text:
\section{Introduction}

The goal of quantum metrology~\cite{Holevo1982, Demkowicz2015, Degen2017, Pezze2018} is to enable and enhance measurements of parameters of physical systems using quantum systems that serve as probes. Quantum physics limits the performance of sensors employing uncorrelated (classical) input states, establishing the so-called standard quantum limit (SQL). At the same time, quantum physics provides us with entanglement as a resource to overcome the SQL and approach the ultimate limits for measurement precision in quantum sensing, which define the Optimal Quantum Sensor (OQS). Identifying and eventually building such OQS is one of the outstanding challenges in quantum metrology. This involves first of all the identification of optimal entangled input states and measurements for a given sensing task. Subsequently, the question arises how quantum sensors operating close to the limits defined by the OQS can be implemented on specific quantum sensor platforms given the experimental (entangling) resources.

For {\em single-parameter} quantum metrology,
there is a clear theoretical understanding of the OQS~\cite{Macieszczak2014,Demkowicz2015, Kaubruegger2021}, and recent experiments with programmable quantum sensors have demonstrated close to optimal $N$-atom Ramsey interferometry with low-depth entangling quantum circuits~\cite{Marciniak2021}. In contrast, we will be interested below in {\em multi-parameter} quantum metrology~\cite{Gessner2018, Demkowicz2020, Liu2020, Sidhu2020, Sidhu2021}, which deals with the precise estimation of several parameters \emph{simultaneously}, and is hence relevant for numerous practical applications~\cite{Humphreys2013,Gessner2020, Huang2021, Belliardo2022, Baamara2022, Conlon2023, Eriksson2023}, e.g.~vector field sensing~\cite{Vaneph2013, Baumgratz2016, Gorecki2022}, which is related to magnetometry~\cite{Budker2007, Budker2013, LeSage2013, Behbood2013, Lee2015, Thiele2018, Zheng2020,Wang2021} or electrometry~\cite{Fan2015, Brownutt2015, Sedlacek2013}.

\begin{figure*}
\includegraphics[width=\linewidth]{./Sensors.pdf}
\caption{\textbf{Quantum sensing of a vector field ${\bf B}$ using $\mathbf{SU(2)}$ interferometry.} We consider Ramsey interferometry with entangled atoms, where the vector of phases $\bm{\phi}$ are encoded in the quantum system via the $SU(2)$ unitary $U(\bm \phi)$ that carries information about the vector field $\textbf{B}$
(see Sec.~\ref{sec:SU2}). Panels (a,b,c) define three interferometers, discussed in more detail in Secs.~\ref{sec:CircuitModel} and \ref{sec:LimitedControl}, which differ in the level of control available for manipulating atomic entanglement in an experimental setup.
Panels (d,e,f) provide operational definitions of the interferometers in terms of quantum circuits and projective measurements. The entangler $\Uen$ and decoder $\Ude$ transform the quantum state of the system before and after the phase encoding. The projective measurement outcomes $\bm{\sigma}$, $m$, $m^{A(B)}$ are converted into an estimate $\bm{\xi}$ of the true phase vector $\bm{\phi}$. (a, d): The most general $N$-atom sensor assumes universal control of atoms, which allows to realize the optimal quantum sensor (see section ~\ref{subsec:OQS}).
(b, e): The atoms in the 1-partite quantum sensor (see Sec.~\ref{subsec:1p-QS}) are controlled uniformly. The sensor dynamics are thus constrained to the permutation symmetric subspace of $N$ atoms, i.e., the atomic ensemble can be represented by a single macrospin with angular momentum $J=N/2$. 1-partite control is realized by circuits composed of collective rotation $\mathcal{R}_{x(y)}$ and the one-axis twisting (OAT) gate $\mathcal{T}$.  (c, d): The atoms in the 2-partite quantum sensor (see Sec.~\ref{subsec:2p-QS}) are grouped into two partitions $A$ and $B$ and controlled uniformly within each partition, which includes uniform control of the entire atomic ensemble. The system is thus equivalent to two entangled macrospins with angular momenta $J^A=J^B=N/4$. 2-partite control is realized by circuits composed of collective  rotations $\mathcal{R}^{A(B)}_{x(y)}$ and  OAT $\mathcal{T}^{A(B)}_z$ acting on each partition, and  OAT $\mathcal{T}_z$ on the full ensemble.}
\label{fig:Sensors}
\end{figure*}

A fundamental difference between multi-parameter and single-parameter metrology arises from the potential incompatibility of optimal measurements for different parameters encoded by non-commuting Hamiltonians. This distinctive feature of multi-parameter quantum metrology has been studied primarily within the framework of quantum Fisher information, where the sensor precision is tightly lower bounded by the Holevo Cram\`er-Rao bound~\cite{Holevo1982, Albarelli2019, Demkowicz2020} as the number of measurement repetitions of the same signal tends to infinity. Instead, we are interested here in measurement scenarios that are limited to one or a few measurement repetitions, as is relevant, e.g., for sensing time-varying signals~\cite{Tsang2011}.
The problem of finding a sensor that maximizes the precision of a single measurement in a given range of parameter values can be formulated in the framework of Bayesian quantum metrology~\cite{Holevo1982, Demkowicz2020}. However, solving Bayesian multi-parameter quantum metrology for non-commuting Hamiltonians remains a longstanding challenge. So far, only a handful of special problems with an underlying group symmetry have been addressed in this framework~\cite{Bagan2001, Chiribella2004a, Bagan2004, Kahn2007}.

In the present work, we study the paradigmatic multi-parameter quantum metrology problem of vector field sensing~\cite{Vaneph2013,Baumgratz2016,Gorecki2022} within the Bayesian framework.
While entangled field sensors for 1D sensing have already been experimentally realized~\cite{Wasilewski2010, Sewell2012, Ockeloen2013, Arias2019}, we consider the open problem of sensing 2D and 3D fields in the context of entangled Ramsey interferometry with $N$ atoms. We establish a method to determine the OQS, that sets the ultimate achievable precision limit for simultaneous estimation of the components of the vector field from a single measurement with a given number of atoms. The latter is of particular importance when the number of atoms cannot be arbitrarily increased, e.g., to maintain a high spatial resolution. Keeping current experimental capabilities in mind, we consider sensors with limited experimental quantum control that nevertheless approach the OQS in terms of performance and significantly outperform the SQL, defined by sensors operating without entanglement. Programmable sensors can be realized on a variety of sensing platforms, including trapped ions~\cite{Marciniak2021, Conlon2023}, laser-excited Rydberg atoms~\cite{Facon2016, Omran2019}, atoms in optical cavities~\cite{Robinson2022}, polar molecules~\cite{Carr2009, Loic2019}, color centers~\cite{Bermudez2011, Yao2012} and photons with orbital angular momentum~\cite{Bouchard2017,Brandt2020,Eriksson2023}. Considering the example of one-axis twisting (OAT) interactions~\cite{Kitagawa1993}, which is available e.g.~for trapped ions, we optimize variational quantum circuits~\cite{Cerezo2021} that point to a practical route to implement entangled quantum sensors~\cite{Marciniak2021}.

This work is organized as follows. In Sec.~\ref{sec:BayesianMDFS} we introduce $SU(2)$-interferometry to measure the components of a vector field, and we set the stage for OQS's in the framework of Bayesian multi-parameter estimation. Section~\ref{sec:OQS} discusses theoretical aspects and provides (numerical) algorithms for finding the OQS, and defines quantum circuit models. In Sec.~\ref{sec:LimitedControl} we introduce one- and two-partite quantum sensors as quantum sensors with limited  entangling capabilities. This leads to the discussion of variational low-depth quantum circuits approximating the optimal quantum sensor in Sec.~\ref{sec:VariationalCircuits}. While most of the results presented in this work are for 2D field sensing, Sec.~\ref{sec:3D} discusses aspects of 3D field sensing, followed by conclusions.

\section{Bayesian multi-dimensional field sensing}
\label{sec:BayesianMDFS}

In this section, we review the theory of multi-parameter estimation using multi-dimensional field sensing as an example. We focus on the Bayesian framework of phase estimation, which accounts for prior uncertainty about the parameters being measured.
We define the OQS as the sensor that minimizes a Bayesian cost function.

\subsection{Vector field sensing with $SU(2)$-interferometers}
\label{sec:SU2}

We study the problem of estimating all components of a vector field $\mathbf{B}=(\text{B}_x, \text{B}_y, \text{B}_z)^T$ in $SU(2)$ interferometry \footnote{We refer to an interferometer with a unitary phase encoding that belongs to the group of SU(2) as an SU(2) interferometer.}, where we assume that the field is uniformly coupled to an ensemble of $N$ identical two-level atoms.
The physical sensing scenarios we have in mind can be, e.g., DC magnetic fields or AC magnetic and electric fields. In Appendix~\ref{app:vfs_implement}, we elaborate on exemplary experimental settings that realize the model system consider below.

In our model, the atoms couple to the vector field
according to the Hamiltonian
\begin{align}
H=\mathbf{B}\cdot\bm{J}\equiv\sum_{\nu}\text{B}_{\nu}J_{\nu}.
    \label{eq:H}
\end{align}
Here, $J_{\nu}=\sum_{k=1}^{N}\sigma_{k}^{\nu}/2$ are the collective spin operators and $\sigma_{k}^{\nu}$ are the Pauli operators of the $k$-th atoms with $\nu=x,y,z$.
After an evolution time $T$, the phases $\bm{\phi}=\mathbf{B}\,T$ are imprinted on the input state $\ket{\psi_{{\rm in}}}$ by the unitary $SU(2)$ operator
\begin{align}
    U(\bm{\phi})=\exp\left[-i\,\bm{\phi}\cdot\bm{J}\right]
    \label{eq:PhaseEncoding}
\end{align}
such that $\ket{\psi_{\bm{\phi}}}=U(\bm{\phi})\ket{\psi_{{\rm in}}}$. Here we assume that the coherence time of the field $\tau \gg T$ is long compared to $T$, i.e.~the field remains effectively constant during the evolution.
Finally, a measurement of the two-level atoms in the state~$\ket{\psi_{\bm{\phi}}}$ is performed to estimate the phases $\bm{\phi}$ and thus infer the field $\textbf{B}$. While we focus here on measurements at discrete times, an alternative approach based on continuous measurements is discussed in~\cite{Yang2022}.

The most general measurement is described by a positive operator-valued measure (POVM), i.e., a set $\{M_{\mu}\}$ of positive Hermitian operators, $M_{\mu}\succeq0$, such that $\sum_{\mu}\,M_{\mu}=\openone$. In the following, we contrast POVMs with the less general but experimentally more accessible projective measurements, denoted by $\Pi_\mu$. For a given vector of phases $\bm{\phi}$, the measurement outcomes, labeled $\mu$, are realized with probabilities
\begin{align}
	\label{eq:conditional_probability}
    p(\mu|\bm{\phi})=\mathrm{Tr}\{M_{\mu}\ket{\psi_{\bm{\phi}}}\bra{\psi_{\bm{\phi}}}\}.
\end{align}
Based on the result~$\mu$, one estimates the phases using an estimator $\bm{\xi}_{\mu}=(\xi_{\mu}^{x},\xi_{\mu}^{y},\xi_{\mu}^{z})^T$. The deviation of the estimation from the actual value $\bm\phi$ is characterized by an error~(or loss) function $\epsilon(\bm{\phi}, \bm{\xi}_{\mu})$, commonly chosen to be the squared error
\begin{equation}
\label{eq:error_function}
\epsilon(\bm{\phi}, \bm{\xi}_{\mu}) = \left(\bm{\phi}-\bm{\xi}_{\mu}\right)\cdot \left(\bm{\phi}-\bm{\xi}_{\mu}\right).
\end{equation}
Finally, the figure of merit, which accounts for measurement statistics, is given by the average estimation error~(or risk) for a given $\bm{\phi}$. In our case, it takes the form of the mean squared error (MSE)
\begin{equation}
    \text{MSE}(\bm{\phi})=\sum_{\mu}\epsilon(\bm{\phi}, \bm{\xi}_{\mu})\, p(\mu|\bm{\phi}).
\label{eq:MSE_POVM}
\end{equation}

Note that the $SU(2)$ phase encoding $U(\bm{\phi})$ is periodic which means that $\abs{\bm{\phi}}$ can only be estimated modulo $2\pi$. Since we want to infer the vector field from the estimated phases according to ${\textbf{B}^{\rm est} \sim \bm{\xi}/ T}$, it is crucial to take this ambiguity into account. Therefore, we consider phases in the interval ${-\infty < \bm{\abs{\phi}} < \infty}$, and we explicitly choose the squared error~\eqref{eq:error_function}, which is a non-periodic error function to strongly penalize phases that slip outside the $d$-dimensional sphere $|\bm{\phi}|\leq \pi$ of unambiguous phase estimation.

