###### Abstract

An electron-positron collider designed for precision studies of the Higgs
boson, a so-called Higgs factory is the highest-priority next collider of the
particle physics community. This contribution summarises the key physics goals
of such a Higgs factory and reviews the status of the various proposed
realisations from mature concepts to very recent ideas. The commonalities and
special advantages of circular and linear approaches will be discussed,
respectively, before highlighting some recent developments regarding the key
technologies, the operation scenarios and sustainability aspects for future
colliders.

##  1 Higgs Factory Projects

In the last update of the European Strategy for Particle Physics, an
$e^{+}e^{-}$ collider serving as a Higgs factory was identified as the highest
priority next collider [1]. Last year, this was re-emphasised in the final
report of the Snowmass Community Study in the US, recognising that the Higgs
boson is intimately connected to nearly all our fundamental questions about
the origin and development of the universe, as illustrated in Fig. 1, and that
thus a significantly more precise characterisation of its properties and its
potential is an essential ingredient to addressing these questions [2].

Figure 1: The Higgs boson is at the center of many fundamental questions on
our universe and its evolution. From [3].

This contribution summarises the status of the main $e^{+}e^{-}$ Higgs factory
projects, FCCee, CEPC, CLIC and ILC, as well as of more recent proposals like
C3, following a series of recent dedicated workshops on these projects [4, 5,
6, 7]. The most mature among these proposals, the International Linear
Collider, ILC [8, 9, 10, 11, 12, 13], is based on superconducting RF as used
successfully in e.g. the EU-XFEL and LCLS-II. In its Higgs Factory stage at a
center-of-mass energy ($\sqrt{s}$) of $250$ GeV, it would be about $20$ km
long and could be operated also at lower energies down to the $Z$ pole, while
being upgradable to $\sqrt{s}=1$ TeV. Since several years, it is under
political consideration for construction in Japan, where most recently the ILC
Technology Network was launched to address the last remaining R&D questions.
The Compact Linear Collider, CLIC [14, 15], could reach $\sqrt{s}=380$ GeV
with a footprint of only $11$ km thanks to the drive-beam technology pioneered
at CERN, and could eventually be upgraded to $3$ TeV.

CERN’s first priority in terms of future projects, however, is the Future
Circular Collider, FCC [16], a $90$ km circular facility which could host an
$e^{+}e^{-}$ collider running at the $Z$ pole, the $WW$ threshold, the $ZH$
threshold and eventually the $t\bar{t}$ threshold, before being replaced by a
proton-proton collider. Currently, the feasibility of such a project is being
studied, with a final report expected by the end of 2025. A similar project,
the Circular Electron Positron Collider, CEPC [17, 18] is being developed in
China. CEPC is in the final phases of publishing a Technical Design Report,
including a costing reviewed by international experts, and aims for an
approval in the next 5-year plan in 2025. If indeed a decision to construct
CEPC or ILC is taken in the next couple of years, these machines could start
data-taking in the late 2030ies, with overlap to HL-LHC operation. Due to the
latter, any new project at CERN could not start data-taking before the end of
2040ies.

Figure 2: Envisioned timeslines of various future collider projects as
discussed in the Snowmass process. For most projects, they are technically
driven and would allow a start of data-taking in the late 2030ies – if a
decision for construction is made soon. The timeline of the CERN-based
projects is coupled to the completion of the HL-LHC programme, shifting the
possible start of data-taking into the late 2040ies.

More recently, a linear collider based on cooled copper cavities, the Cool
Copper Collider, C3 [19], has been proposed in the US. Thanks to much higher
gradients than possible in SCRF, it requires only a footprint of about $4$ km
to reach $\sqrt{s}=250$ GeV, and even a $550$ GeV version would still fit into
the FNAL site. Assuming a significant investment in the required R&D over the
next five years, C3 could start operation in the early 2040ies. Figure 2
summarizes the current view on potential timelines of various future
colliders.

There is even a newer kid on the block: only in spring this year, a Hybrid
Asymmetric Linear Higgs Factory, HALHF [20], has been proposed based on
colliding a high-energy, plasma-wakefield accelerated electron beam with a
lower energy conventional positron beam. This idea motivates a significant
plasma acceleration R&D program targeting collider applications, in particular
to demonstrate the concatenation of several plasma cells.

While all these proposals differ in the accelerating technology and the site-
specific design, they fall into two basic categories: circular colliders and
linear colliders, each with their own advantages. Circular colliders offer
several interaction regions, very clean experimental conditions and very high
luminosity and power-efficiency at their lower energy stages. Linear colliders
have a smaller footprint, offer polarised beams and give high luminosity and
power-efficiency at high energies. Both also come with visions for the long-
term future: circular facilities could host a proton-proton collider, while
linear facilities could be upgraded in energy – either by extending the
tunnel, or by replacing the accelerating structures with advanced accelerator
technologies.

##  2 Physics Considerations

Figure 3: Constraints on Higgs and triple gauge couplings from SMEFT fit to
Higgs and electroweak sector for HL-LHC and various proposed lepton colliders
[21]. Compared to HL-LHC and the MuonCollider, all $e^{+}e^{-}$ projects
deliver a comparable performance.

Despite these different features, all the proposed projects perform
surprisingly similarly with respect to the core Higgs factory programme, as
shown in the most recent SMEFT evaluation [21] performed for the Snowmass
Community Study, and displayed in Fig. 3: the Higgs boson’s couplings to the
$Z$ and $W$, to the gluons as well as to $b$ quarks and $\tau$ leptons will be
constrained at the level of a few permille, and the couplings to $c$ quarks
and photons at the level of one percent.

The most striking gain over the HL-LHC capabilities, however, is that the
absolute couplings become accessible without any assumptions on the total
width or the absence of exotic decay modes. Thanks to the known initial-state
four-momentum, $e^{+}e^{-}$ colliders offer the unique possibility to detect
$e^{+}e^{-}\to ZH$ events only via the recoil against the $Z$ boson,
independently of the decay modes of the Higgs boson, as illustrated by a
simulated event display in Fig. 4. This model-independent determination of the
absolute couplings represents a qualitative leap in terms of knowledge and in
our ability to test extensions of the Standard Model.

Beyond the minimal Higgs programme, however, circular and linear colliders do
place emphasis on different areas of physics. The very high luminosity of
circular machines at the $Z$ pole offers a rich programme of physics at the
intensity frontier, ranging from electroweak and QCD precision measurements to
$b$\- and $\tau$-physics and searches for very light exotic particles, but
also creates challenges for the detectors. Linear colliders, on the other
hand, offer a unique programme at higher centre-of-mass energies, comprising
di-Higgs production, direct searches for new particles and a full precision
programme for the top-quark, including the full CP structure of its
electroweak and Yukawa couplings. A recent SMEFT fit of the top sector based
on projections for the HL-LHC and various $e^{+}e^{-}$ projects clearly showed
that $e^{+}e^{-}$ collisions at energies of $500$ GeV and beyond with
polarised beams are required in order to lift degeneracies between operators
[22].

Figure 4: Event display of $e^{+}e^{-}\to ZH\to\mu^{+}\mu^{-}b\bar{b}$ in the
ILD detector [25]

##  3 Recent Developments

During the Snowmass process, a task force analysed the technical readiness,
costs and timelines of the various proposed future projects [23]. The radio-
frequency systems and the positron source were identified as critical
components limiting the technological readiness level of the main Higgs
factory contenders — and at the same time, a lot of progress is being achieved
in these areas.

Concerning SCRF cavities, which are not only needed for the ILC main linac,
but also for the booster and collider rings of circular colliders as well as
the damping rings of linear colliders, very interesting progress has been
shown recently in the new PAPS facility [24] at IHEP, reaching very high
quality factors of about $5\cdot 10^{10}$ at a gradient of $31$ MV/m with a
simplified surface treatment receipe [26]. The performance surpasses the CEPC
specs and promises lower power consumption and reduced production costs at the
same time. An other important means to reduce the power consumption is the
development of more efficient klystrons, where both CERN and IHEP are reaching
efficiencies beyond $80\%$ based on new design methods [27, 28]. The power
estimates for CLIC, FCCee and CEPC are based on these developments, while the
power estimates for ILC still assume an efficiency of 65% based on
commercially available klystrons.

