Valid lower bound for all estimators in quantum parameter estimation

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1 PAPER OPEN ACCESS Valid lower bound for all estimators in quantum parameter estimation To cite this article: Jing Liu and Haidong Yuan 06 New J. Phys View the article online for updates and enhancements. Related content - Super-additivity in communication of classical information through quantum channels from a quantum parameter estimation perspective Jan Czajkowski, Marcin Jarzyna and Rafa Demkowicz-Dobrzaski - Quantum-enhanced multi-parameter estimation for unitary photonic systems Nana Liu and Hugo Cable - Noisy frequency estimation with noisy probes Agnieszka Górecka, Feli A Pollock, Pietro Liuzzo-Scorpo et al. Recent citations - Quantum-enhanced measurements without entanglement Daniel Braun et al - Quantum Fisher information for unitary parametrization processes governed by arbitrary Lie algebras Hongbin Liang et al - Control-enhanced multiparameter quantum estimation Jing Liu and Haidong Yuan This content was downloaded from IP address on 07/09/08 at 9:36

2 New J. Phys. 8 (06) doi:0.088/ /8/9/ OPEN ACCESS RECEIVED 3 May 06 REVISED July 06 ACCEPTED FOR PUBLICATION 9 August 06 PUBLISHED 6 September 06 PAPER Valid lower bound for all estimators in quantum parameter estimation Jing Liu and Haidong Yuan Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Shatin, Hong Kong, People s Republic of China hdyuan@mae.cuhk.edu.hk Keywords: quantum metrology, mean square error, Cramér Rao bound Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Abstract The widely used quantum Cramér Rao bound (QCRB) sets a lower bound for the mean square error of unbiased estimators in quantum parameter estimation, however, in general QCRB is only tight in the asymptotical limit. With a limited number of measurements biased estimators can have a far better performance for which QCRB cannot calibrate. Here we introduce a valid lower bound for all estimators, either biased or unbiased, which can serve as standard of merit for all quantum parameter estimations.. Introduction An important task in quantum metrology is to find out the ultimate achievable precision limit and design schemes to attain it. This turns out to be a hard task, and one often has to resort to various lower bounds to gauge the performance of heuristic approaches, such as the quantum Cramér Rao bound [ 4], the quantum Ziv Zakai bound [5], quantum measurement bounds [6] and Weiss Weinstein family of error bounds [7]. Among these bounds the quantum Cramér Rao bound (QCRB) is the most widely used lower bound for unbiased estimators [8 30]. However, with a limited number of measurements many practical estimators are usually biased. For eample the minimum mean square error (MMSE) estimator, which is given by the posterior mean ˆ ( y) = p( y) d [34], is in general biased in the finite regime, here denotes the parameter and y denotes measurement results, the posterior probability distribution py ( ) can be obtained by the Bayes rule py p( y ) p( ) ( ) =, with p ( )as the prior distribution of and p( y ) = Tr( r M p( y ) p( ) d y)given by the Bornʼs rule. The MMSE estimator provides the minimum mean square error n å MSE( ˆ) = p( ) (ˆ ( y) - ) p( y ) d. ( ) k= 0 The performance of this estimator, however, cannot be calibrated by quantum Cramér Rao bound in the finite regime as with limited number of measurements it is usually biased. This is also the case for many other estimators including the commonly used maimum likelihood estimator [7 30]. In this paper we derive an optimal biased bound (OBB) which sets a valid lower bound for all estimators in quantum parameter estimation, either biased or unbiased. This bound works for arbitrary number of measurements, thus can be used to gauge the performances of all estimators in quantum parameter estimation. And the difference between this bound and the quantum Cramér Rao bound also provides a way to gauge when quantum Cramér Rao bound can be safely used, i.e., it provides a way to gauge the number of measurements needed for entering the asymptotical regime that the quantum Cramér Rao bound works. The classical optimal biased bound has been used in classical signal processing [35, 36]. 06 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft

