Dynamical phase transitions as a resource for quantum enhanced metrology
Katarzyna Macieszczak, Madalin Guta, Igor Lesanovsky, Juan P. Garrahan
DDynamical phase transitions as a resource for quantum enhanced metrology
Katarzyna Macieszczak,
1, 2
M˘ad˘alin Gut¸˘a, Igor Lesanovsky, and Juan P. Garrahan School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UK School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, UK (Dated: November 17, 2014)We consider the general problem of estimating an unknown control parameter of an open quantumsystem. We establish a direct relation between the evolution of both system and environment andthe precision with which the parameter can be estimated. We show that when the open quantumsystem undergoes a first-order dynamical phase transition the quantum Fisher information (QFI),which gives the upper bound on the achievable precision of any measurement of the system andenvironment, becomes quadratic in observation time (cf. “Heisenberg scaling”). In fact, the QFIis identical to the variance of the dynamical observable that characterises the phases that coexistat the transition, and enhanced scaling is a consequence of the divergence of the variance of thisobservable at the transition point. This identification allows to establish the finite time scaling ofthe QFI. Near the transition the QFI is quadratic in time for times shorter than the correlationtime of the dynamics. In the regime of enhanced scaling the optimal measurement whose precisionis given by the QFI involves measuring both system and output. As a particular realisation of theseideas, we describe a theoretical scheme for quantum enhanced phase estimation using the photonsbeing emitted from a quantum system near the coexistence of dynamical phases with distinct photonemission rates.
PACS numbers: 05.30.Rt,05.30.-d,64.70.P-
Introduction.
The estimation of unknown parameters isa crucial task for quantum technology applications suchas state tomography [1], system identification [2], andquantum metrology [3–5]. Enhancement in precision canbe achieved by using highly correlated/entangled quan-tum states which encode the unknown parameter, likethe Greenberger-Horne-Zeilinger (GHZ) state | GHZ (cid:105) = | (cid:105) ⊗ N + | (cid:105) ⊗ N constructed out of N qubits. With sucha state as a resource an unknown parameter g can beencoded as | GHZ g (cid:105) = | (cid:105) ⊗ N + e − iNg | (cid:105) ⊗ N . Since thephase effectively encoded in the state is N g , the estima-tion error on g scales as N − (referred to as Heisenbergscaling [6]) instead of the standard N − scaling for anon-correlated state ( | (cid:105) + e − ig | (cid:105) ) ⊗ N .The key property that makes correlated states such as | GHZ (cid:105) useful for enhanced metrology is that they can bethought of as “bimodal”, in the sense that the proba-bility of an appropriate observable is peaked in two (ormore) “phases” (the states | (cid:105) ⊗ N and | (cid:105) ⊗ N in the case of | GHZ (cid:105) ). This bimodality is reminiscent of what occursnear a (first-order) phase transition. In fact, enhancedparameter estimation can be achieved with pure statesat quantum phase transitions [7]. For large N , highlycorrelated pure states are challenging to prepare in prac-tice [8], either as the ground state of a closed many-bodysystem, or as the stationary state of some dissipative dy-namics [9]. Typically, the latter requires careful systemengineering, since generic open quantum systems have mixed rather than pure stationary states. In general onetherefore has to deal with mixed states. These howeverhave an additional complication since the best possiblemeasurement is difficult to formulate in general, exceptfor particular cases such as thermal states [10]. This means that with mixed states it is often difficult to com-pute the best possible precision of parameter estimation.In this paper we show theoretically how to exploit thedynamics of open quantum systems (for example, drivenatomic or molecular ensembles emitting photons [11], orquantum dots [12]) to generate states for quantum en-hanced metrology. Our approach connects to recent workon parameter estimation with single stationary states ofopen quantum systems [17–19]. We overcome the prob-lem of mixed states by considering the combined state ofthe system and output. This is a pure quantum state—actually a matrix product state (MPS) [13, 14]—whichencodes the state of the system as well as the record ofemissions for the whole observation time. This allows usto find the best estimation precision using the system-output as a resource.This approach has several advantages. First, it pro-vides improved precision due to the fact that the effec-tive “size” of the system and output is now N t where t is the observation time and N is the system size. Thesecond advantage arises from the fact that open systemscan feature dynamical phase transitions (DPTs) [14–16]which, in contrast to static transitions, are characterisedby singular changes in observables on the whole dynam-ical evolution and not just on the state of the system.We show that at a first-order DPT [15, 16] the quantumFisher information (QFI) of the system-and-output maybecome quadratic in t giving rise to Heisenberg scaling.We also clarify the behaviour away from the transitionpoint. Here Heisenberg scaling of the QFI is present fortimes shorter than the correlation time of the dynamics,while asymptotically linear scaling is recovered. More-over, due to the pure form of the system-output state a r X i v : . [ qu a n t - ph ] N ov FIG. 1. (a) Open quantum system with a dynamics that features two dynamical phases of different activity and depends onthe unknown parameter g . Near a first-order DPT the output (e.g. photons) shows strong intermittency where the temporallength of active/inactive periods is approximately given by the correlation time τ . (b) In the vicinity of a DPT the QFI ofthe combined system-output state scales quadratically for observation times t (cid:28) τ . In the example of photon counting thisregime features a bimodal count distribution, i.e. the two phases can be resolved. For t (cid:29) τ this is no longer the case andthe distribution becomes unimodal. Consequently, the QFI acquires a linear scaling with t . (c) Wigner distribution W ( Q, P )of the state (12) after being projected on an appropriate system state, e.g. | I (cid:105) + | A (cid:105) . The two peaks are located at radii thatcorrespond to the count rates µ I , A of the inactive/active phase. The count distribution is not sensitive to the parameter φ andhence counting is not an appropriate measurement for phase estimation. Still, this state features an enhanced QFI with respectto changes in the parameter φ due to the highly oscillatory fringe pattern [with period ∝ t ( µ A − µ I )] in between the two peakswhich is characteristic for a Schr¨odinger cat state. we can always (formally) construct the optimal mea-surement. We discuss our ideas in a specific setting forquantum enhancement in phase estimation using an in-termittent system near a dynamical first-order transitionas shown in Fig. 1(a). Elements of quantum metrology.
