Quantum-limited metrology with product states
Sergio Boixo, Animesh Datta, Steven T. Flammia, Anil Shaji, Emilio Bagan, Carlton M. Caves
aa r X i v : . [ qu a n t - ph ] O c t Quantum-limited metrology with product states
Sergio Boixo,
1, 2
Animesh Datta, Steven T. Flammia,
1, 3, ∗ Anil Shaji, Emilio Bagan,
4, 1 and Carlton M. Caves
1, 5 Department of Physics and Astronomy, MSC07-4220,University of New Mexico, Albuquerque, New Mexico 87131-0001 Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada Grup de F´ısica Te`orica, Universitat Aut`onoma de Barcelona, 08193 Bellaterra (Barcelona), Spain Department of Physics, University of Queensland, Brisbane, Queensland 4072, Australia (Dated: May 29, 2018)We study the performance of initial product states of n -body systems in generalized quantum metrologyprotocols that involve estimating an unknown coupling constant in a nonlinear k -body ( k ≪ n ) Hamiltonian.We obtain the theoretical lower bound on the uncertainty in the estimate of the parameter. For arbitrary initialstates, the lower bound scales as /n k , and for initial product states, it scales as /n k − / . We show thatthe latter scaling can be achieved using simple, separable measurements. We analyze in detail the case of aquadratic Hamiltonian ( k = 2 ), implementable with Bose-Einstein condensates. We formulate a simple model,based on the evolution of angular-momentum coherent states, which explains the O ( n − / ) scaling for k = 2 ;the model shows that the entanglement generated by the quadratic Hamiltonian does not play a role in theenhanced sensitivity scaling. We show that phase decoherence does not affect the O ( n − / ) sensitivity scalingfor initial product states. I. INTRODUCTION
Parameter estimation is a fundamental physical task. It typ-ically involves picking a physical system whose state, throughits evolution, depends on the value of the parameter. In mostquantum metrology schemes [1–18], this system, which wecall the “probe,” is a composite made up of n elementaryquantum constituents. The influence of the unknown parame-ter γ on the probe is described by an n -body Hamiltonian H γ = γH , (1.1)in which γ appears as a coupling constant and H is a dimen-sionless coupling Hamiltonian (we use units with ~ = 1 , so γ has units of frequency). The precision with which γ canbe determined depends on the initial state of the probe, thenature of the parameter-dependent Hamiltonian, and the mea-surements that are performed on the probe to extract informa-tion about the parameter. Other factors, such as decoherencein the probe [3, 19–21], also have an effect on the achievablesensitivity.The appropriate measure of the precision with which γ canbe determined is the units-corrected mean-square deviation ofthe estimate γ est from the true value γ [22, 23]: δγ = (cid:28)(cid:18) γ est | d h γ est i /dγ | − γ (cid:19) (cid:29) / . (1.2)This estimator uncertainty is inversely proportional to the dis-placement in Hilbert space of the state of the probe corre-sponding to small changes in γ . The fundamental limit onthe precision of parameter estimation, δγ ≥ √ ν t ∆ H , (1.3) ∗ Electronic address: [email protected] called the quantum Cram´er-Rao bound (QCRB) [22–25], isan expression of the maximum amount the state can changeunder the evolution due to H γ . In Eq. (1.3), ν is the numberof trials with independent, identical probes, t is the time forwhich each probe evolves under H γ , and ∆ H = ( h H i −h H i ) / denotes the uncertainty in H for each probe, whichdoes not change under the evolution due to H γ . The QCRBis independent of the choice of estimator and is achievableasymptotically in the limit of a large number of trials, pro-vided the initial state of the probe is a pure state. If the initialstate of the probe is not pure or if nonunitary processes de-stroy the purity of the initial state, the bound (1.3) is not tight,and a stricter version of the QCRB, given in [22–25], can beused, but we have no need for this stricter bound in this paper.The uncertainty in H is bounded above by ∆ H ≤ Λ max − Λ min , (1.4)where Λ max and Λ min are the maximum and minimum eigen-values of H . The difference between the largest and leasteigenvalues, denoted by k H k = Λ max − Λ min , (1.5)is an operator semi-norm of H . Using this semi-norm, we canwrite a state-independent version of the QCRB [22]: δγ ≥ √ ν t k H k . (1.6)This bound can be achieved by using the initial state ( | Λ max i + | Λ min i ) / √ , which evolves after time t to e − iH γ t √ | Λ max i + | Λ min i )= 1 √ e − i Λ max γt | Λ max i + e − i Λ min γt | Λ min i ) . (1.7)Measurement in a basis that includes the states |±i =( | Λ max i ± | Λ min i ) / √ yields outcomes ± with probabilities p + = cos ( k H k γt/ and p − = sin ( k H k γt/ . Letting σ denote an observable with the outcome values ± , we have h σ i = cos( k H k γt ) and ∆ σ = | sin( k H k γt ) | . An appropriateestimator is defined in terms of the mean of the outcomes, ν ν X k =1 σ k ≡ cos( k H k γ est t )= cos( k H k γt ) − k H k t sin( k H k γt )( γ est − γ ) , (1.8)where the second expression is the linear approximation to therelation between γ est and the mean of the outcomes, holdingstatistically in the limit of a large number of trials, specifically, ν ≫ tan ( k H k γt ) . Now it is easy to see that h γ est i = γ and δγ = (cid:10) ( γ est − γ ) (cid:11) / = ∆ σ √ ν k H k t | sin( k H k γt ) | = 1 √ ν t k H k , (1.9)showing that the bound (1.6) can be achieved and thus makingit the fundamental limit to quantum metrology.The √ ν factor in Eqs. (1.3) and (1.6) is the well-understoodstatistical improvement available from averaging over manyprobes. In the remainder of the paper, we sometimes do notinclude this factor explicitly, referring to the remaining termon right-hand side of Eq. (1.3) or Eq. (1.6) as the QCRB, al-ways remembering, of course, that generally the bound canonly be achieved asymptotically in the limit of large ν .It is clear from Eqs. (1.3) and (1.6) that strategies for im-proving the precision in estimating a parameter include chang-ing the initial state of the probe, the coupling Hamiltonian H ,or both. The case that has received most attention in the pastis the one in which the probe constituents are coupled inde-pendently to the parameter: H = n X j =1 h j . (1.10)Here h j is a single-body operator acting on the j th constituentof the probe (hence, all these operators commute). In thiscase, the QCRB (1.6) scales like δγ = O (1 /n ) , a scalingknown as the Heisenberg limit [1, 10]. This scaling outper-forms that attainable with classical statistics, which goes as δγ = O (1 / √ n ) , a scaling known as the standard quantumlimit or, sometimes, as the shot-noise limit . The / √ n scalingis the optimum sensitivity allowed by the QCRB (1.3) whenthe coupling Hamiltonian has the form (1.10) and the opti-mization of the probe state is restricted to product states.Achieving the quantum-enhanced Heisenberg sensitivitywith the linear coupling Hamiltonian (1.10) requires the probeto be initialized in a highly entangled state, which is aformidable challenge using current technology [1]. Much progress has been made in preparing such