the system. In this way information about the wave function of the system is passed to the meter, creating an entangled state of the composed system. Reading off the position of the meter obeys the statistics described above. A detailed description of the model can be found in [CM87].
2.4 The Weak Influence of Unsharp Measurements
An unsharp measurement of an ordinary observable possesses a lower accuracy (higher vari-ance12) than a sharp measurement of . Still sometimes it is preferable to carry out an unsharp measurement instead of a sharp measurement. This is the case if one of the objectives is to keep the state change caused by the measurement small. Unsharp measurements fulfil two requirements for a small state change:
☞ Pure states change into (other) pure states, i.e., there is no external noise produced by entanglement to the environment or classical randomness (cp. the case of a “loose pointer”
in subsection 2.2.6).
☞ Low accuracy is a necessary condition for small state changes. High accuracy measure-ments destroy superpositions of eigenstates of the measured observable. This decoher-ence can be interpreted as being caused by the extraction of information, cp. section 2.7.
The more information is extracted, the higher is the decoherence.
The influence of the extraction of information on the state of a quantum system can be isolated and identified with a certain part of the corresponding Kraus operator as follows. A general Kraus operator
(and in fact any bounded operator) can be decomposed like a complex num-ber into “phase” and “modulus” (polar decomposition, cp. [Ped89], p.96):
with ;
( (2.84)
where the “phase” is an unitary operator acting on the Hilbert space of the m-system. The modulus of
equals the square root of the corresponding Effect:
(
(2.85) One method to process the information contained in the results of unsharp measurements was introduced in the previous two sections. It employs the fact that the expectation value of an
12There are several measures for the accuracy, one of them is the variance of the measurement results, cp. section 2.3. Which measure should be employed depends on the context. In the context of the measurement of ordinary observables an alternative to the variance is given by the mean decrease of entropy (cp. section 2.7). For quantum state estimation appropriate measures of accuracy are the error probability [Bus97] and the estimation fidelity [Ban01, ADK02a].
62 Unsharp Measurements observable corresponding to the unsharp measurement is determined (and also all higher mo-ments of the observable) by the probabilities of the results of the unsharp measurement. These probabilities, on the other hand, depend — apart from the initial state of the m-system — only on the Effects and thus on the squared moduli of the corresponding Kraus operators:
%8"
$/
tr
(2.86)
The approximation of probabilities via relative frequencies of measurement results and thus the estimation of the moments of the ordinary observable is independent from the unitary part of the Kraus operators. In general the following statement holds: Since the pre-measurement state of the m-system is coupled to the probabilities of the results only via the Effects of the unsharp measurement, any efficient, non-trivial way of estimating a property of the pre-measurement state must depend on the moduli and must be independent from the unitary parts of the Kraus operators.
Due to the polar decomposition of the Kraus operators, a state change
can be interpreted as a state change due to the extraction of information followed by an additional unitary evolution which is due to the particular properties of the measurement apparatus (coupling of meter to system, detection of meter observable etc.), compare Wiseman [Wis95b] and Doherty et al.[DJJ00]. Since the influence of the measurement apparatus ideally should be kept minimal, ought to be (up to a physically irrelevant phase factor , ) the identity
or it should compensate the state change inflicted by
. Such a compensation seems to be impossible at least in the case of a totally unknown pre-measurement state, where given the Effects the smallest average state change is achieved for measurements with Kraus operators
(
, thus
cf. [Ban01].
Definition 8 A measurement with pure operations and Kraus operators
is called a minimal measurement.
Proposition 2.4.1 A minimal unsharp measurement is of first kind, i.e. the probability%&": $
for any measurement event, !#" $ (cp. section 1.2) and any initial state satisfies:
%8":B $ This means if two measurements are carried out on the same quantum system the probability to measure in the first measurement is the same as the probability to measure in the second measurement. Please observe that the second measurement takes place, no matter what came out of the first measurement (non-selective regime of measurement). A measurement with this property is called measurement of first kind, cf. [BLM91].
Proof: The non-selective operation of a minimal measurement can be expressed by means of the the Kraus operators ( according to
":
$/ 5 2
(2.88)
2.4 The Weak Influence of Unsharp Measurements 63 Since the measurement considered is also unsharp, all its Effects commute. Hence
%&":
where in the last line the completeness of the Effects, i.e.
2
, has been exploited.
Analogously one can show that the expectation value of ordinary observables are not changed in a non-selective minimal and unsharp measurement, if all its Effects commute with the observable
of a minimal unsharp measurement commute, there is a com-mon basis of eigenstates
1 of the Kraus operators. These eigenstates
1 are not changed due to the measurement. For any possible composed result-,.!#" $ one finds:
1 1 2 1 1
%&":
1 $ 1 1 if %&":
1 $ ? *
(2.90) Proposition 2.4.2 The Bloch vector of a qubit moves due to a minimal unsharp measurement with result “ ” on a great circle of the Bloch sphere towards the Bloch vector corresponding to the eigenvector of the Kraus operator
. Then the state change
%&"
$
corresponds to a change of the Bloch vector
. Expressed in spherical coordinates "
$
the radius is equal to one for both Bloch vectors. The azimuth angle
does not change due to the measurement:
Hence, the only spherical coordinate which changes due to a minimal unsharp measurement is the angle . Such a change of spherical coordinates corresponds to a movement on a meridian of the Bloch sphere, cp. Fig.2.3. It remains to show that
64 Unsharp Measurements
PSfragreplacements
Figure 2.3: Shift of the Bloch vector due to a minimal unsharp measurement with two results “ ” and “ ”.
The Bloch vector of is shifted along a great circle containing the eigenvectors of the Kraus operator. Here
, moves towards the eigenvector belonging to the largest eigenvalue of the relevant Kraus operator
.
*
occurs in both cases for eigenstates of
The modulus of the angle
between the Bloch vector after the measurement and the Bloch vector
before the measurement decreases the closer the initial state is to an eigenstate of . Fig.2.4 illustrates the dependence of on .