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# Early Universe Physics PHYS 610

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PHYS 610 Recent Developments in Quantum Mechanics and Quantum Information Spring 2009 Notes Master Equation 1 Master Equation A main and obvious advantage of the densityoperator formalism is that it provides a method for handling nonunitary evolution of the quantum state This generally occurs in the treatment of open quantum systems quantum systems coupled to external systems that we do not directly track1 We will now study the evolution of a quantum system described by Hamiltonian HS interacting with a reservoir described by Hamiltonian Hal We will assume the systemireservoir interaction described by HSR to e weak causing slow evolution on the uncoupled time scales of the system and reservoir separately The evolution of the total system is unitary given by i atPsR vapsRlv 1 where pm is the combined state of the system and reservoir and the total Hamiltonian is HHSHRHSR 2 Our goal is to derive an equation of motion for the state of the system alone given by a partial trace over the reservoir degrees of freedom P1 Tral sal 3 Note that so long as we are interested in operators that act solely on the systems Hilbert space this reduced density operator is suf cient to compute any appropriate expectation values We will derive the master equation with a number of approximations and idealizations mostly related to the reservoir having many degrees of freedoml The approximations here typically work extremely well in quantum optics though not necessarily in other areas such as condensedmatter physics where for example the weakcoupling idealization may break down Examples of reservoirs include the quantum electromagnetic eld in a vacuum or thermal state or the internal degrees of freedom of a composite objectl 11 Interaction Representation The rst step is to switch to the interaction representation in effect hiding the fast dynamics of the uncoupled system and reservoir and focusing on the slow dynamics induced by HSR We do this via the transformations iHSHRth HS FHR h e sa PSRt 4 HSRO eiHsHRthHSRei HerHR h so that the formerly timeindependent interaction becomes explicitly timedependent The equation of motion then becomes i Mis t gTHSRab PSROHA 5 1For further reading see William H Louisell Quantum Statistical Properties of Radiation Wiley 1973 Chapter 6 Claude eniTannoudji Jacques DuponteRoc and Gilbert Grynberg Atomephoton Interactions Basic Processes and Applications Wiley 1992 Chapter IV and Howard Carmichael An Open Systems Approach to Quantum Optics Springer 1993 Chapter 1 Integrating this from t to t At 2 tAt am At am 7 g dt Hawmam lt6 1 lterating this equation by using it as an expression for SRt Z tAt HA1 1 gammaawake dt leaO LSsaOHi dt WarsawHsaanmsacnii lt7 Now in taking the trace over the reservoir In doing so we will assume that the rst term on the righthand side vanishes More speci cally we assume TrRljvgsizlt7ygtf7s ltltlgtl 0 8 This follows by assuming that the total systemireservoir state always approximately factorizes sa W W ay 9 where 2R is the stationary state of the reservoir This amounts to assuming that the reservoir is large and complex and weak coupling of the system to the reservoir so that the perturbation to the reservoir by the system is small In this case the time interval At gtgt TC where TC is the correlation time of the reservoirithe time for reservoir and systemireservoir correlations to decay away This also amounts to a coarsegraining approximation which means that we are smoothing out any fast dynamics on time scales of the order of TC or shorter Thus any correlations that have arisen in past time intervals have decayed awayi Of course new correlations arise due to the coupling in the present time interval which will give rise to nonunitary terms in the evolution equation for the reduced statei Then the assumption 8 amounts to nRHSRt3R or 10 This assumption means essentially that there is no dc component to the systemireservoir couplingithat is the systemireservoir coupling consists of uctuations about a zero mean This can always be arranged by absorbing any nonzero mean into the system Hamiltoniani 12 BorniMarkov Approximation Since the rst term vanishes under the partial trace with the trace Eq 7 becomes tAt 2 A ti dt dwTrRiHsm Hsacnmsacnit 11gt with A t 2t At 7 ti Now we will make the BorniMarkov approximation by setting SROH m Mt R 12 In fact there is a pair of approximations at work here The Born approximation amounts to assuming the factorization in 9 which we have justi ed in terms of a large complex reservoir with a short coherence timer The Markov approximation amounts to setting pt to pt in 12 which will result in an evolution equation that only depends on pt and not the past history of the density operator We can justify this approximation by noting that At is small and HSR induces a weak perturbation so that Mt ptOAti Then this amounts to a lowestorder expansion in At