VII. Quantum Entanglement

Transcription

1 VII. Quantum Entanglmnt Quantum ntanglmnt is a uniqu stat of quantum suprposition. It has bn studid mainly from a scintific intrst as an vidnc of quantum mchanics. Rcntly, it is also bing studid as a basic lmnt for quantum information procssing such as quantum computing and quantum communication. This chaptr dscribs quantum ntanglmnt. Rviw of polarization stat of a photon Th polarization stat of a photon is a good matrial for xplaining quantum ntanglmnt. Classically, light is an lctromagntic wav. Elctric/magntic fild oscillats along th transvrsal dirction whil propagating. Sinc th transvrsal dirction is two-dimnsional, th dirction of th oscillation has two dgrs of frdom. Th way of th oscillation in th transvrsal two-dimnsional plan is calld polarization stat. horizontal linar polarization vrtical linar polarization y y z: propagating dirction z: propagating dirction x, Ey ) (,0) x, Ey ) (0,) 45dg. right diagonal right circular E y E y E x E x, E y ) (,), E y ) i / (, ) (, i) 45dg. lft diagonal lft circular, E y ) i (, ) (, ), E y ) i / (, ) (, i)

2 Th abov dscription is for classical lightwav. How about th polarization stat of a photon? att. on photon polarization bam splittr () stat in which on photon is horizontally polarizd. H> Y> = c H H> + c V V> stat in which on photon is vrtically polarizd. H> and V> ar ignstats of th physical quantity oprator of whthr a photon is horizontal or vrtical. Thus, an arbitrary photon stat is a suprposition of a horizontal photon and a vrtical photon, i.., a linar combination of H> and V>, whr H> and V> ar th bass forming a Hilbrt spac. V> For xampl, 45dg. right diagonal: D H V 45dg. lft diagonal: D H V right circular: R H i V lft circular: L H i V Gnrally spaking, bass ar ignstats of a physical quantity oprator. Anothr physical quantity oprator has anothr st of ignstats (bass), and thn a quantum stat can b xprssd by th othr bass, dpnding on th physical quantity. att. l/4 plat R> Th physical quantity is { R>, L>} L> Y> = c R R> + c L L>

3 Quantum ntanglmnt Suppos that two photons ar rgardd as on quantum stat (i.., a product stat). For a two-dimnsional Hilbrt spac, a product stat is gnrally xprssd as 3 Y Y Y c { ch H cv V } { ch H cv V } H H c H V c V H c V V H> : photon # is stat H> V> : photon # is stat V> H> : photon # is stat H> V> : photon # is stat V> Howvr, a particular product stat xprssd as blow can xist. Y { H H V V } Y { H H V V } Y3 { H V H V } Y4 { H V H V } Such a stat is calld quantum ntanglmnt or quantum ntangld stat, which has uniqu proprtis much diffrnt from th classical sns. Proprtis of quantum ntanglmnt Whn th abov stat is obsrvd, th rsultant stat is H> H> or V> V> (in cas of Y >). - Th obsrvd stat is random for ach photon. - Thr is a corrlation btwn th obsrvd stats of ach photon. Whn photon # is H>, photon # is always H>. Whn photon # is V>, photon # is always V>. - This corrlation is irrspctiv of th distanc btwn th two photons. Th stat of photon # is dtrmind instantly at a momnt whn photon # is obsrvd. # ntanglmnt # sourc nonlocality of wavfunction

4 - A corrlation is also mad for diffrnt bass. Y { H H V V } R ( H i V ) L ( H i V ) H ( R L ) i V ( R L ) 4 {( { { R L )( R L ) ( R L )( R L R R R L L R L L R R R L L R L L } R L L R Diffrnc from classical ntanglmnt A corrlation looks similar to th abov can b classically obtaind by a systm blow. } signal sourc )} pol. mod. # # pol. mod. H> or V> H> or V> Th gnratd stat is Y V Y H H or V - Th obsrvd stat is random for on photon. - Th obsrvd stats of ach photon hav a corrlation. Whn photon # is H>, photon # is always H>. Whn photon # is V>, photon # is always V>. looks lik ntanglmnt at a first glanc But, how about in diffrnt bass,,,,,, Y { R R R L L R L L } or Y { R R R L L R L L } no corrlation A corrlation is mad irrspctiv of basis in quantum ntanglmnt.

5 About probability amplitud of ntangld stat A stat blow looks lik quantum ntanglmnt. Y c H H c V V Th obsrvd stat is random for on photon. Whn photon # is H>, photon # is always H>. Whn photon # is V>, photon # is always V>. 5 But, in anothr bass,,,,, Y { c( R L )( R L ) c ( R L )( R L )} {( c c ) R R ( c c ) R L ( c c) L R ( c c) L L } No corrlation for arbitrary c and c. For a corrlation to b mad, or c = c c = - c Y> = c { R> L> + L> R> } Y> = c { R> R> + L> L> } (normalizd) Y { R L L R } Y { R R L L } Quantum-ntangld stat is Y { H H V V } Similarly, Y { H V V H } is also quantum ntanglmnt. Gnrally spaking, prfct quantum ntanglmnt is Y { a b a b} Y { a b a b} { >, >}: (two-dimnsional) basis How to gnrat quantum ntanglmnt Whn intns pump light is injctd into a nonlinar mdium, signal and idlr photons ar simultanously gnratd from a pump photon via nonlinar paramtric intraction. pump light nonlinar mdium pump photon (f p ) signal photon (f s ) idlr photon (f i )

6 Two nonlinar mdia ar prpard, and th stup blow is constructd. 6 pump light NL H s H i signal idlr NL V s V i signal idlr Y Suppos that th probability of on-pair gnration = r. Thn, th probability that both NLs gnrat a photon pair = r, th probability that ithr on of NLs gnrats a photon pair = r( r). r << r( r) >> r : A photon pair is gnratd ithr at NL or NL. Whn a photon pair is output, w do not know which NL gnrats it. suprpositiond stat i Y { H s H i V s V i} ( : phas diffrnc through th two paths) normalization from NL = 0 from NL Y { H s H i V s V i} ntanglmnt Applications of quantum ntnaglmnt Quantum cryptography Quantum Computing Quantum tlportation Quantum rpatr Quantum ntworks tc.