Limited-control metrology approaching the Heisenberg limit without entanglement preparation
Benedikt Tratzmiller, Qiong Chen, Ilai Schwartz, Susana F. Huelga, Martin B. Plenio
LLimited-control metrology approaching the Heisenberg limit without entanglement preparation
Benedikt Tratzmiller, Qiong Chen,
1, 2
Ilai Schwartz,
1, 3
Susana F. Huelga, and Martin B. Plenio Institut f¨ur Theoretische Physik and IQST, Albert-Einstein-Allee 11, Universit¨at Ulm, D-89081 Ulm, Germany College of Physics and Electronics, Hunan Normal University, Changsha 410081, China NVision Imaging Technologies GmbH, Albert-Einstein-Allee 11, Universit¨at Ulm, D-89081 Ulm, Germany (Dated: April 6, 2020)Current metrological bounds typically assume full control over all particles that are involved in the protocol.Relaxing this assumption we study metrological performance when only limited control is available. As anexample, we measure a static magnetic field when a fully controlled quantum sensor is supplemented by particlesover which only global control is possible. We show that even for a noisy quantum sensor, a protocol that mapsthe magnetic field to a precession frequency can achieve transient super-Heisenberg scaling in measurementtime and Heisenberg scaling in the particle number. This leads to an estimation uncertainty that approachesthat achievable under full control to within a factor independent of the particle number for a given total time.Applications to hybrid sensing devices and the crucial role of the quantum character of the sensor are discussed.
INTRODUCTION
The use of quantum resources in sensing and metrology hasa longstanding history which originated with the use of single-mode squeezed states [1] and multi-particle spin-squeezing [2,3], i.e., entanglement, to enhance precision in interferometryand atomic spectroscopy.The goal of quantum metrology is the optimisation of thescaling of metrological precision with the available physicalresources [4, 5]. Notably, in a noiseless setting, independentpreparation and measurement of M particles in parallel re-sults in a / √ M scaling of the uncertainty, the so called stan-dard quantum limit (SQL), while the collective preparationof the particles in an entangled state leads to a /M -scaling,commonly referred to as Heisenberg scaling (HS) [2, 3] (see[6, 7] for more general upper bounds obtained via the quantumFisher information). The use of entangled states is necessaryto achieve the optimal precision and exact HS but sequences ofprobe states with an asymptotically vanishing amount of en-tanglement can reach a scaling arbitrarily close to the Heisen-berg limit (HL) [8]. Environmental noise is known to have anon-trivial impact on metrology [9] and a meaningful compar-ison of different schemes needs to specify carefully the con-ditions under which the metrological protocol is carried out,such as the number of particles or the total amount of timeavailable [9, 10]. A wide variety of setting has been anal-ysed [9–12] and noise models have been found to result inmetrological scaling intermediate between SQL and the HL[12–16]. However, these results depend on access to perfectand arbitrarily fast control and feedback operations [17, 18].In practice, however, only limited control is possible overexperimental resources and the asymptotic regime of largenumbers of fully controlled particles is not accessible. Whatcan be achieved in metrology for systems where, for exam-ple, particles cannot be addressed individually, multi-particlequantum gates are not available or the rate of measurements,feedback and the number of accessible particles is limited?In order to initiate investigations of this type in a concretesetting, we allow ourselves to be motivated by the recentlydeveloped concept of quantum-hybrid sensors [19, 20]. These are devices that integrate at least two components, one being afully controlled quantum sensor and another, typically an as-sembly of quantum particles, mutually interacting or not, thatare coupled to the quantum sensor but over which there is noindividual control. This second component acts as a trans-ducer of a signal to a form that is then detected by the quan-tum sensor. An example is a device composed of a piezo-magnetic material deposited on a diamond surface that trans-lates a force into a stray magnetic field which is then detectedby a shallowly implanted nitrogen-vacancy (NV) center indiamond[21–23]. Another possibility, motivated by recentlyrealised nanoscale NMR measurements [24–26], consists ofan ensemble of M nuclear spins and an NV center. Here anapplied magnetic field can be measured as it induces a nuclearLarmor precession which can be monitored by an NV center.Similarly, atoms in microfabricated vapour cells [27] couldallow for observation of their Larmor precession by an NVcenter rather than a classical laser field. We assume that thequantum sensor is subject to noise, and cannot exert individualcontrol over the noise-free auxiliary spins.While we are motivated by concrete settings, our analysisyields more compact expressions, without affecting the scal-ing properties, by assuming that all involved spins have thesame magnetic moment. Furthermore, all the following con-siderations will neglect direct interactions between the auxil-iary spins.By means of analyzing the scaling of the Fisher informationwith respect to particle number M and measurement time T we will show that, despite the limitations of partial control, itis possible to get close to the Heisenberg limit, we will discussthe origin of this scaling and compare our scheme to a singlequantum sensor without auxiliary spins. APPROACHING THE HEISENBERG LIMIT WITHOUTENTANGLEMENT PREPARATION
Before stating the main results, we briefly recapitulate theachievable uncertainties under unconstrained metrology tomake it available for later comparison with our schemes. a r X i v : . [ qu a n t - ph ] A p r Freeev. φ Freeev. φ Int. k Int. k ... | i π x -DD- π y | i π x -DD- π y | i π | i π | i π Local osc.Local osc.
