Optimal estimation of a physical observable s expectation value for pure states

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1 Optiml estimtion of physicl observble s expecttion vlue for pure sttes A. Hyshi, M. Horibe, n T. Hshimoto Deprtment of Applie Physics, University of Fukui, Fukui , Jpn Receive 5 December 2005; publishe 4 June 2006 We stuy the optiml wy to estimte the quntum expecttion vlue of physicl observble when finite number of copies of quntum pure stte re presente. The optiml estimtion is etermine by minimizing the squre error verge over ll pure sttes istribute in unitry invrint wy. We fin tht the optiml estimtion is bise though the optiml mesurement is given by successive projective mesurements of the observble. The optiml estimte is not the smple verge of observe t, but the rithmetic verge of observe n efult nonobserve t, with the ltter consisting of ll eigenvlues of the observble. DOI: 0.03/PhysRevA PACS number s : Hk I. ITRODUCTIO One of the funmentl tsks in quntum physics is to etermine the expecttion vlue of physicl observble of n unknown quntum stte. With only single copy of the quntum stte given, we cnnot etermine the expecttion vlue of physicl observble becuse of the sttisticl nture of quntum mesurement. Suppose we re presente with certin finite number of copies of n unknown quntum stte. We cnnot increse the number of the copies since the no-cloning theorem forbis it. Then wht is the optiml wy to etermine the expecttion vlue of the observble for given? An intuitively plusible optiml estimte is given by the rithmetic verge of the t prouce by successive projective mesurements of the observble on the iniviul systems. This problem, however, is by no mens trivil. Given quntum system compose of subsystems, we cn consier two types of mesurement. One is seprte mesurements: sequence of mesurements on the iniviul subsystems, possibly epenent on the outcomes of erlier mesurements. The other is joint mesurement: single mesurement on the system s whole. Recent stuies on quntum-stte iscrimintion n estimtion 2,3 provie consierble instnces in which joint mesurements perform better thn seprte mesurements even for stte compose of mutully uncorrelte subsystems. Peres n Wootters showe tht certin set of three biprtite prouct sttes cn be better istinguishe by joint mesurement 4,5 see lso 6 8. An even stronger exmple ws provie by Bennett et l. 9, which shows tht certin orthogonl set of biprtite prouct sttes cnnot be relibly istinguishe by ny seprte mesurement though joint mesurement perfectly istinguishes them becuse of their mutul orthogonlity. The superiority of joint mesurement hs lso been iscusse in the problem of quntumstte estimtion for ienticlly prepre copies of n unknown stte see 0 4,6, for exmple. In 5, D Arino, Giovnnetti, n Perinotti rise the question of whether the stnr proceure of verging the outcomes of repete mesurements of n observble over eqully prepre systems is the best wy of estimting the expecttion vlue of the observble, or whether joint mesurement cn improve the estimtion. They showe tht the stnr proceure is inee optiml if one is restricte to the clss of unbise estimtion for ny generlly mixe stte. Here n estimtor is si to be unbise if the verge over mny inepenent estimtes gives the true vlue to be estimte. An unbise result is certinly one of the esirble properties for estimtion but not necessry conition. A nturl question is then whether bise estimtion performs better thn the stnr unbise estimtion. Let us tke simple exmple, in which we estimte the expecttion vlue of the observble z for single-qubit system in n unknown pure stte. We ssume tht the stte of the qubit is chosen ccoring to the uniform istribution on the Bloch sphere. Suppose tht the projective mesurement of z prouce the outcome, which mens the smple verge is. ow, one cn sk if it is resonble to conclue tht the expecttion vlue of z is most likely equl to. ote tht the expecttion vlue of z is only if the qubit lies exctly t the north pole of the Bloch sphere. On the other hn, the mesurement of z cn prouce the outcome with some probbility unless the qubit is exctly t the south pole. Therefore, it