The relationship between thermodynamics and statistical physics is valid in the thermodynamic limit-when the number of particles becomes very large. Here we study thermodynamics in the opposite regime-at both the nanoscale and when quantum effects become important. Applying results from quantum information theory, we construct a theory of thermodynamics in these limits. We derive general criteria for thermodynamical state transitions, and, as special cases, find two free energies: one that quantifies the deterministically extractable work from a small system in contact with a heat bath, and the other that quantifies the reverse process. We find that there are fundamental limitations on work extraction from non-equilibrium states, owing to finite size effects and quantum coherences. This implies that thermodynamical transitions are generically irreversible at this scale. As one application of these methods, we analyse the efficiency of small heat engines and find that they are irreversible during the adiabatic stages of the cycle.

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 ...

The ideas of thermodynamics have proved fruitful in the setting of quantum information theory, in particular the notion that when the allowed transformations of a system are restricted, certain states of the system become useful resources with which one can prepare previously inaccessible states. The theory of entanglement is perhaps the best-known and most well-understood resource theory in this sense. Here, we return to the basic questions of thermodynamics using the formalism of resource theories developed in quantum information theory and show that the free energy of thermodynamics emerges naturally from the resource theory of energy-preserving transformations. Specifically, the free energy quantifies the amount of useful work which can be extracted from asymptotically many copies of a quantum system when using only reversible energy-preserving transformations and a thermal bath at fixed temperature. The free energy also quantifies the rate at which resource states can be reversibly interconverted asymptotically, provided that a sublinear amount of coherent superposition over energy levels is available, a situation analogous to the sublinear amount of classical communication required for entanglement dilution. Quantum resource theories are specified by a restriction on the quantum operations (state preparations, measurements, and transformations) that can be implemented by one or more parties. This singles out a set of states which can be prepared under the restricted operations. If the parties facing the restriction acquire a quantum state outside the restricted set of states, then they can use this state to implement measurements and transformations that are outside the class of allowed operations, consuming the state in the process. Thus, such states are useful resources.A few prominent examples serve to illustrate the idea: if two or more parties are restricted to communicating classically and implementing local quantum operations, then entangled states become a resource [1]; if a party is restricted to quantum operations that have a particular symmetry, then states that break this symmetry become a resource [2][3][4]; if a party is restricted to preparing states that are completely mixed and performing unitary operations, then any state that is not completely mixed, i.e., any state that has some purity, becomes a resource [5].In this Letter, we develop the quantum resource theory of states that are T athermal, i.e., not thermal at temperature T. This provides a useful new formulation of equilibrium and nonequilibrium thermodynamics for finite-dimensional quantum systems, and allows us to apply new mathematical tools to the subject. The restricted class of operations which defines our resource theory includes only those that can be achieved through energy-conserving unitaries and the preparation of any ancillary system in a thermal state at temperature T, as first studied by Janzing et al. [6] in the context of Landauer's principle. Here, the ancillary systems can have an arbitrary Hilbert space and an a...

We consider the amount of work which can be extracted from a heat bath using a bipartite state shared by two parties. In general it is less then the amount of work extractable when one party is in possession of the entire state. We derive bounds for this ''work deficit'' and calculate it explicitly for a number of different cases. In particuar, for pure states the work deficit is exactly equal to the distillable entanglement of the state. A form of complementarity exists between physical work which can be extracted and distillable entanglement. The work deficit is a good measure of the quantum correlations in a state and provides a new paradigm for understanding quantum nonlocality.Strong connections exist between information and thermodynamics. Work is required to erase a magnetic tape in an unknown state [1] and bits of information can be used to draw work from single heat bath [2,3]. The second law of thermodynamics forbids the drawing of work from a single heat bath, however, if one has an engine which contains ''negentropy'' (bits of information) then one can draw work from it. The process does not violate the second law because the information is depleted as entropy from the heat bath accumulates in the engine. Typically, the source of information is particles in known states, and these states can be thought of as a type of fuel or resource [4]. In particular, quantum states can be used as fuel [5], and recently, physically realizable microengines have been proposed [6].The field of quantum information theory has also yielded tantalizing connections between entanglement and thermodynamics [7]. Bipartite states (jointly held by two parties) such as the maximally entangled state AB 1 2 p j00i j11i;(1) exhibit mysterious nonlocalities which can be exploited to perform quantum useful logical work [8] such as teleporting qubits [9]. For many states, one can distill singlets in order to perform logical work, but there is also bound entanglement [10] which cannot be distilled from the state and it has been proposed that this is analogous to heat [11]. Pure bipartite states can be reversible transformed into each other in a manner which is reminiscent of a Carnot cycle [12,13]. Furthermore, the preparation of certain jointly held states appears to result in a greater loss of information when the state is prepared by two separated observers than when the entire state is prepared by a single party [14]. Connections between Landauer erasure, measurement, error correction, and distillation have also been explored [15].In this paper we ask how much work can be drawn from a single heat bath if the information is distributed between two separated parties Alice and Bob. It turns out that in general their engines will be more efficient when information is localized, and that the degree to which this is the case provides a powerful new paradigm to understand and quantify nonlocality in quantum mechanics.As with the distant labs scenario for entanglement analysis, we allow Alice and Bob to perform local operations on their st...