of the righthand side of Eq 11 which is appropriate in view of the limit At A 0 to obtain a differential equation though in a coarsegrained sense since strictly speaking we always require At gtgt TC Next we change integration variables by setting 739 tit 13 tAt 2 A tAt dt dt dT dt 0 Wm 14 N d7 dti 0 so that the integration becomes 2 In writing down the nal approximate form for the integrals we have used the fact that the integrand involves an expectation value of the interaction Hamiltonian taken at times that differ by 739 as we will explore further shortlyi That is the integrand involves reservoir correlation functions which decay away on the time scale TC 13 Interact ion Now we make a reasonably general assumption regarding the interaction Hamiltonian namely that it can be written as a sum of products over system and reservoir operators HSR hSaRw 15 Recall that repeated indices imply summationi The interpretation here is that if SD is a Hermitian operator then it represents an observable that is being effectively or actually monitored via coupling 39 39 quot of the form to the environment For example a position 1s A t by an HSR zRi Alternately the operators need not be Hermitiani For example an interaction of the form HSR SRT SlR represents the exchange of quanta eg of energy between the system and reservoir and would thus represent dissipation or loss of energy to the reservoir Such interactions occur in spontaneous emission and cavity decayi With the interaction of the form 15 and the change of integration in Eqs 14 the change 11 in the quantum state becomes 00 HA Am 7 dr dt Sat 3at 7 mac 7 w 7 rgt lttgtsaltt gtaa ltrgt 0 z 16 7 WWW 7 Two 7 Sat 3t5 t 7 7 saw where we have de ned the reservoir correlation functions Gum 7 m RaegtR w 7 7 7 ltRaltt gtRgltt 7 rgtgtR 7 ltRaltrgtRgltogtgt 17gt which depend only on the time difference because the reservoir is in a stationary state Now we make the 7 R further assumption SOLO eiHsthgae7iHsth eiwat 18 about the interactionpicture system operators This is not necessarily a restrictive assumption since multiple frequencies for a given system operator may be separated in the sum in 15 Then Eq 16 becomes A t m 7 047 WM Sas m 7 S mmlaaw 19gt l ms sa WOW GMJei a eiw ltr7Tgti Now de ning At 1 I 41206 mg I dt el ltw3 z 10 0 d7 e inGaMT 20 wga 0 dTe iMNGgJiT we can write A t 7 Sas m 7 ammole ma a 7 Sa t3gwga1wa W 21 Under the assumption of fast uncoupled system and reservoir dynamics At gtgt ma Marl 22 the integral 004 Lug averages to zero unless ma Lug 0 Thus we may replace the integral with a Kronecker delta 004 Lug At 6wa flag 23 Now formally taking the limit of small At A t W 7 76 7wggt Swagac 7 Sgplttgtsalwt 7 Ma a 7 Saplttgt8 lw a 24gt where again we must keep in mind that this differential equation is coarsegrained in the sense of not representing dynamics on time scales as short as TC or ma wg 1 for different frequencies Now transforming out oft e 39 quot 39 using t e 18 and ma flag i amt 7 lm20gt 7 6m 7wgt swagw 7 wash 7 puma 7 Saplttgt8 lw a 25gt Now we use the fact that HSR is Hermitian so terms of the form SR in 15 that are not Hermitian must be accompanied by their adjoint terms SlRli Clearly terms where SD S satisfy 6wa 7M3 1 so we can explicitly combine these pairs of terms to write the master equation in terms of only a single sum W 7 7stt 2 some 7 5350420 w eaaw 7 plttgt85aw 26gt Of course terms of the same form carry through when SD is Hermitiani In the expression above we have also de ned the reduced integrals w I de 67m ltI3LL739I3040gtR oo 27 g 0 dTEiwaTltR0RaTgt S 7 wxr R Note that other crossterms could in principle occur in Eq 25 that satisfy ma 7M3 which we appear to be missing here However if we end up with terms like 5125 this can always be absorbed into terms of the form 51 Sgp51 Sgl representing interferences in the couplings represented by 513 The cross terms are weighted by a crosscorrelation function between R1 and R2 representing the cross terms of the coherecei In the absence of cross coherence only terms of the form 51251 and 5225 should appear Weighted combinations of these terms with 51 Sgp51 52yr terms can account for any degree of coherence Now separating out the real and imaginary parts of the integrals 27 in 26 239 amt ths7 tlZkaSaJtsii sLsaplttgtplttgtssaenmiwx sismmwl 28gt where 160 I 2Rewi 29 Since the last term has the form of Hamiltonian evolution we may drop it assuming it is accounted for by H3 or arranging the de nition of the system such that it does Now separating out the real and imaginary parts of the integrals amt 7HS7 M Z legged2w 7 g swag2m wow BorniMarkov master equation 30 We have thus arrived at the general Lindblad form of the master equation in the BorniMarkov approxi mation Again the system operators SD represent the coupling channel of the system to the reservoir and thus the channel by which the system may be observed Thus for example if SD 8 I then we have the master equation for a position measurement whereas if SD 8 a where a is the annihilation