This topical review article gives an overview of the interplay between quantum information theory and thermodynamics of quantum systems. We focus on several trending topics including the foundations of statistical mechanics, resource theories, entanglement in thermodynamic settings, fluctuation theorems and thermal machines. This is not a comprehensive review of the diverse field of quantum thermodynamics; rather, it is a convenient entry point for the thermo-curious information theorist. Furthermore this review should facilitate the unification and understanding of different interdisciplinary approaches emerging in research groups around the world. CONTENTS

2 These authors contributed equally to this work.While neural networks have been remarkably successful for a variety of practical problems, they are often applied as a black box, which limits their utility for scientific discoveries. Here, we present a neural network architecture that can be used to discover physical concepts from experimental data without being provided with additional prior knowledge. For a variety of simple systems in classical and quantum mechanics, our network learns to compress experimental data to a simple representation and uses the representation to answer questions about the physical system. Physical concepts can be extracted from the learned representation, namely: (1) The representation stores the physically relevant parameters, like the frequency of a pendulum.(2) The network finds and exploits conservation laws: it stores the total angular momentum to predict the motion of two colliding particles. (3) Given measurement data of a simple quantum mechanical system, the network correctly recognizes the number of degrees of freedom describing the underlying quantum state. (4) Given a time series of the positions of the Sun and Mars as observed from Earth, the network discovers the heliocentric model of the solar systemthat is, it encodes the data into the angles of the two planets as seen from the Sun. Our work provides a first step towards answering the question whether the traditional ways by which physicists model nature naturally arise from the experimental data without any mathematical and physical pre-knowledge, or if there are alternative elegant formalisms, which may solve some of the fundamental conceptual problems in modern physics, such as the measurement problem in quantum mechanics.Problem: Predict the position of a one-dimensional damped pendulum at different times. Physical model: Equation of motionSolution:Observation: Time series of positions: o = x(t i ) i∈{1,...,50} ∈ R 50 , with equally spaced t i . Mass m = 1kg, amplitude A 0 = 1m and phase δ 0 = 0 are fixed; spring constant κ ∈ [5, 10] kg/s 2 and damping factor b ∈ [0.5, 1] kg/s are varied between training samples. Question: Prediction times: q = t pred ∈ R.Correct answer: Position at time t pred : a cor = x(t pred ) ∈ R .Implementation: Network depicted in Figure 1b with 3 latent neurons. Key findings:• SciNet predicts the positions x(t pred ) with a root mean square error below 2% (with respect to the amplitude A 0 = 1m) (Figure 2a).• SciNet stores κ and b in two of the latent neurons, and does not store any information in the third latent neuron (Figure 2b).

The heat generated by computations is not only an obstacle to circuit miniaturization but also a fundamental aspect of the relationship between information theory and thermodynamics. In principle, reversible operations may be performed at no energy cost; given that irreversible computations can always be decomposed into reversible operations followed by the erasure of data, the problem of calculating their energy cost is reduced to the study of erasure. Landauer's principle states that the erasure of data stored in a system has an inherent work cost and therefore dissipates heat. However, this consideration assumes that the information about the system to be erased is classical, and does not extend to the general case where an observer may have quantum information about the system to be erased, for instance by means of a quantum memory entangled with the system. Here we show that the standard formulation and implications of Landauer's principle are no longer valid in the presence of quantum information. Our main result is that the work cost of erasure is determined by the entropy of the system, conditioned on the quantum information an observer has about it. In other words, the more an observer knows about the system, the less it costs to erase it. This result gives a direct thermodynamic significance to conditional entropies, originally introduced in information theory. Furthermore, it provides new bounds on the heat generation of computations: because conditional entropies can become negative in the quantum case, an observer who is strongly correlated with a system may gain work while erasing it, thereby cooling the environment.

To clarify the implications of our result, we note that, although the erasure processes we considered in our Letter can have negative work cost (that is, they can yield work), they do not violate the second law of thermodynamics, because they are not cyclic processes. A negative work cost is associated with the consumption of entanglement, which can only be restored by doing work. Our results are also consistent with the original unconditional form of Landauer's principle, which says that if there is no information available about the data being erased, the cost of erasure is always positive. Similarly, because in a computation with deterministic classical output the joint entropy of all registers conditioned on the output cannot be negative, the overall work cost of such a computation is always positive or zero (even though temporary quantum correlations may be created and exploited during the course of the computation). In fact, standard techniques of reversible information processing allow any deterministic classical algorithm to be performed, on a classical or quantum computer, in a thermodynamically reversible fashion, with work cost arbitrarily close to zero 1,2 . These clarifications are developed in more detail in the Supplementary Information to this Addendum. We thank Charles H. Bennett for remarks on reversible computation.Supplementary Information is linked to the online version of the Addendum at www.nature.com/nature.1. Bennett, C. H. The thermodynamics of computation-a review. Int. J. Theor. Phys. 21, 905-940 (1982). 2. Watrous, J. Quantum computational complexity.

