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This is basically what you're asking:
https://arxiv.org/pdf/cond-mat/0101157
A global momentum measurement of the screen only yields...
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Like before, a global measurement of the screen’s total momentum gives which-path information only if the two recoil states are...
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In the paper, the experimental setup is asymmetric in the sense that the which–path measurement effectively takes place in only one arm...
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Consider the standard double-slit experiment with photons (or massive particles), where a screen containing two narrow slits is...
Like.
This is basically what you're asking: https://arxiv.org/pdf/cond-mat/0101157 A global momentum measurement of the screen only yields...