Photons and Quantum Information. Stephen M. Barnett

Transcription

1 Photons and Quantum Informaton Stehen M. Barnett

2 . bt about hotons. Otcal olarsaton 3. Generalsed measurements 4. State dscrmnaton Mnmum error Unambguous Maxmum confdence

3 . bt about hotons Photoelectrc effect - Ensten 95 ev h W BUT Modern nterretaton: resonance wth the atomc transton frequency. We can descrbe the henomenon quanttatvely by a model n whch the matter s descrbed quantum mechancally but the lght s descrbed classcally. Photons?

4 Sngle-hoton (?) nterference - G. I. Taylor 99 Lght source Smoked glass screens Screen wth two slts Longest exosure - three months Photograhc late ccordng to Sr J. J. Thomson, ths sets a lmt on the sze of the ndvsble unts.

5 Sngle hotons (?) Hanbury-Brown and Twss P(,) RI TI RT I P() R I P() T I I () g () I Blackbody lght g () () = Laser lght g () () = Sngle hoton g () () =!!! volaton of Cauchy-Swartz nequalty

6 Sngle hoton source - sect 986 Detecton of the frst hoton acts as a herald for the second. Second hoton avalable for Hanbury-Brown and Twss measurement or nterference measurement. Found g () () ~ (sngle hotons) and frnge vsblty = 98%

7 Two-hoton nterference - Hong, Ou and Mandel 987 Each hoton then nterferes only wth tself. Interference between dfferent hotons never occurs Drac 5/5 beam sltter R=T=/ P = R T = / Two hotons n overlang modes Boson clumng. If one hoton s resent then t s easer to add a second.

8 Two-hoton nterference - Hong, Ou and Mandel 987 Each hoton then nterferes only wth tself. Interference between dfferent hotons never occurs Drac 5/5 beam sltter R=T=/ P = R T = / Two hotons n overlang modes Boson clumng. If one hoton s resent then t s easer to add a second.

9 Two-hoton nterference - Hong, Ou and Mandel 987 Each hoton then nterferes only wth tself. Interference between dfferent hotons never occurs Drac 5/5 beam sltter R=T=/ P =!!! Two hotons n overlang modes Destructve quantum nterference between the amltudes for two reflectons and two transmssons.

10 . bt about hotons. Otcal olarsaton 3. Generalsed measurements 4. State dscrmnaton Mnmum error Unambguous Maxmum confdence

11 Maxwell s equatons n an sotroc delectrc medum take the form: E B E B B t E c t E, B and k are mutually orthogonal For lane waves (and lab. beams that are not too tghtly focussed) ths means that the E and B felds are constraned to le n the lane erendcular to the drecton of roagaton. E S E B B k

12 Consder a lane EM wave of the form E B E B ex ( kz ex ( kz t) t) If E and B are constant and real then the wave s sad to be lnearly olarsed. B E Polarsaton s defned by an axs rather than by a drecton: B E

13 If the electrc feld for the lane wave can be wrtten n the form E E( j)ex ( kz t) Then the wave s sad to be crcularly olarsed. B For rght-crcular olarsaton, an observer would see the felds rotatng clockwse as the lght aroached. E

14 The Jones reresentaton We can wrte the x and y comonents of the comlex electrc feld amltude n the form of a column vector: y x y x y x e E e E E E The sze of the total feld tells us nothng about the olarsaton so we can convenently normalse the vector: Horzontal olarsaton Vertcal olarsaton Left crcular olarsaton Rght crcular olarsaton

15 One advantage of ths method s that t allows us to descrbe the effects of otcal elements by matrx multlcaton: Lnear olarser (orented to horzontal): 45, 9, Quarter-wave late (fast axs to horzontal): 45, 9, Half-wave late (fast axs horzontal or vertcal): The effect of a sequence of n such elements s: B d c b a d c b a B n n n n

16 We refer to two olarsatons as orthogonal f * E E Ths has a smle and suggestve form when exressed n terms of the Jones vectors: f to orthogonal s * * * * B B B B B B B B There s a clear and smle mathematcal analogy between the Jones vectors and our descrton of a qubt.

17 Sn and olarsaton Qubts Poncaré and Bloch Sheres Two state quantum system Bloch Shere Electron sn Poncaré Shere Otcal olarzaton

18 We can realse a qubt as the state of sngle-hoton olarsaton Horzontal Vertcal Dagonal u Dagonal down Left crcular Rght crcular

19 . bt about hotons. Otcal olarsaton 3. Generalsed measurements 4. State dscrmnaton Mnmum error Unambguous Maxmum confdence

20 Probablty oerator measures Our generalsed formula for measurement robabltes s P( ) Tr The set robablty oerators descrbng a measurement s called a robablty oerator measure (POM) or a ostve oerator-valued measure (POVM). The robablty oerators can be defned by the roertes that they satsfy:

21 Proertes of robablty oerators n n I. They are Hermtan Observable II. They are ostve n Probabltes III. They are comlete n n Î Probabltes IV. Orthonormal j j??

