Partial Trace - Details

Details

Suppose V, W are finite-dimensional vector spaces over a field, with dimensions m and n, respectively. For any space A let L(A) denote the space of linear operators on A. The partial trace over W, TrW, is a mapping

It is defined as follows: let

and

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

relative to the basis

of

.

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

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.

Among physicists, this is often called "tracing out" or "tracing over" W to leave only an operator on V in the context where W and V are Hilbert spaces associated with quantum systems (see below).

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