Optimal and Secure Measurement Protocols for Quantum Sensor Networks

Zachary Eldredge,1, 2 Michael Foss-Feig,1, 2, 3 Jonathan A. Gross,4 S. L. Rolston,1 and Alexey V. Gorshkov1, 2 1Joint Quantum Institute, NIST/University of Maryland, College Park, MD 20742, USA 2Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, MD 20742, USA 3United States Army Research Laboratory, Adelphi, MD 20783, USA 4Center for Quantum Information and Control, University of New Mexico, Albuquerque, NM 87131, USA (Dated: March 26, 2018) Studies of quantum metrology have shown that the use of many-body entangled states can lead to an enhancement in sensitivity when compared to unentangled states. In this paper, we quantify the metrological advantage of entanglement in a setting where the measured quantity is a linear function of parameters individually coupled to each qubit. We ﬁrst generalize the Heisenberg limit to the measurement of non-local observables in a quantum network, deriving a bound based on the multi-parameter quantum Fisher information. We then propose measurement protocols that can make use of Greenberger-Horne-Zeilinger (GHZ) states or spin-squeezed states and show that in the case of GHZ states the protocol is optimal, i.e., it saturates our bound. We also identify nanoscale magnetic resonance imaging as a promising setting for this technology.

I. INTRODUCTION

Entanglement is a valuable resource in precision measurement, as measurements using entangled probe systems have fundamentally higher optimal sensitivity than those using unentangled states [1]. A generic measurement using N unentangled probes will have a standard deviation√ from the true value asymptotically proportional to 1/ N. By using N maximally entangled probes, a single parameter coupled independently to each probe system can be measured with an uncertainty proportional to 1/N. This is the best possible scaling con- FIG. 1. (a) An illustration of the network setup in a nanoscale sistent with the Heisenberg uncertainty principle and is NMR setting. Nodes, located at different points relative to a known as the Heisenberg limit [1,2]. The procedure can large molecule, share an entangled state; at each node there is also be reversed–enhanced sensitivity to disturbances can both an unknown parameter θi and a known relative weight provide experimental evidence of entanglement [3–5]. αi. We are concerned with estimating α · θ. (b) Illustration of the partial time evolution protocol for three qubits. Solid Measurements making use of entanglement usually green segments of the timeline represent periods when a qubit couple one parameter to N different systems [1,6,7]. is evolving due to coupling to the local parameter θi, while However, the emerging potential of long-range quantum dashed red segments represent periods after the qubit stops information opens new avenues for metrology [8,9] and evolving. The switches occur at times corresponding to the entanglement distribution [10]. The ability to distribute qubits’ weights in the final linear combination. The weight of the last qubit is α = 1. entanglement across spatially separated regions has al- 3 ready been used for recent loophole-free tests of Bell’s

arXiv:1607.04646v3 [quant-ph] 23 Mar 2018 inequality [11–13]. In this work, we are interested in cou- by a known real number α , pling N parameters to N different systems, which may i be spatially separated, and measuring a linear function N of all of them (see Fig.1a) such as a single mode of a X q = αiθi = α · θ. (1) spatially varying field. Such measurements may be of i=1 interest in geodesy, geophysics, or medical imaging [14– 18], but in this paper we focus on potential application In this paper, we characterize the advantage entangle- to nanoscale nuclear magnetic resonance (NMR) imag- ment provides in this setting and construct an optimal ing. Later in this paper we will discuss precisely how our strategy equivalent to turning some qubits’ evolution method might apply in this setting. “on” and “off” for time proportional to the weight with which their parameter contributes to the function q (see The function q we wish to measure is a weighted sum Fig.1b). With this scheme of “partial time evolution,” of the deterministic individual parameters θi, where i we can measure a linear function with the minimum vari- indexes the individual systems and each weight is denoted ance permitted by quantum mechanics, which can be 2 viewed as an extension of the Heisenberg limit to linear We consider this the standard quantum limit for net- combinations. We will also show that our method can works. To compare to the typical case where N inde- protect the secrecy of the result, allowing the network as pendent qubits measure a single parameter, consider the ¯ a whole to perform a measurement without eavesdrop- average θ, which is equivalent to setting all αi = 1 and pers learning any details of α · θ. then using Θ¯ = Q/N to obtain Var Θ¯ = 1/Nt2. It is our goal in this paper to present a means to improve on the limit in Eq. (4). II. SETUP