3 New J. Phys. 8 (06) Main result Based on different assumptions there eists different ways of deriving lower bounds, for eample some Bayesian quantum Cramér Rao bounds, which are based on a quantum type of Van Tree inequality, have been obtained [3 33]. These bounds require the differentiability of the prior distribution at the boundary of the support region, thus may not apply, for eample, to the uniform prior distribution. The optimal biased bound does not require the differentiability of the prior distribution at the boundary, thus can be applied more broadly. For the completeness, we will first follow the treatment of Helstrom [] to derive a lower bound for estimators with a fied bias, from which we then derive a valid lower bound for all estimators by optimizing the bias. We consider the general case of estimating a function f ( ) for the interested parameter with a given prior distribution. To make any estimation, one needs to first perform some measurements on the state r, which are generally described by a set of Positive Operator Valued Measurements (POVM), denoted as { P y }. The measurements have probabilistic outcomes y with probability p( y ) = Tr( Py r ). An estimator fˆ ( y ), based on the measurement results y, has a mean E ( f ˆ ( y) ) = fˆ ( y ) Tr ( r P y ) d y = f ( ) + b ( ), where b ( ) represents the bias of the estimation. This equation can be written in another form ( f ˆ ( y) - E( )) Tr ( r P ) dy = 0. ( ) where we use E ( ) as a short notation for E ( f ˆ ( y) )which equals to f ( ) + b( )and only depends on. Assuming the prior distribution is given by p ( ), the mean square error is then in the form r y MSE f = d p f y - f ( ˆ) ( )[ ˆ ( ) ( )] Tr ( P ) dy = p( )[ df ˆ + b( )] d, ( 3) where dfˆ = ( f ˆ ( y) - E( )) Tr( r Py) dyis the variance of fˆ ( y ). Differentiating equation () with respect to and use the fact that E ( ) Tr( r P y) dy = E ( ), with E ( ) E/ we get y f y r - E ( ˆ ( ) ( )) Tr P y d y = E ( ). ( 4) Now multiply p ( )at both sides of equation (4) and substitute the following equation into it r r r = ( L + L ), ( 5) here L is known as the symmetric logarithmic derivative of r which is the solution to equation (5). We then obtain Re p ( )( f ˆ ( y) - E ( )) Tr ( r LP ) d y= E ( p ) ( ), ( 6) where Re( ) represents the real part. Multiply both sides again with a real function z( )then integrate with respect to, r y y Re d p ( )( f ˆ ( y) - E ( )) Tr ( z ( ) LP ) dy = pe ( ) ( z ) ( ) d. ( 7) Now we denote A = p ( )( f ˆ ( y) - E ( )) r Py and B = p( ) z( ) r L Py, then the left side of above Re d Tr( A B) dy. Therefore, equation (7) now has the form equation can be rewritten as Using Schwarz inequality we have Re d Tr( A B) dy = p( ) E ( ) z( ) d. ( 8) ( )( ) Re d Tr( A B) dy d Tr( A A) dy d Tr( B B) dy = p ( ) df ˆ d pz ( ) ( J ) ( r ) d,

4 New J. Phys. 8 (06) the last equality we used the fact that and d Tr( A A) dy = d p( )( f ˆ ( y) - E( )) Tr( r P ) dy = = = p ( ) df ˆ d, () 9 d Tr( B B) dy p( ) z( ) Tr( r L) d pz ( ) ( J ) ( r ) d. ( 0) Here J ( r ) = Tr( r L ) is the quantum Fisher information [, ]. Based on above equations, we can obtain pe z d p ( ) df ˆ ( ) ( ) ( ) d pz Jr d, ( ) ( ) ( ) ( ) which is valid for any z( )that satisfies the inequality pz ( ) ( J ) ( r ) d> 0. Assuming J ( r )is complete positive, i.e., J ( r ) > 0, let z( ) = E ( ) J( r )we obtain + pdf p E f b ( ) ˆ ( ) d ( ) d = J r p( ) [ ( ) ( )] d. ( ) ( ) J ( r ) From equation (3) we then get the lower bound for the mean square error f + b MSE( f ˆ) [ ( ) ( )] p( ) + b( ) d. ( 3) J ( r ) When b ( ) = 0, i.e., for unbiased estimators