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ARTICLE OPEN Quantum parameter estimation with general dynamics

Haidong Yuan1 and Chi-Hang Fred Fung2

One of the main quests in quantum metrology, and quantum parameter estimation in general, is to find out the highest achievable precision with given resources and design schemes to attain it. In this article we present a general framework for quantum parameter estimation and provide systematic methods for computing the ultimate precision limit, which is more general and efficient than conventional methods. npj Quantum Information (2017) 3:14 ; doi:10.1038/s41534-017-0014-6

INTRODUCTION is up to our choice and can be optimized. A pivotal task in fi ρ A pivotal task in science and technology is to identify the highest quantum metrology is to nd out the optimal initial state 0 and achievable precision in measurement and estimation and design the corresponding maximum quantum Fisher information under schemes to reach it. Quantum metrology, which exploits quantum any given evolution ϕx. When ϕx is unitary the GHZ-type of states mechanical effects such as entanglement, can achieve better are known to be optimal, which leads to the Heisenberg limit. precision than classical schemes and has found wide applications However when ϕx is noisy, such states are in general no longer in quantum sensing, gravitational wave detection, quantum- optimal. Finding the optimal probe states and the corresponding enhanced reading of digital memory, quantum imaging, atomic highest precision limit under general dynamics has been the main clock synchronization, etc.;1–11 this has gained increasing attention quest of the field. Recently using the purification approach much in recent years.12–26 progress has been made on developing systematical methods of 12, 13, 15, 18, 19 A typical situation in quantum parameter estimation is to calculating the highest precision limit. These estimate the value of a continuous parameter x encoded in some methods, however, require smooth representations of the Kraus quantum state ρx of the system. To estimate the value, one needs operators, which is not intrinsic to the dynamics. to first perform measurements on the system, which, in the In this article, we provide an alternative purification approach general form, are described by Positive Operator Valued that does not require smooth representations of the Kraus Measurements (POVM), {Ey}, which provides a distribution for the operators. This framework provides systematic methods for measurement results p(y|x) = Tr(Eyρx). According to the computing the ultimate precision limit, which can be formulated Cramér–Rao bound in statistical theory,2, 3, 27, 28 the standard as semi-definite programming and solved more efficiently than deviation for any unbiased estimator of x, based on the conventional methods. We also extend the Bures angle on measurement results y, is bounded below by the Fisher quantum states to quantum channels, which is expected to find fi information: δ^x p1ffiffiffiffiffiffi ; where δ^x is the standard deviation of wide application in various elds of quantum information science. IxðÞ x I x the estimation of , and ( ) is the Fisher information of the P 2 ∂lnpðÞ yj x 29 RESULTS measurement results, IxðÞ¼ y pyxðÞj . The Fisher ∂x Ultimate precision limit information can be further optimized over all POVMs, which gives The precision limit of measuring x from a set of quantum states ρx 1 1 is determined by the distinguishability between ρx and its δ^x qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffi ; ρ 30, 32 J ρ ð1Þ neighboring states x + dx. This is best seen if we expand maxE IxðÞ ðÞx y the Bures distance between the neighboring states ρx and ρx + dx up to the second order of dx:30 where the optimized value J(ρx) is called quantum Fisher information.2, 3, 30, 31 If the above process is repeated n times, ÀÁ 2 1 2 d ρx; ρx dx ¼ JðÞρx dx ; ð2Þ then the standard deviation of the estimator is bounded by Bures þ 4 δ^x pffiffiffiffiffiffiffiffiffi1 : nJðρx Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d ρ ; ρ F ρ ; ρ F ρ ; ρ To achieve the highest precision, we can further optimize the qwhereffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiBuresðÞ¼1 2 2 2 ðÞ1 2 ; here ðÞ¼1 2 x→ρ J ρ 1=2 1=2 encoding procedures x so that ( x) is maximized. Typically Tr ρ ρ ρ is the fidelity between two states. Thus maximizing the encoding is achieved by preparing the probe in some initial 1 2 1 ρ the quantum Fisher information is equivalent as maximizing the state 0, then let it evolve under a dynamics that contains the fi ϕ Bures distance, which is equivalent as minimizing the delity ρ x ρ ϕ ρ ρ ϕ ρ ϕ ρ interested parameter, 0 ! x. Usually x is determined by a between x and x + dx. If the evolution is given by x, x = x( ) given physical dynamics which is then fixed, while the initial state and ρx + dx = ϕx + dx(ρ), the problem is then equivalent to finding

1Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Shatin, Hong Kong and 2Munich Research Center, Huawei Technologies Düsseldorf GmbH, München, Germany Correspondence: Haidong Yuan ([email protected]) Received: 2 February 2016 Revised: 10 August 2016 Accepted: 26 September 2016

Published in partnership with The University of New South Wales Quantum parameter estimation H Yuan and C-H F Fung 2 Âà ÀÁ N out minρ F ϕx ρ ; ϕx dx ρ . We now develop tools to solve this iHtdx ðÞ þ ðÞ θmax θmin ¼ NðÞλmaxjjdx t λminjjdx t .ThusΘ I; e ¼ problem for both unitary and open quantum dynamics. θ θ Nλ dx Nλ dx t ϕ ϕ fi max min ¼ ðÞmaxjj min jj; Eq. (6) then recovers the Heisenberg Given two evolutionÀÁx and x + dx,wedeÂÃneÀÁ the Bures angle 2 2 Θ ϕ ; ϕ 1 F ϕ ρ ; ϕ ρ limit between them as x xþdx ¼ maxρ cos xðÞ xþdxðÞ . This generalizes the Bures angle on quantum states33 to 1 1 δ^x pffiffiffi : quantum channels. Θ(ϕx, ϕx dx) can be seen as an induced ð8Þ + nðÞλmax λmin t N measure on quantum channel from the Bures angle on quantum states, it thus also defines a metric on quantum 40–42 channels. From theÂà definition of the Bures distanceÀÁ it is easy to This also has close connection to the quantum speed limit, 2 essentially the optimal probe state in this case, which is the equal see maxρ d ϕxðÞρ ; ϕx dxðÞρ ¼ 2 2 cos Θ ϕx; ϕx dx , thus Bures þ þ λ λ from Eq. (2) we have superposition of the eigenvectors corresponding to max and min,is ÂÃÀÁ also the state that has the fastest speed of evolution. Θ ϕ ; ϕ 81 cos x xþdx max J½¼ϕxðÞρ lim : ð3Þ ρ dx!0 dx2 Ultimate precision limit for noisy quantum channels. For a general quantum channel that maps from a m1-dimensional to m2- The ultimate precision limit under the evolution ϕx is thus ϕ dimensional Hilbert space,P the evolution can be represented by a determined by the Bures angle between x and the neighboring d † Kraus operation KðÞ¼ρS FjρSF ; here the Kraus operators Fj, channels j¼1 P j d † ≤ j ≤ d m m F Fj Im 1 1 are of the size 2 × 1, j¼1 j ¼ 1 . The channel can δ^x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; Θ ϕ ;ϕ ffiffiffi 4 be equivalently represented as follows: 81½cos ðÞx xþdx p ð Þ dx n ÀÁ lim !0 jjdx † KðÞ¼ρS TrE UESðÞji0E hj0E ρS UES ; ð9Þ where n is the number of times that the procedure is whereji 0E denotes some standard state of the environment, and repeated. If ϕx is continuous with respect to x, then when dx→0, UES is a unitary operator acting on both system and environment, Θ(ϕx, ϕx + dx)→Θ(ϕx, ϕx) = 0, in this case which we will call as the unitary extension of K. A general UES can be written as follows: Θ ϕ ;ϕ 81½cos ðÞx xþdx 2 3 max J½¼ϕxðÞρ lim 2 F ρ dx dx 1 !0 6 7 6 F2 7 Θ ϕ ;ϕ 6 7 2 ðÞx xþdx ð5Þ . . . 16 sin 2 6 . . . 