traced" bijection . Partial trace for operators on Hilbert spaces The partial trace generalizes to operators on infinite dimensional Hilbert spaces. Suppose V, W are Hilbert spaces, and let be an orthonormal basis for W. Now there is an isometric isomorphism Under this decomposition, any operator can be regarded as an infinite matrix of operators on V where . First suppose T is a non-negative operator. In this case, all the diagonal entries of the above matrix are non-negative operators on V. If the sum converges in the strong operator topology of L(V), it is independent of the chosen basis of W. The partial trace TrW(T) is defined to be this operator. The partial trace of a self-adjoint operator is defined if and only if the partial traces of the positive and negative parts are defined. Computing the partial trace Suppose W has an orthonormal basis, which we denote by ket vector notation as . Then The superscripts in parentheses do not represent matrix components, but instead label the matrix itself. Partial trace and invariant integration In the case of finite dimensional Hilbert spaces, there is a useful way of looking at partial trace involving integration with respect to a suitably normalized Haar measure μ over the unitary group U(W) of W. Suitably normalized means that μ is taken to be a measure with total mass dim(W). Theorem. Suppose V, W are finite dimensional Hilbert spaces. Then commutes with all operators of the form and hence is uniquely of the form . The operator R is the partial trace of T. Partial trace as a quantum operation The partial trace can be viewed as a quantum operation. Consider a quantum mechanical system whose state space is the tensor product of Hilbert spaces. 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. In symbols, To show that this is indeed a sensible way to assign a state on the A subsystem to ρ, we offer the following justification. Let M be an observable on the subsystem A, then the corresponding observable on the composite system is . However one chooses to define a reduced state , there should be consistency of measurement statistics.