# A Natural Field-like Entanglement Resource in Trapped-Ion Chains

Natalie Klco<EMAIL_ADDRESS>Duke Quantum Center and Department of
Physics, Duke University, Durham, NC 27708, USA  D. H. Beck
<EMAIL_ADDRESS>Department of Physics and Illinois Quantum Information
Science and Technology (IQUIST) Center, University of Illinois at Urbana-
Champaign, Urbana, IL 61801, USA

###### Abstract

The electromagnetic trapping of ion chains can be regarded as a process of
non-trivial entangled quantum state preparation within Hilbert spaces of the
local axial motional modes. To begin uncovering properties of this
entanglement resource produced as a byproduct of conventional ion-trap quantum
information processing, the quantum continuous-variable formalism is herein
utilized to focus on the leading-order entangled ground state of local
motional modes in the presence of a quadratic trapping potential. The decay of
entanglement between disjoint subsets of local modes is found to exhibit
features of entanglement structure and response to partial measurement
reminiscent of the free massless scalar field vacuum. With significant
fidelities between the two, even for large system sizes, a framework is
established for initializing quantum field simulations via “imaging” extended
entangled states from natural sources, rather than building correlations
through deep circuits of few-body entangling operators. By calculating
probabilities in discrete Fock subspaces of the local motional modes,
considerations are presented for locally transferring these pre-distributed
entanglement resources to the qudits of ion internal energy levels, improving
this procedure’s anticipated experimental viability.

††preprint: IQuS_WKSHP@UW-23-001

###### Contents

  1. I Introduction
  2. II Local Axial Motional Modes
  3. III Entanglement Resource
  4. IV Preliminary Applications
    1. IV.1 Vacuum State Preparation
    2. IV.2 Transferring Distributed Entanglement
  5. V Summary and Outlook
  6. A Small-Chain Examples
  7. B Local Motional Modes in Fock-space
  8. C Numerical Tables

> “ _In endeavors of discovery
>  it is wise to listen
> as Nature speaks for herself._”

##  I Introduction

Spatially distributed quantum correlations not only play an important role in
the structure, observed dynamics, and apparent thermalization of quantum many-
body systems [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
19], but act as a fundamental resource in designing and operating quantum
computation, simulation, sensing, and communication technologies [20, 21, 22,
23, 24, 25, 26, 27, 28, 29, 30, 31]. For such applications, the necessary
entanglement is commonly produced via a series of direct interactions
generated between two or more quantum degrees of freedom, or distributed after
production through quantum networking techniques that maintain coherence
throughout transmission. A third possibility, however, relies upon connecting
locally to a shared system of naturally distributed entanglement. For example,
the vacuum state of a quantum field is entangled at spacelike separations [32,
33, 34, 35, 36], and a pair of locally-interacting detectors can transfer that
entanglement into basic quantum degrees of freedom [37, 38, 39] to drive
subsequent quantum information processing protocols. While the raw
entanglement production through this type of extraction is expected to be far
smaller than can be generated through artificially constructed logic gates,
“imaging” natural entanglement can provide a valuable structure of quantum
correlations, e.g., as a starting point for large-scale quantum simulations of
physical systems or the generation of entangled sensing arrays. The present
manuscript focuses on the local axial motional modes of a trapped ion chain,
quantifying their entanglement properties and utilizing this quantum
correlation information to guide opportunities for applications in the quantum
simulation of quantum fields.

With each global motional mode in its ground state, the vacuum of ion-chain
zero-point fluctuations is an unentangled, tensor-product state with respect
to partitions among the normal modes. However, for partitions among the
associated local motional modes, this vacuum exhibits entanglement and, to a
good approximation for practical trapping potentials, constitutes a Gaussian
continuous-variable (CV) system. Note that the ion chain supports both
transverse and axial modes (see Ref [40] for a recent review). While the
global motional modes are commonly utilized to mediate all-to-all entangling
gates among ions within the trap [41, 42, 43, 44, 45, 46], the local
transverse motional modes have been proposed to represent computational
bosonic degrees of freedom in the design of lattice gauge theory quantum
simulations [47, 48].

For a pair of disjoint regions within the global vacuum state, each comprising
the fluctuations associated with one or more consecutive ions, it is herein
found that many entanglement characteristics are remarkably similar to those
of the massless non-interacting lattice scalar field vacuum. In particular,
the logarithmic negativity between the regions decreases exponentially with
increasing separation, whereas the corresponding two-point functions decrease,
at most, as a power law. As with the scalar field vacuum, the stronger
entanglement represented by the two-point functions can also be recovered with
classical communication of measurements on parts of the system external to the
two regions [49].

After presenting the local axial modes of the trapped ion vacuum as a Gaussian
CV system in Section II, its entanglement characteristics are described with
reference to those of the scalar field vacuum in Section III. Two possible
applications are subsequently discussed in Section IV—using the trapped ion
vacuum both as a state preparation resource for the scalar field vacuum, and
as a more general resource from which entanglement can be transferred into the
discrete-variable framework of ion internal states.

In pursuing these applications, the entanglement structure was found to
provide essential guidance: indicating that these two systems inherently share
prominent features of distributed entanglement, and governing beneficial
entanglement rearrangements in preparation for entanglement transfer. While
the connections explored here represent only early observations, this concrete
example suggests significant opportunities for incorporating natural sources
of distributed entanglement as active components in quantum technologies.

##  II Local Axial Motional Modes

The Hamiltonian describing $N$ ions in a 1D-linear trap is the sum of the ion
kinetic energies ($T$) and Coulomb interactions within the trapping potential
($U$),

$\displaystyle T$ $\displaystyle=\sum_{i=1}^{N}\frac{m\dot{z}_{i}^{2}}{2}$ (1)
$\displaystyle U$
$\displaystyle=\sum_{i=1}^{N}q\kappa_{2}z_{i}^{2}+\frac{1}{2}\sum_{\stackrel{{\scriptstyle
i,j=1}}{{i\neq j}}}^{N}\frac{q^{2}}{4\pi\epsilon_{0}|z_{i}-z_{j}|}\ \ \ ,$ (2)

where $q,m$ are the charge and mass of each ion (assuming homogeneous ion
species), and $z_{i}$ is the position of the $i^{\text{th}}$ ion along the
one-dimensional trap axis [50, 51]. Though the incorporation of
anharmonicities in the trapping potential [52, 53] will be of future interest,
e.g., for opportunities to design ion spacings that produce local motional
modes with a regularity that mirrors conventional latticizations of field
theories, the current work utilizes a quadratic trapping potential.

As usual, the equilibrium positions of ions within the trap may be calculated
by solving the set of equations enforcing vanishing first derivatives of the
potential with respect to displacements,

$\frac{\partial U}{\partial
z_{i}}\Big{\rvert}_{\mathbf{z}=\mathbf{z}_{0}}=\vec{0}\ \ \ ,$ (3)

to place each ion in a stable local minimum. For small systems, these
equations can be solved exactly (see Appendix A). For larger systems, the
equilibrium positions can be reliably calculated numerically [54, 55]. The
central ions of these large chains develop an approximately evenly spaced
distribution, with the edges systematically relaxing to larger separations as
the soft harmonic potential is countered by the center-dominated Coulomb
repulsion.

The above Hamiltonian has two significant scales, one associated with the
inter-ion spacing and the other associated with oscillations of the ions about
their equilibrium positions. Isolating these characteristic scales to identify
a dimensionless description of the system, the Hamiltonian may be written as

$\displaystyle H$ $\displaystyle=\frac{m\omega_{z}^{2}\ell^{2}_{\mu
m}}{2}\sum_{i=1}^{N}\left(\dot{\bar{z}}_{i}^{2}+\bar{z}_{i}^{2}+\sum_{\stackrel{{\scriptstyle
j=1}}{{j\neq i}}}^{N}\frac{1}{|\bar{z}_{i}-\bar{z}_{j}|}\right)$ (4)
$\displaystyle=\frac{m\omega_{z}^{2}\ell^{2}_{\mu
m}}{2}\left(\bar{T}+\bar{U}\right)\ \ \ ,$ (5)

where $\bar{z}=\frac{z}{\ell_{\mu m}}$ and
$\dot{\bar{z}}=\frac{\dot{z}}{\omega_{z}\ell_{\mu m}}$. The characteristic
spacing and oscillation parameters are $\ell_{\mu
m}~{}=~{}\left(\frac{q}{8\pi\epsilon_{0}\kappa_{2}}\right)^{\frac{1}{3}}$ and
$\omega_{z}^{2}=\frac{2q\kappa_{2}}{m}$, where the latter is the center-of-
mass oscillation frequency for an ion chain. The root-mean-square of
fluctuations from equilibrium in the ground state of an oscillator of this
frequency yields the length, $\ell_{nm}=\sqrt{\frac{\hbar}{m\omega_{z}}}$. To
provide a sense of scale for these two lengths, a single 171Yb+ ion in a trap
with $\omega_{z}/2\pi=1$ MHz yields fluctuations characterized by the scale
$\ell_{nm}=7.7$ nm, while the separation between such ions is governed by the
larger scale, $\ell_{\mu m}=2.7~{}\mu$m. As such, the lengths have been
labeled for their canonical magnitudes in a broad regime of trapped ion
applications.

