$\begin{split}\hat{\mathcal{F}_{\textnormal{Q}}}(rW)&=\sup_{|{\psi}\rangle}\big{\\{}r\langle{W}\rangle_{\psi}-4(\Delta
J_{z})^{2}_{\psi}\big{\\}}\\\
&=\sup_{|{\psi}\rangle}\big{\\{}r\langle{W}\rangle_{\psi}-4\langle{J_{z}^{2}}\rangle_{\psi}+4\langle{J_{z}}\rangle^{2}_{\psi}\big{\\}}\\\
&=\sup_{|{\psi}\rangle}\big{\\{}\langle{rW-4J_{z}^{2}}\rangle_{\psi}+\langle{2J_{z}}\rangle^{2}_{\psi}\big{\\}},\end{split}$
(4.7)

where we have used the fact that the QFI can be expressed as a convex roof of
$(\Delta J_{z})^{2}$ and we arrive at the problem of an optimization over a
single parameter for simplicity on the following derivations. Equation (4.7)
can be rewritten as an optimization linear in operator expectation values and
over a parameter $\mu$ as

$\hat{\mathcal{F}}_{\text{Q}}(rW)=\sup_{|{\psi}\rangle,\mu}\big{\\{}\langle{rW-4J_{z}^{2}}\rangle_{\psi}+8\mu\langle{J_{z}}\rangle_{\psi}-4\mu^{2}\mathbbm{1}\big{\\}},$
(4.8)

which, making use of $\max\\{\langle{A}\rangle\\}=\lambda_{\max}[A]$ for any
observable, can be reformulated as

$\begin{split}\hat{\mathcal{F}}_{\text{Q}}(rW)&=\sup_{|{\psi}\rangle}\big{\\{}\lambda_{\max}[rW-4J_{z}^{2}+8\mu
J_{z}-4\mu^{2}]\big{\\}}\\\
&=\sup_{|{\psi}\rangle}\big{\\{}\lambda_{\max}[rW-4(J_{z}-\mu)^{2}]\big{\\}},\end{split}$
(4.9)

where we omitted in writing $\mathbbm{1}$ for clarity and $\lambda_{\max}[A]$
stands for the maximum eigenvalue of the operator $A$. At the extremum, the
derivative with respect to $\mu$ must be zero, hence at the optimum
$\mu=\langle{J_{z}}\rangle_{\text{opt}}$ which represents the expectation
value of $J_{z}$ should have considering the optimal state in Eq. (4.7). This
also means that we have to test $\mu$ values in the interval
$-N/2\leqslant\mu\leqslant N/2$ only for spin-half systems.

The full optimization problem to be solved consists of Eqs. (4.2) and (4.9)
substituting $g(\rho)$ by
$\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{z}}]$,

$\mathcal{B}_{\,\mathcal{F}}(w)=\sup_{r}\big{\\{}rw-\sup_{\mu}\\{\lambda_{\max}[rW-4(J_{z}-\mu)^{2}]\\}\big{\\}}.$
(4.10)

It is crucial that the optimization over $r$ is a concave function, since the
theory tells us that $\hat{\mathcal{F}}_{\text{Q}}(rW)$ is a convex function
[101], even when the multi-parameter case is considered. Thus the optimum can
be determined easily with simple methods, e.g., the gradient method, looking
for the maximum in $r$. Based on Eq. (4.2), we can see that even if we do not
find the global optimum in $r$, we obtain a valid lower bound. The extension
of this bound to the multi-parameter case is done using the recipe given in
Eq. (4.6). On the other hand, the function to be optimized for $\mu$ does not
have a single maximum in general. Moreover, not finding the optimal $\mu$
leads to an overestimating of the bound. Thus, a large care must be taken when
optimizing over $\mu$.

We stress again the generality of these findings beyond linear interferometers
covered in the following sections. For nonlinear interferometers [66, 82, 85,
84, 81, 102], the phase $\theta$ must be estimated assuming unitary dynamics
$U=\exp{-iG\theta}$, where $G$ is not a sum of single spin operators, hence,
it is different from the angular momentum components.

####  4.1.4 Exploiting the symmetries

When making calculations for quantum systems with an increasing number of
qubits, we soon run into difficulties when computing the largest eigenvalue of
Eq. (4.9). The reason is that for $N$ qubits, we need to handle $2^{N}\times
2^{N}$ size matrices, hence we are limited to systems of 10 to 15 qubits.

We can obtain bounds for much larger particle numbers, if we restrict
ourselves to the symmetric subspace [95, 96]. This approach can give optimal
bounds for many systems, such as Bose-Einstein condensates of two-level atoms,
which are in a symmetric multiparticle state. The bound computed for the
symmetric subspace might not be correct and generally might overestimate the
real bound for general cases.

Finally, it is important to note that if the operators $W_{k}$ are
permutationally invariant and the eigenstate with the maximal eigenvalue in
Eq. (4.9) is non-degenerate, then we can do the computations on the symmetric
subspace only. The resulting maximal eigenvalue is the maximal eigenvalue even
when the whole Hilbert space is taken into account for the maximization.
Hence, the lower bound obtained in the symmetric subspace is valid even for
the general case.

For completeness, we follow presenting the proof of the observation mentioned
above. Let us denote the ground state of a permutationally invariant
Hamiltonian by $|{\Psi}\rangle.$ This is at the same time the $T=0$ thermal
ground state, hence it must be a permutationally invariant pure state. For
such states
$S_{kl}|{\Psi}\rangle{\\!}\langle{\Psi}|S_{kl}=|{\Psi}\rangle{\\!}\langle{\Psi}|$,
where $S_{kl}$ is the swap operator exchanging qubits $k$ and $l$. Based on
this, follows that $S_{kl}|{\Psi}\rangle=c_{kl}|{\Psi}\rangle$, and
$c_{kl}\in{-1,+1}$. There are three possible cases to consider:

  1. i)

All $c_{kl}=+1$. In this case, for all permutation operator $\Pi_{j}$ we have

$\Pi_{j}|{\Psi}\rangle=|{\Psi}\rangle,$ (4.11)

since any permutation operator $\Pi_{j}$ can be constructed as
$\Pi_{j}=\prod_{i}S_{k_{i}l_{i}}$. Equation (4.11) means that the state
$|{\Psi}\rangle$ is symmetric.

  2. ii)

All $c_{kl}=-1$. This means that the state is anti-symmetric, however this
state exists only for $N=2$ qubits.

  3. iii)

Not all $c_{kl}$ are identical to each other. In this case, there must be
$k_{+},l_{+},k_{-},k_{-}$ such that

$\begin{split}S_{k_{+},l_{+}}|{\Psi}\rangle&=+|{\Psi}\rangle,\\\
S_{k_{-},l_{-}}|{\Psi}\rangle&=-|{\Psi}\rangle.\end{split}$ (4.12)

Let us assume that $k_{+},l_{+},k_{-},l_{-}$ are index different from each
other. In this case,
$|{\Psi^{\prime}}\rangle=S_{k_{+},k_{-}}S_{l_{+},l_{-}}|{\Psi}\rangle$ another
ground state of the Hamiltonian $H$ such that

$\begin{split}S_{k_{+},l_{+}}|{\Psi^{\prime}}\rangle&=-|{\Psi^{\prime}}\rangle,\\\
S_{k_{-},l_{-}}|{\Psi^{\prime}}\rangle&=+|{\Psi^{\prime}}\rangle.\end{split}$
(4.13)

Comparing Eqs. (4.12) and (4.13) we can conclude that
$|{\Psi^{\prime}}\rangle\neq|{\Psi}\rangle$, while due to the permutational
invariance of $H$ we have that
$\langle{H}\rangle_{\Psi^{\prime}}=\langle{H}\rangle_{\Psi}$. Thus,
$|{\Psi}\rangle$ is not a non-degenerate ground state. The proof works in an
analogous way for the only nontrivial case $k_{+}=k_{-}$, when
$S_{k_{+},k_{-}}=\mathbbm{1}$.

Hence, if $N>2$ then only i) is possible and $|{\Psi}\rangle$ must be
symmetric.

Next, we will demonstrate the use of our approach for several experimentally
relevant situations. In the many-particle case, often symmetric operators can
be used to describe accurately the system, which makes it possible to carry
out calculations for thousand of particles, as will be shown later in this
chapter.

###  4.2 Examples

In this section, we show how to obtain lower bounds based on the fidelities
with respect to the GHZ state and the unpolarized Dicke state as well as with
different sets of powers of collective angular momentum operators, e.g., the
set
$\\{\langle{J_{y}}\rangle,\langle{J_{x}}\rangle,\langle{J_{x}^{2}}\rangle\\}$.

####  4.2.1 Fidelity measurements

Let us consider the case when $W$ is a projector onto a pure quantum state.
First, we consider GHZ states. Hence $W$ is the projector
$|{\textnormal{GHZ}}\rangle{\\!}\langle{\textnormal{GHZ}}|$, where

$|{\textnormal{GHZ}}\rangle=\tfrac{1}{\sqrt{2}}(|{0\cdots 0}\rangle+|{1\cdots
1}\rangle)$ (4.14)

for spin-$\frac{1}{2}$ particles, and $\langle{W}\rangle=F_{\textnormal{GHZ}}$
is the fidelity with respect to the GHZ state. Based on knowing
$F_{\textnormal{GHZ}}$, we would like to estimate
$\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{z}}]$§§§ Not tight lower
bounds on the quantum Fisher information based on the fidelity have been
presented in [103]..

Using Eq. (4.10), we will obtain an analytical tight lower bound on the QFI
based on the fidelity $F_{\textnormal{GHZ}}$. The calculation that we have to
carry out is computing the bound

$\mathcal{B}_{\,\mathcal{F}}(F_{\textnormal{GHZ}})=\sup_{r}\big{\\{}rF_{\textnormal{GHZ}}-\sup_{\mu}\\{\lambda_{\max}[r|{\textnormal{GHZ}}\rangle{\\!}\langle{\textnormal{GHZ}}|-4(J_{z}-\mu)^{2}]\\}\big{\\}}.$
(4.15)

We will make our calculations in the $J_{z}$ orthonormal basis, which is
defined with the $2^{N}$ basis vectors $b_{0}=|{00\dots 000}\rangle$,
$b_{1}=|{00\dots 001}\rangle$, …, $b_{(2^{N}-2)}=|{11\dots 110}\rangle$, and
$b_{(2^{N}-1)}=|{11\dots 111}\rangle$, as it can be found in Eq. (A.2) for
$j=\frac{1}{2}$. It is easy to see that the matrix in the argument of
$\lambda_{\max}$ in the Eq. (4.15) is almost diagonal in the $J_{z}$ basis. To
be more specific, the only non-diagonal matrix block comes from
$|{\textnormal{GHZ}}\rangle{\\!}\langle{\textnormal{GHZ}}|$, which has non-
trivial matrix elements only in the $\\{b_{0},b_{(2^{N}-1)}\\}$ basis. Thus,
we have to diagonalize the following matrix

$r|{\textnormal{GHZ}}\rangle{\\!}\langle{\textnormal{GHZ}}|-4(J_{z}-\mu)^{2}=\begin{pmatrix}\frac{r}{2}-4(\frac{N}{2}-\mu)^{2}&\frac{r}{2}\\\
\frac{r}{2}&\frac{r}{2}-4(\frac{N}{2}+\mu)^{2}\end{pmatrix}\oplus D,$ (4.16)

where $D$ is already a $(2^{N}-2)\times(2^{N}-2)$ diagonal matrix with
$D_{k}=-4(\langle{J_{z}}\rangle_{b_{k}}-\mu)^{2}$ negative eigenvalues for
$k=1,2,\dots,(2^{N}-2)$. This means that the Eq. (4.16) can be diagonalized as
$\text{diag}[\lambda_{+},\lambda_{-},D_{1},D_{2},\dots,D_{2^{N}-2}]$, where
the two eigenvalues $\lambda_{\pm}$ are

$\lambda_{\pm}=\frac{r}{2}-N^{2}-4\mu^{2}\pm\sqrt{16\mu^{2}N^{2}+\frac{r^{2}}{4}}.$
(4.17)

Next, we show a way that can simplify our calculations considerably. As
indicated in Eq. (4.15), we have to look for the maximal eigenvalue and then
optimize it over $\mu$. We exchange the order of the two steps, that is, we
look for the maximum of each eigenvalue over $\mu$, and then find the maximal
one. The eigenvalues of $D$ are negative and for some $\mu$’s some of them can
be zero. Due to this, the problem can be simplified to the following equation

$\begin{split}\sup_{\mu}\\{\lambda_{\max}[r|{\textnormal{GHZ}}\rangle{\\!}\langle{\textnormal{GHZ}}|-4(J_{z}-\mu)^{2}]\\}:=&\max\\{0,\sup_{\mu}(\lambda_{+})\\}\\\
=&\left\\{\begin{aligned} &0,&&\text{ if }r<0,\\\
&\frac{r}{2}+\frac{r^{2}}{16N^{2}}&&\text{if }0\leqslant r\leqslant 4N^{2},\\\
&-N^{2}+r&&\text{if }r>4N^{2},\end{aligned}\right.\end{split}$ (4.18)

where we did not have to have to look for the maximum of $\lambda_{-}$ over
$\mu$ since clearly $\lambda_{+}\geqslant\lambda_{-}$. Finally, we have to
substitute Eq. (4.18) into Eq. (4.15), and carry out the optimization over
$r$, considering $F_{\textnormal{GHZ}}\in[0,1]$.

This way we arrive at a lower bound of the QFI based on the fidelity with
respect to the GHZ state as

$\mathcal{B}_{\,\mathcal{F}}(F_{\textnormal{GHZ}})=\left\\{\begin{aligned}
&N^{2}(1-F_{\textnormal{GHZ}})^{2}&&\text{if }F_{\textnormal{GHZ}}<1/2,\\\
&0&&\text{if }F_{\textnormal{GHZ}}\leqslant 1/2.\end{aligned}\right.$ (4.19)

This equation is plotted in Figure 4.1-(a). Note that in the figure the plot
is normalized by $N^{2}$ and that the resulting semi-parabola is independent
of the number of particles. Moreover, the bound is zero for
$F_{\textnormal{GHZ}}\leqslant 1/2$. This is consistent with the fact that for
the product states $\rho=|{111\dots 11}\rangle$ or $\rho=|{000\dots
00}\rangle$ we have $F_{\textnormal{GHZ}}=1/2$, while
$\mathcal{F}_{\text{Q}}[\rho,J_{z}]=0$.

\begin{overpic}[scale={.65}]{img/LT_fidGHZ.pdf} \put(5.0,10.0){\small(a)}
\end{overpic}\begin{overpic}[scale={.65}]{img/LT_fidDicke.pdf}
\put(5.0,10.0){\small(b)} \end{overpic}

Figure 4.1:  (a) Analytical solution of the bound
$\mathcal{B}_{\,\mathcal{F}}$ for different values of the fidelity with
respect to the GHZ state. (b) Numerical results for the minimum quantum Fisher
information as a function of the fidelity with respect of unpolarized Dicke
states perpendicular to the magnetic field, $|\text{D}_{N}^{0}\rangle$. (line)
For systems with 4 particles and (dashed) for systems with 40 particles. One
may note that when the fidelity is at its maximum the bound approaches 0.5 as
it is the quantum Fisher information for a large particle number.

Next, let us consider a symmetric unpolarized Dicke state with even $N$
particles along the $x$-direction $|{\textnormal{D}_{N}}\rangle_{x}$, given by
Eq. (3.1). This state is known to be highly entangled [95, 104] and allows for
a Heisenberg limited interferometry [97]. In the following we may omit the
subscript $x$ since this Dicke state will be always at the center of our
attention, the unpolarized Dicke state perpendicular to the magnetic field in
this case along the $z$-direction. The witness operator that can be used for
noisy Dicke states is
$W=|{\textnormal{D}_{N}}\rangle{\\!}\langle{\textnormal{D}_{N}}|$, hence the
expectation value of the witness is just the fidelity with respect to the
Dicke state, i.e., $\langle{W}\rangle=F_{\text{Dicke}}$. In Figure 4.1-(b), we
plotted the results for symmetric Dicke states of various particle numbers.
$F_{\text{Dicke}}=1$ corresponds to
$\mathcal{F}_{\text{Q}}[\rho,J_{z}]=N(N+2)/2$. At this point, note that for
the examples presented above, the QFI bound scales as $\mathcal{O}(N^{2})$ in
the asymptotic limit if the quantum state has been prepared
perfectly¶¶¶$\mathcal{O}(x)$ is the usual Landau notation used to describe the
asymptotic behavior for large $x$ [77, 92]..

Note that estimating $\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{z}}]$
based on $F_{\text{Dicke}}$ was possible for 40 qubits for Figure 4.1-(b),
since we carried out the calculations for the symmetric subspace. For our
case, the witness operator $W$ is permutationally invariant and it has a non-
degenerate eigenstate corresponding to the maximal eigenvalue. Hence, based on
the arguments of the Section 4.1.4 the bound is valid even for the general
case, i.e., non-symmetric states.

We now compute several quantities for the large $N$ case. We show that if the
fidelity with respect to the Dicke state is larger than a bound then
$\mathcal{B}_{\,\mathcal{F}}>0$, where we omit the arguments for brevity.
Moreover, we have seen in Figure 4.1-(b) that the lower bound on
$\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{z}}]$ as a function of the
fidelity $F_{\text{Dicke}}$ normalized by $N^{2}$ is not the same curve for
all $N$. Next, we will demonstrate by numerical evidence that the lower bound
normalized by $N^{2}$ collapses to a nontrivial curve for large $N$.