7 ¼ lim dx2 6 . . . 7 dx!0 6 7 U W I 6 F 7; ES ¼ð E m2 Þ 6 d 7 Θ2 ϕ ;ϕ 6 7 ð10Þ 4 ðÞx xþdx ; ¼ lim dx2 6 0 7 dx!0 6 7 6 . . . 7 the ultimate precision limit is then given by 4 . . . 5 δ^x 1 : |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}0 Θ ϕ ;ϕ ffiffiffi ðÞx xþdx p ð6Þ U dx n lim !02 jjdx where only the first m1 columns of U are fixed and WE∈U(p)(p × The problem is thus reduced to determine the Bures angle p fi unitaries) only acts on the environment and can be between quantum channels. We will rst show how to compute chosen arbitrarily; here p ≥ d as p − d zero Kraus operators can the Bures angle between unitary channels, then generalize to be added. noisy quantum channels. Given a channel an ancillary system can be used to Ultimate precision limit for unitary channels U improve the precision limit, this can be described as the extended . GivenÀÁ two unitaries 1 channel U FUρU†; U ρU† X andÀÁ2 ofthesamedimension,sinceÀÁ1 1 2 2 ÀÁÀÁ† † † † F ρ; U U ρU U Θ U ; U Θ I; U U ðÞK IA ðÞ¼ρSA Fj IA ρSA Fj IA ; ¼ 1 2 2 1 , we have ðÞ¼1 2 1 2 , i.e., the Bures angle between two unitaries can be reduced to the Bures angle j m m U between the identity and a unitary. For a × unitary matrix ,let where ρSA represents a state of the original and ancillary systems. iθj e be the eigenvalues of U,whereθj ∈(−π, π], 1 ≤ j ≤ m,whichwe Without loss of generality, the ancillary system can be assumed to will call the eigen-angles of U.Ifθmax = θ1 ≥ θ2≥⋯≥θm = θmin are have the same dimension as the original system. θ θ Θ I; U max min θ − K K arranged in decreasing order, then ðÞ¼ 2 when max Given two quantum channels 1 and 2 of the same dimension, – −iHt λ λ t U U K K θ ≤ π,34 39 specifically if U = e ,thenΘðÞ¼I; U ðÞmax min if let ES1 and ES2 as unitary extensions of 1 and 2, respectively, min 2 we have43 ðÞλmax λmin t π,whereλmax(min) is the maximal (minimal) eigenvalue of H.ThisprovideswaystocomputeBuresangleson unitary channels. For example, suppose the evolution takes the form −ixHt ⊗N −ixHt ΘðÞ¼K1 IA; K2 IA min ΘðÞUES1; UES2 U(x)=(e ) (tensor product of e for N times, which means UES1;UES2 e−ixHt N ð11Þ the same unitary evolution acts on all probes). Then ¼ min ΘðÞUES1; UES2 UES1 ¼ min ΘðÞUES ; UES : † U 1 2 Θ½¼UxðÞ; UxðÞþ dx Θ½I; U ðÞx UxðÞþ dx ES2 ð7Þ hiÀÁ 44 iHtdx N This extends Uhlmann’s purication theorem on mixed states ¼ Θ I; e : to noisy quantum channels. Furthermore, Θ(K1⊗IA, K2⊗IA) can be explicitly computed from the Kraus operators of K1 and K2 K ρ It is easy to see that the difference between the maximal (pleaseP see supplementalP material for detail): if 1ðÞS e−iHtdx ⊗N d F ρ F† K ρ d F ρ F† Θ eigen-angle and the minimal eigen-angle of ( ) is ¼ j¼1 1j S 1j, 2ðÞ¼S j¼1 2j S 2j, then cos

