partial trace

In linear algebra and functional analysis, the partial trace is a generalization of the trace. Whereas the trace is a scalar valued function on operators, the partial trace is an operator-valued function. The partial trace has applications in the relative state interpretation of quantum mechanics.

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

\operatorname{L}(V \otimes W) \ni T \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

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, $\operatorname{Tr}_V(R \otimes S) = R \, \operatorname{Tr}(S) \quad \forall R \in \operatorname{L}(V) \quad \forall S \in \operatorname{L}(W)$

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

such that $\operatorname{Tr}_V (1_{V \otimes W}) = \dim W \ 1_{V}$

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

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 $\bigoplus_{\ell \in I} (V \otimes \mathbb{C} f_\ell) \rightarrow V \otimes W$

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 \\ \vdots & \vdots & & \vdots \\ T_{k1}& T_{k2} & \ldots & T_{k j} & \ldots \\ \vdots & \vdots & & \vdots
\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
Suppose W has an orthonormal basis, which we denote by ket vector notation as $\{| \ell \rangle\}_\ell$ . Then $\operatorname{Tr}_V\left(\sum_{k,\ell} T_{k \ell} \, \otimes \, | k \rangle \langle \ell |\right) = \sum_j T_{j j}$

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 $\int_{\operatorname{U}(W)} (1_V \otimes U^*) T (1_V \otimes U) \ d \mu(U)$

commutes with all operators of the form $1_V \otimes S$ and hence is uniquely of the form $R \otimes 1_W$ . The operator R is the partial trace of T.

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