\subsection{Bayesian multi-parameter estimation}
\label{sec:BMPE}

In the Bayesian approach, the parameter vector $\bm{\phi}$ to be estimated is assumed to be a random variable. Knowledge about the phases to be estimated is represented by the prior probability density ${\cal P}(\bm{\phi})$. By estimating the phases, we infer information, and the prior density is updated to the posterior probability density.

The quantity central to our discussion is the Bayes Mean Squared Error (BMSE), defined as the MSE~\eqref{eq:MSE_POVM} averaged over the prior probability density,
 \begin{align}
    \text{BMSE} & \equiv {\cal C} = \int d\bm{\phi}\, \text{MSE}(\bm{\phi})\, \mathcal{P}(\bm{\phi}).
\label{eq:BMSE_POVM}
\end{align}
Below, the BMSE will play the role of a cost function ${\cal C}$, which depends on the input state $\ket{\psiin}$, POVMs $\{M_{\mu}\}$, and estimators $\{\bm{\xi}_{\mu}\}$.

We use Bayes theorem ${p(\bm{\phi}|\mu) = p(\mu|\bm{\phi}) \mathcal{P}(\bm{\phi}) / p(\mu)}$ to interpret $\mathcal{C}$ as the posterior expected error of the phase estimation. In particular, $\mathcal{C}$ can be rewritten as
\begin{align}
    \mathcal{C} = \sum_{\mu} p(\mu)\, \int d\bm{\phi} \, \epsilon(\bm{\phi}, \bm{\xi}_{\mu})\, p(\bm{\phi}|\mu),
\end{align}
where $p(\mu) = \int d\bm{\phi}\, p(\mu| \bm{\phi}) \mathcal{P}(\bm{\phi})$ is the probability of measuring $\mu$. The integral over the phase vector $\bm{\phi}$ represents the expected squared error, i.e., the squared error averaged with respect to the posterior probability density $p(\bm{\phi}|\mu)$, where the latter represents our knowledge of $\bm{\phi}$ given that the outcome of the measurement is $\mu$.

The minimum MSE estimator, defined by minimizing the expected squared error and thus the BMSE, is given by
\begin{align}
\label{eq:MMSEe}
	\bm{\xi}_{\mu}^{*} = \int d\bm{\phi}\,\bm{\phi}\, p(\bm{\phi}|\mu) = \langle \bm{\phi} \rangle _{p(\bm{\phi}|\mu)},
\end{align}
which corresponds to the mean value of the posterior probability density $p(\bm{\phi}|\mu)$.

If the estimators $\{\bm{\xi}_\mu\}$ are chosen to be the minimum MSE estimators $\{\bm{\xi}_\mu^*\}$, the BMSE is the posterior variance
\begin{align}
    \Delta^{2}&\equiv\left.{\cal C}\right|_{\bm{\xi}=\bm{\xi}^{*}} = \sum_{\mu }p(\mu)\,\Big\langle \epsilon\left(\bm{\phi}, \langle\bm{\phi}\rangle_{p(\bm{\phi}|\mu)}\right) \Big\rangle_{p(\bm{\phi}|\mu)},
\end{align}
i.e.~the variances of the posterior densities averaged according to the probability $p(\mu)$ to measure $\mu$.

In the following, the posterior uncertainty $\Delta^2$ will be the quantity of interest that evaluates performance and allows comparison of various quantum sensors for a given prior density $\mathcal{P}(\bm{\phi})$. By minimizing the metrological cost function~\eqref{eq:BMSE_POVM}, we implicitly minimize the posterior variance of the sensor. Therefore, our results will be presented as plots, where we compare the ratio between posterior uncertainty $\Delta$ and prior uncertainty for the different sensor models (see Figs.~\ref{fig:2DCost}, \ref{fig:2DScaling},
\ref{fig:3DCost}). The more information we can gain about the phases $\bm{\phi}$ in a single measurement, the smaller the value of this ratio.

\begin{figure}
\includegraphics[]{./2DOptimalCostScan.pdf}
\caption{
\textbf{Bayesian quantum sensing of a 2D field.} (a) Ratio of the posterior width $\Delta$ to the prior width $\sqrt{2}\delta$ vs. $\delta$ for $N = 8 $ atoms. The OQS  sets the ultimate performance limit; the gray-shaded region is unattainable for any quantum sensor with $N$ atoms. The optimal 1-partite quantum sensor, the 2-partite quantum sensor, and the 2-partite classical sensor are shown for comparison. The asymptotic bounds for the classical (blue dotted line) and quantum (black dotted line) sensors are predicted by the Bayesian Cram\'er-Rao bounds ~\eqref{eq:BayesianQL} and \eqref{eq:BayesianSQL}. In Figs.~\ref{fig:1p-QS} and \ref{fig:2p-QS} we illustrate the working of the optimal sensors indicated here by the colored triangles. Magnified panel: Performance of variational circuits (purple lines) defined in Sec.~\ref{sec:VariationalCircuits} for an increasing number of decoding layers $\nde$, and a fixed entangler depth of $\nen = 2$.}
\label{fig:2DCost}
\end{figure}

Before proceeding, we comment on our choice of the prior probability density $\mathcal{P}(\bm{\phi})$. In Bayesian quantum metrology, the $\textbf{B}$ field to be sensed has a prior density which translates into a density for the phases $\mathbf{\phi}$. To give a specific example, we assume below a prior for the field in the form of a Gaussian density centered around the origin with prior width $({\delta {\abs{\mathbf{B}}}})$. At the same time, there is no knowledge of the orientation of the field. The corresponding prior probability density for the phases is thus
\begin{eqnarray}
\mathcal{P}_{\delta}(\bm{\phi}) = \frac{1}{\left(\sqrt{2 \pi}\, \delta\right) ^d } \exp \left[-\frac{\bm{\phi}\cdot \bm{\phi}}{2\delta^2}\right],
\label{eq:GaussianPrior}
\end{eqnarray}
where $d$ is the dimension of the vector $\bm{\phi}$. For 2D field sensing we take $d=2$ and assume that $\text{B}_z = {\phi}_z = 0$. In the present example, the prior is thus parameterized by a single parameter $\delta\sim (\delta {\abs{\mathbf{B}}})\,T$.  However, the following discussion can be adapted (see Appendix~\ref{app:PriorDensities}) to more general priors motivated by particular quantum sensor settings and tasks. As noted before, in mapping from fields to phases via $\bm{\phi}=\mathbf{B}\,T$, we must ensure with a proper choice of $T$, that there is a one-to-one map of estimated phases to the vector field, i.e.~there are no phase slips. Implicit in our discussion is the assumption that the field $\mathbf{B}$ to be measured is essentially static during time $T$.

\subsection{Optimal quantum sensors}
\label{subsec:OQS}
We define the OQS as the minimum of the metrological cost function BMSE~\eqref{eq:BMSE_POVM} over all input states $\ket{\psi_{{\rm in}}}$, measurements $\{M_{\mu}\}$, and estimators $\{\bm{\xi}_{\mu}\}$  for a given prior density $\mathcal{P}_\delta(\bm{\phi})$. Thus, the posterior variance of the OQS,
\begin{equation}
\Delta^2_{\rm OQS}=\min_{\ket{\psi_{{\rm in}}}, M_{\mu}, \bm{\xi}_{\mu}} \mathcal{C},
\label{eq:Delta2_OQS}
\end{equation}
represents the fundamental limit on the posterior variance achievable by any $N$-atom sensor for a given prior $\mathcal{P}_\delta(\bm{\phi})$.

As a first result, in Sec.~\ref{sec:OQS} we will develop a method that allows us to efficiently find the OQS for vector field sensing, i.e. to perform a numerical optimization that scales only polynomially with the number of particles $N$.

While the OQS is defined via unconstrained optimization over (entangled) input states and measurements, the practical realization of the `best possible' quantum sensors will be constrained by limited experimental quantum resources and imperfections. This will lead us to discuss optimal quantum sensors with limited (experimental) entangling capabilities, i.e. identifying the best possible quantum sensors given experimental constraints. Similar to the OQS, the performance of these sensors is characterized by the minimum posterior variance $\Delta^2$ defined as in Eq.~\eqref{eq:Delta2_OQS}, but evaluated by imposing relevant constraints on the input states and measurements.

Figure~\ref{fig:Sensors} gives an overview of quantum sensors with different levels of control and entanglement capabilities, whose performance we compare in Fig.~\ref{fig:2DCost}. An operational definition of these quantum sensors is given in terms of a quantum circuit model, see Sec.~\ref{sec:CircuitModel}. This includes in Fig.~\ref{fig:Sensors}(a) a quantum sensor with universal ($N$-partite) control of the $N$-particle quantum system, which subsumes the OQS. We contrast this in Fig.~\ref{fig:Sensors}(b) to a 1-partite quantum sensor (1p-QS) defined by uniform control of the $N$ atoms and in Fig.~\ref{fig:Sensors}(c) to a 2-partite quantum sensor (2p-QS) with uniform control on the level of two partitions of the atomic ensemble (for details see Secs.~\ref{sec:1pQS} and~\ref{sec:2pQS}, respectively). We will see below that the optimal 2p-QS operates close to the OQS for 2D field sensing independent of the width of the prior density, and this will motivate our search for variational approximations to the relevant quantum circuits.

Finally, we comment on the relationship between the Bayesian and Fisher information approaches to multi-parameter quantum metrology~\cite{Demkowicz2020} in relation to the present work, see Appendix~\ref{app:Fisher} for a more detailed discussion. The metrological cost function in the latter approach is the MSE of a locally unbiased estimator, which is lower bounded by the Cram\'er-Rao bound. In other words, the sensitivity is considered {\em locally} around a given value $\bm{\phi}_0$ of the phase vector, which is in contrast to the BMSE, which averages the MSE over a {\em range} of phase values. However, in the limiting case of small prior widths $\sqrt{d}\delta$, when the Bayesian approach becomes {\it local}, the two approaches can be connected via the van Trees inequality \eqref{eq:VanTrees}. In the Fisher information framework, the sensitivity of any $N$-atom multi-dimensional field sensor is lower bounded by a Heisenberg-like limit, which in turn limits the BMSE through the van Trees inequality; see Appendix~\ref{app:Fisher}. For the examples considered in the following, the OQS saturates these bounds~\cite{Kolenderski2008, Vaneph2013, Gorecki2022} in the small~$\delta$ limit, which highlights that the OQS is also optimal from the Fisher information point of view.

\section{Theory of the optimal quantum sensor}
\label{sec:OQS}

In this section, we solve the mathematical problem of minimizing the BMSE~\eqref{eq:BMSE_POVM} as the metrological cost function, which defines the OQS and underlies the numerical results for the OQS presented in Figs.~\ref{fig:2DCost} and~\ref{fig:2DScaling}. We conclude the section by introducing a quantum circuit model that describes practical realizations of (optimal) quantum sensors.

