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.

Contents

1 Details 2 Partial trace for operators on Hilbert spaces 2.1 Computing the partial trace 3 Partial trace and invariant integration

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

<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 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,

<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 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 <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.