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# A geometric perspective: experimental evaluation of the quantum Cramér-Rao
bound

Changhao Li gbsn(李长昊)  Research Laboratory of Electronics, Massachusetts
Institute of Technology, Cambridge, Massachusetts 02139, USA  Department of
Nuclear Science and Engineering, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139, USA  Mo Chen gbsn(陈墨)  Research Laboratory of
Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts
02139, USA  Institute for Quantum Information and Matter, California Institute
of Technology, Pasadena, CA 91125, USA  Thomas J. Watson, Sr., Laboratory of
Applied Physics, California Institute of Technology, Pasadena, CA 91125, USA
Paola Cappellaro  Research Laboratory of Electronics, Massachusetts Institute
of Technology, Cambridge, Massachusetts 02139, USA  Department of Nuclear
Science and Engineering, Massachusetts Institute of Technology, Cambridge,
Massachusetts 02139, USA  Department of Physics, Massachusetts Institute of
Technology, Cambridge, Massachusetts 02139, USA

###### Abstract

The power of quantum sensing rests on its ultimate precision limit, quantified
by the quantum Cramér-Rao bound (QCRB), which can surpass classical bounds. In
multi-parameter estimation, the QCRB is not always saturated as the quantum
nature of associated observables may lead to their incompatibility. Here we
explore the precision limits of multi-parameter estimation through the lens of
quantum geometry, enabling us to experimentally evaluate the QCRB via quantum
geometry measurements. Focusing on two- and three-parameter estimation, we
elucidate how fundamental quantum uncertainty principles prevent the
saturation of the bound. By linking a metric of “quantumness” to the system
geometric properties, we investigate and experimentally extract the attainable
QCRB for three-parameter estimations.

Quantum sensing stands as a key application for quantum technologies Degen
_et al._ (2017). Quantum sensors promise to achieve better sensitivity or
precision than classical systems and have been utilized in many fields,
ranging from material science Marchiori  _et al._ (2022); Thiel  _et al._
(2019); Yip  _et al._ (2019); Lesik  _et al._ (2019); Hsieh  _et al._ (2019)
to biology Kucsko  _et al._ (2013); Choi  _et al._ (2020); Li  _et al._
(2022); Marais  _et al._ (2018). Quantum metrology quantifies the precision
limit of quantum sensing with the quantum Cramér-Rao bound (QCRB) Helstrom
(1967); Yuen and Lax (1973); Belavkin (1976); Holevo (1977); Carollo  _et al._
(2019); Liu  _et al._ (2019); Albarelli  _et al._ (2020). For unbiased
estimation of an unknown system parameter, the QCRB is given by the inverse of
the quantum Fisher information Holevo (1977); Carollo  _et al._ (2019); Liu
_et al._ (2019); Albarelli  _et al._ (2020). The estimator is chosen to be
optimal, therefore the QCRB only depends on the quantum state. In turn, the
precision of estimating a parameter can be linked to the “distance” between
two nearby states differing by an infinitesimally small parameter change
Braunstein and Caves (1994); Guo  _et al._ (2016); Kolodrubetz  _et al._
(2017); Liu  _et al._ (2019). This naturally connects quantum sensing to the
quantum geometric properties of the system, as the quantum metric tensor is
closely related to the ultimate estimation precision quantified by the quantum
Fisher information.

This picture becomes more involved when extending the goal from estimating a
single parameter to multiple parameters, in analogy to the complexity of
multi-dimensional system versus 1D systems. The complication arises from the
incompatibility between each parameter’s optimal estimators–a signature of
quantum mechanics Albarelli  _et al._ (2020); Liu  _et al._ (2019); Guo  _et
al._ (2016). This leads to trade-offs among the estimation precisions of
different parameters when picking the initial state. This interplay among
parameters makes the multi-parameter estimation problem more intriguing than
single parameter estimation. Hence developing tools to investigate the
ultimate precision and the attainable QCRB in a multi-parameter estimation
setting is of great interest for quantum metrology and sensing applications. A
promising strategy, similar to the single-parameter scenario, is to link
quantum geometric properties of the (multi-)parametrized state to the
estimation precision. Specifically, in addition to the link between the
quantum metric tensor and the quantum Fisher information, we can link the non-
commutativity of optimal measurement operators for different parameters to the
Uhlmann curvature Carollo  _et al._ (2019); Liu  _et al._ (2019); Albarelli
_et al._ (2020), as we explain below. Characterizing the geometry of the
parametrized quantum states by measuring the quantum geometric tensor would
thus help explore the problem of quantum multi-parameter estimation and
evaluate the attainable QCRB from a geometric perspective.

In this work, we explore the relation between quantum geometry and multi-
parameter estimation to gain novel insight into the corresponding precision
bound. In particular, we show that geometric and parameter estimation metrics
are connected via the characterization number $\gamma$, which characterizes
the “quantumness” of the system. We find that the characterization number
$\gamma$ is linked to both metrology and geometry: $\gamma$ is upper-bounded
by the fundamental uncertainty relation and it can be a signature of exotic
magnetic monopoles in parameter space. Beyond its theoretical insight, this
relation provides a strategy to experimentally evaluate quantum estimation
bounds. We can thus provide an experimental demonstration of these results by
focusing on a three-parameter, three-level pure state model, which is
synthesized using a single nitrogen-vacancy (NV) center in diamond. Based on
measuring spin-1 Rabi oscillations upon parameter-modulated driving, we
develop experimental tools to extract the geometric quantities and determine
the quantum Fisher information and Uhlmann curvature of the quantum state.
Then the system’s quantumness, i.e., the incompatibility among parameters to
be estimated, is characterized with the help of the characterization number
$\gamma$. We finally evaluate the attainable QCRB for the parameterized state,
as well as the two-parameter states in the subspace of the three-parameter
model.

## Relating quantum geometry to quantum multi-parameter sensing

We consider a generic quantum multi-parameter estimation problem given by a
quantum statistic model $\rho_{\boldsymbol{\theta}}$, a family of density
operator labelled by
$\boldsymbol{\theta}=(\theta_{1},\theta_{2},...,\theta_{d})^{T}$ that is the
set of unknown parameters to be estimated. Performing POVM
$\mathcal{M}=\\{\mathcal{M}_{k}|\sum_{k}\mathcal{M}_{k}=I,\mathcal{M}_{k}\geq
0\\}$, we obtain the measurement conditional probabilities
$p(k|\boldsymbol{\theta})=\textrm{Tr}[\rho_{\boldsymbol{\theta}}\mathcal{M}_{k}]$.
With $N$ repeated measurements, the unknown parameters are estimated via the
estimator $\tilde{\boldsymbol{\theta}}(k)$. The accuracy of the estimation is
quantified by the mean-square error matrix
$\boldsymbol{V(\boldsymbol{\theta})}=\sum_{k}p(k|\boldsymbol{\theta})[\tilde{\boldsymbol{\theta}}(k)-\boldsymbol{\theta}][\tilde{\boldsymbol{\theta}}(k)-\boldsymbol{\theta}]^{T}$.
Obtaining the lower bounds of this error matrix is of critical importance in
quantum sensing and metrology.

The bounds usually depend on the choice of POVM. The symmetric logarithmic
derivative Helstrom (1967); Yuen and Lax (1973); Belavkin (1976) (SLD) can be
used to define a bound that only depends on the quantum statistic model
$\rho_{\boldsymbol{\theta}}$, having optimized over the measurement operators.
The SLD $L_{\mu}$ ($\mu\in\boldsymbol{\theta}$) is implicitly defined via the
equation

$\partial_{\mu}\rho_{\boldsymbol{\theta}}=\frac{L_{\mu}\rho_{\boldsymbol{\theta}}+\rho_{\boldsymbol{\theta}}L_{\mu}}{2}.$
(1)

The corresponding quantum Fisher information matrix (QFIM),
$\boldsymbol{J}(\boldsymbol{\theta})$, is given by

$J_{\mu\nu}(\boldsymbol{\theta})=\textrm{Tr}[\rho_{\boldsymbol{\theta}}\frac{L_{\mu}L_{\nu}+L_{\nu}L_{\mu}}{2}],$
(2)

which yields the matrix SLD QCRB
$\boldsymbol{V(\boldsymbol{\theta})}\geq\boldsymbol{J}(\boldsymbol{\theta})^{-1}$.
It is convenient to introduce a scalar QCRB,
$C^{S}(\boldsymbol{\theta},\boldsymbol{W})=\textrm{Tr}\left[{\boldsymbol{W}\boldsymbol{J}(\boldsymbol{\theta})^{-1}}\right]$.
For a weight matrix $\boldsymbol{W}$, the inequality becomes
$\textrm{Tr}\left[{\boldsymbol{W}\boldsymbol{V}}\right]\geq
C^{S}(\boldsymbol{\theta},\boldsymbol{W})$. This SLD-CRB is generally not
attainable due to the incompatibility of generators of different parameters.
That is, the optimal measurement operators corresponding to different
parameters do not commute with each other, making this scalar bound
unreachable.

