Quantum gravitational control of temporal order

According to the Einstein equations, a massive object gives rise to a space-time metric g μν , μ, ν = 0, ..., 3, which in isotropic coordinates and a post-Newtonian expansion reads31: \(g_{{\mathrm{00}}}(r) = - \left( {1 + 2\frac{{{\mathrm{\Phi }}(r)}}{{c^2}}} \right)\), \(g_{ij}(r) = \delta _{ij}\left( {1 - 2\frac{{{\mathrm{\Phi }}(r)}}{{c^2}}} \right),\) i, j = , 2, 3, where r denotes the distance to the location of the mass. In other words, if a test mass or a clock is positioned at a spatial coordinate R a as described by a far-away agent (as in Fig. 1) and the massive object is at a coordinate r M , then r = |R a − r M |, which for clarity we denote below by R a − r M . It is important to note that the same coordinates describe scenarios where the mass is placed at different locations at a finite distance from r M , as long as it remains far from an asymptotic region so that the spatial and temporal coordinates of the far-away agent remain unaffected (i.e. are those of flat Minkowski space-time). In these coordinates, the Hamiltonian of a clock—a particle with internal DOFs—reads

$$H_{\mathrm{a}} = \sqrt { - g_{{\mathrm{00}}}(R_{\mathrm{a}} - r_{M})({\mathrm{\Omega }}_{\mathrm{a}}^2 + c^2g_{ij}(R_{\mathrm{a}} - r_{M})P^iP^j)} ,$$ (9)

(see e.g. refs. 57,58,59) where Pi, i = 1, 2, 3 are the components of the momentum operator, and Ω a is the internal Hamiltonian, describing the local time evolution of the internal DOFs. Note that we can restrict ourself to an effectively one-dimensional scenario, so only one of the spatial coordinates has been kept in the above expression. In the first post-Newtonian expansion and considering that both the mass and the clock follow fixed world lines at constant R a and r M , respectively, Eq. (9) becomes

$$H_{\mathrm{a}} \approx {\mathrm{\Omega }}_{\mathrm{a}}\left( {1 + \frac{{{\mathrm{\Phi }}(R_{\mathrm{a}} - r_{M})}}{{c^2}}} \right).$$ (10)

The asymptotic time coordinate t defines space-like hypersurfaces that are independent of the location of the mass and on which one can define states of all the involved systems (the clocks, the target systems and the mass itself) and Hamiltonian (10) describes their time evolution of with respect to t. Due to the interactions between the mass and the clocks—effected by the space-time metric, which contains the potential Φ(R a − r M )—the time evolution of the clocks depends on their relative distance R a − r M to the mass. Crucially, by the definition of t and the Hamiltonian our description includes both considered different mass configurations: the mass can be semi-classically localised around a single spatial coordinate r or in superposition of different spatial coordinates and the associated states belong to the same Hilbert space associated with a space-like hypersurface labelled by t. We thus have all the tools to analyse time evolution in the presence of a superposition state of the mass, even though it leads to a quantifiably non-classical causal structure.

With respect to t and the associated foliation of space-time, the evolution of the clock, which at t = 0, is in an internal state |s a (τ 0 )〉, where τ 0 denotes the clock’s proper time at t = 0, reads

$$e^{ - i{\mathrm{\Omega }}_{\mathrm{a}}t\left( {1 + \frac{{{\mathrm{\Phi }}(R_{\mathrm{a}} - r_{M})}}{{c^2}}} \right)}|R_{\mathrm{a}}\rangle |s_{\mathrm{a}}(\tau _0)\rangle = |R_{\mathrm{a}}\rangle |s_{\mathrm{a}}(\tau _0 + \tau (R_{\mathrm{a}} - r_{M},t))\rangle ,$$ (11)

where \(\tau (R_{\mathrm{a}} - r_{M},t): = t\left( {1 + \frac{{{\mathrm{\Phi }}(R_{\mathrm{a}} - r_{M})}}{{c^2}}} \right)\) is the proper time elapsing for the clock at a radial distance |R a − r M | from the mass when the elapsed coordinate time is t; and for clarity we set ħ = 1.