operator for the harmonic 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3 lo iquot FOL 2 C109 53 L Ls I de Q amp 9 smhk V39 Ob loot lt1I v 9 Qqub elfw n m 3907 addu5w V PM ago 39 41 1 J 739 u i W31quot mud Cm Loci lt14 9 th O JI N quot 451d ccz 4 gig 513 Q de lcolt Iltd ct i 3 M sptu wr 0amp3 Mink 3 WskmkGquot 3 is Q quot54 cc flun 5mg cuJan mummy ye jg thwsdc fun5 Jim 21 2w sea 539 34quot 121730 anal lt2qu gc io to 59071 e i t J 1 39 sz W139 3 m 1133quot 8amp0 02403 4 39 I I M m n fg lo75uUo7 3 OH ream SE4 9 mac Wu pee Nam 07 PHYS 610 Recent Developments in Quantum Mechanics and Quantum Information Spring 2009 Notes Measurement and POVMs 1 Positive OperatorValued Measures To help handle generalized measurements we will now introduce the somewhat mathematical concept of a positive operatorvalued measure POV i By referring to generalized measurements we mean to differ entiate these measurements from the usual projective or von Neumann measurements which is what you normally nd in introductory quantummechanics texts The usual description goes like this for a quantum system in state a measurement of the observable Q leaves the system in an eigenstate lqgt of Q with probability in which case the result of the measurement is the eigenvalue 4 We can see that this notion of a measurement is lacking in two situations First it does not properly describe the situation in photodetection of atomic radiation where each detection event results in the loss of energy lie the atom is always found to be in the ground state and we gain information even during instants when a photon is not detected Thus POVMs are crucial to the formal de nition of a continuous measurement The second situation is when the observable is the position operator where eigenstate collapse is unphysical a position eigenstate is a state of in nite energyl POVMs allow us to de ne an imprecise or partial measurement of an observable which will be a stepping stone on the way to de ning a continuous measurement of position 11 Discrete Finite Spaces Consider a discrete nite Hilbert space of dimension Ni That is the Hilbert space is spanned by the set of eigenstates lqgtrq17m7N 1 of the observable Qt Then we can de ne a positive operatorvalued measure POVM as a set of positivesemide nite operators 999 that sum to the identity operator Nq 2939 1 2 q1 Note that we are writing the qth positive operator 9591 as a factorization in terms of the Kraus operator Qq since any positive operator always has such a factorizationl We also note that the number Nq of positive operators is not necessarily the same as the dimension N of the Hilbert space Now the important physical point here is that a POVM de nes a quantum measurement on the Hilbert space The qth possible outcome of the measurement is that the state vector changes according to the replacement WW W A 7 3 law Sig29 1 m L911 7 4 Trl qJ ql ltngnqgt or in terms of the density operator That is in the qth outcome the state is hit by the operator g and then renormalized if necessary The probability that the qth outcome occurs is 134 mnqpngj 394 5 The classical result of the quantum measurement in this case is simply q or some physically meaningful function of g This notion may seem rather abstract but we can note that the usual projective measurement comes out as a special case of the POVMbased measurementi In particular the usual measurement arises from a projectionvalued measure where we partition the Hilbert space according to a set of Hermitian projection operators Pq I l4gtlt4l 6 that also sum to the identity N Z Pf 1i 7 Of course P Pq but we have written the sum in this form to emphasize the similarity with Eq 2 by taking Qq RI and Nq Ni Then the standard projective measurement of the observable Q results in the qth outcome of a reduction to the qth eigenstate lq P W l gt 172 0 gt 4 8 or in terms of the density operator Pij Pij qiqf 332quot l4gtltql 9 TrlPt qu lt qgt This outcome happens with probability 134 TrquJpqll ltPq2gt ltPqgt7 10 which for a pure state becomes the familiar Born rule 134 WM 11 Thus the POVMbased measurement above is a reasonably straightforward generalization of the usual projective measurements at least when the standard measurements are cast in the proper way 12 Measure Why is a POVM called a POVM The answer requires an excursion into mathematics and so the short answer if you feel the need to skip forward is that a measure is usually something that assigns numbers to sets and so a positive operatorvalued measure is a measure that instead associates positive operators with sets and thence probabilities to the same sets via the expectation value as above To really answer this question we need to de ne what we usually mean by a measure and then adapt it to the operator casei lnformally a measure is a rule for assigning numbers to subsets of some set or space This is a very useful notion in probability theory where you would consider the set of all possible outcomes or events and the measure would assign probabilities