Metrology with entangled coherent states - a quantum scaling paradox
MMetrology with entangled coherent states - a quantum scaling paradox
Michael J. W. Hall
Centre for Quantum Computation and Communication Technology (Australian Research Council),Centre for Quantum Dynamics, Griffith University, Brisbane, QLD 4111, Australia
There has been much interest in developing phase estimation schemes which beat the so-calledHeisenberg limit, i.e., for which the phase resolution scales better than 1 /n , where n is a measureof resources such as the average photon number or the number of atomic qubits. In particular, anumber of nonlinear schemes have been proposed for which the resolution appears to scale as 1 /n k or even e − n , based on optimising the quantum Cramer-Rao bound. Such schemes include the useof entangled coherent states. However, it may be shown that the average root mean square errorsof the proposed schemes (averaged over any prior distribution of phase shifts), cannot beat theHeisenberg limit, and that simple estimation schemes based on entangled coherent states cannotscale better than 1 /n / . This paradox is related to the role of ‘bias’ in Cramer-Rao bounds, and isonly partially ameliorated via iterative implementations of the proposed schemes. The results arebased on new information-theoretic bounds for the average information gain and error of any phaseestimation scheme, and generalise to estimates of shifts generated by any operator having discreteeigenvalues. PACS numbers: 42.50.St, 03.67.-a, 06.20.Dk, 42.50.Dv
I. INTRODUCTION
Phase estimation is ubiquitous in metrology, and formsthe underlying principle for the estimation of physicalquantities via interferometry. For example, a physicalvariable of interest, such as temperature or refractive in-dex or pressure or gravitational wave strength, may in-fluence the relative phase between two pathways in anoptical interferometer, and estimation of the change inphase then allows estimation of the physical variable.At the fundamental quantum level, the accurate es-timation of (relative) phase requires being able to dis-tinguish between members of a set of overlapping quan-tum states, where each member has undergone a differentphase shift. For optical phase shifts, the ability to do sovery much depends on photon number properties. For ex-ample, the overlap of two pure states | ψ (cid:105) and e − iNφ | ψ (cid:105) ,having a small relative phase shift φ (cid:28) π , is easily cal-culated to be |(cid:104) ψ | e − iNφ | ψ (cid:105)| = 1 − φ (∆ N ) + O ( φ ) . Hence, a correspondingly large photon number variance,on the order of 1 /φ , is necessary for the overlap to besmall enough to allow an accurate distinction betweenthese states. More generally, the maximum possible res-olution of a given phase estimation scheme will dependon the resources available, such as the average photonnumber per probe state for optical probe states, or thenumber of atomic qubits of the probe state in Ramseyinterferometry.The optimal scaling of the resolution with the re-sources, for a given estimation scheme, is of considerableinterest. A common tool for determining such scalingsis the quantum Cramer-Rao bound [1, 2], which sug-gests, for example, that using the NOON state [ | n (cid:105)| (cid:105) + | (cid:105)| n (cid:105) ] / √ /n [3], while using the n -qubit state ⊗ n | z (cid:105) can achieve a phase accuracy scaling as 2 − n [4].The quantum Cramer-Rao bound similarly suggests thatentangled coherent states can achieve a similar 1 /n scal-ing as for NOON states [5, 6], and surpass NOON statesfor small values of n [5, 7].However, recent results on the accuracy of phase esti-mation schemes, based on quantum information proper-ties of the probe state, raise a scaling paradox [8]. Forexample, they imply that a NOON state cannot achievemore than one bit of phase information per use, indepen-dently of n , and that the corresponding accuracy doesnot approach zero as n is increased. More generally, theresults imply that the accuracy promised by the quantumCramer-Rao bound, for a given probe state, often cannotbe achieved, unless the phase shift is already known toabout that accuracy!The scaling paradox arises essentially because thequantum Cramer-Rao bound has a restricted applica-tion, to regions in which the estimation scheme is lo-cally unbiased. For the above-mentioned phase estima-tion schemes, these regions are comparable in size to theaccuracy promised by the Cramer-Rao bound. This lim-itation can be overcome, to some extent, via consideringiterative implementations of the proposed