PRL 112, 120405 (2014)
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Quantum Metrology in Open Systems: Dissipative Cramér-Rao Bound S. Alipour, M. Mehboudi, and A. T. Rezakhani Department of Physics, Sharif University of Technology, Tehran 14588, Iran (Received 26 July 2013; revised manuscript received 10 November 2013; published 26 March 2014) Estimation of parameters is a pivotal task throughout science and technology. The quantum Cramér-Rao bound provides a fundamental limit of precision allowed to be achieved under quantum theory. For closed quantum systems, it has been shown how the estimation precision depends on the underlying dynamics. Here, we propose a general formulation for metrology scenarios in open quantum systems, aiming to relate the precision more directly to properties of the underlying dynamics. This feature may be employed to enhance an estimation precision, e.g., by quantum control techniques. Specifically, we derive a Cramér-Rao bound for a fairly large class of open system dynamics, which is governed by a (time-dependent) dynamical semigroup map. We illustrate the utility of this scenario through three examples. DOI: 10.1103/PhysRevLett.112.120405
PACS numbers: 03.65.Ta, 03.65.Yz, 03.67.Lx, 06.20.Dk
Introduction.—Metrology and parameter estimation lie at the heart of science, and are prevalent in any aspect of technology. The basic task of identification or estimation of a set of unknown parameters essentially requires an inference from a pool of observed data about the parameters or the system to which they are attributed. As errors and imperfections are unavoidable in practice, increasing the precision of the underlying tasks of data acquisition and inference—hence, improving the quality of estimation—is an important goal of metrology [1]. Improving the quality of measurement instruments and removing sources of systematic errors aside, statistics provides useful suggestions for enhancing metrology, such as increasing data size and repeated measurements on an ensemble of N “probe” systems. Additionally (and more interestingly), the underlying physics of the system of interest may also dictate some restrictions or bounds on the ultimate achievable precision (usually described through a Cramér-Rao inequality [2]), or even may offer new possibilities to exploit. In quantum mechanics, measurements act differently than in classical systems. In addition, interactions with an environment or other systems as well as (quantum) correlations can each affect observed data [3], hence introducing new playing factors in estimation theory. For example, it has been shown that entanglement in a probe ensemble can be exploited to the advantage of a quantum metrology task [4], so that it enables the estimation error of Oð1=NÞ (the “Heisenberg limit”), pffiffiffiffi in contrast to the classical statistical limit of Oð1= N Þ (the “shot-noise limit”). Alternatively, enabling k-body (k ≥ 2) interactions among quantum p probe ffiffiffiffiffiffi systems has been shown to allow an error of Oð1= N k Þ [5], or, it has been argued that application of a suitable entangling operator may even offer an error as small as Oð2−N Þ [6] (beyond the Heisenberg limit). Moreover, nonclassicality has been examined as a potential resource for increasing the metrology resolution in quantum optics [7] (for a general framework of resource analysis, see, 0031-9007=14=112(12)=120405(6)
e.g., Ref. [8]). It thus seems natural to expect that some properties of quantum systems can be employed as a useful “resource” for metrology. Numerous experiments have indeed demonstrated the achievability of sub-shot-noise limit error by using aspects of quantum mechanics; see, e.g., Ref. [9]. In open quantum systems, due to interaction with an environment, the underlying dynamics becomes “noisy.” As a result, formulation and analysis of quantum estimation also becomes more involved [10,11]. In general, dynamics of an open system can be described as ϱS ðτÞ ¼ TrE ½USE ðτ; τ0 ÞϱSE ðτ0 ÞU †SE ðτ; τ0 Þ, where ϱSE is the state of the systems and environment (SE), and USE ðτ; τ0 Þ is the corresponding unitary evolution [12,13]. Thereby, one can argue that in general there may exist a flow of information between the system and the environment [14]. Under some conditions, this dynamics can feature quantum Markovian or non-Markovian properties [15]. The former case typically appears when the environment has a small decoherence time during which correlations disappear, whereas in the latter correlations (both classical and/or quantum [16]) with the environment would form and persist. Such correlations are in practice inevitable, which necessitates investigation of noisy quantum metrology [10,17–21], and may in turn offer new resources for enhancing estimation tasks. However, developing relatively general frameworks for open-system metrology is still needed and is of fundamental and practical importance. Here, we first lay out a fairly general formalism for open quantum system metrology. This (re)formulation of the problem (e.g., cf. Ref. [10]) has this advantage that here precision of estimation is more directly related to the underlying dynamics; besides, it is in some sense analogous to the closed system formulation. This formulation also obviates the need for optimization, whereas it provides efficient and reliable estimation of the error scaling with system size, which is always achievable (and often close
