Partial Trace - Partial Trace For Operators On Hilbert Spaces

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

\begin{bmatrix} T_{11} & T_{12} & \ldots & T_{1 j} & \ldots \\ T_{21} & T_{22} & \ldots & T_{2 j} & \ldots \\ \vdots & \vdots & & \vdots \\ T_{k1}& T_{k2} & \ldots & T_{k j} & \ldots \\ \vdots & \vdots & & \vdots \end{bmatrix},

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.

Read more about this topic:  Partial Trace

Famous quotes containing the words partial, trace and/or spaces:

    “There is no luck in literary reputation. They who make up the final verdict upon every book are not the partial and noisy readers of the hour when it appears; but a court as of angels, a public not to be bribed, not to be entreated, and not to be overawed, decides upon every man’s title to fame. Only those books come down which deserve to last.”
    —Ralph Waldo Emerson (1803–1882)

    “Superstition saw
    Something it had never seen before:
    Brown eyes that loved without a trace of fear,
    Beauty so sudden for that time of year.”
    —Jean Toomer (1894–1967)

    “through the spaces of the dark
    Midnight shakes the memory
    As a madman shakes a dead geranium.”
    —T.S. (Thomas Stearns)