\subsection{Identifying the optimal quantum sensor}\label{subsec:identify_OQS}

To find the OQS, one minimizes the metrological cost function~\eqref{eq:BMSE_POVM} over an unconstrained set of input states, measurements, and estimators, as in Eq.~\eqref{eq:Delta2_OQS}. In general, solving such a mathematical problem is difficult, given the exponentially large number of degrees of freedom associated with a quantum system of $N$ particles, the corresponding measurements, and estimators. Here, we show that optimal Bayesian solutions can be found using computational resources that scale only polynomially in the size of the system $N$. Our approach is based on three key observations presented in the following and further detailed in the Appendices~\ref{subsec:Irreducible_Hilbert_space}--\ref{subsec:covariant_POVM}.

First, we note that the dynamics of a sensor comprising $N$ atoms, which is uniformly coupled to the field $\mathbf{B}$ according to Eq.~\eqref{eq:H}, is restricted to an effective Hilbert space with dimension scaling as~$\sim N^3$. This is in contrast to the full $2^N$-dimensional Hilbert space describing $N$ spin-1/2 atoms. The simplification originates in the $SU(2)$ and particle permutation symmetry of the unitaries $U(\bm\phi)$ that encode the phases. The idea is to consider the $N$-qubit space using irreducible representations of the $SU(2)$ group. For example, by adding angular momenta one can represent two spin-1/2's as a direct sum of two orthogonal subspaces corresponding to a spin-1 (triplet) and a spin-0 (scalar), $\bm{\frac12}\otimes\bm{\frac12}=\bm1\oplus\bm0$. The addition of more than 2 angular momenta leads to multiple orthogonal subspaces, or equivalent representations, corresponding to the same spin-$j$, e.g., for three qubits we have 2 equivalent spin-1/2 representations, $\bm{\frac12}^{\otimes 3}=\bm{\frac32}\oplus\bm{\frac12}\oplus\bm{\frac12}$. The number of equivalent representations $\ell_j$ of spin-$j$ grows exponentially with the number of added qubits $N$ and thus exceeds the number of degrees of freedom of a single spin-$j$ subspace, $d_j=2j+1$. Since the $SU(2)$ unitaries do not couple different angular momenta $j$ and their equivalent representations, we can utilize not more than $d_j\leq\ell_j$ equivalent representations by entangling them. The other $\ell_j-d_j$ equivalent representations remain unpopulated and cannot be used for sensing, thus reducing the effective Hilbert space dimension of a $SU(2)$ sensor to $\sim N^3$~\cite{Bagan2004, Chiribella2004a, Chiribella2004b}. We present details in Appendix~\ref{subsec:Irreducible_Hilbert_space}.

The second observation is the multi-convex nature of the sensor optimization problem. As we show in Appendix~\ref{subsec:Minimization-of-the-Cost}, this means that minimization of the metrological cost Eq.~\eqref{eq:BMSE_POVM} is a convex optimization problem~\footnote{Convexity of optimization problem guarantees that every local minimum is a global minimum.} with respect to each of the three sets ($\ket{\psi_{{\rm in}}}$,
$\{M_{\mu}\}$, and $\{\bm{\xi}_{\mu}\}$) of variables, if the other two are fixed. The hardest of the corresponding subproblems is the optimization over POVMs for fixed input state and estimators. It can be recast as a semidefinite program, which is solvable in polynomial time for our problem of polynomial size in $N$~\footnote{The upper bound on the POVM complexity for $d_N\sim N^3$ dimensional Hilbert space is given by a POVM comprising $d_N^2\sim N^6$ measurement operators of full rank. We find the OQS solution to be significantly simpler than the upper bound.}, as discussed above.

The optimization algorithm is summarized as follows: Starting from a random initial state, POVM, and estimators, one iteratively solves the convex subproblems until the solution converges to the optimal sensor (with $\ket{\psi_{\rm in}^{\star}}$, $\{M^{\star}_\mu\}$, and $\{\bm{\xi}_{\mu}^{\star}\}$) that minimizes the metrological cost~\eqref{eq:BMSE_POVM}. We present formal definitions of the corresponding convex subproblems in Appendix~\ref{subsec:Minimization-of-the-Cost}.

Finally, on the more technical side, we observe that a reliable convergence to the optimum solution can be achieved when symmetric degrees of freedom are eliminated from the search space. To this end, we have identified a covariant ansatz for input states, generalized measurements, and estimators, which describes the optimal sensor in the case of {\em a priori} unknown field direction as reflected in the isotropic prior density, Eq.~\eqref{eq:GaussianPrior}, that we consider. The covariant ansatz is detailed in Appendix~\ref{subsec:covariant_POVM} for the optimal 2-partite 2D field sensor. The general case of the OQS for 2D and 3D field sensing will be considered in~\cite{Vasilyev23}.

\subsection{Quantum circuit model of a quantum sensor}
\label{sec:CircuitModel}

Practical realizations of a quantum sensor will rely on unitary quantum circuits that manipulate the quantum state of the sensor, and projective measurements performed on the atoms, as illustrated in Fig.~\ref{fig:Sensors}. In our quantum circuit model of a sensor, the initial state
\begin{align}
    \ket{\psiin} = \Uen \, \ket{\psi_{0}}
\end{align}
is prepared by an entangling unitary $\Uen$ from a product state, e.g., $\ket{\psi_0} = \ket{\downarrow}^{\otimes N}$. On the other hand, the decoding unitary $\Ude$ can be viewed as transforming the projective measurement basis given by $\ket{\bm{\sigma}}=\ket{\sigma_1, \cdots, \sigma_N}$, where each atom $j$ is projected into $\sigma_j=\ket{\uparrow}$ or $\sigma_j=\ket{\downarrow}$. Therefore, the conditional probability $p(\bm{\sigma}|\bm{\phi}) = \tr\{\Pi_{\bm{\sigma}} \ket{\psiphi}\bra{\psiphi}\}$ to obtain the measurement outcome $\bm{\sigma}$ can be expressed in terms of the effective measurement projectors
\begin{align}
    \Pi_{\bm{\sigma}} = \Ude^{\dagger} \ket{\bm{\sigma}}\bra{\bm{\sigma}} \Ude.
\end{align}

A general POVM underlying the discussion of OQS can be implemented in the circuit model with additional ancillary atoms that are not involved in the actual sensing process. Interestingly, we observe that, for sufficiently large $N$, the optimal POVM acting on the OQS subspace of polynomial dimension, $\sim N^3$, can be realized as projective measurements on the full $N$-atom space. The $N$-atom Hilbert space has an exponential number of degrees of freedom, $2^N$, which can be used instead of the ancillary atoms. This suggests that the OQS can be faithfully represented as a quantum circuit if we assume universal quantum control at the level of individual atoms, as shown in Fig.~\ref{fig:Sensors}(a).
Our numerical results suggest that the OQS for 2D fields can be realized without using ancillary atoms for sensors comprising $N\gtrsim8$ atoms.

\section{Optimal few-partite quantum sensors}
\label{sec:LimitedControl}

In the previous section, we discussed general aspects of the OQS for multi-dimensional field sensing in $SU(2)$-interferometry with $N$ atoms.
In the present section, we will introduce and discuss few-partite quantum sensors, and compare their performance to the OQS. These sensors are defined by partitioning $N$ atoms representing the sensor into subensembles. In particular, we consider the 1p-QS and the 2p-QS indicated in Figs.~\ref{fig:Sensors}(b,e) and (c, f), respectively.
For the 1p-QS, we assume that the atomic ensemble is uniformly controlled, while the 2p-QS assumes uniform control at the level of partitions $A$ and $B$.

The motivation for introducing few-partite quantum sensors is two-fold: First, we observe that the wave function of the OQS does not always occupy the full Hilbert space of an $N$-atom $SU(2)$ interferometer, see Appendix~\ref{appendix:1p_and_2p_subspaces}. The specific structure of the occupied subspace suggests that the OQS for 2D field sensing can be approximated by few-partite quantum sensors.
Second, few-partite quantum sensors require limited quantum control to implement entangling and decoding unitaries in our quantum circuit model [see Fig.~\ref{fig:Sensors}(e,f)]. We will discuss such implementations in Sec.~\ref{sec:VariationalCircuits}, where we study variational approximations to the corresponding quantum circuits which can be realized with current experimental techniques.

\subsection{Few-partite quantum sensors}

\subsubsection{One-partite quantum sensor (1p-QS)}
\label{subsec:1p-QS}

The 1p-QS and the corresponding quantum circuit model are sketched in Figs.~\ref{fig:Sensors}(b, e), respectively. We assume global control so that the dynamics of the 1p-QS, i.e. the action of the 1p-QS unitaries $\mathcal{U}_{\rm en(de)}^{\rm 1p}$ and the phase encoding $U(\bm{\phi})$ on $\ket{\psi_0}$, are confined to the permutation invariant subspace $\mathcal{H}_{\text{1p}}= \mathbb{C}^{N + 1}$ of the $N$-atom Hilbert space $(\mathbb{C}^2)^{\otimes N}$. Therefore, the system is equivalent to a single `macrospin' with angular momentum $J = N /2$~\cite{Chalopin2018}. The eigenstates $\ket{m}$ of $J_z$, which satisfy $J_z \ket{m} = m \ket{m}$ with $m = -J,-J+1,\dots,J$, form a basis of the 1p-QS Hilbert space.

In this basis, the initial state and the measurement projectors in Fig.~\ref{fig:Sensors}(e) are defined as
\begin{align}
    \ket{\psiin} &= \mathcal{U}_{\rm en}^{\text{1p}} \ket{m = - J},
    \label{eq:1pQS_psi}\\
    \Pi_m &= \left. \mathcal{U}_{\rm de}^{\text{1p}}\right.^{\dagger}\ket{m }\bra{m} \mathcal{U}_{\rm de}^{\text{1p}}.
    \label{eq:1pQS_pi}
\end{align}
The optimal 1p-QS is characterized by the posterior variance
\begin{equation}
\Delta^2_{\text{1p-QS}}=\min_{\mathcal{U}_{\rm en}^{\text{1p}},\, \mathcal{U}_{\rm de}^{\text{1p}},\, \bm{\xi}_{m}} \mathcal{C}.
\label{eq:PosteriorVariance1pQS}
\end{equation}
Details of the optimization are presented in Appendix~\ref{app:projective_optimization}.

\subsubsection{Two-partite quantum sensor (2p-QS)}
\label{subsec:2p-QS}

The 2p-QS and the corresponding quantum circuit are displayed in Figs.~\ref{fig:Sensors}(c, f), respectively. Here the sensor dynamics are confined to a subspace ${\mathcal{H}_{\text{2p}} = \mathbb{C}^{N/2 + 1}\otimes \mathbb{C}^{N/2 + 1}}$. This setup is equivalent to a pair of macrospins $J^{A} = N/4$ and $J^{B} = N/4$, which can be entangled through bi-partite uniform control.

Here, the basis states $\ket{m^A, m^B}$,  are defined as the simultaneous eigenstates of $J_z^A,J_z^B$ satisfying $ J_z^{A(B)}\ket{m^A,m^B} = m^{A(B)}\ket{m^A, m^B}$ with $m^{A(B)}=-J^{A(B)},\dots,J^{A(B)}$. The initial state and the measurement projectors are now
\begin{align}
    \ket{\psiin} &= \mathcal{U}_{\rm en}^{\text{2p}} \ket{m^A \!=\! -\! J^A,m^B\! =\! -\! J^B},
        \label{eq:2pQS_psi}\\
    \Pi_{m^A, m^B} &= \left.\mathcal{U}_{\rm de}^{\text{2p}}\right.^{\dagger} \ket{ m^A, m^B}\bra{m^A, m^B} \mathcal{U}_{\rm de}^{\text{2p}}.
        \label{eq:2pQS_pi}
\end{align}

The optimal 2p-QS is characterized by the posterior variance (see Appendix~\ref{app:projective_optimization})
\begin{equation}
\Delta^2_{\text{2p-QS}}=\min_{\mathcal{U}_{\rm en}^{\text{2p}},\, \mathcal{U}_{\rm de}^{\text{2p}},\, \bm{\xi}_{m^A, m^B}} \mathcal{C}.
\label{eq:PosteriorVariance2pQS}
\end{equation}

\subsubsection{Two-partite classical sensor (2p-CS)}

We contrast the 2p-QS to a 2-partite classical sensor (2p-CS), defined by further restricting $\mathcal{U}_{\rm en (de)}^{\text{2p}}$ to allow only collective rotations of all atoms within each ensemble. This implies that there is no entanglement between the atoms. We use the corresponding $\Delta^2_{\text{2p-CS}}$ to define the SQL for two-partite sensors.

