# Simulating Indefinite Causal Order With Rindler Observers

^{1}Faculty of Physics, University of Belgrade, Belgrade, Serbia^{2}Departamento de Física, Universidade Federal de Minas Gerais, Belo Horizonte, Brazil^{3}Physics Department and Research Center Optimas, Technische Universität Kaiserslautern, Kaiserslautern, Germany^{4}Faculty of Physics, Vienna Center for Quantum Science and Technology (VCQ), University of Vienna, Vienna, Austria^{5}Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Vienna, Austria

Realization of indefinite causal order (ICO), a theoretical possibility that even causal relations between physical events can be subjected to quantum superposition, apart from its general significance for the fundamental physics research, would also enable quantum information processing that outperforms protocols in which the underlying causal structure is definite. In this paper, we start with a proposition that an observer in a state of quantum superposition of being at two different relative distances from the event horizon of a black hole, effectively resides in ICO space-time generated by the black hole. By invoking the fact that the near-horizon geometry of a Schwarzschild black hole is that of a Rindler space-time, we propose a way to simulate an observer in ICO space-time by a Rindler observer in a state of superposition of having two different proper accelerations. By extension, a pair of Rindler observers with entangled proper accelerations simulates a pair of entangled ICO observers. Moreover, these Rindler-systems might have a plausible experimental realization by means of optomechanical resonators.

## 1. Introduction

The principle of causality is an implicit assumption of every physical theory and it is universally supported by our experience of nature. From an operational point of view, causality can be understood as a system of signaling or communication relations between physical systems; an information flow whose properties are intimately related to the nature of space and time. One may even say that *the very essence of the classical structure of space and time is to impose a physical constraint on information processing*.

In the old Newtonian picture of the World, space and time are two generically different entities, universal for all observers. There is a single, three-dimensional flat Euclidean space and a single global time that enable us to unambiguously distinguish between past, present and future. Together, they constitute an absolute, independent background structure relative to which every physical event takes place. Signals can propagate in space with unlimited speed (action at a distance) and, consequently, each event can be caused by any other in its present or past. The special theory of relativity (SR) changed this paradigm: space and time became united into the *space-time continuum*—Minkowski space - in which signals cannot travel faster than the speed of light, enforcing them to stay within the local light cone. Nevertheless, the structure of Minkowski space adhered to the character of an independent, fixed background on which dynamical matter fields propagate.

The radical change came with Einstein’s general theory of relativity (GR). The gravitational field came to be understood as the curved space-time itself, gravity being encoded in a metric tensor coupled to dynamical matter fields. There is no fixed, independent metric structure, no absolute background stage relative to which locations of physical events are to be defined, there are just dynamical fields, the metric being one of them, and physical events can only be located *relative to each other*. The possibility of communication between different observers, i.e. the causal order, is entirely determined by the dynamical configuration of light cones, and so, although dynamical, space-time, as a landscape of physical events, has definite causal order (DCO). Thinking about Quantum Mechanics (QM) of gravitating objects, the question arises whether there is a way to relax the restrictions of classical space-time structure and enable processes that do not obey definite causal relations, i.e., can there be a quantum superposition of different causal orders—an *indefinite causal order* (ICO)?

It is generally expected that unification of QM and gravitational physics will provide us with some deeper insights concerning the nature of space and time and their relationship with matter. However, the standard methods of quantization of matter fields employed in Quantum Field Theory (QFT) do not seem to work for Einstein’s gravity; it holds a status of a non-renormalizable effective theory with undetermined high-energy degrees of freedom. In order to surpass the traditional concepts of GR and QFT, various ways of “quantizing” gravity were proposed so far, such as String Theory, Quantum Loop Gravity, Noncommutative Geometry, Supergravity, etc. However, to date, there has been no conclusive empirical evidence that would support or disprove any of the proposed “high-energy theories.” This state of affairs motivates us to reconsider in which sense and to what extent can the seemingly contradictory principles of QM and GR be reconciled, while adhering to the tenets of both theories [1, 2].

