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# Early Universe Physics PHYS 610

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Date Created: 09/08/15

PHYS 610 Recent Developments in Quantum Mechanics and Quantum Information Spring 2009 Notes Bayesian Statistics 1 A Third Prelude Bayesian View of Quantum Measurement With the introduction of POVMs as generalized measurements we will now compare quantum measurements with classical Bayesian inferenceigaining some insight into quantum measurements as processes of re ning quantum information We will only do so at a fairly simple level more details on this modern View of quantum measurement may be gleaned from more extreme Bayesians1 11 Bayes7 Rule Bayes7 rule is the centerpiece of statistics from the Bayesian point of View and is simple to derive Starting from the de nition of conditional probability PB A the probability of event B occuring given that event A occured PAB PBlAPA l we can use this to write what seems to be a simple formula PA AB 7 PBAPA 7B 1 MB H M 2 This is Bayes7 Rule which we will rewrite by replacing B by one possible outcome DD out of a set D04 of all possible disjoint outcomes PDalAPA 3 P AlDa W Bayesl Rule Again while this rule seems to be a fairly simple generalization of the de nition of conditional probability the key is in the intemretation of the various elements in this formula The basic idea is that in learning that an outcome DD actually occurred out of a set of possible measurement outcomes Du allows us to re ne the probability we assign to A based on this new knowledge The various factors are H The prior PA represents the probability assigned to event Aiprior to knowing the outcome of the measurementibased on any knowledge or A 39 This p quot39 is not quot39 on D04 to The probability of the measurement outcome PDalA is the probability that the particular measurement outcome or event DD would occur given that A actually happened 9 The renormalization factor PDa is the probability of the measurement outcome Dad by which we must divide for the result to come out correctly This is computed most simply by summing over the probabilities of a complete nonintersecting set of outcomes Ag conditioned on D0 weighted by the probabilities that the Ag occur PDa ZPDalA PA 4 1Christopher A ichs Quantum Mechanics as Quantum Information and only a little more arXiVorg preprint quantephOZOSOSQV This paper is interesting overall but also worth reading just for the quote from Hideo Mabuchi on p 13 4 The posterior PAlDa is the re ned probability of A now that we know that the measurement outcome DD has occurred The posterior probability thus reflects the information gained or revealed by the outcome event D04 12 Example The Monty Hall Problem77 One standard example of applying Bayes7 rule is the Monty Hall problem This is standard almost to the point of being painfully trite but still this is a useful example in setting up our comparison to quantum measurement We will de ne the rules as follows 1 Youlre a contestant on the game show Let 5 Make a Deal and you are shown three doors we will call them doors 1 2 and 3 E0 Behind one door is a brandnew car and behind the other two are goats zonk prizes We will suppose that you like cars very much but you aren t especially fond of goats they smell funny and make you sneeze We will also suppose that they are randomly placed one behind each of the three doors and the problem is invariant under any permutation of the door labels CA3 You pick a door we will call that one door 177 without loss of generality You stand to gain whatever is behind that door F The host opens up one of the other two doors to reveal a goat without loss of generality we will call this door 3 We will assume the host knowingly and intentionally revealed a goat and if he could do this in multiple ways he would pick a door at random 5 The problem is is it to your advantage to switch to door 2 or should you stay with door 1 The answer somewhat counterintuitively is that you double your chances of successfully winning the car if you switch doors This result is not hard to work out using Bayes7 rule Prior we will de ne the three events Ca which is the event where the car is behind door 1 Since the arrangement is random 1 13006 g Vae123 5 a Data the outcome event or data that gives us information is D3 which will be our shorthand for the goat being behind door 3 and the host chose to reveal door 3 if there were multiple choices for revealing a goat If the car were behind door 1 then there are goats behind doors 2 and 3 so the host would have a 50 chance of opening door 3 PD3l01 6 If the car were behind door 2 then opening door 3 would be the only choice PD3l02 17 7 while if the car were behind door 3 opening door 3 wouldnlt be an option PD3l03 0 8 The probability for D3 to occur is given by summing over all conditional probabilities for D3 weighted by the probability of each conditioning event to occur 1 PD3 ZPDglCaPCa 0 g 7 9 1 1 1 2 3 3 Posterior Now given the information revealed by the hosts choice we can compute the posterior probabilities of the car being behind each door Plt011D3gt PD LCBC Plt021D3gt g 10gt Plt031D3gt PDCS 0 Clearly it is to your advantage to switch to door 2 since the probability of nding the car there is double what it was before Note that in accounting for the action of the host the probability distribution for nding the car behind each door changed discontinuously the distribution was initially uniform then changed to a different situation where one possibility has the maximum probability and another possibility has been ruled out This is quite reminiscent of wavefunction collapse after a quantum measurement 121 Quantum Language