to each outcome or collection of outcomes Alternately a measure is an abstraction of the notion of volume where the measure represents the volume of subsets of the main seti Before formally de ning a measure though we should rst note that for a given space it is problematic to try to de ne a measure on every subseti Instead we will de ne the measure on only a limited collection of subsets chosen to make the de nition of the measure consistenti Formally this collection is a aalgebra which we de ne as a collection 5 of subsets of the space X such that l The empty set is included 0 E 5 2 Countable disjoint unions are included with countable here meaning nite or countably in nite if 9 C 5 with A N B D for any AB E 9 and 9 is countable then U A e 54 12 Ae 3 Complements are included if A E 5 then X 7 A E 5 Any element of a aalgebra is said to be a measurable set This de nition can be contrasted with the possibly familiar de nition for a topology on a space X which is a collection 9 of subsets of X such that1 l The empty set and the whole space are included 0 E y X E 9 2 Arbitrary unions are included if 9 C 5 then U A e 54 13 Ae 3 Finite intersections are included if 9 C 5 with 9 nite then A e 5e 14 Ae Any element of a topology is said to be an open set while the complement of an open set is said to be a Closed set Thus while topologies contain in general only open sets aalgebras contain both open and closed sets For example on the real line R the standard topology is the topology consisting of all open intervals of the form a b and all possible unions of such intervals and the empty set It turns out there is a unique aalgebra associated with the standard topology which is the smallest aalgebra containing it This is called the Borel aalgebra on R which would contain all open intervals as well as all closed intervals of the form ab and many other sets The notion of a aalgebra may not be intuitively clear at this stage but the de nition is basically concocted to make the de nition of measure work out as we will now see A measure is a function M Y a 0 00 de ned on a aalgebra 5 on a space X which satis es 1 The empty set has zero measure 03 0 2 The measure for countable disjoint unions adds if 9 C 5 with A N B D for any A B E 9 and 9 is countable then Mlt U A 2 AA 15gt Ae z Ae These two requirements are sensible considering our analogies to probabilities and volumes and we can also see how the requirements for a aalgebra guarantee that we donlt have any problems in de ning a measure the last axiom for a aalgebra imposes the sensible constraint that if A is a measureable subset then so is X 7 A Note that the point 00 is explicitly included in the range of a measure which is intuitively a good measure for something like the entire real line Also strictly speaking we have de ned a positive measure since we have only allowed nonnegative values in the range of M As an example of measure the Lebesgue measure on the real line is de ned on the Borel aalgebra We can de ne it in several cases as follows 1For further reading see eg James R Munkres Topology a First Course PrenticeHall 1975 1 It turns out that any open set A can be written as the union ofa countable set of open intervals aj bj in which case the Lebesgue measure of A is the sum of the interval lengths MA 21 7 aj 16 j 2 It turns out that any closed set E can be written as a closed interval a b with the union of a countable set of open intervals ajbj removed from it B a b 7 aha 17 where every aj gt a and every bj lt b in which case the Lebesgue measure of B is the length of the closed interval minus the Lebesgue measure of the removed component A1b 7 a 7 2071 7 a1 18 9 For any other set C in the Borel aalgebra the Lebesgue measure is the in mum greatest lower bound of the set of Lebesgue measures of all open sets containing C MC infMA A is open and C C A 19 Note that there exist sets that do not have Lebesgue measures according to the above de nitions and thus they are excluded by considering only the aalgebra The Lebesgue measure is useful in that it extends the notion of length to more complicated and subtle sets the set of rational numbers being countable is a set of Lebesgue measure zero on the real line and the Cantor middlethirds set a fractal set constructed by starting with the interval 01 removing the open middle third77 interval 1323 removing the middlethirds of the two remaining closed intervals and so on ad in nitum is an uncountable set but of zero Lebesgue measure For measurements the concept ofa probability measure is more useful and it is simply that of a measure but where the range of the measure is 01 rather than 000 with a measure of the whole space being unity For example the Lebesgue measure on the space 0 1 is a probability measure and corresponds to a uniform probability density on the same interval 13 General De nition Now with the above mathematical concepts we can now give a more general de nition of a POVM than in the nite case above In more general terms a positive operatorvalued