Partial trace

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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.

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

<math> \operatorname{L}(V \otimes W) \ni T \mapsto \operatorname{Tr}_V(T) \in \operatorname{L}(V)<math>

It is defined as follows: let

<math>e_1, \ldots, e_m <math>

and

<math>f_1, \ldots, f_n <math>

be bases for V and W respectively; then T has a matrix representation

<math> \{a_{k \ell, i j}\} \quad 1 \leq k, i \leq m, 1 \leq \ell,j \leq n <math>

relative to the basis

<math> e_k \otimes f_\ell <math>

of

<math> V \otimes W<math>.

Now for indices k, i in the range 1, ..., m, consider the sum

<math> b_{k, i} = \sum_{j=1}^n a_{k j, i j}. <math>

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

For example,

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

The partial trace operator can be characterized invariantly as follows: It is the unique linear operator

<math> \operatorname{Tr}_V: V \otimes W \rightarrow V <math>

such that

<math> \operatorname{Tr}_V (1_{V \otimes W}) = \dim W \ 1_{V} <math>
<math> \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)<math>

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

<math> \{f_i\}_{i \in I} <math>

be an orthonormal basis for W. Now there is an isometric isomorphism

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

Under this decomposition, any operator <math> T \in \operatorname{L}(V \otimes W)<math> can be regarded as an infinite matrix of operators on V

<math> \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}<math>

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

<math> \sum_{\ell} T_{\ell \ell} <math>

converges in the strong operator topology of L(V), it is independent of the chosen basis of W. The partial trace TrV(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 <math> \{| \ell \rangle\}_\ell <math>. Then

<math> \operatorname{Tr}_V\left(\sum_{k,\ell} T_{k \ell} \, \otimes \, | k \rangle \langle \ell |\right) = \sum_j T_{j j} <math>

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

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

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

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