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:

    “America is hard to see.
    Less partial witnesses than he
    In book on book have testified
    They could not see it from outside....”
    —Robert Frost (1874–1963)

    “Yet ere I can say where—the chariot hath
    Passed over them—nor other trace I find
    But as of foam after the ocean’s wrath”
    —Percy Bysshe Shelley (1792–1822)

    “Surely, we are provided with senses as well fitted to penetrate the spaces of the real, the substantial, the eternal, as these outward are to penetrate the material universe. Veias, Menu, Zoroaster, Socrates, Christ, Shakespeare, Swedenborg,—these are some of our astronomers.”
    —Henry David Thoreau (1817–1862)