2.7 Information Gain and Disturbance
2.7.7 Quantum State Estimation and the Balance between Information Gain
which is not related to the information gain, but rather to physical properties of the measurement apparatus.
2.7.7 Quantum State Estimation and the Balance between Information Gain and Disturbance
A common approach to investigate the relation between information gain and disturbance due to generalised measurements stems from the theory of state estimation in Quantum Mechanics.
Pioneering work on statistical inference and state estimation in Q.M. has been done by Helstrom [Hel76] and Holevo [Hol82]. Over the last decade this field has attracted much interest in quan-tum information theory in general and particularly in the context of quanquan-tum cryptography, see references in [Ban01]. In this subsection I want to sketch a particular state estimation method, illustrate it at the example of a sequence of N unsharp measurements (N-series) and briefly introduce to the view point estimation theory produces on information gain and disturbance.
In general it is not possible to determine the state a single quantum system possesses before a measurement (pre-measurement state) from the result of the measurement. Nevertheless the probability with which the result occurs depends on the pre-measurement state. Therefore the measurement result supports an estimation of the pre-measurement state. But in general also the state after the measurement (post-measurement state) is unknown and therefore subject to estimation. Only in case of a von Neumann measurement, where the state is projected onto an eigenstate of the observable, the post-measurement state is determined by the measurement result. For general measurements the state after the measurement has to be estimated if only the measurement result but not the initial state is known.
The study of state estimation usually involves three stages:
1. Find a measure for the quality of the estimate and a measure for the disturbance intro-duced to the system by the measurement on the result of which the estimate relies.
2. Find the optimal estimate with respect to the measure of quality of the estimate.
3. Compare the quality of the estimates to the disturbance for different measurements and try to establish a criterion for an “optimal” measurement.
It is not within the scope of this treatise to discuss the different existing measures (first item) and estimation methods (second item). I just want to mention two kinds of measures for the quality of the estimate: the probability of correct inference19, cp. [Bus97] and measures of distance between states as e.g. trace distance or fidelity [NC00]. Fidelity can serve as measure of quality and as measure of disturbance. It induces a particular state estimator which is discussed below.
19If there are two alternatives: the state belongs to a certain set of states or to its complement this measure can be employed. It originates from the test of statistical hypothesis by means of samples cp. chapter 14 of [Kre70].
82 Unsharp Measurements Stage 1: Estimation- and Operation Fidelity
The distance between the estimated state and the state to be estimated (index “ ” stands for “desired”) in terms of the Uhlmann fidelity is given by
": $ is invariant under a change of the order of arguments: ":
$/
it reduces to the squared “transition amplitude”:
"
$/
(2.136) Analogously, one obtains the distance between the state
before the measurement and the state
after the measurement (with measurement result ):
" $
(2.137) Since both – and – vary from measurement to measurement depending on the measurement result, it is advantageous to consider the mean fidelities gained by averaging over all possible outcomes of the measurement:
5
A further (weighted) averaging is sensible if more than one pre-measurement state can occur.
is a measure on the space of pure states which is invariant under unitary transformations and the probability density%8"
$
reflects the a priori knowledge about the pre-measurement state. The normalisation of the integration measure is such that
%&"
$@
. and are called the estimation fidelity and the operation fidelity, respectively.
In case the pre-measurement state is a totally unknown pure state and involve an inte-gration over the set of all pure states which are equally weighted. The fidelity of estimating the pre-measurement state then amounts to [Ban01]:
pre
" $ 5 " $ " $
with
(2.142)
2.7 Information Gain and Disturbance 83 where the
are the Kraus operators of the measurement and
represents the dimension of the Hilbertspace of the measured system. On the other hand the state immediately after the measurement can be estimated with the mean fidelity [ADK02b]:
post
5 " $
" $ (2.143)
And finally, the operation fidelity reads [Ban01]:
" $ 5
tr
(2.144)
Stage 2: Best Estimator
The second stage of the estimation process is now easily accomplished: The optimal estimate maximises the estimation fidelity (as a measure of the accuracy of the estimate). For the pre-measurement state one obtains as optimal estimator
" $ the eigenstate of which
cor-responds to the highest eigenvalue of . Analogously the optimal estimator
" $ for the
post-measurement state is given by the eigenvector associated with the highest eigenvalue of
Estimation- and Operation Fidelity of a N-series
Let us, for example, consider pre and of a N-series20 of unsharp measurements on a qubit.