schemes. Suchimplementations use multicomponent probe states, eachcomponent of which estimates a different bit of the phaseshift. In many cases (but not all), one can recover thepromised scaling of the Cramer-Rao bound in this way,albeit with a large scaling factor in general [8].The new information-theoretic bounds for phase es-timation yield general bounds on phase resolution andinformation gain, applicable to any phase estimationscheme, and take into account any prior informationavailable about the phase shift. They are reviewed below,and compared with the quantum Cramer-Rao bounds forthe case of entangled coherent states in particular. It isshown that using such a state as a probe state cannot a r X i v : . [ qu a n t - ph ] J u l achieve an average resolution that asymptotically scalesbetter than n − / , whereas, in contrast, unentangled co-herent states can achieve a scaling of n − / . This is incontrast to the quantum Cramer-Rao bound scalings of n − and n − / respectively. Moreover, it is shown thatit is not straighforward to achieve the latter n − scalingfor entangled coherent states (and may not even be pos-sible to do so), due to the presence of a strong vacuumbackground contribution to the phase properties of suchstates. The behaviour of the two types of bound for smallaverage photon number n is also examined. II. PHASE ESTIMATION SCHEMES
As shown in Fig. 1, a generic phase estimation schemeinvolves application of a phase shift to a probe state,and subsequent estimation of the phase shift via somemeasurement. Such schemes can be very general, andinclude the case of complex (possibly adaptive) measure-ments made across the components of an entangled mul-ticomponent probe state. The generator G of the phaseshift will be taken throughout this paper to have integereigenvalues, ensuring that the phase-shifted state ρ φ sat-isfies the fundamental phase-shift property ρ φ +2 π = ρ φ .However, many of the results below hold for more generaloperators G [8]. FIG. 1: Generic structure of phase estimation schemes. Aprobe state ρ undergoes a phase shift φ , generated by someoperator G , and a measurement on the probe state is usedto make an estimate, φ est , of φ . The phase shift may arisefrom the probe state passing through a particular environ-ment (e.g., in one arm of an interferometer), or via some ex-ternal modulation (e.g., in a communication scenario). Theprobe state may comprise, for example, a single-mode opti-cal field, a multimode field, an atomic qubit, or several suchqubits. The generator G may be any suitable Hermitian op-erator defined on the Hilbert space of the probe state: forexample N or N for a single-mode field having photon num-ber operator N ; N + N + · · · + N m for a multimode fieldcomprising m single-mode fields; or σ (1) z + σ (2) z + · · · + σ ( n ) z for a probe state comprising n atomic qubits. The measure-ment may comprise, for example, individual measurementson components of the probe state (with some type of aver-aging of the individual outcomes to form an estimate), or acomplex measurement across all components of an entangledprobe state. A central question of interest is how good can a givenestimation scheme be? The answer will in general notonly depend on the probe state, the generator, and the measurement, but also on the measurement of perfor-mance used. Several performance bounds are discussedin this section.
III. PHASE ESTIMATION BOUNDSA. Quantum Cramer-Rao bound
One can first ask how well a given phase estimationscheme performs for a particular phase shift value φ . Anatural measure of the performance for this case is thelocal root mean square error, RMSE φ , defined byRMSE φ := (cid:20)(cid:90) dφ est p ( φ est | φ ) ( φ est − φ ) (cid:21) / , (1)where p ( φ est | φ ) denotes the conditional probability of es-timating φ est for an applied phase shift φ . Note thatthere is an ambiguity in defining the phase reference in-terval over which the integration is performed; however,the bounds given below are independent of the choice ofthis interval [9].The quantum Cramer-Rao bound is valid for the spe-cial case that the estimate is locally unbiased at φ , i.e.,when (cid:104) φ est (cid:105) φ (cid:48) := (cid:90) dφ est p ( φ est | φ (cid:48) ) φ est = φ (cid:48) for all φ (cid:48) in some neighbourhood of φ . In particular, forlocally unbiased estimates one has [1, 2]RMSE φ ≥ (cid:112) F φ ≥ G , (2)where F φ denotes the Fisher information of the estimate,and ∆ G denotes the root mean square spread of the gen-erator for the probe state. The second inequality is some-times referred to as the Helstrom-Holevo bound [10], andis saturated for pure states [2].It is easy to show, for example, that if the probestate comprises a NOON state of a single-mode field,then Eq. (2) reduces to RMSE φ ≥ /n [3]. Note alsothat if the probe state comprises a tensor product of m identical components, with G = G + . . . G m , then(∆ G ) = (∆ G ) + . . . (∆ G m ) = m (∆ G ) , and hencethe righthand term scales as 1 / √ m [2]. B. Local unbiasedness
