Entanglement is not very useful for estimating multiple phases

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1 PHYSICAL REVIEW A 70, (2004) Entanglement is not very useful for estimating multiple phases Manuel A. Ballester* Department of Mathematics, University of Utrecht, Box 80010, 3508 TA Utrecht, The Netherlans (Receive 8 April 2004; publishe 13 September 2004) The problem of the estimation of multiple phases (or of commuting unitaries) is consiere. This is a submoel of the estimation of a completely unknown unitary operation where it has been shown in recent works that there are consierable improvements by using entangle input states an entangle measurements. Here it is shown that, when estimating commuting unitaries, there is practically no avantage in using entangle input states or entangle measurements. DOI: /PhysRevA PACS number(s): Mn I. INTRODUCTION A unitary operation is a map that transforms a ensity operator 0 on C to another ensity operator =U 0 U on C, where USU is a special unitary matrix. Suppose one is given a evice that performs an unknown U. One can learn something about U by learning about how it transforms a known state 0. In orer to completely etermine a unitary operation one woul nee to know how it transforms a basis of C plus some linear combinations thereof. This is known as quantum process tomography [5]. More precisely, to estimate U one prepares many copies of the necessary input states an performs a measurement on the output that they prouce. As a result, some classical ata are obtaine an from that one can estimate U. This is shown schematically in Fig. 1. Another approach (use in Refs. [1 4]) is to prepare a bipartite entangle input state 0 on C C an then use one of the parts as input for U, while nothing is one to the other part, as shown in Fig. 2. The effect of the operation is to transform the state 0 to U 1 0 U 1. This output state is then measure an estimate, an, since in this case there is a one-to-one relation between the output state an U, one gets an estimate for U as well. The avantages of this metho with respect to quantum process tomography are that only one input state is neee, an that there is potentially a better accuracy in the estimation (if nonseparable measurements are use) [4]. In this paper a relatively less ifficult problem will be stuie, the estimation of unitary operations that commute with one another, that is, only a maximal Abelian subgroup of SU is consiere instea of the whole group. In this case, the number of unknown parameters ecreases from 2 1 to 1. This problem has alreay been aresse in Ref. [6] where it is given the name of multiple phase estimation (MPE). One woul like to know whether it is also avantageous to use an entangle input in MPE. In what follows the MPE moel that uses entanglement (Fig. 2) will be referre to as MPEE an the one that oes not use it (Fig. 1) will be referre to as MPEU. *Electronic aress: ballester@math.uu.nl; URL: There are two things that nee to be optimize here, the input state an the measurement that is to be performe. Therefore one nees a quantitative measure of how goo an input state is an of how goo a measurement is. II. PRELIMINARIES In this section, the necessary concepts of quantum Fisher information (QFI) an Fisher information (FI) will be introuce, an the quantum Cramér-Rao boun (QCRB) will be state. The QFI an the FI will be use as measures of the performance of an input state an a measurement, respectively. The QCRB relates these two quantities in a nice way. A. QFI Suppose that the quantum state ensity matrix on C is parametrize by R p where p is the number of parameters. In our case woul be the output state an p= 1. Define the symmetric logarithmic erivatives (SLDs) 1,..., p as the self-ajoint operators that satisfy,i = i = 1 2 i + i. For pure states, =, they simply are i =2,i. The QFI is efine as the p p matrix with elements H ij =Retr i j which for pure states reuces to H ij =Rel i l j, where l i = i. The l vectors have a simple geometric interpretation. Suppose one has a pure state moel parametrize by R. For simplicity, the parameter has been taken to be one imensional. Denote the set of all state vectors by H = an the set of all ensity operators by M =. State vectors an ensity operators are relate by the map :, H M. This map is many to one since =e i where is an arbitrary real phase. This means that 1 0 is a circle in H an that a curve in M is mappe through 1 to a tube in H. Conversely, all curves that lie on the surface of the tube, are /2004/70(3)/032310(6)/$ The American Physical Society

