# Schwarzschild and Ledoux are equivalent on evolutionary timescales

Evan H. Anders CIERA, Northwestern University, Evanston IL 60201, USA  Kavli
Institute for Theoretical Physics, University of California, Santa Barbara, CA
93106, USA  Adam S. Jermyn Center for Computational Astrophysics, Flatiron
Institute, New York, NY 10010, USA  Kavli Institute for Theoretical Physics,
University of California, Santa Barbara, CA 93106, USA  Daniel Lecoanet CIERA,
Northwestern University, Evanston IL 60201, USA  Department of Engineering
Sciences and Applied Mathematics, Northwestern University, Evanston IL 60208,
USA  Kavli Institute for Theoretical Physics, University of California, Santa
Barbara, CA 93106, USA  Adrian E. Fraser Department of Applied Mathematics,
Baskin School of Engineering, University of California, Santa Cruz, CA 95064,
USA  Kavli Institute for Theoretical Physics, University of California, Santa
Barbara, CA 93106, USA  Imogen G. Cresswell Department Astrophysical and
Planetary Sciences & LASP, University of Colorado, Boulder, CO 80309, USA
Kavli Institute for Theoretical Physics, University of California, Santa
Barbara, CA 93106, USA  Meridith Joyce Space Telescope Science Institute, 3700
San Martin Drive, Baltimore, MD 21218, USA  Kavli Institute for Theoretical
Physics, University of California, Santa Barbara, CA 93106, USA  J. R. Fuentes
Department of Physics and McGill Space Institute, McGill University, 3600 rue
University, Montreal, QC H3A 2T8, Canada Evan H. Anders
<EMAIL_ADDRESS>
###### Abstract

Stellar evolution models calculate convective boundaries using either the
Schwarzschild or Ledoux criterion, but confusion remains regarding which
criterion to use. Here we present a 3D hydrodynamical simulation of a
convection zone and adjacent radiative zone, including both thermal and
compositional buoyancy forces. As expected, regions which are unstable
according to the Ledoux criterion are convective. Initially, the radiative
zone adjacent to the convection zone is Schwarzschild-unstable but Ledoux-
stable due to a composition gradient. Over many convective overturn timescales
the convection zone grows via entrainment. The convection zone saturates at
the size originally predicted by the Schwarzschild criterion, although in this
final state the Schwarzschild and Ledoux criteria agree. Therefore, the
Schwarzschild criterion should be used to determine the size of stellar
convection zones, except possibly during short-lived evolutionary stages in
which entrainment persists.

Stellar convective zones (301), Stellar physics (1621); Stellar evolutionary
models (2046)

††journal: ApJ

##  1 Introduction

The treatment of convective boundaries in stars is a long-standing problem in
modern astrophysics. Models and observations disagree about the sizes of
convective cores (Claret & Torres, 2018; Joyce & Chaboyer, 2018; Viani & Basu,
2020; Pedersen et al., 2021; Johnston, 2021), the depths of convective
envelopes (inferred from lithium abundances; Pinsonneault, 1997; Sestito &
Randich, 2005; Carlos et al., 2019; Dumont et al., 2021), and the sound speed
at the base of the Sun’s convection zone (see Basu, 2016, Sec. 7.2.1).
Inaccurate convective boundary specification can have astrophysical impacts by
e.g., affecting mass predictions of stellar remnants (Farmer et al., 2019;
Mehta et al., 2022) and the inferred radii of exoplanets (Basu et al., 2012;
Morrell, 2020).

In order to resolve the many uncertainties involved in treating convective
boundaries, it is first crucial to determine the boundary location. Some
stellar evolution models determine the location of the convection zone
boundary using the _Schwarzschild criterion_ , by comparing the radiative and
adiabatic temperature gradients. In other models, the convection zone boundary
is determined by using the _Ledoux criterion_ , which also accounts for
compositional stratification (Salaris & Cassisi, 2017, chapter 3, reviews
these criteria). Recent work states that these criteria _should_ agree on the
location of the convective boundary (Gabriel et al., 2014; Paxton et al.,
2018, 2019), but in practice they can disagree (see Kaiser et al., 2020,
chapter 2) which can affect asteroseismic observations (Silva Aguirre et al.,
2011). Efforts to properly choose convective boundaries locations have
produced a variety of algorithms in stellar evolution software instruments
(Paxton et al., 2018, 2019).

Multi-dimensional simulations can provide insight into the treatment of
convective boundaries. Such simulations show that a convection zone adjacent
to a Ledoux-stable region can expand by entraining material from the stable
region (Meakin & Arnett, 2007; Woodward et al., 2015; Jones et al., 2017;
Cristini et al., 2019; Fuentes & Cumming, 2020; Andrassy et al., 2020, 2021).
However, past simulations have not achieved a statistically-stationary state,
leading to uncertainty in how to include entrainment in 1D models (Staritsin,
2013; Scott et al., 2021).

In this letter, we present a 3D hydrodynamical simulation with a convection
zone that is adjacent to a Ledoux-stable but Schwarzschild-unstable region.
Convection entrains material until the adjacent region is stable by both
criteria. Our simulation demonstrates that the Ledoux criterion
_instantaneously_ describes the size of a convection zone. However, when the
Ledoux and Schwarzschild criteria disagree, the Schwarzschild criterion
correctly predicts the size at which a convection zone saturates. Therefore,
when evolutionary timescales are much larger than the convective overturn
timescale (e.g., on the main sequence; Georgy et al., 2021), the Schwarzschild
criterion properly predicts convective boundary locations. When correctly
implemented, the Ledoux criterion should return the same result (Gabriel et
al., 2014). We discuss these criteria in Sec. 2, describe our simulation in
Sec. 3, and briefly discuss the implications of our results for 1D stellar
evolution models in Sec. 4.

Figure 1:  Volume renderings of the composition $\mu$ at early (left) and late
(right) times. A stable composition gradient is denoted by the changing color
from top of the box (white) to the top of the convection zone (dark purple).
The convection zone is well-mixed, so we expand the colorbar scaling there;
black low-$\mu$ fluid is mixed into the yellow high-$\mu$ convection zone.
Orange and purple horizontal lines respectively denote the heights at which
$\mathcal{Y}_{\rm{L}}=0$ and $\mathcal{Y}_{\rm{S}}=0$. The Schwarzschild and
Ledoux criteria are equivalent in the right panel, so the orange line is not
visible. The simulation domain spans $z\in[0,3]$, but we only plot
$z\in[0,2.5]$ here. A movie version of this figure is available online in the
HTML version of the paper and in the supplementary materials (Anders et al.,
2022b); in the movie version, the initial Ledoux boundary height is denoted as
a dotted orange line.

##  2 Theory & Experiment

The Schwarzschild criterion for convective stability is

$\mathcal{Y}_{\rm{S}}\equiv\nabla_{\rm{rad}}-\nabla_{\rm{ad}}<0,$ (1)

whereas the Ledoux criterion for convective stability is

$\mathcal{Y}_{\rm{L}}\equiv\mathcal{Y}_{\rm{S}}+\frac{\chi_{\mu}}{\chi_{T}}\nabla_{\mu}<0.$
(2)

The temperature gradient $\nabla\equiv d\ln P/d\ln T$ (pressure $P$ and
temperature $T$) is $\nabla_{\rm{ad}}$ for an adiabatic stratification and
$\nabla_{\rm{rad}}$ if all the flux is carried radiatively. The Ledoux
criterion includes the effects of the composition gradient
$\nabla_{\mu}=d\ln\mu/d\ln P$ (mean molecular weight $\mu$), where
$\chi_{T}=(d\ln P/d\ln T)_{\rho,\mu}$ and $\chi_{\mu}=(d\ln
P/d\ln\mu)_{\rho,T}$ (density $\rho$).

Stellar structure software instruments assume that convective boundaries
coincide with sign changes of $\mathcal{Y}_{\rm{L}}$ or $\mathcal{Y}_{\rm{S}}$
(Paxton et al., 2018, sec. 2). The various stability regimes that can occur in
stars are described in section 3 and figure 3 of Salaris & Cassisi (2017), but
we note four important regimes here:

  1. 1.

Convection Zones (CZs): Regions with ${\mathcal{Y}_{\rm{L}}>0}$ are
convectively unstable.

  2. 2.

Radiative Zones (RZs): Regions with
$\mathcal{Y}_{\rm{L}}\leq\mathcal{Y}_{\rm{S}}<0$ are always stable to
convection. Other combinations of $\mathcal{Y}_{\rm{L}}$ and
$\mathcal{Y}_{\rm{S}}$ _may_ also be stable RZs, as detailed below in #3 and
#4.

  3. 3.

“Semiconvection” Zones (SZs): Regions with ${\mathcal{Y}_{\rm{S}}>0}$ but
$\mathcal{Y}_{\rm{L}}<0$ are stablized by a composition gradient despite an
unstable thermal stratification. These regions can be stable RZs or linearly
unstable to oscillatory double-diffusive convection (ODDC, see Garaud, 2018,
chapters 2 and 4).

  4. 4.

“Thermohaline” Zones: A stable thermal stratification can overcome an unstable
composition gradient in regions with
$\mathcal{Y}_{\rm{S}}<\mathcal{Y}_{\rm{L}}<0$. These regions can be stable RZs
or linearly unstable to thermohaline mixing (see Garaud, 2018, chapters 2 and
3).

In this letter, we study a three-layer 3D simulation of convection. The
initial structure of the simulation is an unstable CZ (bottom, #1), a
compositionally-stabilized SZ (middle, #3), and a thermally stable RZ (top,
#2). We examine how the boundary of the CZ evolves through entrainment. In
particular, we are interested in determining whether the heights at which
$\mathcal{Y}_{\rm{S}}=0$ and $\mathcal{Y}_{\rm{L}}=0$ coincide on timescales
that are long compared to the convective overturn timescale.

Our simulation uses the Boussinesq approximation, which is formally valid when
motions occur on length scales much smaller than the pressure scale height.
This approximation fully captures nonlinear advective mixing near the CZ-SZ
boundary, which is our primary focus. Our simulations use a height-dependent
$\nabla_{\rm{rad}}$ and buoyancy is determined by a combination of the
composition and the temperature stratification, so $\mathcal{Y}_{\rm{S}}$ and
$\mathcal{Y}_{\rm{L}}$ are determined independently and self-consistently. Our
simulation length scales are formally much smaller than a scale height, but a
useful heuristic is to think of our 3D convection zone depth (initially 1/3 of
the simulation domain) as being analogous to the mixing length in a 1D stellar
evolution model. For details on our model setup and Dedalus (Burns et al.,
2020) simulations, we refer the reader to appendices A and B.

