Time-local optimal control for parameter estimation in the Gaussian regime
TTime-local optimal control for parameter estimation in the Gaussian regime
Alexander Predko a , Francesco Albarelli b,c , Alessio Serafini a a Department of Physics & Astronomy, University College London, Gower Street, WC1E 6BT, London, United Kingdom b Faculty of Physics, University of Warsaw, 02-093 Warszawa, Poland c Department of Physics, University of Warwick, Gibbet Hill Road, CV4 7AL, Coventry, United Kingdom
Abstract
Information about a classical parameter encoded in a quantum state can only decrease if the state undergoes a non-unitaryevolution, arising from the interaction with an environment. However, instantaneous control unitaries may be usedto mitigate the decrease of information caused by an open dynamics. A possible, locally optimal (in time) choice forsuch controls is the one that maximises the time-derivative of the quantum Fisher information (QFI) associated with aparameter encoded in an initial state. In this study, we focus on a single bosonic mode subject to a Markovian, thermalmaster equation, and determine analytically the optimal time-local control of the QFI for its initial squeezing angle(optical phase) and strength. We show that a single initial control operation is already optimal for such cases andquantitatively investigate situations where the optimal control is applied after the open dynamical evolution has begun.
1. Introduction
In analysing the accuracy of high precision metrologicalsetups, one must take into account quantum mechanics,since it enforces an intrinsically statistical description ofexperiments. The field of quantum parameter estimationwas born to address the challenges that arise when combin-ing concepts from classical statistics with the formalism ofquantum mechanics [1, 2]. This line of research has leadto the realisation that peculiar properties of quantum sys-tems can be employed to build high precision sensors, withperformances not available to purely classical systems [3–7].However, when the quantum system is subjected to anoisy (non-unitary) dynamics the promised advantage caneasily be lost [8–10]. Several approaches have been pro-posed to suppress or at least mitigate the effect of noise inquantum metrology. Some of those assume (partial) accessto the environment causing the non-unitary dynamics [11–16], possibly also applying measurement-based feedback.Another popular way of tackling this issue is the use oferror correcting codes, which have to be specifically tailoredfor metrological applications [17–26].The form of quantum control that is closest to exper-imental implementations is focused on devising optimaltime-dependent pulses given an available set of controlHamiltonians [27]. In this context, optimal pulses for noisymetrology have been investigated by numerical methods [28–30]. Dynamical decoupling [31, 32] is a particularly usefulapproach in choosing pulses to protect a quantum systemfrom an environment, and it has also been studied in thecontext of quantum metrology [33, 34].
Email addresses: [email protected] (FrancescoAlbarelli), [email protected] (Alessio Serafini)
We mention that the interplay between quantum metrol-ogy and quantum control is a vast subject that goes wellbeyond the aim of counteracting and mitigating the effectof noise. Optimal control pulses have been employed toachieve a nonclassical time-scaling for the precision in theestimation of parameters characterizing time-dependentfields [35–37], and also to enhance the performance of re-mote parameter estimation [38].In some physical systems, it is reasonable to assume thatthe control operations can act essentially instantaneously(relative to the timescales of the free dynamics). Underthis assumption, we can think of the controlled dynamicsas the free nonunitary evolution interspersed by unitaryoperations; this is the framework we will be working in.The use of such control unitaries was shown to be useful torestore a nonclassical time scaling when sensing nontrivialHamiltonian parameters [39–41], and the same idea hasbeen applied to Gaussian multimode interferometry [42].In this paper, we focus on a particular flavour of op-timal control that is “time-local’, meaning that the time-derivative of some figure of merit is optimised over in-stantaneous unitary operations. This approach has beenapplied to thermodynamical quantities, both in finite-dimensional [43] and Gaussian quantum systems [44, 45]and also to counteract the decay of entanglement in two-mode Gaussian systems [46]. Here we follow the sameapproach but with the aim of counteracting the decay ofthe metrological usefulness of a quantum state, quantifiedby the quantum Fisher information (QFI). In particular, wefocus on a simple but paradigmatic model: a single bosonicmode evolving in a Markovian thermal environment. Fur-thermore, we concentrate on Gaussian states and Gaussiancontrol operations, since they are readily implementable inmany physical platforms [47–50].
Preprint submitted to Elsevier January 22, 2020 a r X i v : . [ qu a n t - ph ] J a n e consider a situation in which the encoding of theunknown parameter happens before the free open dynam-ics of the system. This is a reasonable model when alsothe parameter encoding happens very quickly comparedto the time scale of the free evolution. In other words,the noisy dynamics can be considered as an unavoidablepart of the detection stage, which does not affect the pa-rameter encoding stage. This approach is different fromconsidering the encoding on the parameter as part of thedynamical evolution, as done in many of the studies wehave mentioned.One of our main results and our starting point forthis analysis is a compact analytical formula for the timederivative of the QFI. We find that the information about aparameter encoded in the first moments is unaffected by thecontrol strategies we consider, and thus we restrict to pa-rameters encoded in the covariance matrix (CM), i.e., angleand strength of squeezing. Not surprisingly, the estimationof squeezing with Gaussian states has been considered byvarious authors [51–57]. Here, we find the optimal uni-tary controls to preserve the QFI associated with both thesqueezing angle (i.e., an optical phase parameter) and thesqueezing strength. Interestingly, the optimal strategy wefind is analogous to the one found in [39, 40] for noiselessHamiltonian estimation, i.e., applying the inverse of the pa-rameter encoding unitary. This inverse transformation alsoappears when studying optimal measurements for noiselessmultiparameter estimation [58, 59]. Since these optimaloperations depend on the unknown true value of the pa-rameter, their effectiveness only make sense in the contextof an adaptive estimation scheme [60].Remarkably, we also find that repeated control opera-tions are unnecessary and the optimal control operationis equivalent to an optimal encoding of the state before itundergoes the open dynamics. We also study the effect ofapplying the optimal control operations with some delayafter the start of the open evolution of the state. Applyingan initial control operation is equivalent to optimally en-coding information before the action of a noisy channel, anidea that has been proposed to preserve the multipartiteentanglement and the QFI of multiqubit systems [61, 62]and recently tested experimentally [63]. In a continuousvariables setting, one may, along similar lines, use squeezingto minimise the decoherence of non-Gaussian states evolv-ing in a Gaussian environment, which is formally equivalentto squeezing the state of the environment [64–67].This paper is structured as follows. In Section 2, weintroduce Gaussian systems and the dynamical model wewill consider in the following. In Section 3, we revisethe basics of quantum parameter estimation, in particularapplied to Gaussian systems. Section 4 contains our resultsabout time-local optimal control applied to the preservationof the QFI of Gaussian quantum states. Section 5 ends thepaper with some remarks and possible directions for futurestudies.