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to exact ultimate precision). Specifically, we derive a quantum Cramér-Rao bound (QCRB) for open system dynamics generated through a dynamical map with the semigroup property. We next illustrate this setting through several examples. Open system dynamics.—Under some specific conditions, the dynamical equation describing the state of an open system to ∂ τ ϱS ðτÞ ¼ ϱS [defined on a Hilbert space HS ] reduces R τ
L 0 dτ0
Lτ ½ϱS ðτÞ, or, equivalently, ϱS ðτÞ ¼ Te τ0 τ ½ϱS ðτ0 Þ, in which T denotes time ordering, and Lτ ½∘ ¼ −i½H S ðτÞ;∘ þ P † † k ηk ðτÞðAk ðτÞ∘Ak ðτÞ − ð1=2ÞfAk ðτÞAk ðτÞ;∘gÞ (for some set of operators fAk ðτÞg) is the (Lindbladian) generator of the dynamical map, with HS ðτÞ being the system Hamiltonian up to a Lamb shift term (we omit the subscript S henceforth). We have also assumed ℏ ≡ 1. In (timedependent) Markovian evolutions, we have ηk ðτÞ ≥ 0 ∀k, τ, while if some ηk becomes negative for some intervals, the associated dynamics would be non-Markovian [12,13,15]. Let us assume that a set of unknown parameters x ¼ ðx1 ; …; xl Þ are to be estimated in a quantum system subject to interaction with an environment. For simplicity of our analysis, here we consider the single-parameter case, while generalization of our framework to the multiparameter case is also straightforward (see example II in the sequel). In the closed-system scenario, this parameter x is usually assumed to enter into the dynamics as a linear coupling in the Hamiltonian HðxÞ ¼ xH acting on some known initial state. In the open-system scenario, similarly, the devised dynamics would, in general, depend on x as ∂ τ ϱðx; τÞ ¼ Lτ ðxÞ½ϱðx; τÞ. For our later use, we vectorize ~ τ ðxÞjϱii, where L ~ τ is this equation, which yields ∂ τ jϱii ¼ L the matrix representation of Lτ [22,23]. Next, we define the normalized pure state ϱ~ ≡ jϱiihhϱj=Tr½ϱ2 (in H⊗2 ), and ~ where L~ does not depend on ~ τ ðx; τÞ ¼ xðτÞL, assume L time; hence, Rτ Rτ ~ ~† e 0 xðsÞdsL ϱ~ ð0Þe 0 xðsÞdsL Rτ Rτ : (1) ϱ~ ðx; τÞ ¼ ~ ~† Tr½e 0 xðsÞdsL ϱ~ ð0Þe 0 xðsÞdsL The initial preparation ϱ~ ð0Þ may itself depend on x, but here we do not assume such generality. QCRB for open system metrology.—Given a data set D ≡ fγ i g constituted from some measurement outcomes γ i over N (identical) probe systems, an estimator xest ðDÞ is chosen for the true value x. By repeating this scenario M times and averaging, the ffiprecision of the estimated x, pffiffiffiffiffiffiffiffiffiffiffiffi evaluated by δx ¼ varðxÞ, is then fundamentally limited by the QCRB qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi δx ≥ 1= MF ðQÞ ðx; NÞ:
denoting the average with respect to the underlying quantum probability distribution), and F ðQÞ ðx; NÞ is the so-called “quantum Fisher information” (QFI) [17,24,25]. By assuming the state of each N-probe set to be ϱðNÞ ðx; τÞ (hereafter, we omit superscript N for brevity) and assigning the corresponding symmetric logarithmic derivative Lϱ through ∂ x ϱ ¼ ðLϱ ϱ þ ϱLϱ Þ=2, the QFI is defined as F ðQÞ ðx; τ; NÞ ≡ Tr½ϱðx; τÞL2ϱðx;τÞ . Note that in closed systems, with ϱðx; τÞ ¼ Uðx; τÞϱð0ÞU † ðx; τÞ P½with i∂ τ U ¼ HU, the P spectral decomposition ϱ ¼ ri jri ihri j and Lϱ ¼ 2 ij hri j∂ x ϱjrj i= ðri þ rj Þjri ihrj j (valid for general dynamics) lead to a direct relation between F ðQÞ and the interaction H. In particular, when HðxÞ ≡ xH and ϱ is pure, we have F ðQÞ ¼ 4τ2 Covϱ ðH; HÞ;
(with equality replaced with ≤ for mixed ϱ), where Covϱ ðX; YÞ ≡ hXYiϱ − hXiϱ hYiϱ is the covariance of a pair of observables X and Y with respect to the state ϱ, which here is the very quantum standard deviation Δ2 H (with h∘iϱ ≡ Tr½ϱ ∘). The resulting relation h pffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii pffiffiffiffiffi δx ≥ 1= 2τ M Covϱ ðH; HÞ ¼ 1= 2τ M ΔH ; (4) pffiffiffiffiffiffiffiffiffi where ΔH ≡ Δ2 H, is more in the spirit of an uncertaintylike relation [24], and shows explicitly how the precision is dictated by the interaction. In open-system cases, however, deriving a similar, direct relation is hardly possible since, e.g., calculating Lϱðx;τÞ is involved as it requires the knowledge of the spectral decomposition of the density matrix. Thus, it is difficult to capture how interaction with an environment affects and the precision. To partially alleviate this issue, here we follow an alternative approach working with the vectorized state ϱ~ instead, which enables a bound somewhat akin to Eq. (4)—with H replaced with L. Although our method gives bounds on the QFI, we demonstrate that this formalism retains significant utility in suggesting correct behavior (e.g., scaling) for the estimation error, and show this explicitly in various examples. Now, from the symmetric logarithmic derivative Lϱ~ ¼ 2∂ x ϱ~ , one can define an associated QFI F~ ðQÞ by replacing ðϱ; Lϱ Þ → ð~ϱ; Lϱ~ Þ in F ðQÞ . After some straightforward algebra [23], using the dynamical equation, Eq. (1), ~ it can ~ τ ðxÞ ≡ xL, and assuming a linear x dependence as L be seen that F~ ðQÞ ¼
(2)
Here, varðxÞ is the variance of any unbiased estimator xest ðDÞ (for which, by definition, hxest i ¼ x, with h∘i
(3)
4 ~ Covϱ~ ðL~ † ; LÞ; ½∂ τ ln xðτÞ2
(5)
which for the time-independent case reduces to ~ This relation is analogous to Eq. (3), where 4τ2 Covϱ~ ðL~ † ; LÞ.