the bound reduces to a Bayesian Cramér Rao bound [3] (another Bayesian QCRB using a left logarithmic derivative is in [3]). Furthermore, if f ( ) =, the bound reduces to the well-used Cramér Rao form [3]. If we only consider f ( ) = and take the prior distribution as a uniform one, above bound can be treated as the quantum version of the biased Cramér Rao bound []. The bound given in equation (3) vividly displays the tradeoff between the variance and the bias of the estimate: at one etreme by letting b ( ) = 0 the unbiased estimates minimize the term b ( ), while the first term is fied; at the other etreme by letting b ( ) =-f( )we can minimize the the first term, but now with a fied bias b( ) = f ( ). The actual minimum of this bound lies somewhere between these two etremes, which provides a lower bound for all estimators. To obtain a valid lower bound for all estimators we use the variational principle to find the optimal b ( )that minimizes the bound in equation (3) which follows the treatment in Ref. [36]. Suppose the support of the prior distribution p ( )is in ( a, a), i.e., p ( ) = 0 for any outside ( a, a). Denote G ( b, ) = p( ){[ f ( ) + b ( )] J( r ) + b( )}, and using variation of calculus, the optimal b ( )that a minimizes Gb (, ) dshould satisfy the Euler Lagrange equation a G G - G b b = G y 0, ( 4) with the Neumann boundary condition = = 0. Substituting the epression of G ( b, ) into b = a b = a the equation, one can obtain = + pb p f ( ) b ( ) ( ) ( ) ( ), ( 5) J ( r ) which gives the following differential equation for the optimal b ( ) which can be reorganized and written compactly as p ( )[ b ( ) + f ( )] + p ( )[ f ( ) + b ( )] pj ( ) ( r) = [ b ( ) + f ( )] + J( r) p( ) b( ), ( 6) J ( r ) p ( ) J( r) b( ) = [ b ( ) + f ( )] ln + b ( ) + f ( ), ( 7) J ( r ) with boundary conditions b ( a) = - f ( a)and b ( a) = - f ( a). Note that the obtained solution of b ( )may not correspond to an actual bias of an estimator, it is just used as a tool to get the lower bound [35]. The optimal 3

5 New J. Phys. 8 (06) bias b ( )can then be obtained by solving this equation, either numerically or analytically. Net, substituting it back to equation (3), one can get a valid lower bound for all estimates. If the prior distribution p ( )and the quantum Fisher information J ( r )are independent of, then the equation simplifies to Jb ( ) = b ( ) + f ( ), ( 8) which can be analytically solved. For eample consider a uniform prior distribution on ( 0, a), and we would like to estimate the unknown parameter itself, i.e., f ( ) =. In this case we can obtain an analytical solution for the optimal bias b ( ) = cosh [ J( a - )] - cosh( J). ( 9) J sinh( Ja) Substituting it back to the right side of the inequality (3), we obtain a valid lower bound for all estimates a MSE ( ˆ) - tanh J. ( 0) J aj3 Compare to the quantum Cramér Rao bound, this bound has an etra term which is then always lower. 3. Eamples In this section, we give four eamples for the valid lower bound. In the first three eamples, the QFI is independent of the parameter under estimation. In these eamples, taking the prior distribution as uniform, the MSE can be directly obtained via equation (0). However, in some cases, the QFI is actually dependent on the estimated parameter. The fourth eample is such a case. In this eample, the optimal bias has to be solved via equation (7). Eample. As the first eample, we consider N spins in the NOON state, ( 00 0ñ+ ñ), which evolves under the dynamics U ( ) = ( e- is3 t ) ÄN (same unitary evolution e- is3 t acts on each of the N spins) with s = 0ñá + ñ0, s y =-i0 ñá + i ñá0 and s 3 = 0ñá0 - ñá as Pauli matrices. After t units of time it evolves to i y ñ= e Nt 00 0ñ+ e- Nt ñ. i ( ) ( ) ( ) We can take the time as a unit, i.e., t =. This NOON state has the quantum Fisher information J = N [4]. For n times repeated measurements, the quantum Fisher information is nn. If the prior distribution p ( )is uniform on ( 