By a simple transformation of the results of an N-series:
; +
%
%
(2.146) one obtains for the N-series in good approximation Gaussian effects
20cp. section 2.3.3. This example is of a particular importance since N-series play an important role in several contexts throughout this treatise.
84 Unsharp Measurements is the product of decoherence rate and duration
of a N-series. Here the range of the measurement results has been extended to the whole of . A derivation of which contains the details about the decoherence rate, the continuous approximation etc. can be found in section 5.4. In order to find an optimal estimation of the state before an N-series it is advantageous to look at the spectral decomposition of :
the eigenvalue corresponding to the eigenstate
* is maximal, otherwise the eigenvalue corresponding to the eigenstate is maximal. The estimator thus reads:
* ; *
;
otherwise (2.150)
The estimation fidelity (2.142) now amounts to
pre The operation fidelity (2.144) of a N-series can be calculated taking into account that an N-series is a minimal measurement, i.e.
( . In Fig. 2.7 both fidelities are plotted as function of the number of single measurements within a N-series. The values of and converge for to the values of a von Neumann measurement on a qubit: + .
Stage 3: Optimal Measurement
Once the question which is the best estimator has been decided one can ask which is the optimal measurement (stage three). The answer depends on the context. If e.g. the disturbance does not matter, von Neumann measurements lead to the highest estimation fidelities: post
the operation fidelity given, one could search for the measurement with the highest estimation fidelity and an operation fidelity
(
. This task could arise, e.g., for an eavesdropper who tries to obtain as much information as possible from a quantum channel without getting detected, which will happen if he inflicts state changes with
. In fact any weighting of the
”costs” represented by and the ”performance” represented by can be expressed in a trade-off function " $ . The optimal measurement is then the measurement which maximises . There is not a unique trade-off function which is superior to all others. Therefore there is neither a single measurement which is better than all other measurements. This can be seen at the example of the eavesdropper: his choice of measurement depends on how much risk of being caught he is prepared to take in order to possibly obtain certain informations. There are constraints between and pre [Ban01] and thus also between and post [ADK02b] which limit the performance/cost ratio. These constraints are illustrated in Fig. 2.8 and Fig. 2.9.
2.7 Information Gain and Disturbance 85
200 400 600 800 1000
0.5 0.6 0.7 0.8 0.9
F,G
N F(0.01)
F(0.05) F(0.1)
G(0.1)
G(0.05) G(0.01)
Figure 2.7: Fidelity of the estimation of the pre-measurement state and operation fidelity for a sequence of unsharp measurements on a qubit. Values in brackets represent the value of of the single unsharp measurement, e.g. F(0.01) means operation fidelity for
. All measurements with .
0.1 0.2 0.3 0.4 0.5 0.6 0.2
0.4 0.6 0.8 1 F
d=2 d=4
d=8 d=16
PSfragreplacements
pre
Figure 2.8: Maximal operation fidelity for a given estimation fidelity pre of the pre-measurement state in Hilbertspaces with dimension . Dashed lines mark the domain of possible combinations of and
pre for dimension .
86 Unsharp Measurements
0.2 0.4 0.6 0.8 1 0.2
0.4 0.6 0.8 1 F
d=2
d=4 d=8 d=16
PSfragreplacements
post
Figure 2.9: Maximal operation fidelity for a given estimation fidelity post of the post-measurement state in Hilbertspaces with dimension . Dashed lines: domain of possible combinations of and post for dimension .
Chapter 3
Unsharp Measurements and the Detection of Dynamics
3.1 Introduction
Is it possible to detect the dynamics of a single quantum system in real time? In this chapter we discuss this question at the example of a single qubit which performs Rabi-oscillations under the influence of a resonant external field. Our goal is to detect the Rabi-oscillations.
