A crash course in real-world quantum mechanics

Transcription

1 A crash course n real-world quantum mechancs Basc postulates for an solated quantum system Pure states (mnmum-uncertanty states) of a physcal system are represented by vectors n a complex Hlbert space. may be thought of as the most effcent and complete mathematcal representaton of everythng that you know about the state of the system. A system s dynamcs are specfed by a Hermtan operator H, and tme-evoluton s gven by the Schrödnger Equaton Ò H. Mutually exclusve measurement outcomes correspond to orthogonal projecton operators P 0,P 1,Ê, and the probablty of a partcular outcome s gven by P 2. For a complete measurement specfcaton we must have P 1. Note that f each of the P s rank one, the outcome probabltes are just the square magntudes of the components of n some orthonormal bass. Note that tself s NOT drectly observable! But t does represent suffcent nformaton to compute the samplng dstrbuton for any possble measurement we mght wsh to perform... Drac notaton for quantum states States may be wrtten as ket or bra a è a 0 a 1, 2 a 2 è a a 0 a 1 a 2. 3 By conventon, state vectors are assumed to be normalzed: a 2 1. Bras and kets are related by Hermtan conjugaton: a è è a, è a a è. 4 The nner product of a bra and a ket s a complex number: 1

2 è a b è a 0 a 1 a 2 b 0 b 1 b 2 a 0 b 0 a 1 b 1 a 2 b 2. 5 One typcally drops one of the vertcal bars and wrtes è a b è. Note that è a b è 0 mples that the two vectors are orthogonal. Normalzed states satsfy è è 1. The outer product of a ket and a bra s a lnear operator (matrx): a èè b a 0 a 1 b 0 b 1 b 2 a 2 a 0 b 0 a 0 b 1 a 0 b 2 a 1 b 0 a 1 b 1 a 1 b 2 a 2 b 0 a 2 b 1 a 2 b 2 It s often convenent to work wth bass kets or bras. For example, a è a 0 0 è a 1 1 è a 2 2 è 0 è 1 0 0, 1 è 0 1 0, 2 è 0 0 1,. 6 è a a 0 è 0 a 1 è 1 a 2 è 2, where è jè, j. Hence, a 0 0 è a 2 2 è b 0 è 0 b 1 è 1 a 0 b 0 0 èè 0 a 2 b 0 2 èè 0 a 0 b 1 0 èè 1 a 2 b 1 2 èè 1 a 0 b 0 a 0 b a 2 b 0 a 2 b Of course, the same physcal state a è can be expressed n (uncountably) many dfferent bases. For example, f a è a 0 0 è a 1 1 è, 9 then n the bass xè 1 0 è 2 1 è, y è 1 0 è 1 è, 2 a è 1 2 a 0 a 1 x è 1 2 a 0 a 1 y è. 10 Hermtan conjugate of operators n Drac notaton: aèèb bèèa. Operators act on kets from the left and on bras from the rght. Usng orthonormal bass vectors, t s easy to compute the results. For example, O 0 èè 1 1 èè 0, O a è 0 èè 1 1 èè 0 a 0 0 è a 1 1 è a 2 2 è a 1 0 è a 0 1 è, è a O a 0 è 0 a 1 è 1 a 2 è 2 0 èè 1 1 èè 0 a 0 è 1 a 1 è The outer product of any vector wth tself s a (rank one) projecton operator: # 7 2