\subsection{Results: 2D field sensing with few-partite quantum sensors}

Our main results on 2D sensing, where $\mathbf B=(\text{B}_x,\text{B}_y, 0)^T$, are summarized in Fig.~\ref{fig:2DCost}. We plot the ratio of posterior and prior widths, $\Delta/(\sqrt{2}\,\delta)$, as a function of the prior width $\sqrt{2}\delta$ for the OQS representing the ultimate quantum limit, the optimal 2p-CS defining the SQL, and the optimal 1p-QS and 2p-QS. The results are presented for $N=8$ atoms, which is large enough to demonstrate a reasonable sensitivity gain \footnote{Results for small atom numbers are relevant for field sensing in scenarios where only few atoms can be used for the sensing task, e.g., in order to maintain a good spatial resolution.}.

In this figure, there are two limits where the Bayesian update following the measurement does not provide information about the signal, i.e., the ratio $\Delta/(\sqrt{2}\,\delta)$ goes to~$1$. The first is the limit $\delta\to 0$, where quantum measurement fluctuations overwhelm the signal, and the second is the regime $\delta\gg1$, where the signal is dwarfed by a large estimation error due to phases that slip outside the unambiguous estimation region $|\bm{\phi}|\lesssim\pi$. As a result, each sensor has an optimal operating point $\delta^*$ at which the information gain per measurement is maximized.

The results in Fig.~\ref{fig:2DCost}(a) show that the  OQS reaches its optimal operating point in the large prior width regime $\delta \sim1$, corresponding to an interrogation time $T\sim(\delta{\abs{\mathbf{B}}})^{-1}$. Another relevant regime is that of small prior widths, $\smash{\delta\lesssim 1/N}$, where the OQS performance is limited by the Bayesian Cram\`er-Rao bound (black dotted line), which follows from the van Trees inequality (see Appendix~\ref{app:Fisher}). The corresponding small evolution times $T\sim(\delta{\abs{\mathbf{B}}}N)^{-1}$ allow us to monitor more rapidly time-varying signals since the requirement $\tau \gg T$ on the coherence time $\tau$ of the fields is consequently relaxed.

Although the 1p-QS can realize the OQS for small prior widths, it fails to improve over the classical sensor (blue curve) in the regime of large prior widths. This is in contrast to the 2p-QS, which can perform close to the OQS for the full range of prior widths. The near-optimal 2D field-sensing performance of the 2p-QS and the limited experimental control it requires motivate a practical implementation on a programmable quantum sensor, which we discuss in Sec.~\ref{sec:VariationalCircuits} in terms of variational quantum circuits.

In the following, we will take a closer look at the interplay between the optimal input states, measurement projectors, and estimators to understand how these sensors achieve (near-)optimal sensitivity at various prior widths. In particular, we will rely on the Wigner quasi-probability distributions~\cite{Dowling1994}
to represent states $\ket{\psiin}\bra{\psiin}$ and measurement projectors $\Pi_m$ on a generalized Bloch sphere corresponding to the macrospin representation of the sensor. Wigner distributions rotate as rigid objects under the action of $\Uphi$, and the overlap between the Wigner distribution of the rotated state $\ket{\psiphi}\bra{\psiphi}$ and the projective measurement $\Pi_m$ integrated over the Bloch sphere corresponds to the probability $p(m|\bm{\phi})$ of observing the corresponding measurement outcome $m$.

\subsubsection{Optimal 1p-QS}
\label{sec:1pQS}

The 1p-QS is optimal for sensing in the regime of small prior widths $\delta\lesssim 1/N$. In this regime, the optimal entangled input state is $\ket{m=0}$, i.e.~a permutation symmetric state with an equal number of atoms in $\ket{\uparrow}$ and $\ket{\downarrow}$ (we consider even $N$).
The Wigner distribution of this state is shown in Fig.~\ref{fig:1p-QS} and represents a ring around the equator of the generalized Bloch sphere, which implies that the state is invariant under $z$-rotations and is equally sensitive to rotations around any axis in the $xy$-plane. Remarkably, the optimal measurement in this regime of 2D field sensing is projective. Figure~\ref{fig:1p-QS} shows a representative projector $\Pi_m$, which has the greatest overlap with the input state rotated about the $x$-axis. The corresponding measurement outcome $m$ is consequently mapped to a phase estimate $\bm\xi_m$ that indicates a field oriented along the $x$ axis.

The $N+1$ possible measurement outcomes are mapped to phase estimates that are uniformly spaced in a circle. This means that, in the small prior width limit, the 1p-QS operates like a ``compass'', that is, a \emph{single} measurement provides information only on the direction but not the magnitude of $\bm{\phi}$, and hence also of $\mathbf{B}$. The high sensitivity of the 2D ``compass'' to small rotations is particularly suitable for monitoring rapidly changing signals. This is analogous to the GHZ state interferometer~\cite{Bollinger1996} in 1D field sensing, which in a \emph{single} measurement can estimate only the sign but not the strength of a 1D field. Note that the field magnitude is encoded in the probabilities of the different measurement outcomes. Thus, repeated measurements with the ``compass'' will enable us to access full information about the vector field.

\subsubsection{Optimal 2p-QS}
\label{sec:2pQS}

In Fig.~\ref{fig:2p-QS}, we visualize the state, projectors and estimators at the optimal operating point of the 2p-QS, i.e., at the minimum of the green curve in Fig.~\ref{fig:2DCost}, around $\delta\approx 0.75$. To show the Wigner distributions, we use the addition of angular momentum to represent the Hilbert space of the two equal-sized macrospins as the direct sum of spin-$j$ representations, i.e. $\mathcal{H}_{\rm 2p} = {\mathbb{C}^{N/2 + 1} \otimes \mathbb{C}^{N/2 + 1}=\bigoplus_{j = 0}^{N /2} \mathbb{C}^{2j +1}}$. We project the operators onto the different angular momentum manifolds and show Wigner distributions corresponding to each of the $j$-manifolds. This means that coherences between different $j$-manifolds are not represented in the resulting Wigner distributions. Nevertheless, the distributions give an intuitive picture of how the sensor works. The optimal input state of the 2p-QS consists of equatorial rings in all spin representations, which is an approximation of the optimal covariant 2p-QS (see Appendix~\ref{subsec:covariant_POVM}) that is a superposition of $\ket{j, m = 0}$ states. The use of different spin manifolds enables the the 2p-QS to extend the range of detectable field strengths.

The projective measurement outcomes $m^A$ and $m^B$ corresponding to the entangled projectors $\Pi_{m^A, m^B}$ are assigned to one of the evenly distributed estimators, which approximate $N/2 + 1$ concentric rings. The rings shown in the estimator plot in Fig.~\ref{fig:2p-QS} are predicted from the optimal covariant 2p-QS. In contrast to the 2D ``compass'', a single measurement by the optimal 2p-QS provides information on both field strength and direction. The three representative projectors visualized in Fig.~\ref{fig:2p-QS} detect rotations around the $x$-axis with increasing strength, i.e., they have a large overlap with $\ket{\psiin}\bra{\psiin}$ after it is increasingly rotated around the $x$-axis.
\begin{figure}[t]
    \centering
    \includegraphics[]{./1p-QS.pdf}
    \caption{
    \textbf{Illustration of the optimal 1p-QS for 2D fields.} The sensor is optimal for the prior width $\delta$ indicated by the red triangle in Fig.~\ref{fig:2DCost}.  We show the Wigner distributions of the input state $\ket{\psiin}$ [prepared by the unitary $\mathcal{U}_{\rm en}^{\text{1p}}$~\eqref{eq:1pQS_psi}] and a representative projective measurement operator $\Pi_m$ [realized by the unitary $\mathcal{U}_{\rm de}^{\text{1p}}$~\eqref{eq:1pQS_pi}]. For visual clarity, here we show the Wigner distributions for $N = 32$, which are qualitatively the same for any particle number in the regime $\delta \lesssim 1 /N$. The measurement outcomes $m$ are mapped to estimators $\bm{\xi}_{m}$ in the $\phi_x\phi_y$-plane, and the one corresponding to the displayed $\Pi_m$ is highlighted by a purple frame.}
    \label{fig:1p-QS}
\end{figure}

\section{Variational quantum circuits}
\label{sec:VariationalCircuits}

Today's programmable quantum sensors are NISQ devices implemented on various platforms that
support a reduced instruction set of native gates, which can nevertheless be executed with high fidelity and are typically scalable to tens, and potentially hundreds of particles. A practical implementation of a quantum sensor, and in particular a quantum circuit model, must therefore rely on a decomposition of the entangling and decoding unitaries into this native set of gates. This can be achieved in an approximate manner via variational quantum circuits. Here, an ansatz is made for ${\cal U}_{\rm en}$ and ${\cal U}_{\rm de}$ in terms of low-depth quantum circuits, which are parametrized by a set of variational parameters $\bm{\theta}$ ($\bm{\vartheta}$) for the entangler (decoder). Given such a variational ansatz, the optimal entangler and decoder within a class of variational quantum circuits correspond to the minimum of the metrological cost function
\begin{align}\label{eq:varCost}
    \Delta^2_{\rm var } = \min_{\bm{\theta}, \bm{\vartheta}, \bm{\xi}} \mathcal{C}\quad (\ge \Delta ^2_{\rm OQS}),
\end{align}
thus defining the best possible quantum sensor for given (entangling) resources, i.e. resource gate set, circuit design, and circuit depth. The practical aspects of the variational minimization are discussed in Appendix~\ref{app:VQ_optimization}.
This optimization can be performed within the theoretical model of the sensor or by optimizing the variational parameters in a quantum feedback loop on the actual physical device~\cite{Marciniak2021}. The latter case provides us with the opportunity to optimize the quantum sensor even when the corresponding quantum circuits are challenging to simulate classically, and in the presence of control errors and noise.

\begin{figure}[t]
    \centering
    \includegraphics[]{./2p-QS.pdf}
    \caption{
    \textbf{Illustration of the optimal 2p-QS for 2D fields.} The sensor is optimal for the prior width $\delta$ indicated by the green triangle in Fig.~\ref{fig:2DCost}.  We show
    the Wigner distributions of the input state $\ket{\psiin}$ [prepared by the unitary $\mathcal{U}_{\rm en}^{\text{2p}}$~\eqref{eq:2pQS_psi}] and representative projective measurement operators $\Pi_{m^A, m^B}$ [realized by the unitary $\mathcal{U}_{\rm de}^{\text{2p}}$~\eqref{eq:2pQS_pi}]. We visualize projections on to  the spin-$j$ representations of the 2p-QS space, with  $j=0, 1, \dots, N/2$ (for further details see Sec.~\ref{sec:2pQS}). The measurement outcomes $m^A$ and $m^B$ are mapped to estimators $\bm{\xi}_{m^A, m^B}$ in the $\phi_x\phi_y$-plane, and the ones corresponding to the displayed $\Pi_{m^A, m^B}$'s are highlighted by purple frames.}
    \label{fig:2p-QS}
\end{figure}

Below we present a study of variational quantum circuits of increasing depth, which allow us to approach the performance of the OQS in multi-parameter quantum metrology. As a native gate set, we assume one-axis twisting (OAT) operations as entangling gates and uniform spin rotations as single qubit gates. This is motivated by the feasibility of realizing programmable quantum sensors using strings of trapped ions, where OAT is natively implemented as a M\o{}lmer-S\o{}rensen gate
\cite{Leibfried2005, Monz2011}, and spin rotations are performed via laser or microwave driving. The 1p-QS and 2p-QS can be implemented directly as circuits of the natively available gates in such a setup. We note that a similar set of quantum resources is also available in Cavity QED setups~\cite{Leroux2010} and spin mixtures of Bose-Einstein condensates~\cite{Riedel2010, Gross2010}.