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PHYSICAL REVIEW LETTERS 28 MARCH 2014 4λmax ðϱÞ=Tr½ϱ2 ; ϱ mixed instead of the Hamiltonian we have the generator of the KðϱÞ ≡ (9) open dynamics. 2; ϱ pure; ðQÞ has a natural interpretation. Recall that The QFI F~ and for the latter case the inequality in Eq. (8) is replaced F ðQÞ indeed emerges from the optimization of the Fisher with equality. This bound only needs the knowledge of information over all possible quantum measurements on the generators of the dynamics (Lτ ) and the instantaneous the system [24]. Similarly then, F~ ðQÞ is obtained if any state (ϱ), without need to calculate Lϱ or to do any quantum measurement on the “extended system” is optimization. allowed. Note, however, that a natural extension of the ⊗2 A desirable property of F ðQÞ is that for a fully product measurements in H to H does not necessarily translate estimation scenario with N product input states, we have into most general measurements there. ðQÞ ðQÞ ~ F ðQÞ ðx; τ; NÞ ¼ NF ðQÞ ðx; τ; 1Þ, which naturally carries Let us see how F compares with F . First, we remark that, from vectorizing the very definition of the over to F~ ðQÞ ðx; τ; NÞ [23]. Thus, for this special case, at symmetric logarithmic derivative, we have Lϱ~ ¼ Lϱ ⊗ 1þ the left- and right-hand sides of Eq. (7), we must replace ϱ 1 ⊗ LTϱ − ∂ x ln Tr½ϱ2 . This in turn yields the following [the N-probe state] with ϱð1Þ [the single-probe state] and expression [23]: which exhibits the F~ ðQÞ ðx; τ; NÞ with N F~ ðQÞ ðx; τ; 1Þ,pffiffiffiffi expected shot-noise scaling Oð1= N Þ for the estima 2 2 ðTr½ϱ L Þ 2 ϱ ðQÞ 2 2 tion error. F~ Tr½ϱLϱ ϱLϱ þ Tr½ϱ Lϱ − 2 : ¼ Tr½ϱ2 Tr½ϱ2 Example I.—We assume N probe particles each of which only interacts with a common bath such that the inter(6) actions induce all possible k-body terms (Fig. 1) [26,27] in the Lindbladian as follows: This form is not yet directly related to F ðQÞ . However, X using λmin ðXÞTr½Y ≤ Tr½XY ≤ λmax ðXÞTr½Y (valid for L ½ϱ ¼ σ i1 σ ik ϱσ i1 σ ik − CN;k ϱ; (10) τ any pair of positive matrices X and Y) [λminðmaxÞ ðXÞ denotes i1 ik the minimum (maximum) eigenvalues of X], we obtain PRL 112, 120405 (2014)
Tr½ϱ2 ~ ðQÞ Tr½ϱ2 ~ ðQÞ ≤ F ðQÞ ≤ F F þ FðϱÞ; 4λmax ðϱÞ 4λmin ðϱÞ
(7)
where FðϱÞ ≡ ðTr½ϱ2 Lϱ Þ2 =ðλmin ðϱÞTr½ϱ2 Þ. Note that the upper bound would be vacuous when λmin ðϱÞ ¼ 0; thus, this case needs special care if one wants to use this bound. Another special case is when the evolution is unitary with a pure initial state, i.e., jΨðx; τÞi ¼ Uðx; τÞjΨð0Þi. Here, however, a significant simplification occurs due to hΨðx; τÞjLΨ jΨðx; τÞi ¼ 0, whence Eq. (6) reduces to F~ ðQÞ ¼ 2F ðQÞ [whereas the lower bound of Eq. (7) gives F~ ðQÞ ≤ 4F ðQÞ ]. Equation (7) provides lower and upper bounds on the exact QFI F ðQÞ . To obtain the scaling of F ðQÞ , it suffices to find the scaling of the lower bound of Eq. (7), since if this bound scales as OðN p Þ (for some p ≥ 0), it is guaranteed that F ðQÞ ¼ OðN q Þ with some q ≥ p. However, an upper bound on the QFI might result in an unachievable (hence, unreliable) estimation error; thus, care must be taken with such bounds. This is another distinctive feature of our method in comparison to the methods of Refs. [10,18] that here we use a lower bound on the QFI to predict the scaling of the estimation error. Putting everything together, in general, we have obtained 1=F ðQÞ ≤ K=F~ where
ðQÞ
;
(8)
where σ ij are all the same Pauli matrix (e.g., σ z ), subscript ij is the particle index, and the factor CN;k ¼ N!=½K!ðN − KÞ! counts the number of k-body operators. This is a generalization of the scenario considered in the closedsystem context of Ref. [5], and is beyond the scope of the analysis in Ref. [18] for estimation scenarios with separable channels. We choose the initial state of the whole N-probe system to be the maximally entangled pure ϱð0Þ ¼ jΨihΨj, pffiffistate ffi ⊗N ⊗N where jΨi ¼ ðjEM i − jEm i Þ= 2, and Em (EM ) is the smallest (largest) eigenvalue of σ. For odd ks, σ i1 σ i2 σ ik ⊗ σ i1 σ i2 σ ik ðjΨi ⊗ jΨ iÞ ¼ jΨ⊥ ijΨ⊥ i, p ffiffi ffi where jΨ⊥ i ¼ ðjEM i⊗N þ jEm i⊗N Þ= 2. It is straightforpffiffiffi ward to see that [23] ðjΨ⊥ ijΨ⊥ i − jΨijΨ iÞ= 2 is a
FIG. 1. N probes, initially well isolated from each other, all interact with a common bath through two-body interactions H BPi . Here HPi and HB are the free Hamiltonians of probe i and the bath, respectively. These two-body interactions may induce a many-body quantum correlation among the probes [26,27].
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5
normalized eigenvector of L~ corresponding to the eigenvalue −2CN;k , whence, K F~ ðQÞ
¼
ðe−2CN;k τx þ 1Þðe−4CN;k τx þ 1Þ : 4τ2 C2N;k e−4CN;k τx
C2
C1
x1
(11)
2 1
F~ ðQÞ Nτ2 e−3Γ=2 p ðx1 Þ ¼ ; K 2 chðΓ=2Þ
(12)
N 2 τ2 e−3NΓ=2 F~ ðQÞ e ðx1 Þ ¼ ; K 2 chðNΓ=2Þ
(13)
0 0.0
ðQÞ
F e ðx1 Þ ¼ N 2 τ2 e−2NΓ :
(14)
0.4
0.6
0.8
1.0
1.2
1.4
although with a different noise model in the Markovian case the e scenario has been shown to be advantageous [30]. In the non-Markovian case, however, here the e scenario may outperform p for our specific noise model. Estimation of x2 .—Similar calculations [23] yield F~ ðQÞ Ne−Γ=2 p ðx2 Þ ¼ ; K 4½∂ τ ln x2 2 chðΓÞchðΓ=2Þ
(16)
N 2 e−NΓ=2 F~ ðQÞ e ðx2 Þ ¼ : K 4½∂ τ ln x2 2 chðNΓÞchðNΓ=2Þ
(17)
On the other hand, here the exact QFIs are obtained as ðQÞ
1 Ne−Γ ; 2 2 shðΓÞ ½∂ τ ln x2
(18)
1 N 2 e−NΓ : 2 2 shðNΓÞ ½∂ τ ln x2
(19)
F p ðx2 Þ ¼
(15)
R Here, ΓðτÞ ≡ 0τ x2 ðsÞds, ch ¼ cosh, and sh ¼ sinh. It is evident that the ratio of the bound (12) and the exact value (14) is always equal to ð1 þ e−Γ Þ−1 ; and for large Ns, the ratio of the bound (13) and the exact value (15) goes to 1. Note that when x2 ¼ 0, the ratios both are 1=2, which is consistent with what we expect in the unitary case [F~ ðQÞ ¼ 2F ðQÞ ]. Therefore, our framework correctly captures the scaling of the error in this example. The very problem of estimating x1 (with product input states) has already been discussed in Refs. pffiffiffiffi [10,18] too, where it has been found that δx1 ¼ OðC= N Þ, with a given constant C. However, interestingly, here our formalism gives a more improved scaling in that it compares with the exact solution more favorably and with a better C; see Fig. 2 and the discussion in Ref. [23]. A more exhaustive comparison of the p and e scenarios necessitates finding optimized measurement times for either. We have performed this analysis in Ref. [23] and shown that in the Markovian case of this estimation task no relative advantage is offered by the e scenario,
0.2
pffiffiffiffi FIG. 2 (color online). Factor CðΓÞ in the scaling OðC= N Þ of the x1 estimation with product states. The values C1 [down, purple], Cexact [middle, dot-dashed], and C2 [up, orange] are given through Refs. [10,18], exact calculation, and our method, respectively.
