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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 ...
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 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.
Then the partial trace of , with respect to either system A or B, is a diagonal matrix whose non-zero diagonal elements are | |. In other words, the Schmidt decomposition shows that the reduced states of ρ {\displaystyle \rho } on either subsystem have the same spectrum.
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 normalization condition that the trace of be equal to 1 defines the partition function to be () = (). If the number of particles involved in the system is itself not certain, then a grand canonical ensemble can be applied, where the states summed over to make the density matrix are drawn from a Fock space .
The Schatten 1-norm is the nuclear norm (also known as the trace norm, or the Ky Fan n-norm [1]). The Schatten 2-norm is the Frobenius norm . The Schatten ∞-norm is the spectral norm (also known as the operator norm , or the largest singular value).
In Matlab/GNU Octave a matrix A can be vectorized by A(:). GNU Octave also allows vectorization and half-vectorization with vec(A) and vech(A) respectively. Julia has the vec(A) function as well. In Python NumPy arrays implement the flatten method, [note 1] while in R the desired effect can be achieved via the c() or as.vector() functions.