Effects of Systematic Error on Quantum-Enhanced Atom Interferometry

State preparation error occurs when we believe that we have prepared the state

but have actually prepared the state in Equation 1, with $\mathbf{\Lambda}\neq\mathbf{\Lambda^{\prime}}$. We use the notation that the dashed superscript denotes that a variable is assumed, while the absence of a dash implies the actual physical value. Given a state preparation error, the $\lambda_{p}$ value used in our estimator will be $\lambda_{p}=\lambda^{\prime}$ instead of $\lambda_{p}=\lambda$, as this will correspond to the estimator we think will be unbiased. As our estimator is then parameterised by the wrong value, it will output an expected value $\langle\tilde{\varphi}(\mathbf{X}|\lambda^{\prime})\rangle=\varphi^{*}$, where the true phase value $\varphi\neq\varphi^{*}$ in most instances. In addition to the introduction of bias, the biased estimator will lead to different single shot quantum noise $Q$ relative to an unbiased estimator. Furthermore, assuming the wrong state preparation degree of freedom may bias other state preparation parameters dependent on $\lambda$, thus changing the measurement data as well as the estimator. These distinct measurement data may cause the state preparation error to further change $Q$ relative to an unbiased quantum-enhanced atom interferometry sequence. Figure 2 demonstrates how each of the actual, assumed, and estimator state preparation values interact to produce a value for the error in the parameter of interest.

In many cases, such as when using spin-squeezed states, bias induced by state preparation error will be linearly proportional to the parameter $\varphi$ for $\varphi\ll 1$, and the total MSE will be characterised by the parameters $B$ and $Q$:

where $B$ is defined by $\sigma_{B}=B\varphi$. As the bias decreases when operating close to the bias-free operating point of $\varphi=0$, it is desirable to operate as close to this point as possible. Adaptive schemes will be ultimately limited by the resolution of the measurement. As such, we can obtain a self-consistent lower bound on the MSE by setting $\varphi=Q/\sqrt{m}$ in Equation 8, which yields $\text{MSE}\geq E^{2}/mN$, where

is the total error of the measurement, normalised by the precision attainable in a single bias-free, uncorrelated atomic ensemble. Alongside $B$ and $Q$, $E>1$ is a useful indicative quantity as it implies that the presence of state preparation error has worsened the precision of atom interferometry relative to using an unentangled state - one would have been better off dispensing with entangled state preparation entirely.

We consider two spin-squeezing methods to study spin-squeezed states under state preparation error, namely two-axis twisting (TAT) [46, 55] and one-axis twisting (OAT) [56, 57, 39, 32, 20]. TAT has yet to be experimentally realised, but provides the “simplest” example of an entangled state preparation sequence, given that its optimal $\hat{J}_{x}$ rotation parameter is independent of $\lambda$. It has been shown that TAT dynamics can be produced in cavity-mediated atom-atom interactions [58], and can produce a state similar to that prepared by quantum-nondemolition measurement (QND) squeezing [48, 32, 59, 34, 49, 35]. For an initial $\hat{J}_{x}$ eigenstate, the TAT Hamiltonian and state preparation sequence are

where $\lambda=\chi t$. OAT has also been experimentally demonstrated [19, 20, 31]. The OAT Hamiltonian is given by

and the OAT state preparation sequence is given by

where the optimal $\hat{J}_{x}$ rotation $\theta$ is dependent on $\lambda$, and thus a suboptimal rotation will be applied if $\lambda^{\prime}\neq\lambda$.

