# Qudits for Witnessing Quantum Gravity Induced Entanglement of Masses Under
Decoherence

Jules Tilly<EMAIL_ADDRESS>Department of Physics and Astronomy,
University College London, WC1E 6BT London, United Kingdom  Ryan J. Marshman
Department of Physics and Astronomy, University College London, WC1E 6BT
London, United Kingdom  Anupam Mazumdar  Van Swinderen Institute, University
of Groningen, 9747 AG Groningen, The Netherlands  Sougato Bose  Department of
Physics and Astronomy, University College London, WC1E 6BT London, United
Kingdom

###### Abstract

Recently a theoretical and an experimental protocol known as quantum gravity
induced entanglement of masses (QGEM) has been proposed to test the quantum
nature of gravity using two mesoscopic masses each placed in a superposition
of two locations. If, after eliminating all non-gravitational interactions
between them, the particles become entangled, one can conclude that the
gravitational potential is induced via a quantum mediator, i.e. a virtual
graviton. In this paper, we examine a range of different experimental set-ups,
considering different geometries and the number of spatially superposed states
taken, in order to determine which would generate entanglement faster. We
conclude that without decoherence, and given a maximum distance $\Delta x$
between any two spatial states of a superposition, a set of two qubits placed
in spatial superposition parallel to one another will outperform all other
models given realistic experimental parameters. Furthermore, when a
sufficiently high decoherence rate is introduced, multi-component
superpositions can outperform the two-qubit set-up. This is further verified
with an experimental simulation, showing that $O(10^{3})$ measurements are
required to reject the no entanglement hypothesis with a parallel qubits set-
up without decoherence at a 99.9$\%$ confidence level. The number of
measurements increases when decoherence is introduced. When the decoherence
rate reaches $0.125$ Hz, 6-dimensional qudits are required as the two-qubit
system entanglement cannot be witnessed anymore. However, in this case,
$O(10^{6})$ measurements will be required. One can group the witness operators
to measure in order to reduce the number of measurements (up to ten-fold).
However, this may be challenging to implement experimentally.

††preprint: APS/123-QED

##  I Introduction

Testing the quantum aspect of gravity is a central question of modern physics.
While many theories of quantum gravity have been developed, there remains no
consensus on how to unify the theories of general relativity and quantum
physics. However, the lack of experimental evidence to gravity being quantum
still remains an impediment in ongoing research [1].
A number of recent experimental proposals have focused on trying to unveil
General-relativity (GR) and post GR evidence [2, 3, 4, 5, 6, 7, 8, 9, 10], and
there has also been an initial attempt to rule out the semi-classical
treatment of quantum gravity in [11]. Even attempts of detecting the B-mode
polarisation of stochastic gravitational waves have uncertainties in the
initial conditions for the Universe, which does not provide a concrete test
for quantum nature of gravity [12].

In this regard, there has been recent progress in providing a razor-sharp
witness to test the existence of the quantum nature of a graviton in a table-
top experiment based on the observation that [13]:

  * •

The mediator of the universal gravitational interaction occurs via a spin-2
massless graviton, and if the graviton is quantum, it will entangle the two or
more quantum matter states, and provide the static gravitational potential at
the lowest order in the graviton/matter loop expansion.

  * •

The above statement strictly relies on two main assumptions: special
relativity and perturbative quantum field theory, which allows an off-
shell/virtual exchange of a graviton to mediate the gravitational force.

Based on this fact, a bonafide test for quantum nature of the gravitational
interaction has been proposed in [14], where the two mesoscopic masses were
allowed to interact in a spatially superposed quantum state via gravity. A
similar proposal has also been made in [15]. This has attracted significant
interest from the research community [16, 17, 18, 21, 19, 20, 21, 22, 23, 24,
25, 26], and an experimental initiative in creating macroscopic superposition
with Stern-Gerlach set-up [27]. The above proposal has been coined as Quantum
Gravity induced Entanglement of Masses (QGEM), which exploits the loophole
that as local operations and classical communications are unable to entangle
the two quantum states if they were not entangled, to begin with, quantum
communication is required to generate the entanglement as highlighted in [13].
How a non-local gravitational interaction [28, 29] entangles the two quantum
states of matter is also shown in [13].

This paper aims at analysing different possible set-ups for the QGEM proposal
in order to determine which will be most efficient to implement in a real
experiment. In particular, we consider how quickly different set-ups can
generate entanglement according to a generalised model of the QGEM experiment
and how many measurements would be required to witness that entanglement. In
addition, proposals surrounding the QGEM experiment have been limited to
qubits, so far. In this paper, we assess the implications of using quantum
objects of higher dimensions (three: qutrits, $D$: qudits). We test our
findings in the presence of decoherence, furthering the analyses presented in
[19], [21], [20] and [22].
Our key findings are that: the parallel set-ups [19] of the experiment
entangle faster than any other set-up considered; in the presence of
decoherence, using qudits may be beneficial for the experiment, and could even
be necessary; We provide an order of magnitude for the number of measurements
required to reach a 99.9% level of confidence (about 3.4$\sigma$) for
different decoherence rates (up to $0.125Hz$).
This paper will proceed as follows: Section II presents a generalised version
of the QGEM experiment, with arbitrary geometries and allowing for the use of
spatial qudits. In Section III we analyse the different set-ups proposed using
the entanglement entropy under the assumption of no decoherence. More
practical entanglement witnesses are introduced in Section IV and used to re-
analyse the experiment allowing for decoherence in Section V. Finally, a
statistical simulation of the results is presented in Section VI to
demonstrate how the required number of runs varies with the dimension of the
qudits, and the decoherence rate assumed.

##  II QGEM with Qudits

Linear set-up.

Parallel set-up.

Figure 1: Schematic representation of the experiment, shown here using qubits.
On the left, the liner set-up presented in [14, 15, 20]. On the right, a
schematic for the parallel set-up initially presented in [19]