Figure 1. The proposed measurement scheme uses a control se-quence on the sensor spin (blue) to weakly measure the auxiliaryspins (black) without the need of further control after initialisationin either the pure state | + + + ... (cid:105) as shown here or the completelymixed state. The weak measurement is realised by initialising a sen-sor spin (blue) into | (cid:105) , applying π/ -pulses and a dynamical de-coupling (DD) sequence to weakly entangle the sensor spin with theauxiliary spins and finally measuring the sensor spin, applying aneffective CPTP map onto the auxiliary spins. In between these mea-surements that are synchronised with an external local oscillator, theauxiliary spins acquire a phase φ ∝ µ n B , leading to a total phase nφ after n cycles. The ideal case of full quantum control
Consider a quantum sensor and M auxiliary spins, all withthe same magnetic moment µ n , over which we can exert ar-bitrary and fast control. Then the optimal uncertainty for theestimation of the magnetic field in a time T in the absence ofnoise is obtained via Ramsey spectroscopy using the M + 1 accessible particles prepared in a highly entangled state of theform ( | . . . (cid:105) + | . . . (cid:105) ) / √ and is given by [9] ∆ B = (cid:126) µ n ( M + 1) T . (1)We observe a linear decrease in the uncertainty, i.e. HS, bothin the total measurement time T and the number of spins M . The case of limited control setting
We consider a perfectly controlled quantum sensor supple-mented by M auxiliary spins all having the same magneticmoment. These M auxiliary spins can be controlled by aglobal field and interact weakly with the quantum sensor, i.e.the product of sensor-auxiliary spin interaction strength andinteraction time is much smaller than unity. For simplicitywe assume that each auxiliary particle is interacting with thesame strength and phase with the quantum sensor, howeverthe basic findings remain the same in the more general case.We measure a static external magnetic field B with this hy-brid sensor and determine the achievable uncertainty ∆ B . Tothis end we initialise the auxiliary spins in a fully polarisedstate [28] and then subject them to a π/ -pulse. We deter-mine the resulting rate of precession of these M spins by pe-riodically measuring the time-dependent magnetic field gener-ated by the precessing spins in regular time intervals using the quantum sensor and a dynamical decoupling (DD) sequenceto weakly entangle it with the auxiliary spins. This allows forcomparison of the precession frequency to that of a local os-cillator [24, 25] as sketched in figure 1. This DD sequencedoes not only cancel static noise on the sensor spin, but alsoallows to accumulate signal over several auxiliary spin Lar-mor periods by creating an effective σ z ⊗ σ x interaction. Inorder to see this, we start with a Hamiltonian of the form H = Ω( t )2 σ x + M (cid:88) m =1 ω L σ ( m ) z (2) + A ( m ) ⊥ σ z ⊗ (cid:16) σ ( m ) x cos( φ m ) + σ ( m ) y sin( φ m ) (cid:17) that describes a NV center coupled to M nuclear spins inthe interaction picture where Ω( t ) is the Rabi frequency with Ω( t ) = 0 during the free evolution, ω L is the nuclear Larmorfrequency, A ( m ) ⊥ is the perpendicular coupling of the nuclearspins and φ m the corresponding phase. The operators σ i acton the electron spin (NV center), σ ( m ) i act on the mth nuclearspin.When we fulfil τ = π/ω L in the DD (e.g. XY-8) sequence,the sequence produces a modulation function that modulatesthe NV σ z operator to σ z f ( t ) where f ( t ) = 4 π cos( ω L t ) + rot . (3)and rot. denotes terms that vanish after the rotating-wave ap-proximation in a frame rotating with the nuclear Larmor fre-quencies. Using cos ( ω L t ) = 12 (1 + cos(2 ω L t )) , (4)we obtain the effective interaction Hamiltonian H eff = (cid:88) m A ( m ) ⊥ π σ z ⊗ (cid:16) σ ( m ) x cos( φ m ) + σ ( m ) y sin( φ m ) (cid:17) (5)that will be the starting point for the following discussion.We use k = µ n B s / (cid:126) where B s is the field generated byone of the M auxiliary spins at the position of the NV center Transient super-Heisenberg scaling in measurement time
In a first step we analyse the Fisher information scalingwith the measurement time. For this purpose we derive theprobability p n of finding the internal state of a spin- / quan-tum sensor in the spin-down state in the n-th measurementin leading order assuming the length T s of each instance ofa magnetic field measurement is short and if we apply thesemeasurements every τ m .We simplify to the case that all couplings A ( m ) ⊥ = k and allphases φ m are equal, see the appendix for a more general dis-cussion. Here k = µ n B s / (cid:126) where B s is the field generatedby one of the M auxiliary spins at the position of the NV cen-ter. In the described protocol the auxiliary spins gain a phase φ = δτ m in each step, where δ = 2 π ( ν − ν loc ) is the differ-ence of precession frequency ν and local oscillator frequency ν loc . For a nuclear spin with an already accumulated phasefrom n cycles equalling nφ the readout probability is calcu-lated for the NV measurement in basis Y , i.e. the eigenbasisof the Pauli , σ y operator, and NV preparation in | + (cid:105) . p n −