SCRF cavities and the positron source are also two important topics of the ILC
Technology Network [29, 30], which recently was launched by an agreement
between CERN and KEK. Within the ITN, CERN will act as a European hub to
facilitate money transfer to other labs and universities. As a first project,
design and prototyping of a new capture device, a pulsed solenoid, for the
positron source is being pursued, since simulations showed recently the
potential to increase the positron yield substantially with such a device
[31].

There are also new ideas on how to operate these machines: FCCee is now
considering the possibility of starting operation with the Higgs physics run,
as illustrated in Fig. 5 and later installing the dedicated cavities for the
high-luminosity run at the $Z$ pole, which adds important flexibility to the
running programme. CEPC is investigating the possibility of providing a
longitudinally polarised electron beam to the experiments [32]. Simulations
show that when using a polarised source with $|P|>85\%$, up to $70\%$
transverse polarisation could be maintained through the booster ring and the
top-up injection into the collider ring. The next steps will be to integrate
spin rotators and polarimeters into the lattice.

Figure 5: Alternative running scenario for FCCee starting with a Higgs run.

When looking at the running scenarios for linear colliders, a peculiar
situation can be found: While ILC historically picked $500$ GeV, it has been
known ever since the discovery of the Higgs boson at a mass of $125$ GeV that
this is slightly too low for an optimal study of $t\bar{t}H$ production [33].
Therefore, C3 chose $550$ GeV as the baseline, which immediately improves the
projections for the achievable precision on the top Yukawa coupling by more
than a factor of two. The CLIC running programme doesn’t consider data-taking
in the range of $500$ to $600$ GeV at all, and instead proposes to jump
directly from $380$ GeV to $1.5$ TeV. This seems to indicate an emerging need
to re-discuss the optimal physics choice of energy stages beyond the Higgs
run.

##  4 Sustainability

Last but not least, the particle physics community has realised that any
future project needs to be designed and chosen taking into account its overall
resource consumption and ecological footprint. Based large share of the global
warming potential (GWP) of future colliders is due to the construction of the
tunnel, in particular the production of concrete and steel. As future circular
colliders have a rather large circumference, the tunnel construction seems to
dominate the GWP, while for linear colliders, which have a shorter footprint,
the operation dominates [34]. On the positive side, a full life-cycle
assessment according to ISO standards recently performed for CLIC and ILC [35]
showed the potential to reduce the GWP in the construction by up to $40\%$ by
using low-CO2 materials and by optimising the thickness of the tunnel walls.
Also the tunnel shape plays an important role, as illustrated in Fig. 6: While
the 5.6 m diameter round tunnel for the CLIC drive-beam baseline has a very
similar CO2 footprint per kilometer as the kamaboko-shaped ILC tunnel with a
span of 9.5 m, the 10 m diameter tunnel foreseen for the klystron option of
CLIC produces more than twice as much CO2 per kilometer.

Figure 6: Comparison of the tunnel shapes, length, tunneling techniques for
ILC250 and the driven-beam and klystron-based options for CLIC380.

Realising the importance of sustainability considerations for future projects,
the European Laboratory Directors Group recently established a task force to
develop coherent criteria and guidelines for an assessment of new
infrastructures. This group is expected to deliver a final set of standards by
summer 2024, to be used by all proposals for the next update of the European
Strategy for Particle Physics [36].

##  5 Conclusions

There is a strong agreement in the world-wide particle physics community
precision studies of the Higgs boson are crucial to answer our burning
questions about the universe and its evolution. An electron-positron collider
operating as a Higgs factory as therefore been identified as the highest-
priority next collider. Several concrete projects have been proposed, all
being able to deliver the core Higgs factory program, but being complementary
in their further ambitions. Studies of the expected physics performance are
essential to understand better the implications of the choices to be made. An
active R&D program is addressing the last remaining technological challenges,
as well as working towards reducing the ecological footprint of these
machines.

The approval of any new large-scale collider, however, requires a strong HEP
community behind it, and an excellent communication of the scientific case to
experts from other fields, funding agencies and the general public,
disseminated by a growing community of supporters. Thus: get engaged and make
it happen!

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# Using Simulations to Highlight the Role of Electromagnetism in Special
Relativity

Refath Bari

###### Abstract

Physics Education Research (PER) studies have demonstrated that undergraduate
students struggle with fundamental principles from Electromagnetism (EM) and
Special Relativity (SR). However, few studies have approached the intersection
of the two subjects: the role which EM played in inspiring the development of
SR. To address this issue, this paper presents two simulations which highlight
the fundamental role of electromagnetism in special relativity. The first
simulation demonstrates the Lorentz Transformations and the origin of the
Gamma Factor from Maxwell’s Equations. The second simulation offers an
experiential introduction to the Biot-Savart Law, from which the Displacement
Current in Ampere-Maxwell’s Law can be derived. These programs may be useful
in an undergraduate electromagnetism course. The simulations discussed in this
paper are available at the link given in the
footnote111https://refath.notion.site/Using-Simulations-to-Highlight-the-Role-
of-Electromagnetism-in-Special-
Relativity-107a01c7bd394fdab5b6e7bbfe158fe4?pvs=4.

##  I Introduction

A significant program in Physics Education Research (PER) has been research on
student difficulties with undergraduate electromagnetism (Rachel et al.,,
2010, 2012; Eder et al.,, 2023; Mahima and Das,, 2023). Furthermore, the
counter-intuitive features of Special Relativity have also been well
documented (Paolo,, 2014; de Hosson Cécile et al.,, 2010; Dimitrios et al.,,
2015). However, the subtle insight which inspired Einstein to make the leap
from Classical Electrodynamics to Special RelativityDunstan, (2008), namely
the frame dependence of electric and magnetic fields, is not emphasized in
most undergraduate electromagnetism textbooks (Zafer,, 2014). As a result,
many students struggle to understand how relativistic phenomenon such as time
dilation and the Lorentz gamma factor emerge from classic electromagnetism.

Undergraduate textbooks seldom highlight the historical connection between
Special Relativity and Electromagnetism Jackson, (2007); Purcell and Morin,
(2013); Griffiths, (1999); Fleisch, (2019). Many studies have demonstrated the
effectiveness of simulations in clarifying student confusions of abstract
concepts such as space-time intervals and causality (Paul et al.,, 2023; M. et
al.,, 2007; Dadan et al.,, 2020). Although a few emerging texts and individual
papers are paving the way for experiential-based electromagnetism via
computational visualizationsChabay and Sherwood, (2018); Xuan, (2023); Morris
and Styer, (2012); Franklin and Ryder, (2019), most textbooks have yet to take
advantage of simulations to better highlight the intimate relationship between
the two subjects. To address this issue, this paper presents two simulations
which highlight the role of electromagnetism in special relativity,
supplemented with the prerequisite theory for completeness. The two
simulations are as follows: (1) A simulation of the Lorentz Transformation in
1+1 Dimensions and (2) A simulation of the Biot-Savart Law. These simulations
may be useful in an undergraduate course on Electromagnetism.

The above two principles were selected due to previously documented student
difficulties associated with them. For instance, the Lorentz transformation is
a subject of considerable consternation amongst students Boudreaux, (2002);
Castells et al., (2010); Tanel, (2014). Its formulation as a symmetric matrix
transformation may seem to be a mere accidentScherr et al., (2002).
Furthermore, it may seem to be an unmotivated ansatz, spuriously satisfying
lorentz invarianceSelçuk, (2010). The application of the lorentz
transformation to transition between reference frames is another well-
documented student difficulty Sherin et al., (2016); Weiskopf et al., (2006).
To address this issue, we present a simulation of the Lorentz transformations
in 1+1 dimensions. We also present a theoretical motivation for the Lorentz
transformation, showing why anything less than a full transformation of both
space and time (i.e., Galilean Transformation and Fitzgerald Transformation),
fails to preserve the invariance of the wave equation.