While $\mu$ represents the mean molecular weight in stellar modeling (e.g.,
Eqn. 2), throughout the rest of this manuscript we will use $\mu$ to denote
the composition field in our dynamical model. In stellar modeling, the
quantity that determines convective stability (the $B$ term in e.g., Unno et
al., 1989; Paxton et al., 2013) is obtained by accounting for the variation of
pressure with composition in the full equation of state. In our simulation, we
employ an ideal equation of state in which compositional stability is
determined by the gradient of $\mu$.

Figure 2:  Horizontally-averaged profiles of the composition (top), the
discriminants $\mathcal{Y}_{\rm{S}}$ and $\mathcal{Y}_{\rm{L}}$ (middle, Eqns.
1 & 2), and important frequencies (bottom, the Brunt–Väisälä frequency
$N^{2}=-\mathcal{Y}_{\rm{L}}$ and the square convective frequency
$f_{\rm{conv}}^{2}$, see Eqn. 3). Positive and negative values are
respectively solid and dashed lines. We show the initial (left) and evolved
(right, time-averaged over 100 convective overturn times) states. There are no
motions in the initial state, so $f_{\rm{conv}}^{2}=0$ and does not appear.
The background color is orange in the CZ, green in the SZ, and purple in the
RZ. The lightly hashed background region in the evolved RZ is the mechanical
overshoot zone.

##  3 Results

In Fig. 1, we visualize the composition field in our simulation near the
initial state (left) and evolved state (right). Thick horizontal lines denote
the convective boundaries per the Ledoux (orange, $\mathcal{Y}_{\rm{L}}=0$)
and Schwarzschild (purple, $\mathcal{Y}_{\rm{S}}=0$) criteria. Initially, the
bottom third of the domain is a CZ, the middle third is an SZ, and the top
third is an RZ. Convection motions extend beyond $\mathcal{Y}_{\rm{L}}=0$ at
all times; we refer to these motions as overshoot (which is discussed in Korre
et al., 2019). Overshoot occurs because the Ledoux boundary is not the
location where convective velocity is zero, but rather the location where
buoyant acceleration changes sign due to a sign change in the entropy
gradient.

The difference between the left and right panels demonstrates that the CZ
consumes the SZ. Overshooting convective motions entrain low-composition
material into the CZ where it is homogenized. This process increases the size
of the CZ and repeats over thousands of convective overturn times until the
Ledoux and Schwarzschild criteria predict the same convective boundary. After
this entrainment phase, the convective boundary stops moving. The boundary is
stationary because the radiative flux renews the stable temperature gradient;
there is no analogous process to reinforce the composition gradient111Nuclear
timescales are generally much longer than dynamical timescales and can be
neglected as a source of composition..

In Fig. 2, we visualize vertical profiles in the initial state (left) and
evolved state (right). Shown are the composition $\mu$ (top), the
discriminants $\mathcal{Y}_{\rm{L}}$ and $\mathcal{Y}_{\rm{S}}$ (middle), and
two important frequencies (bottom): the square Brunt–Väisälä frequency $N^{2}$
and the square convective frequency,

$f_{\rm{conv}}^{2}=\frac{|\mathbf{u}|^{2}}{\ell_{\rm{conv}}^{2}},$ (3)

with $|\mathbf{u}|$ the horizontally-averaged velocity magnitude and
$\ell_{\rm{conv}}$ the depth of the Ledoux-unstable layer.

The composition is initially uniform in the CZ (${z\lesssim 1}$) and RZ
($z\gtrsim 2$), but varies linearly in the SZ ($z\in[1,2]$). We have
$\mathcal{Y}_{\rm{L}}(z\approx 1)=0$ but $\mathcal{Y}_{\rm{S}}(z\approx 2)=0$.
An unstable boundary layer at the base of the CZ drives the instability and
has negative $N^{2}$. For $z\gtrsim 1$, we have positive $N^{2}$, which is
larger in the RZ than the SZ. We found similar results in simulations where
$N^{2}$ was constant across the RZ and SZ.

In the evolved state (right panels), the composition (top) is well-mixed in
the CZ and hashed overshoot zone, but decreases rapidly above the overshoot
region. We take the height where the horizontally-averaged kinetic energy
falls below 10% of its bulk-CZ value to be the top of the overshoot zone. Rare
convective events provide turbulent diffusion above the overshoot zone and
smooth the profile’s transition from its CZ value to its RZ value. In this
evolved state, the Schwarzschild and Ledoux criteria agree upon the location
of the convective boundary (middle).

The rate at which the CZ entrains the SZ depends on the stiffness of the
radiative-convective interface,

$\mathcal{S}=\frac{N^{2}|_{\rm{RZ}}}{f_{\rm{conv}}^{2}|_{\rm{CZ}}},$ (4)

which is related to the Richardson number ${\rm{Ri}=\sqrt{\mathcal{S}}}$. The
time to entrain the SZ is roughly ${\tau_{\rm{entrain}}\sim(\delta
h/\ell_{\rm{c}})^{2}\rm{R}_{\rho}^{-1}\mathcal{S}\tau_{\rm{dyn}}}$ (per
Fuentes & Cumming, 2020, eqn. 3), where $\delta h$ is the depth of the SZ,
$\ell_{\rm{c}}$ is the characteristic convective length scale,
$\mathrm{R}_{\rho}\in[0,1]$ is the density ratio (see Garaud, 2018, eqn. 7),
and $\tau_{\rm{dyn}}$ is the dynamical timescale which in our simulation is
the convective overturn timescale. In Fig. 2, bottom right panel, we have
$f_{\rm{conv}}^{2}|_{\rm{CZ}}\approx 3\times 10^{-3}$ and
$N^{2}|_{\rm{RZ}}\approx 10^{2}$, so $\mathcal{S}\approx 3\times 10^{4}$.
Convective boundaries in stars often have $\mathcal{S}\gtrsim 10^{6}$, so our
simulation is in the same high-$\mathcal{S}$ regime as stars. The value of
$\mathrm{R}_{\rho}$ can vary greatly throughout the depth of an SZ in a star;
we use $\rm{R}_{\rho}=1/10$. The relevant evoluationary timescale during the
main sequence is the nuclear time $\tau_{\rm{nuc}}$. Since
$\tau_{\rm{nuc}}/\tau_{\rm{dyn}}\gg(\delta
h/\ell_{\rm{c}})^{2}\mathcal{S}/\mathrm{R}_{\rho}$ even for $\mathcal{S}\sim
10^{6}$, SZs should be immediately entrained by bordering CZs on the main
sequence and during other evolutionary stages in which convection reaches a
steady state. Note that while values of $\mathrm{R}_{\rho}\ll 1$ increase
$\tau_{\rm{entrain}}$, they also support efficient mixing by ODDC (see Sec.
4).

Finally, in Fig. 3 we display a Kippenhahn-like diagram of the simulation’s
evolution. This diagram demonstrates how the CZ, SZ, and RZ boundaries evolve.
The convective boundary measurements are shown as orange
($\mathcal{Y}_{\rm{L}}=0$) and purple ($\mathcal{Y}_{\rm{S}}=0$) lines. The CZ
is colored orange and fills the region below the Ledoux boundary, the RZ is
colored purple and fills the region above the Schwarzschild boundary, and the
SZ is colored green and fills the region between these boundaries. Convection
motions overshoot beyond the Ledoux boundary into a hashed overshoot zone,
which we define identically to the one displayed in Fig. 2. The top of the
overshoot zone (black line) correspond with the edge of the well-mixed region
(Fig. 2, upper right). While the Schwarzschild and Ledoux boundaries start at
different heights, 3D convective mixing causes them to converge.

We briefly note that we performed additional simulations with the same initial
stratification as in Fig. 2 (left), but with lower values of $\mathcal{S}$,
higher and lower values of $\mathrm{R}_{\rho}$, and less turbulence (lower
Reynolds number), and the evolutionary trends described here are present in
all simulations.

Figure 3:  A Kippenhahn-like diagram of the simulation evolution. The $y$-axis
is simulation height and the $x$-axis is simulation time. The orange line
denotes the Ledoux convective boundary ($\mathcal{Y}_{\rm{L}}=0$); the CZ is
below this and is colored orange. The purple line denotes the Schwarzschild
convective boundary ($\mathcal{Y}_{\rm{S}}=0$); the RZ is above this and is
colored purple. The SZ between these boundaries is colored green. The black
line denotes the top of the overshoot zone, which is hashed. The simulation
has an “entrainment phase”, in which the CZ expands, and a pure “overshoot
phase”, in which the convective boundary remains stationary.

##  4 Conclusions & Discussion

In this letter, we present a 3D simulation of a convection zone adjacent to a
compositionally stable and weakly thermally unstable region. This region is
stable according to the Ledoux criterion, but unstable according to the
Schwarzschild criterion. Overshooting convective motions entrain the entire
Schwarzschild-unstable region until the Schwarzschild and Ledoux criteria both
predict the same boundary of the convection zone.

This simulation demonstrates that the Ledoux criterion _instantaneously_
predicts the location of the convective boundary, but the Schwarzschild
criterion correctly predicts its location on evolutionary timescales (for
$t_{\rm{evol}}\gg(\delta
h/\ell_{\rm{c}})^{2}\mathrm{R}_{\rho}^{-1}\mathcal{S}t_{\rm{dyn}}$, see Sec.
3). Our 3D simulation supports the claim that logically consistent
implementations of mixing length theory (Gabriel et al., 2014; Paxton et al.,
2018, 2019) should have convective boundaries which are Schwarzschild-stable.
E.g., the MESA software instrument’s “convective pre-mixing” (CPM, Paxton et
al., 2019) is consistent with our simulation. Given our results, the
predictions made by 1D stellar evolution calculations should not depend on the
choice of stability criterion used if/when convective boundary treatments are
properly implemented and $t_{\rm{evol}}\gg(\delta
h/\ell_{\rm{c}})^{2}\mathrm{R}_{\rho}^{-1}\mathcal{S}t_{\rm{dyn}}$.

In stars, SZs should often be unstable to oscillatory double-diffusive
convection (ODDC). Mirouh et al. (2012) show that convective layers often
emerge from ODDC, and thus mix composition gradients more rapidly than
entrainment alone; ODDC is discussed thoroughly in Garaud (2018). Moore &
Garaud (2016) apply ODDC to SZs which form outside core convection zones in
main sequence stars, and their results suggest that ODDC should rapidly mix
these regions. Our simulation demonstrates that entrainment should prevent SZs
from ever forming at convective boundaries.