The requirement of local unbiasedness for the valid-ity of Eq. (2) is surprisingly strong. Suppose, for exam-ple, that one in fact knows beforehand that the phaseshift applied to the probe state is φ . Clearly, there isthen no need for any physical measurement to make aperfect estimate: one simply takes φ est ≡ φ . The lo-cal root mean square error in Eq. (1) then vanishes, i.e.,RMSE φ = 0. At first sight this appears to contradict thequantum Cramer-Rao bound in Eq. (2), since the latterimplies that in the case the probe state is a NOON state,then RMSE φ ≥ /n . There is, of course, no real contra-diction: the ‘perfect estimate’ is not locally unbiased, sothat Eq. (1) does not apply (one has (cid:104) φ est (cid:105) φ = φ , ratherthan φ ). However, this example shows that the restric-tion to locally unbiased estimates can give misleadingindications as to what may be possible. More practicalexamples will be given in later sections.One may remove the requirement for local unbiased-ness via a more general form of the quantum Cramer-Raoinequality due to Helstrom [11]:(RMSE φ ) ≥ [ (cid:104) φ est (cid:105) φ − φ ] + [ ∂ (cid:104) φ est (cid:105) φ /∂φ ] F φ ≥ [ (cid:104) φ est (cid:105) φ − φ ] + [ ∂ (cid:104) φ est (cid:105) φ /∂φ ] G ) , (3)where (cid:104) φ est (cid:105) φ is defined above, and the second inequal-ity follows via the corresponding inequality in Eq. (2).Note this formula implies that Eq. (2) also holds in thecase of estimates with a locally constant bias (i.e., with ∂ (cid:104) φ est (cid:105) φ /∂φ = 1).More significantly, Eq. (3) implies, for example, thatone cannot obtain a 1 /n scaling of the local root meansquare error for NOON states, if the estimate is biasedat φ . Thus, only locally unbiased estimates tend to beconsidered when using Cramer-Rao bounds to comparethe accuracies of various phase estimation schemes. Thisrestriction, however, greatly limits the applicability ofsuch bounds to many of the schemes proposed in theliterature, which are sometimes only locally unbiased fora finite set of phase shift values.Finally, another way remove the requirement for localunbiasedness is to replace the measure of performance bya different quantity, the local precision , defined by [2] P φ := (cid:34)(cid:90) dφ est p ( φ est | φ ) (cid:18) φ est | ∂ (cid:104) φ est (cid:105) φ /∂φ | − φ (cid:19) (cid:35) / . This quantity satisfies the same inequalities as RMSE φ in Eq. (2), but is valid for any estimate, whether biasedor unbiased [2]. In particular, the local precision scales as1 /n for NOON states, for all values of φ [3]. However, theoperational meaning of the local precision as a measureof performance is not clear, due to the nonlinear scalingterm in the denominator. The only exception appears tobe if this term is a constant, k say, over the range of ofphase shifts of interest, as it is then natural to replace theestimate φ est by the rescaled estimate φ (cid:48) est = φ est /k . Butfor this case the local precision of φ est is just the localroot mean square error of φ (cid:48) est , and the bound reduces toa particular case of the general Helstrom bound above. C. Information-theoretic bounds
Recently, new bounds on phase estimation have beenderived, which are valid for both biased and unbiasedestimates, and which allow prior information about thephase shift (such as used to make the ‘perfect estimate’in the above example) to be taken into account. Thefirst bound limits the Shannon mutual information be-tween the phase shift and the estimate, H ( φ est | φ ), i.e.,the amount of information which can be gained per esti-mate about the phase shift, and is given by [8] H ( φ est | φ ) ≤ A G ( ρ ) ≤ H ( G | ρ ) . (4)Here A G ( ρ ) is the increase in von Neumann entropycorresponding to a projective measurement of G on theprobe state, also known as the G -asymmetry of the probestate [12], and H ( G | ρ ) denotes the Shannon entropy ofthe probability distribution of G for the probe state.Note, for example, that estimation using a NOON statein an interferometer, with G = N where N and N are the number operators of the modes, it follows that A G ( ρ ) = H ( G | ρ ) = log 2. Hence no more than onebit of information about the phase shift can be extractedfrom a NOON state, independently of n . A similar resultholds if G is replaces by any (possibly nonlinear) functionof N and N [8]. Note, however, that this one bit canstill be very useful, as part of a more complex ‘bit-by-bit’ phase estimation scheme based on multicomponentprobe states, as will be seen in the next section.One may also derive an information-theoretic boundfor the average estimation error, i.e., for the root meansquare errorRMSE := (cid:20)(cid:90) dφ dφ est p ( φ est , φ ) ( φ est − φ ) (cid:21) / (5)of the estimate, where p ( φ est , φ ) the joint probabilitydistribution for φ est and φ . Note that if ℘ ( φ ) denotesthe prior probability density for the applied phase shift,then one has the relations p ( φ est , φ ) = p ( φ est | φ ) ℘ ( φ ) and(RMSE) = (cid:82) dφ ℘ ( φ ) (RMSE φ ) . The latter relationshows that the average estimation error is a suitable mea-sure of the average performance of the estimate, whichtakes prior information about the phase shift into accountvia ℘ ( φ ).As shown in Ref. [8], Eq. (4) implies the lower boundsRMSE ≥ (2 πe ) − / e H ( φ ) e − A G ( ρ ) ≥ (2 πe ) − / e H ( φ ) e − H ( G | ρ ) , (6)for the root mean square error, which strengthen andgeneralise earlier results in the literature [13, 14]. Here H ( φ ) := − (cid:82) dφ ℘ ( φ ) log ℘ ( φ ) denotes the Shannon en-tropy of the prior distribution, and reduces to log 2 π inthe case of random phase shifts, i.e., when ℘ ( φ ) ≡ / π .The second inequality is saturated for pure probe states.For example, again taking a NOON state with G = N one finds the lower bound RMSE ≥ (8 πe ) − / e H ( φ ) , forall values of n . Hence, no scaling of the root mean squareerror with n is possible in this case — contrary to whatis suggested by the corresponding quantum Cramer-Raobound RMSE φ ≥ /n following from Eq. (2). This ap-parent scaling paradox arises because of the restriction ofthe latter bound to locally unbiased measurements, andis examined more closely in the following sections, andwith particular reference to entangled coherent states inSec. V.Finally, note that, for the ‘perfect estimate’ discussedat the beginning of Sec. III B, one has the prior distribu-tion ℘ ( φ ) = δ ( φ − φ ). Hence H ( φ ) = ∞ , and the lowerbound in Eq. (6) yields RMSE ≥ IV. SCALING PARADOX AND ITERATIVEESTIMATION SCHEMES
The quantum Cramer-Rao bound in Eq. (2) sug-gests that, all else being equal, probe states shouldhave as large a variance of G as possible, whereas theinformation-theoretic bound in Eq. (6) suggests that theyshould have as large an entropy of G as possible. Thesecan lead to conflicting results in various cases, as seen inthe NOON state examples above, which have large vari-ance but small entropy. More generally, for generatorswith integer eigenvalues, if follows from Eq. (6) that [8] RM SE ≥ (cid:112) πe [(∆ G ) + 1 / , which scales as 1 / ∆ G for ∆ G (cid:29)
1. While this is con-sistent with the quantum Cramer-Rao bound in Eq. (2),the bound in Eq. (6) is typically much stronger.For example, for a single-mode field with average pho-ton number (cid:104) N (cid:105) , and a nonlinear phase shift generator G = N k , one can easily find states with the scalings∆ G ∼ (cid:104) N k (cid:105) / ∼ (cid:104) N (cid:105) k , suggesting via the quantum Cramer-Rao bound that alocal root mean square error scaling as 1 / (cid:104) N (cid:105) k is possible[7, 15] — greatly improving on the Heisenberg scalinglimit 1 / (cid:104) N (cid:105) for k >
1. However, noting that the entropyof N and N k are identical for any state, the information-theoretic bound may be used to show that the averageestimation error is always bounded by [8]RMSE ≥ e H ( φ ) √ πe (cid:104) N + 1 (cid:105) . (7)Hence, no better than Heisenberg scaling is possible forthe average performance of the estimate, in such nonlin-ear scenarios. The paradoxical differences in scalings between the twobounds arise from the restriction of the quantum Cramer-Rao bound to locally unbiased measurements. In partic-ular, many of the proposed phase estimation schemes inthe literature are only locally unbiased, or approximatelyso, for a small range of phase shift values. Indeed, thisrange is often of the same order of magnitude as thelower bound in Eq. (2), as is discussed in more detailin Refs. [8] and [9]. In such cases, the quantum Cramer-Rao bound can only be applied if the phase shift is knownto fall within this narrow range, i.e., if the phase is al-ready known to an accuracy comparable to that promisedby the bound! Thus, while these schemes appear