5.3 An Elementary Cell
5.3.3 Decoherence and Dissipation in an Elementary Cell
t applied to "
$
propagates this state to the state at time :
" $
Here we have simply translated the operators with superscripts "
$
, "
$
back into operators which act from the left and from the right, respectively. In order to shorten the transformation, the rules given in Eqs. (5.26) can be applied. Another useful identity in this context is given by
+"
$
(5.44)
We thus obtained a difference equation up to second order in time between two consecutive measurements. It describes the motion of the state of the system under the combined influ-ences of the Schr¨odinger dynamics and the elementary measurement in one elementary cell of the sequential measurement.
5.3.3 Decoherence and Dissipation in an Elementary Cell
Let us interprete the terms influencing the state motion as they appear in difference equation (5.43).
☞ " $
This term looks like the lowest order approximation of a unitary state transformation gen-erated by the “effective Hamiltonian ” . In the Barchielli limit (cp. section 5.5), in which the sequential measurement becomes a continuous measurement, it generates a unitary transformation with this effective Hamiltonian. The supplementary Hamiltonian
% " % $
resembles an average of the back-action Hamiltonians with the statistical weights% and % rather then the actual probabilities% that is applied.%
depends on the state and therefore cannot appear in the Hamil-tonian. This would lead to a non-linear equation of motion and violate the superposition principle of Quantum Mechanics. The actual statistical weights% and .% represent
144 Non-selective Discription of Unsharp Measurements with Unitary Backaction the probabilities% and% respectively to apply
averaged over all possible pure states
6. The first term of (5.43) indicates the modification of the unitary dynamics due to the unitary part of the back-action. Taking into account terms of higher order in time the purely unitary part of the dynamics during a sequential measurement amounts to the operation "% " % $ $ . This can be seen by inserting % and
" % $
in Eq. (5.30). A geometrical interpretation of the first term of Eq.
(5.43) is an approximated rotation of the Bloch sphere about the axis specified by each of the Bloch vectors corresponding to the eigenvectors of the effective Hamiltonian.
☞ " $ "' * * $
Condition (5.12) implies . Due to this term the off-diagonal elements
and
of the density matrix expressed with respect to the eigenstates * , of are reduced.
The second term of difference equation (5.43) thus resemblesdecoherence with respect to the basis of eigenstates of . This decoherence is due to the unsharp measurement of
, we are dealing with. In a geometrical picture the transition of the Bloch vector
":
corresponding to the state transformation which neglects all other influences,
The whole Bloch sphere is thus contracted in - and -direction and mapped onto a rota-tional ellipsoid with major- and minor axis of lengths and '+ , respectively. The left part of Fig. 5.2 illustrates the contraction of the Bloch sphere in and direction caused by decoherence with respect to the basis (
* , ).
☞ ": $); %
" $
A" $
.
can be expressed with respect to the basis constituted by the identity and the Pauli operators
1 , which are real numbers because is self-adjoint.
With this notation the fourth term of Eq. (5.43) reads:
"'
where represents a normed measure on the space of states. A similar calculation shows that is averaged over all states.
5.3 An Elementary Cell 145 Please note, that ":
$
const but still is a convex-linear mapping on the space of states
as it has to be in order to not break the convex-linearity of the operation . This can be seen by multiplying the right-hand side of equation (5.48) by tr . The multiplication does not change ":
$
, because tr
but shows that is convex linear because the mapping tr is convex linear.
What is the physical interpretation of this term? Starting with any state the summand
"'
$
in (5.43) shifts the expectation value of or/and , depending on , . We note
that ": $ does not change . If tr "' $ ? * , being the Hamiltonian of the system
without measurement, then the weak measurement with unitary back-action changes the expectation value of energy of the system. Figuratively speaking: Energy can be dissi-pated or pumped into the system depending on the components , of . We will refer to "' $ as “generalised friction”, because its structure is reminiscent of a term dis-covered in the context of Quantum Brownian motion, which describes the friction of a particle moving through a background of thermal particles [CL83, Di´o93, Vac00]. For an introduction to Quantum Brownian Motion see [GJK 96]. The term “generalised”
denotes that it is not always the expectation value of energy, which is changed by ":
$
, but in fact the expectation value of any observable with tr "' $ ? * . By choosing the appropriate positive and unitary parts of the measurement back-action the expectation value of any observable of the qubit can be shifted by means of ":
$ 7. For the sake of concreteness let us discuss the shift of expectation values at the example of energy. Oper-ations with general friction are unlike unitary transformOper-ations or minimal measurements of energy, which do not transfer energy in or out of the system in the following sense. A unitary transformation can increase the expectation value of energy for certain states but at the same time it would decrease the expectation value of other states. The net effect, when averaged over all states, is zero: the expectation value of energy stays the same.
