- Partial trace
linear algebraand functional analysis, the partial trace is a generalization of the trace. Whereas the trace is a scalarvalued function on operators, the partial trace is an operator-valued function. The partial trace has applications in quantum informationand decoherencewhich is relevant for quantum measurementand thereby to the decoherent approaches to interpretations of quantum mechanics, including consistent historiesand the relative state interpretation.
Suppose "V", "W" are finite-dimensional vector spaces over a field of
dimension"m", "n" respectively. The partial trace Tr"V" is a mapping
It is defined as follows: let
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 "b""k", "i". The associated linear operator on "V" is independent of the choice of bases and is by definition the partial trace.
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: From this abstract definition, the following properties follow:
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
orthonormal basisfor "W". Now there is an isometric isomorphism
Under this decomposition, any operator can be regarded as an infinite matrixof operators on "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 topologyof 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 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 . 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 and hence is uniquely of the form . 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 of Hilbert spaces. A mixed state is described by a density matrixρ, 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". 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 . However one chooses to define a reduced state , there should be consistency of measurement statistics. The expectation value of "M" after the subsystem "A" is prepared in and that of 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 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
is completely positive and trace-preserving.
The partial trace map as given above is induces a dual map between the
C*-algebras of bounded operators on and given by
maps observables to observables and is the
Heisenberg picturerepresentation of .
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 theoremcorresponds 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.
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