For stages in stellar evolution where ${t_{\rm{evol}}\sim(\delta
h/\ell_{\rm{c}})^{2}\mathrm{R}_{\rho}^{-1}\mathcal{S}t_{\rm{dyn}}}$, time-
dependent convection (TDC, Kuhfuss, 1986) implementations can be used to
improve accuracy. These implementations should include time-dependent
entrainment models to properly advance convective boundaries (e.g., Turner,
1968; Fuentes & Cumming, 2020).

Anders et al. (2022a) showed convective motions can extend significantly into
the radiative zones of stars via “penetrative convection.” In this work, we
used parameters which do not have significant penetration. This can be seen in
the right panels of Fig. 2, because the composition is well-mixed above the
convective boundary, but the thermal structure is not.

We assume that the radiative conductivity and $\nabla_{\rm{rad}}$ do not
depend on $\mu$ for simplicity. The nonlinear feedback between these effects
should be studied in future work, but we expect that our conclusions are
robust.

In summary, we find that the Ledoux criterion provides the instantaneous
location of the convective boundary, and the Schwarzschild criterion provides
the location of the convective boundary in a statistically stationary state;
in this final state, the Ledoux and Schwarzschild criteria agree.

We thank Anne Thoul, Dominic Bowman, Jared Goldberg, Tim Cunningham, Falk
Herwig, and Kyle Augustson for useful discussions which helped improve our
understanding. We thank the anonymous referee for their constructive feedback
which improved the clarity of this manuscript. EHA is funded as a CIERA
Postdoctoral fellow and would like to thank CIERA and Northwestern University.
The Flatiron Institute is supported by the Simons Foundation. DL and IGC are
supported in part by NASA HTMS grant 80NSSC20K1280. AEF acknowledges support
from NSF Grant Nos. AST-1814327 and AST-1908338. IGC acknowledges the support
of the University of Colorado’s George Ellery Hale Graduate Student
Fellowship. MJ acknowledges support from the Barry M. Lasker Data Science
Fellowship awarded by the Space Telescope Science Institute. JRF acknowledges
support from a McGill Space Institute (MSI) Fellowship. This research was
supported in part by the National Science Foundation under Grant No.
PHY-1748958, and we acknowledge the hospitality of KITP during the Probes of
Transport in Stars Program. Computations were conducted with support from the
NASA High End Computing (HEC) Program through the NASA Advanced Supercomputing
(NAS) Division at Ames Research Center on Pleiades with allocation GID s2276.

##  Appendix A Model & Initial Conditions

We study incompressible, Boussinesq convection in which we evolve both
temperature $T$ and concentration $\mu$. The nondimensional equations of
motion are

$\displaystyle\mathbf{\nabla}\mathbf{\cdot}\mathbf{u}=0$ (A1)
$\displaystyle\partial_{t}\mathbf{u}+\mathbf{u}\mathbf{\cdot}\mathbf{\nabla}\mathbf{u}+\mathbf{\nabla}\varpi=\left(T-\frac{\mu}{\mathrm{R}_{\rho}}\right)\hat{z}+\frac{\mathrm{Pr}}{\rm{Pe}}\mathbf{\nabla}^{2}\mathbf{u},$
(A2)
$\displaystyle\partial_{t}T+\mathbf{u}\mathbf{\cdot}(\mathbf{\nabla}T-\hat{z}\,\partial_{z}T_{\rm{ad}})=\mathbf{\nabla}\mathbf{\cdot}[\kappa_{T,0}\mathbf{\nabla}\overline{T}]+\frac{1}{\rm{Pe}}\mathbf{\nabla}^{2}T^{\prime},$
(A3)
$\displaystyle\partial_{t}\mu+\mathbf{u}\mathbf{\cdot}\mathbf{\nabla}\mu=\frac{\tau_{0}}{\rm{Pe}}\mathbf{\nabla}^{2}\overline{\mu}+\frac{\tau}{\rm{Pe}}\mathbf{\nabla}^{2}\mu^{\prime},$
(A4)

where $\mathbf{u}$ is velocity. Overbars denote horizontal averages and primes
denote fluctuations around that average such that $T=\overline{T}+T^{\prime}$.
The adiabatic temperature gradient is $\partial_{z}T_{\rm{ad}}$ and the
nondimensional control parameters are

$\begin{split}&\mathrm{Pe}=\frac{u_{\rm{ff}}h_{\rm{conv}}}{\kappa_{T}},\qquad\mathrm{R}_{\rho}=\frac{|\alpha|\Delta
T}{|\beta|\Delta\mu},\qquad\\\
&\mathrm{Pr}=\frac{\nu}{\kappa_{T}},\qquad\tau=\frac{\kappa_{\mu}}{\kappa_{T}},\qquad\end{split}$
(A5)

where the nondimensional freefall velocity is
$u_{\mathrm{ff}}=\sqrt{|\alpha|gh_{\rm{conv}}\Delta T}$ (with gravitational
acceleration $g$), $h_{\rm{conv}}$ is the initial depth of the convection
zone, the constant $\Delta\mu$ is the initial composition change across the
Ledoux stable region, the constant $\Delta
T=h_{\rm{conv}}(\partial_{z}T_{\rm{rad}}-\partial_{z}T_{\rm{ad}})$ is the
initial superadiabatic temperature scale of the convection zone,
$\alpha\equiv(\partial\ln\rho/\partial T)|_{\mu}$ and
$\beta\equiv(\partial\ln\rho/\partial\mu)|_{T}$ are respectively the
coefficients of expansion for $T$ and $\mu$, the viscosity is $\nu$,
$\kappa_{T}$ is the thermal diffusivity, and $\kappa_{\mu}$ is the
compositional diffusivity. In stellar structure modeling,
$\mathrm{R}_{\rho}=|N_{\rm{structure}}^{2}/N_{\rm{composition}}^{2}|$ is the
ratio of respectively the thermal and compositional components of the
Brunt–Väisälä frequency as measured in a semiconvection zone or thermohaline
zone. Eqns. A1-A4 are identical to Eqns. 2-5 in Garaud (2018), except we
modify the diffusion coefficients acting on $\overline{T}$ ($\kappa_{T,0}$)
and $\overline{\mu}$ ($\tau_{0}$). By doing this, we keep the turbulence (Pe)
uniform throughout the domain while also allowing the radiative temperature
gradient $\partial_{z}T_{\rm{rad}}=-\rm{Flux}/\kappa_{T,0}$ to vary with
height. We furthermore reduce diffusion on $\overline{\mu}$ to ensure its
evolution is due to advection.

We define the Ledoux and Schwarzschild discriminants

$\mathcal{Y}_{\rm{S}}=\left(\frac{\partial T}{\partial
z}\right)_{\rm{rad}}-\left(\frac{\partial T}{\partial
z}\right)_{\rm{ad}},\,\,\mathcal{Y}_{\rm{L}}=\mathcal{Y}_{\rm{S}}-\mathrm{R}_{\rho}^{-1}\frac{\partial\mu}{\partial
z},$ (A6)

and in this nondimensional system the square Brunt–Väisälä frequency is
$N^{2}=-\mathcal{Y}_{\rm{L}}$.

We study a three-layer model with $z\in[0,3]$,

$\displaystyle\left(\frac{\partial T}{\partial
z}\right)_{\rm{rad}}=\left(\frac{\partial T}{\partial
z}\right)_{\rm{ad}}+\begin{cases}-1&z\leq 2\\\
10\mathrm{R}_{\rho}^{-1}&z>2\end{cases},$ (A7)
$\displaystyle\frac{\partial\mu_{0}}{\partial z}=\begin{cases}0&z\leq 1\\\
-1&1<z\leq 2\\\ 0&2>z\end{cases},$ (A8)

We set $(\partial T/\partial z)_{\rm{ad}}=-1-10\mathrm{R}_{\rho}^{-1}$. The
intial temperature profile has $\partial_{z}T_{0}=\partial_{z}T_{\rm{rad}}$
everywhere except between $z=[0.1,1]$ where
$\partial_{z}T_{0}=\partial_{z}T_{\rm{ad}}$. Step functions are not well
represented in pseudospectral codes, so we use smooth heaviside functions
(Eqn. B1) to construct these piecewise functions. To obtain $T_{0}$, we
numerically integrate the smooth $\partial_{z}T_{0}$ profile with
$T_{0}(z=3)=1$. To obtain $\mu_{0}$, we numerically integrate the smooth Eqn.
A8 with $\mu_{0}(z=0)=0$.

For boundary conditions, we hold ${\partial_{z}T=\partial_{z}T_{0}}$ at $z=0$,
$T=T_{0}$ at $z=3$, and we set
${\partial_{z}\mu=\hat{z}\mathbf{\cdot}\mathbf{u}=\hat{x}\mathbf{\cdot}\partial_{z}\mathbf{u}=\hat{y}\mathbf{\cdot}\partial_{z}\mathbf{u}(z=0)=\hat{y}\mathbf{\cdot}\partial_{z}\mathbf{u}(z=3)=0}$
at $z=[0,3]$. The simulation in this work uses $\rm{Pe}=3.2\times 10^{3}$,
$\mathrm{R}_{\rho}^{-1}=10$, $\rm{Pr}=\tau=0.5$, $\tau_{0}=1.5\times 10^{-3}$,
and ${\kappa_{T,0}=\rm{Pe}^{-1}[(\partial T/\partial
z)_{\rm{rad}}|_{z=0}]/(\partial T/\partial z)_{\rm{rad}}}$ The convective
cores of main sequence stars with ${2M_{\odot}\lesssim M_{*}\lesssim
10M_{\odot}}$ have $\rm{Pe}=\mathcal{O}(10^{6})$,
$\tau\approx\rm{Pr}=\mathcal{O}(10^{-6})$, and stiffnesses of
$\mathcal{S}=\mathcal{O}(10^{6-7})$ (see Jermyn et al. 2022, “An Atlas of
Convection in Main-Sequence Stars”, in prep). Our simulation is as turbulent
as possible while also achieving the long-term entrainment of the Ledoux
boundary, and is qualitatively in the same regime as stars ($\rm{Pe}\gg 1$,
$\rm{Pr}<1$, $\mathcal{S}\gg 1$). Unfortunately, stars are both more turbulent
and have stiffer boundaries than can be simulated with current computational
resources.