We consider a system in which there are N sensor III. HEISENBERG LIMIT FOR SENSOR nodes. Each sensor node i possesses a single qubit cou- NETWORKS pled to an unknown parameter θi unique to each node. We suppose that the state evolves unitarily under the A. Using Fisher Information Matrix Hamiltonian N Our task is to perform parameter estimation on a quan- ˆ ˆ X 1 z H = Hc(t) + θiσˆ . (2) tum system evolving under some set of parameters {θ } 2 i i i=1 linearly coupled to sensor qubits as in Eq. (2)[19–22]. ˆ Although we are only interested in measuring a single Here, Hc(t) is a time-dependent control Hamiltonian cho- number, we still need to treat a system that has many sen by us, which may include coupling to additional an- parameters in the evolution, necessitating the use of a cilla qubits andσ ˆx,y,z are the Pauli operators acting on i multi-parameter theory as in Refs. [23–31]. It is known qubit i. We wish to measure the quantity q defined in from classical estimation theory that, given a probability Eq. (1). We assume that ∀i : |αi| ≤ 1 and additionally distribution p(z) over a set of outcomes z that depends on that there is at least one αi such that αi = 1. These a number of parameters, all estimators of the parameters conditions simply set a scale for the function, and for an obey the Cramér-Rao inequality [32, 33], arbitrary α all that is needed is division by the largest αi to meet this requirement. As an example, a network with F −1 two nodes interested in measuring the contrast between Σ ≥ . (5) M those nodes would set α = (1, −1) to measure θ1 − θ2. We would like to establish how well an arbitrary measure- Here, M is the number of experiments performed, F is ment of α·θ can be made and what the best measurement the Fisher information matrix (see below), and Σ is the protocol is for doing so. By “protocol” we mean three dif- covariance matrix, where Σij = E [(Θi − θi) (Θj − θj)]. ferent choices: (1) which input state we begin with, (2) The inequality is a matrix inequality, meaning that MΣ− ˆ −1 what auxiliary control Hamiltonian Hc(t) we implement, F is positive semidefinite. We will concern ourselves and (3) how the final measurement is made. with the single-shot Fisher information, and set M = 1 We define the quality of measurement in terms of from now on. The Fisher information matrix captures an estimator, Q, constructed from experimental data. how each parameter changes the probability distribution (Throughout this paper, we denote operators with hats, of outcomes, vectors by boldface, quantities to be estimated by low- ercase, and corresponding estimators by uppercase.) We Z ∂ ln p(z) ∂ ln p(z) Fij = p(z) dz. (6) assume that the estimator is unbiased, so that its expec- ∂θi ∂θj tation value is the true value E [Q] = q. Then our metric for the quality of the measurement is the average squared This bound is a purely classical statement about proba- error, or variance, of the estimator, bility distributions, and is saturated asymptotically using a maximum-likelihood estimator [34]. Note that al- h 2i Var Q = E (Q − q) . (3) though we have presented the formulas for the Fisher information matrix, in the case of a single parameter the If measurements of θi can be made locally with accuracy Fisher information will be a scalar which can be obtained Var Θi for an estimator Θi, then we could compute the by setting i = j in Eq. (6). linear combination by local measurements and classical Quantum theory bounds the probability distributions computation. In this case, the variance is given by clas- that can result from a state evolved under a parameter- 2 sical statistical theory as Var Q = kαk Var Θ0 assuming dependent unitary operation [19]. We thus define the that Var Θi is identical at each site and equal to Var Θ0. quantum Fisher information FQ for a process with a given A measurement of an individual θi in Eq. (2) can be initial state as the maximization of the Fisher informa- 2 made in time t with a variance of 1/t [2]. Therefore, our tion over all possible measurement schemes. This gives entanglement-free figure of merit is rise to the quantum Cramér-Raobound (QCRB), which kαk2 simply replaces F with FQ in Eq. (5). A matrix element Var Q ≥ . (4) ˆ t2 of FQ for a pure state evolving under a Hamiltonian H 3

is given by Here, F˜Q is the quantum Fisher information for a bb single parameter, as defined by Eq. (7). In Ref. [21], it (F ) = 4t2 [hgˆ gˆ i − hgˆ i hgˆ i] , (7) Q ij i j i j was shown that for any time-dependent control Hamilto- ˆ nian Hc(t), including those with ancilla qubits, whereg î = ∂H/∂θˆ i is the generator corresponding to ˜ 2 2 parameter i. For instance, in Eq. (2) the generatorg î FQ ≤ t kgˆbks. (14) 1 z bb is the operator 2 σî . Unlike the Cramér-Raobound, the QCRB cannot always be satisfied, even asymptotically. Here kgˆbks is the operator seminorm (difference between However, in the setting of this paper, where all generators the largest and smallest eigenvalues) of the generator cor- commute, it can be [23]. Equation (5) then takes the responding to parameter θb. Our final bound comes from form: applying this condition and recognizing that the formula

