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The Second Laws of Quantum Thermodynamics

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Mathematical Challenges in Quantum Information

The second law of thermodynamics tells us which state transformations are so statistically unlikely that they are effectively forbidden, and applies to systems composed of many particles. However, using tools from quantum information theory, we are seeing that one can make sense of thermodynamics in the regime where we only have a small number of particles interacting with a heat bath, or when we have highly correlated systems and wish to make non-statistical statements about them. Is there a second law of thermodynamics in this regime? Here, we find that for processes which are cyclic or very close to cyclic, the second law for microscopic or highly correlated systems takes on a very different form than it does at the macroscopic scale, imposing not just one constraint on what state transformations are possible, but an entire family of constraints. In particular, we find a family of quantum free energies which generalise the traditional ones, and show that they can never increase. The ordinary second law corresponds to the non-increasing of one of these free energies, with the remainder, imposing additional constraints on thermodynamic transitions of quantum systems. We further find that there are three regimes which govern which family of second laws govern state transitions, depending on how cyclic the thermodynamical 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. By making precise the definition of thermal operations, the laws of thermodynamics take on a simple form with the first law defining the class of thermal operations, the zeroeth law emerging as a unique equilibrium condition, and the remaining laws being a monotonicity property of our generalised free energies based on the Renyi-divergence. The derivations use tools from majorisation theory.

This talk is part of the Isaac Newton Institute Seminar Series series.

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