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.

Suppose ${\displaystyle V}$, ${\displaystyle W}$ are finite-dimensional vector spaces over a field, with dimensions ${\displaystyle m}$ and ${\displaystyle n}$, respectively. For any space ${\displaystyle A}$ let ${\displaystyle L(A)}$ denote the space of linear operators on ${\displaystyle A}$. The partial trace over ${\displaystyle W}$, ${\displaystyle \operatorname {Tr} _{W}:\operatorname {L} (V\otimes W)\to \operatorname {L} (V)}$, is a mapping

...
Wikipedia