2. System and dynamics
We shall consider a system of one bosonic, mode associ-ated to a vector of canonical operators ˆ r = (ˆ x, ˆ p ) T obeyingthe canonical commutation relations (CCR) [ˆ x, ˆ p ] = i ˆ ,where we have set (cid:126) = 1. One may also express the CCRin terms of the symmetrised commutator [47] [ˆ r , ˆ r T ] =ˆ r ˆ r T − (ˆ r ˆ r T ) T = i Ω, where Ω is a 2 × (cid:18) − (cid:19) . For a quantum state ˆ (cid:37) , the expectation value of the observ-able ˆ x is given by (cid:104) ˆ x (cid:105) = Tr[ˆ (cid:37) ˆ x ]. Using vector notation, thiscan be generalised to give the first and second statisticalmoments of a state: r = Tr[ˆ (cid:37) ˆ r ] , σ = Tr[ { (ˆ r − r ) , (ˆ r − r ) T } ˆ (cid:37) ] . (1)The above definition leads to a real, symmetric CM σ ,satisfying σ ≥ i Ω.We will consider the encoding of classical information(a real parameter’s value) in Gaussian states, which mayin general be defined as the ground and thermal states ofquadratic Hamiltonians. Such states are fully characterisedby first and second statistical moments, as defined above.Unitary operations which map Gaussian states into Gaus-sian states are those generated by a quadratic Hamiltonian.The effect of such operations on the vector of operators isa symplectic transformation ˆ r → S ˆ r where S is a 2 × , R ,i.e. S Ω S T = Ω. The corresponding effect on the CM ofthe system is the transformation σ −→ S σ S T , while firstmoments are transformed according to r (cid:55)→ S r . It will beconvenient to parametrise single-mode symplectic trans-formations though their singular value decomposition [47]: S = (cid:18) cos ϕ sin ϕ − sin ϕ cos ϕ (cid:19) (cid:18) z z (cid:19) (cid:18) cos χ sin χ − sin χ cos χ (cid:19) , (2)for χ, ϕ ∈ [0 , π [ and z ≥
1. From the normal modedecomposition of σ and Eq. (2), it follows that the mostgeneral CM of a single-mode system may be written as arotated and squeezed thermal state of the free Hamiltonian(ˆ x + ˆ p ) [47]: σ = ν (cid:18) cos θ sin θ − sin θ cos θ (cid:19) (cid:18) y y (cid:19) (cid:18) cos θ − sin θ sin θ cos θ (cid:19) , (3)with squeezing parameter y ≥
1, optical phase θ ∈ [0 , π [,we will also use the terms squeezing strength for the pa-rameter r , defined as y = e r , and squeezing angle θ .These parameters appear in the unitary transformationˆ S ξ = e ξ ∗ ˆ a − ξ ˆ a † , with the complex parameter ξ = r e iθ ,where we have introduced the bosonic annihilation operatorˆ a = (ˆ x + i ˆ p ) / √
2. The symplectic eigenvalue ν ≥ ν = 1are pure, while the ground state of the free Hamiltonian(the “vacuum” state) is obtained by setting y = ν = 1, sothat σ = .The free, uncontrolled dynamics of our system willbe the diffusion induced by contact with a white noise(Markovian) environment at finite temperature, describedby a Lindblad master equation˙ˆ (cid:37) = L ¯ N ˆ (cid:37) = (cid:0) ¯ N + 1 (cid:1) D [ˆ a ]ˆ (cid:37) + ¯ N D [ˆ a † i ]ˆ (cid:37) , (4)where we have introduced the superoperator D [ˆ o ]ˆ (cid:37) =ˆ o ˆ ρ ˆ o † − (cid:8) ˆ o † ˆ o, ˆ ρ (cid:9) and the Lindbladian L ¯ N , i.e. the gen-erator of the dynamical semigroup [68, 69]. We work withdimension-less time, expressed in units of the inverse lossrate, that thus does not appear in (4); ¯ N is the meannumber of excitations in the environment, related to itsinverse temperature β by the Bose law. This dynamicsdescribes loss in a thermal environment and is ubiquitous inquantum optics. Since the generator is time-independent,the solution from a time t to a time t is formally givenby the map e ( t − t ) L ¯ N . This evolution is also known as thequantum attenuator channel.At the level of Gaussian states this dynamics is de-scribed by the following equations of motion for first andsecond moments: ˙ σ = − σ + N , (5)˙ r = − r , (6)where N = (2 ¯ N + 1). For a more general and detaileddescription of Gaussian quantum systems and dynamicssimilar to the one presented here see [47, 70].We shall assume the ability to intersperse such an opendynamics with instantaneous Gaussian (IG) unitary opera-tions ˆ U IG that will enact our control, which correspond tosymplectic transformations in the phase-space descriptionsuited to Gaussian states. In practice, this is a reasonableassumption if the inverse loss rate is much larger thanthe time it takes to perform a control operation. Albeitsuch times may vary widely in specific cases, there existoptical set-ups with typical decoherence times around 10 µ s and operating times around 1 ns, where the hypothesisof instantaneous controls is very reasonable.