In fact we can recast this same problem in the notation ofv1 39 t as A t by POVMs quite easily This is the identical problem though so there will in fact be nothing quantum mechanical about the treatment of this example except for the notation This is simply an exercise to emphasize the similarity of quantum measurements to Bayes7 rule 1 We will label the three outcomes by the states lagt with projection operators Pa lagtltal The initial state is equiprobable and for a classical mixture the density operator is thus simply proportional to the identity operator 1 p P1P2P3 11 We can then represent the revelation of a goat behind door 3 by guessing the operator 1 Q i131 P2 12 Thus QTQ P1 P2 and a POVM could be completed for example by the alternate possibility Q 2131 P3 We can verify that the operator 9 gives the right conditional probabilities for the each of the outcomes given by the appropriate trace TEN25 setting the density operator equal to the appro priate projector p Dquot 1 P2 1 PD3l01 Trl piml gTr i PD3102 mnpgm Tr P22 1 13 PD3103 mnpgm 0 These are of course the same classical probabilities as in Eqs 678 and this is precisely the justi cation for de ning this operator Now the conditioned state pc is given by the POVM transformation P l P P ltAP2gt P1P2P3 lt P2gt lt P2 9m 1 2 JCT rligm 121 P 1P P P121 P THE P P1 P2 14gt dr 2gt 1 2 3lt 2 2 Finally the posterior or conditioned probabilities of nding the car behind each of the doors is given by a similar trace where the projector Pa de nes the outcome of nding the car behind door a in a future measurement Pclt01gt TrlPlJCH 13402 TrP2pCP2 g 15 Paws TrlPsacPsi 0 These are the same probabilities that we obtained using Bayes7 rule in standard form 13 Quantum Measurement as Inference from Data To generalize the Monty Hall example we can recast the POVM reduction as a quantum Bayes7 Rule77 Assume we have a set D0 of Krause operators that are comprised in a POVM Then the ath measurement outcome converts the quantum state p into the conditioned state pa according to DapDL TrDapDL 16 Pa We can identify elements here that are very similar to the classical Bayes7 Rule H The prior in this case is the initial density operator p E0 The reduction the operators DD and D act like the conditional probability PDalA in the classical case which effects the change in the probability in response to the occurrence of D0 regarded as an event As we saw in the Monty Hall example these quantum operators can be constructed to be equivalent to the classical conditional probabilities 9 The renormalization factor we then renormalize the probability by dividing by TrlDapDL which is just the probability PDDP in the classical case This step of course ensures a normalized conditioned density operator which we of course need for a sensible probability distribution F The posterior the conditioned state pa then re ects our knowledge of the quantum state given the ath outcome of the measurement in the same way that PAlDa re ects the probability for outcome A given the event D0 The obvious but super cial difference here is that the classical rule describes the change in the assigned probability for a single event A whereas the quantum rule handles all possible outcomes of a future mea surement all at once While similar the quantum and classical rules can t quite be cast in the same form since the quantum rule is both more general in handling coherent superpositions quantum probabilities and different in that measurements on some aspects of a system must disturb complementary aspects quan tum backaction We can conclude this interlude by noting a number of points regarding how one can use the quantum Bayes7 rule as a framework for thinking about quantum measurementi2 While bordering on the philosophical this is a very useful framework for thinking about measurements in modern experiments particularly where single quantum systems and multiple sequential measurements are involved 2 o The quantum state p is the information about a quantum system according to a particular observeri A quantum measurement re nes the observers information about the system and thus modi es the density operatori This removes any problems with collapse of the wave function77 as a discontinuous process The wave function is in fact literally in the observers head and the collapse is just an update of information o This View is particularly useful in considering multiple observers for the same system both performing their own weak measurements but possibly not sharing their results We treated this for example in the case of stochastic master equations for photodetection where each observer ends up with a different conditioned density operatori Each density operator incorporates the measurement information of the corresponding observer but also a trace over the unknown measurement results of the other observeri The price of all this is the potentially distasteful feature of subjective or observerdependent quantum states Actually this shouldnlt be unreasonable in the case of multiple observers however even multiple observers with access to all the same measurement results could disagree on the details of the quantum state because they may have begun with different prior statesi There are a number of important objective features howeveri For example the data measurement results are objectiveias are the rules for incorporating dataiand as the observers continue to incorporate more data their states should converge at least in the aspects reflected by the measurements with suf cient data eventually the information from the data should completely swamp the prior Further in constructing priors both observers should either agree that the probability