Quantum metrology in the presence of spatially correlated noise: Restoring Heisenberg scaling

Environmental noise can hinder the metrological capabilities of entangled states. While the use of entanglement allows for Heisenberg-limited resolution, the largest permitted by quantum mechanics, deviations from strictly unitary dynamics quickly restore the standard scaling dictated by the central limit theorem. Product and maximally entangled states become asymptotically equivalent when the noisy evolution is both local and strictly Markovian. However, temporal correlations in the noise have been shown to lift this equivalence while fully (spatially) correlated noise allows for the identification of decoherence free subspaces. Here we analyze precision limits in the presence of noise with finite correlation length and show that there exist robust entangled state preparations which display persistent Heisenberg scaling despite the environmental decoherence, even for small correlation length. Our results emphasize the relevance of noise correlations in the study of quantum advantage and could be relevant beyond metrological applications.

How small can experimental error bars become? The use of entangled states in quantum metrology shows that the answer of this fundamental question can rely on playing with quantum advantage, that is, exploiting a quantitative (quantum) resource to outperform classical strategies. In a typical metrological set up, an atomic transition frequency is estimated from the phase relation accumulated between the two components in a superposition state. In purely unitary evolution, the relative phase of a Greenberger-Horne-Zeilinger (GHZ) state of the form (|0 ⊗n + |1 ⊗n )/ √ 2 advances n times faster than that of a single qubit and parity measurements allow one to saturate the Heisenberg limit [1,2]. That is, their associated measurement uncertainty decreases as 1/n, which provides a 1/ √ n improvement over the standard quantum limit (SQL) obtained by performing n independent queries on uncorrelated particles. Signal-to-noise ratios (SNR) overcoming the spectroscopic resolution achievable in ideal experiments using single qubits have been demonstrated using three entangled ions [3]. When the dimension of the probe state grows, decoherence effects will no longer be negligible and the question arises of whether Heisenberg scaling can still be attainable under a non unitary (noisy) evolution. In the case of the dynamics being strictly Markovian, and provided the noise is local, both pure dephasing and dissipative losses restore the standard scaling dictated by the central limit theorem even in the limit of arbitrarily small noise levels [4]. Rigorous bounds showing standard scaling under this type of noise have been recently put forward [5,6]. These noise models though make two important assumptions, namely, that the noise stems from a Markovian bath and that this noise acts locally on each subsystem. Relaxing the assumption of Markovianity has been shown to result in a new fundamental limit [7] which lifts the previous metrological equivalence of maximally en-tangled and product states under time-correlated noise and predicts a novel scaling of the form ∼ 1/n 3/4 . In this paper we relax the assumption of noise locality and consider a general model for bath correlation length [8] to show the persistence of Heisenberg scaling under correlated noise of finite length.
In the presence of Markovian dephasing noise, frequency measurements on entangled states generally yield probabilities for population measurements of the form p = [1 + cos(nω 0 t) exp(Γ(n, ξ)t)]/2, where ω 0 is the atomic transition frequency and Γ(n, ξ) the dephasing rate of the state, dependent on the number n of considered qubits and the spatial correlation length ξ of the environmental noise. Considering as fixed resources the total number of particles n and the total duration of the experiment T , the duration of the optimal interrogation time t, and with it the number of repetitions T /t for a given noise source has to be determined [4]. This procedure yields the time-optimized frequency uncertainty of the measured frequency For spatially uncorrelated Markovian decoherence, the dephasing rate of GHZ states scales as Γ(n, ξ) → Γ uc = nγ uc , where γ uc is the dephasing rate of a single qubit superposition state. This yields a resolution ∼ 1/ √ n.
The persistence of the standard scaling under Markovian decoherence is actually valid for optimized (entangled) initial states and generalized measurements, with the optimal achievable resolution ∆ω opt 0 = 2γ uc /(nT ) [4,5]. While the derivation above relies on an independent noise model [9,10], recent experiments with trapped ions have proven to be dominated by spatially correlated dephasing [11,12]. Particularly, measurements of the dephasing rate of GHZ states have shown a clear n 2 de-arXiv:1307.6301v1 [quant-ph] 24 Jul 2013 pendence [12] which can only be explained by strongly correlated noise. When the noise acts globally on all qubits, it is possible to identify suitable decoherence free subspaces (DFS). This allows for the accurate determination of frequency shifts, as illustrated in [13] by using an entangled state of two ions for the determination of the quadrupole moment of 40 Ca + , a quantity of relevance for the calibration of optical frequency standards [14]. It remains unclear, however, what the situation would be when considering larger qubit arrays so that the noise exhibits a correlation length that is smaller than the total length of the system. Using a formalism to consider realistic partially correlated noise, where the correlations decay over a certain correlation length ξ [8], we will show that, even for small ξ, Heisenberg scaling prevails when using certain types of entangled states for the estimation of small frequency shifts, as those involved in the precise estimation of an atomic quadrupole moment. We consider a system of n hydrogen-like ions with a Zeeman splitting term of the sublevels J z of the total angular momentum and a small correction term due to the interaction of the atomic electric quadrupole moment with the external electric field gradient. These correction terms are quadratic in J z [13,15].
