# Characterisation of multi-level quantum coherence without ideal measurements

Benjamin Dive∗,1,2, Nikolaos Koukoulekidis∗,1, Stefanos Mousafeiris1, and
Florian Mintert1 ∗These two authors contributed equally to this work
1Department of Physics, Imperial College, London SW7 2AZ, UK  2Institute of
Quantum Optics and Quantum Information, Austrian Academy of Sciences, Vienna
1090, Austria

###### Abstract

Coherent superpositions are one of the hallmarks of quantum mechanics and are
vital for any quantum mechanical device to outperform the classically
achievable. Generically, superpositions are verified in interference
experiments, but despite their longstanding central role we know very little
about how to extract the number of coherently superposed amplitudes from a
general interference pattern. A fundamental issue is that performing a phase-
sensitive measurement is as challenging as creating a coherent superposition,
so that assuming a perfectly implemented measurement for verification of
quantum coherence is hard to justify. In order to overcome this issue, we
construct a coherence certifier derived from simple statistical properties of
an interference pattern, such that any imperfection in the measurement can
never over-estimate the number of coherently superposed amplitudes. We
numerically test how robust this measure is to under-estimating the coherence
in the case of imperfect state preparation or measurement, and find it to be
very resilient in both cases.

##  I Introduction

The superposition principle allows wave mechanics, in particular quantum
mechanics, to feature dynamics that are unthinkable for classical particles.
The prospect of exploiting quantum coherence for applications in quantum
computation, communication, metrology, and thermodynamics Stahlke (2014);
Knill  _et al._ (2000); Zhang  _et al._ (2017); Lostaglio  _et al._ (2015);
Korzekwa  _et al._ (2016) has resulted in numerous activities towards the
classification and quantification of quantum coherence Levi and Mintert
(2014); Baumgratz  _et al._ (2014); Girolami (2014); von Prillwitz  _et al._
(2015); Winter and Yang (2016); Marvian and Spekkens (2016); Streltsov  _et
al._ (2015, 2017).

Those developments are inspired by earlier work in the theory of entanglement.
There is, however, a central difference between entanglement and coherence
that poses a fundamental challenge in its experimental characterisation. To
create entanglement it is necessary to use coherent interactions between
particles that go beyond Local Operations and Classical Communications (LOCC).
It can however be detected using only local measurements and classical
processing of the resulting data, e.g., in terms of Bell inequalities,
witnesses or state tomography Horodecki  _et al._ (2009); Friis  _et al._
(2018). Thus, verifying entanglement requires less challenging experimental
tools than to prepare it.

This distinction between resources needed for preparation and detection does
not typically exist for coherence. Coherence is always defined with respect to
a basis and this is generically the only basis in which measurements can be
performed. Creating coherence requires an operation that maps a basis state
into a coherent superposition of basis states; detecting coherence requires a
measurement in such a superposition basis. As the latter typically cannot be
done, it is instead replaced with an operation that maps the state back to an
incoherent one (essentially the reverse of the preparation step), followed by
a projection onto one of the basis states. This results in the awkward
situation that any measurement that is supposed to verify the successful
preparation of a coherent superposition is reliable only under the assumption
that coherent superpositions can be created.

As we show here, this is not an insurmountable obstacle. We can find suitable
figures of merit that offer a detailed characterisation of coherence
properties, but that do not require any assumption on the ability to realise
operations that can create coherent superpositions.

Doing this first requires a rigorous definition of the aspects of coherence
that we want to certify. For any given reference basis $\\{\ket{j}\\}$, one
can define pure states $\ket{\psi}=\sum_{j}\psi_{j}\ket{j}$ with at least $k$
non-vanishing amplitudes $\psi_{j}$ to be $k$-coherent. Extending this, a
mixed state $\rho$ is $k$-coherent if all decompositions
$\rho=\sum_{i}p_{i}|\psi_{i}\rangle\langle\psi_{i}|$ into pure states
$\ket{\psi_{i}}$ with $p_{i}\geq 0$ contains at least one $k$-coherent pure
state Levi and Mintert (2014). We denote the set of $k$-coherence state for a
given Hilbert space by $C_{k}$, and have the natural relation that
$C_{k+1}\subset C_{k}$ for $k\geq 1$ and where $C_{1}$ is the full state
space.

Following this definition, the concept of $k$-coherence is closely analogous
to genuine $k$-partite entanglement. Most of the prior literature on quantum
coherence has not yet addressed this fine classification of different classes
of coherence, but there are figures of merit that characterize $k$-coherence
quantitatively Levi and Mintert (2014); Ringbauer  _et al._ (2018) or
qualitatively von Prillwitz  _et al._ (2015). Almost all existing approaches
do rely on the assumption that measurements can be performed reliably in a
basis other than that of the $1$-coherent states, which is highly problematic
for the reasons described above. The only exception we are aware of is from
one of the proposals in Ringbauer  _et al._ (2018) where the coherence is
instead bounded by the probability of success of a quantum game, which comes
with its own assumptions about the dynamics on the system and the measurements
performed. Our method does away with these different assumptions and instead
requires only the acquisition of relative phases and the ability to perform
some rank-$1$ measurement afterwards.

We envision an experiment similar to the famous Ramsey sequence. This involves
a preparation unitary $U_{p}$ such that
$U_{p}\ket{0}=\sum_{j}\psi_{j}\ket{j}=\ket{\psi}$, followed by an evolution
$U(t)$ generated by the system Hamiltonian $H$ for a time $t$. This is
followed by an effective projection onto a state
$\ket{\chi}=\sum_{j}\chi_{j}\ket{j}$ which is realised by the unitary
evolution $U_{r}$, defined by $U_{r}^{\dagger}\ket{0}=\ket{\chi}$, and a
subsequent projection onto the basis state $\ket{0}$. As such, the probability
of getting a ‘click’ in the detector for an initial pure state $\ket{0}$ is
given by $p(t)=|\langle\chi|U(t)|\psi\rangle|^{2}$. This defines the
interference pattern that is observed.

The coherence of $\ket{\psi}$, with respect to the eigenbasis of $H$, can be
characterised in terms of the statistical moments of this probability
distribution, $M_{q}=\langle p^{q}\rangle$, where the average is taken over
the period of the dynamics. When $\ket{\chi}$ is promised to be an equal
superposition of all the eigenstates of $H$ (a state denoted by $\ket{W}$),
these moments provide a rigorous indicator of $k$-coherence. That is, there is
a threshold value such that moments above this threshold value can only be
achieved with states that are at least $k$-coherent von Prillwitz  _et al._
(2015). The intuition behind this is that the interference pattern of higher
coherent states exhibit higher peaks and deeper troughs than low coherent
states; in an analogous way to how the interference pattern of a diffraction
grating and a double slit differ. This behaviour can be detected with the
statistical moments, with higher moments being more sensitive to the more
extreme peaks and troughs.

As argued above, it is highly problematic to assume that the desired
projection onto the state $\ket{W}$ can be performed reliably. Assuming that
such a projection was performed when a different measurement was realised can
suggest a higher degree of coherence than there is. This can easily be seen
with the extreme case of $\ket{\chi}=\ket{0}$. In this case $p(t)$ is
maximised with the incoherent initial state $\ket{0}$, and since this holds
for all $t$, also all moments adopt their maximum value for this state.
Erroneously implementing a measurement including the projection onto the state
$\ket{0}$ rather than the projection onto a balanced superposition of all
basis states is certainly not a realistic experimental scenario, but it helps
to illustrate that uncontrollable experimental imperfections can result in
wrong conclusions if assumptions on the type of measurement are made. In order
to have trusted certification, we require a function that can identify
coherence in the case of suitable measurements, but that does not result in
false positives.

In this paper we introduce a family of functions which do this, based on the
ratio of moments of an interference pattern. We will show that those are
convex functions of a quantum state, which makes them directly applicable to
mixed states. The maximum value that such functions can adopt for a
$k$-coherent state will be shown to be bounded from above independently of the
Hamiltonian $H$ and the projector $\ket{\chi}\bra{\chi}$. Experimental
limitations in the realization of the desired measurement will thus not result
in wrong conclusions on the coherence properties of the state, but will in the
worst case only result in the failure to exceed the threshold.

The construction of these coherence certifiers is presented in Sec.II, where
their properties are also discussed. The technical aspects of the proofs are
left to the appendices. In the cases where the exact threshold values are not
known, we use numerical methods to approximate them; a discussion of these
results is given in Sec.III. This is followed in Sec.IV by a discussion of the
ability of the proposed framework to verify $k$-coherence in the presence of
various imperfections, and we conclude in Sec.V.

##  II Coherence certifier

To talk in precise terms about the coherence certifiers we introduce, it is
necessary to specify exactly the range of systems under consideration. The
coherence of a state is defined with respect to a basis, and the natural basis
to use for a Ramsey-like experiment is the eigenbasis of the system
Hamiltonian. We make no restrictions on this Hamiltonian other than it being
time-independent and having a discrete and commensurate spectrum (all finite
Hamiltonians are discrete and $\epsilon$-close to being commensurate). It may
contain some degeneracies but, as degenerate levels always have the same
relative phases, these will never get picked up by the interference pattern
and so the amount of coherence would be underestimated. As we are only lower
bounding the coherence, this is not a problem. In order to simplify the
analysis it is therefore convenient to ignore these degeneracies and,
furthermore, expand the Hilbert space of the system by adding new levels such
that the spectrum of the Hamiltonian is equally spaced. As this does not
affect the evolution of the physical state, there is no loss of generality in
only considering Hamiltonians

$\displaystyle H=\sum_{n}n\ket{n}\bra{n}\ ,$ (1)

with the spectrum of a harmonic oscillator. For the certifier of coherence we
introduce below, any anharmonicity in the physical Hamiltonian will lead to
less coherence being measured, and therefore cannot result in a false
certification of the amount of coherence present in the state.