To be specific, our discussion will focus on the 2p-QS illustrated in Fig.~\ref{fig:Sensors}c. In the previous section, we have shown that the 2p-QS has close to optimal performance for 2D field sensing. Here, we show that low-depth variational circuits built from OAT and spin rotations provide good approximations to the optimal 2p-QS, and enable a significant enhancement over unentangled sensors (see Fig.~\ref{fig:2DCost}).

\subsection{Resource gate set for 2p-QS}

The 2p-QS assumes that we have a native set of gate operations available that act independently on partitions $A$ and $B$ [see Fig.~\ref{fig:Sensors}(c)]. This includes, first of all, uniform rotation of spins in the partitions,
\begin{align}
    \mathcal{R}^{A(B)}_{\nu}(\theta) = e^{-i \theta J^{A(B)}_{\nu}}\qquad \text{with}\ \nu = x,y,z.
\end{align}
Furthermore, we assume entanglement operations in the form of OAT. These include
\begin{align}
    \mathcal{T}^{A(B)}_z(\theta) = e^{-i \theta \left(J_{z}^{A(B)}\right)^2 }
    \label{eq:TwistingPartition}
\end{align}
which respectively entangle the atoms within the partitions $A$ and $B$, while the entanglement between $A$ and $B$ is generated with
\begin{align}
    \mathcal{T}_z(\theta) = e^{-i \theta \left(J_{z}^A + J_{z}^{B}\right)^2 },
    \label{eq:TwistingGate}
\end{align}
which completes the gate set~\footnote{Note that extensions of this gate set, i.e. allowing for twisting around arbitrary axes~\cite{Schulte2020, Thurtell2022} can lead to more efficient parameterization in terms of the number of gates required to achieve a certain performance.}. Such gates can be realized on trapped ion platforms, such as the one used to demonstrate optimal 1D field sensing~\cite{Marciniak2021}.

\subsection{Circuit design for 2p-QS}

\begin{figure}[t]
    \centering
    \includegraphics[]{./2D_LargePrior_Convergence.pdf}
    \caption{\textbf{Convergence of estimators of optimized variational circuits.} Estimators $\bm{\xi}_{m^A, m^B}$ of the optimized $N = 8$ atoms variational circuits for $\delta \approx 0.75$, with $\nen = 2$ and $\nde = 1, 10, 32$. The estimators are colored according to the corresponding measurement outcomes $m^A$ and $m^B$. The corresponding sensitivity is shown in Fig.~\ref{fig:2DCost}. }
    \label{fig:2DLargePriorEstimators}
\end{figure}

Variational circuits approximating the optimal entangler and decoder of the 2p-QS (see Fig.~\ref{fig:Sensors}c)
 are compiled from the above elementary gates. In particular, we consider circuits that are constructed by stacking multiple layers. Each layer is described by a unitary
\begin{align}
    \mathcal{L}(\bm{\theta_k}) = &\,
    \mathcal{R}^A_y\left(\theta_{k}^{(4)}\right)
    \mathcal{R}^B_y\left(-\theta_{k}^{(4)}\right)
    \mathcal{R}^A_x\left(\theta_{k}^{(3)}\right)
    \mathcal{R}^B_x\left(\theta_{k}^{(3)}\right)
    \notag \\ &\, \times
    \mathcal{T}^A_z\left(\theta_{k}^{(2)}\right)
    \mathcal{T}^B_z\left(\theta_{k}^{(2)}\right)
    \mathcal{T}_z\left(\theta_{k}^{(1)}\right),
\end{align}
that depends on the four parameters contained in the vector $\bm{\theta}_{k}= \left(\theta_{k}^{(1)}, \cdots, \theta_{k}^{(4)}\right)^T$. The parameters within each layer are correlated (i.e. $\theta_{k}^{(4)}, \theta_{k}^{(3)}, \text {and } \theta_{k}^{(2)}$ respectively appear in two consecutive gates), so that the resulting circuits preserve a symmetry that we identified in the optimal 2p-QS solution. Therefore, the unitaries will not explore the entire Hilbert space $\mathcal{H}_{\text{2p}}$ of the 2p-QS, which reduces the number of variational parameters required for a given circuit depth of the 2p-QS.

We parameterize entangling unitaries consisting of $\nen$ layers through a $4\nen$ dimensional parameter vector $\bm{\theta}= \left(\bm{\theta}_{1}, \cdots, \bm{\theta}_{\nen}\right)^T$ as
\begin{align}
	\mathcal{U}_{\rm en}^{\text{2p}}(\bm{\theta}) = \mathcal{L}\left(\bm{\theta}_{\nen}\right) \cdots \mathcal{L}\left(\bm{\theta}_{n_{\text{1}}}\right)
 \mathcal{R}^A_y\left(\tfrac{\pi}{2}\right)
 \mathcal{R}^B_y\left(\tfrac{\pi}{2}\right),
\label{eq:2p_Uen}
\end{align}
and decoding unitaries consisting of $\nde$ layers through a $4\nde+2$ dimensional parameter vector $\bm{\vartheta}= \left(\vartheta_0^{(1)}, \vartheta_0^{(2)}, \bm{\vartheta}_{1} ,\cdots, \bm{\vartheta}_{\nen}\right)^T$ as
\begin{align}
    \mathcal{U}_{\rm de}^{\rm 2p}(\bm{\vartheta}) = &\,
    \mathcal{R}^A_x\big(\vartheta_0^{(1)}\big)\,
    \mathcal{R}^B_x\big(\vartheta_0^{(1)}\big)\,
    \mathcal{R}^A_y\big(\vartheta_0^{(2)}\big)\,
    \mathcal{R}^B_y\big(- \vartheta_0^{(2)}\big)\notag \\\
    &\, \times\mathcal{L^{\dagger}}(\bm{\vartheta}_{1}) \cdots \mathcal{L}^{\dagger}(\bm{\vartheta}_{n_{\text{de}}}).
	\label{eq:2p_Ude}
\end{align}
Therefore, a sensor with an entangler and decoder depth of $(\nen, \nde)$ depends on a total number of $4(\nen + \nde) + 2$ variational parameters. The parameters of the circuit are variationally optimized using the cost function Eq.~\eqref{eq:varCost} to find the best-performing sensor for a given circuit with depth $(\nen,\nde)$.

\subsection{Results: variational optimization of 2p-QS}

In the magnified panel of Fig.~\ref{fig:2DCost}, we examine how closely the optimized variational circuits approximate the performance of the optimal 2p-QS comprising $N=8$ atoms. With increasing decoder depth, the variational circuits consistently approach the optimal 2p-QS and eventually converge to it for large $\nde$. Interestingly, optimal sensitivity is achieved for the shallow depth of the entangler $\nen = 2$, indicating that it is significantly more challenging to implement optimal decoding unitaries than to implement optimal entangling unitaries. This is due to the fact that the entangler unitary only needs to transform $\ket{\psi_0}$ to $\ket{\psiin}$, while the decoder needs to transform all measurement bases to the desired bases simultaneously, imposing significantly more constraints on the optimal unitary.

To understand how the variational circuits approach the 2p-QS, it is insightful to study the optimal estimators and see how they change as the depth of the decoder increases. In Fig.~\ref{fig:2DLargePriorEstimators}, we show the estimators of the optimized variational circuits. The minimal depth circuit $\nde = 1$ results in an asymmetric distribution of estimators in the $\phi_x\phi_y$-plane, indicating that the sensor is not equally sensitive to $\phi_x$ and $\phi_y$. Increasing the depth to $\nde = 10$, the estimators become more evenly distributed, whereas the estimator pattern for $\nde = 32$ is very close to the optimal 2p-QS shown in Fig.~\ref{fig:2p-QS}.

An important question for currently available quantum sensors is how much sensitivity gain we can expect for a fixed circuit depth. We address this by determining the minimum value of the ratio $\Delta/\sqrt{2} \delta$ at the optimal operating value of $\delta$ for a given circuit depth. In Fig.~\ref{fig:2DScaling}, this value is shown as a function of the system size $N$. Interestingly, we find that even for $\nde=1$, there is an increase in performance that appears to scale with $N$. As the depth of the circuit increases, the sensitivity and scaling improve steadily.

\begin{figure}[t]
    \centering
    \includegraphics[]{./2DScaling.pdf}
    \caption{\textbf{Performance scaling of variational 2D field sensors.}
        Scaling of the minimum value of the posterior to prior uncertainty ratio as a function of system size $N$. The purple lines correspond to variational quantum sensors with different numbers of decoding layers $\nde$ at a fixed entangler depth of $\nen = 2$. For comparison, we also show the performance of the 2p-CS (blue line) and the optimal 2p-QS (green line).}
    \label{fig:2DScaling}
\end{figure}

\section{3D field sensing}
\label{sec:3D}

While so far our focus has been on 2D field sensing, the techniques developed in the previous sections are readily extended to 3D. Here, we present first results and observations for optimal and variational 3D field quantum sensors.

We observe that the OQS solution in 3D makes use of the full effective Hilbert space of an $SU(2)$ sensor~($\text{dimension}\sim N^3$), see Appendix~\ref{appendix:1p_and_2p_subspaces}. This suggests that sensors made of a few partitions do not closely approximate the OQS for 3D sensing of fields with arbitrary prior widths. Nevertheless, few-partite sensors can provide a significant improvement over classical sensors and even perform optimally at small prior widths, as we demonstrate below.

Figure~\ref{fig:3DCost} shows the performance of various sensors with $N=6$ atoms, quantified by the ratio $\Delta/\sqrt{3}\delta$, as  $\delta$ is scanned.  The 1p-QS and 2p-QS sensors, are both significantly less sensitive than the OQS in the large prior regime and only the full $N$-partite quantum sensor ($N$p-QS), relying on universal control, approaches the OQS. Here, we extend the discussion to a 3-partition quantum sensor (3p-QS), with uniform control at the level of three partitions. However, the 3p-QS performance is still appreciably different from that of the OQS, highlighting the added complexity of the 3D OQS near its optimal operating point. To compare the few-partite sensors with a classical reference, we study a 3-partite classical sensor, where the atoms are unentangled. Here, we find that each partition is primarily sensitive to one component. In comparison to the 3p-CS, the 2p-QS and 3p-QS show a considerable improvement by using entanglement between the sensor atoms.

\begin{figure}
    \centering
    \includegraphics[]{./3DCostScan.pdf}
    \caption{\textbf{Bayesian quantum sensing of a 3D field.}
    Left panel: Ratio of the posterior width $\Delta$ to the prior width $\sqrt{3}\delta$ vs. $\delta$ for $N=6$ atoms. The black marker shows one preliminary data point obtained for the OQS at $\delta=0.75$. This point matches well with the sensitivity of a $N$p-QS (dark gray line). The asymptotic bound on quantum (black dotted line) and classical (blue dotted line) sensors are predicted by the Bayesian Cram\'er-Rao bounds~\eqref{eq:BayesianQL} and \eqref{eq:BayesianSQL}. In Fig.~\ref{fig:3D_1p-QS}, we illustrate the working of the optimal sensor indicated here by the red triangle.
    Magnified panel (right): Performance of variational circuits (purple lines) with an increasing total number of layers $n_{\rm l} = n_{\rm en} + n_{\rm de}$. }
    \label{fig:3DCost}
\end{figure}

\subsubsection{Optimal 1p-QS}
In the regime of small prior widths $\delta\lesssim 1/N$, the performance of the 1p-QS is indistinguishable from the optimal performance. To gain insight into the 1p-QS solution, we visualize the optimal state, measurement projectors, and estimators in Fig.~\ref{fig:3D_1p-QS}. The optimal state shows a regular distribution of maxima and minima on the generalized Bloch sphere, highlighting the high degree of symmetry in this state. Notably, the estimators are found to lie on a sphere. As a result, they provide only directional information and therefore the sensor acts as a 3D ``compass''. The visualized measurement projectors resemble the input state rotated around the axis specified by the respective estimator. In other words, each projector is sensitive to rotations around a different axis.

