††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.

# Supplemental Material: Optimal Generators for Quantum Sensing

Jarrod T. Reilly  JILA, NIST, and Department of Physics, University of
Colorado, 440 UCB, Boulder, CO 80309, USA  John Drew Wilson  JILA, NIST, and
Department of Physics, University of Colorado, 440 UCB, Boulder, CO 80309, USA
Simon B. Jäger  Physics Department and Research Center OPTIMAS, Technische
Universität Kaiserslautern, D-67663, Kaiserslautern, Germany  Christopher
Wilson  Department of Physics, Cornell University, Ithaca, New York 14853, USA
Murray J. Holland  JILA, NIST, and Department of Physics, University of
Colorado, 440 UCB, Boulder, CO 80309, USA

###### pacs:

Valid PACS appear here

###### Contents

  1. I Operator Basis for Symmetric $\mathrm{SU}(n)$ Systems
  2. II Unitary Connection Between Operators
  3. III Example Realization and Additional Analysis of the $\mathrm{SU}(4)$ System
    1. III.1 Example Realization of the Hamiltonian
    2. III.2 Example Realization of the Basis Operators
  4. IV Geometric Formalism
    1. IV.1 The Lie Group
    2. IV.2 The Hilbert Space
    3. IV.3 The Fisher Information Metric

##  I Operator Basis for Symmetric $\mathrm{SU}(n)$ Systems

We now outline how to form the operator basis for general $\mathrm{SU}(n)$
systems. We want our basis operators to be linearly independent
$\Tr[\hat{G}_{\mu}\hat{G}_{\nu}]=\mathcal{C}\delta_{\mu\nu}$, traceless
$\Tr[\hat{G}_{\mu}]=0$, and all have the same norm $\mathcal{C}$ defined in
Eq. (S47). The norm condition ensures the basis operators have the same
spectrum $\sigma(\hat{G}_{\mu})=\sigma(\hat{G}_{\nu})$, so that they can be
connected unitarily without need for rescaling.

Since we are considering states which are symmeterized over the constituent
particles, we may utilize the Schwinger boson representation [1]. For $n$
states $\ket{a_{1}},\ket{a_{2}},\ket{a_{3}},\ldots,\ket{a_{n}}$, we define the
annihilation operators
$\hat{a}_{1},\hat{a}_{2},\hat{a}_{3},\ldots,\hat{a}_{n}$ which can then form
the collective operators

$\displaystyle\hat{\mathcal{A}}_{12}^{+}$
$\displaystyle=\hat{a}_{1}^{\dagger}\hat{a}_{2},\hat{\mathcal{A}}_{13}^{+}=\hat{a}_{1}^{\dagger}\hat{a}_{3},\ldots,\hat{\mathcal{A}}_{1n}^{+}=\hat{a}_{1}^{\dagger}\hat{a}_{n},$
(S1) $\displaystyle\hat{\mathcal{A}}_{23}^{+}$
$\displaystyle=\hat{a}_{2}^{\dagger}\hat{a}_{3},\hat{\mathcal{A}}_{24}^{+}=\hat{a}_{2}^{\dagger}\hat{a}_{4},\ldots,\hat{\mathcal{A}}_{(n-1)n}^{+}=\hat{a}_{n-1}^{\dagger}\hat{a}_{n}.$

Defining $\hat{\mathcal{A}}_{ij}^{-}=(\hat{\mathcal{A}}_{ij}^{+})^{\dagger}$,
we can find $\binom{n}{2}=n!/[2!(n-2)!]=n(n-1)/2$ closed $\mathfrak{su}(2)$
sub-algebras by the relations

$\hat{\mathcal{A}}_{ij}^{x}=\frac{\hat{\mathcal{A}}_{ij}^{+}+\hat{\mathcal{A}}_{ij}^{-}}{2},\;\hat{\mathcal{A}}_{ij}^{y}=\frac{\hat{\mathcal{A}}_{ij}^{+}-\hat{\mathcal{A}}_{ij}^{-}}{2i},\;\hat{\mathcal{A}}_{ij}^{z}=\frac{\left[\hat{\mathcal{A}}_{ij}^{+},\hat{\mathcal{A}}_{ij}^{-}\right]}{2}.$
(S2)

We have, by definition, that the $x$ and $y$ operators from the different sub-
algebras are linearly independent, leading to $2\binom{n}{2}=n(n-1)$ basis
operators. This leaves $n^{2}-1-2\binom{n}{2}=n-1$ basis operators to be made
up from the $\hat{\mathcal{A}}_{ij}^{z}$ operators. However, one can clearly
see that more care must be taken for the $\hat{\mathcal{A}}_{ij}^{z}$
operators when they are rewritten as

$\hat{\mathcal{A}}_{ij}^{z}=\frac{\hat{a}_{i}^{\dagger}\hat{a}_{i}-\hat{a}_{j}^{\dagger}\hat{a}_{j}}{2},$
(S3)

since the number operator $\hat{a}_{i}^{\dagger}\hat{a}_{i}$ appears in
multiple $z$ operators. Thus, these operators will not be linearly independent
in general, with
$\Tr[\hat{\mathcal{A}}_{ij}^{z}\hat{\mathcal{A}}_{ik}^{z}]\neq 0$ as an
example.

To choose the remaining $n-1$ basis operators, one can first select $n/2$
linearly independent $\hat{\mathcal{A}}_{ij}^{z}$ operators if $n$ is even or
$(n-1)/2$ operators if $n$ is odd:

$\displaystyle\hat{\mathcal{A}}_{12}^{z},\hat{\mathcal{A}}_{34}^{z},\ldots,\hat{\mathcal{A}}_{(n-1)n}^{z},$
$\displaystyle n\in 2\mathbb{N},$ (S4)
$\displaystyle\hat{\mathcal{A}}_{12}^{z},\hat{\mathcal{A}}_{34}^{z},\ldots,\hat{\mathcal{A}}_{(n-2)(n-1)}^{z},$
$\displaystyle n\in 2\mathbb{N}-1,$

where $\mathbb{N}$ represents the natural numbers. Then, one makes normalized
operators from alternating indices:

