2.2 What are Unsharp Measurements?
2.2.1 Example: Stern-Gerlach Measurement of Spin
In order to introduce unsharp measurements of ordinary observables let us look at a concrete example: the spin measurement by means of an inhomogeneous magnetic field. A current of silver atoms is sent through an inhomogeneous magnetic field . Behind the magnetic field the silver atoms hit a screen (cp. Fig. 2.1). The current of silver atoms is so weak, that there is no interaction between them on their flight.
For two reasons it is interesting to look at the original experiment which was conducted by Stern and Gerlach. Firstly, from the dimensions of the measurement record one obtains a good idea why projection measurements in realistic setups are the exception and not the rule. And secondly, this experiment is seen as the discovery of electronic spin. The original experiment was suggested by Stern [Ste21] in 1921 and carried out by Stern and Gerlach [SG22] in Frank-furt (Germany) in 1922. The setup of the experiment was as sketched in Fig. 2.1. The silver atoms were produced in a furnace and a glass plate served as screen. Although the glass plate was exhibited to the current of impinging silver atoms for a period of eight hours, the deposit of atoms on the plate was so thin, that it had to be chemically treated in order to show traces of the silver current. It showed two lines in the shape of a pair of lips with length mm, cp. Fig.2.1.
The maximal separation between the lower boundary of the upper lip and upper boundary of the lower lip amounted to* mm. The maximal distance between the middle of the upper lip and the middle of the lower lip was
* + mm.
This result lead to the postulation of a new quantity: electronic spin. The electronic spin along a distinguished direction (here the direction of the magnetic field) shows in a projection
2.2 What are Unsharp Measurements? 39
B
magnet oven
z
x
screen
Figure 2.1: Stern-Gerlach measurement of spin .
measurement only two discrete values: . In the Stern-Gerlach experiment these values were associated with atoms being detected in the upper and lower lip, respectively. In the area where the lips overlap there is no unambiguous association of spin values possible and the measure-ment is not a projection measuremeasure-ment. But also where the lips appear to be separated there might be a small overlap which cannot be resolved by the applied technics. Depending on the strength of the overlap the measurement is more or less “unsharp”.
The Stern-Gerlach experiment is believed to be well understood. It is described in terms of POVM’s in [BGL95], pp. .
The coupling of spin and position
Silver atoms posses a single valence electron with zero orbital angular momentum in its ground state and spin quantum number s=1/2. Only the electronic spin of the valence electron con-tributes to the total magnetic moment of the silver atom
(2.1)
where is the elementary charge,
the velocity of light and the mass of the electron. The spin can be represented by the selfadjoint operator '+ , where the components
1
are the three Pauli spin operators given in Eq. (1.36). The magnetic moment expressed by means of Bohr’s magneton and the Pauli spin operators reads
with ;
(2.2)
40 Unsharp Measurements The unitary evolution of the state of a silver atom during its passage through the magnetic field is given by the Hamiltonian1
It is this coupling of the spin to the magnetic field which enables a measurement of the third component '+ of the spin. As we shall see, the atom is accelerated due to this coupling and the acceleration depends on the spin part of the atomic state. It is the motion of the centre of mass which is finally detected and this motion then allows to draw conclusions about the spin state before the atom entered the magnetic field. Apart from the magnetic moment, the internal structure of the atom does not influence its centre of mass motion and can be neglected here.
The atomic state can thus be represented by a spatial part
referring to the centre of mass motion and a spin part
which are independent from each other before the atom enters the magnetic field:
(2.5)
Spin 1/2 means the spin state can be viewed as a qubit:
*
(2.6)
* , represent the eigenstates of . Due to the coupling of magnetic moment with the magnetic field the spatial- and the spin part of the state of each silver atom get entangled:
1On their way through the magnet towards the screen the silver atoms experience also dispersion of their centre-of-mass wavefunctions due to the free Hamiltonian ! #". However, if the magnetic field is strong enough and the period between magnet and screen is sufficiently short this influence can be neglected.
2.2 What are Unsharp Measurements? 41 In (2.7) I have expanded the centre of mass part
of the state
with respect to the eigen-states of the momentum operator % and in the third line made use of the fact, that the z-component of the position operator is the generator of a translation in z-direction of momentum space. The change of the wave function
* aquire momentum in direction of the gradient of the -field, while atoms entering in are accelerated in the opposite direction. This is a typical measurement correla-tion, the centre of mass states
and
resemble the states of the pointer of a measurement apparatus pointing out the measurement result. In state
the pointer (the silver atom) has not yet a definite position. But a definite position of the silver atom is observed on the screen.
While in the first phase of the measurement the coupling of meter and system via a common unitary development takes place (this phase is sometimes called premeasurement) the last phase incorporates the objectification of the pointer observable, which in our case is the position of the silver atoms. It is still unknown how this objectification proceeds (this is the content of the mea-surement problem). Mathematically the result of the state change caused by the objectification process is modelled by projecting on an eigenstate of the pointer observable.
Detection of the position
Obviously the resolution of the position measurement of the silver atoms on the screen is lim-ited. Therefore the pointer observable is not the exact position on the screen represented by the position operator with eigenvectors . Rather it is a coarse grained version of the posi-tion which is measured. The coarse graining can here be understood literally since one can think of the surface of the screen as consisting of small grains which show a trace when hit by an incoming particle2. The upper limit of the resolution is thus given by the area cov-ered by such a grain. The result of the measurement with the highest resolution using a certain screen thus corresponds to such an area on the surface of the screen. The probability to find a trace of an impinging silver atom on is modelled by means of the projection operator
and accordingly. As will be discussed below these expectation values of are the key to sharpness or unsharpness of the measurement.
2Instead of a glass plate also a photographic plate with silver bromide grains on the surface can serve as screen.
