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# Entanglement of quantum clocks through gravity

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

## Significance

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.

## Abstract

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.

A crucial aspect of any physical theory is to describe the behavior of systems with respect to the passage of time. Operationally, this means establishing a correlation between the system itself and another physical entity, which acts as a clock. In the context of general relativity, time is specified locally in terms of the proper time along world lines. It is believed that clocks along these world lines correlate to the metric field in such a way that their readings coincide with the proper time predicted by the theory—the so-called “clock hypothesis” (1). A common picture of a reference frame uses a latticework of clocks to locate events in space–time (2). An observer, with a particular split of space–time into space and time, places clocks locally, over a region of space. These clocks record the events and label them with the spatial coordinate of the clock nearest to the event and the time read by this clock when the event occurred. The observer then reads out the data recorded by the clocks at his/her location. Importantly, the observer does not need to be sitting next to the clock to do so. We will call an observer who measures time according to a given clock, but not located next to it, a far-away observer.

In the clock latticework picture, it is conventionally considered that the clocks are external objects that do not interact with the rest of the universe. This assumption does not treat clocks and the rest of physical systems on equal footing and therefore is artificial. In the words of Einstein: “One is struck [by the fact] that the theory [of special relativity]… introduces two kinds of physical things, i.e., (1) measuring rods and clocks, (2) all other things, e.g., the electromagnetic field, the material point, etc. This, in a certain sense, is inconsistent…” (3). For the sake of consistency, it is natural to assume that the clocks, being physical, behave according to the principles of our most fundamental physical theories: quantum mechanics and general relativity.

In general, the study of clocks as quantum systems in a relativistic context provides an important framework for investigating the limits of the measurability of space–time intervals (4). Limitations to the measurability of time are also relevant in models of quantum gravity (5, 6). It is an open question how quantum mechanical effects modify our conception of space and time and how the usual conception is obtained in the limit where quantum mechanical effects can be neglected.

In this work, we show that quantum mechanical and gravitational properties of the clocks put fundamental limits to the joint measurability of time as given by clocks along nearby world lines. As a general feature, a quantum clock is a system in a superposition of energy eigenstates. Its precision, understood as the minimal time in which the state evolves into an orthogonal one, is inversely proportional to the energy difference between the eigenstates (7⇓⇓⇓–11). Due to the mass–energy equivalence, gravitational effects arise from the energies corresponding to the state of the clock. These effects become nonnegligible in the limit of high precision of time measurement. In fact, each energy eigenstate of the clock corresponds to a different gravitational field. Because the clock runs in a superposition of energy eigenstates, the gravitational field in its vicinity, and therefore the space–time metric, is in a superposition. We prove that, as a consequence of this fact, the time dilation of clocks evolving along nearby world lines is ill-defined.We show that this effect is already present in the weak gravity and slow velocities limit, in which the number of particles is conserved. Moreover, the effect leads to entanglement between nearby clocks, implying that there are fundamental limitations to the measurability of time as recorded by the clocks.

The limitation, stemming from quantum mechanical and general relativistic considerations, is of a different nature than the ones in which the space–time metric is assumed to be fixed (4). Other works regarding the lack of measurability of time due to the effects the clock itself has on space–time (5, 6) argue that the limitation arises from the creation of black holes. We will show that our effect is independent of this effect, too. Moreover, it is significant in a regime orders of magnitude before a black hole is created. Finally, we recover the classical notion of time measurement in the limit where the clocks are increasingly large quantum systems and the measurement precision is coarse enough not to reveal the quantum features of the system. In this way, we show how the (classical) general relativistic notion of time dilation emerges from our model in terms of the average mass–energy of a gravitating quantum system.

From a methodological point of view, we propose a *gedanken* experiment where both general relativistic time dilation effects and quantum superpositions of space–times play significant roles. Our intention, as is the case for *gedanken* experiments, is to take distinctive features from known physical theories (quantum mechanics and general relativity, in this case) and explore their mutual consistency in a particular physical scenario. We believe, based on the role *gedanken* experiments played in the early days of quantum mechanics and relativity, that such considerations can shed light on regimes for which there is no complete physical theory and can provide useful insights into the physical effects to be expected at regimes that are not within the reach of current experimental capabilities.

