One of the basic tools in any quantum information theorist is the completely positive trace preserving map, or quantum channel. They provide a general framework for discussing the dynamics and information content of quantum systems with uncontrolled degrees of freedom, i.e. open systems. In the context of quantum computing in particular they are used to provide models of ‘errors’ in quantum computers. For this reason, much attention has been given in recent years to providing metrics that quantify the ‘error-free-ness’ of a quantum channel, preferably in a way that is efficiently measureable in the lab. Some of these measures, such as the average fidelity to the identity and the unitarity have by now become standard benchmarks for the quality of new quantum devices.
One of the central challenges in relating such measures to the operational capacity of a quantum device (e.g. the ability to perform quantum error correction and fault-tolerant quantum computation) lies in the difficulty of relating the quality of a set of elementary quantum channels to the quality of their composition. This is so because the set of quantum channels is complicated and it is generally difficult to make strong statements about the composition of several quantum channels based on limited information (such as the average fidelity) about the constituent channels. It is known that given elementary channels of fixed fidelity the fidelity of the composed channel can change radically based on the structure of the elementary channels. This divergence has important consequences for quantum error correction, where the relation between the average fidelity of an error channel and the performance of a quantum error correction code (with respect to this error channel) strongly depends on the nature of this channel. Over the last few years a somewhat folkloristic knowledge arose that there are two types of quantum channels: coherent channels (i.e. unitary rotations) and decohering channels (e.g. a depolarizing channel), that have a different behaviour under composition and thus impact the behaviour of e.g. error correction codes in different ways.
However a clear elucidation of this dichotomy remained lacking until the present work by Carignan-Dugas et al. . Their fundamental insight lies in realising that it is fruitful not to pursue an exact classification of channels in to ‘coherent’ and ‘decoherent’ types, but rather construct an approximate dichotomy, where ‘approximation’ is expressed in average fidelity to the identity. The first realisation of Carignan-Dugas et al is that a quantum channel is well-approximated by its largest (in Euclidean norm) Kraus operator (the so-called Leading Kraus Approximation) as long as it is close enough to the identity. This in itself is not that surprising, since it is equivalent to approximating the Choi matrix of a quantum channel by the rank one projector associated to its largest eigenvalue. However, Carignan-Dugas et al. turn this insight into a powerful tool by first proving that the average fidelity and unitarity of the composition of quantum channels are well approximated by the composition of their leading Kraus approximations. Moreover they argue that through this leading Kraus approximation the quantum channel can be decomposed into coherent and decoherent factors by considering the polar decomposition of the leading Kraus operator.
This vindicates the folkloristic notion that all error processes are either coherent or decohering or a composition of both. Moreover, this insight is leveraged to prove a powerful statement about how quantum channels compose with respect to the fidelity and unitarity. Carignan-Dugas et al. prove that both the fidelity and the average fidelity of a composition of quantum channels are essentially dictated by the level of coherence of the underlying quantum channels, as captured by their unitarity. In particular for the average fidelity they prove that the average fidelity of a composition is upper bounded by the product of the fidelities of the decoherent factors (as defined by the polar decomposition) of the elementary channels and that any deviation from that upper bound is accounted for by a unitary rotation. This means that, at least with respect to fidelity, coherence and decoherence are essentially all there is.
It must be noted that all the above statements do not hold exactly, but only up to some small correction factors. And I think here the power of the core idea of the paper reveals itself, namely that things often become a lot clearer if you squint a little.
 Arnaud Carignan-Dugas, Matthew Alexander, and Joseph Emerson. A polar decomposition for quantum channels (with applications to bounding error propagation in quantum circuits). Quantum 3, 173 (2019), 10.22331/q-2019-08-12-173.
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