3. Quantum parameter estimation
Let ˆ (cid:37) θ be a set of quantum states whose exact formdepends on an unknown parameter θ . The problem of theoptimal estimation of θ (i.e., of obtaining an estimatorwith minimal variance) through a fixed POVM Π = { ˆΠ x ≥ , (cid:80) x ˆΠ x = } , is a classical problem associated with theprobability distribution p ( x | θ ) = Tr[ ˆΠ x ˆ (cid:37) θ ]. The optimalsolution is given by the classical Fisher information I Π ,θ (where we emphasise the dependence on the chosen POVM in the context pictured above), given by I Π ,θ = (cid:88) µ p ( µ | θ ) [ ∂ θ ln ( p ( µ | θ ))] = (cid:88) µ ( p (cid:48) ( µ | θ )) p ( µ | θ ) (7)(where the prime denotes the partial derivative with respectto the parameter θ ), and by the associated Cram´er–Raobound, which may be saturated by unbiased estimators:∆ θ ≥ (cid:112) nI Π ,θ , (8)where ∆ θ is the standard deviation on the estimate of θ and n is the number of measurements carried out (in thelanguage of statistics this corresponds to the sample size).We stress that, while an unbiased estimator might not existfor finite n , it is usually possible to find an estimator thatsaturates (8) in the limit of large n [71, 72].The optimisation of the classical Fisher information overall possible POVMs gives rise to the QFI I θ [1, 2, 73, 74]: I θ = max Π I Π ,θ = lim (cid:15) → − F [ˆ (cid:37) θ , ˆ (cid:37) θ + (cid:15) ]) (cid:15) , (9)where, as above, the symbol Π stands for the whole setof POVMs and the QFI is expressed in terms of the fi-delity F [ (cid:37), σ ] = (cid:13)(cid:13) √ (cid:37) √ σ (cid:13)(cid:13) , where (cid:107) A (cid:107) = Tr (cid:104) √ AA † (cid:105) isthe trace norm. One can prove that the QFI is monotoni-cally decreasing when parameter independent channels areapplied to the state, while it is invariant if the transforma-tion is a unitary [75, 76]. As one would expect, the QFIenters the quantum Cram´er–Rao bound∆ θ ≥ √ nI θ . (10)Since this inequality may be shown to be achievable, itrepresents the ultimate bound to quantum parameter es-timation. The QFI is therefore the most fundamentalquantity in assessing the sensitivity of a system to a certainparameter, which might reflect environmental or technicalfactors. Just as in the classical case, the inequality (10) isusually saturated only in the asymptotic limit. Howeverthe optimal POVM arg max Π I Π ,θ might depend on thetrue unknown value θ , thus some kind of adaptive strategyis needed in general [60]. In other words, the informationquantified by the QFI makes sense in a local estimation scenario, where we assume to have prior knowledge aboutthe parameter value and to be in a neighbourhood of thetrue value of the parameter.If the quantum states ˆ (cid:37) θ are all Gaussian states, andtherefore the dependence on θ is entirely contained in theCM σ and in the vector of first moments r , one mayobtain an analytical formula for the QFI. Such a formula isparticularly wieldy for single-mode Gaussian states [47, 77–80], on which we shall focus in this paper: I θ = 12 Tr[( σ − σ (cid:48) ) ]1 + µ + 2 µ (cid:48) − µ + 2 r (cid:48) T σ − r (cid:48) , (11)3here µ = Tr[ˆ (cid:37) θ ] = 1 / √ Det σ = 1 /ν is the purity of thequantum states and the prime (cid:48) denotes differentiationwith respect to the parameter θ . Notice that, in all ofthe formulae above, the derivatives are taken at the ‘true’value ¯ θ of the parameter θ , so that both the QFI and theoptimal POVM will in general depend on ¯ θ (more on thisissue later). In the following, it will be convenient to makethe distinction between θ and ¯ θ explicit and clear.
4. Locally optimal control to protect the quantumFisher information
Later on, we shall assume Gaussian states dependingon an unknown optical phase or squeezing parameter andsubject to the diffusive dynamics (5,6), and assess theperformance of instantaneous control symplectics towardsthe task of parameter estimation. We would like thereforeto determine controls that maximise the evolving QFIassociated with phase estimation. Notice that, althoughthe QFI is obviously invariant under unitary, and hencesymplectic, operations, its time-derivative under the freeopen dynamics we are considering need not be, and in factis not. Therefore, symplectic controls may enhance theQFI during the time evolution.Our first step is then to obtain a general expression forthe time-derivative of the QFI (11) under Eqs. (5) and (6),which is derived in Appendix A and turns out to be rathercompact:˙ I θ = µ Tr[( σ − σ (cid:48) ) ]( N Tr[ σ − ] − µ ) − N Tr[( σ − σ (cid:48) ) σ − ]1 + µ − µ Tr[ σ − σ (cid:48) ] (cid:0) ( N Tr[ σ − ] − σ − σ (cid:48) ] + 2 N Tr[ σ − σ (cid:48) ] (cid:1) − µ )+ µ (1 − µ ) Tr[ σ − σ (cid:48) ] (2 − N Tr[ σ − ]) − N r (cid:48) T σ − r (cid:48) . (12)The “locally” (in time) optimal control will be determinedby letting σ (cid:55)→ S σ S T (and likewise for σ (cid:48) ) and r (cid:55)→ S r inthe expression above, and by maximising it with respectto the parameters χ , ϕ and z that parametrise the control S as per Eq. (2).Inspection reveals that the controlled ˙ I θ does not de-pend on the first rotation in the singular value decomposi-tion of S , so that we can set χ = 0 in what follows withoutloss of generality.Also note that the control can never help preservingthe QFI associated with first moments, since the only termwhich depends on them is clearly invariant under any S .We will therefore neglect first moments hereafter. As a first case study, let us consider the estimation ofthe optical phase θ of a squeezed state with first momentsindependent from θ and CM of the form of Eq. (3). Noticethat this also encompasses the case of a state undergoing the dynamics described by Eqs. (5,6) for any initial tran-sient time, since the dynamics is phase-covariant, i.e. itcommutes with the action of the phase shifter imprintingthe dependence on the parameter θ . In other words, theonly difference between considering the instantaneous con-trol acting at the beginning or at some intermediate time isreflected in different values for the parameters ν and y , butthe optimization problem to be solved remains identical.It is now convenient to define R θ = (cid:18) cos θ sin θ − sin θ cos θ (cid:19) (13)and observe that R (cid:48) θ = R ¯ θ Ω = Ω R ¯ θ , (14) R T (cid:48) θ = R (cid:48)− θ = − R − ¯ θ Ω = − Ω R − ¯ θ (15)(recall that all the derivatives with respect to the parameter θ must be taken at the true value ¯ θ ), where Ω is the 2 × Y = diag( y , /y ), which characterises the ini-tial state, and Z = diag( z, /z ), characterising the controloperation.Including the transformation (2) for χ = 0, one hasthe following expressions for the controlled σ − and σ (cid:48) associated to the estimation of the phase of squeezing: σ − = ν − Z − R (¯ θ + ϕ ) Y − R − (¯ θ + ϕ ) Z − , (16) σ (cid:48) = νZR (¯ θ − ϕ ) [Ω , Y ] R ( ϕ − ¯ θ ) Z = ν (cid:18) y − y (cid:19) ZR (¯ θ − ϕ ) σ x R ( ϕ − ¯ θ ) Z , (17)where σ x is the Pauli x matrix. Whence σ − σ (cid:48) = (cid:18) y − y (cid:19) Z − R (¯ θ + ϕ ) Y − σ x R − (¯ θ + ϕ ) Z (18)and therefore, in this case,Tr[ σ − σ (cid:48) ] = ν − (cid:18) y − y (cid:19) Tr[ Y − σ x ] = 0 , (19)which simplifies our task greatly, since it sets to zerothe third and fourth term in Eq. (12) which, noticingthat Tr[( σ − σ (cid:48) ) ] = 2 (cid:16) y − y (cid:17) and Tr[( σ − σ (cid:48) ) σ − ] = (cid:16) y − y (cid:17) Tr[ σ − ], reduces to:˙ I θ = − (cid:18) y − y (cid:19) ν + N ν (2 ν + 1)Tr[ σ − ]2( ν + 1) . (20)The optimal control operation is therefore the one thatminimises Tr[ σ − ], whose coefficient above is negative. Itis apparent from the general expression of a single-modeCM (3) that, up to the symplectic eigenvalue that is not af-fected by a symplectic transformation, Tr σ − is minimised,obtaining the value 2 /ν , by making the CM proportional to4 ( t t c ) L ¯ N
0. This setup isschematically represented in Fig. 1.Before the control is applied (i.e., for t < t c ), one has σ = e − t R ¯ θ Y R − ¯ θ + (1 − e − t ) N , (21) σ (cid:48) = e − t (cid:18) y − y (cid:19) R ¯ θ σ x R − ¯ θ , (22)which may be inserted into Eq. (11) to obtain the QFI ofthe evolved state (as shown in Appendix A, the secondterm on the RHS of Eq. (11) is proportional to Tr[ σ − σ (cid:48) ]and hence vanishes in our case, and so does the third asthe first moments do not contribute to the estimation). I θ I θ Figure 2: The QFI for the estimation of the squeezing angle θ as afunction of time (both dimensionless, the latter in units of inverse lossrate), shown for different times at which the locally optimal controlis applied: the top (blue) curve depicts the case where the controlis applied at t = 0; at each subsequent curve, from top to bottom,the control is applied 0 .