12 = Tr (cid:104) ˆ O measurement U ρ n U † (cid:105) = Tr (cid:104)(cid:16) σ y ⊗ ⊗ M (cid:17) U (cid:18) + σ x ⊗ (cid:18) + cos( φn ) σ x + sin( φn ) σ y (cid:19)(cid:19) ⊗ M U † (cid:35) = i (cid:34)(cid:18) cos 4 k T s π − i sin 4 k T s π cos( φn ) (cid:19) M − (cid:18) cos 4 k T s π + i sin 4 k T s π cos( φn ) (cid:19) M (cid:35) . (6)For M k T s (cid:28) we can approximate (cid:18) cos 4 k T s π ± i sin 4 k T s π cos( φn ) (cid:19) M ∼ = exp (cid:18) ± iM sin 4 k T s π cos( φn ) (cid:19) (7)to derive the signal p n = cos (cid:18) M k T s π cos ( φn ) − π (cid:19) . (8)Imperfect polarisation P of the auxiliary spins can be in-corporated via k = µ n P B s / (cid:126) .By eq. (8) we estimate the frequency πν = µ n B/ (cid:126) andhence the magnitude of the magnetic field B . For N mea-surements, the achievable uncertainty in the estimate of ν isobtained via the classical Fisher information I N = N (cid:88) n =1 p n (1 − p n ) (cid:18) ∂p n ∂φ (cid:19) (cid:18) ∂φ∂δ (cid:19) . (9)We can describe decoherence processes in terms of a decayrate γ = γ + γ b (see appendix), where γ refers to T pro-cesses on the nuclear spins and γ b refers to measurement back-action from the quantum sensor. Then the effective couplingafter n measurements is k n = k e − γn and we find I N ( γ ) = N (cid:88) n =1 (cid:18) τ m M k n T s nπ (cid:19) sin φn. (10)Under the assumptions max [ γ, N ] (cid:28) πφ and M k T s (cid:28) ,i.e. when we sample at least one full oscillation of the signal of frequency δ , eq. (10) is well approximated by I N ( γ ) ∼ = 16 M τ m k T s π N (cid:88) n =1 n e − γn (1 − cos 2 φn ) ∼ = 2 M τ m k T s π − e − γN (1 + 2 γN (1 + γN )) γ . (11)For γN (cid:28) (for γN < . errors are smaller than 3%) I N ( γ ) ∼ = 2 M τ m k T s π ( 4 N − γN + 6 γ N (12)and hence ∆ B ≤ (cid:115) π (cid:126) µ n M τ m k T s N . (13)As a result, for small γN our limited control procedure ex-hibits a scaling in the number of measurements N or, equiv-alently, the total measurement time T = N τ m that exceedsthe standard HS of eq. (1) while the scaling in the num-ber of particles M achieves the HL. Note that unlike thecase of interaction-based quantum metrology [29] this super-Heisenberg scaling is not due to interactions between theauxiliary spins. Intuitively the scaling can be understood toemerge due to the fact that, without noise, the last measure-ment of the measurement record alone would already givequadratic Fisher information scaling, and the linear number ofintermediate measurements leads to a cubic scaling in total. Asymptotic SQL scaling in measurement time
However, the super-Heisenberg scaling identified in theprevious subsection has to be transient and cannot persist forarbitrarily long times as this would be in violation of the fun-damental limit of sensitivity that is imposed by the full controlscheme in the absence of any noise.For γ = 0 , the remaining contribution to the decay rate γ isdue to the measurement backaction of the quantum sensor onthe auxiliary spins which is negligible only for γN (cid:28) . Ourmeasurement scheme then yields (see appendix for a deriva-tion) γ b = 4 k T s π . (14)Due to the measurement backaction, the signal weakens withincreasing number of measurements N and the rate of in-crease of the Fisher information slows. When determiningthe scaling in this regime, a note of caution is in order as thecalculation of the measurement backaction in eq. (11) deter-mines the Fisher information of the averaged density matrixof the auxiliary spins. However, as we have access to and useall the intermediate measurements, the correct Fisher infor-mation of the protocol is obtained by weighted averaging overmeasurement trajectories. For γN (cid:29) this results in a scal-ing linear in N (see also [30–32]), as indicated analyticallyin the appendix and numerically in figure 2. For M = 100 nuclei in an initial product state (red data) the transient super-Heisenberg I N ∝ N ∝ T scaling evolves into the shotnoise scaling (SQL) of I N ∝ N ∝ T (blue asymptote) whileeq. (11) was calculated with the average density matrix andtherefore would yield a constant. Remarkably, this linear scal-ing is independent of the initial state and we can achieve thesame scaling using a completely mixed initial state (orangetriangles) [33]. In the limit of small interaction strength k T s and decay, the asymptotic ( γN (cid:29) ) value of the Fisher in-formation can be estimated to be I N = sin (4 k T s /π )16 ( γ b + γ ) M τ m N (15) Heisenberg scaling in particle number and relation to theHeisenberg limit
