Locality and nonlocality in the interaction-free measurement

Abstract

Abstract. We present a paradox involving a particle and a mirror. They exchange a nonlocal quantity, modular angular momentum L z mod 2 , but there seems to be no local interaction between them that allows such an exchange. We demonstrate that the particle and mirror do interact locally via a weak local current L z mod 2 w . In this sense, we transform the "interaction-free measurement" of Elitzur and Vaidman, in which two local quantities (the positions of a photon and a bomb in the two arms of a Mach-Zehnder interferometer) interact nonlocally, into a thought experiment in which two nonlocal quantities (the weak modular angular momentum of the particle and of the mirror) interact locally. The quantum Cheshire CatSo-called "weak values" [1] have taken their place alongside eigenvalues and expectation values as possible measured values in quantum mechanics. But while an ordinary ensemble suffices for measuring eigenvalues and expectation values, weak values require a "pre-and post-selected" (PPS) ensemble. Though unconventional, a PPS ensemble with initial state |ψ in and final state |ψ f in is (in principle) easy to prepare: we measure an operator that has |ψ in as an eigenstate, and then an operator with |ψ f in as an eigenstate, on as many systems as we like; and then we keep only those systems with those respective eigenstates. In between, we measure whatever we like, but with a measurement interaction weak enough to be consistent with the PPS ensemble. If the interaction is weak enough, the result of measuring an operator A is the weak value A w of A:In this way, weak values enable us to answer questions about quantum systems that we otherwise cannot even ask. An example of a weak value is the "quantum Cheshire cat" [2,3], named after the Cheshire Cat in Alice in Wonderland [4] who could disappear while leaving its grin behind. In the weak-value version, a photon takes one path through a Mach-Zehnder interferometer while its net polarization vanishes on that path but not on the other. In this experiment, the photon and its polarization separate at a welldefined moment as the Cat passes through the first beam-splitter of the interferometer. There is also [5] an experiment in which the separation is continuous: the Cat is confined to one side of a potential a

Hilbert Mathematics Versus Gödel Mathematics III- Hilbert Mathematics by Itself, and Gödel Mathematics Versus the Physical World within It: both as Its Particular Cases

“…The conflict of locality and nonlocality can be also illustrated by the "interaction-free measurement"(Rohrlich, Aharonov, Landsberger 2018), by the "pure-state identification problems"(Ishida, Hashimoto, Horibe, Hayashi 2008), or a two-proton EPR experiment(McWeeny, Amovilli 1995).…”