1 Mixed Quantum State. 2 Density Matrix. CS Density Matrices, von Neumann Entropy 3/7/07 Spring 2007 Lecture 13. ψ = α x x. ρ = p i ψ i ψ i.

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1 CS 94- Desty Matrces, vo Neuma Etropy 3/7/07 Sprg 007 Lecture 3 I ths lecture, we wll dscuss the bascs of quatum formato theory I partcular, we wll dscuss mxed quatum states, desty matrces, vo Neuma etropy ad the trace dstace betwee mxed quatum states Mxed Quatum State So far we have dealt wth pure quatum states ψ α x x x Ths s ot the most geeral state we ca th of We ca cosder a probablty dstrbuto of pure states, such as 0 wth probablty / ad wth probablty / Aother possblty s the state { wth probablty / 0 wth probablty / I geeral, we ca th of mxed state {p, ψ } as a collecto of pure states ψ, each wth assocated probablty p, wth the codtos 0 p ad p Oe cotext whch mxed states arse aturally s quatum protocols, where two players share a etagled pure quatum state Each player s vew of ther quatum regster s the a probablty dstrbuto over pure states acheved whe the other player measures ther regster Aother reaso we cosder such mxed states s because the quatum states are hard to solate, ad hece ofte etagled to the evromet Desty Matrx Now we cosder the result of measurg a mxed quatum state Suppose we have a mxture of quatum states ψ wth probablty p Each ψ ca be represeted by a vector C, ad thus we ca assocate the outer product ψ ψ ψ ψ, whch s a matrx a a a N ā ā ā N a ā a ā a ā N a ā a ā a ā N a N ā a N ā a N ā N We ca ow tae the average of these matrces, ad obta the desty matrx of the mxture {p, ψ }: ρ p ψ ψ We gve some examples Cosder the mxed state 0 wth probablty of / ad wth probablty / The , CS 94-, Sprg 007, Lecture 3

2 ad 0 Thus ths case ρ / 0 0 / Now cosder aother mxed state, ths tme cosstg of + wth probablty / ad wth probablty / Ths tme we have + + /, ad / Thus ths case the offdagoals cacel, ad we get ρ / 0 0 / Note that the two desty matrces we computed are detcal, eve though the mxed state we started out was dfferet Hece we see that t s possble for two dfferet mxed states to have the same desty matrx Noetheless, the desty matrx of a mxture completely determes the effects of mag a measuremet o the system: Theorem 3: Suppose we measure a mxed state {p, ψ } a orthoormal bases β The the outcome s β wth probablty β ρ β Proof: We deote the probablty of measurg β by Pr[] The Pr[] p ψ β p β ψ ψ β β p ψ ψ β β ρ β We lst several propertes of the desty matrx: ρ s Hermta, so the egevalues are real ad the egevectors orthogoal If we measure the stadard bass the probablty we measure, P[] ρ, Also, the egevalues of ρ are o-egatve Suppose that λ ad e are correspodg egevalue ad egevector The f we measure the egebass, we have Pr[e] e ρ e λ e e λ CS 94-, Sprg 007, Lecture 3

3 3 trρ Ths s because f we measure the stadard bass ρ, Pr[] but also Pr[] so that ρ, Pr[] Cosder the followg two mxtures ad ther desty matrces: cos θ 0 +s θ wp / cθ cθ sθ sθ cθ cos θ 0 s θ wp / cθ sθ sθ c θ cθsθ cθsθ s θ c θ cθsθ cθsθ s θ cos θ 0 0 s θ 0 wpcos θ cos 0 0 wp s θ s 0 θ 0 0 cos 0 0 s cos θ 0 0 s θ Thus, sce the mxtures have detcal desty matrces, they are dstgushable 3 Vo Neuma Etropy We wll ow show that f two mxed states are represeted by dfferet desty measuremets, the there s a measuremet that dstgushes them Suppose we have two mxed states, wth desty matrces A ad B such that A B We ca as, what s a good measuremet to dstgush the two states? We ca dagoalze the dfferece A B to get A B EΛE, where E s the matrx of orthogoal egevectors The f e s a egevector wth egevalue λ, the λ s the dfferece the probablty of measurg e : Pr A [] Pr B [] λ We ca defe the dstace betwee two probablty dstrbutos wth respect to a bass E as D A D B E Pr A [] Pr B [] If E s the egebass, the D A D B E λ tr A B A B tr, whch s called the trace dstace betwee A ad B Clam Measurg wth respect to the egebass E of the matrx A B s optmal the sese that t maxmzes the dstace D A D B E betwee the two probablty dstrbutos Before we prove ths clam, we troduce the followg defto ad lemma wthout proof Defto Let {a } N ad {b } N be two o-creasg sequeces such that a b The the sequece {a } s sad to maorze {b } f for all, a b CS 94-, Sprg 007, Lecture 3 3

