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 , are finite-dimensional vector spaces over a field, with dimensions and , respectively. For any space , let denote the space of linear operators on . The partial trace over is then written as , where denotes the Kronecker product. It is defined as follows: For , let , and , be bases for V and W respectively; then T has a matrix representation relative to the basis of . 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). The partial trace operator can be defined invariantly (that is, without reference to a basis) as follows: it is the unique linear map such that To see that the conditions above determine the partial trace uniquely, let form a basis for , let form a basis for , let be the map that sends to (and all other basis elements to zero), and let be the map that sends to . Since the vectors form a basis for , the maps form a basis for . From this abstract definition, the following properties follow: It is the partial trace of linear transformations that is the subject of Joyal, Street, and Verity's notion of . A traced monoidal category is a monoidal category together with, for objects X, Y, U in the category, a function of Hom-sets, satisfying certain axioms.