to of-fer increased phase resolution with increasing resources n , they can only achieve this if the phase shift beforemeasurement is correspondingly known more and moreaccurately with increasing n .In contrast, the information-theoretic bound in Eq. (6)explicitly takes prior information about the phase shiftbefore measurement into account, via the term e H ( φ ) .For example, if the phase shift is known to be randomlytaken from an interval of width W , then this term gives ascaling factor of W for the RMSE. Thus, the bound ex-plicitly separates out the scaling contributions from thechoice of probe state on the one hand, and prior infor-mation about the phase shift on the other. FIG. 2: Iterative phase estimation schemes. In this concep-tual diagram, each circular region represents a component of amulticomponent probe state. The components correspondingto the five largest regions, on the left of the figure, are used toestimate the first bit of φ/ (2 π ); the next five components areused to estimate the second bit, and so forth. For example,each size region may correspond to a different NOON state,with n = 1 , , , ,
16 from left to right, or to a different en-tangled coherent state, with exponentially increasing α fromleft to right. Finally, it is of interest to note that the scaling promisesuggested by quantum Cramer-Rao bounds can some-times (but not always) be achieved via the implementa-tion of iterative versions of the proposed schemes, basedon multicomponent probe states. The idea is that dif-ferent components, each having a different number ofresource n , are used to estimate successive bits of thephase shift, as depicted in Fig. 2. When the total num-ber of resources required for the probe state are addedup (e.g., the total photon number or the total number ofatomic qubits), and compared with the resolution of theestimate, one can often obtain the scaling suggested bythe quantum Cramer-Rao bound for a single-componentprobe state, although generally with a larger scaling con-stant. Several examples have been discussed elsewhere[8], including a case where a 2 − n scaling, suggested viathe quantum Cramer-Rao bound [4], cannot be achievedeven with an iterative implementation. V. ENTANGLED COHERENT STATES
Entangled coherent states (ECS) may be defined invarious ways (see Ref. [16] for a recent review of suchstates), but for our purposes will be taken to be super-positions of tensor products of Glauber coherent states.In particular, consider an ECS of the form | ψ α (cid:105) = 1 (cid:112) e −| α | ) ( | α (cid:105)| (cid:105) + | (cid:105)| α (cid:105) ) . (8)Such two-mode states have been shown to share precisely1 bit of entanglement [17], and maximally violate a Bellinequality of the Clauser-Horne-Shimony-Holt type [18].The above ECS is seen to be similar in form to NOONstates, with number states being replaced by coherentstates, and indeed it has been shown [5–7] that for large | α | the quantum Cramer-Rao bound scales in the sameway with the total average photon number, n := (cid:104) N + N (cid:105) , for both types of state [5–7]. Further, for small values of n , ECS outperform NOON states, as quantified by theCramer-Rao bound, both for linear and nonlinear phaseshifts [5, 7]. This suggests that ECS may be valuablestates for quantum metrology.However, in light of the preceding sections, it is seenthat it is important to take into account that the quan-tum Cramer-Rao bound is restricted in operational sig-nificance, to the case of estimation schemes that are (atleast approximately) locally unbiased over the phase shiftrange of interest. Hence, the performance of the averageestimation error for schemes based on entangled coherentstates is carefully assessed below, using the more generalinformation-theoretic bound in Eq. (6). A. Large average photon number
Consider an inteferometric setup in which the secondcomponent of the ECS | ψ α (cid:105) is subjected to a phase shift φ [5]. For a linear phase shift the generator is then G = N .To evaluate the bound for the RMSE in Eq. (6), one thenneeds to determine the entropy of N for the probe state.Taking α > p m := e − α α m /m ! denote the photon number probability dis-tribution for a single-mode coherent state. It is straight-forward to calculate from Eq. (8) that the probabilitydistribution of N for the ECS | ψ α (cid:105) is then p m = ˜ p m + δ m (1 + 2˜ p )2(1 + e − α ) ≈
