Partial trace

The partial trace has applications in quantum information and decoherence which is relevant for quantum measurement and thereby to the decoherent approaches to interpretations of quantum mechanics, including consistent histories and the relative state interpretation.

are finite-dimensional vector spaces over a field, with dimensions

⁠, be bases for V and W respectively; then T has a matrix representation relative to the basis

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).

The partial trace operator can be defined invariantly (that is, without reference to a basis) as follows: it is the unique linear map such that To see that the conditions above determine the partial trace uniquely, let

From this abstract definition, the following properties follow: It is the partial trace of linear transformations that is the subject of Joyal, Street, and Verity's notion of Traced monoidal category.

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.

there exists a corresponding "partially traced" bijection

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

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 TrW(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.

Suppose W has an orthonormal basis, which we denote by ket vector notation as ⁠

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).

Suppose V, W are finite dimensional Hilbert spaces.

Consider a quantum mechanical system whose state space is the tensor product

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

The partial trace of ρ with respect to the system B, denoted by

, is called the reduced state of ρ on system A.

To show that this is indeed a sensible way to assign a state on the A subsystem to ρ, we offer the following justification.

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

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 density matrix ρ is Hermitian, positive semi-definite, and has a trace of 1.

It has a spectral decomposition: Its easy to see that the partial trace

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.