We first review someessential aspects of quantum parameter estimation. Sup-pose that we wish to estimate a parameter g encodedin a quantum state ρ g , by measuring an observable M . The estimation precision is given by the signal tonoise ratio [20], SNR g ( M ) = ( d (cid:104) M (cid:105) g /dg ) / ∆ g M , where (cid:104) M (cid:105) g = Tr( ρ g M ) and ∆ g M = Tr( ρ g M ) − (cid:104) M (cid:105) g arethe mean and variance of measuring M on ρ g , respec-tively. The observable with the optimal SNR is (up tolinear transformations) given by the so-called symmetriclogarithmic derivative, D g , defined by the relation [21] dρ g dg = 12 ( D g ρ g + ρ g D g ) . (1)Except for very particular forms of ρ g , the optimal mea-surement D g is difficult to engineer. Nevertheless, theSNR for this observable is given by the quantum Fisherinformation [21], F ( ρ g ), which bounds the precision ofany measurement that can be performed in practice.This bound is in fact given by the variance of D g , i.e. F ( ρ g ) = ∆ g D g .In general, the QFI is hard to compute, but for a purestate, | ψ g (cid:105) , it can be obtained from the fidelity (cid:104) ψ g | ψ g (cid:105) [7, 17] according to, F ( | ψ g (cid:105) ) = 4 ∂ g g log (cid:104) ψ g | ψ g (cid:105) (cid:12)(cid:12) g = g = g . (2)A situation which is relevant for what follows is when theparameter is encoded as a phase in a unitary transforma-tion on a pure state, | ψ g (cid:105) = e − ig G | ψ (cid:105) . Here the fidelity (cid:104) ψ g | ψ g (cid:105) is the characteristic function of G at g − g , andthe QFI is given by its variance, F ( | ψ g (cid:105) ) = 4∆ g G . Notethat while the QFI is given by the variance of both D g and G , these two operators play very different roles. Theoptimal measurement to recover the parameter g is D g ,and its SNR is maximal, SNR g ( D g ) = F ( | ψ g (cid:105) ). In con-trast, G encodes g in the quantum state, but measuringit provides no information about g since SNR g ( G ) = 0.For example, for the state | GHZ g (cid:105) the generator is G = (cid:80) j (1 + σ ( j ) z ) / D g = (cid:80) j e − igG σ ( j ) y e igG , where σ ( j ) a are Pauli operatorsacting on qubit j . The QFI for the GHZ state then obeysHeisenberg scaling, F ( | GHZ g (cid:105) ) = N . This is related tothe fact that the distributions of both G and D g are bi-modal. In contrast, the QFI of the uncorrelated state isstandard, F (( | (cid:105) + e − ig | (cid:105) ) ⊗ N ) = N , given by the factthat the corresponding distributions are unimodal. Be-low we show that an analogous change from bimodal tounimodal also accompanies a change in the scaling withtime of the QFI when approaching a first-order DPT. Open dynamics, MPS and DPTs.
Our goal is to exploreopen quantum systems as resources for parameter esti-mation. We consider systems whose reduced dynamics,after tracing out the environment, is given by a Marko-vian master equation [22] dρdt = L ρ = − i [ H, ρ ] + k (cid:88) j =1 (cid:18) L j ρL † j − { L † j L j , ρ } (cid:19) (3)where H is the system’s Hamiltonian, and L j are thejump operators ( j = 1 , . . . , k ). In the input-outputformalism [23], the joint system and output state isgiven by a continuous MPS (CMPS) [13, 14]. For clar-ity, we discretise time by δt , and the CMPS is ap-proximated by a regular MPS [13, 14, 24], | Ψ( t ) (cid:105) = (cid:80) kj n ,...,j =0 K j n · · · K j | χ (cid:105) ⊗ | j , ..., j n (cid:105) , where n = t/δt , K = e − iδtH (cid:113) − δt (cid:80) j L † j L j , K j> = e − iδtH √ δtL j ,and | χ (cid:105) is the initial state of the system. The outputstate | j , ..., j n (cid:105) describes the time record of emissionsinto the environment, as sketched in Fig. 1(a).The state | Ψ( t ) (cid:105) can have a singular change when vary-ing a parameter in (3). This could either correspond toa static phase transition in the stationary state of thesystem, or to a dynamical phase transition in the systemand output. Both kinds of transitions are captured bydiscontinuities in the average, or a higher cumulant, ofan observable that acts on the whole of | Ψ( t ) (cid:105) . Relation between DPTs and QFI.
We now assume thatthe dynamics depends on the parameter g to be esti-mated, see Fig. 1(a). This means that the Hamiltonian, H g and jump operators, L j,g , and consequently the mas-ter operator, L g , Eq. (3), may depend on g . It followsthen that the MPS, | Ψ g ( t ) (cid:105) , also depends on g , and sodoes the fidelity [24], (cid:104) Ψ g ( t ) | Ψ g ( t ) (cid:105) = Tr { e t L g ,g | χ (cid:105)(cid:104) χ |} , (4)where L g ,g is a deformation of the Master operator [24], L g ,g ρ = − iH g ρ + iρH g (5)+ k (cid:88) j =1 (cid:20) L j,g ρL † j,g −
12 ( L † j,g L j,g ρ + ρL † j,g L j,g ) (cid:21) . Thus, in the long time limit the QFI of | Ψ g ( t ) (cid:105) is re-lated to the largest eigenvalue λ ( g , g ) of L g ,g ,lim t →∞ t − F ( | Ψ g ( t ) (cid:105) ) = 4 ∂ g g λ ( g , g ) (cid:12)(cid:12) g = g = g . (6)One can already see that something interesting will occuras the system approaches a DPT, so that the gap betweenthe two leading eigenvalues of L g closes at some g .When the gap is small, for example close to a DPT,there is a time regime where the QFI is quadratic in time, F ( | Ψ g ( t ) (cid:105) ) = 4 t ∂ g g Re Tr {L g ,g PL g,g P| χ (cid:105)(cid:104) χ |} g = g = g − (cid:12)(cid:12)(cid:12) t∂ g Tr {L g ,g P| χ (cid:105)(cid:104) χ |} g = g (cid:12)(cid:12)(cid:12) + t O ( tλ ) + O ( t ) , (7)where P is a projection onto the first two eigenvectors of L g corresponding to the two eigenvalues with the largestreal part, ( λ = 0 , λ ). The gap is given by − Re λ . Thisapproximation of Eq. (7) is valid for τ (cid:48) (cid:28) t (cid:28) τ , where τ is the correlation time given by the gap, τ ≡ ( − Re λ ) − ,while τ (cid:48) is the longest timescale associated with the restof the spectrum, τ (cid:48) ≡ ( − Re λ ) − . The quadratic timedependence of the QFI (7) is a consequence of time-correlations in the system-output MPS. Furthermore, ata DTP λ → Heisenberg scaling is given exactly by the Eq. (7) forall t (cid:29) τ (cid:48) . Enhanced phase estimation and intermittency.