With a second set of variables defined and scaled for excursions from
equilibrium positions,
$\bar{\boldsymbol{\zeta}}=\frac{\mathbf{z}-\mathbf{z}_{0}}{\ell_{nm}}=(\bar{\mathbf{z}}-\bar{\mathbf{z}}_{0})\frac{\ell_{\mu
m}}{\ell_{nm}}$, the potential may be expanded about the equilibrium as

$U=\frac{\hbar\omega_{z}}{2}\frac{\ell_{\mu
m}^{2}}{\ell_{nm}^{2}}\left(\frac{1}{2!}\frac{\partial^{2}\bar{U}}{\partial\bar{z}_{i}\partial\bar{z}_{j}}\Big{\rvert}_{\bar{\mathbf{z}}=\bar{\mathbf{z}}_{0}}\left(\frac{\ell_{nm}}{\ell_{\mu
m}}\right)^{2}\bar{\zeta}_{i}\bar{\zeta}_{j}\ +\right.\\\
\left.\frac{1}{3!}\frac{\partial^{3}\bar{U}}{\partial\bar{z}_{i}\partial\bar{z}_{j}\partial\bar{z}_{k}}\Big{\rvert}_{\bar{\mathbf{z}}=\bar{\mathbf{z}}_{0}}\left(\frac{\ell_{nm}}{\ell_{\mu
m}}\right)^{3}\bar{\zeta}_{i}\bar{\zeta}_{j}\bar{\zeta}_{k}+\cdots\right)\ \
,$ (6)

where a shift has been introduced to place the equilibrium configuration at
zero potential, $\bar{U}\left(\bar{\mathbf{z}}_{0}\right)=0$. With higher
order contributions suppressed by powers of $\frac{\ell_{nm}}{\ell_{\mu m}}$,
the convergence of this expansion is commonly rapid. For the representative
171Yb+ example considered above, this ratio is $2.8\times 10^{-3}$.
Implementing a harmonic approximation of the potential, the leading-order
dimensionless Hamiltonian describing small oscillations about equilibrium is

$\bar{H}_{2}=\sum_{i=1}^{N}\dot{\bar{\zeta}}^{2}_{i}+\sum_{i,j=1}^{N}\bar{\zeta}_{i}\bar{L}_{ij}\bar{\zeta}_{j}\
\ \ ,$ (7)

where $H_{2}=\frac{\hbar\omega_{z}}{2}\bar{H}_{2}$ and
$\bar{L}_{ij}=\frac{1}{2}\frac{\partial^{2}\bar{U}}{\partial\bar{z}_{i}\partial\bar{z}_{j}}\rvert_{\bar{\mathbf{z}}=\bar{\mathbf{z}}_{0}}$
is the Hessian matrix of second derivatives. In the following, this
harmonicity supports the use of the Gaussian continuous variable formalism
[56, 57], allowing the leading-order entanglement information of the ground
state to be succinctly captured by the covariance matrix alone.

By diagonalizing the potential, axial normal modes of motion may be calculated
as position space eigenvectors, $e_{\alpha}$, of the Hessian with eigenvalues,
$\bar{\omega}_{\alpha}^{2}$, in units of the center-of-mass oscillation
frequency, ${\bar{\boldsymbol{\omega}}=\boldsymbol{\omega}/\omega_{z}}$. In
this basis of normal modes, the dimensionless Hamiltonian operator is that of
a set of independent oscillators,

$\hat{\bar{H}}_{2}=\sum_{\alpha}\hat{\bar{\nu}}_{\alpha}^{2}+\bar{\omega}_{\alpha}^{2}\hat{\bar{\xi}}_{\alpha}^{2}\
\ \ ,$ (8)

with $\hat{\bar{\xi}}_{\alpha}$ and $\hat{\bar{\nu}}_{\alpha}$ the
dimensionless position and momentum operators, respectively, of the chain’s
normal modes. As usual, these operators may be written in terms of normal mode
creation and annihilation operators,

$\displaystyle\hat{\bar{\xi}}_{\alpha}$
$\displaystyle=\frac{\hat{\xi}_{\alpha}}{\ell_{nm}}=\frac{\hat{a}_{\alpha}+\hat{a}_{\alpha}^{\dagger}}{\sqrt{2\bar{\omega}_{\alpha}}}$
(9) $\displaystyle\hat{\bar{\nu}}_{\alpha}$
$\displaystyle=\frac{\ell_{nm}}{\hbar}\hat{\nu}_{\alpha}=-i\sqrt{\frac{\bar{\omega}_{\alpha}}{2}}\left(\hat{a}_{\alpha}-\hat{a}^{\dagger}_{\alpha}\right)\
\ \ ,$ (10)

with
$\left[\hat{a}_{\alpha},\hat{a}^{\dagger}_{\beta}\right]=\delta_{\alpha\beta}$.
Naturally, the matrix of normal mode eigenvectors, $\mathbf{e}$, constructed
in rows as $e_{\alpha,i}=\left(e_{\alpha}\right)_{i}$, provides the unitary
basis transformation between global and local motional modes

$\hat{\boldsymbol{\phi}}=\mathbf{e}^{-1}\hat{\boldsymbol{\xi}}\quad,\quad\hat{\boldsymbol{\pi}}=\mathbf{e}^{-1}\hat{\boldsymbol{\nu}}\
\ \ ,$ (11)

where $\hat{\boldsymbol{\phi}},\hat{\boldsymbol{\pi}}$ are the vectors of
position and conjugate-momentum operators for the local motional modes,
respectively.

In the global normal mode vacuum, $\hat{a}_{\alpha}|0\rangle=0$, with two-
point expectation values of $\bar{\omega}_{\alpha}\langle
0|\hat{\bar{\xi}}_{\alpha}\hat{\bar{\xi}}_{\beta}|0\rangle=\frac{1}{\bar{\omega}_{\alpha}}\langle
0|\hat{\bar{\nu}}_{\alpha}\hat{\bar{\nu}}_{\beta}|0\rangle=\frac{\delta_{\alpha\beta}}{2}$,
the two-point functions of the local oscillators are

$\displaystyle\frac{\langle
0|\hat{\phi}_{i}\hat{\phi}_{j}|0\rangle}{\ell_{nm}^{2}}$
$\displaystyle=\frac{1}{2}\sum_{\alpha=1}^{N}\frac{\left(\mathbf{e}^{-1}\right)_{i\alpha}\left(\mathbf{e}^{-1}\right)_{j\alpha}}{\bar{\omega}_{\alpha}}$
(12) $\displaystyle\frac{\ell_{nm}^{2}}{\hbar^{2}}\langle
0|\hat{\pi}_{i}\hat{\pi}_{j}|0\rangle$
$\displaystyle=\frac{1}{2}\sum_{\alpha=1}^{N}\left(\mathbf{e}^{-1}\right)_{i\alpha}\left(\mathbf{e}^{-1}\right)_{j\alpha}\bar{\omega}_{\alpha}\
\ \ .$ (13)

In the local phase space basis of
$\hat{\bar{\mathbf{r}}}=\left\\{\hat{\bar{\phi}}_{1},\hat{\bar{\pi}}_{1},\hat{\bar{\phi}}_{2},\hat{\bar{\pi}}_{2},\cdots,\hat{\bar{\phi}}_{N},\hat{\bar{\pi}}_{N}\right\\}$,
where the symplectic matrix of commutation relations is
$\Omega=\bigoplus_{j=1}^{N}iY$ with $Y$ the second Pauli matrix, the
dimensionless ground state covariance matrix is

$\bar{\sigma}_{i,j}=\langle
0|\left\\{\hat{\bar{r}}_{i}-\langle\hat{\bar{r}}_{i}\rangle,\hat{\bar{r}}_{j}-\langle\hat{\bar{r}}_{j}\rangle\right\\}_{+}|0\rangle\
\ \ .$ (14)

Examples of these covariance matrices for the local axial motional modes in
small ion chains are provided in Appendix A. Because the vacuum expectation
values of position-momentum anticommutators vanish,

$\langle 0|\left\\{\hat{\phi}_{i},\hat{\pi}_{j}\right\\}_{+}|0\rangle=0,$ (15)

the two point functions of Eqs. (12) and (13) fully determine the covariance
matrix, and thus the entanglement properties of the local motional modes in
the leading-order Gaussian ground state.

The independence of the dimensionless CM, Eqs. (12)-(14), with respect to both
trap and ion experimental parameters indicates a universality of the harmonic
approximation (Eq. (6)) of the trapped ion local motional mode correlation
structure. With a single dimensionless CM to describe all ion choices and
quadratic trap strengths, this universality results in leading-order
insensitivity to experimental implementations. This may prove useful for error
robustness, or, conversely, yield challenges as all modifications to the CM,
e.g., for optimizing connections to the scalar field vacuum as discussed in
Sec. IV.1, must be implemented through explicit quantum operations.
Furthermore in this context, the dimensionful CM incorporates all ion and trap
parameters through the length scale
$\ell_{nm}^{2}~{}=~{}\frac{\hbar}{\sqrt{2mq\kappa_{2}}}$, which functions
numerically as a global set of single-mode squeezing parameters with zero
entanglement power, i.e., scaling the
$\langle\hat{\boldsymbol{\phi}}\hat{\boldsymbol{\phi}}\rangle$ expectation
values inversely from those of conjugate-momentum operators
$\langle\hat{\boldsymbol{\pi}}\hat{\boldsymbol{\pi}}\rangle$, c.f., Eqs. (12)
and (13). As such, leading-order entanglement quantities among local modes
with a quadratic trapping potential are pure numbers independent of the
variables of experimental design.

##  III Entanglement Resource

For assessing the entanglement naturally distributed among the local motional
modes, consider entanglement properties between two disjoint regions, $A$ and
$B$, each composed of $d$ contiguous ions and separated by $\tilde{r}$ ions.
For this bipartite state, three logarithmic negativity values are of current
interest: that when the local modes outside the regions are traced, resulting
in an $A$-$B$ mixed state, or are projectively (ideal homodyne) measured in
the $\phi$\- or $\pi$-basis (see Ref. [49] for further technical details),
resulting in $A$-$B$ pure states. With differing levels of classical
information provided from possible external observers, these measures quantify
here the operational amount of entanglement that could be consolidated into
two-mode pairs spanning the disjoint regions [58, 59, 60].

Figure 1: Logarithmic negativity between two regions (of sizes $d=1,3,5$ modes
each) as a function of their separation in one spatial dimension. In the ion
application, each mode is a local axial motional mode, with the pair of
regions centered within a 150-ion chain. In the scalar field application, each
mode is a field lattice site within an infinite volume in the massless regime
($m=10^{-10}$). Each system is shown when the volume external to the regions
is traced or measured in the local position ($\phi$) or local conjugate-
momentum ($\pi$) basis.

As demonstrated in Fig. 1 for disjoint regions of local motional modes
separated across the center of a 150-ion chain, the logarithmic negativity
decays exponentially with the separation between regions upon tracing of the
modes in the remaining chain [61, 62, 63]. Originating from a system with
polylog two-point functions, the emergence of this strong decay is a result of
the classical noise introduced in the tracing procedure [49], i.e., resulting
from the loss of information about an entangled quantum system. These features
are common to the massless scalar field vacuum, for which Fig. 1 shows the
analogous calculation in grayscale for reference. Beyond the magnitude and
decay, the calculation of Fig. 1 indicates also close agreement in the
separation at which the classical noise causes these regions of finite
pixelation to transition into a regime of vanishing negativity [64, 65, 66,
61, 67, 68, 69, 70, 71, 72, 73, 74, 62, 63] that is also separable [75, 63].
With no free parameters, the externally traced logarithmic negativity is
remarkably similar between these two distinct physical systems, one at the
center of a quadratically trapped ion chain and the other within an infinite
volume of a free massless scalar field vacuum.

