Indefinite temporal order without gravity

For later reference. we will briefly review the salient aspects of the protocol introduced in Ref. Bell4time . The goal of the protocol is to realise four events, $A_{1}$, $B_{1}$, $A_{2}$, $B_{2}$, whose pair-wise order is ‘entangled’: $A_{1}$ is in the causal past/past lightcone of $B_{1}$, denoted $A_{1}\prec B_{1}$, when $A_{2}\prec B_{2}$, and $A_{1}$ is in the causal future/future lightcone of $B_{1}$, denoted $A_{1}\succ B_{1}$ when $A_{2}\succ B_{2}$; the full scenario is arranged such that this entanglement leads to correlations that do not admit a classical explanation, which is formalised in analogy to Bell-like scenarios for local classical properties. Crucially, an “event” is here understood operationally as “something that happens at a particular time and place” and thus is defined by some physical reference system. In the protocol, the reference systems are four clocks, $a_{1}$, $b_{1}$, $a_{2}$, $b_{2}$, while the events are associated with quantum operations performed on some additional system when the corresponding clock reaches a specified proper time: $A_{1}$ takes place (an operation is performed) when $a_{1}$ reaches time $\tau^{*}$, $B_{1}$ takes place when $b_{1}$ reaches time $\tau^{*}$, and so on.

In flat spacetime, if the clocks are initially synchronised (in an arbitrary chosen reference frame), all events are space-like separated. However, introducing a massive body closer to some clocks than others causes a differential time dilation, which can ‘push’ some events into the future light-cone of other events. Thereby, the position of the mass provides a control of the time order of events, see Fig. 1.

If the control mass is prepared in a semiclassical configuration denoted $K_{A}$, event $A_{j}$, $j=1,2$ is in the past of $B_{j}$, while for a different configuration, $K_{B}$, event $B_{j}$ is in the past of $A_{j}$. Thus, $A_{j}$ is time-like from $B_{j}$, for each mass configuration, but their order is interchanged. Moreover, for both mass configurations the pair $A_{1}$, $B_{1}$ is space-like from $A_{2}$, $B_{2}$.

The full protocol features also two of the above mentioned additional systems on which the operations are performed, $S_{1}$ and $S_{2}$, referred to as ’targets’. The operations at events $A_{1}$ and $B_{1}$ are applied only on the target system $S_{1}$, while those at $A_{2}$ and $B_{2}$ are applied only on $S_{2}$. The two target systems are initially in a product state, $\ket{\psi}^{S_{1}}\ket{\psi}^{S_{2}}$, and the considered operations are unitaries: $\hat{U}^{A_{1}}$ is applied on $S_{1}$ at event $A_{1}$, etc. By preparing the control mass in a superposition state, $\ket{K}=\frac{1}{\sqrt{2}}\left(\ket{K_{A}}+\ket{K_{B}}\right)$, one obtains the final state

Next, the control mass is measured in the basis $\ket{\pm}=\frac{1}{\sqrt{2}}\left(\ket{K_{A}}\pm\ket{K_{B}}\right)$, leaving the target system in

where $\ket{\psi_{A}}=\hat{U}^{B}\hat{U}^{A}\ket{\psi}$, $\ket{\psi_{B}}=\hat{U}^{A}\hat{U}^{B}\ket{\psi}$, and we are using the same unitaries in the two ‘wings’: $\hat{U}^{A_{1}}=\hat{U}^{A_{2}}\equiv\hat{U}^{A}$, $\hat{U}^{B_{1}}=\hat{U}^{B_{2}}\equiv\hat{U}^{B}$.

In general, $\ket{\Psi^{\textrm{post}}}$ is an entangled state, unless $\ket{\psi_{A}}=\ket{\psi_{B}}$ (and it is elementary to find examples of unitaries $\hat{U}^{A}$, $\hat{U}^{B}$ such that this is not the case). In the final step of the protocol, the entangled state is measured in appropriate bases, that lead to a violation of a Bell inequality.

The argument presented in Ref. Bell4time is that, given the initial product state of the target systems, local operations that are performed in a definite order would not be able to produce entanglement, even after conditioning on the control system. Therefore, if a set of conditions is satisfied, a violation of Bell inequalities implies that the operations were not performed in a definite order.