npj Quantum Information (2017) 14 Published in partnership with The University of New South Wales Quantum parameter estimation H Yuan and C-H F Fung ÀÁ ÀÁ 3 K I ; K I 1 λ K K† λ K K† obvious in the region of high η, i.e., low noises. It is also found that ðÞ¼1 A 2 A maxkkW 1 2 min W þ W ; here min W þ W K K† there exists a threshold for η, above the threshold the GHZ state is denotesP the minimum eigenvalue of W þ W , where † the optimal state that achieves the maximal quantum Fisher KW ¼ wijF F j, with wij as the ij-th entry of a d × d matrix W, ij 1i 2 information, but with the decreasing of η the optimal state which satisfies kkW 1(kk denotes the operator norm, which is K K gradually changes from GHZ state to separable state, and this equal to the maximum singularP value). If we substitute 1 = x and d † threshold increases with the number of qubits. K = Kx dx, where KxðÞ¼ρS FjðÞx ρSF ðÞx and Kx dxðÞ¼ρS P2 + j¼1 j þ In Fig. 2 the quantum Fisher information for the optimal state, d F x dx ρ F† x dx x GHZ state, and the separable state are plotted. j¼1 jðÞþ S j ðÞþ with being the interested para- meter, then Θ K I ; K I Parallel scheme cos ðÞx A ÀÁxþdx A † Previous results on the SQL (standard quantum limit)-like scaling 1 λ K K ; ð12Þ 12, 13, 15, 18, 45 ¼ max 2 min W þ W kkW 1 for certain independent noise processes can also be P recaptured in this framework.P In ref. 43 we showedP that given † d † d † where KW wijF x Fj x dx . K ρ F ρ F K ρ F ρ F ¼ ij i ðÞ ðÞþ any two channels 1ðÞ¼S j¼1 1j S 1j, 2ðÞ¼S j¼1 2j S 2j, By substituting ϕx = Kx⊗IA and ϕx + dx = Kx + dx⊗IA in Eq. (3), we we have then get the maximal quantum Fisher information for the K ⊗I extended channel x A, ÀÁ ÂÃÀÁ N N 1 † 2 2 cos Θ K IA; K IA 81 max W 1 λmin KW þ KW 1 2 max J ¼ lim kk2 : ð13Þ ð15Þ dx!0 dx2 † 2 N 2I KW KW þ NNðÞ 1 kkI KW ; fi The maximization in Eq. (13)ÀÁ can be formulated as semi-de nite ⊗N 1 † where K denote N channels in parallel as in Fig. 3, programming: max W λ KW K P kk1 2 min þ W ¼ K w F† F w ij d d W W ¼ ij ij 1i 2j, with ij as the -th entry of a × matrix which satisfies kkW 1. This inequality is valid for any W with maximize 1 t kkW 1, the smaller the right side of the inequality, the tighter 2 2 the bound is. In the asymptotical limit, NNðÞ 1 kkI KW is the 0 1 dominating term, in that case we would like to choose a W I K B C minimizing kk W for a tighter bound. This can be formulated B IW† C ð14Þ s:t: B C 0; @ A 25 Optimal state WI GHZ stat e |+++++> 20 † KW þ KW tI 0:

15 For example, consider two qubits with independent dephasing noises, which can be represented by four Kraus operators: F1(x) ⊗F (x), F (x)⊗F (x), F (x)⊗F (x), F (x)⊗F (x) with F ðÞ¼x 10 qffiffiffiffiffiffiffi1 1 2 qffiffiffiffiffiffiffi2 1 2 1 ÀÁ1 η η σ 1þ Ux F x 1 σ Ux Ux i 3 x : 2 ðÞ, 2ðÞ¼ 2 3 ðÞ; here ðÞ¼exp 2 Figure 1 shows the maximal quantum Fisher information and the quantum Fisher information 5 Fisher information for the separable input state jiþþ , where 0 1 jiþ ¼ jiþpffiffiji. It can be seen that the gain of entanglement is only 2 0

4 Optimal input state −5 Separable input state 3.5 0 0.2 0.4 0.6 0.8 1 η 3 Fig. 2 Quantum Fisher information for optimal probe states, GHZ state, and separable state for five qubits under independent 2.5 dephasing noises

2

1.5 Fisher information

1

0.5

0 0 0.2 0.4 0.6 0.8 1 η Fig. 1 Quantum Fisher information for the optimal input state and separable input state jiþþ for two qubits with independent Fig. 3 N probes with independent noisy processes dephasing noises