whereas the exact QFIs are argued to be [21,29] F p ðx1 Þ ¼ Nτ2 e−2Γ ;
Cexact
4 3
An immediate implication of this relation and CN;k ¼ OðN k Þ is that for small values of the x parameter a polynomial precision in the estimation can be achieved. Example II.—Consider a dephasing channel acting separately on an N-qubit P system, described by Lτ ½ϱ ¼ ix1 ½H; ϱ þ ð1=2Þx2 ðτÞð Nm¼1 σ zm ϱσ zm − NϱÞ, in whichP x1 is the gap of the Hamiltonian H ¼ Nm¼1 j1im h1j, whose ground-state energy is zero [28]. We assume two different pffiffiffi ⊗N initial states; the product state jΨp i ¼ ½ðj0i þ j1iÞ= 2 and the entangled “GHZ” pffiffiffi state jΨe i ¼ ðj0i⊗N þ j1i⊗N Þ= 2. Estimation of x1 .—Using Eq. (8) and after some algebra [23], it is obtained that in the case of the product (p) and entangled (e) states we have
ðQÞ
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ðQÞ
F e ðx2 Þ ¼
Again it is evident that the ratio of the bound (16) and the exact value (18) is always ðe2Γ − eΓ Þ=ðe2Γ þ 1Þ, and the ratio of the bound (17) and the exact value (19), for large Ns, goes to 1. These results also exhibit correct scalings and behaviors. Example III.—Consider a lossy bosonic channel descri^ þ ϱnÞ=2, ^ bed by Lτ ½ϱ ¼ x½aϱa† − ðnϱ where a (a† ) is the bosonic annihilation (creation) operator, nˆ ¼ a† a, and x is the loss parameter. The QCRB for estimation of φ—defined through tan2 ½φðx; τÞ ¼ exτ − 1—has been ffi obtained as pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi ¯ , and whereby δx ≥ x=ðnτÞ ¯ , where n¯ ¼ δφ ≥ 1= 4nτ ˆ Tr½nϱð0Þ [31]. Particularly, it has been shown that Fock states are optimal for this estimation [19]. Here we revisit this example and demonstrate that the behavior of the error is captured correctly in our framework.
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6
8
2
0.10 0.08 0.06 0 100 N
200
FIG. 3 (color online). ðδφÞmin [or K=F~ ðQÞ ] vs φ and N in example III [Eq. (21)]. Black curves represent values p φffiffiffiffi∈ fπ=20; 2π=20; …; 9π=20g all showing the scaling cðφÞ= N , where cðφÞ ≈ 1=2 as in Ref. [31].
The evolution of this system, when the initial state is ϱð0Þ ¼ jNihNj (whence, n¯ ¼ N), is given by jϱðx; τÞii ¼
N X
s2m c2ðN−mÞ CN;m jN − mijN − mi; (20)
m¼0
in which s ¼ sin φ and c ¼ cos φ. The analytic expression of K=F~ ðQÞ can be found as F~
K ðQÞ
ðφÞ
¼ ð1=4Þcot2 φ max ½CN;m s2m c2ðN−mÞ 0≤m≤N
×
X N
s4m c4ðN−mÞ C2N;m A2N;m
m¼0
−
ð
PN
4m 4ðN−mÞ 2 CN;m AN;m Þ2 −1 m¼0 s c ; PN 4m 4ðN−mÞ 2 CN;m m¼0 s c
(21)
where AN;m ¼ mð1 þ cot2 φÞ − N. Using F ðQÞ ðxÞ ¼ ð∂ x φÞ2 F ðQÞ ðφÞ, one can relate the lower bound for estimation of x to that of φ. Figure 3 depicts ðδφÞmin , which verifies that our bound gives the correct behavior of the error. Summary and outlook.—Here we have outlined a fairly general formalism for open quantum system metrology. In this formulation, the precision of estimation is more directly related to the underlying dynamics, in some sense similar to the closed-system formulation. This property may enable us to enhance metrology in noisy systems by employing quantum or classical control methods to partially engineer or manipulate the system. We have derived a quantum Cramér-Rao bound for open system dynamics generated through a dynamical map with the semigroup property. It has been shown that this method always gives an achievable precision (which is mostly close or equal to the ultimate bound), while it also offers other advantages over existing methods such as providing an efficient method for deriving bounds based on dynamics. This setting was then illustrated through several examples. The first example implied the possibility of exploiting induced correlations of probe quantum systems through a common environment in
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order to achieve a relatively higher precision. The other two examples illustrated that our bound could indeed give the correct scaling of the estimation error. Our formalism may introduce novel methods for utilizing some of the resources offered in open quantum dynamics, such as induced many-body correlations and memory, to, hopefully, enhance a quantum estimation task in the presence of noise. This in turn can spur applications in, e.g., quantum sensing [17,32] and quantum control of optomechanical devices for advanced means and technologies [33]. Supported by Sharif University of Technology’s Office of Vice President for Research.
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[23]
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Phys. Rev. Lett. 105, 050403 (2010); S. Alipour, A. Mani, and A. T. Rezakhani, Phys. Rev. A 85, 052108 (2012); K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Vedral, Rev. Mod. Phys. 84, 1655 (2012). A. Pernice, J. Helm, and W. T. Strunz, J. Phys. B 45, 154005 (2012). V. Giovannetti, S. Lloyd, and L. Maccone, Nat. Photonics 5, 222 (2011). R. Demkowicz-Dobrzański, J. Kołodynski, and M. Gut¸ă, Nat. Commun. 3, 1063 (2012). G. Adesso, F. Dell’Anno, S. De Siena, F. Illuminati, and L. A. M. Souza, Phys. Rev. A 79, 040305(R) (2009). A. Monras and F. Illuminati, Phys. Rev. A 83, 012315 (2011). A. W. Chin, S. F. Huelga, and M. B. Plenio, Phys. Rev. Lett. 109, 233601 (2012). C. M. Caves, J. Supercond. 12, 707 (1999); E. Andersson, J. D. Cresser, and M. J. W. Hall, J. Mod. Opt. 54, 1695 (2007). See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.112.120405 for details of the calculations and examples.