2 MANUEL A. BALLESTER PHYSICAL REVIEW A 70, (2004) FIG. 1. Quantum process tomography. mappe through to the same curve in M. Then out of all curves H satisfying 0= 0 an = ( is a given curve in M) there is a minimal curve, efine as the one that at every goes from the circle 1 to the circle 1 + using the shortest path. l is a vector pointing in the irection of that shortest path. It can then be calculate as l=2. B. FI Take a positive operator value measure (POVM) with elements M 1,...,M n. This POVM inuces a probability istribution given by p =trm, the probability to obtain outcome if the parameter has the value. The Fisher information for this measurement is efine as the p p matrix with elements I ij M, = E M, i ln p j ln p. For an estimator ˆ an a measurement M, locally unbiase at 0, 1 the (classical) Cramér-Rao boun is satisfie VM, 0,ˆ IM, 0 1, i.e., the FI is the smallest variance that a locally unbiase estimator base on this measurement can have. This also means that, if one of the eigenvalues of I is zero, then the variance of the function of the parameters corresponing to that eigenvalue is infinity an therefore cannot be estimate. If one has N copies of the quantum state an performs the same measurement on each of the copies then the FI of the N copies, I N, satisfies I N M,=NIM, where IM, is the FI of one system. It follows that V N M, 0,ˆ I N M, 0 1 = IM, 0 1 /N. It is a well known fact in mathematical statistics that the maximum likelihoo estimator in the limit of large N is asymptotically unbiase an saturates the classical Cramér-Rao boun. Moreover no other reasonable estimator (unbiase or not) can o better. C. QCRB The QCRB states that for any measurement M IM, H. 1 In other wors, H IM, is a positive semiefinite matrix. 1 This means that the expectation of the estimator satisfies E M,o ˆ i= 0i an j E M, ˆ i =0 = ij FIG. 2. Entanglement is use, C C moel. This boun is not achievable in general. A theorem ue to Matsumoto [7] states that for pure states the boun is achievable at = 0 if an only if Iml i 0 l j 0 =0. If a moel satisfies the above conition, it is sai to be quasiclassical at 0. Furthermore, if conition (2) hols, there is a measurement with p+2 elements that achieves the boun. In fact, any measurement of the type M = b b, =1,...,p +1, m+1 M p+2 = 1 M, =1 p+1 b = o m, =1 m k = H 1/2 kl l l, m p+1 =, 3 l with o a p+1p+1 real orthogonal matrix satisfying o,p+1 0 achieves IM, 0 =H 0. Now one can see why these quantities are a goo measure of the performance of input states an measurements. From VI 1 /NH 1 /N one can see that a goo input state is one that achieves an H as large as possible an a goo measurement is one that achieves an I as large as possible. Since I an H are matrices the best input state an best measurement can not always be ecie unambiguously. This ambiguity will not be completely resolve in the case of choosing an input state. However, in the case of choosing an optimal measurement there will be no ambiguity. In the next section, it will be shown that for every input state there is a measurement on the output state that achieves equality in Eq. (1). Therefore, an optimal measurement is one that achieves equality in the QCRB. III. MPE IS A QUASICLASSICAL MODEL It will be shown here that both MPEE an MPEU are quasiclassical everywhere (for all R 1 ) an for any input state. In particular, this means that only one input state is necessary in MPEU also. In this respect MPE is quite ifferent from the estimation of a completely arbitrary U. Actually, since MPEU can be consiere as a special case of MPEE, one nees to show quasiclassicality only for MPEE. The MPEE moel is

