Thermodynamics is a highly successful macroscopic theory widely used across the natural sciences and for the construction of everyday devices, from car engines to solar cells. With thermodynamics predating quantum theory, research now aims to uncover the thermodynamic laws that govern finite size systems which may in addition host quantum effects. Recent theoretical breakthroughs include the characterisation of the efficiency of quantum thermal engines, the extension of classical non-equilibrium fluctuation theorems to the quantum regime and a new thermodynamic resource theory has led to the discovery of a set of second laws for finite size systems. These results have substantially advanced our understanding of nanoscale thermodynamics, however putting a finger on what is genuinely quantum in quantum thermodynamics has remained a challenge. Here we identify information processing tasks, the so-called projections, that can only be formulated within the framework of quantum mechanics. We show that the physical realisation of such projections can come with a non-trivial thermodynamic work only for quantum states with coherences. This contrasts with information erasure, first investigated by Landauer, for which a thermodynamic work cost applies for classical and quantum erasure alike. Repercussions on quantum work fluctuation relations and thermodynamic single-shot approaches are also discussed.

We consider an isolated autonomous quantum machine, where an explicit quantum clock is responsible for performing all transformations on an arbitrary quantum system (the engine), via a time-independent Hamiltonian. In a general context, we show that this model can exactly implement any energy-conserving unitary on the engine, without degrading the clock. Furthermore, we show that when the engine includes a quantum work storage device we can approximately perform completely general unitaries on the remainder of the engine. This framework can be used in quantum thermodynamics to carry out arbitrary transformations of a system, with accuracy and extracted work as close to optimal as desired, while obeying the first and second laws of thermodynamics. We thus show that autonomous thermal machines suffer no intrinsic thermodynamic cost compared to externally controlled ones.Recently there has been a great deal of interest in the application of thermodynamics to individual quantum systems, which may be composed of just a few atoms or qubits [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. Given that thermodynamics was invented before quantum theory was even envisaged, and typically applies to macroscopic objects, it is perhaps surprising how close an analogy can be drawn between the quantum and classical case. In [1][2][3][4], thermal engines are constructed out of quantum mechanical parts, incorporating an explicit system, thermal bath and work storage system. In other approaches [5,6], the thermal engine is a system with externally controlled Hamiltonian and access to a thermal bath.So far, these frameworks all involve the external application of discrete transformations to the thermal engine. An interesting open question, raised by several authors [1,2,7,8], is whether this external control should carry a thermodynamic cost, and how to include this control explicitly in the framework.The issue of how to implement transformations of one or more systems via interactions with another 'controlling' system has been addressed before from different perspectives, for example in interactions between atoms and a field [16,17] or in quantum driving [18]. In this paper, we address this issue by describing how an explicit quantum clock can control the evolution of a completely arbitrary quantum engine, thus allowing any unitary protocol to be carried out via a time-independent global Hamiltonian.We first show that any energy-conserving unitary operation can be exactly implemented on a quantum system (the engine) by attaching a quantum clock to it via the correct time-independent interaction Hamiltonian. Furthermore, this process is essentially independent of the initial state of the clock, requiring only that it lies within a known finite region. In particular, it is not necessary for the clock to precisely specify the 'time'. After the unitary has been fully implemented, the clock is not correlated with the system and could be used to perform further operations.Next, we show that we can also approximately implement any unita...

Optimal (reversible) processes in thermodynamics can be modelled as step-bystep processes, where the system is successively thermalized with respect to different Hamiltonians by an external thermal bath. However, in practice interactions between system and thermal bath will take finite time, and precise control of their interaction is usually out of reach. Motivated by this observation, we consider finite-time and uncontrolled operations between system and bath, which result in thermalizations that are only partial in each step. We show that optimal processes can still be achieved for any non-trivial partial thermalizations at the price of increasing the number of operations, and characterise the corresponding tradeoff. We focus on work extraction protocols and show our results in two different frameworks: A collision model and a model where the Hamiltonian of the working system is controlled over time and the system can be brought into contact with a heat bath. Our results show that optimal processes are robust to noise and imperfections in small quantum systems, and can be achieved by a large set of interactions between system and bath.

Rate-distortion theory provides bounds for compressing data produced by an information source to a specified encoding rate that is strictly less than the source's entropy. This necessarily entails some loss, or distortion, between the original source data and the best approximation after decompression. The so-called Information Bottleneck Method is designed to compress only 'relevant' information. Which information is relevant is determined by the correlation between the data being compressed and a variable of interest, so-called side information. In this paper, an Information Bottleneck Method is introduced for the compression of quantum data. The channel communication picture is used for compression and decompression. The rate of compression is derived using an entanglement assisted protocol with classical communication, and under an unproved conjecture that the rate function is convex in the distortion parameter. The optimum channel achieving this rate for a given input state is characterised. The conceptual difficulties arising due to differences in the mathematical formalism between quantum and classical probability theory are discussed and solutions are presented.

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