Indefinite causal order and quantum coordinates

Abstract

Classically the causal order of two timelike separated events A and B is fixed -- either A before B or B before A. This is no longer true in quantum theory, where it is possible to encounter superpositions of causal orders. The quantum switch is one of the most prominent processes with indefinite causal order. Optical realizations of the quantum switch have been successfully implemented in experiments, but, some argue this merely simulates a process with indefinite causal order and that a superposition of spacetime metrics is required for a true realization. Here, we provide a relativistic definition of causal order between operationally defined events that defines a meaningful observable in both the general relativistic and quantum mechanical sense. Importantly, this observable does not distinguish between the indefinite causal order implemented on an optical bench and the gravitational quantum switch, a gedankenexperiment where the indefinite causal order is achieved by a quantum superposition of gravitational fields. Therefore, our results support the thesis that the optical quantum switch is just as much a realization of indefinite causal order as its gravitational counterpart, which makes use of the quantum mechanical behavior of spacetime.

Figures (4)

• Figure 1: We consider a scenario of two systems and a test particle on a fixed spacetime. The timelike worldline $\gamma_0$ of the test particle is depicted in black, while the timelike curves of the systems are depicted by the blue line $\gamma_A$ and the red line $\gamma_B$. The initial and final points defining the curve $\gamma_0$ are denoted by $p_{\gamma_0,i}$ and $p_{\gamma_0,f}$, respectively. The worldlines $\gamma_0$ and $\gamma_A$ coincide once and their crossing defines the event ${\mathcal{E}_A}$, while ${\mathcal{E}_B}$ is defined by the single crossing of the worldlines $\gamma_0$ and $\gamma_B$. We can use the proper time of $\gamma_0$ together with fixed initial and final points to define a causal order for the events.
• Figure 2: A superposition of two scenarios with different causal orders $s^{(1)}$ and $s^{(2)}$. In each branch, we consider a manifold $\mathcal{M}_{1,2}$, a metric $g^{(1),(2)}$, and the worldlines of three systems. The black, blue, and red line depict $\gamma_0$, $\gamma_A$, and $\gamma_B$, with different transparencies illustrating the superposition. There are two events $\mathcal{E}_A$ and $\mathcal{E}_B$ defined by the crossing of the test particle with one of the other trajectories. First, we apply a quantum-controlled diffeomorphism $\phi^{(1)}: \mathcal{M}_1 \to \mathcal{M}$ together with a diffeomorphism $\phi^{(2)}: \mathcal{M}_2\to\mathcal{M}$ chosen such that each of the events is associated with a single point on $\mathcal{M}$. The metric and causal order may still be in superposition.
• Figure 3: In order to encode the causal order operationally, we consider a particle with an internal spin degree of freedom moving along $\gamma_0$, and two agents in laboratories with worldlines $\gamma_A$ and $\gamma_B$. These are placed in a superposition of causal orders between events $\mathcal{E}_{A}$ and $\mathcal{E}_B$. The setup is chosen such that the particle always enters the first laboratory at proper time $\tau_1^\ast$ while it crosses the second laboratory at $\tau_2^\ast$. Through a careful choice of these proper times, we ensure that the agents can perform non-disturbing measurements on the particle whenever it enters their laboratory and thus encode the causal order in a memory register.
• Figure 4: Bloch representation of the state space of operationally encoded causal order. Along the $z$-axis, the causal order is (a) definite, that is, in a state $\mathinner{|{s=\pm 1}\rangle}$, or (b) in a classical mixture of such states. (c) On the surface of the Bloch ball (except the points $\mathinner{|{s=\pm 1}\rangle}$), the system is in a coherent superposition of causal orders and, finally, (d) the remaining states inside the ball represent mixed indefinite causal orders.

Theorems & Definitions (2)

• Definition 1: Causal order between events
• Definition 2