of a particular measurement outcome is either zero or nonzero even if they disagree on the exact probability an assigned probability of zero is the only really claim that is absolutely falsi able by future measurementsi inally there are objective ways of constructing prior states such as the maximumentropy principle which chooses the state with the least information that is consistent with all known constraints3 though in practice determining and implementing constraints can be a tricky business o In any quantum or classical measurement the knowledge should increase or at least it shouldnlt decrease For example in a quantum projective measurement a mixed state always transforms to a pure state with correspondingly less uncertainty ie the uncertainty reduced to the quantum mechanical minimum though of course once in a pure state a projection can only modify the state without increasing knowledgei Essentially the same is true of general POVMsi5 The information gained in a quantum measurement is not about some preexisting reality iiei hidden variables but rather in the measurement the uncertainty for predictions of future measurements decreasesi Exercise Suppose that a ctitious serious disease called Bayes7 syndrome affects 01 of the population Suppose also that you are tested for Bayes7 syndrome The test has a falsepositive rate of 01 and a falsenegative Phy 5 2see Christopher A ichs op cit for much additional detail E Jaynes Probability Theory The Logic of Science Cambridge 2003 nk The Constraint Rule of the Maximum Entropy Principle Studies in History and Philosophy of Modern Jo sics 27 47 1996 doi 101016135572198950002274 see Christopher A ichs op cit for proofs rate of 01 ie for either outcome the test is correct 999 of the time If you test positive for Bayes7 syndrome what is the probability that you actually have the disease Use Bayes7 rule to calculate the answer identifying the various factors in Bayes7 ruler Surprisingly the answer turns out to be 50 or The reason is that the prior knowledge of the situation is that you are very unlikely to have the disease and because it is so skewed the prior strongly in uences the posterior expectation 35941 fanz L w mum39s t911 an n quotWW I W0 Iquot I kugt Cy ub O W vol 010 nrvoal7d 7J 1S 1 7quot WM Q wwcv lvquot Vlmwyol ma wisp Iw mm W 337 aqwmnt39w 7W iimas b W 39 J wwwlg flv quotml vpolg 37 un rtww3 annaquw BF 1 rib 39 nng cww Mcow Mb i quot 7 3 L wgrnwn ifth LroqLvaW In 0 can 31 911J quot5 9 WI WNW quotU WM I L7l7 1 4Hf39w vvnor MCIS 730 AU rm ow Kr y wavy L Vawnvmpa v W s 99 arrivan 71912 ju n z 4 lt91 my HvIWMWnM cfan WCCS WPpr 39quot 7 M 341 K 7 quotW VQ WVOVC JXa t jwo39 rump 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Course Notes The Density Operator and Entanglement 1 Multiple Degrees of Freedom 11 Merging Hilbert Spaces Suppose two degrees of freedom are prepared in two quantum states completely independently of each other This could happen say for two particles prepared in separate distant galaxies We will refer to the two degrees of freedom as particles even though they could correspond to different degrees of freedom of the same system such as the spin and centerofmass position of an atom or the spin and spatial pro le of a photon Labeling the two particles as A and B if the individual states of the particles are WM and W g then we can write the composite state as W WM Eli1h7 1 where 8 denotes the tensor product or direct product Often this is product is written without an explicit tensorproduct symbo WM WM E l gtAWgtB E WAN13 2 The particle labels can even be dropped since the ordering determines which state applies to which particlel We can also see the meaning of the tensor product in component forml Let each separate state be expressed in an orthonormal basis as B A WA 0 WM WE 20 WBA 3 0 Then we can express the composite state as W an laA Bgt7 4 043 where M we lt5 Note that CD43 is still understood to be a vectorlike object with a single indexi Thus there is an implicit bijective mapping of the ordered index pair a to a single index which we simply denote as a i Similarly we can write a density operator for two independent particles by the same tensor product P PM MB 6 We can also write this in component form for the density matrices as A paw pi i 7 where again 04 and u are to be taken as composite indices The same tensorproduct notation applies to Hilbert spaces That is we can write WAT13gt E i 3 8 ifl112gtA E and km E if 12 Entanglement The above composite states described by tensor products of separated states are called separable states However not all states are separable and those that are not separable are called entangled In some sense entanglement is the most quantum77 of all quantum effects Thus we can see that a composite state is entangled if and only if it cannot be written in the separable form W WA We 9 The de nition for density operators is somewhat more general a density operator for a composite system is separable if and only if it can be written in the form P Z P0472 tB 10 Unfortunately given an arbitrary mixed density operator it is dif cult to tell if it corresponds to an entangled state in fact this turns out to be an NPhard problem The point is that two entangled systems do not have local states that can be treated independently This is in con ict with the apparently reasonable assumption of local realism which states that distant systems should have independent observerindependent realities in particular they should not directly in uence each other Herein lies the importance of the famous Bell inequalities and their experimental veri cation local realism contradicts quantum mechanics and so we must either give up locality or realism