05 inverse loss rates later; the bottom curve(green) depicts the uncontrolled QFI. In the upper panel we considera system with initial squeezing y = 3 and an open dynamics with N = 1 (“pure loss”), while in the lower panel we consider y = 10 and N = 2. After the control is applied, for t ≥ t c , one has instead σ = e − t Y − c R (¯ θ − ϕ c ) Y R − (¯ θ − ϕ c ) Y − c + (1 − e − t ) N Y c = (cid:16) e − ( t − t c ) ν c + (1 − e − ( t − t c ) ) N (cid:17) , (23) σ (cid:48) = e − t e − t (cid:18) y − y (cid:19) Y − c σ x Y − c = e − t (cid:18) y − y (cid:19) σ x , (24)where ϕ c = ¯ θ (the distinction between the two has beenmaintained to make the derivation of σ (cid:48) clearer), the di-agonal squeezing transformation Y c is chosen ad hoc tomake the CM proportional to the identity, and ν c = (cid:114)(cid:104) e − t c y + (1 − e − t c ) N (cid:105)(cid:104) e − tc y + (1 − e − t c ) N (cid:105) is the sym-plectic eigenvalue of the evolving state at the moment thecontrol is enacted.The calculations detailed in the two previous paragraphsyield the following expression for the “controlled” QFI asa function of time: I θ = e − t (cid:16) y − y (cid:17) (cid:104) e − t y +(1 − e − t ) N (cid:105)(cid:104) e − ty +(1 − e − t ) N (cid:105) +1 , t < t c , e − t (cid:16) y − y (cid:17) ( e − ( t − tc ) ν c +(1 − e − ( t − tc ) ) N ) +1 , t ≥ t c . (25)The effect of such a control scheme on the QFI is illustratedin Fig. 2, where one may appreciate the advantage gained5y activating the control protocol after different intervalsfrom the initial time. By comparing the two panels we canalso appreciate on a qualitative level that an initial controloperation is more useful when the initial squeezing is high,since in this case the uncontrolled QFI drops more steeply. The same analysis carried out above for the squeezingphase may be repeated for the squeezing strength of theinitial state. To this aim, it is convenient to re-parametrisethe state, setting y = e r , so that Y = e rσ z ( σ z beingthe Pauli z matrix), and consider the estimation of theparameter r . The problem becomes then completely phase-invariant, so we can omit the rotation in the evolving statewithout loss of generality, and just assume the CM ν e rσ z ,for r > ν ≥ σ − and σ − associated to the estima-tion of the squeezing parameter r take the form σ − = ν − ZR − ϕ e − rσ z R ϕ Z , (26) σ (cid:48) = νZ − R − ϕ e rσ z σ z R ϕ Z − . (27)Whence σ − σ (cid:48) = ZR − ϕ σ z R ϕ Z − , (28)such that, once again, Tr[ σ − σ (cid:48) ] = 0, and Tr[( σ − σ (cid:48) ) ] =2, Tr[( σ − σ (cid:48) ) σ − ] = Tr[ σ − ], leading to˙ I r = − ν + N ν (2 ν + 1)Tr[ σ − ]2( ν + 1) . (29)In this case too, the optimal control operation is the onethat minimises Tr[ σ − ] that, as above, is obtained by set-ting Z = e rσ z and ϕ = 0. The optimal strategy requiresa single control, since the free dynamics preserve the opti-mal form of the CM (of minimal Tr[ σ − ] upon symplecticaction).Notice that the formulae for the estimation of the squeez-ing parameter are identical to what found above, exceptfor the absence of the factor ( y − − y ) in σ (cid:48) (clearly, forthe optical phase, the QFI is always equal to 0 for y = 1,because the initial state is then rotationally invariant; thisis not an issue for squeezing, since such an operation admitsno invariant states). Assuming, like above, to start with apure squeezed vacuum state ˆ ρ ξ = ˆ S ξ | (cid:105)(cid:104) | ˆ S † ξ , and that thecontrol is applied at some time t c ≥
0, one has the rescaledformula for the QFI as a function of time: I r = e − t (cid:2) e − t y +(1 − e − t ) N (cid:3)(cid:2) e − ty +(1 − e − t ) N (cid:3) +1 , t < t c , e − t (cid:2) e − ( t − tc ) ν c +(1 − e − ( t − tc ) ) N (cid:3) +1 , t ≥ t c . (30)with ν c = (cid:2) e − t c y + (1 − e − t c ) N (cid:3)(cid:2) e − tc y + (1 − e − t c ) N (cid:3) ,exactly as in the previous section. We plot this quantityin Fig. 3: the behaviour in time is qualitatively the sameas the previous case of phase estimation, albeit the scale is I r I r Figure 3: The QFI for the estimation of the squeezing strength as afunction of time (both dimensionless, the latter in units of inverse lossrate), shown for different times at which the locally optimal controlis applied: the top (blue) curve depicts the case where the controlis applied at t = 0; at each subsequent curve, from top to bottom,the control is applied 0 .
05 inverse loss rates later; the bottom curve(green) depicts the uncontrolled QFI. In the upper panel we considera system with initial squeezing y = 3 and an open dynamics with N = 1 (“pure loss”), while in the lower panel we consider y = 10 and N = 2. different because of the absence of the factor ( y − − y ) inthe expression for I r . In this case the initial QFI at t = 0is independent from the parameter and equal to 1 / r = 0.However, as one would expect, in this particular case thereis no need for control operations since the state is alreadythermal; this can also be checked explicitly from (30) bysetting y = 1.
