Maximum information gain in weak or continuous measurements of qudits: complementarity is not enough

1.9 MB·English

by  Joshua Combes

Maximum information gain in weak or continuous measurements of qdits:
complementarity is not enough
Joshua Combes1,2 and Howard M. Wiseman1
1ARC Centre for Quantum Computation and Communication Technology, and
Centre for Quantum Dynamics, Griffith University, Brisbane, 4111, Australia
2Center for Quantum Information and Control, University of New Mexico, Albuquerque, NM 87131-0001, USA
(Dated: January 20, 2013)
Tomaximizeaverageinformationgainforaclassicalmeasurement,alloutcomesofanobservation
must be equally likely. The condition of equally likely outcomes may be enforced in quantum
theory by ensuring that one’s state ρ is maximally different, or complementary, to the measured
observable. Thisrequirestheabilitytoperform unitaryoperationsonthestate,conditionedonthe
2
results of prior measurements. We consider the case of measurement of a component of angular
1
momentum for a qudit (a D-dimensional system, with D = 2J +1 where J is the total angular
0
momentum). For weak or continuous-in-time (i.e. repeated weak) measurements, we show that
2
the complementarity condition ensures an average improvement, in the rate of purification, of

n only 2. However, we show that by choosing the locally optimal control protocol of this type, one
a can attain the best possible scaling, O(D2), for the average improvement. For this protocol the
J
acquisition of information is nearly deterministic. Finally we contrast these results with those for

5 complementarity-based protocols in a register of qbits.
1

] PACSnumbers: 03.67.-a,02.30Yy,02.50.-r,89.70.-a
h
p
-
t I. INTRODUCTION mentary basis led to a speed-up in the purification by a
n
factoroftwoforatwolevelsystem[4]. (Practically,such
a
arapidpurificationprotocolcanbeaprimitiveforarapid
u In the classical world predictability is associated with
q intimate knowledge of a system while unpredictability state preparation protocol [6, 7].) Later this was gen-
[ eralized to complementary bases for D-dimensional sys-
implies surprise. In information theory, the surprisal of

tems. InRef.[8]the lowerboundonthe speed-upforD-
2 an outcome k is defined as I(k) log(Pk), were Pk is
≡ − dimensionalsystemswasfoundtobeS =(2/3)(D+1)
v the probability of that outcome [1]. The suprisal is an LB
when the monitoring was performed in an complemen-
1 important quantity as it quantifies the amount of infor-
6 mation one learns from an outcome of a measurement. tary basis. Recently in Ref. [9] the upper bound SUB
9 was proven to be bounded above by D2/2 for measure-
When the average surprisal of an experiment is maxi-
0 ment in any complementary basis.
mized the observermaximizes the extractionofinforma-
.
5 tion from the experiment.
0
In quantum theory the relationship between pre-
1
dictabilityandknowledgeisnotclearcut. Evenwhenan
1
: observer has maximal knowledge about a quantum sys- In this paper we give an explicit construction of an
v
tem (that is, the observer’s state of knowledge is pure), complementary measurement protocol for a qudit (a D-
i
X measurement outcomes may be unpredictable. This is dimensional system, with D = 2J + 1 where J is the
r the caseifmeasurementsonthe systemareperformedin totalangularmomentum)basedonthequantumFourier
a a basis which is not the eigenbasis of the state. (The transform (QFT) which achieves a speed-up of 0.2D2.
eigenbasis means the basis in which the operator un- Furtherwe showthattransformingthe state’s eigenbasis
der consideration, e.g. ρ, only has entries on the diago- sothatitiscomplementarywithrespecttothe measure-
nal.) Neverthelesspurequantumstatesarethemostpre- menteigenbasisisnot sufficienttoachievethis speed-up;
dictable states available. It is for this reason,and others the speed-up can be as low as 2. In addition to this we
eloquently explained in Ref. [2], that we take the impu- show, numerically, that unbiased bases can achive the
ritytobeameasureofinformation. Theseconsiderations upper bound, saturating the inequality S D2/2, for D
haveleadto information[2]andcontrol[3]theoretic for- even. To define these speed-ups rigorous≤ly, it is neces-
mulations of information–disturbance relations. sarytoderiverigorousboundsoninformationextraction
Inspired by this work,Jacobsaskedthe question: how rateforcommutingmeasurementswithoutcontrol,sowe
quickly on average can information be extracted from also do that in this paper. (For the rate of commuting
a quantum system using continuous quantum measure- measurements with control see Refs. [5, 7]). Finally we
ments in an complementary basis [4]. (Other ways of consider measurements in commuting and complemen-
quantifying the rate of informationgaincan alsobe con- tary bases for a register of qbits. The complementary
sidered [5, 6].) He found, for a widely applicable contin- measurement scheme based on the QFT appears only to
uous measurement model, that measuring in an comple- give a speed-up of 2 in the case of a register.
2
II. FURTHER BACKGROUND AND ments provide maximal information gain and maximal
STRUCTURE OF THIS PAPER disturbance through the following picture. Consider a
mixed qbit state with its Bloch vector aligned along the
In this section we explain the structure of our paper +z direction. By measuring in the x basis the Bloch
| i | i
by reviewing the relevant prior work on the subject and vector will elongate regardless of the measurement out-
relating it to the work presented in this article. come. ThispointisillustratedinFig.1(a). Furthermore
both outcomes are equally likely, which means the aver-
In the quantum information context, information–
age information extraction is maximal. Unfortunately
disturbance relations were inspired by the epistemic in-
the post-measurement state is not pointing in the same
terpretation of quantum states [10] as applied to quan-
direction as the pre-measurement state, so it will not
tum cryptography [2]. The idea was that the informa-
commute with the original state. Conversely if the mea-
tion gathering and disturbance should be groundedwith
surement basis and the state are along the z direction
respect to an observer’s state of knowledge, not some
| i
then, due to the stochastic nature of the measurement
pre-existingpropertyofthesystem. Toformalizethisin-
outcomes,theaverageelongationoftheBlochvectorwill
tuitionFuchsandJacobsinRef.[2]consideredsomething
be minimal [6, 11]. In any particular measurement the
like the following game. Two observer’s Alice and Bob
impurity may increase or decrease. But the disturbance
initially agree that the state of some quantum system is
tothestatewillbeminimal[2]. InFig.1(b)weillustrate
ρ , animpure state. Atsomelatertime Charliewillper-
0
that one of the possible measurement outcomes causes
form a measurement along a randomly chosen axis and
the Bloch vector to shrink whist the other elongates it.
askAliceandBobtopredicttheoutcomeofthemeasure-
ment. After repeating this many times Charlie rewards
z z
the individual who makes the most correct guesses. Al- (a) (b)
ice, being clever and resourceful, has the ability to per- M
+
form one unitary operation followed by a two-outcome
M
-
measurement. Her aim is to refine her state of knowl-
edge about the quantum system, thus increasing its pre-
M M x x
- +
dictabilityandaccordinglythelikelihoodofreceivingthe
money. However, Alice’s morals prevent her from inten-
tionally sabotaging Bob for the pecuniary reward. Thus
Alice would like to increase the accuracy of her predic-
tions without affecting those of Bob.