We test the principles of classical modal logic in fully quantum settings. Modal logic models our reasoning in multi-agent problems, and allows us to solve puzzles like the muddy children paradox. The Frauchiger-Renner thought experiment highlighted fundamental problems in applying classical reasoning when quantum agents are involved; we take it as a guiding example to test the axioms of classical modal logic. In doing so, we find a problem in the original formulation of the Frauchiger-Renner theorem: a missing assumption about unitarity of evolution is necessary to derive a contradiction and prove the theorem. Adding this assumption clarifies how different interpretations of quantum theory fit in, i.e., which properties they violate. Finally, we show how most of the axioms of classical modal logic break down in quantum settings, and attempt to generalize them. Namely, we introduce constructions of trust and context, which highlight the importance of an exact structure of trust relations between agents. We propose a challenge to the community: to find conditions for the validity of trust relations, strong enough to exorcise the paradox and weak enough to still recover classical logic.Draco said out loud, "I notice that I am confused." Your strength as a rationalist is your ability to be more confused by fiction than by reality... Draco was confused. Therefore, something he believed was fiction. Eliezer Yudkowsky, Harry Potter and the Methods of Rationality

Relativistic protocols have been proposed to overcome certain impossibility results in classical and quantum cryptography. In such a setting, one takes the location of honest players into account, and uses the signalling limit given by the speed of light to constraint the abilities of dishonest agents. However, composing such protocols with each other to construct new cryptographic resources is known to be insecure in some cases. To make general statements about such constructions, a composable framework for modelling cryptographic security in Minkowski space is required. Here, we introduce a framework for performing such a modular security analysis of classical and quantum cryptographic schemes in Minkowski space. As an application, we show that (1) fair and unbiased coin flipping can be constructed from a simple resource called channel with delay; (2) biased coin flipping, bit commitment and channel with delay through any classical, quantum or post-quantum relativistic protocols are all impossible without further setup assumptions; (3) it is impossible to securely increase the delay of a channel, given several short-delay channels as ingredients. Results (1) and (3) imply in particular the non-composability of existing relativistic bit commitment and coin flipping protocols.

We extend the tools of quantum resource theories to scenarios in which multiple quantities (or resources) are present, and their interplay governs the evolution of physical systems. We derive conditions for the interconversion of these resources, which generalise the first law of thermodynamics. We study reversibility conditions for multi-resource theories, and find that the relative entropy distances from the invariant sets of the theory play a fundamental role in the quantification of the resources. The first law for general multi-resource theories is a single relation which links the change in the properties of the system during a state transformation and the weighted sum of the resources exchanged. In fact, this law can be seen as relating the change in the relative entropy from different sets of states. In contrast to typical single-resource theories, the notion of free states and invariant sets of states become distinct in light of multiple constraints. Additionally, generalisations of the Helmholtz free energy, and of adiabatic and isothermal transformations, emerge. We thus have a set of laws for general quantum resource theories, which generalise the laws of thermodynamics. We first test this approach on thermodynamics with multiple conservation laws, and then apply it to the theory of local operations under energetic restrictions.

When studying thermalization of quantum systems, it is typical to ask whether a system interacting with an environment will evolve towards a local thermal state. Here, we show that a more general and relevant question is "when does a system thermalize relative to a particular reference?" By relative thermalization we mean that, as well as being in a local thermal state, the system is uncorrelated with the reference. We argue that this is necessary in order to apply standard statistical mechanics to the study of the interaction between a thermalized system and a reference. We then derive a condition for relative thermalization of quantum systems interacting with an arbitrary environment. This condition has two components: the first is state-independent, reflecting the structure of invariant subspaces, like energy shells, and the relative sizes of system and environment; the second depends on the initial correlations between reference, system and environment, measured in terms of conditional entropies. Intuitively, a small system interacting with a large environment is likely to thermalize relative to a reference, but only if, initially, the reference was not highly correlated with the system and environment. Our statement makes this intuition precise, and we show that in many natural settings this thermalization condition is approximately tight. Established results on thermalization, which usually ignore the reference, follow as special cases of our statements. I. THE CASE FOR RELATIVE THERMALIZATION A. Subjectivity in thermodynamicsThermodynamics was originally developed to study and improve the performance of steam engines: to turn the heat of a gas into work, as efficiently as possible. Today, it is also being applied to study heat and work flows in the micro and nano regimes. In fact, advances in the manipulation of small systems have allowed us to extract work from systems such as quantum dots and trapped ions [1,2]. Yet, thermodynamics as a science is still adapting to this new regime, and it still bears some of the traits of the gaseous systems for which it was first designed. For example, the information available about the state of a gas used to be limited and objective: we would measure the temperature, pressure and volume of a gas, but we could not keep track of each individual particle. Crucially, all conceivable observers had access to the same information about the state of the system, and could manipulate it in equivalent ways-like letting a gas expand to obtain work. And yet, since very early on, several thought experiments have challenged the idea that thermodynamics should be objective. In 1871 James Maxwell realized that a "demon" able to measure the position and velocity of the particles of a gas could extract more work from it than the typical observer implicit in standard thermodynamics [3]. Picking up on Maxwell's idea on the power of information, Leó Szilárd imagined a partitioned box with a single-particle gas on one side. Depending on their information on the location of the particle, tw...