states toward pre-cisely this end, e.g., by using measurement-induced squeez-ing [26], but it is still currently infeasible to use Heisenberg-limited experiments to outperform the best measurements op-erating at the standard quantum limit, although Heisenberg-like scalings have been achieved in related serial protocolsthat involve repeated interactions with a single constituentand thus do not involve entanglement [27]. Practical propos-als [28] for reaching the Heisenberg limit have also been madein the context of measurements on a harmonic oscillator pre-pared in a state that displays sub-Planck phase-space struc-ture [29].More general families of Hamiltonians, containing nonlin-ear couplings of the constituents to the parameter, in contrastto the independent, linear coupling of Eq. (1.10), can performbetter than the /n scaling of the Heisenberg limit, while re-specting the QCRB [6, 9, 11, 12, 14, 15, 18]. In particular,when the coupling Hamiltonian has symmetric k -body termsin it, it is possible to achieve the scaling δγ = O ( n − k ) , asshown in [12]. This O ( n − k ) scaling requires entangled inputstates, but we will show here that the optimal scaling with ini-tial product states is O ( n − k +1 / ) . Thus scalings better thanthe /n Heisenberg scaling are possible for k ≥ , even withinitial product states for the probe, a result found for k = 2 in [6, 9, 18].In this paper we investigate the theoretical and practicalbounds on precision in the generalized quantum metrologyscheme introduced in [12], which allows for nonlinear cou-plings of the probe constituents to the parameter. In particular,we study the nonlinear coupling Hamiltonian [12] H = (cid:18) n X j =1 h j (cid:19) k = n X a ,...,a k h a · · · h a k . (1.11)For simplicity, we assume that the probe constituents are iden-tical and that the single-body operators h j are the same for allthe constituents. We review in Sec. II the optimal precisionthat can be achieved when the probe can be prepared in anyinitial state, particularly, entangled states of the probe con-stituents, but our emphasis in this paper is on the precisionthat can be attained when the initial state is restricted to be aproduct state.In Sec. III we show that the optimal precision with product-state inputs scales as O ( n − k +1 / ) , and we find the corre-sponding optimal input state for the probe constituents. Thesensitivity that can be achieved in practice depends, as men-tioned above, on the measurements that are performed on theprobe to extract information about γ . By analyzing the short-time limit, we show in Sec. IV that when the probe is ini-tialized in a product state, simple, separable measurements onthe probe constituents can achieve the optimal sensitivity. Theconclusion, reached in [6, 9, 18] for specific cases, is that scal-ings better than the standard quantum limit—indeed, betterthan the Heisenberg limit—can be had without the need to in-vest in the generation of fragile initial entangled states. Ourscheme thus circumvents a major bottleneck in attaining, inpractice, scalings superior to / √ n .In Sec. V, we analyze in detail a quadratic coupling Hamil-tonian ( k = 2 ) for effective qubits, with h j = Z j / , where Z j is the Pauli z operator for the j th qubit. This case can be im-plemented in Bose-Einstein condensates [15, 18], as was sug-gested in [12], and in fermionic atoms in optical lattices [15].We show that the optimal sensitivity for input product states,scaling as O ( n − / ) , can be achieved by using the optimalproduct-state input found in Sec. III and making separablemeasurements of equatorial components of the total angularmomentum of the effective qubits. Moreover, we show that,as was found independently by [18], the O ( n − / ) scaling isattainable with these measurements starting with almost anystate of the constituents except the equatorial states, whichwere the subject of the analysis in [15]. We formulate a sim-ple model, based on the evolution of angular-momentum co-herent states, that explains the origin of the O ( n − / ) scaling.The model indicates that the entanglement generated by thequadratic Hamiltonian does not play a role in the enhancedsensitivity, and it suggests that, unlike protocols based on theuse of entangled inputs, the product-state scheme should notbe extremely sensitive to decoherence. We verify this sugges-tion by a brief analysis of the effect of phase decoherence inSec. V B.Section VI concludes with a brief summary of our results,including an extension of the coherent-state model to arbitrary k , and a discussion of our perception of the field of quantum-enhanced metrology. II. GENERALIZED QUANTUM METROLOGY AND THEQUANTUM CRAM ´ER-RAO BOUND
Attaining the QCRB (1.6) requires using an appropriate ini-tial state and making appropriate measurements to extract theinformation about γ , but changing the optimal scaling with n requires changing the dependence of the coupling Hamil-tonian H on n . This can be done by replacing the linearcoupling Hamiltonian (1.10), which has just n terms, withthe nonlinear Hamiltonian (1.11). The coupling Hamilto-nian (1.11) describes a system with symmetric k -body cou-plings, including self-interactions, and it has n k terms. Forexample, if the constituents are spin- particles and the oper-ator h j is the z -component of the the j th particle’s spin, then H describes a coupling of the parameter to the k th power ofthe z -component of the total angular momentum.The eigenvectors of H are products of the eigenvectors ofthe h j s. The eigenvectors can be labeled by a vector of single-body eigenvalues, | λ i ≡ | λ , . . . , λ n i . The correspondingeigenvalues of h are given by the polynomial Π k ( λ ) = h λ | H | λ i = (cid:18) n X j =1 λ j (cid:19) k = n X a ,...,a k λ a · · · λ a k , (2.1)which is symmetric under permutation of its arguments andis known in the mathematical literature as the k th-degree ele-mentary symmetric polynomial (on n variables).To calculate the QCRB (1.6) for the k -body couplingHamiltonian (1.11), we have to calculate the maximum andminimum eigenvalues of H in terms of the eigenvalues ofthe single-body operators h j , the total number of constituents n , and the degree of the coupling k . Let λ max and λ min bethe largest and smallest eigenvalues of h j . We consider fourcases.1. k odd. The largest (smallest) eigenvalue of H is Λ max = ( nλ max ) k [ Λ min = ( nλ min ) k ], correspond-ing to the eigenvector | Λ max i = | λ max , . . . , λ max i ( | Λ min i = | λ min , . . . , λ min i ).2. k even, λ min ≥ . The same conclusions apply as inCase 1.3. k even, λ max ≤ . The same conclusions apply asin Case 2, except that the roles of λ max and λ min arereversed: the largest (smallest) eigenvalue, Λ max =( nλ min ) k [ Λ min = ( nλ max ) k ], corresponds to theeigenvector that has every constituent in the state | λ min i ( | λ max i ).4. k even, λ min < < λ max . Since all the eigenvalues of H are nonnegative, the maximum eigenvalue is Λ max =( n | λ | max ) k , where | λ | max ≡ max {| λ max | , | λ min |} ,corresponding to all the constituents being in either | λ max i or | λ min i . The minimum eigenvalue comesfrom the string