While the quadratic scaling with the number of nuclei M is obvious for γN (cid:28) , it also persists for γN (cid:29) as con-firmed in equation (15) and in Fig. 3 (see Fig. 4 and therelated discussion for details). In the full control scenariosuch a quadratic scaling can be traced back to the prepara-tion of an macroscopically entangled resource state of theform ( | ... (cid:105) + | ... (cid:105) ) / √ including the quantum sensorand the auxiliary spins which contain one ebit of entangle-ment [34, 35]. This entanglement is destroyed with the finalmeasurement and represents the resource that is required toachieve HS. This is in sharp contrast with our scenario, whichstarts from an initial product state. Here the quadratic scal-ing arises because the auxiliary spins are interrogated by aquantum sensor which results in a readout operator that is notparticle local, i.e. cannot be represented by a product of singleparticle operators as would be the case when using a classicalreadout device. Indeed, every measurement applies a CPTP(completely positive and trace-preserving) map to the auxil-iary spins that is represented by the Kraus operators (see also[30] for the case M = 1 and our appendix) U ± = (cid:104)± y | U | + (cid:105) = e − i k Tsπ (cid:80) σ ( m ) x ∓ ie i k Tsπ (cid:80) σ ( m ) x (16)Crucially, this operator is diagonal in a basis different thanthe eigenbasis of the free evolution operator - the latter be-ing a tensor product of unitary evolutions on every auxiliaryspin. If the same held true for the readout operator, it is triv-ial to see that every spin would be measured independentlyand therefor SQL scaling would apply. The readout opera-tor here, however, combines all particle states in a nontrivial,nonlocal, manner. This is similar to the metrological advan-tage obtained for indistinguishable particles [5] (section III).In our work, this advantage is achieved by the sensor spin that,unlike a classical sensor, allows to apply the same, particlenon-local, CPTP map in every measurement. We would like Figure 2. Upper graph: Numerical Fisher information scaling (red:initial product state/ orange: initial mixed state), the analytical ap-proximation for small N eq. (11) (red line, invalid for large N due tothe average density matrix calculation being invalid for γN > ) andthe asymptotic ( γN (cid:29) ) behaviour (blue) for φ = 0 . and M=100nuclear spins, each coupled with k T s = 0 . to the NV center.The HL (black) for M + 1 spins is shown for comparison. The re-sults are averaged over 96 (32 for mixed state) runs with N = 2 measurements each. Lower graph: Same data (red) and results for k T s = 0 . in green. The Fisher information is initially larger butdominated by backaction earlier, resulting in a smaller prefactor forthe asymptotic regime. In both graphs and all other simulations theachievable Fisher information approaches the HL to within a factor (cid:46) . This Fisher minimal distance is independent of the interactionstrength k T s and the number of nuclei M , see appendix. to stress that the entanglement build-up due to the backactionvia a non-local measurement operator does not contribute tothe quadratic scaling with the number of auxiliary spins, nordoes it support the transient N super-Heisenberg scaling.In the limit of large nuclear coherence times ≈ γ (cid:28) γ b the Fisher information approaches the HL achievable underfull control for N opt ≈ (2 k T s /π ) − = 1 /γ b measurementsafter which the N -scaling turns into a scaling ∝ N . Notethat the fundamental Heisenberg limit is not violated as withdecreasing interaction k T s both the backaction and the infor-mation gain per measurement decrease at the same rate thuscompensating each other. Remarkably, at this point the ra-tio of ultimate sensitivity under global control and the lim-ited control scheme used here only depends on the interaction Figure 3. For the same parameters as in figure 2: Numerical Fisherinformation scaling (red: initial product state/ orange: initial mixedstate ( k T s = 0 . at N = 2 each)/ green: initial product state( k T s = 0 . at N = 2 )) with the number of nuclei M . The bluelines show the predicted scaling ∝ M from eq. (15) and Figure 2. strength and therefore can be tuned to approach the HL for N opt measurements to within a Fisher minimal distance of afactor independent of T , see figure 2. Furthermore it is natu-ral to assume that this holds independently of M as the Fisherinformation of both the HL and the asymptote for γN (cid:29) exhibit the same quadratic in M scaling. This was confirmednumerically (see appendix for confirmation). DISCUSSION
