O tempo pode ser impreciso: se assim for, a ideia de causalidade pode estar em apuros!

sexta-feira, junho 09, 2017

Entanglement of quantum clocks through gravity

Esteban Castro Ruiz a,b,1, Flaminia Giacomini a,b, and Časlav Brukner a,b

Author Affiliations

a Vienna Center for Quantum Science and Technology, Faculty of Physics, University of Vienna, A-1090 Vienna, Austria;

b Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, A-1090 Vienna, Austria

Edited by Abhay V. Ashtekar, The Pennsylvania State University, University Park, PA, and approved January 30, 2017 (received for review October 4, 2016)


We find that there exist fundamental limitations to the joint measurability of time along neighboring space–time trajectories, arising from the interplay between quantum mechanics and general relativity. Because any quantum clock must be in a superposition of energy eigenstates, the mass–energy equivalence leads to a trade-off between the possibilities for an observer to define time intervals at the location of the clock and in its vicinity. This effect is fundamental, in the sense that it does not depend on the particular constitution of the clock, and is a necessary consequence of the superposition principle and the mass–energy equivalence. We show how the notion of time in general relativity emerges from this situation in the classical limit.


In general relativity, the picture of space–time assigns an ideal clock to each world line. Being ideal, gravitational effects due to these clocks are ignored and the flow of time according to one clock is not affected by the presence of clocks along nearby world lines. However, if time is defined operationally, as a pointer position of a physical clock that obeys the principles of general relativity and quantum mechanics, such a picture is, at most, a convenient fiction. Specifically, we show that the general relativistic mass–energy equivalence implies gravitational interaction between the clocks, whereas the quantum mechanical superposition of energy eigenstates leads to a nonfixed metric background. Based only on the assumption that both principles hold in this situation, we show that the clocks necessarily get entangled through time dilation effect, which eventually leads to a loss of coherence of a single clock. Hence, the time as measured by a single clock is not well defined. However, the general relativistic notion of time is recovered in the classical limit of clocks.

quantum clocks entanglement gravity classical limit


1To whom correspondence should be addressed. Email: esteban.castro.ruiz@univie.ac.at.

Author contributions: E.C.R., F.G., and Č.B. designed research; E.C.R., F.G., and Č.B. performed research; and E.C.R. wrote the paper with input from F.G. and C̆.B.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1616427114/-/DCSupplemental.

Freely available online through the PNAS open access option.