Upon measurement of the external modes in the local position $\phi$-basis,
removing a particular form of classical noise upon classical communication,
the two systems again exhibit comparable underlying entanglement resources.
Upon measurement of the external modes in the conjugate-momentum $\pi$-basis,
removing an alternate form of classical noise 111Explicit forms of the
classical noise removed upon measurement in each basis are provided in Ref.
[49]., the distinction between these two physical systems becomes apparent.
This difference can be understood as a result of the field gradients and
quadratically balanced Coulomb repulsion being inequivalent methods of
generating entanglement among local modes in the scalar field and trapped ion
motional modes, respectively.

Beyond the magnitudes of these entanglement measures, the local motional modes
are found to have further structural entanglement features in common with the
free lattice scalar field. Though the spatial parity symmetry, present when
regions are centered within the ion chain, is generally insufficient for the
techniques of Ref. [60] to assure consolidation of negativity into a separable
set of two-mode entangled pairs governed by the partially transposed
symplectic eigenvalues, it is found that the local motional modes within the
trapped ion chain do share this consolidation feature with the free scalar
field vacuum. As such, the PT symplectic eigenvectors provide guidance for
locally transforming the accessible many-body entanglement between disjoint
regions of ions into a series of two-mode squeezed states. The availability of
this two-mode organization of many-body mixed-state entanglement may prove to
be useful experimentally in transferring entanglement from the local motional
modes into discrete ion internal energy levels, as first proposed in Ref. [76]
and further discussed in section IV.2.

##  IV Preliminary Applications

###  IV.1 Vacuum State Preparation

For digital quantum algorithms, studies establishing the efficiency (in e.g.,
particle number, coupling strength, etc.) of simulating scattering in
interacting quantum fields [77, 78] show that the dominant use of quantum
computational resources is expected to be required not in the real-time
dynamics, but in the process of initial state preparation, e.g., of the
quantum field vacuum. From a complementary perspective, exploring the low
energy subspace and preparing states within it from a trivial tensor-product
fiducial state can be beyond the expected asymptotic efficiency of quantum
hardware and software, even for few-body spatially local Hamiltonians [79, 80,
81, 82]. These considerations, as well as the presence of phase transitions in
lattice gauge theories, motivate the exploration of diverse methods of
entangled quantum field vacuum state preparation [83, 84, 85, 86, 87, 73, 88,
89, 90, 91, 92, 93, 94, 95, 96, 97, 98], with strategies ranging from
variational techniques and guidance by tensor networks to adiabatic flows and
the design of reservoirs for dynamical cooling procedures. To this literature,
the present manuscript introduces the possibility of replacing the trivial-
complexity fiducial state with an experimentally available state of naturally
distributed entanglement.

When generating entanglement through interleaved layers of few-body operators,
entanglement is created with an expansion rate governed by the Lieb-Robinson
bound [99, 100]. However, with natural many-body entangled quantum resources,
e.g., the extended quantum system of ion local axial motional degrees of
freedom, incorporated as active elements of a computational framework,
different approaches become available. Analogous to the way in which providing
a computational framework with quantum degrees of freedom allows more
efficient representation and imitation of quantum systems themselves [20],
providing a computational framework with access to naturally distributed
arrays of entangled degrees of freedom could provide similar advantages for
quantum field simulations.

For scalar field applications suggested by the entanglement observations
above, consider quantifying the provided leading-order approximation with the
quantum fidelity [101, 102, 103, 22],
$\mathcal{F}(\rho_{1},\rho_{2})=\mathcal{F}(\rho_{2},\rho_{1})=\text{tr}\left(\sqrt{\sqrt{\rho_{1}}\rho_{2}\sqrt{\rho_{1}}}\right)$,
where $\rho_{1,2}$ are the density matrices of local axial motional modes in
the center of an ion chain and of CV lattice sites within an infinite-volume
scalar field. The fidelity saturates to unity when the two density matrices
are equivalent, vanishes when they are composed of orthogonal subspaces, and
may be directly related to metrics providing a formal sense of distance
between quantum states [104, 105]. Furthermore, many preparation algorithms
rely upon the availability of an input initial state with non-vanishing
fidelity with respect to a desired state, impacting both figures of merit and
necessary statistics.

For mixed Gaussian CV states, Ref. [106] provides several techniques for
calculating the fidelity from the set of first and second moments of each
Gaussian state 222For example, when the first moments of both states are
vanishing, the following formulation [106],
$\mathcal{F}(\sigma_{1},\sigma_{2})=\left(\frac{\det\left(2\left[\sqrt{\mathbb{I}+\frac{\left(V\Omega\right)^{-2}}{4}}+\mathbb{I}\right]V\right)}{\det\left(\sigma_{1}+\sigma_{2}\right)}\right)^{\frac{1}{4}}\
\ \ ,$ (16) with
$V=\Omega^{T}(\sigma_{1}+\sigma_{2})^{-1}\left(\frac{\Omega}{4}+\sigma_{2}\Omega\sigma_{1}\right)\
\ \ ,$ (17) is straightforward to compute from the covariance matrices in the
present harmonic approximation of the combined ion trapping and Coulomb
potentials.. A fidelity analysis is shown in Fig. 2 between the local modes of
the trapped-ion motional ground state and the free, massless, lattice scalar
field vacuum. Unfortunately, due to the increasing differences between quantum
states exacerbated by the growth of Hilbert space dimensionality,
approximations that may be sufficient for small systems commonly experience
exponential decay in the fidelity as a function of system size, becoming
rapidly impractical starting points for large systems.

Figure 2: Fidelity between central local axial modes of trapped ion chains
cooled to their motional ground state and the same-sized region of the
infinite-volume, lattice scalar field vacuum in the massless regime
($m=10^{-10}$). Shown for regions of up to 50 local modes/lattice sites in the
setting of 30-, 50-, and 150-ion chains, a one-parameter optimization over a
globally implemented single-mode squeezing parameter allows the creation of
approximations to the latticized field vacuum with enduring fidelity.

As shown by the open points in Fig. 2, such catastrophic loss of fidelity
occurs for the ion local motional modes naturally produced in the hardware.
However, inspired by the similar distribution of disjoint two-region
entanglement in both systems demonstrated in Fig. 1, performing a one-
parameter optimization of a single-mode squeezing applied to each local
motional mode (a unitary with zero entanglement power) results in substantial
fidelities even for large systems. The values of this squeezing parameter as a
function of region and chain size are recorded in the numerical tables of
Appendix C. For chain lengths comparable to the field region size, these
squeezing magnitudes are realistic given current capabilities and ongoing
developments in the experimental squeezing of ion motional modes [107, 108,
109, 110, 111, 112, 113, 114]. Note that the universality of the dimensionless
CM, Eqs. (12)-(14), indicates that this squeezing must be introduced through
external quantum operations on each local motional mode.

It is encouraging that a simply calculated set of local squeezings can produce
a state of local motional modes with similar disjoint entanglement structure
and $\mathcal{O}(1)$ fidelity with respect to large instances of the massless
lattice scalar field vacuum. The remaining fundamental distinctions between
the small oscillation approximation of ion local motional modes and the scalar
vacuum suggest that the calculated fidelity will not asymptote to unity for
larger ion chains. However, note that this calculation considered only the
flexibility of one single-mode squeezing parameter; for a given system, all
entanglement features are thus generated only by the natural ion local
motional modes themselves. As such, this calculation is best interpreted as a
leading-order result, subject to systematic improvement upon the action of a
broader array of CV quantum circuitry.

These preliminary results motivate dedicated future inquiry exploring the
replacement of canonical tensor-product initial states by experimentally
convenient states of naturally distributed entanglement.

###  IV.2 Transferring Distributed Entanglement

For purposes of simulation with finite-dimensional quantum systems or for
broader computational applications of spatially distributed quantum
correlations, it is of interest to consider transferring the entanglement
among local CV motional modes into the Hilbert space composed of the
associated ions’ internal energy levels. This process was first proposed as an
opportunity for experimentally detecting vacuum entanglement and for
entangling trapped ions with quantum gates utilizing only local phase-space
excitations [76]. The local motional modes then provide an entangled fabric of
spatially distributed entanglement that may be strategically accessed. That
the naturally available entanglement structure is remarkably similar to that
of a simple latticized quantum field suggests further possible applications
from quantum simulation of Standard Model physics to large-scale quantum error
correction. Motivated by recent theoretical and experimental advances in
qudit-based quantum information processing [115, 116, 117, 118, 119, 120, 121,
122, 123, 124, 125, 126, 127], the following section details one important
consideration affecting the magnitude and efficiency of CV-to-qudit
entanglement transfer in the context of trapped ions.

There are many strategies for transferring entanglement from CV local motional
modes to finite dimensional qudit systems [128, 129, 130, 76, 131, 132]. One
basic paradigm involves writing the CV system in an indexed orthonormal basis
and performing an approximate swapping operation between $n$ basis states and
the $n$-dimensional qudit system. For example, Ref. [76] presents an
approximation of such a swap operator for an expression of the CV local
motional modes in the Fock basis, transferring the quantum information present
in the low-energy Fock space to qubits of ion internal energy levels.
Generalizing to qudits, the population of the CV system in states external to
the chosen subspace (e.g., above the lowest energy Fock states) yields excess
population in the qudit initial state, reducing the transferred entanglement.
For the Fock-space transfer, this external-population figure of merit is shown
in Fig 3 of Appendix B for the local motional modes of a two-ion chain as a
function of increasing qudit dimension.

Though the entanglement between two Hilbert spaces is not impacted by local
operations, the entanglement accessible from within a Hilbert space subset can
be highly sensitive to local restructuring. As such, local CV operations
applied prior to qudit extraction into the ion internal states can impact the
probability in the qudit subspace and thus the transferred entanglement.
Figure 3 indicates that first including a pair of single-mode squeezings to
transform the local motional mode CM into normal form with balanced
$\langle\boldsymbol{\phi}\boldsymbol{\phi}\rangle$ and
$\langle\boldsymbol{\pi}\boldsymbol{\pi}\rangle$ matrix elements, reduces the
population external to the low-energy Fock states by several orders of
magnitude, with an impact increasing with increasing qudit dimension. By
calculating directly the motional mode CM naturally produced in the ion chain,
such operations can be designed to optimize the localization of the ion modes
in an experimentally convenient orthonormal basis for entanglement transfer.