λ , perhaps containing all eigenvalues,that makes Π k ( λ ) as close to zero as possible.In Cases 1–3, the QCRB (1.6) takes the form δγ ≥ tn k | λ k max − λ k min | , (2.2)displaying the O ( n − k ) scaling found in [12].Case 4 requires further discussion regarding Λ min . We canbound Λ min from above by considering strings λ that containonly λ max and λ min . If λ max appears a fraction p of the time,the corresponding eigenvalue is Π k ( λ ) = [ npλ max + n (1 − p ) λ min ] k . (2.3)This eigenvalue can be minimized by making pλ max + (1 − p ) λ min as close to zero as possible, i.e., by choosing np = (cid:24) n | λ min |k h k (cid:23) , (2.4)where ⌈ x ⌋ denotes the nearest integer to x and k h k = λ max − λ min is the semi-norm of the single-particle operators. Theresulting eigenvalue can be written as Π k ( λ ) = ( δ k h k ) k ,where δ ≤ / is the magnitude of the difference between n | λ min | / k h k and the integer closest to it. The minimumeigenvalue can be written as Λ min = [( δ − ǫ ) k h k ] k , (2.5)where ǫ ( ≤ ǫ ≤ / ) accounts for the fact that stringscontaining other eigenvalues of the h j s can generally make Λ min smaller. If the constituents are qubits, ǫ = 0 . Since k h k / | λ | max ≤ , we have Λ min Λ max = (cid:18) δ − ǫn k h k| λ | max (cid:19) k ≤ n − k . (2.6)The upshot is that for even k and λ min < < λ max , theQCRB (1.6) is given by δγ ≥ tn k | λ | k max − Λ min / Λ max . (2.7)Thus the symmetric k -body coupling leads to a QCRB scalingas O ( n − k ) in all four cases.A closely related k -body coupling Hamiltonian is the sameas Eq. (1.11), except that the self-interaction terms are omit-ted, which might be more appropriate in some physical situ-ations. We analyze this alternative k -body coupling Hamilto-nian in App. A and show that it also leads to a O ( n − k ) QCRBscaling when arbitrary input states are allowed.Luis and collaborators [6, 9] showed that O ( n − ) scalingscan be achieved in principle using a Kerr-type optical non-linearity, and Luis generalized these results to optical non-linearities of arbitrary order in [14]. Reference [17] pro-poses a method for synthesizing a quadratic ( k = 2 ) Hamilto-nian from a linear Hamiltonian by passing a light beam twicethrough an atomic medium and finds a O ( n − ) scaling for thismethod. III. ATTAINABLE PRECISION WITH PURE PRODUCTSTATESA. General bound
The QCRB (1.6) gives the best possible measurement pre-cision, but can only be achieved for an optimal probe initialstate, i.e., one of the form ( | Λ max i + e iφ | Λ min i ) / √ , whichthe results of Sec. II show is typically highly entangled. In thissection we obtain lower bounds on δγ in the situation wherethe initial state is a pure product state, | Ψ i = | ψ i ⊗ · · · ⊗ | ψ n i ; (3.1)for this purpose, we start from the state-dependentQCRB (1.3). Since all the one-body operators h j in the cou-pling Hamiltonian are assumed to be identical, it is reason-able to expect that the optimal initial product state will haveall constituents in the same state, but we do not assume thisat the outset, instead allowing its moral equivalent to emergefrom the analysis.The trick to evaluating ∆ H is to partition the unrestrictedsum in Eq. (1.11), in which terms in the sum contain differentnumbers of duplicate factors, into sums such that each termhas the same sort of duplicate factors. Thus we write H = X ( a ,...,a k ) h a · · · h a k + (cid:18) k (cid:19) X ( a ,...,a k − ) h a · · · h a k − h a k − + · · · , (3.2)where a summing range with parentheses, ( a , . . . , a l ) , de-notes a sum over all l -tuples that have no two elements equal.The two sums in Eq. (3.2) are the leading- and subleading-order terms in an expansion in which successive sums have fewer terms. The first sum in Eq. (3.2), in which the termshave no duplicate factors, has n ! / ( n − k )! = O ( n k ) terms,and the second sum, in which one factor is duplicated in eachterm, has n ! / ( n − k − O ( n k − ) terms. The binomialcoefficient multiplying the second sum accounts for the num-ber of ways of choosing the factor that is duplicated. The nextsums in the expansion, involving terms with factors h j and h j h l , have n ! / ( n − k − O ( n k − ) terms. These expan-sions require that n ≥ k , which we assume henceforth, andthe scalings we identify further require that n ≫ k .Given the expansion (3.2), the expectation value of H hasthe form h H i = X ( a ,...,a k ) h h a i · · · h h a k i + (cid:18) k (cid:19) X ( a ,...,a k − ) h h a i · · · (cid:10) h a k − (cid:11) D h a k − E + O ( n k − ) . (3.3)The expression for (cid:10) H (cid:11) follows by replacing k with k : (cid:10) H (cid:11) = X ( a ,...,a k ) h h a i · · · h h a k i + (cid:18) k (cid:19) X ( a ,...,a k − ) h h a i · · · (cid:10) h a k − (cid:11) D h a k − E + O ( n k − ) . (3.4)The rest of the analysis is based on an artful switching be-tween restricted and unrestricted sums. By changing the ini-tial sum in Eq. (3.3) to an unrestricted sum, we can rewrite h H i to the required order as h H i = X a ,...,a k h h a i · · · h h a k i + (cid:18) k (cid:19) X ( a ,...,a k − ) h h a i · · · (cid:10) h a k − (cid:11) ∆ h a k − + O ( n k − ) . (3.5)Squaring this expression and changing the unrestricted sumsback to restricted ones, again keeping only the leading- andsubleading-order terms, gives h H i = X ( a ,...,a k ) h h a i · · · h h a k i + (cid:18) k (cid:19) X ( a ,...,a k − ) h h a i · · · (cid:10) h a k − (cid:11) (cid:10) h a k − (cid:11) + 2 (cid:18) k (cid:19) X ( a ,...,a k − ) h h a i · · · (cid:10) h a k − (cid:11) ∆ h a k − + O ( n k − ) . (3.6)We can now find (∆ H ) by subtracting Eq. (3.6) fromEq. (3.4): (∆ H ) = k X ( a ,...,a k − ) h h a i · · · (cid:10) h a k − (cid:11) ∆ h a k − + O ( n k − )= k (cid:18) n X j =1 h h j i (cid:19) k − (cid:18) n X j =1 ∆ h j (cid:19) + O ( n k − ) . (3.7)In the final form, we take advantage of the fact that in the nowleading-order sum, we can convert the restricted sum to anunrestricted one.To make the QCRB (1.3) as small as possible, we need tomaximize the variance (∆ H ) of Eq. (3.7). We can imme-diately see that for fixed expectation values h h j i , we shouldmaximize the variances ∆ h j , and this is done by using foreach constituent a state that lies in the subspace spanned by | λ max i and | λ min i . Letting p j be the probability associatedwith | λ max i for the j th constituent, we have x j ≡ h h j i = p j λ max + (1 − p j ) λ min = λ min + p j k h k , ∆ h j = p j λ + (1 − p j ) λ − x j = k h k p j (1 − p j )= ( λ max − x j )( x j − λ min ) . (3.8)Thus we should maximize (∆ H ) = k (cid:18) n X j =1 x j (cid:19) k − n X j =1 ( λ max − x j )( x j − λ min ) (3.9)within the domain defined by λ min ≤ x j ≤ λ max , j =1 , . . . , n .Discarding potential extrema of (∆ H ) given by P j x j , since these either are minima or lie outside the rel-evant domain, we find that the conditions for extrema of (∆ H ) imply immediately that x j = x (and thus p j = p )for j = 1 , . . . , n . Thus the optimal states in the initial productstate (3.1) have the form | ψ j i = √ p | λ max i + e iφ j p − p | λ min i . (3.10)The only possible difference between the states for differ-ent constituents is in the relative phase between | λ max i and | λ min i .Since the optimal constituent states live and evolve in a two-dimensional subspace, we can regard the constituents effec-tively as qubits, with standard basis states | i = | λ max i and | i = | λ min i , serving as the basis for constructing Pauli oper-ators X , Y , and Z . Restricted to this subspace, the operator h takes the form h = λ max | ih | + λ min | ih | = ¯ λ + k h k Z/ , (3.11)where ¯ λ ≡ ( λ max + λ min ) / is the arithmetic mean of thelargest and smallest eigenvalues of h . In the analyses in Secs. IV and V, we assume that all theconstituents have zero relative phase ( φ j = 0 ), giving an ini-tial state | Ψ β i = | ψ β i ⊗ n , where | ψ β i = e − iβY/ | i = cos( β/ | i + sin( β/ | i ,p = cos ( β/