−1 −1 must hold for all b: Σ ≥ F ≥ FQ . (8) α2 Var Q ≥ max b . (15) To formulate the appropriate Cramér-Raobound in 2 2 b t kgˆbks the case where the quantity we wish to estimate is a linear combination of the θi, we simply use the fact that We emphasize that Eq. (15) remains true no matter what the variance of a linear combination α · θ can be written time-dependent control Hˆc(t) is applied. as αT Σα. It follows immediately from Eq. (8) that 1 z In Eq. (2), allg ˆb = 2 σˆb , kgˆbks = 1, and we find a T −1 bound, Var Q ≥ α FQ α. (9) 2 αi 1 Note that although we began by considering the full co- Var Q ≥ max = . (16) i t2 t2 variance matrix, we now focus on just a single scalar T −1 α FQ α because our quantity of interest is a single lin- Here we have used the fact that the largest αi = 1. If we ear transformation of the original parameters. want to estimate the average of the θi, then all qubits are In order to properly define the Cramér-Raobound, it equally weighted and the desired quantity is θ¯ = q/N, so 2 2 is necessary to consider the fact that F and FQ are only Var Θ¯ ≥ 1/N t and we reproduce the desired Heisenberg positive semi-definite and not necessarily invertible. For scaling which is more precise than the 1/N in Eq. (4). instance, if a parameter has no effect on probabilities at However, note that if we wanted to estimate only a sin- all, then it cannot be estimated from experimental re- gle θi, then we would not benefit from the entanglement. sults and the bound on the variance of its estimator is In general, we can, for some situations, greatly improve undefined. To sidestep this issue, we can instead look the precision of parameter esitmation with nonlocal tech- at F˜Q, the quantum Fisher information projected onto niques if the parameter itself is also non-local. Our bound its own image [31], assuming that α has no overlap with allows us to explore the full range of possible α between the kernel of FQ. This matrix (and its inverse) are now these two extremes. Compared to the bound on unen- both positive definite, meaning they can always be in- tangled states [Eq. (4)], Eq. (15) simply picks out the verted. Equation (9) is therefore always well-defined if largest contribution due to uncertainty from a single site. F˜Q is used. Equation (15) can be viewed as an extension of the usual q ˜ ˜ Heisenberg bound to linear combinations of parameters. Since FQ is Hermitian and positive definite, FQ is We can illustrate the above argument by optimizing Hermitian. We can then write the following for an arbi- over the space of all control Hamiltonians Hˆc(t). As this trary real b by invoking the Cauchy-Schwarz inequality: is computationally expensive, we limit ourselves to a two- q q qubit sensor network with no ancillas. The Hamiltonians ˜−1 2 ˜ 2 k FQ αk k FQbk we optimize over include enough operators to provide αT F˜−1α = (10) universal control on two qubits, meaning we can effec- Q T ˜ b FQb tively modify the input state as well as the final measure- q q T ˜−1 ˜ 2 ment basis in order to optimize the Fisher information. kα FQ FQbk ≥ (11) In order to test the form of our bound, Eq. (15), which T b F˜Qb depends both on relative weights of each parameter and z kαT bk2 the underlying generator, we couple θ1 to a generatorσ ˆ1 ≥ . (12) z T which has kσˆ1 ks = 2. We leave the second qubit coupled b F˜Qb 1 z to a generator 2 σˆ2 as in Eq. (2). The bound correspond- 2 2 ing to the first qubit from Eq. (15) is α1/4t and that Taking b to be the bth element of the standard basis gives 2 2 of the second qubit is α2/t . In our numerics, we set 2 2 α1 = t = 1, meaning the two bounds are 1/4 and α . T ˜−1 αb 2 Var Q ≥ α FQ α ≥ . (13) Our analytic result leads us to believe therefore that if F˜Q 2 2 bb α2 > 1/4, the minimum possible variance should be α2. 4

corresponds to the parameter of interest.) The advantage 1.0 of rewriting θ this way is that we can now identify the term in the Hamiltonian which is proportional only to 0.8 α · θ. The generator corresponding to the quantity α · θ is:

1 0.6 Q

F ∂Hˆ β · σˆ T gˆ = = . (18) 0.4 ∂ (α · θ) 2

0.2 To obtain the quantum Fisher information corresponding to this generator, we consider the variance of the 0.0 operatorg ˆ. The maximum variance of this generator is 0.0 0.2 0.4 0.6 0.8 1.0 given by the operator seminorm [21]. Using this fact, we 2 can write: 2 T −1 ! FIG. 2. Numerical optimization of α FQ α for two qubits X F ≤ t2kgˆk2 = t2 |β | . (19) with α1 = 1 compared to the bound predicted by our analytic Q s i result. Each point is generated by running a gradient descent i algorithm until convergence; the control parameters begin at small random values. The dashed (dotted) line is the analytic In general the bound on Var Q derived from Eq. (19) bound derived from the ﬁrst (second) qubit. As α increases, is a looser lower bound than Eq. (16). For example, with 2 1 1 the second qubit becomes the source of the relevant bound. α = (1, 2 ) and α1 = ( 2 , −1), this implies that β = 4 2 ( 5 , 5 ). Equation (19) would suggest that