FIG. 1: A Bloch sphere depiction of the extremes of
Thisformulationsuggests(intuitively)thatweakmea- information–disturbance tradeoffs. In both figures the dark
surements are the right way to probe the system. From arrow represents the Bloch vector of the pre-measurement
the point of view of the weak measurement formalism, state. The arrows labeled with M± correspond to the post
aprojectivemeasurementcorrespondstoameasurement measurement states conditioned on the random outcome of
of infinite strength [2]. For finite strength measurements a weak measurement. Figure (a) is measurement of a state
in a complementary or unbiased basis, in this case the mea-
on a single qbit it was found that there is a non-trivial
surement axis is along the x direction. When this occurs the
tradeoff between Alice’s information gathering actions
Bloch vector always elongates, as indicated by the light grey
(therefinementofherpredictability)andthedisturbance
arrow(theplusoutcomerotated foreasyvisualcomparison).
to Bob’s predictability (his state of knowledge).
Figure (b) depicts thecommuting measurement. In this case
For a fixed measurement rate the interval of the non- if the measurement result M− is obtained the Bloch vector
trivial tradeoff may be characterized by the angle be- decreases in length.
tweentheBlochvectorandthemeasurementaxis. There
are two extremes of this tradeoff. Minimal disturbance
and minimal average information gain occurs when Al- While information–disturbance relations are interest-
ice’saxisisalignedwiththeBlochvector,thatisthemea- ing for single measurements the subject has been thor-
surement eigenbasis and the state’s eigenbasis commute. oughly explored. Less exploration has been undertaken
Maximal disturbance and maximal average information in continuous measurement setting due to the increased
gain, is attained when Alice’s axis is orthogonal to the difficultlyoftheanalysis. Inordertomakesomeprogress
Bloch vector. This strategy corresponds to making the the focus on disturbance has mostly been ignored (for
outcomes equi–likely, that is maximizing the suprisal [1] good reasons as we shall see). That is, the focus has
ofthemeasurementresult. Inthissituationthebasesare been on information gathering. This restriction has led
saidtobecomplementary,maximallynon-commuting,or toaninterestingdevelopmentinthetheoryofcontinuous
unbiased. Thus we conclude that quantum theory does quantum measurements [12–14] .
notallowAlicetosucceed. Theabilitytomeasureindif- To demonstrate an improvement in the extraction of
ferentbasesisanessentialfeatureofquantummechanics informationthe extractionrate of a standardcontinuous
which provides the richness to the problem considered measurementmustbedefined. We followJacobsandde-
here. fine the standard measurement as a measurement in a
Itiseasytounderstandhownon-commutingmeasure- commuting basis [4]. This is sensible for a number of
3
reasons. Firstly measurement will rapidly destroy the inginanycomplementaryis(2/3)(D+1)[8]. Thesecond
coherences of any state that does not commute with the study showedthis speed-up is upper bounded by a factor
measuredobservable[15]. Thiscanbeunderstoodasthe D2/2 [9]. We summarise these results in Sec. IVA. In
measurementprojectingthestatetotheeigenbasisofthe Secs. IVB to IVD of this paper we give an explicit con-
observable. To counter this effect an adaptive measure- struction of an unbiased basis protocol for an arbitrary
mentorclosedloopfeedbackcontrolisrequired. Thesec- finite dimensional system based on the quantum Fourier
ondreasonis acontinuousmeasurementina commuting transform. We derive upper and lower bounds on the
eigenbasis is essentially a classical continuous measure- speed-up of the QFT protocol. Surprisingly the numer-
ment. ics show that the protocol saturates the upper bound
The analysis of disturbance and information gain is – for even D. For completeness we mention the recent
significantly harder for continuous measurements of D- workofRuskovetal. [21],whereitwasshownthatmon-
dimensional systems [8, 16]. Minimal disturbance still itoring a qbit in three complementary bases where they
occurs when measuring in the same eigenbasis as the claimed a speed-up S 3. We are currently exploring
∼
state (a commuting basis). A full analysis of this sit- this relationship so we will not discuss it further in this
uation would require characterizationof the distribution article.
of information extraction rates, allowing calculation of Motivated by some of the above investigations Jacobs
the average information extraction, median, worst case and Lund argued that the task of state stabilisation is
and best case. In Sec. III we rigorously derive the av- bestaccomplishedbymeasurementinancomplementary
erage purification rate for a commuting measurement of basis; in the regime of strong feedback [19]. This is one
a D-dimensional system and provide an asymptotic ex- reason to continue investigations into maximal informa-
pression. Additionally we give exact results for bounds tion extraction, foundational issues aside. In Ref. [19]
onthespreadofthedistributionofpurities,inSec.IIID, the authorsraisedtwointerestingquestions: first,areall
which were heuristically derived in previous work [16]. complementary bases are equally good for information
Now that we have defined what a standard measure- extraction when D 4? Second, is measurement in a
≥
mentiswecanexplorewhatitmeanstoincreasetherate complementary basis is best for information extraction?
of extraction of information. In the context of control We reexamine these questions in Sec. V and Sec. VI.