Before continuing on to the gravitational quantum control, we give an example of an internal Hamiltonian, state, and evolution. Let us take Ω a = E 0 |0〉〈0| + E 1 |1〉〈1| and \(|s_{\mathrm{a}}(\tau _0 = 0)\rangle = \frac{1}{{\sqrt 2 }}(|0\rangle + |1\rangle )\), which describe, for example, an atom in an equal superposition of some two electronic energy levels |0〉,|1〉 with energies E 0 , E 1 , respectively. Under H a from Eq. (10) internal state |s a (0)〉 from Eq. (11) evolves as

$$\begin{array}{*{20}{l}} {e^{ - i{\mathrm{\Omega }}_{\mathrm{a}}t\left( {1 + \frac{{{\mathrm{\Phi }}(R_{\mathrm{a}} - r_{M})}}{{c^2}}} \right)}\left| {s_{\mathrm{a}}(0)} \right\rangle } \hfill & = \hfill & {\frac{1}{{\sqrt 2 }}e^{ - iE_0t\left( {1 + \frac{{{\mathrm{\Phi }}(R_{\mathrm{a}} - r_{M})}}{{c^2}}} \right)}\left| 0 \right\rangle + \frac{1}{{\sqrt 2 }}e^{ - iE_1t\left( {1 + \frac{{{\mathrm{\Phi }}(R_{\mathrm{a}} - r_{M})}}{{c^2}}} \right)}\left| 1 \right\rangle } \hfill \\ {} \hfill & \equiv \hfill & {\frac{1}{{\sqrt 2 }}e^{ - iE_0\tau (R_{\mathrm{a}} - r_{M},t)}\left| 0 \right\rangle + \frac{1}{{\sqrt 2 }}e^{ - iE_1\tau (R_{\mathrm{a}} - r_{M},t)}\left| 1 \right\rangle ,} \hfill \end{array}$$ (12)

which is simply |s a (τ(R a − r M , t))〉.

We now use the above to show how the quantum superposition principle and general relativity lead to the prediction that quantised matter acts as a quantum control of temporal order. To this end, we assume conditions (a)–(c) from the Results section and consider two clocks positioned at R A and R B , respectively. The Hamiltonian of clock a is thus Eq. (10) and fully analogously for b, \(H_{\mathrm{b}} \approx {\mathrm{\Omega }}_{\mathrm{b}}\left( {1 + \frac{{{\mathrm{\Phi }}(R_{\mathrm{b}} - r_{M})}}{{c^2}}} \right)\). The clocks are initially synchronised with each other and with a clock of the distant agent so that at t 0 = 0 both clocks are at τ 0 = 0. We further consider a target system, for example, a mode of the electromagnetic field, initially in a state |ψ〉S, on which an operation \({\cal{O}}_{\mathrm{A}}\) is performed at an event A = (R a , τ a = τ*) and an operation \({\cal{O}}_{\mathrm{B}}\) at an event B = (R b , τ b = τ*), where τ a , τ b refer to the proper times of the clock A, B, respectively. We effectively represent these operations as \({\cal{O}}_{\mathrm{A}} = \delta (\tau _{\mathrm{a}} - \tau ^ \ast ,r - R_{\mathrm{a}}){\mathrm{O}}_{\mathrm{A}}\), where δ(τ A − τ*, r − R a ) is a Dirac delta distribution and O A is an operator (e.g. describing rotation of the polarisation of an electromagnetic field mode by a particular half-wave plate) independent of time and location. The total Hamiltonian reads

$$H_{{\mathrm{tot}}} = H_{\mathrm{a}} + H_{\mathrm{b}} + {\cal{O}}_{\mathrm{A}} + {\cal{O}}_{\mathrm{B}},$$ (13)

which for simplicity assumes trivial time evolution of the mass and of the target system between the application of the operations. We furthermore consider the following initial (at t 0 = 0) state of the mass, clocks and the target system:

$$\left| {\psi (0)} \right\rangle ^{{\mathrm{MSab}}} = |R_{\mathrm{a}}\rangle |R_{\mathrm{b}}\rangle |s_{\mathrm{a}}(\tau _0 = 0)\rangle |s_{\mathrm{b}}(\tau _0 = 0)\rangle \left| \psi \right\rangle ^{\mathrm{S}}\left( {\left| {r_{\mathrm{L}}} \right\rangle ^{\mathrm{M}} + \left| {r_{\mathrm{R}}} \right\rangle ^{\mathrm{M}}} \right),$$ (14)