Most modern practitioners of quantum mechanics choose to give up realism which says that systems have observer independent realities in favor of locality The Bohm formulation of quantum mechanics is a wellknown realistic but nonlocal theory 121 Cloning With the language of entanglement it is relatively simple to demonstrate the no Cloning theorem1 which says that the state of a single quantum system cannot be copied to another particle This turns out to be a simple consequence of unitary evolution Let7s examine just a simple case Suppose that cloning is possible on a twostate system from particle A to particle B Particle B must be in a particular state to begin with and without loss of generality we may take this to be the 0 state Then to copy the eigenstates of A we see that there must be a unitary transformation U such that U0gtA0gtB i0gtAiogtB7 U1gtA0gtB i1gtAilgtBA 11 However if particle A is in the superposition state m i200 m 12 then we see that the cloning operator gives L 2 1W K Wootters and W H Zurek A single quantum cannot be cloned Natme 299 802 1982 D Dieks Communication by EPR devices Physics Letters A 92 271 1982 mum 0gtA0gtB 1gtA1gtB lt13 which is the entangled Schrodingercat state However what we wanted for cloning to work properly is the separable state Iowa 00M l1gtAl0gtB m3 14gt We can see that the problem in this particular example is that U acts nonlocally and thus induces entangle ment between the two particles In fact the controlledNOT CNOT gate is a quantum operation that effects the transformations in Eqs 11 if A and B are in eigenstates the CNOT ips the state of system B if and only if system A is in state 1 Of course it is possible to clone a state if you already know everything about it ie you have classical knowledge of the state or if you have an in nite ensemble of copies Copying a state is possible to within some delity tolerance for a nite ensemble of copies In this case enough measurements may be made to reconstruct the state of the system arbitrarily well and of course this procedure does not correspond to a unitary transformation The problem with the single system is that in general a measurement of the system destroys its state and a single measurement is not enough to determine the state of the systemi Of course there is no problem with the cloning of the basis states as in Eqs 11 the problem is in cloning general states that are not orthogonal to the basis states In particular this means that with a bit of extra information beyond what is contained in the quantum state eg the state of particle A is either l0A or llA but not any coherent superposition of the two cloning may in fact be possible 13 Open Systems Church of the Larger Hilbert Space One important function use of the density operator is in describing open quantum systemsisystems inter acting with auxiliary systems environments or reservoirs that we don7t have access to We will treat open quantum systems in great detail but for now let s examine a simple model for why the density operator is usefuli Consider the entangled state 1 between particles qubits A and Bi Suppose that we have access to particle A but particle B is locked up in a box so that we don7t know anything about it The density operator for the composite system is W l0AgtlOBgtl1AgtllBgt 15gt P WW l0AgtlOBgtlt0AlltOBl l1AgtllBgtlt1Allt1Bl l1AgtllBgtlt0AlltOBl l0AgtlOBgtlt1Allt1Bl 16 We can de ne the reduced density operator that describes only particle A by performing a partial trace over the state of particle B m mm Baamm OAgtlt0A1Agtlt1A reduced density operator 1 7 Thus we can see that the reduced state of particle A corresponds to a completely incoherent superposition of the two states even though the composite system carried a completely coherent superpositioni This is a simple model for the process of decoherencei A quantum system can start in a local state of coherent superpositioni But if it interacts with the environment the coupling causes entanglement between the system and environment since the interaction is nonlocali Because we don7t have access to the state of the environment we must trace over it which reduces the purity of the reduced density operatori Note that we can t keep track of the environment even in principle since it generally has many degrees of freedomi As the interaction continues the entanglement progresses driving the reduced density operator towards a completely incoherent superposition This is at a simple level why classical macroscopic things behave classically coupling to the environment destroys quantum coherencei Conversely whenever we have a system described by a mixed density operator PZpal agtlt al7 18 0 we can always think of it as part of a larger systemi We can see this as follows We will introduce a ctitious environment with orthonormal basis states laEgti Then we can write the state vector for the composite system as mm Z mwim 19 puri cation of mixed state When we compute the total density operator for the composite pure state and trace over the environment we recover the original density operator 18 as the reduced density operator of the larger state This procedure of switching to a larger pure state is referred to as puri cation or the doctrine of the Church of the larger Hilbert space The extra environment degree of freedom is often referred to as the ancillai Often this is a useful picture for thinking about mixed quantum states especially in quantuminformation problemsi 2Terminology introduced by John Smolin see Daniel Gottesman and HoieKwong Lo From Quantum Cheating to Quantum Security Physics Today 53 no 11 22 2000 Online link httpwwwaiporgptvoli53issillp22htmlv

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