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[24] S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72, 3439 (1994); S. L. Braunstein, C. M. Caves, and G. J. Milburn, Ann. Phys. (N.Y.) 247, 135 (1996); M. G. A. Paris, Intl. J. Quant. Inf. 07, 125 (2009); D. Braun, Eur. Phys. J. D 59, 521 (2010). [25] M. Hayashi, Quantum Information: An Introduction (Springer-Verlag, Berlin, Heidelberg, 2006). [26] D. Braun, Phys. Rev. Lett. 89, 277901 (2002). [27] F. Benatti, R. Floreanini, and M. Piani, Phys. Rev. Lett. 91, 070402 (2003). [28] S. F. Huelga, C. Macchiavello, T. Pellizzari, A. K. Ekert, M. B. Plenio, and J. I. Cirac, Phys. Rev. Lett. 79, 3865 (1997). [29] A. del Campo, I. L. Egusquiza, M. B. Plenio, and S. F. Huelga, Phys. Rev. Lett. 110, 050403 (2013). [30] R. Chaves, J. B. Brask, M. Markiewicz, J. Kołodynski, and A. Acín, Phys. Rev. Lett. 111, 120401 (2013). [31] A. Monras and M. G. A. Paris, Phys. Rev. Lett. 98, 160401 (2007). [32] V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev. Lett. 108, 260405 (2012). [33] J. Abadie et al., Nat. Phys. 7, 962 (2011).
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Quantum Metrology in Open Systems: Dissipative Cramér-Rao Bound S. Alipour, M. Mehboudi, and A. T. Rezakhani Department of Physics, Sharif University of Technology, Tehran 14588, Iran (Received 26 July 2013; revised manuscript received 10 November 2013; published 26 March 2014) Estimation of parameters is a pivotal task throughout science and technology. The quantum Cramér-Rao bound provides a fundamental limit of precision allowed to be achieved under quantum theory. For closed quantum systems, it has been shown how the estimation precision depends on the underlying dynamics. Here, we propose a general formulation for metrology scenarios in open quantum systems, aiming to relate the precision more directly to properties of the underlying dynamics. This feature may be employed to enhance an estimation precision, e.g., by quantum control techniques. Specifically, we derive a Cramér-Rao bound for a fairly large class of open system dynamics, which is governed by a (time-dependent) dynamical semigroup map. We illustrate the utility of this scenario through three examples. DOI: 10.1103/PhysRevLett.112.120405
PACS numbers: 03.65.Ta, 03.65.Yz, 03.67.Lx, 06.20.Dk
Introduction.—Metrology and parameter estimation lie at the heart of science, and are prevalent in any aspect of technology. The basic task of identification or estimation of a set of unknown parameters essentially requires an inference from a pool of observed data about the parameters or the system to which they are attributed. As errors and imperfections are unavoidable in practice, increasing the precision of the underlying tasks of data acquisition and inference—hence, improving the quality of estimation—is an important goal of metrology [1]. Improving the quality of measurement instruments and removing sources of systematic errors aside, statistics provides useful suggestions for enhancing metrology, such as increasing data size and repeated measurements on an ensemble of N “probe” systems. Additionally (and more interestingly), the underlying physics of the system of interest may also dictate some restrictions or bounds on the ultimate achievable precision (usually described through a Cramér-Rao inequality [2]), or even may offer new possibilities to exploit. In quantum mechanics, measurements act differently than in classical systems. In addition, interactions with an environment or other systems as well as (quantum) correlations can each affect observed data [3], hence introducing new playing factors in estimation theory. For example, it has been shown that entanglement in a probe ensemble can be exploited to the advantage of a quantum metrology task [4], so that it enables the estimation error of Oð1=NÞ (the “Heisenberg limit”), pffiffiffiffi in contrast to the classical statistical limit of Oð1= N Þ (the “shot-noise limit”). Alternatively, enabling k-body (k ≥ 2) interactions among quantum p probe ffiffiffiffiffiffi systems has been shown to allow an error of Oð1= N k Þ [5], or, it has been argued that application of a suitable entangling operator may even offer an error as small as Oð2−N Þ [6] (beyond the Heisenberg limit). Moreover, nonclassicality has been examined as a potential resource for increasing the metrology resolution in quantum optics [7] (for a general framework of resource analysis, see, 0031-9007=14=112(12)=120405(6)
e.g., Ref. [8]). It thus seems natural to expect that some properties of quantum systems can be employed as a useful “resource” for metrology. Numerous experiments have indeed demonstrated the achievability of sub-shot-noise limit error by using aspects of quantum mechanics; see, e.g., Ref. [9]. In open quantum systems, due to interaction with an environment, the underlying dynamics becomes “noisy.” As a result, formulation and analysis of quantum estimation also becomes more involved [10,11]. In general, dynamics of an open system can be described as ϱS ðτÞ ¼ TrE ½USE ðτ; τ0 ÞϱSE ðτ0 ÞU †SE ðτ; τ0 Þ, where ϱSE is the state of the systems and environment (SE), and USE ðτ; τ0 Þ is the corresponding unitary evolution [12,13]. Thereby, one can argue that in general there may exist a flow of information between the system and the environment [14]. Under some conditions, this dynamics can feature quantum Markovian or non-Markovian properties [15]. The former case typically appears when the environment has a small decoherence time during which correlations disappear, whereas in the latter correlations (both classical and/or quantum [16]) with the environment would form and persist. Such correlations are in practice inevitable, which necessitates investigation of noisy quantum metrology [10,17–21], and may in turn offer new resources for enhancing estimation tasks. However, developing relatively general frameworks for open-system metrology is still needed and is of fundamental and practical importance. Here, we first lay out a fairly general formalism for open quantum system metrology. This (re)formulation of the problem (e.g., cf. Ref. [10]) has this advantage that here precision of estimation is more directly related to the underlying dynamics; besides, it is in some sense analogous to the closed system formulation. This formulation also obviates the need for optimization, whereas it provides efficient and reliable estimation of the error scaling with system size, which is always achievable (and often close
