![]() However, their von Neumann entropy, as defined in Equation ( 1), can be different. Thus, density matrices define the same state as long as they have the same diagonal elements. Consider the algebra of diagonal n × n matricesĪnd depends only on the diagonal elements of ρ. We can get a glance of the ambiguity by the following example. In this context, in general, the set of observables is not the full operator algebra, but a subalgebra. In fact, the proper mathematical formalization of a quantum field theory requires the introduction of C ∗-algebras. In, this ambiguity is traced back to the ambiguity in the definition of a density matrix associated with a state in an algebraic theory. 74), as ambiguities in this definition arise from the dependence on the cutoffs introduced to regularize the theory. However, “it is never hard to find trouble in field theory” (, p. Von Neumann entropy is used to evaluate the entropy of a black hole, which originates from the lack of knowledge of the system inside of it. Īs quantum field theory developed, attempts to extend Equation ( 1) to a broader scheme have been made. This measure is then extended to mixed states by exploiting the convex structure of the set of quantum states, and taking the infimum over all the possible decompositions into pure states. Besides its importance from a fundamental point of view, von Neumann entropy is useful also to answer practical questions in quantum information theory, for example when dealing with multipartite systems and one wants to characterize the entanglement between them in this context, it has been shown that a particularly meaningful measure of the entanglement contained in a pure state shared by two parties is the von Neumann entropy of the reduced state of one party, since it allows characterizing the usefulness of such entanglement in the thermodynamic limit when multiple copies of the state are available.
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