A completely depolarising quantum channel always outputs a fully mixed state and thus cannot transmit any information. In a recent Letter\cite{ebler18}, it was however shown that if a quantum state passes through two such channels in a quantum superposition of different orders---a setup known as the ``quantum switch''---then information can nevertheless be transmitted through the channels. Here, we show that a similar effect can be obtained when one coherently controls between sending a target system through one of two identical depolarising channels. Whereas it is tempting to attribute this effect in the quantum switch to the indefinite causal order between the channels, causal indefiniteness plays no role in this new scenario. This raises questions about its role in the corresponding effect in the quantum switch. We study this new scenario in detail and we see that, when quantum channels are controlled coherently, information about their specific implementation is accessible in the output state of the joint control-target system. This allows two different implementations of what is usually considered to be the same channel to therefore be differentiated. More generally, we find that to completely describe the action of a coherently controlled quantum channel, one needs to specify not only a description of the channel (e.g., in terms of Kraus operators), but an additional ``transformation matrix'' depending on its implementation.

The concept of causal nonseparability has been recently introduced, in opposition to that of causal separability, to qualify physical processes that locally abide by the laws of quantum theory, but cannot be embedded in a well-defined global causal structure. While the definition is unambiguous in the bipartite case, its generalisation to the multipartite case is not so straightforward. Two seemingly different generalisations have been proposed, one for a restricted tripartite scenario and one for the general multipartite case. Here we compare the two, showing that they are in fact inequivalent. We propose our own definition of causal (non)separability for the general case, which-although a priori subtly different -turns out to be equivalent to the concept of 'extensible causal (non)separability' introduced before, and which we argue is a more natural definition for general multipartite scenarios. We then derive necessary, as well as sufficient conditions to characterise causally (non)separable processes in practice. These allow one to devise practical tests, by generalising the tool of witnesses of causal nonseparability. J Wechs et ali.e. the unitaries are applied in a 'superposition of orders'. The output state is then sent to a third party C (Charlie) who can measure the control qubit, and possibly also the target system. The protocol just described can straightforwardly be generalised to the case where A and Bʼs operations are general quantum instruments instead of unitaries. This so-called quantum switch can be understood as a quantum supermap [23], or higher order transformation, that maps A and Bʼs local operations to the overall global transformation. It cannot be realised by inserting the local operations into a circuit with a well-defined causal order, and therefore constitutes a new resource for quantum computation that goes beyond causally ordered quantum circuits [4]. It has attracted particular interest as a consequence of being readily implementable, and indeed several implementations have been experimentally realised [24-28]. Consequent work has sought to clarify whether such implementations can New J. Phys. 21 (2019) 013027 J Wechs et al New J. Phys. 21 (2019) 013027 J Wechs et al 12 1 2 (which may be empty), respectively, one must have 18 New J. Phys. 21 (2019) 013027 J Wechs et al W k M k 1 1 is causally separable. 30 New J. Phys. 21 (2019) 013027 J Wechs et al