Holevo derived Holevo (1977) a tighter scalar bound, the Holevo Cramér-Rao
bound (HCRB) $C^{H}(\boldsymbol{\theta},\boldsymbol{W})$, which is however
only obtained as an optimization. More recently, it has been shown that these
two scalar bounds satisfy the following inequality Carollo  _et al._ (2019)

$\displaystyle C^{S}(\boldsymbol{\theta},\boldsymbol{W})\leq
C^{H}(\boldsymbol{\theta},\boldsymbol{W})$ (3) $\displaystyle\leq$
$\displaystyle
C^{S}(\boldsymbol{\theta},\boldsymbol{W})+||\sqrt{\boldsymbol{W}}\boldsymbol{J}^{-1}\boldsymbol{F}\boldsymbol{J}^{-1}\sqrt{\boldsymbol{W}}||_{1}$
$\displaystyle\leq$
$\displaystyle(1+\gamma)C^{S}(\boldsymbol{\theta},\boldsymbol{W})\leq
2C^{S}(\boldsymbol{\theta},\boldsymbol{W}),$

where $||\cdot||_{1}$ denotes the trace norm and $\boldsymbol{F}$ is known as
the mean Uhlmann curvature with element
$F_{\mu\nu}=-\frac{i}{4}\textrm{Tr}[\rho_{\boldsymbol{\theta}}[L_{\mu},L_{\nu}]]$
(for pure states this reduces to the Berry curvature.) The characterization
number $\gamma$ that appears in Eq. (3) is defined as

$\gamma=||i2\boldsymbol{J}^{-1}\boldsymbol{F}||_{\infty},$ (4)

where $||\boldsymbol{A}||_{\infty}$ denotes the largest eigenvalue of
$\boldsymbol{A}$. $\gamma$ characterizes the “quantumness” of the system,
i.e., it quantifies the amount of incompatibility of the statistic model
$\rho_{\boldsymbol{\theta}}$. As implicit in Eq. (3) and explained hereafter,
it can be proved that $0\leq\gamma\leq 1$ Carollo  _et al._ (2019).

An explicit expression for an attainable scalar QCRB with weight matrix
$\boldsymbol{W}=\boldsymbol{J}$, which monotonically increases with $\gamma$,
was derived in Ref. Matsumoto (2000, 2002)

$\displaystyle C(\boldsymbol{\theta})$
$\displaystyle=\textrm{Tr}\left[{\textrm{Re}\left[\sqrt{\openone_{d}+2i\boldsymbol{J}(\boldsymbol{\theta})^{-1/2}\boldsymbol{F}(\boldsymbol{\theta})\boldsymbol{J}(\boldsymbol{\theta})^{-1/2}}\right]^{-2}}\right]$
$\displaystyle=\sum_{i}\frac{2}{1+\sqrt{1-|\gamma_{i}|^{2}}},$ (5)

where $\openone_{d}$ is the d-dimensional identity matrix and $\gamma_{i}$ is
the i-th eigenvalue of $i2\boldsymbol{J}^{-1}\boldsymbol{F}$. The evaluation
of this attainable QCRB in terms of geometric quantities, and its relation
with SLD-CRB and HCRB, is our key result.

Figure 1:  (a). Diagram showing the “distance” between nearby states
$\ket{\psi(\boldsymbol{\theta})}$ and
$\ket{\psi(\boldsymbol{\theta}+d\boldsymbol{\theta})}$ in parameter space,
from which the metric tensor $\boldsymbol{g}$ (proportional to quantum Fisher
information $\boldsymbol{J}$) and Berry curvature $\boldsymbol{F}$ naturally
arise. (b). SLD-CRB and attaianable QCRB for different $\gamma$ values. At
$\gamma=0$ the model is quasi-classical and the SLD-CRB is reachable, i.e.,
$C^{S}=C^{H}=C$, while the model is called coherent at $\gamma=1$ with
$C^{S}<C^{H}=C$. Here we take the weight matrix
$\boldsymbol{W}=\boldsymbol{J}$ and denote
$C^{S(H)}(\boldsymbol{\theta},\boldsymbol{W})$ as $C^{S(H)}$ for simplicity.

To highlight the geometric interpretation of $\boldsymbol{J},\boldsymbol{F}$
(so-far introduced via the SLD) and thus the connection between estimation and
geometry, in the following we consider pure state models, i.e.,
$\rho_{\boldsymbol{\theta}}=\ket{\psi(\boldsymbol{\theta})}\\!\bra{\psi(\boldsymbol{\theta})}$.
The geometric properties of this state are captured by the quantum geometric
tensor (QGT), which naturally appears when one considers the “distance”
between nearby states $\ket{\psi(\boldsymbol{\theta})}$ and
$\ket{\psi(\boldsymbol{\theta}+d\boldsymbol{\theta})}$ (Fig. 1):

$ds^{2}\equiv
1-|\bra{\psi(\boldsymbol{\theta})}{\psi(\boldsymbol{\theta}\rangle+d\boldsymbol{\theta})}|^{2}=d\theta_{\mu}\chi_{\mu\nu}d\theta_{\nu}+O(|d\boldsymbol{\theta}|^{3}).$

Here we defined the QGT, given by
$\chi_{\mu\nu}=\bra{\partial_{\mu}\psi}(I-\ket{\psi}\bra{\psi})\ket{\partial_{\nu}\psi}=g_{\mu\nu}+\frac{i}{2}F_{\mu\nu}$.
Its imaginary (antisymmetric) part $F_{\mu\nu}$ is the Berry curvature. The
symmetric real part $g_{\mu\nu}$, also known as Fubini-Study metric tensor,
characterizes the overlap between the nearby states. A larger $g_{\mu\nu}$
indicates a larger distinguishability between two states differing by an
infinitesimal change of parameters, and it is thus related to a larger quantum
Fisher information. The role of the Berry curvature is subtler, as it links
the attainability of the QFI bound in multi-parameter scenarios to the
observable compatibility.

To show the relation between QGT and parameter estimation, one can assume the
parameter $\boldsymbol{\theta}$ to be introduced by evolving an initial state
$\ket{\psi_{0}}$ via a unitary evolution, i.e.,
$\ket{\psi(\boldsymbol{\theta})}=U(\boldsymbol{\theta})\ket{\psi_{0}}$. Then,
the generator for parameter $\mu\in\boldsymbol{\theta}$ is the operator
$\mathcal{G}_{\mu}\equiv i(\partial_{\mu}U)U^{\dagger}$. It can be shown that
the QFIM $J_{\mu\nu}$ and the metric tensor $g_{\mu\nu}$ are related by SOM

$J_{\mu\mu}\\!=\\!4\left\langle\Delta\mathcal{G}_{\mu}^{2}\right\rangle\\!=\\!4g_{\mu\mu},\quad
J_{\mu\nu}\\!=\\!4\mathrm{Cov}(\mathcal{G}_{\mu},\mathcal{G}_{\nu})\\!=\\!4g_{\mu\nu},$
(6)

where $\left\langle\Delta\mathcal{G}_{\mu}^{2}\right\rangle$ is the generator
variance and $\mathrm{Cov}(\mathcal{G}_{\mu},\mathcal{G}_{\nu})$ their
covariance. Similarly, the Berry curvature is given by the commutator between
the parameter generators

$F_{\mu\nu}=i\left\langle[\mathcal{G}_{\mu},\mathcal{G}_{\nu}]\right\rangle.$
(7)

It is now clear that the Berry curvature matrix characterizes the
incompatibility of the system as it arises from the non-commutativity between
generators of different parameters. When $F_{\mu\nu}=0$
($\forall\mu,\nu\in\boldsymbol{\theta}$), the characterization number is
$\gamma=0$ according to Eq. 4 and the system is quasi-classical. The
discrepancy between the attainable QCRB and the SLD-CRB now vanishes, i.e.,
$C(\boldsymbol{\theta})=C^{S}(\boldsymbol{\theta},\boldsymbol{J})=C^{H}(\boldsymbol{\theta},\boldsymbol{J})=\textrm{Tr}\left[{\openone_{d}}\right]=d$.

As one of the main contributions of this letter, we next show that in the
opposite limit the characterization number $\gamma$ is upper-bounded by the
fundamental quantum uncertainty principles. We consider the two-parameter case
and start from the Robertson-Schrödinger uncertainty relation for two
operators $\hat{A},\hat{B}$ Robertson (1934), which is the stronger version of
the better-known Heisenberg uncertainty relation. In the quantum parameter
estimation case, the operators $\hat{A},\hat{B}$ can be replaced by the
generators of parameters $\mu,\nu$, giving

$\begin{split}\left\langle\Delta\mathcal{G}_{\mu}^{2}\right\rangle\left\langle\Delta\mathcal{G}_{\nu}^{2}\right\rangle\geq\frac{1}{4}|\left\langle[\mathcal{G}_{\mu},\mathcal{G}_{\nu}]\right\rangle|^{2}+\frac{1}{4}\langle\\{\mathcal{G}_{\mu},\mathcal{G}_{\nu}\\}\rangle^{2}\\\
=\frac{1}{4}|\left\langle[\mathcal{G}_{\mu},\mathcal{G}_{\nu}]\right\rangle|^{2}+\mathrm{Cov}(\mathcal{G}_{\mu},\mathcal{G}_{\nu})^{2}.\end{split}$
(8)

With Eq. (4,6,7) we have

$\gamma=2\frac{F_{\mu\nu}}{\sqrt{\mathrm{det}(\boldsymbol{J})}}=\frac{1}{2}\frac{F_{\mu\nu}}{\sqrt{\mathrm{det}(\boldsymbol{g})}}\leq
1,$ (9)

which relates $\gamma$ to half the Berry’s phase along the curve that encloses
a unit area induced by metric tensor. This relation holds for any two-
parameter space in a multi-parameter setting. The above relation can also be
derived from the fact that the QGT $\chi_{\mu\nu}$ is a positive semi-definite
matrix and has been explored in various models Ozawa and Mera (2021); Mera and
Ozawa (2021); Mera  _et al._ (2022a). Reversely, generalization of the two-
operator product uncertainty relations Eq. 8 to multi-operators can be
performed using the fact that $\gamma\leq 1$, as we show in the supplementary
material SOM .

As a concrete example, we consider a two-parameter model in a qubit system
$\ket{\psi(\theta,\phi)}=\cos\left(\frac{\theta}{2}\right)\ket{0}-\sin\left(\frac{\theta}{2}\right)e^{-i\phi}\ket{1}$.
The QFIM and Berry curvature matrix,

$\boldsymbol{J}=\begin{pmatrix}1&0\\\ 0&\sin^{2}\theta\\\
\end{pmatrix},\quad\boldsymbol{F}=\begin{pmatrix}0&\sin\theta/2\\\
-\sin\theta/2&0\\\ \end{pmatrix},$ (10)

yield $\gamma=1$, implying that the model is coherent and the two parameters
$\theta,\phi$ here are informationally exclusive. Indeed, they are associated
with the non-commuting generators $\sigma_{z}$ and $\sigma_{y}$, respectively.