The measurement scheme which will be proposed involves a sequence of consecutive unsharp measurements (from now on also referred to as ”sequential measurement”).
Parts of the content of this chapter are published in [AKS01]. The version given here is updated and contains various supplements, a.o. about QND measurements, the Zeno effect for rotating observables, the decoherence rate due to a sequence of unsharp measurements and the statistics of sequential measurements.
Consider a qubit with basis - states * and under the influence of a periodic driving potential " $ . The resulting motion of the normalised state
*
(3.1)
involves oscillations of the probabilities
" $
and
" $
called Rabi oscillations. One could think of a single two-level atom in a trap driven by a resonant laser field or a resonant oscillating magnetic field. But also a quantum dot in a double well potential or a photon shared between two coupled microwave cavities (cp. chapter 4) undergo such oscillations. The usual way to measure the dynamics of
" $
employs many projection measurements of an observable with eigenstates
* and . For this purpose an ensemble is prepared in the initial state and a projection measurement is carried out at time on each member of the ensemble leading to the determination of
" $
. Repeating the procedure for different times one obtains the Rabi oscillations of
" $
.
This procedure fails if there is only one single quantum system available and the objective is to visualise its motion in real time. Imagine, e.g., there is a single two-level system performing
88 Unsharp Measurements and the Detection of Dynamics only once a hundred Rabi oscillations. Is there a method to record them in real time? There are two complementary conditions such a method must meet:
☞ The measurement must not disturb the system too much, so that its evolution remains close to the undisturbed case.
☞ On the other hand, the coupling of the measurement apparatus to the qubit should never-theless be so strong that the readout shows accurately enough the modified dynamics of the system (including the disturbance by the measurement).
This seems impossible to achieve by standard measurements which project onto an eigenstate of the measured observable. They disturb the system too strongly. For instance, the well known Quantum Zeno effect [MS77, IHBW90] shows that it is possible to detect perfectly the modified dynamics of a system at the price of loosing all information about the undisturbed motion. The measurement disturbs the dynamics so much as to make the system stay in one of the eigenstates of the observable.
In order to keep the disturbance of the system minimal the back-action of the measurement apparatus on the system should be as small as possible. Quantum Non-Demolition (QND) measurements ([CTD 80, BK92]) can monitor the evolution of the state of a single quantum mechanical system avoiding back-action of the measurement apparatus. The characteristic trait of QND measurements is represented by the fact, that the system at all times is in an eigenstate of the (ordinary) observable which is measured:
" $ " $ " $ ,# (3.2) time. In this case has to possess a continuous spectrum and therefore cannot represent an observable of a system with finite dimensional Hilbertspace1.
A sequence of projective measurements of the observables
" $ " $ " $
with
" $ " $ *
" $ (3.3)
which are carried out consecutively on a single quantum system lead in the limit of a continuous measurement2 for a large class of dynamics to the Zeno effect: the system is forced into the state
" $ . In case the undisturbed evolution of the system is already given by
" $ it is not
changed at all by the sequence of measurements. On the other hand these measurements cannot be used to distinguish between any of the undisturbed dynamics
" $ which are projected
onto
" $ . In this sense this sequential measurement scheme is trivial: it does not convey
information about which of these dynamics would occur without measurement. Therefore it is not suitable for our purpose to detect the undisturbed dynamics in real time.
The Zeno effect for rotating observables " $ " $ " $ " $ can be reduced to the usual Zeno effect by transforming into the rotating frame, with respect to which "
$
is
1see also appendix of [Per89].
2
and (where is the time between two consecutive measurements), such that const.
3.1 Introduction 89 constant. This can be seen at the example of a qubit by expressing the state of the system with respect to the basis given by
*
" $ which is constant in the non-rotating frame (given by
* and ), e.g.
