text:
\section{Introduction}

There are several types of uncertainty relations in quantum mechanics. To start with, if two non-commuting observables are each measured separately, `` sharply"  (each observable measured on an ensemble of identically prepared quantum systems), then the product of their variances is bounded from below by uncertainty relations~\cite{Heisenberg1927anschaulichen,robertson1929uncertainty,EScrhodinger1930About}.
In addition, measurements generally disturb a measured quantum state. This leads to further limitations on how well two observables can be measured {\em jointly} on the {\em same} quantum system. Different criteria for exactly what is to be optimised lead to different uncertainty or tradeoff relations for joint measurements, see e.g.~\cite{Heisenberg1927anschaulichen, arthurskelly1965, arthursgoodman1988, stenholm1992simultaneous, hall2004prior, ozawa2003universally, busch2013proof, Busch:1986kxba, Busch:2001ue, busch2014Heisenberg}.

Uncertainty relations apply to measurements of any non-commuting observables, such as position and momentum, and spin-1/2 (qubit) observables. Aside from their fundamental interest, uncertainty relations are relevant for quantum technology, including for quantum state estimation and quantum metrology. For example, they limit how much we can learn about different properties of quantum systems, and are related to why one can bound the information held by an eavesdropper in quantum key distribution.

In this paper, we present the realisation of a tradeoff relation for joint measurements of a spin-1/2 system, given in~\cite{Busch:1986kxba,Busch:2001ue}. Our realization uses the polarization of heralded single photons.
Several experimental tests of different kinds of uncertainty relations for joint measurements have been reported, see for example~\cite{Erhart2012,Weston2013, Sulyok2013, rozema2012violation,Ringbauer:2014gk,Kaneda:2014emba, xiong2017optimal}.
Many of these realisations have used weak measurements, even if any optimal quantum measurement will necessarily be described by a specific generalised quantum measurement (POM or POVM) which always can be realized in a single shot, with no need to resort to the framework of weak measurements or postselection.  Joint measurements can also be accomplished through quantum cloning~\cite{Brougham:2006kz,thekkadath2017determining}. This usually requires   entangling operations, thereby imposing practical limitations, e.g., for  photonic quantum technologies where deterministic entangling gates are lacking.

One might also expect that in order to realise a joint measurement of two non-commuting observables, it would be necessary to couple the quantum system to be measured to an ancillary system.
For two qubit observables, however, it turns out that this is not necessary, and that an optimal measurement can be implemented by probabilistically selecting to perform one or the other of two projective measurements~\cite{Andersson:2005kvbacadaea}. Such a setup was also suggested for measurement along two orthogonal spin directions in \cite{busch1987some,Barnett1997}. This leads to the simplest possible realisation of an optimal joint measurement, requiring no entangling operations, and is therefore the technique we employ here.

An early example of a tradeoff relation for joint measurements was given in~\cite{Busch:1986kxba,Busch:2001ue}. This relation holds for measurements on spin-1/2 systems. A related relation~\cite{busch2014Heisenberg} has been experimentally realized on a single trapped ion~\cite{xiong2017optimal}. Here, we aim to test the original relation given in~\cite{Busch:1986kxba,Busch:2001ue} using single photons.

\section{Theoretical framework}
{\bf The sharpness tradeoff relation.}
In~\cite{Busch:1986kxba,Busch:2001ue}, the joint measurement is assumed to have marginal measurement operators of the form
\begin{equation}
\label{eq:margoper}
\Pi^a_\pm
{1\over 2}(\mathbf{\hat 1}\pm \alpha {\bf a}\cdot\hat{\sigma}), ~~~
\Pi^b_\pm
{1\over 2}(\mathbf{\hat 1}\pm \beta {\bf b}\cdot\hat{\sigma}),
\end{equation}
for the jointly measured spin-1/2 observables $\hat A = {\bf a}\cdot\hat\sigma $ and $\hat B = {\bf b}\cdot\hat\sigma$, where $\bf a$ and $\bf b$ are Bloch vectors, each corresponding to a qubit (e.g., polarization) state.
It always holds that $0\le \alpha, \beta\le 1$. If $\alpha=1$, then
$\Pi^a_\pm $ correspond to a projective measurement of \mbox{${\bf a}\cdot\hat\sigma$}, while $\alpha=0$ corresponds to a random guess, and similarly for $\Pi^b_\pm$. The coefficients $\alpha$ and $\beta$ can be referred to as the {\em sharpnesses} of the measurements of ${\bf a}\cdot\hat\sigma$ and ${\bf b}\cdot\hat\sigma$, and the closer they are to 1, the sharper the measurements.

The tradeoff relation for $\alpha$ and $\beta$ given in \cite{Busch:1986kxba,Busch:2001ue} is
\begin{equation}
\label{alphabound}
 |\alpha {\bf a} + \beta {\bf b}| + |\alpha {\bf a} - \beta {\bf b}|\leq 2,
\end{equation}
which can be rewritten as~\cite{Andersson:2005kvbacadaea}
\begin{equation}
\label{eq:alphaunc}
\Delta_\alpha^2\Delta_\beta^2 \equiv {(1-\alpha^2)(1-\beta^2)\over {\alpha^2\beta^2}}\geq\sin^2(2\theta),
\end{equation}
where $2 \theta$ is the angle between $\bf a$ and $\bf b$, and $\theta$ would be the angle between the equivalent polarization- or qubit-state vectors.
The bound in \eqref{alphabound} and \eqref{eq:alphaunc} is tight, in the sense that a joint quantum measurement with marginal measurement operators given by \eqref{eq:margoper}, for any $\alpha$ and $\beta$ saturating the bound, can always be realised.

Note that the bound
does not depend on the measured state, nor on what values we assign to the measurement outcomes. In this sense, the bound in \eqref{alphabound} and \eqref{eq:alphaunc} can be said to be more ``fundamental" than relations which depend on what values are assigned for measurement outcomes, which is the case for typical error-disturbance relations. In return, we assume
that ${\bf a}\cdot\hat\sigma$ and ${\bf b}\cdot\hat\sigma$ are jointly measured using measurement operators of the form in \eqref{eq:margoper}. More generally, however, measurement operators for a joint measurement of ${\bf a}\cdot\sigma$ and ${\bf b}\cdot\sigma$ do not have to be of the form in \eqref{eq:margoper}~\cite{hall2004prior, busch2014Heisenberg}, in which case the bound in \eqref{alphabound} and \eqref{eq:alphaunc} also retains its relevance. In fact, any dichotomic spin-1/2 observable will have measurement operators of the form
\begin{equation}
\label{eq:margopergen}
\Pi_\pm={\gamma_\pm\over 2}\mathbf{\hat 1}\pm{{\gamma_k\over 2} \bf k}\cdot\hat{\sigma},
\end{equation}
where $\gamma_+, \gamma_- \ge \gamma_k$ (since measurement operators must be positive) and $\gamma_+ + \gamma_- = 1$ (since $\Pi_++\Pi_-=\hat {\mathbf 1}$ must hold). Thus, the marginal measurement operators of a joint measurement must more generally be of this form. It turns out that if we choose $\gamma_+=\gamma_- = 1/2$, then the measurement can be made sharper, in the sense that $\gamma_k$ in the joint measurement is as large as possible, while keeping the direction $\bf k$ the same. Therefore, no matter what the measurement results are used to estimate, measurement operators of the form in \eqref{eq:margoper} can be said to be optimal. In this sense, \eqref{alphabound} and \eqref{eq:alphaunc} retain their relevance even more generally.
For example, in~\cite{busch2014Heisenberg}, one essentially ends up with measurement operators of the form \eqref{eq:margoper}, but for two ``new" spin directions other than $\bf a$ and $\bf b$, and hence one obtains a relation of the form in \eqref{alphabound} and \eqref{eq:alphaunc}, just for these two ``new" spin directions.