where positions r L , r R of the mass refer to the configurations in the left and the right panel of Fig. 1, respectively, that is, they realise configurations \({\mathrm{K}}_{{\mathrm{A}} \prec {\mathrm{B}}}\) and \({\mathrm{K}}_{{\mathrm{B}} \prec {\mathrm{A}}}\): for |r L 〉 the mass is at a distance r a = r L − R a from clock a and at r b = r a − h from b, while for |r R 〉 the relative distances are swapped and the mass is at a distance r a − h from a and at r a from b. After coordinate time t such that τ(r a , t) > τ* (where \(\tau ^ \ast > \frac{{2r_{\mathrm{b}}^2c}}{{GM}}\), see main text) the state evolves to

$$\begin{array}{*{20}{l}} {|\psi (t)\rangle ^{{\mathrm{MSab}}}} \hfill & = \hfill & {|R_{\mathrm{a}}\rangle |R_{\mathrm{b}}\rangle \left( {|s_{\mathrm{a}}(\tau (r_{\mathrm{a}},t))\rangle |s_{\mathrm{b}}(\tau (r_{\mathrm{a}} - h,t))\rangle e^{ - i{\mathrm{O}}_B}e^{ - i{\mathrm{O}}_{\mathrm{A}}}\left| \psi \right\rangle \left| {r_{\mathrm{L}}} \right\rangle ^{\mathrm{M}}} \right.} \hfill \\ {} \hfill & {} \hfill & {\left. { + |s_{\mathrm{a}}(\tau (r_{\mathrm{a}} - h,t))\rangle |s_{\mathrm{b}}(\tau (r_{\mathrm{a}},t))\rangle e^{ - i{\mathrm{O}}_{\mathrm{A}}}e^{ - i{\mathrm{O}}_{\mathrm{B}}}\left| \psi \right\rangle ^{\mathrm{S}}\left| {r_{\mathrm{R}}} \right\rangle ^{\mathrm{M}}} \right).} \hfill \end{array}$$ (15)

The order of applying unitary transformations \(U_{\mathrm{A}} = e^{ - i{\mathrm{O}}_{\mathrm{A}}}\) and \(U_{\mathrm{B}} = e^{ - i{\mathrm{O}}_{\mathrm{B}}}\) to the target system is controlled by the position of the mass, which due to time-dilation changes causal relations between events A and B. Swapping the mass distribution: |r L 〉 → |r R 〉, |r R 〉 → |r L 〉 and letting the state evolve for another time interval t results in the final state where the clocks become synchronised again

$$\left| {\psi (t)} \right\rangle ^{{\mathrm{MSab}}} = |R_{\mathrm{a}}\rangle |R_{\mathrm{b}}\rangle |s_{\mathrm{a}}(\tau _{\mathrm{f}})\rangle |s_{\mathrm{b}}(\tau _{\mathrm{f}})\rangle \left( {U_{\mathrm{B}}U_{\mathrm{A}}\left| \psi \right\rangle ^{\mathrm{S}}\left| {r_{\mathrm{R}}} \right\rangle ^{\mathrm{M}} + U_{\mathrm{A}}U_{\mathrm{B}}\left| \psi \right\rangle ^{\mathrm{S}}\left| {r_{\mathrm{L}}} \right\rangle ^{\mathrm{M}}} \right),$$ (16)

where τ f = τ(r a , t) + τ(r a − h, t). Measuring the mass in a superposition basis |r L 〉M ± |r R 〉M prepares the target system in the corresponding superposition state U B U A |ψ〉S ± U A U B |ψ〉S.

The above example demonstrates that under very conservative assumptions a spatial superposition of a mass generates a quantum-controlled application of unitary operations. More fundamentally, this effect stems from the superposition of different causal structures associated with the superposed states of the mass.