The QGEM experimental protocol [14, 20] ensured that the gravitational
interaction would dominate over the electromagnetic interactions and the
Casimir induced vacuum fluctuations. This also provides one of the primary
constraints on the experimental set-up, which we will not seek to modify in
this paper. Specifically, there must be some minimum distance $d$ maintained
between the two particles. A schematic of two potential forms of the
experiment is presented in Figure 1.
The superposition width (or distance between the left-most and right-most
superposition instance of each qubit) is labelled $\Delta x$, and was
originally suggested to be $\sim 250\mu m$[14]. While the superposition is
held, the qubits are maintained at a distance such that their inner-most
superposition instances are a least $d$ apart (with $d\sim 200\mu m$ [14]). If
$d<200\mu m$ then forces such as Casimir-Polder forces and van der Waals
forces can affect the overall state of the system: gravity is no longer the
only possible quantum mediator for interactions between the two objects. There
has been further work considered to mediate this (for instance, see: [20]),
however, as it is beyond the primary considerations made here, we will not
look to include this here as it does not affect any final conclusions.
The two qubits are held in this superposition state for a time $\tau$ ([14]
suggests $\tau=2.5s$) after which the spatial states are brought together. The
qubits are then measured to determine whether they were entangled by their
gravitational interactions during the superposition period.
Nguyen and Bernards proposed a nearly identical scheme [19], in which the
superposition positions of each qubit are aligned parallel to each other as
opposed to linearly in [14, 20]. A schematic for this can be found in Figure
1. This scheme was motivated by the fact that maintaining the distance between
the two qubits would be easier in the parallel case than in the linear case.
In the remainder of this paper, we take the experimental parameters to match
those proposed in Bose et al., that is, $d$ is always set to $\sim 200\mu m$
and the masses of the two qudits are always $\sim 10^{-14}kg$. Furthermore,
unless otherwise stated $\Delta x\sim 250\mu m$.
While both [14] and [19] have discussed the implementation of the respective
set-ups, none of the intermediary models has been considered, and no direct
comparison has been drawn with respect to their impact on how fast the qubit
pair entangles or whether qudits would result in faster entanglement.
We present a generalised model for the QGEM experiment, considering rotations
of each interferometer, centred on the innermost spatial states of the two
qudits (see Figure 2). The allowed space of the rotation angles, $\theta_{1}$
and $\theta_{2}$ must be restricted so that at no point the two masses come
within a distance $d$ of one another. The linear set-up and the parallel set-
up are special cases of the above, using the angles $\theta_{1}=\theta_{2}=0$
and $\theta_{1}=\frac{3\pi}{2},\theta_{2}=\frac{\pi}{2}$ respectively.
Our model also allows for the use of qudits, rather than qubits. That is,
using $D$ spatial superposition states (and equivalently spin states, assuming
Stern-Gerlach interferometry is used as previously proposed) where $D\geq
2|D\in\mathbb{N}$. Under these conditions, the generalised state of the system
resulting from the QGEM experiment can be written as:

$\Ket{\psi(t=\tau)}=\frac{1}{D}\sum_{p=0}^{D-1}(\Ket{p}\otimes\sum_{q=0}^{D-1}e^{i\phi_{pq}}\Ket{q}).$
(1)

Where: $\phi_{pq}$ is defined by:

$\phi_{pq}\sim\frac{Gm_{1}m_{2}\tau}{\hbar C_{pq}}.$ (2)

Figure 2: Top view for generalised QGEM model showing the maximum spread of
the qudits (Distance $C_{pq}$ in red). The parametrization of the set-up is
controlled by $\theta_{1}$ and $\theta_{2}$.

The value $C_{pq}$ for each superposition pair can be derived using simple
trigonometric rules and is given by

$C_{pq}=\sqrt{A_{p}^{2}+B_{q}^{2}-2A_{p}B_{q}\cos(\theta_{3})},$ (3)

where

$A_{p}=((D-1)-p)\frac{\Delta x}{(D-1)},$ (4) $\displaystyle B_{q}$
$\displaystyle=[d^{2}+\bigg{(}q\frac{\Delta x}{(D-1)}\bigg{)}^{2}$
$\displaystyle-2d\bigg{(}q\frac{\Delta
x}{(D-1)}\bigg{)}\cos{(\pi-\theta_{2})}]^{\frac{1}{2}},$ (5)
$\theta_{3}=\pi-\theta_{1}+\arcsin{\bigg{(}\frac{q\frac{\Delta
x}{(D-1)}\sin(\theta_{2})}{B_{q}}\bigg{)}}.$ (6)

##  III Entanglement entropy test

To assess different set-ups for the QGEM experiment, we begin by comparing the
von Neumann entropy (VNE), or entanglement entropy, of the output state. We
recall that noting $\rho_{1}$ the partial trace over the first qudit of the
two-qudit system $\rho$, the entanglement entropy is given by [30]:

$S(\rho)=-Tr[\rho_{1}log_{2}\rho_{1}],$ (7)

or, using the eigen-decomposition of $\rho_{1}$:
$\rho_{1}=\sum_{j}\lambda_{j}\Ket{j}\Bra{j}$ we can re-write $S(\rho)$ as:

$S(\rho)=-\sum_{j}\lambda_{j}log_{2}(\lambda_{j}).$ (8)

For a $D$ level system, the entanglement entropy is bound by:

$0\leq S(\rho)\leq log_{2}(D),$ (9)

From which it follows that more entanglement can be produced from higher
dimension systems. However, this does not provide any information on how
quickly such entanglement is created. Here we focus on the linear and parallel
set-ups, a discussion regarding all other alternatives is presented in
Appendix A.

Parallel set-up

Linear set-up

Figure 3: Entanglement entropy scalling with the state dimension for $D=2$ to
$D=6$. On the left, we present the results for the parallel set-up, on the
right we present the results for the linear set-up.

In order to assess whether qubits perform better or worse than higher
dimensional qudits, we compute the entanglement entropy for $\rho_{para}$ and
$\rho_{lin}$ using the generalised version of the model for 2 to 6 dimensional
qudits. The resulting entropy scaling with time is plotted in Figure 3. The
parallel set-up appears to perform significantly better than the linear set-up
in realistic experiment times.
At the proposed experimental time of $2.5$s, the qubit parallel case achieves
a entanglement entropy of $0.152$, much larger than that for the qutrit case
of $0.084$. Going to higher dimensions further reduces the entanglement
entropy to $0.068$, $0.060$ and $0.056$ for 4, 5 and 6 dimensions
respectively111Note that, at $\tau=2.5s$, as $D\to\infty$ the entanglement
entropy converges to $\sim 0.039$ suggesting that, under this model, multi-
component superpositions do not entangle faster through gravity.
Of course, given the superposition phases for the state resulting from the
QGEM experiment are periodic, the linear set-up will achieve a higher
entanglement entropy than the parallel set-up for sufficiently large
experiment times.
The entanglement entropy is not a valid metric for entanglement if classical
mixing or decoherence are affecting the system as these are indistinguishable
from entanglement as a source of entanglement entropy. Therefore, entanglement
entropy is only used when assuming the system is at all times in a pure state.
As such, for more realistic experiments, it is necessary to consider other
methods of witnessing the entanglement.