     \begin{figure}[t!]
\centering{\includegraphics[width=0.7\linewidth]{Fig1_Bloch_ExpSetupv3a2}}
\caption{{\bf Joint measurement of incompatible observables  ${\hat{A}={\bf a} \cdot\hat\sigma}$ and ${\hat{B}={\bf b}\cdot\hat\sigma}$}. The  joint measurement is implemented by doing a projective measurement \emph{either} of ${\bf c}\cdot\hat\sigma$ {\em or} of ${\bf d}\cdot\hat\sigma$, with probabilities $p$ and $1-p$ respectively, where ${\bf c}$ and ${\bf d}$ will lie in the plane defined by $\bf a$ and $\bf b$. The measurements correspond to projective measurements of photon polarization in appropriate bases. Here ${\bf a}, {\bf b}, {\bf c}, {\bf d}$ are Bloch vectors, denoted in the figure using ket representations of the corrrepsonding polarization states $|{ a}\rangle$, $|{ b}\rangle$,  $|{ c}\rangle$, $|{ d}\rangle$; each Bloch vector of unit length corresponds to a pure state.
   } \label{fig1:exptsetupa} \end{figure}

We can also connect \eqref{eq:alphaunc} with uncertainty relations. The total uncertainties in the joint measurement, denoted $\Delta A_j$ and $\Delta B_j$, arise from two sources: the ``intrinsic uncertainties" $\Delta A$ and $\Delta B$ in the quantum observables when they are measured sharply (measured separately, not jointly) on some quantum state, and ``extra" uncertainty coming the fact that they are measured jointly.
If we assume that the measurement results both for the sharp and the joint measurements are said to be  $\pm 1$, then the variance in the joint measurement of ${\bf a}\cdot\hat\sigma$, scaled with $\alpha^{-2}$, can be written
\begin{equation}
\Delta^2 A_j /\alpha^2= (1-\alpha^2\langle\hat{A}\rangle^2)/\alpha^2 =
(1-\alpha^2)/\alpha^2 + 1-\langle\hat{A}\rangle^2,
\end{equation}
and similarly for $\Delta^2 B_j$\footnote{Equivalently, we can assume that the measurement results are $\pm 1/\alpha$}.
Here $1-\langle{\hat{A}}\rangle^2=\Delta^2A$ is the variance of $\hat{A}$, when measured separately and sharply, and similarly for $\hat{B}$. The quantities $(1-\alpha^2)/\alpha^2\equiv \Delta_\alpha^2$ and $(1-\beta^2)/\beta^2\equiv \Delta_\beta^2$ are seen to be contributions coming from the fact that the measurement is a joint measurement. A lower bound on their product is given by (\ref{eq:alphaunc}), which can now be interpreted as an uncertainty relation giving a lower bound on the uncertainty associated purely with the fact that quantum observables $\hat{A}$ and $\hat{B}$ are measured {\it jointly}.

{\bf Optimal joint-measurement scheme.} It turns out that an optimal joint measurement along spin directions ${\bf a}$ and ${\bf b}$ can be realized by doing a projective measurement \emph{either} along $\bf{c}$ {\em or} along $\bf{d}$
 with probability $p$ or $1-p$ respectively, as illustrated in Fig.~\ref{fig1:exptsetupa}~\cite{Andersson:2005kvbacadaea}.  The results of  the joint measurement are assigned
 as follows. If measurement along $\bf{c}$ is chosen, and the outcome is  $C=+1$, then the result of the joint measurement is  $A_j=+1 $ and $B_j=+1 $. If the outcome is  $C=-1$, the result of the joint measurement is  $A_j=-1 $ and $B_j=-1 $. However, if the selected measurement is along $\bf{d}$, and the outcome is $D=+1$, then result of the joint measurement is $A_j=+1 $ and $B_j=-1$, while $D=-1$ corresponds to $A_j=-1 $ and $B_j=+1 $.

The expectation values for this joint measurement are then
\begin{align}
\overline{A_j}  = p\langle{{\bf c}\cdot\hat\sigma}\rangle +  (1-p)\langle {{\bf d}\cdot\hat{\sigma}}\rangle,\nonumber\\
\overline{B_j}  = p\langle{{\bf c}\cdot\hat{\sigma}}\rangle -  (1-p)\langle {{\bf d}\cdot\hat{\sigma}}\rangle.
\label{eq1:jmconstr}
\end{align}
On the other hand, the expectation values for a joint measurement with marginal measurement operators given by \eqref{eq:margoper} are $\overline{A_j}=\alpha\langle {\bf a}\cdot\hat\sigma\rangle$ and $\overline{B_j}=\beta\langle {\bf b}\cdot\hat\sigma\rangle$. We therefore obtain
\begin{align}
 {\bf c} =\frac{\left( \alpha  {\bf a} + \beta  {\bf b} \right)}{2p} {\rm ~~~and~~~}
 {\bf d} = \frac{\left( \alpha  {\bf a} - \beta  {\bf b} \right)}{2(1-p)}.
\label{eq1:jmconstruct}
\end{align}
Since ${\bf c},~{\bf d}$ are unit vectors, it holds that
\begin{eqnarray}
p &=&|(\alpha {\bf a} + \beta {\bf b})|/2\nonumber\\
1-p &=&|(\alpha {\bf a} - \beta {\bf b})|/2.
\label{eq:pexpr}
\end{eqnarray}
Adding these two equations, we see that this measurement satisfies equality in \eqref{alphabound}, meaning that the joint measurement realized through measuring either $\hat{C}={\bf c}\cdot\hat\sigma$ or $\hat{D}={\bf d}\cdot\hat\sigma$ with probabilities $p$ and $1-p$ is indeed optimal.

\begin{figure}[t!]
\includegraphics[width=1\linewidth]{Fig1_Setup_ExpSetupv3av3}
\caption{{ {\bf Schematic of experimental realization of a deterministic scheme for joint quantum measurements:}
Single signal photons (623~nm) heralded by idler photons (871~nm) were generated from a four-wave mixing source in a photonic-crystal fiber pumped at 726~nm. To realize the joint measurement of observables ${{\bf a}\cdot\hat{\sigma}}$ and ${\bf b}\cdot\hat{\sigma}$,
the signal photons, prepared in a well defined polarization state, are measured in either a polarization basis corresponding to a measurement of  ${\bf c}\cdot\hat\sigma$ or to that of ${\bf d}\cdot\hat\sigma$,
with probabilities $p,~1-p$ respectively.  The random selection probability corresponding to the splitting ratio of the beam splitter is $p\sim 0.7$, but could also be implemented with a toss of a unbalanced classical coin.
The source generates photon pairs cross-polarised to the pump which are filtered by both the PBS and additional wideband filters (not shown) resulting in pure horizontally polarised heralded photons entering the measurement stage.} Components: Pulsed laser (Ti-Sapph), half-wave plate (HWP), quarter-wave plate (QWP), polarizing beam splitter (PBS), non-polarizing beam splitter (BS), dichroic mirror (DM), multi-mode-fiber-coupled single-photon avalanche diode (APD). } \label{fig1:exptsetup} \end{figure}

Solving \eqref{eq:pexpr} and using equality in (\ref{eq:alphaunc}), we can express the corresponding optimal $\alpha$ and $\beta$ in terms of $\theta$ and $p$ as
\begin{align}
 \label{eq:albeSolns}
\alpha_{\rm opt}=&\frac{(2 p-1) }{\beta_{\rm opt} \cos (2 \theta )}, \text{ where} \nonumber\\
\beta_{\rm opt} =&\pm \{\sqrt{\pm[2 (p-1) p+1]^2-(1-2 p)^2 \sec ^2(2 \theta)}+\\\nonumber
 &~~~~~~~~+2 (p-1) p+1\}^{\frac{1}{2}}.
\end{align}

\section{Experiment}

The schematic of the experimental measurement setup realizing the strategy is shown in Fig.~\ref{fig1:exptsetup}.
We prepare the input state using a combination of waveplates (not shown) on the input arm before the beam splitter, and implement the random selection between measurement directions  $\bf{c}$,  $\bf{d}$ using a fixed, non-polarizing beam splitter with a splitting ratio corresponding to $p\sim0.7$.
The reason for choosing $p\sim 0.7$ is that this choice allows us to investigate a range of angles between the directions $\bf a$ and $\bf b$, by varying the directions $\bf c$ and $\bf d$. In return, the maximum possible angle between $\bf a$ and $\bf b$ we can achieve is about $2\theta = 50^\circ$.
If one would want to perform a joint measurement of maximally complementary observables, this can only be achieved with $p=1/2$. Conversely, $p=1/2$ would always result in a measurement of two maximally complementary spin-1/2 observables; by varying the directions $\bf c$ and $\bf d$, one can in that case vary the relative sharpnesses of the measurements of ${\bf a}\cdot\hat\sigma$ and ${\bf b}\cdot\hat\sigma$ in the joint measurement.