$\displaystyle\mathcal{N}_{2}\left[\hat{\mathcal{A}}_{13}^{z}+\hat{\mathcal{A}}_{24}^{z}\right],\mathcal{N}_{4}\left[\hat{\mathcal{A}}_{15}^{z}+\hat{\mathcal{A}}_{26}^{z}+\hat{\mathcal{A}}_{35}^{z}+\hat{\mathcal{A}}_{46}^{z}\right],\ldots,$
(S5)
$\displaystyle\mathcal{N}_{n-2}\left[\sum_{i=1}^{\frac{n}{2}-1}\hat{\mathcal{A}}_{(2i-1)(n-1)}^{z}+\sum_{j=1}^{\frac{n}{2}-1}\hat{\mathcal{A}}_{(2j)n}^{z}\right],\quad
n\in 2\mathbb{N},$
$\displaystyle\mathcal{N}_{2}\left[\hat{\mathcal{A}}_{13}^{z}+\hat{\mathcal{A}}_{24}^{z}\right],\mathcal{N}_{4}\left[\hat{\mathcal{A}}_{15}^{z}+\hat{\mathcal{A}}_{26}^{z}+\hat{\mathcal{A}}_{35}^{z}+\hat{\mathcal{A}}_{46}^{z}\right],\ldots,$
$\displaystyle\mathcal{N}_{n-3}\left[\sum_{i=1}^{\frac{n-3}{2}}\hat{\mathcal{A}}_{(2i-1)(n-2)}^{z}+\sum_{j=1}^{\frac{n-3}{2}}\hat{\mathcal{A}}_{(2j)(n-1)}^{z}\right],\quad
n\in 2\mathbb{N}-1,$

where the normalization for operator $\hat{O}$ is given by
$\mathcal{N}_{i}=\sqrt{\mathcal{C}/\Tr[\hat{O}^{2}]}=\sqrt{\Tr[(\hat{\mathcal{A}}_{12}^{z})^{2}]/\Tr[\hat{O}^{2}]}$.
This gives another $n/2-1$ basis operators for even $n$ and $(n-1)/2-1$ more
operators for odd $n$. The final linearly independent basis operator for odd
$n$ is then given by

$\mathcal{N}_{n-1}\left[\hat{\mathcal{A}}_{1n}^{z}+\hat{\mathcal{A}}_{2n}^{z}+\ldots+\hat{\mathcal{A}}_{(n-1)n}^{z}\right],\quad
n\in 2\mathbb{N}-1.$ (S6)

##  II Unitary Connection Between Operators

In this section, we outline a method to find the unitary rotation that takes
the optimal generator $\hat{\mathcal{G}}$ to one of physical interest
$\hat{\mathcal{Z}}$ in order to sense some specific parameter. In the case of
$\mathrm{SU}(2)$, one can always find the unitary connection between these
operators
$\hat{\mathcal{Z}}=\hat{U}_{c}^{\dagger}\hat{\mathcal{G}}\hat{U}_{c}$.
However, when larger group structures are considered, there is no such
guarantee. A unitary connection can be found if and only if the operators have
the same spectrum $\sigma(\hat{\mathcal{G}})=\sigma(\hat{\mathcal{Z}})$. In
this case, one can find the operator, $\hat{\mathcal{R}}$, that generates
$\hat{U}_{c}=\exp[-i\hat{\mathcal{R}}\pi/2]$ by solving a special case of
Sylvester’s equation [2]:

$i[\hat{\mathcal{R}},\hat{\mathcal{G}}]=\hat{\mathcal{Z}}.$ (S7)

A non-unitary connection can be found when
$\sigma(\hat{\mathcal{G}})\neq\sigma(\hat{\mathcal{Z}})$, and there may be
some cases where there is experimental relevancy to such solutions [3, 4, 5,
6, 7], but we do not pursue these connections here.

To solve Eq. (S7), one vectorizes $\hat{\mathcal{R}}$ and $\hat{\mathcal{Z}}$
via the mapping

$\hat{O}=\sum_{a,b}o_{a,b}\outerproduct{a}{b}\longleftrightarrow\lvert\hat{O}\rangle\rangle=\sum_{a,b}o_{a,b}\ket{a}\otimes\ket{b},$
(S8)

and then solves

$i\left[\hat{\mathbb{I}}\otimes\hat{\mathcal{G}}^{T}-\hat{\mathcal{G}}\otimes\hat{\mathbb{I}}\right]\lvert\hat{\mathcal{R}}\rangle\rangle\equiv
i\hat{\vphantom{\rule{1.0pt}{8.5pt}}\smash{\hat{\mathbb{G}}}}\lvert\hat{\mathcal{R}}\rangle\rangle=\lvert\hat{\mathcal{Z}}\rangle\rangle,$
(S9)

with identity operator $\hat{\mathbb{I}}$. The superoperator
$\hat{\vphantom{\rule{1.0pt}{8.5pt}}\smash{\hat{\mathbb{G}}}}$ can not be
inverted as
$\absolutevalue{\hat{\mathbb{I}}\otimes\hat{\mathcal{G}}^{T}}=\absolutevalue{\hat{\mathcal{G}}\otimes\hat{\mathbb{I}}}$.
However, there are techniques to solve this linear set of equations
numerically. For example, one can sometimes utilize the Moore-Penrose
pseudoinverse $\hat{A}^{+}$ [8, 9] which satisfies the conditions

$\displaystyle\hat{A}\hat{A}^{+}\hat{A}=\hat{A},$
$\displaystyle\quad\hat{A}^{+}\hat{A}\hat{A}^{+}=\hat{A}^{+}$ (S10)
$\displaystyle\left(\hat{A}\hat{A}^{+}\right)^{\dagger}=\hat{A}\hat{A}^{+},$
$\displaystyle\quad\left(\hat{A}^{+}\hat{A}\right)^{\dagger}=\hat{A}^{+}\hat{A}.$

Note that $\hat{A}^{+}=\hat{A}^{-1}$ when $\hat{A}$ is invertible. The
pseudoinverse exists and is unique for any matrix, but it does not always have
a simple algebraic form.

In the case that
$\hat{\vphantom{\rule{1.0pt}{8.5pt}}\smash{\hat{\mathbb{G}}}}$ has linearly
independent columns, the pseudoinverse becomes

$\hat{\vphantom{\rule{1.0pt}{8.5pt}}\smash{\hat{\mathbb{G}}}}^{+}=(\hat{\vphantom{\rule{1.0pt}{8.5pt}}\smash{\hat{\mathbb{G}}}}^{\dagger}\hat{\vphantom{\rule{1.0pt}{8.5pt}}\smash{\hat{\mathbb{G}}}})^{-1}\hat{\vphantom{\rule{1.0pt}{8.5pt}}\smash{\hat{\mathbb{G}}}}^{\dagger},$
(S11)

and corresponds to a left inverse
$\hat{\vphantom{\rule{1.0pt}{8.5pt}}\smash{\hat{\mathbb{G}}}}^{+}\hat{\vphantom{\rule{1.0pt}{8.5pt}}\smash{\hat{\mathbb{G}}}}=\hat{\mathbb{I}}\otimes\hat{\mathbb{I}}$,
such that

$\lvert\hat{\mathcal{R}}\rangle\rangle=-i\left(\hat{\vphantom{\rule{1.0pt}{8.5pt}}\smash{\hat{\mathbb{G}}}}^{\dagger}\hat{\vphantom{\rule{1.0pt}{8.5pt}}\smash{\hat{\mathbb{G}}}}\right)^{-1}\hat{\vphantom{\rule{1.0pt}{8.5pt}}\smash{\hat{\mathbb{G}}}}^{\dagger}\lvert\hat{\mathcal{Z}}\rangle\rangle.$
(S12)

One can then map this back into the original space via Eq. (S8) and then
retrieve the coefficients $\mathcal{R}^{\mu}$ that compose
$\hat{\mathcal{R}}=\mathcal{R}^{\mu}\hat{G}_{\mu}$ in order to determine the
physical implementation of $\hat{\mathcal{R}}$. In the more general case, the
pseudoinverse can be calculated using the single value decomposition of
$\hat{\vphantom{\rule{1.0pt}{8.5pt}}\smash{\hat{\mathbb{G}}}}$ [9].