theory one might be interested in revealing or acquir- We show that complementary, i.e. equi-likelihood con-
ing the value of some parameter; that is the domain of dition, is not sufficient to guarantee maximum informa-
adaptive measurements [14, 17]. But more typical con- tionextraction. Ouranalysisprovidesaninsightintothe
trol objectives are to drive the evolution of the state or mechanismforinformationextraction,whichisquantum
to stabilize the state against noise [14, 16, 18–20]. Thus mechanical backaction. From this we explain how maxi-
disturbing the prior state is not a problem. This why muminformationextractionis achievedby enforcingthe
disturbanceisnotusuallyconsideredinfeedbackcontrol. complementary condition and arranging the eigenvalues
What becomes important is the noise introduced by the inphasespacesomeasurementbackactioncanhavemax-
measurementto the system. Doherty, Jacobsand, Jugn- imum effect. We provide further evidence for the claim
man in Ref. [3] showed, for a continuously monitored that not all bases are equally good. Because Jacobs and
qbit, that the measurement noise is maximal when the Lund did not examine permutations of the unbiased ba-
eigenbases of the state and measurement axis are unbi- sis it was still unclear from their analysis whether all
ased. unbiased bases are equally powerful for information ex-
traction. So we also consider the role of permutations in
ThestudyofWiseman,ManciniandWanginRef.[20],
Sec.VI. Our results showthat evenafter maximizing in-
exploredstatestabilisationinaqbit. The relevantresult
formation extraction over permutations of the unbiased
from Ref. [20] to the present discussion relates to the
basis, not all unbiased bases are equally good at infor-
difficulty of stabilising a state which is complementary
mation extraction.
to the measurement basis. They showed that this can
In Sec. VII we examine complementary measurements
be achieved in the limit of infinite strength “Bayesian”
in a register of qbits using the QFT. We start our anal-
feedback, but not in simpler “Markovian” feedback pro-
ysis,in Sec. VIIA, with the case ofcommuting measure-
cedures. This is a consequenceofthe increasedmeasure-
ment noise as predicted by Doherty et al. [3]. Later Ja- ments [9]. Then, in Sec. VIIB, we review the results of
Ref.[9]whereitwasshownthatthespeed-upboundedby
cobs showed how continuously monitoring the state of a
2n(2n 1) 1 S 2n. InSec.VIICweconstructapro-
qbitinacomplementarybasis(assuminginfinitestrength −
− ≤ ≤
tocolbasedontheQFTandpresentresultswhichsuggest
feedback)leads to a doubling rate of informationextrac-
the speed-up is at most 2. We conclude in Sec. VIII.
tionovermonitoringinacommutingbasisasymptotically
[4].
Recentlyanumberofstudieshaveexaminedtheuseof
weak(andcontinuous)measurementina complementar- III. COMMUTING MEASUREMENTS
itybasisinsystemsofdimensionDgreaterthantwo. The
first study showedthat the lower bound on the improve- Later in this article we will investigate strategies that
mentin the rateofinformationextractionwhen measur- increase the rate at which information can be extracted
4
fromquantumsystems. Beforethesestrategiesareexam- Wewilldenotethecontinuousmeasurementrecordob-
ined, we must fully characterize this rate in the absence tained by the observer integrated up until time t as
of feedback. This will provide a benchmark to assessthe
t
performance of the strategies that aim to increase the
R(t)= dR(t′). (3)
rate of information extraction. Z
0
In this section we restrict our attention to continuous
TheunnormalisedversionofEq.(1)isknowasthelinear
measurement (CM) of a single D-dimensional quantum
SME. We denote this linear SME as [14, 30–32]
system, also called a qdit, monitored by a single output
channel. Thisoutputchannelismonitoredinsuchaway
dρ¯=dt¯ [X]ρ¯ 2γdt [X]ρ¯+ 2γdR ¯[X]ρ¯. (4)
that it results in a diffusive quantum trajectory [12, 14]. Lc ≡ D H
p
This can be thought of as a sequence of short-duration, Here ¯[A]ρ Aρ+ρA , andthe baroverρ denotes the
†
weak measurements. Following Fuchs and Jacobs[2] our H ≡
lack ofnormalizationat all but the initial time. We take
notion of information is characterised by the impurity: the initial state to be ρ¯ = ρ(0) = I/D where I is the
L[ρ(t)] 1 Tr[ρ(t)]. Accordingly we derive an exact
D D identity matrix. Because the initial state matrix
≡ −
expression for the average impurity L(t) and a simpler ×
commutes with X at all times this makes obtaining the
h i
approximate expression in Sec. IIIA. In Sec. IIIB and
solution to Eq. (4) simple. This solution is [24]
Sec. IIIC we examine the validity of our approximate
expression. To completely characterise the rate of in- ρ¯(R,t) = exp( 4γX2t)exp(2 2γXR(t))I/D. (5)
formation extraction from commuting measurements we −
p
lookatboundingthedistributionofpuritiesinSec.IIID. Usingthelineartrajectory(seeRefs.[30–32])solution,
ThenfinallywesummarisethekeymessagesinSec.IIIE. i.e.Eq.(5),wenowexplicitlycalculatetheformoftheD-
dimensional state matrix. We take the observable X to
be the z component of angular momentum (represented
by the operator J ), thus D = 2J +1. In matrix form
A. Qdit - impurity analysis z
the J operator is J = diag(J,J 1,..., J +1, J).
z z
− − −
UsingthisrepresentationofJ theunnormalizedsolution
z
Thestartingpointforthecurrentanalysisissimilarto
of the state evolution can be written as matrix elements
that presented in Refs. [8, 9, 22]. We use a widely appli-
exponentiated:
cablemodelthatdescribesthe evolutionofthe stateρof
a quantum system due to repeated weak measurements
of a dimesionless observable X. This model is described ρ¯(R,t) = 1diag e 4γJ2te2√2γJR,
−
by the stochastic master equation (SME) [14, 23–25] D (cid:16)
e 4γ(J 1)2te2√2γ(J 1)R,
− − −
dρ=dt [X]ρ 2γdt [X]ρ+ 2γdW [X]ρ, (1)
c
L ≡ D p H ,e 4γ( J)2te 2√2γJR . (6)
− − −
where D[A]ρ ≡ AρA† − 12(A†Aρ + ρA†A), H[A]ρ ≡ ··· (cid:17)
Aρ+ρA† Tr (A†+A)ρ ρ[14],anddW istheincrement The normalization factor can be written with a non-
−
of a Wiener n(cid:2)oise proces(cid:3)s [26]. It should be noted that symmetric sum or a symmetric sum
we have moved to a frame that has enabled us to factor
γou,tdettheermHianmesiltthoeniraanteeavtolwuthiiocnh.inTfohremmateiaosnurisemexetnrtacrtaetde,, Tr[ρ¯] = 1 2J e−4γ(J−m)2te2√2γ(J−m)R (7)
N ≡ D
andthustherateatwhichthesystemisprojectedontoa mX=0
mτsine≫galseγu−reei1gmewenensttmaoateyf otshafyeXtoh[b1as6te,rw2vea7–bh2lae9v]eX.p(.eT)rhfoTisrhmmeeedmanaesapsfruoorrjeetmcimteivneest = D1 sX=J Je−4γs2te2√2γsR. (8)
−
result in a small time interval [t,t+dt) is denoted
Fordiscussionslaterwealsocalculatetheprobabilitydis-
tribution of the result R and choose the symmetric sum
dR(t)= 4γ X(t) dt+dW(t), (2)
for the normalization
h i
p
wheredW isthesameWienernoiseprocessthatappears dR (R) = dRTr[ρ¯(R,t)]P(R) (9)
P
in Eq.(1), and X(t) = Tr[Xρ(t)]. A quantum trajec-
J
toryisacontinuhous–iin–timedescriptionofthestatecon- = dR 1 e (R 2√2γst)2/2t. (10)
− −
ditioned on the measurement result. Thus, to specify a D√2πtsX= J
trajectory over the interval [0,T) we must be given the −
measurementresultsoverthatinterval. ThentheIto¯def- HereP(R)hasbeencalledthe ostensibledistributionfor
initionρ(t+dt)=ρ(t)+dρ(t)maybeusedrecursivelyto R [14, 31]. It is given by P(R,t)= exp( R2/2t)/√2πt.
