measurements. An experimental realization of unsharp measurements as specified above may be performed in quantum optics or in condensed matter physics. We therefore give some details, cp also sections 2.5 and 2.6.
The two-level system in the state
" * $ *
(3.20)
interacts with another two-level system with orthonormed states
which acts as the me-ter M. Its initial state is
" * $ . Hence the initial state of the compound system is
" * $
" * $
" * $ . It is then assumed that the interaction lasting for the time results in the
partic-ular unitary development
After a sharp measurement on the meter alone is performed which, depending on the outcome
and are taken as the measurement outcomes of Section 3.2.1. The readout
which agrees with% of (3.17). If the measurement result reads ” ”, the state of the compound system after the measurement is the product state
as given in equations (3.10) with (3.11) and (3.12). In order to obtain the final state for the measurement readout one simply has to interchange the indices
and . The class of unsharp measurements of Section 3.2.1 is thus reconstructed.
The device specified above shows clearly that the unsharpness in question is of genuine quantum nature. It cannot be traced back to an imperfect measurement procedure, which itself may for example be caused by an imperfectly fixed pointer (cp. subsection 2.2.6) showing from time to time although the meter state is
. The reason is instead that the quantum dynamics leading to the unitary development (3.21) does not correlate * only with
and only with
but allows the appearance of the states
and
*
in the superposition.
3.3 N-series of Successive Unsharp Measurements
3.3.1 One N-series
We will now perform on the same single system unsharp measurements of the type specified above with the same parameters in an immediate succession. This is called an N-series. Later
94 Unsharp Measurements and the Detection of Dynamics on we will allow the measurements to be separated in time by . A motivation to use N-series will be given below. Since the Kraus operators and commute, the final state after the N-series will not depend on the order of and results. For any particular sequence of results with a total number of results
the state is transformed for
* Now we discard the information about the order of the results, i.e., we restrict ourselves to the information that the total number of results is , regardless when they occurred in the sequence. As measurement outcome attributed to the total N-series we take in the following the relative frequency of positive results
;=
(3.26) In order to transcribe for this type of unsharp measurement the scheme of Section 3.2.1, which is based on the equations (3.10), (3.15) and (3.17) for non-normalised state vectors, we have to work out the related Kraus operators and Effects. The probability%&"
$
that positive results are measured in an N-series equals
times the probability that a particular ordered sequence of positive and negative results is obtained:
%&"
The corresponding probability amounts to
%&"
To complete the scheme we write the pure state resulting after the N-series in the form
" $ " * $ (3.30)
An alternative derivation of the Kraus operators of a N-series can be found in subsection 2.3.3.
3.3 N-series of Successive Unsharp Measurements 95 Gaussian Approximation of Binomial Effects
It is interesting to note, that the Effects (and the Kraus operators) can be approximately rep-resented by operator-valued densities of Gaussian form. A transition from the results of a N-series to results leads in good approximation to the Gaussian Effect
is the decoherence rate due to a N-series and
represents its duration. The correspond-ing Kraus operators are given by ( . Details can be found in subsection 5.4.
Operation Fidelity of N-series
In order to quantify by how much an N-series of measurements affects the state, we calculate for arbitrary values of % and%
the fidelity, which shows how closely the post-measurement state resembles the pre-measurement state. The fidelity " $ between the pure state
" * $
before the measurements and the state
5
(3.34) after the N-series is equal to the square of the overlap between
" * $ and
"
$<;= (3.35)
Please note, that
is unnormalised. represents the so-called “operation fidelity” of a N-series. After some algebra one finds
" Equation (3.36) shows that for all choices of the parameters% and%
'+ and its maximum for
and
*
(or with indices interchanged). This means, that the state which is in the average most sensitive to the influence of the measurement is the equally weighted superposition
" * $ + .
96 Unsharp Measurements and the Detection of Dynamics Operation fidelity may serve as a direct measure for the weakness of the influence of an N-series of weak measurements on the pre-measurement state. This becomes evident by looking at the limiting cases: for extremely sharp measurements with% and%
*
(or vice versa) the operation fidelity equals the operation fidelity
+
of a projection measurement.
The maximal fidelity is obtained for infinitely weak measurements with% %
. For an infinite N-series the limit
+
is equal for all values of
% % , *
with%
?
%
to the operation fidelity of an projection measurement. That such an iteration of measurements produces an eigenstate has theoretically been discussed in [Son98].
For an experimental scheme which realizes a projection measurement by an iteration of unsharp measurements see [BHL 90] and chapter 4. The dependence of the operation fidelity on the number of measurements within a N-series averaged over all possible initial states is plotted in Fig. 2.7.
3.3.2 Best Guess for the Outcome of One N-series
For the moment we refer again to repeated measurements on the same initial state
"* $
(en-semble approach). The statistical expectation value of reads
The latter equation follows with (3.29). The variance of amounts to:
Based on (3.38) it is easy to relate
of a N-series of unsharp measurements starting with
" * $ : Please note, that both quantities
and
"
$
can only be measured on a large ensemble of systems prepared in state
" * $ .
If only one N-series carried out on a single system is available, as will be the case below, only a best guess
for the quantity
can be obtained. Equation (3.39) suggests to choose it as
represents an unbiased estimator of
, i.e. "
$
. Unbiased estimators lead to the exact determination of the estimated quantity in the limit of an infinitely large ensemble4.
4In classical statistical theory unbiased estimators are preferred to biased ones. Even general methods have been invented to correct the bias of estimators, cp. [HP89], pp. 181. Nevertheless the necessity of unbiasedness is discussed controversially among statisticians [HP89], ibidem.
3.4 Measurement of a Dynamically Driven State by a Sequence of N-series 97