Towards a measurement theory in QFT: “Impossible” quantum measurements are possible but not ideal

Nicolas Gisin and Flavio Del Santo

Group of Applied Physics, University of Geneva, 1211 Geneva, Switzerland
Constructor University, Geneva, Switzerland

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Naive attempts to put together relativity and quantum measurements lead to signaling between space-like separated regions. In QFT, these are known as $\textit{impossible measurements}$. 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 $\textit{principle of no-nonphysical communication}$). 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 $localizable$ (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 $ideal$, i.e., are not immediately reproducible. These considerations could help guide the development of a complete theory of measurement in QFT.

Naïve attempts to merge relativity with quantum measurements theoretically leads to instantaneous communication across distant regions. This work shows that such an issue, known in quantum field theory (QFT) as "impossible measurements," also appears in non-relativistic quantum physics, where certain joint measurements on spatially separated systems could enable signaling even if no physical carrier is traveling between the parties.
Research in non-relativistic quantum information has paralleled the dilemmas seen in QFT, suggesting a common underlying challenge. The crucial question is identifying which nonlocal (i.e. performed on two or more systems without bringing them in the same place) quantum measurements are feasible without breaking the no-signaling principle. It turns out that nonlocal measurements can be made without violating no-signaling, but cannot always be ideal (i.e., they cannot be perfectly repeated immediately). Moreover, they can be performed at the cost of using additional entangled states as resources.
These insights are key to advancing our understanding of quantum measurement both in non-relativistic settings and in QFT, nudging us closer to a unified theory of quantum measurement.

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