##  IV Entanglement Witness tests

Multiple external factors can affect the system state throughout the
experiment, these include decoherence and classical uncertainties introduced
by the hardware used to implement the experiment. Entanglement Witnesses
provide a convenient testing system in the context of an experiment.
The Positive Partial Transpose (PPT) entanglement witness is an appropriate
witness for two for Negative Partial Transpose (NPT) entangled qudits:

$\mathcal{W}_{ppt}=\Ket{\lambda_{-}}\Bra{\lambda_{-}}^{T}$ (10)

In the parallel qubit case, this witness is simply [21]:

$\mathcal{W}_{ppt}=\frac{1}{4}[\mathbb{I}-X\otimes X-Z\otimes Y-Y\otimes Z]$
(11)

It is worth noting however that although this witness is not optimal in the
linear set-up, having fewer terms to measure results in less variance when
conducting the experiment (for a given number of measurements) (see Section
VI). Besides, all of the operators in the witness commute, as a result, there
exist a measurement basis in which one can derive the expectation value of all
these operators, reducing the number of terms to measure to one.
When considering the qudit case, there exist states which are entangled but
can not be detected by a PPT witness (see for example [32, 33]). These states
have also been referred to as bound entanglement, and multiple methods have
been developed to assess them [34, 35, 36, 37, 38, 39]. The witnesses
described above may therefore not be sufficient to test entanglement of
$D\otimes D$ systems (with $D>2$) as they would, by construction, fail to
detect Positive Partial Transpose entangled states (PPTES). These can
nonetheless be useful in case the theoretical density matrix of the state
created by the experiment is indeed NPT.
There is currently no general witness construction strategy for detecting
PPTES (also known as entangled bound states) and research is focused on
designing witnesses specific to certain families of states. Thankfully bound
entanglement represents only a small proportion of all entangled states, and
remains unlikely to occur in the proposed experiment.
We computed the PPT entanglement witness expectation value for $D$ dimensional
qudits with $D=2$ to $6$. The results are presented in Figure 4 for both the
parallel and linear cases.

Figure 4: Expectation value of PPT entanglement witness for the parallel set-
up with state dimension for $D=2$ to $D=6$

The expectation value of the PPT entanglement witness is nearing $-0.148$ at
$\tau=2.5s$ for qubits. The witness values are slightly higher for higher
dimensional qudits, echoing the findings of the entanglement entropy test.
These witnesses also appear to be finest for the set of states that are
produced by each set-up. A witness $\mathcal{W}_{A}$ is set to be finer than
another witness $\mathcal{W}_{B}$ if it detects as entangled all the states
detected by $\mathcal{W}_{B}$ and at least one more (see Ref. [40]). In this
case, the witnesses computed are clearly able to detect all entangled states
produced (for any $\tau$) albeit with values very close to zero for low values
of $\tau$.
Following the above, we can also note that the high dimension states generated
by the QGEM experiment set-ups considered must be NPT entangled states;
otherwise, $\mathcal{W}_{ppt}$ would fail to detect them as entangled.
A further consideration regarding the entanglement witness is how many
operators it needs to be broken into to be measured experimentally. For this,
we can consider the scaling of generalised Gell-Mann matrices, which is a
generalisation of the Pauli basis to quantum states of dimensions higher than
2. There are $D^{2}$ element for a set of Gell-Mann matrices for a $D$
dimensional quantum state. Because we build a system composed of two quantum
objects, we are therefore looking at a maximum total number of operators of
$D^{4}$ for the entanglement witness.
We found that the entanglement witness derived from the PPT principle has, in
general, a slightly lower than that number (due to the weights of certain
operators being negligible or equal to zero in the decomposition of the
entanglement witness).
We can further significantly improve on this number by grouping together the
operators that can be jointly measured. In general, operators can be jointly
measured if they can be diagonalised together in a specific Tensor Product
Basis (TPB) (for an overview of the method: [41]. Conveniently this is
equivalent to saying that operators can be jointly measured if they commute.
Commutation is not transitive, and as such, there may be many different
solutions to grouping operators together. To find a good solution, we use the
Largest Degree-first Colouring (LDFC) algorithm (for a good summary of the
LDFC algorithm, use the Supplementary materials in [42]), whereby group are
composed starting from one of the operators which have the highest number of
commuting operators. The results we obtained are summarised in Table 1.

D | PPT witness | PPT witness (grouped)
---|---|---
2 | 4 | 1
3 | 77 | 14
4 | 244 | 28
5 | 613 | 53
6 | 1272 | 94
Table 1:  Number of operators in the generalised Pauli decomposition of the
witnesses, that must be measured to estimate the expectation value of the
entanglement witness in the parallel case for $D=2$ to $D=6$. The grouped
column present the number of operator groups that can be jointly measured in a
single Tensor Product Basis, obtained using the Largest Degree First Coloring
(LDFC) algorithm. Given this algorithm is a heuristic, one could find a
different set and number of groups.

One point to note is that measuring operators jointly requires finding and
implementing a joint measurement basis. While straightforward in some cases,
this could yield some significant complications in an actual experiment.

##  V Testing models with decoherence

So far we have considered the case where both qubits are only coupled with
each other through their positional superposition. Real experimental setting
cannot however fully remove the potential for coupling of the studied quantum
system with the environment.
The particles’ coherence and hence joint entanglement erodes over time due to
interaction with the environment. This results in decoherence of the
positional qudits into a single, defined position or a classical mixture of
differing but well defined positions.
We schematically incorporate this in our model by adding a time dependent
exponential decay to all off diagonal term of each qudit’s density matrix,
parametrised by the decoherence rate $\gamma$. This is under the assumption
that in any experiment $\frac{\Delta x}{D}\gg\lambda_{\textrm{dB}}$ where
$\lambda_{\textrm{dB}}$ is the masses de Broglie wavelength. For a generic
qudit, with dimension $D$ and density matrix $\rho_{d}$, we can write:

$\leavevmode\resizebox{173.44534pt}{}{{\hbox{ \rho_d = \begin{bmatrix}c_{11}
&c_{12} &\vdots&c_{1(d-1)} &c_{1d} \\\ c_{21} &c_{22} &\vdots&c_{2(d-1)}
&c_{2d}\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\ c_{(d-1)1} &c_{(d-1)2}
&\vdots&c_{(d-1)(d-1)} &c_{(d-1)d} \\\ c_{d1} &c_{d2} &\vdots&c_{d(d-1)}
&c_{dd} \end{bmatrix} }}},$ (12)

as such, following the model described above, decoherence is incorporated as:

$\leavevmode\resizebox{195.12767pt}{}{{\hbox{ \rho_d
\rightarrow\begin{bmatrix}c_{11} &c_{12}e^{-\gamma\tau}
&\vdots&c_{1(d-1)}e^{-\gamma\tau} &c_{1d}e^{-\gamma\tau} \\\
c_{21}e^{-\gamma\tau} &c_{22} &\vdots&c_{2(d-1)}e^{-\gamma\tau}
&c_{2d}e^{-\gamma\tau}\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\
c_{(d-1)1}e^{-\gamma\tau} &c_{(d-1)2}e^{-\gamma\tau} &\vdots&c_{(d-1)(d-1)}
&c_{(d-1)d}e^{-\gamma\tau} \\\ c_{d1}e^{-\gamma\tau} &c_{d2}e^{-\gamma\tau}
&\vdots&c_{d(d-1)}e^{-\gamma\tau} &c_{dd} \end{bmatrix} }}}.$ (13)

The overall two-qudit system density matrix is then computed using:
$\rho_{system}$ = $\rho_{d}^{(1)}\otimes\rho_{d}^{(2)}$.
We computed the expectation value of $\mathcal{W}_{ppt}$ for dimensions 2 to 6
for incremental values of $\gamma$. The results are plotted in Figure 5.