We nowuse the ideas above for the case of a system with in-termittent dynamics [12, 16, 25] used as a resource forparameter estimation, see Fig. 1(a). The parameter hereis a phase g = φ encoded in the jump operator L , bydefining L ,φ = e − iφ L . For concreteness, note that thequantum jump associated with L is the emission of aphoton. This means that a phase φ is imprinted on eachoutgoing photon. As we now show, if the system dis-plays intermittent photon emission associated to a (first-order) DPT in counting statistics [14–16], then it will bean efficient resource for quantum metrology. With theabove choice, the Master operator is independent of φ , L φ = L . In turn, the deformed generator L φ,φ (cid:48) , Eq. (5),from which the QFI is obtained, reads (∆ φ = φ − φ (cid:48) ) L φ,φ (cid:48) ρ = L ρ + (cid:0) e − i ∆ φ − (cid:1) L ρL † . (8)With these definitions there is a direct connection toa photon counting problem [15, 23]. The phase φ isencoded in a unitary transformation of the MPS withgenerator G = Λ( t ), where Λ( t ) is the operator thatcounts the number of photons emitted up to time t , sothat | Ψ φ ( t ) (cid:105) = e − iφ Λ( t ) | Ψ( t ) (cid:105) . The fidelity (cid:104) Ψ φ ( t ) | Ψ φ (cid:48) ( t ) (cid:105) is the characteristic function of Λ( t ), the logarithm ofwhich encodes all its cumulants. The cumulants arealso encoded in the cumulant generating function (CGF),Θ t ( s ) = log (cid:80) Λ e − s Λ P (Λ , t ), where P (Λ , t ) is the prob-ability of observing Λ photons in time t . The CGFcan be related to a deformation of the Master opera-tor, Θ t ( s ) = Tr { e t L ( s ) | χ (cid:105)(cid:104) χ |} , where L ( s ) is the same as(8) with ∆ φ = − is . The long time limit of the CGF, θ ( s ) = lim t →∞ t − Θ( s, t ), plays the role of a dynami-cal free-energy for the ensemble of trajectories of photonemissions [15]. A singularity of θ ( s ) at some s c is an indi-cation of a phase transition in the ensemble of quantumjump trajectories, and when s c = 0 we have what weterm a DPT, i.e., a singular change in the actual dynam-ics of the open system associated with a vanishing of thespectral gap λ [14, 15].The asymptotic QFI (6) becomes,lim t →∞ t − F ( | Ψ g ( t ) (cid:105) ) = 4 ∂ s θ ( s ) (cid:12)(cid:12) s =0 . (9)When the function θ ( s ) has a first-order singularity atsome | s c | (cid:38)
0, i.e. we are near a DPT, Eq. (9) will belarge at s = 0. In this a case the system will display anintermittent dynamics that switches between long peri-ods with very distinct emission characteristics. Such asituation can be understood in terms of the coexistenceof dynamical phases with significantly different photoncount rates [15], see Fig. 1(a). The QFI of | Ψ φ ( t ) (cid:105) isproportional to the variance of the photon counting gen-erator G = Λ( t ). For times shorter than the correlationtime τ the system is mostly in one of the two phases,the distribution of the photon count is approximately bi-modal, and the dynamics displays large fluctuations inthe total photon emission, see Fig. 1(b). This impliesa quadratic increase of the QFI with time, with Eq. (7)reducing to, F ( | Ψ g ( t ) (cid:105) ) ≈ t p A p I ( µ A − µ I ) + O ( t ) . (10)Here µ A and µ I are the average counting rates, (cid:104) Λ( t ) (cid:105) /t ,in the two phases (which we term “active” and “inac-tive” as we assume µ A > µ I ), while p A and p I are theprobability of the initial state | χ (cid:105) being in either phase.The above approximation holds for t < τ , and becomesvalid for all times at a DPT. For times longer than τ ,dynamics switches between the two phases, giving rise tointermittent behaviour, and eventual normal (unimodal)distribution of the photon count around the overall aver-age (9); see Fig. 1(b) and derivations in [24].The above shows that an intermittent system near aDPT can be used as a photon source for quantum en-hanced phase estimation. The situation is then similarto that of GHZ states: the total photon count distribu-tion is bimodal for times up to the correlation time τ and imprints an effective macroscopic phase difference of t ( µ A − µ I ) φ between the active and inactive dynamicalphases; see discussion after Eq. (12). Enhanced metrology and DPT in general.
We now extendthe above discussion to the case where the dynamics hasan arbitrary dependence on the parameter g to be es-timated. In this case, g is encoded in the action of a“generator” G g ( t ), G g ( t ) | Ψ g ( t ) (cid:105) = − i∂ g | Ψ g ( t ) (cid:105) , (11)where G g ( t ) is the time-integral of a local-in-time ob-servable, just like Λ( t ) in the photon counting case.In terms of G g ( t ) the fidelity reads, (cid:104) Ψ g ( t ) | Ψ g ( t ) (cid:105) = (cid:104) Ψ g ( t ) |T e − i (cid:82) g g dg (cid:48) G g (cid:48) ( t ) | Ψ g ( t ) (cid:105) , where T is the g -ordering (cf. time-ordering) operator, see also [26, 27].The QFI is then the variance of G g ( t ). It follows that ifwe have a system which displays a first-order DPT wherethe dynamical phases are characterised by G g ( t ), then,in the τ (cid:48) (cid:28) t (cid:28) τ time regime, the QFI follows Eq. (10),where µ A , I are the averages of G g ( t ) per unit time in thetwo coexisting dynamical phases [28]. Again this empha-sises the connection between dynamical bimodality andenhanced quantum sensitivity.The t behaviour of the QFI is an intrinsically quan-tum feature. This behaviour cannot occur in systems forwhich the associated MPS is real and therefore cannotaccumulate any quantum phase. Note that this includesall classical systems. In such a case the average of G g ( t )is zero, cf. Eq. (11), and only terms linear in t will survivein Eq. (10). Measurement schemes.
We have shown that near a DPTthe system-output state can have a large QFI. But to ex-ploit this, and achieve quantum enhanced sensitivity, itis necessary to measure an appropriately chosen observ-able. The optimal observable is known to be the sym-metric logarithmic derivative D g defined by (1), which for pure states can be written explicitly as D g = 2 ∂ g | ψ g (cid:105)(cid:104) ψ g | .However, the measurement of D g will be difficult to en-gineer in most practical situations. One needs thereforeto find an alternative which is both practical and whoseSNR is as close as possible to the QFI.Despite the fact that the intricacy of the optimal mea-surement makes it impractical, we can still formulategeneral characteristics for a measurement that achievesenhanced precision. The first consideration is whetherthe measurement should be on the system or output, orboth. In fact, in the regime of enhanced scaling the op-timal measurement whose precision is given by the QFIinvolves measuring both system and output. The reasonis that the precision achievable by measuring only theoutput is bounded by p A F ( | Ψ A ( t ) (cid:105) )+ p I F ( | ψ I ( t ) (cid:105) ), whichscales linearly in time. Here | Ψ A , I ( t ) (cid:105) are the MPS statesassociated to the individual active/inactive stationarystates, and p A , I are their probabilities, see Eq. (10). Thislast result is the precision of an idealised protocol givenby a first measurement of the system to project onto oneof the subspaces associated with the competing station-ary states, followed by an optimal measurement of theconditioned system-output state | Ψ A , I ( t ) (cid:105) . The secondconsideration is what should be the time extension t of asingle measurement run. Here we imagine that the totaltime available to the experiment is T and one performs n = T /t independent repetitions of an efficient system-output measurement of the state | Ψ g ( t ) (cid:105) . This corre-sponds to a measurement of the joint state | Ψ g ( t ) (cid:105) ⊗ n ,and the optimal time t is that which maximises the QFIof the joint state, F ( | Ψ g ( t ) (cid:105) ⊗ n ) = n F ( | Ψ g ( t ) (cid:105) ). Equa-tion (7) tells us that this optimal time is of the order ofthe correlation time, t = O ( τ ).For the case of phase estimation at a DPT, the bi-modality of the system-output state means that it is es-sentially of the form of a “Schr¨odinger cat” state. Assum-ing for simplicity that the competing stationary states arepure, it reads, | Ψ φ ( t ) (cid:105) = √ p I | I (cid:105) ⊗ | α I ( φ ) (cid:105) + √ p A | A (cid:105) ⊗ | α A ( φ ) (cid:105) (12)where | α A ( φ ) (cid:105) are coherent states with amplitudes α I , A ( φ ) = e iφ √ t µ I , A , where µ I , A are the photon emis-sion rates of the phases, see Eq. (10) and Fig. 1(c). Infact, as shown in Ref. [29], the state (12) is approximatelya GHZ state with relative phase t ( µ A − µ I ) φ . Note thatfor (12) neither counting nor homodyne measurementsachieve Heisenberg scaling, which highlights the generalchallenge of identifying optimal measurements. However,one might think of instead employing interferometric pro-tocols, related to the ones put forward in Refs. [29–31]for superpositions of coherent states, in order to exploitthe enhanced precision scaling. Conclusions.