We simulate the TAT and OAT interferometry sequence according to Equations 22 and 20 for $N=100$ atoms in an initial $\hat{J}_{x}$ eigenstate given a sample number $m=10^{4}$, a phase $\varphi=\frac{Q}{\sqrt{m}}$ encoded anti-clockwise around the $\hat{J}_{y}$ axis, and a range of actual and assumed $\lambda$ and $\lambda^{\prime}$ values. For each $\lambda$, $\lambda^{\prime}$ pair, we calculate $B$, $Q$, and $E$ according to Equations 16, 18, and 9 respectively. In the case of both OAT and TAT, we find that $\lambda^{\prime}\neq\lambda$ leads to a biased estimate of the encoded phase as $\tilde{\varphi}$ is parameterised by $\lambda_{p}=\lambda^{\prime}$ instead of $\lambda_{p}=\lambda$. In both cases, $\lambda^{\prime}<\lambda$ leads to negative bias and $\lambda^{\prime}>\lambda$ leads to positive bias, as can be seen in Figure 4 (b) and (f). The magnitude and sign of the bias depend on the discrepancy between the assumed and actual $\langle\hat{J}_{x_{0}}\rangle$ values, as per Equation 16, with the dependency of $\langle\hat{J}_{x_{0}}\rangle$ for both TAT and OAT displayed in Figure 4 (a) and (e) respectively. For both OAT and TAT in the spin-squeezing regime, the $\langle\hat{J}_{x_{0}}\rangle$ value begins to decrease at an increasing rate as $\lambda$ increases. This results in the slope of the graph of $B$ increasing for both OAT and TAT with respect to the assumed state preparation parameter.

The plots of $E$ for OAT and TAT are qualitatively distinct. In the case of TAT in Figure 4 (d), we find that $E$ increases uniformly as the assumed state preparation value $\lambda^{\prime}$ increases, with $E$ for $\lambda^{\prime}<\lambda$ lower than for $\lambda^{\prime}>\lambda$. This has a surprising implication: that the “wrong” estimator may be capable of improving the precision of an atom interferometry experiment. Whilst $E$ is a lower bound on the achievable error, given the small contribution of the $\varphi$ term in Equation 18 and the neighbouring plot of $Q$ in (c), we expect that $E$ will be close to the actual error achievable for some $m$ and $\varphi$ values. We will investigate this in the next subsection. The decrease in $Q$ comes because the variance of the parameter estimate is reduced as a result of underestimating $\langle\hat{J}_{x_{0}}\rangle$ according to Equation 18. The increased rate of $Q$ increase for $\lambda^{\prime}>\lambda$ owes to the increased rate of decrease in $\langle\hat{J}_{x_{0}}\rangle$ in the same expression. For OAT, we find that $E$ is optimal when $\lambda^{\prime}=\lambda$. This is because state preparation error in OAT leads to the “wrong” $\hat{U}_{\theta}$ being applied, meaning that the axis of maximal spin-squeezing is not along the measurement axis $\hat{J}_{z}$ - the data $\mathbf{X}$ produced are suboptimal. As a result, greater single shot variance ensues, as can be seen in (g). We note that $Q$ climbs much more steeply for $\lambda^{\prime}<\lambda$ than for $\lambda^{\prime}>\lambda$. This owes to the fact that the optimal $\theta$ angle changes much more quickly for low $\lambda$ values than for higher $\lambda$ values.

The qualitative distinction between $E$ for OAT and TAT can be understood by considering the Q-sphere phase space representations of TAT and OAT states before parameter encoding in Figure 4 (i), (j), and (k). In (i), we see a TAT quantum state for $\lambda=0.02$ for all $\lambda^{\prime}$ values. As the $\hat{J}_{x}$ rotation angle does not depend on $\lambda$, the axis of maximal spin squeezing is always aligned to the $\hat{J}_{z}$ axis. In (j), we see an OAT state with $\lambda=0.034$ prepared with an assumed value $\lambda^{\prime}=0.014$, which, in contrast to (i), results in the axis of maximal spin squeezing not aligning to the readout axis $\hat{J}_{z}$ as $\theta$ is dependent on $\lambda$. The corresponding OAT state for $\lambda=\lambda^{\prime}=0.034$ is displayed in (k), and is aligned to the $\hat{J}_{z}$ readout axis.