##  Appendix B Simulation Details & Data Availability

We time-evolve equations A1-A4 using the Dedalus pseudospectral solver (Burns
et al., 2020, git commit 1339061) using timestepper SBDF2 (Wang & Ruuth, 2008)
and CFL safety factor 0.3. All variables are represented using a Chebyshev
series with 512 terms for $z\in[0,2.25]$, another Chebyshev series with 64
terms for $z\in[2.25,3]$, and Fourier series in the periodic $x$ and $y$
directions with 192 terms each. Our domain spans $x\in[0,L_{x}]$,
$y\in[0,L_{y}]$, and $z\in[0,L_{z}]$ with $L_{x}=L_{y}=4$ and $L_{z}=3$. To
avoid aliasing errors, we use the 3/2-dealiasing rule in all directions. To
start our simulations, we add random noise temperature perturbations with a
magnitude of $10^{-6}$ to the initial temperature field.

Spectral methods with finite coefficient expansions cannot capture true
discontinuities. To approximate discontinuous functions such as Eqns. A7 & A8,
we define a smooth Heaviside step function centered at $z=z_{0}$,

$H(z;z_{0},d_{w})=\frac{1}{2}\left(1+\mathrm{erf}\left[\frac{z-z_{0}}{d_{w}}\right]\right).$
(B1)

where erf is the error function and we set $d_{w}=0.05$.

We produced figures 2 and 3 using matplotlib (Hunter, 2007; Caswell et al.,
2021). We produced figure 1 using plotly (Inc., 2015) and matplotlib. The
Python scripts used to run the simulation and to create the figures in this
paper are publicly available in a git repository
(https://github.com/evanhanders/schwarzschild_or_ledoux); the data in the
figures is available online in a Zenodo repository (Anders et al., 2022b).

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# Note on Connes spectral distances of qubits

Bing-Sheng Lin1,2,†, Han-Liang Chen1, Tai-Hua Heng 3
1School of Mathematics, South China University of Technology,
Guangzhou 510641, China
2Laboratory of Quantum Science and Engineering,
South China University of Technology, Guangzhou 510641, China
†Email<EMAIL_ADDRESS>
3School of Physics and Optoelectronic Engineering, Anhui University,
Hefei 230601, China

###### Abstract

By virtue of the mathematical tools in noncommutative geometry, we study the
Connes spectral distances between one- and two-qubit states. We construct a
spectral triple corresponding to the $2D$ fermionic phase spaces, and
calculate the Connes spectral distance between one qubits. Based on the Connes
spectral distance, we define a coherence measure of quantum states, and
calculate the coherence of one-qubit states. We also study some simple cases
about two-qubit states, and the corresponding spectral distances satisfy the
Pythagoras theorem. We find that the Connes spectral distances are different
from quantum trace distances. The Connes spectral distance can be considered
as a significant supplement to the trace distances in quantum information
sciences. These results are significant for the study of physical relations
and geometric structures of qubits and other quantum states.

Keywords: Connes spectral distance; Qubit; Coherence; Noncommutative geometry.

##  1 Introduction

In quantum mechanics, a physical system is represented by some kind of quantum
state. In order to study properties of the physical systems and also the
relations between quantum states, or measure the distinguishability between
the states, one can define some kinds of abstract distance measures between
quantum states. For example, quantum trace distance and quantum fidelity [1].
In quantum information science, one can use quantum trace distance or quantum
fidelity to quantify the differences between quantum states.

One usually consider the quantum systems with two levels (namely, qubits) in
quantum information and quantum computation. To describe a quantum system with
two levels, one can use the Grassmann representation of Fermi operators in
fermionic phase spaces. Since quantum phase spaces are some kinds of
noncommutative spaces, one can also use the mathematical tools in
noncommutative geometry to study the geometric structures of quantum states in
phase spaces [2]. In a noncommutative space, a pure state is the analog of a
traditional point in a normal commutative space, and the Connes spectral
distance between pure states corresponds to the geodesic distance between
points [3]. The Connes spectral distances in some kinds of noncommutative
spaces have already been studied in the literatures [4, 5, 6, 7, 8, 9, 10, 11,
12, 13, 14, 16, 15]. For example, Dai _et. al._ have studied Connes’ distance
in $1D$ lattices [5]. Cagnache _et. al._ computed Connes spectral distances
between the pure states which corresponding to eigenfunctions of the quantum
harmonic oscillators in the Moyal plane[6]. Martinetti _et. al._ obtained the
spectral distance between coherent states in the so-called double Moyal plane
[7]. Scholtz and his collaborators have studied the Connes spectral distances
of harmonic oscillator states and also coherent states in Moyal plane and
fuzzy space [11, 12, 13]. In the present work, we will study the Connes
spectral distance between qubits which can be represented by fermionic Fock
states in phase spaces.

This paper is organized as follows. In Sec. 2, we consider the $2D$ fermionic
phase space and construct a corresponding spectral triple based on the
Hilbert-Schmidt operatorial formulation. In Sec. 3, we review the definition
of Connes spectral distance, and derive the explicit expressions of the
spectral distances between one qubits with respect to the corresponding Bloch
vectors. In Sec. 4, we calculate the Connes spectral distances between some
simple cases of two-qubit states. Some discussions and conclusions are given
in Sec. 5.

##  2 $2D$ fermionic phase space and spectral triple

First, let us consider the simplest $2D$ fermionic phase space
$(\theta_{1},\theta_{2})$, and the coordinate operators $\hat{\theta}_{1}$,
$\hat{\theta}_{2}$ satisfy the following anticommutation relation

$\\{\hat{\theta}_{i},\,\hat{\theta}_{j}\\}\equiv\hat{\theta}_{i}\hat{\theta}_{j}+\hat{\theta}_{j}\hat{\theta}_{i}=\delta_{ij}\hbar,\qquad
i,j=1,2.$ (1)

The Grassmann parity of a function $f$ is denoted by $\varepsilon(f)$, for
example, $\varepsilon(\hat{\theta}_{i})=1$ and
$\varepsilon(\hat{\theta}_{i}\hat{\theta}_{j})=0$. One can define the
following annihilation and creation operators,

$\hat{f}=\frac{1}{\sqrt{2\hbar}}\left(\hat{\theta}_{1}+\mathrm{i}\hat{\theta}_{2}\right),\qquad\hat{f}^{\dagger}=\frac{1}{\sqrt{2\hbar}}\left(\hat{\theta}_{1}-\mathrm{i}\hat{\theta}_{2}\right).$
(2)

These operators satisfy the commutation relations
$\\{\hat{f},\hat{f}^{{\dagger}}\\}=1$, and
$\\{\hat{f},\hat{f}\\}=\\{\hat{f}^{{\dagger}},\hat{f}^{{\dagger}}\\}=0$. Let
$|0\rangle$ be the vacuum state, there are $\hat{f}|0\rangle=0$,
$\hat{f}^{{\dagger}}|0\rangle=|1\rangle$, $\hat{f}|1\rangle=|0\rangle$,
$\hat{f}^{{\dagger}}|1\rangle=0$. One can also use the convenient matrix
representations, $|0\rangle=(1,0)^{\prime}$, $|0\rangle=(0,1)^{\prime}$, and

$\hat{f}=|0\rangle\langle 1|=\left(\begin{array}[]{cc}0&1\\\
0&0\end{array}\right),\qquad\hat{f}^{\dagger}=|1\rangle\langle
0|=\left(\begin{array}[]{cc}0&0\\\ 1&0\end{array}\right).$ (3)

The fermionic phase space $(\theta_{1},\theta_{2})$ is some type of
noncommutative space. In general, a noncommutative space corresponds to a
spectral triple $(\mathcal{A},\mathcal{H},\mathcal{D})$ [2], $\mathcal{A}$ is
an involutive algebra acting on a Hilbert space $\mathcal{H}$ through a
representation $\pi$, and the Dirac operator $\mathcal{D}$ is a self-adjoint,
densely defined operator on $\mathcal{H}$ which satisfies:

1\. $\mathcal{D}$ can be unbounded operator but $[\mathcal{D},\pi(a)]$ is
bounded;

2\. $\mathcal{D}$ has compact resolvent, for
$\lambda\in\mathds{C}/\mathds{R}$, $(\mathcal{D}-\lambda)^{-1}$ is compact
when the algebra $\mathcal{A}$ is unital or $\pi(a)(\mathcal{D}-\lambda)^{-1}$
be compact if it is non-unital.

In the present work, we will use the Hilbert-Schmidt operatorial formulation
developed in Refs. [13, 17] to construct a spectral triple corresponding to
the fermionic phase space. One can define a fermion Fock space
$F=\mathrm{span}\left\\{|0\rangle,|1\rangle\right\\}$ and a quantum Hilbert
space $Q=\mathrm{span}\left\\{|i\rangle\langle j|\right\\}$, $i,j=0,1$. In the
followings, we will also denote the elements
$\psi(\hat{\theta}_{1},\hat{\theta}_{2})$ of the quantum Hilbert space $Q$ by
$|\psi)$. A spectral triple $(\mathcal{A},\mathcal{H},\mathcal{D})$ for the
$2D$ fermionic phase space $(\theta_{1},\theta_{2})$ can be constructed as
follows,

$\mathcal{A}=Q,\qquad\mathcal{H}=F\otimes\mathds{C}^{2},$ (4)

and an element $e\in\mathcal{A}$ acts on
$\Psi=\left(|\psi_{1}\rangle,|\psi_{2}\rangle\right)^{\prime}\in\mathcal{H}$
through the diagonal representation $\pi$ as

$\pi(e)\Psi=\left(\begin{array}[]{cc}e&0\\\ 0&e\\\
\end{array}\right)\left(\begin{array}[]{c}|\psi_{1}\rangle\\\
|\psi_{2}\rangle\\\
\end{array}\right)=\left(\begin{array}[]{c}e|\psi_{1}\rangle\\\
e|\psi_{2}\rangle\\\ \end{array}\right).$ (5)

In order to construct the Dirac operator for the fermionic phase space, one
can consider the following extended auxiliary noncommutative space in which
the coordinate operators $\hat{\Theta}_{i}$ and $\hat{\Lambda}_{i}$ satisfy
the following anticommutation relations,

$\\{\hat{\Theta}_{i},\,\hat{\Theta}_{j}\\}=\\{\hat{\Lambda}_{i},\,\hat{\Lambda}_{j}\\}=\delta_{ij}\hbar,\qquad\\{\hat{\Theta}_{i},\,\hat{\Lambda}_{j}\\}=\delta_{ij}\lambda,\qquad
i,j=1,2.$ (6)