## Clock Model

Any system that is in a superposition of energy eigenstates can be used as a reference clock with respect to which one defines time evolution. The simplest possible case is that in which the clock is a particle with an internal degree of freedom that forms a two-level system. In the following, we assume the clock to follow a semiclassical trajectory that is approximately static, that is, it has (approximately) zero velocity with respect to the observer who uses the clock to define operationally his/her reference frame, in the sense stated in the Introduction. In this way, special relativistic effects can be ignored. We stress the fact that the observer does not need to be located next to the clock. He/she can perform measurements on it by sending a probe quantum system to interact with the clock and then measuring the probe in his/her location. In the following, we focus only on the clock’s internal degrees of freedom, which are the only ones relevant to our model. The internal Hamiltonian of the particle in its rest reference frame,

An operational meaning of the “passage of a unit of time,” in which, by definition, the system goes through a noticeable change from an initial state to a final state, can be given in terms of the “orthogonalization time” of the clock, that is, the time it takes for the initial state to become orthogonal to itself. For a two-level system, the orthogonalization time is equal to *Clocks in the Classical Limit*, when studying how the general relativistic notion of time dilation emerges in the classical limit.

The gravitational effects due to the energies involved are to be expected at a fundamental level. In particular, for a given energy of the clock, there is a time dilation effect in its surroundings, due to the mass–energy equivalence. However, because the mass–energy corresponding to the amplitude of **2** is uncertain (Fig. 1). Consider a second clock localized at a coordinate distance *SI Appendix, Analysis of the Coordinate* t, we quantify the minimum distance between the observer and the clocks such that he/she cannot operationally distinguish between the different gravitational fields.

As a consequence of these considerations, there is a fundamental trade-off between the accuracy of measuring time at the location of the clock and the uncertainty of time dilation at nearby points. This trade-off can be succinctly described by the relation

So far, our treatment of time dilation in the vicinity of the clock has been classical and nonoperational. In *Two Clocks*, we explain the above effect in terms of gravitational interaction between quantum clocks.

## Two Clocks

Consider two gravitationally interacting clocks, labeled by **4** based on the superposition principle and the mass–energy equivalence, see *SI Appendix, Heuristic Derivation of the Two-Clock Hamiltonian*. The same Hamiltonian can be obtained from a field theory perspective by the restriction to the two-particle sector of the field (16) and the use of the mass–energy equivalence, as we sketch in *SI Appendix, Two-Clock Hamiltonian from Quantum Field Theory Approach*. Although the methods presented here suffice to describe the entanglement of clocks arising from gravitational interaction, a full description of the physics with no background space–time would require a fundamental quantum theory of gravity. In the works of Rovelli (17) and Isham (18), for example, it is suggested that time itself emerges from the dynamics of more fundamental degrees of freedom.

Let us assume that the energies of both Hamiltonians

We see from Eq. **5** that the clocks get entangled through gravitational interaction: The rate at which time runs in one clock is correlated to the value of the energy of the other clock. The state gets maximally entangled for the time

It is important to point out that, for this effect to arise, it is crucial that we consider the internal energy of the clocks as a quantum operator, instead of just taking into account the expectation value of the energy, as is done in semiclassical gravity. To explain this point, let us describe the evolution of clock *Clocks in the Classical Limit* that this situation is effectively recovered in the classical limit of clocks.

Note that, after

The effect presented here has a fundamental influence on the measurement of time that follows only from quantum mechanics and general relativity in the weak-field limit. It is independent of the usual argument concerning limitations of the measurability of space–time intervals due to black hole formation (5, 6). As we will see in *N + 1 Clocks*, the effect is significant in a parameter regime that occurs long before formation of black holes becomes relevant. To strengthen the effect, we next consider

*N* + 1 Clocks

Now suppose there are *Two Clocks*,

In our case,

We now give an estimate of the parameter regime where decoherence is significant. The calculations are done ignoring all effects external to our model and should be understood in terms of a *gedanken* experiment. The intention is to contrast the predictions given by our model with the usual predictions given by quantum gravity models, which do not expect limitations due to the combined effects of quantum mechanics and general relativity before the Planck scale. (For a discussion of the role of the Planck scale in the possibility of defining time, see refs. 5, 19, and 20). Fig. 2 shows the decoherence time ^{4}He (21), and a macroscopic number of particles

To end this section, we note that, despite the fact that this effect is not large enough to be measured with the current experimental capabilities, it might be possible to perform experiments on analog systems to test this effect. Specifically, in ref. 22, the authors consider an atom traversing an oscillating quantum reference frame, and show that the phase of the wave function of the atom has an uncertainty that can be related to the uncertainty in the atom’s elapsed proper time. By the equivalence principle, it is possible to interpret the acceleration that the oscillating reference frame induces on the atom as the gravitational effect that one clock suffers as a consequence of the presence of another nearby clock.