As a first step, let us consider the state completely polarized along
$z$-direction $|{1}\rangle_{y}^{\otimes N}$. This state does not change under
rotations around the $z$-axis, hence
$\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{z}}]=0$. Its fidelity with
respect to the Dicke state $|{\textnormal{D}_{N}}\rangle_{x}$ is

$F_{\text{Dicke}}(|{1}\rangle_{y}^{\otimes
N})=\frac{1}{2^{N}}\binom{N}{N/2}\approx\sqrt{\frac{2}{\pi N}}.$ (4.20)

From convexity of the bound on the quantum Fisher information in
$F_{\text{Dicke}}$, it immediately follows that for $F_{\text{Dicke}}$ smaller
than Eq. (4.20) the optimal bound on
$\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{z}}]$ will give zero.

Next, we examine what happens if the fidelity is larger than Eq. (4.20). For
that we note first that $\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{z}}]$
is the convex roof of $4(\Delta J_{z})^{2}$ [88, 89]. Hence, if we have a
mixed state for which $\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{z}}]$
is zero, then it can always be decomposed into the mixture of pure states for
which $\mathcal{F}_{\textnormal{Q}}[{\textstyle|{\Psi}\rangle,J_{z}}]$ is zero
too. As a consequence, the extremal states of the set of states for which
$\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{z}}]=0$ are pure states, and
we can restrict our search for pure states. The optimization problem we have
to solve is given as

$F_{\text{opt}}=\big{\\{}\max_{\Psi}|\langle{\Psi}|{\textnormal{D}_{N}}\rangle_{x}|^{2}\,:\,\mathcal{F}_{\textnormal{Q}}[{\textstyle|{\Psi}\rangle,J_{z}}]=0\big{\\}}.$
(4.21)

Hence, we have to carry out the optimization over pure states $|{\Psi}\rangle$
that are invariant under $U_{\theta}=\exp(-iJ_{z}\theta)$ for any $\theta$.
Such states are the eigenstates of $J_{z}$. In order to maximize the overlap
with the Dicke state $|{\textnormal{D}_{N}}\rangle_{x}$, we have to look for
symmetric eigenstates of $J_{z}$. These are the symmetric Dicke states in the
$z$-basis $|{\textnormal{D}_{N,m}}\rangle_{z}$. Then, using the following
identity

$\sum_{k=0}^{q}(-1)^{k}\binom{n}{k}\binom{n}{q-k}=\left\\{\begin{aligned}
&\binom{n}{q/2}(-1)^{q/2}&&\text{for even }q,\\\ &0&&\text{for odd
}q,\end{aligned}\right.$ (4.22)

one finds that the squared overlap is given by

$|\langle{\textnormal{D}_{N,m}}|{{}_{z}}|{\textnormal{D}_{N}}\rangle_{x}|^{2}=\left\\{\begin{aligned}
&\frac{\binom{N/2}{m/2}^{2}\binom{N}{N/2}}{2^{N}\binom{N}{m}}&&\text{for even
}m,\\\ &0&&\text{for odd }m,\end{aligned}\right.$ (4.23)

which is maximal in the case of even $N$ when $m=N$ or $m=0$, the state
totally polarized along the $+z$-direction or along the $-z$-direction
respectively. We skip the case in which $N$ is odd. For detailed calculations
of Eq. (4.23) see Appendix F.

Next, we will examine the behavior of our lower bound on
$\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{z}}]$ based on the fidelity
$F_{\text{Dicke}}$ for large $N$. In Figure 4.2, the calculations up to
$N=500$ present a strong evidence that for fidelity values
$F_{\text{Dicke}}=0.2,0.5,0.7$ the lower bound on QFI has a
$\mathcal{O}(N^{2})$ scaling for increasing $N$. If this is correct then
reaching a fidelity larger than a certain bound for large $N$ would imply
Heisenberg scaling for the bound on the quantum Fisher information. Note that
it is difficult to present a similar numerical evidence for small values of
$F_{\text{Dicke}}$ since in that case the bound for QFI is nonzero only for
large $N$ due to Eq. (4.20).

Figure 4.2:  Lower bound on QFI normalized by $N^{2}$ for various particle
numbers $N=50,100,200,300,$ $400,500$. (circles) Lower bound for
$F_{\text{Dicke}}=0.2$, (stars) for $F_{\text{Dicke}}=0.5$, and (triangles)
for $F_{\text{Dicke}}=0.7$. For a better visibility we use a logarithmic scale
for the $y$-axis.

####  4.2.2 Spin-squeezed states

In the case of spin squeezing, the quantum state has a large spin in the
$y$-direction, while a decreased variance in the $x$-direction. By measuring
$\langle{J_{y}}\rangle$ and $(\Delta J_{x})^{2}$ we can estimate the lower
bound on the quantum Fisher Information by Eq. (2.50). However, this formula
does not necessarily give the best lower bound for all values of the
collective observables. With our approach we can find the best bound.

To give a concrete example, we choose $W_{1}=J_{y}$, $W_{2}=J_{x}^{2}$ and
$W_{3}=J_{x}$ for the operators to be measured. We vary $w_{1}$ and $w_{2}$ in
some interval. We also require that $w_{3}=0$, since we assume that the mean
spin points into the $y$-direction111 Due to symmetries of the problem, when
minimizing $\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{z}}]$ with the
constraints on $\langle{J_{z}}\rangle$ and $\langle{J_{x}^{2}}\rangle$, we do
not have to add explicitly the constraint $\langle{J_{x}}\rangle=0$. The
optimization with only the first two constraints will give the same bound..
This is reasonable since in most spin-squeezing experiments we know the
direction of the mean spin.

Our result can be seen in Figure 4.3. We chose $N=4$ particles since for small
$N$ the main features of the plot are clearly visible. The hatched area
corresponds to non-physical combination of expectation values. States at the
boundary can be obtained as ground states of
$H_{\text{bnd}}^{(\pm)}(\lambda)=\pm J_{x}^{2}-\lambda J_{y}$, see Appendix D.
In Figure 4.3, the state fully polarized in the $y$-direction, and initial
state for spin-squeezing experiments, corresponds to point T. The unpolarized
Dicke state along the $x$-direction Eq. (3.1) corresponds to point D.

Figure 4.3:  We show as a function of the expectation value,
$\langle{J_{y}}\rangle$, and the variance in the perpendicular direction,
$(\Delta J_{x})^{2}$, the minimum sensitivity for a 4-qubit system. (hatched)
The physically forbidden region is indicated. Interesting quantum states: (M)
Mixed state defined in the text, (T) totally polarized state, (S) singlet
state, and (D) Dicke state. (W) Any mixture of the singlet state and the
completely mixed state of the symmetric subspace. Other states can be found on
this line, for instance, the completely mixed state of the whole Hilbert
space. (dashed) Shot-noise threshold. Below this line non-classical
sensitivities can be achieved. (cross) In Figure 4.4, we compute the bound
when an additional expectation value is measured.

We add that outside the symmetric subspace, there are other states with
$\langle{J_{y}}\rangle=\langle{J_{x}^{2}}\rangle=0$, which also correspond to
the point D, e.g the singlet state labeled by the point S. However, usual
spin-squeezing procedures remain in the symmetric subspace, thus we discuss
only the Dicke state. Spin-squeezing makes $(\Delta J_{x})^{2}$ decrease,
while $\langle{J_{y}}\rangle$ also decreased somewhat. Hence, at least for
small squeezing it corresponds to moving down from point T to point D
following the boundary, while the metrological usefulness is increasing. Below
the dashed line $\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{z}}]>N$,
hence the state possesses metrologically useful entanglement [3]. The equal
mixture of $|{000\dots 00}\rangle_{x}$ and $|{111\dots 11}\rangle_{x}$
corresponds to point M, with
$\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{z}}]=N$. Finally, the
completely mixed state rests on the line W. It cannot be used for metrology,
hence $\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{z}}]=0$.

We now compare the difference between our bound and the bound of L. Pezzè and
A. Smerzi Eq. (2.50). First, we consider the experimentally relevant region
for which $(\Delta J_{x})^{2}\leqslant 1$. We find that for points away from
the physical boundary at least by 0.001 on the vertical axis, the difference
between the two bounds is smaller than $2\times 10^{-6}$. Hence, Eq. (2.50)
practically coincides with the optimal bound for $(\Delta J_{x})^{2}<1$.

For points at the boundary, the difference is somewhat larger, but still
small, the relative difference is smaller than $2\%$ for 4 particles. We
compute the difference between the Eq (2.50) and our bound for different
number of particles and for states at the boundary from the state totally
polarized T to the unpolarized Dicke state at D, see Figure 4.4-(a).

\begin{overpic}[scale={.65}]{img/LT_edge_diff.pdf} \put(5.0,10.0){\small(a)}
\end{overpic}\begin{overpic}[scale={.65}]{img/LT_ss_add_jx4.pdf}
\put(5.0,10.0){\small(b)} \end{overpic}

Figure 4.4:  (a) Difference between the bound of Pezzè-Smerzi and the optimal
bound for the quantum Fisher information normalized by the value of the
optimal bound itself for the bosonic ground states of $H=J_{x}^{2}-\lambda
J_{y}$ for $\forall\lambda\in[0,\infty)$. From dark to lighter colors (solid-
dark, dotted, dash-dotted, dashed, pointed, solid-light), results for
different particle numbers, $N=4,6,10,20,1000$ respectively. For large
particle numbers the difference is largest when the polarization is around two
thirds of the maximal polarization and that this difference is less than
$2.6\%$. (b) Lower bound on QFI for $\langle{J_{y}}\rangle=1.5$, $(\Delta
J_{x})^{2}=0.567$, as a function of $\langle{J_{x}^{4}}\rangle$. The
corresponding point in Figure 4.3 is denoted by a cross. (gray-area) Lower
bound on precision below the shot-noise limit. (dashed) Lower bound without
constraining $\langle{J_{x}^{4}}\rangle$. (dash-dotted) Lower bound when
bosonic symmetry is considered. As can be seen, an additional constraint or
assuming symmetry improves the bound.

We now consider regions on Figure 4.3 for which $(\Delta J_{x})^{2}>1$. The
difference between the two bounds is now larger. It is larger at point M, for
which the bound Eq. (2.50) is zero. Hence for measurement values corresponding
to points close to M, our method improve the formula Eq. (2.50).

It is important from the point of view of applying our method to spin-
squeezing experiments that the bound Eq. (2.50) can be substantially improved
for $(\Delta J_{x})^{2}<1$, if we assume bosonic symmetry for the system, or
we measure an additional quantity, such as $\langle{J_{x}^{4}}\rangle$ as
shown in Figure 4.4-(b).

####  4.2.3 Dicke states

In this section, we use our method to find lower bounds on the QFI for states
close to the Dicke states (3.1) along the $x$-direction, based on collective
measurements. We discuss what operators have to be measured to estimate the
metrological usefulness of the state. In Section 4.3.2, we will test our
approach for a realistic system with very many particles.

In order to estimate the metrological usefulness of states created in such
experiments, we choose to measure $W_{1}=J_{x}^{2}$, $W_{2}=J_{y}^{2}$ and
$W_{3}=J_{z}^{2}$ since the expectation values of these operators uniquely
define the ideal Dicke state, and they have been already used for entanglement
detection [24]. In cold gas experiments of nowadays, the state created is
invariant under transformations of the type $U_{x}(\phi)=\exp(-iJ_{x}\phi)$
[105]. For such states $\langle{J_{y}^{2}}\rangle=\langle{J_{z}^{2}}\rangle$,
which we also use as a constraint in our optimization.

Let us demonstrate how our method works in an example for small systems.
Figure 4.5 shows the result for 6 qubits for symmetric states for which

$\langle{J_{x}^{2}+J_{y}^{2}+J_{z}^{2}}\rangle=\frac{N(N+2)}{4}=:\mathcal{J}_{N/2},$
(4.24)

which was introduced in Chapter 3. It can be seen that the lower bound on
quantum Fisher Information is the largest for $\langle{J_{x}^{2}}\rangle=0$.
It reaches the value corresponding to the ideal Dicke state,
$\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{z}}]/N=(N+2)/2=4$. It is
remarkable that the state is also useful for metrology if
$\langle{J_{x}^{2}}\rangle$ is very large. In this case
$\langle{J_{y}^{2}}\rangle$ and $\langle{J_{z}^{2}}\rangle$ are smaller than
$\langle{J_{x}^{2}}\rangle$.

Figure 4.5:  Optimal lower bound on the quantum Fisher Information for
symmetric states with $\langle{J_{y}^{2}}\rangle=\langle{J_{z}^{2}}\rangle$.
Even if it is metrologicaly useful for a wide range of
$\langle{J_{x}^{2}}\rangle$, the numerics shows us a tiny region where the
metrological gain is surpassing the shot-noise limit.

###  4.3 Calculations for experimental data

In this section, we use our method to find tight lower bound on the QFI based
on experimental data. In particular, we will determine the bound for several
experiments in photons and trapped ions creating GHZ states and Dicke states,
in which the fidelity has been measured [60, 52, 54, 49, 55, 56, 106, 107,
108, 109], which is much easier than obtaining the quantum Fisher Information
from the density matrix [77], or estimation it from a metrological procedure
[8]. We will obtain a bound on the QFI for a spin-squeezing experiment with
thousand of particles [7]. Based on numerical examples, we see that the bound
Eq. (2.50) is close to the optimal even for not completely polarized states.
Assuming symmetry or knowing additional expectation values can improve the
bound Eq. (2.50). Finally, we will also obtain the bound for the QFI for a
recent experiment with Dicke states [24]. The estimate of the precision based
on considering the particular case when $\langle{J_{x}^{2}}\rangle$ is
measured for parameter estimation [105] is close to the optimal bound computed
by our method.

####  4.3.1 Few-particle experiments

Now, we will estimate the quantum Fisher information based on the fidelity
with respect to Dicke states and GHZ states for several experiments with
photons and trapped cold ions, following the ideas of Section 4.2.1.

Our results are summarized in Table 4.1. The experiments in [54, 109] are with
hyperentangled qubits, while in the rest of experiments a single qubit is
stored in a particle. Ref. [56] describes experiments with 2-14 ions, we
presented only results of two of them. Finally, for the experiment of Ref.
[110] we used the fidelity estimated using reasonable assumptions discussed in
that paper, while the worst case fidelity is lower.

Physical | Target | Fidelity | $\mathcal{B}_{\,\mathcal{F}}/N$ | Ref.
---|---|---|---|---
system | quantum state
photons | $|{\textnormal{D}_{4}}\rangle$ | $0.844\pm 0.008$ | $1.432\pm 0.044$ | [106]
$0.78\pm 0.008$ | $1.124\pm 0.236$ | [109]
$0.8872\pm 0.0055$ | $1.680\pm 0.036$ | [60]
$0.873\pm 0.005$ | $1.44\pm 0.024$ | [34]
$|{\textnormal{D}_{6}}\rangle$ | $0.654\pm 0.024$ | $0.564\pm 0.076$ | [107]
$0.56\pm 0.02$ | $0.304\pm 0.048$ | [108]
photons | $|{\textnormal{GHZ}_{4}}\rangle$ | $0.840\pm 0.007$ | $1.848\pm 0.076$ | [110]
$|{\textnormal{GHZ}_{5}}\rangle$ | $0.68$ | $0.65$ | [110]
$|{\textnormal{GHZ}_{8}}\rangle$ | $0.59\pm 0.02$ | $0.256\pm 0.128$ | [111]
$|{\textnormal{GHZ}_{8}}\rangle$ | $0.776\pm 0.06$ | $2.4376\pm 0.1072$ | [54]
$|{\textnormal{GHZ}_{10}}\rangle$ | $0.561\pm 0.019$ | $0.15\pm 0.11$ | [54]
trapped-ions | $|{\textnormal{GHZ}_{3}}\rangle$ | $0.89\pm 0.03$ | $1.824\pm 0.291$ | [49]
$|{\textnormal{GHZ}_{4}}\rangle$ | $0.57\pm 0.02$ | $0.08\pm 0.052$ | [55]
$|{\textnormal{GHZ}_{6}}\rangle$ | $0.509\pm 0.004$ | $0.0018\pm 0.0018$ | [112]
$|{\textnormal{GHZ}_{8}}\rangle$ | $0.817\pm 0.004$ | $3.21\pm 0.08$ | [56]
$|{\textnormal{GHZ}_{10}}\rangle$ | $0.626\pm 0.006$ | $0.64\pm 0.06$ | [56]
Table 4.1:  Fidelity values and the corresponding bound for the QFI for
several experiments with Dicke states and GHZ states. Bounds normalized with
$N$ are shown. The ones surpassing the value one in the fourth column show
quantum entanglement enhanced metrological usefulness. For Dicke states the
maximum is achieved at $(N+2)/2$, i.e., $3$ for the
$|{\textnormal{D}_{4}}\rangle$ case and $4$ for the
$|{\textnormal{D}_{6}}\rangle$ case. For the case in which GHZ states are used
the limit for the normalized bound is $N$, the particle number.

We can compare our estimate to the quantum Fisher information of the state for
the experiment of Ref. [60], where the QFI for the density matrix was obtained
as $\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{z}}]/N=(10.326\pm
0.093)/N=(2.5816\pm 0.02325)$. As can be seen in Table 4.1, this value is
larger than we obtained, however, it was calculated by knowing the entire
matrixm, while our bound is obtained from the fidelity alone.

