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Whereas the trace is a scalar-valued function on operators, the partial trace is an operator-valued function. The partial trace has applications in quantum information and decoherence which is relevant for quantum measurement and thereby to the decoherent approaches to interpretations of quantum mechanics , including consistent histories and ...
The trace of a Hermitian matrix is real, because the elements on the diagonal are real. The trace of a permutation matrix is the number of fixed points of the corresponding permutation, because the diagonal term a ii is 1 if the i th point is fixed and 0 otherwise. The trace of a projection matrix is the dimension of the target space.
In mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a trace may be defined, such that the trace is a finite number independent of the choice of basis used to compute the trace. This trace of trace-class operators generalizes the trace of matrices studied in linear algebra.
The trace operator can be defined for functions in the Sobolev spaces , with <, see the section below for possible extensions of the trace to other spaces. Let Ω ⊂ R n {\textstyle \Omega \subset \mathbb {R} ^{n}} for n ∈ N {\textstyle n\in \mathbb {N} } be a bounded domain with Lipschitz boundary.
The corresponding operation for density matrices is the partial trace. So one can assign to ρ a state on the subsystem A by = where Tr B is partial trace with respect to system B. This is the reduced state of ρ AB on system A.
When m = 1, that is when f : R n → R is a scalar-valued function, the Jacobian matrix reduces to the row vector; this row vector of all first-order partial derivatives of f is the transpose of the gradient of f, i.e. =.
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A trace is a weight with ω(aa*) = ω(a*a) for all a. A tracial state is a trace with ω(1) = 1. Any factor has a trace such that the trace of a non-zero projection is non-zero and the trace of a projection is infinite if and only if the projection is infinite. Such a trace is unique up to rescaling.
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