There are two main incentives for this paper. The first came from the work of Oreshkov, Costa and Brukner [3], where it was found that it is possible to formulate quantum mechanics without any reference to a global causal structure, i.e. without predefined space-time. The resulting framework - the *process matrix formalism* - allows for processes incompatible with any definite causal order between operations performed on quantum systems. These abstract indefinite causal structures are shown to be advantageous for quantum computing [4, 5] and quantum communication [6–8]. One particular example that has an experimental demonstration is the so called “quantum switch” [4, 9–14], where the main idea is to use an auxiliary quantum system that can coherently control the order in which certain operations are applied. In the case of the so-called gravitational quantum switch (GQS) [15] the role of the control system is played by a gravitating object prepared in a state of quantum superposition of being at two different spatial locations. The second incentive comes from the intriguing idea of *quantum reference frames* (QRF) [16, 17] where one regards reference frames not as abstract systems of coordinates, but as actual physical objects subjected to the laws of quantum mechanics and describes the world from their perspective.

In this paper, we propose a way to simulate ICO processes by considering the fact that the near-horizon geometry of a Schwarzschild black hole (BH) is that of a Rindler space-time. Namely, a Rindler observer in a state of superposition of having two different proper accelerations corresponds to a near-horizon Schwarzschild observer in a state of superposition of being at two different locations along a single radial direction. From the viewpoint of such an observer the geometry of space-time is indefinite. This correspondence can be extended to a pair of Rindler observers with entangled proper accelerations simulating a pair of entangled ICO observers, as we illustrate by means of a simple example. Although they represent idealizations, these Rindler-systems could become a valuable resource for studying ICO processes in laboratory conditions, with plausible experimental realization in the form of opto-mechanical oscillators [18–21].

## 2. Rindler Observers

In order to set the stage, consider the (1 + 1)-dimensional Minkowski space *Rindler coordinates* *right Rindler wedge* or simply the R-wedge [see Figure 1 (left panel)]. The family of coordinate lines of constant ρ are the branches of hyperbolas *Rindler observers*. The proper time of a Rindler observer, with a given

**FIGURE 1**. *Rindler observers.* **(Left panel)**: *Coordinate lines in the right (R) Rindler wedge of the Minkowski space.* Coordinate lines of constant ρ are hyperbolas and they correspond to worldlines of Rindler observers. Regions L and R, called the left and the right Rindler wedge, respectively, are causally disconnected due to the presence of the Rindler horizon **(Right panel)**: *Photon’s worldline intersects the Rindlers.* A pair of Rindler observers in the R-wedge with different proper accelerations *S*, located at

Rindler observer with greater proper acceleration has a more curved worldline (closer to the Rindler horizon). Note also that, due to the presence of the Rindler horizon, Rindler observers in the R-wedge are causally disconnected from the ones in the L-wedge, meaning that they are unable to communicate with each other.

Consider now a pair of Rindler observers in the R-wedge, with different proper accelerations *S*, located at *S*, implying that *a unique* value of

for which

## 3. Indefinite Causal Order via Rindler Observers

Let us assume that we have a pair of Rindler observers in the R-wedge, Rindler-Amber (*Amber* and *Blue* are the colors by which we distinguish the two observers, see Figure 2. Note, however, that these “observers” need not be actual macroscopic measuring devices of any sort, nor sentient beings; they could be microscopic physical systems with some internal degrees of freedom (like spin). On the other hand, they have definite worldlines since they are confined within accelerating laboratories (imagine well-enough localized classical “boxes” each carrying an atom). We assume that these internal degrees of freedom are such that they do not get affected by the accelerated motion of the Rindler laboratory, which we also assume to be completely isolated. Source *S* emits a photon whose worldline intersects the worldlines of

**FIGURE 2**. *Entangled Rindler observers.* Worldline of a photon emitted by the source *S* intersects worldlines of the two Rindler observers, *a* and *b*) occur at the same proper time *A*) or blue (*B*), that distinguishes them. Observer *C*.

When their worldlines intersect, *a*. In general, *b*. By choosing a suitable values of the proper accelerations *a* and *b*) occur *at the same proper times*

Here we want to stress that physical events are not regarded as pure geometrical points that constitute space-time manifold (modulo diffeomorphisms) with some definite set of causal relations defined by the metric. Rather they are defined operationally, through application of a specific unitary transformation, or more generally a specific completely positive trace-preserving (CPTP) map. Taking quantum mechanics into account, we consider the possibility that the same physical event can be in a superposition of occurring at different space-time locations. This would enable the realization of indefinite causal order between pairs of events, such as *a* and *b* in the above discussion. One example of this situation is the already mentioned quantum switch [9–14].