As discussed in the introduction, the basic objects we use to study coherence
are the moments of the interference pattern. The $n$th moment is

$\displaystyle M_{n}(\rho,\ket{\chi})$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}p(t)^{n}\,dt$ (2)
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}\braket{\chi}{e^{-iHt}\,\rho\,e^{iHt}}{\chi}^{n}\,dt,$

where the duration of the integral is due to the energy scale picked in
Eq.(1). The key object of interest is the ratio

$R_{n}(\rho,\ket{\chi})=\frac{M_{n}}{M_{1}^{n-1}}$ (3)

of the moments $M_{n}$ and $M_{1}^{n-1}$ for $n>2$. In particular, we will
focus on $R_{3}$ as it is the lowest order which can act as a coherence
certifier.

$k$-coherence |  $R_{3}$ Threshold |  $R_{3}$ Best Known
---|---|---
$1$ | $1$ | $1$
$2$ | $5/4$ | $1.25$
$3$ | $179/96\approx 1.86$ | $1.77$
Table 1: The maximum values that $R_{3}(\rho,\ket{\chi})$ can attain, under
any Hamiltonian, for any $\ket{\chi}$ and for any $\rho\in C_{k}$ as a
function of $k$. As such, exceeding these values means that the state must be
at least $(k+1)$-coherent. The middle column is an upper bound to this highest
value obtained analytically. The last column is the highest value we found
after conducting a thorough numerical optimisation.

A central property of these functions is their convexity under the mixing of
states

$R_{n}\left(\lambda\rho_{1}+(1-\lambda)\rho_{2}\right)\leq\lambda
R_{n}\left(\rho_{1}\right)+(1-\lambda)R_{n}\left(\rho_{2}\right),$ (4)

with the same $\ket{\chi}$ throughout, as proven in App.A. As $C_{k}$ is
itself convex, it is highly desirable for our certifier to also have this
property as it implies that $R_{n}$ is maximised for pure states, i.e.

$\max_{\ket{\psi}\bra{\psi}\in
C_{k},\,\ket{\chi}}R_{n}(\ket{\psi},\ket{\chi})\geq\max_{\rho\in
C_{k},\,\ket{\chi}}R_{n}(\rho,\ket{\chi}),$ (5)

where the ket in the first argument of $R_{n}$ stands for the corresponding
pure state. Because of this, the maximum found for pure states also applies to
mixed states directly.

Another useful feature of $R_{n}$ is that its maximum is reached when the
measurement projector and the initial state are the same, i.e.

$\max_{\ket{\psi}\bra{\psi}\in
C_{k}}R_{n}(\ket{\psi},\ket{\psi})\geq\max_{\ket{\psi}\bra{\psi}\in
C_{k},\,\ket{\chi}}R_{n}(\ket{\psi},\ket{\chi}).$ (6)

This is not necessary for a coherence certifier, but is nevertheless desirable
for two reasons. Firstly it aligns with the intuition of a Ramsey-like
interferometer, where the highest contrast is obtained by projecting onto the
initial state, which is also what was found in prior work where $\ket{\chi}$
was assumed to be the equal superposition state $\ket{W}$ von Prillwitz  _et
al._ (2015). Secondly it further simplifies calculating the threshold values,
rather than maximising over the $4d$ real variables that define $\ket{\psi}$
and $\ket{\chi}$: it is enough to consider only the $d$ variables,
$\psi_{i}\chi_{i}^{*}$, which can always be chosen such that they are real.
This is proven in App.B.

Of particular importance is the need for $R_{n}$ to be hierarchical, such that
it obeys the strict inequality

$\max_{\rho\in C_{k+1},\,\ket{\chi}}R_{n}(\rho,\ket{\chi})>\max_{\rho\in
C_{k},\,\ket{\chi}}R_{n}(\rho,\ket{\chi}),$ (7)

where the maximum for a given $k$ is known as the threshold value for $k+1$.
As proven in App.C, this holds for $k=1,2$ and $3$ independently of the
dimension of the system Hilbert space. Observing a higher value than those
thresholds, given in Tab.(1), therefore proves that the state is at least
$2,3$, or $4$-coherent respectively.

The assumption so far is that the measurement is projective. In practice,
however, the realization of the unitary $U_{r}$ can be affected by noise, and
repetitions of the experiment that are required to obtain good statistics will
suffer from fluctuations in $U_{r}$.

The signal on the measurement device will thus not reliably indicate
projection onto the state $\ket{\chi}$, but rather randomly a projection onto
one out of several states $\ket{\chi_{j}}$ occurring with probability $q_{j}$.
In this case the recorded interference pattern reads

$\displaystyle p(t)$ $\displaystyle=\sum_{j}q_{j}p_{j}(t),\quad\text{where}$
(8) $\displaystyle p_{j}(t)$
$\displaystyle=\bra{\chi_{j}}\,U(t)\rho\,U^{\dagger}(t)\ket{\chi_{j}}\ ,$ (9)

and the definition of moments given above in Eq.(2) generalizes to

$\displaystyle M_{n}(\rho,\sigma_{\chi})$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}(\mbox{Tr}\
{e^{-iHt}\,\rho\,e^{iHt}\,\sigma_{\chi}})^{n}\,dt\,$

with $\sigma_{\chi}=\sum_{j}q_{j}|\chi_{j}\rangle\langle\chi_{j}|$. In exactly
the same way that
$R_{n}(\rho,\sigma_{\chi})=M_{n}(\rho,\sigma_{\chi})/M_{1}(\rho,\sigma_{\chi})^{n-1}$
is convex in the first argument $\rho$ for any given $\sigma_{\chi}$, it is
also convex in the second argument for any given $\rho$ such that

$R_{n}(\rho,\sigma_{\chi})\leq\sum_{j}q_{j}R_{n}(\rho,\ket{\chi_{j}})\ ,$ (10)

for any state $\sigma_{\chi}$ and convex decomposition into pure states
$\sum_{j}q_{j}|\chi_{j}\rangle\langle\chi_{j}|=\sigma_{\chi}$. Since no
projective measurement can overestimate the degree of coherence, no
fluctuations in the realisation of such a measurement can result in a false
positive either.

##  III Numerical threshold values

While the previous section details analytically proved results about the
threshold values for $k$ up to $4$, we can go to much higher coherence levels
numerically. We do this by maximising the value of $R_{n}$ over all $\rho\in
C_{k}$ and all $\ket{\chi}$, for given values of $n$ and $k$. This problem is
substantially simplified using the results of the previous section, which lets
us set $\rho=\ket{\psi}\bra{\psi}$ and $\ket{\psi}=\ket{\chi}$ which only
contain real coefficients in the eigenbasis of the Hamiltonian. We are
confident that the results found this way are an excellent approximation of
the true maxima as they are stable under different parameterisations of the
problem and for different initial conditions in the numerical optimisation.
These numerical results can also be compared to the upper bounds given by the
analytic results, thereby illustrating how tight they are.

$R_{n}$ | $k$ | $R_{n}(\ket{\Psi_{k}})$ | $R_{n}(\ket{W_{k}})$ | $\boldsymbol{\Psi}_{k}$
---|---|---|---|---
$R_{3}$ | 2 | 1.25 | $1.25$ | $(0.50,0.50)$
| 3 | 1.77 | $1.74$ | $(0.31,0.38,0.31)$
| 4 | 2.32 | $2.27$ | $(0.22,0.28,0.28,0.22)$
| 5 | 2.88 | $2.80$ | $(0.17,0.21,0.23,0.21,0.17)$
$R_{4}$ | 2 | 2.19 | 2.19 | $(0.50,0.50)$
| 3 | 4.61 | 4.56 | $(0.32,0.36,0.32)$
| 4 | 8.02 | 7.90 | $(0.23,0.27,0.27,0.23)$
| 5 | 12.42 | 12.21 | $(0.18,0.21,0.22,0.21,0.18)$
$R_{5}$ | 2 | 3.94 | 3.94 | $(0.50,0.50)$
| 3 | 12.39 | 12.28 | $(0.32,0.36,0.32)$
| 4 | 28.71 | 28.39 | $(0.24,0.26,0.26,0.24)$
| 5 | 55.52 | 54.84 | $(0.19,0.21,0.21,0.21,0.19)$
Table 2: Numerical results for the first three hierarchical ratios for up to
$5$-coherent states; showing their behaviour as coherence certifiers. The
values of $R_{n}$ are given for the equal superposition state $\ket{W_{k}}$,
and for the state $\ket{\Psi_{k}}$ which maximises the value (in all cases the
basis for the projector $\ket{\chi}$ is equal to the state itself as we know
that this maximises $R_{n}$). The states $\ket{W_{k}}$ and $\ket{\Psi_{k}}$
only have adjacent energy levels populated, spacing these levels out always
results in a decrease in $R_{n}$ (unless they are spaced out equally in which
case they are effectively adjacent levels for a different harmonic
Hamiltonian). $\ket{\Psi_{k}}$ is found through numerical optimisation and is
stable through different parametrisations of the problem and from different
initial points. The amplitudes squared of $\ket{\Psi_{k}}$ are also listed as
a vector to show how it differs from the uniform case of $\tfrac{1}{k}$.

