Quantum mechanics allows events to happen with no definite causal order: this can be verified by measuring a causal witness, in the same way that an entanglement witness verifies entanglement. Here, we realize a photonic quantum switch, where two operations andB act in a quantum superposition of their two possible orders. The operations are on the transverse spatial mode of the photons; polarization coherently controls their order. Our implementation ensures that the operations cannot be distinguished by spatial or temporal position-further it allows qudit encoding in the target. We confirm our quantum switch has no definite causal order by constructing a causal witness and measuring its value to be 18 standard deviations beyond the definite-order bound. DOI: 10.1103/PhysRevLett.121.090503 In daily experience, it is natural to think of events happening in a fixed causal order. Strikingly, it has been proposed that quantum physics allows for nonclassical causal structures where the order of events is indefinite [1,2]. It has been theoretically shown that such a possibility provides an advantage for computation [3], communication complexity [4,5], and other information processing tasks [6][7][8]. Furthermore, investigations of indefinite causal orders suggest a promising route towards a theory that combines general relativity and quantum mechanics [9,10].Indefinite causal orders can be studied using a framework that distinguishes whether some experimental situationcalled a "process"-is compatible with a fixed causal order of the events or not. An example of a process with indefinite causal order is the "quantum switch" [1]. In the quantum switch, the order in which two quantum operation andBconsidered as "black box operations"-are performed on a target system is coherently controlled by a control quantum system (Fig. 1). This can also be seen as a particular case of "superposition of time evolution" [11]. The advantages provided by the quantum switch arise from the fact that it cannot be reproduced by an ordinary quantum circuit which uses the same number of black box operations [3][4][5][6][7].Here, we present an optical implementation of the quantum switch where the control system is the photon's polarization and the target is the transverse spatial mode. We verify indefinite causal order by introducing a causal witness [14,15], for which we obtain a value 18 standard deviations beyond the bound for definite ordering. One notable achievement of our experiment is that it opens the possibility of encoding more than two levels in the target system-transverse spatial mode can indeed be highdimensional and hence can act as a qudit.In previous implementations [12,13], the location of each black box-the spot where photons go through a set of wave plates-was different depending on the order, resulting in four distinct locations in space [ Fig. 1(c)]. Furthermore, the photons had a coherence length much shorter than the distance between the two sets of wave plates: in effect, the operations could also be distinct in...

Classically, no information can be transmitted through a depolarising, which is a completely noisy, channel. We show that by combining a depolarising channel with another channel in an indefinite causal order-that is, when there is a superposition of the order that these two channels were applied-it becomes possible to transmit significant information. We consider two limiting cases. When both channels are fully depolarising, the ideal limit is communication of 0.049 bits; experimentally we achieve (3.4 ± 0.2) × 10 −2 bits. When one channel is fully depolarising, and the other is a known unitary, the ideal limit is communication of 1 bit. We experimentally achieve 0.64 ± 0.02 bits. Our results offer intriguing possibilities for future communication strategies beyond conventional quantum Shannon theory.

Quantum causality extends the conventional notion of the fixed causal structure by allowing channels and operations to act in an indefinite causal order. The importance of such an indefinite causal order ranges from the foundational—e.g., towards a theory of quantum gravity—to the applied—e.g., for advantages in communication and computation. In this review, the authors will walk through the basic theory of indefinite causal order and focus on experiments that rely on a physically realizable indefinite causal ordered process—the quantum switch.