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Computational complexity theory is a branch of computer science dedicated to classifying computational problems in terms of their difficulty. While computability theory tells us what we can compute in principle, complexity theory informs us regarding what is feasible. In this chapter I argue that the science of quantum computing illuminates the foundations of complexity theory by emphasising that its fundamental concepts are not model-independent. However this does not, as some have suggested, force us to radically revise the foundations of the theory. For model-independence never has been essential to those foundations. The fundamental aim of complexity theory is to describe what is achievable in practice under various models of computation for our various practical purposes. Reflecting on quantum computing illuminates complexity theory by reminding us of this, too often under-emphasised, fact.

Entropy is an important information measure. A complete understanding of entropy flow will have applications in quantum thermodynamics and beyond; for example it may help to identify the sources of fidelity loss in quantum communications and methods to prevent or control them. Being nonlinear in density matrix, its evaluation for quantum systems requires simultaneous evolution of more-than-one density matrix. Recently in [1] a formalism for such an evolution has been proposed and [2] shows that the flow of entropy between two systems corresponds to the full counting statistics of physical quantities that are exchanged between them. Interestingly, in quantum systems with heat dissipations this will not be equivalent to the second law of thermodynamics. In this talk I will describe a consistent formalism to evaluate entropy and show how to measure it in some quantum systems; for example in quantum point contacts in nanoelectronics and in the quantum heat engines introduced to describe photosynthesis and photovoltaic cells. The entropy flow is made of two parts: 1) an incoherent part, which can be re-evaluated semiclassically from the second law, and 2) a coherent part, which has no semiclassical analogue and appear as a result of extending Kubo-Martin-Schwinger (KMS) correlations.

[1] M.H.A. and Y. Nazarov, Phys. Rev. B 91, 104303 (2015)

[2] M.H.A. and Y. Nazarov, Phys. Rev. B 91, 174307 (2015)