2 25 However, if α2 < 1/4, then the lower bound should be Var Q ≥ , (20) 1/4. That behavior is precisely what we find through 36t2 the numerical optimization shown in Fig.2, confirming which is looser than the 1/t2 given by Eq. (15). This Eq. (15). discrepancy can be addressed by thinking more closely about the process of choosing a new basis. We will use the seminorm condition again to bound the maximum pos- B. Using Single-Parameter Bounds sible Fisher information. To start calculating the seminorm, we express it in terms of the elements of β: It is tempting to dismiss the above argument as un- necessarily complicated, as the ultimate quantity of in- X 1 X kgˆk = k β σˆzk = |β | . (21) terest is only a single parameter. Why not simply apply s j 2 j s j j j the Cramér-Rao bound directly to α · θ instead of using the matrix approach? We will now show that the We will now show that it is possible to choose a basis single-parameter bound that arises from naive applica- such that the seminorm in Eq. (21) goes to infinity. This tion of the Cramér-Raobound is looser than Eq. (15). shows that the approach which led us to Eq. (19) should This gap occurs because the single-parameter bound can not be applied blindly, and we will then discuss how to only be applied if there is only one unknown parameter control for this issue. First, an illustration of the bound controlling the evolution of the input state, which im- diverging. Suppose that in a two-parameter problem, plicitly places a constraint on the other components of the basis vectors we choose are α and α0. It can then be the field. Later, we will discuss how the single-parameter shown by direct computation of the matrix inverse that approach can be amended to take this into account and yields the dual basis that the implied maximum Fisher agree with Eq. (15). information from Eq. (21) is: To apply the single-parameter Cramér-Raobound to 0 0 our evolution Hamiltonian Eq. (2), we consider the α2 + α1 1 F ≤ kgˆks = 0 0 . (22) Hamiltonian as 2 θ · σˆ where σˆ is simply a vector of op- |α1α2 − α1α2| erators whose ith element isσ ˆz. We then rewrite θ in a i 0 new basis, If we then choose α = (α1/α2 + ε, 1), it follows that:

α1 N−1 1 + ε + X F ≤ α2 . (23) θ = (αi · θ) βi. (17) εα2 i=0 As ε → 0, this becomes arbitrarily large. From this we We assume that α0 = α and that the other αi>0 make conclude that our previous approach was ill-advised as it up a basis. The set of vectors βi is then a dual basis can yield arbitrarily small lower bounds on the estimator such that αi · βj = δij. (For this basis as well, we will variance – using this basis, we would conclude that the drop the subscript 0 to indicate that this particular vector right-hand side of Eq. (19) could be ∞. 5

In order to produce a useful bound from Eq. (21), we bounded by 1/t2 as shown in Sec.IV. Then: recognize that any possible choice of basis must yield a P |α | 1 valid bound. Therefore, rather than look at one particu- Var (Q0) = Var Q i ≥ , (28) lar basis (as we did in deriving Eq. (19)) we instead need kαk2 t2 to optimize for the highest lower bound over all possible kαk4 choices of basis. Finding the tightest bound on Fisher =⇒ Var Q ≥ 2 . (29) t2 (P |α |) information will then produce the highest lower bound i on parameter uncertainty. To do this, we first write the This saturates the bound in Eq. (19). The reason we following chain of inequalities using the relationship of α are able to outperform Eq. (15) is that we have assumed and β: something about the structure of the field which reduces X X X it to a lower-dimensional problem. This is only possible 1 = αjβj ≤ |αjβj| ≤ |βj| , (24) by using knowledge about components of θ not parallel j j j to α. Otherwise, there is no guarantee that θ will be proportional to α. In general cases w · θ will contain where the last line follows due to the fact that |αj| ≤ 1. noise from “undesired” components. Note that we can achieve equality, In many situations where the field structure is known, X new strategies can be introduced which may outperform |βj| = 1, (25) our previous results, even asymptotically. Consider as an j example a case with a field: by taking the other N − 1 basis vector αj to be unit vec- α θ = θ 2 + θγ γ, (30) tors ej in the standard basis, making sure that the j that kαk does not appear has αj 6= 0 to ensure the entire space is spanned. Now we look to the minimum possible value of where α·γ = 0 and θγ is a nuisance parameter describing kgˆk . The minimum possible value is interesting to us the field magnitude orthogonal to α. Any field can be s written in this way to separate out the α component. because the minimum kgˆks will be the choice of basis for which the bound on Fisher information is tightest. Suppose we measure w · θ. We know that: It follows from Eq. (21) and Eq. (24) that the min- 1 Var Q0 ≥ (31) imum seminorm kgˆks is equal to 1, implying that the t2 maximum value for Varg ˆ is 1/4 [21]. Using this to optimize the bound in Eq. (19) over all possible choices of is an achievable bound. By writing w = cαα + cγ γ, re-parameterization implies that Var α · θ ≥ 1/t2, just as decomposing w into its only relevant components, we we found in Eq. (16). can obtain the following bound on Var Q: The single-parameter bound is applicable in our situ- 1 Var Q ≥ + c2 kγk2. (32) ation, but it requires careful accounting of the influence t2c γ of other parameters in the problem. The reason that our α previous results such as Eq. (15) do not hold in this case Therefore, the optimal strategy is to pick a w which max- is that Cramér-Raobound does not apply if we can take imizes w · α while minimizing (preferably to zero) w · γ. advantage of constraints on the signal field θ to improve However, in general, learning the structure of the field our estimation strategy. perpendicular to α is just as difficult as learning the These naive single-parameter bounds can be applied component parallel to α, so beginning from a state of and saturated if the field structure is known before the ignorance, it is still optimal to measure α · θ rather than measurement takes place. To demonstrate, suppose that a different linear combination. for a set of fields θ where we wish to learn α · θ, we know The bound in Eq. (19) can actually be found by that the fields are proportional to αi. Then we can write other statistical methods which fully treat the initial the total field as: multi-parameter structure, for instance, the constrained α Cramér-Raobound of Ref. [35]. It can also be derived θ = q , (26) from the Van Trees inequality [36] by assuming that we kαk2 have pre-existing knowledge that the components of θ and our goal is to estimate q = α · θ. This is now a truly perpendicular to α have a normal distribution of width one-parameter problem, enabling a new strategy which ε and then taking the limit ε → 0. saturates Eq. (19). By defining w as a new vector such If rather than a constraint we simply have some initial information in the form of a prior distribution, the Van that wi = sgn (αi), we can measure the quantity Trees inequality (which takes into account that prior in- P |α | formation) will reduce to the Cramér-Raobound in the w · θ = q i . (27) kαk2 limit of many measurements. This is because the information gained from measurements scales linearly with Since w is a linear combination which satisfies the con- the number of measurements while the prior information 0 dition |wi| ≤ 1, we can estimate q = w · θ with accuracy is static. 6