22 Generalsed measurements as comarsons S Preare an ancllary system n a known state: S S + Perform a selected untary transformaton to coule the system and anclla: U S S Perform a von Neumann measurement on both the system and anclla: S

23 The robablty for outcome s P( ) U S S U U S S U POM rules: I. Hermtcty: U U U U II. Postvty: U S The robablty oerators act only on the system state-sace. III. Comleteness follows from: Î,S

24 Generalsed measurements as comarsons We can rewrte the detecton robablty as P( ) P U S U P s a rojector onto correlated (entangled) states of the system and anclla. The generalsed measurement s a von Neumann measurement n whch the system and anclla are comared. n m P P n S P m

25 Smultaneous measurement of oston and momentum The smultaneous erfect measurement of x and would volate comlementarty. Poston measurement gves no momentum nformaton and deends on the oston robablty dstrbuton. x

26 Smultaneous measurement of oston and momentum The smultaneous erfect measurement of x and would volate comlementarty. Momentum measurement gves no oston nformaton and deends on the momentum robablty dstrbuton. x

27 Smultaneous measurement of oston and momentum The smultaneous erfect measurement of x and would volate comlementarty. Jont oston and measurement gves artal nformaton on both the oston and the momentum. Poston-momentum mnmum uncertanty state. x

28 POM descrton of jont measurements Probablty densty: ), ( ), ( m m m m x Tr x Mnmum uncertanty states: Î,, 4 ) ( ex, 4 / m m m m m m m m m m x x d dx x x x x dx x

29 Ths leads us to the POM elements: m m m m m m x x x,, ), ( The assocated oston robablty dstrbuton s 4 ) Var( & ) Var( ) ( ex ) ( x x x x x x dx x m m m m Increased uncertanty s the rce we ay for measurng x and.

30 . bt about hotons. Otcal olarsaton 3. Generalsed measurements 4. State dscrmnaton Mnmum error Unambguous Maxmum confdence

31 The communcatons roblem lce reares a quantum system n one of a set of N ossble sgnal states and sends t to Bob selected. rob. Prearaton devce Measurement devce Measurement result j P( j ) Bob s more nterested n Tr j P( j) Tr j Tr( j )

32 In general, sgnal states wll be non-orthogonal. No measurement can dstngush erfectly between such states. Were t ossble then there would exst a POM wth Comleteness, ostvty and What s the best we can do? Deends on what we mean by best. and ostve

33 Mnmum-error dscrmnaton We can assocate each measurement oerator wth a sgnal state. Ths leads to an error robablty ny POM that satsfes the condtons wll mnmse the robablty of error. j j j N j e Tr P j k j j j N k k k k k k k j j j, ) (

34 For just two states, we requre a von Neumann measurement wth rojectors onto the egenstates of wth ostve () and negatve () egenvalues: mn P e Tr Consder for examle the two ure qubt-states cos sn cos( ) cos sn

35 The mnmum error s acheved by measurng n the orthonormal bass sanned by the states and. We assocate wth and wth : P e The mnmum error s the Helstrom bound mn P e 4 /

36 sngle hoton only gves one clck P = + P = But ths s all we need to dscrmnate between our two states wth mnmum error.

37 more challengng examle s the trne ensemble of three equrobable states: It s straghtforward to confrm that the mnmum-error condtons are satsfed by the three robablty oerators

38 Smle examle - the trne states Three symmetrc states of hoton olarsaton 3 Mnmum error robablty s /3. 3 Ths corresonds to a POM wth elements j 3 j j 3 How can we do a olarsaton measurement wth these three ossble results?

39 Polarsaton nterferometer - Sasak et al, Clarke et al. / 6 / 3/ 6 / 6 / 6 / 3 PBS 3 / 3 / PBS PBS / 3/ 6 / 6 / / 3 / 3 / / / / 3 / / 3 / / 6 / 6

40 Unambguous dscrmnaton The exstence of a mnmum error does not mean that error-free or unambguous state dscrmnaton s mossble. von Neumann measurement wth P P wll gve unambguous dentfcaton of : result error-free result? nconclusve

41 There s a more symmetrcal aroach wth? Î Result Result Result? State State

42 How can we understand the IDP measurement? Consder an extenson nto a 3D state-sace b a a b

43 Unambguous state dscrmnaton - Huttner et al, Clarke et al.? a b a b smlar devce allows mnmum error dscrmnaton for the trne states.

44 Maxmum confdence measurements seek to maxmse the condtonal robabltes for each state. For unambguous dscrmnaton these are all. Bayes theorem tells us that so the largest values of those gve us maxmum confdence. ) ( P k k k k P P P ) ( ) ( ) (

45 The soluton we fnd s where j j j j Croke et al Phys. Rev. Lett. 96, 74 (6)

46 Examle 3 states n a - dmensonal sace Maxmum Confdence Measurement: Inconclusve outcome needed

47 Otmum robabltes Probablty of correctly determnng state maxmsed for mnmum error measurement Probablty that result obtaned s correct maxmsed by maxmum confdence measurement:

48

49 Results: Maxmum confdence Mnmum error

50 Conclusons Photons have layed a central role n the develoment of quantum theory and the quantum theory of lght contnues to rovde surrses. True sngle hotons are hard to make but are, erhas, the deal carrers of quantum nformaton. It s now ossble to demonstrate a varety of measurement strateges whch realse otmsed POMs The subject of quantum otcs also embraces atoms, ons molecules and solds...