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to exact ultimate precision). Specifically, we derive a quantum Cramér-Rao bound (QCRB) for open system dynamics generated through a dynamical map with the semigroup property. We next illustrate this setting through several examples. Open system dynamics.—Under some specific conditions, the dynamical equation describing the state of an open system to ∂ τ ϱS ðτÞ ¼ ϱS [defined on a Hilbert space HS ] reduces R τ
L 0 dτ0
Lτ ½ϱS ðτÞ, or, equivalently, ϱS ðτÞ ¼ Te τ0 τ ½ϱS ðτ0 Þ, in which T denotes time ordering, and Lτ ½∘ ¼ −i½H S ðτÞ;∘ þ P † † k ηk ðτÞðAk ðτÞ∘Ak ðτÞ − ð1=2ÞfAk ðτÞAk ðτÞ;∘gÞ (for some set of operators fAk ðτÞg) is the (Lindbladian) generator of the dynamical map, with HS ðτÞ being the system Hamiltonian up to a Lamb shift term (we omit the subscript S henceforth). We have also assumed ℏ ≡ 1. In (timedependent) Markovian evolutions, we have ηk ðτÞ ≥ 0 ∀k, τ, while if some ηk becomes negative for some intervals, the associated dynamics would be non-Markovian [12,13,15]. Let us assume that a set of unknown parameters x ¼ ðx1 ; …; xl Þ are to be estimated in a quantum system subject to interaction with an environment. For simplicity of our analysis, here we consider the single-parameter case, while generalization of our framework to the multiparameter case is also straightforward (see example II in the sequel). In the closed-system scenario, this parameter x is usually assumed to enter into the dynamics as a linear coupling in the Hamiltonian HðxÞ ¼ xH acting on some known initial state. In the open-system scenario, similarly, the devised dynamics would, in general, depend on x as ∂ τ ϱðx; τÞ ¼ Lτ ðxÞ½ϱðx; τÞ. For our later use, we vectorize ~ τ ðxÞjϱii, where L ~ τ is this equation, which yields ∂ τ jϱii ¼ L the matrix representation of Lτ [22,23]. Next, we define the normalized pure state ϱ~ ≡ jϱiihhϱj=Tr½ϱ2 (in H⊗2 ), and ~ where L~ does not depend on ~ τ ðx; τÞ ¼ xðτÞL, assume L time; hence, Rτ Rτ ~ ~† e 0 xðsÞdsL ϱ~ ð0Þe 0 xðsÞdsL Rτ Rτ : (1) ϱ~ ðx; τÞ ¼ ~ ~† Tr½e 0 xðsÞdsL ϱ~ ð0Þe 0 xðsÞdsL The initial preparation ϱ~ ð0Þ may itself depend on x, but here we do not assume such generality. QCRB for open system metrology.—Given a data set D ≡ fγ i g constituted from some measurement outcomes γ i over N (identical) probe systems, an estimator xest ðDÞ is chosen for the true value x. By repeating this scenario M times and averaging, the ffiprecision of the estimated x, pffiffiffiffiffiffiffiffiffiffiffiffi evaluated by δx ¼ varðxÞ, is then fundamentally limited by the QCRB qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi δx ≥ 1= MF ðQÞ ðx; NÞ:
denoting the average with respect to the underlying quantum probability distribution), and F ðQÞ ðx; NÞ is the so-called “quantum Fisher information” (QFI) [17,24,25]. By assuming the state of each N-probe set to be ϱðNÞ ðx; τÞ (hereafter, we omit superscript N for brevity) and assigning the corresponding symmetric logarithmic derivative Lϱ through ∂ x ϱ ¼ ðLϱ ϱ þ ϱLϱ Þ=2, the QFI is defined as F ðQÞ ðx; τ; NÞ ≡ Tr½ϱðx; τÞL2ϱðx;τÞ . Note that in closed systems, with ϱðx; τÞ ¼ Uðx; τÞϱð0ÞU † ðx; τÞ P½with i∂ τ U ¼ HU, the P spectral decomposition ϱ ¼ ri jri ihri j and Lϱ ¼ 2 ij hri j∂ x ϱjrj i= ðri þ rj Þjri ihrj j (valid for general dynamics) lead to a direct relation between F ðQÞ and the interaction H. In particular, when HðxÞ ≡ xH and ϱ is pure, we have F ðQÞ ¼ 4τ2 Covϱ ðH; HÞ;
(with equality replaced with ≤ for mixed ϱ), where Covϱ ðX; YÞ ≡ hXYiϱ − hXiϱ hYiϱ is the covariance of a pair of observables X and Y with respect to the state ϱ, which here is the very quantum standard deviation Δ2 H (with h∘iϱ ≡ Tr½ϱ ∘). The resulting relation h pffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii pffiffiffiffiffi δx ≥ 1= 2τ M Covϱ ðH; HÞ ¼ 1= 2τ M ΔH ; (4) pffiffiffiffiffiffiffiffiffi where ΔH ≡ Δ2 H, is more in the spirit of an uncertaintylike relation [24], and shows explicitly how the precision is dictated by the interaction. In open-system cases, however, deriving a similar, direct relation is hardly possible since, e.g., calculating Lϱðx;τÞ is involved as it requires the knowledge of the spectral decomposition of the density matrix. Thus, it is difficult to capture how interaction with an environment affects and the precision. To partially alleviate this issue, here we follow an alternative approach working with the vectorized state ϱ~ instead, which enables a bound somewhat akin to Eq. (4)—with H replaced with L. Although our method gives bounds on the QFI, we demonstrate that this formalism retains significant utility in suggesting correct behavior (e.g., scaling) for the estimation error, and show this explicitly in various examples. Now, from the symmetric logarithmic derivative Lϱ~ ¼ 2∂ x ϱ~ , one can define an associated QFI F~ ðQÞ by replacing ðϱ; Lϱ Þ → ð~ϱ; Lϱ~ Þ in F ðQÞ . After some straightforward algebra [23], using the dynamical equation, Eq. (1), ~ it can ~ τ ðxÞ ≡ xL, and assuming a linear x dependence as L be seen that F~ ðQÞ ¼
(2)
Here, varðxÞ is the variance of any unbiased estimator xest ðDÞ (for which, by definition, hxest i ¼ x, with h∘i
(3)
4 ~ Covϱ~ ðL~ † ; LÞ; ½∂ τ ln xðτÞ2
(5)
which for the time-independent case reduces to ~ This relation is analogous to Eq. (3), where 4τ2 Covϱ~ ðL~ † ; LÞ.