−
generateatrajectory. Clearlyasingleweakmeasurement By choosing the symmetric summation for the normal-
is described by one application of this relation. izationonecanseethattheprobabilitydistributionforR
5
contains D peaks centred around the values s, where 0.6
s [J,J 1,..., J+1, J]. Fortimest γ∝1,thedis- V = -1±½ Region I
−
∈ − − − ≫
tribution is sharply peaked about these D values. Each Region II
y
of these peaks has a FWHM of 2√2ln2√t 2.35√t. nsit0.4
≈ e
This seems paradoxical; one would expect that as time D
y
increases the distribution would get narrower, reflecting blit
the fact that the observeris more confident about which oba0.2
eigenstate they actually have. This paradox is resolved Pr
when one realizes that the distance between the peaks
increasesata rategreaterthanthe increaseinthe width
0
ofthepeaks. Thistime-dependentscalingofthedistance
-3 -2 -1 0 1 2 3
betweenthe peakscanbe removedbychangingvariables V
to V R/(2√2γt). Under this transformationthe prob-
≡
ability distribution is FIG. 2: The dashed curve is the probability distribution
P(V), Eq. (11); the solid line is Λ(V,t), Eq. (14); shown for
dVP(V) = dV D1 r4πγt J e−(V−s)2/(1/4γt). (11) mDu=ltip5liaetdtby=240γ(−so1.thTahteitdcoatntebdecvuiervweedisoΛn(tVh,et)sNam(Ve)sPca(lVe)).
sX= J Alsoplotted areregions forthetwoeigenvalueexpression for
− theimpurity integral.
Now the FWHM of each peak scales as ln2/(γt)
≈
0.83/√γt, which clearly illustrates that thpe probability
distribution becomes sharplypeaked about the D values
for long-times. In Fig. 2 the dashed curve is a plot of expression for L(t) in the D-dimensional case. Now
h i
Eq. (11) for D = 5 and t = 4γ 1. All of the peaks are we will make some approximations to obtain a simple
−
clearlydistinguishablewithameasuredFWHM 0.418. analytic expression for Eq. (15). We will show that the
This agrees with the prediction given by 0.83/√≈γt. approximations allow one to place a bound on the full
Returningtothe non-symmetricsums,i.e. Eq.(7),for expression. Then we will confirm our approximations
the remainder of this calculation, the purity is with numerical simulations in Sec. IIIB and Sec. IIIC.
The first approximation involves truncating the state
2J matrix to two eigenvalues and then renormalizing. We
Tr ρ2 = 1 e−8γ(J−m)2te16γt(J−m)V.(12) call this the two eigenvalue approximation. The moti-
D2 2
(cid:2) (cid:3) N mX=0 vation for the truncation stems from the following ob-
servations. In the long-time limit (t γ 1) the true
The impurity, for a particular value of V, is given by ≫ −
probability distribution, (V) = (V)P(V), is sharply
P N
2J e 8γ(J m)2te16γt(J m)V peaked in D places; see Fig. 2. Also in this limit one
Λ(V,t) = 1 m=0 − − − (13) finds that Λ(V,t), Eq. (14), is sharply peaked in (D 1)
(cid:18) − P D2 2 (cid:19) places; the peaks are between peaks of (V). Wisem−an
N
D2 2 2J e 8γ(J m)2te16γt(J m)V andRalphhavepointedout,foraqbit,tPhatthisarrange-
= N − m=0 − − − . ment leads to poorly purifying trajectories dominating
(cid:18) P D2 2 (cid:19)
N L(t) [6]. In a qbit the poorly purifying trajectories are
(14) h i
those trajectories with eigenvalues of the same magni-
tude (V(t) 0) in the long-time limit. Physically this
The solid curve in Fig. 2 is a plot of Eq. (14) for D = 5 ≈
means that the measurement has not projected, or the
and t = 4γ 1. To find the average value of impurity for
− filter has not decided, if the state is in z = 1. In qdits
a continuous measurement of Jz one must integrate this there are (D 1) regions like this, where V±(2t) r for
functionoverthescaledmeasurementrecordV,weighted r [J 1,J −3,...,J+ 3, J+ 1]. In any of th≈ese re-
by P(V) [8]: ∈ − 2 − 2 2 − 2
gionsonly twoeigenvaluescontribute significantly to the
integrand. Consequently it is reasonable to truncate the
state to the two eigenvalues in a particular region and
∞
L(t) = Λ(t,V) (V)P(V)dV. (15)
h i Z N renormalize.
