#### Bell's theorem for temporal order

To figure a unification of quantum theory and gravity, the physics community has recently started pursuing a new direction. The core idea of this direction is rather simple. Suppose, an object with mass is placed in a quantum superposition of two location, then presumably the gravitational interaction of this object with other nearby objects should also be in a quantum superposition. As an example, consider placing a massive object in a superposition of being on the left and right of a test mass. The test mass should fell a superposition of being pulled to the left or right.

What does it even mean to be in a superposition of being pulled to the left or right? First of all, you can check this with electromagnetism, by imagining the massive object is a proton and the test mass is an electron. The formalism of quantum theory tells you already how the electron will move in such a scenario. Gravity is different from electromagnetism (or the strong and weak forces), and the mechanics of the test mass's motion can be more complicated when gravity is fully accounted for. The difference is that the gravity of a massive object not only exerts forces on the test mass, but also determines in which causal order other objects interact with this test mass, or even if they even interact with it at all.

According to Einstein's theory of gravity, mass changes the metric of spacetime, which determines how far different spacetime points (called events) are. It also asserts that there is a maxiumum speed at which an object may move in spacetime (the speed of light). Two events are said to be causally connected if information (moving at the speed of light) can transmit from one event to the other. If it can, then the earlier event can causally influence physical systems at the later event. Otherwise, it cannot.

Now, by placing a massive object just so, one can change the distance between two events. For example, suppose two objects were not causally connected, but the placement of the massive object, reduces the distance between the two objects so that now they are causally connected. Or vice versa, two causally connected events can be made to be causally disconnected. Even more interestingly, consider two causally disconnected events $A$ and $B$, such that $A$ and $B$ are simultaneous in some reference frame. Now, one placement of a massive object forces $A$ to be in the causal past of $B$, and another placement of the massive object forces $B$ to be in the past of $A$.

Imagine that there is some physical system traversing through spacetime, that passes through this region where $A$ and $B$ are. Now, depending on the placement of massive object, this physical system will either pass through $A$ first or $B$. Morever, there is someone sitting at $A$ and someone sitting at $B$, whose job is to transform the state of the physical system by $U_A$ and $U_B$ respectively. So, the physical system will either be transformed by $U_BU_A$ or $U_AU_B$, depending on the placement of the massive object.

Finally, to complete the picture, imagine the massive object is in the superposition of the two locations, so the physical system is effected by a superposition of $U_BU_A$ and $U_AU_B$. Such a scenario is called one of indefinite causal order. The causal order in which the state of the physical system is transformed cannot be partially ordered. Note that superpositions are different from statistical mixture of causal orders. Deterministic or statistical mixtures are termed classical, while others are termed non-classical.

I am reading this fascinating paper, titled Bell's theorem for temporal order, which develops this idea much further. The fundamental question the authors try to tackle is, whether there is an experimental test which can distinguish a classically ordered causal order from that of non-classical one. Meaning, the experiment should be such that no classical system can reproduce the results of one consisting of superpositions of causal orders.

The regular Bell theorem specifies an experimental test that distinguishes between local hidden variable theories from quantum theories. Two entangled particles are distributed to two distant agents, who subsequently measure these particles in certain pre-agreed upon ways. The correlatons between the results of their measurements are higher than what would be possible in a universe governed by classical laws.

Here, the authors construct a similar test. Instead of starting with an entangled state, they start a separable state. The separable state cannot be used in the regular Bell test. What they do is use the superposition of causal orders to convert this separable state into an entangled state. Then they perform the regular Bell test on this entangled state. If the state is genuinly entangled, then the Bell test will be passed, otherwise it will be failed. Since, the source of the entangled state is the indefinite causal order, success of the subsequent test is evidence that indefinite causal order working as expected.

What I would like to know is what other information processing tasks can be designed whose possibility or difficulty will be determined by the laws of the universe we live in. One way forward is to look at the list of currently known information processing tasks in which operations occur in some well defined order and modify them so they involve indefinite causal orders.