The quantum state that is most robust to state preparation error in Figure 4 is the TAT state for small $\lambda$ ($\lambda=0.005$). This is not only because $\theta$ is independent of $\lambda$, but also because its $\langle\hat{J}_{x_{0}}\rangle$ moment changes negligibly with $\lambda$ around this region. By Equations 16 and 18, this means bias will be small, and $Q$ will nearly be equal to $Q(\Delta\lambda=0)$. In general, thsee equations suggest spin-squeezed quantum states around which $\partial_{\lambda}\langle\hat{J}_{x_{0}}\rangle=0$ (and where $\Delta\lambda\neq 0$ doesn’t influence later state preparation parameters) will be robust to state preparation error.

In Figure 4 (d), we see for $\lambda^{\prime}<\lambda$ that $E$ decreases below the error for a TAT interferometry sequence free from state preparation error (for $\Delta\lambda=0$, $E$ is saturated). Using an estimator $\tilde{\varphi}(\lambda_{p}=\lambda^{\prime})$ rather than $\tilde{\varphi}(\lambda_{p}=\lambda)$ led to a decreasing lower bound on the state preparation error. Although this did not demonstrate that a biased estimator would necessarily reduce MSE, it indicated that such a scenario may arise for TAT, since we expect $E$ to be close to actual MSE. To investigate how using a biased estimator may decrease MSE, we fix the parameter of interest at $\varphi=0.001$, and plot the ratio of the lowest mean squared error $\text{MSE}_{\text{opt}}$ to MSE$(\lambda_{p}=\lambda)$ given a set of estimators $\tilde{\varphi}(\lambda_{p})$ with $\lambda_{p}$ spanning from $0$ to $\lambda$, over a range of state preparation parameters and sample sizes.

In Figure 5 (a), we see that for every $\lambda$, the unbiased estimator never produces the least error - the optimal ratio between the best biased MSE and the unbiased MSE is always less than 1. We thus see that atom interferometry with TAT states would be amenable to the bias variance trade-off, where a biased estimator decreases overall error relative to an unbiased estimator because of its lower variance.

The bias variance trade-off becomes more useful for higher $\lambda$ values. This owes to the fact that $\langle\hat{J}_{x_{0}}\rangle$ is smaller for higher $\lambda$ as shown in Figure 4 (a), meaning the denominator in Equation 18 can be artificially increased by picking an estimator that “overestimates” $\langle\hat{J}_{x_{0}}\rangle$. We note that the larger $m$ is, the lower the achievable bias variance trade-off. This is because as $m$ increases, the non-zero bias term from the biased estimator stays constant, whilst $\sigma_{Q}$ decreases as $\frac{1}{\sqrt{m}}$ - the decrease in the variance possible from choosing a biased estimator is counteracted by the bias. In Figure 5 (b), we see the $\lambda_{p}$ value parameterising the MSE-optimising estimator for a range of $\lambda$ and $m$ values. The $m$ value at which the optimal $\lambda_{p}\approx\lambda$ decreases for more highly squeezed states. This is because more highly squeezed states are more susceptible to the biasing effects of state preparation error. As per Figure 4 (b), this bias is reduced by having $\lambda_{p}$ closer to $\lambda$, where we interpret $\lambda^{\prime}$ as being $\lambda_{p}$ in this subsection. The above results suggest that biased estimators may lead to lower measurement error if TAT can be realised and manipulated in an atom interferometry experiment. For other state preparation schemes where the measurement results $\mathbf{X}$ are not affected by $\lambda^{\prime}$, it is likely that similar bias variance trade-offs exist.

We now investigate the effect of state preparation error on atomic ensembles large enough to produce useful measurements ($N>10^{4}$). Numerically simulating an interferometry sequence with this number of atoms is intractable, due to the size of the Hilbert space. To probe the sensitivity of larger number of atoms under systematic state preparation error, we use the analytic expressions for the moments of OAT states in Appendix VI.2, combined with Equations 16 and 18. We assume that an experimenter seeks to produce maximally spin-squeezed states. The maximally spin-squeezed OAT state is reached at a $\lambda$ value of [46]

where the above expression is derived in the limit of small $\lambda$ which is valid so long as $N>>1$. We define $\Delta\lambda_{\text{crit}}$ as the value of $|\lambda^{\prime}-\lambda|$ such that $E$ passes a certain threshold (in the case of our results, $\Delta\lambda<0$ when these thresholds are reached). This places a bound on the maximum sensitivity achievable with a $N$-atom OAT ensemble under state preparation error at the $\lambda^{\prime}$ where this threshold is passed for all $\varphi$ operating points.

We first consider a set of thresholds relative to the quantum noise of an unbiased state preparation scheme - $E(\Delta\lambda=0)$. We also consider a set of thresholds relative to the amount of error achievable with an unentangled state, denoted $E_{\text{un}}$.

We plot $\frac{\Delta\lambda_{\text{crit}}}{\lambda}$ for a range of thresholds as a function of $N$. In Figure 6 (b), we see $\frac{\Delta\lambda_{\text{crit}}}{\lambda}$ for $E(\Delta\lambda)/E(\Delta\lambda=0)=2$, $4$, $6$, and $8$. For each of these thresholds, less percentage error in $\lambda$ is required to cause an equivalent error factor as $N$ increases. This is because as $N$ increases, so does the maximal spin-squeezing one can generate, as shown in Figure 6 (a). More squeezed states are more susceptible to suboptimal $\hat{J}_{x}$ rotation angles, which means a lower percentage of systematic error is required to decrease the precision by a particular factor. In (c), we see that roughly equivalent magnitudes of state preparation error at varying $N$ values lead to the same precision decrease relative to constant fractions of shot noise. This suggests that squeezing under state preparation does not get much worse for large $N$ relative to the precision achievable with unentangled ensembles of atoms.

The values in Figure 6 (b) and (c) suggest that entangled state preparation error is a significant hindrance to precision in quantum-enhanced atom interferometry. Given the multitude of ways of realising OAT, we will not attempt to analyse state preparation error in the context of a specific experimental configuration. However, our results provide a useful starting point for an experimenter wishing to do so. More generally, they offer a schematic for how to calculate error in parameter resolutions under a particular state-generation Hamiltonian for atomic ensembles large enough to produce metrologically useful measurements.

We investigate OAT and TNT non-Gaussian states under state preparation error using Monte Carlo techniques for small state preparation errors, and compare them to the Taylor expansion-based estimate. The TNT Hamiltonian is defined by

where $\Omega=\frac{\chi N}{2}$ [26, 60, 61]. Non-Gaussian OAT states have yet to be produced, whilst non-Gaussian TNT states have been produced but have yet to yield subshot noise precision in an atom interferometry experiment [24]. The QFI for a non-Gaussian OAT is maximised without a $\hat{J}_{x}$ rotation, and thus the state preparation sequence is

In the case of TNT, the QFI can be increased by applying an appropriate $\hat{J}_{x}$ rotation just before parameter encoding. Whilst the optimal rotation is dependent on the state preparation parameter $\lambda$, highly metrologically useful states can be obtained with $\theta=-1$, with the upside of a system that is simpler to experimentally implement and analyse. This results in a TNT state preparation sequence of

where $\lambda=\chi t$ and $\delta=\Omega t$, with the optimal value of $\delta$ being $\delta=\frac{\lambda N}{2}$. For both OAT and TNT, we seek to perform an optimal readout, i.e. the readout that saturates the QCRB. The optimal readout basis for OAT and TNT non-Gaussian states is $\hat{J}_{x}$ [62, 61, 63]. A readout in this basis can be realised through a $\pi/2$ rotation clockwise about the $\hat{J}_{y}$-axis, followed by a standard number-difference measurement. Thus, we attempt to find the value $\varphi$ that maximises the value of the log-likelihood function

for OAT and TNT respectively.