2) = (1 + cos β ) / . (3.12)Here we describe the one-body state | ψ β i in terms of the ro-tation angle β about the y axis that produces it from | i . Thecorresponding initial density operator is ρ β = | Ψ β ih Ψ β | = n O j =1
12 ( j + X j sin β + Z j cos β ) . (3.13)The variance of H now takes the simple form (∆ H ) = k n k − h h i k − (∆ h ) = k n k − x k − ( λ max − x )( x − λ min ) , (3.14)which leads, in the QCRB (1.3), to a sensitivity that scales as /n k − / for input product states. This should be comparedwith the O ( n − k ) scaling that can be obtained by using initialentangled states [12]. Notice that for k ≥ , the O ( n − k +1 / ) scaling is better than the /n scaling of the Heisenberg limit,which is the best that can be achieved in the k = 1 case evenwith entangled initial states.The coupling Hamiltonian that has the self-interactionterms of Eq. (1.11) removed is analyzed in App. A. We showthat for initial product states, this modified Hamiltonian hasthe same leading-order behavior in the variance of H ; it thushas O ( n − k +1 / ) scaling for initial product states and the sameoptimal product states as we now find for the coupling Hamil-tonian (1.11). B. Optimal product states
The problem of finding the optimal input product state isnow reduced to maximizing the k -degree polynomial f ( x ) ≡ x k − ( λ max − x )( x − λ min )= x k − (cid:0) k h k / − ( x − ¯ λ ) (cid:1) (3.15)with respect to the single variable x = h h i on the domain λ min ≤ x ≤ λ max . The condition for an extremum is f ′ ( x ) = 2 x k − (cid:2) ( k − (cid:0) k h k / − ( x − ¯ λ ) (cid:1) − x ( x − ¯ λ ) (cid:3) . (3.16)We assume k ≥ , because the k = 1 case is already wellunderstood. For k = 1 , there is a single maximum at x =¯ λ , corresponding to equal probabilities for | λ max i and | λ min i and to (∆ H ) = n k h k / .The polynomial f vanishes at x = λ min and x = λ max . Wecan make some general statements about the extrema of f inthree cases.1. If λ min < < λ max , f has a minimum at x = 0 and two maxima within the allowed domain, one ata positive x + > ¯ λ and one at a negative x − < ¯ λ .The global maximum is at x + ( x − ) if | λ | max = λ max ( | λ | max = | λ min | ).2. If λ max > λ min > , f has a maximum at x = 0 , aminimum for a positive x − < λ min , and a maximumwithin the allowed domain at x + > ¯ λ . Only the last ofthese lies in the relevant domain.3. If λ min < λ max < , f has a maximum at x = 0 , aminimum for a negative x + > λ max , and a maximumwithin the allowed domain at x − < ¯ λ . Only the last ofthese lies in the relevant domain.These general observations are perhaps more enlighteningthan the form of the (nonzero) solutions of Eq. (3.16): x ± = (cid:18) − k (cid:19) ¯ λ ± s ¯ λ k + (cid:18) − k (cid:19) k h k . (3.17)The ± here means the same thing as in the discussion of thethree cases above.As k increases, x + approaches λ max , and x − approaches λ min . Indeed, as k → ∞ , we have x + = (1 − / k ) λ max , corresponding to p + = 1 − λ max / k k h k and (∆ H ) = ( k/ e )( nλ max ) k − k h k , and x − = (1 − / k ) λ min , corresponding to p − = − λ min / k k h k and (∆ H ) = ( k/ e )( − nλ min ) k − k h k .An important limiting case, not covered in the discussionabove, occurs when λ min = − λ max . Then the maxima occursymmetrically at x ± = ± k h k p − /k , (3.18)corresponding to probabilities p ± = + x ± / k h k = (1 ± p − /k ) = 1 − p ∓ and to sin β ± = p /k . The twomaxima lead to the same variance, (∆ H ) = k (1 − /k ) k − n k − ( k h k / k , (3.19)thus yielding a QCRB δγ ≥ k − k / (1 − /k ) ( k − / t n k − / k h k k . (3.20)Of course, when λ min = − λ max , we can always choose unitssuch that λ max = 1 / ( k h k = 1 ), which means that the single-body operators are h j = Z j / . It is this situation that weanalyze in the remainder of this paper. IV. SEPARABLE MEASUREMENTS
In the previous section we obtained the theoretical limitson the measurement uncertainty with symmetric k -body cou-plings and initial product states for the probe. The theoretical bound is saturated by a measurement of the so-called sym-metric logarithmic derivative [22–25]; this measurement, ingeneral, is entangled and depends on the value of the param-eter that we are attempting to estimate. In this section weshow that for some Hamiltonians of interest, standard separa-ble measurements lead to uncertainties for small γ that havethe same scaling as the theoretical bounds. The restriction tosmall values of γ is not a strong limitation, because we canalways use feedback to operate in this regime, as we discussin more detail in Sec. V A.We consider the special case in which the single-body op-erators are h j = Z j / , leading to a coupling Hamiltonian H = (cid:18) X j Z j / (cid:19) k = J kz . (4.1)Here we introduce J z as the z component of a “total angularmomentum” corresponding to the effective qubits. We assumean initial state of the form (3.13), and we let this state evolvefor a very short time, i.e., φ ≡ γt ≪ . In the remainder of thepaper, we often work in terms of the parameter φ instead of γ .After the time evolution, we measure the separable observable J y = X j Y j . (4.2)Over ν trials, we estimate φ as a scaled arithmetic mean of theresults of the J y measurements.The expectation value of any observable at time t is givenby h M i t = Tr (cid:0) U † M U ρ β (cid:1) = (cid:10) U † M U (cid:11) , (4.3)where U = e − iH γ t = e − iHφ , and where we introduce theconvention that an expectation value with no subscript is takenwith respect to the initial state. For small φ , we have U † M U = M − iφ [ M, H ] + O ( φ ) . (4.4)Thus the expectation value and variance of J y at time t takethe form h J y i t = h J y i − iφ h [ J y , H ] i + O ( φ ) , (4.5a) (∆ J y ) t = (∆ J y ) − iφ (cid:10) ( J y − h J y i )[ J y , H ] + [ J y , H ]( J y − h J y i ) (cid:11) + O ( φ ) . (4.5b)The initial expectation value and variance of J y are thoseof an angular-momentum coherent state in the x - z plane: h J y i = 0 , (4.6a) (∆ J y ) = (cid:10) J y (cid:11) = 14 X j,l h Y j Y l i = n . (4.6b)In evaluating the other expectation values in Eqs. (4.5), wecan avail ourselves of the expansions used in Sec. III A, sincewe are only interested in the leading-order behavior in n . Toleading order, the coupling Hamiltonian has the form H = 12 k X ( a ,...,a k ) Z a · · · Z a k + O ( n k − ) . (4.7)In this section we use ≈ to