To determine the optimal measurement directions ${\bf c}$ and ${\bf d}$, we solve the equations \eqref{eq:pexpr}  for $\alpha,\beta$ satisfying equality in \eqref{eq:alphaunc} for each combination of $\bf{a},\bf{b}$. For each $\bf{a},\bf{b}$, we then use these solutions $\alpha_{\rm opt}$, $ \beta_{\rm opt}$, i.e., \eqref{eq:albeSolns}, in   \eqref{eq1:jmconstruct} to get the  $\bf{c},\bf{d}$ that are subsequently used as the measurement settings. Of course, due to experimental imperfections, the actual experimental values ${\alpha}_{\rm exp},{\beta}_{\rm exp}$ determined from the measurements of $\bf{c},\bf{d}$ chosen in this way may not necessarily saturate the bound in \eqref{eq:alphaunc}. However, we are able to saturate this bound within experimental error bars from Poissonian photon counting statistics.
Note that for this scheme it would suffice to use a classical random selection of the measurement, and this is equivalent to the toss of an unbalanced classical coin.

\begin{figure}[ht!]
\centering{\includegraphics[width=0.8\linewidth]{Fig_JM_MainResultsv6a.pdf}}
\caption{{{\bf Experimental results}: {\bf (a) Input state tomography.}
Tomography has been performed in the the ``C measurement" arm of the apparatus as well as the ``D measurement" arm resulting in state reconstructions having fidelity $\mathcal{F}=99.9993(2)\%$ confirming that the measurement bases are well calibrated to  each other.
{\bf (b) Sharpness of the joint measurement.} RHS of \eqref{eq:alphaunc} as a function of $\theta$, where $\theta = \arccos(|{\bf a}\cdot{\bf b}|)/2 $, with error bars determined only by Poisson statistics of raw count rates for three sets of experiments each corresponding to ${\bf a}$, ${\bf b}$ defining a distinct plane on the Bloch sphere (see Fig.~\ref{figS1:BlochABCDstates}).  Each data point is an average of $100$ runs, each of which involved the detection of $\sim1.5\times 10^4$ heralded single photons.
 The solid black line represents the quantum limit. Data below this line are forbidden by quantum theory.
 } }
\label{fig3:VarVSoverlap}
\end{figure}
\begin{figure}[ht!]
\centering{\includegraphics[width=1\linewidth]{Fig_JM_SideResultsv6a2}}
\caption{Experimental values of (a) $\alpha$ and (b) $\beta$ as functions of $\theta$, plotted with the corresponding theoretical values. (c) Product of experimental variances for the sharp measurements. (d) Product of experimental variances for the joint measurements and their comparison with theory. }
\label{figS4:gRowAlphasBetas}
\end{figure}

We perform three sets of experiments using pairs ${\bf a},{\bf b}$, with $\bf a$ kept constant as the ${\bf z}$ direction, and varying $\bf  b$  to traverse $\theta=1,\ldots,25$\textdegree, along a different plane on the Bloch sphere for each experiment corresponding to azimuthal angles $\phi_1=-160.7$\textdegree,
$\phi_2=-51.6$\textdegree,
$\phi_3=83.7$\textdegree,
for experiments 1, 2 and 3,  respectively.
For our experiments, we chose as input state the eigenstate of ${\bf a}\cdot \hat{\sigma}$, denoted $|a\rangle$, which coincides with the ``$|0\rangle$" state.
 We carry out the  measurements of ${\bf c}\cdot\hat\sigma$ and ${\bf d}\cdot\hat\sigma$,
 by measuring in the correpondng polarization bases   using appropriate  settings of the half-wave plates and quarter-wave plates and subsequent measurement in the $\{|H\rangle,|V\rangle\}$ polarization basis using a polarizing beam splitter and fiber-coupled single-photon detectors.
Using coincidence detection with idler photons as  heralds, we are able to register any of the  four possible outcomes for each heralded photon going through the measurement circuit (see Fig.~\ref{fig1:exptsetup}).

\begin{figure*}[ht!]
\centering{\includegraphics[width=0.8\linewidth]{ABCsAndDsOnBlochv2}}
\caption{ {\bf Examples of pairs of incompatible observables used in the experiments.} Spin directions  $\bf{a}$,$\bf b$ (states $|a\rangle$,$|b\rangle$) defining the incompatible observables, along with $\bf{c}$, $\bf d$ (states $|c\rangle$,$|d\rangle$) for implementing their joint measurements with  (a) $\theta = 13$\textdegree~in experiment 1 (b) $\theta = 25$\textdegree~in experiment 1 (c) $\theta = 13$\textdegree~in experiment 2 (d) $\theta = 25$\textdegree~in experiment 2 (e) $\theta = 13$\textdegree~in experiment 3 (f) $\theta = 25$\textdegree~in experiment 3.
For the experiments,
$\bf a$  is kept constant and $\bf b$  is varied such that $\theta \equiv \arccos(|{\bf a}\cdot{\bf b}|)/2 = 1,4,7,\ldots,25$\textdegree, for each of the three sets of experiments, corresponding to azimuthal angles $\phi_1=-160.7$\textdegree,
$\phi_2=-51.6$\textdegree,
$\phi_3=83.7$\textdegree,
for experiments 1, 2 and 3, respectively.
}
\label{figS1:BlochABCDstates}
\end{figure*}

\section{Methods}
We ensure a true single-photon implementation by exploiting a heralded source of single photons consisting of a microstructured  photonic crystal fibre (PCF) exploiting birefringent phase-matching~\cite{halder2009nonclassical,clark2011intrinsically} to produce  photon pairs via  spontaneous four-wave mixing (SFWM).  This source is pumped by a pulsed Ti-Sapphire laser with a repetition rate of 80~MHz. The fiber is highly birefringent ($\Delta n = 4 \times 10^{-4}$) with  phase-matching conditions leading to generation of signal-idler pairs with polarization orthogonal to that of the pump.
In addition to this birefringence,  the waveguide contributions to the dispersion can be used to tailor the SFWM for generation of naturally narrowband, spectrally uncorrelated photons when pumped with Ti-Sapphire laser pulses at the flat region  of the phase-matching curves ($\lambda_{\rm pump} \simeq 726$~nm) where the idler photons ($\lambda_i = 871$~nm) are group-velocity matched to the pump pulse  so that they become spectrally broad ($\Delta \lambda_i = 2.2$~nm) while the signal photons ($\lambda_s = 623$~nm) are intrinsically narrow-band ($\Delta \lambda_s = 0.3$~nm). This narrowband phase-matching results in a highly separable joint-spectral amplitude for a wide range of pump bandwidths, thereby enabling the generation of single photons of high state purity.
Although the fibre source is in a Sagnac-loop configuration allowing for generation of entangled states when the pump pulse is set to diagonal polarization and split at the polarising beam-splitter (Fig.~\ref{fig1:exptsetup}), we use horizontally polarised pump pulses so that the PCF is pumped in only one direction for use as a heralded single-photon source.

\begin{figure*}[ht!]
\includegraphics[width=1\linewidth]{Fig5_AllExpValuesv2.pdf}
\caption{ {\bf Expectation values for the individual ``sharp" and joint measurements}. For the three experiments, the plots show expectation values for sharp measurements of
(a) ${\bf a}\cdot \hat{\sigma}$ and (b) ${\bf b}\cdot \hat{\sigma}$, (c) ${\bf c}\cdot\hat\sigma$ and (d) ${\bf d}\cdot\hat\sigma$, and expectation values for the  joint measurements (e) $ \overline{A_j}$ and (f) $\overline{B_j}$.  Also plotted  for comparison are the ideal theoretical predictions, which do not include the effects of experimental imperfections.}
\label{figS2:AllExpVals}
\end{figure*}

\section{Results}
Let the heralded detector count rates corresponding to ${ C}=\pm1$ and ${ D}=\pm1$ be  $\mathcal{C}_{\pm}$ and $\mathcal{D}_{\pm}$, respectively. From these, we determine the experimental expectation values as
\begin{equation}
\langle{\hat{C}}\rangle = \frac{\mathcal{C}_{+}-\mathcal{C}_{-}}{\mathcal{C}_{+}+\mathcal{C}_{-}},~~~\langle{\hat{D}}\rangle = \frac{\mathcal{D}_{+}-\mathcal{D}_{-}}{\mathcal{D}_{+}+\mathcal{D}_{-}}.
\label{eq:Cexpvalues}
\end{equation}
Using this in \eqref{eq1:jmconstr}, and using the value of $p=0.670(1)$ obtained from total count rates in the $C$, and $D$ channels,
the experimental joint-measurement expectations values  are then  obtained directly according to \eqref{eq1:jmconstr}.