Information--be it classical or quantum--is measured by the amount of communication needed to convey it. In the classical case, if the receiver has some prior information about the messages being conveyed, less communication is needed. Here we explore the concept of prior quantum information: given an unknown quantum state distributed over two systems, we determine how much quantum communication is needed to transfer the full state to one system. This communication measures the partial information one system needs, conditioned on its prior information. We find that it is given by the conditional entropy--a quantity that was known previously, but lacked an operational meaning. In the classical case, partial information must always be positive, but we find that in the quantum world this physical quantity can be negative. If the partial information is positive, its sender needs to communicate this number of quantum bits to the receiver; if it is negative, then sender and receiver instead gain the corresponding potential for future quantum communication. We introduce a protocol that we term 'quantum state merging' which optimally transfers partial information. We show how it enables a systematic understanding of quantum network theory, and discuss several important applications including distributed compression, noiseless coding with side information, multiple access channels and assisted entanglement distillation.

Two central concepts of quantum mechanics are Heisenberg's uncertainty principle and a subtle form of nonlocality that Einstein famously called "spooky action at a distance." These two fundamental features have thus far been distinct concepts. We show that they are inextricably and quantitatively linked: Quantum mechanics cannot be more nonlocal with measurements that respect the uncertainty principle. In fact, the link between uncertainty and nonlocality holds for all physical theories. More specifically, the degree of nonlocality of any theory is determined by two factors: the strength of the uncertainty principle and the strength of a property called "steering," which determines which states can be prepared at one location given a measurement at another.

In spite of many results in quantum information theory, the complex nature of compound systems is far from being clear. In general the information is a mixture of local, and non-local ("quantum") information. It is important from both pragmatic and theoretical points of view to know relationships between the two components. To make this point more clear, we develop and investigate the quantum information processing paradigm in which parties sharing a multipartite state distill local information. The amount of information which is lost because the parties must use a classical communication channel is the deficit. This scheme can be viewed as complementary to the notion of distilling entanglement. After reviewing the paradigm in detail, we show that the upper bound for the deficit is given by the relative entropy distance to so-called pseudo-classically correlated states; the lower bound is the relative entropy of entanglement. This implies, in particular, that any entangled state is informationally nonlocal i.e. has nonzero deficit. We also apply the paradigm to defining the thermodynamical cost of erasing entanglement. We show the cost is bounded from below by relative entropy of entanglement. We demonstrate the existence of several other non-local phenomena which can be found using the paradigm of local information. For example, we prove the existence of a form of non-locality without entanglement and with distinguishability. We analyze the deficit for several classes of multipartite pure states and obtain that in contrast to the GHZ state, the Aharonov state is extremely nonlocal (and in fact can be thought of as quasi-nonlocalisable). We also show that there do not exist states, for which the deficit is strictly equal to the whole informational content (bound local information). We discuss the relation of the paradigm with measures of classical correlations introduced earlier. It is also proven that in the one-way scenario, the deficit is additive for Bell diagonal states. We then discuss complementary features of information in distributed quantum systems. Finally we discuss the physical and theoretical meaning of the results and pose many open questions. Contents

We consider a quantum state shared between many distant locations, and define a quantum information processing primitive, state merging, that optimally merges the state into one location. As announced in [Horodecki, Oppenheim, Winter, Nature 436, 673 (2005)], the optimal entanglement cost of this task is the conditional entropy if classical communication is free. Since this quantity can be negative, and the state merging rate measures partial quantum information, we find that quantum information can be negative. The classical communication rate also has a minimum rate: a certain quantum mutual information. State merging enabled one to solve a number of open problems: distributed quantum data compression, quantum coding with side information at the decoder and sender, multi-party entanglement of assistance, and the capacity of the quantum multiple access channel. It also provides an operational proof of strong subadditivity. Here, we give precise definitions and prove these results rigorously.