3 ENTANGLEMENT IS NOT VERY USEFUL FOR PHYSICAL REVIEW A 70, (2004) where = U 0 U, 1 U = expi m T m 1, 4 1,..., (this basis is consiere to be known). From this point, all calculations will be mae in this basis. The input state is 0 = 0 0, an 0 can be expane as 0 = kl R kl kl. The partial trace is then tr B 0 0 =RR, an since RR is self-ajoint an has trace 1, it can be written as 0 on C C is a pure state ensity matrix, 2 an T 1,...,T 1 are selfajoint traceless matrices that commute with one another. They are chosen so that they satisfy an orthonormality conition an The SLDs are tr T m T n = nm. m =2 m =2iT m 1, tr m n =4tr 0 T m T n 1 tr 0 T m 1 0 T n 1. It is easy to see that ue to the commutativity of the T s this quantity is real an therefore the moel is quasiclassical, i.e., it satisfies the conition (2). Therefore for every there exists a measurement M (which may epen on ) such that IM,=H. Furthermore, if one has a large number N of copies, performs the same (optimal) measurement on all N, an calculates the MLE, the mean square error shoul behave as V N M,,ˆ MLE = H 1 /N + o1/n. IV. OPTIMAL INPUT STATE It is now clear from Eq. (6) that one nees to choose the input state so that the QFI is large. Suppose there is an input state that has a QFI that is larger than or equal to the QFI of any other input state. In that case one can unambiguously choose that state as the optimal one. Unfortunately, in our case there is not such a state. Furthermore, there are situations in which one has two input states 1 an 2 with QFI H 1 an H 2, respectively, which satisfy neither H 1 H 2 nor H 1 H 2. This is resolve by maximizing a quantity like Tr GH, with G a real positive semiefinite matrix. By oing this one assigns relative weights to the mean square errors of the ifferent parameters. Furthermore, achieving maximum Tr GH ensures that no other input state can have a larger QFI. In this paper a particular choice is mae: all parameters are given the same importance, i.e., the input state will be chosen so that it maximizes Tr H. With this particular choice it is possible to obtain nice analytic results an, since the trace of the QFI is parametrization invariant, the results obtaine will not epen on the chosen parametrization. A. MPEE Since the self-ajoint matrices T 1,...,T 1 commute with one another, there is a basis where all of them are iagonal 2 0 can be taken to be pure since it was shown in [8] that the QFI is convex. 5 6 RR = t T, =1 where, in general, the last sum inclues all generators of the Lie algebra su. Then the QFI is H mn =4 0 T m T n T m T n 1 0 =4trRR T m T n tr RR T m tr RR T n =4trRR T m T n t m t n an its trace is 1 1 Tr H =4trRR T m 2 t m 2. The commuting T s can be written as T m = c mk kk. The tracelessness conition implies c mk =0, while the orthonormality conition (5) implies c mk c nk = mn. These two together lea to an then to 1 1 T m 2 = c mk c ml = kl 1 1 c 2 mk kk = 1 1. Substituting this in the equation for the trace, one gets Tr H =4 1 1 t m 2, an one immeiately sees that tr H is maximal if an only if t m =0,,..., 1. Of course the rest of the t s can be anything (as long as RR remains positive). Concluing, any pure input state 0 satisfying tr 0 T m 1 =0, m =1,..., 1, achieves a QFI H mn =4 mn / which has the maximum possible trace among all QFIs. In particular, maximally entangle states satisfy this conition. In the full SU moel one obtains a similar result, an the maximum is attaine at an only at the maximally entangle state; this will be shown in the Appenix. B. MPEU It will be shown here that for every entangle input state 0 = kl R kl klc C an every value of the

4 MANUEL A. BALLESTER PHYSICAL REVIEW A 70, (2004) 1-imensional parameter there is an input state 0 C that achieves the same QFI everywhere. This means that in this moel, unlike in the full SU moel, there is no improvement in the accuracy of the estimation by using entangle inputs. The moel is now where = U 0 U, 1 U = expi m T m an 0 = 0 0 is a pure state ensity matrix. Of course this moel is also quasiclassical, an by a calculation ientical to the one mae in the C C case one gets that the QFI is H mn =4 0 T m T n 0 0 T m 0 0 T n 0. It is not ifficult to see that any input state of the form 0 = krr ke i kk, where the s are arbitrary phases, achieves H mn =4 0 T m T n T m T n 1 0, the QFI when the input is the entangle state 0. In particular, any input state of the form 0 = 1 e i kk achieves the maximum trace of the QFI. A state of this form (with all the s set to zero) was use in Ref. [6]. V. OPTIMAL MEASUREMENT A. MPEU This moel (an actually also the MPEE) has the property that I,U MU =I0,M, which is easy to prove; the only necessary ingreient is that U k U = k U =0. Therefore, if one has an optimal measurement M at =0, then the measurement U MU will be optimal at. One can now use the recipe given by Eq. (3) to fin an optimal measurement at =0. The optimal input state 3 given by Eq. (8) is use. At the origin one has l n = 2i = 1 c nk e i kk, e i kk. From these vectors one can form an orthonormal set 3 That is, the one that achieves maximum trace of the QFI. 8 m n = i c nk e i kk, n =1,..., 1, m = 1 e i kk an with the choice o kl = kl 2/ one gets the set of orthonormalize states onto which the measurement elements will project, b k = m k 2 m l, k =1,...,. 