indicate equalities that are good toleading order in n . We can now write [ J y , H ] ≈ k +1 n X j =1 X ( a ,...,a k ) [ Y j , Z a · · · Z a k ]= i k k X l =1 X ( a ,...,a k ) Z a · · · Z a l − X a l Z a l +1 · · · Z a k = ik k X ( a ,...,a k ) X a Z a · · · Z a k , (4.8)from which it follows that h [ J y , H ] i ≈ ik k X ( a ,...,a k ) h X a i h Z a i · · · h Z a k i≈ ik h J x i h J z i k − . (4.9)Elaborating this procedure one step further, we can show thatto leading order in n , the expectation value in the second lineof Eq. (4.5b) vanishes. Our results to this point are summa-rized by h J y i t ≈ φk h J x i h J z i k − + O ( φ )= φk ( n/ k sin β cos k − β + O ( φ ) , (4.10a) (∆ J y ) t ≈ √ n/ O ( φ ) . (4.10b)If we let our estimator φ est be the arithmetic mean of the ν measurements of J y , scaled by the factor ( d h J y i t /dφ ) − =1 /k ( n/ k sin β cos k − β , we have h φ est i = h J y i t d h J y i t /dφ ≈ φ + O ( φ ) , (4.11) δφ ≈ √ ν (∆ J y ) t | d h J y i t /dφ | + O ( φ ) ≈ √ ν k − kn k − / sin β | cos k − β | + O ( φ ) . (4.12)This scheme thus attains the O ( n − k +1 / ) scaling that is thebest that can be achieved by initial product states. In an anal-ysis of optical nonlinearities of arbitrary order, Luis [14] re-ported finding this O ( n − k +1 / ) scaling.The minimum of δφ , occurring when sin β = p /k , givesan optimal sensitivity δφ ≈ √ ν k − k / (1 − /k ) ( k − / n k − / + O ( φ ) , (4.13)which is identical to the optimal QCRB sensitivity for initialproduct states. For k = 2 , the case that is the subject of the next section, the two optimal values of β are β = π/ and β = 3 π/ , and the sensitivity becomes δφ ≈ √ ν n / + O ( φ ) . (4.14)Aside from showing that the QCRB scaling for initial prod-uct states can be achieved, the analysis in this section servesto illustrate how the product-state scheme works in a regimethat has a singularly simple description. The J kz couplingHamiltonian induces a nonlinear rotation about the z axis,which rotates the probe through an angle h J y i t / h J x i ≈ φk h J z i k − . This rotation induces a signal in J y of size ≈ φk h J x i h J z i k − , which is k h J z i k − times bigger than for k = 1 , yet is to be detected against the same coherent-stateuncertainty √ n/ in J y as for k = 1 . To take advantageof the nonlinear rotation, we can’t make the J x lever arm ofthe rotation as large as possible, because the nonlinear rota-tion vanishes when the initial coherent state lies in the equa-torial plane. Nonetheless, we still win when we make theoptimal compromise between the nonlinear rotation and thelever arm. The optimal compromise comes from maximizing h J x i h J z i k − , which turns out to be exactly the same as find-ing the optimum in the QCRB analysis of Sec. III B because h X i = sin β = ∆ Z .A more careful consideration of the terms neglected in thisanalysis suggests that, as formulated in this section, the small-time approximation requires that φ ≪ /n k − . Nonethe-less, the analysis is consistent because φ can be resolved morefinely than this scale, i.e., δφ n k − = O (1 / √ n ) . This conclu-sion is confirmed by the more detailed analysis of the k = 2 case in Sec. V. On the other hand, the simple model ofcoherent-state evolution, developed for k = 2 in Sec. V, sug-gests the description of the preceding paragraph can be ex-tended to much larger times. We return to this point in theConclusion. V. SEPARABLE MEASUREMENTS FOR THEINTERACTION H γ = γJ z We focus now on the symmetric, k = 2 coupling Hamilto-nian H γ = γJ z = γ (cid:18)X j Z j / (cid:19) . (5.1)This is perhaps the most important example for practicalapplications of nonlinear Hamiltonians to quantum metrol-ogy [12], since it occurs naturally whenever the strength oftwo-body interactions is modulated by a parameter γ . As sug-gested in [12], one good place to look for this kind of cou-pling Hamiltonian is in Bose-Einstein condensates. Indeed,in [15, 18], it is shown how this Hamiltonian can be imple-mented using the internal atomic states of BECs. In analyz-ing the BEC scenario, Ref. [15] finds a sensitivity that scalesas O (1 /n ) for separable measurements made on a probethat evolves from an an initial product state chosen to be anangular-momentum coherent state in the equatorial plane. Theresults in the previous sections show that we should be able toimprove this scaling to O ( n − / ) through a wiser choice ofthe initial coherent state. In this section we analyze this situa-tion in some detail.We take the initial state of the n -qubit probe to be anangular-momentum coherent state that is at an angle β fromthe z axis in the x - z plane. This state is obtained from thecoherent state along the z axis, | J, J i = | i ⊗ n , by a rotationthrough β about the y axis: | Ψ β i = e − iβJ y | J, J i = ( e − iβY/ | i ) ⊗ n . (5.2)The rotation about the y axis and the nonlinear rotation underthe interaction Hamiltonian (5.1) both leave the state in the (2 J + 1) -dimensional subspace with angular momentum J = n/ , so we can use the basis | J, m i of J z eigenstates for thissubspace, with m = − J, . . . , J . The initial probe state usedin [15] is a special case, β = − π/ .The state | Ψ β i = P Jm = − J d m | J, m i , can be expanded inthe basis | J, m i using a reduced Wigner rotation matrix [30] d m ≡ d JmJ ( β ) = h J, m | e − iβJ y | J, J i = s (2 J )!( J + m )!( J − m )! [cos( β/ J + m [sin( β/ J − m . (5.3)At time t = φ/γ , the state of the probe becomes | Ψ β ( t ) i = e − iφJ z | Ψ β i = J X m = − J d m e − iφm | J, m i . (5.4) A. Measurements
We now look at the attainable measurement uncertaintiesusing both J x = P X j / and J y = P j Y j / measurementson the final state of the probe. It turns out that J x and J y measurements are on nearly the same footing, with J y mea-surements being marginally better, for all β except β = π/ .For very short times, the superiority of J y measurements for β = π/ is clear from the analysis in Sec. IV, since thechange in h J y i is linear in φ , whereas the change in h J x i isquadratic. What happens for longer times and for β = π/ cannot be addressed by the short-time analysis in Sec. III.What we find in this section is that both J x and J y measure-ments can achieve the optimal scaling obtained in Sec. III .For β = π/ , J y measurements provide no information about φ , but J x measurements achieve the O ( n − ) scaling foundin [15]. These conclusions assume no decoherence, and inSec. V B we explore the impact of decoherence on the abil-ity to achieve super-Heisenberg scalings with the symmetric, k = 2 coupling Hamiltonian.For measurements of J x or J y , the sensitivity is given by δφ x,y = t δγ x,y = (∆ J x,y ) φ | d h J x,y i φ /dφ | (5.5) (for this subsection, we revert to our practice of omitting the / √ ν statistical factor from our sensitivity formulas).The expressions needed to calculate δφ for J x and J y mea-surements are derived in Appendix B. These results are con-veniently expressed in terms of the raising and lowering oper-ators J ± = J x ± iJ y , (5.6)since we can write h J x i φ = Re (cid:0) h J + i φ (cid:1) , h J y i φ = Im (cid:0) h J + i φ (cid:1) , (5.7) (cid:10) J x,y (cid:11) φ = 14 h J + J − + J − J + i φ ±