0, a), then from equation (0), we have a MSE ( ˆ) - tanh nn. ( ) nn an 3n3 We will compare these bounds with an actual estimation procedure using the MMSE estimator. Consider the measurements in the basis of y0ñ= ( 00 0ñ+ ñ) and yñ= ( 00 0ñ- ñ), which has the measurement results 0 and with probability distribution p0 = áy0 yñ = cos( N )and p = - p0 = sin ( N ). Assuming the measurement is repeated n times, the probability that has k outcomes as is given by n - pk = = - k p k p n k n N N ( ) sink cos( n k), ( 3) 0 k where ()is n the binomial coefficient. From which we can then obtain the MMSE estimator as eplained in the k introduction. To compare the QCRB, MMSE and OBB with the mean square error of this procedure, we plot these three quantities as functions of measurement number n in figure. The solid red, dashed blue lines and black dots in this figure represent the mean square error for the MMSE estimator, the QCRB and the OBB, respectively. From which we can see that while QCRB fails to calibrate the performance of the MMSE estimator, the optimal biased bound provides a valid lower bound. And from the closeness between the MMSE estimator and the optimal biased bound, one can gauge that in this case the MMSE estimator is almost optimal. The bias for the MMSE estimator is also plotted in figure. It can be seen that when n is small, the MMSE estimator is indeed biased, for this reason the QCRB fails to calibrate the performance, while when n gets larger, the estimator becomes more unbiased, indicating a transition into the asymptotical regime where the QCRB starts to be valid. 4

6 New J. Phys. 8 (06) Figure. Mean square error for the minimum mean square error estimator (MMSE, solid red line, equation ()), optimal biased bound (OBB, black dots, equation ()) and quantum Cramér Rao bound (QCRB, dashed blue line) with different number of repeated measurements n. Here we consider a NOON state of N = 0 particles. The prior distribution p ( )is taken as the uniform distribution on ( 0, p 0). Figure. Bias for posterior mean in minimum mean square error estimator for different number of measurements. n = : dotted green line; n = : dash black line; n = 3: dash-dotted red line; n = 5: solid blue line; n = 0: yellow triangulars. Here we consider a NOON state of N = 0 particles. The prior distribution p ( )is taken as the uniform distribution on ( 0, p 0). Eample. We consider a qubit undergoing an evolution with dephasing noise. The master equation for the density matri ρ of the qubit is s g r =- i z r + srs -r, ( z z ), ( 4) where γ is the decay rate and is the parameter under estimation. Take the initial state as y0ñ= ( 0ñ+ ñ), then after time t, which we normalize to, the evolved state reads h - e i r =, ( 5) hei where h = ep( -g). The quantum Fisher information in this case is given by J = h. The quantum Cramér Rao bound for n repeated measurements then gives MSE( ˆ). ( 6 ) n h For the optimal biased bound we again takes the prior distribution p ( )as uniform on ( 0, p). Based on equation (0), one can get the optimal biased bound as 5

7 New J. Phys. 8 (06) Figure 3. Mean square error for the minimum mean square error estimator (MMSE, solid red line, equation ()), optimal biased bound (OBB, dash-dotted black line, equation (7)) and quantum Cramér Rao bound (QCRB, dashed blue line) for a qubit at different rate of dephasing noise η, with the measurements number n = 5. The prior distribution is taken as the uniform distribution on ( 0, p). MSE( ˆ) p - tanh n h. 7 h p h 3 n ( n ) We also use this bound to gauge the performance of a measurement scheme, which measures in the basis of y ñ= ( 0ñ+ ñ) and y ñ= ( 0ñ- ñ). The distributions of the measurement results are given by 0 ( ) p =áyryñ= + h cos( ) ( ) 0, ( 8) p =áy r y ñ= - h cos( ) ( ) 0 0. ( 9) The probability that has k outcomes as among n repeated measurements is pk ( ) = pk( ) pn-k( ) 0. k Again using the minimum mean square error estimator, which is given by