The Quantum Cheshire Cat Effect in the Presence of Decoherence

Even the subtle and apparently strange quantum effects can sometimes survive otherwise lethal influence of an omnipresent decoherence. We show that an archetypal quantum Cheshire Cat, a paradox of a separation between a position of a quantum particle, a photon, and its internal property, the polarization, in a two-path Mach-Zehnder setting, is robust to decoherence caused by a bosonic infinite bath locally coupled to the polarization of a photon. Decoherence affects either the cat or its grin depending on which of the two paths is noisy. For a pure decoherence, in an absence of photon-environment energy exchange, we provide exact results for weak values of the photon position and polarization indicating that the information loss affects the quantum Cheshire Cat only qualitatively and the paradox survives. We show that it is also the case beyond the pure decoherence for a small rate of dissipation