5. Conclusions and remarks
We have only started to uncover the usefulness of time-local quantum control for the task of parameter estimation.In particular, Gaussian systems proved to be a useful testbed for these strategies, since their simplicity allowed us toderive closed form expressions for the relevant quantitiesat play.Essentially, we have shown that, for the considered opendynamics, the locally-optimal (in time) way to delay thedecay of QFI about a parameter unitarily encoded in theCM of a single-mode Gaussian state is to unsqueeze it,thereby transforming it into a thermal state. This meansthat such a thermal state is best suited to withstand a lossyevolution in a thermal environment, which is mathemati-cally described by a phase-covariant channel. Intriguingly,our results might be related to the fact that the minimum6utput entropy of a phase-covariant Gaussian channel isachieved by a thermal input state [81, 82], although welack a deeper understanding of this connection.Several extensions of the ideas we have proposed herecan be envisioned, most notably dropping the assumptionof fast parameter encoding. This means that, instead ofonly delaying the demise of the QFI of an initial state,we should consider the situation where the parameter isencoded simultaneously to the open dynamics, e.g., byadding an Hamiltonian term in the Lindblad master equa-tion or estimating parameters of the non-unitary part ofthe Gaussian dynamics, see, e.g., [83–89]. In such scenariostime-local control would be used to increase the rate atwhich information about the parameter is acquired duringthe dynamics.Finally, let us briefly mention that there exist dynam-ical decoupling schemes tailored for continuous variablesystems [90, 91] that could in principle be used to com-pletely remove the effect of the environment. However, indynamical decoupling the control operations must be notonly instantaneous, but they have to be applied with arate greater than the environment cut-off frequency andare thus very hard to implement in practice. On the otherhand our proposed strategy, while only capable of miti-gating the effect of noise, can be readily implemented onexperimental platforms through a single control operation.
Acknowledgements
We are grateful to D. Branford, A. Datta and M. G. Genonifor useful discussions. FA acknowledges financial supportfrom the UK National Quantum Technologies Programme(EP/M013243/1) and from the National Science Center(Poland) grant No. 2016/22/E/ST2/00559.
Appendix A. Derivation of the time derivative ofthe QFI
The starting point of this derivation is Eq. (11) for theQFI of a single mode state together with the equations ofmotion of the first an second statistical moments of theGaussian state (5) and (6).The time derivative of the QFI (11) is˙ I θ = 12 (1 + µ )Tr[ ∂ t (cid:0) σ − σ (cid:1) ] − Tr[ (cid:0) σ − σ (cid:1) ] ∂ t µ (1 + µ ) + 2(1 − µ ) ∂ t ( µ (cid:48) ) + 2 µ (cid:48) ∂ t µ (1 − µ ) + 2( ∂ t r (cid:48) ) T σ − r (cid:48) + 2 r (cid:48) T σ − ( ∂ t r ) + 2 r (cid:48) T ( ∂ t σ − ) r (cid:48) (A.1)Finally, we need the equation of motions for the deriva-tives of the first moment vector and of the CM, obtained by differentiating (5) and (6) with respect to the parameter:˙ σ (cid:48) = − σ (cid:48) (A.2)˙ r (cid:48) = − r (cid:48) . (A.3)In this calculation we will use the the following for-mula for the derivative of the determinant of an invertiblematrix A ( t ): dd t Det A ( t ) = Det A ( t )Tr (cid:2) A ( t ) − t A ( t ) (cid:3) , aswell as the formula for the derivative of the inverse matrix dd t A ( t ) − = − A ( t ) − (cid:0) dd t A ( t ) (cid:1) A ( t ) − . First term
We first start by noticing thatTr (cid:104) ∂ t (cid:0) σ − σ (cid:48) (cid:1) (cid:105) = 2Tr (cid:2) ( σ − σ (cid:48) ) ∂ t ( σ − σ (cid:48) ) (cid:3) , (A.4)where we have used the cyclicity of the trace. This termcan be simplified as follows( σ − σ (cid:48) ) ∂ t ( σ − σ (cid:48) ) = ( σ − σ (cid:48) )( ∂ t σ − σ (cid:48) + σ − ∂ t σ (cid:48) )= ( σ − σ (cid:48) )( − σ − ( ∂ t σ ) σ − σ (cid:48) + σ − ∂ t σ (cid:48) )= ( σ − σ (cid:48) )( − σ − ( − σ + N ) σ − σ (cid:48) + σ − ( − σ (cid:48) ))= − N σ − σ (cid:48) σ − σ (cid:48) , (A.5)so that we obtainTr (cid:104) ∂ t (cid:0) σ − σ (cid:48) (cid:1) (cid:105) = − N Tr (cid:2) ( σ − σ (cid:48) ) σ − (cid:3) . (A.6)The second part of the first term’s numerator can beexpanded using the following identity ∂ t µ = ∂ t (Det σ ) − = − σ ) ( ∂ t Det σ )= − σ Tr (cid:2) σ − ∂ t σ (cid:3) = − σ Tr (cid:2) σ − ( − σ + N ) (cid:3) = − µ (cid:0) N Tr[ σ − ] − (cid:1) . (A.7)Using Eqs. (A.6) and (A.7), from the first term in (A.1)we obtain the first line of Eq. (12). Second term
First of all, using the definition of the purity µ =(Det σ ) − / we find that µ (cid:48) = −
12 (Det σ ) − / Tr (cid:2) σ − σ (cid:48) (cid:3) = − µ (cid:2) σ − σ (cid:48) (cid:3) (A.8)and analogously ∂ t µ = − µ (cid:2) σ − ∂ t σ (cid:3) . (A.9)The first term we need to evaluate from the second lineof Eq. (A.1) is the following ∂ t ( µ (cid:48) ) = 2 µ (cid:48) ( ∂ t µ (cid:48) ) = − µ Tr (cid:2) σ − σ (cid:48) (cid:3) ( ∂ t µ (cid:48) ) , (A.10)7here now we need to evaluate this last term ∂ t µ (cid:48) = −