##  III Example Realization and Additional Analysis of the $\mathrm{SU}(4)$
System

We now briefly describe an example realization of the $\mathrm{SU}(4)$
Hamiltonian from Eq. (10) of the Main Text. This theoretical model was first
presented in Ref. [10].

###  III.1 Example Realization of the Hamiltonian

We consider a packet of $N$ two-level atoms with uniform velocity that pass
through a dispersive cavity. The atoms can be prepared in the $\ket{\hbar
k/2}$ state utilizing Kapitza-Dirac pulses [11], and assuming $\hbar
Ng^{2}/(4\absolutevalue{\Delta})\ll(\hbar k)^{2}/m$, they will remain in the
momentum subspace spanned by $\ket{\pm\hbar k/2}$. Here, $k$ is the wavenumber
of the cavity, $g$ is the cavity coupling strength, $\Delta$ is the atom-
cavity detuning, and $m$ is the atomic mass. After the cavity is adiabatically
eliminated based on large detuning $\absolutevalue{\Delta}\gg\sqrt{N}g$, the
Hamiltonian is given by [10]

$\hat{H}=\hbar\chi\sum_{i,j=1}^{N}\hat{s}_{i}^{x}\hat{s}_{j}^{x}\hat{\sigma}_{i}^{+}\hat{\sigma}_{j}^{-},$
(S13)

where $\chi=g^{2}/(4\Delta)$, $\hat{\sigma}_{i}^{+}=\outerproduct{e}{g}_{i}$,
and $\hat{s}_{i}^{x}=(\outerproduct{\hbar k/2}{-\hbar
k/2}_{i}+\outerproduct{-\hbar k/2}{\hbar k/2}_{i})/2$. This Hamiltonian can
generate both particle-particle and spin-motion entanglement when the atoms
are prepared in the state
$\ket{\psi_{0}}=2^{-\frac{N}{2}}(\ket{g}+\ket{e})^{\otimes N}\otimes\ket{\hbar
k/2}^{\otimes N}$, where $\ket{g}$ and $\ket{e}$ are the atoms’ internal
states.

We note the system has an underlying $\mathrm{SU}(4)$ structure and so, in
analogy to Ref. [12], we label

$\displaystyle\ket{u}$ $\displaystyle\equiv\ket{e,\hbar
k/2},\quad\ket{d}\equiv\ket{g,-\hbar k/2},$ (S14) $\displaystyle\ket{s}$
$\displaystyle\equiv\ket{e,-\hbar k/2},\quad\ket{c}\equiv\ket{g,\hbar k/2}.$

If one begins in the symmetric subspace of $\mathrm{SU}(4)$,

$\ket{P_{\alpha,\beta,\gamma,\delta}}=\sqrt{\frac{\alpha!\beta!\gamma!\delta!}{N!}}\mathcal{S}(\ket{u}^{\otimes\alpha}\ket{d}^{\otimes\beta}\ket{s}^{\otimes\gamma}\ket{c}^{\otimes\delta}),$
(S15)

for $\alpha+\beta+\gamma+\delta=N$ and symmetrizer $\mathcal{S}$, then the
system will remain in this subspace. The subspace scales as
$(N+1)(N+2)(N+3)/6\sim\mathcal{O}(N^{3})$ making exact numerical calculations
feasible. Using the associated Schwinger boson representation, Eq. (S13) can
be rewritten as

$\hat{H}=\hbar\chi(\hat{u}^{\dagger}\hat{d}+\hat{s}^{\dagger}\hat{c})(\hat{u}\hat{d}^{\dagger}+\hat{s}\hat{c}^{\dagger}).$
(S16)

We can now introduce three subgroups $\mathfrak{J}$, $\mathfrak{K}$, and
$\mathfrak{E}$, with corresponding algebras $\\{\hat{J}^{\pm},\hat{J}_{z}\\}$,
$\\{\hat{K}^{\pm},\hat{K}_{z}\\}$, and $\\{\hat{E}^{\pm},\hat{J}_{z}\\}$,
respectively. These subgroups represent the spin and momentum dipoles and
entanglement generating processes between the two subsystems. Here, we have
defined the collective operators

$\displaystyle\hat{J}^{\pm}$
$\displaystyle=\frac{1}{\sqrt{2}}\sum_{j=1}^{N}\hat{\sigma}_{j}^{\pm},\quad\hat{J}_{z}=\frac{1}{2\sqrt{2}}\sum_{j=1}^{N}\hat{\sigma}_{j}^{z},$
(S17) $\displaystyle\hat{K}^{\pm}$
$\displaystyle=\frac{1}{\sqrt{2}}\sum_{j=1}^{N}\hat{s}_{j}^{\pm},\quad\hat{K}_{z}=\frac{1}{2\sqrt{2}}\sum_{j=1}^{N}\hat{s}_{j}^{z},$
$\displaystyle\hat{E}^{\pm}$
$\displaystyle=\frac{1}{\sqrt{2}}\sum_{j=1}^{N}\hat{\sigma}_{j}^{\pm}\hat{s}_{j}^{x},$

and noted $\hat{E}_{z}=[\hat{E}^{+},\hat{E}^{-}]/2=\hat{J}_{z}$. Writing
$\hat{E}^{+}=(\hat{u}^{\dagger}\hat{d}+\hat{s}^{\dagger}\hat{c})/\sqrt{2}$,
the Hamiltonian becomes

$\hat{H}=2\hbar\chi\hat{E}^{+}\hat{E}^{-}=2\hbar\chi\left(\hat{E}^{2}-\hat{J}_{z}^{2}+\hat{J}_{z}\right),$
(S18)

where $\hat{E}^{2}=\hat{E}_{x}^{2}+\hat{E}_{y}^{2}+\hat{J}_{z}^{2}$. Notice
that the factor of $2$ difference with respect to the Hamiltonian in Ref. [10]
is because of the $\sqrt{2}$ in Eq. (S17). This factor is included so that the
normalization conditions $\sum_{\mu}(\mathcal{O}^{\mu})^{2}=1$ and Eq. (S47)
are met with respect to the operator basis defined in Eq. (9) of the Main
Text.