the posterior mean k ˆ ( ) = pk ( ) d, we can get the mean square error via equation ().Infigure 3, we plotted the mean square error for the MMSE estimator, the optimal biased bound and quantum Cramér Rao bound at different strength of dephasing noise. It can be seen that while the quantum Carmér Rao bound fails to provide a valid lower bound, the optimal biased bound provides pretty tight bound at all ranges of dephasing noise, which indicates that the MMSE estimator is close to be optimal even at the presence of dephasing noises. Eample 3. In this eample, we consider a SU() interferometer described via a unitary transformation ep( -is). Here S is a Schwinger operator defined as S = ( a b - b a ) with a ( a ), bb ( ) the annihilation i (creation) operators for ports A and B. is the parameter under estimation. Now we take the import state as a coherent state bñfor port A and a cat state a( añ+ -añ) for port B. Here a = + e - a ( )is the normalization number. Taking into account the phase-matching condition, the quantum Fisher information for in this case is in the form [37] n () J = n n + n + n + n a, ( 30) A B A B A where n A = b and nb = a tanh a are photon numbers in port A and B. Based on above epression, the quantum Fisher information J is independent of. Thus, for the optimal biased estimation, the mean square error MSE( ˆ) satisfies equation (0). The maimum Fisher information with respect to n A and n B for a fied yet large total photon number in this case can be achieved when photon numbers for both ports are equal, which is J = N m + N [37], with N the total photon number in the interferometer. Using the optimal biased bound and taking the prior distribution as uniform on ( 0, a), for n times repeated measurements, MSE( ˆ) then satisfies MSE ( ˆ) - nj a( nj) a ( ) tanh nj. 3 3 Figure 4 shows the quantum Cramér Rao bound (dashed blue line), the optimal biased bound (dash-dotted black line) and the minimum mean square error for the MMSE estimator (solid red line). The prior distribution taken as uniform in ( 0, p 5). In this figure, na = nb =. For the MMSE estimator, we measure along the state 6

8 New J. Phys. 8 (06) Figure 4. Optimal biased bound (OBB, solid red line, equation (3)), quantum Cramér Rao bound (QCRB, dashed blue line), the minimum mean square error for the MMSE estimator(mmse, solid red line) for the phase estimation in the interferometer. Here we consider a SU() interferometer with na = nb =. The prior distribution is uniform in ( 0, p 5). ñ. We can see that the optimal biased bound provides a valid lower bound at all range of n, however the gap between the mean square error of the MMSE estimator and the bound indicates that the measurement along the state ñmay not be optimal. Eample 4. The quantum Fisher information in above eamples is independent of the estimating parameter. We give another eample with the quantum Fisher information depending on. Consider a qubit system with the Hamiltonian B H = ( scos + s3sin ), ( 3) which describes the dynamics of a qubit under a magnetic field in the XZ plane, the interested parameter denotes the direction of the magnetic field. The quantum Fisher information of this system has been recently studied with various methods [38 40]. For a pure initial state ( 0ñ+ ñ), the quantum Fisher information is given by (with the evolution time normalized as t = ) B B J = 4sin - cos sin, 33 ( ) ( ) which depends on. In this case, we have to solve equation (7). Like previous eamples, we take the prior distribution p ( )as uniform on ( 0, p ). If we take B = p, with n repeated measurements, J = n( - sin ), then equation (7) reduces to n( - sin) b + sin( ) b = ( -sin) b -sin( ). ( 34) This equation can be numerically solved and by substituting the obtained b ( )into equation (3), the optimal biased bounds can be obtained which is plotted in figure 5. Again we use this bound to gauge the performance of a measurement scheme which takes measurements along