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PHYSICAL REVIEW LETTERS 28 MARCH 2014 4λmax ðϱÞ=Tr½ϱ2 ; ϱ mixed instead of the Hamiltonian we have the generator of the KðϱÞ ≡ (9) open dynamics. 2; ϱ pure; ðQÞ has a natural interpretation. Recall that The QFI F~ and for the latter case the inequality in Eq. (8) is replaced F ðQÞ indeed emerges from the optimization of the Fisher with equality. This bound only needs the knowledge of information over all possible quantum measurements on the generators of the dynamics (Lτ ) and the instantaneous the system [24]. Similarly then, F~ ðQÞ is obtained if any state (ϱ), without need to calculate Lϱ or to do any quantum measurement on the “extended system” is optimization. allowed. Note, however, that a natural extension of the ⊗2 A desirable property of F ðQÞ is that for a fully product measurements in H to H does not necessarily translate estimation scenario with N product input states, we have into most general measurements there. ðQÞ ðQÞ ~ F ðQÞ ðx; τ; NÞ ¼ NF ðQÞ ðx; τ; 1Þ, which naturally carries Let us see how F compares with F . First, we remark that, from vectorizing the very definition of the over to F~ ðQÞ ðx; τ; NÞ [23]. Thus, for this special case, at symmetric logarithmic derivative, we have Lϱ~ ¼ Lϱ ⊗ 1þ the left- and right-hand sides of Eq. (7), we must replace ϱ 1 ⊗ LTϱ − ∂ x ln Tr½ϱ2 . This in turn yields the following [the N-probe state] with ϱð1Þ [the single-probe state] and expression [23]: which exhibits the F~ ðQÞ ðx; τ; NÞ with N F~ ðQÞ ðx; τ; 1Þ,pffiffiffiffi expected shot-noise scaling Oð1= N Þ for the estima 2 2 ðTr½ϱ L Þ 2 ϱ ðQÞ 2 2 tion error. F~ Tr½ϱLϱ ϱLϱ þ Tr½ϱ Lϱ − 2 : ¼ Tr½ϱ2 Tr½ϱ2 Example I.—We assume N probe particles each of which only interacts with a common bath such that the inter(6) actions induce all possible k-body terms (Fig. 1) [26,27] in the Lindbladian as follows: This form is not yet directly related to F ðQÞ . However, X using λmin ðXÞTr½Y ≤ Tr½XY ≤ λmax ðXÞTr½Y (valid for L ½ϱ ¼ σ i1 σ ik ϱσ i1 σ ik − CN;k ϱ; (10) τ any pair of positive matrices X and Y) [λminðmaxÞ ðXÞ denotes i1 ik the minimum (maximum) eigenvalues of X], we obtain PRL 112, 120405 (2014)
Tr½ϱ2 ~ ðQÞ Tr½ϱ2 ~ ðQÞ ≤ F ðQÞ ≤ F F þ FðϱÞ; 4λmax ðϱÞ 4λmin ðϱÞ
(7)
where FðϱÞ ≡ ðTr½ϱ2 Lϱ Þ2 =ðλmin ðϱÞTr½ϱ2 Þ. Note that the upper bound would be vacuous when λmin ðϱÞ ¼ 0; thus, this case needs special care if one wants to use this bound. Another special case is when the evolution is unitary with a pure initial state, i.e., jΨðx; τÞi ¼ Uðx; τÞjΨð0Þi. Here, however, a significant simplification occurs due to hΨðx; τÞjLΨ jΨðx; τÞi ¼ 0, whence Eq. (6) reduces to F~ ðQÞ ¼ 2F ðQÞ [whereas the lower bound of Eq. (7) gives F~ ðQÞ ≤ 4F ðQÞ ]. Equation (7) provides lower and upper bounds on the exact QFI F ðQÞ . To obtain the scaling of F ðQÞ , it suffices to find the scaling of the lower bound of Eq. (7), since if this bound scales as OðN p Þ (for some p ≥ 0), it is guaranteed that F ðQÞ ¼ OðN q Þ with some q ≥ p. However, an upper bound on the QFI might result in an unachievable (hence, unreliable) estimation error; thus, care must be taken with such bounds. This is another distinctive feature of our method in comparison to the methods of Refs. [10,18] that here we use a lower bound on the QFI to predict the scaling of the estimation error. Putting everything together, in general, we have obtained 1=F ðQÞ ≤ K=F~ where
ðQÞ
;
(8)
where σ ij are all the same Pauli matrix (e.g., σ z ), subscript ij is the particle index, and the factor CN;k ¼ N!=½K!ðN − KÞ! counts the number of k-body operators. This is a generalization of the scenario considered in the closedsystem context of Ref. [5], and is beyond the scope of the analysis in Ref. [18] for estimation scenarios with separable channels. We choose the initial state of the whole N-probe system to be the maximally entangled pure ϱð0Þ ¼ jΨihΨj, pffiffistate ffi ⊗N ⊗N where jΨi ¼ ðjEM i − jEm i Þ= 2, and Em (EM ) is the smallest (largest) eigenvalue of σ. For odd ks, σ i1 σ i2 σ ik ⊗ σ i1 σ i2 σ ik ðjΨi ⊗ jΨ iÞ ¼ jΨ⊥ ijΨ⊥ i, p ffiffi ffi where jΨ⊥ i ¼ ðjEM i⊗N þ jEm i⊗N Þ= 2. It is straightforpffiffiffi ward to see that [23] ðjΨ⊥ ijΨ⊥ i − jΨijΨ iÞ= 2 is a
FIG. 1. N probes, initially well isolated from each other, all interact with a common bath through two-body interactions H BPi . Here HPi and HB are the free Hamiltonians of probe i and the bath, respectively. These two-body interactions may induce a many-body quantum correlation among the probes [26,27].