12 (˜ p m + δ m )where the approximation is valid for sufficiently large α .Thus, the distribution of N is well approximated byan equal mixture of the Poisson distribution { ˜ p m } withthe trivial distribution { δ m } – where these distributionsare effectively nonoverlapping since ˜ p (cid:28) α . Hence, as is easily verified, the entropy of this dis-tribution is approximately equal to the average of theentropies of the Poisson and trivial distributions, pluslog 2. The above approximation for p m also implies that n = 2 (cid:104) N (cid:105) ≈ α . Finally, since the Poisson distribution { ˜ p m } is well approximated by a Gaussian distribution ofthe same mean ( α ) and variance ( α ) for large α , wherethe entropy of a Gaussian of variance v is known to be(1 /
2) log 2 πev , it follows that H ( G | ψ α ) ≈
14 log 2 πeα + log 2 ≈
14 log 2 πen + log 2 . Substitution into Eq. (6) gives the approximate lowerboundRMSE(
ECS ) (cid:38) e H ( φ ) πe ) / n / ≈ . e H ( φ ) n / (9)for phase estimation based on an ECS, in the limit oflarge n .This scaling as 1 /n / strongly contrasts with the 1 /n scaling of the quantum Cramer-Rao bound for ECS [5,6]. Indeed, as seen in Fig. 3, the scaling in Eq. (9) issurpassed by the corresponding scaling for unentangled coherent probe states of the form | α (cid:105)| α (cid:105) , for which onefinds the approximate lower boundRMSE( COH ) (cid:38) e H ( φ ) πe √ n ≈ . e H ( φ ) n / (10)in the limit of large n , via a similar calculation (a nu-merical check shows that both approximate bounds areaccurate for n (cid:38) N OON ) ≥ e H ( φ ) √ πe ≈ . e H ( φ ) (11)for any (integer) value of n . FIG. 3: Dependence of quantum Cramer-Rao andinformation-theoretic bounds, on the total average photonnumber n , for entangled coherent states and factorisable co-herent states. The dotted curves show the quantum Cramer-Rao bound for the local performance, RMSE φ in Eq. (2), asa function of n , with the upper (green) dotted curve corre-sponding to probe states of the form | α (cid:105)| α (cid:105) , and the lower(blue) dotted curve to ECS probe states as per Eq. (8).Thus the Cramer-Rao bound suggests that ECS have a bet-ter local performance. However, the solid curves show theinformation-theoretic bound for the average performance,RMSE in Eq. (6), with the upper (blue) solid curve corre-sponding to ECS probe states and the lower (green) solidcurve corresponding to factorisable coherent states (for thechoice of a random prior distribution, ℘ ( φ ) = 1 / π ). Hence,it is seen that the bound on average performance is very sim-ilar for both types of state, for small values of n ( n < n in-creases – as expected from the asymptotic scalings in Eqs. (9)and (10). B. Iterative implementations
It is known that, despite Eq. (11), iterative implemen-tations based on NOON-state components can recoverthe 1 /n scaling suggested by the quantum Cramer-Raobound, albeit with a larger scaling constant [8]. It isalso shown below that iterative implementations basedon factorisable coherent states can achieve the 1 / √ n scaling of the corresponding quantum Cramer-Rao andinformation-theoretic bounds. This is because the canon-ical phase distribution for the second mode of the state | α (cid:105)| αe − iφ (cid:105) (this distribution is expected to be optimalin the case of an unknown phase shift on such a state[14, 19]) is approximately Gaussian, with mean φ andvariance 1 / α = 1 / n [19], and hence can resolve phaseto an accuracy on the order of 1 / √ n , where typically M repetitions will be required to do so with near certainty,for M typically in the range 4-8 [8]. However, as will beseen, it is not clear whether a similar conclusion holds foriterative implementations based on ECS.In particular, consider first a multicomponent probestate ρ comprising: M copies of such factorisable stateswith total average photon number n = 1, to estimatethe first bit of φ/ π ; M copies with n = 2 to esti- mate the second bit; etc., culminating with M copieswith n m = 2 m − to estimate the m th bit (see Fig. 2).The total average photon number of this multicomponentprobe state is then n = M (cid:80) m − j =0 j = M (4 m − / G is the sum of the photon num-ber operators over all components of ρ . Since each com-ponent has an approximately Gaussian photon numberdistribution (with variances n / , n / , . . . , n m / H ( G | ρ ) can beapproximated as H ( G | ρ ) ≈