###  III.2 Example Realization of the Basis Operators

Once one diagonalizes the QFIM with respect to the operator basis from Eq. (9)
of the Main Text, one may wish to rotate the state to make certain operators
the optimal generators for single or multiparameter estimation. We therefore
discuss how one could realize such rotations in the experimental setup
discussed in the previous subsection. We assume the atomic beam passes through
the cavity to become entangled, and then one performs metrology once the atoms
leave the cavity. The atoms will spatially separate into two packets due to
their opposite momentum with respect to the cavity axis, which we designate as
the $x$ axis, in a similar fashion to recent network node interferometry
experiments [13, 14].

We begin with rotations on the $\mathcal{M}$ and $\mathcal{N}$ subalgebras,
which have the raising operators
$\hat{\mathcal{M}}^{+}=\hat{u}^{\dagger}\hat{c}$ and
$\hat{\mathcal{N}}^{+}=\hat{s}^{\dagger}\hat{d}$. Once the two packets are
spatially separated, one can address them individually. Therefore, these
rotations can be achieved by addressing the $\hbar k/2$ packet for
$\mathcal{M}$ and the $-\hbar k/2$ packet for $\mathcal{N}$. The lasers that
perform these rotations should be oriented in an orthogonal direction from the
$x$ axis, which we designate as the $z$ axis. This ensures that these lasers
can perform rotations on the internal states without changing the atoms’
$x$-momentum.

We next discuss $\mathcal{Q}$ and $\Sigma$ rotations, generated by the raising
operators $\hat{\mathcal{Q}}^{+}=\hat{u}^{\dagger}\hat{d}$ and
$\hat{\Sigma}^{+}=\hat{s}^{\dagger}\hat{c}$. These can be accomplished by
laser pulses oriented in the $x$-direction. Pulses in the $+x$-direction will
cause $\mathcal{Q}$ rotations while pulses in the $-x$-directions drives
$\Sigma$ rotations. One must take care that the laser Rabi frequency $\Omega$
for these pulses is small enough that one does not drive excitations to higher
motional states, $\absolutevalue{\Omega}\ll\hbar k^{2}/m$.

Lastly, we turn to $\mathcal{U}$ and $\mathcal{V}$ rotations, generated by the
raising operators $\hat{\mathcal{U}}^{+}=\hat{u}^{\dagger}\hat{s}$, and
$\hat{\mathcal{V}}^{+}=\hat{c}^{\dagger}\hat{d}$. These rotations can be
accomplished using Bragg pulses. Consider an auxiliary internal excited state
$\ket{f}$ from which one can independently drive the
$\ket{e}\leftrightarrow\ket{f}$ and $\ket{g}\leftrightarrow\ket{f}$
transitions using different laser wavelengths for $\mathcal{U}$ and
$\mathcal{V}$ rotations, respectively. One orients a laser with Rabi frequency
$\Omega_{1}$ in the $z$-direction and a laser with Rabi frequency $\Omega_{2}$
oriented in the $x$-$y$ plane. For the second laser to address transitions
distinguishable from the $\ket{g}\leftrightarrow\ket{e}$ transition while
still achieving the desired momentum impulses $\pm\hbar k$, the $x$-component
of this laser’s wave number must satisfy $\vec{k}_{2}\cdot\hat{x}\approx k$.
Assuming both lasers have the same wavelength that addresses the
$\ket{i}\leftrightarrow\ket{f}$ transition for $i\in\\{g,e\\}$, the
Hamiltonian in the rotating frame of the lasers is given by [15]

$\displaystyle\hat{H}_{i}=$
$\displaystyle\sum_{j=1}^{N}-\hbar\Delta_{i}\outerproduct{f}{f}_{j}+\frac{\hbar}{2}\left(\Omega_{1}\outerproduct{i}{f}_{j}+\mathrm{H.c.}\right)$
(S19)
$\displaystyle+\frac{\hbar}{2}\left(\Omega_{2}\hat{s}_{j}^{+}\outerproduct{i}{f}_{j}+\mathrm{H.c.}\right),$

where $\Delta_{i}$ is the detuning of the lasers from the respective
transition’s frequency and $\hat{s}_{j}^{+}=\outerproduct{\hbar k/2}{-\hbar
k/2}_{j}=(\hat{s}_{j}^{-})^{\dagger}$ is the momentum shift operator. This
assumes that $\absolutevalue{\Omega_{2}}\ll\hbar k^{2}/m$ such that the atoms
remain in the $p_{x}=\pm\hbar k/2$ manifold. We take
$\absolutevalue{\Delta_{i}}\gg\absolutevalue{\Omega_{1}},\absolutevalue{\Omega_{2}}$
such that we can eliminate the auxiliary state to arrive at [15]

$\displaystyle\tilde{\hat{H}}_{i}=$
$\displaystyle\sum_{j=1}^{N}\frac{\hbar}{4\Delta_{i}}\left(\absolutevalue{\Omega_{1}}^{2}+\absolutevalue{\Omega_{2}}^{2}\hat{s}_{j}^{+}\hat{s}_{j}^{-}\right)\outerproduct{i}{i}_{j}$
(S20)
$\displaystyle+\frac{\hbar}{4\Delta_{i}}\left(\Omega_{1}^{*}\Omega_{2}\hat{s}_{j}^{+}+\Omega_{1}\Omega_{2}^{*}\hat{s}_{j}^{-}\right)\outerproduct{i}{i}_{j}.$

The first line represents ac Stark shifts while the second line is the desired
momentum exchange interaction. Since the effective identity operator is now
$\hat{\mathbb{I}}_{i}=\sum_{j}\left(\hat{s}_{j}^{+}\hat{s}_{j}^{-}+\hat{s}_{j}^{-}\hat{s}_{j}^{+}\right)\outerproduct{i}{i}_{j}$,
we can shift all energies by
$\hbar(\absolutevalue{\Omega_{1}}^{2}-\absolutevalue{\Omega_{2}}^{2}/2)/(4\Delta_{i})$
to arrive at

$\tilde{\hat{H}}_{i}=\frac{\hbar}{4\Delta_{i}}\sum_{j=1}^{N}\left[\frac{\absolutevalue{\Omega_{2}}^{2}}{2}\hat{s}_{j}^{z}+\left(\Omega_{1}^{*}\Omega_{2}\hat{s}_{j}^{+}+\Omega_{1}\Omega_{2}^{*}\hat{s}_{j}^{-}\right)\right]\outerproduct{i}{i}_{j},$
(S21)

where we have defined
$\hat{s}_{j}^{z}=\hat{s}_{j}^{+}\hat{s}_{j}^{-}-\hat{s}_{j}^{-}\hat{s}_{j}^{+}$.
We thus have a Hamiltonian that only drives collective momentum interactions
for a desired internal level, allowing one to perform the potentially needed
$\mathcal{U}$ and $\mathcal{V}$ rotations.