The outcome of a minimal measurement of energy, on the other hand, is a random pro-cess. In the average over all outcomes the expectation value of energy will not change.
There is another interesting point about the term "'
$ all states. There is no eigenstate of as there are eigenstates of the operations corre-sponding to unitary transformations or minimal measurements. This is due to the fact, that the operation elements of
posses basises of eigenstates if and only if
% *
or
*
. This claim follows from the fact that the have to be normal in order to posses a basis of eigenstates (cf. [Ped89]), thus -* . The normality is equiv-alent to
*
, which in turn leads to the claim. Since the
are simultaneously diagonalisable, also the operation elements posses a common basis of eigenstates, if they are diagonalisable at all. So if there is a common basis of eigenstates of the operation elements the friction term vanishes because of
% *
or
*
. On the other hand if the friction term vanishes there does not have to exist a basis of eigenstates. If for example and thus * but ? * and % ? * , the friction term vanishes but there is no basis of eigenstates.
7This becomes clear by recalling that we defined the pseudo spin observable via the eigenbasis of the positive part .
146 Non-selective Discription of Unsharp Measurements with Unitary Backaction In order to have a geometrical picture of what happens let us look at the transformation
"'
$
(5.49) and its equivalent on the Bloch-sphere:
This mapping represents a translation of the Bloch sphere. Although the density operators are mapped under (5.49) on operators with trace equal to , these operators are in general not positive anymore, and therefore the mapping (5.49) is not an operation. This can be seen by observing that there always exist Bloch vectors of pure states (
) which are mapped under (5.50) for a finite translation onto vectors with a norm greater than :
Such vectors correspond to non-positive operators since they posses a positive and a neg-ative eigenvalue :
+
(5.52) For precisely this reason an operation has to correspond to a map of the Bloch sphere
onto or into itself, i.e.,
t containing the term ":
$
is of course positive and even completely positive by construction, because it is built up by operation elements (c.f. [Kra83, NC00]). As a consequence there are terms in t corresponding to a contraction of the Bloch sphere, such that the translated and contracted Bloch sphere still fits inside its pre-image. These terms are up to second order in the small quantities and
represented by those double commutators of Eq. (5.43) which do not contain from the original evolution8. For an exact identification of the contracting (decoherent) part of the double commutators containing see subsection 5.3.4. The translation of the Bloch sphere, i.e. the shift of the expectation value of observables, comes always together with a contraction of the Bloch sphere, which in our case is connected to decoherence.
This will become obvious in subsection 5.3.4, where the decoherent part of the operation due to is isolated. In figure 5.2 the action of the friction term on the Bloch sphere is illustrated and compared to the action of pure decoherence.
Generalised friction is a useful tool to manipulate statistical mixtures of states and drive them in the vicinity of a desired state, without making selective measurements and dis-carding samples. Therefore it is especially valuable in experiments with single quantum systems as in quantum computing, or if it is desirable not too dilute the volume of samples too much as in the cooling of quantum gases .
8
can be set to zero and the remaining operation is still positive.
5.3 An Elementary Cell 147
Figure 5.2: On the left: Contraction of the Bloch sphere due to decoherence caused by a minimal unsharp measurement of , i.e. . In order to illustrate the effect I used a relatively high accuracy of the measurement and . On the right: Translation of the Bloch sphere in -direction. The translation is generated by generalised friction (see above), which is connected to the unitary part of the back-action of the measurement. Here ". The parameters of the unsharp measurement of are
and . Along with the translation there comes a contraction of the Bloch sphere in - and - direction (cp.
decoherence terms containing
in (5.43) and an additional small contraction in and -direction due to the unsharp measurement.