####  4.3.2 Many-particle experiments

In this section, we will estimate the quantum Fisher information based on
collective measurements for experiments aiming to create spin-squeezed states
and Dicke states.

Spin-squeezing experiment

We turn our attention to a recent many-particle spin-squeezing experiment in
cold gases to use our method to find lower bounds on the quantum Fisher
information, following the ideas of Section 4.2.2. With that we show that the
lower bound given in Eq. (2.50) is close to the optimal. We also demonstrate
that we carry out calculations for real systems.

In particular, for our calculations we use the data from spin-squeezing
experiments of Ref. [7]. The particle number is $N=2300$, and the spin-
squeezing parameter defined as

$\xi_{\textnormal{s}}^{2}=N\frac{(\Delta
J_{x})^{2}}{\langle{J_{y}}\rangle^{2}}$ (4.25)

has the value
$\xi_{\textnormal{s}}^{2}=-8.2\textnormal{dB}=10^{-8.2/10}=0.1514$. The spin
length $\langle{J_{y}}\rangle$ has been close to maximal. In our calculations,
we choose

$\langle{J_{y}}\rangle=\alpha\frac{N}{2},$ (4.26)

where we will test our method with various values for $\alpha$. For each
$\alpha$ we use $(\Delta J_{x})^{2}$ will be given such that we get the
experimentally obtained spin-squeezing parameter Eq. (4.25). Moreover, we
assume $\langle{J_{x}}\rangle=0$, as the $y$-direction was the direction of
the mean spin in the experiment. Based on Eq. (2.50), the bound for the
quantum Fisher information is obtained as

$\frac{\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{z}}]}{N}\geqslant\frac{1}{\xi_{\textnormal{s}}^{2}}=6.605.$
(4.27)

For our computations we need a tool to handle large systems. We will carry out
the calculations for symmetric states. this way we obtain a lower bound on the
QFI that we will denote by $\mathcal{B}_{\,\textnormal{sym}}$. As already
mentioned, we could obtain a bound for the QFI that is valid even for general
case, not necessarily symmetric states if the matrix from which compute the
maximum eigenvalue Eq. (4.9) has a non-degenerated largest eigenvalue. This is
not the case in general for the spin-squeezing problem. However, we still know
that our bound obtained with our calculations in the symmetric subspace cannot
be smaller than the optimal bound $\mathcal{B}_{\,\mathcal{F}}$, which must be
larger or equal to the Eq. (2.50) since it cannot be smaller than the optimal
bound for general states. These relations can be summarized as

$\mathcal{B}_{\,\textnormal{sym}}\geqslant\mathcal{B}_{\,\mathcal{F}}\geqslant\frac{\langle{J_{y}}\rangle^{2}}{(\Delta
J_{x})^{2}},$ (4.28)

where on the right-hand side we just used the bound in Eq. (2.50).

Our calculations lead to

$\frac{\mathcal{B}_{\,\textnormal{sym}}(\langle{J_{y}}\rangle,(\Delta
J_{x})^{2})}{N}=6.605$ (4.29)

for a wide range of values of $\alpha$. That is, based on numerics, the left-
hand side and the right-hand side of Eq. (4.29) seem to be equal. This implies
that the lower bound Eq. (2.50) is optimal for estimating the QFI for the
system.

We follow giving the details of our calculations for $\alpha=0.5,0.85$ and we
show examples in which we can improve the bound Eq. (2.50) with our approach,
if symmetry is assumed. We present a simple scheme that can be used to handle
large systems, and make calculations for larger particle number. Thus, we need
fewer steps for the numerical optimization for large system sizes, which makes
our computations faster. Second, while we will be able to carry out the
calculation for the particle number of the experiment, we will also see that
we could even extrapolate the results from the results obtained for lower
particle numbers. This is useful for future application of our method for very
large systems.

The basic idea is that we transform the collective quantities from $N$ to a
smaller particle number using the scaling relation

$\displaystyle\langle{J_{y}}\rangle$
$\displaystyle=\frac{N^{\prime}}{2}\alpha,$ (4.30) $\displaystyle(\Delta
J_{x})^{2}$
$\displaystyle=\xi_{\textnormal{s}}^{2}\frac{N^{\prime}}{4}\alpha^{2}.$ (4.31)

We see that for the scaling we consider, for all $N^{\prime}$ the bound in Eq.
(2.50) is valid, i.e., is obtained as

$\frac{\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho_{N^{\prime}},J_{z}}]}{N^{\prime}}\geqslant\frac{1}{\xi_{\textnormal{s}}^{2}}=6.605.$
(4.32)

Let us first take $\alpha=0.85$, which is somewhat smaller than the
experimental value, however, it helps us to see various characteristics of the
method. At the end of the section we will also discuss the results for other
values of $\alpha$. Based on these ideas, we compute the bound
$\mathcal{B}_{\,\textnormal{sym}}$ for the quantum Fisher information for an
increasing system size $N^{\prime}$.

The results can be seen in Figure 4.6-(a). The bound obtained this way is
close to the bound in Eq. (4.27) even for small $N^{\prime}$. For larger
particle number it is constant and coincides with the bound in Eq. (4.27) This
also strongly supports the idea that we could use the result for small
particle numbers to extrapolate the bound for $N$. Since for the experimental
particle number we obtain that $\mathcal{B}_{\,\text{sym}}$ equals the bound
in Eq. (4.27), we find that all three lower bounds in Eq. (4.29) must be
equal. Hence, Eq. (2.50) is optimal for the experimental system and $\alpha$
considered before in this section. Besides, these results present also a
strong argument for the correctness of our approach.

\begin{overpic}[scale={.65}]{img/LT_spsq_scaling_1.pdf}
\put(5.0,10.0){\small(a)}
\end{overpic}\begin{overpic}[scale={.65}]{img/LT_spsq_scaling_2.pdf}
\put(5.0,10.0){\small(b)} \end{overpic}

Figure 4.6:  (Color line) Lower bound on the QFI based on
$\langle{J_{y}}\rangle$ and $(\Delta J_{x})^{2}$ obtained for the symmetric
subspace for different particle numbers $N^{\prime}$. $N{=}2300$ corresponds
to the spin-squeezing experiment [7]. (a) Almost fully polarized spin-squeezed
state. Even for a moderate $N^{\prime}$, the bound is practically identical to
the right-hand side of the Eq. (4.32). (b) Spin-squeezed state that is not
fully polarized. For large $N^{\prime}$, the bound converges to the right-hand
side of the Eq. (4.32), represented by a dashed line. (dots) Results of our
calculations, which are connected by straight lines.

We now give more details of the calculation. We were able to carry out the
optimizations up to $N^{\prime}=2300$ with a usual laptop computer using
MATLAB programming language∥∥∥ For MATLAB R2015a, see
http://www.mathworks.com.. We started the calculation for each given particle
number with the $r_{k}$ parameters obtained for the previous simulation with a
smaller particle number. This allows for faster finding of the solution than
if we would start the $r_{k}$ parameters arbitrarily.

Let us consider a spin-squeezed state that is not fully polarized and
$\alpha=0.5$. In Figure 4.6-(b), we can see that for small particle numbers we
have a larger bound on $\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{z}}]$
than the one obtained from Eq (2.50). Thus for the case in which the particle
number would be smaller we could improve the bound Eq. (2.50) by assuming
symmetry. On the other hand, for large particle number we recover the bound
Eq. (2.50).

Finally, we add a note on the technical details. We carried out our
calculations with the constraints on $(\Delta J_{x})^{2}$ and
$\langle{J_{y}}\rangle$, with the additional constraint
$\langle{J_{x}}\rangle=0$. For the experimental particle numbers, one can show
that our results are valid even if we constrained only $(\Delta J_{x})^{2}$
and $\langle{J_{y}}\rangle$, and did not use the $\langle{J_{x}}\rangle=0$
constraint. This way, in principle, we could only get a lower bound that is
equal to the one we obtained before or lower. However, we obtained before a
value identical to the analytical bound Eq. (2.50). The optimal bound cannot
be below the analytic bound, since then the analytic bound would overestimate
the quantum Fisher information, and it would not be a valid bound. Hence, even
an optimization without the $\langle{J_{x}}\rangle=0$ constraint could not
obtain a smaller value than our results.

Experiment creating Dicke states

In this section, we present our calculations for an experiment aiming at
creating Dicke states in cold gases [24]. The basic ideas are similar to the
ones explained in Section 4.2.3 for small systems. The experimental data, as
in previous Section 3.4, are $N=7900$, $\langle{J_{y}^{2}}\rangle=112\pm 31$,
$\langle{J_{x}^{2}}\rangle=\langle{J_{z}^{2}}\rangle=6\times 10^{6}\pm
0.6\times 10^{6}$ [105]. Applying some simple transformations, we can make
calculations for a very large numbers of particles, and obtain results even
for general, non-symmetric systems.

In the general, non-symmetric case, we can handle only small problems. Thus,
we have to transform the collective quantities such that the corresponding
quantum state, i.e., it has to fulfill

$\langle{J_{x}^{2}}\rangle_{\text{sym}}+\langle{J_{x}^{2}}\rangle_{\text{sym}}+\langle{J_{x}^{2}}\rangle_{\text{sym}}=\mathcal{J}_{N/2},$
(4.33)

where $\mathcal{J}_{N/2}$ is defined on Eq. (4.24). A mapping of this type can
be realized equally scaling all second moments of the angular momentum
projections as

$\langle{J_{l}^{2}}\rangle_{\text{sym},N}=\gamma\langle{J_{l}^{2}}\rangle_{N},$
(4.34)

where we now added the label $N$ to avoid confusions in upcoming equations,
$l=x,y,z$ and where we used the coefficient $\gamma$ to be

$\gamma=\frac{\mathcal{J}_{N/2}}{\langle{J_{x}^{2}}\rangle_{N}+\langle{J_{y}^{2}}\rangle_{N}+\langle{J_{z}^{2}}\rangle_{N}}.$
(4.35)

Note that $\gamma=1$ if the original state is symmetric.

Next, based on the ideas of this chapter, we calculate the lower bound on the
quantum Fisher information for symmetric systems, which we denote
$\mathcal{B}_{\,\text{sym},N}$. To obtain the results for the original non-
symmetric case, the convex nature of the $\mathcal{B}_{\,N}$ implies that

$\mathcal{B}_{\,N}\leqslant\frac{1}{\gamma}\mathcal{B}_{\,\text{sym},N},$
(4.36)

where $\mathcal{B}_{\,\text{sym},N}$ corresponds to the bound one would obtain
in the symmetric subspace for expectation values given by the Eq. (4.34). This
can also be seen using an auxiliary state $\tilde{\rho}$ that mixes the
symmetric state that has the expectation values computed with Eq. (4.34) and
the singlet state that has zero value for all these expectation values. Hence,
if we construct a mixture of this type as follows

$\tilde{\rho}_{N}=(1-\gamma^{-1})\rho_{\text{singlet},N}+\gamma^{-1}\rho_{\text{sym},N},$
(4.37)

we have that $\tilde{\rho}_{N}$ has the same expectation values as the
original non-symmetric case. This way, we can relate the bound for general
systems to the quantum Fisher information for symmetric cases as

$\mathcal{B}_{\,N}\leqslant\mathcal{F}_{\textnormal{Q}}[{\textstyle\tilde{\rho}_{N},J_{z}}]=\frac{1}{\gamma}\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho_{\text{sym,N}},J_{z}}].$
(4.38)

Here, the inequality comes due to that our bound cannot be larger than the QFI
of any state having the given set of expectation values. On the other hand,
the equality holds due to the fact that both $\tilde{\rho}$ and $J_{z}$ can be
written as block-diagonal matrix of blocks corresponding to different
eigenvalues of $\boldsymbol{J}^{2}$. In particular, $\rho_{\text{singlet},N}$
has non-zero elements only in the blocks for which
$\langle{\boldsymbol{J}^{2}}\rangle=0$, while $\rho_{\text{sym},N}$ has
nonzero elements only in the blocks in which
$\langle{\boldsymbol{J}^{2}}\rangle$ is maximal. Note that
$\boldsymbol{J}^{2}$ is a shorthand of $J_{x}^{2}+J_{y}^{2}+J_{z}^{2}$. Then
we can use the general formula [68]

$\mathcal{F}_{\textnormal{Q}}[{\textstyle\bigoplus_{k}p_{k}\rho_{k},\bigoplus_{k}A_{k}}]=\sum_{k}p_{k}\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho_{k},A_{k}}],$
(4.39)

where $\rho_{k}$ are density matrices with unit trace, $\sum_{k}p_{k}=1$ and
the $k$ index represent the block subspaces of the system and the operators
$A_{k}$.

Extensive numerics for small systems show that the inequality in Eq. (4.38) is
very close to an equality within the numerical precision

$\mathcal{B}_{\,N}\approx\frac{1}{\gamma}\mathcal{B}_{\,\text{sym},N}.$ (4.40)

To obtain the lower bound $\mathcal{B}_{\,N}$ we also use an increasing system
size $N^{\prime}$ as we have done in at the beginning of this section.
However, in this case we will not be able to do the calculation for the
experimental particle number, and we will use extrapolation from the results
obtained for smaller particle numbers.

First, we transform the measured second moments to values corresponding to a
symmetric system using Eqs. (4.34) and (4.35). For our case, $\gamma=1.301$.
This way, we obtain

$\begin{split}\langle{J_{y}^{2}}\rangle_{\text{sym},N}&=145.69,\\\
\langle{J_{x}^{2}}\rangle_{\text{sym},N}&=\langle{J_{z}^{2}}\rangle_{\text{sym},N}=7.8\times
10^{6}.\end{split}$ (4.41)

Next, we will carry out calculations for symmetric systems. We will consider a
smaller system $N^{\prime}$ that keeps expectation values such that the
corresponding quantum state must be symmetric. Hence, we will use the
following relation to find the target expectation values for smaller systems

$\begin{split}\langle{J_{y}^{2}}\rangle_{\text{sym},N^{\prime}}&=\langle{J_{y}^{2}}\rangle_{\text{sym},N},\\\
\langle{J_{x}^{2}}\rangle_{\text{sym},N^{\prime}}&=\langle{J_{z}^{2}}\rangle_{\text{sym},N^{\prime}}=\frac{1}{2}(\mathcal{J}_{N^{\prime}/2})-\langle{J_{y}^{2}}\rangle_{\text{sym},N^{\prime}}),\end{split}$
(4.42)

where $\mathcal{J}_{N^{\prime}/2}$ is defined in Eq. (4.24). Note that with
Eq. (4.24) holds for all $N^{\prime}$, hence the state must be symmetric.
Hence, the main characteristics of the scaling relation can be summarized as
follows, $\langle{J_{y}^{2}}\rangle_{\text{sym},N^{\prime}}$ remains constant
for all $N^{\prime}$ while $\langle{J_{x}^{2}}\rangle_{\text{sym},N^{\prime}}$
and $\langle{J_{z}^{2}}\rangle_{\text{sym},N^{\prime}}$ are chosen such that
they are equal to each other and the state is symmetric. For large N, this
implies
$\langle{J_{x}^{2}}\rangle_{\text{sym},N}=\langle{J_{z}^{2}}\rangle_{\text{sym},N}\sim
N(N+2)/8$.

Let us now turn to central quantities of our chapter, the lower bounds on the
quantum Fisher information. A central point in our scheme is that due to the
scaling properties of the system, we can obtain the value for the particle
number $N$ from the values of a smaller particle number $N^{\prime}$ as [98]

$\mathcal{B}_{\,\text{sym},N}\approx\frac{\mathcal{J}_{N/2}}{\mathcal{J}_{N^{\prime}/2}}\mathcal{B}_{\,\text{sym},N^{\prime}},$
(4.43)

which we will verify numerically. Note that for large $N$, we have
$\mathcal{J}_{N/2}/\mathcal{J}_{N^{\prime}/2}\sim N^{2}/(N^{\prime})^{2}$.

As last step, we have to return from the symmetric system to our real system,
not fully symmetric one. Based on Eq. (4.43) and assuming Eq. (4.40), a
relation for the lower bound for the original problem can be obtained from the
bound on the symmetric problem with $N^{\prime}$ particles as

$\mathcal{B}_{\,N}\approx\frac{1}{\gamma}\frac{\mathcal{J}_{N/2}}{\mathcal{J}_{N^{\prime}/2}}\mathcal{B}_{\,\text{sym},N^{\prime}}=\frac{\langle{J_{x}^{2}}\rangle_{N}+\langle{J_{y}^{2}}\rangle_{N}+\langle{J_{z}^{2}}\rangle_{N}}{\mathcal{J}_{N^{\prime}/2}}\mathcal{B}_{\,\text{sym},N^{\prime}}.$
(4.44)

In Figure 4.7, we plotted the right-hand side of Eq. (4.44) as the function of
$N^{\prime}$ divided by $N$. We can see that $\mathcal{B}_{\,N^{\prime}}/N$ is
constant or slightly increasing for $N^{\prime}>400$. This is a strong
evidence that Eq. (4.43) is valid for relatively large particle numbers. With
this, we arrive at the result for the experimental system

$\frac{\mathcal{B}_{\,N}(\langle{J_{y}^{2}}\rangle,\langle{J_{x}^{2}}\rangle=\langle{J_{z}^{2}}\rangle)}{N}\approx
2.94.$ (4.45)

The $\approx$ sign is used referring to the fact that we assume that the
inequality in Eq. (4.38) is close to be saturated and that we did sufficient
numerics for an increasing system size $N^{\prime}$ to have a good asymptotic
approach to the real value Eq. (4.43).