On the left panel of Figure 2, the proper acceleration of *entangled state* that is a superposition of the two previous ones, at

It is important to realize that the event *a* has to be *one and the same* in both “branches” of the superposition. The meeting of the photon with

For

By the time the photon went through the laboratories, unitary transformations,

where

Finally, we need to disentangle the state of the photon from the state of the Rindler observers. To this end, at some moment

Postselection on any pair of possible measurement results leads to the final state of the photon

Subsequently, the photon may be observed at *C*. However, being in the state 7, there is no way to distinguish, given the photon alone, which of the two events (*a* and *b*) lies in the causal future of the other, and the information about the causal order is lost.

## 4. Gravitational Scenario

Imagine now that we have a system that involves a Schwarzschild BH and an observer (outside the horizon) in a state of superposition of being at two different relative proper distances from the horizon. The observer is well-enough localized and has a negligible effect on the gravitational field. Also, we do not assume the existence of a fixed background geometry with reference to which we could define positions; only the *relative* distance between the BH (its horizon) and the observer has physical meaning.

In Schwarzchild coordinates

with *M* being the mass of the BH) is

Therefore, our bipartite system can be interpreted as a situation where we have an observer in the state

The idea that a gravitating object in a state of quantum superposition of being at two different locations somehow “induces” a quantum superposition of different geometries, dates back to Feynman [22] and it has been successfully promoted recently [15, 23–26]. Although this seems as a natural way to combine GR and the linearity of QM, it remains unclear in which sense can a gravitational field (i.e., space-time geometry) be in a state of quantum superposition, see for example [27]. Here, we propose a way of looking at this situation based on the relational character of quantum superposition [28].

Einstein’s equivalence principle states that a gravitational field is *locally equivalent* to an accelerating reference frame in flat space-time. As a consequence, for every well-enough localized stationary

By extending this reasoning, we propose a “quantum” version of Einstein’s equivalence principle by stating that ICO space-time is locally equivalent to a non-inertial reference frame with superposed proper accelerations. In particular, we can relate a Schwarzschild observer in the state

Consider now a pair of observers in the near-horizon region of a Schwarzschild BH, Schwarzschild-Amber *entangled proper accelerations*. On the other hand, from the point of view of the pair of observers, we have a BH in a state of superposition of being “at two different sides” of the observers, symmetrically. These states of the BH are denoted by

**FIGURE 3**. *Gravitational quantum switch*. The system involves a photon source *S*, two observers,

We can perform a photon experiment, similar to the one described in Section 3, that involves the observers *S* that emits a photon in the direction that depends on the position of the BH relative to the observers. The position of the BH plays the role of a quantum control for the whole process (gravitational quantum switch). Due to the gravitational time dilation, we can arrange things so that the photon passes through both laboratories at the same moment of their local proper time (see Supplementary Material for details). This is analogous to the case of the Rindler quantum switch from Section 2. When the photon gets inside the laboratory, a unitary transformation, *event a*, and likewise, the meeting of the photon and the laboratory *event b*. After performing a projective measurement in the superposition basis

## 5. Conclusion

In this paper, we proposed a way to characterize ICO space-time as a space-time associated with the reference frame of a quantum observer - quantum observer perceives ICO space-time. As an illustration, we considered a bipartite system that involves a Schwarzschild BH and an observer outside the horizon, in a state of quantum superposition of being at two different relative distances. By invoking the fact that near-horizon geometry is that of a Rindler space-time, we can relate this ICO observer to the Rindler observer in a state of superposition of having two different proper accelerations. By extension, a pair of ICO observers with entangled proper distances from the BH horizon corresponds to a pair of Rindler observers with entangled proper accelerations. As an example, we analyzed Rindler quantum switch and the related gravitational quantum switch.

Furthermore, a Bell’s inequality for temporal order of events was found in [15]. The same kind of inequality can be derived by using two pairs of Rindler observers, one in the left and the other in the right Rindler wedge. In the corresponding gravitational scenario we would have to take into account the Kruskal extension of the Schwarzschild solution. In this case, the gravitational quantum switch would involve two pairs of observers residing in conformally flat space-times connected by Einstein-Rosen bridge. We postpone this interesting analysis for future work.