is more resonble to consier tht the expecttion vlue of z is not, but somewhere between 0 n. In fct, the optiml estimte turns out to be /3 in this cse, s we will see in the next section. In this pper, without ssuming unbiseness of the estimtion, we stuy the optiml proceure for the expecttion vlue of physicl observble of n unknown pure stte, when copies of the stte re presente. We ssume tht the unknown pure stte is chosen from the pure-stte spce ccoring to unitry invrint priori istribution. The optiml estimtion is etermine by minimizing the squre error verge over the priori istribution. II. OPTIMAL ESTIMATIO We etermine the optiml wy to estimte the expecttion vlue of physicl observble when copies of n unknown pure stte = on -imensionl Hilbert spce H re given. Let E be positive-opertor-vlue mesure POVM on the totl system H, with outcome lbele proviing n estimte for the expecttion vlue given by /2006/73 6 / The Americn Physicl Society

2 HAYASHI, HORIBE, AD HASHIMOTO tr. For given, the men squre error in the estimte is written s = tr E tr 2. We will first verge this over ll pure sttes n then minimize it with respect to the POVM E n the estimte. The istribution of the pure sttes is specifie in the following wy. Expn pure stte s = i= c i i in terms of n orthonorml bse i of H. The istribution is then efine to be the one in which the 2-component rel vector x i =Rec i, y i =Imc i is uniformly istribute on the 2 -imensionl hypersphere of rius. The istribution is unitry invrint in the sense tht it is inepenent of the orthonorml bse i chosen to efine it. Let us enote the verge over this istribution by. All we nee in the following clcultion is useful reltion for the verge of n given in Ref. 6, tht is, n = S n n, where S n is the projection opertor onto the totlly symmetric subspce of H n n n is its imension given by n =tr S n = n+ C. It my be instructive to see how this formul comes out in some simple cses of qubits =2, in which the bove istribution mens tht the Bloch vector n is uniformly istribute on the surfce of the Bloch sphere. Then we cn esily verify 2 = n 4 n 2 = /3 S =, 3 0 S =0, where S is the eigenvlue of the totl spin. This is specil cse of the formul 2, since the stte is symmetric if S= n ntisymmetric if S=0. The cse of three qubits provies nother exmple. +n 4 n = 3 /4 S = 3/2, S = /2. ow going bck to the generl-imensionl cse, we expn Eq. n perform verging over by use of the formul 2 : = tr E 2 2 tr + tr 2 = , 5 where we enote the three terms in by, 2, n 3, n we evlute ech seprtely. The first term is reily clculte s = 2 tr E S. 6 For 3, we first use the completeness of the POVM by summing over n perform the verge in the following wy: 3 = tr 2 = tr 2 2 = tr S 2 2 = 2 + tr 2 +tr 2, 7 where 2 is unerstoo to be the tensor prouct of two s in spces n 2, n the spce on which the opertor cts is specifie by the number in the prentheses. Herefter we will use this convention in more generl cses, nmely, n n. Evlution of the secon term 2 is more involve. Introucing nother system on H, which we cll system +, we hve 2 = 2 tr E tr = 2 tr E + + = 2 + tr E S + +, where the trces in the secon n thir equlities is unerstoo to be over systems,2,...,, n +. The opertor + cts on system +. The projection opertor S + is the sum of ll permuttion opertors of + systems ivie by fctor of +!. Any permuttion of + objects is either just permuttion mong the first objects or the prouct of permuttion mong the first objects n the trnsposition between the + th object n one of the first objects. With this observtion we fin, for ny opertor, S tr + S + + = + n, + tr where tr + is the trce over the + st system. We use this formul to trce out the newly introuce system + in the expression 2 given by Eq. 9. The result is given by 2 = 2 tr E S ˆ, where we efine the symmetric one-boy opertor ˆ to be ˆ + n. + tr 2 Combining the three verges, 2, n 3, we obtin