In designing local operators for more than two ions to optimize this form of
CV to finite-dimensional entanglement transfer, the observation discussed at
the end of Section III—that the ion local motional modes in this context
satisfy requirements allowing the techniques of Ref. [60] to reorganize the
available entanglement between regions—provides an additional advantage. Such
consolidation, informed by the eigensystem of the partially transposed
$2d$-ion mixed-state, reorganizes the bipartite entanglement between disjoint
many-body field or ion-chain regions into a tensor-product series of
$(1_{A}\times 1_{B})$ entangled pairs. As such, this reorganization pre-
conditions the CV entanglement to be conducive to subsequent entanglement
transfer, ultimately allowing this entanglement to be consolidated into fewer
ion pairs and improving the entanglement yield.

##  V Summary and Outlook

Considering entanglement structure from multiple operational perspectives can
serve to quantify resources available for quantum information protocols and to
guide the design of quantum simulations. This manuscript considers whether the
local axial motional modes of a simple trapped ion system can serve as a
useful starting point for generation of entangled vacuum states of the
massless scalar field. Recognizing that several region-region entanglement
measures exhibit strong similarities between the two systems, analyses of
fidelities and localization in Fock-space constitute initial steps toward
utilizing naturally distributed entanglement resources for computational
applications. This perspective highlights common elements among quantum
simulation, sensing, and networking as quantum fields are considered for
incorporation directly into quantum architectures.

The present analysis, however, only modestly addresses a subset of the
important questions driving research toward quantum technology goals. For
scientific application to quantum state preparation, the approximate
correspondence observed in Section IV.1 must be promoted to a systematically
improvable protocol, possibly through the introduction of a hierarchy of
higher-complexity quantum operations [73, 90]. For the application of Section
IV.2, further work is needed to determine practical techniques for
implementing an approximate swapping operator between the local motional modes
and ion internal levels, including the assessment of many choices of basis and
the design of pre-conditioning procedures to strategically organize
entanglement. Better understanding of successful procedures for imaging
naturally distributed CV entanglement to finite dimensions may further inform
the design of entanglement-efficient bosonic truncation and digitization
schemes, e.g., in designing computational bases for the practical quantum
simulation of quantum field theories [133, 134, 135, 136, 137, 138, 139, 140,
141, 142, 118, 143, 144, 145, 123, 146, 147, 148, 149, 150].

The present connection between the scalar field vacuum and local motional
modes of a trapped-ion chain, guided by entanglement structure, brings present
research still closer to the vision of quantum simulation [20]—leveraging the
mutual intersimulatability of diverse quantum systems, even those widely
separated in natural energy scales, e.g., the connection between atomic and
subatomic physics arising in present quantum simulations of quantum
chromodynamics.

###### Acknowledgements.

We thank Rainer Blatt, Crystal Noel, Guido Pagano, Martin J. Savage, and Peter
Zoller for inspiring discussions. The creative initiation of several aspects
of this work were established during the workshop _At the Interface of Quantum
Sensors and Quantum Simulation_ at the InQubator for Quantum Simulation (IQuS)
hosted by the Institute for Nuclear Theory (INT). We thank Rainer Blatt and
Peter Zoller for challenging the organizers with homework throughout this
workshop. DB is supported in part by NSF Nuclear Physics grant PHY-2111046.

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oneΔ

##  Appendix A Small-Chain Examples

With a quadratic trapping potential, the equilibrium positions for two- and
three-ion chains can be evaluated exactly to be

$\mathbf{z}^{0}_{N=2}=\left\\{-\frac{2^{1/3}\ell_{\mu
m}}{2},\frac{2^{1/3}\ell_{\mu
m}}{2}\right\\}\quad,\quad\mathbf{z}_{N=3}^{0}=\left\\{-\frac{10^{1/3}\ell_{\mu
m}}{2},0,\frac{10^{1/3}\ell_{\mu m}}{2}\right\\}\ \ \ ,$ (18)

where $\ell_{\mu m}=\left(\frac{q}{8\pi\epsilon_{0}\kappa_{2}}\right)^{1/3}$
is a characteristic ion-separation length scale. The resulting frequencies and
normal modes are

$\bar{\boldsymbol{\omega}}=\frac{\boldsymbol{\omega}}{\omega_{z}}=\left\\{1,\sqrt{3}\right\\}\quad,\quad
e=\begin{pmatrix}\mathbf{e}_{1}\\\
\mathbf{e}_{2}\end{pmatrix}=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\\
-1&1\end{pmatrix}\ \ \ $ (19)

for a two-ion chain and

$\bar{\boldsymbol{\omega}}=\frac{\boldsymbol{\omega}}{\omega_{z}}=\left\\{1,\sqrt{3},\sqrt{\frac{29}{5}}\right\\}\quad,\quad
e=\begin{pmatrix}\mathbf{e}_{1}\\\ \mathbf{e}_{2}\\\
\mathbf{e}_{3}\end{pmatrix}=\begin{pmatrix}\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{3}}\\\
-\frac{1}{\sqrt{2}}&0&\frac{1}{\sqrt{2}}\\\
\frac{1}{\sqrt{6}}&-\frac{2}{\sqrt{6}}&\frac{1}{\sqrt{6}}\end{pmatrix}\ \ \ $
(20)

for a three-ion chain. Interestingly, the ground state center-of-mass motion
frequency and gap to the excited state breathing mode are known to be
independent of the number of ions in the chain [41].

Utilizing Eqs. (12)-(14), the dimensionless covariance matrices for two and
three ions are directly calculated to be

$\bar{\sigma}_{N=2}=\frac{1}{2}\left(\begin{array}[]{cccc}\frac{3+\sqrt{3}}{3}&0&\frac{3-\sqrt{3}}{3}&0\\\
0&1+\sqrt{3}&0&1-\sqrt{3}\\\ \frac{3-\sqrt{3}}{3}&0&\frac{3+\sqrt{3}}{3}&0\\\
0&1-\sqrt{3}&0&1+\sqrt{3}\\\
\end{array}\right)\quad,\quad\sigma_{N=2}=\frac{\ell_{nm}^{2}}{2}\left(\begin{array}[]{cccc}\frac{3+\sqrt{3}}{3}&0&\frac{3-\sqrt{3}}{3}&0\\\
0&\frac{\hbar^{2}\left(1+\sqrt{3}\right)}{\ell_{nm}^{4}}&0&\frac{\hbar^{2}\left(1-\sqrt{3}\right)}{\ell_{nm}^{4}}\\\
\frac{3-\sqrt{3}}{3}&0&\frac{3+\sqrt{3}}{3}&0\\\
0&\frac{\hbar^{2}\left(1-\sqrt{3}\right)}{\ell_{nm}^{4}}&0&\frac{\hbar^{2}\left(1+\sqrt{3}\right)}{\ell_{nm}^{4}}\\\
\end{array}\right)$ (21)

and

$\bar{\sigma}_{N=3}=\scalebox{0.8}{\mbox{$\displaystyle\left(\begin{array}[]{cccccc}\frac{29\sqrt{3}+\sqrt{145}+58}{174}&0&\frac{1}{3}-\frac{\sqrt{\frac{5}{29}}}{3}&0&\frac{-29\sqrt{3}+\sqrt{145}+58}{174}&0\\\
0&\frac{15\sqrt{3}+\sqrt{145}+10}{30}&0&\frac{5-\sqrt{145}}{15}&0&\frac{-15\sqrt{3}+\sqrt{145}+10}{30}\\\
\frac{1}{3}-\frac{\sqrt{\frac{5}{29}}}{3}&0&\frac{1}{3}+\frac{2\sqrt{\frac{5}{29}}}{3}&0&\frac{1}{3}-\frac{\sqrt{\frac{5}{29}}}{3}&0\\\
0&\frac{5-\sqrt{145}}{15}&0&\frac{2\sqrt{145}+5}{15}&0&\frac{5-\sqrt{145}}{15}\\\
\frac{-29\sqrt{3}+\sqrt{145}+58}{174}&0&\frac{1}{3}-\frac{\sqrt{\frac{5}{29}}}{3}&0&\frac{29\sqrt{3}+\sqrt{145}+58}{174}&0\\\
0&\frac{-15\sqrt{3}+\sqrt{145}+10}{30}&0&\frac{5-\sqrt{145}}{15}&0&\frac{15\sqrt{3}+\sqrt{145}+10}{30}\\\
\end{array}\right)$}}\ \ \ .$ (22)

As demonstrated at the right of Eq. (21), restoring the dimensions of the CM
is achieved, according to Eq. (12) and (13), through insertion of
$\ell_{nm}^{2}$ or $\frac{\hbar^{2}}{\ell_{nm}^{2}}$ on the
$\langle\hat{\boldsymbol{\phi}}\hat{\boldsymbol{\phi}}\rangle$ or
$\langle\hat{\boldsymbol{\pi}}\hat{\boldsymbol{\pi}}\rangle$ submatrices,
respectively. Note that the dimensionless covariance matrix of this harmonic
approximation is universal, i.e., no trap or ion experimental parameters are
present. Therefore, all entanglement properties are pure numbers, independent
of the specifics of the chosen ion and strength of the quadratic trapping
potential. In particular, the two-mode logarithmic negativities are found to
be

$\displaystyle\mathcal{N}^{N=2}_{1|2}$ $\displaystyle=\frac{\ln 3}{\ln
16}=0.396241...$ (23) $\displaystyle\mathcal{N}^{N=3}_{1|2}$
$\displaystyle=\frac{\ln\left(-\frac{5220}{-145\left(4\sqrt{3}+9\right)-2\sqrt{145}\left(11\sqrt{3}+153\right)+\sqrt{435\left(-5952\sqrt{3}+4\left(105\sqrt{3}+299\right)\sqrt{145}+13571\right)}}\right)}{\ln
4}=0.384585...$ (24) $\displaystyle\mathcal{N}^{N=3}_{1|3}$
$\displaystyle=\frac{\ln\left(\frac{5}{3}\left(49-4\sqrt{145}\right)\right)}{\ln
16}=0.118608...$ (25)

where the third mode in the $N=3$ examples has been traced. For the two-ion
chain, the single-mode entanglement entropies are found to be

$E_{1}^{N=2}=\frac{1}{\ln
4096}\left[\left(\sqrt{6\left(2\sqrt{3}+3\right)}+6\right)\ln\left(\sqrt{6\left(2\sqrt{3}+3\right)}+6\right)\right.\\\
\left.-12\ln
12-\left(\sqrt{6\left(2\sqrt{3}+3\right)}-6\right)\ln\left(\sqrt{6\left(2\sqrt{3}+3\right)}-6\right)\right]=0.13618...$
(26)

consistent with Ref. [76] where the two-mode state of the local motional modes
was taken to be a two mode squeezed vacuum state state in normal form, i.e.,
consistent with Eq. (21) up to local operations.