Figure 2:  (a). Experimental results on elements of quantum Fisher information
matrix and Berry curvature matrix. The solid lines are theoretical
calculations. (b-c). Characterization number $\gamma$ (b) and corresponding
attainable QCRB (c) for the three-parameter model. The solid lines are
theoretical predictions in both plots. The dashed line in (c) is the
attainable QCRB when $\gamma=0$, i.e., the SLD-CRB for the model. The
errorbars are propagated from the errors in quantum geometric quantities. We
note that the measured $\gamma$ values can deviate from or even be above one
when $\alpha\in(0,\pi/2)$ and this is due to the fact that the geometric
quantities are measured in separate experiments.

Eq. (6-7) show the connection between quantum geometry and quantum multi-
parameter estimation (Fig. 1). The value of $\gamma$ characterizes both the
quantum multi-parameter estimation scenario, where it is a measure of system’s
incompatibility, and the geometry of the quantum state manifold, where it
relates the metric tensor determinant to the Berry curvature. This ratio is
deeply connected to the existence of topological invariants. Indeed, more
generally $\gamma$ can be expressed as the ratio between an $n-$form
generalized Berry curvature $F_{\alpha\beta...}$ and the determinant of the
metric tensor, $\gamma=c\frac{F_{\alpha\beta...}}{\sqrt{det(\boldsymbol{g})}}$
with $c$ being a constant normalization factor. Accordingly, $\gamma$ is
gauge-invariant and for pointlike or extended monopoles Palumbo and Goldman
(2018); Chen  _et al._ (2022), we have $\gamma(\boldsymbol{\theta})=1,$
$\forall\boldsymbol{\theta}$ in parameter space, indicating the largest
quantumness of these monopoles. The isotropy of $\gamma$ is due to the radial
nature of the field emanating from monopole sources; when $\gamma=1$, the flux
(i.e., the generalized Berry curvature) through an unit area (given by the
metric tensor determinant) is unit and thus the flux integral over a closed
sphere $S^{n}$ gives the quantized topological number, upon normalization.

As an example, the previous two-parameter, qubit model
$\ket{\psi(\theta,\phi)}$ describes pure states on a $S^{2}$ sphere such that
the integral of their Berry curvature $F_{\mu\nu}$ over the sphere yields the
first Chern number, corresponding to a Dirac monopole at the origin of
parameter space.

In the following, we experimentally explore the relation between quantum
geometry and quantum multi-parameter estimation. In particular, we measure the
geometric quantities using a three-level system and demonstrate the relation
(3) and obtain the attainable QCRB. By embedding a two-parameter model into a
3-level system, we can further explore scenarios away from the two bounds
$\gamma=0,1$.

## Experiments

We engineer a pure state model using a single nitrogen-vacancy (NV) center in
diamond at room temperature. The degeneracy of $\ket{m_{s}=\pm 1}$ is lifted
by an external magnetic field $B$=490 G along the NV quantization axis, which
also polarizes the intrinsic nitrogen nuclear spin during optical
initialization. This qutrit system can be fully controlled using dual-
frequency microwave pulses on-resonance with the transitions between
$\ket{m_{s}=0}$ and $\ket{m_{s}=\pm 1}$. We can thus create any state
parametrized by three angles,

$\ket{\psi(\alpha,\beta,\phi)}=\frac{1}{\sqrt{2}}\begin{pmatrix}\cos(\alpha)e^{-i\beta}\\\
-1\\\ \sin(\alpha)e^{-i\phi}\\\ \end{pmatrix}.$ (11)

We measure the QGT of the this state to determine the QFIM and Berry curvature
matrix to characterize both its geometry and multi-parameter estimation
properties. Specifically, we measure all the components of the real and
imaginary part of the QGT separately using weak modulations of the parameters
$\mu,\nu\in\\{\alpha,\beta,\phi\\}$, a method that has been explored in
previous work Chen  _et al._ (2022); Ozawa and Goldman (2018); Yu  _et al._
(2019). While more details can be found in ref Chen  _et al._ (2022), here we
briefly illustrate the principles of the measurement.

To extract the real part of the QGT, we first prepare the state in Eq. (11)
with a two-tone microwave pulse after initialization to $\ket{m_{s}=0}$. We
then apply driving with in-phase modulation of the amplitudes $\mu,\nu$, i.e.,
$\mu(t)=\mu_{0}+m_{\mu}\sin(\omega t)$ and $\nu(t)=\nu_{0}+m_{\nu}\sin(\omega
t)$ with $|m_{\mu}|,|m_{\nu}|\ll 1$. Similarly, the imaginary part of the QGT
can be extracted with out-of-phase modulations, where
$\mu(t)=\mu_{0}+m_{\mu}\sin(\omega t)$ and $\nu(t)=\nu_{0}+m_{\nu}\cos(\omega
t)$. These weak periodic modulations of the parameters will drive Rabi
oscillations between the state in Eq. (11) and the other two eigenstates of
the qutrit system, when the driving frequency $\omega$ is on-resonance with
the energy splitting between $\ket{\psi(\alpha,\beta,\phi)}$ and the other two
eigenstates $\ket{\psi_{1}}=(-\sin\alpha e^{-i\beta},0,\cos\alpha
e^{-i\phi})^{T}$ and $\ket{\psi_{2}}=(\cos\alpha e^{-i\beta},1,\sin\alpha
e^{-i\phi})^{T}$. Following the technique introduced in Ref. Chen  _et al._
(2022), we can extract the QGT, and thus the QFIM, by measuring the resulting
Rabi frequencies SOM .

The QGT components reconstructed using this technique are shown in Fig. 2(a).
The measurement results are in good agreement with the theoretical predictions
derived from the definition of quantum geometry tensor,

$\begin{split}\boldsymbol{J}(\alpha,\beta,\phi)=\begin{pmatrix}2&0&0\\\
0&2\cos^{2}(\alpha)-\cos^{4}(\alpha)&-\frac{1}{4}\sin^{2}(2\alpha)\\\
0&-\frac{1}{4}\sin^{2}(2\alpha)&2\sin^{2}(\alpha)-\sin^{4}(\alpha)\\\
\end{pmatrix}\end{split}$ (12)

and

$\begin{split}&\boldsymbol{F}(\alpha,\beta,\phi)=\frac{1}{2}\begin{pmatrix}0&\sin(2\alpha)&-\sin(2\alpha)\\\
-\sin(2\alpha)&0&0\\\ \sin(2\alpha)&0&0\\\ \end{pmatrix}.\\\ \end{split}$ (13)

The characterization number $\gamma$ can then be determined as in Fig. 2(b).
When $\alpha=0$ or $\frac{\pi}{2}$, the parameterized model Eq. 11 effectively
becomes a qubit system in a one-parameter (sub)space. For example, when
$\alpha=0$ we have $J_{\phi\phi}=0$, meaning that we have no information on
$\phi$, and $\boldsymbol{F}=0$, indicating that there are no non-commuting
generators, and the parameter $\beta$ can be estimated trivially. The model is
thus quasi-classical. When $\alpha\in(0,\frac{\pi}{2})$, however, the model is
coherent with $\gamma=1$, indicating the maximal non-commutativity of the
generators corresponding to the three parameters.

We can further measure the attainable QCRB, Fig. 2(c). Again, when $\alpha=0$
or $\frac{\pi}{2}$, the attainable QCRB is equal to the SLD-CRB and HCRB,
i.e., $C=C^{S}=C^{H}$ due to the quasi-classical nature of the system. In all
other cases, when $\gamma=1$, the attainable QCRB is equal to the HCRB
Matsumoto (2002) and is higher than SLD-CRB due to the incompatibility of the
three parameters. From a geometric point of view, this three-level model
corresponds to the ground state of a 4D Weyl-type Hamiltonian in spherical
coordinates, that hosts a tensor monopole at the origin Chen  _et al._ (2022).
We can rewrite the characterization number as

$\gamma=2\frac{\sqrt{J_{\phi\phi}F_{\alpha\beta}^{2}+J_{\beta\beta}F_{\alpha\phi}^{2}-2J_{\beta\phi}F_{\alpha\beta}F_{\alpha\phi}}}{\sqrt{\det\boldsymbol{J}}}\equiv\frac{2\mathcal{H}_{\alpha\beta\phi}}{\sqrt{\det\boldsymbol{J}}}.$
(14)

When $\gamma=1$, one recovers the relation between the generalized 3-from
curvature $\mathcal{H}$ and metric tensor $\boldsymbol{g}$,
$\mathcal{H}=\frac{1}{2}\sqrt{\det\boldsymbol{J}}=4\sqrt{\det\boldsymbol{g}}$.
The integral of the curvature over the $S^{3}$ sphere yields the Dixmier-
Douady invariant, which is the topological number for a tensor monopole and
its related Kalb-Ramond field Chen  _et al._ (2022); Palumbo and Goldman
(2018).

Figure 3: . Characterization number (a) and attainable QCRB (b) for two-
parameter estimation in the subspace of three-parameter model. The solid
colored lines are theoretical calculations in both figures. At
$\alpha=0(\pi/2)$, the predicted value for $\gamma$ (attainable QCRB) is 0
(2). The solid (dashed) black lines in (b) are attainable QCRB values when
$\gamma=1$ ( $\gamma=0$).