Metrology assisted by environmental spins have been con-sidered before, see e.g. [36, 37]. There, however, the empha-sis was placed on spins that are strongly interacting with thequantum sensor and the measurement protocol creates a jointentangled state of the quantum sensor and the auxiliary spinswhich then evolves for some time followed by an inversionof the entangling operation and the subsequent measurementof the state of the quantum sensor. In this approach the HSis achieved in the number of strongly coupled auxiliary spinswhile we assume no such spins in our set-up. Furthermore,this protocol suffers from the drawback that it is fundamen-tally limited by the coherence time of the quantum sensor andhence does not take full advantage of the long coherence timeof the auxiliary spins. In contrast, the measurements in ourprotocol can be made shorter than the coherence time withoutadversely affecting the achievable sensitivity.Besides the theoretical interest in the novel scaling regimes,we stress that the proposed scheme employing auxiliary spinsunder limited control provides enhanced sensitivity as com-pared to the quantum sensor alone. This advantage is theresult of two processes. First, the transduction of the staticmagnetic field to a time-dependent Larmor precession whichis then detected by the quantum sensor facilitates the use ofdynamical decoupling schemes to filter out noise without ad-versely affecting the signal. Secondly, as each auxiliary spin contributes to the signal, the overall signal strength scales withthe number of spins and hence leads to a considerable signalenhancement. Remarkably, magnetometry schemes such asatoms in gas cells which are probed independently by a clas-sical field lead to a M − scaling of the variance with the par-ticle number M . In sharp contrast, it is the transduction of thesignal to a quantum sensors, e.g. an NV center, which resultsin a particle non-local measurement which causes the M − Heisenberg scaling. This suggests a practical route for en-hancing the measurement capacity of gas cell magnetometers.Furthermore, for an NV center as quantum sensor, even whenconsidering nuclei with their small magnetic moment as aux-iliary spins, we may obtain an increased sensitivity. To thisend, let us consider the | m = 0 (cid:105) ↔ | m = +1 (cid:105) transition ofan NV center in an external magnetic field B and assume thatthe NV center is dominated by pure dephasing which resultsin a coherence time T ( NV )2 . For perfect readout efficiency, theoptimal interrogation scheme yields ∆ B = (cid:115) e (cid:126) µ e T ( NV )2 T = (cid:118)(cid:117)(cid:117)(cid:116) e (cid:126) µ e N T ( NV )2 2 (17)[9] where N = 2 T /T ( NV )2 . The maximum of equation (15)that takes the form γ b / ( γ b + γ ) is obtained by choosing γ b = 2 γ . This allows to compare with our indirect measure-ment scheme using M hydrogen nuclear spins and assuming (cid:16) µ e µ n (cid:17) T ( NV )2 T n ≈ we find that for M > (cid:118)(cid:117)(cid:117)(cid:116) e µ e T ( NV )2 µ n T ( n )2 ≈ (18)the auxiliary spin assisted sensor outperforms the bare NVcenter. Nuclear spins couple more weakly to both noise andsignal due to smaller gyromagetic ratio resulting in typicallylarger coherence times µ e µ n ≈ T n T ( NV )2 , so similar results ap-ply for other systems. While our protocol makes use of a farsmaller magnetic moment compared to even a single electronspin M µ n (cid:28) µ e , this is compensated by the longer coherencetime and the possibility to measure during the signal accumu-lation.Furthermore the obtained expression also highlights the ad-vantage of the M − Heisenberg scaling over the SQL thatindividual measurements on the nuclear spins give, providinghigher Fisher Information for
M > / e ≈ . .Finally, we note that our analysis also covers the case M =1 corresponding to the detection of the Larmor frequency of asingle nuclear spin via an NV center. Super-Heisenberg scal-ing applies for as long as the measurement backaction is weak.This applies for distant nuclear spins or for measurements thatare designed to be weak, i.e. not obtaining a full bit of infor-mation in each single measurement. Conclusions –
We have examined metrology in a real-istic setting of limited control and found transient super-Heisenberg scaling in the total measurement time and a metro-logical precision approaching that of the same number of par-ticles under full experimental control. This is despite the ab-sence of initial entanglement in the system. In fact, in thisscheme entanglement emerges only with increasing numberof measurements and adversely affects the metrological scal-ing. Furthermore, the proposed set-up, which employs auxil-iary spin under limited control, also represents an hybrid sen-sor that may outperform a bare quantum sensor thus providingnew design principles for quantum sensors.