##  Appendix B Local Motional Modes in Fock-space

A significant convenience of Gaussian CV quantum states is that they are
completely described by a covariance matrix of second moments, $\sigma$, and a
vector of first moments, $\bar{\mathbf{r}}$, in phase space. The
dimensionality of these two objects scales only linearly with the number of
modes in the system, allowing states of significant size to be computationally
practical. However, transferring entanglement from CV to finite-dimensional
quantum systems—e.g., from local motional degrees of freedom to internal
energy levels of trapped ions as first proposed by Ref. [76]—inspires
departing from the succinct phase-space language by expanding the CV state in
the infinite tower of Fock states. From this basis, Ref. [76] discussed the
experimental viability of dynamically swapping the lowest-excitation motional
Fock states with an internal state ion qubit.

Before evaluating the Fock basis decomposition for an arbitrary covariance
matrix, consider first the Fock representation of a two mode squeezed vacuum
state [57],

$|\psi_{r}\rangle=e^{r\left(\hat{a}_{1}^{\dagger}\hat{a}_{2}^{\dagger}-\hat{a}_{1}\hat{a}_{2}\right)}|0,0\rangle\
\ \ ,$ (27)

where $\hat{a}_{j},\hat{a}_{j}^{\dagger}$ are bosonic creation and
annihilation operators,
$\left[\hat{a}_{i},\hat{a}_{j}^{\dagger}\right]=\delta_{ij}$. As usual, the
associated covariance matrix can be calculated via application of the two-mode
squeezing symplectic operation on the vacuum (identity) covariance matrix,

$\sigma_{r}=S_{r}S_{r}^{T}\qquad,\qquad S_{r}=\begin{pmatrix}\cosh r&0&\sinh
r&0\\\ 0&\cosh r&0&-\sinh r\\\ \sinh r&0&\cosh r&0\\\ 0&-\sinh r&0&\cosh
r\end{pmatrix}\ \ \ .$ (28)

Evaluation of Eq. (27) can be performed utilizing the su(1,1) algebra, with
generators $K_{\pm},K_{0}$ satisfying the following commutation relations,

$\left[K_{+},K_{-}\right]=-2K_{0}\qquad,\qquad\left[K_{0},K_{\pm}\right]=\pm
K_{\pm}\ \ \ .$ (29)

Transformations in this context can be simplified by using so-called
disentanglement theorems [151, 152],

$e^{v_{+}K_{+}+v_{-}K_{-}+v_{0}K_{t}}=e^{t_{+}K_{+}}e^{\left(\ln
t_{0}\right)K_{0}}e^{t_{-}K_{-}}\ \ \ .$ (30)

Determination of the coefficients $\vec{t}$ from those of $\vec{v}$ can be
achieved by utilizing a small-dimensional representation of the
generators—e.g., $K_{+}=\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}$,
$K_{-}=\begin{pmatrix}0&0\\\ -1&0\end{pmatrix}$, $K_{0}=\frac{1}{2}Z$, with
$Z$ the third Pauli matrix—and equating each matrix element of the two
transformations. The resulting coefficients are

$\displaystyle t_{0}$
$\displaystyle=\left(\cosh(f)-\frac{v_{0}}{2f}\sinh(f)\right)^{-2}$ (31)
$\displaystyle t_{\pm}$
$\displaystyle=\frac{v_{\pm}\sinh(f)}{f\Big{\lvert}\cosh(f)-\frac{v_{0}}{2f}\sinh(f)\Big{\rvert}}\
\ \ ,$ (32)

with parameter $f^{2}=\frac{1}{4}v_{0}^{2}-v_{+}v_{-}$. For application to the
two-mode squeezed vacuum state, one may identify
$K_{+}=\hat{a}_{1}^{\dagger}\hat{a}_{2}^{\dagger}$ and
$K_{-}=\hat{a}_{1}\hat{a}_{2}$, leading to
$K_{0}=\frac{1}{2}\left(\hat{a}_{1}^{\dagger}\hat{a}_{1}+\hat{a}_{2}^{\dagger}\hat{a}_{2}+1\right)$.
The squeezing operation may thus be written in the well-known Fock basis form,

$\displaystyle
e^{r\left(\hat{a}_{1}^{\dagger}\hat{a}_{2}^{\dagger}-\hat{a}_{1}\hat{a}_{2}\right)}|0,0\rangle$
$\displaystyle=e^{\tanh(r)\hat{a}_{1}^{\dagger}\hat{a}_{2}^{\dagger}}e^{-\left(\ln\cosh{r}\right)\left(\hat{a}_{1}^{\dagger}\hat{a}_{1}+\hat{a}_{2}^{\dagger}\hat{a}_{2}+1\right)}e^{-\tanh(r)\hat{a}_{1}\hat{a}_{2}}|0,0\rangle$
(33)
$\displaystyle=\frac{1}{\cosh(r)}\sum_{n=0}^{\infty}\tanh^{n}(r)|n,n\rangle\ \
\ $ (34) $\displaystyle\equiv\sqrt{1-e^{-2\beta}}\sum_{n=0}^{\infty}e^{-\beta
n}|n,n\rangle\ \ \ ,$ (35)

with $\beta\equiv-\ln\tanh(r)$ commonly introduced. When acting on the vacuum
state in Eq. (33), only the identity survives from the first exponential on
the right, the second exponential becomes simply a normalization factor, and
the last exponential produces a tower of states with equal occupation and
dominant support in the space of low excitations. As stronger squeezing is
applied, the distribution in Fock space becomes increasingly delocalized and
the entanglement increases. For example, the logarithmic negativity, which is
a necessary and sufficient measure of entanglement in this two-mode context,
is directly proportional to the squeezing parameter as
$\mathcal{N}=\frac{2|r|}{\ln 2}$.

For general Gaussian states, it is valuable to calculate Fock-basis density
matrix elements directly. Each Fock state,
$|\mathbf{n}\rangle=\left(n_{1},\cdots,n_{N}\right)$, is labeled by a vector
of numbers corresponding to the quantized excitations present in each of the
$N$ modes. With diagonal elements of the density matrix,
$\langle\mathbf{n}|\rho|\mathbf{n}\rangle$, commonly calculated for
applications in Gaussian boson sampling [153, 154], computational techniques
for the generic matrix element, $\langle\mathbf{m}|\rho|\mathbf{n}\rangle$
applicable to mixed quantum states have been developed recently [155]. For
states with $\bar{\mathbf{r}}=\mathbf{0}$, the techniques of Ref. [155] are
summarized by the relations,

$\langle\mathbf{m}|\rho|\mathbf{n}\rangle=\frac{\text{haf}\left(\tilde{A}\right)}{\sqrt{\det\left(\sigma_{Q}\right)\prod_{j=1}^{N}m_{j}!n_{j}!}}\qquad,\qquad\tilde{A}=\bar{A}-\text{diag}(\bar{A})\quad,\quad
A=X\left(\mathbb{I}_{2N}-\sigma_{Q}^{-1}\right)$ (36)

with

$X=\begin{pmatrix}0&\mathbb{I}_{N}\\\
\mathbb{I}_{N}&0\end{pmatrix}\quad,\quad\sigma_{Q}=\frac{1}{2}\left(U_{\hat{a}}\sigma_{xxpp}U_{\hat{a}}^{\dagger}+\mathbb{I}_{2N}\right)\quad,\quad
U_{\hat{a}}=\frac{1}{\sqrt{2}}\begin{pmatrix}\mathbb{I}_{N}&i\mathbb{I}_{N}\\\
\mathbb{I}_{N}&-i\mathbb{I}_{N}\end{pmatrix}$ (37)

where $U$ is the matrix transferring phase space operators into bosonic
creation and annihilation operators and $\sigma_{xxpp}$ is the position-
momentum reordered covariance matrix according to the basis,
$\hat{\mathbf{r}}^{\prime}=\left\\{\hat{\phi}_{1},\hat{\phi}_{2},\cdots,\hat{\phi}_{N},\hat{\pi}_{1},\hat{\pi}_{2},\cdots,\hat{\pi}_{N}\right\\}$.
Remaining to be discussed are the hafnian and $\bar{A}$. As usual, the hafnian
is a matrix property that can be calculated as

$\text{haf}(B)=\sum_{\vec{j}\in\mathcal{M}}B_{j_{1},j_{2}}\cdots
B_{j_{n-1},j_{n}}$ (38)

where $\mathcal{M}$ is the set of all ways to partition the $n$ elements of
$B$ into non-repeating pairs. For example, $\text{haf}\left(B_{4\times
4}\right)=B_{12}B_{34}+B_{13}B_{24}+B_{14}B_{23}$, and the hafnian of an odd-
dimensioned matrix is zero. The hafnian of a zero-dimensional matrix is set to
be unity. While this computational form of the hafnian makes manifest its
combinatoric nature, note that Wick’s theorem also allows
$\text{haf}\left(B\right)$ to be calculated as a Gaussian expectation value of
the product of all phase space operators subject to a covariance matrix
governed by $B$, and first moment displacements, $\text{diag}(B)$. Finally,
the matrix $\bar{A}$ is constructed from $A$ through repetition of its
$j^{\text{th}}$ rows and columns $n_{j}$($m_{j}$) times for the first(second)
$N$ dimensions to create a matrix of dimension,
$\left[\bar{A}\right]=\sum_{j}n_{j}+m_{j}$, scaling with the total number of
Fock space excitations involved in the matrix element.