The above three-parameter model enables us to study both quasi-classical
($\gamma=0$) and coherent ($\gamma=1$) cases. That model, as well as the two-
level model
$\ket{\psi(\theta,\phi)}=\cos\left(\frac{\theta}{2}\right)\ket{0}-\sin\left(\frac{\theta}{2}\right)e^{-i\phi}\ket{1}$,
is indeed one of the eigenstates of a gapless Weyl-type Hamiltonian that hosts
monopoles at the origin of parameter space Mera  _et al._ (2022a); Chen  _et
al._ (2022), leading to $\gamma=1$ (or 0 at special points). To study
intermediate cases where $\gamma\leq 1$, one can consider different
geometrical models Mera  _et al._ (2022b) or, as done here, rely on the two-
parameter subspaces of the above three-parameter model where $\gamma$ is a
function of $\alpha$. From the viewpoint of the Hamiltonian, this corresponds
to taking a slice of the original gapless Weyl-type system, which yields a
gapped topological insulator that can have $\gamma\leq 1$. We thus now revisit
the problem of two-parameter estimation in the subspace of the qutrit system.
Specifically, we focus on the subspace spanned by parameters $(\alpha,\beta)$
and $(\alpha,\phi)$ (the $(\beta,\phi)$ subspace is trivial since all the
corresponding Berry curvature components are zero and the SLD-CRB is
attainable.) In Fig. 3 we plot the characterization number $\gamma$ as well as
the corresponding attainable QCRB. When $\alpha=0(\frac{\pi}{2})$, the three-
level system effectively reduces to a two-level system and the parameter
$\phi$($\beta$) cannot be estimated due to the null population in
$[0,0,1]^{T}$ ($[1,0,0]^{T}$). We observe $\gamma$ vary continuously from 0 to
1 when sweeping $\alpha$ in the two subspaces. The HCRB thus ranges from 2,
equal to the scalar SLD-CRB, (asymptotically) to 4, where the attainable
scalar QCRB is largest (two times larger than the SLD-CRB, as bounded by Eq.
3). These results show how $\gamma$ directly predicts the attainable QCRB for
subspace models, and thus sets the estimation precision when evaluating a
subset of parameters. The analysis above can then be applied to scenarios
where there are nuisance parameters, i.e., parameters that are not of interest
but nevertheless affect the precision of estimating other parameters of
interest Yang  _et al._ (2019); Suzuki (2020); Suzuki  _et al._ (2020).

While we consider a three-parameter model here, the techniques can be extended
to higher dimensional models (beyond three parameters.) For example, we find
that $\gamma=1$ for the pure-state, four-parameter space model

$\ket{\psi(\alpha,\phi_{1},\phi_{2},\phi_{3})}=\frac{1}{\sqrt{3}}\begin{pmatrix}\cos\alpha
e^{i\phi_{1}}\\\ e^{i\phi_{2}}\\\ \sin\alpha e^{i\phi_{3}}\\\ 1\\\
\end{pmatrix},$ (15)

and similarly, in the subspace spanned by $(\alpha,\phi_{1},\phi_{2})$ and
$(\alpha,\phi_{2},\phi_{3})$, $\gamma$ varies from 0 to 1. Similar to Eq.
(14), we find the relation
$\gamma=2\frac{\mathcal{E}_{\alpha\phi_{1}\phi_{2}\phi_{3}}}{\sqrt{det{\boldsymbol{J}}}}=1$
where $\mathcal{E}_{\alpha\phi_{1}\phi_{2}\phi_{3}}$ is the 4-form generalized
Berry curvature SOM . Its integral over the $S^{4}$ sphere will yield a
topological number which is directly proportional to the second Chern number
of a 5D tensor monopole Yang (1978); Hasebe (2014); Palumbo and Goldman
(2019).

In conclusion, we explore the relation between quantum geometry and quantum
multi-parameter estimation. In addition to further insight into the ultimate
precision limits, this relation enables us to experimentally extract the
attainable scalar quantum Cramér-Rao bound for a three-parameter pure state
model synthesized in a single spin system. We show that the SLD-CRB is not
reachable due to the non-commutativity of parameter generators that affect the
uncertainty relations of quantum mechanics. This bound is clarified through
the link to quantum geometry, where the system properties is characterized not
only the distance between states (the Fubini-Study metric tensor), but also by
the space curvature. The present technique and discussions can provide tools
and insights for finding optimal measurement strategies and evaluating the
ultimate precision of multi-parameter models Albarelli  _et al._ (2020); Yu
_et al._ (2020); Hou  _et al._ (2020, 2021).

## Acknowledgements

This work is supported in part by ARO grant W911NF-11-1-0400 and NSF grant
PHY1734011.

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See pages 1 of SOM.pdf See pages 2 of SOM.pdf See pages 3 of SOM.pdf See pages
4 of SOM.pdf See pages 5 of SOM.pdf See pages 6 of SOM.pdf See pages 7 of
SOM.pdf


(see Introduction for notation). For convenience, we set $R_{0}=S_{0}:=eAe$
and $B_{0}:=(1-e)A(1-e).$

######  Definition 4.6.

For $n\geq 1$, the algebras $R_{n}$, $S_{n}$, $A_{n}$ and $B_{n}$ are called
the $n$-th mirror-reflective, reduced mirror-reflective, gendo-symmetric and
reduced gendo-symmetric algebras of $(A,e)$, respectively.

By Propositions 4.2 and 4.3(1), the algebras $R_{n}$ and $S_{n}$ are
symmetric. Thus $A_{n}$ and $B_{n}$ are gendo-symmetric. They are
characterized in terms of Morita context algebras in Section 2.1. Moreover, it
follows from Theorem 1.1(2) that ${\rm dom.dim}(A_{n+1})\geq{\rm
dom.dim}(A_{n})+2$ and ${\rm dom.dim}(B_{n+1})\geq{\rm dom.dim}(B_{n})+2$.
Thus $\min\\{{\rm dom.dim}(A_{n}),{\rm dom.dim}(B_{n})\\}\geq{\rm
dom.dim}(A)+2(n-1)\geq 2n.$

######  Lemma 4.7.

$(1)$ Let $I_{n}:=R_{n}\overline{e}_{n}R_{n}$ and
$J_{n}:=R_{n}(e_{n}-\overline{e}_{n})R_{n}$ with
$\overline{e}_{n}=e_{n}\otimes e_{n}\in R_{n}$ for $n\geq 1$. Then $A_{n+1}$
is derived equivalent and stably equivalent of Morita type to the Morita
context algebra $M_{l}(R_{n},I_{n},J_{n})$.

$(2)$ Let $K_{n}:=S_{n}\overline{f}_{n}S_{n}$ and
$L_{n}:=S_{n}\cap\big{(}R(B_{n})(f_{n}-\overline{f}_{n})R(B_{n})\big{)}$ for
$n\geq 1$. Then $B_{n+1}$ is derived equivalent and stably equivalent of
Morita type to the Morita context algebra $M_{l}(S_{n},K_{n},L_{n})$.

Proof. $(1)$ Recall that there is a surjective algebra homomorphism
$\pi_{1,n}:R_{n}\to A_{n}$ with ${\rm Ker}(\pi_{1,n})=I_{n}$ which induces an
isomorphism $R_{n}(e_{n}-\overline{e}_{n})\simeq A_{n}e_{n}$ of
$R_{n}$-modules. Thus $I_{n}\simeq\Omega_{R_{n}}(A_{n})\oplus Q_{n}$ with
$Q_{n}$ a projective $R_{n}$-module, and $A_{n}e_{n}$ is a projective
$R_{n}$-module. Hence $A_{n+1}$ is Morita equivalent to
$A_{n+1}^{\prime}:={\rm End}_{R_{n}}(R_{n}\oplus A_{n})$. Let $C_{n+1}:={\rm
End}_{R_{n}}(R_{n}\oplus I_{n})$. Since $R_{n}$ is symmetric, it follows from
[17, Corollary 1.2] and [16, Theorem 1.1] that $A_{n+1}^{\prime}$ and
$C_{n+1}$ are both derived equivalent and stably equivalent of Morita type.
Consequently, $A_{n+1}$ and $C_{n+1}$ are both derived equivalent and stably
equivalent of Morita type. It remains to show $C_{n+1}\simeq
M_{l}(R_{n},I_{n},J_{n})$ as algebras.

In fact, since $I_{n}^{2}=I_{n}$, the inclusion
$\lambda_{n}:I_{n}\hookrightarrow R_{n}$ induces ${\rm
End}_{R_{n}}(I_{n})\simeq{\rm Hom}_{R_{n}}(I_{n},R_{n})$. As $R_{n}$ is
symmetric and $J_{n}={\rm Ann}_{R_{n}^{\rm op}}(I_{n})$ by Lemma 3.4(1), we
get $R_{n}/J_{n}\simeq{\rm End}_{R_{n}}(I_{n})$ as algebras via the
restriction of $\lambda_{n}$. This yields a series of isomorphisms

$C_{n+1}\simeq\left(\begin{array}[]{lc}R_{n}&I_{n}\\\ {\rm
Hom}_{R_{n}}(I_{n},R_{n})&{\rm
End}_{R_{n}}(I_{n})\end{array}\right)\simeq\left(\begin{array}[]{lc}R_{n}&I_{n}\\\
{\rm End}_{R_{n}}(I_{n})&{\rm
End}_{R_{n}}(I_{n})\end{array}\right)\simeq\left(\begin{array}[]{lc}R_{n}&I_{n}\\\
R_{n}/J_{n}&R_{n}/J_{n}\end{array}\right),$

of which the composition is an isomorphism from $C_{n+1}$ to
$M_{l}(R_{n},I_{n},J_{n})$ of algebras. This shows $(1)$.