Here $\lambda$ is some real parameter. It is easy to verify that, a unitary
representation of the algebra (6) can be obtained by the following actions on
the quantum Hilbert space $Q$:

$\hat{\Theta}_{i}|\phi)=|\hat{\theta}_{i}\phi),\qquad\hat{\Lambda}_{i}|\phi)=\frac{\lambda}{\hbar}|\hat{\theta}_{i}\phi)+(-1)^{\varepsilon(\phi)}\frac{\sqrt{\lambda^{2}-\hbar^{2}}}{\hbar}|\phi\,\hat{\theta}_{i}).$
(7)

One can define the following useful operators

$\hat{B}=\hat{\Lambda}_{1}+\mathrm{i}\hat{\Lambda}_{2},\qquad\hat{B}^{\dagger}=\hat{\Lambda}_{1}-\mathrm{i}\hat{\Lambda}_{2}.$
(8)

It is easy to verify that,

$\displaystyle\hat{B}|\phi)=\lambda\sqrt{\frac{2}{\hbar}}|\hat{f}\phi)-{\rm
i}(-1)^{\varepsilon(\phi)}\sqrt{\frac{2(\hbar^{2}-\lambda^{2})}{\hbar}}|\phi\hat{f}),$
$\displaystyle\hat{B}^{\dagger}|\phi)=\lambda\sqrt{\frac{2}{\hbar}}|\hat{f}^{\dagger}\phi)-{\rm
i}(-1)^{\varepsilon(\phi)}\sqrt{\frac{2(\hbar^{2}-\lambda^{2})}{\hbar}}|\phi\hat{f}^{\dagger}).$
(9)

By virtue of the result in Ref. [20], one may express the Dirac operator
$\mathcal{D}$ as

$\mathcal{D}=\frac{1}{\lambda}\sum_{i=1,2}\sigma_{i}\hat{\Lambda}_{i},$ (10)

where $\sigma_{i}$’s are the Pauli matrices,

$\displaystyle\sigma_{1}=\left(\begin{array}[]{cc}0&1\\\ 1&0\\\
\end{array}\right),\qquad\sigma_{2}=\left(\begin{array}[]{cc}0&-{\rm i}\\\
{\rm i}&0\\\ \end{array}\right).$ (15)

So the Dirac operator (10) can be written as

$\mathcal{D}=\frac{1}{\lambda}\left(\begin{array}[]{cc}0&\hat{\Lambda}_{1}-\mathrm{i}\hat{\Lambda}_{2}\\\
\hat{\Lambda}_{1}+\mathrm{i}\hat{\Lambda}_{2}&0\\\
\end{array}\right)=\frac{1}{\lambda}\left(\begin{array}[]{cc}0&\hat{B}^{\dagger}\\\
\hat{B}&0\\\ \end{array}\right).$ (16)

Consider the graded commutator
$[\mathcal{D},\pi(a)]_{gr}\equiv\mathcal{D}\pi(a)-(-1)^{\varepsilon(a)}\pi(a)\mathcal{D}$
with $a\in\mathcal{H}$. After some straightforward calculations, one can
obtain $[\mathcal{D},\pi(a)]_{gr}$ acting on an element $\Phi\in
Q\otimes\mathds{C}^{2}$ as

$\displaystyle[\mathcal{D},\pi(a)]_{gr}\Phi$ $\displaystyle=$
$\displaystyle[\mathcal{D},\pi(a)]_{gr}\left(\begin{array}[]{c}|\phi_{1})\\\
|\phi_{2})\\\ \end{array}\right)$ (19) $\displaystyle=$
$\displaystyle\frac{1}{\lambda}\left(\begin{array}[]{cc}0&[\hat{B}^{\dagger},a]_{gr}\\\
{}[\hat{B},a]_{gr}&0\\\
\end{array}\right)\left(\begin{array}[]{c}\\!|\phi_{1})\\\ \\!|\phi_{2})\\\
\end{array}\\!\right)$ (24) $\displaystyle=$
$\displaystyle\sqrt{\frac{2}{\hbar}}\left(\begin{array}[]{cc}0&[\hat{f}^{\dagger},a]_{gr}\\\
{}[\hat{f},a]_{gr}&0\\\
\end{array}\right)\left(\begin{array}[]{c}\\!|\phi_{1})\\\ \\!|\phi_{2})\\\
\end{array}\\!\right).$ (29)

Therefore, one can identify the Dirac operator $\mathcal{D}$ for the $2D$
fermionic phase space $(\theta_{1},\theta_{2})$ as

$\mathcal{D}=\sqrt{\frac{2}{\hbar}}\left(\begin{array}[]{cccc}0&\hat{f}^{\dagger}\\\
\hat{f}&0\\\ \end{array}\right).$ (30)

##  3 Connes spectral distances between one-qubit states

Using the Dirac operator constructed above, one can calculate the Connes
spectral distances between quantum states in the fermionic phase space. For
the quantum states $\omega$ which are normal and bounded, they can be
represented by density matrices $\rho$. The action of the state $\omega$ on an
element $e\in\mathcal{A}$ can be written as

$\omega(e)=\mathrm{tr}_{F}(\rho e),$ (31)

where $\mathrm{tr}_{F}(\cdot)$ denotes the trace over $F$. Suppose the quantum
states $\omega_{1}$ and $\omega_{2}$ correspond to the density matrices
$\rho_{1}$ and $\rho_{2}$, respectively. The Connes distance between the
quantum states $\omega_{1}$ and $\omega_{2}$ are [3]

$d(\omega_{1},\omega_{2})=\sup_{e\in
B}|\mathrm{tr}_{F}(\rho_{1}e)-\mathrm{tr}_{F}(\rho_{2}e)|=\sup_{e\in
B}|\mathrm{tr}_{F}(\Delta\rho\,e)|,$ (32)

where $\Delta\rho=\rho_{1}-\rho_{2}$. The set
$B=\\{e\in\mathcal{A}:\|[\mathcal{D},\pi(e)]\|_{op}\leqslant 1\\}$, and
$\|A\|_{op}$ is the operator norm of $A$,

$\|A\|_{op}\equiv\sup_{\tiny\psi\in\mathcal{H},\|\psi\|=1}\|A\psi\|,\qquad\|A\|^{2}\equiv\mathrm{tr}_{F}(A^{{\dagger}}A).$
(33)

The inequality $\|[\mathcal{D},\pi(e)]\|_{op}\leqslant 1$ is the so-called
ball condition.

Since Hermitian elements can give the supremum in the Connes spectral distance
functions [18], one only need to consider the element $e$ being Hermitian. For
a Hermitian element $e\in\mathcal{A}$, using the Dirac operator $\mathcal{D}$
(30), we have

$\displaystyle[\mathcal{D},\pi(e)]$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2}{\hbar}}\left(\begin{array}[]{cc}0&[\hat{f}^{\dagger},e]\\\
{}[\hat{f},e]&0\\\ \end{array}\right)$ (36) $\displaystyle=$
$\displaystyle\sqrt{\frac{2}{\hbar}}\left(\begin{array}[]{cc}0&-[\hat{f},e]^{\dagger}\\\
{}[\hat{f},e]&0\\\ \end{array}\right).$ (39)

Any Hermitian element $e\in\mathcal{A}$ can be expressed as the following
matrix,

$e=\left(\begin{array}[]{cc}s&w^{*}\\\
w&t\end{array}\right)=\left(\begin{array}[]{cc}s&u-{\rm i}v\\\ u+{\rm
i}v&t\end{array}\right),$ (40)

where $w=u+{\rm i}v$, and $s,t,u,v$ are real numbers. After some
straightforward calculations, one can obtain

$[\hat{f},e]=\left(\begin{array}[]{cc}w&t-s\\\ 0&-w\end{array}\right).$ (41)

Since $\|[\mathcal{D},\pi(e)]\|_{op}$ is just the square root of the largest
eigenvalue of the matrix $[\mathcal{D},\pi(e)]^{\dagger}[\mathcal{D},\pi(e)]$,
using the ball condition and the above matrix representations (36) and (41),
one can obtain

$2|w|^{2}+(s-t)^{2}+|s-t|\sqrt{4|w|^{2}+(s-t)^{2}}\leqslant\hbar.$ (42)

So there are

$2|w|^{2}\leqslant\hbar,\qquad 2(s-t)^{2}\leqslant\hbar,$ (43)

or

$|w|\leqslant\sqrt{\frac{\hbar}{2}},\qquad|s-t|\leqslant\sqrt{\frac{\hbar}{2}}.$
(44)

In general, the density matrice $\rho$ for a qubit can be expressed as [1]

$\rho=\frac{I+\vec{r}\cdot\vec{\sigma}}{2}=\frac{1}{2}\left(\begin{array}[]{cc}1+z&x-{\rm
i}y\\\ x+{\rm i}y&1-z\end{array}\right),$ (45)

where the real vector $\vec{r}=(x,y,z)$ are the so-called Bloch vector,
$|\vec{r}|\leqslant 1$, and $\vec{\sigma}=(\sigma_{1},\sigma_{2},\sigma_{3})$,
$\sigma_{i}$ are the Pauli matrices. Consider the states $\rho_{1}$ and
$\rho_{2}$ corresponding to the Bloch vectors
$\vec{r}_{1}=(x_{1},y_{1},z_{1})$ and $\vec{r}_{2}=(x_{2},y_{2},z_{2})$,
respectively. Using the matrix representation (40), one can obtain

$\mathrm{tr}_{F}(\Delta\rho\,e)=\frac{1}{2}(s-t)\Delta z+u\Delta x+v\Delta y,$
(46)

where $\Delta x=x_{1}-x_{2}$, $\Delta y=y_{1}-y_{2}$, $\Delta z=z_{1}-z_{2}$.
So we have

$\displaystyle d(\rho_{1},\rho_{2})$ $\displaystyle=$ $\displaystyle\sup_{e\in
B}|\mathrm{tr}_{F}(\Delta\rho\,e)|=\sup_{e\in B}\left|\frac{1}{2}(s-t)\Delta
z+u\Delta x+v\Delta y\right|$ (47) $\displaystyle\leqslant$
$\displaystyle\sup_{e\in B}\left(\frac{1}{2}|(s-t)\Delta z|+|u\Delta x+v\Delta
y|\right)$ $\displaystyle\leqslant$ $\displaystyle\sup_{e\in
B}\left(\frac{1}{2}|(s-t)\Delta z|+\sqrt{u^{2}+v^{2}}\sqrt{\Delta x^{2}+\Delta
y^{2}}\right)$ $\displaystyle=$ $\displaystyle\sup_{e\in
B}\left(\frac{1}{2}|(s-t)\Delta z|+|w|\sqrt{\Delta x^{2}+\Delta
y^{2}}\right).$

In the second inequality above, we have used the Cauchy-Schwartz inequality,
$(a_{1}b_{1}+a_{2}b_{2})^{2}\leqslant(a_{1}^{2}+a_{2}^{2})(b_{1}^{2}+b_{2}^{2}),$
where the equality holds if $a_{1}b_{2}=a_{2}b_{1}$.