3 OPTIMAL ESTIMATIO OF A PHYSICAL¼ = tr E S 2 2 ˆ + + tr 2 +tr 2. 3 To minimize we complete the squre with respect to in this expression. Owing to the completeness of the POVM, this is reuce to the clcultion of tr S ˆ 2, which cn be performe by using the following formuls: tr S n = tr, 4 tr S n m = tr 2 + tr 2 + tr 2 n = m, n m. 5 After some clcultion we fin tr S ˆ 2 = + + tr tr 2. 6 We thus finlly obtin the men squre error in the complete squre form = tr E S ˆ tr 2 tr 2. 7 ow note tht the first term in Eq. 7 is positive. This is becuse ˆ is symmetric uner exchnge of component subsystems n therefore S ˆ 2 =S ˆ 2 S is positive opertor. The hs lower boun given by the secon term of Eq. 7. Let us enote the eigenvlue of by i i=,..., n the corresponing eigenstte by i. Itis then reily seen tht this lower boun cn be chieve if the inex of the POVM element collectively represents the set of i,i 2,...,i, the POVM element is tken to be the projector E i,i 2,...,i = i i 2 i i i 2 i, 8 n the estimte to be the corresponing eigenvlue of ˆ, i,i 2,...,i = + in. + tr 9 Thus we conclue tht the men squre error in the estimtion for the expecttion vlue of the observble tkes its minimum vlue given by opt = + + tr 2 tr 2, 20 if one mesures the observble inepenently for ech system n mkes the estimte given by opt + in, + tr 2 where i, i2,..., i re the t observe by the mesurement. The optiml estimte opt is not the rithmetic verge of observe t the smple verge, though the optiml mesurement is projective n inepenent. For finite, it is not unbise either since tr E =tr ˆ = + tr + tr, 22 which only symptoticlly pproches tr. In Sec. IV we will iscuss the biseness of opt n present n interprettion of its structure. Wht o we obtin for the men squre error if we tke the smple verge of the vlues of observe by the successive mesurements on ech copy? In this cse the POVM is given by Eq. 8 n the estimte by i,i 2...,i = in v, 23 which cn be esily shown to be unbise. The squre error for given given in Eq. tkes the form v = tr 2 tr 2. After the verge over we hve 24 v = v = + tr 2 tr 2, 25 where we use Eq. 7. Compring opt n v, we fin tht the only ifference between them is in the fctor in the enomintors, + in opt n in v. While opt is certinly less thn v, both show the sme symptotics when the number of copies goes to infinity. The ifference becomes importnt when the number of copies is comprble to the imension of the system. Let us exmine the exmple iscusse in Sec. I, in which z is mesure with the result for single qubit in n unknown pure stte =2 n =. In this cse the observe t is. The estimte by the smple verge gives v = for the expecttion vlue of z with the men squre error v =2/3, wheres the optiml estimtion preicts opt =/3 with the men squre error opt =2/9. III. ESTIMATIO WITH THE UBIASEDESS CODITIO In Ref. 5, D Arino, Giovnnetti, n Perinotti consiere the estimtion for the expecttion of observbles uner the unbiseness conition for ny generlly mixe stte n showe tht the optiml estimte uner the constrint is given by the smple verge obtine by the inepenent successive mesurement of the observble on ech copy. In

4 HAYASHI, HORIBE, AD HASHIMOTO this section we briefly iscuss the sme problem in the pure stte cse n show the sme conclusion hols. The unbiseness conition is written s tr E =tr. 26 ote tht tr on the right-hn sie cn be expresse s tr =tr ˆ v, 27 ˆ v n. 28 If the unbiseness conition 26 is ssume for ny generlly mixe stte, then it cn be shown 5 tht E = ˆ v 29 for ny permuttion-invrint POVM E. If we require the unbiseness conition for ny pure stte, we cn still show tht the reltion 29 hols in the totlly symmetric subspce of H, nmely, S E ˆ v S =0. 30 This follows from lemm for n opertor A on H : tr A = 0 for ny pure stte, if n only if S AS =0. The if prt is trivil n we sketch the proof of the only if prt. We write = i= c i i in terms of bsis i of H, where =. Then we hve tr A = i i,j j c i * c * i2 c * i c j c j2 c j i i 2 i A j j 2... j *n = c *n2 *n c 2 c m c m2 m c 2 c n i,m i n n 2 n A m m 2 m =0, 3 where the summtion over integers n i 0 n m i 0 shoul be tken uner the conitions i n i = i m i =, n the stte n n 2 n is the occuption-number representtion of symmetric sttes generlly not normlize, with n i being the occuption number of stte i. Eqution 3 shoul hol for ny complex c i, implying n n 2 n A m m 2 m =0. The ifference between the two unbise conitions 29 n 30 is the projection opertor