IV. PROTOCOLS We can also directly evaluate FQ and F for this protocol. FQ can be found by noting that this protocol is 1 P z We now present two protocols that saturate the bound identical to evolution under the Hamiltonian 2 αiθiσî . of Eq. (16) and are therefore optimal. The first be- Therefore the quantum Fisher information matrix FQ is gins from the conceptually simple Greenberger-Horne- simply Zeilinger (GHZ) state or a spin-squeezed state and uses (F ) = t2 α σˆzα σˆz − hα σˆzi α σˆz = α α t2. time-dependent control during phase accumulation to Q ij i i j j i i j j i j produce an output state sensitive to the desired α · θ, (37) while the second method uses a more complicated initial Furthermore, we can show that this FQ satisfies the sec- ˜ state but requires no control during the phase accumula- ond inequality in Eq. (13). The inverse of FQ can be tion. easily written, as FQ simply projects onto α. In order to ˜−1 ˜ T 2 get FQ FQ = α α/kαk (identity on the image of FQ), we must have A. Protocols Involving Time-Dependent Control α α F˜−1 = i j . (38) Q t2kαk4 1. Using GHZ Input State T ˜−1 2 α FQ α is then equal to 1/t , saturating the second We start by considering an N-qubit GHZ state: inequality in Eq. (13) for the basis vector b corresponding to the largest α component, αb= 1. 1 To evaluate the classical Fisher information in this √ |0i⊗N + |1i⊗N . (33) 2 case, we note that the final measurement [37] projects onto one of two outcomes with probability sin2 (α · θt/2) Underσ ˆz evolution, each |1i accumulates a phase relative and cos2 (α · θt/2). Therefore the classical Fisher infor- to |0i. By allowing qubits to accumulate phase propor- mation is simply: tional to the desired weight αi, we obtain a final state in 2 2 ⊗N ∂ sin2 α·θt ∂ cos2 α·θt which |1i has accumulated a total phase of α · θt rel- ( 2 ) ( 2 ) ⊗N ∂α·θ ∂α·θ ative to |0i . We refer to our protocol as “partial time F = + (39) evolution” because it relies on a qubit undergoing evo- sin2 α · θt/2 cos2 α · θt/2 lution for a fraction of the total measurement time (see = t2. (40) x Fig.1). We can realize this by applying ˆσi to a qubit at time ti = t (1 + αi) /2 so that the qubit evolution will be This Fisher information also implies the variance bound identical to evolving it for a time αit. Note that if there in Eq. (16). is a fixed experimental time t, this scheme can realize val- It may seem surprising that an optimal measurement ues of αi ∈ [−1, 1], which motivates our restrictions on can be one in which most qubits spend some of the mea- the values of individual αi. Specifying this sequence of surement time idle. Since more time yields more sig- gates identifies the Hˆc(t) which defines the protocol. The nal, intuition suggests that the most effective strategy result of this protocol is an effective evolution according would make better measurements on the less-weighted to the unitary operator qubits rather than keep them off for much of the measurement time. For example, by disentangling a qubit −i t PN α θ σˆz Uˆ(t) = e 2 i=1 i i i . (34) from the larger state halfway through the protocol, a sep- 1 1 arate measurement could be made on θ1 + 2 θ2 and 2 θ2, Under this evolution, the final state is: which appears to yield more information than just mea- 1 suring the quantity of interest θ1 + 2 θ2. This reasoning 1 t ⊗N t ⊗N −i 2 q i 2 q fails because there is no way to use information about θ2 √ e |0i + e |1i . (35) 1 2 to improve an estimate of θ1 + 2 θ2 without also knowing about θ1. Because we do not know about the individual Now we make a measurement of the overall parity of the parameters, only a measurement of the entire function is ˆ NN x state, P = i=1 σî . The details of this measurement usable and our scheme is optimal in this case. However, and calculation of hPî are given in Ref. [1]; notably, the once we account for pre-existing knowledge about the measurement can be performed locally at each site. Mea- parameter values (drawn from physically-motived esti- surement of the time-dependent expectation value hPî(t) mates or less-precise previous measurements) our bound allows for the estimation of Q with accuracy [37] will instead apply in the regime of asymptotically many measurements (M 1) and in that setting our scheme Var Pˆ(t) sin2 qt 1 will also saturate it [38]. This is because the value of Var Q = = = , (36) D E 2 t2 sin2 qt t2 prior knowledge becomes increasingly low as we accumu- ∂ Pˆ /∂q late measurement data. One advantage of this protocol is that an eavesdropper saturating the bound in Eq. (16) and Fig.2. cannot learn the result of the network measurement by 7 capturing a subset of the nodes’σ ˆx measurement results. Suppose that we perform Ramsey interferometry on This privacy can be shown by tracing out the first qubit such a state [2, 37]. The protocol includes both partial in Eq. (35), which leaves no phase information in the ˆ ˆ π time evolution U(t) and a final rotation pulse Rx 2 = resulting mixed state. The central node can receive the π P x exp −i σî . A final measurement is made of the measurement outcomes from all other nodes but keep 4 i total spin projection Jˆ after applying these operations: its own secret, and no eavesdropper is able to extract z information from the broadcasted results. This is true D E D π π E Jˆ (t) = Uˆ †(t)Rˆ† Jˆ (0)Rˆ Uˆ(t) , (42) even if the central node’s qubit is unweighted (i.e., αi = z x 2 z x 2 0), which follows simply from the properties of the GHZ * N + 1 X state. = σˆx sin α θ t +σ ˆy cos α θ t . (43) 2 i i i i i i i=1