Figure 5: Expectation value of PPT entanglement witness as a function of the
decoherence rate in the parallel set-up and with $D$ ranging from 2 to 6.

Interestingly, higher dimension models appear more resilient to decoherence
than the qubit case. It is worth noting that increasing decoherence in the
model also reduces the ‘optimal’ time for the experiment, that is, the time at
which the entanglement witness is most negative, and the ability to detect
entanglement with longer experiment times. To be better understand the
interplay between time, decoherence and the number of dimensions, the expected
value of $\mathcal{W}_{ppt}$ was computed at two different values for the
decoherence rate: $0.1$Hz and $0.125$Hz.

$\gamma=0.1$Hz

$\gamma=0.125$Hz

Figure 6: Entanglement witness expectation value over time for the parallel
set up and with $D$ ranging from 2 to 6. We present two different possible
values of $\gamma$: 0.1Hz, and 0.125Hz

The advantage of higher-dimensional models, when decoherence is increased,
become significant when $\gamma\gtrsim 0.1$Hz. This suggest that in a real run
of the experiment, multi-component superposition may be preferable, or even
necessary, if decoherence is sufficiently high. We can also observe as
expected that longer time becomes detrimental for high decoherence rates. We
also considered the problem of optimising the experiment run-time to maximise
the decoherence rate for which the entangled witness could detect the state as
entangled. In Appendix C we plot the expectation value of $\mathcal{W}_{ppt}$
for different values of $\tau$, including decoherence. The result is that
$\tau=2.5$s is near optimal for qubits and qudits, though lower time performs
marginally better for the latter. Of course the experiment can be run for
shorter times without any negative impacts provided a detectable level of
entanglement has developed.
While higher dimension set-ups are more resilient to decoherence, one key
question in this analysis is to determine what number of measurements will be
required to reject that the experiment state is not entangled. In order to
further this analysis, we now need to fully simulate the experiment in order
to determine the required number of measurements and run-time of the QGEM
experiment. This is the object of the next section.

##  VI Experiment simulation

In this section, we present experiment simulations used to estimate the number
of measurements that will be required to reject the hypothesis that the qubit
pair is not entangled. Failure to reject only means that it is impossible at
this stage to confidently test for gravitationally mediated entanglement.
To compare the entanglement witnesses, and the different model proposed, we
simulate an experiment, as described below:

  1. 1.

The entanglement witness is decomposed into a weighted sum of generalised
Pauli tensors terms (Gell-Mann matrices for qutrits, and generalised $d$
dimension Pauli operators for qudits). (We used the method for generalisation
of Pauli operators described in Thew et al. [43]). In some cases, we group the
operators (following the groups described in Table 1).

  2. 2.

The quantum state resulting from the experiment is measured a pre-determined
number of times against each of the Pauli terms or group, to compute (1) their
expectation value, (2) the standard error of the measurement series. To
minimise the variance of the observable for a given number of measurements, we
have weighted the number of measurement in proportion to the weight of each
Pauli tensor terms (or group) in the decomposition of the witness.

  3. 3.

Details on how confidence levels are computed can be found in appendix D.

The plots presented in this section are an average over many single runs of
each numerical experiment simulation. As such, these should be representative
of a typical run of the experiment. The conclusions drawn from these are only
meant to indicate the order of magnitude of the number of measurements
required in order to define the most adequate experiment set-up.
We first describe the result obtained from the experiment simulation with
qubits. The simulation is first run comparing the linear and parallel set-ups
with no decoherence (Figure 7 followed by plots of the simulation for
increasing decoherence rate $\gamma$ in Figure 8. We have used
$\mathcal{W}_{ppt}$ with full basis decomposition in this first experiment
simulations, holding $\Delta x=250\mu m$ and $\tau=2.5s$.

Figure 7: QGEM experiment simulation - linear and parallel set-ups (no
decoherence) - Expected level of confidence (probability of the witness value
actually being negative) in the qubit case. We added the case in which the
parallel witness is used for the linear set-up showcasing the benefits of
having a lower number of terms to compute

As we can see from Figure 7, with no decoherence, it takes about 500
measurements to reject the hypothesis that the two states are not entangled at
a 99.9% confidence level in the parallel case and at least 3,000 measurements
for the linear set-up. Using the witness derived in the parallel set-up
marginally improves the results of the linear case, although not sufficient
for it to be comparable to the parallel version of the experiment. This
further confirms that the parallel set-up will be preferable in a real
experiment and we, therefore, discard the linear set-up in the remainder of
the simulations.
We can expect that incorporating decoherence will increase the number of
measurements required as it pushes the expectation value of
$\mathcal{W}_{ppt}$ upward. The result of the qubit experiment simulations
with decoherence are presented in Figure 8. It illustrates the rapid increase
in the number of measurements required as the decoherence rate is raised. For
$\gamma=0.05$Hz, the parallel set-up still only require about $\sim 2,000$
measurements. This figure goes up to $\sim 6,000$ measurements for
$\gamma=0.075$Hz. At $\gamma=0.1$Hz, the experiment would required at least
$25,000$ measurements. At this decoherence rate, qubits have fewer negative
expectation values for $\mathcal{W}_{ppt}$ than some qudits, and at
decoherence rate above $\gamma=0.12$Hz, the expectation value of the witness
is positive (that is, no entanglement is detected and therefore the results
are not plotted).

Figure 8: QGEM Experiment simulation with qubits - Confidence levels for three
decoherence level (0.05Hz, 0.075Hz and 0.1Hz) as a function of the number of
measurements available. Here the witness as expectation value of -0.074,
-0.043, and -0.016, respectively

We repeated the experiment simulations as described above with differing
decoherence rates in the case of qudits. We have used the $D=6$ qudit case for
illustration. For brevity, we will denote them as 6-qudits, and use an
analogous nomenclature for other dimensions. As for the qubits simulations,
all the experiments use the parallel set-up, $\Delta x=250\mu$m and
$\tau=2.5$s.

Figure 9: QGEM Experiment simulation with D=6 qudits - Confidence levels for
three decoherence level (0.05Hz, 0.075Hz and 0.1Hz) as a function of the
number of measurements available. Here the witness as expectation value of
-0.045, -0.032, and -0.021, respectively

As expected, for the 6-qudits case, the number of measurements required is
significantly higher. This is primarily due to the large number of additional
terms to compute. Our results are presented in Figure 9 and show that for
$\gamma=0.05$Hz, over 200,000 measurements would be needed, while nearly
400,000 would be required if $\gamma=0.075$Hz, and about 600,000 for
$\gamma=0.1$Hz.
One last test that is worth considering is the situation in which grouping of
terms is allowed (this is subject to being able to produce the relevant
measurement bases experimentally, as mentioned in Section IV).
Figure 10 presents the results for the qubits case. We can already see the
drastic reduction in the total number of measurements required, resulting from
the reduction of terms to measure from 4 (in reality 3 since the identity term
does not need to be measured), to 1. Less than 1,000 measurements and 2,000
measurements are necessary when $\gamma=0.05$Hz and $\gamma=0.075$Hz,
respectively. Similarly, only about 12,000 measurements is required if
$\gamma=0.1$Hz.