Figure 4.7:  Asymptotic behavior of the bound as a function of $N^{\prime}$.
The bound is first obtained for a symmetric subspace of $N^{\prime}$ particles
and then the bound for $N$ particles is computed using the Eq. (4.44). The
function is monotonically increasing. Hence, with $N^{\prime}\approx 200$ we
already obtain a good lower bound. This approach does not overestimate the
precision bound.

It is instructive to compare the value of Eq. (4.45) to the one obtained in
Section 3.4, where the same system was characterized base on its metrological
usefulness. Such result implies
$\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{z}}]/N\geqslant 3.3$ which is
somewhat larger than our recent result as we did not use the knowledge of the
fourth moments, only the second moments. The closeness of the two results is a
strong argument for the correctness of our calculations.

###  4.4 Scaling of $\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{z}}]$
with $N$

Recent important works examine the scaling of the quantum Fisher information
with the particle number for metrology under the presence of decoherence [61,
62]. They consider the QFI defined now for the non-unitary, noisy evolution.
They find that for small $N$ it is close to the value obtained by considering
coherent dynamics. Hence, even the Heisenberg scaling, $\mathcal{O}(N^{2})$,
can be reached. However, if $N$ is sufficiently large, then, due to the
decoherence during the parameter estimation, the QFI scales as
$\mathcal{O}(N)$.

We have to stress that the findings of B. M. Escher et al [61] and R.
Demkowicz-Dobrzański et al [62] are not applicable to our case. Our methods
estimate the quantum Fisher information assuming a perfect unitary dynamics.
The quantum Fisher information can be smaller that what we expect ideally only
due to the imperfect preparation of the state****** This is also relevant for
Ref. [103], where $\mathcal{F}_{\textnormal{Q}}=\mathcal{O}(N^{2})$ is reached
with weakly entangled states.. We can even find simple conditions on the state
preparation that lead to a Heisenberg scaling. Based on Eq. (4.18), if we
could realize quantum states $\rho_{N}$ such that
$F_{\text{GHZ}}(\rho_{N})\geqslant 0.5+\epsilon$ for $N\rightarrow\infty$ for
some $\epsilon>0$, then we would reach
$\mathcal{B}_{\,\mathcal{F}}(F_{\text{GHZ}})=\mathcal{O}(N^{2})$. Strong
numerical evidence suggest that a similar relation holds for fidelity
$F_{\text{Dicke}}$ and $\mathcal{B}_{\,\mathcal{F}}(F_{\text{GHZ}})$, see
Section 4.2.3. From another point of view, our method can estimate
$\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{z}}]$ for large particle
numbers, while a direct measurement of the metrological sensitivity
considerably underestimates it.

##  5 Precision bound for gradient field estimation with atomic ensembles
Precision bound for gradient field estimation with atomic ensembles

"To consult the statistician after an experiment is finished is often merely
to ask him to conduct a post mortem examination. He can perhaps say what the
experiment died of."

Ronald Fisher

In this chapter, one of the most fundamental two-parameter estimation tasks in
magnetometry is considered, namely gradient magnetometry. We will add the
gradient of the magnetic field as the second parameter beside the constant
(homogeneous) part of the field. While most works in magnetometry with a
single ensemble focus only on the determination of the strength and direction
of the magnetic field, certain measurement schemes for the gradient have
already been proposed and tested experimentally. We study gradient
magnetometry with an ensemble of atoms described by a very general probability
distribution function for the position of the atoms, and considering atoms
with an arbitrary spin. Some schemes use an imaging of the ensemble with a
high spatial resolution, however, they do not count as single-ensemble methods
in the sense we use this expression in our paper, since in this case not only
collective observables are measured [18, 19, 17]. There is a method based on
collective measurements of the spin length of a fully polarized ensemble [37].
Finally, there is a scheme where they use as a trial state a many-body singlet
states, which is described in Ref. [57].

We calculate precision bounds for estimating the gradient of the magnetic
field based on the quantum Fisher information. For quantum states that are
invariant under the homogeneous magnetic field, a single parameter estimation
is sufficient. In contrast to this, for states that are sensitive to the
homogeneous fields, a two-parameter estimation problem must be solved to
obtain the gradient parameter, since the homogeneous field must also be taken
into account. We use our method to calculate precision bounds for gradient
estimation with a chain of atoms and even with two spatially separated atomic
ensembles which feel different magnetic fields. As we said, we also consider a
single atomic ensemble with an arbitrary density profile, in which atoms
cannot be addressed individually, which is a very relevant case for
experiments. Our model can take into account even correlations between
particle positions.

The atoms will be distributed along the $x$-axis, so $y=z=0$, and in principle
they will be able to feel differences in the magnetic field at different
points of the axis. The magnetic field at the atoms will be given by a linear
function of the position $x$

$\boldsymbol{B}(x,0,0)=\boldsymbol{B}_{0}+x\boldsymbol{B}_{1}+O(x^{2}),$ (5.1)

where we will neglect the terms of order two or higher. We will consider the
magnetic field pointing along the $z$-direction direction only,
$\boldsymbol{B}_{0}=B_{0}\boldsymbol{k}$ and
$\boldsymbol{B}_{1}=B_{1}\boldsymbol{k}$, where $\boldsymbol{k}$ is the
unitary vector pointing on the $z$-direction. For this configuration, due to
the Maxwell equations, with no currents or changing electric fields, we have

$\displaystyle\nabla\cdotp\boldsymbol{B}$ $\displaystyle=0,$
$\displaystyle\nabla\times\boldsymbol{B}$ $\displaystyle=\boldsymbol{0},$
(5.2)

where $\boldsymbol{0}\equiv(0,0,0)$ is the 3-dimensional null vector. This
implies $\sum_{l=x,y,z}\partial_{l}B_{l}=0$ and
$\partial_{m}B_{l}-\partial_{l}B_{m}=0$ for all $l\neq m$, where
$\partial_{m}\equiv\partial/\partial_{m}$ stands for the partial derivative
over the variable $m$. Thus, the spatial derivatives of the field components
are not independent of each other. However, in the case of a linear arranged
particle ensemble only the derivative along the $x$-axis has an influence on
the quantum dynamics of the atoms.

We will determine the precision bounds for the estimation of the magnetic
field gradient $B_{1}$ based on the quantum Fisher information [2, 76, 31, 87,
113, 33]. In this context the Heisenberg and shot-noise scaling are defined as
usual. The achievable precision in terms of the number of particles scales as
$(\Delta\theta)^{-2}\sim N$ and $(\Delta\theta)^{-2}\sim N^{2}$ for shot-noise
scaling and Heisenberg scaling, respectively. We will show that with spin
chains or two ensembles at different positions the Heisenberg scaling is
possible. Concerning the case of a single ensemble, we will show the
following. Since in such systems the atoms cannot be be individually
addressed, we will assume that the quantum state is permutationally invariant.
We will show that for states insensitive to the homogeneous magnetic field,
one can reduce the problem to a one-parameter estimation scenario. Such states
can arise in a single-ensemble scheme, however, it will be shown that the
Heisenberg limit cannot be reached in this case. When the state is sensitive
to the homogeneous field, the spatial correlation between the atoms must be
taken into account in order to show whether the system can overcome the shot-
noise scaling and achieve the Heisenberg scaling. Nevertheless, single-
ensemble measurements have certain advantages since the spatial resolution can
be higher and the experimental requirements are smaller since only a single
ensemble must be prepared.

On the other hand, for states sensitive to the homogeneous field, the
classical limit can be overcome only if the particle positions are highly
correlated with each other. Our calculations are generally valid for any
measurement, thus they are relevant to many recent experiments [13, 14, 15,
16, 17, 18, 19, 37]. We note that in the case of the singlet, our precision
bounds are saturated by the metrological scheme presented in Ref. [57].

We can also connect our results to entanglement theory [65, 35, 36]. We find
that even in the case of gradient magnetometry the shot-noise scaling cannot
be surpassed with separable states, while the Heisenberg scaling can be
reached with entangled states. However, in the single-ensemble scenario, the
shot-noise scaling can be surpassed only if the particle positions are
correlated, which is the case if the particles attract each other. We will go
into the details in Section 5.4.

The chapter is organized as follows. In Section 5.1, we will present the setup
of the system. In Section 5.2, the metrological basic concepts used in the
chapter are presented. In Section 5.3, we will show the results for the chain
of ions and for when two distant ensembles are considered In Section 5.4, we
restrict our calculations to single permutationally invariant atomic ensembles
and we develop some particular cases, such as the singlet spin state or the
totally polarized state.

###  5.1 The setup

In this section, we will present the characteristics of our setup. For
simplicity, as well as following recent experiments (e.g., Ref. [17]), we will
consider an ensemble of spin-$j$ particles placed in a one-dimensional trap or
a chain. Furthermore, we will assume that the particles are point-like and
distinguishable. On the other hand, we assume that the particles have a spin,
which is a quantum degree of freedom. Such a model is used typically to
describe experiments with cold atomic ensembles.

Based on these considerations, we assume that the state is factorizable into a
spatial and a spin part as

$\rho=\rho^{(\text{x})}\otimes\rho^{(\text{s})},$ (5.3)

and that the spatial part can be characterized as an incoherent mixture of
point-like particles that can be written as

$\rho^{(\text{x})}=\int\text{Pr}(\boldsymbol{x})|{\varphi_{\boldsymbol{x}}}\rangle{\\!}\langle{\varphi_{\boldsymbol{x}}}|\,\text{d}^{N}\boldsymbol{x},$
(5.4)

where $|{\varphi_{\boldsymbol{x}}}\rangle$ is a pure state of each particle
been placed at $\boldsymbol{x}=(x_{1},x_{2},\dots,x_{N})$, respectively. Each
part of the state acts on different Hilbert spaces denoted by
$\mathcal{H}^{(\text{x})}$ and $\mathcal{H}^{(\text{s})}$, respectively. Note
that we skip to write the superscript $(\text{x})$, denoting the Hilbert space
to which $|{\varphi_{\boldsymbol{x}}}\rangle$ belongs, for simplicity.

In order to write the operators, including the state $\rho^{(\text{x})}$,
acting on the Hilbert space $\mathcal{H}^{(\text{x})}$, we will invoke the
completeness relation found in the literature [70, 69] for the spatial
continuous Hilbert space

$\int|{\boldsymbol{x}}\rangle{\\!}\langle{\boldsymbol{x}}|\,\text{d}^{N}\boldsymbol{x}=\mathbbm{1},$
(5.5)

where
$|{\boldsymbol{x}}\rangle=|{x_{1}}\rangle^{(1,\text{x})}\otimes|{x_{2}}\rangle^{(2,\text{x})}\cdots\otimes|{x_{N}}\rangle^{(N,\text{x})}$
is the tensor product of the position eigenvectors of each particle which obey

$\langle{\boldsymbol{x}}|{\boldsymbol{y}}\rangle=\delta(\boldsymbol{x}-\boldsymbol{y}),$
(5.6)

where $\delta(\boldsymbol{x}-\boldsymbol{y})$ is the Dirac delta found in the
literature [70, 69].

To see how our notation works, let us write the vector of the position
operators for each particle as
$\hat{\boldsymbol{x}}\equiv(x^{(1)},x^{(2)},\dots,x^{(N)})$, where we used the
$\hat{\cdot}$ notation on top of $\boldsymbol{x}$ to distinguish it from the
vector of position variables. It is known that $\hat{\boldsymbol{x}}$ acting
on $|{\boldsymbol{x}}\rangle$ will give
$\boldsymbol{x}|{\boldsymbol{x}}\rangle$ [70, 69]. Hence,

$\hat{\boldsymbol{x}}=\hat{\boldsymbol{x}}\mathbbm{1}=\hat{\boldsymbol{x}}\int|{\boldsymbol{x}}\rangle{\\!}\langle{\boldsymbol{x}}|\,\text{d}^{N}\boldsymbol{x}=\int\hat{\boldsymbol{x}}|{\boldsymbol{x}}\rangle{\\!}\langle{\boldsymbol{x}}|\,\text{d}^{N}\boldsymbol{x}=\int\boldsymbol{x}|{\boldsymbol{x}}\rangle{\\!}\langle{\boldsymbol{x}}|\,\text{d}^{N}\boldsymbol{x}.$
(5.7)

We can follow similar arguments to rewrite the pure state
$|{\varphi_{\boldsymbol{x}}}\rangle$. First, note also that the expectation
value of the position operator for such a state is always $\boldsymbol{x}$.
Hence, such a state must be proportional to $|{\boldsymbol{x}}\rangle$. On the
other hand, a pure state must be normalized to one, hence, we can construct
such a state by dividing the eigenvector $|{\boldsymbol{x}}\rangle$ by the
square-root of its norm as

$|{\varphi_{\boldsymbol{x}}}\rangle\equiv\frac{|{\boldsymbol{x}}\rangle}{\sqrt{\langle{\boldsymbol{x}}|{\boldsymbol{x}}\rangle}}.$
(5.8)

The meaning of Eq. (5.8) is clear, while a rigorous form of how various limits
are taken could overcomplicate our discussion. For illustrative purposes, we
compute in this basis the expectation value of the position operator for these
pure states as

$\langle{\hat{\boldsymbol{x}}}\rangle_{\varphi_{\boldsymbol{x}}}=\langle{\varphi_{\boldsymbol{x}}}|{\int\boldsymbol{y}|{\boldsymbol{y}}\rangle{\\!}\langle{\boldsymbol{y}}|\,\text{d}^{N}\boldsymbol{y}}|{\varphi_{\boldsymbol{x}}}\rangle=\int\boldsymbol{y}\frac{\langle{\boldsymbol{x}}|{\boldsymbol{y}}\rangle\langle{\boldsymbol{y}}|{\boldsymbol{x}}\rangle}{\langle{\boldsymbol{x}}|{\boldsymbol{x}}\rangle}\,\text{d}^{N}\boldsymbol{y}=\boldsymbol{x}\frac{\langle{\boldsymbol{x}}|{\boldsymbol{x}}\rangle}{\langle{\boldsymbol{x}}|{\boldsymbol{x}}\rangle}=\boldsymbol{x},$
(5.9)

where in the step before the last we have used
$\langle{\boldsymbol{y}}|{\boldsymbol{x}}\rangle=\delta(\boldsymbol{y}-\boldsymbol{x})$
to compute the integral. We can rewrite the spatial state Eq. (5.4) as

$\rho^{(\text{x})}=\int\frac{\text{Pr}(\boldsymbol{x})}{\langle{\boldsymbol{x}}|{\boldsymbol{x}}\rangle}|{\boldsymbol{x}}\rangle{\\!}\langle{\boldsymbol{x}}|\,\text{d}^{N}\boldsymbol{x}.$
(5.10)

Note also that if the eigen-decomposition of the internal state is
$\rho^{(\text{s})}=\sum_{\lambda}p_{\lambda}|{\lambda}\rangle{\\!}\langle{\lambda}|$,
then the whole state is decomposed as

$\rho=\int\sum_{\lambda}\frac{\text{Pr}(\boldsymbol{x})}{\langle{\boldsymbol{x}}|{\boldsymbol{x}}\rangle}p_{\lambda}|{\boldsymbol{x},\lambda}\rangle{\\!}\langle{\boldsymbol{x},\lambda}|\,\text{d}^{N}\boldsymbol{x},$
(5.11)

where $|{\boldsymbol{x},\lambda}\rangle$ is a shorthand for
$|{\boldsymbol{x}}\rangle^{(\text{x})}\otimes|{\lambda}\rangle^{(\text{s})}$,
the eigenstates, where their corresponding eigenvalues are in this case
$\frac{\text{Pr}(\boldsymbol{x})}{\langle{\boldsymbol{x}}|{\boldsymbol{x}}\rangle}p_{\lambda}$.

At this point, we want to emphasize that our method could easily be extended
to the case of Bose-Einstein condensates, or any other spatial configuration,
not considered in this paper. In the case of BECs, the spatial state of the
particles would be a pure state, and we would have
$\rho^{(\text{x})}=(|{\Psi}\rangle{\\!}\langle{\Psi}|)^{\otimes N},$ where
$|{\Psi}\rangle$ is a spatial single-particle state.

Although in our case the parameter to be estimated is $B_{1}$, the time-
evolution of the state is usually also affected by the second unknown
parameter, the homogeneous field $B_{0}$, which means that we generally have
to consider a two-parameter estimation problem. The angular momentum of an
individual atom is coupled to the magnetic field, yielding the following
interaction term

$h^{(n)}=\gamma B_{z}^{(n,\text{x})}\otimes j_{z}^{(n,\text{s})},$ (5.12)

where the operator $B_{z}^{(n)}=B_{0}+B_{1}x^{(n)}$ acts on the spatial part
of the $n^{\text{th}}$ particle Hilbert space $\mathcal{H}^{(n,\text{x})}$.

The sum of all single-particle interactions with the magnetic field provide
the total Hamiltonian

$H=\gamma\sum_{n=1}^{N}B_{z}^{(n,\text{x})}\otimes j_{z}^{(n,\text{s})},$
(5.13)

which will generate the time evolution of the system.

