Proceedngs of he Inernaonal MulConference of Engneers and Compuer Scenss 018 Vol II IMECS 018, March 14-16, 018, Hong Kong Real-me Quanum Sae Esmaon Based on Connuous Weak Measuremen and Compressed Sensng Jngbe Yang, Shuang Cong, and Sen Kuang Absrac We propose he proocol of real-me quanum sae esmaon based on connuous weak measuremen and compressed sensng (CS). We consder a pure ensemble of dencal qub spns whch neracs wh an opcal mode and s measured connuously by homodyne deecon. Assumng ha he compung me s gnored, he sae of he qub ensemble can be esmaed n real me. We verfy he real-me esmaon by smulaon expermens on MALAB. he evoluon rajecores of he acual saes and he correspondng esmaed resuls are shown n Bloch sphere, and he nfluence of he parameers on he performance of he esmaon resuls s analyzed. he expermenal resuls show ha when he sysem s under a consan appled magnec feld and he seleced weak measure operaor does no concde wh or be orhogonal o he Hamlonan of he sysem, he sae of he qub ensemble can be accuraely esmaed n real me wh only consecuve measuremens records. Index erms compressed sensng, connuous weak measuremen, quanum rajecory, real-me quanum sae esmaon Q I. INRODUCION UANUM sae omography (QS), also called quanum sae esmaon or quanum sae reconsrucon, s one of he mporan conens n he research of quanum nformaon processng and quanum conrol [1][]. By esmang he sae of he quanum sysem, people can effecvely oban he curren nformaon of he sysem and desgn he correspondng conrol scheme [3][4]. In order o fully esmae a quanum sae, one needs o oban he measuremen values on a complee se of observables on he sae. For an n-qub quanum sae, he number of complee observables s n n d = = 4 n. In general, one needs o oban he complee observaon values by desrucvely measurng a leas d ensembles of dencally prepared copes of he sae. As a resul, he number of measuremens requred for convenonal QS ncreases exponenally as he number of qubs n ncreases, whch brngs grea dffcules o he reconsrucon of hgh-qub quanum saes. Manuscrp receved December 8, 017, revsed January 10, 018. hs work was suppored n par by he Naonal Scence Foundaons of Chna under Gran No. 61573330 and No. 6170106009. J. Yang, S. Cong and S. Kuang are wh he Deparmen of Auomaon, Unversy of scence and echnology of Chna, Hefe, 3007, Chna (correspondng auhor s phone: 86-551-63600710; fax: 86-551-6360344; e-mal: scong@usc.edu.cn). ISSN: 078-0958 (Prn); ISSN: 078-0966 (Onlne) In some physcal plaforms, s dffcul o perform desrucve srong measuremens drecly on he sysem. In order o oban nformaon of he arge quanum sysem, people need o use a probe o assocae wh he quanum sysem and hen measure he probe srongly. he nformaon of he arge quanum sae can be nferred by he measuremen records of he probe. If he neracon srengh beween he sysem and he probe s raher weak, hs knd of measuremen s called weak measuremen [5] (Do no confuse hs wh anoher weak measuremen concep relaed o he weak value [6]). Unlke he srong measuremen ha wll compleely desroy he sae, weak measuremen wll only brng slgh changes o he sae and he mpac o he sysem s also weak. Usng weak measuremens, one can measure a quanum ensemble and drecly oban he expecaon value of an observable as he measuremen record. Snce he weak measuremen s no compleely desrucve, one can make connuous measuremens of he quanum sysem wh weak measuremens. In connuous weak