To benchmark the performance of the implemented joint measurements, we also perform separate sharp  measurements of the incompatible observables ${\bf a}\cdot\hat{\sigma}$, ${\bf b}\cdot\hat{\sigma}$.
 Again, if we denote the detector count rates corresponding to ${ A}=\pm1$ and ${B}=\pm1$ as  $\mathcal{A}_{\pm}$ and $\mathcal{B}_{\pm}$, respectively, the expectations values for the sharp measurements are
\begin{equation}
\langle{{\bf a}\cdot\hat{\sigma}}\rangle = \frac{\mathcal{A}_{+}-\mathcal{A}_{-}}{\mathcal{A}_{+}+\mathcal{A}_{-}},
~~~\langle{{\bf b}\cdot\hat{\sigma}}\rangle = \frac{\mathcal{B}_{+}-\mathcal{B}_{-}}{\mathcal{B}_{+}+\mathcal{B}_{-}}.
\label{eq:Aexpvalues}
\end{equation}
This allows us to obtain experimental values of  $\alpha$, $\beta$ which directly indicates by how much the sharpnesses are worsened solely by the fact that the measurement is joint, and to evaluate the LHS of relation \eqref{eq:alphaunc}, which we plot as a function of $\theta$ in Fig.~\ref{fig3:VarVSoverlap}(b) as our main result.

Fig.~\ref{figS4:gRowAlphasBetas} shows $\alpha,~\beta$ and the product of total variances for separate (sharp) and joint measurements.  The ideal theoretical product of total ``intrinsic" variances for sharp measurements is zero, as indicated by the solid black line in Fig.~\ref{figS4:gRowAlphasBetas}(c),  since the measured state is an eigenstate of ${\bf a}\cdot\hat\sigma$, while that for the joint measurements (determined by the incompatibility of the jointly measured observables, and parametrised by $\theta$) is plotted with the black filled circles in  Fig.~\ref{figS4:gRowAlphasBetas}(d). Fig.~\ref{figS1:BlochABCDstates} shows examples of pairs of spin directions ${\bf a},{\bf b},{\bf c},{\bf d}$ used in the sets of experiments.
Also, shown in Fig.~\ref{figS2:AllExpVals} are the expectation values for the individual ``sharp" measurements of ${\bf a}\cdot\hat{\sigma}$, ${\bf b}\cdot\hat{\sigma}$, and expectation values resulting from the implemented joint-measurement strategy.

 \section{Discussion}
The non-ideal experimental values of $\Delta^2A \Delta^2B$, rather than the joint measurement strategy, is accountable for the deviation of the $\Delta^2A_j \Delta^2B_j$ from the ideal value  because, as seen in Fig.~\ref{fig3:VarVSoverlap}, the contribution purely due to the jointedness of the measurement is at the quantum limit.

We see that, even without making any other corrections for experimental imperfections,  our results verge on the quantum mechanical limit of how much variances must increase due to performing the quantum measurements jointly. This is thanks to the simplicity of the scheme, the brightness of the heralded single-photon source which reduced the effect of Poissonian noise, and the precise calibration of the two measurement setups with each other with a fidelity of $99.9993(2)\%$.
We emphasise that
the quantity $\Delta^2_\alpha\Delta^2_\beta$ is extremely sensitive to experimental error and has no upper bound.

In conclusion,
we have demonstrated an optimal joint measurement scheme that does not use filtering or postselection, nor does it need entangling interactions with an ancilla.
A true joint measurement of two observables performed on a single qubit or spin-$\tfrac{1}{2}$ system should have four possible outcomes for each qubit measured, as is now demonstrated here.
 The non-requirement for 2-dimensional arrays of (single)-photon detectors, as often used in weak-measurement-based setups, also makes our scheme easier to implement.
 The implemented scheme can easily be applied to other qubit degrees of freedom and other two-level quantum systems, since standard projective measurements and flips of unbalanced classical ``coins'"are generally easy to implement. It would be of interest to extend this scheme to joint measurement of incompatible observables of higher-dimensional systems (qudits), or of multiple incompatible observables.
This work  demonstrates how to fully implement joint measurements of non-commuting spin-1/2 (or qubit) observables in an optimal way, and with less quantum resources than often employed previously.

{\bf Acknowledgements:} This  work  was  funded by  the  Future  Emerging  Technologies (FET)-Open  FP7-284743  project  Spin  Photon  Angular  Momentum  Transfer  for  Quantum  Enabled  Technologies  (SPANGL4Q) and the Engineering and Physical Sciences Research
Council (EPSRC) (EP/L024020/1, EP/M024156/1,   EP/N003381/1 and EP/M024458/1).


text:
\section{Introduction}
Searching for topological superconductors (TSc) and the corresponding self-dual charge neutral Majorana zero modes associated with their excitations at edges has become one of the central problem in condensed matter physics \cite{Liang-RMP-2011,Alicea-MF-2012}. Theoretical proposals and experimental realizations of TSc are mostly heterostructure combining strong spin-orbit coupled materials and conventional superconductors by proximity effect \cite{Sau-PRL-MF,Oreg-PRL-MF,Paaske-PRL-MF}. The emergence of the topological edge states in such systems can be explained in terms of the single-particle band structure without considering many-body electron correlations. Recently, the search for topological phases of matter  has focused on a more intriguing class of materials that exist in nature.  Their topological properties are driven by strong electron correlations instead of the proximity effect. Kondo effect, describing the screening of a local spin moment by conduction electrons, is a well-known strong correlation between electrons existing in heavy electron compounds. The Kondo-mediated topological phases of matter have been studied in the context of topological Kondo insulators \cite{dzero-Ann-TKI,dzero-TKI-PRB,dzero-TKI-PRL} and topological Kondo semi-metals \cite{Weyl-kondo-PNAS}, where the topological properties are driven by either the odd-parity Kondo hybridization or by the Kondo hybridization with strong spin-orbit coupling.

Spin-triplet $p$-wave superconductors are known to be the prime candidates for TSc. However, they are scarce in nature. While it is still debatable for SrRu$_2$O$_4$ \cite{Mackenzie-RMP-SrRuO,Maeno-JPSJ-SrRuO,Kallin-JPCM-2009}, more convincing evidence for $p$-wave triplet superconductivity  was observed in noncentrosymmetric superconductor BiPd from  phase-sensitive measurement \cite{CLChen-BiPd-PRL}. More recently, signatures of triplet chiral $p$-wave  superconductivity were observed in heavy-electron Kondo lattice compound UTe$_2$ at the edge of ferromagnetism, possibly marking the first example of topological superconductor induced by the strongly correlated  Kondo effect \cite{Ran-FM-UTe,Ran-UTe-Nature-ExtremeMag,Aoki-2019-JPSJ,Jiao-2020-UTe2}.

