In linear algebra and functional analysis, the partial trace is a generalization of the trace. 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 relative state interpretation.
Details Suppose V {\displaystyle V} , W {\displaystyle W} are finite-dimensional vector spaces over a field, with dimensions m {\displaystyle m} and n {\displaystyle n} , respectively. For any space A {\displaystyle A} , let L ( A ) {\displaystyle L(A)} denote the space of linear operators on A {\displaystyle A} . The partial trace over W {\displaystyle W} is then written as Tr W : L ( V ⊗ W ) → L ( V ) {\displaystyle \operatorname {Tr} _{W}:\operatorname {L} (V\otimes W)\to \operatorname {L} (V)} , where ⊗ {\displaystyle \otimes } denotes the Tensor Product. It is defined as follows: For T ∈ L ( V ⊗ W ) {\displaystyle T\in \operatorname {L} (V\otimes W)} , let e 1 , … , e m {\displaystyle e_{1},\ldots ,e_{m}} , and f 1 , … , f n {\displaystyle f_{1},\ldots ,f_{n}} , be bases for V and W respectively; then T has a matrix representation
{ a k ℓ , i j } 1 ≤ k , i ≤ m , 1 ≤ ℓ , j ≤ n {\displaystyle \{a_{k\ell ,ij}\}\quad 1\leq k,i\leq m,\quad 1\leq \ell ,j\leq n}
relative to the basis e k ⊗ f ℓ {\displaystyle e_{k}\otimes f_{\ell }} of V ⊗ W {\displaystyle V\otimes W} . Now for indices k, i in the range 1, ..., m, consider the sum
b k , i = ∑ j = 1 n a k j , i j {\displaystyle b_{k,i}=\sum _{j=1}^{n}a_{kj,ij}}
This gives a matrix bk,i. The associated linear operator on V is independent of the choice of bases and is by definition the partial trace. Among physicists, this is often called "tracing out" or "tracing over" W to leave only an operator on V in the context where W and V are Hilbert spaces associated with quantum systems (see below).
Invariant definition The partial trace operator can be defined invariantly (that is, without reference to a basis) as follows: it is the unique linear map
Tr W : L ( V ⊗ W ) → L ( V ) {\displaystyle \operatorname {Tr} _{W}:\operatorname {L} (V\otimes W)\rightarrow \operatorname {L} (V)}
such that
Tr W ( R ⊗ S ) = Tr ( S ) R ∀ R ∈ L ( V ) ∀ S ∈ L ( W ) . {\displaystyle \operatorname {Tr} _{W}(R\otimes S)=\operatorname {Tr} (S)\,R\quad \forall R\in \operatorname {L} (V)\quad \forall S\in \operatorname {L} (W).}
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