measuremens, he observables are no orhogonal and he correspondng nformaon beween dfferen observables overlaps, so he number of observables requred o ensure complee nformaon s usually greaer han d. Real-me quanum sae esmaon refers o he connuous sae esmaon of any momen based on he connuous weak measuremen records of he quanum sae. I s usually mpossble o drecly calculae he sae of he quanum sysem by usng non-orhogonal observables n connuous weak measuremens. One can calculae he opmal resul based on he measuremen records usng an esmaor and an approprae algorhm as he sae esmaon resul. Real-me quanum sae esmaon s he bass of quanum feedback conrol, bu here s ye no complee heorecal framework n hs respec. Slberfarb e al. frs proposed a quanum sae esmaon scheme and employed a connuous measuremen proocol o perform QS on he seven-dmensonal, F = 3 aomc hyperfne spn manfold, n an ensemble of cesum aoms [7]. Smh e al. acheved sae-o-sae quanum mappng performance esmaon based on he connuous weak measuremen, as well as he opmal conrol echnology desgn and mplemenaon [8]. And he quanum sae of he spns undergong he quanum chaoc dynamc of a nonlnear kcked op s measured based on he connuous weak measuremen [9]. Compressed sensng (CS) provdes a new soluon o he problem of reducng he number of measuremens n IMECS 018
Proceedngs of he Inernaonal MulConference of Engneers and Compuer Scenss 018 Vol II IMECS 018, March 14-16, 018, Hong Kong quanum sae esmaon [10] [1]. CS clams ha: f he rank of a sae densy marx r s much smaller han s dmenson d (r d ), hen he sae densy marx can be reconsruced wh only a small amoun of randomly sampled measuremen records. Gross proved ha one can reconsruc he sae densy marx wh only Ord ( log d ) measuremen records when usng he Paul measuremen operaors [13]. CS can be appled o real-me quanum sae esmaon and mprove he effcency of calculaon. Deusch e al. frs appled CS o he esmaon of he saes of a conrolled quanum sysem under connuous weak measuremens and realzed he rapd reconsrucon of he sae of a 16-dmensonal cesum aomc spn ensemble [14]. In hs paper, we sudy he quanum sae real-me esmaon based on CS and he connuous weak measuremen of one qub spn ensemble. We verfy he real-me esmaon by smulaon expermens on MALAB usng dfferen conrol felds and dfferen nal weak measuremen operaors. he evoluon rajecory of he acual sae and he correspondng esmaed resuls n Bloch sphere are shown, and he nfluence of he parameers on he performance of he esmaon resuls s analyzed. he srucure of hs paper s organzed as follows: In Sec. II, we gve a dealed revew of he prncples of real-me quanum sae esmaon based on he connuous weak measuremen and CS. In Sec. III, we verfy he real-me esmaon of he seleced qub spn ensemble by smulaon expermens on MALAB, and he nfluence of parameers, such as he conrol feld and nal weak measuremen operaors, on he esmaon performance s analyzed. Fnally, a bref concluson s gven n Sec. VI. II. PRINCIPLES OF REAL-IME QUANUM SAE ESIMAION A. Quanum Sae Esmaon Quanum sae esmaon refers o he process of reconsrucng he densy marx of he quanum sae accordng o he measuremen record and he correspondng measuremen operaors. In quanum mechancs, a measuremen operaor s a marx ha can reflec nformaon or some mechancal quanes of he sysem, usually denoed by he operaor M. he measuremen of a d-dmensonal quanum sae ρ acually means measurng he expecaon probables of ρ projecng on a se of measuremen operaors M. People can oban one of he measuremen values M j ( ρ ) correspondng o M j by each measuremen: M j( ρ ) = r( ρ M j), j = 1,,, d (1) In order o esmae he densy marx ρ, s necessary for people o measure he expecaon probables over mulple observables o oban suffcen nformaon. For an arbrary quanum sae ρ of a d-dmensonal Hlber space, assume here s no pror nformaon, s generally necessary o measure on a leas d 1 muually orhogonal observables so as o accuraely reconsruc ρ. M = M, M,... M denoe a se of complee Le { 1 } d observables of ρ, ncludng d orhogonal observables d { } M j ( j = 1,,, ). M = M1, M,... M may be d regarded as a se of bass correspondng o he Hlber spaces. In hs case, he densy marx ρ can be drecly calculaed by he followng formula: 1 d d = 1 ρ = M M () When he seleced se of observables s no complee, he measuremen of ρ s called an nformaonally ncomplee measuremen. In hs case, ρ canno be drecly calculaed by (), bu can only be calculaed by usng an opmzaon algorhm o calculae he closes esmaon resul under he exsng condons. In addon, he drec calculaon of he densy marx usng () s based on he assumpon ha he observables M are orhogonal o each oher, whereas he j observables n acual end o be non-orhogonal, such as generalzed measuremens, posve operaor valued measuremens (POVMs) and quanum weak measuremens. And s necessary o subsue he measuremen records of he observables no an opmzaon algorhm o fnd he opmal esmaon value hrough erave calculaon. B. Quanum Weak Measuremen Quanum weak measuremen s a mehod of measurng quanum sae by usng he weak couplng effec beween he sae and he probe [7]. I s usually used o measure quanum ensembles. Unlke he srong measuremen ha always causes nsananeous collapse of he arge sae, weak measuremen s a non-ransen measuremen process, and he mpac on he quanum sysem s weak. Weak measuremen generally ncludes wo pars: deecon and readou. he process of weak measuremen s shown n Fgure 1, n whch he lef vrual box s for he deecon par and he rgh vrual box s for he readou par. Weak measuremen process nroduces a probe P whch becomes coupled wh he arge ensemble S for a shor me for he deecon par. hen he probe P s srongly measured. Par of he nformaon of he arge S can be nferred wh he measuremen records of P. Probe P Inpu φ deecon par φ coupled wh ψ arge Sysem S Inal sae ψ jon sysem Ψ( Δ) Sae of S : Projecve measuremen ψ ( Δ) Projecor I couplng removed Fg. 1. he process of weak measuremen readou readou par When weakly measurng he arge sysem S, he frs hng s o prepare a probe P. P becomes coupled wh S resulng n a jon sysem S P. Suppose he nal sae of he probe P s φ, and he nal sae of he arge sysem S s ρ0 = ψ ψ. H S and H P are he Hamlonan of sysem S and P, respecvely, and H = HP HS s he Hamlonan of he jon sysem. he nal sae of he coupled sysem s: Ψ = φ ψ. Afer he jon evoluon of S and P for me Δ, he sae Ψ becomes oupu ISSN: 078-0958 (Prn); ISSN: 078-0966 (Onlne) IMECS 018