 Motivated by these discoveries, in this paper, we propose  a distinct class of triplet $p$-wave superconductors in the absence of spin-orbit coupling or proximity effect/heterostructure \cite{Kim-Kondo-Kitaev} in a two-dimensional Kondo lattice model driven by odd-parity Kondo hybridization. We start from the Anderson lattice model (ALM) with odd-parity hybridization, which occurs between $d$- and $f$-orbital electrons in various heavy-fermion compounds \cite{dzero-Ann-TKI,dzero-TKI-PRB,dzero-TKI-PRL}. Via the Schrieffer-Wolff transformation \cite{SW-transformation,hewson1997kondo}, we  derive an effective Kondo lattice model with odd-parity hybridization. Furthermore, by integrating out the conduction electron degrees of freedom, an effective ferromagnetic RKKY interaction is generated. We explore the mean-field phase diagram of this ferromagnetic Kondo-Heisenberg model.  In the fermionic mean-field approach, the ferromagnetic RKKY coupling describes the $p$-wave ($S_z=\pm 1$) $t$-RVB spin-liquid state. A time-reversal invariant topological superconducting phase is reached when the Kondo effect co-exists with the $p$-wave $t$-RVB order parameter. The topological nature of this superconducting phase is manifested by the non-trivial $Z_2$ topological  Chern number of the bulk band and by the existence of helical Majorana zero modes at the edges of a finite-sized ribbon. Our results offer a qualitative and some quantitative understanding of the spin-triplet superconductivity recently observed in UTe$_2$ (see Discussions).

\section{Model}
\subsection{Anderson lattice model with odd-parity hybridization}
We start with the odd-parity Anderson lattice model  (ALM) on a two-dimensional (2D) square lattice, which has been shown to exhibit topologically non-trivial states \cite{dzero-Ann-TKI,dzero-TKI-PRB,dzero-TKI-PRL}:
\begin{align}
H_{PAM}= H_c +H_f+ H_{cf} , \label{eq:PAM}
\end{align}
where $H_c = \sum_{\mathbold{k},\sigma = \uparrow,\downarrow} \varepsilon_\mathbold{k} c^\dagger_{\mathbold{k}\sigma}c_{\mathbold{k}\sigma}$ describes the hopping of electrons in the $d$ orbits with orbital angular momentum $l=2$ and  dispersion $\varepsilon_\mathbold{k}=-2t (\cos k_x + \cos k_y)-\mu$.   The Hamiltonian $H_f$ of the more localized electron in the $f$ orbits with orbital angular momentum $l=3$ is given by
\begin{align}
    H_f = \sum_{i,\sigma}\left[ \varepsilon_f f^\dagger_{i\sigma}f_{i\sigma}+\frac{U}{2}n_{i\sigma}^{f}n_{i,-\sigma}^{f}\right],
\end{align}
 where $\varepsilon_f$ denote the energy level of the $f$-electron, and $U$ is the repulsive on-site Coulomb potential (the Hubbard-$U$ term). Hybridization of the local and conduction electrons is described by
 \begin{align}
     H_{cf} = \sum_{\langle i,j\rangle}\sum_{ \sigma,\sigma^\prime = \uparrow\downarrow} V_{ij}^{\sigma \sigma^\prime} c^\dagger_{i\sigma}f_{j\sigma^\prime}+H.c..
 \end{align}
To conserve the parity symmetry of hybridization between electrons with their angular momentum quantum numbers differing by one, $V_{ij}^{\sigma\sigma^\prime}$ have to be odd under parity transformation. This restriction results in the hybridization having to depend on sites and spins \cite{dzero-Ann-TKI,dzero-TKI-PRB,dzero-TKI-PRL}:
 \begin{align}
 V_{ij}^{\sigma\sigma^\prime} \equiv V_{\hat{\alpha}}^{\sigma\sigma^\prime} =  iV\nu_{\hat{\alpha}}\sigma_{{\alpha}}^{\sigma\sigma^{\prime}},
 \label{eq:V-ij}
 \end{align}
 distinct from the well-known onsite and spin-conserving  Anderson hybridization.  In Eq. (\ref{eq:V-ij}), $\nu_{ij}$ satisfies $\nu_{ij} \equiv \nu_{\hat{\alpha}} = -\nu_{ji}$ with  $\hat{\alpha} \equiv i-j \in \ \hat{x}, \hat{y} \, (\alpha \in x,y)$  on a  2D square lattice, and $\sigma_{\alpha}$ denotes the Pauli matrix of the $\alpha$ component.

 \subsection{The effective odd-parity ferromagnetic Kondo lattice model}

 \begin{figure}[t]
     \centering
     \includegraphics[width = 0.45 \textwidth]{JH-2.pdf}
     \caption{The effective RKKY coupling $J_H$ (normalized with $J_K^2$) as a function of $R/a$ for different chemical potentials $\mu$. $J_H$ is computed by Eq. (\ref{eq:J_H}) with $\mathbold{R}_{ij} \parallel (1,1) $ and $a=1$ being chosen.}
     \label{fig:JH}
 \end{figure}

\added{In this paper, we focus on the competition of the Kondo and the magnetic interaction among impurities--the Doniach scenario \cite{Doniach-scenario}. We, therefore, derive the effective Kondo-Heisenberg lattice Hamiltonian from ALM in the Kondo limit  where the vacant and doubly-occupied states are projected out from the entire Hilbert space, namely $1 = \sum_\sigma  f^\dagger_{i\sigma}f_{i\sigma}$. The low-energy effective Kondo term  from the odd-parity ALM of Eq. (\ref{eq:PAM}) can be derived by applying the Schrieffer-Wolff transformation (SWT) \cite{SW-transformation,hewson1997kondo,ETH-Phd-thesis}, yielding}

  \begin{align}
      H_K =  (-J_K) & \sum_i\sum_{\sigma\sigma^{\prime}}\sum_{\sigma^{\prime\prime}\sigma^{\prime\prime\prime}}\sum_{\alpha,\alpha^\prime}\left(i\nu_{\hat{\alpha}}\sigma_{\alpha}^{\sigma\sigma^{\prime}}c_{i+\hat{\alpha},\sigma}^{\dagger}f_{i\sigma^{\prime}}\right)
      \nonumber \\
      & \times \left(i\nu_{\hat{\alpha}^{\prime}}\sigma_{\alpha^{\prime}}^{\sigma^{\prime\prime}\sigma^{\prime\prime\prime}}f_{i\sigma^{\prime\prime}}^{\dagger}c_{i-\hat{\alpha}^{\prime},\sigma^{\prime\prime\prime}}\right)
      \label{eq:HK}
  \end{align}

\added{with $J_{K}=\frac{V^{2}}{U+\varepsilon_{f}-\varepsilon_{F}}+\frac{V^{2}}{\varepsilon_{F}-\varepsilon_{f}}>0$ (see Appendix \ref{app:SWT}). The Kondo-like term of Eq. (\ref{eq:HK}) describes the screening of an impurity by its neighboring conduction electrons, distinct from the conventional (on-site) Kondo term.}  \deleted{Via using the Schrieffer-Wolff transformation (SWT) to generate the Kondo term and via applying the perturbative approach to generate the antiferromagnetic RKKY interaction,}