12 ( ∂ t µ )Tr (cid:2) σ − σ (cid:48) (cid:3) − µ (cid:0) ∂ t Tr (cid:2) σ − σ (cid:48) (cid:3)(cid:1) = µ (cid:2) σ − ∂ t σ (cid:3) Tr (cid:2) σ − σ (cid:48) (cid:3) − µ (cid:0) − Tr (cid:2) σ − ( ∂ t σ ) σ − σ (cid:48) (cid:3) + Tr (cid:2) σ − ( ∂ t σ (cid:48) ) (cid:3)(cid:1) = µ (cid:2) σ − ( − σ + N ) (cid:3) Tr (cid:2) σ − σ (cid:48) (cid:3) − µ (cid:0) − Tr (cid:2) σ − ( − σ + N ) σ − σ (cid:48) (cid:3) + Tr (cid:2) σ − ( − σ (cid:48) ) (cid:3)(cid:1) = µ (cid:8)(cid:0) N Tr (cid:2) σ − (cid:3) − (cid:1) Tr (cid:2) σ − σ (cid:48) (cid:3) + 2 N Tr (cid:2) σ − σ (cid:48) (cid:3)(cid:9) (A.11)We thus get to ∂ t ( µ (cid:48) ) = − µ (cid:2) σ − σ (cid:48) (cid:3)(cid:26)(cid:0) N Tr (cid:2) σ − (cid:3) − (cid:1) Tr (cid:2) σ − σ (cid:48) (cid:3) + 2 N Tr (cid:2) σ − σ (cid:48) (cid:3)(cid:27) (A.12)For the next part of the second line of Eq. (A.1) weneed to evaluate the following term µ (cid:48) ∂ t µ = (cid:16) − µ (cid:2) σ − σ (cid:48) (cid:3)(cid:17) µ (cid:0) − µ Tr (cid:2) σ − ∂ t σ (cid:3)(cid:1) = − µ Tr (cid:2) σ − σ (cid:48) (cid:3) (cid:0) N Tr (cid:2) σ − (cid:3) − (cid:1) . (A.13)From Eqs. (A.12), (A.13) and (A.1) we obtain the firstterm on the third line of Eq. (12). Third term
We consider the three terms on the last line of (A.1)together and we find2( ∂ t r (cid:48) ) T σ − r (cid:48) + 2 r (cid:48) T σ − ( ∂ t r ) + 2 r (cid:48) ( ∂ t σ − ) r (cid:48) = − r (cid:48) T σ − r (cid:48) − r (cid:48) T σ − ( ∂ t σ ) σ − r (cid:48) = − r (cid:48) T σ − r (cid:48) − r (cid:48) T σ − ( − σ + N ) σ − r (cid:48) = − N r (cid:48) T σ − r (cid:48) , (A.14)which is the last term in Eq. (12). References [1] Carl W. Helstrom,
Quantum detection and estimation theory (Academic Press, New York, 1976).[2] Alexander S. Holevo,
Probabilistic and Statistical Aspects ofQuantum Theory , 2nd ed. (Edizioni della Normale, Pisa, 2011).[3] Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone, “Ad-vances in quantum metrology,” Nat. Photonics , 222–229 (2011).[4] Rafa(cid:32)l Demkowicz-Dobrza´nski, Marcin Jarzyna, and JanKo(cid:32)lody´nski, “Quantum Limits in Optical Interferometry,” in Prog. Opt. Vol. 60 , edited by Emil Wolf (Elsevier, Amsterdam,2015) Chap. 4, pp. 345–435.[5] C. L. Degen, F Reinhard, and Paola Cappellaro, “Quantumsensing,” Rev. Mod. Phys. , 035002 (2017). [6] Daniel Braun, Gerardo Adesso, Fabio Benatti, Roberto Flore-anini, Ugo Marzolino, Morgan W. Mitchell, and Stefano Piran-dola, “Quantum-enhanced measurements without entanglement,”Rev. Mod. Phys. , 035006 (2018).[7] Luca Pezz`e, Augusto Smerzi, Markus K. Oberthaler, RomanSchmied, and Philipp Treutlein, “Quantum metrology withnonclassical states of atomic ensembles,” Rev. Mod. Phys. ,035005 (2018).[8] Susana F. Huelga, Chiara Macchiavello, T. Pellizzari, Artur K.Ekert, Martin Bodo Plenio, and J. Ignacio Cirac, “Improvementof Frequency Standards with Quantum Entanglement,” Phys.Rev. Lett. , 3865–3868 (1997).[9] B. M. Escher, R. L. de Matos Filho, and Luiz Davidovich,“General framework for estimating the ultimate precision limitin noisy quantum-enhanced metrology,” Nat. Phys. , 406–411(2011).[10] Rafa(cid:32)l Demkowicz-Dobrza´nski, Jan Ko(cid:32)lody´nski, and M˘ad˘alinGut¸˘a, “The elusive Heisenberg limit in quantum-enhancedmetrology,” Nat. Commun. , 1063 (2012).[11] Qiang Zheng, Li Ge, Yao Yao, and Qi-jun Zhi, “Enhancingparameter precision of optimal quantum estimation by directquantum feedback,” Phys. Rev. A , 033805 (2015).[12] Tuvia Gefen, David A. Herrera-Mart´ı, and Alex Retzker, “Pa-rameter estimation with efficient photodetectors,” Phys. Rev. A , 032133 (2016).[13] Martin Bodo Plenio and Susana F. Huelga, “Sensing in thepresence of an observed environment,” Phys. Rev. A , 032123(2016).[14] Francesco Albarelli, Matteo A. C. Rossi, Dario Tamascelli, andMarco G. Genoni, “Restoring Heisenberg scaling in noisy quan-tum metrology by monitoring the environment,” Quantum ,110 (2018).[15] Yao Ma, Mi Pang, Libo Chen, and Wen Yang, “Improving quan-tum parameter estimation by monitoring quantum trajectories,”Phys. Rev. A , 032347 (2019).[16] Francesco Albarelli, Matteo A. C. Rossi, and Marco G.Genoni, “Quantum frequency estimation with conditional statesof continuously monitored independent dephasing channels,”arXiv:1910.12549 (2019).[17] G. Arrad, Yuval Vinkler, Dorit Aharonov, and Alex Retzker,“Increasing Sensing Resolution with Error Correction,” Phys. Rev.Lett. , 150801 (2014).[18] Wolfgang D¨ur, Michalis Skotiniotis, Florian Fr¨owis, and BarbaraKraus, “Improved Quantum Metrology Using Quantum ErrorCorrection,” Phys. Rev. Lett. , 080801 (2014).[19] E. M. Kessler, Igor Lovchinsky, A. O. Sushkov, and Mikhail D.Lukin, “Quantum Error Correction for Metrology,” Phys. Rev.Lett. , 150802 (2014).[20] Xiao-Ming Lu, Sixia Yu, and C. H. Oh, “Robust quantummetrological schemes based on protection of quantum Fisherinformation,” Nat. Commun. , 7282 (2015).[21] David Layden and Paola Cappellaro, “Spatial noise filteringthrough error correction for quantum sensing,” npj QuantumInf. , 30 (2017).[22] Sisi Zhou, Mengzhen Zhang, John Preskill, and Liang Jiang,“Achieving the Heisenberg limit in quantum metrology usingquantum error correction,” Nat. Commun. , 78 (2018).[23] David Layden, Sisi Zhou, Paola Cappellaro, and Liang Jiang,“Ancilla-Free Quantum Error Correction Codes for QuantumMetrology,” Phys. Rev. Lett. , 040502 (2019).[24] Theodoros Kapourniotis and Animesh Datta, “Fault-tolerantquantum metrology,” Phys. Rev. A , 022335 (2019).