−∞ The effect ofthe two eigenvalue approximationis two-
When using this result to calculate the measurement- fold: it increases the purity of the state, and makes it
induced evolution of the mean linear entropy one should possible to derive a fully analytic expression for the im-
use the kernel in Eq. (14) rather than the kernel in purity. The eigenvalues of truncated state matrix are
Eq. (13). This is because it will reduce the numerical (λ ,λ )/(λ +λ )[58], where λ is the largesteigenvalue
0 1 0 1 0
error that occurs when subtracting from one a number of ρ and λ > λ when a < b. It is easy to show that ρ
a b
very close to one. is majorized [33] by ρ (that is, ρ ρ ), meaning that
2 2
≺
To calculate this integral one must perform numerical the originalstate is more mixed than our two eigenvalue
integration. It would be useful to have a fully analytic approximation to it. From the fact that the purity is
6
Schur-convex [34] it follows that Tr ρ2 Tr ρ2 . This Ascouldreasonablybeexpectedforaqditunderthetwo-
≤ 2
means the impurity of the two eigen(cid:2)valu(cid:3)eappr(cid:2)oxi(cid:3)mation eigenvalueapproximation(effectivelyatwo-levelsystem),
is a lower bound on the true impurity: L(ρ) L(ρ ). theresultingaverageimpurityisproportionaltotheqbit
2
≥
In the long-time limit it is reasonable to expect that impurity with the proportionalitydepending on the qdit
L(ρ) L(ρ ) from the arguments above. dimensionality. In Sec. IIIB we examine the accuracy
2
≈
Now we calculate the impurity of the truncated state and validity of this approximation.
matrix L (t) . We split the integral in Eq. (15) into
2
h i
regionssuchthatthesametwoeigenvaluesarethelargest
two eigenvalues for all values of V in each region. The B. Comparison with numerics
integrationisthenperformedineachoftheseregionsand
then the regions are summed. Figure (2) illustrates how In Fig. 3 (a) one can see that for t & γ 1 the impu-
−
the regionsshouldbe split forD =5. After a coordinate rity, found by numerically evaluating Eq. (15), is better
transform(for example in regionI it would be V′ =V approximated by Eq. (18) than Eq. (21). Moreover, for
−
1/2) the integral in region one becomes all times the qbit impurity Eq. (21) is a lower bound on
the true impurity. However,when calculating the speed-
RI = e−Dγtr4ππtZ 121 coesh−4(γ4tγVt′V2 )dV′. (16) upplototnede iins iFnitge.re3st(ebd).in times where γt ≫ 1, which are
′
−2
After a similar transformation the integral in region II
becomes
R = e−γt 4πt ∞ e−4γtV′2 dV . (17)
II D r π Z 1 cosh(4γtV ) ′
′
−2
The total integral is thus
L (t) =2R +(D 3)R . (18)
2 II I
h i −
Fort γ 1theintegrandsbecomesharplypeakedabout
≫ − FIG. 3: A comparison of approximations to the evolution of
V =0withnegligiblecontributionstotheintegralsfrom
′ hLi, for D = 3. (a) (starting from the upper-most curve):
theregionoutsidetheintervalV [ 1,1],sothatR
R . In this limit we have ′ ∈ −2 2 II ≃ hLi i.e. Eq. (15) shown by green circles; hL2i i.e. Eq. (18)
I with the integrals evaluated numerically, shown by magenta
upward pointing triangles; hLi i.e. Eq. (21) shown by
e−γt 4πt 12 e−4γtV′2 solid black line. (b) (starting fqrboimt the upper-most curve):
L (t) (D 1) dV . (19)
h 2 i≃ − D r π Z 1 cosh(4γV ) ′ hL2iLT i.e. Eq. (20) shown by magenta downward pointing
′
−2 triangles; hLi i.e.Eq.(15) shown bygreen circles; hLiqbit−LT
i.e.Eq.(22) shown bydashedblack line;hLi i.e.Eq.(21)
The final approximation is arrived at by noting that qbit
shown by solid black line. Eq. (18) is not plotted in (b) be-
in the long-time limit the distribution in the nu-
cause on thisscale it is indistinguishable from hLi.
merator is broad compared to the distribution in
the numerator for t γ 1. Thus we can make
−
the approximation: ≫∞dVe−4γtV′2/cosh(√2γV) Eq. (18) is not plotted in Fig. 3 (b) because on
≈
∞dV/cosh(√2γV) R=−∞π/√2γ. The final expression this scale it is practically indistinguishable from L
fRo−r∞the impurity is thus at long-times (it approaches L from below). Bhuti
h i
L (t) , that is Eq. (20), approaches L from above.
2(D 1) πe γt h 2 iLT h i
L (t) = − − . (20) Thus the approximations made to obtain Eq. (20) from
2 LT
h i D √16γtπ Eq. (18) increase the impurity. This effect is eas-
ily understood once one realizes that the approxima-
This is the analytic expressionfor L(t) that we set out
h i tions amount to: throwing away parts of the integral
to find, and is the key result of this section.
Eq.(19);andover-estimatingtheintegralbyapproximat-
Because previous work has used the exact qbit results
ing exp( 4γtV 2)=1. The next closest curve is that of
we write it out for future reference: − ′
the qbit long-time limit expression Eq. (22), and below
e−γt 4πt ∞ e−4γtV′2 that is the qbit exact result Eq. (21).
hL(t)iqbit = 2 r π Z cosh(4γtV )dV′. (21) From these curves we infer the following: the long-
′ time expression for the two eigenvalue approximation,
−∞
The long time limit of this expression is L2(t) LT, is closer to L than the qbit long-time
h i h i
limit; L (t) is approaching L in the same way
2 LT
hL2(t)iqbit−LT = √π1e6−γγttπ. (22) thaLn(tt,)iqabhsit−prLeTviiiosuaslpyprionacthhienglitheLra(tth)uirqeibiti.t hTahsisbeisenimcpoomr--
7
monto usethe qbitlong-timelimitexpressionasalower
bound for L any dimension [5, 8]. It is clear that for
h i
all times t γ 1, L (t) is a better approximation
− 2 LT
≫ h i
to L than the qbit expression. Of course asymptoti-
h i
cally the expressions decay at the same rate. Therefore
thereisnothinginthisworktosuggestthatthespeed-up
calculated in previous works are incorrect.
C. Stochastic simulations
We now compare the analytic solutions from Sec-
tion IIIA and the exact numerics from Section IIIB, to
stochastic non-linear trajectory simulations of Eq. (1).
We used an Euler integrationmethod with the following
parameters: δt = 1 10 4γ 1; D = 5; ensemble size
− −
×
=20. The ensemble size is small because we will plot all
of the trajectories so that we may gain some qualitative
understanding from them.