9 l=1 The optimal measurement at =0 has elements M k =b k b k an, since the above vectors form an orthonormal basis of C, they satisfy M k =1. The optimal measurement at has elements U b k b k U. Note that the above choice of the orthogonal matrix o works for 3; for =2 another matrix must be chosen. The =2 case will be treate in the following example. Example V1 =2. The orthonormal set forme from the input state an l is m 1 = i 0 1, 2 m 2 = These vectors are then rotate to obtain b 1 = cos m 1 sin m 2 = i 2 e i 0 e i 1, 10 b 2 = sin m 1 + cos m 2 = 1 e i 0 + e i 1, 2 11 where must satisfy sin 0 an cos 0. The measurement elements are b 1 b 1 an b 2 b 2. The FI of this measurement at the origin is equal to the QFI (equal to 2) as expecte. Furthermore, this equality happens to hol everywhere an not only at the origin. This feature is very useful in practice, it means that the optimal measurement oes not epen on the actual (unknown) value of the parameter an therefore an aaptive scheme is not necessary. Whether it is possible to fin measurements with this characteristic for 2 is an open problem. B. MPEE An optimal measurement in this case can also be erive using the recipe given by Eq. (3). In general, such a measurement is a joint measurement on the two output systems. One coul ask whether it is possible to achieve the boun with local measurements an classical communication. In what follows, it will be shown that this is inee possible

5 ENTANGLEMENT IS NOT VERY USEFUL FOR PHYSICAL REVIEW A 70, (2004) The input state is taken to be the maximally entangle state kk/. Alice measures the system coming out of U an Bob measures the other one. The strategy is the following. Bob performs the von Neumann measurement B k =w k w k with k B k =1 on his system where w k = 1 l=1 exp 2kl il, k =1,...,. He obtains outcome k with probability 1/; he then phones Alice an tells her the outcome of his measurement. The net result of this is that Bob prepares the state 1 l=1 exp 2kl il at the input of U. It is easy to recognize this state as one of the optimal states in the C case. Now Alice can perform the measurement A kl with l A kl =1, which is the optimal measurement escribe above for the C case, an where the arbitrary phases are now fixe to l =2k/. It is not ifficult to check that this measurement inee achieves equality in the QCRB at =0. The measurement on C C is then kl A kl B k =1 1 an the measurement kl U A kl U Bk =1 1 is optimal at. If this is applie to the = 2 case an one uses the -inepenent optimal measurement erive for MPEU, a -inepenent optimal measurement is obtaine for MPEE. VI. ASYMPTOTIC FIDELITY From the results obtaine previously, one can infer the asymptotic behavior of the average fielity. Inee, in Ref. [9], it was establishe that the fielity between nearby states is given by 4 F, + = tr + 2 p =1,=1 H + o 2, 4 12 where p is the number of parameters an H is the QFI. In the case stuie here, this fielity woul be between output states. Denote by ˆ the guess for if the outcome of the measurement was, then the fielity, average over all possible outcomes, is F,ˆ = trm F,ˆ tr HVM,,ˆ =1 + o 2, 13 4 where VM,,ˆ = trm ˆ ˆ is the mean square error of measurement M an estimator ˆ. Therefore, using Eq. (6) one gets 4 Actually, this fielity is the square of the fielity use in Ref. [9]. FIG. 3. The points are N1 F N as a function of N for =2,...,5, where F N is the optimal fielity obtaine in Ref. [6]. The continuous lines are at the value 1/4 for =2,...,5, (0.25, 0.5, 0.75, an 1, respectively.) lim N1 F,ˆ N Tr HH 1 = = 1 N This result can be compare with the optimal fielities obtaine in Ref. [6]. These optimal fielites were also average with respect to a uniform prior