##  II Motivating the Lorentz Transformation

The motivating basis for the Lorentz Transformations is that the Galilean
Transformations fails to maintain the invariance of Maxwell’s Equations. Many
students grapple with the inconsistency of the Galilean Transformations, but
to no avail. Indeed, many struggle with the basic Galilean Principle of
Relativity itselfde Hosson Cécile et al., (2010). To address this problem, we
demonstrate why transforming neither space or time (Galilean Transformation)
or transforming only space (Fitzgerald Transformation) fails to preserve the
invariance of the wave equation under a transformation. The wave equation is
trivial to obtain from Maxwell’s Equations:

$\displaystyle\begin{array}[]{l}\nabla\times B=\frac{\partial E}{\partial
t}+{\mu}_{0}J\rightarrow-\nabla(\nabla\cdot E)+\nabla^{2}E=E_{tt}\end{array}$
(2)

We now demonstrate why the Galilean and Fitzgerald Transforms fail to preserve
this wave equation, and thus motivate why a transformation of both the space
and time is required.

###  II.1 Application of GT to Wave Equation

To simplify our analysis, we only consider a wave dependent on $(z,t)$.

$\displaystyle\frac{\partial^{2}E}{\partial^{2}t}=\nabla^{2}E\rightarrow{E}_{tt}-{E}_{zz}=0$
(3)

The Galilean Transformation has $\zeta=z+vt,\tau=t$

$\displaystyle E(t,z)=\varepsilon(t,z+vt)=\varepsilon(\tau,\zeta)$ (4)

We use the multivariable chain rule to verify whether ${E}_{tt}-{E}_{zz}=0$.
We begin with

$\displaystyle{E}_{t}=\frac{\partial\varepsilon}{\partial\tau}\cdot\frac{\partial\tau}{\partial
t}+\frac{\partial\varepsilon}{\partial\zeta}\cdot\frac{\partial\zeta}{\partial
t}={\varepsilon}_{\tau}+{\varepsilon}_{\zeta}\cdot\frac{\partial(z-vt)}{\partial
t}={\varepsilon}_{\tau}-v{\varepsilon}_{\zeta}\rightarrow{E}_{tt}=\frac{\partial}{\partial
t}({\varepsilon}_{\tau}-v{\varepsilon}_{\zeta})$ (5)

Finding ${E}_{tt}$ requires evaluating $\frac{\partial}{\partial
t}({\varepsilon}_{\tau})$ and $\frac{\partial}{\partial
t}(-v{\varepsilon}_{\zeta})$. We find that ${E}_{tt}$ becomes

$\displaystyle{E}_{tt}=\frac{\partial}{\partial
t}({\varepsilon}_{\tau})+\frac{\partial}{\partial
t}(-v{\varepsilon}_{\zeta})=({\varepsilon}_{\tau\tau}-v{\varepsilon}_{\tau\zeta})-(-v{\varepsilon}_{\zeta\tau}+{v}^{2}{\varepsilon}_{\zeta\zeta})={\varepsilon}_{\tau\tau}-v{\varepsilon}_{\tau\zeta}+v{\varepsilon}_{\zeta\tau}-{v}^{2}{\varepsilon}_{\zeta\zeta}$
(6)

Likewise, we find ${E}_{zz}$. Note that $\uptau=t=0$ and that $\uptau$ has no
dependence on $z$, so that ${\uptau}_{z}=0$ and
${\mathcal{E}}_{\uptau}{\uptau}_{z}=0$. By the Galilean Transformation,
$\zeta=z-vt$, and thus we have

$\displaystyle{E}_{z}=\frac{\partial\varepsilon}{\partial\tau}\cdot\frac{\partial\tau}{\partial
z}+\frac{\partial\varepsilon}{\partial\zeta}\cdot\frac{\partial\zeta}{\partial
z}={\varepsilon}_{\tau}\cdot
0+{\varepsilon}_{\zeta}\cdot\frac{\partial(z-vt)}{\partial
z}={\varepsilon}_{\zeta}$ (7)
$\displaystyle{E}_{zz}=\frac{\partial^{2}E}{\partial
z^{2}}=\frac{\partial}{\partial
t}({\varepsilon}_{\zeta})=\frac{\partial({\varepsilon}_{\zeta})}{\partial\tau}\cdot\frac{\partial\tau}{\partial
z}+\frac{\partial({\varepsilon}_{\zeta})}{\partial\zeta}\cdot\frac{\partial\zeta}{\partial
z}={\varepsilon}_{\zeta\tau}\cdot
0+{\varepsilon}_{\zeta\zeta}\cdot\frac{\partial(z-vt)}{\partial
z}={\varepsilon}_{\zeta\zeta}$ (8)

Thus, the Galilean Transformation fails to preserve the Wave Equation:

$\displaystyle{E}_{tt}-{E}_{zz}={\varepsilon}_{\tau\tau}-v{\varepsilon}_{\tau\zeta}+v{\varepsilon}_{\zeta\tau}-{v}^{2}{\varepsilon}_{\zeta\zeta}-{\varepsilon}_{\zeta\zeta}\neq
0$ (9)

The next significant transformation came from George Francis FitzGerald, who
made conjectured that length itself may contract. We will now implement the
Fitzgerald Transformation and find that it also fails to preserve the wave
equation.

###  II.2 Application of FT to Wave Equation

The Fitzgerald Transformation (FT) was coined as an ad-hoc correction
Michelson-Morley to the Galilean Transformation by George Francis Fitzgerald
in his 1889 paper on the Ether and the Earth’s AtmosphereEther; Fitzgerald.
Fitzgerald published a colloquial paper, stating that length contracted ”by an
amount depending on the square of the ratio of their velocities to that of
light”:

$\displaystyle
E(t,z)=\varepsilon(t,\frac{z}{\sqrt{1-{v}^{2}}}-\frac{vt}{\sqrt{1-{v}^{2}}})=\varepsilon(\tau,\zeta)$
(10)

To verify whether the Fitzgerald Transformations holds the Wave Equation
invariant, we must verify whether ${E}_{tt}-{E}_{zz}=0$ by expanding ${E}_{t}$
using the multivariable chain rule.

$\displaystyle{E}_{t}={\varepsilon}_{\tau}+{\varepsilon}_{\zeta}\cdot\frac{\partial(\frac{z}{\sqrt{1-{v}^{2}}}-\frac{vt}{\sqrt{1-{v}^{2}}})}{\partial
t}={\varepsilon}_{\tau}+{\varepsilon}_{\zeta}\cdot(-\frac{v}{\sqrt{1-{v}^{2}}})\rightarrow{E}_{t}={\varepsilon}_{\tau}-\frac{v{\varepsilon}_{\zeta}}{\sqrt{1-{v}^{2}}}$
(11) $\displaystyle{E}_{tt}=\frac{\partial^{2}E}{\partial
t^{2}}=\frac{\partial}{\partial
t}({\varepsilon}_{\tau}-\frac{v{\varepsilon}_{\zeta}}{\sqrt{1-{v}^{2}}})={\varepsilon}_{\tau\tau}-\frac{2v{\varepsilon}_{\tau\zeta}}{\sqrt{1-{v}^{2}}}+\frac{v^{2}{\varepsilon}_{\zeta\zeta}}{1-{v}^{2}}$
(12)

Likewise, we find ${E}_{zz}$ using the chain rule to be

$\displaystyle{E}_{z}={\varepsilon}_{\tau}\cdot
0+{\varepsilon}_{\zeta}\cdot\frac{\partial(\frac{z}{\sqrt{1-{v}^{2}}}-\frac{vt}{\sqrt{1-{v}^{2}}})}{\partial
z}=\frac{{\varepsilon}_{\zeta}}{\sqrt{1-{v}^{2}}}\rightarrow{E}_{zz}=\frac{{\varepsilon}_{\zeta\zeta}}{1-{v}^{2}}$
(13)