We simulate the generation, readout, and estimation of a parameter encoded onto the quantum states through Monte-Carlo sampling. We perform the simulation for states in both the spin-squeezed and non-Gaussian regimes, with the non-Gaussian regime starting for OAT at $\lambda\approx 0.1$, and for TNT at $\lambda\approx 0.045$. For OAT, we consider $\lambda^{\prime}=\lambda\pm 0.0025$ for $45$ $\lambda$ values between $0.01$ and $0.17$, and for TNT, we consider $\lambda^{\prime}=\lambda\pm 0.0015$ for $45$ $\lambda$ values between $0.01$ and $0.11$. For each $\lambda,\lambda^{\prime}$ pair, we perform a maximum likelihood estimate parameterised by $\lambda_{p}=\lambda^{\prime}$ given $10^{5}$ measurements of the prepared quantum states, and repeat these $5$ and $25$ times for each $\lambda,\lambda^{\prime}$ pair for OAT and TNT respectively. We calculate the bias, and the ratio of the quantum noise $Q$ to the QCRB, comparing the biased estimate to that derived from Equation 26. Given that bias is not proportional to the parameter of interest for an MLE estimator, a similarly useful lower bound to $E$ for spin-squeezed states cannot be devised, and the parameter of interest is fixed at $\varphi=0.02$ for the simulations.

In the case of OAT in Figure 7 (a), we see that the covariance matrix ratio in Equation 26 reliably predicts the bias in both the Gaussian and non-Gaussian regimes, where the start of the non-Gaussian regime is denoted by the dotted red line. In (b), we see that under small state preparation error in the non-Gaussian regime, OAT is essentially immune to state preparation error - $Q/\text{QCRB}$ remains constant. This is because the likelihood function being optimised changes negligibly from the unbiased likelihood function despite $\lambda_{p}\neq\lambda$, which can be inferred from the small $\frac{F_{\varphi\lambda}}{F_{\varphi\varphi}}$. In the case of TNT, we see that the approximation correctly predicts bias in the Gaussian regime, but is inaccurate past early in the non-Gaussian regime. This can be attributed to higher-order derivatives being required to characterise the likelihood function even at this small amount of state preparation error - the metric of Equation 26 is of limited use in the absence of likelihood functions that change sufficiently slowly with respect to the state preparation parameter. We see that in contrast to OAT, $Q$ in the case of non-Gaussian TNT states strays significantly from the QCRB even for small state preparation error. Currently, the only non-Gaussian atomic states that have been generated with a significant ($N>10^{3}$) atomic ensemble are non-Gaussian TNT state. Our results indicate that state preparation error is likely to degrade the overall precision of metrology with non-Gaussian TNT states.

OAT’s robustness to state preparation error under a maximum likelihood estimator raises an interesting proposition - that under a drifting state preparation parameter, such as in the case of phase diffusion, one would be better off dispensing with a method of moments of estimator in favour of a maximum likelihood estimator, even in the spin-squeezing regime. To investigate this, we compare the $\sigma_{B}$ value for a MOM estimator and an MLE estimator for constant $\Delta\lambda$ in the absence of any $\hat{J}_{x}$ rotation. We calculate $\sigma_{B}$ for the MLE estimator through the first order Taylor expansion in Equation 26, as this approximation is reliable in the case of OAT for larger $\Delta\lambda$, and calculate $\sigma_{B}$ for the spin-squeezed state with Equation 16. For the OAT states where an MLE estimator is used, the measurement occurs in the $\hat{J}_{x}$ basis as opposed to the $\hat{J}_{z}$ basis.

In Figure 8 (a) and (b), we see the $\sigma_{B}$ value for an OAT state for varying $\lambda$ and constant $\Delta\lambda$ given a fixed $\varphi$ value of $0.02$, for an MOM and MLE estimator respectively. Whilst $\sigma_{B}$ is smaller in the case of the MOM estimator for $\lambda<0.015$, the MOM estimator dramatically reduces the total bias for larger $\lambda$ values. This suggests that under considerable phase diffusion for larger $\lambda$ values, it is wise to use the MLE estimator instead of the canonical MOM estimator.