##  IV Geometric Formalism

In this section, we provide an introduction to quantum state geometry. Quantum
mechanics is often phrased algebraically, but there are important formulations
in terms of Riemannian geometry [16, 17]. Within these formulations, the
coefficients of state vectors provide coordinates for a manifold, whereupon
the vector space structure of the Hilbert space is used to derive notions of
curvature and distance. In finite dimensional quantum systems with $d$
discrete states, there is an implicit map from the Hilbert space $\mathcal{H}$
to the real numbers $\mathbb{R}^{2d}$ as a differential manifold. When
considering normalized states with no global phase, $\mathcal{H}$ then maps to
to $\mathbb{C}\text{P}^{d}\ $ as a manifold. Here, $\mathbb{C}\text{P}^{d}\ $
is the complex projective space of $d$-dimensional c-numbers with unit norm
and no global phase, i.e. $\mathbb{C}\text{P}^{d}\ \subset\mathbb{R}^{2d}$ is
the space of rays representing physical quantum states. The vector space
structure of $\mathcal{H}$ provides the inner product between two states which
may then be used to define tensors on $\mathcal{H}$ as a manifold. From this,
one may use the coefficients of basis states $\ket{e_{\mu}}$ to expand any
state as $\ket{\psi(\textbf{z})}=z^{\mu}\ket{e_{\mu}}$, with
$\textbf{z}\in\mathbb{C}\text{P}^{d}\ $ or $\mathbb{C}^{d}$ depending on
normalization and global phase. Now, the inner product provides a notion of
differential path length

$\displaystyle\quad f^{2}=$
$\displaystyle|\innerproduct{\psi(\textbf{z})}{\psi(\textbf{z}+d\textbf{z})}|^{2}$
(S22) $\displaystyle d\zeta^{2}\equiv$ $\displaystyle 1-f^{2}=Q_{\mu\nu}\
dz^{\mu}dz^{\nu}$ $\displaystyle Q_{\mu\nu}\equiv$
$\displaystyle\bra{\partial_{\mu}\psi(\textbf{z})}\left(\hat{\mathbb{I}}-\outerproduct{\psi(\textbf{z})}{\psi(\textbf{z})}\right)\ket{\partial_{\nu}\psi(\textbf{z})},$

where $Q_{\mu\nu}$ is the quantum geometric tensor [18, 19]. The real part of
$Q_{\mu\nu}$ gives the canonical Fubini-Study metric via
$ds^{2}\propto\real(d\zeta^{2})$, while the imaginary part provides an exact
symplectic form corresponding to the Berry curvature [19].

While these formulations are powerful in their generality, dynamics are
rephrased from algebraic equations of motion into $2d$ many differential
equations. Our formulation instead inherits the coordinates for $\mathcal{H}$
from a Lie group with unitary representation over $\mathcal{H}$. This
inheritance is equivalent to the construction using a $C^{*}$ algebra [16],
but here we reformulate much of the construction to instead naturally fit the
language of quantum metrology. Now, the group coordinates which define unique
states provide the coordinates on the the quantum state manifold, and locally
return a sub-manifold of $\mathbb{R}^{2d}$. However, nearly all important
calculations can be done on the group level. In other words, the underlying
manifold structure is inhereted through a chosen representation of the Lie
group, where the group-manifold provides the underlying geometry to the
Hilbert space. Furthermore, for the case that one chooses the unitary group
for which $\mathcal{H}$ is the defining representation, e.g. a two dimensional
Hilbert space and $\mathfrak{G}=\mathrm{SU}(2)$, these two approaches become
mechanically identical.

We now outline the mathematical objects we use with the goal of providing
intuition for why they’re useful and how they’re connected. We explore the
case of a finite $d$-dimensional Hilbert space, $\mathcal{H}$, and a Lie
group, $\mathfrak{G}$, that describes quantum evolution on this Hilbert space.
The Hilbert space $\mathcal{H}$ provides a representation space for
$\mathfrak{G}$, giving a unitary representation
$\Pi:\mathfrak{G}\rightarrow\text{U}(\mathcal{H})$. Now, we identify an
initial state $\ket{\Psi}$, whereupon one can act on the state using
$g\in\mathfrak{G}$ through the representation $\Pi[g]$. In the language of
quantum metrology, $\ket{\Psi}$ is referred to as the probe state or the
fiducial state [16]. Since $\mathfrak{G}$ is a manifold, each group element
$g$ may be identified with coordinates
$\textbf{x}\in\mathbb{R}^{\text{dim}(\mathfrak{G})}$, such that
$g(\textbf{x})\in\mathfrak{G}$. Through the representation of the group
action, these coordinates may be mapped into $\mathcal{H}$ via
$\ket{\psi(\textbf{x})}\equiv\Pi[g(\textbf{x})]\ket{\Psi}$. Now, each state is
parameterized by the coordinates of its associated group element. The set of
all states realizable in this way forms our manifold
$\mathcal{M}_{\Psi}\subseteq\mathcal{H}$, as defined explicitly in Sec. IV.2.
These coordinate dependencies, and the maps from one manifold to another are
shown in Fig. 1, with the map $\iota_{\Psi}(g)=\Pi[g]\ket{\Psi}$ being the
explicit map from the group to the Hilbert space.

Figure 1:  The commutative diagram of maps between each manifold.
$\mathrm{Proj}$ stands for projecting onto the manifold.

By doing measurements on states reached from group evolution on the probe
state, one may construct probability distributions over $\mathcal{M}_{\Psi}$.
The precision for which one may infer the parameters x with a given
measurement basis is diagnosed by the Classical Fisher Information Matrix
(CFIM). If one further optimizes over all possible measurement bases, we
recover the Quantum Fisher Information Matrix (QFIM) $\bm{\mathcal{F}}$.

Intuitively, the CFIM diagnoses how well one may distinguish probability
distributions differentially far away from one another in parameter space. The
CFIM may have a non-trivial kernel, in which case measurements on
$\mathcal{H}$ don’t resolve a corresponding coordinate change in
$\mathfrak{G}$. This is the case when one has phases which may not be measured
by in the chosen measurement basis, or to a global phase which may never be
measured. By removing the kernel of the CFIM, one recovers a well-defined
metric on these probability distributions over $\mathcal{M}_{\Psi}$.