We now verify if FT preserves the invariance of the Wave Equation:

$\displaystyle{E}_{tt}-{E}_{zz}=({\varepsilon}_{\tau\tau}-\frac{2v{\varepsilon}_{\tau\zeta}}{\sqrt{1-{v}^{2}}}+\frac{v^{2}{\varepsilon}_{\zeta\zeta}}{1-{v}^{2}})-(\frac{{\varepsilon}_{\zeta\zeta}}{1-{v}^{2}})={\varepsilon}_{\tau\tau}-{\varepsilon}_{\zeta\zeta}-\frac{2v{\varepsilon}_{\tau\zeta}}{\sqrt{1-{v}^{2}}}\neq
0$ (14)

We find that even length contraction has failed to uphold the invariance of
the wave equation. However, it is instructive to examine the matrix
formulation of $F(R^{2}\rightarrow R^{2})$:

$\displaystyle F:{R}^{2}\rightarrow{R}^{2}=\begin{pmatrix}1&0\\\
\frac{-v}{\sqrt{1-{v}^{2}}}&\frac{1}{\sqrt{1-{v}^{2}}}\end{pmatrix}\begin{pmatrix}t\\\
z\end{pmatrix}=\begin{pmatrix}t\\\
\frac{z-vt}{\sqrt{1-{v}^{2}}}\end{pmatrix}=\begin{pmatrix}\tau\\\
\zeta\end{pmatrix}$ (15)

Hendrik Lorentz independently made the next conjecture, that both space and
time must be transformed to preserve $E_{tt}=E_{zz}$. We observe that the
matrix formulations for the FT and LT are quite similar. Students will thus be
able to understand the origin of the Lorentz Transformation, not as an ansatz,
but as an evolution from GT and FT.

$\displaystyle
L:{R}^{2}\rightarrow{R}^{2}:\begin{pmatrix}\frac{1}{\sqrt{1-{v}^{2}}}&\frac{-v}{\sqrt{1-{v}^{2}}}\\\
\frac{-v}{\sqrt{1-{v}^{2}}}&\frac{1}{\sqrt{1-{v}^{2}}}\end{pmatrix}$ (16)

Immediately, we find the transformation to be symmetric such that $L={L}^{T}$
and $L={L}^{-1}$. But does it maintain the invariance of the wave equation? To
verify, we briefly apply LT to the Wave Equation.

###  II.3 Application of LT to Wave Equation

We now consider the Lorentz Transformations (LT) and test whether
$E_{tt}=E_{zz}$ under $L$.

$\displaystyle
L:{R}^{2}\rightarrow{R}^{2}:\begin{pmatrix}\frac{1}{\sqrt{1-{v}^{2}}}&\frac{-v}{\sqrt{1-{v}^{2}}}\\\
\frac{-v}{\sqrt{1-{v}^{2}}}&\frac{1}{\sqrt{1-{v}^{2}}}\end{pmatrix}\begin{pmatrix}t\\\
z\end{pmatrix}=\begin{pmatrix}\frac{t-vz}{\sqrt{1-{v}^{2}}}\\\
\frac{z-vt}{\sqrt{1-{v}^{2}}}\end{pmatrix}$ (17)

$\displaystyle
E(t,z)=\varepsilon(\tau,\zeta)=\varepsilon(\frac{t-vz}{\sqrt{1-{v}^{2}}},\frac{z-vt}{\sqrt{1-{v}^{2}}})\rightarrow{E}_{t}=\frac{\partial\varepsilon}{\partial\tau}\cdot\frac{\partial\tau}{\partial
t}+\frac{\partial\varepsilon}{\partial\zeta}\cdot\frac{\partial\zeta}{\partial
t}$ (18)

However, the two observers’ times are no longer equivalent. By the Lorentz
Transformation, we now have time dilated as one’s velocity $v$ approaches the
speed of light $c$.

$\displaystyle{E}_{t}={\varepsilon}_{\tau}\cdot\frac{\partial(\frac{t}{\sqrt{1-{v}^{2}}}-\frac{vz}{\sqrt{1-{v}^{2}}})}{\partial
t}+{\varepsilon}_{\zeta}\cdot\frac{\partial(\frac{z}{\sqrt{1-{v}^{2}}}-\frac{vt}{\sqrt{1-{v}^{2}}})}{\partial
t}=\frac{{\varepsilon}_{\tau}-{\varepsilon}_{\zeta}v}{\sqrt{1-{v}^{2}}}$ (19)

Taking the second partial derivative of the Electric Field in respect to time,
we have

$\displaystyle{E}_{tt}=\frac{\partial}{\partial\tau}({E}_{t})=\frac{\partial}{\partial\tau}(\frac{{\varepsilon}_{\tau}-{\varepsilon}_{\zeta}v}{\sqrt{1-{v}^{2}}})=\frac{{\varepsilon}_{\tau\tau}-{\varepsilon}_{\zeta\tau}v}{1-{v}^{2}}+\frac{-v{\varepsilon}_{\tau\zeta}+{\varepsilon}_{\zeta\zeta}v^{2}}{1-{v}^{2}}=\frac{{\varepsilon}_{\tau\tau}-2{\varepsilon}_{\zeta\tau}v+{\varepsilon}_{\zeta\zeta}v^{2}}{1-{v}^{2}}$
(20)

Likewise, we find ${E}_{zz}$ using the chain rule.

$\displaystyle{E}_{z}=\frac{\partial\varepsilon}{\partial\tau}\cdot\frac{\partial\tau}{\partial
z}+\frac{\partial\varepsilon}{\partial\zeta}\cdot\frac{\partial\zeta}{\partial
z}={\varepsilon}_{\tau}\cdot-\frac{v}{\sqrt{1-{v}^{2}}}+{\varepsilon}_{\zeta}\cdot\frac{1}{\sqrt{1-{v}^{2}}}=\frac{{\varepsilon}_{\zeta}-v{\varepsilon}_{\tau}}{\sqrt{1-{v}^{2}}}$
(21) $\displaystyle{E}_{zz}=\frac{\partial}{\partial
z}(\frac{{\varepsilon}_{\zeta}-v{\varepsilon}_{\tau}}{\sqrt{1-{v}^{2}}})=\frac{\partial(\frac{{\varepsilon}_{\zeta}-v{\varepsilon}_{\tau}}{\sqrt{1-{v}^{2}}})}{\partial\tau}\cdot\frac{\partial\tau}{\partial
z}+\frac{\partial(\frac{{\varepsilon}_{\zeta}-v{\varepsilon}_{\tau}}{\sqrt{1-{v}^{2}}})}{\partial\zeta}\cdot\frac{\partial\zeta}{\partial
z}=\frac{-2v{\varepsilon}_{\zeta\tau}+v^{2}{\varepsilon}_{\tau\tau}+{\varepsilon}_{\zeta\zeta}}{1-{v}^{2}}$
(22)

Completing the proof, we find that

$\displaystyle{E}_{tt}-{E}_{zz}=\frac{{\varepsilon}_{\tau\tau}-2{\varepsilon}_{\zeta\tau}v+{\varepsilon}_{\zeta\zeta}v^{2}}{1-{v}^{2}}-\frac{-2v{\varepsilon}_{\zeta\tau}+v^{2}{\varepsilon}_{\tau\tau}+{\varepsilon}_{\zeta\zeta}}{1-{v}^{2}}=0$
(23)

We have just proved the Wave Equation does indeed hold invariant under the
Lorentz Transformation $L$. From this transformation alone, rises all major
tenets of Special Relativity, including Time Dilation and Length Contraction.