Proceedngs of he Inernaonal MulConference of Engneers and Compuer Scenss 018 Vol II IMECS 018, March 14-16, 018, Hong Kong ΨΔ ( ) = U( Δ) Ψ, where U( Δ ) s he jon evoluon operaor U( Δ ) = exp( -ξδh / ), and ξ represens he neracon srengh beween sysem S and P (he un s 1/s ). ΨΔ ( ) s an enangled sae composed of S and P whch canno be separaely descrbed wh he saes of S and P. A me Δ, a projecve measuremens X = s performed on P, where s he egensae of he sysem P, and s he egenvalue correspondng o. Afer he projecve measuremen he enanglemen beween S and P dsappears, and he weak measuremen process s over. Le ψ( Δ ) denoe he sae of S a me Δ and he sae of he jon sysem afer he weak measuremen s Ψ ( Δ ) = ψ (Δ ). he weak measuremen process can be regarded as a measuremen operaon on he sysem S, and he Kraus operaor M s used o represen he weak measuremen operaor. herefore, 0 Mρ0M ρ = (3) P ( ρ ) where ρ = ψ ( Δ ) ψ ( Δ ) s he sae densy marx of S afer he weak measuremen. he weak measuremen operaor M s M = I U( Δ) φ I (4) MM= 1. P ( ρ 0) s he probably of measurng oucome : he weak measuremen operaors { M } sasfy Le 0 0 P ( ρ ) = r( M Mρ ) = ψ M M ψ (5) λ = ξδ denoe he weak measuremen srengh, where boh he neracon srengh ξ and he evoluon me of he jon sysem Δ are small values. herefore λ s a small amoun approachng 0 and M φ I whch s closed o 0 when and φ are orhogonal or approxmaely orhogonal. Le measuremen value of M denoe he weak M. Usng he operaors { M } and he correspondng measuremen values { M }, one can esmae he pre-measuremen sae ρ 0 and he pos-measuremen sae ρ of he sysem S. C. Connuous Weak Measuremen and Quanum rajecory Connuous weak measuremen means measurng he seleced quanum sysem connuously usng weak measuremens. hs s a dynamc process. One can oban he nformaon of he sysem based on he measuremen records. Connuous weak measuremen s usually used for he quanum feedback sysem. Based on he connuous measuremen records, people can esmae he sae of he sysem n real me and desgn a proper conrol law of he feedback. akng he aomc ensemble ρ () n vacuum under a magnec feld as an example, he schemac of he expermen for connuous weak measuremen and he ISSN: 078-0958 (Prn); ISSN: 078-0966 (Onlne) correspondng process srucure dagram are shown n Fg., n whch he probe s a connuous laser beam. he nal sae of he phoon n he laser beam s φ, whch coupled wh he aomc ensemble ρ () resulng he oupu of he probe laser n he enanglemen. he oupu probe s measured connuously by homodyne deecon by he measuremen operaor X =. Snce he srengh of weak measuremen λ s very small, he back-acon of he weak measuremen s gnored. he weak measuremen operaors of hs process are M ( d) wh dfferen, where d represens he very shor me nerval requred for he weak measuremen and s one of he possble measuremen oupus. he probably of obanng he oupu on he probe laser s P ( ρ ()) = r( M Mρ()). here exs sho noses (SN) n he deecon process, whch wll lead o a Gaussan dsrbuon flucuaon n he acual measuremen records. he measuremen record can be modeled hrough a Wener process W() wh zero mean and un varance and he acual measuremen record can be expressed as y () = r( Mρ()) + σw () (6) u () ρ ( ) (a) Y () ˆ( ρ ) (b) Fg.. (a) Schemac of he expermen under connuous weak measuremen and (b) he process srucure dagram of he schemac n (a) In he expermen shown as Fg., he sae ρ () canno be compleely esmaed wh only one measuremen record y () of a ceran momen because he measuremen operaor relaed wh y () only covers a small par of he nformaon of ρ (). Accordng o he prncple of quanum omography, a se of nformaonally complee measuremen records s needed o fully esmae he densy marx of he sae ρ (). herefore, he sysem saes mus be esmaed by usng measuremen records a dfferen momens. he sysem dynamcs s Markovan when gnorng he effecs of weak measuremens and sho noses. Assumng ha all he parameers of he sysem excep he sae are known, f he sae of a momen ρ ( j ) can be esmaed accuraely, he sysem dynamcs rajecory wh me can be esablshed accordng o he known parameers. One can calculae he sae of he sysem a any me wh ρ ( j ) and he dynamcs rajecory, whch solves he problem ha he sae changes IMECS 018