We conclude that the optimal sensor at small prior widths displays a universal compass-like behavior for 1D, 2D, and 3D fields. However, a key difference is that while GHZ ($\ket{m=0}$) states are optimal independent of $N$ in 1D (2D), the optimal state for 3D field sensing changes with $N$ similar to the states discussed in~\cite{Zimba2006, Kolenderski2008, Giraud2010, Bjork2015,Bouchard2017}.

\subsubsection{Variational optimization of 1p-QS}
The performance of the 1p-QS motivates us to study variational circuits to approximate the optimal 1p-QS. As sketched in Fig.~\ref{fig:Sensors}(e), we consider circuits that parameterize $\mathcal{U}_{\rm en(de)}^{\rm 1p}$ according to
\begin{align}
    \mathcal{U}_{\rm en}^{\text{1p}}(\bm{\theta}) &= \mathcal{L}\left(\bm{\theta}_{\nen}\right) \cdots \mathcal{L}\left(\bm{\theta}_{n_{\text{1}}}\right)\\
    \mathcal{U}_{\rm de}^{\rm 1p}(\bm{\vartheta}) &= \mathcal{L^{\dagger}}(\bm{\vartheta}_{1}) \cdots \mathcal{L}^{\dagger}(\bm{\vartheta}_{n_{\text{de}}}).
\label{eq:1p_U}
\end{align}
Here, the unitaries are constructed by repeated application of layers containing three elementary gates
\begin{align}
\mathcal{L}(\bm{\theta}) = \mathcal{T}_z\left(\bm{\theta}^{(3)}\right)\,
    \mathcal{R}_y\left(\bm{\theta}^{(2)}\right)\,
    \mathcal{R}_{x}\left(\theta^{(1)}\right).
\end{align}
The number of variational parameters at a given circuit depth is therefore $3(\nen + \nde)$. In the magnified panel of Fig.~\ref{fig:3DCost}, we plot the performance of variational circuits with varying depths and indeed find that circuits with modest depths can approach the optimal 1p-QS performance. These results highlight the potential for implementing a 3D quantum ``compass'' with currently available resources.

\begin{figure}
    \centering
    \includegraphics[]{./1p-QS_3D.pdf}
    \caption{\textbf{Illustration of the optimal 1p-QS for 3D fields.} The sensor is optimal for the prior width $\delta$ indicated by the red triangle in Fig.~\ref{fig:3DCost}.  We show the Wigner distributions of the input state $\ket{\psiin}$ [prepared by the unitary $\mathcal{U}_{\rm en}^{\text{1p}}$~\eqref{eq:1pQS_psi}] and representative projective measurement operators $\Pi_m$ [realized by the unitary $\mathcal{U}_{\rm de}^{\text{1p}}$~\eqref{eq:1pQS_pi}]. The measurement outcomes $m$ are mapped to estimators $\bm{\xi}_{m}$ in the $\phi_x\phi_y$-plane, and the ones corresponding to the displayed $\Pi_m$ are highlighted by colored frames. The purple arrows illustrate that the Wigner distribution of the projectors resemble approximate rotated versions of the Wigner distribution of the input state, when it is rotated about the axis along the vector specified by the corresponding estimator $\bm{\xi}_{m}$.}
    \label{fig:3D_1p-QS}
\end{figure}

\section{Conclusion and Outlook}

In this work, we have identified fundamental bounds and studied the performance of practically feasible sensors for the sensing of 2D and 3D vector fields within the Bayesian framework for $SU(2)$-quantum interferometry. We established a procedure to determine the OQS that sets the fundamental bound on the performance of an $N$-atom sensor. Furthermore, we introduced sensors with limited experimental control, namely the 1p-QS and 2p-QS, and studied their ability to approach the OQS. As a further step towards practical realization, we also simulated the variational optimization of sensors with limited entangling capabilities, considering the resources available on state-of-the-art trapped ion systems.

Our study of 2D field sensing reveals important similarities and differences between single- and multi-parameter metrology. In the case of 1D sensing, the OQS operates in the particle permutation symmetric subspace of the $N$ atoms, and hence the 1p-QS is optimal at all prior widths. Here, we find that the 1p-QS is optimal for 2D sensing only in the regime of small prior widths. In this regime, it operates like a 2D ``compass'', providing information only on the field direction. This is similar to the GHZ-state interferometer, optimal for 1D sensing with small prior widths, which only provides information on the sign of the 1D field. In contrast to the 1D case, we find that the 2p-QS approaches the OQS performance for 2D field sensing at large prior widths.

Although such a sensor requires more experimental control than a 1p-QS, it can nevertheless be realized on state-of-the-art sensing platforms, as we demonstrate by simulating a variational circuit optimization of the 2p-QS. Our simulation reveals that sensors made of low-depth circuits provide appreciable improvements over classical (unentangled) sensors, while deeper circuits may be required to achieve optimal performance. This is in contrast to 1D sensing, where low-depth variational circuits have previously been shown theoretically and experimentally to achieve optimal performance. Nevertheless, these results have promising implications for small-scale sensors as well as for those that involve a macroscopic number of atoms; current technology provides excellent coherent control of a small number of atoms, enabling the realization of deep circuits  that can reach the 2p-QS limit for small quantum sensors. Such sensors may be relevant for practical applications where spatial resolution is important. On the other hand, for sensors operating with a large number of atoms, our results demonstrate that shallow-depth circuits can significantly improve performance compared to unentangled sensors.

Although our study has focused on using infinite-range OAT as an entangling resource, many promising quantum sensing platforms host finite-range interactions, e.g. solid-state systems
 or neutral atoms in tweezer arrays
\cite{Young2020, Graham2019, Madjarov2020, Schine2022}. Variational optimization of the metrological cost function can be directly generalized to finite-range interactions, as done in~\cite{Kaubruegger2019, Koczor2020, Kaubruegger2021, Zheng2022} for other metrological cost functions. However, in that case, the classification into few-partite quantum sensors does not apply, and simulating the full many-body problem is challenging for classical algorithms, which makes feedback loop optimization on the physical quantum device even more desirable. Furthermore, the $N$-atom 1p-QS and 2p-QS can also be realized, respectively, as a single large macrospin \cite{Chalopin2018} of length $J=N/2$ or two entangled macrospins \cite{Kruckenhauser2023} of length $J=N/4$ each, allowing for a route to optimal sensing with qudit-based quantum sensors.

We also briefly extended our study to the problem of 3D field sensing. Here, we found an appreciable difference in the performance of sensors with a few partitions and the optimal sensor. Therefore, it is an open challenge to find circuits that efficiently approximate the OQS for 3D field sensing. However, in the regime of small prior widths, we once again find that the 1p-QS is essentially optimal. Similar to the 1D and 2D cases, it operates as a 3D ``compass'' in this regime, with the estimators arranged on the surface of a sphere. Furthermore, we showed that such a 3D ``compass'' can be well approximated with low-depth variational circuits, paving the way for their experimental realization.

In conclusion, we have highlighted the importance of finding Optimal Quantum Sensors, which operate at the ultimate precision allowed by quantum physics, and implementing them as Variational Quantum Sensors. This is crucial not just for uncovering new physics through precision measurements \cite{Gilmore2021} but also for practical applications such as quantum computing. Programmable atomic sensors, discussed in this article, can be integrated with atomic quantum computing systems by utilizing common quantum resources, like entangling quantum gates. In this way, an optimal quantum sensor for electromagnetic fields embedded in the system can precisely detect and monitor noise fields, contributing to the stability of quantum devices through an optimal feedback mechanism.

\begin{acknowledgments} We thank C. Marciniak, and Q. Liu for comments on the manuscript. The computational results were obtained using the University of Innsbruck's LEO HPC infrastructure.  Research  in  Innsbruck  is supported by the US Air Force Office of Scientific Research (AFOSR) via IOE Grant No.~FA9550-19-1-7044 LASCEM,
the European Union's Horizon 2020 research and innovation program under PASQuanS 2, and by the Simons Collaboration on Ultra-Quantum Matter, which is a grant from the Simons Foundation (651440, P.Z.),
and by the Institut f\"ur Quanteninformation. Innsbruck theory is a member of the NSF Quantum Leap Challenge. AS acknowledges the support of a CV Raman Post-Doctoral Fellowship, IISc.
\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


Kohal on live esinejad ja oma dj-d.
NB!! LOOSIPAKID! Külla tuleb ka jõuluvana ja et kingikotist oleks midagi võtta palume kõigil kaasa võtta kingipakk väärtusega kuni 5 eurot.
NB vol 2! Peol avatud ka baar! Tasumine sulas!
Christmas party for all students of landscape architecture!
There will be live performances and also our own djs.
NB!! Santa Claus will visit also our party and to make sure that everyone receives a gift, we ask You to prepare a small gift (up to 5 euros).
To receive a gift, please prepare a small performance (poem, song, play) for Santa, alone or together with your course mates.
NB vol 2! There's also bar at the party! Cash only!
NB vol 3! EMÜS's members and alumnus and ofcourse lecturers are have free entrance, for all the others the ticket will be 2 € from advance sale and 3€ on the spot.