###  II.4 Simulation of Lorentz Transformation in 1+1 Dimensions

The simulation below demonstrates the lorentz transformation in 1+1
dimensions. Students may use the simulation to transform any event $E:(x,t)$
between reference frames. Three sliders are presented at the bottom of the
simulation, which students may use to control the location of an event in
space-time and the relative velocity between two observers. Students will
subsequently observe an animation of the coordinate axes of the simulation
bending to illustrate the new coordinates.

Figure 1: Lorentz Transformation Simulation in 1+1 Dimensions

##  III $\frac{\partial\vec{E}}{\partial{t}}$ from Biot-Savart and STR

The traditional introduction of the Displacement Current term in standard
electromagnetics textbooks is via an inconsistency in Ampere’s Law when
applied to Parallel Plate Capacitors Buschauer, (2014) (an Amperian Loop can
be chosen such that a current does not penetrate the boundary of the surface).
This presentation may lead one to believe the Displacement Current to be an
ad-hoc correction to Ampere’s Law, when it is actually a natural consequence
of a moving test charge Sato, (1980).

To motivate the existence of the displacement current, we chose an alternative
route – not a parallel plate capacitor, but a simple test charge located at
the origin.

According to Maxwell himself, the Displacement Current term is
’electrostatically’ analogous to a normal current element or a moving test
charge. We thus examine a test charge from a stationary and moving observer’s
point of view. Whereas a stationary observer only finds an electric field, an
observer moving with velocity $-v$ with respect to the stationary frame will
witness both an electric and magnetic field. Such a test charge would thus
exhibit the displacement current, due to the moving electric field. We take
the relativistic form of the electromagnetic fields of the test charge and
derive the Biot Savart Law from it. Hidden implicitly within the Biot Savart
Law is the Displacement Current Term, which we reformulate using the Partial
Derivative of a Cross Product to conclude with Ampere’s Law, corrected with
the Displacement Current Term. Before doing so, however, it is crucial to
understand the Biot-Savart Law as it is traditionally applied to a current-
carrying wire.

###  III.1 Visualization of Biot-Savart

We seek to create the magnetic field vector at any distance $R$ from the
current-carrying wire. To do so, we require both Magnitude and Direction. The
magnitude will be supplied by the Biot-Savart Law as

$\displaystyle\|\vec{B}\|=\frac{{mu}_{0}I}{2\pi R}$ (24)

he Right Hand Rule gives the direction of the magnetic field to be
counterclockwise, since the current has dir$(I)=\hat{j}$. The unexpected
challenge, however, is to compute a consistently counterclockwise magnetic
field $\vec{B}$ vector around the wire. To do so, we construct three auxiliary
vectors: a vector $\vec{u}$ from the origin to the $y$ axis, a vector
$\vec{p}$ from the origin to the cursor position, and a vector $\vec{v}$ as
the difference between the two vectors. From that difference, a dot product
$\vec{v}\cdot\vec{w}$ is calculated such that the result is $0$, and thus we
have that $\vec{w}$ is indeed the magnetic field vector at the cursor’s
position.

Figure 2: Simulation of Magnitude and Direction of $\vec{B}$ for a Current-
Carrying Wire

Prior to constructing this in code, we must declare three functions – each
corresponding to three of the user’s actions: impressing the track-pad,
dragging, and releasing. We thus define our down() function to be In the next
auxiliary function Pythonmove(), the program updates the magnitude and
direction of all four vectors based on the position of the cursor. We must
also declare a few global variables so they may be used outside the
Pythonmove() function.

Finally, we have the release function Pythonup(), which generates a graph of
the magnitude of the magnetic field by distance and updates real-time, based
on the user’s placement of Magnetic Field vectors. We may verify the graph and
indeed find that the magnetic field drops proportional to the distance $R$.

###  III.2 Time Dilation from EM

We now pivot from the Biot-Savart Law to a moving test charge. In doing so, we
will find that the Law we have just visually demonstrated implicitly contains
within it the Displacement Current, without the need for any capacitors.
Consider once again our test charge $Q$, centered at the origin and moving in
the $\hat{i}$ direction. As Maxwell himself states the Displacement Current to
be ”electrostatically” equivalent to a traditional current formed by a moving
Electric Field, it makes sense to find the change in the Electric Field of the
charge in respect to time. To do so, we employ the chain rule as

$\displaystyle\frac{\partial E}{\partial t}=\frac{\partial E}{\partial
x}\cdot\frac{\partial x}{\partial t}=\frac{\partial E}{\partial
t}=(-v)\cdot\frac{\partial}{\partial
x}(\frac{1}{4\pi{\varepsilon}_{0}}\frac{Q}{{r}^{2}})$ (25)

Here’s where relativity comes in. Through the Electromagnetic Field Strength
Tensor, we find the equations of a relativistic electric field to be

$\displaystyle{E}_{\parallel}={E^{\prime}}_{\parallel},{B}_{\parallel}=0,{E}_{\perp}=\gamma{E^{\prime}}_{\perp},B=+\gamma\frac{1}{{c}_{2}}v\times
E^{\prime}$ (26)

We thus have the Magnetic Field at any given distance $R$ to be

$\displaystyle B(r)=\gamma\frac{1}{{c}^{2}}v\times
E=\gamma{\mu}_{0}\frac{qv\times\hat{r}}{4\pi{r}^{2}}$ (27)

We thus find that the Displacement Current is implicit within the Biot-Savart
Law.

$\displaystyle
B(r)=\gamma\frac{1}{{c}^{2}}(\vec{v}\times\vec{E})=\gamma{\mu}_{0}r\times({\epsilon}_{0}\frac{\partial
E}{\partial t})\rightarrow\frac{\partial B}{\partial
t}=\gamma{\mu}_{0}v\times({\epsilon}_{0}\frac{\partial E}{\partial t})$ (28)

We thus find that the Biot Savart Law implicitly contains the Displacement
Current Neuenschwander and Turner, (1992).

##  IV Conclusion

We have demonstrated how computational programming visualizations can work
symbiotically with theoretical derivations to solidify the concepts behind the
bridge between EM and SR and make them tangible – indeed, computational
visualizations such as the ones above offer rich pedagogical opportunities for
teachers and students alike to intuitively grasp the ideas of Maxwell’s
Equations and Einstein’s Special Relativity.

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supmat_mana.pdf

# Critical behaviors of non-stabilizerness in quantum spin chains

Poetri Sonya Tarabunga  The Abdus Salam International Centre for Theoretical
Physics (ICTP), Strada Costiera 11, 34151 Trieste, Italy  International School
for Advanced Studies (SISSA), via Bonomea 265, 34136 Trieste, Italy  INFN,
Sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy

###### Abstract

Non-stabilizerness - commonly known as magic - measures the extent to which a
quantum state deviates from stabilizer states and is a fundamental resource
for achieving universal quantum computation. In this work, we investigate the
behavior of non-stabilizerness around criticality in quantum spin chains. To
quantify non-stabilizerness, we employ a monotone called mana, based on the
negativity of the discrete Wigner function. This measure captures non-
stabilizerness for both pure and mixed states. We introduce Rényi
generalizations of mana, which are also measures of non-stabilizerness for
pure states, and utilize it to compute mana in large quantum systems. We
consider the three-state Potts model and its non-integrable extension and we
provide strong evidence that the mutual mana exhibits universal logarithmic
scaling with distance in conformal field theory, as is the case for
entanglement.

Introduction.– Over the last two decades, remarkable advancements in our
understanding of many-body physics have been achieved through the exploration
of concepts originating from quantum information theory and their application
to quantum many-body systems. A prominent example is quantum entanglement [1,
2, 3], quantified by von Neumann and Rényi entropies, which has emerged as a
powerful tool for investigating various many-body phenomena [4, 5], one of
them is to identify universality classes in one-dimensional quantum critical
points [6, 7].