These numerical results are listed in Tab.(2), which also shows the state
$\ket{\Psi_{k}}$ that gives the maximum value of $R_{n}$ over all states in
$C_{k}$, and how this value compares to the value given by the equally
balanced state $\ket{W_{k}}=\tfrac{1}{\sqrt{k}}\sum_{i}^{k}\ket{i}$. These
states are, surprisingly, not the same, although they both share the property
of having $k$ adjacent basis states populated while the others have zero
amplitude. $\ket{\Psi_{k}}$ has a concentration of population towards the
middle of the occupied energy levels. One way to understand this is to note
that interferences between basis states with small energy differences
contribute more to $R_{n}$ than those with large energy differences. As the
basis states in the middle of the spectrum are closer to more of the basis
states, the function is maximised by populating them more than the others.
This intuition is more visible in the re-parametrisation of $R_{n}$ done in
App.C. Furthermore, the larger $k$ is and the smaller $n$ is, the more
pronounced the difference between $\ket{W_{k}}$ and $\ket{\Psi_{k}}$ is.

Figure 1: Comparison of numerical and analytical threshold values. The crosses
show the maximum value that we found for $R_{3}$ for $\rho\in C_{k}$ as a
function of $k$. The solid blue line is a linear fit for these, showing how
they are equally spaced. The dashed orange line shows what value the equal
superposition state $\ket{W_{k}}$ has for the optimal measurement for
comparison and is given by $\frac{4+5k^{2}+11k^{4}}{20k^{3}}$ (derived in
App.D), which is asymptotically linear. The horizontal lines are the analytic
threshold values. For the 2-coherent case, the equal superposition and optimal
states overlap, and lie immediately below the threshold for certifying
3-coherence. For the 3-coherent case and higher, there is a finite but small
gap between the equally balanced and optimal states. The threshold for
4-coherence also does not lie exactly above the maximum for 3-coherence, but
the gap is again very small and, as we are lower bounding the amount of
coherence present, this only means that $R_{3}$ is occasionally too cautious.

In all cases of interest, however, the difference in the $R_{n}$ value between
$\ket{\Psi_{k}}$ and $\ket{W_{k}}$ is relatively small, which can be seen in
Fig.(1). This figure also compares these to the analytic thresholds which
shows how tight they are. Furthermore, the maximal values grow linearly
(tested up to $k=30$, not shown on the graph). This constant interval means
that $R_{3}$ would also be able to distinguish between more highly coherent
states. The functions $R_{4}$ and $R_{5}$ seem to have even faster growth,
potentially making them more useful in such circumstances, although the
additional experimental difficulty in accurately reconstructing higher moments
should not be neglected Flusser  _et al._ (2009).

##  IV Verification of $k$-coherence in the presence of imperfections

In this section we demonstrate that the present approach can verify coherence
properties, even in the presence of substantial imperfections in the
projective measurement and that coherence can be detected even in highly mixed
states.

###  IV.1 Measurement tolerance

Having proved that an imperfect measurement will never overestimate the
coherence of a state, it is important to demonstrate that it does not
underestimate it too strongly either. Therefore, we quantify this implication
of measurement imperfections here. To achieve this, we produce a sample of
random faulty measurements and estimate the deviation from perfect measurement
required to reduce the value of the maximum $k$-coherent state below the
threshold below which $k$-coherence is not verified anymore.

We define the states $\ket{\chi_{k}(\tau)}$ that define a projective
measurement in terms of a random Hamiltonian $H_{r}$ via the relation

$\ket{\chi_{k}(\tau)}=\mathcal{U}(\tau)\ket{\Psi_{k}}\coloneqq
e^{iH_{r}\tau}\ket{\Psi_{k}}\ ,$ (11)

with $\ket{\Psi_{k}}$ given in Tab. 2; the random Hamiltonians $H_{r}$ are
drawn from the Gaussian Unitary Ensemble (GUE) Fyodorov (2004).

The degree to which the projective measurement deviates from the ideal
measurement can be quantified by the norm

$D(\tau)\coloneqq||\ket{\chi_{k}(\tau)}-\ket{\chi_{k}(0)}||\equiv\sum\limits_{i=1}^{d}\left[\mathcal{U}_{ij}(\tau)\chi_{j}-\chi_{i}\right]^{2}\
,$ (12)

for each realisation of $H_{r}$.

Fig.(2) depicts the ensemble average of
$R_{3}(\ket{\Psi_{k}},\ket{\chi_{k}(\tau)})$ with the average performed over
$100$ random Hamiltonians as function of $D(\tau)$ with black lines for $k=3$
and $k=4$. The blue and pink lines depict the width of the underlying
distribution, and the horizontal black dashed lines depict the threshold
values for the detection of 3-coherence and 4-coherence. As one can see, a
substantial value of $\tau$ is required before the recorded values of $R_{3}$
drop below the threshold values. As one might have expected the verification
of 3-coherence can tolerate a large amount of deviations, but even for the
verification of 4-coherence, a deviation $D\leq 0.3$ is typically good enough.

Figure 2: Ensemble average (black) of $R_{3}$ for the states $\ket{\Psi_{k}}$
($k=4$ at the top, $k=3$ at the bottom) obtained with faulty measurements as
function of the measurement deviation $D$ defined in Eq.(12). The standard
deviation of the random distribution is depicted with solid (blue) and dotted
(pink) lines centred around the average. The values of $D$, for which the
threshold values (Tab. 1) are reached, are depicted in turqoise crosses
(ensemble average) with solid horizontal red lines for the width of the
distribution.

###  IV.2 Decoherence tolerance

Since our central aim is the ability to verify coherence in the presence of
experimental imperfections, the big remaining question is on the degree of
decoherence that can be present, before the present criteria fail to verify a
desired level of coherence. Repetitions of the experiment may result in some
instances of initially mixed states which are then evolved through the system.
The decoherence effects of such a faulty state preparation is to reduce the
visibility of the interference pattern, thus rendering the task of bounding
coherence more challenging.

We explore the impact of decoherence by introducing the Werner-like state
Werner (1989)

$\rho_{W}=(1-\lambda)\ket{W_{k}}\bra{W_{k}}+\frac{\lambda}{k}\mathbb{I}_{k}\
,$ (13)

and exploring the ability of the ratios to distinguish its level of coherence.
The Werner-like state is given by a mixture of the equal superposition
$k$-coherent state and the totally incoherent state $\mathbb{I}_{k}/d$. The
degree of mixedness is varied with the parameter $\lambda\in[0,1]$. For
$\lambda=0$, the system is pure and $k$-coherent, while $\lambda=1$
corresponds to a completely mixed state. Therefore, there must be a
theoretical upper bound $\lambda_{\text{dec}}(q)$ above which the system is in
$C_{q}$, but not $C_{q+1}$, and as the noise increases further there must be
another bound above which the coherence drops further. For a $k$-dimensional
system, these bounds $\lambda_{\text{dec}}(q)$ are given by

$\lambda_{\text{dec}}(q)=\frac{k-q}{k-1},\quad 1\leq q\leq k,$ (14)

as proved in App.E and also discussed in Ref. Ringbauer  _et al._ (2018).
Similarly, we can define threshold values $\lambda_{\text{thr}}^{(n)}(q)$ at
which a given certifier $R_{n}$ fails to verify $(q+1)$-coherence in a system
from its interference pattern. The values $\lambda_{\text{thr}}^{(n)}(k-1)$ at
which a given certifier $R_{n}$ fails to identify $k$-coherence are depicted
in Tab.(3) for $R_{3}$, $R_{4}$ and $R_{5}$, and numerical expressions for
$\lambda_{\text{dec}}(q)$ are given for comparison.

As one can see, the threshold values for the detection of $k$-coherence are
larger the smaller $k$ is. $k$-coherence can thus be identified for rather
strongly mixed states as long as $k$ is sufficiently low. $R_{5}$ can identify
coherence for larger values of $\lambda$ (i.e. more strongly mixed states)
than $R_{4}$ for any value of $k$, and $R_{4}$ outperforms $R_{3}$ in the same
sense. If a given $R_{n}$ fails to verify $k$-coherence in a strongly mixed
state, one can thus resort to a certifier $R_{n}$ with a larger value of $n$,
and find better performance. Even for $R_{5}$, however, the threshold value
$\lambda_{\text{dec}}(k-1)$ is about $50\%$ larger than
$\lambda_{\text{thr}}^{(5)}(q)$, and higher moments would be required in order
to identify the $k$-coherence in very strongly mixed states.