S in the pure-stte cse. This, however, oes not hmper the subsequent rgument since the support of the opertor for pure is the totlly symmetric subspce. We go bck to the expne form of s in Eq. 5, but before being verge over. By using the unbise conition 30 we reily fin 2 = 2 3 so tht we hve = 2 tr = 2 tr E tr 2. E tr It cn be shown tht 2 tr E tr ˆ v 2, 33 since in the symmetric subspce we hve 0 ˆ v E ˆ v = 2 E ˆ v It is evient tht the equlity hols if the POVM element E is the projector of the eigenstte of ˆ v n the estimte is the corresponing eigenvlue, which is the smple verge of the observe vlues of for ech copy. Thus the minimum vlue of the squre error in the unbise estimtion is given by tr ˆ v 2 tr 2 = tr 2 tr 2 = v, 35 which shows tht the conclusion of Ref. 5 hols if we restrict ourselves to the pure-stte input ensemble. Averging over gives v given in Eq. 25. IV. DISCUSSIO AD COCLUDIG REMARKS We hve seen tht the optiml estimtion of the expecttion vlue of physicl observble is bise, though the optiml mesurement is given by the successive projective mesurement of the observble. The optiml estimte opt is not given by the rithmetic verge of observe t. We cn interpret the expression 2 of the optiml estimte opt in the following wy. First of ll, we shoul remember tht we hve full knowlege on properties of the observble incluing its eigenvlues. Otherwise we cnnot perform mesurement ssocite with. Then wht cn we expect for outcomes of the mesurement before performing the mesurement? The stte is given to us ccoring to the unitry invrint istribution on the pure-stte spce, implying tht we expect tht ech eigenvlue i occurs with equl probbilities s the outcome of the mesurement. This priori knowlege shoul be somehow tken into ccount in the estimtion. We cn see tht this priori knowlege is incorporte into the optiml estimte opt in nturl wy. It is just the rithmetic verge of observe t points in n the efult nonobserve t points i i=, the ltter of which up to the trce of the observble. One my still woner why the weights of the verge for the observe n nonobserve t re equl. Actully this is feture of the pure-stte ensemble consiere in this pper. To see this, let us tke the simplest exmple of =2 n =, but this time the stte is generlly mixe. We ssume tht the Bloch vector n is istribute isotropiclly insie the Bloch sphere. The ensemble is chrcterize by

5 OPTIMAL ESTIMATIO OF A PHYSICAL¼ the verge n 2, which is for the pure-stte ensemble, but generlly less thn. After some clcultion, the men squre error turns out to be opt = 3 3 n2 tr + n 2 i, 2 38 = tr E ˆ 2 + n2 n2 2 2tr tr 2, 36 where ˆ = 3 3 n2 tr + n This implies tht the optiml mesurement is the projective mesurement of, n the optiml estimte is given by where i is the observe eigenvlue of. The miniml men squre error is given by the secon term of Eq. 36. We cn see tht the weight for the observe t ecreses s the egree of mixing of the ensemble increses. When n 2 =0, this opt implies we shoul isregr the observe t. The reson is tht we know tht the expecttion vlue is given by tr /2 for completely mixe stte. The generliztion of our nlysis to n ensemble of mixe sttes, incluing the etils of the bove iscussion, will be presente elsewhere. W. K. Wootters n W. H. Zurek, ture Lonon 299, C. W. Helstrom, Quntum Detection n Estimtion Theory Acemic Press, ew York, A. S. Holevo, Probbilistic n Sttisticl Aspects of Quntum Theory orth-holln, Amsterm, A. Peres n W. K. Wootters, Phys. Rev. Lett. 66, W. K. Wootters, e-print qunt-ph/ M. Bn, K. Kurokw, R. Momose, n O. Hirot, Int. J. Theor. Phys. 36, M. Sski, K. Kto, M. Izutsu, n O. Hirot, Phys. Rev. A 58, Y. C. Elr n G. D. Forney, Jr., IEEE Trns. Inf. Theory 47, C. H. Bennett, D. P. DiVincenzo, C. A. Fuchs, T. Mor, E. Rins, P. W. Shor, J. A. Smolin, n W. K. Wootters, Phys. Rev. A 59, S. Mssr n S. Popescu, Phys. Rev. Lett. 74, R. Derk, V. Bužek, n A. K. Ekert, Phys. Rev. Lett. 80, E. Bgn, M. Big, n R. Muñoz-Tpi, Phys. Rev. A 64, ; Phys. Rev. Lett. 87, M. Hyshi, J. Phys. A 3, A. S. Holevo, e-print qunt-ph/ G. M. D Arino, V. Giovnnetti, n P. Perinotti, J. Mth. Phys. 47, A. Hyshi, T. Hshimoto, n M. Horibe, Phys. Rev. A 72,