Acknowledgements –
The authors thank Liam McGuin-ness and Jan F. Haase for discussions and comments on themanuscript. This work was supported by the ERC SynergyGrant BioQ, the EU projects AsteriQs, HYPERDIAMOND,the BMBF projects NanoSpin and DiaPol, the DFG CRC 1279and the DFG project 414061038. The authors acknowledgesupport by the state of Baden-W¨urttemberg through bwHPCand the German Research Foundation (DFG) through grant noINST 40/467-1 FUGG (JUSTUS cluster). The first results ofthis work have been presented at the Workshop On QuantumMetrology, 22nd - 23rd June 2017 in Ulm, Germany. [1] C. M. Caves. Quantum-mechanical noise in an interferometer.
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Appendix: Limited-control metrology approaching the Heisenberg limit without entan-glement preparation
Derivation of the signal for few measurements
This section presents a more general calculation of equations (6-8) of the main text. For a nuclear spin (initial state de-scribed by polarisation P) with an already accumulated phase from n cycles φ m = δτ m n + φ the readout probability is (NVmeasurement in basis Xc α + Y s α and NV preparation in X ) p n −
12 = Tr (cid:104) ˆ O measurement U ρ n U † (cid:105) (19) = Tr (cid:34)(cid:18) σ x cos α + σ y sin α ⊗ ⊗ M (cid:19) U (cid:32) + σ x ⊗ M (cid:89) m =1 (cid:32) + P cos φ m σ ( m ) x + P sin φ m σ ( m ) y (cid:33)(cid:33) U † (cid:35) (20) = 14 (cid:34) cos α (cid:32) M (cid:89) m =1 (cid:32) cos 4 A ( m ) ⊥ T s π − i sin 4 A ( m ) ⊥ T s π P cos φ m (cid:33) + M (cid:89) m =1 (cid:32) cos 4 A ( m ) ⊥ T s π + i sin 4 A ( m ) ⊥ T s π P cos φ m (cid:33)(cid:33) (21) + i sin α (cid:32) M (cid:89) m =1 (cid:32) cos 4 A ( m ) ⊥ T s π − i sin 4 A ( m ) ⊥ T s π P cos φ m (cid:33) − M (cid:89) m =1 (cid:32) cos 4 A ( m ) ⊥ T s π + i sin 4 A ( m ) ⊥ T s π P cos φ m (cid:33)(cid:33)(cid:35) (22)where φ m = φ m − φ m .For M (cid:80) m =1 A ( m ) ⊥ T s /π (cid:28) we can approximate M (cid:89) m =1 (cid:32) cos 4 A ( m ) ⊥ T s π ± i sin 4 A ( m ) ⊥ T s π P cos φ m (cid:33) = M (cid:89) m =1 exp ± i sin 4 A ( m ) ⊥ T s π P cos φ m + O (cid:32) A ( m ) ⊥ T s π (cid:33) (23) ∼ = exp (cid:32) ± i M (cid:88) m =1 sin 4 A ( m ) ⊥ T s π P cos φ m (cid:33) (24)to derive the signal p n = 12 + 12 cos (cid:32) M (cid:88) m =1 sin 4 A ( m ) ⊥ T s π P cos φ m − α (cid:33) = cos (cid:32) M (cid:88) m =1 sin 2 A ( m ) ⊥ T s π P cos φ m − α (cid:33) . (25)The generalisation of this case described in the main text to different coupling A ( m ) ⊥ (cid:54) = A ( m ) ⊥ can be described by an effectivecoupling. Different phases cos φ m (cid:54) = cos φ m can produce additional features, but in the asymptotic case these effects areirrelevant as the initial state becomes less important, see next section. Derivation of the Fisher Information from the probabilities for a full measurement record
The outcome of the kth measurement is denoted by every individual X k ∈ { , −} , X k is a measurement record ofthe form { , , , , , , ... } with k components, X l is the lth component of it and β ≡ A x T s /π is the coupling achieved by theXY-sequence. Furthermore U φ = exp (cid:16) − iφ (cid:80) σ ( m ) z / (cid:17) and U ± = (cid:104)± y | U | + (cid:105) = e − iβ (cid:80) σ ( m ) x ∓ ie iβ (cid:80) σ ( m ) x (26)with the coupling from the first section U = exp( − iβσ NVz (cid:80) σ ( m ) x ) . Here we assumed all coupling constants to be equal,however different β m don’t change the structure of the result.Each measurement probability can be described