3 èè P, P 2 èè èè P. 12 The Schrödnger Equaton The dynamcs of a quantum system s specfed by a Hermtan operator H, called the Hamltonan. Tme-evoluton of quantum states s gven by the Schrödnger Equaton, Ò dt d è H è, 13 where Òh/2 and h à [J sec] s Planck s constant. For fnte-dmensonal systems, (13) s a coupled system of lnear ordnary dfferental equatons. If the physcal system s truly solated (autonomous), then H must be constant and we may wrte the formal soluton t è exp H t 0 è. 14 Ò In some cases t s actually possble to compute the operator exponental, whch s defned (as usual) va Taylor expanson: exp O 1 O 2 2 O2 3 3! O3 4 4! O4. 15 Here s an arbtrary (real) scalar. Note that f O s a Hermtan operator, exp O 1 O 2 2 O2 3 3! O3 4 4! O4 16 and exp O exp O exp O exp O That s, exp O s a untary operator. In the case of the Schrödnger Equaton, we wrte T t 2,t 1 exp H t 2 t 1 18 Ò and refer to T t 2,t 1 as the system s untary propagator or tme development operator from tme t 1 to t 2. Note that T t 2,t 1 1 T t 2,t 1 ß T t 1,t 2 19 can be thought of as an operator that evolves a state backwards n tme. Recall that untary operators may be thought of as the complex generalzaton of rotaton operators n a real vector space. Hence quantum evoluton for an solated system corresponds to a rgd rotaton of the state space. As a consequence, tme evoluton preserves the norms of ndvdual state vectors, and preserves the nner product (angle) between arbtrary pars of state vectors. Note that by takng the Hermtan conjugate of the entre Schrödnger Equaton, we get a tme evoluton equaton for bras: 3

4 Ò d dt è è H, Accordngly, as noted above. è t 2 è t 1 T t 1,t è a t 2 b t 2 è è a t 1 T t 1,t 2 T t 2,t 1 b t 1 è 21 è a t 1 b t 1 è, Quantum computaton Let the ntal state of a quantum regster be gven by x è Ü span 0 è, 1 è, 2 è, 3 è, 4 è,.... We can thnk of ths as somethng lke the nput to a functon F x. Physcal mplementaton of the functon evaluaton corresponds to untary evoluton: x è U x è â F x è. Note that the last equvalence s an assocaton we make n order to vew the untary evoluton as a computaton. Snce quantum mechancal evoluton s untary, we clearly have U 1 2 è j è 1 2 U è U j è 1 2 F è F j è. Ths type of parallelsm appears to be a key ngredent n quantum computaton. Note that snce the fnal state vector s not drectly observable, we cannot smply read-out one or more evaluatons of our choosng. Rather, we must do somethng sneaky to extract global nformaton about the behavor of F. As one mght magne, ths nvolves makng a fnal measurement n a bass other than the computatonal bass 0 è, 1 è, 2 è, 3 è, 4 è,.... Ths effectvely creates nterference between dfferent computatonal paths. Intrnsc uncertanty Note that for any pure state è, we are guarenteed to be able to fnd uncountably many measurements P 0,P 1,Ê for whch the result cannot be predcted wth certanty. Hence quantum mechancs contans ntrnsc uncertanty no matter how much care you take n the accurate preparaton of a quantum system, many of ts qualtes are unsharply defned. Nonorthogonalty and mperfect dstngushablty 4

5 The followng smple example s ntended to hghlght our frst genune example of a mysterous property of quantum mechancs that follows drectly from the rules for representaton and predcton. Consder a nce smple two-dmensonal Hlbert space wth bass kets x è and y è. Gven an arbtrary par of states c è c x x è c y y è, d è d x x è d y y è, 28 under what condtons s t possble to fnd a (standard) measurement that can dstngush them wth zero probablty of error? Recall that a standard measurement s specfed by a complete set of orthogonal projectors. Snce for now we are workng n just two dmensons, we actually only need to specfy a sngle ket è. Then unambguously, P 1 èè, P 2 1 P In the current scenaro, we are tryng to pck è such that Pr 1 c è c P 1 c è 1, Pr 1 d è d P 1 d è 0, Pr 2 c è c P 2 c è 0, Pr 2 d è d P 2 d è In order to satsfy the frst condton, we clearly need to choose è c è. 31 As a consequence Pr 1 d è d P 1 d è c x d x c y d y 2, 32 and we fnd that our two states c and d are not perfectly dstngushable unless they have zero overlap: Pr 1 d 0 ff è c d è Hopefully t should be clear that among all possble pars of vectors n a Hlbert space, only a vanshng fracton are orthogonal! And yet, accordng to our quantum representaton rule, every vector n the Hlbert space corresponds to a dstnct physcal state of the system that s, to a dstnct preparaton procedure. Even though quantum measurement theory allows for non-projectve measurements (see thrd term of ths course), t s nonetheless a theorem that No measurement can dstngush nonorthogonal states, wth zero probablty of error, nasngletral. 5