We will calculate lower bounds on the precision of estimating $B_{1}$ based on
measurements on the state after it passed through the unitary dynamics
$U=\exp(-i\frac{H}{\hbar}t)$, where $t$ is the time spent by the system under
the influence of the magnetic field. The unitary operator can be rewritten in
the following way

$U=e^{-i\left(b_{0}H_{0}+b_{1}H_{1}\right)},$ (5.14)

where the $b_{i}=\gamma B_{i}t/\hbar$ and therefore $b_{1}$ encodes the
gradient of the magnetic field $B_{1}$. Here, the generator describing the
effect of the homogeneous field is given as

$H_{0}=\sum_{n=1}^{N}j_{z}^{(n)}=J_{z},$ (5.15)

while the generator describing the effect of the gradient is

$H_{1}=\sum_{n=1}^{N}x^{(n)}j_{z}^{(n)}.$ (5.16)

As in Eq. (5.16), we will usually omit $\otimes$ and the superscripts
$(\text{x})$ and $(\text{s})$ for simplicity, and will use it only if it is
necessary to make our explanation clearer.

Note that the operators $H_{0}$ and $H_{1}$ commute with each other. These two
commuting dynamics are the two simplest in an atomic ensemble as they are
based on collective operators not requiring an individual access to the
particles. This is mainly because the spatial part in Eq. (5.12) is
represented by a single-particle operator and not by a scalar depending on the
position of the particle. The last approach, where the position of the
particle is encoded in a scalar, would require to know in advance the location
of the particles to construct the Hamiltonian $H_{1}$ Eq. (5.16), which would
yield finally to a non-collective operator for $H_{1}$. This approach is
widely adopted by the community, since it simplifies the problem considerably
[57, 114].

Note also that it is not necessarily true that the operators we have to
measure in order to estimate $b_{0}$ and $b_{1}$ must commute with each other.
On the other hand, in schemes in which the gradient is calculated based on
measurements in two separate atomic ensembles or different atoms in a chain,
the measuring operators can always commute with each other [13, 14, 98].

###  5.2 Cramér-Rao bound for gradient estimation

In this section, we show how the Cramér-Rao bound and the QFI help us to
obtain the precision bound that is valid for any measurement scenario. We will
discuss gradient magnetometry using quantum states that are insensitive to
homogeneous fields, which is a single-parameter estimation task. Then, we
discuss the case of quantum states sensitive to homogeneous fields, which is a
two-parameter estimation problem. We show that the precision bound obtained
does not change under spatial translation, which is one of the main tools to
derive our bounds. For the two-parameter estimation task, we will introduce
the two-parameter Cramér-Rao bound and the corresponding two-parameter QFI
matrix, and we adapt those expressions to our problem.

For clarity, we present our main tools in subsequent paragraphs before going
into details. Here we define a functional very similar to the QFI Eq. (2.51).
This expression will be used along this chapter and it will be useful for the
transition to the multi-parameter problem, i.e, it is equivalent to the QFI
for the single parameter estimation problem but still it gives the chance to
switch to the multi-parameter case easily. The function is defined as follows.
For two arbitrary operators $A$ and $B$, it is written as

$\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,A,B}]:=2\sum_{\lambda,\nu}\frac{(p_{\lambda}-p_{\nu})^{2}}{p_{\lambda}+p_{\nu}}{A}_{\lambda,\nu}{B}_{\nu,\lambda},$
(5.17)

where the subscript for $A$ and $B$ stand for the matrix elements on the
eigenbasis of the initial state
$\rho=\sum_{\lambda}p_{\lambda}|{\lambda}\rangle{\\!}\langle{\lambda}|$. If
the two operators are the same, the usual form of the QFI Eq. (2.51) is
recovered [2, 76, 31, 87, 113, 33],

$\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,A,A}]:=\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,A}]=2\sum_{\lambda,\nu}\frac{(p_{\lambda}-p_{\nu})^{2}}{p_{\lambda}+p_{\nu}}|{A}_{\lambda,\nu}|^{2}.$
(5.18)

We mention that in our case the operators $A$ and $B$ will commute in all
situations, making the computations easier. We also make use of the fact that
the QFI as written in Eq. (5.17) is linear in the second and the third
arguments,

$\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,A,\textstyle{\sum}_{i}b_{i}}]=\sum_{i}\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,A,b_{i}}].$
(5.19)

It also holds for commuting $A$ and $B$, that the last two arguments can be
exchanged without affecting the outcome,
$\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,A,B}]=\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,B,A}]$.

Similar to Eq. (2.54), Eq. (5.17) can be rewritten as

$\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,A,B}]=4\langle{AB}\rangle-8\sum_{\lambda,\nu}\frac{(p_{\lambda}-p_{\nu})^{2}}{p_{\lambda}+p_{\nu}}{A}_{\lambda,\nu}{B}_{\nu,\lambda},$
(5.20)

when the operators $A$ and $B$ commute. This form leads to simpler arguments
in our derivations through the following sections.

For pure states it simplifies also to

$\mathcal{F}_{\textnormal{Q}}[{\textstyle|{\psi}\rangle,A,B}]=4\left(\langle{AB}\rangle_{\psi}-\langle{A}\rangle_{\psi}\langle{B}\rangle_{\psi}\right).$
(5.21)

Note that we recover
$\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,A,A}]=4(\Delta A)_{\rho}^{2}$ as
can be found in the Eq. (2.54) for single-parameter estimation with pure
states [2, 88].

Another important feature of the function Eq. (5.17) is that it is convex on
the states. This property is written as follows

$\mathcal{F}_{\textnormal{Q}}[{\textstyle
q\rho_{1}{+}(1{-}p)\rho_{2}}]\leqslant
p\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho_{1}}]{+}(1{-}p)\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho_{2}}],$
(5.22)

where we omit in writing the last two arguments for simplicity. Finally, it is
also useful to note that additive under the tensor product as

$\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho^{(1)}\otimes\rho^{(2)},A^{(1)}\otimes\mathbbm{1}^{(2)}+\mathbbm{1}^{(1)}\otimes
A^{(2)},B^{(1)}\otimes\mathbbm{1}^{(2)}+\mathbbm{1}^{(1)}\otimes
B^{(2)}}]=\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho^{(1)},A^{(1)},B^{(1)}}]+\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho^{(2)},A^{(2)},B^{(2)}}].$
(5.23)

In the following subsections we show the general form for the precision bounds
for states insensitive to the homogeneous fields and for states sensitive to
them. We also show that both bounds are invariant under spatial translation of
the system which makes the computing for particular cases much easier.

####  5.2.1 States insensitive to the homogeneous field: Single-parameter
estimation

Let us consider quantum states that are insensitive to the homogeneous field.
For these states, $[\rho,H_{0}]=0$ and hence the evolved state is a function
of a single unknown parameter, $b_{1}$. For the unitary dynamics we consider,
the QFI for single-parameter estimation problem can be expressed in terms of
the eigenstates and eigenvalues of the density matrix as [2, 76, 31, 87, 113,
33]

$\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,H_{1}}]=2\sum_{\lambda,\nu}\frac{(p_{\lambda}-p_{\nu})^{2}}{p_{\lambda}+p_{\nu}}|\langle{\lambda}|{H_{1}}|{\nu}\rangle|^{2}.$
(5.24)

Note that here the eigenstates $|{\lambda}\rangle$ and $|{\nu}\rangle$ live on
both the external and internal Hilbert spaces. Due to the Cramér-Rao formula,
the QFI gives us an upper bound for the precision

$(\Delta
b_{1})^{-2}|_{\max}=\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,H_{1}}].$
(5.25)

Note that it is _always_ possible to find a measurement that saturates the
precision bound above. Hence, we denote it using the "$|_{\max}=$" notation.
Here, $\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,H_{1}}]$ denotes the QFI
that depends, in the case of unitary transformation of the form Eq. (5.14), on
the state $\rho$ and on the generator of the evolution $H_{1}$.

For the particular case in which the state has the form of Eqs. (5.3) and
(5.10), Eq. (5.24) can be simplified in the following way. Note that we have
to compute the matrix elements of $H_{1}$ which is already diagonal in the
spatial subspace. Therefore, the following holds for the matrix elements of
$H_{1}$

$\begin{split}(H_{1})_{\boldsymbol{x},\lambda;\boldsymbol{y},\nu}&=\langle{\boldsymbol{x},\lambda}|{H_{1}}|{\boldsymbol{y},\nu}\rangle\\\
&=\langle{\boldsymbol{x},\lambda}|{\sum_{n=1}^{N}x^{(n)}j^{(n)}}|{\boldsymbol{y},\nu}\rangle\\\
&=\delta(\boldsymbol{x}-\boldsymbol{y})\sum_{n=1}^{N}x_{n}\langle{\lambda}|{j^{(n)}}|{\mu}\rangle,\end{split}$
(5.26)

where $|{\lambda}\rangle$ and $|{\mu}\rangle$ refer now to eigenstates of the
internal state $\rho^{(\text{s})}$ and we use
$\langle{\boldsymbol{x}}|x^{(n)}|{\boldsymbol{y}}\rangle=\delta(\boldsymbol{x}-\boldsymbol{y})x_{n}$.
We will use the Dirac delta function appearing in Eq. (5.26) to further
simplify the Eq. (5.24).

We show now that spatial translations does not change the sensitivity of
gradient estimation. The translation operator $U_{d}$ moves the state to a new
position at a distance $d$ from its previous location, and it is written as

$U_{d}=e^{-idP_{x}/\hbar},$ (5.27)

where $P_{x}$ is the sum of all single-particle linear momentum operators
$p_{x}^{(n)}$ in the $x$-direction and it only acts on the external degrees of
freedom of the state, i.e., the external Hilbert space
$\mathcal{H}^{(\text{x})}$. To show that the precision is unchanged, we use
the Heisenberg picture in which the operators are transformed instead of the
states. Thus, we compute the transformation of $H_{1}$ as

$\begin{split}U_{d}:H_{1}\rightarrow
H_{1}(d)&=U_{d}^{\dagger}H_{1}U^{\phantom{\dagger}}_{d}\\\
&=\sum_{n=1}^{N}U_{d}^{\dagger}x^{(n)}U^{\phantom{\dagger}}_{d}\otimes
j_{z}^{(n)}\\\ &=\sum_{n=1}^{N}(x^{(n)}-d)j_{z}^{(n)}\\\
&=H_{1}-dH_{0}.\end{split}$ (5.28)

Hence, the new unitary evolution operator to represent the translated system,
instead of Eq. (5.14), is

$U=e^{-i(b_{0}H_{0}+b_{1}H_{1}(d))}=e^{-i((b_{0}-b_{1}d)H_{0}+b_{1}H_{1})}.$
(5.29)

Eq. (5.29) is equivalent to Eq. (5.14) for states insensitive to the
homogeneous fields, since in this case $[\rho,H_{0}]=0$.

To compute the QFI, we used the Dirac delta function appearing in Eq. (5.26),
and the state defined by Eq. (5.11). See Appendix G for details.

The following bound in the precision of the estimation of the gradient
parameter $b_{1}$ holds for states insensitive to the homogeneous magnetic
fields

$(\Delta
b_{1})^{-2}|_{\max}=\sum_{n,m}^{N}\int\text{Pr}(\boldsymbol{x})x_{n}x_{m}\,\text{d}^{N}\boldsymbol{x}\,\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho^{(\text{s})},j_{z}^{(n)},j_{z}^{(m)}}],$
(5.30)

where the integral has the form of a two-point correlation function of the
spatial state.

####  5.2.2 States sensitive to the homogeneous field: Two-parameter
dependence

In order to obtain the precision bound for states sensitive to the homogeneous
field, one has to consider the effect on the state of a second unknown
parameter, in this case $b_{0}$, which represents the homogeneous magnetic
field. The homogeneous field will rotate all the spins in the same way, while
the field gradient rotates the spins differently depending on the position of
the particles. Now, instead to the Cramér-Rao bound Eq. (5.25), we have a
matrix inequality [2]

$\text{Cov}(b_{0},b_{1})\geqslant\mathbfcal{F}^{-1},$ (5.31)

where $\text{Cov}(b_{0},b_{1})$ is the covariance matrix for $b_{0}$ and
$b_{1}$.

For the matrix inequality Eq. (5.31), we have the inverse of QFI matrix
$\mathbfcal{F}$ on one hand, which depends on $\rho$ and the two generators
$H_{0}$ and $H_{1}$, and the covariance matrix on the other hand. In this
section, we are only interested in the variance of the gradient parameter,
$(\Delta b_{1})^{2}$. Since we have to compute the inverse of the QFI matrix
and then look at the element corresponding to the $(\Delta b_{1})^{2}$, the
determinant of $\mathbfcal{F}$ cannot be zero. $H_{0}$ and $H_{1}$ are
Hermitian operators and commute with each other. For unitary dynamics of the
type Eq. (5.14), the QFI matrix elements are computed as
$\mathbfcal{F}_{ij}\equiv\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,H_{i},H_{j}}],$
following the definition given in Eq. (5.17).

In the two-parameter estimation problem, $\mathbfcal{F}$ is a $2\times 2$
matrix and the precision bound for the estimation of the gradient is

$(\Delta
b_{1})^{-2}\leqslant\mathbfcal{F}_{11}-\frac{\mathbfcal{F}_{01}\mathbfcal{F}_{10}}{\mathbfcal{F}_{00}},$
(5.32)

where the inequality is saturated only if there exists a set of compatible
measurements to determine both parameters $b_{0}$ and $b_{1}$, which is not
true in general and must be studied for each particular case [2, 115]. We
distinguish this case from the Eq. (5.25), in which the bound is surely
saturated by some measurement, using an inequality "$\leqslant$" instead of
"$|_{\max}=$".

To compute the bound Eq. (5.32), we will need to simplify the matrix elements
of $H_{0}$ and $H_{1}$ written in the eigenbasis of the state Eq. (5.11), see
Eq. (5.17). Note that the matrix elements for $H_{1}$ were already computed in
Eq. (5.26). Hence, we now calculate
$(H_{0})_{\boldsymbol{x},\lambda;\boldsymbol{y},\nu}$ in a similar way as we
did for Eq. (5.26)

$\begin{split}(H_{0})_{\boldsymbol{x},\lambda,\boldsymbol{y},\nu}&=\langle{\boldsymbol{x},\lambda}|{H_{0}}|{\boldsymbol{y},\nu}\rangle\\\
&=\langle{\boldsymbol{x},\lambda}|{J_{z}}|{\boldsymbol{y},\nu}\rangle\\\
&=\delta(\boldsymbol{x}-\boldsymbol{y})\langle{\lambda}|{J_{z}}|{\nu}\rangle.\end{split}$
(5.33)

With this we are now in the position to compute the missing matrix elements of
$\mathbfcal{F}$. One can find most of the computations of the matrix elements
of $\mathbfcal{F}$ in the Appendix G. First of all, we compute
$\mathbfcal{F}_{11}=\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,H_{1},H_{1}}]$
which turns to be equal to Eq. (5.30) for obvious reasons,

$\mathbfcal{F}_{11}=\sum_{n,m}^{N}\int
x_{n}x_{m}\text{Pr}(\boldsymbol{x})\,\text{d}^{N}\boldsymbol{x}\,\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho^{(\text{s})},j_{z}^{(n)},j_{z}^{(m)}}].$
(5.34)

Second, the most trivial matrix element is $\mathbfcal{F}_{00}$ which turns to
depend only on the internal state $\rho^{(\text{s})}$,

$\mathbfcal{F}_{00}=\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho^{(\text{s})},J_{z}}].$
(5.35)

Finally, we compute both $\mathbfcal{F}_{01}$ and $\mathbfcal{F}_{10}$. To
compute them, note that the two matrix elements are equal
$\mathbfcal{F}_{01}=\mathbfcal{F}_{01}$, due to the properties of Eq. (5.17)
for commuting $H_{0}$ and $H_{1}$. Therefore, we have to compute only one of
them

$\mathbfcal{F}_{01}=\sum_{n=1}^{N}\int
x_{n}\text{Pr}(\boldsymbol{x})\,\text{d}^{N}\boldsymbol{x}\,\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho^{(\text{s})},j_{z}^{(n)},J_{z}}].$
(5.36)

With these results the bound for the precision for states sensitive to the
homogeneous field which have the form of Eq. (5.11) is

$\begin{split}(\Delta b_{1})^{-2}\leqslant&\sum_{n,m}^{N}\int
x_{n}x_{m}\text{Pr}(\boldsymbol{x})\,\text{d}^{N}\boldsymbol{x}\,\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho^{(\text{s})},j_{z}^{(n)},j_{z}^{(m)}}]-\frac{\left(\sum_{n=1}^{N}\int
x_{n}\text{Pr}(\boldsymbol{x})\,\text{d}^{N}\boldsymbol{x}\,\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho^{(\text{s})},j_{z}^{(n)},J_{z}}]\right)^{2}}{\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho^{(\text{s})},J_{z}}]}.\end{split}$
(5.37)

To simplify our calculations, we show that the bound Eq. (5.37) is invariant
under displacements of the system. We use the linearity of the last two
arguments of $\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,A,B}]$, Eq. (5.19),
the fact that $H_{0}$ remains unchanged in the Heisenberg picture, and we also
use the shifted $H_{1}(d)$ operator computed in Eq. (5.28). Hence, the
translated QFI matrix elements are written as

$\displaystyle\mathbfcal{F}_{00}(d)$
$\displaystyle=\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,H_{0}(d)}]=\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,H_{0}}],$
(5.38a) $\displaystyle\mathbfcal{F}_{01}(d)$
$\displaystyle=\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,H_{0}(d),H_{1}(d)}]$
$\displaystyle=\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,H_{0},H_{1}-dH_{0}}]=\mathbfcal{F}_{01}-d\mathbfcal{F}_{00},$
(5.38b) $\displaystyle\mathbfcal{F}_{11}(d)$
$\displaystyle=\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,H_{1}(d),H_{1}(d)}]$
$\displaystyle=\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,H_{1}-dH_{0},H_{1}-dH_{0}}]$
$\displaystyle=\mathbfcal{F}_{11}-2d\mathbfcal{F}_{01}+d^{2}\mathbfcal{F}_{00}.$
(5.38c)

Simple algebra shows that the bound for the precision of the estimation of the
gradient remains constant under spatial translations,

$\begin{split}(\Delta
b_{1})^{-2}_{d}\leqslant\,&\mathbfcal{F}_{11}(d)-\frac{(\mathbfcal{F}_{01}(d))^{2}}{\mathbfcal{F}_{00}(d)}\\\
=\,&\mathbfcal{F}_{11}-2d\mathbfcal{F}_{01}+d^{2}\mathbfcal{F}_{00}\\\
&-\frac{\mathbfcal{F}_{01}^{2}-2d\mathbfcal{F}_{01}\mathbfcal{F}_{00}+d^{2}\mathbfcal{F}_{00}^{2}}{\mathbfcal{F}_{00}}\\\
=\,&\mathbfcal{F}_{11}-\frac{\mathbfcal{F}_{01}^{2}}{\mathbfcal{F}_{00}}.\end{split}$
(5.39)

These observations make the computations of the precision bounds in the next
sections easier, since now we can place the system arbitrarily wherever we
choose. It also allows us, for example, to place the origin of our coordinate
system as well as the system itself where the magnetic field is zero. So, we
can write the linear magnetic field simply as
$\boldsymbol{B}(x)=xB_{1}\boldsymbol{k}$ where $\boldsymbol{k}$ is the unitary
vector pointing on the $z$-direction perpendicular to $x$\- and $y$-axis. The
discourse we have had on the preceding section has a vital importance to
understand properly our results.