So for the given states $\rho_{1}$ and $\rho_{2}$, in order to attain the
supremum of $|\mathrm{tr}_{F}(\Delta\rho\,e)|$, one must choose the Hermitian
element $e$ to make $|s-t|$ and $|w|$ as large as possible. So there should be

$2|w|^{2}+(s-t)^{2}+|s-t|\sqrt{4|w|^{2}+(s-t)^{2}}=\hbar,$ (48)

and

$|s-t|=\frac{1}{\sqrt{2\hbar}}(\hbar-2|w|^{2}).$ (49)

So we have

$\displaystyle|\mathrm{tr}_{F}(\Delta\rho\,e)|$ $\displaystyle\leqslant$
$\displaystyle\frac{1}{2}|(s-t)\Delta z|+|w|\sqrt{\Delta x^{2}+\Delta y^{2}}$
(50) $\displaystyle=$ $\displaystyle\frac{1}{2}\sqrt{\frac{\hbar}{2}}|\Delta
z|-\frac{|\Delta z|}{\sqrt{2\hbar}}|w|^{2}+|w|\sqrt{\Delta x^{2}+\Delta
y^{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sqrt{\frac{\hbar}{2}}\frac{|\Delta\vec{r}|^{2}}{|\Delta
z|}-\frac{|\Delta
z|}{\sqrt{2\hbar}}\left(|w|-\sqrt{\frac{\hbar}{2}}\frac{\sqrt{\Delta
x^{2}+\Delta y^{2}}}{|\Delta z|}\right)^{2}$ $\displaystyle\leqslant$
$\displaystyle\frac{1}{2}\sqrt{\frac{\hbar}{2}}\frac{|\Delta\vec{r}|^{2}}{|\Delta
z|}.$

and $\Delta\vec{r}=\vec{r}_{1}-\vec{r}_{2}=(\Delta x,\Delta y,\Delta z)$,
$|\Delta\vec{r}|=\sqrt{\Delta x^{2}+\Delta y^{2}+\Delta z^{2}}\leqslant 2$.

From Eq. (44), the ball condition for $e\in B$ means
$|w|\leqslant\sqrt{\hbar/2}$. So when $\Delta x^{2}+\Delta
y^{2}\leqslant\Delta z^{2}$, one can choose the optimal element $e$ with

$|w|=\frac{\sqrt{\Delta x^{2}+\Delta y^{2}}}{|\Delta
z|}\sqrt{\frac{\hbar}{2}},\qquad|s-t|=\sqrt{\frac{\hbar}{2}}\frac{\Delta
z^{2}-\Delta x^{2}-\Delta y^{2}}{\Delta z^{2}}.$ (51)

In this case, there is

$d(\rho_{1},\rho_{2})=\frac{1}{2}\sqrt{\frac{\hbar}{2}}\frac{\Delta
x^{2}+\Delta y^{2}+\Delta z^{2}}{|\Delta
z|}=\frac{1}{2}\sqrt{\frac{\hbar}{2}}\frac{|\Delta\vec{r}|^{2}}{|\Delta z|}.$
(52)

When $\Delta x^{2}+\Delta y^{2}>\Delta z^{2}$, one can only choose the optimal
element $e$ with

$|w|=\sqrt{\frac{\hbar}{2}}<\frac{\sqrt{\Delta x^{2}+\Delta y^{2}}}{|\Delta
z|}\sqrt{\frac{\hbar}{2}},\qquad|s-t|=\frac{1}{\sqrt{2\hbar}}(\hbar-2|w|^{2})=0.$
(53)

In this case, there is

$d(\rho_{1},\rho_{2})=\frac{1}{2}|(s-t)\Delta z|+|w|\sqrt{\Delta x^{2}+\Delta
y^{2}}=\sqrt{\frac{\hbar}{2}}\sqrt{\Delta x^{2}+\Delta y^{2}}.$ (54)

It is convenient to use the spherical coordinates in the Bloch sphere, one can
also denote
$\Delta\vec{r}=\vec{r}_{1}-\vec{r}_{2}=r(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$,
$0\leqslant\theta\leqslant\pi$, $0\leqslant\phi<2\pi$ and $0\leqslant
r\leqslant 2$. The Connes spectral distance between the one-qubit states
$\rho_{1}$ and $\rho_{2}$ can be expressed as

$d(\rho_{1},\rho_{2})=\left\\{\begin{array}[]{ll}\sqrt{\dfrac{\hbar}{2}}\sqrt{\Delta
x^{2}+\Delta
y^{2}}=r\sin\theta\sqrt{\dfrac{\hbar}{2}},&\qquad\hbox{$\dfrac{\pi}{4}\leqslant\theta\leqslant\dfrac{3\pi}{4}$;}\\\\[10.00002pt]
\dfrac{1}{2}\sqrt{\dfrac{\hbar}{2}}\dfrac{|\Delta\vec{r}|^{2}}{|\Delta
z|}=\dfrac{r}{2|\cos\theta|}\sqrt{\dfrac{\hbar}{2}},&\qquad\hbox{others.}\end{array}\right.$
(55)

These results are similar to those in Refs. [6, 9], but the spectral triple
constructed in the present work is different from those considered in Refs.
[6, 9]. The method used in the present work is also different from those used
in the literatures.

From the result (55), one can find that these spectral distances are additive
when the corresponding points of the states in the Bloch sphere are collinear,

$d(\rho_{1},\rho_{3})=d(\rho_{1},\rho_{2})+d(\rho_{2},\rho_{3}).$ (56)

Furthermore, when the corresponding points of the states in the Bloch sphere
are on the same horizontal plane, namely $\Delta z=0$, the Connes spectral
distances between the states are proportional to the Euclidean distances in
the Bloch representation with a factor $\sqrt{\hbar/2}$. It is easy to see
that, for the diagonal states, namely $x_{i}=y_{i}=0$ in (45), the optimal
elements $e$ for the Connes spectral distances can also be diagonal, this is
similar to the result in Ref. [19].

Using the formulas (55), one can obtain the Connes spectral distances between
some one-qubit states. For example,

$\displaystyle
d(|0\rangle,|1\rangle)=d(|0\rangle,|x;+\rangle)=d(|0\rangle,|y;+\rangle)=\sqrt{\frac{\hbar}{2}},$
$\displaystyle d(|x;+\rangle,|y;+\rangle)=\sqrt{\hbar},\qquad
d(|x;+\rangle,|x;-\rangle)=\sqrt{2\hbar},$ $\displaystyle
d\left(|x;+\rangle,\frac{I}{2}\right)=d\left(|y;+\rangle,\frac{I}{2}\right)=\sqrt{\frac{\hbar}{2}},$
$\displaystyle
d\left(|0\rangle,\frac{I}{2}\right)=d\left(|1\rangle,\frac{I}{2}\right)=\frac{1}{2}\sqrt{\frac{\hbar}{2}},$
(57)

where the states $|x;\pm\rangle$ and $|y;\pm\rangle$ are

$|x;\pm\rangle=\frac{1}{\sqrt{2}}(|0\rangle\pm|1\rangle),\qquad|y;\pm\rangle=\frac{1}{\sqrt{2}}(|0\rangle\pm\mathrm{i}|1\rangle).$
(58)

These spectral distances are depicted in Figure 1.

Figure 1: Connes spectral distances between one-qubit states in the Bloch
sphere.

The Connes spectral distance between the mixed states

$\rho=p|0\rangle\langle 0|+(1-p)|1\rangle\langle
1|,\qquad\rho^{\prime}=q|0\rangle\langle 0|+(1-q)|1\rangle\langle 1|$ (59)

is

$d(\rho,\rho^{\prime})=|p-q|\sqrt{\frac{\hbar}{2}}.$ (60)

This is just proportional to the quantum trace distance between the states
$\rho$ and $\rho^{\prime}$.

The quantum trace distances between qubits are equal to half the corresponding
Euclidean distances in the Bloch representations. In general, the results of
Connes spectral distances are quite different from those about the quantum
trace distances of one-qubit states. Similar to the trace distance, one can
use the Connes spectral distance to analyse the physical properties and
relations of qubits, such as quantum discord and coherence.

As an example, let us use the Connes spectral distance to study the coherence
of qubits. Quantum coherence is an important resource in quantum information
sciences [21]. Similar to Refs. [21, 22], one can define the coherence of a
state $\rho$ as follows,

$C_{SD}(\rho)=\sqrt{\frac{2}{\hbar}}\min_{\delta\in\mathcal{I}}d(\rho,\delta),$
(61)

where $\mathcal{I}$ is the set of incoherent states. The elements in
$\mathcal{I}$ are just the diagonal states in a fixed basis. Using the results
(55), one can find that, for the one-qubit state $\rho$ (45),

$C_{SD}(\rho)=\sqrt{x^{2}+y^{2}},$ (62)

and the nearest incoherent state is just $\rho_{\textrm{diag}}$ with the Bloch
vector $\vec{r}=(0,0,z)$. This is just the same as the $l_{1}$ norm of
coherence [21] and also the trace norm of coherence defined in Ref. [22].