12 Re (cid:0) (cid:10) J (cid:11) φ (cid:1) , (5.8)where the upper sign in Eq. (5.8) applies to J x and the lowersign to J y . In Appendix B, we show that h J + i φ = J sin β (cos φ + i sin φ cos β ) J − = J sin β r J − e i (2 J − θ , (5.9) h J + J − + J − J + i φ = J + J (2 J − β , (5.10) (cid:10) J (cid:11) φ = J (2 J − β (cos 2 φ + i sin 2 φ cos β ) J − = J (2 J − β R J − e i ( J − , (5.11)where r = (1 − sin φ sin β ) / , (5.12a) θ = tan − (tan φ cos β ) , (5.12b)and R = (1 − sin φ sin β ) / , (5.13a) Θ = tan − (tan 2 φ cos β ) . (5.13b)Plugging these results into Eqs. (5.7) and (5.8), we arrive at h J x i φ = J sin β r J − cos[(2 J − θ ] , (5.14) h J y i φ = J sin β r J − sin[(2 J − θ ] , (5.15)and (cid:10) J x,y (cid:11) φ = J J (2 J − β × (cid:0) ± R J − cos[2( J − (cid:1) . (5.16)In using these results in what follows, it is easier to deal di-rectly with the first forms in Eqs. (5.9) and (5.11) rather thanworking with the functions r , θ , R , and Θ .The expectation values h J x,y i φ change sign when φ ad-vances by π . This means that their squares and absolute val-ues, which are all that appear in the sensitivity (5.5), are peri-odic with period π . The second moments (cid:10) J x,y (cid:11) φ are periodicwith period π/ . The upshot is that the uncertainties ∆ J x,y and the precision δφ x,y are periodic with period π . This π -periodicity is a consequence of periodic revivals in the evolvedstate | Ψ β ( t ) i .The main features of the sensitivity δφ for measurements of J x and J y can be gleaned from Fig. 1. It is clear from theseplots that the best sensitivity is achieved when φ is near zeroand also, because of the periodicity of δφ , when φ is near qπ ,for q any integer.When J is large, we can develop a good approximation forthe entire region of high sensitivity, where φ is small, by writ-ing (cos φ + i sin φ cos β ) J − ≃ e iJφ cos β e − Jφ sin β , (5.17a) (cos 2 φ + i sin 2 φ cos β ) J − ≃ e iJφ cos β e − Jφ sin β . (5.17b)These approximations are good to second order in φ in theexponent. When φ is near qπ , the same approximations canbe had by replacing φ with φ − qπ . The complex exponentialsgive rise to rapidly oscillating fringes in h J x,y i φ and (cid:10) J x,y (cid:11) φ ,with periods ∼ /J cos β ; the slower Gaussian envelopes takethese expressions to zero when φ is a few times | sin β | / √ J .It is not hard to work out the sensitivity in this approxi-mation, but the formulas are sufficiently messy that they arelittle more illuminating than the exact expressions. We can,however, develop a very simple, yet instructive picture of thefringes by keeping them, but assuming that φ is small enoughthat the Gaussian envelopes have yet to become effective, i.e., √ Jφ sin β is somewhat smaller than 1. In this approximation,the fringes are uniform in φ , and we obtain h J x i φ ≃ J sin β cos(2 Jφ cos β ) , (5.18) h J y i φ ≃ J sin β sin(2 Jφ cos β ) , (5.19) (∆ J x ) φ ≃ J − sin β cos (2 Jφ cos β )] , (5.20) (∆ J y ) φ ≃ J − sin β sin (2 Jφ cos β )] . (5.21)These lead to sensitivities δφ x ≃ J − sin β cos (2 Jφ cos β )sin β sin (2 Jφ cos β ) , (5.22) δφ y ≃ J − sin β sin (2 Jφ cos β )sin β cos (2 Jφ cos β ) . (5.23)Within this uniform-fringe approximation, the best sensi-tivities are achieved at the troughs of the fringes: the best op-erating points are, for J x measurements, φ = qπ + ( s + 1 / π J cos β , (5.24)and for J y measurements, φ = qπ + sπ J cos β , (5.25) where q and s are integers. At these operating points, thesensitivity for both measurements becomes δφ x,y ≃ √ J / | sin 2 β | , (5.26)which takes on its optimal value, / √ J / = 2 /n / , when β = π/ or β = 3 π/ , in agreement with the analyses inSecs. III and IV. This O ( J − / ) scaling has been found inde-pendently by Choi and Sundaram [18] and for nonlinear opti-cal systems by Luis and collaborators [6, 9]. For β = π/ and J = 2 500 , Fig. 2 plots the central fringes of the approximatesensitivities (5.22) and (5.23) and compares them with the ex-act sensitivities and the Gaussian approximation for measure-ments of J x and J y .As we show in App. B, within the uniform-fringe approx-imation, the evolved state (5.4) is an angular-momentum co-herent state that makes an angle β with the z axis and thatrotates around the z axis with angular velocity γJ cos β .The enhanced sensitivity available from a quadratic Hamil-tonian is a consequence of this increased rotation rate, whichis greater by a factor of J cos β = 2 h J z i than that avail-able from a linear Hamiltonian. This same conclusion cameout of the short-time analysis of Sec. IV, but it is strongernow because the uniform-fringe approximation is much betterthan the short-time approximation. The short-time approx-imation requires that Jφ ≪ and thus describes correctlyonly the center of the central fringe for J y measurements. Incontrast, the uniform-fringe approximation only requires that φ √ J | sin β | ≪ ; within this requirement, there can be sev-eral fringes, i.e., Jφ cos β can be somewhat larger than π ,provided that J ≫ tan β . The more accurate uniform-fringeapproximation allows us to see the other near-optimal oper-ating points for J y measurements and to see the optimal op-erating points for J x measurements, which lie not at φ = 0 ,but at φ = ± π/ J cos β . As β approaches π/ , the fringesbecome wider and wider, making the uniform-fringe approx-imation reliable only for larger and larger values of J . For β = π/ , the fringes disappear entirely, and a separate analy-sis is required to find the scaling for J x measurements (since h J y i φ = 0 for β = π/ , measurements of J y provide no in-formation about φ ).That the final state (5.4), within the region of high sensi-tivity, is approximately an angular-momentum coherent statetells us two important things. First, even though the quadraticHamiltonian will generate entanglement from a product state,this entanglement plays no role in the enhanced sensitivity.The improved sensitivity comes from the increased rotationrate of the coherent state, which is a product state, having noentanglement among the probe constituents. Indeed, for themeasurements we consider here, the deviation from being acoherent state makes the sensitivity worse. Second, that theprobe state is approximately a product state within the regionof high sensitivity hints that this scheme should not be as frag-ile in the presence of decoherence as schemes that rely on ini-tial entanglement. We investigate the impact of decoherencein Sec. V B and show that the O ( n − / ) scaling is unaffectedby phase decoherence.0 - - - Φ ´ Π (cid:144) H a L ∆ Φ x - - - Φ ´ Π (cid:144) H b L ∆ Φ y FIG. 1: (Color online) Sensitivity (solid red lines) vs. φ ( − π/ ≤ φ ≤ π/ ) using an optimal initial state at angle β = π/ : (a) J x measurements; (b) J y measurements. The total angular momentum J of the probe is 200, corresponding to n = 400 . The lower bound onthe sensitivity, / √ J / , is plotted as the dotted (green) line. The sensitivity is characterized by rapidly oscillating fringes and a decay ofsensitivity away from the best sensitivities near φ = 0 . The sensitivity patterns repeat with periodicity π ; only