###  5.3 Testing our approach

Despite the generality of the tools we developed in Section 5.2, it is always
useful to start with simple but concise examples. For this, we consider two of
the most relevant cases for the external state $\rho^{(\text{x})}$ which we
know that behave well for estimating the gradient parameter. We will study the
chain of atoms, where the atoms are placed one-by-one in a $1$-dimensional
lattice with a constant separation $a$, and the two-ensembles of atoms, where
half of the atoms are in $x=-a$ and the rest in $x=+a$.

####  5.3.1 Distinguishable atoms in a 1D lattice

As we have said, the first spatial state we consider will be given by $N$
particles all placed equidistantly from each other in a one-dimensional spin
chain, i.e., a chain of atoms in a $1$-dimensional lattice [116], see Figure
5.1. The probability density function describing the system is

$\text{Pr}(\boldsymbol{x})=\prod_{n=1}^{N}\delta(x_{n}-na).$ (5.40)

\begin{overpic}[scale={1.15}]{img/GM_evolution_chain_initial.pdf}
\put(5.0,10.0){\small(a)}
\end{overpic}\begin{overpic}[scale={1.15}]{img/GM_evolution_chain_final.pdf}
\put(5.0,10.0){\small(b)} \end{overpic}

Figure 5.1:  (blue-circles) A system of $N$-atoms of spin-$j$ trapped in a
chain configuration. (green-area) Magnetic field gradient. Note that the field
is pointing outwards of the figures. (red-arrows) Spins of the particles
initially all aligned. (a) The ensemble is initially totally polarized along a
perpendicular direction, in this case $y$-direction, of the magnetic field
$B_{z}$. The internal state can be written as $|{+j}\rangle_{y}^{\otimes N}$,
the number represents $m_{y}$ the eigenvalue of the one particle operator
$j_{y}^{(n)}$. (b) One can see how the gradient field affects with a varying
field strength the different spins when they are placed in different
positions.

With this at hand we compute the single-point averages and the two-point
averages corresponding to the ion-chain. For the single-point average, one of
the integrals appearing in Eq. (5.37), we have that

$\int x_{n}\text{Pr}(\boldsymbol{x})\,\text{d}^{N}\boldsymbol{x}=na,$ (5.41)

and for the two-point correlation, which appears in Eqs. (5.30) and (5.37), we
have the following for the case of the chain of atoms,

$\int
x_{n}x_{m}\text{Pr}(\boldsymbol{x})\,\text{d}^{N}\boldsymbol{x}=nma^{2}.$
(5.42)

On the other hand, we use first a totally polarized state in the $y$-direction
for the internal degrees of freedom,
$\rho^{(\text{s})}=(|{+j}\rangle{{}_{y}}\langle{+j}|_{y})^{\otimes N}$
appeared in Eq. (2.33). This state is sensitive to the homogeneous field, so
using that the state is pure and separable

$\mathcal{F}_{\textnormal{Q}}[{\textstyle|{+j}\rangle_{y}^{\otimes},j_{z}^{(n)},j_{z}^{(m)}}]=4(\langle{j_{z}^{(n)},j_{z}^{(m)}}\rangle-\langle{j_{z}^{(n)}}\rangle\langle{j_{z}^{(m)}}\rangle)=2j\delta_{n,m}.$
(5.43)

Since this function is linear in the second and third arguments, we have that
$\mathcal{F}_{\textnormal{Q}}[{\textstyle|{+j}\rangle_{y}^{\otimes},j_{z}^{(n)},J_{z}}]=2j$
and
$\mathcal{F}_{\textnormal{Q}}[{\textstyle|{+j}\rangle_{y}^{\otimes},J_{z}}]=2jN$.
With this, we can now write the precision bound for a chain of atoms when the
internal state is totally polarized along the $y$-axis as

$\begin{split}(\Delta b_{1})^{-2}_{\text{ch,tp}}&\leqslant
a^{2}\left\\{\sum_{n=1}^{N}n^{2}2j-\frac{(\sum_{n=1}^{N}n2j)^{2}}{N2j}\right\\}\\\
&=a^{2}\frac{N^{2}-1}{12}2jN.\end{split}$ (5.44)

Despite that Eq. (5.44) is a third order function of the particle number $N$
and that it seems to overcome the ultimate scaling Heisenberg scaling, note
that the length of the chain increases as we introduce more particles into the
system. Hence, we have to normalize the bound with the effective size of the
system, such that separable states would scale as shot-noise scaling. This
restores the ultimate threshold of the precision to $\sim N^{2}$ as usual.

In this section, we will use the standard deviation of the averaged particle
position as the length measure of the system. We also include in our next
definitions the averaged correlation of two different particle positions,
since it will appear in the following sections and for completeness. They are
computed as

$\displaystyle\mu$
$\displaystyle=\int\frac{\sum_{n=1}^{N}x_{n}}{N}\text{Pr}(\boldsymbol{x})\,\text{d}^{N}\boldsymbol{x},$
(5.45a) $\displaystyle\sigma^{2}$
$\displaystyle=\int\frac{\sum_{n=1}^{N}x_{n}^{2}}{N}\text{Pr}(\boldsymbol{x})\,\text{d}^{N}\boldsymbol{x}-\mu^{2},$
(5.45b) $\displaystyle\eta$ $\displaystyle=\int\frac{\sum_{n\neq
m}x_{n}x_{m}}{N(N-1)}\text{Pr}(\boldsymbol{x})\,\text{d}^{N}\boldsymbol{x}-\mu^{2},$
(5.45c)

where $\mu$ denotes the averaged position of the particles, $\sigma^{2}$ the
variance of position of the particles and $\eta$ the position correlation
between different particles. So, the system effective width for the chain of
atoms, computed by the variance Eq. (5.45b), is given as

$\sigma_{\text{ch}}^{2}=a^{2}\frac{N^{2}-1}{12}.$ (5.46)

It turns out that Eq. (5.46) exactly coincides with one of the factors we have
in Eq. (5.44). Substituting this into the Eq. (5.44) we have that, for ion-
chains where the particles are separated by a constant distance and where the
spin-state $\rho^{(\text{s})}$ is the totally polarized state along the
$y$-direction, the precision bound is given by

$(\Delta b_{1})^{-2}_{\text{ch,tp}}\leqslant\sigma_{\text{ch}}^{2}2jN,$ (5.47)

in terms of $\sigma_{\text{ch}}$, the spin of each particle $j$ and the
particle number $N$.

####  5.3.2 Differential magnetometry with two ensembles

We now turn our attention to the case of two ensembles of distinguishable
atoms. Two ensembles of spin-$j$ atoms spatially separated from each other
have been realized in cold gases (e.g., Ref. [41]), and can be used for
differential interferometry [14, 117]. We will also use an internal state such
the maximal QFI is achieved so the reader gets familiar with our approach and
sees how the best state to measure the gradient parameter looks like in our
framework.

The spatial part is described by the following probability density function,
where for an even number of particles half of the particles are at one
position and the rest at another, both places at a distance of $a$ from the
origin

$\text{Pr}(\boldsymbol{x})=\prod_{n=1}^{N/2}\delta(x_{n}+a)\prod_{n=N/2+1}^{N}\delta(x_{n}-a).$
(5.48)

This could be realized in a double-well trap, where the width of the wells is
negligible compared to the distance of the wells. Note that a state defined by
Eq. (5.48) and Eq. (5.10) is a pure state in the position Hilbert space. To
distinguish the two wells we will use the labels "L" and "R" for the left and
right wells respectively, which is a shorthand which collects the first half
of the particles and the last half into a single index, respectively. With
this we are able to compute the single-point and two-point correlation
functions as

$\displaystyle\int x_{n}\text{Pr}(\boldsymbol{x})\,\text{d}^{N}\boldsymbol{x}$
$\displaystyle=\left\\{\begin{aligned} &{-a}&&\text{if }n\in\text{L},\\\
&a&&\text{if }n\in\text{R},\end{aligned}\right.$ (5.49a) $\displaystyle\int
x_{n}x_{m}\text{Pr}(\boldsymbol{x})\,\text{d}^{N}\boldsymbol{x}$
$\displaystyle=\left\\{\begin{aligned} &{-a^{2}}&&\text{if
}(n,m)\in\text{(L,L) or (R,R),}\\\ &a^{2}&&\text{if }(n,m)\in\text{(L,R) or
(L,R).}\end{aligned}\right.$ (5.49b)

We will try states insensitive to the homogeneous fields. Since the spatial
part is a pure state in the position subspace, we will try pure states in the
spin subspace due to the fact that the QFI is maximized by pure states as it
was shown in Section 2.3. For pure states we have that the QFI is computed
directly as four times the variance of the gradient Hamiltonian,
$\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,H_{1}}]=4(\Delta H_{1})^{2}$.
So, we just choose a spin-state that maximizes the variance of $H_{1}$ when
the spatial part is Eq. (5.48). Hence, the best state in this case is

$|{\Psi}\rangle=\frac{1}{\sqrt{2}}(|{{+j}\cdots{+j}}\rangle^{(\text{L})}|{{-j}\cdots{-j}}\rangle^{(\text{R})}+|{{-j}\cdots{-j}}\rangle^{(\text{L})}|{{+j}\cdots{+j}}\rangle^{(\text{R})}),$
(5.50)

where it can be seen as the superposition of the state with the smallest
energy and the state with the greatest energy when the Hamiltonian is $H_{1}$,
see Figure 5.2. The state $|{\Psi}\rangle$ is indeed insensitive to the
homogeneous field since both states of the superposition in Eq. (5.50) are
eigenstates of $H_{0}$ with the same eigenvalue. Moreover, the state is also
maximally entangled.

Figure 5.2:  (blue-circle) Atoms located at (L) or (R). (red-arrow) Spin state
of each of the atoms. (green-area) Linear magnetic field $B_{z}$. Note that
the $z$-axis is to represent the direction of the spins. On the other hand,
the state is a linear superposition of the upper state and the lower state,
represented by $|{\cdot}\rangle$ and $+$ sign. Note that all particles at (L)
or (R) are assumed to be in the same spatial spot.

We compute first the
$\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho^{(\text{s})},j_{z}^{(n)},j_{z}^{(m)}}]$
for $|{\Psi}\rangle$ as

$\mathcal{F}_{\textnormal{Q}}[{\textstyle|{\Psi}\rangle,j_{z}^{(n)},j_{z}^{(m)}}]=\left\\{\begin{aligned}
&4j^{2}&&\text{if }(n,m)\in\text{(L,L) or (R,R)},\\\ &{-4j^{2}}&&\text{if
}(n,m)\in\text{(L,R) or (R,L)}.\end{aligned}\right.$ (5.51)

Now, if we separate the terms of the sum in Eq. (5.30) into two groups, such
that one of the sums is for indexes $(n,m)\in\text{(L,L) or (R,R)}$, and the
other is for indexes $(n,m)\in\text{(L,R) or (R,L)}$, we can compute the bound
for the best state for the two ensemble case as

$\begin{split}(\Delta
b_{1})^{2}|_{\max}&=\sum_{\genfrac{}{}{0.0pt}{}{(n,m)\in}{\text{(L,L) or
(R,R)}}}a^{2}4j^{2}+\sum_{\genfrac{}{}{0.0pt}{}{(n,m)\in}{\text{(L,R) or
(R,L)}}}-a^{2}(-4j^{2})\\\ &=4a^{2}N^{2}j^{2}.\end{split}$ (5.52)

On the other hand, if we compute now the standard deviation Eq. (5.45b), as we
did before for the case of the chain Eq. (5.46), we have that for the two
ensembles case $\mu=0$ and the standard deviation for the spatial state is

$\sigma_{\text{te}}^{2}=a^{2},$ (5.53)

with which

$(\Delta b_{1})^{2}_{\text{HL}}|_{\max}=4\sigma_{\text{te}}^{2}N^{2}j^{2}.$
(5.54)

For a general $\sigma^{2}$, the Eq. (5.54) can be seen as the Heisenberg limit
for gradient metrology. Note that the state is insensitive to the homogeneous
field, hence this bound is saturable by some measurement, and that the state
$|{\Psi}\rangle$ maximizes the variance of $H_{1}$ for any given $\sigma^{2}$.
Before concluding, we want to show another more usual approach to the same
problem.

Knowing that the QFI is convex in the state and considering the spatial state
to be Eq. (5.48), we reduce our problem to the internal subspace in which the
state that maximizes the QFI is the one that maximizes $(\Delta H_{1})^{2}$.
In this case, taking into account the particle locations are given and that we
have zero magnetic field at the origin, we obtain

$H_{1,\text{eff}}^{(\text{s})}=a(\mathbbm{1}^{(\text{L})}\otimes
J_{z}^{(\text{R})}-J_{z}^{(\text{L})}\otimes\mathbbm{1}^{(\text{R})}),$ (5.55)

where we write the effective Hamiltonian that the particles on the left and
right feel. This proves that we have used the right state, since it maximizes
the variance $(\Delta H_{1,\text{eff}}^{(\text{s})})^{2}$ [117].

States separable into $|{\psi}\rangle^{({\rm L})}\otimes|{\psi}\rangle^{({\rm
R})}$

Now that we have already introduced the reader to the case of the two
ensembles, we take the opportunity to show some more important results for
states of the form of
$|{\psi}\rangle^{(\text{L})}\otimes|{\psi}\rangle^{(\text{R})}$. These states
can reach the Heisenberg limit, while they are easier to realize
experimentally than states in which the particles on the left and particles on
the right are entangled with each other.

First, we will compute the bound for states insensitive to the homogeneous
field. For such states we only have to compute the QFI for $H_{1,\text{eff}}$
Eq. (5.55) such that

$(\Delta
b_{1})^{-2}|_{\max}=\mathcal{F}_{\textnormal{Q}}[{\textstyle|{\psi}\rangle^{(\text{L})}\otimes|{\psi}\rangle^{(\text{R})},a(\mathbbm{1}^{(\text{L})}\otimes
J_{z}^{(\text{R})}-J_{z}^{(\text{R})}\otimes\mathbbm{1}^{(\text{L})})}]=2a^{2}\mathcal{F}_{\textnormal{Q}}[{\textstyle|{\Psi}\rangle^{(\text{L})},J_{z}^{(\text{L})}}],$
(5.56)

where we used the general rule Eq. (2.60) and that any scalar multiplying the
second argument of the QFI can be extracted outside the function squared.

