Matrices and Their (Un)Faithful Fermi-quadratic Forms
We consider the algebra of n × n matrices over ℂ and their corresponding Fermi-quadratic forms. The properties of these operators are studied with respect to the properties of the underlying matrices. It is well known that these Fermi-quadratic forms have a faithful matrix representation. The purpose of this article is to investigate the (un)faithful representation of the matrix algebra by its Fermi-quadratic forms. The preservation of the matrix commutators, anticommutators, and eigenvalues in the Fermi-quadratic forms are discussed. Other matrix functions such as the exponential function are studied, as well as an application to quantum channels where we consider density matrices and operators and the Kraus representation. Lastly, we consider extensions of these quadratic forms and entangled states that arise from these forms.
KeywordsFermi operators Matrix embedding
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