The QFIM, similarly, defines how well one could optimally distinguish between
two states which are differentially far away in parameter space, and thus may
be made into to a Riemannian metric on these states. Similar to the CFIM, some
parameters may correspond to the kernel of the QFIM, such as when a Lie group
element $g$ acts on a state which is an eigenvector of the corresponding Lie
algebra member: $g(\textbf{x})\ket{\psi}=e^{-i\theta(\textbf{x})}\ket{\psi}$
for global phase $\theta(\textbf{x})$. Once again the removal of these
elements returns a well-defined metric which now lives on the manifold
$\mathcal{M}_{\Psi}$.

The group coordinates which correspond to unique states on
$\mathcal{M}_{\Psi}$ become coordinates on the quantum state manifold. For two
coordinates $x^{\mu}$ and $x^{\nu}$ of
$\textbf{x}=(x^{0},...,x^{\mu},...,x^{\nu},...)$, this optimization yields

$\mathcal{F}_{\mu\nu}=\frac{1}{2}\Tr[\hat{\rho}\\{\hat{L}_{\mu},\hat{L}_{\nu}\\}],\quad\partial_{\mu}\hat{\rho}=\frac{1}{2}\left(\hat{\rho}\hat{L}_{\mu}+\hat{L}_{\mu}\hat{\rho}\right),$
(S23)

where $\\{\hat{A},\hat{B}\\}=\hat{A}\hat{B}+\hat{B}\hat{A}$ is the anti-
commutator and $\hat{L}_{\mu}$ is the Symmetric Logarithmic Derivative (SLD)
[20] with respect to the coordinate $x^{\mu}$. Geometrically,
$\mathcal{F}_{\mu\nu}$ provides a metric tensor over the space of density
matrices,

$ds^{2}=\mathcal{F}_{\mu\nu}dx^{\mu}dx^{\nu},$ (S24)

and is equivalent to four times the Bures metric for mixed states. For pure
states, one finds that $\hat{L}_{\mu}=2\partial_{\mu}\hat{\rho}$, and so [21]

$\mathcal{F}_{\mu\nu}=4\real(Q_{\mu\nu}),$ (S25)

matching the Fubini-Study metric (up to rescaling).

###  IV.1 The Lie Group

We now outline the relevant Lie group structures. In general, any quantum
evolution may be described by the appropriate Lie group represented unitarily
on the Hilbert space. Let $\mathfrak{G}$ be some Lie group with
$g\in\mathfrak{G}$ being some group element. For applications to quantum
metrology, it suffices to only consider simply connected Lie groups. We may
choose coordinates on $\mathfrak{G}$ by specifying a basis for the Lie Algebra
of $\mathfrak{G}$ given by the vector space $\mathfrak{g}$. We choose basis
vectors $\\{A_{\mu}\\}\subset\mathfrak{g}$ such that
$\text{span}(\\{A_{\mu}\\})=\mathfrak{g}$. For concreteness, if
$\mathfrak{G}=\text{SU}(n)$ then one could construct, for example, the basis
operators $A_{\mu}$ according to Sec. I. Now, any group element
$g\in\mathfrak{G}$ can be identified via

$g=\exp(x^{\mu}A_{\mu}),$ (S26)

with real numbers $x^{\mu}$. From this, it is clear that any change of basis
on $\mathfrak{g}$ has a corresponding change of coordinates. For example, if
$\\{B_{\nu}\\}\subset\mathfrak{g}$ with
$\text{span}(\\{B_{\nu}\\})=\text{span}(\\{A_{\mu}\\})$ we can find
$\Lambda^{\mu}_{\nu}$ such that $B_{\nu}=\Lambda^{\mu}_{\nu}A_{\mu}$ implies
$x^{\mu}=\Lambda^{\mu}_{\nu}y^{\nu}$, and thus the group element $g$ stays
coordinate independent:

$g=\exp(x^{\mu}A_{\mu})=\exp(y^{\nu}B_{\nu}).$ (S27)

Lastly, we point out that the Lie Group $\mathfrak{G}$ carries with it a
canonical metric in the form of the Killing form, defined on the Lie algebra
as:

$\kappa:\mathfrak{g}\times\mathfrak{g}\rightarrow\mathbb{R},\quad\kappa(A_{\mu},A_{\nu})\equiv\Tr_{\mathfrak{g}}[\mathrm{ad}_{A_{\mu}}\circ\mathrm{ad}_{A_{\nu}}],$
(S28)

for arbitrary basis vectors $A_{\mu},A_{\nu}\in\mathfrak{g}$ and with the
trace taken over the Lie algebra as a vector space. The function
$\mathrm{ad}_{A_{\mu}}$ is the adjoint action of $A_{\mu}$ on the Lie algebra,
which is the usual commutator
$\mathrm{ad}_{A_{\mu}}(A_{\nu})\equiv[A_{\mu},A_{\nu}]$. The general
properties of $\kappa$ are described in Ref. [22], but notably for
$\mathrm{SU}(n)$, one has $\kappa(A_{\mu},A_{\nu})\equiv
2n\Tr[A_{\mu}A_{\nu}]$ with the trace being taken over the vector space of the
defining representation. Thus, $\kappa$ provides a metric on $\mathfrak{G}$
near the identity via

$(ds^{2})_{\mathfrak{G}}\equiv\kappa_{\mu\nu}dx^{\mu}dx^{\nu},$ (S29)

with $\kappa_{\mu\nu}=\kappa(A_{\mu},A_{\nu})$. For
$\mathfrak{G}=\mathrm{SU}(n)$, one has

$(ds^{2})_{\mathrm{SU}(n)}\equiv 2n\Tr[A_{\mu}A_{\nu}]dx^{\mu}dx^{\nu}.$ (S30)

###  IV.2 The Hilbert Space

Now, we show how the structure of the Lie group is inherited to the Hilbert
space through a unitary representation. We consider a Hilbert space
$\mathcal{H}$ of dimension $|\mathcal{H}|=d$. Since $\mathcal{H}$ is a vector
space, we may choose representation map,

$\Pi:\mathfrak{G}\rightarrow\text{U}(\mathcal{H}),$ (S31)

which implicitly carries with it the representation of $\mathfrak{g}$:
$R:\mathfrak{g}\rightarrow\mathfrak{u}(\mathcal{H})$. The representation $R$
acts on our basis vectors $A_{\mu}$ according to

$R(A_{\mu})=-i\hat{G}_{\mu},$ (S32)

where $\hat{G}_{\mu}$ is a Hermitian operator on $\mathcal{H}$. Here, we note
that this factor of $-i$ accounts for the difference in convention between
physics and mathematics and is representation independent, but the relation
from $A_{\mu}$ to $\hat{G}_{\mu}$ is notably representation dependent. The
exact relationship between each of these representation maps is

$\displaystyle\Pi[g]=$
$\displaystyle\Pi[\exp(x^{\mu}A_{\mu})]=e^{x^{\mu}R(A_{\mu})}=e^{-ix^{\mu}\hat{G}_{\mu}}.$
(S33)

The choice of basis vectors on $\mathfrak{g}$ is thus equivalent to a choice
of basis for Hermitian matrices on $\mathcal{H}$.

