Communication through coherent control of quantum channels

Alastair A. Abbott1,2, Julian Wechs2, Dominic Horsman3, Mehdi Mhalla3, and Cyril Branciard2

1Département de Physique Appliquée, Université de Genève, 1211 Genève, Switzerland
2Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, 38000 Grenoble, France
3Univ. Grenoble Alpes, CNRS, Grenoble INP, LIG, 38000 Grenoble France

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A completely depolarising quantum channel always outputs a fully mixed state and thus cannot transmit any information. In a recent Letter[3], 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 standard framework in quantum computing is that of quantum circuits, where quantum operations are applied to physical systems in a definite causal order. Recently, it has been found that one can go beyond this paradigm, and connect quantum operations in more exotic ways – e.g., with no well-defined causal order. Such indefinite orders open up new possibilities for quantum computing and quantum communication.
In that context, a particular quantum communication effect has attracted substantial interest. A completely noisy quantum channel cannot transmit any information by itself. However, information transmission is possible if two such channels are applied in a superposition of orders – or more precisely, in an order that is coherently determined by a control qubit, taken to be in a quantum superposition.
In our work, we show that a similar phenomenon occurs in an even simpler situation where a control qubit determines which of the two channels acts on the target system, rather than their order. This raises interesting questions about how this communication advantage is related to indefinite causal order.
Our study of this example leads us to a more general analysis of the concept of a quantum-controlled channel, which turns out to be ill-defined. We show that for a complete account of the situation one needs more information about the channel implementation than is usually considered.

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