Now, we analyze how the bound would look like for states sensitive to the
homogeneous field. Note that the single-point correlation function for
particles at "(L)" and "(R)" is $a$ and $-a$ respectively, and the two-point
correlation function is $a^{2}$ for both. Thus, in the case of computing the
bound for the states sensitive to the homogeneous fields, we have that
$F_{01}^{(\text{L})}=-F_{01}^{(\text{R})}$, which we used the superscript to
indicate in this case over which subspace is computed the QFI matrix element,
whereas the other two matrix elements we have to compute are equal for both
subspaces "(L)" and "(R)". The precision bound for states sensitive to the
homogeneous fields is obtained as

$\begin{split}(\Delta
b_{1})^{-2}\leqslant&F_{11}+\frac{(F_{01})^{2}}{F_{00}}\\\
=&F_{11}^{(\text{L})}+F_{11}^{(\text{R})}+\frac{(F_{01}^{(\text{L})}+F_{01}^{(\text{R})})^{2}}{F_{00}^{(\text{L})}+F_{00}^{(\text{R})}}\\\
=&2F_{11}^{(\text{L})}+\frac{(F_{01}^{(\text{L})}-F_{01}^{(\text{L})})^{2}}{2F_{00}^{(\text{L})}}\\\
=&2F_{11}^{(\text{L})}\\\
=&2a^{2}\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho^{(L)},J_{z}^{(L)}}],\end{split}$
(5.57)

where we use in the first line the identities for additions under tensor
products Eqs. (2.60) and (5.23), and in the last line we extract the common
factor $a^{2}$ and we use the linearity on the arguments the QFI. Note that
this is the same precision bound we will obtain for states insensitive to the
homogeneous fields. Note that this bound relates how good the state on the
"(L)" or "(R)" subsystem is in sensing the homogeneous field with the
precision achievable for the gradient parameter. This is reasonable because
the state in "(L)" is not interacting neither correlated with "(R)". Hence,
after the homogeneous field is estimated for "(L)" and "(R)" independently,
the gradient can also be estimated as the difference between the two estimates
divided by the square distance $a^{2}$.

In the literature one can find several states that can be used to measure a
homogeneous field, such as the GHZ states [118], unpolarized Dicke states, and
spin-squeezed states. Note that if $|{\Psi}\rangle$ is separable, then based
in Eq. (2.64) and for any of the two bounds Eqs. (5.56) and (5.57), we have

$\mathcal{F}_{\textnormal{Q}}[{\textstyle|{\Psi}\rangle_{\text{sep}}^{(\text{L})}|{\Psi}\rangle_{\text{sep}}^{(\text{R})},\,a(J_{z}^{(\text{L})}\mathbbm{1}^{(\text{R})}-\mathbbm{1}^{(\text{L})}J_{z}^{(\text{R})})}]=2a^{2}\mathcal{F}_{\textnormal{Q}}[{\textstyle|{\Psi}\rangle_{\text{sep}}^{(\text{L})},J_{z}^{(\text{L})}}]=2a^{2}4\frac{N}{2}j^{2}\leqslant
4a^{2}Nj^{2}.$ (5.58)

Note that each of the ensembles has half of the total particle number $N$. Eq.
(5.58) can be seen as the shot-noise limit when two ensembles are used for
gradient metrology. In Table 5.1, we summarized the precision bounds for
states of type $|{\Psi}\rangle^{(\text{L})}\otimes|{\Psi}\rangle^{(\text{R})}$
for the two-ensemble case.

States in (L) and (R) |  $\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{z}}]$ for $N/2$ | $(\Delta b_{1})^{-2}\leqslant$
---|---|---
$|{+j}\rangle_{l}^{\otimes N/2}\otimes|{+j}\rangle_{l}^{\otimes N/2}$ | $Nj$ | $2a^{2}Nj$
$|{\textnormal{GHZ}}\rangle\otimes|{\textnormal{GHZ}}\rangle$ | $N^{2}/4$ | $a^{2}N^{2}/2$
$|{\textnormal{D}_{{N/2}}}\rangle_{x}\otimes|{\textnormal{D}_{{N/2}}}\rangle_{x}$ | $N(N+4)/8$ | $a^{2}N(N+4)/4$
$|{\Psi}\rangle_{\text{sep}}\otimes|{\Psi}\rangle_{\text{sep}}$ | $2Nj^{2}$ | $4a^{2}Nj^{2}$
Table 5.1:  (first column) The complete state as a tensor product of the state
in (L) and the state in (R). Note that for the GHZ state and the unpolarized
Dicke state, the spin is $j=\frac{1}{2}$. The last state $|{\Psi}\rangle$ is
the best separable state for the estimation of the homogeneous field. Hence,
the bound for $|{\Psi}\rangle$ coincides with the shot-noise limit for
gradient metrology with two ensembles. (second column) Precision of the
estimation of the homogeneous magnetic field in one of the ensembles. (third
column) From the second column and based on Eqs. (5.56) and (5.57), we compute
the precision for differential magnetometry for various product quantum states
in two ensembles spatially separated from each other by a distance $a$. Note
that all states are sensitive to the homogeneous field so the saturability of
of the bound is not ensured. This is the reason we use "$\leqslant$" instead
of "$|_{\max}$".

In this section we have shown to the reader how one should handle the spatial
width of the system for classifying it for gradient metrology as well as a
state-of-the-art system in which the Heisenberg limit is achieved. Moreover,
we have shown how to use the tools developed in the previous section to
compute simple bounds. In the next section we will focus on single cold-atom
ensembles since they play an important role in quantum technology, and many
groups are trying to realize them whit great success but with few theoretical
support.

###  5.4 Magnetometry with a single atomic ensemble

In this section, we discuss magnetometry with a single atomic ensemble in more
detail. We consider a one-dimensional ensemble of spin-$j$ atoms placed in a
one dimensional trap, which is elongated in the $x$-direction. The magnetic
field points in the $z$-direction, and has a constant gradient along the
$x$-direction. The setup is depicted in Fig. 5.3. In the last part of this
section, we calculate precision bounds for the gradient estimation for some
important multi-particle quantum states, for instance, Dicke states or GHZ
states. Note that all these states are permutationally invariant, since we
assume that a permutationally invariant procedure prepared the states.

Figure 5.3:  (blue-point) Atomic ensemble with their spins (red-arrow)
pointing randomly in any direction coupled with a linear magnetic field
$B_{z}$ (green). The spatial state $\rho^{(\text{x})}$ is assumed to be
permutationally invariant. The ensemble is centered around the place at which
the magnetic field is zero due to the invariance of the precision bound under
translations of the system.

####  5.4.1 Precision bound

In an atomic ensemble of very many atoms, typically the atoms cannot be
individually addressed. This can be taken into account, if we consider quantum
states that are permutationally invariant. Hence, we will consider states for
which both the internal state $\rho^{(\text{s})}$ and the probability
distribution function $\text{Pr}(\boldsymbol{x})$, appearing in Eq. (5.10),
are permutationally invariant. The permutational invariance of
$\text{Pr}(\boldsymbol{x})$ implies that

$\text{Pr}(\boldsymbol{x})=\tfrac{1}{N!}\sum_{k}\mathcal{P}_{k}[\text{Pr}(\boldsymbol{x})],$
(5.59)

where the summation is over all the possible permutations of the variables
$x_{n}$ denoted by $\mathcal{P}_{k}$.

As we have shown in Section 5.2, the precision bound is invariant under
spatial translations. This allows us to place the "center of mass" of the
system at the origin of the coordinate system. With this simplifying
assumption and based on Eqs. (5.45a), (5.45b) and (5.45c), the single-point
average appearing in Eq. (5.37) is

$\int
x_{n}\text{Pr}(\boldsymbol{x})\,\text{d}^{N}\boldsymbol{x}=\int\frac{\sum_{n=1}^{N}x_{n}}{N}\text{Pr}(\boldsymbol{x})\,\text{d}^{N}\boldsymbol{x}=\mu=0,$
(5.60)

where we used the permutational invariance of $\text{Pr}(\boldsymbol{x})$
substitute $x_{n}$ by the sum appearing in Eq. (5.45a). In a similar way we
obtain

$\int
x_{n}x_{m}\text{Pr}(\boldsymbol{x})\,\text{d}^{N}\boldsymbol{x}=\left\\{\begin{aligned}
&\sigma^{2}&&\text{for }n=m,\\\ &\frac{\eta}{N-1}&&\text{for }n\neq
m,\end{aligned}\right.$ (5.61)

where we used that the system is placed at the origin $\mu=0$. An interesting
property of the covariance of this type is that its a value is bounded from
below and from above by the variance itself and the particle number $N$ in the
following way,

$\frac{-\sigma^{2}}{N-1}\leq\eta\leq\sigma^{2}.$ (5.62)

Note that in the first sum in Eq. (5.37) there are in total $N(N-1)$ terms
proportional to $\eta/(N-1)$ and $N$ terms proportional to $\sigma^{2}$.

From the linearity in the second and third arguments of
$\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,A,B}]$ Eq. (5.17) and for states
insensitive to the homogeneous field, where
$\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{z}}]=0$, we have that

$\sum_{n=1}^{N}\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,j_{z}^{(n)}}]=-\sum_{n\neq
m}^{N}\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,j_{z}^{(n)},j_{z}^{(m)}}].$
(5.63)

From the definition of the QFI for states insensitive to the homogeneous
field, Eq. (5.30), we compute the bound for single ensembles as

$\begin{split}(\Delta b_{1})^{-2}|_{\max}&=\sum_{n,m}^{N}\int
x_{n}x_{m}\text{Pr}(\boldsymbol{x})\,\text{d}^{N}\boldsymbol{x}\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho^{(\text{s})},j_{z}^{(n)},j_{z}^{(m)}}]\\\
&=\sum_{n=1}^{N}\sigma^{2}\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,j_{z}^{(n)}}]+\sum_{n\neq
m}^{N}\eta\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,j_{z}^{(n)},j_{z}^{(m)}}].\end{split}$
(5.64)

Together with Eq. (5.63) we write the precision bound for states insensitive
to the homogeneous fields as

$(\Delta
b_{1})^{-2}|_{\max}=(\sigma^{2}-\eta)\sum_{n=1}^{N}\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho^{(\text{s})},j_{z}^{(n)}}].$
(5.65)

Note that the bound in Eq. (5.65) can be saturated by an optimal measurement.
Nevertheless, it cannot surpass the shot-noise scaling, $\sim N$, because
$\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho^{(\text{s})},j_{z}^{(n)}}]$, the
QFI for the single-particle operator $j_{z}^{(n)}$, cannot be larger than
$j^{2}$.

To compute the bound for states sensitive to the homogeneous field, note that
in the second term appearing in Eq. (5.37) is proportional to the single-point
average Eq. (5.60) which was chosen to be equal to zero. Hence, we only have
to compute the first term of the Eq. (5.37) as

$\begin{split}(\Delta b_{1})^{-2}\leqslant&\sum_{n,m}^{N}\int
x_{n}x_{m}\text{Pr}(\boldsymbol{x})\,\text{d}^{N}\boldsymbol{x}\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho^{(\text{s})},j_{z}^{(n)},j_{z}^{(m)}}]\\\
=&\sum_{n=1}^{N}\sigma^{2}\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,j_{z}^{(n)}}]+\sum_{n\neq
m}^{N}\eta\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,j_{z}^{(n)},j_{z}^{(m)}}]\\\
=&(\sigma^{2}-\eta)\sum_{n=1}^{N}\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,j_{z}^{(n)}}]+\eta\sum_{n,m}^{N}\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,j_{z}^{(n)},j_{z}^{(m)}}],\end{split}$
(5.66)

where in the second line we compute the diagonal and the off-diagonal terms of
the sum separately and in the last line we add
$\eta\sum_{n=1}^{N}\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,j_{z}^{(n)}}]$
to the last term and subtract it from the first term to make the expression
more similar to Eq. (5.65).

Hence, for states sensitive to homogeneous fields, the precision of estimating
the gradient is bounded from above as

$(\Delta
b_{1})^{-2}\leqslant(\sigma^{2}-\eta)\sum_{n=1}^{N}\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho^{(\text{s})},j_{z}^{(n)}}]+\eta\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho^{(\text{s})},J_{z}}].$
(5.67)

The second term on the right-hand side of Eq. (5.67) is new in the sense that
it did not appear in the bound for states insensitive to homogeneous fields.
Note that the bound in Eq. (5.67) is not necessarily saturable if the optimal
measurements to estimate the gradient parameter and the homogeneous parameter
do not commute with each other. Note also that even if the first term cannot
overcome the shot-noise scaling, in the second term the covariance is
multiplied by QFI for estimating the homogeneous field and therefore this
concrete term can make the bound, for extremely correlated particle positions,
to scale as Heisenberg scaling.

####  5.4.2 Precision bounds for different spin-states

In this section, we present the precision limits for different classes of
important quantum states such as the totally polarized state, the state having
the largest precision among separable states, or the singlet state. We will
calculate the precision bounds presented before, Eqs. (5.65) and (5.67), for
these systems. We show first the results for singlets that are insensitive to
homogeneous fields. In this case, the bounds can be achieved by choosing the
appropriate magnitude to measure. The rest of the results are for states
sensitive to homogeneous fields which in general are not necessarily
achievable bounds.

Before going into the details of our computations we present a summary of the
results obtained in this section. The summary for different states can be
found in the Table 5.2.

States | $(\Delta b_{1})^{-2}$
---|---
permutationally invariant singlet states | $|_{\max}=(\sigma^{2}-\eta)\frac{4Nj(j+1)}{3}$
$|{+j}\rangle_{y}^{\otimes N}$ | $\leqslant\sigma^{2}2Nj$
Best separable state $|{\Psi}\rangle$ | $\leqslant\sigma^{2}4Nj^{2}$
$|{\textnormal{D}_{N}}\rangle_{z}$ | $|_{\max}=(\sigma^{2}-\eta)N$
$|{\textnormal{D}_{N}}\rangle_{x}$ | $\leqslant(\sigma^{2}-\eta)N+\eta\frac{N(N+2)}{2}$
$|{\textnormal{GHZ}}\rangle$ | $\leqslant(\sigma^{2}-\eta)N+\eta N^{2}$
Table 5.2:  Precision bounds for differential magnetometry for various quantum
states. For the definition of the states, see the text. If the bound are
proved to be saturable then the "$|_{\max}=$" subscript is used instead of an
inequality.

Permutationally invariant singlet states

We consider now the singlet state, which is invariant under the influence of a
homogeneous field along any direction. So, we have to compute the formula for
the bound of the precision Eq. (5.65). A pure singlet state is an eigenstate
of the collective $J_{z}$ and $J^{2}$ operators, with an eigenvalue zero in
both cases. There are many different singlet states for an ensemble of $N$
spin-$j$ particles, which some of them are permutationally invariant.
Surprisingly the precision bound we compute is the same for any
permutationally invariant singlet. Atomic ensembles in a singlet state have
been experimentally created with cold gases [34, 58].

In an $N$-particle system, there are several singlets pairwise orthogonal to
each other. The number of such singlets, $D_{0}$, depends on the particle spin
$j$ and the number of particles $N$.

The most general singlet state can be written in the total angular momentum
basis, using $D$ to label the degenerate states, see Appendix A. In its
eigenbasis the singlet is written as

$\rho^{(\text{s})}=\sum_{D=1}^{D_{0}}\lambda_{D}|{0,0,D}\rangle{\\!}\langle{0,0,D}|,$
(5.68)

where $\sum_{D}\lambda_{D}=1$. In its complete form the eigenvalues of the
spin density matrix are $\lambda_{J,M_{z},D}=\delta_{0,J}\lambda_{D}$.

Looking at Eq. (5.65), we must compute the QFI for the one-particle operator
$j_{z}^{(n)}$ in order to compute the precision bound for permutationally
invariant singlet states. For that purpose we use the fact that when
$j_{z}^{(n)}$ acts on a singlet state, it produces a state outside of the
singlet subspace. This can be proved by noting that

$e^{i\pi J_{x}}j_{z}^{(n)}e^{-i\pi J_{x}}=-j_{z}^{(n)}$ (5.69)

and that $e^{-i\pi J_{x}}|{0,0,D}\rangle=|{0,0,D}\rangle$ holds for any pure
singlet state. Hence, we can arbitrarily flip the sign of $j_{z}^{(n)}$ so

$\langle{0,0,D}|{j_{z}^{(n)}}|{0,0,D^{\prime}}\rangle=-\langle{0,0,D}|{j_{z}^{(n)}}|{0,0,D^{\prime}}\rangle,$
(5.70)

which implies

$\langle{0,0,D}|{j_{z}^{(n)}}|{0,0,D^{\prime}}\rangle=0,$ (5.71)

for any pair of pure singlet singlet states.