Another assumption of the protocol, which ends up a focal point of the present work, is that any additional evolution of the target systems (including their free evolution) between the events of interest can be neglected.

In the “Methods” and the “Results” sections, we focus on reproducing the gravitational protocol using special relativistic time dilation. We present an operational setting where the dynamics of all the relevant DoFs is incorporated, in particular the DoFs whose interactions realize the four uniatries $\hat{U}^{A},\hat{U}^{B}$. In the “Ambiguity in the signature of indefinite temporal order” section we discuss our main result – that entanglement can be generated and Bell inequality for temporal order can be violated even if temporal order is classically defined – and argue that it is due to the failure of the additional assumption mentioned above, that target systems have trivial evolution apart from the unitaries marking the four spacetime events of interest. In the “Conclusion” section we discuss the implications of our result, including the fundamental question of how to isolate quantum features of a causal structure from other non-classical effects.

Each molecule has two possible trajectories, depending on the spin state. The specific trajectories we propose are shown in Fig. 3. The involved trajectories are analogous between the two wings of the experiment: They start and end at a common event but the proper time elapsing along each trajectory is different. Both trajectories can be constructed by joining four identical hyperbolic segments, characterised by proper accelerations $\mathcal{A}_{\uparrow}$, $\mathcal{A}_{\downarrow}$ for spin up, $\ket{\uparrow}$, or spin down, $\ket{\downarrow}$, respectively. These segments have to be rotated or flipped according to Fig. 3, so that the acceleration for each trajectory switches sign three times. Let us assume that the acceleration for the spin $\ket{\uparrow}$ is bigger than an acceleration for the spin $\ket{\downarrow}$. Knowing the value of proper time along the generic hyperbolic trajectory (see e.g. Ref. Dragan2021 ), we can find the difference between proper times measured at the common end of one such pair of trajectories:

where $\tau_{\uparrow/\downarrow}$ is the proper time measured along trajectory of a $\uparrow/\downarrow$ spin and $T_{\mathcal{A}}$ is the total travel time in a common inertial frame of reference, e.g. the frame used to initially synchronize the clocks.

Note that it is possible to achieve a large value of $\Delta\tau$ using a small value of proper acceleration. For a large value of $T_{\mathcal{A}}$ we have:

It means that we can achieve the required $\Delta\tau$ (to make the events defined by such time dilated pair of clocks timelike) by accelerating for long enough time, even for arbitrarily small accelerations. Finally, taking the initial state of the spin of the two molecules to be entangled, one of the two clocks in each cavity will “be older” than another in a correspondingly correlated manner. Thus, the order of operations, controlled by the clocks, will likewise be “entagled”, as a direct consequence of the initial spin entanglement and of time dilation. In this scenario, the joint spin state of the two molecules plays the role of the control, which is played by the position of a massive object in the gravitational case Bell4time . Table 1 summarises the differences between the gravitational and our cavity-based implementation of the protocol, in terms of control systems, target(s), and local operations.

We now move to the full protocol, whose aim is to demonstrate indefinite causal order in a fully operational setting. We consider two ‘wings’ of the experiment, where in each wing two operations take place. The goal is to correlate the order of each pair of operations with a control system and, after measuring the control, produce an entangled state between the two wings, which would not have been possible had the operations been realised in a definite order.

Recall that we want to entangle the ‘target’ systems which comprise the fields in both cavities as well as the detectors (which are two per cavity), while the four spin-$\frac{1}{2}$ particles play the role of the control. The full protocol involves four molecules (two in each wing), where each molecule contains a clock, a detector, and a spin-$\frac{1}{2}$ particle. Initially, the state of these molecules reads:

where $\ket{\tau_{0}}$ is the initial state of one clock (and thus all four clocks are initially decorrelated and synchronised) and each detector is in its ground state, while the spins are entangled.