One floor of the Swiss National Museum is dedicated to furnishings including furniture, entire rooms and the contents there of. The is one exhibit that displays decorative woodworking techniques including marquetry and intarsia.
One of the exhibits. My feet at the bottom. Wearing white after Labor Day.
The illustration looked familiar and it turns out I knew it.
It is a plate from André Jacob Roubo's L'Art du Menuisier (The Art of the Joiner). Those of us who are even vaguely aware of Chris Schwarz and Lost Art Press know that Roubo was an 18th century carpenter and cabinet maker. His book, L'Art du Menuisier (The Art of the Joiner), is an object of veneration to many woodworkers and scholars. LAP has issued a translation of the section on marquetry, veneer work, wood selection, finishing and some casework, To Make as Perfectly as Possible. They have also issued The Book of Plates containing every single illustration from all of the volumes.
While we are on things close to Herr Schwarz's heart, I offer these images with no explanation. Also in the museum.
Known to us as the Roorkhee chair or facsimile.
My wife's reasons for planning the trip to South Africa was to experience the culture, sample the wine and see the animals. Being the more passive member of the excursion, I just hope to see some unique furniture. All goals were achieved.
Not available at Rooms To Go. Or, internationally, IKEA.
Breadboarding in the African style.
Back legs are shaved or turned down.
I was told this is a Baule chair and saw many similar examples of this chair around Cape Town. The story is that this chair is used by the elder or chief at a low dining table while those of lower status sit on the floor. Having sat in a few of these chairs, I believe that the floor is a fine place to sit. Heavy is the head that wears the crown.
The details are different but basically the same chair. Perhaps more refined, but the construction is the same. Better wood negating the need for breadboard ends.
I was told this is a Senufo chair used by the chief or elder at a low dining table while others sit on the floor. Baule and Senufo are two ethnic groups of the Ivory Coast (Côte d'Ivoire if you prefer the French). The Baule inhabit Central Ivory Coast and the Senufo the North and Mali.
A Google image search for Baule and Senufo chairs leads me to believe that the terms are interchangeable with both type chairs show up in either. Keep in mind using Google as a primary reference is alway problematic but it's what I have.
Possibly a serpent's head? You might not be able to see it but the wedge on the left runs left to right while the right on runs front to back.
The head of a hippo? Or a different serpent? No quick and easy explanation of the 8 pointed star.
I have more pictures of the chairs HERE if you care. They're still there if you don't care, actually.
The finish on the Baule chair had a very waxy feel with a low sheen. I wondered if it was a soap finish that's been all the rage lately. Or possibly just an accumulation of commercial cleaning problems.
If you do your own Google image search, you will see many variations of the chairs with lesser and greater degrees of ornamentation and height. Not all chiefs like sitting that close to the floor I suppose.
Construction is different but it comes down to four legged chairs, front legs stop at the seat and are wedged. Rear legs continue up and support the back. Slightly unique implementation.
I've Run Out of Things to See in the US.
As you might have read, I've recently been to South Africa with stops in Zürich and Munich to break up the time locked in overstuffed aluminum tubes. Jumping off one eight-hour flight to get on another one has never been fun and with age even less so. Since we don't have to, we don't.
We stopped for three nights in Zürich (Switzerland) on the way out. An amazing and expensive city. My wife did a fantastic job of planning and found us a hotel in the middle of most things interesting at a price that was only slightly scary. We've paid more for less.
Don't often get to see them in museums. They're always there, just out of sight.
Can't do that with an iPhone. You can, but you shouldn't. Not at the Swiss Museum.
I like using real cameras from time to time.
This is what 1895 looked like for some.
Was this the precursor of Danish Modern?
I really like the wood grained TV.
Nobody can explain to me late 1970's furniture. And I lived through it.
There will be lots more to come in the next few weeks. I am so backed up on furniture pics. I still have all the furniture from the Met in New York, Bayou Bend in Houston, St. Louis Art Museum and tons of auctions.
More when the laundry is done.
I think Rooms To Go sells one just like it.
The ones from Rooms To Go have carved acanthus leaves instead.
And while this technically isn't a foot, it does support the case.
Axially symmetrical. Must be a Federal piece.
Simple is in the eye of the beholder.
I used to use a lot of these then I grew out of it.
Major wedges. Might have been gappy. Might be German.
To see the balance of the pictures, click HERE.
There is an auction house about 90 minutes from my house. They have two auctions a month most months. One tends to be a bit ordinary while the other is nearly extraordinary. Problem is that it is 90 minutes away. Driving there, taking pictures and driving back will take up most of a day.Even I have other things to do and can only make it a few times a year.
Another dang slant front desk.
Haven't shown secret compartments for a while. It's time.
Might have been three drawers once. I can't tell.
Might have been but there isn't now.
I'm sure you all have a few of these.
I'm not sure where you buy these.
Not shown in any book I own.
And a 150 year old pocket screw.
To see the rest of Part One, click HERE.
I can't spend all my time wandering Missouri, Kansas and the World looking for antiques and other interesting things. Some days I have to stay home to do laundry and change the cat boxes.
When life interferes with my travels, I have to console myself by visiting the local auction houses and shops. They have interesting things and I get to eat dinner at home. With the cats.
Rope handle has been lost.
It exists, really. Most tills are on the left.
As is "Klieg lights". Klieg lights were actually carbon arc spotlights but same company.
Looking at is and its strange fasteners, you wonder if it is a knock-down or ready to assemble kit.
A few fasteners on the back legs.
Your guess is as good as mine. Maybe better.
Click HERE to see the scintillating set.
I drove through West Virginia on the way to Kansas City for Woodworking in America. Paying $6.00 to drive through West Virginia bothers me until I consider what the PA Turnpike costs from end to end.
As I do from time to time, I found an antiques mall and spent some time wandering through for my continuing quest to find the most interesting thing I've ever seen. If I ever find it, what to I do the next day?
We all know this box. More on this coming soon. Yes, there's more.
Interesting to think there was a need/desire to make Hummel knockoffs.
Cupboard or press, you decide.
The third known (to me) example of angled door edges being used to hide the gap.
Angled, really. Trust me. That's my foot. I was wearing brown shoes.
You can check the archive for the other two examples.
Lots more of these coming soon.
I figure there must have been Germans in West Virginia.
I've seen this hinge before. I need to check the archives.
The rest of the antiques from the Switzerland of America are HERE.
Yes, I'm going to make you work again. My wife is putting together a trip to South Africa later this year. Her current notion is to spend time around Cape Town and Port Elizabeth in South Africa with two-day stopovers in Zürich on the way down and Munich on the way home.
I assume that there are more things to see on this trip beyond exotic animals and wineries. I would appreciate any suggestions y'all might have regarding things of interest to me.
After all, you are some of the most interesting people on the web.
A few days back I posted pictures of this plane I found in Cookeville, TN.
Unique plane and the curved blade makes it more interesting.
Monday night, a few of us were sitting around looking at the plane and trying to figure out who made the blade. None of us could read the logo on the blade. The best we could do was the Cast Steel and Sheffield.
Is the first the best? Can anyone offer a guess?
I took three more pictures with various angles of raking light. Are any of these useful?
If any of you have any ideas from whence this came, feel free to share.
Everybody liked the blasting caps' box. Well, some of you liked it. OK, a couple of you liked it. There was some interest in dimensions. I am accommodating you and providing some more pictures and some limited dimensions.
The box is 14.25″ high by 6.25″ wide.
Handle is about 1″ diameter turned down to 0.625″. (5/8″) Handle center is 1.125″ (1 1/8″) down from the top. Sides extend 3.625″ above the vertical. Dovetail is for a latch. It is centered and 0.75″ wide by 2.625″ (2 5/8″) deep. I don't know why the short angled kerfs exist.
Box is about 5″ deep. 3″ radius on the top curve.
Back is rabbeted in on all four sides. Top and bottoms dividers are full sliding dovetails.
Back is pegged at the top.
Lower divider is 1.25″ up from the bottom.
Boards are all ≈ 0.5″ thick. It is 9.4375″ (9 7/16″) from outside to outside of the dividers.
Lid is as wide as it is and around 9″ tall.
No dimensions, just an interesting picture.
If you would like a detailed SketchUp drawing feel free to make one. I'm not going to stop you.
Tonight's new subject is actually a late 19th century ballot box from Vermont, or so the auction house says. My first ever intentional live auction win. I might have paid too much but really only $25 over the underbidder.
Ballot box is 17″ by 10″ by 9″.
It has a slot in the middle of the top. I assume this is where the voter stuffed the ballot.
Another slot on the bottom to remove the ballots. Note, the bottom is not burl.
A sliding door to keep the ballots inside the box until needed.
Nothing much to see here but I like the burl.


text:
Home › Practical Daily Exercises for the Ocarina Volume Five - For Double Ocarina
Practical Daily Exercises for the Ocarina Volume Five - For Double Ocarina
For advanced double ocarina players
Written by master ocarina artist
Enhance your skills with daily exercises
Learn to switch between chambers with ease
Usually ships the same business day
This volume, Volume 5, expands upon the introductory exercises into more advanced exercises. In this volume, the ocarina player learns to put both chambers together in extended, full range (2 octave) exercises. These exercises build a firm foundation in the keys and key signatures that are most commonly expected of the ocarina player.
Before beginning this book, the author highly recommends a review of the Practical Daily Exercises for the Ocarina Volume 4 - For Double Ocarina available through STL Ocarina. Volume Four focused on finger exercises in the second chamber and exercises to develop smooth and efficient switching between chambers.
Master Ocarina Artist Jenna Daum has appeared internationally as soloist and artist-teacher. Passionate about bringing this ancient instrument to a wider audience, Dr. Daum provides a wealth of knowledge to both novice and experienced players in this new series of daily exercise books.
In addition to showcasing the ocarina to enthusiasts around the world, Dr. Daum is an award-winning performer and teacher of the flute and has appeared with orchestras around the country, including The Phoenix Symphony, Colorado Music Festival Orchestra, Paradise Winds, Symphony of the Southwest, and the St. Louis Philharmonic. Jenna holds the Bachelor of Music degree with Highest Honors from the University of Illinois, and the Masters in Music and Doctorate in Music from Arizona State University, where she has served as Faculty Associate and founding director of the Arizona State University Community Flute Ensemble.
Our Music library

redpajama_set_name: RedPajamaCommonCrawl


KrisBaker is not following any shows yet.
I love this community, but I don't know many people in it. I'm looking for new friends to talk about RT with. And everything else, honestly. I haven't seen my friends in 7 months and I'm starting to get bored with life and all the mundane tasks I have to do every day. I'm just looking for something new. Feel free to message me any time.
You can't just wish and hope your dreams will come true. You have to chase it to achieve it. I have to do this for myself. I can't live a life full of "what ifs". I can't handle anymore regrets. Regrets for the things I never did or never said. I don't fully understand this feeling or how I got it. It feels like a beast has been awaken inside of my soul. It's not just passion or desire. It's something more. Something stronger. I believe it's Fate. It'll take time, but I will become the person I'm meant to be. This world will know me. This world will see the Real Kris Baker. No more hiding. My self doubt will wash away and confidence will replace it. I've never had confidence. I never believed in myself. I hated everything I did. I hated myself. My confidence disappeared when I was young. I hated the sound of my voice. That's why I stayed so quiet all the time. That's why I sat alone in school. I didn't like myself. I was a different person back then. That's the amazing thing about time. You grow. You grow into a different person than you were before. Every day is a new day. A new chance to do better and to become your true self. Every minute we have a chance to say something or do something that can change a life. Sometimes it's a stranger in need. Sometimes it's yourself. I'm ready to live my life. I've gone too long without being myself. On August 6th, my life changed. I met amazing people who chased their dreams and made them realities. I met people that I've known for 12 years. Seeing them in person was the best feeling in the world. They made me happy and made me laugh when I needed it the most. And I want to be that for other people. People who are struggling with life just like I was. People who suffer from depression. People who contemplate suicide. People who were bullied or looked down upon. I want to be the light at the end of the tunnel for them. I want them to know that someone cares. Just like Rooster Teeth, I want to be there for people. I want to help fill broken hearts with laughter and joy. I want to replace all the bad thoughts with happiness. I want to help people with beaten down souls and crushed dreams. I'm not sure how I'll do this, but I'll find my way. My own way of helping people. This is Fate. My Fate and my Destiny.
I could only afford my hotel for 2 nights. I am in desperate need of a room for the remainder of RTX. Can anyone help me out? I'll sleep on the floor if I have to. Please.
I could only pay for 2 nights in my hotel. The little money I have is for food. What is the best bridge to sleep under in Austin?
Wanna play? It's been a while.


text:
The Information School or iSchool of the University of Sheffield, in Sheffield, South Yorkshire, England, was founded in 1963 as the University's Postgraduate School of Librarianship and became in 2010 the first UK iSchool. The School is ranked Number One in the World for Library and Information Management.  Other names were the Postgraduate School of Librarianship and Information Science (PGSLIS, 1967–81) and Department of Information Studies (1981-2011). , it employs 33 academic staff, 16 administrative/support staff, 6 affiliated research staff, and has about 65 research students. The current head of school is Professor Val Gillet.

The department opened in 1964 as a library school, becoming only the second university-based department in the UK. Since then, like many information science departments it has grown to encompass teaching and research in cheminformatics, educational informatics, health informatics, information retrieval, information systems, knowledge and information management, as well as libraries and information society. Such is the status of the school, that it has twice been honoured with a special issue of the Journal of Information Science devoted entirely to the department, its staff and its research outputs.

Research achievements
The School is Number One in the World for Library and Information Studies in the QS World University Rankings 2021. The school has ranked highest or joint highest in its subject rating in every Research Assessment Exercise since the running of the first exercise in 1986. In this UK-government sponsored assessment of research outputs, no other department in its subject field (or its University) achieved this consistency; few departments of any subject area in UK universities managed such a high level of continuous research output (see the following links to the 1992, 1996, 2001 RAE results). In 2008, rankings of departments was left to news organizations; the Times Higher Education placed Sheffield at No. 1 again.

In 2008, an analysis of citations showed four of the ten most cited UK information studies academics were working in the Sheffield department. It is also the first UK-based (and 2nd European) department to become an iSchool.