$k$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$
---|---|---|---|---|---|---|---|---
$\lambda_{\text{thr}}^{(3)}(k-1)$ | $0.18$ | $0.13$ | $0.10$ | $0.08$ | $0.06$ | $0.06$ | $0.05$ | $0.04$
$\lambda_{\text{thr}}^{(4)}(k-1)$ | $0.28$ | $0.19$ | $0.14$ | $0.11$ | $0.09$ | $0.08$ | $0.07$ | $0.06$
$\lambda_{\text{thr}}^{(5)}(k-1)$ | $0.33$ | $0.22$ | $0.16$ | $0.13$ | $0.11$ | $0.09$ | $0.08$ | $0.07$
$\lambda_{\text{dec}}(k-1)$ | $0.5$ | $0.33$ | $0.25$ | $0.2$ | $0.17$ | $0.14$ | $0.13$ | $0.11$
Table 3: Numerical expressions for decoherence thresholds for the state
$\rho_{W}$ from Eq.(13), for $R_{3},R_{4}$ and $R_{5}$ between consecutive
levels of coherence for $k=3$ to $10$.

###  IV.3 Best approximations of interference pattern

In addition to the thresholds $\lambda_{\text{dec}}(q)$ and
$\lambda_{\text{thr}}^{(n)}(k-1)$ discussed above, there is also the threshold
$\lambda_{\text{patt}}(q)$ at which a given interference pattern no longer
allows to verify $q$-coherence. As $R_{n}$ is a scalar functional of the
interference pattern, it can contain up to as much information as the pattern
itself, and a small difference between $\lambda_{\text{patt}}$ and
$\lambda_{\text{thr}}^{(n)}$ indicates that only little information is lost by
looking at the ratio of specific moments instead of the full interference
pattern.

These thresholds, for all $n$, satisfy the relation

$0<\lambda_{\text{thr}}^{(n)}(q)\leq\lambda_{\text{patt}}(q)\leq\lambda_{\text{dec}}(q)<1,$
(15)

for all $n$. The value of $\lambda_{\text{patt}}(q)$ is strongly dependent on
the measurement projection $\ket{\chi}\bra{\chi}$, and we show in App.E that
the threshold values $\lambda_{\text{patt}}(q)$ and $\lambda_{\text{dec}}(q)$
nevertheless coincide for Werner-like states, with a projection onto the equal
superposition $k$-coherent state $\ket{W_{k}}$. Strikingly, this verifies that
in this case a single interference pattern can provide enough information for
a complete classification of $q$-coherence.

For $\lambda\geq\lambda_{\text{patt}}(q)$, a $q$-coherent state is mixed
enough to produce a pattern $p(t)$ which can be reproduced by states of lesser
coherence. Patterns $p(t)$ resulting from states with
$\lambda<\lambda_{\text{patt}}(q)$, on the other hand, cannot be reproduced by
states in $C_{q-1}$. In order to exemplify the differences in interference
patterns that the present criteria aim at identifying, for a given
interference pattern $p(t)$ produced by a $k$-coherent state with $k>q$, we
introduce the best $q$-approximation $\bar{p}_{q}(t)$ to $p(t)$, as the
interference pattern resulting from $q$-coherent states only with minimal
deviation from $p(t)$.

Figure 3: (a) Interference patterns $p(t)$ of $\rho_{W}\in C_{3}$ in a
3-dimensional space with different values of $\lambda$ projected under optimal
measurement state $\ket{W_{3}}$ (solid/dotted curves), along with their best
approximations $\bar{p}_{2}(t)$ (dashed curves) reproduced by states in
$C_{2}$. The $R_{3}$ values of the states are, $1.26,0.88,0.60$, with
increasing $\lambda$, so the system corresponding to $\lambda=0.18$ can be
certified by $R_{3}$ as 3-coherent.
(b)-(d) Three linearly independent 2-coherent states that, when mixed with the
given probabilities $p_{m}$, provide the best approximation of $\rho_{W}$ at
$\lambda=0.18$.

In Fig.(3), we focus on $R_{3}$ and we investigate the ability to detect
3-coherence on states $\rho_{W}\in C_{3}$ with optimal projection. The
patterns corresponding to $\rho_{W}$ are plotted for $\lambda=0.18,0.36$ and
$0.54$ along with their best approximations $\bar{p}_{2}(t)$. As long as
$\lambda<\lambda_{\text{dec}}(3)=\frac{1}{2}$, which is the case for the red
and blue curves (corresponding to $\lambda=0.18,0.36$ respectively), the
pattern cannot be reproduced by states in $C_{2}$, as expected, since
$\lambda_{\text{patt}}(3)=\lambda_{\text{dec}}(3)$. The green pattern
($\lambda=0.54$) can be reproduced by states in $C_{2}$ exactly, because in
this case $\lambda>\lambda_{\text{dec}}(3)$. If the projection was
sufficiently far from the optimal, the red and blue patterns would also be
exactly reproducible by patterns of 2-coherent states. The red pattern
corresponds to $\rho_{W}(\lambda=0.18)$, and has a value of $R_{3}$ (when
maximised over $\ket{\chi}$) of $1.26$, which lies above the threshold given
in Tab.(1) to certify a state as $3$-coherent. The bottom part of the plot
gives the patterns of the three 2-coherent states which, when mixed, provide
the best approximation to the red $\rho_{W}$ pattern. Three is the minimum
number of basis states required to form the 3-dimensional Werner-like state.
For $\lambda>\frac{2}{3}$, the patterns could also be decomposed simply into
incoherent states, as Eq.(14) indicates.

##  V Conclusion

Despite the numerous similarities between the theories of entanglement and
coherence, the equality in operation required for creation and verification of
quantum coherence defines a crucial difference between those two theories. Our
proposed solution relies on easily observable quantities such that an
imperfectly implemented verification protocol can never overestimate the
degree of coherence. As such, it offers very practical and robust avenue to
rigorously verify coherence properties beyond the two-level setting.

Beyond the fundamental question ‘when is a triple-slit interference pattern so
washed out, that one can not recognize it anymore?’, the ability to verify the
number of states contributing to a coherent superposition has also very
practical applications in the verification that a potential quantum device is
actually able to operate in the quantum regime that it is supposed to.

##  VI Acknowledgments

We are grateful for stimulating discussions with Nicky Kai Hong Li that
motivated this work. B.D. acknowledges funding from the Engineering and
Physical Sciences Research Council (EPSRC UK) administered by Imperial College
London via the Postdoctoral Prize Fellowship program for the core duration of
this work; and funding from Austrian Science Fund (FWF): P 30947 for the
closing stages. N.K. acknowledges funding from the EPSRC UK through the
Controlled Quantum Dynamics Centre for Doctoral Training for the closing
stages.

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##  Appendix A Proof that $R_{n}$ is convex in either argument

To prove that $R_{n}$ is convex under the mixing of states it suffices to show
that

$R_{n}\left(\lambda\rho_{1}+(1-\lambda)\rho_{2},\ket{\chi}\right)\leq\lambda
R_{n}\left(\rho_{1},\ket{\chi}\right)+(1-\lambda)R_{n}\left(\rho_{2},\ket{\chi}\right),$
(16)

for all pairs of states $\rho_{1},\rho_{2}$, for all projectors
$\ket{\chi}\bra{\chi}$, and for all $\lambda\in[0,1]$.

This property holds for the moments themselves, which are convex and positive
by construction. Products and sums of such functions stay convex, but this is
not necessarily the case for ratios of them. We prove that this particular
function is indeed convex, for $n\geq 2$, by taking the second derivative of
Eq.(16) with respect to $\lambda$ and showing that it is always non-negative.

This second derivative is

$\displaystyle\partial_{\lambda}^{2}R_{n}=\frac{M_{1}^{3n-5}}{M_{1}^{4n-4}}\big{[}M_{1}^{2}\partial_{\lambda}^{2}M_{n}-2(n-1)M_{1}(\partial_{\lambda}M_{1})\partial_{\lambda}M_{n}+n(n-1)(\partial_{\lambda}M_{1})^{2}M_{n}\big{]}.$

Denoting the integrand of $M_{n}$ in Eq.(2) by $p^{n}$ and the time average by
$\langle\cdot\rangle$, allows the derivatives to be calculated according to

$\displaystyle\partial_{\lambda}M_{n}$ $\displaystyle=\langle
n(\partial_{\lambda}p)p^{n-1}\rangle,$
$\displaystyle\partial_{\lambda}^{2}M_{n}$ $\displaystyle=\langle
n(n-1)(\partial_{\lambda}p)^{2}p^{n-2}\rangle.$

Substituting these expressions into Eq.(A) gives

$\displaystyle\partial_{\lambda}^{2}R_{n}=\frac{M_{1}^{3n-5}}{M_{1}^{4n-4}}\langle
n(n-1)p^{n-2}\left[p\langle\partial_{\lambda}p\rangle-\langle
p\rangle\partial_{\lambda}p\right]^{2}\rangle,$ (17)

where the fraction at the front is non-negative, as is the squared term in the
time average and its pre-factor (for $n\geq 2$), thereby showing that $R_{n}$
is convex as desired.