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5
normalized eigenvector of L~ corresponding to the eigenvalue −2CN;k , whence, K F~ ðQÞ
¼
ðe−2CN;k τx þ 1Þðe−4CN;k τx þ 1Þ : 4τ2 C2N;k e−4CN;k τx
C2
C1
x1
(11)
2 1
F~ ðQÞ Nτ2 e−3Γ=2 p ðx1 Þ ¼ ; K 2 chðΓ=2Þ
(12)
N 2 τ2 e−3NΓ=2 F~ ðQÞ e ðx1 Þ ¼ ; K 2 chðNΓ=2Þ
(13)
0 0.0
ðQÞ
F e ðx1 Þ ¼ N 2 τ2 e−2NΓ :
(14)
0.4
0.6
0.8
1.0
1.2
1.4
although with a different noise model in the Markovian case the e scenario has been shown to be advantageous [30]. In the non-Markovian case, however, here the e scenario may outperform p for our specific noise model. Estimation of x2 .—Similar calculations [23] yield F~ ðQÞ Ne−Γ=2 p ðx2 Þ ¼ ; K 4½∂ τ ln x2 2 chðΓÞchðΓ=2Þ
(16)
N 2 e−NΓ=2 F~ ðQÞ e ðx2 Þ ¼ : K 4½∂ τ ln x2 2 chðNΓÞchðNΓ=2Þ
(17)
On the other hand, here the exact QFIs are obtained as ðQÞ
1 Ne−Γ ; 2 2 shðΓÞ ½∂ τ ln x2
(18)
1 N 2 e−NΓ : 2 2 shðNΓÞ ½∂ τ ln x2
(19)
F p ðx2 Þ ¼
(15)
R Here, ΓðτÞ ≡ 0τ x2 ðsÞds, ch ¼ cosh, and sh ¼ sinh. It is evident that the ratio of the bound (12) and the exact value (14) is always equal to ð1 þ e−Γ Þ−1 ; and for large Ns, the ratio of the bound (13) and the exact value (15) goes to 1. Note that when x2 ¼ 0, the ratios both are 1=2, which is consistent with what we expect in the unitary case [F~ ðQÞ ¼ 2F ðQÞ ]. Therefore, our framework correctly captures the scaling of the error in this example. The very problem of estimating x1 (with product input states) has already been discussed in Refs. pffiffiffiffi [10,18] too, where it has been found that δx1 ¼ OðC= N Þ, with a given constant C. However, interestingly, here our formalism gives a more improved scaling in that it compares with the exact solution more favorably and with a better C; see Fig. 2 and the discussion in Ref. [23]. A more exhaustive comparison of the p and e scenarios necessitates finding optimized measurement times for either. We have performed this analysis in Ref. [23] and shown that in the Markovian case of this estimation task no relative advantage is offered by the e scenario,
0.2
pffiffiffiffi FIG. 2 (color online). Factor CðΓÞ in the scaling OðC= N Þ of the x1 estimation with product states. The values C1 [down, purple], Cexact [middle, dot-dashed], and C2 [up, orange] are given through Refs. [10,18], exact calculation, and our method, respectively.
whereas the exact QFIs are argued to be [21,29] F p ðx1 Þ ¼ Nτ2 e−2Γ ;
Cexact
4 3
An immediate implication of this relation and CN;k ¼ OðN k Þ is that for small values of the x parameter a polynomial precision in the estimation can be achieved. Example II.—Consider a dephasing channel acting separately on an N-qubit P system, described by Lτ ½ϱ ¼ ix1 ½H; ϱ þ ð1=2Þx2 ðτÞð Nm¼1 σ zm ϱσ zm − NϱÞ, in whichP x1 is the gap of the Hamiltonian H ¼ Nm¼1 j1im h1j, whose ground-state energy is zero [28]. We assume two different pffiffiffi ⊗N initial states; the product state jΨp i ¼ ½ðj0i þ j1iÞ= 2 and the entangled “GHZ” pffiffiffi state jΨe i ¼ ðj0i⊗N þ j1i⊗N Þ= 2. Estimation of x1 .—Using Eq. (8) and after some algebra [23], it is obtained that in the case of the product (p) and entangled (e) states we have
ðQÞ
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ðQÞ
F e ðx2 Þ ¼
Again it is evident that the ratio of the bound (16) and the exact value (18) is always ðe2Γ − eΓ Þ=ðe2Γ þ 1Þ, and the ratio of the bound (17) and the exact value (19), for large Ns, goes to 1. These results also exhibit correct scalings and behaviors. Example III.—Consider a lossy bosonic channel descri^ þ ϱnÞ=2, ^ bed by Lτ ½ϱ ¼ x½aϱa† − ðnϱ where a (a† ) is the bosonic annihilation (creation) operator, nˆ ¼ a† a, and x is the loss parameter. The QCRB for estimation of φ—defined through tan2 ½φðx; τÞ ¼ exτ − 1—has been ffi obtained as pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi ¯ , and whereby δx ≥ x=ðnτÞ ¯ , where n¯ ¼ δφ ≥ 1= 4nτ ˆ Tr½nϱð0Þ [31]. Particularly, it has been shown that Fock states are optimal for this estimation [19]. Here we revisit this example and demonstrate that the behavior of the error is captured correctly in our framework.
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6
8
2
0.10 0.08 0.06 0 100 N
200
FIG. 3 (color online). ðδφÞmin [or K=F~ ðQÞ ] vs φ and N in example III [Eq. (21)]. Black curves represent values p φffiffiffiffi∈ fπ=20; 2π=20; …; 9π=20g all showing the scaling cðφÞ= N , where cðφÞ ≈ 1=2 as in Ref. [31].
The evolution of this system, when the initial state is ϱð0Þ ¼ jNihNj (whence, n¯ ¼ N), is given by jϱðx; τÞii ¼
N X
s2m c2ðN−mÞ CN;m jN − mijN − mi; (20)
m¼0
in which s ¼ sin φ and c ¼ cos φ. The analytic expression of K=F~ ðQÞ can be found as F~
K ðQÞ
ðφÞ
¼ ð1=4Þcot2 φ max ½CN;m s2m c2ðN−mÞ 0≤m≤N
×
X N
s4m c4ðN−mÞ C2N;m A2N;m
m¼0
−
ð
PN
4m 4ðN−mÞ 2 CN;m AN;m Þ2 −1 m¼0 s c ; PN 4m 4ðN−mÞ 2 CN;m m¼0 s c
(21)
where AN;m ¼ mð1 þ cot2 φÞ − N. Using F ðQÞ ðxÞ ¼ ð∂ x φÞ2 F ðQÞ ðφÞ, one can relate the lower bound for estimation of x to that of φ. Figure 3 depicts ðδφÞmin , which verifies that our bound gives the correct behavior of the error. Summary and outlook.—Here we have outlined a fairly general formalism for open quantum system metrology. In this formulation, the precision of estimation is more directly related to the underlying dynamics, in some sense similar to the closed-system formulation. This property may enable us to enhance metrology in noisy systems by employing quantum or classical control methods to partially engineer or manipulate the system. We have derived a quantum Cramér-Rao bound for open system dynamics generated through a dynamical map with the semigroup property. It has been shown that this method always gives an achievable precision (which is mostly close or equal to the ultimate bound), while it also offers other advantages over existing methods such as providing an efficient method for deriving bounds based on dynamics. This setting was then illustrated through several examples. The first example implied the possibility of exploiting induced correlations of probe quantum systems through a common environment in
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order to achieve a relatively higher precision. The other two examples illustrated that our bound could indeed give the correct scaling of the estimation error. Our formalism may introduce novel methods for utilizing some of the resources offered in open quantum dynamics, such as induced many-body correlations and memory, to, hopefully, enhance a quantum estimation task in the presence of noise. This in turn can spur applications in, e.g., quantum sensing [17,32] and quantum control of optomechanical devices for advanced means and technologies [33]. Supported by Sharif University of Technology’s Office of Vice President for Research.
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