Figure 4 (a) shows that the L of all of the trajectories
isboundedfromaboveby 1 fort γ 1. Intuitivelythis
2 ≫ −
effectcanbeunderstoodfromthetwoeigenvalueapprox-
imation; in the long-time limit a poorly purifying trajec-
tory has two eigenvalues and thus these trajectories are
upper bounded by L = 1. A more rigorous explanation FIG. 4: Short time impurity dynamics for D = 5. We plot
2
is givenin Sec. IIID. Onthis Figurewe alsoplotthe nu- 20 impurity trajectories (grey); the ensemble average of the
merically calculated ensemble average impurity and the trajectories (blue); the error bars for the ensemble average,
±σ , are smeared around the mean (light blue). Time is
linear trajectories solution, Eq. (15). mean
measured in units of γ−1. Fig. (a) Linear abscissa. The
There is some difficulty in obtaining convergence for
dashed and dot-dashed lines are approximate upper bounds
L in the stochastic simulations, even with very large
h i on the L trajectories; see Sec. IIID for further explanation.
ensemble sizes. The convergence problem is due to a
The solid black line is the linear trajectories solution for the
small number of poorly purifying trajectories first noted qbit, that is Eq. (21). The circles are hLi for D = 5, i.e.
for a qbit in Ref. [6]. These poorly purifying trajectories Eq. (15). Fig. (b) the impurity is plotted on a logarithmic
are evident in Fig. 4 (b) in that many trajectories touch axis;thishelpstoseetheapproximatelyexponentialdecrease
the dot-dashed line at L = 1. For moderate ensemble in impurity predicted by Eq. (20). The approximate upper
2
sizes the whole region of possible impurities is filled by boundonthespreadofLtrajectoriesisthedotdashline. The
the ensemble even at t = 2γ 1. As noted above the dashedlinecorrespondstothemostlikelytrajectories. Below
−
the solid (magenta) line there is only 1/D of the ensemble
major contributions to L are from the regions where
h i of trajectories at long-times. These lines are explained in
V = r (see Fig. 2); that is, for results V that are as far
Sec. IIID.
aspossiblefromthe mostlikelyvalues ofV [thepeaksof
(V)]. Thankfully one can use Eq. (15) to calculate the
P
evolution of the mean impurity. The advantages of this
method are two fold: the computation time is greatly it is easy to obtain an analytic expression,but the exact
reduced, and it is exact, to numerical precision. formoftheexpressiondependsonD. Thereforewequote
Figure 4 (b) shows that many trajectories also only the long-time asymptotic scaling of the bounds.
touch the dashed line corresponding to a scaling L Centralto our explanationof the bounds of the distri-
∼
exp( 4γt). The reason for this clustering, and that at butionoftrajectoriesistheintuitiongainedbyexamining
L= −21, will be explained in Sec. IIID. Fig.5,wherewe havereplottedportionsofFig.2 ontwo
separateaxes. InFig.5(a) (V,t)isplotted. Thisprob-
P
ability density plot is significant in the current analysis
D. The distribution of impurities because it shows which records, V, are likely (the peaks
of ) and which are unlikely (the troughs). In Fig. 5
P
In Ref. [16] Stockton, van Handel, and Mabuchi gave (b) the kernel Λ(V,t), that is Eq. (14), is plotted on a
reasonablebutnon-rigorousargumentsforabound[their log scale. The key features of this plot are: 1) the peaks
Eq. (41)] on the degree of spreading of trajectories (that at V = 3, 1,1,3 which represent very impure tra-
−2 −2 2 2
is, on the width of the distribution of trajectories) for jectories and correspond to unlikely records, and 2) the
the model of Eq. (1). We now use linear trajectories to troughs at V = 1,0,1 coincide with some of the purest
−
rigorouslyderivetheboundsthattheyfound. Inallcases trajectoriesandcorrespondtominimaof (V,t). Wewill
P
8
elaborate on other relevant features of this graph below 2. A pseudo lower bound: When V(t) = s for s
∈
as each bound is explained. [J 1,..., J +1], the record corresponds to the inner
− −
peaksoftheprobabilitydistribution,asdepictedinFig.5
(a) (the peaks at J are not included in this analysis).
nsity 0.4 (a) These peaks of (±V) are the most likely records and co-
e 0.3
D P
bablity 00..12 The inner peaks iinncFidieg.w5it(hb)s.omFoeroDf t=he5ptuhreesptetarkasjeocftoinriteesr,estthearmeitnhimosae
o
Pr 0 corresponding to V = 1,0,1 in Fig. 5. By substituting
−
V(t) = s into Eq. (6), normalizing, and then calculat-
100
y 10−3 (b) ing the impurity, one obtains the lower bound without
purit 10−6 any approximations. These solutions can be obtained
Im 10−9 for any dimension D. By making a two-eigenvalue ap-
10−12 proximation one can solve the evolution of the impurity
analytically giving
-3 -2 -1 0 1 2 3
V
L exp( 4γt). (24)
LT
∼ −
FIG. 5: (a) probability distribution P(V,t), Eq. (11), for
D = 5 and t = 4γ−1. The shaded regions of the probabil- Equation(24)correspondstothe lowerboundquotedby
ity distribution contain 1/D of the total probability at long- Stockton and coworkers [16]. In their case, their bound
times. (b)thesolidbluelineisΛ(V,t),Eq.(14),plottedwith appeared to be a true lower bound as their initial state
logarithmic scale on the abscissa for D = 5 and t = 4γ−1. was a coherent spin state (a collection of D/2 spin one
The red dashed line is a projection onto the abscissa of the half particles where D = 2J +1 with mean spin vector,
minimaofΛ(V,t)atV =−1,0,1. Theredsolid lineisapro-
J,alignedalongthex-axisinthiscase). Thishasasmall
jection onto the abscissa of the Λ(V,t) evaluated at V =±2;
population in J , and it is therefore unlikely in their
noticethatthisprojection ispurerthanthedashedline. The |± i
case that V > J . For our analysis we see clearly that
shaded regions in this figure depict the impurity of records | | | |
the bound is not a true lower bound (see Fig. 4 (a)),
that have |V|≥J. As these regions concur with the regions
in Fig. (a) they can be interposed as saying that only a pro- because a maximally mixed state has equal populations
portion 1/D of the trajectories will have impurities smaller inalleigenstates. However,becausetheserecordsV(t)=
than those given bythe projection Λ(±J,t). j are the most likely records many of the trajectories
touch or cluster around this bound. This will be further
explored in point 4. below.