##  Appendix B Proof that $R_{n}$ is maximised for equal preparation and
projection

We begin by noting that the expression for the probability distribution in
Eq.(2) for pure states is given by the double sum

$\displaystyle
p(\ket{\psi},\ket{\chi},t)=\sum_{p,q}\chi_{p}^{*}\psi_{p}\psi_{q}^{*}\chi_{q}e^{-i(p-q)t},$
(18)

where $\ket{\psi}=\sum_{p}\psi_{p}\ket{p}$,
$\ket{\chi}=\sum_{q}\phi_{q}\ket{q}$ and the basis states are eigenkets of the
Hamiltonian of Eq.(1) $H=\sum_{n}n\ket{n}\bra{n}$. By defining
$\psi_{p}\chi_{p}^{*}=\alpha_{p}e^{i\phi_{p}}$, $\phi_{pq}=\phi_{p}-\phi_{q}$
and $\omega_{pq}=p-q$ this can be recast as

$\displaystyle
p(\ket{\psi},\ket{\chi},t)=\sum_{p}\alpha_{p}^{2}+2\sum_{p>q}\alpha_{p}\alpha_{q}\cos(\omega_{pq}t+\phi_{pq}),$
(19)

where the $\alpha$ are real and non-negative by construction.

We now show that the maximum of this over $k$-coherent $\ket{\psi}$ and any
$\ket{\chi}$ is reached when the phases $\phi_{pq}$ are all zero, for all $k$.
Firstly, because integrating cosines over an integer number of periods gives
zero, the first moment is independent of them,

$\displaystyle M_{1}=2\pi\sum_{p}\alpha_{p}^{2}.$ (20)

It is therefore clear that changes in $\phi_{pq}$ (arising from different
phases between the state and the projector) affect the numerator of $R_{n}$
but not the denominator. The terms of $M_{>1}$ which depend non-trivially on
the phases are inside the integral over time and are of the form

$\displaystyle\int_{0}^{2\pi}\left(\sum_{p>q}\alpha_{p}\alpha_{q}\cos(\omega_{pq}t+\phi_{pq})\right)^{m}\,dt.$
(21)

To see which terms do not vanish when integrated over, it is useful to look at
the products of cosines individually

$\displaystyle\int_{0}^{2\pi}$
$\displaystyle\alpha_{p_{1}}\alpha_{q_{1}}\cos(\omega_{p_{1}q_{1}}t+\phi_{p_{1}q_{1}})\times\alpha_{p_{2}}\alpha_{q_{2}}\cos(\omega_{p_{2}q_{2}}t+\phi_{p_{2}q_{2}})\times...\,dt.$
(22)

which can themselves be expanded into a sum of cosines, where each term is of
the form

$\displaystyle\propto\int_{0}^{2\pi}\cos\left[(\omega_{p_{1}q_{1}}\pm\omega_{p_{2}q_{2}}...)t+\phi_{p_{1}q_{1}}\pm\phi_{p_{2}q_{2}}...\right]\,dt.$
(23)

If the sum (for the different permutations of signs) of frequencies do not sum
to $0$, then the integral vanishes. If they do sum to $0$, the term is
proportional to the cosine of the sum (for the different permutations of
signs) of the phases. One of the solutions which maximises this is to pick all
the $\phi_{pq}=0$, which simultaneously maximises every such integral no
matter the number of terms or the sign configuration. This itself increases
$M_{n}$ and therefore the value of $R_{n}$.

In this case that there are no relative phases, $R_{n}$ can be written in
terms of a simplified Eq.(19) as

$\displaystyle
R_{n}(\ket{\psi},\ket{\chi})=\frac{\int_{0}^{2\pi}\left(\sum_{p}\alpha_{p}^{2}+2\sum_{p>q}\alpha_{p}\alpha_{q}\cos(\omega_{pq}t)\right)^{n}\,dt}{2\pi\left(\sum_{p}\alpha_{q}^{2}\right)^{n-1}}.$
(24)

From this it can be seen that the mapping $\alpha_{p}\to x\alpha_{p}$ changes
the function $R_{n}\to x^{2}R_{n}$. It is therefore desirable to scale the
$\alpha$ to be as large as possible. The extent to which this can be done is
bounded by the normalisations of the states, using Cauchy-Schwarz we can
express this as:

$\displaystyle\left(\sum_{p}\alpha_{p}\right)^{2}=\left(\sum_{p}\psi_{p}\chi_{p}\right)^{2}$
$\displaystyle\leq\left(\sum_{p}\psi_{p}^{2}\right)\;\left(\sum_{p}\chi_{p}^{2}\right)=1,$
(25) $\displaystyle\implies\sum_{p}\alpha_{p}$ $\displaystyle\leq 1,$

Furthermore, any set of $\\{\alpha_{p}\\}$ that satisfy this bound can be
realised by the normalised states $\ket{\psi}$, $\ket{\chi}$ by picking their
amplitudes according to $\psi_{p}=\chi_{p}=\sqrt{\alpha_{p}}$.

Taking a step back, what we have shown by parametrising the function $R_{n}$
in terms of $\\{\alpha_{p},\phi_{p}\\}$, is that the maximum of $R_{n}$ occurs
when $\phi_{p}=0$ and $\sum\alpha_{p}$ = 1. These two conditions are
equivalent, in terms of the physical state and measurement projector, to
having $\ket{\psi}=\ket{\chi}$. Thus, we know that $R_{n}$ is maximised when
the input state is pure and the projective measurement is equal to it, thereby
greatly shrinking the space over which we have to optimise. Note that this is
not the same as the subtly different question of whether the optimal
$\ket{\chi}$ that should be picked for a given $\ket{\psi}$ is for them to be
the same. Here, we are only interested in the overall bound $R_{n}$ can have
over any input state with a fixed $k$-coherence.

##  Appendix C Derivation of analytic threshold values

We now compute the maximum of Eq.(24) for $n=3$ as a function for $k$ where
$\ket{\psi}\bra{\psi}\in C_{k}$. For $k=2$, this is easily done by using the
previously found constraint of $\sum\alpha_{p}=1$. We denote the two non-zero
$\alpha$’s as $x$ and $1-x$ and can perform the integration over time
explicitely to arrive at

$\max_{\rho\in
C_{2},\ket{\chi}}R_{3}(\rho,\ket{\chi})=\max_{x\in[0,1]}\frac{1+2x(x-1)(5x^{2}-5x+2)}{1+2x(x-1)}.$
(26)

The right hand side is easily solved analytically and gives a value of $5/4$.
Therefore, measuring an $R_{3}$ of greater than that value implies that the
state must be at least $3$-coherent.

To deal with higher $k$, it is highly advantageous to reparametris the
optimsation problem. the starting point is Eq.(24) and we now make another
simplification in the notation by grouping together terms with the same
frequency $\omega_{pq}=p-q$. This allows the sum over the cosines to be
expressed as

$\displaystyle\sum_{p>q}\alpha_{p}\alpha_{q}\cos(\omega_{pq}t)$
$\displaystyle=\sum_{n}D_{n}\cos(\omega_{n}t)$ (27)

where the new variables are given by

$\displaystyle\quad D_{n}$
$\displaystyle=\sum_{p}\alpha_{p+n}\alpha_{p},\quad\omega_{n}=n,$ (28)

which also lets us rewrite the term $\sum_{p}\alpha_{p}^{2}=D_{0}$, thereby
unifying the notation. We also recall that, from previous arguments, that
$\sum_{p}\alpha_{p}=1$ for the maximum of the function. Using this notation in
Eq.(24) for the case $n=3$ we obtain

$\displaystyle R_{3}(\ket{\psi},\ket{\chi})=\int_{0}^{2\pi}\big{[}$
$\displaystyle
D_{0}+6\sum_{i}D_{i}\cos(\omega_{i}t)+\frac{12}{D_{0}}\sum_{ij}D_{i}D_{j}\cos(\omega_{i}t)\cos(\omega_{j}t)+$
(29)
$\displaystyle\frac{8}{D_{0}^{2}}\sum_{ijk}D_{i}D_{j}D_{k}\cos(\omega_{i}t)\cos(\omega_{j}t)\cos(\omega_{j}t)\big{]}\;dt$

Performing the integrals in the way described earlier, only terms where the
$\omega$ sum to $0$ contribute, which yields

$\displaystyle R_{3}(\ket{\psi},\ket{\chi})$
$\displaystyle=6D_{0}\left(\frac{1}{6}+\sum_{ij}\frac{D_{i}D_{j}}{D_{0}^{2}}\delta_{ij}+\sum_{ijk}\frac{D_{i}D_{j}D_{k}}{D_{0}^{3}}\sigma_{ijk}\right)$
$\displaystyle=D_{0}\left(\frac{1}{6}+\sum_{i}\tilde{D}_{i}^{2}+\sum_{ijk}\tilde{D}_{i}\tilde{D}_{j}\tilde{D}_{k}\sigma_{ijk}\right)$
(30)

where $\sigma_{ijk}$ is a phase matching condition which is $1$ if $i+j=k$ and
$0$ otherwise, and $\tilde{D}_{i}=D_{i}/D_{0}$.

Finding the maximum value of $R_{3}$ over all $k$-coherent states as a
function of $k$ has proved very difficult. What we have found is a method to
calculate an analytic upper bound for this quantity for a given $k$. We have
evaluated this bound for small $k$ and, although the method is applicable in
general, it may be too laborious to be practical for high $k$.