$(2)$ By Lemma 3.3(4), $K_{n}=R(B_{n})\overline{f_{n}}R(B_{n})$ and
$S_{n}/K_{n}\simeq(1_{B_{n}}-f_{n})B_{n}(1_{B_{n}}-f_{n})$. By the proof of
Proposition 3.5(1), ${\rm Ann}_{S_{n}^{\rm op}}(K_{n})=L_{n}$. Similarly,
since $S_{n}$ is symmetric, we can show that $B_{n+1}$ and ${\rm
End}_{S_{n}}(S_{n}\oplus K_{n})$ are both derived equivalent and stably
equivalent of Morita type, and that ${\rm End}_{S_{n}}(S_{n}\oplus K_{n})$ is
isomorphic to $M_{l}(S_{n},K_{n},L_{n})$ as algebras. $\square$

######  Remark 4.8.

By the proof of Lemma 4.7, $B_{n+1}$ and ${\rm
End}_{S_{n}}(S_{n}\oplus\,S_{n}/K_{n})$ are isomorphic, while $A_{n+1}$ and
${\rm End}_{R_{n}}(R_{n}\oplus A_{n})$ are Morita equivalent. It follows from
Proposition 3.5(1) that there are recollements of derived module categories
$\big{(}{\mathscr{D}}(A_{n}),{\mathscr{D}}(A_{n+1}),{\mathscr{D}}(A_{n})\big{)}$
and
$\big{(}{\mathscr{D}}(B_{n}),{\mathscr{D}}(B_{n+1}),{\mathscr{D}}(B_{0})\big{)}$,
which are induced by finitely generated and right-projective idempotent ideals
of $A_{n+1}$ and $B_{n+1}$, respectively.

Proof of Theorem 1.2. We keep all the notation introduced in Lemma 4.7 and its
proof.

$(1)$ By Lemma 2.6, there is a recollement
$\big{(}{\mathscr{D}}(R_{n}/I_{n}),{\mathscr{D}}(M_{l}(R_{n},I_{n},J_{n})),{\mathscr{D}}(R_{n}/J_{n})\big{)}$
induced by a finitely generated, left-projective idempotent ideal of
$M_{l}(R_{n},I_{n},J_{n})$. Thus the recollement restricts to a recollement of
bounded-above derived categories. Since $R_{n}/I_{n}\simeq A_{n}\simeq
R_{n}/J_{n}$ as algebras and since $A_{n+1}$ and $M_{l}(R_{n},I_{n},J_{n})$
are derived equivalent by Lemma 4.7(1), there is a recollement
$\big{(}{\mathscr{D}}^{-}(A_{n}),{\mathscr{D}}^{-}(A_{n+1}),{\mathscr{D}}^{-}(A_{n})\big{)}$.

Similarly, we can apply Lemma 4.7(2) and Lemma 2.6 to show the existence of
the recollement
$\big{(}{\mathscr{D}}^{-}(S_{n}/K_{n}),{\mathscr{D}}^{-}(B_{n+1}),{\mathscr{D}}^{-}(S_{n}/L_{n})\big{)}$.
Note that there are isomorphisms of algebras $S_{n}/L_{n}\simeq B_{n}$ and

$S_{n}/K_{n}\simeq(1_{B_{n}}-f_{n})B_{n}(1_{B_{n}}-f_{n})\simeq(1_{B_{n-1}}-f_{n-1})B_{n-1}(1_{B_{n-1}}-f_{n-1})\simeq\cdots\simeq(1-f_{1})B_{1}(1-f_{1})=B_{0}.$

This implies the existence of the second recollement in $(1)$.

$(2)$ Note that $R_{0}$ is symmetric, $A\simeq{\rm End}_{R_{0}}(eA)$ and
$D(eA)\simeq Ae$. Suppose ${\rm dom.dim}(A)=\infty$. By [22, Lemma 3], ${\rm
Ext}_{R_{0}}^{i}(eA,eA)=0$ for all $i\geq 1$. It follows from ${\rm
Ext}_{R_{0}}^{i}(eA,eA)\simeq{\rm Ext}_{R_{0}}^{i}(eA,D(Ae))\simeq D{\rm
Tor}_{i}^{R_{0}}(Ae,eA)$ that ${\rm Tor}_{i}^{R_{0}}(Ae,eA)=0$ for all $i\geq
1$. By Proposition 3.6(3), the recollements in $(3)$ exist for $n=1$. If
$n\geq 1$, then $R_{n}$ and $S_{n}$ are symmetric algebras, while $A_{n}$ and
$B_{n}$ are gendo-symmetric algebras. Moreover, ${\rm
dom.dim}(A_{n})=\infty={\rm dom.dim}(B_{n})$ by $(2)$ and
$(1_{B_{n}}-f_{n})B_{n}(1_{B_{n}}-f_{n})\simeq B_{0}$ as algebras. Thus, by
induction we can show the existence of recollements for $n\geq 1$. $\square$

Theorem 1.2 can be applied to investigate homological dimensions and higher
algebraic $K$-groups. As usual, for a ring $R$ and $m\in\mathbb{N}$, we denote
by $K_{m}(R)$ the $m$-th algebraic $K$-group of $R$ in the sense of Quillen,
and by $nK_{m}(R)$ the direct sum of $n$ copies of $K_{m}(R)$ for $n\geq 0$.
If $R$ is an Artin algebra, then $K_{0}(R)$ is a finitely generated free
abelian group of rank $\\#(R)$.

######  Lemma 4.9.

Let $R$ be a ring with $f^{2}=f\in R$ such that $I:=RfR$ is a strong
idempotent ideal of $R$. Suppose that one of the following conditions holds:

$(a)$ Either ${{}_{R}}I$ or $I_{R}$ is finitely generated and projective.

$(b)$ There is a ring homomorphism $\lambda:R/I\to R$ such that the
composition of $\lambda$ with the canonical surjection $R\to R/I$ is an
isomorphism.

Then $K_{n}(R)\simeq K_{n}(fRf)\oplus K_{n}(R/I)$ for each $n\in\mathbb{N}$.

Proof. When $(a)$ holds, the isomorphisms of algebraic $K$-groups in Lemma 4.9
follow from [5, Corollary 1.3] or [7, Corollary 1.2].

Let $\pi:R\to R/I$ be the canonical surjection. Clearly, $\pi$ is the
universal localization of $R$ at the map $0\to Rf$. Since $I$ is a strong
idempotent ideal of $R$, $\pi$ is a homological (also called _stably flat_)
ring epimorphism. By [24, Theorem 0.5] and [5, Lemma 2.6], the tensor functors
$Rf\otimes_{fRf}-:(fRf)\mbox{{\rm-proj}}\to R\mbox{{\rm-proj}}$ and
$(R/I)\otimes_{R}-:R\mbox{{\rm-proj}}\to(R/I)\mbox{{\rm-proj}}$ induce a long
exact sequence of algebraic $K$-groups of rings

$\cdots\cdots\to K_{n+1}(R/I)\to K_{n}(fRf)\to K_{n}(R)\to
K_{n}(R/I)\to\cdots\to K_{0}(fRf)\to K_{0}(R)\to K_{0}(R/I).$

Suppose $(b)$ holds. Then the composition of the functors
$R\otimes_{R/I}-:(R/I)\mbox{{\rm-proj}}\to R\mbox{{\rm-proj}}$ with
$(R/I)\otimes_{R}-:R\mbox{{\rm-proj}}\to(R/I)\mbox{{\rm-proj}}$ is an
equivalence. This implies that the composition of the maps
$K_{n}(R\otimes_{R/I}-):K_{n}(R/I)\to K_{n}(R)$ with
$K_{n}((R/I)\otimes_{R}-):K_{n}(R)\to K_{n}(R/I)$ induced from tensor functors
is an isomorphism. Consequently, $0\to K_{n}(fRf)\to K_{n}(R)\to K_{n}(R/I)\to
0$ is split-exact. Thus $K_{n}(R)\simeq K_{n}(fRf)\oplus K_{n}(R/I)$.
$\square$

######  Corollary 4.10.

Let $n$ be a positive integer. Then

$(1)$ ${\rm fin.dim}(A_{n})\leq{\rm fin.dim}(A_{n+1})\leq 2{\rm
fin.dim}(A_{n})+2$ and ${\rm fin.dim}(B_{0})\leq{\rm fin.dim}(B_{n+1})\leq{\rm
fin.dim}(B_{0})+{\rm fin.dim}(B_{n})+2$. Thus

${\rm fin.dim}(A_{n+1})\leq 2^{n}{\rm fin.dim}(A)+2^{n+1}-2\mbox{ and }{\rm
fin.dim}(B_{n+1})\leq{\rm fin.dim}(A)+n({\rm fin.dim}(B_{0})+2).$

These inequalities hold true for global dimensions.

$(2)$ $K_{*}(A_{n+1})\simeq 2^{n}\,K_{*}(A)$ and $K_{*}(B_{n+1})\simeq
nK_{*}(B_{0})\oplus K_{*}(A)$ for $*\in\mathbb{N}$.

$(3)$ If ${\rm dom.dim}(A)=\infty$, then $K_{*}(R_{n})\simeq
K_{*}(\Lambda)\oplus(2^{n}-1)K_{*}(A)$ and $K_{*}(S_{n})\simeq
K_{*}(\Lambda)\oplus nK_{*}(B_{0})$ for any $*\in\mathbb{N}$.

Proof. $(1)$ By Lemma 4.7(1), $A_{n+1}$ and $M_{l}(R_{n},I_{n},J_{n})$ are
stably equivalent of Morita type. Since global and finitistic dimensions are
invariant under stably equivalences of Morita type, $A_{n+1}$ and
$M_{l}(R_{n},I_{n},J_{n})$ have the same global and finitistic dimensions.
Now, the statements on $A_{n+1}$ in $(1)$ hold by applying [8, Corollary 3.12
and Theorem 3.17] to the recollement
$\big{(}{\mathscr{D}}(R_{n}/I_{n}),{\mathscr{D}}(M_{l}(R_{n},I_{n},J_{n})),{\mathscr{D}}(R_{n}/J_{n})\big{)}$
(see the proof of Theorem 1.2(1)). In a similar way, we show the statements on
$B_{n}$ by the recollement
$\big{(}{\mathscr{D}}(B_{0}),{\mathscr{D}}(B_{n+1}),{\mathscr{D}}(B_{n})\big{)}$
in Theorem 1.2(1).