\replaced{Here, we go beyond the topological Kondo insulating phase by  further deriving the magnetic RKKY interaction among the local $f$-fermions.  By perturbatively expanding the Kondo term to second order  \cite{rkky-kittel,rkky-vleck,ETH-Phd-thesis}}{Applying the second-order perturbation in $J_K$}, we obtain the effective RKKY-like interaction between the local $f$ fermions $f_{i\sigma}$,
  \begin{align}
      H_J =& \sum_{ i,j }\sum_{\sigma ,\sigma^\prime} J_{ij}f_{i\sigma}^{\dagger}f_{j\sigma^\prime}^{\dagger}f_{j\sigma}f_{i\sigma^\prime}\nonumber \\
      =& \sum_{\left\langle i,j\right\rangle }J_{ij}\left(f_{i\uparrow}^{\dagger}f_{j\uparrow}^{\dagger}f_{j\uparrow}f_{i\uparrow}+f_{i\downarrow}^{\dagger}f_{j\downarrow}^{\dagger}f_{j\downarrow}f_{i\downarrow}\right)\nonumber \\
    &+ \sum_{\left\langle i,j\right\rangle }\frac{J_{ij}}{2}\left(f_{i\uparrow}^{\dagger}f_{j\downarrow}^{\dagger}+f_{i\downarrow}^{\dagger}f_{j\uparrow}^{\dagger}\right)\left(f_{j\downarrow}f_{i\uparrow}+f_{j\uparrow}f_{i\downarrow}\right) \nonumber \\
    & - \sum_{\left\langle i,j\right\rangle }\frac{J_{ij}}{2} \left(f_{i\uparrow}^{\dagger}f_{j\downarrow}^{\dagger}-f_{i\downarrow}^{\dagger}f_{j\uparrow}^{\dagger}\right)\left(f_{j\downarrow}f_{i\uparrow}-f_{j\uparrow}f_{i\downarrow}\right),
      \label{eq:HJ}
  \end{align}
 where
 \begin{align}
     J_{ij} \equiv J_H(& R)=\frac{16J_{K}^{2}}{\mathcal{N}_{s}^{2}}\sum_{\varepsilon_{\mathbold{k}}<\mu}\sum_{\varepsilon_{\mathbold{k}^{\prime\prime}}>\mu} \frac{e^{i\left(\mathbold{k}-\mathbold{k}^{\prime\prime}\right)\cdot\mathbold{R}_{ij}}}{\varepsilon_{\mathbold{k}}-\varepsilon_{\mathbold{k}^{\prime\prime}}} \nonumber \\
     &\times \left(\sin^{2}k_{x}+\sin^{2}k_{y}\right)\left(\sin^{2}k_{x}^{\prime\prime}+\sin^{2}k_{y}^{\prime\prime}\right)
     \label{eq:J_H}
 \end{align}
denotes the effective coupling of the spinons of sites $i$ and $j$ with $R \equiv |\mathbold{R}_{ij}| \equiv |\mathbold{r}_i -\mathbold{r}_j|$. \added{The $H_J$ term of Eq. (\ref{eq:HJ}) can be  re-expressed as a linear combination of a spinon pair wave function with total spin $S  = 0$ (spin-singlet) and $S = 1$ (spin-triplet). Note that  the associated effective spinon coupling of the spin-triplet channel is opposite to that of the spin-singlet}. \added{When $H_J$ is expressed in terms of fermion pair with different spins,   Eq. (\ref{eq:HJ}) is reminiscent of the conventional Heisenberg interaction $\mathbold{S}_{i}\cdot \mathbold{S}_{j} = -\frac{1}{2}\left(f_{i\uparrow}^{\dagger}f_{j\downarrow}^{\dagger}-f_{i\downarrow}^{\dagger}f_{j\uparrow}^{\dagger}\right)\left(f_{i\downarrow}f_{j\uparrow}-f_{i\uparrow}f_{j\downarrow}\right)+\frac{1}{4}n_{i}^{f}n_{j}^{f}$, except for the difference in the constant coefficients of the pair operators.} \deleted{As compared to the conventional RKKY coupling obtained from the perturbation of the Kondo term with constant Anderson hybridization, we expect that} \added{As expected, the RKKY coupling $J_{ij}$ in Eq. (\ref{eq:J_H}) shows an oscillatory behavior in  $R$, accompanied by a decrease in its magnitude with increasing  $R$}\deleted{of the coupling strength $J_{ij}$ in Eq. (\ref{eq:J_H}) tends to}, \added{similar to the behavior of the conventional RKKY coupling}. \deleted{Here,} \added{Due to the rapid attenuation of  $J_{ij}$, we only consider} \deleted{We thus only retain} \added{the dominated nearest-neighbor interaction and  assume $J_{ij}$ to be  spatially  homogeneous, i.e.} \deleted{with} $J_{ij} \to J (R = a) \equiv J_H$\deleted{denoted the coupling of the nearest-neighbor sites}. \added{Furthermore, when $R = a$, we find the effective RKKY coupling is attractive (or of the ferromagnetic type), i.e., $J_H<0$ (see Fig. \ref{fig:JH}), which energetically favors the spin-triplet pairing of spinons. On the other hand, the effective RKKY coupling in the spin-singlet channel shows repulsive interaction and  can be neglected here since it is not energetically favorable.} Lastly, on a two-dimensional lattice, the \added{triplet spin state  $|\uparrow\downarrow\rangle + | \downarrow \uparrow \rangle$ does not exist since the corresponding structure factor is proportional to $k_z$, and $k_z = 0$ is fixed here}. Therefore, based on the above arguments, only the equal-spin states,  $|\uparrow \uparrow \rangle$ and $|\downarrow \downarrow \rangle$, survive, and the $H_J$ term is reduced to
 \begin{align}
     H_{J}\approx-\left|J_{H}\right|\sum_{\left\langle i,j \right\rangle }\left(f_{i\uparrow}^{\dagger}f_{j\uparrow}^{\dagger}f_{j\uparrow}f_{i\uparrow}+f_{i\downarrow}^{\dagger}f_{j\downarrow}^{\dagger}f_{j\downarrow}f_{i\downarrow}\right).
     \label{eq:HJ-triplet}
 \end{align}

Combining $H_K$ and $H_J$ of Eqs. (\ref{eq:HK}), (\ref{eq:HJ}) and (\ref{eq:HJ-triplet}), the  effective Kondo-Heisenberg lattice model with odd-parity Kondo hybridization reads $H_{FKH}=H_{0}+H_\lambda+H_{K}+H_{J}$. Here, $H_\lambda =- \sum_{i} i\lambda_i \left[ \sum_\sigma (f^\dagger_{i\sigma}f_{i\sigma})-1 \right]$ enforces the singly occupied local $f$-spinons with $\lambda_i$ being the Lagrange multiplier. \deleted{$H_K$ and $H_J$ describes the Kondo screening and the antiferromagnetic exchange coupling of the impurity spins.} \added{The Hamiltonian $H_{FKH}$ offers a platform for discovering a distinct class of topological superconducting states induced by electron correlations  via collaboration between the ferromagnetic RKKY coupling and the Kondo effect. To facilitate our numerical calculations of the mean-field phase diagram, we treat $J_K$ and $J_H$ as independent couplings here since it is more convenient to explore the phase diagram by tuning the ratio of $J_K/J_H$ \cite{RMP-Kirchner,sm-phase-PNAS}. In experiments, varying the non-thermal parameter can be expected to follow a certain trajectory of $J_K/J_H$ in the phase diagram.}

\section{Mean-field treatment of the effective Kondo-Heisenberg-like model}

\begin{figure}[t]
    \centering
    \includegraphics[width=0.45\textwidth]{T-0-x-Dt.pdf}
    \caption{The zero-temperature mean-field solutions of $t$-RVB order parameter $\Delta_t$ (brown) and the Kondo correlation $x$ (black) as a function of $J_H$. We fix $J_K = 0.3$ and doping of the conduction band $\delta = -0.3$ ($30$ percent hole doping). Without loss of generality, \added{we set} $t = 1$. This plot reveals a (co-existing) superconducting ground state with  $x \neq 0,\,\Delta_t \neq 0$ for $0 < J_H \lesssim 2.5$ and a pure $t$-RVB phase where  $x = 0,\,\Delta_t \neq 0$ when $J_H \gtrsim 2.52$. A pure Kondo phase ($x \neq 0,\,\Delta_t = 0$) only exists at $J_H = 0$.}