2. Using Spin Squeezed States If we specify that this expectation is to be taken over x y a squeezed state with hσî i ≈ 1 and hσî i = 0, then our signal will be sensitive only to α · θ if each individual The perfect security of the GHZ state arises because phase is small: obtaining the measurement result requires every qubit, but this also implies an extreme sensitivity to noise. This N N noise can be a serious problem for metrological applica- D E 1 X t X Jˆz(t) ≈ sin αiθit ≈ αiθi. (44) tions [39, 40]. Because the GHZ state decoheres faster squeezed 2 2 i=1 i=1 than an individual qubit, the advantage provided by entanglement is nullified if the interrogation time of the This shows that a squeezed state can be used for mea- qubit is limited by its coherence time [41]. However, in surements of linear functions. The sensitivity can then many settings the time spent on a single measurement be calculated just as in Eq. (36), will be much shorter than the decoherence time, for instance, to gather data on short timescales. In these cases, GHZ states still provide a metrological advantage. Note Var Jˆ (t) Var Jˆ Nξ2 Var Q = z = y = . (45) that dynamical decoupling [42] or quantum error correc- D E 2 t2/4 t2 ∂ Jˆz(t) /∂q tion [43, 44] could be used to lengthen the effective de- q=0 coherence time in some cases. In other situations, however, it may be that decoher- We evaluate the sensitivity at q = 0 because we are in- ence is the dominant concern. In these situations, the terested in small signals. Partial time evolution with best strategy uses a highly-symmetric entangled state spin-squeezed√ input beats the standard quantum limit which is more robust to noise than the GHZ state [41]. if ξ ≤ kαk/ N. Note that there are N components√ of α Under dephasing, these states can still offer a constant and therefore kαk will generally be of order N assum- factor improvement over unentangled metrology. In this ing that the moments of the field being measured are well section, we show that spin-squeezed states can also func- distributed. Squeezed states can achieve squeezing pro- tion as inputs to the partial time evolution protocol, and portional to N −1/2 [2, 45], which approaches the bound so may be good candidates for a sensor network operating in Eq. (16) up to numerical prefactors not scaling with in a situation where decoherence limits the interrogation N. time. Squeezed states are collective spin states which, Other highly-entangled states such as Dicke states also due to entanglement, have reduced variance along one have metrological value in the presence of noise and could axis of the collective Bloch sphere at the cost of increased also serve as input states to partial time evolution with variance along an orthogonal axis [2, 45]. Recently, it similarly favorable scaling [30, 47–50]. has been shown that these states may allow Heisenberg- scaling measurements even without single-particle detec- tion, which makes them very attractive for experimental implementations [46]. B. Time-Independent Protocols We consider a state whose overall spin vector is aligned x In this section, we present two other possible mea- along +x, such that hσî i ≈ 1. We assume that the other spin components have zero expectation value, but surement schemes for linear combinations of parameters. that the variance of the collective spin projection Jˆy = Both of these differ from the protocols of Sec.IVA be- 1 P y ˆ cause they prepare a particular state and then allow for 2 i σî is decreased while the variance of Jz is increased. We quantify this effect through the spin-squeezing pa- free evolution during phase accumulation, rather than us- rameter ξ [2], ing pulses to evolve for an effective time of αit on qubit i. We will present time-independent schemes that begin s with both a GHZ-like state and the spin-squeezed state. Var Jˆ ξ = y . (41) Note that these protocols rely on assumptions about the N/4 size of signals θi or the evolution time t. 8