Proceedngs of he Inernaonal MulConference of Engneers and Compuer Scenss 018 Vol II IMECS 018, March 14-16, 018, Hong Kong due o he evoluon [7]. he dynamcs rajecory of a quanum sysem evolvng over me s also known as he quanum rajecory, whch s usually represened by he quanum sae maser equaon (SME) [8]. he dynamcs rajecory of he sysem n Schrödnger pcure can be represened by he Lndblad maser equaon: ρ() = L [ ρ()] 1 1 = [ H, ρ()] LL ρ() + ρ() L L + Lρ() L where L [ ρ( )] s a super operaor, and he operaor L represens he dsspaon or decoherence caused by he measuremen or he envronmen. he soluon of (7) s ρ() = V[ ρ(0)] = Vρ(0) V (8) where V s a super operaor, and dv d = LV,. V = ( exp L s d s), where s he me-orderng 0 operaor. he sae of he sysem changes wh me because of he evoluon. In order o faclae he calculaon, we ransform sysem of Schrödnger pcure no ha of Hesenberg pcure, where he measuremen operaor evolves connuously over me and he quanum sae keep consan [7]. If he super operaor L [ ρ( )] s me-ndependen, hen he evoluon equaon of he measuremen operaor M () under Hesenberg pcure s: M () = L [ M()] 1 1 = [ HM, ()] + MLL () + LLM () (9) LM () L And he soluon of (9) s: M() = V [ M(0)] (10) In parcular, f he measuremen s he deal non-desrucve measuremen, he effec of measuremen on he sysem s neglgble and L = 0. When he sysem Hamlonan does no change wh me, he dynamcs of he sysem under Schrödnger pcure can be descrbed by he Louvlle-von Neumann equaon: ρ () = [ H, ρ()] (11) he soluon of (11) s: ρ( ) = V [ ρ(0)] = exp( -H) ρ(0)exp( H) (1) where he super operaor V = exp( -H). Ignorng he sho noses and he Wener process W () n (6) s always 0, hen he connuous weak measuremen n Schrödnger pcure s equvalen o he measuremen of a consan quanum sae ρ (0) wh a connuously evolvng measuremen operaor M () n Hesenberg pcure. he evolvng measuremen operaor M () n Hesenberg pcure s: = = M ( ) V [ M (0)] exp( H) M exp( -H) (13) And he correspondng measuremen record s: ISSN: 078-0958 (Prn); ISSN: 078-0966 (Onlne) (7) y ( ) = r( Mρ( )) = r( M( ) ρ0) (14) If he super operaor L [ ρ( )] s me-dependen, hen he evolvng measuremen operaor M () n Hesenberg pcure s dfferen from (13) because V L L V. In hs case he operaor M () s M ( ) = V [ L[ M(0)]] (15) Due o he complexy of he calculaon, s dffcul o gve he soluon of (15) drecly. Deusch e al. presen a mehod of numercal compuaon wh pecewse consan [7]: Suppose ha Hamlonan and super-operaor L are consan over any perod of me, f s small enough, he measuremen operaor mees ( M = M0 V, where δ + = L ] ( ] V 1 e V. I should be noed ha, n Hesenberg pcure, he measuremen operaors of dfferen momens are a non-orhogonal. Snce non-orhogonal operaors are no ndependen of each oher, he number of he operaors for nformaonally complee s usually greaer han d. D. Real-me Quanum Sae Esmaon Based on Compressed Sensng he heory of compressed sensng (CS) clams ha f he densy marx ρ of a quanum sae s a low-rank marx, hen he sae densy can be reconsruced wh only Odr ( ln d ) measuremen of random observables by solvng an opmzaon