Published in partnership with The University of New South Wales npj Quantum Information (2017) 14 Quantum parameter estimation H Yuan and C-H F Fung 4 as semi-definite programming with when η ≠ 1(η = 1 corresponds to the case of no dephasing error) so the first-order term in I cancels. In this case up to the second min W kk¼I KW kk1 order min t  s:t: IW† 0; I K R WI ð16Þ kk W ¼ jj ! ð20Þ † tI I K 2 ðÞ W : η dx2 Odx3 ; 0 ¼ 81ðÞη2 þ ðÞ I KW tI N N 1 cos ΘQC K IA;K IA 2 ½ðÞx xþdx η thus max J ¼ limdx 8 2 2 N, and the For quantumP parameter estimation with the noisy channel !0 ffiffiffiffiffiffiffiffi dx 1η d † p K ρ F x ρF x K K K K 1 1η2 1 xðÞ¼ i¼1 iðÞ i ðÞ, we can substitute 1 = x and 2 = x + dx precision limit δx pffiffiffiffi pffiffiffiffi , which scales as pffiffiffi for any η ≠ 1. d d W W nJ η nN N into Eq. (15). If there exists a × matrixP with kk 1 such 12, 13, 18, 19 I K Ddx2 K w F† x F x dx This is consistent with previous studies but here with a that kk W , where W ¼ ij ij i ðÞjðÞþ , then ξ N clear procedure to obtain the value for . K p1ffiffiffi the precision limit of x will scale at most N. As by substituting K = Kx and K = Kx dx into Eq. (15), 1 2 + ÀÁ DISCUSSION N N 2 2 cos Θ K IA; K IA x xþdx We discuss how our results are related to previous studies. † 2 12, 13 N 2I KW K þ NNðÞ 1 kkI KW Previous studies show that for an extended channel Kx⊗IA the ÀÁ W ð17Þ † 2 4 maximal quantum Fisher information is given by NIkk KW þ I KW þ NNðÞ 1 D dx Xd 2DNdx2 þ NNðÞ 1 D2dx4: ^_ † ^_ max J ¼ 4 min Fj ðÞx FjðÞx ð21Þ F^ x The quantum Fisher information is then bounded by fgj ðÞ j¼1 where the minimization is over all smooth representations of N N 1cos Θ K IA;K IA K J ½ðÞx xþdx equivalent Kraus operators of the channel x. Note that this can be max ¼ lim 8 dx2 dx!0 equivalently written as DN; 8 Pd thus the precision limit has SQL scaling ^_ † ^_ max J ¼ 4 min Fj ðÞx FjðÞx 1 1 F^ x j δx pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi : fgj ðÞ ¼1 nJ nDN 8 Pd F^† x dx F^† x F^ x dx F^ x ðÞj ðÞþ j ðÞ ðÞj ðÞþ j ðÞ For example, consider the dephasing channel with ¼ 4 min lim  ^ dx dx dx fgFj ðÞx j¼1 !0 ð22Þ 1 þ η 1 η † KxðÞ¼ρ UxðÞ ρ þ σ3ρ σ3 U ðÞx ; P 0 0 0 d ^† ^ ^† ^ 2 2 2I F ðÞx Fj ðÞþxþdx F ðÞxþdx Fj ðÞx j¼1 ðÞj j   ¼ 4 min 2 ÀÁ ^ dx σ 01 0 i fgFj ðÞx where UxðÞ¼exp i 3 x , σ ¼ , σ ¼ and 2 1 10 2 i 0 hi  qffiffiffiffiffiffiffi Pd λ F^† x F^ x dx F^† x dx F^ x η 2max ^ min j ðÞj ðÞþþ j ðÞþ j ðÞ 10 1þ Fj ðÞx j¼1 ðÞ σ3 ¼ , η∈[0, 1]. In this case F1ðÞ¼x UxðÞ, F2ðÞ¼x fg ; 0 1 2 ¼ 4 dx2 qffiffiffiffiffiffiffi  ξdx i ξdx 1η cosð Þ sinð Þ where the optimization is over all smooth representations of σ3UxðÞ: We choose W ¼ and vary ξ 2 i sinðξdxÞ cosðξdxÞ equivalent Kraus operators. In previous studiesP the equivalent I K F^ x d ω x F x to minimize kk W . In this case Kraus operatorsP are represented by jðÞ¼ i¼1 jiðÞiðÞand F^ x dx d ω x dx F x dx ω x ji jðÞ¼þ i¼1 jiðÞþ iðÞþ , where ji( )is -entry of

