Naive attempts to put together relativity and quantum measurements lead to signaling between space-like separated regions. In QFT, these are known as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mtext class="MJX-tex-mathit" mathvariant="italic">impossible measurements</mml:mtext></mml:mrow></mml:math>. We show that the same problem arises in non-relativistic quantum physics, where joint nonlocal measurements (i.e., between systems kept spatially separated) in general lead to signaling, while one would expect no-signaling (based for instance on the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mtext class="MJX-tex-mathit" mathvariant="italic">principle of no-nonphysical communication</mml:mtext></mml:mrow></mml:math>). This raises the question: Which nonlocal quantum measurements are physically possible? We review and develop further a non-relativistic quantum information approach developed independently of the impossible measurements in QFT, and show that these two have been addressing virtually the same problem. The non-relativistic solution shows that all nonlocal measurements are <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>l</mml:mi><mml:mi>o</mml:mi><mml:mi>c</mml:mi><mml:mi>a</mml:mi><mml:mi>l</mml:mi><mml:mi>i</mml:mi><mml:mi>z</mml:mi><mml:mi>a</mml:mi><mml:mi>b</mml:mi><mml:mi>l</mml:mi><mml:mi>e</mml:mi></mml:math> (i.e., they can be carried out at a distance without violating no-signaling) but they (i) may require arbitrarily large entangled resources and (ii) cannot in general be <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>i</mml:mi><mml:mi>d</mml:mi><mml:mi>e</mml:mi><mml:mi>a</mml:mi><mml:mi>l</mml:mi></mml:math>, i.e., are not immediately reproducible. These considerations could help guide the development of a complete theory of measurement in QFT.
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