1. The upper bound: This is obtained by substituting 3. Physically likely trajectories: In our case there is
V(t)=r(recallthatr [J 1,J 3,...,J+3, J+1]) no lower bound, although a trajectory that purifies in-
into Eq. (6), normalizi∈ng, a−n2d th−en2 calculatin2g−the i2m- finitely fast is infinitely unlikely [59]. It is possible to
purity. This procedure gives the upper bound because givea naturalbound onthe physically likely trajectories
these records correspond to the most impure trajecto- basedonthe probability distribution (V). In the lower
P
ries, that is peaks in Fig. 5 (b). Because this procedure bound section above we did not include the peaks of the
also corresponds to a a worst case scenario of the two probability distribution found at V = J. It is clear
±
eigenvalue approximation it can also be interpreted as from Fig. 5 (b) that there is no corresponding minimum
the filter being unable to decided between the eigenval- of the kernel Λ(V,t) in this region. When Λ(V,t) is eval-
ues λ and λ . For D = 5 the records of interest uated at V(t) = J one finds that it gives impurities
r−12 r+12 smaller (see the so±lid red line in Fig. 5 (b)) than those
correspond to V = 3, 1,1,3 (see Fig. 5 and Fig. 2).
−2 −2 2 2 whichcorrespondtotheinnerpeaks,[thedashedredline
In the long-time limit all solutions give
in Fig. 5 (b)]. From the shaded regions in Fig. 5 (a) it
is apparent that only 1/D of the probability density is
L = 1, (23)
LT 2 found for V J. This means that, in the long-time
| | ≥
limit, only a proportion1/D of the trajectorieswillhave
which is also the bound predicted by Stockton and
purities greater than the bound we now present. Using
coworkers [16]. For small times there is some varia-
the linear trajectory solution with V(t) = J and solv-
tion between the bounds given by V = r = 1 and ±
±2 ing for the impurity one obtains the bound. By making
V = r = 3. This is because (V) is broad at short-
±2 P atwoeigenvalueapproximationonefindsthatthebound
timessothatmoreeigenvaluescontributetotheimpurity
scales asymptotically as
when for example V = 1. Conversely when V = 3 the
2 2
is only one eigenvalue to the left of this record. Accord- L exp( 4γt). (25)
LT
ingly it is reasonable to expect that at short-times the ∼ −
bound given by V = 1 will produce a bound that is ThescalinginEq.(25)andEq.(24)ispreciselythesame.
±2
more impure than the bound given by V = 3. This The difference in the bounds is only apparent when the
±2
behaviour is confirmed in Fig. 4 (a), where the dashed calculation is exact; see Fig. 4 (b). The dashed line
line is V = 3 and the dot dashed line is V = 1. (black) is Eq. (24), while the thin solid line (magenta)is
±2 ±2
9
Eq. (25) (also see Fig. 6). Obviously for a qbit (D = 2) regions are chosen because ℓ is always monotonically in-
the bound given by V = 1 corresponds to the median creasing or decreasing in these regions; see Fig. 5 (b).
±2
ofthedistributionofimpurites,denotedby℘(L,t)dL,at This is required to perform the numerical interpolation
all times, and the mode at long-times. required by the following procedure. In each region one
4. Distribution of Impurities: The distribution of im- discretizes the range of ℓ and then finds the values of V
puritescanbecalculatedanalyticallyforD =2[6]. How- that correspond to this ℓ. The probability distribution
ever we will change variables to the log-impurity so that (V)isthenintegratedoverthis regiontogivedℓ℘(ℓ,t).
P
we may clearly see the featured discussed above. The This is then summed over all regions.
required change of variables from Eq. (24) in Ref. [6] is In Fig.6 the solidcurve is the probabilitydistribution
ℓ≡log10L=log10 21(1−z2) sothatz =√1−2×10ℓ. for D = 5 at t = 2γ−1; this distribution was obtained
The distribution of(cid:2)log-impuri(cid:3)tes is thus using the above method. The upper bound and pseudo-
lower bound described in the text above are quite ap-
e γtln(10)cosh arctanh2(√1 2 10ℓ) parent in distribution; they correspond to the two sharp
−
dℓ℘(ℓ,t)=dℓ h − × i featuresatℓ= 0.301andℓ= 2.87. Itisalsoapparent
2√8πγt√1 2 10ℓ − −
− × that ℓ isamorefaithfulcentraltendencymeasurethan
h i
×exp(cid:16)−arctanh2(p1−2×10ℓ)/4γt(cid:17).(26) lbougl1k0hoLfithaesdthisetrliibnuetcioonrr.esponding to hℓi is closer to the
In Fig. 6 this distribution is plotted at t = 2γ 1 (the
−
red dashed curve). The peak of this distribution at
ℓ = 0.301 becomes less prominent at long-times. This E. Summary
−
is when, as Wiseman and Ralph pointed out [6], consid-
ering the average log-impurity is a good way to find an We conclude our study of information extraction
approximation to the mean time to a fixed L. from commuting measurements by summarising the key
Thepoorlypurifyingtrajectoriesaffect L andalarge points. It is possible to characterise the average amount
h i
number of these trajectories cluster around the upper of information extracted using the average impurity.
bound (as seen in Fig. 4). This motivates the con- We found a simple expression for the average impurity,
sideration of another measure of mixedness which de- L (t) exp( γt), which is valid when t γ 1.
2 LT −
emphasisesthe peaks ofthe distribution [5, 6]. However, Ah dditiional∼ly we f−ound that in this asymptotic≫limit a
in this article, we will remain focused on L . portion (1 1/D) of the trajectories would be bounded
h i by cexp( −4γt) L 1 for a constant c that we can
4.5 − ≤ ≤ 2
determine for a particular D. It is the existence of rare
but poorly purifying trajectories which explains the up-
nsity 3.5 per bound of 12, and which explains the difference be-
e tween the scaling of the mean and that of the lower
D
blity 2.5 bound, cexp(−4γt). Indeed, other measures of central
a tendency, such as the median, or the exponential of the
b
Pro 1.5 mean-log, exhibit the same scaling as the lower bound,
exp( 4γt), which is also reflected in the mean time to
∼ −
0.5 attain a given purity [7].
