The second law of thermodynamics places constraints on state transformations. It applies to systems composed of many particles, however, we are seeing that one can formulate laws of thermodynamics when only a small number of particles are interacting with a heat bath. Is there a second law of thermodynamics in this regime? Here, we find that for processes which are approximately cyclic, the second law for microscopic systems takes on a different form compared to the macroscopic scale, imposing not just one constraint on state transformations, but an entire family of constraints. We find a family of free energies which generalize the traditional one, and show that they can never increase. The ordinary second law relates to one of these, with the remainder imposing additional constraints on thermodynamic transitions. We find three regimes which determine which family of second laws govern state transitions, depending on how cyclic the process is. In one regime one can cause an apparent violation of the usual second law, through a process of embezzling work from a large system which remains arbitrarily close to its original state. These second laws are relevant for small systems, and also apply to individual macroscopic systems interacting via long-range interactions. By making precise the definition of thermal operations, the laws of thermodynamics are unified in this framework, with the first law defining the class of operations, the zeroth law emerging as an equivalence relation between thermal states, and the remaining laws being monotonicity of our generalized free energies.quantum thermodynamics | quantum information theory | statistical physics | resource theory | free energy T he original formulation of the second law, due to Clausius (1), states that "Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time." In attempting to apply Clausius's statement of the second law to the microscopic or quantum scale, we immediately run into a problem because it talks about cyclic processes in which there is no other change occurring at the same time, and at this scale it is impossible to design a process in which there is no change, however slight, in our devices and heat engines. Interpreted strictly, the Clausius statement of the second law applies to situations which never occur in nature. The same holds true for other versions of the second law, such as the KelvinPlanck statement, where one also talks about cyclic processes, in which all other objects beside the system of interest are returned back to their original state. Here, we derive a quantum version of the Clausius statement, by looking at processes where a microscopic or quantum system undergoes a transition from one state to another, whereas the environment and working body or heat engine is returned back to their original state. Whereas macroscopically only a single second law restricts transitions, we find that there is an entire family of more fundamental restrictions at the quantum ...

Quantum thermodynamics is a research field that aims at fleshing out the ultimate limits of thermodynamic processes in the deep quantum regime. A complete picture of thermodynamical processes naturally allows for auxiliary systems dubbed 'catalysts', i.e., any physical systems facilitating state transformations while remaining essentially intact in their state, like an auxiliary system, a clock, or an actual catalyst. In this work, we present a comprehensive analysis of the power and limitation of such thermal catalysis. Specifically, we provide a family of optimal catalysts that can be returned with minimal trace distance error after facilitating a state transformation process. To incorporate the genuine physical role of a catalyst, we identify very significant restrictions on arbitrary state transformations under dimension or mean energy bounds, using methods of convex relaxations. We discuss the implication of these findings on possible thermodynamic state transformations in the quantum regime.Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. 5 Since Gibbs preserving maps in [1] induce the same pre-order structure in the state space as thermal operations for this classical regime (block-diagonal states), our results would apply to the paradigm of Gibbs preserving maps [28] as well.

Fundamental primitives such as bit commitment and oblivious transfer serve as building blocks for many other two-party protocols. Hence, the secure implementation of such primitives is important in modern cryptography. Here we present a bit commitment protocol that is secure as long as the attacker's quantum memory device is imperfect. The latter assumption is known as the noisy-storage model. We experimentally executed this protocol by performing measurements on polarization-entangled photon pairs. Our work includes a full security analysis, accounting for all experimental error rates and finite size effects. This demonstrates the feasibility of two-party protocols in this model using real-world quantum devices. Finally, we provide a general analysis of our bit commitment protocol for a range of experimental parameters.

The generation of random numbers via quantum processes is an efficient and reliable method to obtain true indeterministic random numbers that are of vital importance to cryptographic communication and large-scale computer modeling. However, in realistic scenarios, the raw output of a quantum random-number generator is inevitably tainted by classical technical noise. The integrity of the device can be compromised if this noise is tampered with, or even controlled by some malicious party. To safeguard against this, we propose and experimentally demonstrate an approach that produces side-information independent randomness that is quantified by min-entropy conditioned on this classical noise. We present a method for maximizing the conditional min-entropy of the number sequence generated from a given quantum-to-classical-noise ratio. The detected photocurrent in our experiment is shown to have a real-time random-number generation rate of 14 (Mbit/s)/MHz. The spectral response of the detection system shows the potential to deliver more than 70 Gbit/s of random numbers in our experimental setup.

Cryptography's importance in our everyday lives continues to grow in our increasingly digital world. Oblivious transfer has long been a fundamental and important cryptographic primitive, as it is known that general two-party cryptographic tasks can be built from this basic building block. Here we show the experimental implementation of a 1-2 random oblivious transfer protocol by performing measurements on polarization-entangled photon pairs in a modified entangled quantum key distribution system, followed by all of the necessary classical postprocessing including one-way error correction. We successfully exchange a 1,366 bit random oblivious transfer string in B3 min and include a full security analysis under the noisy storage model, accounting for all experimental error rates and finite size effects. This demonstrates the feasibility of using today's quantum technologies to implement secure two-party protocols.

Apart from their foundational significance, entropic uncertainty relations play a central role in proving the security of quantum cryptographic protocols. Of particular interest are thereby relations in terms of the smooth min-entropy for BB84 and six-state encodings. Previously, strong uncertainty relations were obtained which are valid in the limit of large block lengths. Here, we prove a new uncertainty relation in terms of the smooth min-entropy that is only marginally less strong, but has the crucial property that it can be applied to rather small block lengths. This paves the way for a practical implementation of many cryptographic protocols. As part of our proof we show tight uncertainty relations for a family of Rényi entropies that may be of independent interest.

Sadi Carnot's theorem regarding the maximum efficiency of heat engines is considered to be of fundamental importance in thermodynamics. This theorem famously states that the maximum efficiency depends only on the temperature of the heat baths used by the engine, but not on the specific structure of baths. Here, we show that when the heat baths are finite in size, and when the engine operates in the quantum nanoregime, a revision to this statement is required. We show that one may still achieve the Carnot efficiency, when certain conditions on the bath structure are satisfied; however if that is not the case, then the maximum achievable efficiency can reduce to a value which is strictly less than Carnot. We derive the maximum efficiency for the case when one of the baths is composed of qubits. Furthermore, we show that the maximum efficiency is determined by either the standard second law of thermodynamics, analogously to the macroscopic case, or by the non increase of the max relative entropy, which is a quantity previously associated with the single shot regime in many quantum protocols. This relative entropic quantity emerges as a consequence of additional constraints, called generalized free energies, that govern thermodynamical transitions in the nanoregime. Our findings imply that in order to maximize efficiency, further considerations in choosing bath Hamiltonians should be made, when explicitly constructing quantum heat engines in the future. This understanding of thermodynamics has implications for nanoscale engineering aiming to construct small thermal machines.

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