$(2)$ Note that derived equivalent algebras have isomorphic algebraic
$K$-groups (see [12]). By Lemma 4.9 and the proof of Theorem 1.2(1), we have
$K_{*}(A_{n+1})\simeq K_{*}(M_{l}(R_{n},I_{n},J_{n}))\simeq
K_{*}(R_{n}/I_{n})\oplus K_{*}(R_{n}/J_{n})\simeq 2K_{*}(A_{n})$ and
$K_{*}(B_{n+1})\simeq K_{*}(M_{l}(S_{n},K_{n},L_{n}))\simeq
K_{*}(S_{n}/K_{n})\oplus K_{*}(S_{n}/L_{n})\simeq K_{*}(B_{0})\oplus
K_{*}(B_{n})$. Starting with $A_{1}=A=B_{1}$, we can show the isomorphisms in
$(2)$ by induction.

$(3)$ By Lemma 4.9 and Theorem 1.2(2), $K_{*}(R_{n})\simeq
K_{*}(R_{n-1})\oplus K_{*}(A_{n})$ and $K_{*}(S_{n})\simeq
K_{*}(S_{n-1})\oplus K_{*}(B_{0})$ for each $n\geq 1$. Together with $(2)$, we
can show the isomorphisms in $(3)$ by induction. $\square$

######  Remark 4.11.

Without ${\rm dom.dim}(A)=\infty$, the isomorphisms in Corollary 4.10(3) still
hold for $*=0$. This follows from Corollary 4.10(2) and the fact that if $R$
is a finite-dimensional algebra over a field and $f^{2}=f\in R$, then
$K_{0}(R)\simeq K_{0}(fRf)\oplus K_{0}(R/RfR)$. Thus
$\\#(R_{n})=\\#(\Lambda)+(2^{n}-1)\,\\#(A)$ and
$\\#(S_{n})=\\#(\Lambda)+n\;\\#(B_{0})$.

As a consequence of Theorem 1.2, we obtain bounds for the stratified
dimensions and ratios of iterated mirror-reflective algebras of gendo-
symmetric algebras which are not symmetric. This provides a new approach to
attack the Tachikawa’s second conjecture.

######  Corollary 4.12.

Let $n$ be a positive integer, and let $(A,e)$ be a gendo-symmetric algebra
with ${\rm dom.dim}(A)=\infty$. If $A$ is not symmetric, then

$(1)$ $2^{n}-1\leq{\rm st.dim}(eAe)+(2^{n}-1)({\rm st.dim}(A)+1)\leq{\rm
st.dim}(R_{n})\leq\\#(eAe)+(2^{n}-1)\,\\#(A)-1\mbox{ and }$

$n\leq{\rm st.dim}(eAe)+n({\rm st.dim}(B_{0})+1)\leq{\rm
st.dim}(S_{n})\leq\\#(eAe)+n\,\\#(B_{0})-1.$

$(2)$ $\;\frac{{\rm
st.dim}(A)+1}{\\#(A)}\leq\mathop{\varliminf}\limits_{n\to\infty}{\rm
sr}(R_{n})\leq 1\;\mbox{ and }\;\frac{{\rm
st.dim}(B_{0})+1}{\\#(B_{0})}\leq\mathop{\varliminf}\limits_{n\to\infty}{\rm
sr}(S_{n})\leq 1.$ In particular, if $B_{0}$ is local, then
$\lim\limits_{n\to\infty}{\rm sr}(S_{n})=1$, where $\varliminf$ means the
limit inferior.

Proof. $(1)$ By Theorem 1.2(2) and Proposition 2.9(3), ${\rm
st.dim}(R_{n})\geq{\rm st.dim}(R_{n-1})+{\rm st.dim}(A_{n})+1$ and ${\rm
st.dim}(S_{n})\geq{\rm st.dim}(S_{n-1})+{\rm st.dim}(B_{0})+1$. Similarly, by
Remark 4.8 and Proposition 2.9(3), we have ${\rm st.dim}(A_{n+1})\geq 2\,{\rm
st.dim}(A_{n})+1$, that is, ${\rm st.dim}(A_{n+1})+1\geq 2({\rm
st.dim}(A_{n})+1)$. Moreover, by Proposition 2.9(1), ${\rm
st.dim}(R_{n})\leq\\#(R_{n})-1$ and ${\rm st.dim}(S_{n})\leq\\#(S_{n})-1$.
Combining these inequalities with Remark 4.11, we get $(1)$ by induction.

$(2)$ follows from $(1)$ and Remark 4.11. $\square$

Proof of Theorem 1.3. $(1)\Rightarrow(2)$ Assume that (TC2) holds for all
symmetric algebras over $k$. Let $S$ be an indecomposable symmetric
$k$-algebra and $I$ a strong idempotent ideal of $S$. Then $0={\rm
Ext}_{S/I}^{i}(S/I,S/I)\simeq{\rm Ext}_{S}^{i}(S/I,S/I)$ for all $i\geq 1$.
This means that ${{}_{S}}S/I$ is orthogonal. Then the $S$-module $S/I$ is
projective by (1), and therefore ${{}_{S}}S\simeq I\oplus S/I$. It follows
from $I^{2}=I$ that ${\rm Hom}_{S}(I,S/I)=0$. Since $S$ is symmetric and
${{}_{S}}I$ is projective, ${\rm Hom}_{S}(S/I,I)\simeq D{\rm
Hom}_{S}(I,S/I)=0$. Consequently, $S\simeq{\rm End}_{S}(I)\oplus{\rm
End}_{S}(S/I)$ as algebras. Since $S$ is indecomposable, either ${\rm
End}_{S}(I)$ or ${\rm End}_{S}(S/I)$ vanishes. In other words, $I=0$ or $I=S$.
This implies that $S$ has no nontrivial strong idempotent ideals. So $(1)$
implies $(2)$.

$(2)\Rightarrow(3)$ An algebra $S$ has no nontrivial strong idempotents if and
only if ${\rm st.dim}(S)=0$ if and only if ${\rm sr}(S)=0$. Thus $(3)$
follows.

$(3)\Rightarrow(1)$ Suppose that (TC2) does not hold for an indecomposable
symmetric algebra $S$ over $k$. Then there exists an indecomposable, non-
projective orthogonal $S$-module $M$. Then $A:={\rm End}_{S}(S\oplus M)$ is a
gendo-symmetric, but not a symmetric algebra. Let $S_{n}$ be the $n$-th
reduced mirror symmetric algebra of $A$ for $n\geq 1$. Then $S_{n}$ is
symmetric by Proposition 4.3(1). As $M$ is indecomposable, ${\rm End}_{S}(M)$
is local. Since $M$ contains no nonzero projective direct summands, $S_{1}$ is
indecomposable by Proposition 4.3(3). Further, by the proof of Theorem 1.2(1),
${\rm End}_{S}(M)\simeq(1_{B_{n}}-f_{n})B_{n}(1_{B_{n}}-f_{n})$ as algebras
for any $n\geq 1$. Combining this fact with Proposition 4.3(2), we show that
$S_{n}$ is indecomposable by induction. Since $M$ is orthogonal, we see ${\rm
dom.dim}(A)=\infty$ by [22, Lemma 3]. It follows from Corollary 4.12 that
$\lim\limits_{n\to\infty}{\rm sr}(S_{n})=1$. Thus the supreme in $(3)$ must be
$1$, a contradiction to the assumption (3). This shows that $(3)$ implies
$(1)$. $\square$

Acknowledgements. The research was supported partially by the National Natural
Science Foundation of China (Grant 12031014, 12122112 and 12171457).

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Hongxing Chen,

School of Mathematical Sciences & Academy for Multidisciplinary Studies,
Capital Normal University, 100048 Beijing, P. R. China;

Email<EMAIL_ADDRESS>(H.X. Chen)

Ming Fang, Academy of Mathematics and Systems Science, Chinese Academy of
Sciences, 100190 & School of Mathematical Sciences, University of Chinese
Academy of Sciences, 100049 Beijing, P. R. China;

Email<EMAIL_ADDRESS>(M. Fang)

Changchang Xi,

School of Mathematical Sciences, Capital Normal University, 100048 Beijing; &
School of Mathematics and Statistics, Shaanxi Normal University, Xi’an 710062,
Shaanxi, P. R. China

Email<EMAIL_ADDRESS>(C.C. Xi)


# Virtual walks and phase transitions in two dimensional BChS model with
extreme switches

Kathakali Biswas<EMAIL_ADDRESS>Department of Physics, Victoria
Institution (College), 78B Acharya Prafulla Chandra Road, Kolkata 700009,
India.  Department of Physics, University of Calcutta, 92 Acharya Prafulla
Chandra Road, Kolkata 700009, India.  Parongama Sen<EMAIL_ADDRESS>Department of Physics, University of Calcutta, 92 Acharya Prafulla Chandra
Road, Kolkata 700009, India.

###### Abstract

We have studied a walk in a one-dimensional virtual space corresponding to an
extended version of the three-state BChS model of opinion formation,
originally proposed in Physica A 391, 3257 (2012), in which the agents are
located on a two dimensional lattice. The opinions are designated by the
values $\pm 1$ and zero. Here we also consider switches between the extreme
states $\pm 1$. The model involves two noise parameters representing the
fraction of negative interactions $p$ and the probability of extreme switch
denoted by $q$. The study shows that the nature of the walks changes
drastically as the noise parameters exceed certain threshold values. The
order-disorder phase transitions are independently obtained using the finite
size scaling method showing that these threshold values are indeed consistent
with those values of the parameters where a phase transition exists. The
criticality is found to be Ising-like even when extreme switches are allowed.
A new critical exponent associated with the probability distribution of the
displacement is also obtained independent of the values of the critical
parameters. The nature of the walks is compared to similar virtual walks
studied earlier.

##  I Introduction

In the past few decades, extensive research has been made to study the problem
of opinion formation in a society, using the tools of statistical physics
sociophysics ; soc_rmp ; galam_book . Several models of opinion dynamics have
been proposed in the past. One class of models, namely, the kinetic exchange
(KE) models involves an interaction between two agents at any instant
kem_eco_sco . A particular KE model, popularly called the BChS model,
incorporates negative interactions as well BCS . This model yields an
interesting phase transition. It has also been possible to obtain some results
from this model that correspond to realistic scenarios us_elec_1 ; kbps1 ;
brexit .

