Indefinite Causal Order in a Quantum Switch

Abstract

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...

On the definition and characterisation of multipartite causal (non)separability

“…From definition 5 we recover the natural, intuitive definition of Araújo et al [14] in the particular tripartite case where one party has a trivial outgoing system-a case of practical relevance, as the quantum switch is the first example of a causally nonseparable process that has been demonstrated and studied in laboratory experiments [24,25,27]. One can also readily verify that process matrices that are causally separable by definition 5 cannot generate noncausal correlations (as defined in [17,22]); an explicit proof is given in appendix E.…”

“…Many natural such constraints can also be imposed as SDP constraints, as described in [15], allowing one to find implementable causal witnesses. A particularly natural such constraint is to restrict B and Cʼs operations to unitary operations (as in the experimental implementation of the tripartite switch in [24,27]); we find that the tolerable white noise on W switch to witness its causal nonseparability is reduced, under such a restriction, to 0.746.…”

“…0, 1 and where It was shown that the process matrix describing the quantum switch is causally nonseparable as per definition 2 [14], and this definition has subsequently been used e.g. in [15,18,27].…”

“…in the definition of the conditional matrix. nonseparability to be verified experimentally by having each party perform appropriately chosen measurements [25,27].…”

“…This approach has been used, e.g. to verify experimentally the causal nonseparability of two different implementations of the quantum switch [25,27].…”