Partial trace

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 , Tr W : L ⁡ ( V ⊗ W ) → L ⁡ ( V ) , 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 b_{k, 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).

Invariant definition

The partial trace operator can be defined invariantly (that is, without reference to a basis) as follows: it is the unique linear operator

such that

To see that the conditions above determine the partial trace uniquely, let v 1 , … , v m form a basis for V , let w 1 , … , w n form a basis for W , let E i j : V → V be the map that sends v i to v j (and all other basis elements to zero), and let F k l : W → W be the map that sends w k to w l . Since the vectors v i ⊗ w k form a basis for V ⊗ W , the maps E i j ⊗ F k l form a basis for L ⁡ ( V ⊗ W ) .

From this abstract definition, the following properties follow:

Category theoretic notion

It is the partial trace of linear transformations that is the subject of Joyal, Street, and Verity's notion of Traced monoidal category. A traced monoidal category is a monoidal category ( C , ⊗ , I ) together with, for objects X, Y, U in the category, a function of Hom-sets,

satisfying certain axioms.

Another case of this abstract notion of partial trace takes place in the category of finite sets and bijections between them, in which the monoidal product is disjoint union. One can show that for any finite sets, X,Y,U and bijection X + U ≅ Y + U there exists a corresponding "partially traced" bijection X ≅ Y .

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

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

Under this decomposition, any operator T ∈ L ⁡ ( V ⊗ W ) can be regarded as an infinite matrix of operators on V

where T k ℓ ∈ L ⁡ ( V ) .

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

converges in the strong operator topology of L(V), it is independent of the chosen basis of W. The partial trace Tr_W(T) is defined to be this operator. The partial trace of a self-adjoint operator is defined if and only if 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 { | ℓ ⟩ } ℓ . Then

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

commutes with all operators of the form I V ⊗ S and hence is uniquely of the form R ⊗ I W . The operator R is the partial trace of T.

Partial trace as a quantum operation

The partial trace can be viewed as a quantum operation. Consider a quantum mechanical system whose state space is the tensor product H A ⊗ H B of Hilbert spaces. A mixed state is described by a density matrix ρ, that is a non-negative trace-class operator of trace 1 on the tensor product H A ⊗ H B . The partial trace of ρ with respect to the system B, denoted by ρ A , is called the reduced state of ρ on system A. In symbols,

To show that this is indeed a sensible way to assign a state on the A subsystem to ρ, we offer the following justification. Let M be an observable on the subsystem A, then the corresponding observable on the composite system is M ⊗ I . However one chooses to define a reduced state ρ A , there should be consistency of measurement statistics. The expectation value of M after the subsystem A is prepared in ρ A and that of M ⊗ I when the composite system is prepared in ρ should be the same, i.e. the following equality should hold:

We see that this is satisfied if ρ A is as defined above via the partial trace. Furthermore, it is the unique such operation.

Let T(H) be the Banach space of trace-class operators on the Hilbert space H. It can be easily checked that the partial trace, viewed as a map

is completely positive and trace-preserving.

The partial trace map as given above induces a dual map Tr B ∗ between the C*-algebras of bounded operators on H A and H A ⊗ H B given by

Tr B ∗ maps observables to observables and is the Heisenberg picture representation of Tr B .

Comparison with classical case

Suppose instead of quantum mechanical systems, the two systems A and B are classical. The space of observables for each system are then abelian C*-algebras. These are of the form C(X) and C(Y) respectively for compact spaces X, Y. The state space of the composite system is simply

A state on the composite system is a positive element ρ of the dual of C(X × Y), which by the Riesz-Markov theorem corresponds to a regular Borel measure on X × Y. The corresponding reduced state is obtained by projecting the measure ρ to X. Thus the partial trace is the quantum mechanical equivalent of this operation.