6 It s a profound mystery that quantum mechancs presents us wth such a huge space of possble physcal states wthout allowng us perfectly to dstngush between them. Why s t that we can put more nformaton nto the preparaton of a quantum system than we can get back out n measurements of that very same system? In a sense t s embarassng that we don t yet have a good answer, but some of the most nsghtful scentsts I know beleve that ths s the sngle most mportant queston for contemporary research n quantum theory. Ensembles of quantum states; the densty operator Let s say we re workng n a nce, smple two-dmensonal Hlbert space and that we ve chosen orthonormal bass kets x è and y è. For the followng dscusson, we ll need to defne two state vectors A è a x x è a y y è, B è b x x è b y y è. 25 Suppose I ask you to perform a seres of N measurements on ths system (here N s just a large nteger), correspondng to the projectors P x x èè x, P y y èè y. 26 The trck s, n ths seres of measurements I wll sometmes prepare the ntal state A è (wth probablty p) and sometmes B è (wth probablty 1 p). That s, I wll be gvng you a mxed ensemble of quantum states. How shall we predct the overall number of tmes n x we expect to obtan the measurement outcome x? Accordng to smple probablty theory, n x N Pr A Pr x A Pr B Pr x B N p è A P x A è 1 p è B P x B è N p a x 2 1 p b x Note that snce 0 â p â 1, the quantty n x /N s bounded from below by the smaller of a x 2 and b x 2. In partcular, f both of these quanttes are nonzero than n x must also be greater than zero. A very dfferent expresson for n x would be obtaned f, nstead of a mxed ensemble, I were to present you wth a coherent superposton of the states A è and B è, correspondng to the pure state p A,p B è p A,p B è p A A è p B B è p A a x p B b x x è p A a y p B b y y è. 28 (Note that we should be careful n choosng p A and p B such that p A,p B è s normalzed.) In ths case, we would predct 6

7 n x N è p A,p B P x p A,p B è N p A a x p B b x Note that n certan cases, e.g. p A a x p B b x â 0, t s possble to choose p A,p B such that n x 0 a x 2, b x 2 through the phenomenon of destructve nterference. Ths s the truly mportant dstncton between coherent superpostons (of the type that produce a sngle pure state) and ncoherent admxtures (of the type that produce a mxed ensemble of quantum states). A smplfyng notaton can be ntroduced for performng computatons wth mxed quantum ensembles. If the states composng the ensemble are labelled è and have probabltes p, then the densty operator for ths ensemble s defned as p èè. 30 Note that s ndeed an operator on the Hlbert space, and has the form of a lnear combnaton of projecton operators. However, the ensemble representaton used to defne a densty operator s not necessarly also a spectral decomposton, as the varous è that consttute an ensemble do not need to be mutually orthogonal. The densty operator s automatcally Hermtan, and furthermore has the property that Tr Here Tr denotes the trace operaton Tr è k k è, 32 k where k è s any orthonormal bass for the Hlbert space the numercal result s ndependent of choce of bass. In partcular, snce s Hermtan we can chose to take the trace n ts own egenbass, whch makes t clear that Tr, 33 where are the egenvalues of. Note that f happens to be avalable n matrx form, we can further make use of the fact that the sum of the egenvalues of a matrx s equal to the sum of ts dagonal elements. Densty operators can represent ether pure states, èè, 34 or mxed states p èè 35 where there s more than one p 0. Note that n the former (pure state) case s a true projecton operator, so pure : In partcular, Tr 2 1 for a pure state. For a mxed state, however, we can use the spectral decomposton to show that Tr 2 1. We start by wrtng 7