## Clocks in the Classical Limit

Given the ill-definedness of time measured by a single clock when it is in the presence of other clocks, how does the classical notion of a clock, including relativistic time dilation effects, arise? In what follows, we answer this question by considering the classical limit of our model. The quantum state that is closest to the classical state of a clock is a spin or atomic coherent state. In general, spin coherent states can be defined as

The Hamiltonian that evolves the state of this clock is the extension to angular momentum **1** is written as

One of the approaches to the classical limit from within quantum mechanics is based on an experimental resolution that is coarse enough not to reveal the quantum features of the system (23). In our case, we consider coarse-grained time measurements characterized by the experimental resolution

Consider now two clocks, labeled by **9**. For the initial state **10**. When **12** evolve with different time dilation factors, given by each of the phases

There are two effects, different in nature, whose relative contributions to the evolution of **12** contribute significantly to the state, due to the binomial distribution

The evolution of the reduced state **14** in *SI Appendix, Derivation of the Master Equation*, following closely ref. 15. We show that the derivation of the master Eq. **14** holds, in general, for any quantum system and any form of the Hamiltonians **14** will be nonzero, as the variance of the energy will not vanish; implying that, irrespective of the nature of the clocks, they will get entangled.

Finally, in the light of the analysis of the present section, let us now return to Eq. **3**, obtained via a heuristic semiclassical argument in *Clock Model*, and show that it can also be derived from the classical limit of two interacting clocks, connecting the heuristic arguments based on the superposition principle and gravitational time dilation to our treatment of interacting clocks in the classical limit. Consider the two-clock scenario at the beginning of this section with **3** up to a factor of

## Discussion

In the (classical) picture of a reference frame given by general relativity, an observer sets an array of clocks over a region of a spacial hypersurface. These clocks trace world lines and tick according to the value of the metric tensor along their trajectory. Here we have shown that, under an operational definition of time, this picture is untenable. The reason does not only lie in the limitation of the accuracy of time measurement by a single clock, coming from the usual quantum gravity argument in which a black hole is formed when the energy density used to probe space–time lies inside the Schwarzschild radius for that energy. Rather, the effect we predict here comes from the interaction between nearby clocks, given by the mass–energy equivalence, the validity of the Einstein equations, and the linearity of quantum theory. We have shown that clocks interacting gravitationally get entangled due to gravitational time dilation: The rate at which a single clock ticks depends on the energy of the surrounding clocks. This interaction produces a mixing of the reduced state of a single clock, with a characteristic decoherence time after which the system is no longer able to work as a clock. Although the regime of energies and distances in which this effect is considerable is still far away from the current experimental capabilities, the effect is significant at energy scales that exist naturally in subatomic particle bound states.

These results suggest that, in the accuracy regime where the gravitational effects of the clocks are relevant, time intervals along nearby world lines cannot be measured with arbitrary precision, even in principle. This conclusion may lead us to question whether the notion of time intervals along nearby world lines is well defined. Because the space–time distance between events, and hence the question as to whether the events are space-like, light-like, or time-like separated, depend on the measurability of time intervals, one can expect that the situations discussed here may lead to physical scenarios with indefinite causal structure (25). The notion of well-defined time measurability is obtained only in the limit of high-dimensional quantum systems subjected to accuracy-limited measurements. Moreover, we have shown that our model reproduces the classical time dilation characteristic of general relativity in the appropriate limit of clocks as spin coherent states. This limit is consistent with the semiclassical limit of gravity in the quantum regime, in which the energy–momentum tensor is replaced by its expectation value, despite the fact that, in general, the effect cannot be understood within this approximation.

The operational approach presented here and the consequences obtained from it suggest that considering clocks as real physical systems instead of idealized objects might lead to new insights concerning the phenomena to be expected at regimes where both quantum mechanical and general relativistic effects are relevant.

## Acknowledgments

We thank F. Costa, A. Feix, P. Hoehn, W. Wieland, and M. Zych for interesting discussions. We acknowledge support from the John Templeton Foundation, Project 60609, “Quantum Causal Structures,” from the research platform “Testing Quantum and Gravity Interface with Single Photons” (TURIS), and the Austrian Science Fund (FWF) through the special research program “Foundations and Applications of Quantum Science” (FoQuS), the doctoral program “Complex Quantum Systems” (CoQuS) under Project W1210-N25, and Individual Project 24621.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: esteban.castro.ruiz{at}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.

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