Figure 10: QGEM Experiment simulation with qubits (operators grouped) -
Confidence levels for three decoherence level (0.05Hz, 0.075Hz and 0.1Hz) as a
function of the number of measurements available. Here the witness as
expectation value of -0.074, -0.043, and -0.016, respectively

A very similar pattern can be seen in the qudit case. The results for 6-qudits
are presented in Figure 11. number of measurements are drastically reduced
when operators are grouped. About 25,000 measurements are needed for
$\gamma=0.05$Hz, while $\gamma=0.075$Hz requires less than 40,000
measurements, and $\gamma=0.1$Hz less than 80,000.

Figure 11: QGEM Experiment simulation with 6-qudits (operators grouped) -
Confidence levels for three decoherence level (0.05Hz, 0.075Hz and 0.1Hz) as a
function of the number of measurements available. Here the witness as
expectation value of -0.045, -0.032, and -0.021, respectively

There are clearly no reasons to use high dimensional qudits unless the
decoherence rate is such that the entanglement witness for qubits always has a
positive expectation value. In the experiment setting, we have assumed, this
happens at about $\gamma\approx 0.12$Hz. We therefore present in Figure 12 an
experiment simulation showing how 6-qudits perform in the window of
decoherence rate in which high dimensional qudits become relevant.

Figure 12: QGEM Experiment simulation with 6-qudits - Confidence levels for a
decoherence rate of 0.125Hz as a function of the number of measurements
available in a case where the operators are grouped, and not.

At $\gamma=1.25$Hz, it is clear that the 6-qudits states still require a very
large number of measurements in order to reject the null hypothesis. However,
grouping the operators to measure in joint measurement bases allow reducing
the number of measurements required from nearly 2,000,000 to slightly above
200,000. It is worth noting that at this rate, the qubits case would not work
as decoherence pushes the expectation value of the witness above zero, and the
4-qudits and 5-qudits case are too close to zero to reach confidence of
$99.9\%$ in a realistic number of measurements.

##  VII Conclusion

This paper considered modifying the set-up proposed in the original QGEM
experiments to determine how best to generate, protect and detect entanglement
in a future experiment. We looked at two aspects, in particular, the geometric
set-up of the experiment, and the number of dimensions of the quantum objects
used and developed a generalised mathematical model of the experiment.
Based on this model, and using entanglement entropy, we concluded that the
parallel qubits set-up generates entanglement the fastest for realistic
experiment run-times ($\tau$ $O\left(1\mathrm{s}\right)$). As entanglement
entropy cannot account for entanglement once decoherence is introduced, we
presented an entanglement test based on entanglement witnesses. We concluded
that as the decoherence rate is increased higher dimension, qudits finally
out-perform qubits by providing lower expectation values for the witnesses.
To estimate which set-up would require the least number of measurements to
evidence entanglement, we simulated experiments to define a confidence level
for the negativity of the expectation value of the witness given a certain
number of measurements. Without noise, at a 99.9$\%$ certainty level, the
parallel qubits set-up requires less than 2,000 measurements (1,000 when
grouped) to reject the null hypothesis (that the state is not entangled). The
number of measurements required increases rapidly when decoherence is
introduced.
Qudits of higher dimensions only become more useful than qubits when the
expectation value of the witness for qubits becomes non-negative. That is
because the number of basis elements to be estimated to calculate the witness
expectation value increases quadratically in the number of dimensions
($d^{4}$). In the experiment settings proposed, at $\tau=2.5$s and $\Delta
x=250\mu$m, qudits of dimension 6 are more favourable than qubits when the
decoherence rate is $\gamma\sim 0.125$Hz, however, in this case, over
2,000,000 measurements will be required (200,000 when operators are grouped).
Thus to further improve the experiment design, the following points should be
considered. Clearly, reducing the qudit-pair exposure to the environment to
decoherence would render the experiment more economical. The total expected
decoherence rates in specific, realistic experimental designs must be
estimated in order to confirm whether higher dimension qudits will be
required. Finally, any increase in the superposition width $\Delta x$ is
significantly beneficial as it both improves the entanglement generation rate,
allowing a significant reduction in overall run-time, which therefore also
minimises the impact of decoherence.

##  VIII Acknowledgements

JT is supported by the UK EPSRC grount No. EP/R513143/1. RJM is supported by a
University College London departmental studentship. AM’s research is funded by
the Netherlands Organisation for Science and Research (NWO) grant number
680-91-119. SB would like to acknowledge EPSRC grants No. EP/N031105/1 and
EP/S000267/1.