We have shown that, close to a dynamicalphase transition, the output of an open quantum systemcan be seen as a resource for quantum metrology applica-tions. For times of the order of the correlation time, thesystem-output QFI scales quadratically with time, whilein the long time limit the QFI scales linearly in time withrate which diverges when the spectral gap closes, as in aDTP. It remains an open issue how to exploit in a generaland systematic way the large QFI of the system-outputclose to a DPT.This work was supported by The LeverhulmeTrust (Grant No. F/00114/BG), EPSRC (Grant No.EP/J009776/1) and the European Research Councilunder the European Union’s Seventh Framework Pro-gramme (FP/2007-2013) through ERC Grant AgreementNo. 335266 (ESCQUMA) and the EU-FET Grant No.512862 (HAIRS). I.L. acknowledges discussions with K.Mølmer and E.M. Kessler. [1] H. H¨affner, W. H¨ansel, C. F. Roos, J. Benhelm, D. Chek-al-kar, M. Chwalla, T. K¨orber, U. D. Rapol, M. Riebe, P.O. Schmidt, C. Becher, O. G¨uhne, W. D¨ur and R. Blatt,Nature , 643-646 (2005).[2] D. Burgarth and K. Yuasa, Phys. Rev. Lett. , 080502(2012).[3] For a review see, V. Giovannetti, S. Lloyd and L. Mac-cone, Nature Photon. , 222 (2011).[4] LIGO Scientific Collaboration, Nature Phys. , 962(2011).[5] D.J. Wineland, J.J. Bollinger, W.M. Itano, F.L. Mooreand D.J. Heinzen, Phys. Rev. A, , R6797 (1992);D. Leibfried, M.D. Barrett, T. Schaetz, J. Britton, J.Chiaverini, W.M. Itano, J.D. Jost, C. Langer and D.J.Wineland, Science , 1476 (2004); C.F. Roos, M.Chwalla, K. Kim, M. Riebe and R. Blatt, Nature ,316 (2006).[6] C.M. Caves, Phys. Rev. D , 1693 (1981).[7] P. Zanardi, P. Giorda, M. Cozzini, Phys. Rev. Lett. ,100603 (2007); P. Zanardi, M.G.A. Paris, L. CamposVenuti, Phys. Rev. A , 042105 (2008).[8] R. Blatt, G.J. Milburn and A. Lvovksy, J. Phys. B , 878 (2008); B. Kraus,H.P. B¨uchler, S. Diehl, A. Kantian, A. Micheli, and P.Zoller, Phys. Rev. A 78, 042307 (2008).[10] Z. Jiang, Phys. Rev, A , 032128 (2014).[11] J.C. Bergquist, R.G. Hulet, W.M. Itano and D.J.Wineland, Phys. Rev. Lett. , 1699 (1986).[12] E. Barkai, Y.J. Jung, and R. Silbey, Annu. Rev. Phys.Chem. , 457 (2004).[13] M.M. Wolf, G. Ortiz, F. Verstraete, J.I. Cirac, Phys. Rev.Lett. , 110403 (2006); F. Verstraete and J. I. Cirac,Phys. Rev. Lett. , 190405, (2010).[14] I. Lesanovsky, M. van Horssen, M. Guta, and J.P. Gar-rahan, Phys. Rev. Lett. , 150401 (2013).[15] J.P. Garrahan, I. Lesanovsky, Phys. Rev. Lett. ,160601 (2010).[16] C. Ates, B. Olmos, J.P. Garrahan, and I. Lesanovsky,Phys. Rev. A , 043620 (2012).[17] M. Guta, Phys, Rev. A, , 062324 (2011).[18] S. Gammelmark, K. Mølmer, Phys. Rev. Lett. ,170401 (2014).[19] C. Catana, L. Bouten, M. Guta, arXiv:1407.5131.[20] B. M. Escher, arXiv:1212.2533.[21] C. W. Helstrom, Phys. Lett. , 101 (1967); IEEETrans. Inf. Theory , 234 (1968); S. L. Braunstein andC.M. Caves, Phys. Rev. Lett. , 3439 (1994).[22] V. Gorini, A. Kossakowski, and E.C.G. Sudarshan, J.Math. Phys. , 821 (1976); G. Lindblad, Commun.Math. Phys. , 119 (1976).[23] C.W. Gardiner and P. Zoller, Quantum Noise (Springer-Verlag, Berlin, 2004) (3rd edition).[24] See Supplemental Material.[25] M.B. Plenio and P. L. Knight, Rev. Mod. Phys. , 101(1998).[26] A. De Pasquale, D. Rossini, P. Facchi, V. Giovannetti,Phys. Rev. A , 052117 (2013).[27] M. Guta, J. Kiukas, arXiv:1402.3535.[28] K. Macieszczak, M. Guta, I. Lesanovsky, J.P. Garrahan,[in preparation].[29] T.C. Ralph, Phys. Rev. A , 042313 (2002).[30] J. Joo, W.J. Munro and T.P. Spiller, Phys. Rev. Lett. , 083601 (2011).[31] C.C. Gerry, J. Mimih, Phys. Rev. A, , 013831 (2010).[32] B. Gaveau, L. S. Schulman, J. Math. Phys. , 3 (1998). SUPPLEMENTARY MATERIALI. FIDELITY AND QFI OF MPS STATES
In this section we prove Eqs. (2) and (4) from the paper.We have: ∂ g g log (cid:104) ψ g | ψ g (cid:105)| g = g = g = (cid:104) ψ (cid:48) g | ψ (cid:48) g (cid:105)(cid:104) ψ g | ψ g (cid:105) − (cid:104) ψ (cid:48) g | ψ g (cid:105)(cid:104) ψ g | ψ (cid:48) g (cid:105)(cid:104) ψ g | ψ g (cid:105) = (cid:104) ψ (cid:48) g | ψ (cid:48) g (cid:105) − (cid:12)(cid:12) (cid:104) ψ g | ψ (cid:48) g (cid:105) (cid:12)(cid:12) (13)where | ψ (cid:48) g (cid:105) = ∂∂g | g = g | ψ g (cid:105) and we use the normalisation of the state (cid:104) ψ g | ψ g (cid:105) = 1.On the other hand, for a family of pure states ρ g = | ψ g (cid:105)(cid:104) ψ g | the symmetric logarithmic derivative is D g =2( | ψ g (cid:105)(cid:104) ψ (cid:48) g | + | ψ (cid:48) g (cid:105)(cid:104) ψ g | ). Therefore F ( | ψ g (cid:105) ) = Tr( ρ g D g ) = 4( (cid:104) ψ (cid:48) g | ψ (cid:48) g (cid:105) + (cid:104) ψ (cid:48) g | ψ g (cid:105)(cid:104) ψ g | ψ (cid:48) g (cid:105) + (cid:104) ψ (cid:48) g | ψ g (cid:105) + (cid:104) ψ g | ψ (cid:48) g (cid:105) )= 4( (cid:104) ψ (cid:48) g | ψ (cid:48) g (cid:105) − (cid:104) ψ (cid:48) g | ψ g (cid:105)(cid:104) ψ g | ψ (cid:48) g (cid:105) + (cid:0) (cid:104) ψ (cid:48) g | ψ g (cid:105) + (cid:104) ψ g | ψ (cid:48) g (cid:105) (cid:1) )= 4( (cid:104) ψ (cid:48) g | ψ (cid:48) g (cid:105) − |(cid:104) ψ (cid:48) g | ψ g (cid:105)| ) = 4 ∂ g g log |(cid:104) ψ g | ψ g (cid:105)| g = g = g . (14)In order to prove Eq. (4), let us consider the discretisation of the master dynamics described below Eq. (3) of thepaper. We have: (cid:104) Ψ g ( t ) | Ψ g ( t ) (cid:105) = Tr {| Ψ g ( t ) (cid:105)(cid:104) Ψ g ( t ) |} = Tr k (cid:88) j n ,...,j =0 K j n ,g · · · K j ,g | χ (cid:105)(cid:104) χ | K † j n ,g · · · K † j ,g , (15)where n = t/δt , K ,g = e − iδtH g (cid:113) − δt (cid:80) kj =1 L † j,g L j,g , K j> ,g = e − iδtH g √ δtL j,g , and | χ (cid:105) is the initial state of thesystem. In the limit δt →
0, analogously as tracing out the output in | Ψ g ( t ) (cid:105) gives the state of the system ρ g ( t ): ρ g ( n ) = k (cid:88) j n ,...,j =0 K j n ,g · · · K j ,g | χ (cid:105)(cid:104) χ | K † j n ,g · · · K † j ,g −→ δt → ρ g ( t ) = e t L g | χ (cid:105)(cid:104) χ | , the fidelity becomes (cid:104) Ψ g ( t ) | Ψ g ( t ) (cid:105) = Tr { e t L g ,g | χ (cid:105)(cid:104) χ |} , where L g ,g is a modified Master operator defined in Eq.(5) in the paper. The same result can also be derived by using the continuous MPS state which describes the stateof the system and the output in continuous time: | Ψ( t ) (cid:105) = ∞ (cid:88) m =0 k (cid:88) j ,...,j m =1 (cid:90) t d t (cid:90) tt d t · · · (cid:90) tt m − d t m × (cid:16) e − i ( t − t m ) H eff L j m e − i ( t m − t m − ) H eff · · · L j e − i ( t − t ) H eff L j e − it H eff | χ (cid:105) (cid:17) ⊗ | ( j , t ) , ( j , t ) , ..., ( j m , t m ) (cid:105) (16)where H eff = H − i (cid:80) kj =1 L † j L j is the effective Hamiltonian. II. TIME DEPENDENCE OF QFI
In this section we first discuss the general dependence of the QFI of the MPS state | Ψ( t ) (cid:105) on time t . This will enableus to prove the asymptotic linear behaviour of the QFI in the case of dynamics with a unique stationary state, see Eq.(6) in the paper. Using the general time dependence of the QFI, we then prove the existence of a quadratic scalingregime of the QFI (cf. Eq. (7) in the paper) for dynamics near a DPT. Finally, for a system displaying a first-orderDPT in photon emissions, we argue how the quadratic scaling of the QFI for phase estimation with emitted photons,can be related to difference in photon emission rates between dynamical phases, cf. Eq. (10) in the paper. A. General time dependence of the QFI
In order to express the QFI of the MPS state | Ψ( t ) (cid:105) , we use Eqs. (2) and (4) in the paper, and obtain: F ( | Ψ g ( t ) (cid:105) ) = 4 ∂ g g log Tr { e t L g ,g | χ (cid:105)(cid:104) χ |} g = g = g = − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Tr (cid:26)(cid:90) t d t (cid:48) ∂ g L g ,g ρ g ( t (cid:48) ) (cid:27) g = g (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) +4 Tr (cid:26)(cid:90) t d t (cid:48) ∂ g g L g ,g ρ g ( t (cid:48) ) (cid:27) g = g = g +8 Re Tr (cid:40)(cid:90) t d t (cid:48) (cid:90) t − t (cid:48) d t (cid:48)(cid:48) ∂ g L g ,g e t (cid:48)(cid:48) L g ∂ g L g,g ρ g ( t (cid:48) ) (cid:41) g = g = g (17)where ρ g ( t ) := e t L g | χ (cid:105)(cid:104) χ | , L g is the Master operator, see Eq. (3) in the paper, and L g ,g is the modified Masteroperator, see Eq. (5) in the paper.For clarity of further presentation we assume that L g can be diagonalised and has one stationary state ρ ss , i.e L g = 0 | ρ ss (cid:105)(cid:104) | + (cid:80) d k =2 λ k | ρ k (cid:105)(cid:104) l k | , where d is the dimension of the system Hilbert space H and | ρ k (cid:105) , (cid:104) l k | stand for k -thright and left eigenvectors of L g , ordered so that 0 > Re λ ≥ Re λ ≥ ... ≥ Re λ d and normalised as (cid:104) l j | ρ k (cid:105) = δ jk , j, k = 1 , ..., d . The stationary state ρ ss and other eigenvectors {| ρ k (cid:105) , (cid:104) l k |} d k ≥ with corresponding eigenvalues { λ k } d k ≥ depend on g . In general one should consider a Jordan decomposition of L g , but the following discussion would besimilar for that case.Due to the fact that Eq. (17) involves integrals of e t L g , we need to consider the 0-eigenspace of L g , i.e. thestationary state ρ ss , separately from all the rest of eigenvectors, whose eigenvalues are different from 0. We introducethe projection on the complement of the stationary state P := (cid:80) d k =2 | ρ k (cid:105)(cid:104) l k | and denote the restriction of an operator X to the complement of ρ ss by [ X ] P := P X P .We now express the finite time behaviour of QFI using derivatives of the modified Master operator and the diagonaldecomposition of the original Master operator L