12 log [2 πeM ( n + · · · + n m ) / πen ] . Substitution into the information-theoretic bound inEq. (6) then gives precisely the same bound as in Eq. (10)above. Since the iterative scheme has, by construction,a resolution of ≈ (2 π ) / m +1 ≈ π (cid:112) M/ n , it followsthat the scaling of the corresponding Cramer-Rao bound( ≈ / √ n from Eq. (2) can be achieved by such a scheme,albeit with a larger scaling factor.However, it is not at all clear that on can similarlyachieve a 1 /n scaling for entangled coherent states, de-spite the corresponding quantum Cramer-Rao boundhaving such a scaling [5, 6]. This is essentially becausethe first part of the superposition in Eq. (8) does notsee any phase shift, due to the vacuum contribution ofthe second mode. This vacuum contribution adds signifi-cant phase noise. For example, noting that the canonicalphase distribution of a single mode field | ψ (cid:105) is given by p C ( θ ) = (1 / π ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) m (cid:104) m | ψ (cid:105) e − imθ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , it is straightforward to calculate the joint canonicalphase distribution p C ( θ , θ ) for the phase-shifted ECS e − iN φ | ψ α (cid:105) . The relative phase distribution p C ( φ r ), for φ r := θ − θ , is then found by a suitable integration, as p C ( φ r ) = 1 + e − α [1 − cos( φ r − φ )] cos[ α sin( φ r − φ )]2 π (1 + e − α )For large α this is approximated in the neighbourhoodof the phase shift φ by a mixture of a uniform distri-bution 1 / π with a Gaussian distribution of variance1 /α = 2 /n . The presence of this uniform backgrounddistribution arises from the vacuum component of theECS, and means that the above iterative approach can-not be straightforwardly applied in analogous manner di-rectly analogous to the case of factorisable coherent states(where there was no such background term).Hence, unlike NOON states, it is perhaps not possibleto achieve a 1 /n scaling of phase resolution for entan-gled coherent states. However, further investigation isrequired in this regard. C. Small average photon number
The quantum Cramer-Rao bounds and information-theoretic bounds, for RMSE φ and RMSE respectively,are plotted in Fig. 3 for relatively small values of thetotal average photon number n , for both entangled co-herent states and factorisable coherent states. It is seenthat the bound on the average performance for an ECSprobe state (upper blue solid curve) always falls abovethat for the factorisable case. Hence, as for the caseof large n discussed above, it appears that for ECS togain the advantage suggested by the Cramer-Rao bounds(dotted curves), it is necessary to utilise them in more so-phisticated phase estimation schemes (such as iterativeschemes), using multicomponent probe states. VI. CONCLUSIONS
The main conclusion to be drawn from the above isthat the quantum Cramer-Rao bound for RMSE φ inEq. (2) only has direct operational significance for esti-mates that are locally unbiased in some neighbourhood of φ . Hence caution must be taken when using this bound,particularly if the range over which the estimate is unbi-ased is unknown. Further illustration and discussion ofthis point is given elsewhere [8, 9].The information-theoretic bounds for mutual informa-tion and RMSE in Eqs. (4) and (6) are, in contrast,completely general. They show that the relevant quan-tity for maximising average performance is not the vari-ance of the generator, but its entropy (or, more generally,its G -asymmetry). They may also be used to show, in some cases, that iterative implementations of proposedschemes can actually achieve the promise suggested bythe corresponding Cramer-Rao bound — but typicallya larger scaling constant is required. The information-theoretic bounds have been used elsewhere to find reso-lution bounds for general optical probe states and atomicqubit states, and to evaluate various nonlinear schemesproposed in the literature [8].Application of the new bounds to ECS probe statesraises a question as to their average performance in com-parison to unentangled coherent states. As discussed inSec. V, a demonstration of superior performance will re-quire more sophisticated estimation schemes than thosebased on using single-component probe states, and pos-sibly even than those based on simple iterative imple-mentations using multicomponent probe states. Thisis an important challenge for future research in ECS-metrology.Finally, it should be noted that the effects of noise andloss have not been considered here. Entangled coher-ent states appear to have greater resilience than NOONstates in this regard [5, 7], which may give them an ad-vantage in realistic scenarios — particularly if the chal-lenge in the above paragraph can be met. Acknowledgments
I thank H. Wiseman, A. Lund, D. Berry, and M. Zwierzfor helpful discussions. This research was supported bythe ARC Centre of Excellence CE110001027. [1] C. W. Helstrom
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