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 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.
Details.
Suppose formula_1, formula_2 are finite-dimensional vector spaces over a field, with dimensions formula_3 and formula_4, respectively. For any space formula_5, let formula_6 denote the space of linear operators on formula_5. The partial trace over formula_2 is then written as formula_9.
It is defined as follows: 
For formula_10, let formula_11, and formula_12, be bases for "V" and "W" respectively; then "T"
has a matrix representation
relative to the basis formula_14 of formula_15.
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 formula_19 form a basis for formula_1, let formula_21 form a basis for formula_2, let formula_23 be the map that sends formula_24 to formula_25 (and all other basis elements to zero), and let formula_26 be the map that sends formula_27 to formula_28. Since the vectors formula_29 form a basis for formula_30, the maps formula_31 form a basis for formula_32.
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 formula_35 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 formula_37 there exists a corresponding "partially traced" bijection formula_38.
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 formula_41 can be regarded as an infinite matrix
of operators on "V"
where formula_43.
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 formula_45. Then
The superscripts in parentheses do not represent matrix components, but instead label the matrix itself.
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 formula_48 and hence is uniquely of the form formula_49. 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 formula_50 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 formula_51
The partial trace of ρ with respect to the system "B", denoted by formula_52, 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 formula_54. However one chooses to define a reduced state formula_55, there should be consistency of measurement statistics. The expectation value of "M" after the subsystem "A" is prepared in formula_52 and that of formula_54 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 formula_52 is as defined above via the partial trace. Furthermore, such operation is unique.
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 formula_61 between the C*-algebras of bounded operators on formula_62 and formula_50 given by
formula_61 maps observables to observables and is the Heisenberg picture representation of formula_66.
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.