Proof of Bell’s theorem for temporal order

Bell’s theorem in general asserts that, under certain assumptions, the correlations between the outcomes of independent measurements on two subsystems must satisfy a class of inequalities. The two measuring parties are referred to as Alice and Bob. In every experimental run, each of them measures one of two properties of the subsystem they receive. For each of the properties, one of two outcomes is obtained, for convenience chosen to be ±1. Bell’s inequalities follow from the conjunction of the following assumptions: (1) measurement results are determined by properties that exist prior to and independent of the experiment (hidden variables); (2) results obtained at one location are independent of any measurements or actions performed at space-like separation (locality); (3) any process that leads to the choice of which measurement will be carried out is independent from other processes in the experiment (free choice). The outcomes of Alice A(i, λ) and Bob B(i, λ) thus only depend on their own choice of setting, index i, and on the property of the system, variable λ. The correlation between outcomes A(i, λ) and B(i, λ) for the measurement choices i, j is described by \(E(A_i,B_j) = {\int} {\mathrm{d}}\lambda P(\lambda )A(i,\lambda )B(j,\lambda )\), where P(λ) is the probability distribution over the properties of the systems. It is straightforward to check that one possible inequality satisfied by the correlations E(A i , B j ) is the so-called Clauser–Horne–Shimony–Holt inequality: |E(A 1 , B 1 ) + E(A 1 , B 2 ) + E(A 2 , B 1 ) − E(A 2 , B 2 )| ≤ 2. Crucially, quantum theory allows for the left-hand side of this inequality to reach a value >2, and experimental measurements of this (and other inequalities) have confirmed such a violation3,4,5,6. The significance of the violations of Bell’s inequalities is in showing that neither nature nor quantum mechanics obey all three assumptions mentioned above.

The assumption of classical order is sufficient to derive Causal Inequalities16,60: tasks that, without any further assumptions, cannot be performed on a classical causal structure. However, it is not possible to violate causal inequalities using quantum control of order45,61, this is why additional assumptions were required in the present context. It is an open question whether a gravitational implementation of a scenario that does allow for a violation of causal inequalities is possible.

The theorem we have formulated is theory independent, but not fully device-independent, as it requires the notions of a physical state and a physical transformation (in addition to the measured probability distributions), which we introduce below and then proceed to the proof. Discussion of the present work in the context of the theory-dependent framework of causally non-separable quantum processes16,45,61 and the fully theory- and device-independent approach of causal inequalities16,60 is presented in Supplementary Note 1.

We consider a sufficiently broad framework to describe physical systems that can undergo transformations and measurements, similar to generalised probabilistic theories62,63,64. This framework is more general than quantum or classical theory and we thus need to define key notions required in the proof. In this framework, a state ω is a complete specification of the probabilities P(o|i, ω) for observing outcome o given that a measurement with setting i is performed on the system. We are interested in situations where a system can be split up in subsystems, say S 1 and S 2 , with space-like separated agents performing independent operations on S 1 and S 2 . We say ω is a product state, and write ω = ω 1 ⊗ ω 2 , if probabilities for local measurements factorise as P(o 1 , o 2 |i 1 , i 2 , ω) = P(o 1 |i 1 , ω 1 )P(o 2 |i 2 , ω 2 ). If state \(\omega _1^f\) is prepared for system S 1 and state \(\omega _2^f\) is prepared for system S 2 , according to a probability distribution P(f) for some variable f, we write \(\omega = {\int} {\mathrm{d}}f{\kern 1pt} P(f)\omega _1^f \otimes \omega _2^f\) and say the state is separable. Probabilities are then given by the corresponding mixture: \(P(o_1,o_2|i_1,i_2,\omega ) = {\int} {\mathrm{d}}fP(o_1|i_1,\omega _1^f)P(o_2|i_2,\omega _2^f)P(f)\). Note that for such a decomposition Bell inequalities cannot be violated1,65.

A physical transformation of the system is represented by a function \(\omega \mapsto T(\omega )\). To make our arguments precise we need a notion of local transformations, namely, realised at the time and location defined by a local clock. If S 1 is the subsystem on which a local transformation T 1 acts, and S 2 labels the DOFs space-like separated from T 1 , then, by definition, T 1 transforms product states as \(\omega _1 \otimes \omega _2 \mapsto T_1(\omega _1) \otimes \omega _2\) and separable states by convex extension. How local operations act on general, non-separable states can depend on the particular physical theory; however, action on separable states will suffice for our purposes. We further need to define how the local transformations combine. This depends on their relative spatio-temporal locations: if transformations T 1 , T 2 are space-like separated they combine as (T 1 ⊗ T 2 )(ω 1 ⊗ ω 2 ) = T 1 (ω 1 ) ⊗ T 2 (ω 2 ), which follows from the definition above; if T 1 is in the future of T 2 , we define their combination as T 1 ο T 2 (ω) = T 1 (T 2 (ω)). (For simplicity, we omit possible additional transformations taking place between the specified events, as they are of no consequence for our argument.)