KW cos ξdx F† x F x dx i sin ξdx F† x F x dx i sin ξdx F† x F x dx cos ξdx F† x F x dx ¼ ðÞ1 ðÞ01ðÞþþ ðÞ1 ðÞ2ðÞþþ ðÞ2 ðÞ1ðÞþþ ðÞ1 2 ðÞ2ðÞþ hipffiffiffiffiffiffiffiffiffiffiffiffiffi B idx C ð18Þ B ξdx i η2 ξdx e 2 C B cosðÞþ1 sinðÞ 0 C ¼ B C; @ hipffiffiffiffiffiffiffiffiffiffiffiffiffi A idx 0 cosðÞξdx i 1 η2 sinðÞξdx e 2

thus WE(x)∈U(d), and WE(x) is required to be smooth with respect to x.It  is easy to see that in this case Eq. (22) is a special case of Eq. (13) R iI 0 † I KW ¼ ; ð19Þ with W restricted to the form WE ðÞx WEðÞx þ dx . This restriction 0 R þ iI W pffiffiffiffiffiffiffiffiffiffiffiffiffi arises as the operator E in Eq. (10), which originally can be R ξdx dx η2 ξdx dx arbitrary chosen, was assumed to depend on x smoothly in where ¼ 1 cospðÞffiffiffiffiffiffiffiffiffiffiffiffifficos 2 þ 1 sinðÞsin 2 and 12, 13 dx dx previous studies. Such restriction is not intrinsic to the I ¼ cosðÞξdx sin þ 1 η2 sinðÞξdx cos , then kk¼I KW pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 dynamics. Furthermore without the restriction the set R2 þ I2. Expanding R and I to the second order of dx, we can pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi fgWjkkW 1 is a convex set, which allows a direct formulation 1 4ξ2 4ξ 1 η2 get R ¼ þ þ dx2 þ OdxðÞ3 and I ¼ dx þ 1 η2ξdx as the semi-definite programming. While with the restriction W ¼ 8 2 † 3 1 W x W x dx þOðdx Þ. To minimize kkI KW we should choose ξ ¼ pffiffiffiffiffiffiffiffi E ðÞ EðÞþ needs to be unitary which does not form a 2 1η2

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Published in partnership with The University of New South Wales npj Quantum Information (2017) 14 Quantum parameter estimation H Yuan and C-H F Fung 6 45. Kolodynski, J. Precision bounds in noisy quantum metrology. arXiv. 1409.0535, This work is licensed under a Creative Commons Attribution 4.0 https://arxiv.org/abs/1409.0535 (2014). International License. The images or other third party material in this 46. Doherty, A. C., Parrilo, P. A. & Spedalieri, F. M. Complete family of separability article are included in the article’s Creative Commons license, unless indicated criteria. Phys. Rev. A 69, 022308, doi:10.1103/PhysRevA.69.022308 (2004). otherwise in the credit line; if the material is not included under the Creative Commons 47. Yuan, H. & Fung, C. H. F. Optimal feedback scheme and universal time scaling for license, users will need to obtain permission from the license holder to reproduce the Hamiltonian parameter estimation. Phys. Rev. Lett. 115, 110401, doi:10.1103/ material. To view a copy of this license, visit http://creativecommons.org/licenses/by/ PhysRevLett.115.110401 (2015). 4.0/ 48. Yuan, H. Sequential feedback scheme outperforms the parallel scheme for Hamiltonian parameter estimation. Phys. Rev. Lett. 117, 160801, doi:10.1103/ PhysRevLett.117.160801 (2016). © The Author(s) 2017

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npj Quantum Information (2017) 14 Published in partnership with The University of New South Wales