problem, where d and r are he dmenson and rank of he densy marx ρ, respecvely [13]. he reconsrucon problem of densy marx ρ can be ransformed no he followng opmzaon problem: mn ρ s.. y= A vec( ρ), where ρ s he nuclear-norm * * of ρ, vec( ) represens he ransformaon from a marx o a vecor by sackng he marx s columns n order on he op of one anoher. he samplng marx A s he marx form of he all he sampled observables M, and he samplng vecor y s he vecor form of he correspondng observaon values M. he above opmzaon problem of nuclear-norm s equvalen o he opmzaon problem of mnmzng he -norm under he posve defne consran: mn A vec( ρ) y s.. r ρ= 1, ρ 0 (16) Equaon (16) s also called nonnegave leas squares opmzaon. Researchers have shown ha he non-negave leas squares opmzaon mehod also belongs o he CS opmzaon [19]. wo suffcen condons for complee reconsrucon of a marx based on CS are: (1) he densy marx ρ s a low-rank marx; () he samplng marx A sasfes he Resrced Isomery propery (RIP) [13]. he vecor y and marx A can be expressed accordng o he curren measuremen confguraons as: and y = ( M, M,, M ) (17) k1 k km IMECS 018
Proceedngs of he Inernaonal MulConference of Engneers and Compuer Scenss 018 Vol II IMECS 018, March 14-16, 018, Hong Kong where Mk vec( M ) k 1 vec( M k ) A = (18) vec( M k ) m M s an arbrary measuremen operaor and k s he correspondng measuremen value. I can be deduced accordng o CS ha, f he samplng marx A formed by connuous weak measuremen operaors sasfes he RIP, hen one can esmae he quanum sae n real me wh a small amoun of me-evolvng measuremen operaors { M ( )} and correspondng measures records { y ( )}. Solvng he opmzaon problem (17) wh an approprae algorhm, people can oban he reconsruced densy marx ρ. III. SIMULAION EXPERIMEN AND ANALYSIS In hs Secon, we sudy he real-me sae esmaon of a qub sysem by smulaon expermens on MALAB. We sudy he evoluon rajecory and correspondng real-me esmaed sae rajecory of he conrolled sysem. hrough comparave expermens, he nfluence of exernal conrol feld, conrol srengh and weak observaon observer on real-me sae esmaon s analyzed. Consder a 1/ spn parcle ensemble ρ () as he objec of real-me esmaon, whch s under z drecon consan magnec feld Bz and x drecon conrol magnec feld Bx = Acosφ. In Schrödnger pcure, he nal sae of he spn s ρ (0), and ρ () s he sae a momen. A connuous weak measuremen s appled o he sysem. he nal weak measuremen operaor s M. Ignore he sho noses and assume he srengh of weak measuremen s λ = 0. he evoluon equaon of he sysem s gven by (11). he egen-frequency of he spn ensemble ρ () n he magnec feld Bz s ω0 = γ Bz, where γ s he spn-magnec rao of he parcle ensemble, and Ω = γ A s he Rab frequency of he sysem Ω. he Hamlonan of ρ () s: H = H0 + ux Hx (19) where H0 = ( ) ω0σ z s he free Hamlonan, 1 0 σ z = s he Paul operaor of z, 0 1 ( H ) x e φ φ = Ω σ + e σ + s he conrol Hamlonan, 0 0 σ 0 1 =,, and s he 1 0 σ + + = 0 0 u x me-ndependen conrol srengh. In real-me esmaon of he sae ρ (), we frs conver o Hesenberg pcure. he esmaed sae s consan a hs me and he measuremen operaor evolvng over me s as Equaon (13). Assume ha he momen of connuous weak measuremens are j and he nerval beween wo adjacen momens s Δ. he y ( j ) are recorded from 0 momen. Afer each weak measuremen, subsue he recorded y ( j ) and correspondng { M( j) } no (17) and (18) o ge he real-me samplng vecor y and samplng marx