4 Lemma[Schur] Egevalues of ay Hermta matrx maorzes the dagoal etres f both are sorted ocreasg order Now we ca prove clam 3 Proof Sce we ca reorder the egevectors, we ca assume λ λ λ Note that tra B 0, so we must have λ 0 We ca splt the λ s to two groups: postve oes ad egatve oes, we must have Thus A B tr A B tr λ >0 λ <0 max λ A B tr Now cosder measurg aother bass The the matrx A B s represeted as H FA BF, ad let µ µ µ be the dagoal etres of H Smlar argumet shows that max µ µ D A D B F But by Schur s lemma the λ s maorzes µ s, so we must have D A D B F D A D B E A B tr Let HX be the Shao Etropy of a radom varable X whch ca tae o states p p H{p } p log p I the quatum world, we defe a aalogous quatty, Sρ, the Vo Neuma etropy of a quatum esemble wth desty matrx ρ wth egevalues λ,,λ : Sρ H{λ,,λ } λ log λ 4 Two ope questos related to NRM Lqud NMR Nuclear Magetc Resoace quatum computers have successfully mplemeted 7 qubts ad performed a strpped dow verso of quatum factorg o the umber 5 I lqud NMR, the quatum regster s composed of the uclear sps a sutably chose molecule - the umber of qubts s equal to the umber of atoms the molecule We ca th of the computer as cosstg of about 0 6 such molecules a macroscopc amout of lqud, each cotrolled by the same operatos smultaeously Thus we wll have 0 6 copes of our state, each cosstg of say 7 qubts We assume that we ca address the qubts dvdually, so that for example, we could preform a operato such as CNOT o the d ad 4th qubt smultaeously o each copy CS 94-, Sprg 007, Lecture 3 4

5 The catch lqud NMR quatum computg s that talzg the regster s hard Each qubt starts out state 0 wth probablty / + ε ad state wth probablty / ε Here ε depeds upo the stregth of the magetc feld that the lqud sample s placed Usg very strog magets the NMR apparatus, the polarzato ε s stll about 0 5 If ε 0 the the desty matrx descrbg the quatum state of the regster s ρ I Ths meas that f we apply a utary trasformato U, the desty matrx of the resultg state s I U UIU I So you caot perform ay meagful computato The way NMR quatum computato wors s ths: the tal mxed state wth ε 0 5 s preprocessed through a sequece of quatum gates to obta a ew mxed state whch s maxmally mxed I wth probablty δ ad wth probablty δ Now, f we apply a utary trasformato to ths state, we get I wth probablty δ ad U wth probablty δ Thus f we measure the state, we obta a co flp wth bas δ towards the correct aswer Aother way of thg about ths s that the I gves o et sgal the measuremet, whle the δ sgal gets amplfed by the 0 6 copes of the computato beg carred out smultaeously The problem s that δ s expoetally small the umber of qubts Therefore lqud NMR quatum computato caot scale beyod 0-0 qubts Questo : Say we have a sgle clea bt ad maxmally mxed qubts, what ca we do wth ths? We ca do at least oe quatum computato, phase estmato to approxmate the trace of a utary matrx Use the sgle clea qubt as the cotrol bt ad apply the cotrolled utary to the maxmally mxed qubts We ca th of the qubts as beg a uform mxture over the egevectors of the utary! see prevous lecture for detals Is there aythg else that we ca do wth ust oe qubt? Ca you prove lmts o what ca be doe wth oe clea qubt Questo Let ρ be a desty matrx for a mxed state whch taes o x wth probablty /+ε #0 / ε # where #0 ad # are the umber of 0s ad s x respectvely Therefore, ρ ca be wrtte as a probablty dstrbuto over uetagled states Such mxed states are called separable mxed states It turs out, that f the desty matrx ρ s suffcetly close to the detty matrx I the there always exsts some such decomposto to a dstrbuto over uetagled states Usg the boud 0 5 for the polarzato of qubts lqud NMR, t turs out that for 0 the tal state of a -qubt NMR regster s a separable mxed state So f we apply a utary trasformato U to ρ, we get a ew desty matrx ρ UρU whch s also close to I ad thus s also a separable mxed state Gve that there s o etaglemet, ca we smulate such a quatum computato effcetly? We do t ow how, sce to wrte the state as a separable mxture we mght have to perform a chage of bass after each quatum gate O the other had, we do t ow how to obta ay quatum speedup ths model ether CS 94-, Sprg 007, Lecture 3 5