## References

  * [1] C. Kiefer, Quantum gravity.  Cambridge University Press, (2012).
  * [2] R. Colella, A. Overhauser, and S. Werner, “Observation of gravitationally induced quantum interference,” Phys. Rev. Lett., vol. 34, pp. 1472–1474, 1975.
  * [3] V. Nesvizhevsky, A. Petoukhov, H. Borner, K. Protasov, A. Voronin, A. Westphal, S. Baessler, H. Abele, and A. Gagarski, “Reply to: Comment on ‘measurement of quantum states of neutrons in the earth’s gravitational field,” Phys. Rev. D, vol. 68, p. 108702, 2003.
  * [4] B. A. Stickler, B. Papendell, S. Kuhn, B. Schrinski, J. Millen, M. Arndt, and K. Hornberger, “Probing macroscopic quantum superpositions with nanorotors,” New J. Phys. 20 122001, 2018.
  * [5] I. Pikovski, M. Vanner, and M. Aspelmeyer, “Probing planck-scale physics with quantum optics,” Nature Phys 8, 393–397, 2012.
  * [6] T. Krisnanda, G. Tham, and M. Paternostro, “Observable quantum entanglement due to gravity,” npj Quantum Inf 6, 12, 2020.
  * [7] R. Howl, R. Penrose, and I. Fuentes, “Exploring the unification of quantum theory and general relativity with a bose–einstein condensate,” New J. Phys. 21 043047, 2019.
  * [8] M. Carlesso, A. Bassi, M. Paternostro, and H. Ulbricht, “Testing the gravitational field generated by a quantum superposition,” New J. Phys. 21 093052, 2019.
  * [9] N. Altamirano, P. Corona-Ugalde, R. B. Mann, and M. Zych, “Gravity is not a pairwise local classical channel,” Class. Quantum Grav. 35 145005, 2019\.
  * [10] R. J. Marshman, A. Mazumdar, G. W. Morley, P. F. Barker, S. Hoekstra, and S. Bose, “Mesoscopic Interference for Metric and Curvature (MIMAC) $\&$ Gravitational Wave Detection,” New J. Phys., vol. 22, no. 8, p. 083012, 2020.
  * [11] D. N. Page and C. Geilker, “Indirect Evidence for Quantum Gravity,” Phys. Rev. Lett., vol. 47, pp. 979–982, 1981.
  * [12] A. Ashoorioon, P. Bhupal Dev, and A. Mazumdar, “Implications of purely classical gravity for inflationary tensor modes,” Mod. Phys. Lett. A, vol. 29, no. 30, p. 1450163, 2014.
  * [13] R. J. Marshman, A. Mazumdar, and S. Bose, “Locality and entanglement in table-top testing of the quantum nature of linearized gravity,” Physical Review A, vol. 101, no. 5, p. 052110, 2020.
  * [14] S. Bose, A. Mazumdar, G. W. Morley, H. Ulbricht, M. Toros, M. Paternostro, A. Geraci, P. Barker, M. S. Kim, and G. Milburn, “A spin entanglement witness for quantum gravity,” Phys. Rev. Lett. 119, 240401, (2017).
  * [15] C. Marletto and V. Verdal, “Gravitationally-induced entanglement between two massive particles is sufficient evidence of quantum effects in gravity,” Physical Review Letters, 119, (2017).
  * [16] A. Belenchia, R. M. Wald, F. Giacomini, E. Castro-Ruiz, i. c. v. Brukner, and M. Aspelmeyer, “Quantum superposition of massive objects and the quantization of gravity,” Phys. Rev. D, vol. 98, p. 126009, Dec 2018.
  * [17] M. Christodoulou and C. Rovelli, “On the possibility of experimental detection of the discreteness of time,” Frontiers in Physics, vol. 8, Jul 2020.
  * [18] R. Howl, V. Vedral, M. Christodoulou, C. Rovelli, D. Naik, and A. Iyer, “Testing quantum gravity with a single quantum system,” arXiv, 2020.
  * [19] H. C. Nguyen and F. Bernards, “Entanglement dynamics of two mesoscopic objects with gravitational interaction,” The European Physical Journal D volume 74, Article number: 69, (2020).
  * [20] W. van de Kamp, R. J. Marshman, S. Bose, and A. Mazumda, “Quantum gravity witness via entanglement of masses: Casimir screening,” Phys.Rev.A 102, 6, 062807, (2020).
  * [21] H. Chevalier, A. Paige, and M. Kim, “Witnessing the nonclassical nature of gravity in the presence of unknown interactions,” Phys. Rev. A, vol. 102, no. 2, p. 022428, 2020.
  * [22] M. Toroš, T. W. van de Kamp, R. J. Marshman, M. Kim, A. Mazumdar, and S. Bose, “Relative acceleration noise mitigation for entangling masses via quantum gravity,” arXiv preprint arXiv:2007.15029, 2020.
  * [23] D. Miki, A. Matsumura, and K. Yamamoto, “Entanglement and decoherence of massive particles due to gravity,” 10 2020.
  * [24] A. Matsumura and K. Yamamoto, “Gravity-induced entanglement in optomechanical systems,” Phys. Rev. D, vol. 102, no. 10, p. 106021, 2020.
  * [25] S. Rijavec, M. Carlesso, A. Bassi, V. Vedral, and C. Marletto, “Decoherence effects in non-classicality tests of gravity,” 12 2020.
  * [26] M. Toroš, A. Mazumdar, and S. Bose, “Loss of coherence of matter-wave interferometer from fluctuating graviton bath,” 8 2020.
  * [27] Y. Margalit et al., “Realization of a complete Stern-Gerlach interferometer: Towards a test of quantum gravity,” 11 2020.
  * [28] T. Biswas, E. Gerwick, T. Koivisto, and A. Mazumdar, “Towards singularity and ghost free theories of gravity,” Phys. Rev. Lett., vol. 108, p. 031101, 2012.
  * [29] T. Biswas, A. Mazumdar, and W. Siegel, “Bouncing universes in string-inspired gravity,” JCAP, vol. 03, p. 009, 2006.
  * [30] I. Bengtsson and K. Zyczkowski, Geometry of Quantum States: An Introduction to Quantum Entanglement.  Cambridge University Press, 2006.
  * [31] Note that, at $\tau=2.5s$, as $D\to\infty$ the entanglement entropy converges to $\sim 0.039$.
  * [32] P. Horodecki, “Separability criterion and inseparable mixed states with positive partial transposition,” Physical Letters A, 232, (1997).
  * [33] P. Horodecki, M. Horodecki, and R. Horodecki, “Bound entanglement can be activated,” Phys. Rev. Lett. 82, 1056, (1999).
  * [34] M. Horodecki, P. Horodecki, and R. Horodecki, “Mixed-state entanglement and distillation: Is there a bound entanglement in nature?,” Phys. Rev. Lett. 80(24), 5239, (1998).
  * [35] S. Yu and N.-L. Liu, “Entanglement detection by local orthogonal observables,” Phys. Rev. Lett. 95, 150504, (2005).
  * [36] O. Guhne, M. Mechler, G. Toth, and P. Adam, “Entanglement criteria based on local uncertainty relations are strictly stronger than the computable cross norm criterion,” Phys. Rev. A 74(1), 010301(R), (2006).
  * [37] O. Chruscinski and A. Kossakwoski, “Circulant states with positive partial transpose,” Phys. Rev. A 76, 032308, (2007).
  * [38] O. Chruscinski and A. Rutkowskii, “Entanglement witnesses for $d\otimes d$ systems and new classes of entangled qudit states,” The European Physical Journal D 62(2):273-277, (2011).
  * [39] O. Rudolphl, “On the cross norm criterion for separability,” Journal of Physics A: Mathematical and General, Volume 36, Number 21, (2003).
  * [40] M. Lewenstein, B. Kraus, I. Cirac, and P. Horodecki, “Optimization of entanglement witnesses,” Phys. Rev. A 62, 052310, (2000).
  * [41] P. Gokhale, O. Angiuli, Y. Ding, K. Gui, T. Tomesh, M. Suchara, M. Martonosi, and F. T. Chong, “Minimizing state preparations in variational quantum eigensolver by partitioning into commuting families,” arXiv:1907.13623, 2019.
  * [42] I. Hamamura and T. Imamichi, “Efficient evaluation of quantum observables using entangled measurements,” npj Quantum Information volume 6, Article number: 56, 2020.
  * [43] R. T. Thew, K. Nemoto, A. G. White, and W. J. Munro, “Qudit quantum state tomography,” Phys.Rev. A.66.012303, (2002).
  * [44] O. Guhne and G. Toth, “Entanglement detection,” Physical Reports, 474, (2009).
  * [45] M. Bourennane, M. Eibl, C. Kurtseifer, S. Gaertner, and H. Weinfurter, “Experimental detection of multipartite entanglement using witness operators,” Physical Review Letters, 92, (2004).
  * [46] For an overview of several entanglement witnesses, Ref. [44].
  * [47] P. Hyllus, O. Ghune, D. Bruss, and M. Lewenstein, “Relations between entanglement witnesses and bell inequalities,” Phys. Rev. A 72, 012321, (2005).
  * [48] Nguyen and Bernards provide a threshold value for the phase of the parallel set-up for which it would violate CHSH inequalities, including decoherence [19]. Further research would be necessary in order to conduct a similar test for qudits.