To parameterize $\mathcal{H}$, we first choose a starting state,
$\ket{\Psi}\in\mathcal{H}$, and define the map

$\displaystyle\iota:\mathcal{H}\times\mathfrak{G}\rightarrow\mathcal{H},$
(S34) where
$\displaystyle\iota_{\Psi}:\mathfrak{G}\rightarrow\mathcal{H}\quad\text{and}\quad\iota_{g}:\mathcal{H}\rightarrow\mathcal{H}.$

Now, $\iota_{\Psi}$ is a map between manifolds from which we can push forward
vectors or pull back differentials, while $\iota_{g}$ is a map from
$\mathcal{H}$ to itself, which allows relevant objects to be pushed forward
and pulled back to $\Psi$. This definition matches the definition of
$\iota_{Q}$ in Ref. [16, Eq. (2.7)]. The map $\iota_{\Psi}$ takes $g$ as an
argument and is defined as the action of $\Pi[g]$ on
$\ket{\Psi}\in\mathcal{H}$:

$\iota_{\Psi}(g)\equiv\Pi(g)\ket{\Psi}=e^{-ix^{\mu}\hat{G}_{\mu}}\ket{\Psi}.$
(S35)

Meanwhile, $\iota_{g}$ takes any arbitrary state $\psi$ as an argument and is
similarly defined as the action of $\Pi[g]$ on $\ket{\psi}$:

$\iota_{g}(\psi)\equiv\Pi(g)\ket{\psi}.$ (S36)

Clearly, $\iota(\psi,g)=\iota_{\psi}(g)=\iota_{g}(\psi)$, but both
$\iota_{\psi}(g)$ and $\iota_{g}(\psi)$ are needed for properly defined push
forward and pull backs of tensors.

This map now allows us to define the manifold on which one wishes to carry out
quantum metrology,

$\mathcal{M}_{\Psi}=\\{\ket{\psi_{g}}\
\text{s.t.}\ket{\psi_{g}}=\iota_{\Psi}(g)\\}.$ (S37)

This manifold $\mathcal{M}_{\Psi}\subseteq\mathcal{H}$ is the group orbits of
$\ket{\Psi}$ under all possible group actions in $\mathfrak{G}$; i.e., it is
the set of states that may be reached under group actions when $\ket{\Psi}$ is
the probe state.

Within $\mathfrak{G}$, there is a subgroup which will act trivially on any
given state by only adding a global phase. Following Ref. [18], the Lie group
can be divided into this subgroup and the rest of it’s actions. The subgroup
which acts trivially is the stabilizer group of the state, and this
stabilizing group is generated by a sub-algebra for which the has the quantum
state is an eigenvector. The group elements which induce non-trivial evolution
are generated by the remaining members of $\mathfrak{g}$. By removing the
members of the sub-algebra generating the stabilizing group, and their
corresponding coordinates, one is left with well defined coordinates on
$\mathcal{M}_{\Psi}$. Without this, the group coordinates only parameterize
the states and do not become well defined coordinates.

Now, by considering one parameter paths through $\mathfrak{G}$ we can use
$\iota_{\Psi}$ to identify paths in $\mathcal{M}_{\Psi}$. Let

$\gamma:\mathbb{R}\rightarrow\mathfrak{G},\quad\gamma(t)=\exp(x^{\mu}(t)A_{\mu}),$
(S38)

so that the composition of $\iota_{\Psi}$ with $\gamma$ yields
$\iota_{\Psi}\circ\gamma:\mathbb{R}\rightarrow\mathcal{M}_{\Psi}$. Explicitly,

$(\iota_{\Psi}\circ\gamma)(t)=\ket{\psi(t)}\equiv
e^{-ix^{\mu}(t)\hat{G}_{\mu}}\ket{\Psi}$ (S39)

and if our path has $\iota_{\Psi}\circ\gamma(0)=\ket{\Psi}$, then

$\displaystyle\frac{\partial}{\partial t}\ket{\psi(t)}$
$\displaystyle=\frac{\partial x^{\mu}(t)}{\partial
t}\partial_{\mu}\ket{\psi(t)}$ (S40) $\displaystyle=\frac{\partial
x^{\mu}(t)}{\partial t}(-i\hat{G}_{\mu})\ket{\psi(t)}.$

Now, derivatives of $\iota_{\Psi}\circ\gamma$ for any possible path, $\gamma$,
defines all possible tangent vectors to the point $\Psi$. To find tangent
vectors at any other point in $\mathcal{M}_{\Psi}$, one may use $\iota_{g}$ to
push forward the tangent vectors at $\ket{\Psi}$. In the case that
$x^{\mu}=\omega^{\mu}t$ where $\hbar\omega^{\mu}$ is energies corresponding to
the operator $\hat{G}_{\mu}$ in the Hamiltonian, then $\frac{\partial
x^{\mu}}{\partial t}=\omega^{\mu}$ and Eq. (S40) would correspond to the
familiar case of a time-independent Hamiltonian.

Furthermore, from the equation

$\frac{\partial x^{\mu}}{\partial
t}\partial_{\mu}e^{-ix^{\mu}(t)\hat{G}_{\mu}}\ket{\Psi}=\frac{\partial
x^{\mu}}{\partial
t}\left(-i\hat{G}_{\mu}\right)e^{-ix^{\mu}(t)\hat{G}_{\mu}}\ket{\Psi},$ (S41)

we can match terms and use $\gamma(t)$ to define paths through any point,
who’s derivatives define a basis for tangent vectors at any point. This yields
a basis for vectors at each point on $\mathcal{M}_{\Psi}$, and vector fields
are now given by the standard identification

$\partial_{\mu}\equiv-i\hat{G}_{\mu}.$ (S42)

These tangent vector fields could have equivalently been found via push
forward. Any vector $\vec{V}=V^{\mu}\partial_{\mu}\in\mathfrak{g}$ under
$(\iota_{\Psi})_{*}$:

$(\iota_{\Psi})_{*}[\vec{V}]=V^{\mu}(\iota_{\Psi})_{*}[\partial_{\mu}]=-iV^{\mu}\hat{G}_{\mu}\ket{\Psi},$
(S43)

where $(\iota_{\Psi})_{*}[\vec{V}]\in T_{\Psi}\mathcal{M}_{\Psi}$, with
$T_{\Psi}\mathcal{M}_{\Psi}$ being the tangent vector space to the point
$\Psi$ on the manifold $\mathcal{M}_{\Psi}$. We can pushfoward the tangent
vector fields $T\mathfrak{G}$ to tangent vector fields on the state manifold
$T\mathcal{M}_{\Psi}$, as shown in Fig. 1.