" $
* , shows – being not measured– Rabi oscillations in the rotating frame:
are represented in the rotating frame by
;= * *
. The prob-ability %&" $
to project in consecutive, sharp measurements times on the state
*
where is the time between two consecutive measurements. In the limit of a continuous mea-surement one obtains (please note, that in this limit
Thus a continuous measurement of "
$
permanently projects the state of the system with unit probability onto
* " $ . The same happens for all state evolutions with
" $ " $
for (3.8)
and also for all observables with discrete spectrum in higher dimensional Hilbertspaces. This is the Zeno effect for rotating observables. Since all state evolutions satisfying (3.8) lead to the same measurement result they cannot be distinguished by this kind of measurement.
Our goal here is to find a measurement scheme which is sensitive to the proper motion of the system (i.e. the undisturbed motion) during the period of measurement. For this purpose it is advantageous to open the toolbox with generalised measurements (cp. chapter 1) and look for promising candidates. Since we want to detect Rabi oscillations which are represented by the evolution of
it appears natural to look for (unsharp) measurements of the observable
or equivalently of any observable which commutes with
. Unsharp measurements (cp. chapter 2) of the observable
are measurements with Effects with the property:
*
(3.9) for all measurement results “ ”. Another property of unsharp measurements is that pure states are transformed into pure states. This means there is no classical noise which disturbs unnec-essarily the state of the system. For the mathematical representation of the measurement the
90 Unsharp Measurements and the Detection of Dynamics purity of the operations implies, that to each measurement result there is exactly one Kraus operator, see below.
While unsharp measurements have been considered in many contexts3 the detection of dy-namics has mostly been studied by means of continuous measurements [Dav76, Sri77, Men79, Di´o88, Men93, Car93]. A comparison of different methods to describe continuous measure-ments in the selective and the non-selective regime is given in [POU96].
In the context of continuous quantum measurements of the energy of a two-level atom the question of visualisation of oscillations has been addressed in [AM97] in a phenomenologi-cal approach, which was based on the application of restricted path integrals [Men79, BLP82, Di´o88]. In [AM97] it had been indicated, that for continuous measurements a correlation may exist between the time-dependent measurement readout and modified Rabi oscillations between energy eigenstates. This visualisation of a state evolution has been numerically verified in [AMN97], to the best of my knowledge for the first time. Continuous adaptive measurements have been studied numerically in [AMK01]. The results there indicate that the trade off be-tween information gain and disturbance of the state evolution can be improved by choosing the parameters of the continuous measurement at time depending on the measurement outcome from preceding times
.
Oscillations in coupled quantum dots measured by a quantum point contact are treated in an approach to continuous quantum measurements using stochastic master equations by Korotkov [Kor00, Kor01b]. For a quantum trajectory approach see Goan et al. [Ge00].
A realization scheme for the continuous fuzzy measurement of energy by means of a se-quence of unsharp measurements and the monitoring of a quantum transition was studied in [AM98]. The intention was to present a microphysical basis for the phenomenologically moti-vated continuous measurement scheme. Inspired by this article an independent treatment which is solely based on successive single-shot unsharp measurements was given in [AKS01] and is repeated in this chapter. All concepts necessary to introduce an appropriate measurement readout, to justify this choice and to specify different measurement regimes are based on these single measurements. This is also the case for the numerical evaluation which makes use of the simulation of single unsharp measurements and the otherwise undisturbed dynamical evolu-tion (Rabi oscillaevolu-tions) between the measurements. An applicaevolu-tion of the measurement scheme presented here to a particular quantum optical setup is given in chapter 4.
This chapter is organised as follows: In Sect. 3.2 we specify the particular subclass of unsharp measurements on which our considerations are based. In Sect. 3.3 a succession of these unsharp measurements is studied. In addition we introduce the concept of a ’best guess’
based on the outcome of one series of consecutive measurements (N-series). In Sect. 3.4 a Hamiltonian is introduced generating the Rabi oscillations we want to measure by means of a series of unsharp measurements with time between two consecutive measurements. An
3A number of experiments containing unsharp measurements is described in [BGL95]. De Muynck and Hen-drikx [dMH01] investigated several atomic beam experiments related to the Ramsey interference experiment in the framework of generalised measurements. Unsharp measurements are there called non-ideal measurements of standard observables.
3.2 Single Unsharp Measurements 91