where the components of vector
of eigenstates of . Obviously the double commutators with reduce the off-diagonal elements of written in the corresponding eigenbasises. This means decoherence with respect to these basises. But can unitary transformations like lead to decoherence or is this just an artefact of the approximation like in the case of the double commutator containing the Hamiltonian (cp. Eq. (5.43), which reduces the off-diagonal elements of as a second order correction to the first order term ? In fact decoherence is an option. As figure 5.3 illustrates, even an infinitely unsharp measurement ( % * ),
148 Non-selective Discription of Unsharp Measurements with Unitary Backaction where no information can be extracted and
does not change the state at all, can lead to decoherence, which is caused only by the unitary part of the back-action. If we neglect other influences than the unitary part of the back-action, the state suffers the transformation
% " % $ (5.54)
resulting in general in a proper mixture for
* %
, provided that
?
. It is therefore wrong to say that the change in the state motion can be expressed by making the transition AV, then there would be no dephasing as in (5.54). As a consequence difference equation (5.43) would show in second order in
only a double commutator AV
AV .
☞ The rest of the terms of difference equation (5.43)
➱ " $ "+ $ reduces the off-diagonal elements of the density matrix
with respect to the eigenbasis of Hamiltonian . This is a second order correction to the first order commutator in and does not lead to a net decoherence. More about the identification of decoherence in the presence of unitary development will be said in subsection 5.4.1.
➱ " "% " % $$ $ " $ '+ – second order decoherence with
respect to the eigenbasis of .
➱ AV "
$
– part of composition of original unitary evolution fol-lowed by a unitary evolution generated by AV.
➱ " $ " $ – part of the decoherence with respect to the
-eigenbasis of the in first order rotated .
➱ " % $ % " $ " $ – part of the first order term of
a rotation due to the Hamiltonians of the “ -decohered” state .
A lot of the terms of difference equation (5.43) are identical to those obtained from a composition of the original dynamics
followed by first order decoherence
d (cp. Eq.
5.45) and the unitary evolution
where we have neglected higher order terms. But even if t is a good approximation of the actual operation t, differences show up in higher order terms. Compare for example the last term of Eq. (5.55) with the last given in the preceding list. There is also no generalised friction in t. Generalised friction is a characteristic trait of non-minimal measurements.
5.3 An Elementary Cell 149
☞ Translation in z-direction
By a simple geometrical argument (cp. box below) it becomes clear that also translations of the Bloch sphere in z-direction, i.e. a shift of the expectation value of , can be generated by the unitary part of the back-action. These shifts do not appear in difference equation (5.43) because of the truncation of the expansion of operation . The following higher order term to appear in Eq. (5.43) would be
a
where the Hamiltonians of
have been expressed by
. a shifts the expectation values of all observables
of the qubit with tr ? * or tr ? * and thus adds to the generalised friction. The mapping a results in a translation of the Bloch sphere:
": The translation in z-direction is illustrated in figure 5.4.
150 Non-selective Discription of Unsharp Measurements with Unitary Backaction
Translation of the Bloch sphere by feedback
As we have seen in the context of the friction term which appears in the operation associ-ated with an elementary cell it is possible to shift the Bloch sphere in any direction of the x-y plane. There is a simple geometric interpretation of this translation. Let us consider a qubit operation with operation elements , where
and % % + * * % % + (5.58)
This can be interpreted as an unsharp measurement of represented by the operation elements
followed by turning on a unitary transformation which depends on the measurement result (feedback): for both results the Bloch vector is rotated about the x-axis but the sense of rotation is opposite for the results “ ” and “ ”. How does this lead to a translation of the Bloch sphere? Let us look at the image of the origin of the Bloch sphere under the operation. Because we are dealing with an affine map this image is the center of the shifted Bloch sphere. Both results , which occur for " * * * $ with the same probability%
'+ %
, lead to an image with the same y-component but opposite z-components (cp. Fig. 5.5 (a)):
result “ ”: "* * * $ Averaging over the two cases , the image of the origin yields only a non-vanishing y-component: The origin of the Bloch sphere can be shifted onto any point within the Bloch sphere by adapting the parameters
% and% of the measurement of and using just one rotation axis for both feedback transformations (cp. Fig. 5.5 (b)). Translations in z-direction are not represented in Eq. (5.43) because they are of higher order in the small parameters, namely "
$
. A discussion of the lowest order translation in z-direction can be found as last item of the list above this box.