measure POVM de ned on a aalgebra 5 on a space X is a function H that takes as values positive semide nite Hermitian operators on a Hilbert space if such that for any E if the function a X a 01 de ned by MA I lt lHAl12gt 20 for any measurable subset A of X de nes a probability measure on 5 In particular this implies that HX is the identity operator which is the generalization of the sum rule Thus the OVM associates positive operators with measurable subsets of the space of outcomes which are then associated with probabilities by appropriate expectation values In this way we can de ne a family of probability measures parameterized y the quantum state We could of course write the probability measure more generally in terms of the density operator as MWmle m lncidentally a trace of this form is for a Hilbert space of dimension larger than two the only way to construct a quantum probability measure this is essentially the content of Gleason s theorem 14 Realization It is important to note that measurements induced by POVMs while generalizing projective measurements donlt introduce anything fundamentally new to quantum mechanics any of these more general measure ments can be realized by introducing an auxiliary system ancilla performing a unitary transformation on the combined system and then perform a projective measurement on the ancillai Thus generalized mea surements correspond to indirect measurements where information about a system comes from projective measurements on the environment with which the system has interacted and thus become entangled with This result is known as Naimark s theorem or Neumark s theorem3 and we will only sketch the argument for the nite case here Starting with the system in the state we will extend the Hilbert space to contain the environment whose dimension is equal to the number of Kraus operators de ning the POVM a E 0E We will assume the environment to always start in a particular state that we label Note that we are assuming a pure state for the system which we may as well do as long as we are extending the Hilbert space by invoking puri cationi We can thus de ne an operator U that acts on the composite state as 9 W UM 0Egt E qu gtl4Egt E wwminqw q l4Egt7 22 q q lt l l ql gt where the Kraus operators Qq only operate on the original systemi Now computing the norm of the trans formed composite state lt OElUTUWJ 0Egt Elt4Ellt l l QqWld qq anmw m WW7 so that U preserves the norm of states in the subspace of the original systemi The operator U is thus unitary on this subspace but is not xed uniquely by the above argument In principle the action of U on the environment can be chosen to make U unitary on the composite Hilbert space Basically this is because taken as a matrix the columns of U span the subspace of the original system ie a subset of them form an orthonormal basis and the extra degrees of freedom elements of the extra rows in expanding U to the composite Hilbert space may then be chosen to make the columns of U form an orthonormal basis on the entire composite space Now after the transformation a projective measurement of the state of the environment leads to the result mm with probability Tr lqEgtlt4El UM 012W OElUll lt l939ql l 24 2Andrew M Gleason Measures on the Closed Subspaces of a Hilbert Space Journal of Mathematics and Mechanics 6 885 1957 doi 101512iumj1957656060 3 her Peres Quantum Theory Concepts and Methods Springer 1995 Section 976 p 285 For a similar argument to what we present here for the unitary representation of linear positive maps see Benjamin Schumacher op cit Furthermore the projection of the environment into state lqEgt induces the transformation 911W W A T wnqnqw 25 on the original systemr Thus we have constructed the POVMbased measurement based on the larger projective measurementr 15 Example Spontaneous Emission As an example of a POVM we return to the stochastic master equation for photodetection of atomic resonance uorescence with quantum jumps i opal dp 7 haw 7 award Fltalagt m7 Ma 7 pgt dNr 26 In any given time interval of duration dt there are only two possible outcomes no photon is detected or one photon is detected We can de ne this evolution in terms of a POVM as follows Let Udt denote the evolution operator for the combined atomi eld systemr Before each in nitesimal time interval the eld starts in the vacuum state l0gt and after each in nitesimal time interval the detector projectively measures the eld and registers a detection event if a photon is emitted into my model Since the detector does not distinguish modes we will simply denote the eld state as llgt in the case of an emitted photonr Then the two jump operators77 for the two measurement outcomes are new lt0lUdtl0gt 1 7 dt 7 gag dt 27 91dt lt1lUdtl0gt Vthar In the case of no photon detected the state is transformed according to T p A Shaw290 dt p 7 H ldt 7 E alm pl dt Fltalagt pdt 28 h 2 Tr 90dtpngdt keeping terms to rst order in dt and in the case of a detector click the state is transformed according to 