Despite being a truly quantum property, it has been known that entanglement
alone is insufficient to achieve universal quantum computation. Indeed, there
exist states known as the stabilizer states that can be highly entangled, and
yet they can be efficiently simulated on a classical computer [8, 9, 10, 11].
On the other hand, non-stabilizer states, often referred to as “magic” states,
play a fundamental role in realizing genuine quantum advantage [12, 13, 14,
15]. Non-stabilizer states are essential resources for achieving quantum
computation beyond what classical systems can emulate. Much like entanglement,
non-stabilizerness has been quantified within the framework of resource theory
using measures of non-stabilizerness [16]. These measures assess the amount of
resource a state can provide in quantum protocols involving only Clifford
operations, offering insights into the computational power and quantum
capabilities of different states.

In the many-body context, there have been several studies that suggest
connection between non-stabilizerness and criticality [17, 18, 19, 20, 21]. At
the same time, recent studies have also established that non-stabilizerness is
directly linked with entanglement and Shannon (or participation) entropy [22,
23]. Specifically, it was found that the average over the Clifford orbit of
the entanglement spectrum flatness (in any bipartition) [22] and participation
entropy flatness [23] is directly related to the stabilizer linear entropy
[24]. Both mutual information (of entanglement) and Shannon mutual information
have been shown to display the scaling relation [7, 25, 26, 27, 28, 29, 30]

$I=\frac{c}{4}\log\left[\frac{L}{\pi}\sin(\ell\frac{\pi}{L})\right]+\gamma$
(1)

in conformal field theory (CFT). Here $\ell$ is the subsytem size, $c$ is the
central charge of the CFT, and $\gamma$ is a non-universal constant. Given the
connections mentioned above, it becomes natural to question whether the
corresponding mutual information of non-stabilizerness exhibits similar
scaling behavior as in Eq. (1).

Addressing this question poses challenges, particularly due to the inherent
difficulty of evaluating non-stabilizerness for large system sizes (especially
since in principle Eq. (1) holds only for $\ell,L\gg 1$). Previous studies
have been restricted to very small sizes [17, 18], or relied on a quantifier
of non-stabilizerness that may not be proper monotones [20, 21, 31]. In this
work, we study the non-stabilizerness in quantum critical spin chains governed
by CFT, employing a genuine measure of non-stabilizerness known as mana [32,
33]. To compute the full-state mana for large systems, we construct a
classical statistical mechanics systems from the discrete Wigner function,
such that the computation of mana can be recast as a free energy calculation.
Then, we introduce the mutual mana and address the above question about its
scaling in CFT. We demonstrate that the mana is significant at the critical
point, and more importantly, that the mutual mana scales linearly with
$\log\left[\frac{L}{\pi}\sin(\ell\frac{\pi}{L})\right]$, analogous to
entanglement and Shannon entropy. Our results highlight the difficulty of
removing the non-stabilizerness in CFT with finite-depth quantum circuits, and
in turn in classically simulating CFT.

## Preliminaries.–

Mana is a measure of non-stabilizerness that is only defined in terms of
expectation values of operators, and is thus one of measures of non-
stabilizerness that is relatively easy to compute. However, mana is only well-
defined for systems of odd prime power dimensions. To define mana, we first
define the shift and clock operators as

$X=\sum_{k=0}^{d-1}|k+1\rangle\langle k|\quad\textrm{and}\quad
Z=\sum_{k=0}^{d-1}\omega^{k}_{d}|k\rangle\langle k|,$ (2)

where $\omega_{d}=e^{2\pi i/d}$. Here, the addition is defined modulo $d$.
They satisfy the commutation relation $XZ=\omega ZX$.

The generalized Pauli operators (also known as the Heisenberg-Weyl operators)
are defined as

$T_{aa^{\prime}}=\omega^{-2^{-1}aa^{\prime}}Z^{a}X^{a^{\prime}}$ (3)

for $a,a^{\prime}\in\mathbb{Z}_{d}$. Here, $2^{-1}$ is the inverse element of
$2$ in $\mathbb{Z}_{d}$. For a system of $N$ qudits, the Pauli strings are

$T_{\mathbf{a}}=T_{a_{1},a_{1}^{\prime}}T_{a_{2},a_{2}^{\prime}}...T_{a_{N},a_{N}^{\prime}}.$
(4)

We denote the group of all $N$-qudit Pauli strings as $\mathcal{P}_{N}$.

Next, the phase-space point operators are defined in terms of the Pauli
strings as

$A_{\mathbf{0}}=\sum_{\mathbf{u}}T_{\mathbf{u}},A_{\mathbf{u}}=T_{\mathbf{u}}A_{\mathbf{0}}T_{\mathbf{u}}^{\dagger}.$
(5)

These operators are Hermitian with eigenvalues $1$ and $-1$, with multiplicity
$\frac{d+1}{2}$ and $\frac{d-1}{2}$, respectively. Moreover, they are
orthogonal, i.e,
$\Tr(A_{\mathbf{a}}A_{\mathbf{b}})=d^{N}\delta(\mathbf{a},\mathbf{b})$, and
thus they provide an orthogonal basis for an operator in
$\mathbb{C}^{d^{N}\otimes d^{N}}$. Thus, one can expand the density matrix
$\rho$ of a state (pure or mixed) as

$\rho=\sum_{\mathbf{u}}W_{\rho}(\mathbf{u})A_{\mathbf{u}}.$ (6)

where $W_{\rho}(\mathbf{u})$ is known as the discrete Wigner function [34,
35], a discrete analogue of the infinite-dimensional Wigner function [36].
Equivalently, we can write

$W_{\rho}(\mathbf{u})=\frac{1}{d^{N}}\Tr(A_{\mathbf{u}}\rho).$ (7)

The Wigner functions satisfy the following relations

$\displaystyle\sum_{\mathbf{u}}W_{\rho}(\mathbf{u})$ $\displaystyle=1$ (8a)
$\displaystyle\sum_{\mathbf{u}}W_{\rho}(\mathbf{u})^{2}$
$\displaystyle=e^{-S_{2}}/d^{N},$ (8b)

where $S_{2}$ is the 2-Rényi entropy.

Finally, mana is defined in terms of the Wigner functions as

$\mathcal{M}(\rho)=\log(\sum_{\mathbf{u}}|W_{\rho}(\mathbf{u})|).$ (9)

Due to the normalization condition in Eq. (8a), mana measures the negativity
of the Wigner representation of $\rho$. For pure states, the set of states
with positive Wigner representation is exactly the set of pure stabilizer
states [34], in which case the mana vanishes. For mixed states, the set of
states with positive Wigner representation is strictly larger than the convex
hull of stabilizer states. Nevertheless, it is shown that states with positive
Wigner representation (including those outside of the convex hull of
stabilizer states) cannot be distilled [32], and moreover they are efficiently
simulatable [37]. In fact, mana directly quantifies the cost of classical
simulation based on Monte Carlo in Ref. [37]. Thus, mana is a useful measure
to quantify the resources required for classically simulating a quantum
circuit, both for pure and mixed states [33].

## Rényi generalizations of mana: mana entropy (ME).–

In order to compute mana, we find it useful to introduce Rényi generalizations
of mana, following closely the stabilizer entropies (SEs) introduced in Ref.
[24]. We restrict to the case of pure states, where
$d^{N}W_{\rho}(\mathbf{u})^{2}$ can be interpreted as a probability
distribution (see Eq. (8b)). We now consider the $n$-Rényi entropies
associated to this probability distribution in the same spirit as the SEs
[24], as

$\begin{split}\mathcal{M}_{n}&=\frac{1}{1-n}\log\sum_{\mathbf{u}}\left(d^{N}W_{\rho}(\mathbf{u})^{2}\right)^{n}-N\log
d\\\
&=\frac{1}{1-n}\log\frac{1}{d^{N}}\sum_{\mathbf{u}}\left|\tilde{W}_{\rho}(\mathbf{u})\right|^{2n}\\\
\end{split}$ (10)

where we define
$\tilde{W}_{\rho}(\mathbf{u}):=d^{N}W_{\rho}(\mathbf{u})=\langle
A_{\mathbf{u}}\rangle$. Indeed, MEs are just SEs with the Pauli operators
replaced by the phase-space point operators in Eq. (5). It follows that the
MEs possess the same properties as SEs, namely [24] (i) faithfulness, (ii)
stability under Clifford unitaries, and (iii) additivity. Moreover, they are
upper bounded by $\mathcal{M}_{n}\leq N\log d$.