Applying this formalism to the trapped ion local motional modes, consider the
harmonic limit (Eq. (7)) of the two-ion system, characterized by the
covariance matrix of Eq. (21). In the Fock basis, the density matrix elements
for the first three levels in the basis of
$\left\\{|00\rangle,|01\rangle,|02\rangle,|10\rangle,|11\rangle,|12\rangle,|20\rangle,|21\rangle,|22\rangle\right\\}$
will read

$\bar{\rho}_{(0,1,2)}=\scalebox{0.95}{\mbox{$\displaystyle\left(\begin{array}[]{ccccccccc}0.963&0&\pagecolor{TealBlue!25}-0.0913&0&0.129&0&\pagecolor{TealBlue!25}-0.0913&0&0.0259\\\
0&0&0&0&0&0&0&0&0\\\
\pagecolor{TealBlue!25}-0.0913&0&\pagecolor{TealBlue!25}0.00865&0&\pagecolor{TealBlue!25}-0.0122&0&\pagecolor{TealBlue!25}0.00865&0&\pagecolor{TealBlue!25}-0.00246\\\
0&0&0&0&0&0&0&0&0\\\
0.129&0&\pagecolor{TealBlue!25}-0.0122&0&0.0173&0&\pagecolor{TealBlue!25}-0.0122&0&0.00348\\\
0&0&0&0&0&0&0&0&0\\\
\pagecolor{TealBlue!25}-0.0913&0&\pagecolor{TealBlue!25}0.00865&0&\pagecolor{TealBlue!25}-0.0122&0&\pagecolor{TealBlue!25}0.00865&0&\pagecolor{TealBlue!25}-0.00246\\\
0&0&0&0&0&0&0&0&0\\\
0.0259&0&\pagecolor{TealBlue!25}-0.00246&0&0.00348&0&\pagecolor{TealBlue!25}-0.00246&0&0.000698\\\
\end{array}\right)$}}\ \ \ .$ (39)

Note that, as a subset, this portion of the density matrix is neither pure nor
normalized. The presence of the highlighted elements is a clear indication
that the two-ion state of local motional modes is not simply a two-mode
squeezed vacuum state, Eq. (35), as presented in Ref. [76]. However, one can
transform the two-ion local motional modes into a two-mode squeezed vacuum
state through local symplectic operations, leading to agreement with the
reported entanglement properties for the two-ion system.

While the presence of local operations does not impact the value of
entanglement measures, they do impact the distribution of a CV quantum state
in Fock space. If aims of entanglement extraction from the local motional
modes to the ion internal levels are pursued through exchange with the lowest
Fock states, as first proposed by Ref. [76], it is important to assure
dominant support is present in this low-occupation Hilbert subspace. For a
two-ion chain, the left panel of Figure 3 demonstrates the deviation of this
probability from unity, $P_{\overline{\text{qubit}}}$, i.e., the probability
of the local motional modes being outside the two-state subspace of Fock
states $|0\rangle,|1\rangle$, as a function of local operations. To
parameterize single-mode symplectic operations, one may perform a rudimentary
application of the Bloch-Messiah decomposition—allowing a symplectic
transformation to be decomposed into a set of single-mode squeezing ($Z$)
operations surrounded by a pair of passive ($O$) operations (energy-
preserving, e.g., beam-splitters and phase shifters)—with real numbers
$(\phi_{1},\phi_{2},z)$ as [156, 157, 57]

$S=O_{1}ZO_{2}\qquad,\qquad
O_{j}=\begin{pmatrix}\operatorname{Re}(\gamma_{j})&\operatorname{Im}(\gamma_{j})\\\
-\operatorname{Im}(\gamma_{j})&\operatorname{Re}(\gamma_{j})\end{pmatrix}\quad,\quad
Z=\begin{pmatrix}z&0\\\ 0&\frac{1}{z}\end{pmatrix}\ \ \ ,$ (40)

where $\gamma_{j}=e^{i\phi_{j}}$ is a one-dimensional unitary.

Figure 3: For the local motional modes of a two-ion chain, the probability
outside the lowest two-level Fock subspace
($P_{\overline{\text{qubit}}}=1-P_{\text{qubit}}$) is shown as a function of
parameters governing one-mode symplectic operators applied symmetrically. That
expressed directly by the hardware upon trapping without further operations,
$(z,\phi_{1})=(1,0)$, is indicated in blue. The configurations optimally
localizing the domain of support to this qubit-sized subspace are indicated by
solid and dashed black lines, achieved by locally transforming the covariance
matrix to normal form with constant diagonal
$\langle\hat{\ell}_{j}\hat{\ell}_{j}\rangle=\langle\hat{\pi}_{j}\hat{\pi}_{j}\rangle$.
(right) Scaling of probability outside the qudit subspace as a function of
increasing Fock-space qudit dimension.

When squeezing each local motional mode by
$z=\left(\sigma_{2,2}/\sigma_{1,1}\right)^{1/4}$, the quantum state upon
trapping of an ion pair (blue point of Fig. 3) can be shifted to a form that
optimally compacts the support of the quantum state in the low Fock states
(green point, solid line). The alternate solution (dashed line) simply
exchanges the position and momentum components, and thus both optimal points
yield a two-mode squeezed vacuum state with Fock basis characterized by Eq.
(35). It is anticipated that this determination of the true form of the local
motional mode CV quantum state, and its guidance of additional single-mode
squeezing operations, will improve experimental implementations of ion
motional entanglement extraction.

##  Appendix C Numerical Tables

The following Tables 1-7 provide numerical values presented in Figures 1 and 2
of the main text and Figure 3 of Appendix B.