Laser frequency noise and magnetic field noise make dephasing by far the strongest decoherence source, effectively coupling each ion to a fluctuation via J z , The coupling strength v defines the total decoherence strength by the coefficient γ 0 = v 2 in all dephasing rates, so for simplicity we set v = 1. The bath operators' B j spatial and temporal correlations are determined by the function C(ω, where we assume the ions are spatially arranged in a linear array (figure 1). We employ Bloch-Redfield equations with a Markovian approximation and assume homogeneous, decaying spatial correlations C(0, xd) = exp(−|xd|/ξ) with the correlation length ξ, the distance d between ions in the onedimensional array and x ∈ N. We arrive at the master equation for the system density matrix ρ as [8]: As initial states we consider maximally entangled states of the form (|m 1 , m 2 , . . . , m n + |m n+1 , . . . , m 2n )/ √ 2 where the magnetic quantum numbers m j of the operators J (j) z satisfy n j=1 m j = 2n j=n+1 m j . The two parts of the superposition are Zeeman-shifted by the same amount but their quadrupole moment can be different. We restrict ourselves to three sublevels in this paper: For example in ref. [13] the levels m ∈ {3/2, −1/2, −5/2} in 40 Ca + ions were used. We choose for one part of the initial entangled state all ions to be in 0 and in the other part half of the ions in + and half in This simplifies J z = diag( + , 0 , − ) and the coherent evolution is given by where the relative frequency is given by the quadrupole splitting ω 0 = α 2 ∆ . This frequency is measured with a parity measurement and we will regard the uncertainty scaling with n of this transition frequency.
Uncorrelated Markovian decoherence (ξ → 0) always restores the standard quantum limit [4], whereas for the chosen states correlated decoherence leads to Heisenberg scaling: In the limit of infinite correlation length ξ → ∞ one can define a collective operator S = j J (j) z in Eq. 3 which restores coherent evolution for the density matrix element of interest | + , + , . . . , − , − , . . . 0 , 0 , 0 , . . . |, because S acting from the left or from the right on it is equivalent. This decoherence-free subspace of all states with equal excitation number [8] guarantees the coherent Heisenberg scaling.
We will now discuss the persistence of Heisenberg scaling under partially correlated noise and assume a finite but non-vanishing correlation length ξ > d. The timeevolution of the coherence of interest ρ ±,0 is given by: The dephasing rate Γ(n, ξ) depends on the number of ions n involved in the superposition and the correlation length ξ. In stark contrast to uncorrelated decoherence, it also depends on the order of the ions in the initial state. In Eq. 3 all pairs of ions contribute terms to Γ(n, ξ). Both autocorrelations and cross-correlations of those pairs which are in the same state (i.e. both With the assumed correlation function C(0, xd) = exp(−|x|d/ξ) one finds: To judge whether entangled states give an advantage over the standard quantum limit (∝ 1/ √ n) we need to determine whether the dephasing rate Γ(n, ξ) scales faster or slower than n. For comparison, the dephasing rate for uncorrelated decoherence is Γ uc = −n 2 ∆ /2 (see appendix or [4]).
The frequency ω 0 = α 2 ∆ cannot be measured in a single ion; one needs at least an entangled state of two ions to realise it. To obtain a meaningful comparison for the scaling of the frequency uncertainty this must be taken into account. We therefore define two entangled ions as the minimum entanglement resource for measuring a quadrupole moment. We then compare the scaling of the n-entangled state with a product state of n/2 entangled pairs which contribute n/2 more measurements to the statistics. For this minimal entangled array of ion pairs, we find an uncertainty that scales with the SQL as we increase the number n/2 of pairs, ∆ω 0,p = eΓ(2, ξ)/(nT ).
In contrast to uncorrelated decoherence, there is no unique way of increasing the number of entangled ions for a given noise correlation length. When considering the scaling of the dephasing rate with increasing numbers of ions, one can either keep the length L of the ion array fixed, or the density of ions fixed. We now analyze the achievable spectroscopic resolution in both cases ( figure  1).
First we set the correlation length to a fixed number of ions ξ = cd (figure 1A) which means that the array gets longer relative to the correlation length as we increase n (fixed ion density). This will ultimately restore the SQL when nd ξ ( figure 2). The gradient of the dephasing rate in this case can be approximated as 2 ∆ d/(4ξ) for ξ > d. This gradient is closer to zero than for uncorrelated decoherence Γ uc because the finite correlation length reduces the dephasing-rate contribution from each ion slightly. So even though the scaling follows the standard quantum limit, one finds a better coefficient than for uncorrelated decoherence.