The key idea is to treat the $\\{\tilde{D}_{i}\\}$ as independent variables to
optimise over and $D_{0}$ as a ‘free’ parameter. Eq.(28) is used to form
linear constraints on the $\\{\tilde{D}_{i}\\}$, which forms an outer
approximation to the physically allowed region for a choice of $D_{0}$. We
then show that in this region $R_{3}$ has a positive definite Hessian, which
implies that for any line cutting through this region, the maxima of the
function must be reached where the line crosses the bounding surface.
Therefore, the maximum value is attained at one of the vertices. As this
region is defined by linear constraints it is a polytope, and hence has only a
finite number of vertices which can be individually evaluated to see which
produces the largest value of $R_{3}$. The remaining step is then optimise
over $D_{0}$, which is easily done numerically as the problem is reduced to
finding the turning points of a quotient of low order polynomials in one
variable.

Although we could not show that the Hessian is positive in general, we do find
that it is in all the cases of interest. For convenience, it is useful to list
its components here. These are the derivatives of $R_{3}$, which are given by

$\displaystyle\partial_{\tilde{D}_{a}}R_{3}$
$\displaystyle=6D_{0}\left(2D_{a}+2\sum_{jk}\tilde{D}_{j}\tilde{D}_{k}\sigma_{ajk}+\sum_{ij}\tilde{D}_{i}\tilde{D}_{j}\sigma_{ija}\right)$
$\displaystyle\partial_{\tilde{D}_{a}}\partial_{\tilde{D}_{a}}R_{3}$
$\displaystyle=12D_{0}(1+D_{2a})$ (31)
$\displaystyle\partial_{\tilde{D}_{b}}\partial_{\tilde{D}_{a}}R_{3}$
$\displaystyle=12D_{0}(D_{a+b}+D_{|a-b|}).$ (32)

We now apply the method outlined above to $k=3$, treating the case where the
dimension of the Hamiltonian is truncated at $d=3$ and where it is unbounded
separaly. We also calculate the case $k=4,d=4$ to show that the method can be
applied to higher coherence levels.

### $k=d=3$

For states which are at most $3$-coherent in a $3$-dimensional Hamiltonian,
the variables are explicitely given by:

$\displaystyle D_{0}$
$\displaystyle=\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2}$ $\displaystyle
D_{1}$ $\displaystyle=\alpha_{1}\alpha_{2}+\alpha_{2}\alpha_{3}$
$\displaystyle D_{2}$ $\displaystyle=\alpha_{1}\alpha_{3}$ $\displaystyle 1$
$\displaystyle=\alpha_{1}+\alpha_{2}+\alpha_{3}.$

From this we can write some inequalities which constrain the allowed values.
Firstly, as the $\alpha$’s are all positive we have that $0\leq D_{0}$ and
$0\leq\tilde{D}_{i}$. Secondly, the triangle inequality implies that

$\frac{1}{d}\leq D_{0}\leq 1.$ (33)

The first non-trivial constraint comes about from the same starting point

$\displaystyle 1$ $\displaystyle=(\alpha_{1}+\alpha_{2}+\alpha_{3})^{2}$
$\displaystyle=D_{0}+2D_{1}+2D_{2}$
$\displaystyle=D_{0}\left(1+2\sum_{i}\tilde{D}_{i}\right).$ (34)

From these two relations we upper bound the maximum values of any
$\tilde{D}_{i}$

$\displaystyle\frac{1}{d}\left(1+2\sum_{i}\tilde{D}_{i}\right)\leq 1$
$\displaystyle\sum_{i}\tilde{D}_{i}\leq\frac{d-1}{2}$
$\displaystyle\tilde{D}_{i}\leq\frac{d-1}{2}.$ (35)

Other inequalities can be obtained by considering well chosen sums of squares,
the three useful ones are listed here. Firstly

$\displaystyle(\alpha_{1}-\alpha_{3})^{2}+\alpha_{2}^{2}\geq 0$ $\displaystyle
D_{0}-2D_{2}\geq 0$ $\displaystyle 1-2\tilde{D}_{2}\geq 0.$ (36)

Changing the sign gives a different inequality

$\displaystyle(\alpha_{1}+\alpha_{3})^{2}+\alpha_{2}^{2}\geq\tfrac{1}{2}$
$\displaystyle D_{0}+2D_{2}\geq\tfrac{1}{2}$ $\displaystyle
D_{0}(1+2\tilde{D}_{2})\geq\tfrac{1}{2},$ (37)

where the triangle inequality is used in the first line. Lastly, there is

$\displaystyle(\alpha_{1}-\alpha_{2}+\alpha_{3})^{2}\geq 0$ $\displaystyle
D_{0}-2D_{1}+2D_{2}\geq 0$ $\displaystyle 1-2\tilde{D}_{1}+2\tilde{D}_{2}\geq
0.$ (38)

The last three equations (for fixed $D_{0}$) define a triangular region of
interest, while Eq.(34) is a line that cuts through it. They can be expressed
as succinctly as

$\displaystyle\max\left(\frac{1-2D_{0}}{4D_{0}},0\right)$
$\displaystyle\leq\tilde{D}_{2}\leq\frac{1}{2}$ $\displaystyle 0$
$\displaystyle\leq\tilde{D}_{1}=\frac{1-D_{0}}{2D_{0}}-\tilde{D}_{2}\leq 1$
(39) $\displaystyle 0$ $\displaystyle\leq 1-2\tilde{D}_{1}+2\tilde{D}_{2}$

In order to be sure that the maxima of the function in this region is located
at the vertices, we need the Hessian, which is

$\left(\begin{array}[]{c c}1+D_{2}&D_{1}\\\ D_{1}&1\end{array}\right),$

which is strictly positive definite everywhere in the allowed region.
Therefore, the only points that need to be examined are the vertices of the
polytope (in this case, just a line) defined by Eq.(39) for the valid range of
$D_{0}$. It therefore just remains to find these vertices by solving these
equations on the boundary in the $\tilde{D}_{1}-\tilde{D}_{2}$ plane, which
depends on the value of $D_{0}$. They can be summarised as

$\begin{array}[]{c c c | c}\tilde{D}_{1}&\tilde{D}_{2}&D_{0}&\max R_{3}\\\ \hline\cr\frac{1-D_{0}}{2D_{0}}&0&\frac{1}{2}\leq D_{0}\leq 1&1.25\\\ \hline\cr\frac{1-2D_{0}}{2D_{0}}&\frac{1}{2}&\frac{1}{3}\leq D_{0}\leq\frac{1}{2}&1.58\\\ \frac{1}{4D_{0}}&\frac{1-2D_{0}}{4D_{0}}&\frac{1}{3}\leq D_{0}\leq\frac{1}{2}&1.86\end{array}$

where the largest values of $R_{3}$ over all $D_{0}$ in the allowed range are
also given. From this we can conclude that if $R_{3}$ is larger than $1.86$ we
can certify that the state is not a 3-coherent state lying in adjacent energy
levels of an SHO. For comparison, the perfectly balanced state gives $1.74$
and the largest value we could fine numerically was $1.77$. The largest value
found for a 4-coherent state (that we want to distinguish from) is $2.32$,
while for a 2-coherent state it is $1.25$.

### $k=3,d\geq 3$

We now remove the restriction on the dimension and instead restrict ourselves
to a 3-level state, which is to say that only 3 of the $\alpha$’s are non-
zero. Without loss of generality, we have as the three populated levels
$1,p,q$ with $1<p<q$. This means that the only non-zero variables are
$\alpha_{1},\alpha_{p},\alpha_{q}$, which gives

$\displaystyle D_{0}$
$\displaystyle=\alpha_{1}^{2}+\alpha_{p}^{2}+\alpha_{q}^{2},$ (40)
$\displaystyle D_{p-1}$ $\displaystyle=\alpha_{1}\alpha_{p},$ (41)
$\displaystyle D_{q-p}$ $\displaystyle=\alpha_{p}\alpha_{q},$ (42)
$\displaystyle D_{q-1}$ $\displaystyle=\alpha_{1}\alpha_{q}$ (43)

with the assumption that $p-1\neq q-p$. If these are equal, then the energy
levels are equally spaced and we are back to the 3-level case considered in
the first instance. As before, we now find inequalities on the $\tilde{D}$’s
to define a volume. As each one only contains a single term, this can be done
for each independently by considering

$\displaystyle(\alpha_{i}-\alpha_{j})^{2}+\alpha_{k}^{2}\geq 0$ (44)
$\displaystyle(\alpha_{i}+\alpha_{j})^{2}+\alpha_{k}^{2}\geq\tfrac{1}{2},$
(45)

where $i,j,k$ are all different. This and results of Eqs.(33,34) gives

$\displaystyle\max\left(0,\,\frac{1-2D_{0}}{4D_{0}}\right)\leq\tilde{D}_{i}\leq\frac{1}{2}$
(46) $\displaystyle\frac{1}{3}\leq D_{0}\leq 1$ (47)
$\displaystyle\tilde{D}_{p-1}+\tilde{D}_{q-p}+\tilde{D}_{q-1}=\frac{1-D_{0}}{2D_{0}}.$
(48)

The first line defines a cube in $\tilde{D}_{i}$ space and the last two a
family of planes that cut through that space. We show that within the cube the
Hessian is always positive.