by p ± = Tr (cid:104) ( |± y (cid:105) (cid:104)± y | ⊗ ) U U φ ( |±(cid:105) (cid:104)±| ⊗ ρ N − ) U † φ U † (cid:105) = Tr (cid:104) U X U φ ρ U † φ U † X (cid:105) (27)and evolves the nuclear state to ρ ± = 1 p ± Tr NV (cid:104) U X U φ ρ U † φ U † X (cid:105) . (28)The probability for a measurement record X k can be described as p X k = p X p X | X ...p X N − | X N − ...X p X N | X N − (29) = Tr (cid:104) U X U φ ρ U † φ U † X (cid:105) Tr (cid:104) U X U φ U X U φ ρ U † φ U † X U † φ U † X (cid:105) Tr (cid:104) U X U φ ρ U † φ U † X (cid:105) ... (30) = Tr N (cid:89) k =1 ( U X k U φ ) ρ (cid:32) N (cid:89) k =1 ( U X k U φ ) (cid:33) † . (31)Each part of the sum contributes roughly − N , sum over all contributions is 1. To analyse the effect of these operators, weapply them on a permutation-invariant product state U φ (cid:18) a + bσ x + cσ y + dσ z (cid:19) ⊗ M U † φ = (cid:18) a + ( bc φ − cs φ ) σ x + ( cc φ + bs φ ) σ y + dσ z (cid:19) ⊗ M (32) U ± (cid:18) a + bσ x + cσ y + dσ z (cid:19) ⊗ M U †± = (cid:18) a + bσ x + ( cc β + ds β ) σ y + ( dc β − cs β ) σ z (cid:19) ⊗ M (33) + (cid:18) a + bσ x + ( cc β − ds β ) σ y + ( dc β + cs β ) σ z (cid:19) ⊗ M (34) ± i (cid:34)(cid:18) ( ac β + ibs β ) + ( bc β + ias β ) σ x + cσ y + dσ z (cid:19) ⊗ M (35) − (cid:18) ( ac β − ibs β ) + ( bc β − ias β ) σ x + cσ y + dσ z (cid:19) ⊗ M (cid:35) . (36)It is very difficult to calculate the full expression because every measurement multiplies the number of terms by 4. So we wantto find the relevant terms for the Fisher Information for different limits. For a single nucleus M=1, only two terms are createdevery measurement and dσ z can be neglected. Therefore we simplify to U ± (cid:18) a + bσ x + cσ y (cid:19) U †± = 2 (cid:18) a + bσ x + cc β σ y (cid:19) ± (cid:18) − bs β − as β σ x + cσ y (cid:19) (37) = 2 A [ ρ ] + 2 B [ ρ ] . (38)The cos(2 β ) for the c coefficient produces the backaction-induced decay γ b . We can approximate A [ ρ ] = (cid:18) a + bσ x + cc β σ y (cid:19) ≈ (cid:18) a + (1 + c β ) / bσ x + cσ y )2 (cid:19) . (39)which reduces the bloch vector according to (cid:18) c β (cid:19) k = exp (cid:18) k log (cid:18) c β (cid:19)(cid:19) ≈ exp (cid:0) k log (cid:0) β (cid:1)(cid:1) ≈ exp (cid:0) − kβ (cid:1) . (40)0This is valid because the higher orders will be negligible in the further calculation. T2 processes have a similar effect, whywe define the decay of population in the x-y-plane with γ = − log 1 + c β τ m T ( nuc )2 ≈ β + τ m T ( nuc )2 (41)where τ m is the time for each of the N repetitions.When starting with ρ = ( + σ x ) / can expand the probability for a N measurement record X N as N p X N ≈ s β N (cid:88) l =1 ( − X l exp( − γl ) cos( lφ ) (42) + s β (cid:88) (cid:54) l 1) + 14 γ ≈ s β (cid:20) N − γN (cid:21) . (51)For γN > the geometric series is not valid anymore and many higher orders in l need to be considered. To show that termslinear in N exist, we consider the approximated second order ( /p NX ≈ N )1 Figure 4. For M=10 nuclei: Fisher information after measurements for β = 0 . × /π for different γ averaged over 192 runs comparedto a γ − curve with the numerically obtained prefactor. τ m I N = s β (cid:88) (cid:54) l We numerically investigated the minimum difference between our protocol and the Heisenberg limit, which can only beachieved given full control over the nuclei in absence of decoherence γ = 0 .Figure 5 shows the maximum ratio between the Fisher Information of the investigated protocol and the Heisenberg