Notable staff, past and present
 Micheline Beaulieu - Chair of the Computing and Informatics Panel of the European Research Council (2008-2011)
 Sheila Corrall - first President of Chartered Institute of Library and Information Professionals (CILIP), 2002-3:.
 Michael Lynch, chemoinformatics specialist
 Wilfred Saunders, Founding Director of the Postgraduate School of Librarianship, Sheffield
 Bob Usherwood - President of the Library Association, 1998; fellow of the Royal Society of Arts.
 Steve Whittaker - inducted into the SIGCHI academy in 2008.
 Paul Clough - Head of Information Retrieval Research Group
 Peter Willett - leading researcher in cheminformatics.
 Tom Wilson - recipient of the 2009 ASIS&T SIG USE award for "outstanding contributions to information behavior" and Association for Information Science and Technology (ASIS&T) Award of Merit, 2017.

Notable alumni
 Michael Buckland - Emeritus Professor at the University of California—Berkeley School of Information and Co-Director of the Electronic Cultural Atlas Initiative
 John McTernan - Director of Political Operations at 10 Downing Street under Tony Blair.
 Alasdair Paterson - poet, winner of the 1975 Eric Gregory Award for poetry.
Hatoon Kadi - Saudi Arabian comedian and activist who hosts the Noon Al Niswa comedy show on YouTube.

References

Further reading
Saunders, W. L., ed. University and Research Library Studies: some contributions from the University of Sheffield Post-graduate School of Librarianship and Information Science. Oxford: Pergamon Press, 1968

External links
 University of Sheffield Information School home page

University of Sheffield
Information schools
Library science education

redpajama_set_name: RedPajamaWikipedia


Considering the current racial regression – or call it stagnation – occurring around the world, it is perhaps unsurprising that black and Asian women are still struggling to find makeup products that suit their skin. How is it that, in 2016, some of the biggest international beauty brands still haven't formulated foundations for the varying skin tones of the women who make up a significant proportion of their customer base?
I say this as a half West African, half West Indian, 27-year-old makeup enthusiast who mixes three different foundations each morning in order to create a suitable base. In fact, after a recent holiday resulting in me being a little darker, I'm now boycotting foundation entirely as I'm yet to find the correct colour match (bear in mind how big my makeup bag is as a beauty editor.) And judging by the results from a new survey, it is simultaneously a relief and incredibly disheartening to find that a lot of other women in the UK share my daily struggle.
Last week, Superdrug released findings from a survey of 559 women of black and Asian ethnic backgrounds which revealed that over two thirds (70%) of black and Asian women feel the high street does not cater for their beauty needs. Over a third (36%) feel there isn't enough guidance and advice available in high street beauty stores, and research also revealed that black and Asian women spend on average £137.52 more on beauty products per year than anyone else, including over £25 on their foundation, due to lack of choice.
Sarah Gardner, Head of Beauty at Superdrug stated: "As a leading health and beauty retailer, we feel it is our duty to listen to our customers which is why we've launched the Shades of Beauty campaign. This research just goes to show the lack of products and advice available on the high street for black and Asian women, so we have made it our mission to offer women of colour the latest in beauty products at an affordable price. Since launching the Shades of Beauty campaign, we've already seen an increase in foundation shades from Maybelline, L'Oreal and Revlon and will be launching an additional 23 shades for darker skin tones by the end of July."
This campaign comes as a welcome development following a chain of events which culminated in British model Leomie Anderson taking to Twitter in February, during New York Fashion Week, to comment on how uneducated and ill-prepared many makeup artists are to sufficiently work with darker skin. "Why is it that the black makeup artists are busy with blonde, white girls and slaying their makeup and I have to supply my own foundation?" she tweeted. "Why are there more white makeup artists backstage than black when black ones can do all races' makeup?... "We shouldn't have to feel worried sitting in the chair of a professional that we may not look our best when doing our jobs, unlike our white counterparts."
Similarly, model Nykhor Paul garnered global attention in July last year when she shared her experiences via Instagram, asking: "Why do I have to bring my own makeup to a professional show when all the other white girls don't have to do anything but show up?"
While a number of female black public figures have spoken up about their difficulties and lack of representation, the struggle for Asian women to find beauty products suited to their skin tone is even less documented. Few Asian faces front makeup campaigns and Maybelline hit the headlines in May of this year by signing their first Asian global ambassador, Taiwanese model I-Hua Wu who joins the likes of Gigi Hadid, Jourdan Dunn, and Tanzanian model Herieth Paul. Though Wu isn't the first Asian beauty to represent the brand, she is the first to do so on a global scale. While progress is happening slowly but surely, there is still a long way to go.
"From my research and the experience of my family and friends, the lack of colour ranges has been and remains a major issue to this day (there wouldn't be a need for me to write stories like this or this if there wasn't.) As far as my favourite brands, I tend to gravitate toward black-owned companies more so than others (because I feel like they "get it" more than others), like SheaMoisture, Camille Naturals, and I've recently become a fan of Taliah Waajid's hair care products. In terms of makeup, Black Up has been killing it for a while as well as Black Opal and Iman Cosmetics. More widely-known beauty brands like Nars and MAC do a good job of catering to women whose skin tone is deeper than a caramel shade."
"It's improved dramatically, though not at a value price point. The high end brands like YSL, Lancôme and Chanel all have darker foundation. Sadly it's a catch 22 situation. Women of colour aren't going to Boots because Boots haven't been stocking for them so they go to Superdrug or Global Beauty Supply stores who do. So Boots don't stock for them in case it's sat on the shelf. It's supply and demand, it's just a business. Hopefully it will all start to find a flow.
"I see brown girls advertising fashion and beauty in the mainstream now, especially in the last three years. If this continues, it will actually start representing the population as a whole and that's what I hope for."


text:
DENSO Receives a Michigan Breastfeeding-Friendly Workplace Award
BATTLE CREEK, Mich., May 15, 2019 /PRNewswire/ — The Michigan Breastfeeding Network (MIBFN) recently recognized DENSO's thermal manufacturing facility in Battle Creek, Michigan, for creating and maintaining a supportive workplace...
by	vstartye	May 15, 2019
BATTLE CREEK, Mich., May 15, 2019 /PRNewswire/ — The Michigan Breastfeeding Network (MIBFN) recently recognized DENSO's thermal manufacturing facility in Battle Creek, Michigan, for creating and maintaining a supportive workplace for breastfeeding employees. MIBFN evaluated companies operating across the state of Michigan on a number of criteria including facilities, policy and education, scheduling and resources, ultimately recognizing DENSO with its Silver award. The honor is emblematic of DENSO's long-time commitment to support its employees and ensure they have the resources necessary to succeed at work and life.
"We'd be nothing without our associates who bring their skills and passion to work every day and we take pride in delivering them support and amenities that help them balance their professional and personal lives," said Natalie Scott, communications and external affairs. "This award is a direct reflection of DENSO's resolve to help employees provide for themselves and their families, however needed."
DENSO's thermal facility in Battle Creek first established a mother's room and lactation policy in 1999. Each associate – regardless of department and position – is provided 50 minutes of paid time for lactation within their regularly scheduled workday. Associates work with their supervisor to arrange for the breaks they need and use a posted schedule to share the room with other nursing mothers. The mother's room consists of two individual, locking rooms; lockers; a sink for washing; a refrigerator for storage; two breast pumps mounted on the wall and outlets for those who wish to use their own pumps.
Dana Paulson, a team leader and long-time DENSO employee, recently used the facility for her now two-year-old, but also more than a decade ago for her now teenager. "I was really amazed when I started working here years ago to see how great the set-up is," said Paulson. "My daughter was very picky and refused formula. Were it not for DENSO's supportive policies and facilities I would have had to choose between my child's nourishment and my employment. Instead, I was able to do it all."
According to the MIBFN, when companies support breastfeeding, businesses gain a three-to-one return on investment in addition to greater employee retention, lower absenteeism, and a more creative and productive work environment. More information on the MIBFN awards can be found at https://www.mibreastfeeding.org/workplace/.
About DENSO
DENSO is a $48.3 billion global mobility supplier that develops advanced technology and components for nearly every vehicle make and model on the road today. With manufacturing at its core, DENSO invests in its 221 facilities in 35 countries to produce thermal, powertrain, mobility, electrification, & electronic systems, to create jobs that directly change how the world moves. The company's 170,000+ employees are paving the way to a mobility future that improves lives, eliminates traffic accidents, and preserves the environment. Globally headquartered in Kariya, Japan, DENSO spent 9.3 percent of its global consolidated sales on research and development in the fiscal year ending March 31, 2019. For more information about global DENSO, visit https://www.denso.com/global.
In North America, DENSO employs 27,000+ engineers, researchers and skilled workers across 31 sites in the U.S, Canada and Mexico. In the United States alone, DENSO employs 17,000+ employees across 13 states and 25 sites. Headquartered in Southfield, Michigan, in fiscal year ending March 31, 2019, DENSO in North America generated $10.9 billion in consolidated sales. Join us, and craft not only how the world moves, but also your career. For more information, go to https://www.denso.com/us-ca/en/.
SOURCE DENSO
http://www.densocorp-na.com
Enterprise Truck Rental on Track to Open 40 More Locations This Year
by	vstartye	- May 15, 2019
Industry Leaders Appointed to ATRI Board of Directors


Installing security cameras around your home can provide surveillance as well as peace of mind. By connecting a camera to your VCR, you can even record the data for later viewing. But it's no longer necessary to upgrade your good old video cassette recorder in order to have the time and date on screen for these security backups: The Time Date Generator can project the information on the screen and thus on the tape, so you'll know exactly when events occurred.
The Time Date Generator can display one or two lines of the date and time on your video recording. You can determine the extent of the year/month/date and hour/minute/second information available. Additional features include adjustable text brightness.
The Time Date Generator has two BNC connections; connect to your VCR using an RCA cable with a BNC to RCA adapter (both sold separately).
8.6" x 6" x 2.25"


My interest includes blogging, cooking,baking and decorating cakes, coaching, makeup and fashion, freelance writing and photography.
I graduated in 2000 with a Post Graduate Diploma in Education(PGDE) from the NIE/Nanyang Technological University, Singapore.
Prior to that I obtained a Bachelor of Commerce (1999) from the University of Melbourne.
I am constantly upgrading myself and make it a point to keep abreast with the latest methodology and teaching practices that are being promoted in the Education sector.
Obtaining my ACTA certification in 2016 and DACE in 2018 I'm a qualified curriculum developer and instructional designer.
I have always been passionate about kids, and coming from a family of educators, want to share and enrich the lives of my family, friends and readers as we journey together as parents.
Since becoming a mom I have looked for ways to connect with other parents. I have found the internet to be a great source of information and inspiration which helped me in my parenting.
I am administrator of a few online facebook blogging groups, mentor in blogging and facilitate blog transfers from blogger to wordpress..
I have worked with many PR firms since the start of this blog back in 2007 connecting readers with amazing products and remain PR friendly with many local and international companies.
If you are a business or company and would like for me to host a giveaway and/or conduct product reviews here on Dominique's Desk then please let me know!
I am mainly do paid advertorials and are opened to sponsored post which items/services will are beneficial for myself and my family. I have worked on campaigns with major brands since the start of this blog and have been running longstanding campaigns with several of them.
I have worked with major brands like Loreal (L'oreal White Perfect Laser ), Lifebuoy, Comfort , Drypers and Philips among others.
My article on Bike Riding with Singapore Bike School has been written have been featured on SGLinks a local daily magazine which highlights interesting posts from blogs from Singapore.
The article Keeping Motivated Blogging on a Monday was featured on Parent society as one of the featured bloggers for the week of March 5th 2013.
My media kit can be provided upon request.
You can contact me via my Contact page.
Where did the name of the blog come from?
My inspiration for this blog comes from- My Desk in our study room at home.
This blog is an outlet for me to connect with others.
I warmly welcome you to journey together with me as a parent by keeping in contact through one of our subscription methods you see on our sidebar.
I hope you enjoy your visit here at Dominique's Desk.