In both spin-squeezed and non-Gaussian states, we see that systematic state preparation error mostly leads to bias in parameter estimation. Whilst we show that there exist classes of spin-squeezed and non-Gaussian quantum states that are robust to bias from state preparation error, it may not always be possible to manufacture such a state. We present a solution that keeps the estimate of the encoded parameter $\varphi$ unbiased - we discard our assumed value of the state preparation degree of freedom $\lambda^{\prime}$, and estimate both $\lambda$, and $\varphi$ according to two-parameter maximum likelihood estimation.

For a two-parameter maximum likelihood estimate of a state preparation parameter $\lambda$ and a relative phase $\varphi$, the sensitivity of an estimate of $\varphi$ is given by one on the square root of the upper-diagonal term of the Fisher information matrix,

In a non-diagonal Fisher covariance matrix, $Q_{\text{two-param}}>Q_{\text{one-param}}$ - estimating both parameters at once leads to more error in $\varphi$, than using $\lambda_{p}=\lambda$, and estimating $\varphi$. However, under state preparation error that induces bias, overall error may be lower under two parameter estimation for sufficient sample size.

We demonstrate that this two parameter estimation strategy can result in lower error for a sample size of $m=250$ in both the spin-squeezed and non-Gaussian regimes in Figure 9. We plot $\sqrt{N\text{MSE}}$ for TAT with from $\lambda=0.005$ to $\lambda=0.015$ for $\lambda^{\prime}=\lambda+0.005$, and TNT for $\lambda=0.05$ to $\lambda=0.08$ and $\lambda^{\prime}=\lambda+0.01$, alongside the two-parameter CRB. The TAT state is prepared according to Equation 20, and the TNT state is prepared according to Equation 30. In Figure 9 (a) and (b), we see precision for TNT and TAT under systematic state preparation error, and under two parameter maximum likelihood estimation, respectively. In both instances, we see that the two parameter estimate leads to increased precision. This indicates that using two parameter MLE where a state preparation parameter is not perfectly known is a viable method for precise quantum-enhanced atom interferometry, as well as a means of eliminating measurement bias.

We wish to calculate

in terms of the $\hat{J}_{x}$ rotation $\theta$, and the encoded parameter $\varphi$. We take Equation 13, and compute the variance on the Heisenberg evolved operators as

where $\overline{\text{Covar}}(a,b)=\text{Covar}(a,b)+\text{Covar}(b,a)$. Given that the initial spin-squeezed state is roughly symmetric about the $J_{x}$-axis, the covariance terms in $x,z$ and $x,y$ will be approximately zero. This, in combination with taking the expected value of Equation 13 allows us to write the standard error in $\varphi$ based on Equation 33 as

By picking our initial state to be a CSS to be a $\hat{J}_{x}$ eigenstate as before, we can set $\langle\hat{J}_{y_{0}}\rangle=0$, $\langle\hat{J}_{z_{0}}\rangle=0$. Linearising around $\varphi=0$, we obtain

as desired.

We use analytic expressions to evaluate the spin-moments of OAT developed by Kitagawa and Ueda [46]. Using $\bar{S}_{i}$ instead of $\hat{J}_{i}$ to denote a collective spin operator in a direction $i$, for $S=N/2,$ they found that

where $\mu=2\lambda$, $A=1-\cos^{2S-2}\mu,B=4\sin\frac{\mu}{2}\cos^{2S-2}\frac{\mu}{2}$, and $\delta=\frac{1}{2}\arctan\frac{B}{A}$. $\nu$ is the actual rotation angle about the $x$ axis, $\delta$ is the assumed angle that is applied to the OAT state in the simulation. The assumed angle $\nu$ applied in the simulation is $\pi-\delta^{\prime}$, where $\delta$ is based on the assumed state preparation parameter $\lambda$.