The silver bromide grains which are hit by impinging particles show a black colour after a chemical treatment.
42 Unsharp Measurements How is the state of the silver atoms changed due to the detection? When the silver atoms hit the glass plate they become part of the solid body or might be scattered off. Their state thus has to be described in the framework of solid state physics or scattering theory. Further experiments on the silver atoms are easiest accomplished when the screen contains a hole and is used as a filter. If the particle is known to have passed the hole (e.g. it is detected behind the screen) then the state change due to the the passage can be described by
silver atom passes filter
is a constant to ensure the normalisation
. Please note, that in general the state of the silver atoms behind the filter is still entangled. However, if the wave functions
" $
and
" $
do not vary much over the area (e.g. if the area is small enough):
then the state after the filter can be approximated by a product state:
factor states and their product is equal to .
Description without Reference to Meter
If only the spin of the silver atoms is of interest, the measurement can be represented exclusively in terms of the spin part
.
☞ Probability to measure :
%&"
$
with ;= % " $ * >* %
" $
(2.15) Here we used the abbreviations% "
$ ;
(2.15) can be verified by inserting Eq. (2.6), noting that%&"
$
%&"
$
and com-paring Eq. (2.15) with Eq. (2.10).
☞ State change: If the resolution of the position of the silver atom by the filter is high enough the state of the silver atom behind the screen is a product state cp. Eq. (2.14). In this case the spatial part can be neglected. The change of the spin state during the measurement then amounts to
2.2 What are Unsharp Measurements? 43 The state change can thus be represented by means of a single Kraus operator
(cp.
☞ State change if meter and system are entangled: For a general area
as measurement result the state after the measurement (more precisely behind the filter, see discussion above) is still entangled. Therefore the description of the state change during the mea-surement involves density operators :
where tr is the partial trace over an orthonormal basis (complete system of orthonormal vectors) of the Hilbertspace
" $
associated with the spatial part of the state. The result
is a density operator on the Hilbertspace
of the spin part. Let us write it down in form of a matrix w.r.t. the basis ( * , ) Here the matrix elements are expressed by inner products of the meter wave functions on the Hilbertspace
" $
restricted to the area on the screen.
The reduced density matrix in Eq. (2.19) reveals in how far the spin state after the filter is decohered, depending on whether the wavefunctions
and
are orthogonal after the filter. In terms of Kraus operators the state change reads:
is the initial spin state.
Unsharpness of the Spin Measurement
Consider the case where we only distinguish whether a silver atom is detected in the upper half of the screen ( contains all points on the screen with coordinate (-* ) or in the lower half ( with * ). Ideally we would like to have an unambiguous correlation between meter (spatial part) and system (spin part). This means, if the silver atom enters with spin state
* it
will be detected with certainty in the upper half (result ), while if it enters in state the measurement result will certainly read (lower half of the screen). This condition can be
44 Unsharp Measurements expressed mathematically by means of the wave functions
" $
and
" $
corresponding to the state just before the detection, cp. Eq. (2.7):
The correlation condition above is equivalent to the requirement that the wave functions
" $ This is the case if and only if
formulas for . In other words: The ingoing states * and can be distinguished with certainty in a single measurement with result or , if and only if the associated “Effects”
and
are projectors on these states.
Such a measurement can be viewed as a sharp measurement of the z-component of spin.
This can be understood as follows. The states
* and are eigenstates of the selfadjoint oper-ator '+ which represents the (ordinary) observable “z-component of the spin” of a spin 1/2 system. The eigenstates correspond to the values + and '+ of the z-component, respec-tively. A sharp measurement of provides a one-to-one correspondence between measurement results and values of the z-component, such that the ingoing eigenstate * (z-component '+ ) can be distinguished from the ingoing eigenstate (z-component '+ ) with certainty.
Please note, that the state after a filtering corresponding to a sharp measurement in the Stern-Gerlach experiment is the eigenstate of associated with the measurement result. This can be seen from Eq. (2.19). Therefore this realisation of a sharp measurement is also a “Pro-jection measurement” (also called ideal- or von Neumann measurement) which requires that the measurement is sharp and the system ends up in an eigenstate of the measured observable.
In many cases a sharp measurement as described above is not realized. For example if the wavefunctions
" $
and
" $
overlap then a silver atom in the region of overlap cannot be associated either with spin state * or with spin state . But even if the measurement results do no uniquely correspond to a certain spin component, the measurement result still contains information about the initial spin state if it is more “typical” for one of the spin components. If e.g. the probability%&"
* $
to detect a silver atom with initial state * on the lower half of the screen is finite but smaller than to find it there if the initial state was , then it is more likely that the initial state was actually than that it was
* . For the Stern-Gerlach experiment the Effects corresponding to “ambiguous” results , i.e. results which admit the initial state
* as well as , are of the form
2.2 What are Unsharp Measurements? 45 Effects of ambiguous results thus are not rank-one projectors. In addition they are diagonal with respect to the same basis. This is the characteristic trait of unsharp measurements of ordinary observables. They are called unsharp because they convey less information than sharp measure-ments. Furthermore, they convey information about an ordinary observable which possesses the same eigenvectors as the Effects of the unsharp measurement. In the case of the Stern-Gerlach experiment we are dealing with unsharp measurements of the z-component of spin. This can be seen by noting that the probabilities of the results of an unsharp measurement can be employed to calculate the expectation value of the z-component of spin3.
What property of the state of a quantum system is measured by a generalised measure-ment? How can it be interpreted in terms of the usual observables such as energy, position or angular momentum for which we have developed an intuitive meaning? For a special class of generalised measurements the connection to meaningful observables can be established more easily than for the rest. This is the class of unsharp measurements. Before we define unsharp measurements we introduce some other useful notions.