8 2 2 P 2 P, 37 and note that snce Tr 1, each of the must be strctly less than one for a mxed state. Hence the egenvalues of 2, whch are equal to the 2, must add up to less than one. Jont state space for two subsystems Suppose we have two ndependent quantum systems. It seems clear that we can separately consder the representaton of ther physcal states n two ndependent Hlbert spaces. Labellng the systems A and B, we can smply chose state vectors A è Ü H A, 3 and B è Ü H B. 4 What f we need to brng these systems together and let them nteract? The jont state space for two such systems corresponds to the tensor product of H A and H B, denoted H AB H A å H B. Let N A be the dmenson of H A, and N B the dmenson of H B.If 1 A è, 2 A è, 3 A è, Ê s a complete orthonormal bass for H A and 1 B è, 2 B è, 3 B è, Ê s a complete orthonormal bass for H B, then H A å H B s the Hlbert space of dmenson N AB N A N B spanned by the vectors of the form A è å j B è. (Note that by extenson, ths means we can buld a quantum regster whose state lves n a Hlbert space of dmenson 2 L 1 from L consttuent qubts.) Hence arbtrary states n H AB have the form N A AB è 1 j1 N B c j A è å j B è. 5 As long as we fx an orderng for the new bass states A è å j B è, thesetofn A N B complex coeffcents can be used as a vector representaton for kets n H AB. The tensor product operaton between vectors has the followng propertes: 1. Lnearty: A è å B è A è å B è, where s a complex number 2. Dstrbutvty: A è å B 1 è B 2 è A è å B 1 è A è å B 2 è. 3. Commutatvty : formally, A è å B è sthesameas B è å A è. In practce however, t s wse to use consstent orderng. 8

9 4. Adjont: A è å B è è A å è B. 5. Scalar product: è 1 A å è B A è å B è è 1 2 A A èè 1 2 B B è. It s mportant to note that bass kets A è å j B è Ü H AB thus nhert orthogonalty from ther factors n H A and H B. Entanglement The most profound consequence of ths mathematcal rule for representaton of jont states s that there exst AB è Ü H AB that cannot be expressed the tensor product of a state A è Ü H A wth a state B è Ü H B. Such nonfactorzable states are sad to be entangled. For example, let s consder two two-dmensonal systems. Say we have chosen orthonormal bases 0 A è, 1 A è for H A and 0 B è, 1 B è for H B. Then H AB s spanned by the four states 0 A è å 0 B è, 0 A è å 1 B è, 1 A è å 0 B è, 1 A è å 1 B è. 6 Factorzable (nonentangled) states n H AB are all of the form fac AB è c A 0 0 A è c A 1 1 A è å c B 0 0 B è c B 1 1 B è c A 0 c B 0 0 A è å 0 B è c A 0 c B 1 0 A è å 1 B è c A 1 c B 0 1 A è å 0 B è c A 1 c B 1 1 A è å 1 B è. 7 That s, a certan relatonshp exsts between the coeffcents of the four bass states n H AB. A smple example of an entangled state, whose coeffcents do not exhbt the above relatonshp, s AB è A è å 0 B è 1 A è å 1 B è â A è å B è. 8 When the jont state of two subsystems s entangled, there s no way to assgn a pure quantum state to ether subsystem alone. As we shall see below, t s possble to ascrbe mxed quantum states to each of the subsystems consdered alone, but frst we ll need to have a look at operators on H AB. Tensor products of operators If A s an operator on H A and B s an operator on H B, then A å B 9 s a vald operator on H AB. Its acton on an arbtrary state s defned by AB è c j A è å j B è 10,j 9