| Ions | Scalar
---|---|---
separation, $\tilde{r}/d$ | $\mathcal{N}_{A|B}\left(\sigma^{(t)}\right)$ | $\mathcal{N}_{A|B}\left(\sigma^{(m,\phi)}\right)$ | $\mathcal{N}_{A|B}\left(\sigma^{(m,\pi)}\right)$ | $\mathcal{N}_{A|B}\left(\sigma^{(t)}\right)$ | $\mathcal{N}_{A|B}\left(\sigma^{(m,\phi)}\right)$ | $\mathcal{N}_{A|B}\left(\sigma^{(m,\pi)}\right)$
$0$ | $3.66\times 10^{-1}$ | $3.94\times 10^{-1}$ | $9.25\times 10^{-1}$ | $4.44\times 10^{-1}$ | $5.00\times 10^{-1}$ | $2.30$
$1$ | $5.95\times 10^{-2}$ | $1.14\times 10^{-1}$ | $7.38\times 10^{-1}$ | $0$ | $9.63\times 10^{-2}$ | $2.08$
$2$ | $0$ | $5.48\times 10^{-2}$ | $6.43\times 10^{-1}$ | $0$ | $4.12\times 10^{-2}$ | $1.97$
$3$ | $0$ | $3.25\times 10^{-2}$ | $5.81\times 10^{-1}$ | $0$ | $2.29\times 10^{-2}$ | $1.90$
$4$ | $0$ | $2.16\times 10^{-2}$ | $5.36\times 10^{-1}$ | $0$ | $1.46\times 10^{-2}$ | $1.85$
$5$ | $0$ | $1.54\times 10^{-2}$ | $5.00\times 10^{-1}$ | $0$ | $1.01\times 10^{-2}$ | $1.81$
$6$ | $0$ | $1.16\times 10^{-2}$ | $4.71\times 10^{-1}$ | $0$ | $7.40\times 10^{-3}$ | $1.78$
$7$ | $0$ | $9.04\times 10^{-3}$ | $4.46\times 10^{-1}$ | $0$ | $5.66\times 10^{-3}$ | $1.76$
$8$ | $0$ | $7.26\times 10^{-3}$ | $4.25\times 10^{-1}$ | $0$ | $4.47\times 10^{-3}$ | $1.73$
$9$ | $0$ | $5.97\times 10^{-3}$ | $4.06\times 10^{-1}$ | $0$ | $3.62\times 10^{-3}$ | $1.71$
$10$ | $0$ | $4.99\times 10^{-3}$ | $3.90\times 10^{-1}$ | $0$ | $2.99\times 10^{-3}$ | $1.70$
Table 1: Numerical values for A-B region sizes of 1 ion or lattice-site in Fig. 1. | Ions | Scalar
---|---|---
separation, $\tilde{r}/d$ | $\mathcal{N}_{A|B}\left(\sigma^{(t)}\right)$ | $\mathcal{N}_{A|B}\left(\sigma^{(m,\phi)}\right)$ | $\mathcal{N}_{A|B}\left(\sigma^{(m,\pi)}\right)$ | $\mathcal{N}_{A|B}\left(\sigma^{(t)}\right)$ | $\mathcal{N}_{A|B}\left(\sigma^{(m,\phi)}\right)$ | $\mathcal{N}_{A|B}\left(\sigma^{(m,\pi)}\right)$
$0$ | $6.29\times 10^{-1}$ | $6.73\times 10^{-1}$ | $1.22$ | $7.89\times 10^{-1}$ | $8.39\times 10^{-1}$ | $2.71$
$\frac{1}{3}$ | $2.01\times 10^{-1}$ | $2.47\times 10^{-1}$ | $9.91\times 10^{-1}$ | $1.68\times 10^{-1}$ | $2.19\times 10^{-1}$ | $2.43$
$\frac{2}{3}$ | $6.77\times 10^{-2}$ | $1.40\times 10^{-1}$ | $8.73\times 10^{-1}$ | $2.77\times 10^{-2}$ | $1.14\times 10^{-1}$ | $2.29$
$1$ | $1.27\times 10^{-2}$ | $9.25\times 10^{-2}$ | $7.95\times 10^{-1}$ | $4.41\times 10^{-3}$ | $7.21\times 10^{-2}$ | $2.20$
$\frac{4}{3}$ | $2.96\times 10^{-3}$ | $6.65\times 10^{-2}$ | $7.36\times 10^{-1}$ | $2.07\times 10^{-3}$ | $5.03\times 10^{-2}$ | $2.14$
$\frac{5}{3}$ | $1.45\times 10^{-3}$ | $5.05\times 10^{-2}$ | $6.90\times 10^{-1}$ | $1.19\times 10^{-3}$ | $3.73\times 10^{-2}$ | $2.09$
$2$ | $8.58\times 10^{-4}$ | $3.98\times 10^{-2}$ | $6.52\times 10^{-1}$ | $6.99\times 10^{-4}$ | $2.88\times 10^{-2}$ | $2.05$
$\frac{7}{3}$ | $5.31\times 10^{-4}$ | $3.23\times 10^{-2}$ | $6.19\times 10^{-1}$ | $3.86\times 10^{-4}$ | $2.30\times 10^{-2}$ | $2.02$
$\frac{8}{3}$ | $3.20\times 10^{-4}$ | $2.68\times 10^{-2}$ | $5.91\times 10^{-1}$ | $1.65\times 10^{-4}$ | $1.88\times 10^{-2}$ | $1.99$
$3$ | $1.71\times 10^{-4}$ | $2.26\times 10^{-2}$ | $5.66\times 10^{-1}$ | $0$ | $1.57\times 10^{-2}$ | $1.96$
$\frac{10}{3}$ | $5.99\times 10^{-5}$ | $1.93\times 10^{-2}$ | $5.44\times 10^{-1}$ | $0$ | $1.33\times 10^{-2}$ | $1.94$
$\frac{11}{3}$ | $0$ | $1.67\times 10^{-2}$ | $5.24\times 10^{-1}$ | $0$ | $1.14\times 10^{-2}$ | $1.92$
$4$ | $0$ | $1.46\times 10^{-2}$ | $5.06\times 10^{-1}$ | $0$ | $9.88\times 10^{-3}$ | $1.90$
$\frac{13}{3}$ | $0$ | $1.29\times 10^{-2}$ | $4.89\times 10^{-1}$ | $0$ | $8.65\times 10^{-3}$ | $1.89$
$\frac{14}{3}$ | $0$ | $1.15\times 10^{-2}$ | $4.74\times 10^{-1}$ | $0$ | $7.64\times 10^{-3}$ | $1.87$
$5$ | $0$ | $1.03\times 10^{-2}$ | $4.59\times 10^{-1}$ | $0$ | $6.80\times 10^{-3}$ | $1.86$
$\frac{16}{3}$ | $0$ | $9.28\times 10^{-3}$ | $4.46\times 10^{-1}$ | $0$ | $6.09\times 10^{-3}$ | $1.85$
$\frac{17}{3}$ | $0$ | $8.41\times 10^{-3}$ | $4.33\times 10^{-1}$ | $0$ | $5.49\times 10^{-3}$ | $1.83$
$6$ | $0$ | $7.66\times 10^{-3}$ | $4.22\times 10^{-1}$ | $0$ | $4.97\times 10^{-3}$ | $1.82$
$\frac{19}{3}$ | $0$ | $7.01\times 10^{-3}$ | $4.10\times 10^{-1}$ | $0$ | $4.53\times 10^{-3}$ | $1.81$
$\frac{20}{3}$ | $0$ | $6.43\times 10^{-3}$ | $4.00\times 10^{-1}$ | $0$ | $4.14\times 10^{-3}$ | $1.80$
$7$ | $0$ | $5.93\times 10^{-3}$ | $3.90\times 10^{-1}$ | $0$ | $3.80\times 10^{-3}$ | $1.79$
$\frac{22}{3}$ | $0$ | $5.48\times 10^{-3}$ | $3.81\times 10^{-1}$ | $0$ | $3.50\times 10^{-3}$ | $1.78$
$\frac{23}{3}$ | $0$ | $5.09\times 10^{-3}$ | $3.72\times 10^{-1}$ | $0$ | $3.23\times 10^{-3}$ | $1.78$
$8$ | $0$ | $4.73\times 10^{-3}$ | $3.63\times 10^{-1}$ | $0$ | $2.99\times 10^{-3}$ | $1.77$
$\frac{25}{3}$ | $0$ | $4.41\times 10^{-3}$ | $3.55\times 10^{-1}$ | $0$ | $2.78\times 10^{-3}$ | $1.76$
$\frac{26}{3}$ | $0$ | $4.13\times 10^{-3}$ | $3.47\times 10^{-1}$ | $0$ | $2.59\times 10^{-3}$ | $1.75$
$9$ | $0$ | $3.87\times 10^{-3}$ | $3.39\times 10^{-1}$ | $0$ | $2.42\times 10^{-3}$ | $1.75$
Table 2: Numerical values for A-B region sizes of 3 ions or lattice-sites in Fig. 1. | Ions | Scalar
---|---|---
separation, $\tilde{r}/d$ | $\mathcal{N}_{A|B}\left(\sigma^{(t)}\right)$ | $\mathcal{N}_{A|B}\left(\sigma^{(m,\phi)}\right)$ | $\mathcal{N}_{A|B}\left(\sigma^{(m,\pi)}\right)$ | $\mathcal{N}_{A|B}\left(\sigma^{(t)}\right)$ | $\mathcal{N}_{A|B}\left(\sigma^{(m,\phi)}\right)$ | $\mathcal{N}_{A|B}\left(\sigma^{(m,\pi)}\right)$
$0$ | $7.75\times 10^{-1}$ | $8.18\times 10^{-1}$ | $1.33$ | $9.67\times 10^{-1}$ | $1.01$ | $2.89$
$\frac{1}{5}$ | $2.89\times 10^{-1}$ | $3.32\times 10^{-1}$ | $1.09$ | $2.55\times 10^{-1}$ | $3.00\times 10^{-1}$ | $2.59$
$\frac{2}{5}$ | $1.32\times 10^{-1}$ | $2.02\times 10^{-1}$ | $9.69\times 10^{-1}$ | $8.55\times 10^{-2}$ | $1.71\times 10^{-1}$ | $2.44$
$\frac{3}{5}$ | $5.77\times 10^{-2}$ | $1.41\times 10^{-1}$ | $8.85\times 10^{-1}$ | $2.87\times 10^{-2}$ | $1.15\times 10^{-1}$ | $2.35$
$\frac{4}{5}$ | $2.36\times 10^{-2}$ | $1.06\times 10^{-1}$ | $8.22\times 10^{-1}$ | $1.40\times 10^{-2}$ | $8.42\times 10^{-2}$ | $2.28$
$1$ | $1.17\times 10^{-2}$ | $8.32\times 10^{-2}$ | $7.72\times 10^{-1}$ | $8.53\times 10^{-3}$ | $6.48\times 10^{-2}$ | $2.22$
$\frac{6}{5}$ | $7.11\times 10^{-3}$ | $6.74\times 10^{-2}$ | $7.30\times 10^{-1}$ | $5.66\times 10^{-3}$ | $5.17\times 10^{-2}$ | $2.18$
$\frac{7}{5}$ | $4.74\times 10^{-3}$ | $5.60\times 10^{-2}$ | $6.95\times 10^{-1}$ | $3.86\times 10^{-3}$ | $4.24\times 10^{-2}$ | $2.14$
$\frac{8}{5}$ | $3.30\times 10^{-3}$ | $4.73\times 10^{-2}$ | $6.64\times 10^{-1}$ | $2.63\times 10^{-3}$ | $3.54\times 10^{-2}$ | $2.11$
$\frac{9}{5}$ | $2.32\times 10^{-3}$ | $4.06\times 10^{-2}$ | $6.37\times 10^{-1}$ | $1.73\times 10^{-3}$ | $3.01\times 10^{-2}$ | $2.08$
$2$ | $1.62\times 10^{-3}$ | $3.53\times 10^{-2}$ | $6.13\times 10^{-1}$ | $1.06\times 10^{-3}$ | $2.59\times 10^{-2}$ | $2.06$
$\frac{11}{5}$ | $1.09\times 10^{-3}$ | $3.10\times 10^{-2}$ | $5.91\times 10^{-1}$ | $5.78\times 10^{-4}$ | $2.26\times 10^{-2}$ | $2.04$
$\frac{12}{5}$ | $6.87\times 10^{-4}$ | $2.74\times 10^{-2}$ | $5.70\times 10^{-1}$ | $2.65\times 10^{-4}$ | $1.98\times 10^{-2}$ | $2.02$
$\frac{13}{5}$ | $3.86\times 10^{-4}$ | $2.45\times 10^{-2}$ | $5.52\times 10^{-1}$ | $9.64\times 10^{-5}$ | $1.76\times 10^{-2}$ | $2.00$
$\frac{14}{5}$ | $1.83\times 10^{-4}$ | $2.20\times 10^{-2}$ | $5.35\times 10^{-1}$ | $2.62\times 10^{-5}$ | $1.57\times 10^{-2}$ | $1.98$
$3$ | $6.87\times 10^{-5}$ | $1.99\times 10^{-2}$ | $5.19\times 10^{-1}$ | $9.51\times 10^{-6}$ | $1.41\times 10^{-2}$ | $1.97$