A. We use leas-square algorhm o solve he opmzaon problem (16), and he opmal soluon ρ (0) s he esmaon of ρ (0). Afer obanng ρ (0), he sae densy marx a he curren momen n Schrödnger pcure s calculaed accordng o (1), and he resul ρ ( j ) s he real-me esmaon of he sae. In he expermens, he fdely f s used o represen he effec of sae esmaon: 1 1 f () = r ρ() ρ() ρ() (0) where ρ () represens he acual densy marx, and ρ () s he correspondng real-me esmaed densy marx. We choose he nal sae of he 1/ spn sysem as ρ (0) = [3 4 3 4; 3 4 1 4], and he Bloch sphere coordnae of (0) ( 3, 0, 1) ρ s. Le 18 ω 0 =Ω=.5 10, he nal phase of Conrol feld s φ = 0 and he nerval of he weak measuremens s 18 Δ = 0.4* ω0 = 1 10 s 4 a.u.. Assume ha he esmaon me requred s approxmaely 0. We choose hree dfferen conrol srengh as u x 1 = 0, u x = 0.5, u x3 = 1, and choose wo knds of nal weak measuremen 1 0 operaors as M z = σ z = 0 1, 0 1 M x = σ x = for each 1 0 conrol srengh. Fgure 3 shows he evoluon rajecores of he acual sae ρ () and he real-me esmaon sae ρ () n he Bloch sphere under dfferen parameers, where he red sold lne corresponds o he acual sae, he blue doed lne corresponds o he real-me esmaon sae, " ο " represens he poson of he acual nal sae ρ (0), and "* " represens he nal of esmaon sae ρ (0). In Fg. 3, (a) (b) (c), respecvely, correspond o he conrol srengh u x1 = 0, u x = 0.5, and u x3 = 1, and he lef and rgh columns of Fg. 3, respecvely, correspondng o he measuremen operaors M and M. z A me = 0, he nal esmaed sae ρ (0) s calculaed only from once measuremen on he nal acual sae ρ (0). Snce ρ (0) n Fg. 3 s consan, he nal esmaed sae ρ (0) of he same column n (a) (b) (c) are same, bu of dfferen column are dfferen. he fdeles of ρ (0) correspondng o M z and M x are f z (0) = 0.7906, f x (0) = 0.9354, respecvely. I can be seen from Fg. 3 (a) ha, when u x 1 = 0, he acual sysem s under free evoluon and s evoluon rajecory s a crcular moon on x y plane. he esmaon correspondng o M z s a mxed sae whch s he projecon of ρ () ono he z x ISSN: 078-0958 (Prn); ISSN: 078-0966 (Onlne) IMECS 018
Proceedngs of he Inernaonal MulConference of Engneers and Compuer Scenss 018 Vol II IMECS 018, March 14-16, 018, Hong Kong IV. CONCLUSION axs, and he esmaon correspondng o M x s a projecon of ρ ( ) on he x y plane wh z = 0. However when u x = 0.5 and u x3 = 1, he evoluon rajecores of he acual sae have ceran angles beween he x y plane, and he rajecores of esmaon become concde wh he acual sae rajecores n a shor me. he value of conrol srengh does no affec he accuracy of esmaon. Comparng he expermenal resuls of Fg. 3 we can see ha: n Fg. 3 (a), when he measuremen operaors M z and M x are concden wh or orhogonal o he free Hamlonan, he real-me esmaon canno acheve accurae resuls. However, n Fg. 3 (b) and Fg. 3 (c), he measuremen operaors M z and M x are no concden wh or orhogonal o he sysem Hamlonan, and accurae real-me esmaons of he quanum sae can be obaned a he momen 1 = Δ. In hs paper, he real-me quanum sae esmaon based on CS and connuous weakness measuremen s suded. We gve he prncples and schemes of real-me quanum sae esmaon, and analyze he nfluence of parameers on he performance of he esmaon resuls. We verfy he real-me esmaon by smulaon expermens. he real-me esmaon of a 1/ spn quanum sae s realzed, and dfferen conrol srengh and dfferen nal measuremen operaors are respecvely seleced for he