Proof Assumption (1) says that there is a random variable f determining the local states \(\omega _1^f\), \(\omega _2^f\) of systems S 1 , S 2 , respectively. Assumption (3) says there is a random variable λ that determines the order of events. In general, the two variables can be correlated by some joint probability distribution P(λ, f). By assumption (4), events labelled A 1 , B 1 are space-like separated from events A 2 , B 2 and the order between events within each set (A j , B j ), j = 1, 2 can be defined by a permutation σ j . Most generally, there is a probability P(σ j |λ) that the permutation σ j is realised for a given λ. By assumption (2), for each given order the system undergoes a transformation \(T^{\sigma _1} \otimes T^{\sigma _2}\), where \(T^{\sigma _1}\) is the transformation obtained by composing \(T_{{\mathrm{A}}_{\mathrm{1}}}\) and \(T_{{\mathrm{B}}_{\mathrm{1}}}\) in the order corresponding to the permutation σ 1 and similarly for \(T^{\sigma _2}\). (For example, if σ 1 corresponds to the order \({\mathrm{A}}_{\mathrm{1}} \prec {\mathrm{B}}_{\mathrm{1}}\), then \(T^{\sigma _1} = T_{{\mathrm{B}}_{\mathrm{1}}} \circ T_{{\mathrm{A}}_{\mathrm{1}}}\).) Furthermore, at event D an outcome z is obtained with a probability P(z|λ, f, σ 1 , σ 2 ). Finally, using assumption (1), we write the probabilities for all outcomes as

$$ {P\left( {o_1,o_2,z|i_1,i_2,{\Bbb T},\omega } \right)} =\\ {\mathop {\sum}\limits_{\sigma _1\sigma _2} {{\int} } {\mathrm{d}}\lambda \,{\mathrm{d}}fP(o_1|i_1,T^{\sigma _1}(\omega _1^f))P(o_2|i_2,T^{\sigma _2}(\omega _2^f))P(\sigma _1|\lambda )P(\sigma _2|\lambda )P(z|\lambda ,f,\sigma _1,\sigma _2)P(\lambda ,f).}$$ (17)

A simple Bayesian inversion P(σ 1 |λ)P(σ 2 |λ)P(z|λ, f, σ 1 , σ 2 )P(λ, f) = P(λ, f, σ 1 , σ 2 |z)P(z), where we used P(σ j |λ) = P(σ j |λ, f), gives the desired probabilities

$$\begin{array}{*{20}{l}} {P\left( {o_1,o_2|i_1,i_2,z,{\Bbb T},\omega } \right)} \hfill & = \hfill & {\mathop {\sum}\limits_{\sigma _1\sigma _2} {{\int} } {\mathrm{d}}\lambda \,{\mathrm{d}}fP(o_1|i_1,T^{\sigma _1}(\omega _1^f))P(o_2|i_2,T^{\sigma _2}(\omega _2^f))P(\lambda ,f,\sigma _1,\sigma _2|z)} \hfill \\ {} \hfill & = \hfill & {{\int} {\mathrm{d}} \tilde fP(o_1|i_1,T^{\sigma _1})P(o_2|i_2,T^{\sigma _2})P(\tilde f|z)}, \hfill \end{array}$$ (18)

where \(\tilde f\) is a short-hand for the variables λ, f, σ 1 , and σ 2 . The above probability distribution satisfies the hypothesis of Bell’s theorem and thus cannot violate any Bell inequality.

Exemplary scenario realising Bell test for temporal order of events

The protocol allowing for the violation of Bell’s inequalities for temporal order exploits correlations between the clocks of the agents a 1 , b 1 and the agents a 2 , b 2 , created due to time dilation induced by the mass. It should be noted that the protocol allows maximal violation of the Bell inequality if the joint state of the systems S 1 and S 2 is pure (and maximally entangled) when the Bell measurements are realised. Thus, for a maximal violation, the clocks need to decorrelate from the mass after the application of the unitaries. Below we sketch a scenario that can achieve this.