    \label{GS}
\end{figure}

\added{We now employ a mean-field analysis on the above effective Kondo-Heisenberg-like Hamiltonian with an effective ferromagnetic RKKY interaction and odd-parity Kondo hybridization.}

Via performing Hubbard-Stratonovich transformation, $H_K$ and $H_J$ of Eqs. (\ref{eq:HK}) and (\ref{eq:HJ}) can be factorized as
\begin{align}
    &H_K\to  \sum_{i,\alpha}\sum_{\sigma\sigma^{\prime}} \left[\chi^\dagger_i
       \left(i\nu_{\hat{\alpha}}\sigma_{\alpha}^{\sigma\sigma^{\prime}}f_{i\sigma}^{\dagger}c_{i-\hat{\alpha},\sigma^{\prime}}\right) +H.c. \right]\nonumber \\
       & \qquad\qquad+ \sum_i \frac{|\chi_i|^2}{J_K}, \nonumber \\
& H_J \to \sum_{\left\langle i,j\right\rangle }\left[\Delta^{\uparrow}_t(i,j)f_{i\uparrow}^{\dagger}f_{j\uparrow}^{\dagger}+\Delta^{\downarrow}_t(i,j)f_{i\downarrow}^{\dagger}f_{j\downarrow}^{\dagger}+H.c.\right] \nonumber \\
& \qquad\qquad+\sum_{\langle i,j\rangle}\frac{\left|\Delta^{\uparrow}_t (i,j) \right|^{2}+\left|\Delta^\downarrow_t (i,j) \right|^{2}}{J_{H}}
\label{eq: HS}
\end{align}

where the mean-field values of the bosonic Hubbard-Stratonovich fields, $\chi_i$ and $\Delta^{\sigma}_t (i,j)$ ($\sigma = \uparrow,\downarrow$), represent the order parameters of the Kondo correlation and the $S_z = \pm 1$ spin-triplet RVB bonds between two adjacent up/down spins, respectively. \deleted{In $H_J$, we only retain the components with spin-triplet pairings. We argue that the spin-singlet channel can be omitted if the effective Heisenberg coupling $J_H$ is negative, which makes the spin-singlet pairing to be unstable (see  \textit{SI Appendix}).}

To describe the Kondo-screened  Fermi-liquid state, we allow the $\chi_i$ field to acquire uniformly Bose condensation over the real space; hence, $\chi_i$ can be expressed as $\chi_i\to x + \hat{\chi}_i$ with $x= (-J_K/\mathcal{N}_s)\sum_{i\sigma\sigma^\prime\alpha}\langle i\nu_{\hat{\alpha}}\sigma_{\alpha}^{\sigma\sigma^{\prime}}f_{i\sigma}^{\dagger}c_{i-\hat{\alpha},\sigma^{\prime}} \rangle$ being the Bose-condensed stiffness of $\chi_i$ while $\hat{\chi}_i$ represents its fluctuations. The mean-field order parameter of the $t$RVB is given by $\Delta_t^{\sigma}= (-J_H/4\mathcal{N}_s)\sum_{\langle i,j\rangle}\langle f_{j\sigma}f_{i\sigma}\rangle$. \deleted{To acquire the mean-field model of the Hamiltonian,} Since the ferromagnetic coupling is expected to favor spin-triplet $p$-wave pairing similar to superfluid helium-3 \cite{Mineev-SC}, we restrict ourselves to the $p$-wave pairing, i.e., $\Delta_t^\sigma (i,j)$ here is taken the $p$-wave form, see Eqs. (\ref{eq:Delta-up}) and (\ref{eq:Delta-down}) below.  We further fix the Lagrange multiplier at the mean-field level via $i\lambda_i\to \lambda$ and neglect the fluctuations of $\lambda_i$, $\chi_i$, and $\Delta_{t}^{\sigma}$, leading to the following mean-field Kondo-Heisenberg-like Hamiltonian:

 \begin{align}
 H_{MF}&  = \sum_{\mathbold{k},\sigma}\varepsilon_{\mathbold{k}}c_{\mathbold{k}\sigma}^{\dagger}c_{\mathbold{k}\sigma} + \sum_{\mathbold{k}\sigma}\lambda f_{\mathbold{k}\sigma}^{\dagger}f_{\mathbold{k}\sigma}\nonumber \\
 &  + \sum_{\mathbold{k}}\left[V_{1\mathbold{k}}f_{\mathbold{k}\uparrow}^{\ast}c_{\mathbold{k}\downarrow}+V_{2\mathbold{k}}f_{\mathbold{k}\downarrow}^{\ast}c_{\mathbold{k}\uparrow}+H.c.\right]
 \nonumber \\
 & + \sum_{\mathbold{k}}\left[\Delta_{\mathbold{k}}^{\uparrow}f_{\mathbold{k}\uparrow}^{\dagger}f_{-\mathbold{k}\uparrow}^{\dagger}+\Delta_{\mathbold{k}}^{\downarrow}f_{\mathbold{k}\downarrow}^{\dagger}f_{-\mathbold{k}\downarrow}^{\dagger}+H.c.\right]\nonumber \\
 & +\frac{8\mathcal{N}_{s}\Delta_{t}^{2}}{J_{H}}+\frac{\mathcal{N}_{s}x^{2}}{J_{K}}-\mathcal{N}_{s}\lambda,
 \label{eq:MF-H}
 \end{align}

where $V_{1\mathbold{k}}=2x\left(\sin k_{x}-i\sin k_{y}\right)$ and $ V_{2\mathbold{k}}=2x\left(\sin k_{x}+i\sin k_{y}\right)$. The Fourier transformation for the second-quantized operator is defined as $\psi_{i\sigma}=\frac{1}{\sqrt{\mathcal{N}_s}}\sum_{\mathbold{k}}e^{-i\mathbold{k}\cdot \mathbold{r}_i}\psi_{\mathbold{k}\sigma}$. Note that the mean-field Kondo term of Eq. (\ref{eq:MF-H}) is reminiscent of the  topological Kondo insulator shown in Ref. \cite{coleman-book}.  In Eq. (\ref{eq:MF-H}), $\Delta_t^{\sigma}(\mathbold{k})$  represents the gap structure of the spin-triplet $p$-wave RVB pairing in the momentum space for the spin-$\sigma$ sector, defined as $\Delta_{\mathbold{k}}^{\uparrow}=\Delta_{t}\left(-\sin k_{y}-i\sin k_{x}\right)$ and $\quad \Delta_{\mathbold{k}}^{\downarrow}=\Delta_{t}\left(\sin k_{y}-i\sin k_{x}\right)$ with $\Delta_t$ being denoted the mean-field pairing potential (see Appendix,  Section II). This momentum-dependent gap structure for the up- and down-spin sectors correspond to the following real-space patterns of $\Delta^\uparrow_t (i,j)$ and $\Delta^\downarrow_t(i,j)$ of Eq. (\ref{eq: HS}):
  \begin{align}
     &\Delta_t^{\uparrow}(i,j)\to  \Delta^{\uparrow}_t(i,i+\hat{x})=-\Delta^{\uparrow}_t(i,i-\hat{x})=-\Delta_{t},\nonumber \\
     &\qquad \qquad \quad \Delta^{\uparrow}_t(i,i+\hat{y})=-\Delta^{\uparrow}_t(i,i-\hat{y})=i\Delta_{t},
     \label{eq:Delta-up}
 \end{align}
 and
 \begin{align}
     \Delta_t^{\downarrow}(i,j)\to& \Delta^{\downarrow}_t(i,i+\hat{x})=-\Delta^{\downarrow}_t(i,i-\hat{x})=-\Delta_{t},\nonumber \\ & \Delta^{\downarrow}_t(i,i+\hat{y})=-\Delta^{\downarrow}_t(i,i-\hat{y})=-i\Delta_{t}.
     \label{eq:Delta-down}
 \end{align}

Choosing $\Psi_{\mathbold{k}} = \left( \phi_{A\mathbold{k}}, \phi_{B\mathbold{k}} \right)^T$ with the Nambu spinors defined by  $\phi_{A\mathbold{k}}=\left(c_{\mathbold{k}\uparrow},\,\,c_{-\mathbold{k}\uparrow}^{\dagger},\,\,f_{\mathbold{k}\downarrow},\,\,f_{-\mathbold{k}\downarrow}^{\dagger}\right)^T$ and  $\phi_{B\mathbold{k}}=\left(c_{\mathbold{k}\downarrow},\,\,c_{-\mathbold{k}\downarrow}^{\dagger},\,\,f_{\mathbold{k}\uparrow},\,\,f_{-\mathbold{k}\uparrow}^{\dagger}\right)^T$,  the mean-field Hamiltonian $H_{MF} = \sum_{\mathbold{k}}\Psi_{\mathbold{k}}^{\dagger}\mathcal{H}_{\mathbold{k}}\Psi_{\mathbold{k}} + \mathcal{C}$ can be expressed as a summation of two decoupled $4 \times 4$ matrices as follows
 \begin{align}
     H_{MF}& = H_A + H_B  + \mathcal{C}, \nonumber \\
     H_{A(B)}& =  \sum_{\mathbold{k}}\phi_{A(B)\mathbold{k}}^{\dagger}\mathcal{H}_{\mathbold{k}}^{A(B)}\phi_{A(B)\mathbold{k}}