91 dt p91 dt opal p T Tr 91dtpn dt Way 29 These two transformations correspond exactly to the transformations induced by the SME 26 in the cases dN 0 and dN 1 respectively Notice that this POVM tends to drive the atom towards the ground state as compared to the unconditioned Hamiltonian evolution and for either possible outcome 90 1 By involving the atomic annihilation operator we see in this case that the POVM generalizes projective measurements by modeling dissipation due to the measurement process In the case at hand the physical origin of the dissipation in the case at hand is absorption of radiated photons by the photodetector antum Trajectories and Quantum Measurement theory Quantum and Semiclassical Optics 8 205 4H M Wiseman Qu 1996 doi 10108813557511181015 16 Example Gaussian Projectors POVMs can also generalize projective measurements to model partial or imprecise measurements Partial measurements leave some uncertainty in the measured observable whereas projective measurements leave the system in a state where the observable is perfectly de nedithat is an eigenstate of the observable As a simple example we can model partial measurements by de ning the measurement operators Qq to be Gaussianweighted sums over projection operators for the discrete set of eigenstates lq q E Z of the observable Q 1 v 2 NOH 4 39 39 9q 7 jjgme q WM 30 Here 31 so that 00 Z 939 1 32 1700 as required for the operators to form a POVMi The Gaussian weights lead to having only partial infor mation about Q after the measurement For example in a highly uncertain mixed state where qlplq is approximately the same for any 4 and qlplq 0 for any 4 f q the measurement leads to the collapse 91129 1 2 my 7 m 7 5 1711 2 y 33 MWQZ W21 w lt gt The qth possible nal state is thus peaked about the eigenvalue 4 and additionally has an uncertainty AQ 1 ln the limit H a 00 the measurements here reduce to the usual projective measurements Thus for large H the variance in the measurement results taken over an ensemble of measurements on identically prepared systems is dominated by the uncertainty in the quantum state while for small h the measurement variance is dominated by the uncertainty introduced by the measurement operators qu This distinction divides two categories of measurments strong measurements where H is large and weak measurements where H is sma i We can also generalize these Gaussian projectors to the continuousvariable case For example for a position measurement the properly normalized measurement operators have the form ma 514 dzeW W zm 34gt Again if this operator is applied to a an initially uncertain state such as a momentum eigenstate the resulting position variance in the collapsed state is lH ie the uncertainty is ln what follows we will consider sequences of weak position measurements of this form and thus construct continuous quantum 5Yakir Aharonov David Z Albert and Lev Vaidman How the result of a measurement of a component of the spin of a spine particle can turn out to be 100 Physical Review Letters 60 1351 1988 doi 101103PhysRevLett601351 See also comments by A J Leggett Comment on How the result of a measurement of a component of the spin of a spine particle can turn out to be 1007 7 Physical Review Letters 62 2325 1988 doi 101103PhysRevLett622325 and Asher Peres Quantum Measurements with Postselection Physical Review Letters 62 2326 1988 doi 101103PhysRevLett622326 as well as the reply by Y Aharonov and L Vaidman Physical Review Letters 62 2327 1988 doi 101103PhysRevLett622327 measurements of position For this it is useful to consider the product of two operators HH 14 2 2 90 H Qa H ltTwgt2gt dz dz e N E 0 Mlz Xz le NW D 4lzgtltzl HH 14 2 4 2 4 lt2W2gt dxe N 170 e Mm D HH 14 HH oo exp 7mm 7 Dz2 dz exp 35 which corresponds to a sequence of two Gaussian position measurements the rst of strength H and the second of strength H with measurement outcomes 1 and then 0 respectively This operator product is still Gaussian but it is not normalized properly in the sense that 9a is normalized note that the norm vanishes if a 7 0 becomes large but we can see from its form that applying this operator to an initially HH If iaHMH mm uncertain state gives lH Ii for the resulting position variance of the state Hence a sequence of two Gaussian measurements is effectively equivalent to a single Gaussian measurement where the strength is the sum of the individual measurement strengths as long as no other transformation or evolution occurs between the two measurements Notice how the information from the second measurement is incorporated with that of the rst After the rst measurement the best estimate for the position of the quantum system is a with uncertainty After the second measurement where the result is 0 the new best position estimate is an average of the old estimate and the new measurement result aHoH HH ltIgt 7 36 weighted by the respective uncertaintiesi The new uncertainty of the estimate is reduced to lxIi H

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