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A mixed state is described by a density matrix ρ, that is a non-negative trace-class operator of trace 1 on the tensor product . The partial trace of ρ with respect to the system B, denoted by , is called the reduced state of ρ on system A.
For more properties and a generalization of the partial trace, see traced monoidal categories. If A is a general associative algebra over a field k, then a trace on A is often defined to be any map tr : A ↦ k which vanishes on commutators; tr([a,b]) = 0 for all a, b ∈ A. Such a trace is not uniquely defined; it can always at least be ...
The tensor product of two vector spaces is a vector space that is defined up to an isomorphism.There are several equivalent ways to define it. Most consist of defining explicitly a vector space that is called a tensor product, and, generally, the equivalence proof results almost immediately from the basic properties of the vector spaces that are so defined.
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.
For the special case where the tensor product operation is a contraction of one index and the gradient operation is a divergence, and both and are second order tensors, we have = ():. In index notation, ∫ Ω F i j G p j , p d Ω = ∫ Γ n p F i j G p j d Γ − ∫ Ω G p j F i j , p d Ω . {\displaystyle \int _{\Omega }F_{ij}\,G_{pj,p}\,d ...
Consider a vector of the tensor product . in the form of Schmidt decomposition = =. Form the rank 1 matrix =.Then the partial trace of , with respect to either system A or B, is a diagonal matrix whose non-zero diagonal elements are | |.
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 reduced state of ρ on system A, ρ A, is obtained by taking the partial trace of ρ with respect to the B system: =. The partial trace operation is a CPTP map, therefore a quantum channel in the Schrödinger picture. [5] In the Heisenberg picture, the dual map of this channel is