$\frac{16}{5}$ | $2.00\times 10^{-5}$ | $1.80\times 10^{-2}$ | $5.04\times 10^{-1}$ | $5.13\times 10^{-6}$ | $1.27\times 10^{-2}$ | $1.95$
$\frac{17}{5}$ | $7.42\times 10^{-6}$ | $1.65\times 10^{-2}$ | $4.90\times 10^{-1}$ | $3.28\times 10^{-6}$ | $1.16\times 10^{-2}$ | $1.94$
$\frac{18}{5}$ | $4.01\times 10^{-6}$ | $1.51\times 10^{-2}$ | $4.77\times 10^{-1}$ | $2.27\times 10^{-6}$ | $1.05\times 10^{-2}$ | $1.93$
$\frac{19}{5}$ | $2.59\times 10^{-6}$ | $1.39\times 10^{-2}$ | $4.65\times 10^{-1}$ | $1.63\times 10^{-6}$ | $9.66\times 10^{-3}$ | $1.91$
$4$ | $1.82\times 10^{-6}$ | $1.28\times 10^{-2}$ | $4.53\times 10^{-1}$ | $1.18\times 10^{-6}$ | $8.88\times 10^{-3}$ | $1.90$
$\frac{21}{5}$ | $1.33\times 10^{-6}$ | $1.19\times 10^{-2}$ | $4.42\times 10^{-1}$ | $8.56\times 10^{-7}$ | $8.20\times 10^{-3}$ | $1.89$
$\frac{22}{5}$ | $9.99\times 10^{-7}$ | $1.10\times 10^{-2}$ | $4.32\times 10^{-1}$ | $6.05\times 10^{-7}$ | $7.59\times 10^{-3}$ | $1.88$
$\frac{23}{5}$ | $7.55\times 10^{-7}$ | $1.03\times 10^{-2}$ | $4.21\times 10^{-1}$ | $4.05\times 10^{-7}$ | $7.05\times 10^{-3}$ | $1.87$
$\frac{24}{5}$ | $5.69\times 10^{-7}$ | $9.61\times 10^{-3}$ | $4.12\times 10^{-1}$ | $2.41\times 10^{-7}$ | $6.56\times 10^{-3}$ | $1.87$
$5$ | $4.21\times 10^{-7}$ | $9.00\times 10^{-3}$ | $4.03\times 10^{-1}$ | $1.05\times 10^{-7}$ | $6.12\times 10^{-3}$ | $1.86$
$\frac{26}{5}$ | $3.01\times 10^{-7}$ | $8.45\times 10^{-3}$ | $3.94\times 10^{-1}$ | $0$ | $5.73\times 10^{-3}$ | $1.85$
$\frac{27}{5}$ | $2.01\times 10^{-7}$ | $7.94\times 10^{-3}$ | $3.85\times 10^{-1}$ | $0$ | $5.37\times 10^{-3}$ | $1.84$
$\frac{28}{5}$ | $1.17\times 10^{-7}$ | $7.48\times 10^{-3}$ | $3.77\times 10^{-1}$ | $0$ | $5.04\times 10^{-3}$ | $1.83$
$\frac{29}{5}$ | $4.50\times 10^{-8}$ | $7.07\times 10^{-3}$ | $3.69\times 10^{-1}$ | $0$ | $4.75\times 10^{-3}$ | $1.83$
$6$ | $0$ | $6.68\times 10^{-3}$ | $3.62\times 10^{-1}$ | $0$ | $4.48\times 10^{-3}$ | $1.82$
$\frac{31}{5}$ | $0$ | $6.33\times 10^{-3}$ | $3.55\times 10^{-1}$ | $0$ | $4.23\times 10^{-3}$ | $1.81$
$\frac{32}{5}$ | $0$ | $6.00\times 10^{-3}$ | $3.48\times 10^{-1}$ | $0$ | $4.00\times 10^{-3}$ | $1.81$
$\frac{33}{5}$ | $0$ | $5.70\times 10^{-3}$ | $3.41\times 10^{-1}$ | $0$ | $3.79\times 10^{-3}$ | $1.80$
$\frac{34}{5}$ | $0$ | $5.43\times 10^{-3}$ | $3.34\times 10^{-1}$ | $0$ | $3.59\times 10^{-3}$ | $1.80$
$7$ | $0$ | $5.17\times 10^{-3}$ | $3.28\times 10^{-1}$ | $0$ | $3.42\times 10^{-3}$ | $1.79$
$\frac{36}{5}$ | $0$ | $4.93\times 10^{-3}$ | $3.22\times 10^{-1}$ | $0$ | $3.25\times 10^{-3}$ | $1.78$
$\frac{37}{5}$ | $0$ | $4.71\times 10^{-3}$ | $3.16\times 10^{-1}$ | $0$ | $3.10\times 10^{-3}$ | $1.78$
$\frac{38}{5}$ | $0$ | $4.50\times 10^{-3}$ | $3.10\times 10^{-1}$ | $0$ | $2.95\times 10^{-3}$ | $1.77$
$\frac{39}{5}$ | $0$ | $4.30\times 10^{-3}$ | $3.04\times 10^{-1}$ | $0$ | $2.82\times 10^{-3}$ | $1.77$
$8$ | $0$ | $4.12\times 10^{-3}$ | $2.99\times 10^{-1}$ | $0$ | $2.69\times 10^{-3}$ | $1.76$
$\frac{41}{5}$ | $0$ | $3.95\times 10^{-3}$ | $2.93\times 10^{-1}$ | $0$ | $2.58\times 10^{-3}$ | $1.76$
$\frac{42}{5}$ | $0$ | $3.79\times 10^{-3}$ | $2.88\times 10^{-1}$ | $0$ | $2.47\times 10^{-3}$ | $1.75$
$\frac{43}{5}$ | $0$ | $3.65\times 10^{-3}$ | $2.83\times 10^{-1}$ | $0$ | $2.36\times 10^{-3}$ | $1.75$
$\frac{44}{5}$ | $0$ | $3.50\times 10^{-3}$ | $2.78\times 10^{-1}$ | $0$ | $2.27\times 10^{-3}$ | $1.75$
$9$ | $0$ | $3.37\times 10^{-3}$ | $2.73\times 10^{-1}$ | $0$ | $2.18\times 10^{-3}$ | $1.74$
Table 3: Numerical values for A-B region sizes of 5 ions or lattice-sites in Fig. 1. |  | Fidelity
---|---|---
Region Size | Squeezing, $z$ | Raw | Locally Squeezed
$2$ | $4.119$ | $3.49\times 10^{-1}$ | $0.814$
$4$ | $3.666$ | $1.88\times 10^{-1}$ | $0.767$
$6$ | $3.507$ | $1.03\times 10^{-1}$ | $0.738$
$8$ | $3.420$ | $5.79\times 10^{-2}$ | $0.717$
$10$ | $3.362$ | $3.28\times 10^{-2}$ | $0.699$
$12$ | $3.315$ | $1.89\times 10^{-2}$ | $0.682$
$14$ | $3.274$ | $1.10\times 10^{-2}$ | $0.667$
$16$ | $3.234$ | $6.56\times 10^{-3}$ | $0.652$
$18$ | $3.194$ | $3.99\times 10^{-3}$ | $0.636$
$20$ | $3.152$ | $2.49\times 10^{-3}$ | $0.619$
$22$ | $3.106$ | $1.60\times 10^{-3}$ | $0.600$
$24$ | $3.054$ | $1.08\times 10^{-3}$ | $0.576$
$26$ | $2.993$ | $7.59\times 10^{-4}$ | $0.544$
$28$ | $2.919$ | $5.73\times 10^{-4}$ | $0.493$
$30$ | $2.821$ | $4.47\times 10^{-4}$ | $0.368$
Table 4: Numerical values for $N=30$ ion chain shown in Fig. 2. |  | Fidelity
---|---|---
Region Size | Squeezing, $z$ | Raw | Locally Squeezed
$2$ | $5.118$ | $2.83\times 10^{-1}$ | $0.827$
$4$ | $4.579$ | $1.25\times 10^{-1}$ | $0.784$
$6$ | $4.390$ | $5.70\times 10^{-2}$ | $0.758$
$8$ | $4.292$ | $2.62\times 10^{-2}$ | $0.739$
$10$ | $4.229$ | $1.22\times 10^{-2}$ | $0.724$
$12$ | $4.184$ | $5.67\times 10^{-3}$ | $0.710$
$14$ | $4.149$ | $2.67\times 10^{-3}$ | $0.698$
$16$ | $4.118$ | $1.26\times 10^{-3}$ | $0.686$
$18$ | $4.091$ | $6.04\times 10^{-4}$ | $0.675$
$20$ | $4.066$ | $2.91\times 10^{-4}$ | $0.665$
$22$ | $4.041$ | $1.42\times 10^{-4}$ | $0.654$
$24$ | $4.016$ | $6.97\times 10^{-5}$ | $0.644$
$26$ | $3.991$ | $3.47\times 10^{-5}$ | $0.634$
$28$ | $3.965$ | $1.75\times 10^{-5}$ | $0.624$
$30$ | $3.938$ | $9.00\times 10^{-6}$ | $0.613$
$32$ | $3.910$ | $4.70\times 10^{-6}$ | $0.601$
$34$ | $3.879$ | $2.51\times 10^{-6}$ | $0.588$
$36$ | $3.846$ | $1.37\times 10^{-6}$ | $0.574$
$38$ | $3.811$ | $7.67\times 10^{-7}$ | $0.557$
$40$ | $3.771$ | $4.44\times 10^{-7}$ | $0.538$
$42$ | $3.728$ | $2.66\times 10^{-7}$ | $0.515$
$44$ | $3.679$ | $1.67\times 10^{-7}$ | $0.485$
$46$ | $3.623$ | $1.12\times 10^{-7}$ | $0.445$
$48$ | $3.556$ | $8.03\times 10^{-8}$ | $0.387$
$50$ | $3.473$ | $6.09\times 10^{-8}$ | $0.271$
Table 5: Numerical values for $N=50$ ion chain shown in Fig. 2. |  | Fidelity
---|---|---
Region Size | Squeezing, $z$ | Raw | Locally Squeezed
$2$ | $8.262$ | $1.73\times 10^{-1}$ | $0.850$
$4$ | $7.453$ | $4.86\times 10^{-2}$ | $0.813$
$6$ | $7.166$ | $1.40\times 10^{-2}$ | $0.792$
$8$ | $7.018$ | $4.08\times 10^{-3}$ | $0.776$
$10$ | $6.928$ | $1.19\times 10^{-3}$ | $0.763$
$12$ | $6.866$ | $3.51\times 10^{-4}$ | $0.752$
$14$ | $6.821$ | $1.03\times 10^{-4}$ | $0.742$
$16$ | $6.787$ | $3.06\times 10^{-5}$ | $0.733$
$18$ | $6.759$ | $9.05\times 10^{-6}$ | $0.724$
$20$ | $6.736$ | $2.68\times 10^{-6}$ | $0.716$
$22$ | $6.716$ | $7.98\times 10^{-7}$ | $0.709$
$24$ | $6.699$ | $2.37\times 10^{-7}$ | $0.701$
$26$ | $6.684$ | $7.08\times 10^{-8}$ | $0.694$
$28$ | $6.671$ | $2.12\times 10^{-8}$ | $0.688$
$30$ | $6.658$ | $6.34\times 10^{-9}$ | $0.681$
$32$ | $6.646$ | $1.90\times 10^{-9}$ | $0.675$
$34$ | $6.635$ | $5.71\times 10^{-10}$ | $0.669$
$36$ | $6.625$ | $1.72\times 10^{-10}$ | $0.663$
$38$ | $6.615$ | $5.19\times 10^{-11}$ | $0.657$
$40$ | $6.605$ | $1.57\times 10^{-11}$ | $0.652$
$42$ | $6.596$ | $4.76\times 10^{-12}$ | $0.646$
$44$ | $6.586$ | $1.45\times 10^{-12}$ | $0.640$
$46$ | $6.577$ | $4.41\times 10^{-13}$ | $0.635$
$48$ | $6.568$ | $1.35\times 10^{-13}$ | $0.630$
$50$ | $6.558$ | $4.12\times 10^{-14}$ | $0.624$
Table 6: Numerical values for $N=150$ ion chain shown in Fig. 2. | $P_{\overline{qudit}}$
---|---
Fock truncation/qudit dimension | Raw | Locally Squeezed
$2$ | $1.93\times 10^{-2}$ | $3.47\times 10^{-4}$
$3$ | $1.28\times 10^{-3}$ | $6.46\times 10^{-6}$
$4$ | $3.17\times 10^{-4}$ | $1.20\times 10^{-7}$
$5$ | $2.99\times 10^{-5}$ | $2.24\times 10^{-9}$
$6$ | $5.88\times 10^{-6}$ | $4.17\times 10^{-11}$
$7$ | $6.73\times 10^{-7}$ | $7.77\times 10^{-13}$
$8$ | $1.17\times 10^{-7}$ | $1.45\times 10^{-14}$
Table 7: Numerical values shown in Fig. 3.