g . From Eq. (17) it follows: F ( | Ψ g ( t ) (cid:105) ) = − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t Tr { ∂ g L g ,g ρ ss } + Tr (cid:40) ∂ g L g ,g (cid:20) e t L g − IL g (cid:21) P | χ (cid:105)(cid:104) χ | (cid:41)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g = g +4 (cid:32) t Tr (cid:8) ∂ g g L g ,g ρ ss (cid:9) + Tr (cid:40) ∂ g g L g ,g (cid:20) e t L g − IL g (cid:21) P | χ (cid:105)(cid:104) χ | (cid:41)(cid:33) g = g = g +4 t | Tr { ∂ g L g ,g ρ ss }| + 8 Re Tr { ∂ g L g ,g ρ ss } Tr (cid:40) ∂ g L g,g (cid:20) e t L g − I − t L g L g (cid:21) P | χ (cid:105)(cid:104) χ | (cid:41) g = g = g +8 Re Tr (cid:40) ∂ g L g ,g (cid:20) e t L g − I − t L g L g (cid:21) P ∂ g L g,g ρ ss (cid:41) g = g = g − (cid:40) ∂ g L g ,g (cid:2) L − g (cid:3) P ∂ g L g,g (cid:20) e t L g − IL g (cid:21) P | χ (cid:105)(cid:104) χ | (cid:41) g = g = g +8 Re Tr (cid:40) ∂ g L g ,g (cid:20) e t L g L g (cid:18)(cid:90) t d t (cid:48) e − t (cid:48) L g ∂ g L g,g e t (cid:48) L g (cid:19)(cid:21) P | χ (cid:105)(cid:104) χ | (cid:41) g = g = g , (18)and one can show that (cid:20) e t L g L g (cid:18)(cid:90) t d t (cid:48) e − t (cid:48) L g ∂ g L g,g e t (cid:48) L g (cid:19)(cid:21) P = t d (cid:88) k =2 e tλ k λ k (cid:104) l k | ∂ g L g,g | ρ k (cid:105) | ρ k (cid:105)(cid:104) l k | + d (cid:88) j (cid:54) = k,j,k> e tλ j − e tλ k λ k ( λ j − λ k ) (cid:104) l k | ∂ g L g,g | ρ j (cid:105) | ρ k (cid:105)(cid:104) l j | . The first line and the second line in Eq. (18) correspond to the first and the second line in Eq. (17), respec-tively. All other terms in Eq. (18) correspond to the third line in Eq. (17). We see that the quadratic contribution t | Tr { ∂ g L g ,g ρ ss }| cancels out and for one stationary state there is no explicit quadratic behaviour.Eq. (18) will be used for investigating the asymptotic and the quadratic time regime of QFI.We note that as an alternative route, one can use the eigendecomposition of the modified Master operator L g ,g defined in Eq. (5) in the paper: e t L g ,g = d (cid:88) k =1 e tλ k ( g ,g ) | ρ k ( g , g ) (cid:105)(cid:104) l k ( g , g ) | , which givesTr { e t L g ,g | χ (cid:105)(cid:104) χ |} = d (cid:88) k =1 p k ( g , g ) e tλ k ( g ,g ) , where p k ( g , g ) = (cid:104) l k ( g , g ) | | χ (cid:105)(cid:104) χ |(cid:105) × Tr { ρ k ( g , g ) } and in the case of a single stationary state we have p ( g, g ) = 1, p k ( g, g ) = 0, k = 2 , ...d , which follows from the normalisation of the eigenbasis of L g to Tr { ρ ( g, g ) } = Tr ρ ss = 1,i.e. Tr { ρ k ( g, g ) } = 0 for k = 2 , ..., d .From Eqs. (2) and (4) in the paper, for a single stationary state, we obtain: F ( | Ψ g ( t ) (cid:105) ) = − t | ∂ g λ ( g , g ) | + 2 t Re ∂ g λ ( g , g ) d (cid:88) k =1 e tλ k ∂ g p k ( g, g ) + d (cid:88) j,k =1 e t ( λ k + λ j ) ∂ g p j ( g , g ) ∂ g p k ( g, g ) g = g = g +4 t | ∂ g λ ( g , g ) | + t ∂ g g λ ( g , g ) + d (cid:88) k =1 e tλ k ∂ g g p k ( g , g ) + 2 t Re d (cid:88) k =1 e tλ k ∂ g p k ( g , g ) ∂ g λ k ( g, g ) g = g = g (19)where λ k = λ k ( g, g ) and p k = p k ( g, g ), k = 1 , ..., d . The first line correponds to the first line of Eq. (18) and thesecond to the rest of terms in Eq. (18). We see again that quadratic terms t | ∂ g λ ( g , g ) | cancel out and there isno explicit quadratic behaviour. B. Asymptotic QFI for the case of a unique stationary state
Here we assume that the dynamics has a unique stationary state, i.e. the second eigenvalue of the Master operator L g is different from 0, λ (cid:54) = 0. In order to find the asymptotic behaviour of the QFI of the state | Ψ g ( t ) (cid:105) , we considerthe limit t → ∞ when we have lim t →∞ (cid:2) e t L g (cid:3) P = 0 and from Eq. (18) we obtain:lim t →∞ t − F ( | Ψ g ( t ) (cid:105) ) = 4 Tr (cid:8) ∂ g g L g ,g ρ ss (cid:9) − (cid:110) ∂ g L g ,g (cid:2) L − g (cid:3) P ∂ g L g,g ρ ss (cid:111) g = g = g , (20)Since the limit is finite, this shows that the QFI has an asymptotic linear behaviour in the case of a single stationarystate. This result was also obtained using different methods in [19]. We see that Eq. (20) can diverge at a first-orderDPT when λ → g → g c , as (cid:2) L − g (cid:3) has then a diverging eigenvalue λ − .The asymptotic linear behaviour of the QFI can be also obtained from Eq. (19):lim t →∞ t − F ( | Ψ g ( t ) (cid:105) ) = ∂ g g λ ( g , g ) | g = g = g , , (21)which corresponds to Eq. (6) in the paper. By comparing it to Eq. (20), we see that closing of the gap λ → g → g c , can cause non-analyticity of the eigenvalues and eigenvectors of L g ,g at g = g = g c . C. Quadratic time-regime of QFI
In this subsection we describe the quadratic regime in the QFI scaling with time, which can be present for systemsat and near a first-order DPT, and is given by Eq. (7) in the paper.