###  IV.3 The Fisher Information Metric

Lastly, we show the interplay of the Lie group’s geometry and the Fisher
Information Metric. The QFIM has been shown to be a unique Riemannian metric
on the space of pure quantum states and a well defined metric on the space of
mixed states. We will use pure states to show the pull back of the metric to
the Lie group, but thus far all the maps defined would work suitably well for
unitary evolution of an initially mixed state. The case of evolution with
decoherent dynamics is left for future work.

We now derive the equations in the Main Text and show that the normalization
of vectors on $\mathcal{H}$ is a naturally determined via the Killing Form on
$\mathfrak{G}$. Specifically, we use the fact that a metric serves as the
tensorial promotion of the inner product,

$\displaystyle\langle\vec{V},\vec{W}\rangle_{\mathbf{x}}$
$\displaystyle=\langle\partial_{\mu},\partial_{\nu}\rangle_{\mathbf{x}}V^{\mu}W^{\nu}$
(S44) $\displaystyle=\mathcal{F}_{\mu\nu}V^{\mu}W^{\nu},$

where now, for pure states, $\mathcal{F}_{\mu\nu}$ is given by

$\displaystyle\mathcal{F}_{\mu\nu}$
$\displaystyle=4\,\mathrm{cov}_{\psi_{g}}(\hat{G}_{\mu},\hat{G}_{\nu})$ (S45)
$\displaystyle=2\langle\\{\hat{G}_{\mu},\hat{G}_{\nu}\\}\rangle_{\psi_{g}}-4\langle\hat{G}_{\mu}\rangle_{\psi_{g}}\langle\hat{G}_{\nu}\rangle_{\psi_{g}}.$

This formula may be equivalently derived from the fact that derivatives are
pushed forward from $\mathfrak{G}$, or equivalently, the Lie Algebra is
represented as operators on $\mathcal{H}$.

The tensor $\mathcal{F}_{\mu\nu}$ does not yet behave as a matrix, and
therefore one needs to raise an index. Formally, one may use the pull back
$(\iota_{\Psi})^{*}$ to map $\mathcal{F}_{\mu\nu}$ to a tensor on
$\mathfrak{G}$, as shown schematically for covector spaces in Fig. 1(c). By
pulling back the metric from $\mathcal{M}_{\Psi}$ to $\mathfrak{G}$ as a
tensor using $(\iota_{\Psi})^{*}$, one can use the metric defined by
$\kappa_{\mu\nu}$ to raise one index on $\mathcal{F}_{\mu\nu}$. Using
$(\kappa^{-1})^{\alpha\beta}$, which is defined implicitly by
$(\kappa^{-1})^{\mu\alpha}\kappa_{\alpha\nu}=\delta^{\mu}_{\nu}$, one finds

$\mathcal{F}^{\mu}_{\nu}=(\kappa^{-1})^{\mu\alpha}\
(\iota_{\Psi})^{*}[\mathcal{F}_{\alpha\nu}].$ (S46)

This is a (1,1) tensor which now has well defined eigenvectors and
eigenvalues. Diagonalizing this tensor is equivalent to diagonalizing the
QFIM, $\bm{\mathcal{F}}$, but now with proper dependence on the inner product
on $\mathfrak{g}$. This imposes the normalization properties used in Sec. I.

Since $\mathcal{F}_{\mu\nu}$ defines a metric over pure states in
$\mathcal{H}$, one has that the vector whose inner product is maximized under
this inner product is naturally the direction from $\mathfrak{G}$ which the
state will evolve the fastest. By diagonalizing the tensor
$\mathcal{F}^{\mu}_{\nu}$ on $\mathfrak{G}$, one finds the vector whose inner
product is maximized on $\mathcal{H}$ by equivalently finding the largest
eigenvalue of $\bm{\mathcal{F}}$ over the entire Lie Algebra. Here, the basis
of the Lie Algebra must be orthonormal with respect to the inner product
derived from $\kappa$; otherwise, scaling in the QFI could be due to an
arbitrary rescaling of operators. Using the construction given in Sec. I, one
finds that this normalization for $\mathrm{SU}(n)$ is given by

$\Tr[\hat{G}_{\mu}\hat{G}_{\nu}]=\delta_{\mu\nu}\frac{1}{(n+1)!}\prod_{j=1}^{n}(N+j),$
(S47)

which is equivalent to requiring that the basis elements of the Lie Algebra
are each members of an $N+1$ dimensional representation of
$\mathfrak{su}(2)\subset\mathfrak{su}(n)$. Physically, this normalization
guaruntees that subsystems of the Hilbert space which form qubits, or which
represent symmetric collections of qubits, accumulate relative phase at unit
rates. This fixes the standard quantum limit scaling for unentangled
generalized coherent spin states at $N$, and the Heisenberg limit scaling for
GHZ-like states at $N^{2}$. This normalization additionally guarantees that
the QFI may act as an entanglement witness, with the sufficient condition
being that any eigenvalue of $\bm{\mathcal{F}}$, $\lambda_{i}$, scales as
$\lambda_{i}>N$.

In summary, vectors come from $\mathfrak{G}$ where they are plugged into a
state-dependent inner product as vectors on $\mathcal{H}$ via the push
forward. These vectors define all the directions in which a state may evolve.
Many of these directions will be null-directions in $\mathcal{M}_{\Psi}$ and
may therefore be ignored, leaving only well defined coordinate directions on
$\mathcal{M}_{\Psi}$. Of the vectors which are non-zero, many will be non-
orthogonal under $\mathcal{F}_{\mu\nu}$’s inner product Eq. (S44). This means
that those directions of evolution are partially degenerate according to how
they evolve the state. By pulling back $\mathcal{F}_{\mu\nu}$ to
$\mathfrak{G}$ and using the Killing form to treat $\mathcal{F}_{\mu\nu}$ as a
matrix, $\bm{\mathcal{F}}$, one may diagonalize the QFIM. By diagonalizing
$\bm{\mathcal{F}}$, one finds the eigenvalues representing the QFI. In doing
so, one is left with a matrix whose eigenvectors are orthogonal vectors
according to the Information metric’s inner product, and whose eigenvalues are
the distance, $ds$, which one will evolve in for a differential change in the
corresponding coordinate. These vectors are normalized with respect to the
Killing form’s inner product, which for $\mathrm{SU}(n)$ is the trace inner
product of operators. This guarantees that improvements in scaling of the QFI
are truly a result of entanglement and not an arbitrary rescaling. Thus, a
small change in $x^{\mu}$ corresponds to a large pathlength $ds$ over our
Hilbert space, which can be well distinguished by quantum measurement! Finding
the coordinates for which $\mathcal{F}_{\mu\nu}$ is maximum and diagonal
corresponds to the eigenvalues of the QFIM.

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