The space-time arrangement of the mass and the agents in this example is presented in Fig. 4. It can be realised in one spatial dimension: agents acting on the system S 1 are located at distance h from each other, and the mass is placed at distance r (configuration \({\mathrm{K}}_{{\mathrm{B}} \prec {\mathrm{A}}}\)) or r + L (configuration \({\mathrm{K}}_{{\mathrm{A}} \prec {\mathrm{B}}}\)) from agent a 1 . Agents acting on system S 2 are placed symmetrically on the opposite side of the mass, such that the mass is at a distance r + L from a 2 in configuration \({\mathrm{K}}_{{\mathrm{B}} \prec {\mathrm{A}}}\) and r in configuration \({\mathrm{K}}_{{\mathrm{A}} \prec {\mathrm{B}}}\). Here, events A j are defined by the local time τ a that differs from the local time τ b defining B j , j = 1, 2. In such a case, even though the mass is always closer to a j than to b j , the two mass configurations can lead to different event orders—as they induce different relative time dilations. (Equivalently, one can introduce an initial offset in the synchronisation of the clocks.) Note that the time orders between the two groups are here “anti-correlated”: \({\mathrm{A}}_{\mathrm{1}} \prec {\mathrm{B}}_{\mathrm{1}}\) and \({\mathrm{B}}_{\mathrm{2}} \prec {\mathrm{A}}_{\mathrm{2}}\) for \({\mathrm{K}}_{{\mathrm{A}} \prec {\mathrm{B}}}\), and vice versa for \({\mathrm{K}}_{{\mathrm{B}} \prec {\mathrm{A}}}\). Since otherwise the scenario is the same for S 1 and S 2 , we focus on the operations performed on S 1 . The key observation is that swapping the mass distribution, as depicted in Fig. 4, will eventually disentangle the clocks from the mass, and since the clocks must be suitably time dilated when the operations are performed, the operations must not take place in the future light cone of the swapped mass state.

Fig. 4 Space-time diagram of a protocol for disentangling the clocks from the mass. In configuration \({\mathrm{K}}_{{\mathrm{A}} \prec {\mathrm{B}}}\) the mass is at a distance r + L from a 1 , and at r + L + h from b 1 . In \({\mathrm{K}}_{{\mathrm{B}} \prec {\mathrm{A}}}\)—it is at r from a 1 and at r + h from b 1 .The opposite holds for a 2 , b 2 . The initial mass superposition is swapped (after sufficient time to prepare the clocks in the correlated state) so that they finally show the same time. At the local time τ a of a 1 (at event A 1 ) the agent applies \(U_{{\mathrm{A}}_{\mathrm{1}}}\) on S 1 . At the local time τ b of b 1 the agent applies \(U_{{\mathrm{B}}_{\mathrm{1}}}\) on S 1 . For the mass configuration \({\mathrm{K}}_{{\mathrm{A}} \prec {\mathrm{B}}}\) A 1 is before B 1 (orange-coloured events), while for \({\mathrm{K}}_{{\mathrm{B}} \prec {\mathrm{A}}}\) event B 1 is before A 1 (blue-coloured events). The opposite order holds for events A 2 , B 2 occurring on the opposite side of the mass, where agents a 2 , b 2 act on S 2 . Unitary operations should be applied in the future light cone of the event where the clocks get correlated and outside the future light cone of the event when the mass amplitudes are swapped, Bell measurements (at C 1 , C 2 ) should be made when the clocks become disentangled (at future light-like events to when the mass amplitudes are brought together), and the measurement at event D should be space-like to C 1 , C 2 ; dashed yellow lines represent the relevant light cones Full size image

The proper time τ a that has to elapse for the clock of a 1 such that the order of events is \({\mathrm{A}}_{\mathrm{1}} \prec {\mathrm{B}}_{\mathrm{1}}\) for \(|{\mathrm{K}}_{{\mathrm{A}} \prec {\mathrm{B}}}\rangle\) and \({\mathrm{B}}_{\mathrm{1}} \prec {\mathrm{A}}_{\mathrm{1}}\) for \(|{\mathrm{K}}_{{\mathrm{B}} \prec {\mathrm{A}}}\rangle\) for the present case reads

$$\tau _{\mathrm{a}} = \sqrt { - g_{{\mathrm{00}}}(r)} \frac{{T_{c}(r,h) + T_{c}(r + L,h)\sqrt {\frac{{g_{{\mathrm{00}}}(r + L + h)}}{{g_{{\mathrm{00}}}(r + h)}}} }}{{1 - \sqrt {\frac{{g_{{\mathrm{00}}}(r)g_{{\mathrm{00}}}(r + L + h)}}{{g_{{\mathrm{00}}}(r + h)g_{{\mathrm{00}}}(r + L)}}} }},$$ (19)

where T c (r, L/2) is the coordinate travel time of light between radial distances r and r + L/2 from the mass. The coordinate time corresponding to τ a is \(T_{\mathrm{a}} = \tau _{\mathrm{a}}/\sqrt { - g_{{\mathrm{00}}}(r)}\). The proper time of event B 1 is then defined as:

$$\tau _{\mathrm{b}} = \sqrt { - g_{{\mathrm{00}}}(r + L + h)} \left( {\frac{{\tau _{\mathrm{a}}}}{{\sqrt { - g_{{\mathrm{00}}}(r + L)} }} + T_{c}(r + L,h)} \right).$$ (20)