##  4 Connes spectral distances between two-qubit states

Next, let us consider the Connes spectral distances between two-qubit states.
Similarly, one can construct the following Fock space

$\mathcal{F}=\mathrm{span}\left\\{|m,n\rangle\equiv|m\rangle_{\\!1}\otimes|n\rangle_{\\!2},\quad
m,n=0,1\right\\},$ (63)

where $|1\rangle_{\\!i}=\hat{f}_{i}^{\dagger}|0\rangle_{\\!i}$, $i=1,2$, and
$|0\rangle_{\\!i}$ is the vacuum state. The creation and annihilation
operators $\hat{f}_{i}^{\dagger},\hat{f}_{i}$ satisfy the commutation
relations $\\{\hat{f}_{i},\hat{f}_{i}^{{\dagger}}\\}=1$,
$\\{\hat{f}_{i},\hat{f}_{i}\\}=\\{\hat{f}_{i}^{{\dagger}},\hat{f}_{i}^{{\dagger}}\\}=0$,
and
$[\hat{f}_{1},\hat{f}_{2}^{{\dagger}}]=[\hat{f}_{1}^{{\dagger}},\hat{f}_{2}]=[\hat{f}_{1},\hat{f}_{2}]=[\hat{f}_{1}^{{\dagger}},\hat{f}_{2}^{{\dagger}}]=0$.
We have the following matrix representation with respect to the basis
$\\{|00\rangle$, $|01\rangle$, $|10\rangle$, $|11\rangle\\}$,

$\hat{f}_{1}=\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&0&0&1\\\ 0&0&0&0\\\
0&0&0&0\\\
\end{array}\right),\qquad\hat{f}_{2}=\left(\begin{array}[]{cccc}0&1&0&0\\\
0&0&0&0\\\ 0&0&0&1\\\ 0&0&0&0\\\ \end{array}\right).$ (64)

The corresponding quantum Hilbert space is constructed as follow,

$\mathcal{Q}=\mathrm{span}\left\\{|m_{1},n_{1}\rangle\langle
m_{2},n_{2}|,\quad m_{1},n_{1},m_{2},n_{2}=0,1\right\\}.$ (65)

Using the methods similar to those in the $2D$ case above, one can construct a
spectral triple
$(\mathcal{A}^{\prime},\mathcal{H}^{\prime},\mathcal{D}^{\prime})$ as follows,

$\mathcal{A}^{\prime}=\mathcal{Q},\qquad\mathcal{H}^{\prime}=\mathcal{F}\otimes\mathds{C}^{4},\qquad\mathcal{D}^{\prime}=\mathrm{i}\sqrt{\frac{2}{\hbar}}\left(\begin{array}[]{cccc}0&0&\hat{f}_{2}^{\dagger}&\hat{f}_{1}^{\dagger}\\\
0&0&\hat{f}_{1}&-\hat{f}_{2}\\\ -\hat{f}_{2}&-\hat{f}_{1}^{\dagger}&0&0\\\
-\hat{f}_{1}&\hat{f}_{2}^{\dagger}&0&0\\\ \end{array}\right).$ (66)

Now let us study the Connes spectral distances between two-qubit states in
this noncommutative space. Usually, the calculations are much more cumbersome
and complicated than those in the $2D$ case. For simplicity, here we only
study the spectral distances between the states $|ij\rangle$, $i,j=0,1$.

First, let us consider the spectral distance between $|00\rangle$ and
$|10\rangle$,

$\displaystyle d(|00\rangle,|10\rangle)$ $\displaystyle=$
$\displaystyle\sup_{e\in
B}|\mathrm{tr}_{\mathcal{F}}(\rho_{00}e)-\mathrm{tr}_{\mathcal{F}}(\rho_{10}e)|$
(67) $\displaystyle=$ $\displaystyle\sup_{e\in B}|\langle
00|e|00\rangle-\langle 10|e|10\rangle|$ $\displaystyle=$
$\displaystyle\sup_{e\in B}|\langle 00|e\hat{f}_{1}|10\rangle-\langle
00|\hat{f}_{1}e|10\rangle|$ $\displaystyle=$ $\displaystyle\sup_{e\in
B}|\langle 00|[\hat{f}_{1},e]|10\rangle|$ $\displaystyle\leqslant$
$\displaystyle\big{\|}[\hat{f}_{1},e]\big{\|}_{op}.$

Here we have used the Bessel’s inequality [13]: For any operator $A$ with the
matrix elements $A_{ij}$ in some orthonormal bases, there is

$|A_{ij}|^{2}\leqslant\sum_{i}|A_{ij}|^{2}\leqslant\|A\|_{op}^{2}.$ (68)

For a Hermitian element $e\in\mathcal{A}^{\prime}$, using the above Dirac
operator $\mathcal{D}^{\prime}$, one can calculate the commutator
$[\mathcal{D}^{\prime},\pi(e)]$,

$[\mathcal{D}^{\prime},\pi(e)]=\mathrm{i}\sqrt{\frac{2}{\hbar}}\left(\begin{array}[]{cc}0&M_{1}\\\
M_{1}^{\dagger}&0\\\ \end{array}\right),$ (69)

where

$M_{1}=\left(\begin{array}[]{cc}-[\hat{f}_{2},e]^{\dagger}&-[\hat{f}_{1},e]^{\dagger}\\\
{}[\hat{f}_{1},e]&-[\hat{f}_{2},e]\\\ \end{array}\right).$ (70)

Since $\|M_{1}^{\dagger}M_{1}\|_{op}=\|M_{1}M_{1}^{\dagger}\|_{op}$, for any
Hermitian element $e$, using the ball condition, one can obtain the following
inequality,

$\big{\|}[\mathcal{D}^{\prime},\pi(e)]\big{\|}_{op}^{2}=\left\|[\mathcal{D}^{\prime},\pi(e)]^{\dagger}[\mathcal{D}^{\prime},\pi(e)]\right\|_{op}=\frac{2}{\hbar}\|M_{1}M_{1}^{\dagger}\|_{op}\leqslant
1.$ (71)

From the above expression (70), we have

$M_{1}M_{1}^{\dagger}=\left(\begin{array}[]{cc}[\hat{f}_{1},e]^{\dagger}[\hat{f}_{1},e]+[\hat{f}_{2},e]^{\dagger}[\hat{f}_{2},e]&[\hat{f}_{1},e]^{\dagger}[\hat{f}_{2},e]^{\dagger}-[\hat{f}_{2},e]^{\dagger}[\hat{f}_{1},e]^{\dagger}\\\
{}-[\hat{f}_{1},e][\hat{f}_{2},e]+[\hat{f}_{2},e][\hat{f}_{1},e]&[\hat{f}_{1},e][\hat{f}_{1},e]^{\dagger}+[\hat{f}_{2},e][\hat{f}_{2},e]^{\dagger}\\\
\end{array}\right).$ (72)

By virtue of the Bessel’s inequality (68), one can obtain

$\sup_{\phi\in\mathcal{F},\langle\phi|\phi\rangle=1}\\!\\!\\!\\!\\!\\!\langle\phi|[\hat{f}_{1},e]^{\dagger}[\hat{f}_{1},e]+[\hat{f}_{2},e]^{\dagger}[\hat{f}_{2},e]|\phi\rangle\leqslant\|M_{1}M_{1}^{\dagger}\|_{op}\leqslant\frac{\hbar}{2}.$
(73)

Since
$\langle\phi|[\hat{f}_{1},e]^{\dagger}[\hat{f}_{1},e]|\phi\rangle\geqslant 0$,
$\langle\phi|[\hat{f}_{2},e]^{\dagger}[\hat{f}_{2},e]|\phi\rangle\geqslant 0$,
we also have

$\sup_{\phi\in\mathcal{F},\langle\phi|\phi\rangle=1}\\!\\!\\!\\!\\!\\!\langle\phi|[\hat{f}_{1},e]^{\dagger}[\hat{f}_{1},e]|\phi\rangle\leqslant\frac{\hbar}{2},\qquad\sup_{\phi\in\mathcal{F},\langle\phi|\phi\rangle=1}\\!\\!\\!\\!\\!\\!\langle\phi|[\hat{f}_{2},e]^{\dagger}[\hat{f}_{2},e]|\phi\rangle\leqslant\frac{\hbar}{2},$
(74)

or

$\big{\|}[\hat{f}_{1},e]\big{\|}_{op}^{2}\leqslant\frac{\hbar}{2},\qquad\big{\|}[\hat{f}_{2},e]\big{\|}_{op}^{2}\leqslant\frac{\hbar}{2}.$
(75)

Similarly, we also have the following inequalities,

$\displaystyle\sup_{\phi\in\mathcal{F},\langle\phi|\phi\rangle=1}\\!\\!\\!\\!\\!\\!\langle\phi|[\hat{f}_{1},e][\hat{f}_{1},e]^{\dagger}+[\hat{f}_{2},e][\hat{f}_{2},e]^{\dagger}|\phi\rangle\leqslant\frac{\hbar}{2},$
$\displaystyle\sup_{\phi\in\mathcal{F},\langle\phi|\phi\rangle=1}\\!\\!\\!\\!\\!\\!\langle\phi|[\hat{f}_{1},e][\hat{f}_{1},e]^{\dagger}+[\hat{f}_{2},e]^{\dagger}[\hat{f}_{2},e]|\phi\rangle\leqslant\frac{\hbar}{2},$
$\displaystyle\sup_{\phi\in\mathcal{F},\langle\phi|\phi\rangle=1}\\!\\!\\!\\!\\!\\!\langle\phi|[\hat{f}_{1},e]^{\dagger}[\hat{f}_{1},e]+[\hat{f}_{2},e][\hat{f}_{2},e]^{\dagger}|\phi\rangle\leqslant\frac{\hbar}{2}.$
(76)

Combine the results (67) and (75), there must be

$d(|00\rangle,|10\rangle)\leqslant\sqrt{\frac{\hbar}{2}}.$ (77)

Now we will find some optimal elements which can saturate the above
inequality. Similar to the result in Ref. [19], one can firstly consider the
elements $e_{o}$ being diagonal,

$e_{o}=\left(\begin{array}[]{cccc}e_{1}&0&0&0\\\ 0&e_{2}&0&0\\\ 0&0&e_{3}&0\\\
0&0&0&e_{4}\\\ \end{array}\right),$ (78)

where $e_{i}$ are real numbers.