It is often convenient to study a model in statistical physics by mapping it
into another model and studying the latter. In recent times, various models of
statistical physics involving dynamics of spins, opinions, and financial
status have been mapped to walks in a virtual space godre1 ; newman ;
baldasari ; godre2 ; godre3 ; godre4 ; privman ; derrida1 ; derrida2 ; derri2
; howard ; sbprps ; pratik ; surojit1 ; sanchari_walk ; sanchari . In these
mappings, a walker is associated with each spin/agent. The position of the
walker is updated according to the state of the spin/agent it is associated
with, following either a Markovian or a non-Markovian dynamics. The nature of
the walks usually undergoes a change at the phase transition points, if any.
This is indicated by the existence of a Gaussian distribution of the
displacements of the walkers above the critical point while it has a double
peaked non-Gaussian behavior be- low it. New critical exponents associated
with the width of the distribution have been seen to exist as well. It is
possible to conjecture whether the ordered state is an absorbing state by
studying the scaling behavior of this distribution. An interesting crossover
behavior involving diverging timescales, not obtained directly from the model,
has been observed previously also pratik ; surojit1 .

In this paper, we have studied a KE model of opinion formation, an extended
version of the so-called BChS model. In this opinion dynamics model, the
agents are embedded on a two-dimensional (2$d$) lattice while the walks are
defined in a virtual one-dimensional (1$d$) space.

The BChS model is taken with three opinion states denoted by $\pm 1$ and $0$.
The interactions here can be negative with probability $p$. In the original
version BCS , switches between the extreme states (i.e. $+1$ and $-1$) were
not possible according to the dynamical rules. In some recent works, the
present authors introduced the possibility of extreme switches occurring with
probability $q$ kbps2 ; kbps3 . The mean field case was studied to show that a
phase diagram can be obtained in the $p-q$ plane showing the presence of
ordered and disordered regions kbps3 .

The critical behavior of the BChS model on two-dimensional square lattices for
$q=0$ has been studied earlier using numerical simulations sudip ; croki . It
was found that the model belongs to the two-dimensional Ising criticality
class. In this paper, we have simulated the model in 2$d$ with extreme
switches, i.e., $q\neq 0$, and obtained the corresponding walk from which the
phase diagram can be estimated. In addition, for comparison, a few phase
transition points have been found directly by analyzing the relevant physical
quantities and using finite size scaling. From the results obtained in the
mean-field case kbps3 , we expect that $q$ will effectively impart an
additional noise. It is also interesting to see whether the critical behavior
is affected by $q$; in the mean-field case, it is not.

The walks are also analyzed quantitatively by fitting appropriate curves to
the distributions and studying the fitting parameters. This leads to some
further understanding of the system both quantitatively and qualitatively.

In section II, we describe the model and the methods used. The results are
presented in section III and in the last section, discussions and conclusive
statements have been made.

##  II Model and Method

###  II.1 Simulating BChS model on a two dimensional lattice with the
parameters $p$ and $q$

We have simulated an agent-based BChS model where agents are located on the
sites of a 2$d$ square lattice with $N$ sites ($N=L\times L$) having periodic
boundary conditions. $o_{i}$ is considered as the opinion of the $i$th agent.
Each opinion ($o_{i}=\pm 1,0$) is updated upon interaction with one of the
nearest neighbors denoted by $k$, with opinion $o_{k}$, following the
expression,

$o_{i}(t+1)=o_{i}(t)+\mu o_{k}(t).$ (1)

Here $\mu$ is the interaction parameter; $|\mu|=1,2$ with probability $(1-q)$
and $q$ respectively. $\mu$ can be negative with probability $p$ for both
values of $|\mu|$. If after an interaction the opinion exceeds $1$ or becomes
less than $-1$, it is adjusted to $\pm 1$ respectively.

In the simulations, a homogeneous disordered state is considered as the
initial configuration, i.e., equal proportion of the population has opinion
$\pm 1$ and zero.

###  II.2 Mapping the BChS model into a Virtual Walk

Mapping of the opinion dynamics model to a virtual walk is done by associating
a virtual walker with each agent on the 2$d$ lattice. The virtual walk takes
place in one dimension. Hence we have a scenario of $N$ walkers performing
walks on a one dimensional lattice which is strictly speaking, unbounded. The
walks are not independent as they are generated from the interaction of the
agents.

In the mapping scheme, the initial position of a walker is taken to be $0$.
The walks are implemented according to the opinions of the agents which are
updated asynchronously. The total distance travelled from the starting point
(at $t=0$) by the $i$th walker at the $(t+1)$th Monte Carlo (MC) step is
$X_{i}(t+1)$, given by

$X_{i}(t+1)=X_{i}(t)+o_{i}(t+1).$ (2)

Only Markovian walks have been considered in the present work.

We have simulated the system on $L\times L$ lattices with various values of
$L$. For the virtual walk, the results for $L=64$ (maximum size simulated)
only are presented. The walk is incremented by the value of the opinion after
the completion of one Monte-Carlo step (MCS). One agent is selected randomly
in time $\frac{1}{N}$ and allowed to interact with any of her four nearest
neighbour after which her opinion is updated; one MCS step comprises of $N$
such updates.

###  II.3 Finite size scaling analysis

For a system manifesting a continuous phase transition driven by a certain
parameter, the critical point and the exponents can be obtained using finite
size scaling method in a numerical simulation. Consider a physical quantity
$\Phi$, which either goes to zero or diverges in the manner
$\Phi\propto\epsilon^{\phi}$, where $\epsilon$ is the small deviation from the
critical point. In a finite system of size $L$, $\Phi$ can be expressed as

$\Phi=L^{-\phi/\nu}f(\epsilon L^{1/\nu}),$

where $\nu$ is the correlation length exponent. $f(z)$ is expected to vary as
$z^{\phi}$ as $z\to\infty$ such that one recovers the behaviour
$\Phi\propto\epsilon^{\phi}$ for $L\to\infty$.

A data collapse can be obtained (i.e., all data for different system sizes
fall on the same curve) when properly rescaled quantities are plotted with
accurate choices of the values of the critical point and the exponents.

##  III Results

We have determined the probability $S(X,t)$ associated with the distance $X$
covered by a walker at a specific time $t$ by averaging over many
configurations. In the Ising and voter model in 2$d$ and also in the mean-
field KE model godre4 ; pratik ; surojit1 , the nature of $S(X,t)$ has been
observed to change at the phase transition points. Here also we have observed
a similar change beyond threshold values of the parameters $p,q$. We
investigated the characteristics of the distribution both below and above the
threshold points. Additionally, by performing data collapse analysis of the
scaled data from the simulations conducted on 2$d$ lattices, we identified the
critical points and estimated the critical exponents. The latter study has
been made at larger times such that the system definitely reaches equilibrium.

###  III.1  Features studied from the Virtual walk

In the $(p,q)$ parameter space, the threshold points, denoted as
($p_{T},q_{T}$), have been determined using the virtual walk analysis. In
general, we have kept $q=q_{T}$ fixed and varied $p$ to study the behavior of
the walk. It is observed that when $p$ is less than $p_{T}$, the distribution
$S(X,t)$ exhibits a non-Gaussian, double-peaked behavior. Above the threshold
point, the distribution assumes a centrally peaked Gaussian shape. The data
are shown in Fig. 1 for two different $q_{T}$ values.

Figure 1: The probability distribution of $X$ at time step $t=60000$, when the
model has reached a steady state, has been plotted for two different values of
$q$ ($0.2$ in (a) and $0.6$ in (b)) for various $p$ values. These plots are
presented for a system size of $N=64\times 64$.

The determination of the threshold points $p_{T}$ has been carried out for
several values of $q$ within the range of $[0,1)$, and as displayed in Fig. 2
these points can be fitted to the form

$p_{T}\approx C_{0}\exp^{-\lambda q_{T}}-C_{1},$ (3)

where $\lambda\approx 3.14$, $C_{1}\approx 0.0063$ and $C_{0}\approx 0.1220$.
Specifically, one gets $p_{T}\rightarrow 0$ for $q_{T}=1$, exactly as in the
case of the mean-field version of the present model kbps3 .

To determine the nature of the walk, $S(X,t)$ is fitted to the scaling form

$S(X,t)\approx t^{-\alpha}F(X/t^{\alpha}).$ (4)

When $p<p_{T}$, an approximate data collapse has been obtained with
$\alpha=1.0$ (shown in Fig. 3). $\alpha=1$ implies a ballistic walk. For an
absorbing phase, a perfectly ballistic behavior is expected at late times.
However such is not the case here. This indicates that for $p<p_{T}$, where
the ordered phase exists as is concluded from the nature of $S(X,t)$, we have
an active state. Indeed, the snapshots shown in Fig. 4 taken at different time
show that the system is still active. It is understandable why an active phase
is present; in the presence of noise, some agents will change opinion even at
large times. In previous work, only for the noise-less case, it was shown that
an absorbing phase is reached, although for a considerable number of
configurations, it becomes a very slow process brexit . We will come back to
this point in the next section.

On the other hand, the data collapse observed for $p>p_{T}$ yields
$\alpha=0.5$ (as shown in Fig. 5), which is consistent with the behavior
expected in an unbiased random walk. The Gaussian function is written as
$F(z)\propto e^{-\frac{z^{2}}{2\sigma^{2}}}$, where $z=X/t^{\frac{1}{2}}$. The
distribution width $\sigma$, is observed to diverge as $|p-p_{T}|^{-\delta}$,
plotted in Fig. 6 for different $q$ values. $\delta$ shows no systematic
dependence on the exact location on the phase boundary where it is calculated
and lies within a small range; $\delta\approx 0.84$. Snapshots in the
disordered phase taken at different times show that the states of the spins
are changing considerably over time which will show typical oscillations of
the order parameter about zero for a single configuration (Fig. 7). It is
interesting that both below and above the phase boundary, zero opinions are
much less in number in comparison to the others.