Notice that the index $n=1/2$ corresponds to mana of pure states (up to a
prefactor of 2). Mana has been rigorously proven to obey both monotonicity and
strong monotonicity under stabilizer operations, making it a genuine measure
of non-stabilizerness, also for mixed states [33]. In contrast, SEs of all
index have been shown to violate strong monotonicity, while SEs of index
$0<n<2$ violate monotonicity (the other cases are still an open problem) [31].
It is presently unclear if such monotonicity property holds for MEs of index
$n\neq 1/2$, a question that we leave for future investigations. Nonetheless,
they could be useful to provide non-trivial bounds for other known measures of
non-stabilizerness [38].

## Thermodynamics approach to non-stabilizerness.–

We define a classical statistical system with energies
$E_{\mathbf{u}}=-\log|\tilde{W}_{\rho}(\mathbf{u})|$, such that the free
energy is given by
$F_{\rho}(\beta)=-\frac{1}{\beta}\log\sum_{\mathbf{u}}\left|\tilde{W}_{\rho}(\mathbf{u})\right|^{\beta}$
111We note that similar thermodynamic description has been proposed for
entanglement [61, 62, 63].. One can see that the free energy is the same as
the quantity $\frac{n-1}{2n}\mathcal{M}_{n}-\frac{N}{2n}\log d$ (for $n\neq
1$) with $\beta=2n$ 222The case $n=1$ is instead related to the energy at
$\beta=2$: $\mathcal{M}_{1}=2\langle E_{\mathbf{u}}\rangle_{\beta=2}$. As
such, $\mathcal{M}_{1}$ can be directly estimated through perfect sampling
techniques [31, 64] . The calculation of $\mathcal{M}_{n}$ thus amounts to the
computation of free energy of a classical system. Conventionally, this is
commonly done by direct thermodynamics integration from infinite temperature
($\beta=0$). This is applicable when the free energy at infinite temperature
is known, which is not generally true in this case. Luckily, the free energy
at $\beta=2$ is known due to the relation in Eq. (8b). Indeed, for a pure
state ($S_{2}=0$), Eq. (8b) implies $F_{\rho}(\beta=2)=-\frac{N}{2n}\log d$.
Thus, one can perform a direct thermodynamics integration starting from
$\beta=2$,

$\log\frac{|\tilde{W}_{\rho}(\mathbf{u})|^{\beta}}{d^{N}}=\int_{2}^{\beta}\left\langle\log|\tilde{W}_{\rho}(\mathbf{u})|\right\rangle_{\beta}d\beta,$
(11)

where $\langle...\rangle_{\beta}$ denotes the thermal average at inverse
temperature $\beta$.

Numerically, the thermal average can be calculated via Monte Carlo sampling of
the discrete Wigner function [37]. Here we perform the Monte Carlo sampling
using tensor network methods, slightly modifying the method originally
developed to compute SEs in Ref. [21]. In particular, we focus on mana,
corresponding to $\beta=1$.

## Mutual mana.–

We will also consider the “mutual mana” defined as

$I_{\mathcal{M}}(A,B)=\mathcal{M}(\rho_{AB})-\mathcal{M}(\rho_{A})-\mathcal{M}(\rho_{B}).$
(12)

We will use the notation $I_{\mathcal{M}}(\ell,L)$ to denote the case
$A=\\{1,...,\ell\\}$ and $B=\\{\ell+1,...,L\\}$. Notice that the definition of
mutual mana involves the mana of subsystems, which are mixed states.
Crucially, mana is a genuine measure of non-stabilizerness both for pure and
mixed states, so that the mutual mana is a meaningful quantity that quantifies
the amount of resource that resides in the correlations between parts of the
system. It has also been suggested that it quantifies the difficulty of
removing non-stabilizerness with a finite-depth circuit [17].

We note here that mana is typically an extensive quantity. The subtraction in
Eq. (12) thus serves to eliminate the leading extensive term, resulting in
$I_{\mathcal{M}}(A,B)$ being significantly smaller than the mana itself.
Extracting such a quantity through Monte Carlo sampling of wavefunctions is
known to be a challenging task, akin to the challenge of extracting
topological entanglement entropy from entanglement entropy [41, 42, 43].
Indeed, if one tries to compute $I_{\mathcal{M}}(A,B)$ by directly computing
each of the three terms on the right hand side of Eq. (12) separately (e.g.,
using Eq. (11)), the resulting error bar will be prohibitively large. We
overcome this difficulty by writing $I_{\mathcal{M}}(A,B)$ as

$I_{\mathcal{M}}(A,B)=\log\left(\frac{\sum_{\mathbf{u},\mathbf{v}}|W_{\rho_{AB}}(\mathbf{u}\oplus\mathbf{v})|}{\sum_{\mathbf{u}}|W_{\rho_{A}}(\mathbf{u})|\sum_{\mathbf{v}}|W_{\rho_{B}}(\mathbf{v})|}\right).$
(13)

In view of the thermodynamics description in the previous section, the
expression inside the logarithm can be interpreted as a ratio of partition
functions of the classical systems corresponding to $\rho_{AB}$ and
$\rho_{A}\otimes\rho_{B}$. One way to estimate it in Monte Carlo simulations
is by sampling from one classical system and averaging the ratio of the
Boltzmann weights. Concretely, we consider the probability distribution
$\Pi_{\rho_{A(B)}}(\mathbf{u})\propto|W_{\rho_{A(B)}}(\mathbf{u}))|$. We can
estimate $I_{\mathcal{M}}(A,B)$ using

$I_{\mathcal{M}}(A,B)=\log\left\langle\frac{|W_{\rho_{AB}}(\mathbf{u}\oplus\mathbf{v})|}{|W_{\rho_{A}}(\mathbf{u})||W_{\rho_{B}}(\mathbf{v})|}\right\rangle_{\Pi_{\rho_{A}}(\mathbf{u})\Pi_{\rho_{B}}(\mathbf{v})}.$
(14)

## Quantum Potts model.–

In this work, we consider the quantum Potts model, which can be seen as the
generalization of the quantum Ising model with $d$ states per site [44]. The
Hamiltonian is given by

$H_{\text{Potts}}=-J\sum_{\langle
i,j\rangle}\sum_{k=1}^{d-1}X_{i}^{k}X_{j}^{d-k}-h\sum_{i}\sum_{k=1}^{d-1}Z_{i}^{k},$
(15)

where $X,Z$ are the shift and clock operators in Eq. (2). Here we focus on the
case $d=3$. The point $h_{c}=1$ is a critical self-dual point, which is
governed by a CFT for $d\leq 4$. For $d=3$, the central charge is $c=4/5$ in
the ferromagnetic case ($J=1$) and $c=1$ in the antiferromagnetic case
($J=-1$) [45, 46, 47].

We will also consider an extension of the Potts model introduced in Ref. [30].
The Hamiltonian is given by

$\begin{split}H_{\text{Potts}}(p)=&-\sum_{\langle
i,j\rangle}\sum_{k=1}^{d-1}X_{i}^{k}X_{j}^{d-k}-\sum_{i}\sum_{k=1}^{d-1}Z_{i}^{k}\\\
&-p\sum_{\langle\langle
i,j\rangle\rangle}\sum_{k=1}^{d-1}X_{i}^{k}X_{j}^{d-k}-p\sum_{\langle
i,j\rangle}\sum_{k=1}^{d-1}Z_{i}^{k}Z_{j}^{d-k}.\end{split}$ (16)

The model is self-dual at any $p$, and the case $p=0$ corresponds to the self-
dual point $h=1$ in Eq. (15), which is an integrable point. For $p\neq 0$, the
model is not integrable, but it is expected that they are described by the
same CFT at $p=0$ for sufficiently small $p$ [30].