Quadratic behaviour near a DPT . We consider a system near a DPT when the gap is much smaller than thegap associated with the rest of the spectrum, i.e. ( − Re λ ) (cid:28) ( − Re λ ). This allows to introduce a time regime( − Re λ ) − = τ (cid:48) (cid:28) t (cid:28) τ = ( − Re λ ) − . For simplicity let us assume | λ | ≈
0, which gives e λ t ≈
1. Weintroduce the projection P := | ρ ss (cid:105)(cid:104) | + | ρ (cid:105)(cid:104) l | on the subspace spanned by the stationary state ρ ss and the secondeigenvector ρ . We also introduce the projection on their complement P := I − P = (cid:80) d k =3 | ρ k (cid:105)(cid:104) l k | and denote by[ X ] P = ( I − P ) X ( I − P ) the restriction of an operator X to this complement.We would like to simplify Eq. (18) in the regime τ (cid:48) (cid:28) t (cid:28) τ . In this regime we expect the second eigenvector ρ of L g to be almost stationary and determine, both with the stationary state ρ ss , dominant terms in the behaviour of theQFI. We assume that other eigenvectors do not play significant role, by which we understand that (cid:13)(cid:13)(cid:13)(cid:2) e t L g (cid:3) P (cid:13)(cid:13)(cid:13) ≈ (cid:107) τ (cid:107) := Tr {√ τ † τ } is the trace-norm of operators on H , which for density matrices is always 1.The general behaviour of the QFI given by Eq. (18) simplifies to: F ( | Ψ g ( t ) (cid:105) ) = − (cid:12)(cid:12)(cid:12) t Tr { ∂ g L g ,g P | χ (cid:105)(cid:104) χ |} − Tr (cid:110) ∂ g L g ,g (cid:2) L − g (cid:3) P | χ (cid:105)(cid:104) χ | (cid:111)(cid:12)(cid:12)(cid:12) g = g +4 (cid:16) t Tr (cid:8) ∂ g g L g ,g P | χ (cid:105)(cid:104) χ | (cid:9) − Tr (cid:110) ∂ g g L g ,g (cid:2) L − g (cid:3) P | χ (cid:105)(cid:104) χ | (cid:111)(cid:17) g = g = g +4 t Re Tr { ∂ g L g ,g P ∂ g L g,g P | χ (cid:105)(cid:104) χ |} g = g = g − (cid:40) ∂ g L g ,g P ∂ g L g,g (cid:20) I + t L g L g (cid:21) P | χ (cid:105)(cid:104) χ | (cid:41) g = g = g − (cid:40) ∂ g L g ,g (cid:20) I + t L g L g (cid:21) P ∂ g L g,g P | χ (cid:105)(cid:104) χ | (cid:41) g = g = g +8 Re Tr (cid:110) ∂ g L g ,g (cid:2) L − g (cid:3) P ∂ g L g,g (cid:2) L − g (cid:3) P | χ (cid:105)(cid:104) χ | (cid:111) g = g = g + t O ( λ t ) O (cid:0) c ( c + 1) C (cid:1) + t (cid:26) O ( λ t ) (cid:2) O (cid:0) c C C (cid:1) + O ( c C ) (cid:3) + O (cid:16) (1 + c ) C C (cid:13)(cid:13)(cid:13)(cid:2) e t L g (cid:3) P (cid:13)(cid:13)(cid:13) (cid:17) + O (cid:0) c C C (cid:1) O (cid:18) λ λ (cid:19)(cid:27) + O ( λ t ) O ( c C C ) + O (cid:16) (1 + c ) C C (cid:13)(cid:13)(cid:13)(cid:2) e t L g (cid:3) P (cid:13)(cid:13)(cid:13) (cid:17) + O (cid:16) C C (cid:13)(cid:13)(cid:13)(cid:2) e t L g (cid:3) P (cid:13)(cid:13)(cid:13) (cid:17) + O ( c C C ) O (cid:18) λ λ (cid:19) + O C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) d (cid:88) j (cid:54) = k,j,k> e tλ j − e tλ k λ k ( λ j − λ k ) | ρ k (cid:105)(cid:104) l k | ∂ g L g,g | ρ j (cid:105)(cid:104) l j | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) g = g , (22)where corrections in the approximation are given by c = (cid:107) | ρ (cid:105)(cid:104) l | (cid:107) , C = (cid:107) ∂ g | g = g L g ,g (cid:107) , C = (cid:13)(cid:13)(cid:13)(cid:2) L − g (cid:3) P (cid:13)(cid:13)(cid:13) and C = (cid:13)(cid:13) ∂ g ,g | g = g = g L g ,g (cid:13)(cid:13) . We note that the estimate of the approximation error in Eq. (22) is very roughand implies strong conditions on the Master dynamics to be near a DPT, i.e. the corrections to be negligible. Fora particular model one should check the approximation by comparing to the exact results given by Eq. (18). Thequadratic terms in Eq. (22) correspond to Eq. (7) in the paper.Let us note that using Eq. (19) does not provide clear results in the quadratic regime. From comparing Eq.(21) to Eq. (20), we see that when ( − Re λ ) (cid:28) ( − Re λ ), many terms in Eq. (19) can diverge. In order to obtainthe way they diverge one needs to go back to the operators ∂ g L g ,g and ∂ g g L g ,g and therefore to Eqs. (18) and (22). Quadratic behaviour at a first-order DPT . At a first-order DPT we have λ = 0 and the considered time regime isinfinitely long due to τ = ∞ . Since we have lim t →∞ (cid:13)(cid:13)(cid:13)(cid:2) e t L g (cid:3) P (cid:13)(cid:13)(cid:13) = 0, in the limit of long time t Eq. (22) becomesexact, as all the corrections disappear. Therefore, Eq. (22) gives asymptotic quadratic behaviour of the QFI:lim t →∞ t − F ( | Ψ g ( t ) (cid:105) ) = − | Tr { ∂ g L g ,g P | χ (cid:105)(cid:104) χ |}| g = g + 4 Re Tr { ∂ g L g ,g P ∂ g L g,g P | χ (cid:105)(cid:104) χ |} g = g = g , , (23)see also Eq. (7) in the paper.0 D. Quadratic behaviour and bimodality
Here we consider a system at a first-order DPT with respect to photon counting statistics. We show how thequadratic behaviour of the QFI emerges from the bimodality of the total photon number which which the generatorof the phase transformation. This relation is given by Eq. (10) in the paper. A similar behaviour holds in the generalcase of arbitrary parameter defence, but this will be discussed in later work [28].We consider the estimated parameter to be the phase g = φ encoded on photons emitted by a system, which canbe formalised by defining L ,φ = e − iφ L , where L is a jump operator of the Master operator L , see the paper. Inthat case the Master operator does not depend on φ , L φ = L . We consider the system to be at a first-order DPT,and for simplicity we restrict to the case where the zero eigenvalue of the Master operator L has degeneracy two.This means that there exist two stationary states ρ A and ρ I which are supported on orthogonal subspaces H A , H I , sothat H = H A ⊕ H I . Moreover, the jump and hamiltonian operators have a block diagonal form in this decomposition H = H A ⊕ H I , L ,φ = L A1 ,φ ⊕ L I1 ,φ and L j = L A j ⊕ L I j , j = 2 , ..., k , acting on H = H A ⊕ H I . Let P H A , P H I denoteorthogonal projections on H A , H I , respectively.We note that the block-diagonal structure is preserved for ∂ φ | φ = φ L φ,φ ρ = iL ρL † , see Eq. (8) in the paper.Thus, Tr (cid:110) P H I L ρ A L † (cid:111) = 0 = Tr (cid:110) P H A L ρ I L † (cid:111) , which leads to the following simple form of the asymptotic QFI:lim t →∞ t − F ( | Ψ φ ( t ) (cid:105) ) = − | Tr { ∂ φ L φ ,φ P | χ (cid:105)(cid:104) χ |}| φ = φ + 4 Re Tr { ∂ φ L φ ,φ P ∂ φ L φ,φ P | χ (cid:105)(cid:104) χ |} φ = φ = φ = − (cid:16) p A Tr (cid:110) L † L ρ A (cid:111) + p I Tr (cid:110) L † L ρ I (cid:111)(cid:17) + 4 p A (cid:16) Tr (cid:110) L † L ρ A (cid:111)(cid:17) + 4 p I (cid:16) Tr (cid:110) L † L ρ I (cid:111)(cid:17) = 4 p A p I (cid:16) Tr (cid:110) L † L ρ A (cid:111) − Tr (cid:110) L † L ρ I (cid:111)(cid:17) , (24)where p A = Tr {P H A | χ (cid:105)(cid:104) χ |} and p I = Tr {P H I | χ (cid:105)(cid:104) χ |} .We will now show how the stationary states ρ A , ρ I correspond to the bimodal distribution of the generator, whichis the total photon count Λ( t ), when measured on | Ψ( t ) (cid:105) . For | Ψ φ ( t ) (cid:105) = e − iφ Λ( t ) | Ψ( t ) (cid:105) , we have (cid:104) Ψ φ ( t ) | Ψ φ ( t ) (cid:105) = (cid:104) Ψ( t ) | e i ( φ − φ )Λ( t ) | Ψ( t ) (cid:105) . We can then express the photon emission average as follows: (cid:104) Ψ( t ) | Λ( t ) | Ψ( t ) (cid:105) = i∂ φ (cid:104) Ψ φ ( t ) | Ψ φ ( t ) (cid:105) φ = φ = i ∂ φ Tr { e t L φ,φ | χ (cid:105)(cid:104) χ |} φ = φ , and thuslim t →∞ t − (cid:104) Ψ( t ) | Λ( t ) | Ψ( t ) (cid:105) = i Tr { ∂ φ L φ,φ P | χ (cid:105)(cid:104) χ |} φ = φ = − Tr (cid:110) L † L P | χ (cid:105)(cid:104) χ | (cid:111) . We represent the initial state of the system as | χ (cid:105) = √ p | χ A (cid:105) + √ − p | χ I (cid:105) , where | χ A (cid:105) ∈ H A and | χ I (cid:105) ∈ H A and0 ≤ p ≤
1. By setting p = 1 or p = 0 we arrive at − Tr (cid:110) L † L ρ A (cid:111) = µ A and − Tr (cid:110) L † L ρ I (cid:111) = µ I , where µ A , µ I arethe asymptotic rates of Λ( t ) when the system is initially in the state | χ A (cid:105) ∈ H A , | χ I (cid:105) ∈ H I , respectively. ThereforeEq. (24) becomes: lim t →∞ t − F ( | Ψ g ( t ) (cid:105) ) = 4 p A p I ( µ A − µ I ) , (25)which corresponds to Eq. (10) in the paper. Let us assume µ A > µ I . We see that in the asymptotic limit t (cid:29) τ (cid:48) we can define two dynamical phases corresponding to active (A) and inactive (I) mode in total photon count Λ( t )distribution, to be any MPS states | Ψ A ( t ) (cid:105) , | Ψ I ( t ) (cid:105) , which after tracing out the output are supported only on H A , H I , respectively. In order to ensure quadratic scaling of the QFI F ( | Ψ φ ( t ) (cid:105) ), the initial state | χ (cid:105) of the system needsto be a superposition of states from H A and H I , so that both p A , p I >
0. When the asymptotic emission rates areequal µ A = µ I , there is no quadratic scaling of the QFI, as the asymptotic distribution of photon counts Λ( t ) isunimodal with variance scaling linearly with time t . Quadratic behaviour and approximate bimodality for an intermittent system . Near a DPT in photon emissions, thesystem dynamics is intermittent and switches between long time intervals with different emission rates. The typicallength of those intervals is given by the correlation time τ = ( − Re λ ) − . Therefore, we expect to be able to constructapproximate stationary states for the quadratic QFI regime τ (cid:48) (cid:28) t (cid:28) τ . Let us note that the Master operator L hasonly one stationary state ρ ss of L and its second eigenvector ρ fulfills Tr ρ = 0 due to normalisation of eigenvectors.Nevertheless, L is degenerate up to order λ . Below we briefly present a construction of two approximately stationarystates with different emission rates as linear combinations of ρ ss and ρ . The construction closely follows the theory1of classical non-equilibrium first-order phase transitions [32]. We leave rigorous proofs and discussion of Eq. (25) inthat case for later work [28].Consider first the case when a first-order DPT is approached by changing parameters in the Master operator L , which will then guide the more general considerations near a DPT. When approaching the DPT, the first twoeigenvectors converge to ρ and ρ , such that ρ ≥
0, Tr ρ = 1 and Tr ρ = 0. One can show that in that case ρ = p ρ A + (1 − p ) ρ I , ρ = ρ A − ρ I and l = (1 − p ) P H A − p P H I , where 0 < p < ρ A , ρ I are the stationarystates supported on orthogonal subspaces H A , H I , respectively, and P H A , P H I are the orthogonal projections on thesesubspaces. In the case of the system near a DPT, the construction of approximate stationary states is as follows.The Master operator L acts almost block-diagonally, i.e. H , L j , j = 1 , ..., k are approximately block-diagonal withrespect to an orthogonal splitting into subspaces H and H . For pure initial states | χ (1) (cid:105) , | χ (2) (cid:105) supported in thesesubspaces, the corresponding evolved states ρ (1) ( s ), ρ (2) ( s ) will still be supported in the blocks for time s in quadraticregime of QFI, τ (cid:48) (cid:28) s (cid:28) τ , and will be well approximated by linear combinations of ρ ss and ρ (due to s (cid:29) τ (cid:48) ).Moreover, ρ (1) ( s ), ρ (2) ( s ) are almost stationary w.r.t e t L , where t = O ( s ). In order to define the approximate blocks, H , H , and the initial states | χ (1) (cid:105) , | χ (2) (cid:105) using the Master operator L , we assume λ ∈ R for simplicity. In this caseboth ρ and l are Hermitian matrices on H , i.e. they diagonalise and their spectra are real. First, inspired by theform of the second eigenvector at a first-order DPT, l = (1 − p ) P H A − p P H I , we define the subspaces H , H in thefollowing way. H is spanned by the eigenvectors of l which correspond to positive eigenvalues close to the maximaleigenvalue of l , while H is spanned by eigenvectors of l corresponding to negative eigenvalues close to the minimaleigenvalue of l . Next, the initial states | χ (1) (cid:105) and | χ (2) (cid:105) are chosen to be the eigenvectors corresponding to maximaland minimal eigenvalue of l , respectively. Finally, the two approximate dynamical phases in photon emissionsare defined as any MPS states which after tracing out the output are supported mostly on H , H2