10 A å B AB è c j A A è å B j B è. 11,j Note that the usual relatonshp holds between projectors on the jont state space and outer-products of jont state vectors: A è å B è è A å è B A èè A å B èè B P A å P B. 13 Hence any complete set of jont projectors (summng to the dentty operator on H AB ) specfes a complete measurement. As was the case wth state vectors, lnear combnatons of tensor-product operators are also vald opeators on H AB : O AB c m A m å B m. 14 m Hence, not all operators on a jont state space are factorzable. Gven subsystem densty operators A and B, we can form a tensor-product densty operator that descrbes a mxed ensemble of states n H AB : AB A å B. 15 In general, one can form convex combnatons of such AB to construct new jont densty operators, whch may or may not be factorzable. One can also construct jont densty operators drectly from ensembles of pure states n H AB. For nstance, the densty operator correspondng to the entangled state descrbed above s AB è A è å 0 B è 1 A è å 1 B è AB AB èè AB and n general A èè0 A å 0 B èè0 B 0 A èè1 A å 0 B èè1 B 1 A èè0 A å 1 B èè0 B 1 A èè1 A å 1 B èè1 B AB p AB, 16 èè AB. 17 Note that operators on a tensor-product space can be expressed as complex matrces o kl : O AB o kl k AB èè l AB, 18 kl where the summatons both run over a complete set of N AB bass vectors. Gven matrx representatons for subsystem operators A and B, tscustomaryto choose an orderng for the bass states of the jont space such that 10

11 A å B Û a 11 B a 12 B a 13 B a 21 B a 22 B a 23 B a 31 B a 32 B a 33 B. 19 For example f 1 A è, 2 A è, Ê s the orthnormal bass for H A used n defnng the matrx representaton of A, and smlarly for H B, then 1 AB è Û 1 A è å 1 B è, 2 AB è Û 1 A è å 2 B è, 3 AB è Û 1 A è å 3 B è, N B 1 AB è Û 2 A è å 1 B è, 20 As a result, the common class of operators 1 A å B wll have block-dagonal representatons. Partal trace and reduced densty operators We can now defne the partal trace operaton. Let AB be a densty operator on H AB : AB jkl A è å j B èè k A å è l B, 36 jkl where the summatons are take over orthonormal bases for H A and H B. We defne the partal trace of AB over the B subsystem to be A â Tr B AB N B m1,k1 N A,k1 N A mkm A èè k A m1 N B mkm A èè k A 38 Here A s called the reduced densty operator for subsystem A. It provdes the best possble representaton of subsystem A wthn H A, when the jont state of A and B s entangled/nonfactorzable. A notatonally convenent (but mathematcally mprecse) way of computng the partal trace s as follows: 11

12 N B Tr B AB è m B AB m B è m1 N B m1 N B m1 è m B,k1 jkl A è å j B èè k A å è l B jkl m B è N A mkm A èè k A. 39 Note that f we start wth a pure entangled state and take the partal trace over one subsystem, the reduced densty operator for the remanng subsystem wll represent a mxed state! Hence we see that entanglement between a system of nterest and reservor degrees of freedom removes coherence and gves rse to excess uncertantes... Fnally, let us see that a par of systems that starts out n a factorzable ntal state can evolve nto an entangled state va Hamltonan evoluton. Let s work agan wth our same two two-dmensonal quantum systems. Let the ntal state be AB 0 è A è 1 A è å 0 B è 1 B è and suppose the Hamltonan s A è å 0 B è 0 A è å 1 B è 1 A è å 0 B è 1 A è å 1 B è, 21 H where the orderng of bass states n H AB s The untary evoluton operator s After a tme t 1 2 Ò, , 22 0 A è å 0 B è, 0 A è å 1 B è, 1 A è å 0 B è, 1 A è å 1 B è. 23 T t,0 T t,0 exp Ò e / e / e / e /2 Ht. 24 and , 25 12