limit. Fordifferent coupling strength the ratio is independent of the number of nuclei. Note that the peaks are due to the monte carlosimulation. This is confirmed by the curves on the right hand side of figure 5, where the maxima are found at N ≈ β − asexpected. Simulation The normal simulation (without making use of the permutation invariance) repeats the following steps (after initializing thenuclear spins to | ψ (cid:105) = | + (cid:105) ⊗ M , ρ = | ψ (cid:105) (cid:104) ψ | )1. Simulate nuclear spin evolution with the operator U free = exp (cid:32) − iδτ m (cid:88) m σ ( m ) z / (cid:33) (60)where τ m is the time between two measurements.2. Determine probability to measure the NV in | + y (cid:105) after preparing it in | + (cid:105) and evolving it with the nuclear spins accordingto (5) by p = Tr (cid:2) | + y (cid:105) (cid:104) + y | ⊗ ⊗ M U | + (cid:105) (cid:104) + | ⊗ ρ n U † (cid:3) = Tr (cid:104) U + ρ n U † + (cid:105) (61) = (cid:104) ψ n | U † + U + | ψ n (cid:105) (for pure states) (62)where U + = (cid:104) + y | U ⊗ ⊗ M | + (cid:105) 3. Probabilistically choose result according to p , save result and evolve accordingly including normalisation ρ n +1 = N U + / − ρ n U † + / − .By using the subspace resulting from the symmetry in the case of many spins with equal coupling strength, many spins canbe simulated efficiently, as this subspace has dimension M + 1 instead of M .The Fisher Information I N = (cid:88) X N p X p X (cid:18) ∂p X ∂δ (cid:19) (63)3 Figure 6. Logarithmic negativity for M spins, each spin coupled with β = 0 . × /π with negligible decay γ = 0 . The left graph showsthe entanglement of one of the auxiliary spins with the remaining M-1 spins and the right graph shows entanglement in an equal bi-partitionof the auxiliary spins. The results are averaged over 2000 runs. was calculated numerically for many different runs evolving ρ following the recipe above to determine p X Evolving ρ / according to the same measurement outcomes as ρ , but with a different evolution parameter δ ± d δ allows to determine (cid:18) ∂p X ∂δ (cid:19) = p X ( δ + d δ ) − p X ( δ − d δ )2d δ (64)for many measurement records. After calculating the Fisher Information for every measurement record, the average and standarddeviation can be obtainedThe accuracy is limited by the Fisher Information due to the Cramer-Rao bound δω N ≥ √ I N . (65)For pure states, the Logarithmic negativity can be simplified to an expression depending on the Schmidt coefficients α i : LN ( | Ψ (cid:105) (cid:104) Ψ | ) = 2 log (cid:32)(cid:88) i α i (cid:33) , (66)which can be calculated for considerably larger systems than the partial trace.In order to obtain some insights into the entanglement buildup and its potential role as a resource in the metrology scheme,we use the logarithmic negativity [35] as a quantifier of the entanglement between one of the auxiliary spins with the remaining M − spins and between equal bi-partitions of the auxiliary spins. While the entanglement between the nuclei builds up to asteady state after a time /γ b , it does not contribute to the quadratic scaling with the number of auxiliary spins as this effectis related to the readout process, nor does it support the N super-Heisenberg scaling. However, destroying the entanglementafter every measurement would lead to lower prefactor in the asymptotic SQL scaling as it inevitably leads to destruction ofinformation. Figure 6 shows this buildup of entanglement of a scale of N opt ≈ (2 k T s /π ) − = 1 /γ bb