1. Using GHZ-like Input State evaluate the final observable. We can begin with the result of Eq. (43), but with two alterations. First, rather We begin by defining a single-qubit state |τ i, where τ than Uˆ representing a partial time evolution on each is a vector whose elements are τj = −1, 0, 1: qubit, instead it will be the full time evolution operator ˆ P z U = exp (−it θiσî ). Second, we will add an additional N ( operator at the beginning of the protocol, which we write O |0i τj 6= −1 |τ i = . (46) as Qˆ(η): |1i τ = −1 j=1 j N ˆ O i We then define the entangled state |ψ(τ )i as Q (η) = rˆz (ηi) . (51) i=1 1 |ψ(τ )i = √ (|τ i + |−τ i) . (47) i 2 Here,r ˆz is the single-qubit rotation about the z axis. That is, we apply a qubit-dependent rotation about the This state can be understood as a general class that in- z axis before we begin the evolution. The final operator cludes the GHZ state as the case τj = 1 for all j. For Jˆz(t) will be: every τ = −1, spin j is flipped relative to the GHZ state, j while for every τ = 0, spin j is entirely disentangled. ˆ ˆ† ˆ † ˆ† π ˆ ˆ π ˆ ˆ j Jz(t) = Q (η)U (t)Rx Jz(0)Rx U(t)Q(η). In order to measure α · θ, we will evolve |ψ(τ )i under 2 2 (52) the Hamiltonian in Eq. (2) and then measure the follow- The effect of Qˆ is to add an additional phase to the evo- ing observable Π(ˆ τ ): D E lution, meaning the final value for Jˆz(t) can be found ˆ O j τj Π(τ ) = σˆx . (48) by substituting the angles θit + ηi for αiθit in Eq. (43). j As a result, we find that the final expectation value is:

That is, we multiply the outcomes of the individual pro- * N + D E 1 X jective σx measurements for each qubit which was origi- Jˆ (t) = σˆx sin (θ t + η ) +σ ˆy cos (θ t + η ) . z 2 i i i i i i nally entangled with the others (τj 6= 0). It can be shown i=1 that probability distribution of this observable is (53) By using the conditions that hσˆxi ≈ 1 and hσˆyi ≈ 0, we ( i i cos2 (θ · τ t/2) 1, find that: P Πˆ = ±1 τ , θ = . (49) sin2 (θ · τ t/2) −1 D E 1 Jˆ (t) ≈ sin (θ t + η ) . (54) z 2 i i To create a final protocol, we will now randomize the choice of τ , which in turn means we will randomly select Now we introduce a two-step protocol. In the first step, both the initial state and the final measurement. An we perform this sequence (prepare a spin-squeezed state, ˆ overall sensitivity to α·θ can be realized if the probability add qubit-dependent rotations, evolve, measure Jz) with distribution for every individual spin to be τj is given by: ηi = φi, where cos φi = αi. We will call the quantity ˆ+ measured Jz . Then, we repeat the process with ηi = ( α (α ±1) − j j −φi, and call the resulting quantity Jˆ . The expectation 2 τj = ±1 z P (τj) = 2 . (50) value of the sum of these quantities is: 1 − αj τj = 0 N D E D ˆ+ ˆ−E 1 X By then summing over P (τ ), we find that Π(ˆ θ, t) = J + J ≈ sin (θit + φi) + sin (θit − φi) (55) z z 2 2 2 ˆ 2 i=1 1 − t (α · θ) to lowest order in t. Since Π = 1, we N N can use the same approach as Eq. (36) to find that the X X = cos φ sin θ t ≈ α θ t. (56) sensitivity for this measurement is Var Q = 1/t2, leading i i i i i=1 i=1 to the same sensitivity as the time-independent protocol. Here, as in Sec. IV A 2, we have assumed that the phases to be detected, θit, are small enough to make the small- 2. Using Spin-Squeezed States angle approximation. In order to evaluate the sensitivity of this measure- To implement a time-independent protocol that makes ment, we look at the point of zero signal as in Eq. (45). + − P y use of a spin-squeezed input state, we will actually use At zero signal, Jz + Jz gives αiσi . It can be shown P y a two-part measurement protocol. First we will derive a that Var αiσi ≤ 4 Var Jy, and so, by the same calcu- general expression that applies to both parts, and then lations used in Eq. (45), the variance is no more than show how they can be combined. 4Nξ2/t2. Note, however, that this assumes that both Much as in Sec. IV A 2, we will use the Heisenberg evo- Jˆ+ and Jˆ− are measured for time t. For a fairer com- lution of the total angular momentum along one axis to parison, we can replace t with t/2 so the time required 9 for the two-step protocol is the same as for one time- still enhance sensitivity. dependent round. In this case, the sensitivity is no worse Entanglement-enhanced imaging of objects larger than than 16Nξ2/t2. single molecules may also be a fruitful area of research. Interestingly, this two-step protocol requires only An experiment detecting the firing of a single animal neu- single-qubit operations once the initial squeezed state is ron with accuracy near the standard quantum limit has created. This may make it a more tractable scheme for already been performed [64], making exploration of tech- experimental realizations of quantum enhancements in niques surpassing the limit a natural next step. Similar measurements of linear combinations of parameters. experiments could demonstrate an enhancement due to distributed entanglement in the near future.