The function $R_{3}$, and therefore the Hessian, depends on the indices of the
$\tilde{D}$ due to the $\sigma$ “energy matching” term in the triple sum.
There are several triplets that could enter:

$\displaystyle D_{p-1}\;D_{q-p}\;D_{q-1}$ (49) always contributes
$\displaystyle D_{p-1}\;D_{p-1}\;D_{q-1}\text{ or }D_{q-p}\;D_{q-p}\;D_{q-1}$
(50) are ruled out by the condition $p-1\neq q-p$ $\displaystyle
D_{p-1}\;D_{p-1}\;D_{q-p}$ (51) if and only if $q=3p-2$ $\displaystyle
D_{q-p}\;D_{q-p}\;D_{p-1}$ (52) if and only if $q=\tfrac{1}{2}(3p-1)$

The first case is the generic one. The second case happens if the energy
differences are equal, which we explicitly rule out. The third case happens if
the populated levels are $(1,2,4),\,(1,3,7),\,...$ where the energy difference
is in the ratio $1:2$. The third case requires the populated levels to be
$(1,3,4),\,(1,5,7),\,...$ where the energy difference has the ratio $2:1$.
This is therefore identical to the previous case under the Hamiltonian mapping
$H\to-H$, which clearly leaves the interference pattern unchanged.

There are thus 2 different cases to consider. The Hessian in the first case is

$\displaystyle\left(\begin{array}[]{c c c}1&\tilde{D}_{q-1}&\tilde{D}_{q-p}\\\
\tilde{D}_{q-1}&1&\tilde{D}_{p-1}\\\
\tilde{D}_{q-p}&\tilde{D}_{p-1}&1\end{array}\right).$ (56)

This is positive definite as, from Eq.(48) all principle minors of the matrix
are themselves positive definite in the cubic region of interest Horn and
Johnson (1985). The second case has the Hessian

$\displaystyle\left(\begin{array}[]{c c
c}1+\tilde{D}_{q-p}&\tilde{D}_{q-1}+\tilde{D}_{p-1}&\tilde{D}_{q-p}\\\
\tilde{D}_{q-1}+\tilde{D}_{p-1}&1&\tilde{D}_{p-1}\\\
\tilde{D}_{q-p}&\tilde{D}_{p-1}&1\end{array}\right),$ (60)

which is also positive everywhere, except potentially at some of the vertices
of the cube.

The vertices can be found in much the same way as before, except that the
boundaries are now symmetric between the $\tilde{D}_{i}$. We therefore give
them as triplets where all permutations need to be considered separately for
evaluating $R_{3}$.

$\displaystyle\begin{array}[]{c | c c c | c c}D_{0}&\tilde{D}_{i}&\tilde{D}_{j}&\tilde{D}_{k}&\max R_{3}(\text{generic})&\max R_{3}(1:2\text{ ratio})\\\ \hline\cr\frac{1}{2}<D_{0}\leq 1&0&0&\frac{1-D_{0}}{2D_{0}}&1.25&1.25\\\ \hline\cr\frac{1}{3}\leq D_{0}\leq\frac{1}{2}&\frac{1-2D_{0}}{4D_{0}}&\frac{1-2D_{0}}{4D_{0}}&\frac{1}{2}&1.27&1.33\\\ \end{array}$ (64)

Importantly, these values are all lower than for the case of a 3-level system
in adjacent energy levels. Therefore, the previous result we had is very
significantly strengthened: if $R_{3}$ is larger than $1.86$ then we know that
the state is not 3-coherent for any Hamiltonian.

### $k=d=4$

To highlight that this algorithmic way of calculating the threshold values can
be extended to high dimentions, we demonstrate it for the case of $4$-coherent
states. In order to reduce the number of cases to consider, we limit ourselves
to states where the $4$ populated levels are all adjacent basis states of an
harmonic Hamiltonian. For this case, the variables are

$\displaystyle D_{0}$
$\displaystyle=\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2}+\alpha_{4}^{2}$
$\displaystyle D_{1}$
$\displaystyle=\alpha_{1}\alpha_{2}+\alpha_{2}\alpha_{3}+\alpha_{3}\alpha_{4}$
$\displaystyle D_{2}$
$\displaystyle=\alpha_{1}\alpha_{3}+\alpha_{2}\alpha_{4}$ $\displaystyle
D_{3}$ $\displaystyle=\alpha_{1}\alpha_{4}.$

Eqs.(33, 34, 35) hold as before. Other bounds can be obtained in a similar way
to before by considering sums of squares. These are firstly

$\displaystyle(\alpha_{1}-\alpha_{3})^{2}+(\alpha_{2}-\alpha_{4})^{2}\geq 0$
$\displaystyle 1-2\tilde{D}_{2}\geq 0,$ (65)

and

$\displaystyle(\alpha_{1}+\alpha_{3})^{2}+(\alpha_{2}+\alpha_{4})^{2}\geq\tfrac{1}{2}$
$\displaystyle D_{0}(1+2\tilde{D}_{2})\geq\tfrac{1}{2}.$ (66)

Similarly there is

$\displaystyle(\alpha_{1}-\alpha_{4})^{2}+\alpha_{2}^{2}+\alpha_{3}^{2}\geq 0$
$\displaystyle 1-2\tilde{D}_{3}\geq 0,$ (67)

and

$\displaystyle(\alpha_{1}+\alpha_{4})^{2}+\alpha_{2}^{2}+\alpha_{3}^{2}\geq\tfrac{1}{3}$
$\displaystyle D_{0}(1+2\tilde{D}_{3})\geq\tfrac{1}{3}.$ (68)

Finally

$\displaystyle(\alpha_{1}-\alpha_{2}+\alpha_{3}-\alpha_{4})^{2}\geq 0$
$\displaystyle 1-2\tilde{D}_{1}+2\tilde{D}_{2}-2\tilde{D}_{3}\geq 0.$ (69)

Rougher versions of these can be obtained by eliminating $D_{0}$ by taking the
‘worst case’ approach, providing the simple inequalities

$\displaystyle\tilde{D}_{1}\leq 1$
$\displaystyle\qquad\tilde{D}_{1}+\tilde{D}_{3}\leq 1$
$\displaystyle\tilde{D}_{2}\leq\tfrac{1}{2}$
$\displaystyle\qquad\tilde{D}_{3}\leq\tfrac{1}{2},$ (70)

which will be useful in proving the positivity of the Hessian. The Hessian is
given by

$\displaystyle\left(\begin{array}[]{c c c}1+D_{2}&D_{1}+D_{3}&D_{2}\\\
D_{1}+D_{3}&1&D_{1}\\\ D_{2}&D_{1}&1\end{array}\right).$ (74)

The easiest way to prove positivity is, as before, to show that each of the
principle minors is itself positive definite in Horn and Johnson (1985), which
is straightforward to compute in the region of interest defined by the
inequalities Eq.(70). As before, it remains to find the vertices as a function
of $\tilde{D}_{2}$. This is a harder problem than before, which is most easily
tackled by rewriting the tighter inequalities as

$\displaystyle\max\left(0,\frac{1-2D_{0}}{4D_{0}}\right)\leq\tilde{D}_{2}\leq\frac{1}{2}$
(75)
$\displaystyle\max\left(0,\frac{1-3D_{0}}{6D_{0}}\right)\leq\tilde{D}_{3}\leq\frac{1}{2}$
(76) $\displaystyle 0\leq 1-2\tilde{D}_{1}+2\tilde{D}_{2}-2\tilde{D}_{3}$ (77)
$\displaystyle
0\leq\tilde{D}_{1}=\frac{1-D_{0}}{2D_{0}}-\tilde{D}_{2}-\tilde{D}_{3}\leq 1$
(78)

The first two describe a surface in the $\tilde{D}_{2}-\tilde{D}_{3}$, with
the coordinates of the vertices depending on the value of $D_{0}$. The third
equation states how this protrudes in the $\tilde{D}_{1}$ direction. The
fourth describes a plane that cuts this volume, and imposes additional
physical constraints. The way to solve this is therefore, for a given range of
$D_{0}$, to find the vertices in the $\tilde{D}_{2}-\tilde{D}_{3}$ plane (a
maximum of 4), find the corresponding value of $\tilde{D}_{1}$ and check if
any additional constraints on $D_{0}$ arise. The results are summarised below.

$\displaystyle\begin{array}[]{c | c c c | c}D_{0}&\tilde{D}_{1}&\tilde{D}_{2}&\tilde{D}_{3}&\max R_{3}\\\ \hline\cr\frac{1}{2}<D_{0}\leq 1&\frac{1-D_{0}}{4D_{0}}&0&0&1\\\ \hline\cr&\frac{1-2D_{0}}{2D_{0}}&\frac{1}{2}&0&1.58\\\ \frac{1}{3}<D_{0}\leq\frac{1}{2}&\frac{1-2D_{0}}{4D_{0}}&\frac{1-2D_{0}}{4D_{0}}&\frac{1}{2}&1.25\\\ &\frac{1}{4D_{0}}&\frac{1-2D_{0}}{4D_{0}}&0&1.86\\\ \hline\cr&\frac{1-3D_{0}}{2D_{0}}&\frac{1}{2}&\frac{1}{2}&1.33\\\ \frac{1}{4}\leq D_{0}\leq\frac{1}{3}&\frac{2-3D_{0}}{6D_{0}}&\frac{1}{2}&\frac{1-3D_{0}}{6D_{0}}&2.44\\\ &\frac{1-2D_{0}}{4D_{0}}&\frac{1-2D_{0}}{4D_{0}}&\frac{1}{2}&1.93\end{array}$ (87)

We see that the plane can intersect the volume at a single point (one vertex),
in a plane (three vertices) or, in the case that $D_{0}=\tfrac{1}{4}$ in a
line (two vertices). It is to be expected that this geometry becomes far more
complicated in higher dimensions. This sort of analyses ought to generalise,
but doing so is probably difficult. Nevertheless, from the table we can
conclude that if $R_{3}$ is larger than $2.44$ we can certify that the state
is not a 4-coherent state lying in adjacent energy levels of an SHO. For
comparison, the perfectly balanced state gives $2.26$ and the largest value we
could fine numerically was $2.32$. The largest value found for a 5-coherent
state (that we want to distinguish from) was $2.88$.