a quarter of a period is plottedbecause the sensitivity worsens even more outside the plotted region. Part (a) also shows the sensitivity for J x measurements when β = π/ (dashed blue line); notice the absence of fringes in this case and the substantially degraded sensitivity. - - Φ ´ Π (cid:144) H a L ∆ Φ x ´ - - Φ ´ Π (cid:144) H b L ∆ Φ y ´ FIG. 2: (Color online) Central few fringes of the measurement precision for β = π/ and J = 2 500 ( n = 5 000 ): (a) J x mea-surements; (b) J y measurements. The solid (red) lines are the exact sensitivities, the dashed (blue) lines are the sensitivities givenby the uniform-fringe approximation of Eqs. (5.22) and (5.23), δφ x ≃ J / s (2 Jφ cos β ) , δφ y ≃ J / s (2 Jφ cos β ) , and the dotted (green) lines are the Gaussian-envelope approximation of Eqs. (5.17). The uniform-fringe approximation locates the fringesprecisely, but misses entirely the degradation in sensitivity as one moves away from the central fringes and also fails to characterize accuratelythe shape of the fringes. The Gaussian-envelope approximation improves on this performance by capturing the degradation of sensitivity quitewell, but still fails on the fringe shapes. Even the central fringe for J y measurements is noticeably flatter than in the two approximations. Toget the best sensitivity, one should operate right on the central fringe, at φ = qπ , for J y measurements and on one of the two central fringes,centered at φ = qπ ± π/ J cos β for J x measurements. Notice that J x measurements achieve nearly optimal sensitivity at points near theoutside of these two central fringes. To achieve the optimal sensitivity for J x or J y measure-ments, we need to operate within the appropriate centralfringe, of width π/ √ J for β = π/ . This can be done byusing an adaptive feedback procedure, which we discuss inthe context of J y measurements. The feedback procedure iscarried out in several steps, in each of which the quantity thatis estimated is φ − φ est , where φ est is the estimate of φ fromthe previous step. At each step, we choose J = n/ so that φ − φ est is with very high probability close to the center ofthe central fringe, and we use ν probes to determine φ with greater precision for the next step. As we obtain progressivelyrefined estimates of φ , the quantity being estimated becomessmaller and smaller, always lying well within a sequence ofprogressively finer central fringes.To check that this procedure works and to determine itsscaling properties, imagine that we determine φ/ π bit by bit.At step l , we determine the l th bit of φ/ π by choosing J = J l so that the precision is given by δφ l π = 1 √ ν π √ J / l = 1 f l , (5.27)1where the factor f ∼ l th bit with very high probability. This gives J l = 12 ν / (cid:18) f l π (cid:19) / . (5.28)We must, of course, choose J l to be an integer or half-integer,so we choose the nearest one, but this detail does not changethe resource calculation significantly, so we ignore it. Atstep l + 1 , φ − φ est lies well within the central fringe, as wesee from δφ l π/ √ J l +1 = 2 / πν / (cid:18) πf l (cid:19) / . (5.29)Indeed, because of the O ( J − / ) scaling, the quantity beingestimated is buried progressively deeper fractionally in thecentral fringe as we step through the procedure, despite thefact that the central fringe is itself narrowing exponentially.Suppose now that we use this procedure to estimate L bitsof φ . The total number of constituents used, N = ν L X l =1 J l = (cid:18) νfπ (cid:19) / L/ − / − , (5.30)is dominated by the last step, as is typical in these feedbackprocedures. The ultimate precision displays the O ( N − / ) scaling, π − L = 4 f (2 / − / νN / = 2 f (2 / − / √ ν N/ν ) / , (5.31)with a small additional overhead given by the factor f / (2 / − / .As β becomes smaller, the uniform-fringe approxima-tion becomes progressively better, since the fringes oscillaterapidly and the Gaussian envelopes become very broad. Onthe other hand, the signal in J x and J y disappears, making thesensitivity worsen as / sin β .At the other extreme, as β approaches π/ , the uniform-fringe approximation becomes poorer as the fringes becomeas wide as the Gaussian envelopes and loses validity entirelywhen J | cot β | ∼ . When β = π/ , which is the initial probestate analyzed by Rey et al. [15] (Ref. [15] actually uses β = − π/ , but this state is equivalent to β = π/ for purposes ofthese measurements), | J y i φ = 0 , making J y measurementsuseless for extracting information about φ . Thus we have tochoose the J x measurement. The dashed line in Fig. 1 showsthat the optimal operating point is φ = 0 (or, more generally, φ = qπ ). Near φ = 0 , the expectation value and variance of J x are given by h J x i φ ≃ J − J (2 J − φ , (5.32) (∆ J x ) φ ≃ J (2 J − φ , (5.33) where the approximations hold for φ ≪ / √ J . The resultingoptimal sensitivity, δφ = (∆ J x ) φ | d h J x i φ /dφ | ≃ p J (2 J − , (5.34)has the O ( J − ) sensitivity scaling found in [15].We can relate these results to the general lower-bound anal-ysis in Sec. III by noting that β = π/ means that h h i = h Z i / . This means that the dominant sums in the ex-pansions of Eqs. (3.4) and (3.6) are those that contain onlysquares of h j s. The number of terms in these sums scales as O ( n k ) , which yields a sensitivity that scales as O ( n − k/ ) .To gain further insight into the scaling behavior, we plotthe scaling exponent ξ in δφ = O ( n − ξ ) as a function of β for J x measurements (Fig. 3) and J y measurements (Fig. 4),using three very large values of J . For J y measurements wecalculate ξ at the optimal operating point, φ = 0 . For J x mea-surements, the optimal operating point is a function of β , buta good compromise point, which works well over the entirerange of β , is / √ J , so we calculate the scaling exponent atthis point for all values of β . An investigation of nearby op-erating points scaling as /J gives plots with no discernibledifferences for the large values of J under consideration. Themain differences between J x and J y measurements are thefollowing: (i) right at β = π/ , J x measurements have a scal-ing exponent of 1, whereas J y measurements provide no infor-mation about γ ; (ii) for J y measurements, the plot of scalingexponent has two humps, nearly symmetric about β = π/ and β = 3 π/ , whereas for J x measurements, the scalingexponent is better on the outside of the humps. The overalltrend is for both measurements to have a scaling exponent of ξ = 3 / in the limit of large J , except at β = 0 , π/ , and π . Π Π Π Π Β Ξ FIG. 3: (Color online) Scaling exponent ξ for J x measurements. Thedotted (red) line is for J = 10 , the dashed (green) line for J = 10 ,and the solid (blue) line for J = 10 . B. Decoherence
The coherent-state model suggests that our generalizedquantum metrology scheme with initial product states shouldnot display the fragility of entangled protocols in the presence2 Π Π Π Π Β Ξ FIG. 4: (Color online) Scaling exponent ξ for J y measurements. Thedotted (red) line is for J = 10 , the dashed (green) line for J = 10 ,and the solid (blue) line for J = 10 . of decoherence. We can investigate this possibility by consid-ering independent dephasing of the effective qubits, describedby the Lindblad equation ˙ ρ = − Γ2 (