Despite its limitations, we have used L , which scales
0 -1 -2 -3 -4 -5 -6 -7 h i
log-impurity as exp( γt), as our measure of average information in
−
the main text of the paper. Here we briefly comment
FIG. 6: The distribution of log-impurities at t = 2γ−1. The on what would change if we were to have used a more
bluesolidcurveisthedistributionforD=5. Thereddashed faithful measure of centraltendency such as the median.
curveis for D=2, that is Eq.(26). The line with thecircles In Sec. IVB we found that the spread of trajectories
is log (hLi)=−1.46 at t=2γ−1, according to Eq.(15); the for the QFT protocol was small for sufficiently frequent
10
line with the squares is hℓi = −2.41 at t = 2γ−1. Only a feedback. This feature is true in general for measure-
proportion1/D ofthetrajectories havelowerpuritythan(to
ment in an unbiased basis [9]. The main consequence
therightof)thelinewiththecrosses(plottedatℓ=−3.1736)
of this, for complementary continuous measurements, is
as explained in themain text. Thetwo otherfeatures on the
the meanimpurity is approximatelyequalto the median
graphatℓ=−0.3010andℓ=−2.87correspondtotheupper
impurity. Thus we can directly compare the results ob-
andthepseudo-lowerboundsonthedistributionofimpurities
tainedfor L inthemainpartofthepapertothescaling
described in the main text.
h i
for the median impurity, exp( 4γt), for commuting
∼ −
measurement. The result, obviously, is a diminution in
For D > 2 the distribution of ℓ can be numerically the speed-up offered by the latter by a factor of four:
calculated from the probability distribution (V) and 1(D+1) Smedian 1D2. Forsmallsystems(D =2or
P 6 ≤ ≤ 8
Λ(V,t)asfollows. FirstonesplitsthedistributionΛ(V,t) 3) this implies a slow-down, or at best a modest speed-
into regions ( , j),[ j, j+ 1),...,[j, ) . These up. ButforlargeDthescalingwithDimpliesthatusing
{ −∞ − − − 2 ∞ }
10
feedback to constructsuitable measurements in an unbi- gives a lower bound because the sum of a sequence is al-
assed basis will beat the no-feedback protocol even in ways less than or equal to the greatest term in the sum.
terms of the median impurity or mean time. It was shown in Refs. [8, 9] that a tight lower bound
on Eq. (28), for the optimal permutation, is
IV. INFORMATION ACQUISITION IN 2
dL γt(D+1)L(t). (30)
DIMENSION D AND ITS RELATIONSHIP TO ≤−3
COMPLEMENTARITY
While the upper bound was recently found to be
A. Previous results
1
dL γtD2L(t), (31)
≥−2
In this subsection we briefly summarise the results of
previous studies on information acquisition by measure- forD 1[9]. Equations(30)and(31)holdformeasure-
ment in a complementary basis. The original articles on ment ≫in any complementary basis. Using these bounds
the subject are found in Refs. [2, 4, 8, 9]. The basic idea onemaycomparethetimeittakesforacommutingmea-
is to use quantum feedback control to continuously keep surement (i.e. Eq. (20)) to extract a certain amount of
the eigenbasis of the state complementary to the mea- information (i.e. to attain L = ǫ) to how long a com-
surement basis. By analogy with the intuition presented h i
plementary measurement may take to extract that same
in Fig. 1 it is hoped that this procedure will enhance amount of information. That is we equate Eq. (20) and
information extraction.
the solutions of Eqs. (30) and (31), and solve for the ra-
It is easy to show, from Eq. (1), using Ito¯ calculus tio of t and t . We call this ratio the
complementary commute
thattheinformationacquired[2],ascharacterisedbythe speed-upininformationacquisition. Fort γ 1
commute −
change in impurity, about the state due to measurement ≫
the speed-up is bound by
of infinitesimal duration is [9]
2 D2
dL = 8γdt Tr[ρXρX] 2Tr[Xρ]Tr Xρ2 (D+1) S , (32)
− { − 3 ≤ ≤ 2
+Tr[ρX]2Tr ρ2 (cid:2) (cid:3)
} for D 1 [9].
4 2γdw Tr Xρ2 (cid:2) T(cid:3)r[ρX]Tr ρ2 .(27) ≫
While these results show the usefulness of feedback in
− { − }
p (cid:2) (cid:3) (cid:2) (cid:3) rapid purification, there are some important issues that
Wewishtostudytheeffectofmeasuringinacomplemen-
havenot yetbeen addressed. First, does anexplicit con-
tarybasis. Inordertomeasureinacomplementarybasis
structionofsuchacomplementarymeasurementstrategy
throughoutthemeasurementprocessonemustusequan-
exist? Second, is actual performance of such a strategy
tumfeedbackcontrol[14]tocontinuallyadjustthebasis.
O(D) or O(D2)? Third, what is the mechanism of the
Wewillnotlabouronthosedetailshereastheyhavebeen
speed-up? The first and second questions are answered
adequately discussed before [4, 8, 9, 14]. The required
in Sec. IVB and Sec. IVC respectively. We defer the
transformationofthemeasurementbasisisX(t) Xˇ(t)
→ third question until Sec. V.
such that xˇi = 1/√D, for all xˇ and i where i is an
eigenstate|ohf|ρi|and xˇ is an eigenstate of Xˇ. Re|pilacing
all the X’s by Xˇ in|Eiq. (27) considerably simplifies the
B. Complementarity via quantum Fourier
expression to [8, 9]
transform for D=2,3,4
dL = 8γTr XˇρXˇρ dt, (28)
− In this subsection we explore implementing a contin-
(cid:2) (cid:3)
or equivalently uous complementary measurement in systems of dimen-
sion three and four. There are many unbiased bases, so
dL = 8γdt Xˇ 2λ λ . (29) to make our analysis concrete we must choose one. It
r,c r c
− | |
Xr,c has been noted elsewhere [19] that for D >3 not all un-
biased bases are equally good at reducing the impurity.
Observe that the unbiased condition which simplified Nevertheless we choose a particular complementary, the
Eq. (27) to Eq. (28) leaves a permutational degree of one generatedby the quantum Fourier transform(QFT)
freedom in Eqs. (28) and (29) because a permutation of of the logical basis. The construction of the continu-
the eigenstates of ρ does not affect xˇi =1/√D. This ous complementary measurement then consists of: cal-
|h | i|
means we should optimise over permutations in the ex- culationally diagonalizing the state matrix; calculation-
pressiondL= 8γTr XˇP ρP XˇP ρP dt, where P ally ordering (permuting) those eigenvalues to maximize
− m† m m† m m
isapermutationmatr(cid:2)ix,tomaximizethed(cid:3)ecreaseinim- dL after a measurement; and applying the appropri-
|h i|
purity. To obtaina lowerbound dL of the optimal per- ate permutationand the QFT unitary to the state. This
| |
mutation the we average over all possible permutations: wholeprocedureisanexampleofquantumfeedbackcon-
dL = 8γ Tr XˇP ρP XˇP ρP dt. The average trol[14]. TheD-dimensionalQFTcanberepresentedby
− m m† m m† m
P (cid:2) (cid:3)