In order to compute the QFI for the singlet state we use Eq. (5.20). Hence, we
can write the following for the second term of Eq. (5.20),

$8\sum_{D,D^{\prime}}\tfrac{\lambda_{D}\lambda_{D^{\prime}}}{\lambda_{D}+\lambda_{D^{\prime}}}|\langle{0,0,D}|{j_{z}^{(n)}}|{0,0,D^{\prime}}\rangle|^{2}=0.$
(5.72)

It follows that the QFI of $j_{z}^{(n)}$ for any singlet is indeed simply

$\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho^{(\text{s})},j_{z}^{(n)}}]=4\operatorname{tr}({\rho^{(\text{s})}(j_{z}^{(n)})^{2}}).$
(5.73)

Finally, we must compute the expectation value of the operator
$(j_{z}^{(n)})^{2}$. For that we have that

$\operatorname{tr}(\rho^{(\text{s})}(j_{k}^{(n)})^{2})=\operatorname{tr}(\rho^{(\text{s})}(j_{l}^{(n)})^{2}),$
(5.74)

for any pair $k,l\in x,y,z$ due to the rotational invariance of the singlet,
i.e, all the singlets remain invariant under a $SU(2)$ transformation of the
kind $U=e^{i\phi J_{\boldsymbol{n}}}$, where $\boldsymbol{n}$ is an unitary
vector belonging to the positional space. Then we can write that

$\langle{(j_{x}^{(n)})^{2}+(j_{y}^{(n)})^{2}+(j_{z}^{(n)})^{2}}\rangle=j(j+1),$
(5.75)

for any state, since it represents the spin number of the particle, which is
fixed. Hence, the expectation value of $(j_{z}^{(n)})^{2}$ on the singlet is

$\operatorname{tr}(\rho^{(\text{s})}(j_{z}^{(n)})^{2})=\frac{j(j+1)}{3},$
(5.76)

for all the singlets. Inserting this into Eq. (5.73) and using Eq. (5.65), we
obtain

$(\Delta
b_{1})^{-2}_{\text{s}}|_{\max}=\left(\sigma^{2}-\eta\right)\frac{4Nj(j+1)}{3}.$
(5.77)

To conclude, singlet states are insensitive to homogeneous magnetic fields,
hence determining the gradient leads to a single-parameter estimation problem.
This implies that there is an optimal operator that saturates the precision
bound given by Eq. (5.77). However, it is usually very hard to find this
optimal measurement, although a formal procedure for this exists [2, 115]. In
Ref. [57], a particular set-up for determining the magnetic gradient with
permutationally invariant singlet states was suggested by the measurement of
the $J_{x}^{2}$ collective operator. For this scenario the precision is given
by the error propagation formula as

$(\Delta
b_{1})^{-2}=\frac{|\partial_{b_{1}}\langle{J_{x}^{2}}\rangle|^{2}}{\langle{J_{x}^{4}}\rangle-\langle{J_{x}^{2}}\rangle^{2}}.$
(5.78)

Totally polarized state

The totally polarized state can easily be prepared experimentally. It has
already been used for gradient magnetometry with a single atomic ensemble [17,
18]. For the gradient measurement as for the measurement of the homogeneous
field, the polarization must be perpendicular to the field we would like to
measure in order to take advantage of the interaction between the particles
and the field. Here we chose as before the totally polarized state along the
$y$-axis which is written as $|{j}\rangle_{y}^{\otimes N}$. Note that this
state is sensitive to the homogeneous field, hence, we must use the Eq. (5.67)
to compute the bound.

For the pure states we have that
$\mathcal{F}_{\textnormal{Q}}[{\textstyle|{\psi}\rangle,A}]=4(\Delta A)^{2}$.
Together with, $(\Delta j_{z}^{(n)})^{2}=j/2$ and $(\Delta J_{z})^{2}=Nj/2$,
when the polarization is perpendicular to the $z$-direction, the precision
will be computed straightforward from Eq. (5.67).

Therefore, the Cramér-Rao bound fixes the highest value for the precision of
the totally polarized state as

$(\Delta b_{1})^{-2}_{\text{TP}}\leqslant\sigma^{2}2Nj.$ (5.79)

Note that the precision bound for the totally polarized state is smaller than
that of the optimal separable state we present later on. We can see clearly
that the precision scales as $\mathcal{O}(N)$.

Let us now see, which quantities have to be measured to estimate the field
gradient with a totally polarized state. The homogeneous field rotates all
spins by the same angle, while the gradient rotates the spins at different
positions by a different angle. Due to that, the homogeneous field rotates the
collective spin, but does not change its absolute value. On the other hand,
the field gradient decreases the absolute value of the spin, since it has been
prepared to be maximal, which has been used in Ref. [37] for gradient
magnetometry, see Figure 5.1.

The best separable state

We will now turn our attention to the precision bound for all separable spin
states. It is useful to obtain this value so we have a direct comparison on
what the best classically achievable precision is. It will turn out that for
$j>\frac{1}{2},$ it is possible to achieve a precision higher than with the
fully polarized state. One has to take into account that if the state is
insensitive to the homogeneous field the bound can be saturated for sure, and
if the state is sensitive to homogeneous fields, it would depend on the
measurements compatibility and on the system as we discussed before. From
another point of view and instead of using the Eqs. (5.65) and (5.67), what we
have is that the bound is the same
$\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,H_{1}}]$ for both cases. Note
that we can place the system at the point in which the magnetic field is zero
without changing the result. Thus, it is easy to argue that the precision
bound itself is a convex function of the states. Moreover, it is also a convex
function of the states when the external state $\rho^{(\text{x})}$ is fixed
and only the internal $\rho^{(\text{s})}$ is considered.

In the single ensemble configuration, Eq. (5.34) must be computed only, where
the two-point correlation function returns $\sigma^{2}$ or $\eta$ based on Eq.
(5.61). On the other hand, for pure states we have that
$\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho^{(\text{s})},j_{z}^{(n)},j_{z}^{(m)}}]$
is four times the correlation
$\langle{j_{z}^{(n)}j_{z}^{(m)}}\rangle-\langle{j_{z}^{(n)}}\rangle\langle{j_{z}^{(m)}}\rangle$.
If the state is a product state, then we have
$\langle{j_{z}^{(n)}j_{z}^{(m)}}\rangle-\langle{j_{z}^{(n)}}\rangle\langle{j_{z}^{(m)}}\rangle=0$
for all $n\neq m$. Hence, $\eta$ does not play any role in the precision.
Finally, we have to maximize only the variance of each of the single-particle
operators $4(\Delta j_{z}^{(n)})^{2}$. From the definition of the variance,

$(\Delta
j_{z}^{(n)})^{2}=\langle{(j_{z}^{(n)})^{2}}\rangle-\langle{j_{z}^{(n)}}\rangle^{2}.$
(5.80)

Hence, We try a state that approaches to zero its polarization on the
$z$-direction and maximizes $\langle{(j_{z}^{(n)})^{2}}\rangle$.

We have that $|{\Psi}\rangle=(|{+j}\rangle+|{-j}\rangle)/\sqrt{2}$ is ideal
for this, for any $j$. Hence, we write the entire internal state as
$\rho^{(\text{s})}=(|{\Psi}\rangle{\\!}\langle{\Psi}|)^{\otimes N}$. This
state gives $(\Delta j_{z}^{(n)})^{2}=j^{2}$ which can be used in Eq. (5.34)
after multiplying by four. Note that this state is permutationally invariant,
hence we have finished the search for the best separable permutationally
invariant state. Moreover, the state is sensitive to the homogeneous field.

Finally, the best achievable precision for separable states is written as

$(\Delta b_{1})^{-2}_{\text{SNL}}\leqslant\sigma^{2}4Nj^{2},$ (5.81)

where the state itself is sensitive to homogeneous fields and the shot-noise
limit is achieved. Note that in the two ensembles case
$\sigma^{2}_{\text{te}}=a^{2}$ which tells us that both bounds Eqs. (5.58) and
(5.81) are equal. This bound coincides with the totally polarized state
studied before when the spin number $j=\frac{1}{2}$.

In the following we try to find a better precision bound making use of the
presumably better entangled states. Note that the bound for the singlet state,
even if it is entangled, is above the bound for the totally polarized state
but below of the bound defined for the best separable state. Nevertheless,
when the singlet state is used the effect of the homogeneous magnetic field
has not to be compensated since the state is insensitive to it and thus the
bound can be saturated with an optimal estimator for the gradient field.

The unpolarized Dicke states $|{\textnormal{D}_{N}}\rangle_{z}$ and
$|{\textnormal{D}_{N}}\rangle_{x}$

Unpolarized Dicke states play an important role in quantum optics and quantum
information science. The unpolarized Dicke state
$|{\textnormal{D}_{N}}\rangle_{l}$ with a maximal
$\langle{J_{x}^{2}+J_{y}^{2}+J_{z}^{2}}\rangle=\mathcal{J}_{N/2}$, defined in
Eq. (A.3), and $\langle J_{l}\rangle=0$ for any $l\in x,y,z$ is particularly
interesting due to its entanglement properties and its metrological
usefulness. This state has been created in photonic experiments [106, 107,
109] and in cold atoms [8, 59], while a Dicke state with $\langle
J_{z}\rangle>0$ has been created with cold trapped ions [119].

The Dicke state $|{\textnormal{D}_{N}}\rangle_{z}$ is an eigenstate of $J_{z}$
so it is insensitive to homogeneous magnetic field pointing into the
$z$-direction, thus the precision can be saturated by some measurement. In the
following, $|{\textnormal{D}_{N}}\rangle_{z}$ without the subscript $z$ refers
to $|{\textnormal{D}_{N}}\rangle_{z}$. On the other hand, the Dicke state
$|{\textnormal{D}_{N}}\rangle_{x}$ is sensitive to the homogeneous field.
Moreover it is very useful for homogeneous magnetometry as it has been shown
in Ref. [97]. Here we consider large particle numbers, to make the results
simpler.

Since both Dicke states are pure, and following the procedure we used in
previous sections, we have that to compute all the
$\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho^{(s)},j_{z}^{(n)}}]=4(\langle{j_{z}^{(n)}j_{z}^{(m)}}\rangle-\langle{j_{z}^{(n)}}\rangle\langle{j_{z}^{(m)}}\rangle)$
and $\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho^{(s)},J_{z}}]$ appearing in
Eqs. (5.65) and (5.67). Since both states are unpolarized and permutationally
invariant, we have that $\langle{J_{z}}\rangle=0$ and
$\langle{j_{z}^{(n)}}\rangle=0$ for both cases. Therefore, we only need to
compute the second moments to compute the needed variances.

To distinguish between the to cases, $|{\textnormal{D}_{N}}\rangle$ and
$|{\textnormal{D}_{N}}\rangle_{x}$, we will denote their expectation values by
$\langle{\cdot}\rangle_{\textnormal{D}}$ and
$\langle{\cdot}\rangle_{\textnormal{D},x}$, respectively.

First of all, from the definition of the Dicke states we have that

$\langle{J_{x}^{2}+J_{y}^{2}+J_{z}^{2}}\rangle=\mathcal{J}_{N/2}=\frac{N}{2}\left(\frac{N}{2}+1\right),$
(5.82)

for both cases. Moreover, $\langle{J_{l}^{2}}\rangle=0$ holds for
$|{\textnormal{D}_{N}}\rangle_{l}$. The other two second moments of Eq. (5.82)
are equal to the invariance of the states under rotations around the $l$-axis.
Hence, we can write that

$\displaystyle\langle{J_{z}^{2}}\rangle_{\textnormal{D}}$ $\displaystyle=0,$
(5.83a) $\displaystyle\langle{J_{z}^{2}}\rangle_{\textnormal{D},x}$
$\displaystyle=\frac{\mathcal{J}_{N/2}}{2}.$ (5.83b)

For the single spin components

$\langle{(j_{x}^{(n)})^{2}+(j_{y}^{(n)})^{2}+(j_{z}^{(n)})^{2}}\rangle=\mathcal{J}_{1/2}$
(5.84)

holds. Invoking the rotational symmetry again and that
$\langle{J_{l}^{2}}\rangle=\sum_{n,m}^{N}\langle{j_{l}^{(n)}j_{l}^{(m)}}\rangle$,
we arrive at

$\displaystyle\langle{(j_{z}^{(n)})^{2}}\rangle_{\textnormal{D}}$
$\displaystyle=\frac{1}{4},$ (5.85a)
$\displaystyle\langle{(j_{z}^{(n)})^{2}}\rangle_{\textnormal{D},x}$
$\displaystyle=\frac{1}{4},$ (5.85b)

after solving a system of linear equations.

Substituting Eqs. (5.83) and (5.85) into Eqs. (5.65) and (5.67), the bounds
for unpolarized Dicke states insensitive to the homogeneous field and
sensitive to the homogeneous field are

$\displaystyle(\Delta b_{1})^{-2}_{\textnormal{D}}|_{\max}$
$\displaystyle=(\sigma^{2}-\eta)N,$ (5.86a) $\displaystyle(\Delta
b_{1})^{-2}_{\textnormal{D},x}$
$\displaystyle\leqslant(\sigma^{2}-\eta)N+\eta\frac{N(N+2)}{2},$ (5.86b)

where Eq. (5.86b) shows in principal a Heisenberg scaling behavior in the
second term, whenever the particles are very correlated among each other in
the position subspace. This is due to the metrological enhancement of sensing
the homogeneous field. In the next section, we will see another example of a
state useful for homogeneous field estimation that is also useful for gradient
magnetometry.

The GHZ state

The Greenberger-Horne-Zeilinger (GHZ) states are also highly entangled and
play an important role in quantum information theory [48]. They have been
created experimentally in photonic systems [51, 120, 53] and trapped ions [55,
56].

We invoke the definition of the GHZ states Eq. (4.14) given as

$|{\textnormal{GHZ}}\rangle=\tfrac{1}{\sqrt{2}}(|{0\cdots 0}\rangle+|{1\cdots
1}\rangle),$ (5.87)

where $|{0}\rangle$ and $|{1}\rangle$ stands for particles with eigenvalue
$-1/2$ and $+1/2$ respectively for the one-particle $j_{z}^{(n)}$ operator.
The state Eq. (5.87) is very sensitive to the homogeneous field.

In order to calculate the bound explicitly, let us recall that for pure states
the QFI is simplified to
$\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,A}]=4(\Delta A)^{2}$ Eq. (2.53).
Following the Eq. (5.67), for the GHZ state the expectation values of
$j_{z}^{(n)}$ and $J_{z}$ are equal to zero, and
$\langle{(j_{z}^{(n)})^{2}}\rangle=\frac{1}{4}$ and
$\langle{J_{z}^{2}}\rangle=\frac{N^{2}}{4}$. Hence, the variances of
$j_{z}^{(n)}$ and $J_{z}$ can be computed. Finally, we obtain the precision
bound for gradient magnetometry for the GHZ state as

$(\Delta b_{1})^{-2}_{\textnormal{GHZ}}\leqslant(\sigma^{2}-\eta)N+\eta
N^{2}.$ (5.88)

This means that we can reach the Heisenberg-limit with such states, but only
in cases where $\eta$ is positive, i.e, that the particles stay spatially
correlated.

In summary, we have considered the experimentally most relevant spatial
distributions of particles, which could be used for gradient metrology. We
have have also applied our methods to calculate the quantum Fisher information
for various spin states. As we have seen, in some cases the system overcomes
the shot-noise limit, even when the spatial state is a single ensemble of
atoms, which opens up the possibility of ultra-precise gradient magnetometry
with a smaller experimental effort.

##  6 Conclusions Conclusions

In this thesis we have presented some aspects of quantum metrology from three
different perspectives. Besides the introductory part, Chapters 1 and 2, our
main results can be found in Chapters 3, 4 and 5. In Chapter 3, we have
developed the theory of quantum metrology for metrology with noisy Dicke
states. In Chapter 4, we have presented a method for witnessing the QFI with
expectation values of some general observables. Finally in Chapter 5, we have
computed precision bounds for gradient magnetometry.

In Chapters 3 and 4, we were constructing bounds on the quantum Fisher
information based on the expectation values of some observables of the initial
state. It turns out that to compute the quantum Fisher information is not a
trivial task and there is not a measurement scheme to obtain it from the
initial state apart from a complete tomography, which is very demanding for
large system sizes. Hence, some shortcuts to compute the bound of the QFI are
necessary.

In Chapter 3, we computed the precision bound for noisy unpolarized Dicke
states based on some initial expectation values. Moreover, we first reduced
the number of expectation values needed to four. More explicitly, we have to
measure only the second and the fourth moments of the $y$-component and the
$x$-component of the collective angular momentum in order to estimate the
metrological usefulness of the system. In practice, the fourth-order moments
can also be well approximated by the second-order moments.

In Chapter 4, we developed a method based on the Legendre transform. Based on
this method, we are able to obtain a tight lower bound on the quantum Fisher
information as a function of a set of expectation values of the initial state.
Furthermore, we tested our approach on extensive experimental data of photonic
and cold gas experiments, and demonstrated that it works even for the case of
thousands particles. In the future, it would be interesting to use our method
to test the optimality of various recent formulas giving a lower bound on the
quantum Fisher information [98, 121], as well as to improve the lower bounds
for spin-squeezed states and Dicke states allowing for the measurement of more
observables than the ones used in this publication.

On the other hand, in Chapter 5, we have investigated the precision limits for
measuring the gradient of a magnetic field with atomic ensembles arranged in
different geometries and initialized in different states. In particular, we
studied spin-chain configurations as well as the case of two atomic ensembles
localized at two different positions, and also the experimentally relevant
set-up of a single atomic ensemble with an arbitrary density profile of the
atoms was considered. We discussed the usefulness of various quantum states
for measuring the field strength and the gradient. Some quantum states, such
as singlet states, are insensitive to the homogeneous field. Using these
states, it is possible to estimate the gradient and saturate be Cramér-Rao
bound, while for states that are sensitive to the homogeneous magnetic field,
compatible measurements are needed for this task. For spin chains and the two-
ensemble case, the precision of the estimation of the gradient can reach the
Heisenberg limit. For the single ensemble case, only if strong correlation
between the particles is allowed can the shot-noise limit be surpassed and
even the Heisenberg limit be achieved. However, even if the Heisenberg limit
is not reached, single-ensemble methods can have a huge practical advantage
compared to methods based on two or more atomic ensembles, since using a
single ensemble makes the experiment simpler and can also result in a better
spatial resolution.

## Appendices

###  A Multiparticle angular momentum subspaces

As we mentioned in the Section 2.2.1, when dealing with many particle systems,
the Hilbert space is represented by tensor a product of the subspaces with a
fixed spin. So, the final dimension is the product of all single-particle
dimensions which lead to an exponentially large Hilbert space. In order to
simplify our calculations, it is worth to note that some interesting
structures arise from this kind of tensor product construction.

Let us name some basic assumptions with which the problem of adding angular
momentum subspaces can be simplified. First of all, the single-particle
Hilbert space must be discrete and finite, hence it can be represented by a