On a more practical side, there is a growing effort in demonstrating quantum features of nano-to-mesoscale optomechanical systems. This may provide a challenging, yet feasible experimental realizations for the proposed Rindler systems [18]. Recently, mesoscopic mechanical resonators were considered as quantum non-inertial reference frames [19, 20] and entanglement of two massive mechanical oscillators is achieved [21]. It has been proposed to utilize quantum optical fields in order to prepare and measure the quantum states of mechanical resonators, conceivably opening the possibility to quantum-mechanically control the acceleration of such quantum non-inertial reference frames [18].

In an actual experiment, potential decoherance effects can compromise the predicted result [30]. Moreover, QFT effects could also be taken into account. In this context, our Rindler observers could be viewed as Unruh-DeWitt detectors, where an increase of the thermal noise, due to the Unruh effect, may affect the evolution of the system, such that it can no longer be considered as a coherent superposition, but rather a (convex) classical mixture. However, since we can choose proper accelerations of the Rindler observers to be arbitrarily small by putting the photon source at suitable position, the Unruh effect can always be made negligible. Just to put some numbers, if we set

Finally, and perhaps most importantly, we would like to establish a more rigorous framework that would allow us to formally define an ICO space-time related to a general quantum reference frame. This could be an important step towards a better understanding of the quantum nature of space-time.

## Data Availability Statement

All datasets generated for this study are included in the article.

## Author Contributions

AD, MM, DG, and ČB conceived the research. AD, MM, and DG provided main theoretical idea and equally contributed to writing the paper. NM participated in discussion concerning entangled laboratories. ČB supervised the project.

## Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## Acknowledgments

This is a significantly revised version of the paper that can already be found on the arXiv [33]. The authors thank Dejan Simić, Marko Vojinović, Nikola Paunković, and Ämin Baumeler for helpful comments. AD, MM, and DG acknowledge support from the bilateral project SRB 02/2018, *Causality in Quantum Mechanics and Quantum Gravity*, between Austria and Serbia. AD and MM acknowledge support from the project No. ON171035 and D.G. from the project No. ON171031 of Serbian Ministry of Education and Science. Additionally, AD acknowledges support from scholarship awarded from The Austrian Agency for International Cooperation in Education and Research (OeAD-GmbH). AD and DG acknowledge grant FQXi-MGA-1806 that supported their stay in Vienna. AD and DG would also like to thank University of Vienna and IQOQI for hospitality during their stay. NM thanks Marcelo Terra Cunha, Nelson Yokomizo, Raphael Drumond, Leonardo Neves, and Davi Barros for useful discussions. NM acknowledge Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and Fundação de Amparo à Pesquisa do Estado de Minas Gerais (FAPEMIG) for financial support and VI Paraty Quantum Information School for the introduction into the topic. ČB acknowledges the support from the research platform TURIS, from the European Commission *via* Testing the Large-Scale Limit of Quantum Mechanics (TEQ) (No. 766900) project, and from the Austrian-Serbian bilateral scientific cooperation no. 451-03-02141/2017-09/02, and by the Austrian Science Fund (FWF) through the SFB project BeyondC and a grant from the Foundational Questions Institute (FQXi) Fund. This publication was made possible through the support of the ID 61466 grant from the John Templeton Foundation, as part of the The Quantum Information Structure of Space-time (QISS) Project (qiss.fr). The opinions expressed in this publication are those of the author(s) and do not necessarily reflect the views of the John Templeton Foundation.

## Supplementary Material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fphy.2020.525333/full#supplementary-material

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Keywords: Rindler observer, quantum switch, indefinite causal order, equivalence principle, causality

Citation: Dimić A, Milivojević M, Gočanin D, Móller NS and Brukner Č (2020) Simulating Indefinite Causal Order With Rindler Observers. *Front. Phys.* 8:525333. doi: 10.3389/fphy.2020.525333

Received: 08 January 2020; Accepted: 18 September 2020;

Published: 26 October 2020.

Edited by:

Rosa Poggiani, University of Pisa, ItalyReviewed by:

Jonathan S Ben-Benjamin, Texas A&M University, United StatesPaolo Grigolini, University of North Texas, United States

Rainer Kaltenbaek, University of Ljubljana, Slovenia

Copyright © 2020 Dimić, Milivojević, Gočanin, Móller and Brukner. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Aleksandra Dimić, aleksandra.dimic@ff.bg.ac.rs