Consider a likelihood function

where $\mathbf{X}$ represents a set of measurements dependent on the actual state of state parameters $\mathbf{\Lambda}$, $P$ represents a probability distribution parametrised by the estimator state preparation parameter $\lambda_{p}$, and $\varphi$ is the parameter of interest. We can expand $\mathcal{L}$ to second order around its maximum value $\varphi_{0}$ as

The derivative around the maximum with respect to $\varphi$ will be $0$, so we can write

from which we can calculate the bias as

The numerator and denominator turn out to be the off-diagonal and diagonal terms of the Fisher covariance matrix respectively, thus we can write the above equation as

so we can interpret the ratio of the off-diagonal and $\varphi$ elements of the Fisher covariance as a measure of the susceptibility of a quantum state in a particular basis to state preparation error.

Given that $\lambda_{p}=\lambda^{\prime}$, the assumed state preparation degree of freedom, and $\lambda_{0}=\lambda$, the actual state preparation degree of freedom, we can write that

Let us consider random state preparation error with multiple state preparation parameters $\mathbf{\Lambda}$, but only one degree of freedom $\lambda$ that fluctuates randomly with a known standard deviation $\Delta\lambda$. This $\Delta\lambda$ is defined separately from the $\Delta\lambda$ in the main text. The randomness in $\lambda$ results in the transition from a pure to a mixed state. After parameter encoding, the state before readout can be described as

where $\ket{\psi_{\mathbf{\Lambda}}}\bra{\psi_{\mathbf{\Lambda}}}$ represents the density matrix of a pure quantum state entangled according to $\mathbf{\Lambda}$. Assuming that the experimenter has complete knowledge of the mixed state prepared, there will be no systematic error in the parameter estimate, i.e. $B=0$.

We assume that our state preparation parameter vector is $\mathbf{\Lambda}=\{\lambda,\theta\}$, where $\lambda$ is again the only degree of freedom. For a spin-squeezed state, the phase can be estimated based off $\langle\hat{J}_{z}\rangle$ as in Figure 3 under an estimator accounting for the state preparation parameter fluctuations, and the variance can be calculated as

We consider the sensitivity ratio for varying levels of mixed state preparation error for TAT and OAT. In the case of OAT, we apply the optimal $\hat{J}_{x}$ rotation for the mean state preparation value in the distribution, $\lambda_{0}$. We find that increased uncertainty in the $\lambda$ value leads to greater state preparation error for both OAT and TAT under a method of moments estimator. In Figure 10 (a) for TAT and (b) for OAT, as $\Delta\lambda$ increases, the variance in $\hat{J}_{x}$ increases. This is because the state’s total variance takes the possible range of $J_{z}$ and $J_{x}$ distributions given by the randomly distributed $\lambda$. Given that the state is squeezed along the $J_{z}$ axis, the variance of the mixed state for all fixed values of $\lambda_{0}$ will be greater in $\hat{J}_{x}$ than in $\hat{J}_{z}$, and monotonically increasing in both, leading to an increase in error according to Equation 18.

In the cases of both OAT and TAT, we see that a maximum likelihood estimation strategy improves on a method of moments strategy for $\Delta\lambda\neq 0$. This is because more highly mixed states tend away from being Gaussian, where the quantum state can be characterised by a single moment, and thus the CRB can be saturated by a MOM estimator. When using a maximum likelihood estimation strategy, we see a distinction between OAT and TAT. In the case of TAT, increased uncertainly in $\lambda$ does not necessarily decrease precision. This is because as $\lambda$ increases, less spin-squeezed state are mixed in with more highly spin-squeezed states. The $\hat{J}_{z}$ distribution becomes weighted with the more highly distinguishable distributions of spin-squeezed states, resulting in a distribution that is overall more distinguishable, thus having higher Fisher information than mixed states with lower $\Delta\lambda$ values. In the case of OAT on the other hand, we find increased state preparation uncertainty leads to uniformly increasing $Q$. This is because the phase space rotation after squeezing varies for the state preparation parameter, unlike for TAT. The varied state preparation parameter results in a mixture of states that are not all squeezed along the same axis, as the optimal $\hat{J}_{x}$ rotation can not be applied for all states - far fewer highly distinguishable distributions become part of the new probability distribution.