Note that this state is constructed in such a way that the four molecules can be divided into two identical pairs. These pairs are then accelerated as explained in the “Trajectories” section. After the process of acceleration, the state of all four molecules is:

Due to time dilation, after the acceleration takes place, one clock from each pair of molecules will be older than the other from the same pair. We cannot determine which one because the initial state of spins is a superposition of two different possibilities. Next, each pair of molecules enters a cavity. The first pair is placed in cavity $1$ at the respective positions $x_{1}$ and $x_{2}$ (determined relative to the boundary of this cavity at $x=0$). The second pair of molecules is placed in cavity $2$ at the same positions relative to that cavity. Recall that the field in each cavity is initially in the vacuum state. As discussed in the “Two molecules in the cavity” section, we can decorrelate the clock DoF by re-synchronising the clocks. This is done after the molecules interact with the cavities. To keep track of the effect of the clocks on the detector-field interactions, it is enough to note that a detector belonging to a molecule with spin $\downarrow$ interacts first (and with spin $\uparrow$ interacts second). To sum up the protocol, the initial state of the total system including the two cavities is:

where we used $\otimes$ to separate states from the two cavities and showed which degrees of freedom comprise the targets $S_{1}$ and $S_{2}$ introduced in the “Methods” Section.

After the interactions between the detectors and the cavity fields, the state of the system reads:

where the detectors and cavity states, $\ket{\psi_{L/R}}$, are those in Eqs (20), (21).

Next the molecules are ‘broken apart’. All spins are send to a common location where, at an event labelled $D$, they are jointly measured in the basis $\frac{1}{\sqrt{2}}(\ket{\uparrow\mathrel{\mspace{-1.0mu}}\downarrow\uparrow\mathrel{\mspace{-1.0mu}}\downarrow}\pm\ket{\downarrow\mathrel{\mspace{-1.0mu}}\uparrow\downarrow\mathrel{\mspace{-1.0mu}}\uparrow})$. This prepares the remaining systems— cavities and detectors (which stay next to their cavities)—in the state

where the sign $\pm$ depends on the measurement outcome. This state is entangled as long as $\ket{\psi_{L}}\neq\ket{\psi_{R}}$. This is typically shown by finding the scalar product between $\ket{\psi_{L}}$ and $\ket{\psi_{R}}$ which requires the Dyson series up to second order in $\lambda$.

However, there is an equivalent but technically much simpler method to show that the final state is in general entangled, which is to consider another measurement performed on each detector pair in a basis containing the vector $1/\sqrt{2}\left(\ket{ge}+\ket{eg}\right)$. The resulting conditional states of the fields (arising from Eqs (20) and (21)) read

Upon measuring the control in the above defined entangled basis, $\frac{1}{\sqrt{2}}(\ket{\uparrow\mathrel{\mspace{-1.0mu}}\downarrow\uparrow\mathrel{\mspace{-1.0mu}}\downarrow}\pm\ket{\downarrow\mathrel{\mspace{-1.0mu}}\uparrow\downarrow\mathrel{\mspace{-1.0mu}}\uparrow})$, the joint state of two cavities fields up to the leading order is:

Note that the states $\ket{\Phi_{L}}$ and $\ket{\Phi_{R}}$ describe only the fields in the cavities. The conditional state $\ket{\tilde{\Psi}^{\pm}}$ is entangled as long as $\ket{\Phi_{R}}\neq\ket{\Phi_{L}}$ which is easier to verify than the condition $\ket{\psi_{R}}\neq\ket{\psi_{L}}$. We further note that the measurement on the detectors has different possible outcomes, not all of which result in entangled field states. Crucially however, for a product state across $S_{1}$ and $S_{2}$, all local detector measurement would result in product states of the fields. Therefore, obtaining entangled state for one of the outcomes is sufficient to prove that the original state was entangled.

To sum up, we have shown that using two cavities, four molecules, and the basic effects of special relativity, we can obtain the state given in Eq. (29). If this state is entangled, in the next steps of the protocol this entanglement can be used to violate Bell’s inequalities with the goal to prove that the detector–field interaction events were not classically ordered.

We now proceed to prove that the state in Eq. (29) can indeed be entangled by showing that there exist parameters for which $\ket{\Phi_{R}}\neq\ket{\Phi_{L}}$. The scalar product between these states reads:

Explicit derivation is presented in Supplementary Note 3. Therein we further show how to choose parameters for which the states are not just different but orthogonal. To summarise this procedure, the parameters that have to be chosen are:

1. 1.

   The length $L$ of each cavity and the positions $x_{1},x_{2}$ of the molecules relative to their respective cavity.

2. 2.

   The energy gap $\Omega$ of the detectors.

3. 3.

   The duration $T$ of the interaction between each detector and the cavity field.

4. 4.

   The time dilation $\Delta\tau$ between clocks within each pair of molecules (within each cavity), arising from the different accelerations $\mathcal{A}_{\uparrow/\downarrow}$.

Remarkably, it is also possible to find parameters such that $\ket{\Phi_{R}}\neq\ket{\Phi_{L}}$ even when the two field-detector interactions within each cavity are space-like separated, see Fig. 4. This means that it is possible to obtain an entangled state also in the case when, in some reference frame, the relevant operations are performed in the same temporal order. Although from the perspective of such a reference frame the operations would take place over four different coordinate regions, their order would be the same in each of the amplitudes. Obviously then, entanglement generated in this scheme cannot be simply attributed to non-classical temporal order. We discuss implications and argue for generality of this result in the next sections.

We have explicitly modelled a special-relativistic version of a protocol where gravitational time dilation and quantum superposition lead to an indefinite temporal order of events from Ref. Bell4time . This protocol was formulated in terms of a ‘Bell inequality for temporal order’. The idea was to formulate a protocol where operations in a definite order cannot produce an entangled state, if an appropriate set of assumptions is satisfied. (The final step of the protocol requires measurements on the state in order to violate a Bell inequality and verify the entanglement.) The motivation was to find a test of temporal order that is theory independent, which was based on the observation that a violation of a Bell inequality would prove indefinite temporal order without assuming that the final state is described by quantum mechanics. In this section, we reexamine the assumptions made in Ref. Bell4time , showing that the theory-independent nature of the argument is problematic.

We first remark that the issue we have identified does not arise if we believe that quantum mechanics is valid, i.e., if we wish to provide experimental evidence for non-classical temporal order assuming that the involved states and transformations are faithfully described by quantum theory. In such a case, already the one-cavity part of our protocol, the “Methods” section, or the gravitational equivalent, is sufficient. The reason is that at an abstract level, these protocols implement a “quantum switch” chiribella09 – a scenario where two local operations, represented by unitary operators $\hat{U}^{A}$ $\hat{U}^{B}$, act on a target system in an order determined by a control system which is prepared in a superposition, thus producing a final state of the form

Here, $\ket{0}$, $\ket{1}$ are two basis states of the control, and $\ket{\psi}$ is the initial state of the target system. $\hat{V}_{0}$ and $\hat{V}_{1}$ are two arbitrary unitary operators, representing the evolution of the target between the two operations. Most presentations of the switch do not include the intermediate evolution, but we will see shortly that this is important in our context²²2For generality, one can also include a control-dependent initial state, but this is not necessary for our analysis.. Given an implementation of the switch, by making appropriate final measurements for a set of suitably chosen operations $\hat{U}^{A}$, $\hat{U}^{B}$ it is possible to prove that the operations are not performed in a definite order. This procedure is known as measuring a causal witness araujo15 ; Branciard2016a and it could in principle be incorporated within a single-cavity variant of our protocol to demonstrate that, if the quantum description of the experiment is correct, the cavity-detector interactions do not take place in a definite order.

Returning to the Bell inequality approach, it essentially is an ‘entangled’ version of the switch, with the final state of the form

In order to use such a state to disprove classical temporal order among $\hat{U}^{A}$, $\hat{U}^{B}$, the state has to arise in a scenario satisfying all assumptions used to derive the Bell inequality for temporal order, apart from the assumption of classical order itself. All assumptions used in Ref Bell4time for the derivation of the inequality are: the initial state of the target systems $S_{1},S_{2}$ is separable; transformations performed on the targets are local (i.e. the operations on target $S_{j}$ act as identity on other degrees of freedom of the system); events/spacetime regions at which transformations and measurements take place are suitably separated: both interaction events in one wing are spacelike separated from both interaction events in the other wing, and event $\mathrm{D}$ is spacelike from the events at which the Bell measurements are performed (which as, usual in Bell inequalities, are assumed to be spacelike from each other); the choices of bases for Bell measurements are independent of all other aspects of the experiment (often referred to as ‘free choice’ assumption); and, finally, of course the assumption that events at which transformations and measurements are performed are classically ordered.

The surprising result identified in the “Entanglement of the final state” section is that entanglement is produced while all the above assumptions are met, including the assumption of classical order. Clearly, some other assumption was made to derive the inequality and is violated in our implementation. Indeed, as already mentioned, the additional assumption made in Ref. Bell4time is that the target systems do not have non-trivial evolution apart from the transformations $\hat{U}^{A}$ and $\hat{U}^{B}$. Below we explain why this assumption is violated in the present implementation and in the “Ambiguity in the signature of indefinite temporal order” section we argue that this will remain true in a generic dynamical implementation.

In order to see where the assumption of no free evolution enters and why it is the culprit it is again sufficient to look at one wing of our setup. Comparing a single-cavity scenario, Eq. (18), and a generic quantum switch, Eq. (31), one finds that the evolution operators $\hat{U}_{1}$, $\hat{U}_{2}$ in Eq. (18) are not directly representing the local operations $\hat{U}^{A}$, $\hat{U}^{B}$. The reason is that $\hat{U}_{1}$, $\hat{U}_{2}$ are written in the Dirac (interaction) picture, which necessarily includes time evolution with respect to the free Hamiltonian (starting from some initial time established in a common reference frame). By unravelling this time evolution, one finds precisely a state of the form (31), where $\hat{V}_{0}$ and $\hat{V}_{1}$ represent the free evolution of the targets (i.e. cavity and detectors) between the interactions. Note that the time intervals between the events, and thus intervals of free evolution, are equal in the reference frame of the cavity, which means that in that frame $\hat{V}_{0}=\hat{V}_{1}$. On the other hand, $\hat{U}^{A}$, $\hat{U}^{B}$ describe only the field-cavity interactions in the Schrödinger picture³³3Strictly speaking, we should add free-evolution operators also before and after the two local unitaries—not only in between. However, free evolution acts trivially on the vacuum state (our initial state), while the final evolution can be reabsorbed in the definition of the measurement basis..

The reason this is relevant here is that if the free Hamiltonian does not commute with the interaction, the free evolution does not commute with $\hat{U}^{A},\hat{U}^{B}$, and this is how the presence of entanglement can be explained in the state in Eq. (32) in a frame where $\hat{V}_{0}=\hat{V}_{1}$ and the events are spacelike separated. Note that in that case there is a reference frame where $A_{j}\prec B_{j}$ for both states of the control, however, as we mentioned above, in that frame necessarily $\hat{V}_{0}\neq\hat{V}_{1}$, since the time intervals of free evolution along the worldlines of the molecule 1 and 2 are in such a frame necessarily different. In such a frame the final state in the two wing scenario becomes

and the presence of entanglement is thus interpreted as due to overall different dynamics depending on the control, $\hat{U}^{B_{i}}\hat{V}_{0}\hat{U}^{A_{i}}\neq\hat{U}^{B_{i}}\hat{V}_{1}\hat{U}^{A_{i}}$ i.e., in that frame while the order of operations is common, when they take place relative to periods of free dynamics, and thus the overll evolution, depends on the control. Crucially, in this case temporal order among events cannot be even defined as changes depending on the reference frame. A further discussion of the role played by free evolution in this protocol is presented in Supplementary Note 4.

In fact, if free evolution and the applied operations do not commute, even simpler scenarios can illustrate the issue. Consider that one of the operations is trivial, say $\hat{U}^{B}=\hat{\mathbb{I}}$, and so in fact only one operation is applied. The non-commutativity between $\hat{U}_{A}$ and $\hat{V}=\hat{V}_{0}=\hat{V}_{1}$ (we are in the reference frame of the cavity) would again result in different final states depending on when $\hat{U}^{A}$ is applied relative to $\hat{V}$. This would again lead to an entangled final state in a two-wing scenario, even though in this case there is even no time order of events to speak of.

Finally, we note that in all quantum switch scenarios, including the entangled switch, there is an assumption that each local operation is performed ‘only once’. This condition is, however, naturally satisfied in the relativistic implementations such as ours, as each operation is performed at a specific time of a local clock.