Figure 2: The red dots in the $p,q$ parameter space represent threshold values
separating the ordered and disordered phases. The phase boundary can be fitted
to the form $p_{T}=C_{0}\exp^{-\lambda q_{T}}-C_{1}$ with
$\lambda=3.14957~{}\pm 0.02697$, $C_{1}=0.00626~{}\pm 0.00033$ and
$C_{0}=0.12203~{}\pm 0.00036$. Three critical points directly obtained from
the simulation for $q=0,0.2,0.5$ using finite size scaling are also shown by
the green points in the figure. Figure 3: The data collapse of $S(X,t)t$ using
the scaling variable $X/t$ is illustrated for two different points: (a)
$q=0.2,~{}p=0.055$ and (b) $q=0.6,~{}p=0.010$, both of which lie below the
phase boundary (eq. 3).  Figure 4: Snapshots of the 2$d$ lattice space at
different time steps below the phase boundary. Red, blue, yellow dots denote
$\pm 1$ and $0$ states respectively.

###  III.2 Exploring phase transitions by finite size scaling method

Up to this point, we have discussed the results of the virtual walk generated
from the BChS model with extreme switches. In this section, we report the
results for the critical points in the $p-q$ space obtained directly from the
simulation of the BChS model. The most prevalent approach for investigating
order-disorder phase transitions involves the finite-size scaling of the
quantities like Binder cumulant, the order parameter etc. In this particular
model, the critical points can be expressed as ($p_{c},q_{c}$). The order
parameter for the system is the average of all opinions:
$O=\frac{1}{N}\sum_{i}{o_{i}}$ and the fourth order Binder cumulant is
$U=1-\frac{\langle O^{4}\rangle}{3\langle O^{2}\rangle^{2}}$. The angular
brackets indicate the ensemble average.

Figure 5:  The data collapse of $S(X,t)t^{0.5}$ using the scaling variable
$X/t^{0.5}$ is illustrated for two cases (a) $q=0.2,p=0.064$ and (b)
$q=0.6,p=0.015$, both of which lie above the phase boundary (i.e $p<p_{T}$ in
each case). Both sets of data can be fit using a Gaussian function denoted by
$F(z)$ in the figures, where $z=\frac{X}{t^{1/2}}$.  Figure 6:  The variation
of the width $\sigma$ of the Gaussian functions as a function of $p$ above
$p_{T}$ is shown in this figure for several $q$ values. The figure shows that
the nature of the curves is compatible with a variation
$\sigma\sim(p-p_{T})^{-\delta}$ with $\delta\approx 0.84$.  Figure 7:
Snapshots of the 2$d$ lattice space at different time step above the phase
boundary. Red, blue, and yellow dots denote $\pm 1$ and $0$ states
respectively.

We conducted Monte Carlo simulations for various system sizes $L$ varying
between 12 and 64. The simulations were run with a sufficient number of time
steps to allow measurable quantities to reach a steady value. Subsequently, we
calculated the ensemble averages of these values. The number of configurations
ranged from 2000 to 1000 with an increase in system size. It is well known
that the scaling behavior of the Binder cumulant and the order parameter are
as follows: $U=f_{1}((x-x_{c})L^{\frac{1}{\nu}})$ and
$\langle|O|\rangle=L^{-\frac{\beta}{\nu}}f_{2}((x-x_{c})L^{\frac{1}{\nu}})$,
where the phase transition is driven by the parameter $x$ and occurs at
$x=x_{c}$. $\nu$ and $\beta$ are critical exponents associated with the
correlation length and order parameter respectively.

Our aim is to check whether ($p_{T},q_{T}$) and ($p_{c},q_{c}$) are close
enough so that it can be concluded that the virtual walks bear the signature
of the phase transition. Here we have compared the values for three particular
$q$ values. Fixing the value of $q$ as $q_{c}$, we determined $p_{c}$ by
identifying the crossing points of the Binder cumulant for different system
sizes. Additionally, by employing the data collapse technique, we estimated
the critical exponent $\nu$ to be very close to 1 (see Fig. 8). Using this
value of $\nu$, we estimated the critical exponent $\beta\approx 0.125$ from
the data collapse of the scaled order parameter shown in Fig. 9. Both these
exponent values are very close to the exact values known for the 2$d$ Ising
model.

Figure 8:  The finite-size scaling behavior of the Binder cumulant $U$ as a
function of $p$ is illustrated for two cases: (a) $q=0.0$ and (b) $q=0.5$.
Notably, data collapses are observed with critical values of $p_{c}=0.1110$
and $p_{c}=0.0170$ in (a) and (b) respectively. In both cases, the critical
exponent $\nu$ is estimated to be approximately $1.0\pm 0.05$. The inset
displays the unscaled raw data for $U$ versus $p$. Figure 9:  The data
collapse of the scaled order parameter $\langle|O|\rangle$ for different
system sizes is shown for two cases: (a) $q=0.0$ and (b) $q=0.5$ using the
critical values of $p_{c}$ and $\nu$ obtained from the analysis of the Binder
cumulant (Fig. 8). In both instances, the critical exponent $\beta$ is
estimated to be $0.125\pm 0.001$. The inset displays the unscaled raw data
$\langle O\rangle$ vs $p$.

For $q=0.0$ and $0.5$, we obtain $p_{c}\approx 0.1110$ and $\approx 0.0170$
respectively. From the walk picture, the corresponding threshold values are
$p_{T}\approx 0.1150$ and $\approx 0.0185$ respectively, which are fairly
close to the results obtained using finite size scaling.

Figure 10: The variation of the fraction of the number of active agents $A$
among the total number of agents $N$ is plotted against time, for different
values of $q$, above and below the phase boundary.

##  IV Discussions and Conclusions

In the present work, we have obtained a phase diagram in a two-parameter
kinetic exchange model of opinion dynamics with three opinion states. The two
parameters represent the fraction of negative interactions and the probability
of extreme switches of opinions. By simulating the agent-based model on a 2$d$
lattice, we generated the walks corresponding to each agent’s opinion state in
a 1$d$ virtual space. Analysis of the distribution $S(X,t)$ of the
displacements shows a change in the nature of the walk above threshold values
of the parameters. Below these values, the distribution is double peaked and
the data for different times can be approximately collapsed using a scaling
variable $X/t$ which reveals the nearly ballistic nature of the walk. The
nearly ballistic nature and the double peaked structure with peaks occurring
at nonzero values of $X$ indicate that most of the opinions continue in a
state of either $+1$ or $-1$. This is then a partially ordered state. It may
be mentioned here that for the zero opinion, no displacement is occurring in
the walk and it does not significantly affect the walk’s nature. Above the
threshold values, we get Gaussian distributions centered at zero with the
scaling variable $X/t^{1/2}$ indicating a diffusive walk. This means that the
opinions are changing continuously and randomly in time and the system is
disordered. Hence the two regions below and above the threshold values are
identified as ordered and disordered regions as had been observed earlier for
such virtual walks corresponding to other dynamical systems.

To establish that indeed the walk changes its nature at critical points, we
have also located the critical points for three $q$ values, using finite size
scaling. The results agree fairly well, the small discrepancy may be due to
finite size effects. The critical exponents $\nu$ and $\beta$ are estimated
and found to be close to the Ising exponents in two dimensions, observed
already for the one-parameter model without extreme switches sudip . One can
also compare the results with the mean-field case kbps3 , where the phase
boundary was obtained as a straight line. Except for $q=1$, the mean field
phase boundary lies above that of the 2$d$ one which is a logical result. In
both the mean-field and two-dimensional versions, the role of $q$ is to
provide additional noise without changing the universality class.
Interestingly, in both cases, we find that the system becomes totally
disordered when the probability of extreme switches is unity, $i.e.$ when
$q=1$. In the mean-field case, it was shown that for $q=1$, the model becomes
identical to a voter model with binary opinion values, we conjecture that the
same happens for the finite-dimensional case as essentially the presence of
$q$ decreases the probability of having a zero opinion. This is aptly
reflected in the snapshots for a nonzero value of $q$.

From the walk picture, a critical exponent $\delta$ related to the diverging
width of the Gaussian distribution is obtained. Like the static critical
exponents $\delta$, characterising the distribution of the walk in the
disordered phase, is also $q$ independent. Hence the signature of the phase
transition is contained in the distribution in this manner as well. As
$\delta$ apparently cannot be related to any static critical exponent, we
claim it is a new exponent.

We also obtained the result that the ordered phase is not an absorbing one in
general from the results obtained so far. It is known that for $p=0,q=0$, an
absorbing state can be reached but may take a very long time brexit . The
present study indicates that a detailed study of the dynamics including $q$
and the question of reaching an absorbing state could be interesting. for the
case $p=q=0$, the presence of zero opinion states at the boundary of $+1$ and
${-1}$ domains was responsible for the slow dynamics. With a nonzero value of
$q$, the population of zero opinions decreases and such states could be less
significant for the dynamics. A preliminary study of the active bonds show
that indeed it decreases $q$ is made larger in both the disordered and ordered
states (Fig. 10). That the active bond fraction remains non-zero at long times
also supports the fact that both the phases are active. A detailed study of
this and dynamics in general could be interesting, however, at the moment we
restrict to presenting the walk features and the resulting phase diagram only.

One more point is there to be noted. In some earlier studies of the virtual
walks, a crossover behavior in time had been observed in the ordered phase.
Here, however, no such tendency is noted.

In conclusion, we find that a phase diagram for the two dimensional BChS model
with extreme switches can be obtained using the walk picture. A usual finite
size analysis has also been done to show that the criticality is of Ising
class. One more exponent entirely related to the virtual walks has been
obtained from the divergence of the width of the distribution above the
critical points. For the two dimensional voter model, a similar divergence was
found pratik , however, for the mean field BChS model, the width was found to
be independent of $p$ above $p_{c}$ surojit1 . This feature therefore needs to
investigated more by studying other models.

###### Acknowledgements.

PS acknowledges financial support from CSIR scheme 03/1495/23/EMR-II.

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