Partial trace

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. 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. 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} , Tr W : L ⁡ ( V ⊗ W ) → L ⁡ ( V ) {displaystyle operatorname {Tr} _{W}:operatorname {L} (Votimes W) o operatorname {L} (V)} , is a mapping It is defined as follows: 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 Thas a matrix representation relative to the basis e k ⊗ f ℓ {displaystyle e_{k}otimes f_{ell }} of V ⊗ W {displaystyle Votimes W} . Now for indices k, i in the range 1, ..., m, consider the sum 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).