Alternatively, we can scale the correlation length as a fraction c of the whole array ξ = cL (figure 1B), fixing the correlations between the first ion and the last ion in the array to a value C(0, L) = exp(−1/c). Figure 2 shows that in this case the dephasing rate quickly approaches the constant Γ(n, ξ) = [−(L/ξ) + exp(−L/ξ) − 1] 2 ∆ /4, which can be approximated as 2 ∆ L/(2ξ) for long correlation lengths ξ L. With this constant rate the corresponding uncertainty (Eq. 1) displays Heisenberg scaling (figure 3): We now introduce the relative frequency resolution: where the full expression for r is given by equations 1, 6 and 8. We find that with increasing correlation length ξ the uncertainty approaches the noiseless Heisenberg scaling ( figure 3). Even for partial correlations, which decay on the length scale of the array, the Heisenberg scaling of the uncertainty is robust. The previous observations have more general applications beyond quadrupole measurements. Heisenberg scaling in spatially correlated environments can generally be  An n-entangled state scales better by a factor of √ n than a pair-wise entangled state and approaches noiseless scaling for increasing correlation length ξ.
achieved for entangled states which are superpositions of states with the same number of excitations. A counterexample however are GHZ states. In a noiseless environment a frequency measurement will show Heisenberg scaling of the uncertainty 1/(n √ T t). For spatially uncorrelated noise their dephasing rate scales with n, leading (as for the previously considered initial states) to the SQL for the uncertainty 2eγ uc /(nT ). In spatially correlated noise GHZ states are even more fragile, their dephasing rate scales with n 2 , leading to an uncertainty 2eγ/T , which no longer decreases with n at all [16]. In spatially correlated noise environments GHZ states are therefore strongly disadvantageous.
From our master equation (Eq. 3) it follows generally that for ξ → ∞ the dephasing rate between two states with a difference of n e excitations is proportional to n 2 e . States with the same number of excitations have n e = 0 and form a decoherence-free subspace, whereas GHZ states have n e = n and are the most fragile states in spatially correlated environments.
Up to now we have considered perfect spatial correlations, C(0, x) = 1, and decaying spatial correlations, C(0, x) = exp(−x/ξ). Both are positive functions for all x, and the statements of the last two paragraphs are only valid under this condition. The n 2 scaling of the dephasing rates found experimentally for GHZ states indicates that in ion traps these two functional forms are good approximations for the noise correlations. However, it is also physically possible for the spatial correlations to take the homogeneous form C(0, x) = cos(kx), where points at specific distances have noise with negative correlations [8]. In such an environment GHZ states can be engineered to be within a decoherence-free subspace by arranging an array of sites such that the array length L matches the oscillation length of the environmental spatial correlations L = 2π/k (see appendix). Note that uniformly negative correlation functions are impossible due to the requirements of positive autocorrelations and multipartite correlation rules.
Previous experimental evidence suggests that the environmental noise in ion traps is spatially correlated with purely positive correlations. We showed that non-zero spatial correlation length fundamentally changes the decoherence of entangled states. In such environments, a topology dependence emerges so that the order in which the ions are placed in the array changes their decoherence properties. After optimisation in this regard, the entangled states designed to measure the electric quadrupole moment have an approximately constant dephasing rate with increasing number of ions n. Precision frequency measurements with these initial states therefore show Heisenberg scaling of the uncertainty ∆ω 0 ∝ 1/n with the numbers of ions n. Besides providing a prescription to achieve Heisenberg-scaled resolution in linear ion traps subject to partially correlated noise, our results illustrate the fundamental role of noise correlations in precision spectroscopy. While local Markovian noise eliminates quantum advantage, this is restored when the noise displays a spatial structure. Heisenberg resolution becomes then attainable by means of suitable state preparation whose decoherence rate decreases inversely with ξ so that the evolution is decoherence-free in the limit of infinite correlation length (global noise).
We thank A. Greentree, N. Vogt and T. Dubois for helpful discussions and M. B. Plenio for feedback on the manuscript. We acknowledge financial support from the European Commission through the STREP project PICC.  Note that in the appendix we give ξ in units of d and thereby avoid the appearance of d in the enumerator. The density matrix element of interest is: ρ ±,0 = | − , + , − , + , ... 0 , 0 , 0 , ..., 0 | The coherent part is easily calculated: i(ρ ±,0 H s − H s ρ ±,0 ) = i(βn 0 + αn 2 0 − βn 0 − αn( 2 0 + 2 ∆ )ρ ±,0 = −inα 2 ∆ ρ ±,0 = −inω 0 ρ ±,0 To calculate the decoherent part of eq. 15 for the element ρ ±,0 we regard how J (j) z acts from the left and from the right onto element ρ ±,0 (figure 4 ): Next we calculate the sum over each of the three terms in eq. 15 for a fixed distance of ions |j − k| = x and then regard the coefficients of exp (−x/ξ). There are (n − x) pairs of ions with a distance x between them, which can be seen by moving a fixed distance along figure 4. For x > 0 each pair is counted twice because j will be the right one and the left one once. We distinguish three cases: