+|11>)
and (after moving the second qbit a significant distance away) apply the interferometer transformation
|0> -> 0.5(|0>+|1>)
|1> -> 0.5(|0>-|1>)
to the first qbit, we get
0.5/Sqrt(2)((|0>+|1>)|0>+(|0>-|1>)|1>)
=0.5/Sqrt(2)(|00>+|10>+|01>-|11>)
which gives equal probability of the first qbit ending up in |0> or |1>

Lets now start again with the same spatially separated Bell state but first apply the transformation
|0> -> 0.5(|0>+|1>)
|1> -> 0.5(|0>+|1>)
to the second qbit:
0.5/Sqrt(2)(|0>(|0>+|1>)+|1>(|0>+|1>))
=0.5/Sqrt(2)(|00>+|01>+|10>+|11>)
then apply the original (interferometer) transformation to the first qbit:
0.25/Sqrt(2)((|0>+|1>)|0>+(|0>+|1>)|1>+(|0>-|1>)|0>+(|0>-|1>)|1>)
=0.25/Sqrt(2)(|00>+|10>+|01>+|11>+|00>-|10>+|01>-|11>)
=0.5/Sqrt(2)(|00>+|01>)
Now, the first qbit is in state |0> with 100% (as opposed to 50%) probability as a result of what was done to the second one.

So...can anyone tell me if I made any false assumptions or stupid math mistakes here?

Dustin Soodak
 
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dsoodak said:
Lets now start again with the same spatially separated Bell state but first apply the transformation
|0> -> 0.5(|0>+|1>)
|1> -> 0.5(|0>+|1>)


This transformation isn't reversible, so I think you have to measure the state and collapse the wave function before doing the transformation. After collapsing the wave function, the state should be a mixed state, not a pure state, ie. with 50% chance the state is |0>(|0>+|1>), and with 50% chance the state is |1>(|0>+|1>).
 
The way I've always seen it described in this sort of experiment, something counts as a measurement if the information about the state leaks out.
I originally thought this transformation could be done (using photons as qbits) with an interferometer that has one path length 1/4 wavelength longer than the other. However, I suppose you could then get info about the photon's path information from the difference in travel time.
 
dsoodak said:
The way I've always seen it described in this sort of experiment, something counts as a measurement if the information about the state leaks out.

Cant quite follow your idea here - information about state leaks out?

In modern times an observation is generally considered to have occurred just after decoherence.

Thanks
Bill
 

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