If time permits, I also discuss some applications of these result to thermodynamics and speculate about consequences for quantum information theory and holography. Getting rid of the decohering environment yields the "catalytic entropy conjecture", for which I present some supporting arguments. That entropy is called the quantum state's von Neumann entropy. In quantum information the logarithms are usually taken to be base 2, giving a maximum entropy of 1 for a qubit. Von Neumann said to call it Shannon entropy, as it was a special case of von Neumann entropy. Decades later, Shannon developed an information-theoretic formula for use in classical information theory, and asked von Neumann what to call it. Then I show that the von Neumann entropy governs single-shot transitions whenever one has access to arbitrary auxiliary systems, which have to remain invariant in a state-transition ("catalysts"), as well as a decohering environment. The possible outcomes have probabilities whose Shannon entropy Audrey can calculate. The von Neumann entropy of a quantum state is given by the formula, S () Tr (log) and if i are the eigenvalues of then the Von Neumann entropy can be reexpressed as: S () iilog (i). But von Neumann discovered von Neumann entropy first, and applied it to questions of statistical physics. I first present new results that give a single-shot interpretation to the Area Law of entanglement entropy in many-body physics in terms of compression of quantum information on the boundary of a region of space. In this talk, I will discuss new results that give single-shot interpretations to the von Neumann entropy under appropriate conditions. In quanum physics, the von Neumann entropy usually arises in i.i.d settings, while single-shot settings are commonly characterized by smoothed min- or max-entropies. In the framework of Quantum Field Theory, we provide a rigorous, operator algebraic notion of entanglement entropy associated with a pair of open double cones of the spacetime, where the closure of is contained in.
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