We point out formal correspondences between thermodynamics and entanglement. By applying them to previous work, we show that entropy of entanglement is the unique measure of entanglement for pure states.
We show that quantum entanglement can be understood and quantified using the same axiomatic approach to thermodynamics pioneered by Caratheodory and later extended by Buchdahl and Giles. This leads to the existence of a unique measure of entanglement for pure states of systems consisting of two subsystems, analogous to the thermodynamical entropy. Thermodynamics and (quantum) information are seen to be just two different manifestations of ordering of (abstract) states connected by (abstract) processes.
We explore and develop the mathematics of the theory of entanglement measures. After a careful review and analysis of definitions, of preliminary results, and of connections between conditions on entanglement measures, we prove a sharpened version of a uniqueness theorem which gives necessary and sufficient conditions for an entanglement measure to coincide with the reduced von Neumann entropy on pure states. We also prove several versions of a theorem on extreme entanglement measures in the case of mixed states. We analyze properties of the asymptotic regularization of entanglement measures proving, for example, convexity for the entanglement cost and for the regularized relative entropy of entangle ment.
The first step in quantum information theory is the identification of entanglement as a valuable resource. The next step is learning how to efficiently exploit this resource. We learn how to efficiently exploit entanglement by applying analogues of thermodynamical concepts. These concepts include reversibility, entropy, and the distinction between intensive and extensive quantities. We discuss some of these analogues and show how they lead to a measure of entanglement for pure states. We also ask whether these analogues are more than analogues, and note that, locally, entropy of entanglement is thermodynamical entropy.