##  Appendix D Derivation of $R_{3}(\ket{W_{k}},\ket{W_{k}})$

We seek an exact analytical expression for the value of the certifier $R_{3}$
for the maximally coherent state, $R_{3}(\ket{W_{k}},\ket{W_{k}})$. For the
Hamiltonian in Eq. (1), the certifier can be rewritten as

$R_{3}(\ket{W_{k}},\ket{W_{k}})=\frac{1}{k}+\frac{12}{k^{3}}\frac{1}{T}\int_{0}^{T}\left(\sum_{i<j}\cos{(\omega_{i,j}t)}\right)^{2}+\frac{8}{k^{4}}\frac{1}{T}\int_{0}^{T}\left(\sum_{i<j}\cos{(\omega_{i,j}t)}\right)^{3},$
(88)

for energy level differences $\omega_{i,j}=|i-j|$.

The first and second integral involve products of two and three cosinusoidal
terms respectively. Using the trigonometric identity for terms of frequencies
$\alpha>\beta>\gamma\geq 0$,

$\cos{(\alpha)}\cos{(\beta)}\cos{(\gamma)}=\frac{1}{4}\Bigl{[}\cos{(\alpha+\beta+\gamma)}+\cos{(-\alpha+\beta+\gamma)}+\cos{(\alpha-\beta+\gamma)}+\cos{(\alpha+\beta-\gamma)}\Bigr{]},$
(89)

these products are reduced into linear terms. We need to find those that
survive and calculate the integrals for them. The condition
$\alpha=\beta+\gamma$ is equivalent to the statement that at least one, and in
fact exactly one, of the linearised terms survives. In other words, the
largest energy level spacing must be equal to the sum of the two smaller ones.
Once the conditions for non-vanishing terms in the products of cosines have
been identified, it is a matter of counting the number of combinations $A$ and
$B$ of energy levels that obey these conditions and survive in the first and
second integral in Eq. (88) respectively, leading to:

$R_{3}=\frac{1}{k}+\frac{6}{k^{3}}A+\frac{2}{k^{4}}B$ (90)

Calculating $A$ is simple, since in this case $\gamma=0$ and the non-vanishing
terms are the ones with identical cosines multiplied together. Therefore,
summing over all different values of $\omega_{i,j}$ gives

$A=\sum\limits_{n=1}^{k-1}n^{2}=\frac{k(k-1)(2k-1)}{6}.$ (91)

Calculating $B$ requires that cosine terms multiplied together satisfy that
the largest frequency equals to the sum of the smaller ones. Let us label the
largest frequency by $\omega_{i,i+\alpha}$, then it has multiplicity
$(k-\alpha)$ and there are $S_{\alpha}$ ways that two frequencies can sum up
to $\omega_{i,i+\alpha}$. Now, we seek all frequencies
$\omega_{j_{1},j_{1}+\beta}$ and $\omega_{j_{2},j_{2}+\gamma}$ of
multiplicities $(k-\beta)$ and $(k-\gamma)$ respectively, for which
$\omega_{i,i+\alpha}=\omega_{j_{1},j_{1}+\beta}+\omega_{j_{2},j_{2}+\gamma}$,
for all indices $i,j_{1},j_{2}$. The last factor to consider is that the three
cosines may be multiplied together in any order, so there is a combinatorial
coefficient of $3!$ when three different frequencies are multiplied together
and $\frac{3!}{2!}$ when the two shorter frequencies are the same, as in when
$\beta=\gamma$, which can only happen for even $\alpha$. We now reach the
expression

$\displaystyle S_{\alpha}$
$\displaystyle=\begin{cases}3!\sum\limits_{\begin{subarray}{c}\beta+\gamma=\alpha\\\
\beta\neq\gamma\end{subarray}}(k-\beta)(k-\gamma)&(\alpha\text{ is
odd})\\\\[19.37492pt] 3!\sum\limits_{\begin{subarray}{c}\beta+\gamma=\alpha\\\
\beta\neq\gamma\end{subarray}}(k-\beta)(k-\gamma)+\frac{3!}{2!}\left(k-\frac{\alpha}{2}\right)^{2}&(\alpha\text{
is even})\end{cases}$ (92)
$\displaystyle=\frac{1}{2}(\alpha-1)(\alpha+\alpha^{2}-6\alpha k+6k^{2}),$
(93)

for any $0<\alpha<k$. Finally, summing over all allowed energy level
differences,

$B=\sum\limits_{\alpha=1}^{k-1}(k-\alpha)S_{\alpha}=\frac{1}{40}k(k-1)(k-2)(2-7k+11k^{2}).$
(94)

Substituting $A$ and $B$ in Eq. (90), we get the desired sequence

$R_{3}\left(\ket{W_{k}}\bra{W_{k}},\ket{\chi_{0}}\right)=\frac{4+5k^{2}+11k^{4}}{20k^{3}}.$
(95)

##  Appendix E Derivation of decoherence theoretical and pattern thresholds

We first derive the theoretical threshold of coherence for the Werner-like
state $\rho_{W}$ of Eq.(13) and then prove that an interference pattern gives
a threshold equal to the theoretical, under optimal measurement.

We observe that $\rho_{W}\in C_{k}$ is fully symmetric under permutations of
basis states as well as that all the off-diagonal elements are
$\frac{1-\lambda}{k}$, resulting in

$C_{\ell_{1}}(\rho_{W})=(k-1)(1-\lambda),$ (96)

where $C_{\ell_{1}}(\rho)\coloneqq\sum\limits_{i\neq j}|\rho_{ij}|$ is the
$\ell_{1}$-norm as studied by Bera et al Bera  _et al._ (2015). These two
properties define a Werner-like state.

In general, the $\ell_{1}$ norm of a $q$-coherent state is bounded from above.
The bound is obtained when the system state is pure since $C_{\ell_{1}}$ is a
convex measure Bera  _et al._ (2015). Let $\rho=\ket{\alpha}\bra{\alpha}\in
C_{q}$ for a state $\ket{\alpha}$ defined in the reference basis, so that
$\rho_{ij}=\alpha_{i}\alpha_{j}^{*}=\alpha_{i}^{*}\alpha_{j}$ and
$Tr[\rho]=\sum_{i=1}^{q}|\rho_{ii}|=1$.

$\displaystyle(q-1)-C_{\ell_{1}}(\rho)=$
$\displaystyle(q-1)\sum_{i=1}^{q}|\rho_{ii}|-2\sum\limits_{i<j}|\rho_{ij}|$
(97) $\displaystyle=$
$\displaystyle\sum\limits_{i<j}(|\alpha_{i}|-|\alpha_{j}|)^{2}\geq 0.$ (98)

This means that the coherence of the system is bounded above,

$C_{\ell_{1}}\left(\rho\right)\leq q-1,$ (99)

with equality obtained when $\forall i,j,|\alpha_{i}|=|\alpha_{j}|$ in the
reference basis, so that $\ket{\alpha}$ is the maximally $q$-coherent state.

Using Eqs.(96, 99), we obtain for the Werner-like states in $C_{q}$

$\displaystyle\lambda$ $\displaystyle\geq\frac{k-q}{k-1}$ (100)
$\displaystyle\therefore\lambda_{\text{dec}}(q)$
$\displaystyle=\frac{k-q}{k-1},\quad 1\leq q\leq k.$ (101)

Now projecting with the optimal measurement $\ket{W_{q}}$ we get

$\displaystyle p(t)$ $\displaystyle=\langle W_{q}|\rho_{W}|W_{q}\rangle$ (102)
$\displaystyle\leq\frac{1}{k}+\frac{2}{k}\sum_{i<j}|\rho_{ij}|=\frac{1}{k}+\frac{1}{k}C_{\ell_{1}}\left(\rho\right)$
(103) $\displaystyle\leq\frac{1}{k}+\frac{q-1}{k}=\frac{q}{k}$ (104)

Therefore, a pattern with a maximum higher than this boundary value,
$\frac{q}{k}$, cannot be decomposed into patterns arising from states of
$q$-coherence or lower. We get the threshold value $\lambda_{\text{patt}}(q)$
at which the interference pattern can no longer distinguish consecutive
coherence levels, by bounding the interference pattern produced from the
Werner-like state by the probability maximum, so that

$\displaystyle\frac{q}{k}$ $\displaystyle\geq\langle
W_{q}|\rho_{W}|W_{q}\rangle=1-\lambda+\frac{\lambda}{k}$
$\displaystyle\Rightarrow\lambda$ $\displaystyle\geq\frac{k-q}{k-1}$
$\displaystyle\therefore\lambda_{\text{patt}}(q)$
$\displaystyle=\frac{k-q}{k-1},\quad 1\leq q\leq k,$ (105)

which coincides with $\lambda_{\text{dec}}(q)$.