5.4 N-series
5.4.1 From an Elementary Cell to an N-series
(5.66) and
(5.67) In order to truncate the expansion of the operation corresponding to a N-series we also have to demand that times the second small quantity
% stays small:
%
+ % %
(5.68) The difference equation can be derived in two ways. The straight forward one at this point is to calculate the operation N of a N-series based upon the the operation t of an elementary cell, which we evaluated in the last section. This will be done in the following subsection.
The second way to derive a difference equation for N-series consists in reordering the oper-ator products in the operation elements of a N-series, such that the unitary operoper-ators stand on one side of the products and the unitary parts
stand separated from the positive parts
on the other side of the products. The reordering leads to Effects, which are approximately Gaussian and establishes the connection to Effects and weight functionals employed in contin-uous measurements. In particular the connection between the contincontin-uous fuzzy measurements and sequential unsharp measurements emerges. We will follow this second way in subsection 5.4.2. It is also described in [ADK02b].
5.4.1 From an Elementary Cell to an N-series
The operation N of a N-series can be obtained by carrying out times the operation t of an elementary cell:
t does not posses the “composition” property of continuous representations
" It is thus not possible to compute the state after measurements by simply replacing time by
in Eq. (5.43) but only by iteratively inserting the state into the equation:
" $ "
156 Non-selective Discription of Unsharp Measurements with Unitary Backaction At the first glance the lack of the composition property indicated in (5.70) seems to be peculiar. At least for indirect, non-selective measurements as well as for all unitary interactions with a second system the evolution of the reduced density operator is continuous in time. Thus there exists an operation
This is so, because for unitary couplings the reduced density operator evolves continuously since the unitary evolution is continuous:
"
where tr is the partial trace over the degrees of freedom of the second system. In case of an in-direct measurement the reduced density operator is not sensitive to a non-selective measurement of the second system (represented by operation elements
):
t does not fulfill property (5.73) for all times because it only represents the operation of a whole elementary cell (
). It does in particular not resolve what happens during a mea-surement of duration . Therefore the terms in Eq. (5.43) stemming from operation elements
do not depend on time ( =const) and
t"
$
does not take into account the minimal part of the measurement times as required for measurements.
How to propagate the density operator over the duration of a N-series? According to (5.69) we have to compute the n-th power of the propagator
t of an elementary cell. Up to second order in the small quantities and
%
, the task can be conveniently accomplished by means of the binomial formula:
, and the lowest order of the small quantities in is one, cp. Eq. (5.42) together with Eq. (5.39). It is thus not necessary to consider the third and higher orders of . From Eq. (5.42) we obtain up to second order in and
%
5.4 N-series 157 By plugging and
into Eq. (5.76) the operation associated with an N-series can be expressed as follows:
The expression above can be simplified by employing the following identities:
%
These identities are equivalent to Eqs. (5.63) and (5.64), which were used to identify in the difference equation terms generating unitary development and terms leading to decoherence.
The above identities serve the same purpose. We are now ready to compute " $
N"' "
$$
. In order to eliminate from the equation of motion, we will make use of
158 Non-selective Discription of Unsharp Measurements with Unitary Backaction
Physical Meaning of the Terms
Apart from the coefficients and the triple commutators containing
AV, all terms in (5.81) can already be found in difference equation (5.43) for the elementary cell (cp. also Eq. (5.64)).
An extensive discussion of their meaning can be found in the previous section. Let us just recall the main features and put them in the context of a N-series.
There is an effective unitary development composed of two unitary evolutions. The first of these is the original evolution with Hamiltonian . The second is generated by the
“averaged” Hamiltonian corresponding to the unitary parts of the measurements. The effective unitary development acts consecutively times:
" The corresponding terms in (5.81) are the single and the double commutator with
AV
as well as the term containing AV . The double commutator terms with
and the one with lead to decoherence with respect to the eigenbasis of and respectively. The first can be understood as a consequence of noise (see previous section), while the latter decoherence is due to the extraction of information.
There is a shift of the expectation values of observables of the qubit (generalised friction) caused by the N-series. In our approximation the term proportional to " $
5.4 N-series 159 indicates a shift for any observables
with a component perpendicular to
. By an appropriate choice of
the expectation value of any observable can be shifted in a N-series.
The rest of the terms are proportional to double commutators of applied to single commutators containing a Hamiltonian or vice versa. They can be interpreted to generate decoherence of the unitary evolved state or unitary evolution of the partly decohered state, respectively.