13 AB t è T t,0 AB 0 è 2 0 A è å 0 B è 0 A è å 1 B è 1 A è å 0 B è 1 A è å 1 B è, 26 whch s an entangled state. How do we know that ths s an entangled state? Probably by nspecton, but let s also check the trace of 2 A. Frst off, the jont densty operator s AB AB t èè AB t A0 B èè0 A 0 B 0 A 0 B èè0 A 1 B 0 A 0 B èè1 A 0 B 0 A 0 B èè1 A 1 B 0 A 1 B èè0 A 0 B 0 A 1 B èè0 A 1 B 0 A 1 B èè1 A 0 B 0 A 1 B èè1 A 1 B 1 A 0 B èè0 A 0 B 1 A 0 B èè0 A 1 B 1 A 0 B èè1 A 0 B 1 A 0 B èè1 A 1 B 1 A 1 B èè0 A 0 B 1 A 1 B èè0 A 1 B 1 A 1 B èè1 A 0 B 1 A 1 B èè1 A 1 B. 27 Next we take the partal trace over B: A Tr B AB è0 B AB 0 B è è1 B AB 1 B è A èè0 A 0 A èè1 A 1 A èè0 A 1 A èè1 A 0 A èè0 A 0 A èè1 A 1 A èè1 A 1 A èè1 A A èè0 A 1 A èè1 A 1 2 1A Hence A 1 4 1A 2, and Tr A 1, whch s clearly less than one. Snce densty operators 2 that correspond to pure states are projectors, we conclude that no pure state can be assgned to subsystem A when the jont state of the AB system s AB t è. Note that evolutons producng entanglement between a par of qubts represent quantum logc gates. Luckly, arbtrary untary transformatons on a Hlbert space of dmenson 2 L 1 can be decomposed nto crcuts made up of sequences of one- and two-qubt gates. As t turns out, a gate set composed of sngle-qubt gates plus the above controlled sgn gate (a specal case of a quantum phase gate ) s unversal for quantum computng. On the other hand, entanglng evoluton between a system and reservor wll compromse the purty of the system s quantum state. Essentally any Hamltonan that couples system and reservor degrees-of-freedom wll create entanglement, hence the ubquty of envronmental decoherence n real physcal systems. Open systems The evoluton of a real (open) system s generally descrbed by a Master Equaton. In smple cases, ths takes the form of a dfferental equaton for the system densty operator: d t H t t H dt Ò j 2L j t L j Ê L j Ê L j t t L j Ê L j. 13

14 The frst term represents the coherent part of a system s evoluton, whle the terms n the sum represent the decoherng effects of resdual couplngs to the envronment. They can be thought of as arsng from contnual entanglng wth and tracng over reservor degrees of freedom. They re not untary! Next tme, we ll look at an example of a realstc Master Equaton (for atoms trapped n an optcal cavty) and dscuss the sgnfcance of each of the terms that appear there... Overvew of recommended readng for Thursday and Frday Cavty QED 1. Turchette 1995 Orgnal expermental demonstraton of an optcal quantum gate 2. Crac 1997 Proposed scheme for quantum communcaton wth atoms and photons 3. Ye 1999 Recent progress on trappng atoms n cavtes 4. Pellzzar 1995 An early proposed scheme for quantum computng wth atoms n cavtes Trapped ons 1. Kelpnsk 2002 Proposal for scalable on-trap computng, based on demonstrated technques 2. Monroe 1995 Orgnal expermental demonstraton of an on-trap quantum gate 3. Kng 1998 Crucal techncal demonstraton of coolng multple ons 4. Wneland revew A glmpse at the true techncal complexty of experments lke ths! 5. Kelpnsk 2001 Demonstraton of a decoherence-free subspace technque wth ons 6. Crac 1995 Theoretcal proposal that led to Monroe 1995 Ensemble NMR 1. Vandersypen 2001 Crownng glory of NMR thus far: Shor s algorthm to factor Havel 2002 Revew by another leadng group n NMR quantum computng 3. Mencucc 2002 Arguments for a conservatve nterpretaton of NMR experments 4. Braunsten 1999 Earler verson of the above 5. Chuang 1998b An early demonstraton of quantum algorthms va NMR 14