APPENDICES

##  Appendix A Discussion on the geometrically generalised model

There are a number of parameters in the generalised formula that can be
modified and tested for more effective entanglement generation. We have
isolated two which offer particular insights, the superposition width ($\Delta
x$) in Section A.0.2 and the rotation angles ($\theta_{1}$ and $\theta_{2}$)
in section A.0.1. Prior to this, however, there are a few points that are
worth noting:

  1. 1.

It is clear that more time can allow achieving maximally entangled states,
however increasing the experimental run time is unrealistic due to greater
risk of decoherence. This point is covered in more details in Section V.

  2. 2.

Reducing the minimum distance $d$ also clearly generates much faster
entanglement. However, as mentioned before this is not necessarily useful in
practice. This has been considered by others [20], and any results here will
also hold for such modified set-ups.

  3. 3.

Using more massive quantum objects result in higher relative phases and in
faster entanglement generation. However, more massive objects would also mean
more challenging implementation for the interferometry, larger particle radii
and hence higher Casimir-Polder forces. This could, in turn, increase the
minimum distance $d$ and overall negatively impact entanglement growth. This
is not something we will consider further.

###  A.0.1 Comparing rotation angle set-ups

In the main body of the paper, we have only presented the linear and parallel
cases. Based on our results, the parallel set-up entangles the qudits faster.
For completeness, and to allow for further modification in the implementation
by experimentalists, we also considered a range of additional geometries for
the set-up that can be implemented using the generalised QGEM model derived
previously.
The heat-map presented in Figure 13 displays the entanglement entropy for all
possible combinations of $\theta_{1}$ and $\theta_{2}$ as defined in Figure 2,
having set $\Delta x=250\mu$m and $\tau=2.5$s. The dark blue dot represents
the two possible parallel set-ups
($\theta_{1}=\frac{3\pi}{2},\theta_{2}=\frac{\pi}{2}$ and
$\theta_{1}=\frac{\pi}{2},\theta_{2}=\frac{3\pi}{2}$), while the yellow dot
represents the linear set-up ($\theta_{1}=\theta_{2}=0$). On the heat-map,
blue represents low entanglement entropy while white represents entanglement
entropy nearing 1.0. It appears from this figure that higher entanglement
entropy could possibly be achieved with alternative set-ups; however, this is
without considering that some combinations result in some superposition
instances being under $200\mu m$ and therefore subject to non-negligible
Casimir-Polder forces (represented by the shaded areas on the figure).

Figure 13: Entanglement entropy for all possible combinations of $\theta_{1}$
and $\theta_{2}$. The areas of reduced contrast (the region around the center
$+$) are forbidden based on the requirement that the states do not come too
close.

###  A.0.2 Impact of superposition width ($\Delta x$) on entanglement entropy

Figure 14 shows that a larger superposition width results in faster
entanglement entropy growth. To illustrate this point we can consider the
phases for the qudit parallel set-up: $\phi_{pq}$ is smaller or equal to
$\phi$, reaching equality at $\Delta x=0$m, as $\Delta x$ increases,
$\phi_{pq}$ decreases resulting in all the phase factors $\Delta_{\phi_{pq}}$
to become more negative than they already are and as such accelerating
entanglement generation.

Figure 14: Entanglement entropy for the parallel and linear set-ups in the
qubit case as a function of the superposition width ($\Delta x$)

The superposition width, however, is limited in practice by experimental
considerations such as the magnetic field gradient achievable. This suggests
that by further modifying the arrangement into a parallel set-up one can
further reduce the magnetic field gradient and/ or other experimental
parameters while maintaining a detectable level of entanglement.
Using higher dimension spin quantum objects would, in theory, result in larger
possible superposition width due to the spin-dependent nature of the magnetic
field gradient coupling. This suggests that higher-dimensional objects would
perform better than qubits if their maximum superposition width is higher.
However, once a larger spin object is created, one can simply initialise the
state including only the outer spin states and hence recreating a qubit state
which performs better at an equivalent value of $\Delta x$. For illustration
purposes, Figure 15 represents a plot of entanglement entropy against time for
qudits in which $\Delta x$ is scaled to the number of dimensions. This
somewhat unrealistic set-up leads to higher-dimensional objects perform
significantly better.

Figure 15: Entanglement entropy as a function of time for $D$ ranging from 2
to 6. In this case, we have scaled $\Delta x$ to the number of dimensions

As such, the analysis allows to conclude that:

  1. 1.

Qubits entangle faster than qudits for values of $\tau$ and $\Delta x$
realistic for the experiment.

  2. 2.

Parallel set-ups entangle faster for all dimensions.

  3. 3.

The superposition width $\Delta x$ will be the variable of interest in terms
of obtaining faster entanglement or a more easily implementable experiment.

##  Appendix B Alternative witness: Vicinity witness for Negative Partial
Transpose entangled states

We consider an alternative entanglement witness, which is built to detect
entangled states in the vicinity of a known entangled pure state. Constructing
this witness amounts to finding a value for $\alpha$ such that
$Tr(\mathcal{W}_{vic}\varrho)\geqslant 0$ for all separable states, with
$\mathcal{W}_{vic}$ given by:

$\mathcal{W}_{vic}=\alpha\mathbb{I}-\Ket{\psi}\Bra{\psi}$ (14)

The maximum value of $\alpha$ is then derived as the square of the maximum
Schmidt coefficient of the pure state [45]. Noting $\lambda_{m}$ the highest
of these coefficients, we can re-write the witness as:

$\mathcal{W}_{vic}=\lambda_{m}^{2}\mathbb{I}-\Ket{\psi}\Bra{\psi}$ (15)

We can now compare the performance of the PPT based witness
$\mathcal{W}_{ppt}$ and the ‘vicinity’ witness $\mathcal{W}_{vic}$. This
analysis is restricted to the qubit, parallel version of the QGEM experiment
as it appears to be the most optimal set-up.