From the relation (4), using the above matrix representations (64), (70) and
(78), after some straightforward calculations, one can obtain the following
relations,

$\displaystyle(e_{1}-e_{2})^{2}\leqslant\frac{\hbar}{2},\quad(e_{1}-e_{3})^{2}\leqslant\frac{\hbar}{2},\quad(e_{2}-e_{4})^{2}\leqslant\frac{\hbar}{2},\quad(e_{3}-e_{4})^{2}\leqslant\frac{\hbar}{2},$
$\displaystyle\Big{(}|e_{1}-e_{2}-e_{3}+e_{4}|+\sqrt{(e_{1}-e_{4})^{2}+(e_{2}-e_{3})^{2}}\Big{)}^{2}\leqslant\hbar.$
(79)

Now the Connes spectral distance between the states $|00\rangle$ and
$|10\rangle$ can be expressed as

$\displaystyle d(|00\rangle,|10\rangle)$ $\displaystyle=$
$\displaystyle\sup_{e\in B}|\langle 00|e|00\rangle-\langle 10|e|10\rangle|$
(80) $\displaystyle=$ $\displaystyle\sup_{e\in B}|e_{1}-e_{3}|.$

Using the relations (4), one can choose the following optimal element

$e_{o}^{(1)}=\left(\begin{array}[]{cccc}\sqrt{\frac{\hbar}{2}}&0&0&0\\\
0&\sqrt{\frac{\hbar}{2}}&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\
\end{array}\right)=\left(\begin{array}[]{cc}\sqrt{\frac{\hbar}{2}}&0\\\
0&0\end{array}\right)\otimes I_{2},$ (81)

and then obtain

$d(|00\rangle,|10\rangle)=|\langle 00|e_{o}^{(1)}|00\rangle-\langle
10|e_{o}^{(1)}|10\rangle|=\sqrt{\frac{\hbar}{2}}.$ (82)

Similarly, there is

$d(|01\rangle,|11\rangle)=\sqrt{\frac{\hbar}{2}},$ (83)

and $e_{o}^{(1)}$ can still be the corresponding optimal element.

Using the same method, one can also obtain

$d(|00\rangle,|01\rangle)=d(|10\rangle,|11\rangle)=\sqrt{\frac{\hbar}{2}},$
(84)

and the corresponding optimal element can be chosen as

$e_{o}^{(2)}=\left(\begin{array}[]{cccc}\sqrt{\frac{\hbar}{2}}&0&0&0\\\
0&0&0&0\\\ 0&0&\sqrt{\frac{\hbar}{2}}&0\\\ 0&0&0&0\\\
\end{array}\right)=I_{1}\otimes\left(\begin{array}[]{cc}\sqrt{\frac{\hbar}{2}}&0\\\
0&0\end{array}\right).$ (85)

It is easy to verify that, there are

$[\hat{f}_{1},e_{o}^{(2)}]=0,\qquad[\hat{f}_{2},e_{o}^{(1)}]=0.$ (86)

Next, let us consider the spectral distance between $|00\rangle$ and
$|11\rangle$,

$\displaystyle d(|00\rangle,|11\rangle)$ $\displaystyle=$
$\displaystyle\sup_{e\in
B}|\mathrm{tr}_{\mathcal{F}}(\rho_{00}e)-\mathrm{tr}_{\mathcal{F}}(\rho_{11}e)|$
(87) $\displaystyle=$ $\displaystyle\sup_{e\in B}|\langle
00|e|00\rangle-\langle 11|e|11\rangle|$ $\displaystyle=$
$\displaystyle\sup_{e\in B}|\langle 00|e|00\rangle-\langle
10|e|10\rangle+\langle 10|e|10\rangle-\langle 11|e|11\rangle|$
$\displaystyle=$ $\displaystyle\sup_{e\in B}|\langle
00|e\hat{f}_{1}|10\rangle-\langle 00|\hat{f}_{1}e|10\rangle+\langle
10|e\hat{f}_{2}|11\rangle-\langle 10|\hat{f}_{2}e|11\rangle|$ $\displaystyle=$
$\displaystyle\sup_{e\in B}|\langle 00|[\hat{f}_{1},e]|10\rangle+\langle
10|[\hat{f}_{2},e]|11\rangle|$ $\displaystyle\leqslant$
$\displaystyle\sup_{e\in B}\sqrt{2}\sqrt{|\langle
00|[\hat{f}_{1},e]|10\rangle|^{2}+|\langle
10|[\hat{f}_{2},e]|11\rangle|^{2}}.$

In the inequality above, we have also used the Cauchy-Schwartz inequality.

Since for any states $|\phi\rangle,|\varphi\rangle$, there is
$|\langle\phi|[\hat{f}_{i},e]|\varphi\rangle|^{2}\geqslant 0$. It is easy to
see that,

$\displaystyle|\langle 00|[\hat{f}_{1},e]|10\rangle|^{2}$ (88)
$\displaystyle\leqslant$ $\displaystyle|\langle
00|[\hat{f}_{1},e]|10\rangle|^{2}+|\langle
01|[\hat{f}_{1},e]|10\rangle|^{2}+|\langle
10|[\hat{f}_{1},e]|10\rangle|^{2}+|\langle 11|[\hat{f}_{1},e]|10\rangle|^{2}$
$\displaystyle=$ $\displaystyle\langle
10|[\hat{f}_{1},e]^{\dagger}[\hat{f}_{1},e]|10\rangle.$

Here we have used the resolution of the identity,

$\sum_{i,j=0,1}|ij\rangle\langle ij|=\mathds{I}.$ (89)

Similarly, there is

$|\langle 10|[\hat{f}_{2},e]|11\rangle|^{2}=|\langle
11|[\hat{f}_{2},e]^{\dagger}|10\rangle|^{2}\leqslant\langle
10|[\hat{f}_{2},e][\hat{f}_{2},e]^{\dagger}|10\rangle.$ (90)

So by virtue of the inequalities (4), one can obtain

$\displaystyle|\langle 00|[\hat{f}_{1},e]|10\rangle|^{2}+|\langle
10|[\hat{f}_{2},e]|11\rangle|^{2}$ (91) $\displaystyle\leqslant$
$\displaystyle\langle
10|[\hat{f}_{1},e]^{\dagger}[\hat{f}_{1},e]|10\rangle+\langle
10|[\hat{f}_{2},e][\hat{f}_{2},e]^{\dagger}|10\rangle$ $\displaystyle=$
$\displaystyle\langle
10|\big{(}[\hat{f}_{1},e]^{\dagger}[\hat{f}_{1},e]+[\hat{f}_{2},e][\hat{f}_{2},e]^{\dagger}\big{)}|10\rangle$
$\displaystyle\leqslant$
$\displaystyle\\!\\!\\!\\!\sup_{\phi\in\mathcal{F},\langle\phi|\phi\rangle=1}\\!\\!\\!\\!\\!\\!\langle\phi|[\hat{f}_{1},e]^{\dagger}[\hat{f}_{1},e]+[\hat{f}_{2},e][\hat{f}_{2},e]^{\dagger}|\phi\rangle\leqslant\frac{\hbar}{2}.$

Combine the results (87) and (91), we have

$d(|00\rangle,|11\rangle)\leqslant\sqrt{\hbar}.$ (92)

For example, one can choose the following optimal element

$e_{o}=\left(\begin{array}[]{cccc}\sqrt{\hbar}&0&0&0\\\
0&\frac{\sqrt{\hbar}}{2}&0&0\\\ 0&0&\frac{\sqrt{\hbar}}{2}&0\\\ 0&0&0&0\\\
\end{array}\right)=\frac{1}{\sqrt{2}}\big{(}e_{o}^{(1)}+e_{o}^{(2)}\big{)},$
(93)

and then obtain

$d(|00\rangle,|11\rangle)=|\langle 00|e_{o}|00\rangle-\langle
11|e_{o}|11\rangle|=\sqrt{\hbar}.$ (94)

Similarly, there is

$d(|01\rangle,|10\rangle)=\sqrt{\hbar},$ (95)

and the corresponding optimal element can be chosen as

$e_{o}=\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&\frac{\sqrt{\hbar}}{2}&0&0\\\
0&0&-\frac{\sqrt{\hbar}}{2}&0\\\ 0&0&0&0\\\
\end{array}\right)=\frac{1}{\sqrt{2}}\big{(}e_{o}^{(1)}-e_{o}^{(2)}\big{)}.$
(96)

Figure 2: Connes spectral distances between two-qubit states $|ij\rangle$.

These distances are depicted in Figure 2. From Figure 2, one can see that
these spectral distances satisfy the Pythagoras theorem. This is similar to
the result in Ref. [9].

As a comparison, let us calculate the quantum trace distances $d_{T}$ between
these states,

$d_{T}(\rho_{1},\rho_{2})=\frac{1}{2}\mathrm{tr}|\rho_{1}-\rho_{2}|,$ (97)

where $|A|:=\sqrt{A^{\dagger}A}$. After some straightforward calculations, one
can obtain

$d_{T}(|ij\rangle,|kl\rangle)=1,\qquad i\neq k~{}~{}\mathrm{and/or}~{}~{}j\neq
l.$ (98)

We find that the Connes spectral distances between these two-qubit states are
quite different from their trace distances. So one can use Connes spectral
distances to measure the relations between the qubits. In some special cases,
it should give some new results. In this sense, the Connes spectral distance
can be consider as a useful supplement to the trace distances in quantum
information sciences.

Similarly, one can using the above methods to study Connes spectral distances
between $n$-qubit states in higher-dimensional noncommutative spaces.

##  5 Discussions and conclusions

In this paper, we study the Connes spectral distances between some one- and
two-qubit states which can be represented by some fermionic Fock states. We
construct a spectral triple corresponding to the $2D$ fermionic phase spaces,
and calculate the Connes spectral distance between fermionic Fock states. We
also study some simple cases about two-qubit states. In the simple cases, the
spectral distances satisfy the Pythagoras theorem. These results are
significant for the study of physical relations and geometric structures of
qubits and other quantum states.

From the above results of the Connes spectral distances between qubits, one
can find that the Connes spectral distances are quite different from quantum
trace distances. One can use Connes spectral distances to measure the
relations between the qubits. For example, one can use these distances to
study the discord and coherence of qubits. In some special cases, it should
give some new results. So we believe that the Connes spectral distance can be
used as a significant supplement to the trace distances in quantum information
sciences. As an example, we use the Connes spectral distance to calculate the
coherence measure of one-qubit states. We find that the Connes spectral
distance can obtain the same result as the trace distance.

Furthermore, one can also study the spectral distances between other kinds of
pure states and mixed states in higher-dimensional noncommutative spaces. But
the calculations will be much more cumbersome and complicated. Our method used
in the present work is different from those used in the literatures. We hope
that our methods and results can help the studies of mathematical structures
of noncommutative spaces and also physical properties of quantum systems.

## Acknowledgements

This work is partly supported by Key Research and Development Project of
Guangdong Province (Grant No. 2020B0303300001), the National Natural Science
Foundation of China (Grant No. 11911530750), the Guangdong Basic and Applied
Basic Research Foundation (Grant No. 2019A1515011703) and the Natural Science
Foundation of Anhui Province (Grant No. 1908085MA16).

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