ZρZ − ρ ) , (5.35)where τ = Γ − is the dephasing time. Since dephasing com-mutes with the quadratic Hamiltonian, we can shunt its effectsto the final time t , whence it maps the Pauli operators of eacheffective qubit in the following way: X → e − Γ t X , (5.36a) Y → e − Γ t Y , (5.36b) Z → Z . (5.36c)To obtain the effect of the decoherence on the expectation val-ues and variances of the measured operators at the time ofmeasurement, it is easiest to use the adjoint map [3], whichfor this simple case is identical to the map (5.36) and gives h J x,y i Γ = e − Γ t h J x,y i , (5.37) (∆ J x,y ) = e − t (∆ J x,y ) + J − e − t ) . (5.38)Here a subscript Γ denotes the value with dephasing, and asubscript without. It is now easy to see that under this modelof decoherence, for either of the measurements that we areconsidering, the sensitivity takes the form δγ = δγ + J ( e t − ν ( d h J x,y i /dγ ) = δγ (cid:18) J ( e t − J x,y ) (cid:19) . (5.39)To assess the effects of decoherence, we now focus on J y measurements, and we assume that through an adaptive feed-back procedure like that sketched in Sec. V A, we are operat-ing well within the central fringe, i.e., γt is somewhat smaller than π/ J cos β . Inserting the φ = 0 values from Eqs. (5.21)and (5.23) into Eq. (5.39) yields a sensitivity δγ Γ = e Γ t t √ ν √ J / | sin 2 β | . (5.40)If we now let T = νt be the total time available for measure-ments involving ν probes, the optimal value of t , found bymaximizing e Γ t / √ t is t = τ / , gives a sensitivity δγ Γ = r eT τ J / | sin 2 β | . (5.41)This result assumes that each probe can be processed in atime τ / , but within this constraint, the scaling is the same O ( J − / ) scaling that applies in the absence of decoherence.This is to be contrasted with entangled inputs, where uncor-related phase decoherence degrades the scaling from O ( J − ) to the O ( J − / ) characteristic of product inputs.These arguments hold for general symmetric k -bodyHamiltonians, giving a sensitivity scaling O ( n − k +1 / ) for ini-tial product states subjected to uncorrelated phase decoher-ence. This is the same scaling achieved by initial optimalentangled states under this decoherence model [15]. On theother hand, the use of product states with k -body Hamiltoni-ans for k ≥ can surpass both the standard quantum limit andthe Heisenberg limit, even in the presence of phase decoher-ence. VI. CONCLUSION
The possibility of using nonlinear Hamiltonians has the po-tential to open up a new frontier in quantum metrology. Quan-tum metrology has traditionally focused on linear Hamilto-nians of the form γJ z = γ P nj =1 Z j / . The main tech-nical challenge has been to improve on the standard quan-tum limit for determining the parameter γ , which scales as O ( n − / ) and can be attained relatively easily using prod-uct input states and separable measurements. The goal oflinear quantum metrology has been to achieve the Heisen-berg limit for determining γ , which scales as O ( n − ) andrequires the use of highly entangled input states. Nonlinearcoupling Hamiltonians of the form J kz offer the possibility offurther improvements in scaling. With the same highly entan-gled input states, nonlinear Hamiltonians can achieve a scal-ing O ( n − k ) . More importantly, they provide O ( n − k +1 / ) scalings, better than the Heisenberg limit, for input productstates and separable measurements. We expect that the gen-eralized quantum metrology of nonlinear Hamiltonians willlead to new experiments—and, ultimately, to new devices—that take advantage of the enhanced scaling, which is availableusing the experimentally accessible tools of product-state in-puts and separable measurements.A notable feature of generalized quantum metrology isthat the enhanced scalings available with product-state inputsdo not rely on the entanglement produced by the nonlinearHamiltonian. We reach this conclusion in this paper from adetailed analysis of the k = 2 case, in the course of which3we formulate an approximate coherent-state model of the timeevolution, which applies during the period of enhanced sensi-tivity. In the model, a coherent state that makes an angle β to the z axis rotates with angular velocity γJ cos β . The in-creased rotation rate, larger by a factor of J cos β than for k = 1 , accounts for the enhanced sensitivity. Since coher-ent states are product states, this indicates that entanglementplays no role in the enhanced sensitivity, and it accounts forthe robustness we find in the presence of phase decoherence.Although these conclusions emerge here from the k = 2 analysis in this paper, it is not hard to extend the coherent-state model to arbitrary k . Given the input state (5.2), the stateat time t = φ/γ becomes | Ψ β ( t ) i = e − iφJ kz | Ψ β i = X m d m e − iφm k | J, m i . (6.1)The squares of the Wigner rotation-matrix elements d m ofEq. (5.3) are a binomial distribution, which for large J , ap-proaches a narrow Gaussian, centered at m = h J z i = J cos β ,with half-width √ J + 1 sin β . This encourages us to approx-imate m k in the phases of Eq. (6.1) as ( J cos β + ∆ m ) k ≃ ( J cos β ) k + k ( J cos β ) k − ∆ m , giving | Ψ β ( t ) i = e iφ ( k − J cos β ) k X m d m e − iφk ( J cos β ) k − m | J, m i = e iφ ( k − J cos β ) k e − iφk ( J cos β ) k − J z e − iβJ y | J, J i . (6.2)This is an angular-momentum coherent state at angle β tothe z axis, rotating about the z axis with angular velocity γk ( J cos β ) k − , which is the same enhanced rotation rate thatwe found in the very short-time analysis of Sec. IV. The ap-proximation leading to the coherent state (6.2) thus extends toarbitrary k the uniform-fringe approximation, formulated for k = 2 in Sec. V. The fringes have width π/k ( J cos β ) k − ,and the approximation provides a reasonable description ofthe first and second moments of J x and J y so long as ( J cos β ) k − φ √ J | sin β | ≪ .The enhanced rotation rate is responsible for the improvedscaling, and just as for k = 2 , the coherent-state model indi-cates that the entanglement generated by the nonlinear Hamil-tonian plays no role in the enhancement. In separate work, tobe published elsewhere, we extend these ideas. We investi-gate in more detail the entanglement generated by the nonlin-ear Hamiltonian, quantifying it using standard entanglementmeasures and showing that the enhanced sensitivity with ini-tial product states can be achieved with a vanishing amount ofentanglement. Acknowledgments
This work was supported in part by Office of Naval Re-search Grant No. N00014-07-1-0304 and by the NationalNuclear Security Administration of the U.S. Departmentof Energy at Los Alamos National Laboratory under Con-tract No. DE-AC52-06NA25396. EB acknowledges finan-cial support from the the Catalan government, contract CIRIT SGR- 00185, and from the Spanish MEC through contractsFIS2005-01369, QOIT (Consolider-Ingenio 2010) and travelgrant PR2007-0204. He thanks the Department of Physics andAstronomy of the University of New Mexico for hospitality.
APPENDIX A: SYMMETRIC HAMILTONIAN WITHOUTSELF-INTERACTION TERMS