Figure 1: (a) Mana density $\mathcal{M}/L$ in the vicinity of the critical
point $h=1$ in the three-state quantum Potts model. (b) Data collapse of the
mana density $m=\mathcal{M}/L$ with $\gamma\approx 0.83$ and $\nu\approx
0.85$. The correlation-length exponent $\nu$ is close to the known
$\nu_{Potts}=5/6$.

## Numerical results.–

We now present our numerical results on the mana in quantum Potts model. We
obtain the ground state using Tree Tensor Network (TTN) ground state
variational search algorithm [48, 49], and then we sample the discrete Wigner
function of the ground state using Monte Carlo sampling on TTN discussed in
Ref. [21]. Here, we compute the full-state mana using Eq. (11), while the
mutual mana is evaluated using Eq. (14).

The mana density is shown in Fig. 1a. We observe that $\mathcal{M}/L$ reaches
a maximum at the critical point $h_{c}=1$, which confirm the results of Ref.
[17]. More importantly, with the large systems we are able to simulate, we
obtain good data collapse, shown in Fig. 1b. Overall, these results are also
similar to the behavior of SE studied in [21]. Surprisingly, we find that in
this case the mana is indeed identical to the SE with $n=1/2$ 333 The
equivalence between mana and SE with $n=1/2$ is through the following
proposition: Proposition: Let $|\psi\rangle$ be an $N$-qudit pure state. If
$A_{\mathbf{b}}$ is a phase-space operator such that
$A_{\mathbf{b}}|\psi\rangle=\lambda|\psi\rangle$, where
$\lambda\in\\{+1,-1\\}$, then
$\lambda\langle\psi|A_{\mathbf{a}+\mathbf{b}}|\psi\rangle=\langle\psi|T_{2\mathbf{a}}|\psi\rangle\omega^{2(\mathbf{b}\mathbf{a^{\prime}}-\mathbf{b^{\prime}}\mathbf{a})}$
(17) for all $\mathbf{a}\in\mathbb{Z}_{d}^{2N}$.  The proof can be found in
[38]. As a corollary, the MEs and SEs are identical for all order whenever the
state is stabilized by a phase-space operator (up to a sign). In the case of
three-state Potts model, the ground state satisfies
$A_{\mathbf{0}}|\psi\rangle=|\psi\rangle$, due to the global $S_{3}$ symmetry
of permuting the three $X$ eigenstates. A similar statement holds for all
$d$-state Potts model for odd prime $d$. .

Figure 2: Mutual mana $I_{\mathcal{M}}(\ell,L)$ in the ground state of the
quantum Potts model at the critical point $h/J=1$ with (a) $J=1$ and (b)
$J=-1$. The solid line denotes the linear fit obtained for the largest size.
Clearly the data of different system sizes collapse in the straight line. We
observe odd-even effects for $J=-1$, and thus we plot only the results for
even $\ell$ for clarity.

Next, we investigated the scaling of mutual mana (Eq. (12)) at the critical
point $h_{c}=1$. The results are shown in Fig. 2a(b) for $J=1$ ($J=-1$) for
sizes up to $L=64$ ($L=32$) . We observe that the mutual mana is approximately
proportional to $\log\left[\frac{L}{\pi}\sin(\ell\frac{\pi}{L})\right]$,
similarly to the entanglement and Shannon entropy in CFT. However, we cannot
make a direct connection between the slope and the central charge of the
associated CFT 444Actually, there are also disputes regarding the slope of
Shannon mutual information, and whether it is truly equal to $c/4$. See [27]..
This is expected since mana is a basis-dependent quantity, and hence the
proportionality factor would likely depend on the choice of basis.

We now turn to the extension of the Potts model in Eq. (16). Fig. 3 shows the
mutual mana for various values of $p$ in a chain of $L=32$ sites. These
results clearly reveal a linear scaling of the mutual mana with respect to
$\log\left[\frac{L}{\pi}\sin(\ell\frac{\pi}{L})\right]$, which holds true even
at the non-integrable points. Notably, the slope of the linear growth shows
little variation upon increasing $p$. Based on these findings, we conjecture
that the slope is universal and determined by the underlying CFT, although
possibly not by a simple relation with central charge as entanglement and
Shannon entropy.

Since mana depends on the chosen basis, an important question is whether or
not the logarithmic scaling persists under local basis change. To address this
question, we show the mutual mana after applying unitary transformation
$T^{\otimes N}_{\theta}$, where
$T_{\theta}=\text{diag}(1,e^{i\theta},e^{-i\theta})$, to the ground state at
$h=1$ in Fig. 4a. Note that $\theta=2/9$ corresponds to the canonical $T$-gate
for qutrit. We see that the logarithmic scaling remains evident up to
$\theta=2/9$, while it becomes less apparent for $\theta=3/9$, possibly due to
finite-size effects.

Finally, in order to contrast with the behavior away from criticality, we plot
the scaling of mutual mana both at and away from the critical point in Fig.
4b. We see that the logarithmic scaling is observed only at the critical
point, while away from the critical point the mutual mana saturates at large
$\ell$.

Figure 3:  Mutual mana $I_{\mathcal{M}}(\ell,L)$ for various values of $p$ in
the extension of the quantum Potts model (Eq. (16)) with (a) $J=1$ and (b)
$J=-1$. The logarithmic scaling is also observed at the non-integrable points
$p\neq 0$. The system size is $L=32$. Figure 4:  (a) Mutual mana
$I_{\mathcal{M}}(\ell,L)$ after performing the unitary transformation
$T^{\otimes N}_{\theta}$, where
$T_{\theta}=\text{diag}(1,e^{i\theta},e^{-i\theta})$, to the ground state at
$h=1$ and $J=1$. (b) Mutual mana $I_{\mathcal{M}}(\ell,L)$ in the ground state
of the three-state Potts model and at three different transverse field
strength. The logarithmic scaling is only observed at the critical point at
$h=1$. In contrast, the mutual mana saturates both at $h>1$ and $h<1$. The
system size is $L=32$.

## Conclusions and outlook.–

In this work, we investigate the behavior of mana around criticality in
quantum Potts models and its extension. We introduce Rényi version of mana,
which enables us to calculate mana for large system sizes. Our results on
mutual mana provide clear evidence of logarithmic scaling with distance in
CFT, while it reaches saturation in gapped phases.

Our work opens up many interesting directions for future investigations.
Although mana is only defined for odd prime local dimension, several possible
extensions have been proposed for qubits [52, 53, 54, 55, 56]. It would be
interesting to employ them to investigate the qubit case, in particular
regarding its scaling in CFT. A more comprehensive examination of mutual mana
in CFT also warrants futher investigation, for instance by looking at
different partitioning schemes. Furthermore, our methods enable the
exploration of mana in various scenarios, such as quench dynamics [57], open
systems and finite-temperature scenarios. In addition, it would be interesting
to adapt our approach in different classes of tensor network states such as
PEPS [58] to investigate the mana in higher dimensions. Another interesting
direction is to systematically study and compare the behavior of mana entropy
and stabilizer entropy, which may provide insights into how to construct a
genuine measure of non-stabilizerness for qubits that is efficiently
computable. Finally, while here the mana entropy is introduced to facilitate
the numerical computations of mana, it may also be helpful in the analytical
investigation of mana in important classes of states, such as the quantum
hypergraph states [59].

###### Acknowledgements.

We thank M. Dalmonte, E. Tirrito, T. Chanda, C. Castelnovo for insightful
discussions and collaborations on related topics. This work was partly
supported by the PNRR MUR project PE0000023-NQSTI, and by the EU-Flagship
programme Pasquans2. We acknowledge support from the Simons Foundation through
Award 284558FY19 to the ICTP. Our TTN codes have been implemented using C++
Itensor library [60].

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