for mixed states). As a measure of entanglement Entropy provides one tool that can be used to quantify entanglement, although other entanglement measures exist. If the overall system is pure, the entropy of one subsystem can be used to measure its degree of entanglement with the other subsystems. For bipartite pure states, the von Neumann entropy of reduced states is the unique measure of entanglement in the sense that it is the only function on the family of states that satisfies certain axioms required of an entanglement measure. It is a classical result that the Shannon entropy achieves its maximum at, and only at, the uniform probability distribution {1/n,...,1/n}. Therefore, a bipartite pure state is said to be a maximally entangled state if the reduced state of each subsystem of is the diagonal matrix For mixed states, the reduced von Neumann entropy is not the only reasonable entanglement measure. As an aside, the information-theoretic definition is closely related to entropy in the sense of statistical mechanics (comparing the two definitions in the present context, it is customary to set the Boltzmann constant ). For example, by properties of the Borel functional calculus, we see that for any unitary operator , Indeed, without this property, the von Neumann entropy would not be well-defined. In particular, could be the time evolution operator of the system, i.e., where is the Hamiltonian of the system. Here the entropy is unchanged. The reversibility of a process is associated with the resulting entropy change, i.e., a process is reversible if, and only if, it leaves the entropy of the system invariant. Therefore, the march of the arrow of time towards thermodynamic equilibrium is simply the growing spread of quantum entanglement. This provides a connection between quantum information theory and thermodynamics. Rényi entropy also can be used as a measure of entanglement. Nevertheless, on 23 January 2023, physicists reported, that, after all, there is no second law of entanglement manipulation. In the words of the researchers, "no direct counterpart to the second law of thermodynamics can be established". Entanglement measures Entanglement measures quantify the amount of entanglement in a (often viewed as a bipartite) quantum state. As aforementioned, entanglement entropy is the standard measure of entanglement for pure states (but no longer a measure of entanglement for mixed states). For mixed states, there are some entanglement measures in the literature and no single one is standard. Entanglement