Figure 16: Expectation value of PPT and Vicinity entanglement witnesses as a
function of time in the qubit case

Comparing the expectation value of the PPT-based witness and the vicinity
witness, the former exhibits clearly much lower values on short time frames.
As such, it is the recommended entanglement witness for the implemented
experiment. In Table 2, we show that the vicinity witness, in general,
requires to measure fewer terms than the PPT witness - it is not sufficient
however to make it more advantageous than the PPT witness in an experimental
model, and therefore we have mostly not included it in our analysis.
There also exist numerous other entanglement witnesses, often based on
existing separability criteria, which will not be treated here as not directly
relevant to the proposed research.222For an overview of several entanglement
witnesses, Ref. [44]

D | PPT (para.) | PPT (lin.) | Vicinity (para.)
---|---|---|---
2 | 4 | 9 | 6
3 | 77 | 81 | 60
4 | 244 | 256 | 211
5 | 613 | 625 | 547
6 | 1272 | 1296 | 1166
Table 2: Number of operators to be measured to estimate the expectation value
of the entanglement witness in the parallel case for $D=2$ to $D=6$ for the
parallel case and linear case using the PPT entanglement witness, and for the
parallel case using the Vicinity entanglement witness.

A possible way to reduce the number of operators to be measured would be to
find a sub-optimal witness with a lower number of terms to measure. One
example is an approach similar to Bell inequalities. These do not detect all
the entangled state as it is focused only on identifying states which cannot
be explained through Local Hidden Variable (LHV) models. In [47], Hyllus et
al. show that it is possible to convert an optimal entanglement witness into a
CHSH type inequality. The resulting inequality detects non-LHV states
optimally but not entangled LHV states.
Therefore, there is a trade-off between the optimality of the entanglement
witness (the finest entanglement witness being the optimal witness for a given
entanglement detection problem) and the overall number of measurements
required to test entanglement. The conversion method developed in Hyllus et
al. only relates to qubits and does not take into account the impact of
decoherence on the detectability of entanglement [47]. However, for the qubit
case, given the witness used only as three terms that need to be measured,
CHSH types inequality are unlikely to provide a significant benefit.333Nguyen
and Bernards provide a threshold value for the phase of the parallel set-up
for which it would violate CHSH inequalities, including decoherence [19].
Further research would be necessary in order to conduct a similar test for
qudits.

##  Appendix C Time trade-off with decoherence

In the original paper, Bose et al. suggest an experiment run-time of
$\tau=2.5s$. In this appendix, We consider a simple way to estimate which run-
time produce the highest resilience to decoherence (which run-time results in
the expectation value of the entanglement witness reaching $0$ for the highest
decoherence rate). Figure 17 presents the case of qubits, while Figure 18
present the case of 6-dimensional qudits.

Figure 17: Expectation value of PPT entanglement witness for different runtime
of the experiment and as a function of the decoherence rate in the case $D=2$

On the qubit figure we can that additional time as little impact on how high
the decoherence rate can be allowed to be. It does, however, offer more
negative expectation values for the entanglement witness. If the experiment
decoherence rate is estimated, one can then verify whether additional time can
reduce the overall number of measurements required.

Figure 18: Expectation value of PPT entanglement witness for different runtime
of the experiment and as a function of the decoherence rate in the case $D=6$

The effect is more pronounced in the 6-dimensional qudits case as lower time
offer negative expectation values of the witness for higher decoherence rate.

##  Appendix D Computing confidence interval for a quantum observable

In order to compute statistics related to the expectation value of a quantum
observable, we first need to deconstruct the witness we are trying to estimate
into a weighted sum of observable that can be directly measured, i.e. a set of
Pauli strings.
In particular, in the case of our two qudit system we have at most $D^{4}$
terms. These terms and the tensor product composite of the list of Gell-Mann
matrices of dimension $D$ which are numbered $\mathcal{D}=D^{2}$. Any witness
$\mathcal{W}$ can then be written as:

$\mathcal{W}=\sum_{i}^{\mathcal{D}}\sum_{j}^{\mathcal{D}}c_{ij}\lambda_{i}^{(1)}\otimes\lambda_{j}^{(2)}$
(16)

With $\lambda$ representing any Gell-Mann matrix (in the qubit case, these are
the Pauli matrices: $\lambda\in\\{\mathbb{I},X,Y,Z\\}$). Any tensor
$w_{ij}=\lambda_{i}^{(1)}\otimes\lambda_{j}^{(2)}$ is a quantum observable
that can be directly measured in an experiment.
For a given number of measurements $M$, we can partially improve the overall
variance of the observable by distributing these measurements in proportion to
the weight of each term in the decomposition of the witness, such that, noting
$c=\sum_{i}\sum_{j}\lvert c_{ij}\rvert$, we have:

$\displaystyle M=\sum_{i}\sum_{j}M_{ij}$ (17) $\displaystyle
M_{ij}=\frac{\lvert c_{ij}\rvert}{c}M$ (18)

Considering that we conduct $M_{ij}$ measurements we can then determine the
mean and variance of each of the term as follows:

$\overline{w_{ij}}=\sum_{m}^{M_{ij}}\frac{w_{ij}^{(m)}}{M_{ij}}$ (19)
$\sigma_{ij}^{2}=\sum_{m}^{M_{ij}}\frac{(\overline{w_{ij}}-w_{ij}^{(m)})^{2}}{(M_{ij}-1)}$
(20)

From there, we can compute the witness’ mean and variance:

$\overline{\mathcal{W}}=\sum_{i}^{\mathcal{D}}\sum_{j}^{\mathcal{D}}\sum_{m}^{M_{ij}}\frac{c_{ij}w_{ij}^{(m)}}{M_{ij}}$
(21)
$\sigma_{\mathcal{W}}^{2}=\sum_{i}^{\mathcal{D}}\sum_{j}^{\mathcal{D}}\sum_{m}^{M_{ij}}\lvert
c_{ij}\rvert^{2}\sigma_{ij}^{2}$ (22)

Finally, noting $\mathcal{M}$ the average number of measurements per term, we
can compute the standard error of normally distributed measurement population
as:

$s_{\mathcal{W}}=\frac{\sigma_{\mathcal{W}}}{\sqrt{\mathcal{M}}}$ (23)

We then compute the confidence interval as:

$CI_{\mathcal{W}}=[\overline{\mathcal{W}}-\alpha
s_{\mathcal{W}},\overline{\mathcal{W}}+\alpha s_{\mathcal{W}}]$ (24)

With $\alpha$ the t value corresponding to the desired level of confidence.
For computation of the confidence level, we test against the null hypothesis
$\mathcal{W}\geq\mu_{0}$, with $\mu_{0}=0$, and compute the t values following
the traditional methods for one-sided t-test:

$t=\frac{\lvert\mathcal{W}-\mu_{0}\rvert}{s_{\mathcal{W}}}$ (25)

Confidence is then computed as $1-p$, with $p$ the p-value corresponding to
the t value obtained.

##  Appendix E Python model for the QGEM experiment

Our model for computation and modelling of the QGEM experiment is available on
github under the name Generalised QGEM