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:

    “And meanwhile we have gone on living,
    Living and partly living,
    Picking together the pieces,
    Gathering faggots at nightfall,
    Building a partial shelter,
    For sleeping and eating and drinking and laughter.”
    —T.S. (Thomas Stearns)

    “Emancipation should make it possible for woman to be human in the truest sense. Everything within her that craves assertion and activity should reach its fullest expression; all artificial barriers should be broken, and the road towards greater freedom cleared of every trace of centuries of submission and slavery.”
    —Emma Goldman (1869–1940)

    “Deep down, the US, with its space, its technological refinement, its bluff good conscience, even in those spaces which it opens up for simulation, is the only remaining primitive society.”
    —Jean Baudrillard (b. 1929)