[25] Wojciech Gorecki, Sisi Zhou, Liang Jiang, and Rafa(cid:32)l Demkowicz-Dobrza´nski, “Quantum error correction in multi-parameter quan-tum metrology,” arXiv:1901.00896 (2019).[26] Sisi Zhou and Liang Jiang, “Optimal approximate quantum errorcorrection for quantum metrology,” arXiv:1910.08472 (2019).[27] Christiane P. Koch, “Controlling open quantum systems: tools,achievements, and limitations,” J. Phys. Condens. Matter ,213001 (2016).[28] Jing Liu and Haidong Yuan, “Quantum parameter estimation ith optimal control,” Phys. Rev. A , 012117 (2017).[29] Jing Liu and Haidong Yuan, “Control-enhanced multiparameterquantum estimation,” Phys. Rev. A , 042114 (2017).[30] Han Xu, Junning Li, Liqiang Liu, Yu Wang, Haidong Yuan,and Xin Wang, “Generalizable control for quantum parameterestimation through reinforcement learning,” npj Quantum Inf. , 82 (2019).[31] Lorenza Viola, Emanuel Knill, and Seth Lloyd, “DynamicalDecoupling of Open Quantum Systems,” Phys. Rev. Lett. ,2417–2421 (1999).[32] Dieter Suter and Gonzalo A. ´Alvarez, “Colloquium : Protectingquantum information against environmental noise,” Rev. Mod.Phys. , 041001 (2016).[33] Qing-shou Tan, Yixiao Huang, Le-man Kuang, and XiaoguangWang, “Dephasing-assisted parameter estimation in the presenceof dynamical decoupling,” Phys. Rev. A , 063604 (2014).[34] Pavel Sekatski, Michalis Skotiniotis, and Wolfgang D¨ur, “Dy-namical decoupling leads to improved scaling in noisy quantummetrology,” New J. Phys. , 073034 (2016).[35] Shengshi Pang and Andrew N. Jordan, “Optimal adaptive con-trol for quantum metrology with time-dependent Hamiltonians,”Nat. Commun. , 14695 (2017).[36] Tuvia Gefen, Fedor Jelezko, and Alex Retzker, “Control methodsfor improved Fisher information with quantum sensing,” Phys.Rev. A , 032310 (2017).[37] M. Naghiloo, A. N. Jordan, and K. W. Murch, “AchievingOptimal Quantum Acceleration of Frequency Estimation Us-ing Adaptive Coherent Control,” Phys. Rev. Lett. , 180801(2017).[38] Jukka Kiukas, Kazuya Yuasa, and Daniel Burgarth, “Remoteparameter estimation in a quantum spin chain enhanced by localcontrol,” Phys. Rev. A , 052132 (2017).[39] Haidong Yuan and Chi-Hang Fred Fung, “Optimal FeedbackScheme and Universal Time Scaling for Hamiltonian ParameterEstimation,” Phys. Rev. Lett. , 110401 (2015).[40] Haidong Yuan, “Sequential Feedback Scheme Outperforms theParallel Scheme for Hamiltonian Parameter Estimation,” Phys.Rev. Lett. , 160801 (2016).[41] Zhibo Hou, Rui-Jia Wang, Jun-Feng Tang, Haidong Yuan, Guo-Yong Xiang, Chuan-Feng Li, and Guang-Can Guo, “Control-Enhanced Sequential Scheme for General Quantum ParameterEstimation at the Heisenberg Limit,” Phys. Rev. Lett. ,040501 (2019).[42] Teruo Matsubara, Paolo Facchi, Vittorio Giovannetti, andKazuya Yuasa, “Optimal Gaussian metrology for generic multi-mode interferometric circuit,” New J. Phys. , 033014 (2019).[43] Victor Mukherjee, Alberto Carlini, Andrea Mari, TommasoCaneva, Simone Montangero, Tommaso Calarco, Rosario Fazio,and Vittorio Giovannetti, “Speeding up and slowing down therelaxation of a qubit by optimal control,” Phys. Rev. A ,062326 (2013).[44] Alberto Carlini, Andrea Mari, and Vittorio Giovannetti, “Time-optimal thermalization of single-mode Gaussian states,” Phys.Rev. A , 052324 (2014).[45] Uther Shackerley-Bennett, Alberto Carlini, Vittorio Giovannetti,and Alessio Serafini, “Locally optimal symplectic control ofmultimode Gaussian states,” Quantum Sci. Technol. , 044014(2017).[46] Francesco Albarelli, Uther Shackerley-Bennett, and AlessioSerafini, “Locally optimal control of continuous-variable entan-glement,” Phys. Rev. A , 062312 (2018).[47] Alessio Serafini, Quantum continuous variables : a primer oftheoretical methods (CRC Press, Boca Raton, 2017).[48] Alessandro Ferraro, Stefano Olivares, and Matteo G. A. Paris,
Gaussian states in continuous variable quantum information (Bibliopolis, Napoli, 2005).[49] Christian Weedbrook, Stefano Pirandola, Ra´ul Garc´ıa-Patr´on,Nicolas J. Cerf, Timothy Cameron Ralph, Jeffrey H. Shapiro,and Seth Lloyd, “Gaussian quantum information,” Rev. Mod.Phys. , 621–669 (2012).[50] Gerardo Adesso, Sammy Ragy, and Antony R. Lee, “Continuous Variable Quantum Information: Gaussian States and Beyond,”Open Syst. Inf. Dyn. , 1440001 (2014).[51] G. J. Milburn, Wen-Yu Chen, and K. R. Jones, “Hyperbolicphase and squeeze-parameter estimation,” Phys. Rev. A ,801–804 (1994).[52] Giulio Chiribella, Giacomo Mauro D’Ariano, and Massimil-iano Federico Sacchi, “Optimal estimation of squeezing,” Phys.Rev. A , 062103 (2006).[53] Roberto Gaiba and Matteo G. A. Paris, “Squeezed vacuum as auniversal quantum probe,” Phys. Lett. A , 934–939 (2009).[54] Marco G. Genoni, Carmen Invernizzi, and Matteo G. A. Paris,“Enhancement of parameter estimation by Kerr interaction,” Phys.Rev. A , 033842 (2009).[55] Dominik ˇSafr´anek, Antony R. Lee, and Ivette Fuentes, “Quan-tum parameter estimation using multi-mode Gaussian states,”New J. Phys. , 073016 (2015).[56] Dominik ˇSafr´anek and Ivette Fuentes, “Optimal probe states forthe estimation of Gaussian unitary channels,” Phys. Rev. A ,062313 (2016).[57] Luca Rigovacca, Alessandro Farace, Leonardo A. M. Souza,Antonella De Pasquale, Vittorio Giovannetti, and GerardoAdesso, “Versatile Gaussian probes for squeezing estimation,”Phys. Rev. A , 052331 (2017).