expermens. he evoluon rajecores of he acual saes and he correspondng esmaon saes are shown n Bloch spheres by comparave expermens. he expermenal resuls show ha when he nal measuremen operaor s no concden wh or orhogonal o he sysem Hamlonan, an real-me esmaon of he 1/ spn sysem based on connuous weak measuremens and CS s feasble, and accurae real-me esmaons can be obaned wh a mos consecuve measuremen records. REFERENCES [1] [] (a) u x1 = 0 [3] [4] [5] [6] (b) u x = 0.5 [7] [8] [9] [10] (c) u x3 = 1 Fg. 3. he evoluon rajecores of he acual sae ρ ( ) and he real-me esmaon sae ρ ( ) n he Bloch sphere under dfferen parameers. hese expermenal resuls show ha, f he nal measuremen operaor s concden wh or orhogonal o he sysem Hamlonan, connuous measuremen canno measure suffcenly vald nformaon of he sysem and canno acheve accurae real-me esmaon of he sae. On he conrary, f he nal measuremen operaor s no concden wh or orhogonal o he sysem Hamlonan, a successful real-me esmaon of he quanum sae can be acheved, and he sae evolvng over me can be accuraely esmaed wh a mos consecuve measuremens records. ISSN: 078-0958 (Prn); ISSN: 078-0966 (Onlne) [11] [1] [13] [14] G. M. D Arano, M. G. A. Pars, and M. F. Sacch. Quanum omographc mehods, Lecure Noes n Physcs, vol. 649, pp. 7-58, 004. D. D Alessandro, On quanum sae observably and measuremen, Journal of Physcs A: Mahemacal and General, vol. 36, no. 37, pp. 971-9735, 003. C. J. Bardeen, V. V. Yakovlev, K. R. Wlson, S. D. Carpener, P. M. Weber and W. S. Warren, Feedback quanum conrol of molecular elecronc populaon ransfer, Chemcal Physcs Leers, vol. 80, no. 1, pp. 151-158, 1997. Q. Sun, I. Pelczer, G. Rvello, R. B. Wu and H. Rabz, Expermenal observaon of saddle pons over he quanum conrol landscape of a wo-spn sysem, Phys. Rev. A, vol. 91, no. 4: 04341, 015. O. Oreshkov,. A. Brun, Weak Measuremens Are Unversal, Phys. Rev. Le. vol. 95, no. 11: 110409, 005. Y. Aharonov, D. Z. Alber and L.Vadman. How he resul of a measuremen of a componen of he spn of a spn-1/ parcle can urn ou o be 100, Phys. Rev. Le. vol. 60, no. 14: 1351 1354, 1988. A. Slberfarb, P. Jessen and I. H. Deusch, Quanum sae reconsrucon va connuous measuremen, Phys. Rev. Le. vol. 95, no. 3: 03040, 005. G. A. Smh, A. Slberfarb, I. H. Deusch and P. S. Jessen, Effcen quanum-sae esmaon by connuous weak measuremen and dynamcal conrol, Phys. Rev. Le. vol. 97, no. 18: 180403, 006. S. Chaudhury, A. Smh, B. E. Anderson, S. Ghose and P. S. Jessen, Quanum sgnaures of chaos n a kcked op, Naure, vol. 461, no. 765, pp. 768-771, 009. E. J. Candès, J. Romberg and. ao, Robus uncerany prncples: Exac sgnal reconsrucon from hghly ncomplee frequency nformaon, IEEE ransacons on nformaon heory, vol. 5, no., pp. 489-509, 006. K. Zheng, K. L, and S. Cong. A reconsrucon algorhm for compressve quanum omography usng varous measuremen ses, Scenfc Repors, vol. 6: 38497, 016. K. L, H. Zhang, S. Kuang, F. Meng, and S. Cong, An Improved Robus ADMM Algorhm For Quanum Sae omography, Quanum Informaon Processng, vol.15, no.6, pp. 343-358, 016. D. Gross. Recoverng low-rank marces from few coeffcens n any bass, IEEE ransacons on Informaon heory, vol. 57, no. 3, pp. 1548-1566, 011. A. Smh, C. A. Rofrío, B. E. Anderson, H. Sosa-Marnez., I. H. Deusch, and P.S. Jessen, Quanum sae omography by connuous measuremen and compressed sensng, Phys. Rev. A, vol. 87, no. 3: 03010, 013. IMECS 018