y0ñ= ( 0ñ+ ñ) and yñ= ( 0ñ- ñ). The probability distribution of the measurement results are given by B p( ) = sin sin, ( 35) and p( ) 0 = - p( ). When B equals to p, above probability reduces to p( ) = ( sin ). The n probability of having k outcomes as among n repeated measurements is pk ( ) = pk pn-k () ( ) ( ) 0. k Using the posterior mean as the estimator, we can obtain the mean square error for the MMSE estimator which is also plotted in figure 5. From this figure, one can again see that while the quantum Cramér Rao bound (dashed blue line) fails to gauge the performance of the MMSE estimator (solid red line), the optimal biased bound (dashdotted black line) provides a valid lower bound and from the closeness between the mean square error of the MMSE estimator and the optimal biased bound, one can tell that the MMSE estimator is a good estimator here. 7

9 New J. Phys. 8 (06) Figure 5. Mean square error for minimum mean square error estimator (MMSE, solid red line, equation ()), the optimal biased bound (OBB, dash-dotted black line) and quantum Cramér Rao bound (QCRB, dashed blue line) as a function of measurement number n. Here we consider a qubit under a magnetic field in the XZ plane. The prior distribution is taken as uniform in ( 0, p ). 4. Summary The optimal biased bound provides a valid lower bound for all estimators, either biased or unbiased. It can thus be used to calibrate the performance of all estimators in quantum parameter estimation. Asymptotically the widely used quantum Cramér Rao bound provides a lower bound for quantum parameter estimation, however in practice the number of measurements are often constrained by resources, and it is hard to tell when quantum Cramér Rao bound applies. From the difference between the optimal biased bound and quantum Cramér Rao bound it also provides a way to estimate the number of measurements needed to enter the asymptotical regime. Acknowledgements The work was supported by CUHK Direct Grant References [] Helstrom C W 976 Quantum Detection and Estimation Theory (New York: Academic) [] Holevo A S 98 Probabilistic and Statistical Aspect of Quantum Theory (Amsterdam: North-Holland) [3] Braunstein S L and Caves C M 994 Statistical distance and the geometry of quantum states Phys. Rev. Lett [4] Braunstein S L, Caves C M and Milburn G J 996 Generalized uncertainty relations: theory, eamples, and Lorentz invariance Ann. Phys., NY [5] Tsang M 0 Ziv-zakai error bounds for quantum parameter estimation Phys. Rev. Lett [6] Giovannetti V, Lloyd S and Maccone L 0 Quantum measurement bounds beyond the uncertainty relations Phys. Rev. Lett [7] Lu X-M and Tsang M 06 Weiss-Weinstein family of error bounds for quantum parameter estimation Quantum Sci. Technol [8] Fisher R A 95 Theory of statistical estimation Proc. Cambr. Phil. Soc. 700 [9] Cramér H 946 Mathematical Methods of Statistics (Princeton, NJ: Princeton University Press) [0] Rao C R 945 Information and accuracy attainable in the estimation of statistical parameters Bull. Calcutta Math. Soc [] Pezze L and Smerzi A 006 Phase sensitivity of a mach-zehnder interferometer Phys. Rev. A [] Pezze L and Smerzi A 008 Mach-Zehnder interferometry at the Heisenberg limit with coherent and squeezed-vacuum light Phys. Rev. Lett [3] Giovannetti V, Lloyd S and Maccone L 0 Advances in quantum metrology Nat. Photon. 5 [4] Giovannetti V, Lloyd S and Maccone L 006 Quantum metrology Phys. Rev. Lett [5] Fujiwara A and Imai H 008 A fibre bundle over manifolds of quantum channels and its application to quantum statistics J. Phys. A: Math. Theor [6] Escher B M, de Matos Filho R L and Davidovich L 0 General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology Nat. Phys [7] Escher B M, Davidovich L, Zagury N and de Matos Filho R L 0 Quantum metrological limits via a variational approach Phys. Rev. Lett [8] Tsang M 03 Quantum metrology with open dynamical systems New J. Phys

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