istribution on. However, since the result obtaine here oes not epen on, its average with respect to any prior will be itself. The comparison is shown in Fig. 3. One can easily see that for large N the optimal fielity of Ref. [6] agrees with the result obtaine here. Actually, this can also be prove analytically but the proof will not be shown here. VII. CONCLUSIONS Two moels have been compare, the moel of estimating commuting unitaries with an without the use of entangle inputs (MPEE an MPEU, respectively). It has been shown that the quantum Crámer-Rao boun is achievable in both MPEE an MPEU. It has also been shown that any quantum Fisher information matrix that can be attaine in MPEE can also be achieve in MPEU. These two facts imply that an entangle input state is unnecessary. A conition for attaining maximal trace of the QFI has been erive. In the MPEE it has also been shown that there is a separable measurement that achieves equality in the QCRB. In the =2 case, measurements that are optimal everywhere have been foun in both MPEU an MPEE. This is a useful feature in practice since this means that an aaptive scheme woul not be necessary in this case. However, it is unclear whether it is possible to fin measurements with this characteristic in general. It is also an open question whether entanglement coul prove itself useful in this respect (for 2). These facts show that entanglement is, at best, not as useful in estimating commuting unitaries as in the estimation of a completely unknown unitary

6 MANUEL A. BALLESTER PHYSICAL REVIEW A 70, (2004) ACKNOWLEDGMENTS This research was fune by the Netherlans Organization for Scientific Research (NWO); support from the RESQ (Grant No. IST ) project of the IST-FET programme of the European Union is also acknowlege. APPENDIX: GENERALIZATION OF THE RESULT IN SECTION IV TO THE FULL MODEL In this appenix, a result similar to that of Sec. IV will be prove in the moel that inclues the whole of SU an not only a commuting subgroup, i.e., the moel consiere in Ref. [4]. Denote by H 0 the QFI at if the input state is 0, an by H the QFI when the input state is a maximally entangle state; in what follows the epenence on will be omitte. Then the following inequality hols for any input state 0 : TrH 1 H 0 2 1, A1 an equality is attaine if an only if 0 is a maximally entangle state. This will be prove in what follows. Notice that this trace is parametrization invariant, an not just Tr H as in Sec. IV; in that case H was proportional to the ientity so there is no contraiction. The moel is again escribe by Eq. (4) but now U is U = V 1 where V =expi 2 1 =1 T. As before, the T s are traceless self-ajoint matrices chosen so that trt T =. The input state 0 is chosen to be pure because of the convexity of the QFI [8]. The SLDs are =2,, where, means the partial erivative of with respect to. The matrix elements of H 0 are H 0 =Retr = 4 Re tr,,. Since 0 is pure it can be written as 0 = 0 0 an 0 = kl R kl kl. H 0 can then be calculate to be H 0 =4RetrRR V, V, +trrr V V, trrr V V,. Denote S = iv V, ; then H 0 =4RetrRR S S trrr S trrr S. Note that S su. Substituting RR =1/ in the expression for H 0, one gets H = 4 trs S. The matrices S 1,...,S 2 1 can be orthonormalize, tr 2 The operator 2 H 1/2 S 2 H 1/2 S 2 H 1/2 S =. H 1/2 S = 4 H 1 S S is a Casimir operator an therefore proportional to the ientity. The proportionality factor can be foun by taking the trace, an finally one gets H 1 S S = The wante trace is TrH 1 H 0 = 2 1 H 1 trrr S trrr S. This quantity is always less than or equal to 2 1 an, furthermore, this value is attaine if an only if trrr S =0 for all =1,..., 2 1, which implies that RR =1/, i.e., 0 is maximally entangle. In particular, this implies that there is no input state 0 for which H 0 H. [1] A. Fujiwara, Phys. Rev. A 65, (2001). [2] F. De Martini, A. Mazzei, M. Ricci, an G. M. D Ariano, Phys. Rev. A 67, (2003). [3] A. Acín, E. Jané, an G. Vial, Phys. Rev. A 64, (2001). [4] M. A. Ballester, Phys. Rev. A 69, (2004). [5] M. A. Nielsen an I. L. Chuang, Quantum Computation an Quantum Information (Cambrige University Press, Cambrige, U.K., 2000). [6] C. Macchiavello, Phys. Rev. A 67, (2003). [7] K. Matsumoto, J. Phys. A 35, 3111 (2002). [8] A. Fujiwara, Phys. Rev. A 63, (2001). [9] M. Hübner, Phys. Lett. A 163, 239 (1992)