[58] Peter C. Humphreys, Marco Barbieri, Animesh Datta, and Ian A.Walmsley, “Quantum Enhanced Multiple Phase Estimation,”Phys. Rev. Lett. , 070403 (2013).[59] Luca Pezz`e, Mario A. Ciampini, Nicol`o Spagnolo, Peter C.Humphreys, Animesh Datta, Ian A. Walmsley, Marco Barbieri,Fabio Sciarrino, and Augusto Smerzi, “Optimal Measurementsfor Simultaneous Quantum Estimation of Multiple Phases,” Phys.Rev. Lett. , 130504 (2017).[60] Ole E. Barndorff-Nielsen and Richard D. Gill, “Fisher informa-tion in quantum statistics,” J. Phys. A , 4481–4490 (2000).[61] Rafael Chaves, Leandro Aolita, and Antonio Ac´ın, “Robustmultipartite quantum correlations without complex encodings,”Phys. Rev. A , 020301 (2012).[62] Jonatan Bohr Brask, Rafael Chaves, and Jan Ko(cid:32)lody´nski, “Im-proved Quantum Magnetometry beyond the Standard QuantumLimit,” Phys. Rev. X , 031010 (2015).[63] Massimiliano Proietti, Martin Ringbauer, Francesco Graffitti,Peter Barrow, Alexander Pickston, Dmytro Kundys, Daniel Cav-alcanti, Leandro Aolita, Rafael Chaves, and Alessandro Fedrizzi,“Enhanced Multiqubit Phase Estimation in Noisy Environmentsby Local Encoding,” Phys. Rev. Lett. , 180503 (2019).[64] Alessio Serafini, S De Siena, Fabrizio Illuminati, and MatteoG. A. Paris, “Minimum decoherence cat-like states in Gaussiannoisy channels,” J. Opt. B Quantum Semiclassical Opt. , S591–S596 (2004).[65] Alessio Serafini, Matteo G. A. Paris, Fabrizio Illuminati, andS De Siena, “Quantifying decoherence in continuous variablesystems,” J. Opt. B Quantum Semiclassical Opt. , R19–R36(2005).[66] Radim Filip, “Gaussian quantum adaptation of non-Gaussianstates for a lossy channel,” Phys. Rev. A , 042308 (2013).[67] H. Le Jeannic, A. Cavaill`es, K Huang, Radim Filip, and JulienLaurat, “Slowing Quantum Decoherence by Squeezing in PhaseSpace,” Phys. Rev. Lett. , 073603 (2018).[68] Vittorio Gorini, Andrzej Kossakowski, and E. C. George Su-darshan, “Completely positive dynamical semigroups of N-levelsystems,” J. Math. Phys. , 821 (1976).[69] G¨oran Lindblad, “On the generators of quantum dynamicalsemigroups,” Commun. Math. Phys. , 119–130 (1976).[70] Marco G. Genoni, Ludovico Lami, and Alessio Serafini, “Con-ditional and unconditional Gaussian quantum dynamics,” Con-temp. Phys. , 331–349 (2016).[71] Samuel L. Braunstein, “How large a sample is needed for themaximum likelihood estimator to be approximately Gaussian?”J. Phys. A , 3813–3826 (1992).[72] E L Lehmann and George Casella, Theory of point estimation ,2nd ed., Springer texts in statistics (Springer, New York, 1998).[73] Samuel L. Braunstein and Carlton M. Caves, “Statistical distance nd the geometry of quantum states,” Phys. Rev. Lett. , 3439–3443 (1994).[74] Matteo G. A. Paris, “Quantum estimation for quantum technol-ogy,” Int. J. Quantum Inf. , 125–137 (2009).[75] D´enes Petz, “Monotone metrics on matrix spaces,” Linear Alge-bra Appl. , 81–96 (1996).[76] Masahito Hayashi, Quantum Information Theory (Springer,Berlin, Heidelberg, 2017).[77] Olivier Pinel, P. Jian, Nicolas Treps, Claude Fabre, and DanielBraun, “Quantum parameter estimation using general single-mode Gaussian states,” Phys. Rev. A , 040102 (2013).[78] Alex Monras, “Phase space formalism for quantum estimationof Gaussian states,” arXiv:1303.3682 (2013).[79] Zhang Jiang, “Quantum Fisher information for states in expo-nential form,” Phys. Rev. A , 032128 (2014).[80] Leonardo Banchi, Samuel L. Braunstein, and Stefano Pirandola,“Quantum Fidelity for Arbitrary Gaussian States,” Phys. Rev.Lett. , 260501 (2015).[81] Giacomo De Palma, Dario Trevisan, and Vittorio Giovannetti,“Gaussian States Minimize the Output Entropy of One-ModeQuantum Gaussian Channels,” Phys. Rev. Lett. , 160503(2017).[82] Giacomo De Palma, Dario Trevisan, Vittorio Giovannetti, andLuigi Ambrosio, “Gaussian optimizers for entropic inequalitiesin quantum information,” J. Math. Phys. , 081101 (2018).[83] Alex Monras and Matteo G. A. Paris, “Optimal Quantum Es-timation of Loss in Bosonic Channels,” Phys. Rev. Lett. ,160401 (2007).[84] Marco G. Genoni, O. S. Duarte, and Alessio Serafini, “Unrav-elling the noise: the discrimination of wave function collapsemodels under time-continuous measurements,” New J. Phys. ,103040 (2016).[85] Matteo A. C. Rossi, Francesco Albarelli, and Matteo G. A. Paris,“Enhanced estimation of loss in the presence of Kerr nonlinearity,”Phys. Rev. A , 053805 (2016).[86] Sam McMillen, Matteo Brunelli, Matteo Carlesso, Angelo Bassi,Hendrik Ulbricht, Matteo G. A. Paris, and Mauro Paternos-tro, “Quantum-limited estimation of continuous spontaneouslocalization,” Phys. Rev. A , 012132 (2017).[87] Rosanna Nichols, Pietro Liuzzo-Scorpo, Paul A. Knott, andGerardo Adesso, “Multiparameter Gaussian quantum metrology,”Phys. Rev. A , 012114 (2018).[88] Dominic Branford, Christos N. Gagatsos, Jai Grover, Alexan-der J. Hickey, and Animesh Datta, “Quantum enhanced esti-mation of diffusion,” Phys. Rev. A , 022129 (2019).[89] Patrick Binder and Daniel Braun, “Quantum parameter-estimation of frequency and damping of a harmonic-oscillator,”arXiv:1905.08288 (2019).[90] David Vitali and Paolo Tombesi, “Using parity kicks for deco-herence control,” Phys. Rev. A , 4178–4186 (1999).[91] Christian Arenz, Daniel Burgarth, Paolo Facchi, and RobinHillier, “Dynamical decoupling of unbounded Hamiltonians,” J.Math. Phys. , 032203 (2018)., 032203 (2018).