It can directly be checked that when the mass is placed in configuration \({\mathrm{K}}_{{\mathrm{A}} \prec {\mathrm{B}}}\)—at a distance r + L from a 1 —the event A 1 defined by local clock of a 1 showing proper time τ a from Eq. (18) is in the past light cone of event B 1 , which is defined by the local clock of b 1 showing proper time τ b from Eq. (19). When the mass is placed in configuration \({\mathrm{K}}_{{\mathrm{B}} \prec {\mathrm{A}}}\), event B 1 ends up in the past of the event A 1 . The coordinate time required for the application of the operations can be estimated as twice the travel time of light between the agents, T o = 2T c (r + L/2, h).

The world lines of the mass can be arranged such that: (a) the mass is moving slow so that the two amplitudes of the mass are swapped in a time interval longer than T o ; (b) during the application of the operations the distance of each agent to the mass is approximately the same for both mass configurations (as in Fig. 4). The first guarantees that there is enough time to apply the operations after the clocks get correlated, the second—that the slow-down of light in curved space-time, the Shapiro delay66,67, can be neglected.

The coordinate-time duration of the entire protocol can be estimated as T p = 2T a + 4L/2c, where L/2c is the minimal time required to put the mass in superposition of amplitudes separated by the distance L/2. Taking as an example M ~ 0.1 μg, L = h ~ 0.1 μm, r ~ 1 fm, the protocol in Fig. 4 takes T p ~10 h. Furthermore, we note that a quantum treatment of the local clocks is central to our protocol since the application of the operations on the target systems is conditioned on the states of the clocks. The time-energy uncertainty68,69 thus poses a limitation to a single-shot precision with which space-time events can be defined with physical clocks. The optimal clock state in this context—evolving the fastest—is a balanced superposition of energy eigenstates; for an energy gap ħ ⋅ 2πν c , where ν c is the clock frequency, the smallest time that can be resolved by a single quantum system is the so-called orthogonalisation time70,71,72 t ⊥ = 1/2ν c . For the values of parameters quoted above, the coordinate-time difference between the superposed locations of the events A i , i = 1, 2 is ~10−15 s, and we thus need a system with frequency ν c ≥1015 Hz such as a clock based on optical transitions in ytterbium73 or mercury74, which both give t ⊥ ~10−16 s. While this ideal limit is not reached with practical systems, the resolution of current atomic clocks based on such atoms far exceeds this theoretical bound due to averaging over many atoms, with 2.5 × 10−19 uncertainty of the clock frequency recently demonstrated in ref. 75. We further note that by using n entangled atoms, the orthogonalisation time of the entire system becomes t ⊥ /n and can thus be even a few orders of magnitude smaller76 than required. Finally, such atoms have masses ~10−25 kg and their back action on the metric produced by M ~ 10−7 kg would thus be negligible. Since the mass difference between the atom in the two involved energy levels is 2πħν c /c2 ~10−35 kg also quantum effects from the clocks’ mutual gravitational interactions58 can be neglected.

We conclude that it is in principle possible to achieve the required entanglement of orders, swap the mass distribution so as to finally disentangle the clocks from the mass, and satisfy the locality conditions on the events. Although a direct experiment in such a regime is not practical, the above example surprisingly shows that the regime where entangled temporal order arises is in no way related to the Planck scale. It is usually assumed that the Planckscale marks the regime where quantum gravity effects become relevant (first discussed in this context by Bronstein77), but this is not the case for the superposition of temporal order. In terms of a potential experiment, one could also take a different (theory-dependent) approach and explore possible witnesses of entangled temporal order61, in analogy to witnesses of entanglement in quantum-information theory78. A witness would probe the quantum nature of temporal order indirectly and under further assumptions, but in a relaxed parameter range. Such an approach may lead to more feasible experiments, which will be explored in a future study.