V. ENTANGLEMENT-ENHANCED MOLECULAR NMR VI. OUTLOOK Many applications of entangled sensor networks may emerge as distributed entanglement becomes easier to We have presented measurement protocols for quan- achieve. In this section we focus on an application which tum networks which are useful for measuring linear may be viable in the near future: nanoscale nuclear combinations of parameters and developed a Heisen- magnetic resonance (NMR) as a form of molecular mi- berg limit for the optimal estimation of linear combi- croscopy. NMR has long been used to investigate the nations. Our protocol can be considered a generaliza- chemical composition of molecular structures and per- tion of entanglement-enhanced Ramsey spectroscopy, as form medical imaging [51]. The spatial resolution of in Ref. [1], to the measurement of spatially varying quan- NMR had been limited to a few micrometers until the tities. In the future, we hope to search for further pro- recent advent of nitrogen-vacancy (NV) center magne- tocols and to remove the requirements of small signal or tometers [52–54]. These magnetometers are sensitive to evolution time where we have imposed them. We identi- nanotesla magnetic fields with spatial resolution on the fied magnetometry in general and nanoscale NMR in par- nanometer scale and can be used to image molecules or ticular as candidate applications of our protocol, but we single proteins deposited on a diamond layer with em- wish to stress our protocol’s significantly broader scope. bedded NV centers [55–57]. In particular, we expect that our protocol will be use- Nanoscale NMR applications are a promising setting ful for measuring spatially varying quantities in contexts for entanglement-enhanced sensor networks. The elec- such as gravimetry [65, 66], spectroscopy [6], and rota- tronic spin associated with an NV center in diamond can tion sensing [67–69]. Note there is also no requirement be operated as a two-level system whose free evolution that the parameters measured in a linear combination be results in the accumulation of phase dependent on the of the same physical source. For instance, a sensor net- local magnetic field [54]. Because NV centers are use- work could measure a linear combination of both electric ful platforms for quantum information processing, en- and magnetic fields. tangling protocols already exist and have been demon- In general, our protocol can be applied in any setting strated experimentally [58–61]. Our protocol is particu- where Ramsey spectroscopy can be applied if the quan- larly useful for studies of chemical or magnetic dynamics, tity of interest is nonlocal. In addition, recent work [70] such as Ref. [62], because the measurement timescale may indicates that spatial correlations in measurements may be much shorter than the decoherence time of the GHZ be a useful tool for noise-filtering and error correction in state, making our noise-free treatment applicable. quantum sensors. Linear combinations of spatially separated field val- Many schemes for quantum sensing rely on coherence ues are interesting measurement quantities in nanoscale in photonic, rather than atomic, degrees of freedom, such NMR. Reference [63] describes an imaging protocol which as spectroscopic microscopy [71]. A recent manuscript, combines many different Fourier spatial modes, and Ref. [31] provides a general framework for treatment of Ref. [56] similarly combines many signals to perform sensor networks which is applicable to photonic systems molecular microscopy. These measurements could be and others. performed more accurately using entangled NV sensors. We would like to thank P. Barberis Blostein, J. Borre- In addition, our entanglement scheme can perform sim- gaard, T. Brun, C. Caves, M. Cicerone, Z.-X. Gong, M. ple subtraction of the signal between two qubits. This Hafezi, M. Oberthaler, J. Simon, V. Lekic, E. Polzik, K. allows common mode noise subtraction between a sensor Qian, and J. Ye for discussions. This work was supported qubit and another qubit exposed only to environmental by ARL CDQI, ARO MURI, NSF QIS, ARO, NSF PFC noise. In general, even if a full GHZ state of all sensors at JQI, and AFOSR. Z. E. is supported in part by the is not feasible, smaller clusters of entangled sensors can ARCS Foundation.

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