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DETAILS

Suppose ''V'', ''W'' are finite-dimensional vector spaces over a field of Dimension ''m'', ''n'' respectively.. The partial trace Tr''V'' is a mapping

: T \in \operatorname{L}(V \otimes W) \mapsto \operatorname{Tr}_V(T) \in \operatorname{L}(V)

It is defined as follows: let

:e_1, \ldots, e_m

and

:f_1, \ldots, f_n

be bases for ''V'' and ''W'' respectively; then ''T''
has a matrix representation

: \{a_{k \ell, i j}\} \quad 1 \leq k, i \leq m, 1 \leq \ell,j \leq n

relative to the basis

: e_k \otimes f_\ell

of

: V \otimes W.

Now for indices ''k'', ''i'' in the range 1, ..., ''m'', consider the sum
: b_{k, i} = \sum_{j=1}^n a_{k j, i j}.

This gives a matrix ''b''''k'', ''i''. The associated linear operator on ''V'' is independent of the choice of bases and is by definition the partial trace.

For example,
: \operatorname{Tr}_V(R \otimes S) = R \, \operatorname{Tr}(S) \quad orall R \in \operatorname{L}(V) \quad orall S \in \operatorname{L}(W)

The partial trace operator can be characterized invariantly as follows: It is the unique linear operator
: \operatorname{Tr}_V: V \otimes W ightarrow V

such that

: \operatorname{Tr}_V (1_{V \otimes W}) = \dim W \ 1_{V}

: \operatorname{Tr}_V (T (1_V \otimes S)) = \operatorname{Tr}_V ((1_V \otimes S) T) \quad orall S \in \operatorname{L}(W) \quad orall T \in \operatorname{L}(V \otimes W)


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

: \{f_i\}_{i \in I}

be an Orthonormal Basis for ''W''. Now there is an isometric isomorphism

: \bigoplus_{\ell \in I} (V \otimes \mathbb{C} f_\ell) ightarrow V \otimes W

Under this decomposition, any operator T \in \operatorname{L}(V \otimes W) 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 \
dots & dots & & dots \
T_{k1}& T_{k2} & \ldots & T_{k j} & \ldots \
dots & dots & & dots
\end{bmatrix}

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

: \sum_{\ell} T_{\ell \ell}

converges in the Strong Operator Topology of L(''V''), it is independent of the chosen basis of ''W''. The partial trace Tr''V''(''T'') is defined to be this operator. The partial trace of a self-adjoint operator is defined iff the partial traces of the positive and negative parts are defined.


Computing the partial trace