     \label{eq:H-MF}
 \end{align}

 with $\mathcal{C} \equiv \sum_\mathbold{k} \varepsilon_\mathbold{k} + \frac{8\mathcal{N}_{s}\Delta_{t}^{2}}{J_{H}}+\frac{\mathcal{N}_{s}x^{2}}{J_{K}}$, and

 \begin{align}
     \mathcal{H}_{\mathbold{k}}^{A} & =\begin{pmatrix}
\frac{\varepsilon_{\mathbold{k}}}{2} 	& 0	& \frac{V_{2\mathbold{k}}^{\ast}}{2}	& 0				\\
0 & -\frac{\varepsilon_{\mathbold{k}}}{2} 	& 0	& \frac{V_{2\mathbold{k}}}{2}				\\
\frac{V_{2\mathbold{k}}}{2} 		& 0	& \frac{\lambda}{2} 		& \Delta_{\mathbold{k}}^{\downarrow}	\\
0 & \frac{V_{2\mathbold{k}}^{\ast}}{2}	& \Delta_{\mathbold{k}}^{\downarrow\ast} 	& -\frac{\lambda}{2}
\end{pmatrix},\,\,\\
\mathcal{H}_{\mathbold{k}}^{B} & =\begin{pmatrix}
\frac{\varepsilon_{\mathbold{k}}}{2} 	& 0	& \frac{V_{1\mathbold{k}}^{\ast}}{2}	& 0				\\
0 & -\frac{\varepsilon_{\mathbold{k}}}{2} 	& 0	& \frac{V_{1\mathbold{k}}}{2}				\\
\frac{V_{1\mathbold{k}}}{2} 		& 0	& \frac{\lambda}{2} 		& \Delta_{\mathbold{k}}^{\uparrow}	\\
0 & \frac{V_{1\mathbold{k}}^{\ast}}{2}	& \Delta_{\mathbold{k}}^{\uparrow\ast} 	& -\frac{\lambda}{2}.
\end{pmatrix}
 \end{align}

The Hamiltonian Eq. (\ref{eq:H-MF})  possesses time-reversal symmetry: $\mathcal{H}_{A}$ and $\mathcal{H}_{B}$ constitute the time-reversal partner of each other, i.e. $\Theta \mathcal{H}_{A (B)} \Theta^{-1} = \mathcal{H}_{B(A)}$ where the time-reversal operator $\Theta = \rho^0 \otimes (-i\sigma^{y})K$ with $\sigma^y $ being the $y$-component Pauli matrix on the spin subspace, $\rho^0$ being a $2\times 2 $ identity matrix on the orbital subspace while $K$ being the complex-conjugate operator. Under time-reversal transformation, the spin and quasi-momentum of conduction ($c$) and pseudofermion ($f$) operators are flipped: $ \left(c_{\bm{k}\uparrow},\, c_{\bm{k}\downarrow}, \, f_{\bm{k}\uparrow},\, f_{\bm{k}\downarrow}\right) \stackrel{\Theta}{\longrightarrow}  \left(c_{-\bm{k}\downarrow},\, -c_{-\bm{k}\uparrow}, \, f_{-\bm{k}\downarrow},\, -f_{-\bm{k}\uparrow}\right)$. Meanwhile, our Hamiltonian respects charge-conjugation (particle-hole) symmetry:  $\mathcal{P}\mathcal{H}_{\bm{k}}\mathcal{P}^{-1} = -\mathcal{H}_{-\bm{k}}$ where  $\mathcal{P} \equiv \tau^x K$ is the particle-hole operator  with $\tau_x$ being the $x$-component of the Pauli matrices on the particle-hole basis. Due to the odd-parity $p\pm ip^\prime$ RVB pairing of our model, the parity symmetry is broken here. Thus, our model Eq. (\ref{eq:H-MF}) belongs to the DIII class of topological symmetry \cite{Shinsei-PRB-classification}.  \deleted{Therefore, instead of using the ($8 \times 8$) $\mathcal{H}_{\mathbold{k}}$ matrix for self-consistent calculations, we can consider its ($4 \times 4$) components, $\mathcal{H}_{\mathbold{k}}^{A}$ and $\mathcal{H}_{\mathbold{k}}^{B}$, separately.}

\section{Results}
\subsection{Mean-field phase diagram}
\label{sec:MFPD}

The \added{mean-field} ground states \replaced{are}{is} determined by minimizing the mean-field \replaced{free energy per site $\mathcal{F}_{MF} = \frac{\mathcal{C}}{\mathcal{N}_s} - \frac{ k_B T}{\mathcal{N}_s} \sum_{n\bm{k}}\ln \left[ 1+\exp \left( - \frac{E_{n\bm{k}}}{k_B T}\right) \right] $}{ground state energy of $H_{MF}$} \added{with respect to the mean-field variables $q = (\lambda, \, x,\, \Delta_t)$, i.e. $\partial \mathcal{F}_{MF}/\partial q_i = 0$.} \added{ Here, $E_{n\mathbold{k}} < 0$ is the $n$-th band of $\mathcal{H}_{\mathbold{k}}$. The chemical potential $\mu$ is determined by the relation $\partial \mathcal{F}_{MF}/\partial \mu = -(1+\delta)$ with $\delta$ being the chemical doping of the $c$-electrons for which $\delta  \lesseqqgtr 0 $ is for $p/\text{un}/n-$ doped (half-filling corresponds to $\delta = 0$).} This leads to the following saddle-point equations at zero temperature,

 \begin{align}
   & \frac{1}{\mathcal{N}_s}\sum_{n\bm{k}}\frac{\partial E_{n\bm{k}}}{\partial x} + \frac{2x}{J_K} = 0, \nonumber \\
   & \frac{1}{\mathcal{N}_s}\sum_{n\bm{k}}\frac{\partial E_{n\bm{k}}}{\partial \Delta_t} + \frac{16 \Delta_t}{J_H} = 0, \nonumber \\
   & \frac{1}{\mathcal{N}_s}\sum_{n\bm{k}}\frac{\partial E_{n\bm{k}}}{\partial \lambda}  = 0, \nonumber \\
   &  \frac{1}{\mathcal{N}_s}\sum_{n\bm{k}}\frac{\partial E_{n\bm{k}}}{\partial \mu} +\delta = 0 .
   \label{eq:self-const-eq}
 \end{align}

\deleted{and the chemical potential for the conduction electrons can be solved via $\partial\mathcal{F}/\partial\mu_c = - n_c$,  where $\mathcal{F} = \frac{\mathcal{C}_{\mathbold{k}}}{\mathcal{N}_s} - \frac{2k_BT}{\mathcal{N}_s} \sum_{n\mathbold{k}}\ln\cosh\frac{E_{n\mathbold{k}}}{2k_BT}$  is the free energy per site with $E_{n\mathbold{k}} < 0$ being the eigenvalues of $\mathcal{H}_{\mathbold{k}}$ \replaced{where}{and} $n_c = \frac{1}{\mathcal{N}_s} \sum_{i\sigma} c_{i\sigma}^\dagger c_{i\sigma}$ is the average occupation number of conduction electrons per site (i.e. $n_c > 1$ for n-doped, $= 1$ for undoped or half-filled, and $< 1$ for p-doped).}

 \begin{figure}[t]
     \centering
     \includegraphics[width=0.45\textwidth]{Bulk-disp.pdf}
     \caption{Figures (a) (red curves) and (b) show the bulk energy spectrum of the co-existing superconducting state near the Fermi level $\mu$. The Fermi level locates at $E(\bm k) = 0$. The coupling constants are $J_K = 0.3$ and $J_H = 1.0$. Inset of (a) displays the First Brillouin zone of a square lattice with indications of high-symmetry points $\Gamma,\, X, \,M$.}
     \label{fig:3d-disp}
 \end{figure}

\added{The ground-state  phase diagram (Fig. \ref{GS}) of our  model is obtained  by solving the saddle-point equations  self-consistently.} \deleted{By solving the above saddle-point equations self-consistently for both the bulk and the ribbon with $N_y = 44$ chains, we obtain the ground-state phase diagrams of the model as shown in Fig.~\ref{GS}.} \deleted{Obviously, each} \added{The} phase diagram \replaced{contains}{has} three \replaced{distinct}{separate} \added{mean-field} phases\deleted{, i.e.}: a pure Kondo phase \added{is found} at $J_H = 0$ \added{where $x \neq 0$, $\Delta_t = 0$. At the opposite limit where the RKKY interaction dominates, the ground state shows  short-range magnetic correlation with $p$-wave spin-triplet RVB   pairing ($\Delta_t \neq 0,\, x =0$). In the intermediate range of $0<J_H/J_K<(J_H/J_K)_c$, we find a Kondo-$t$RVB  co-existing (superconducting) phase with $x \neq 0$ and $\Delta_{t} \neq 0$, which can be explained via the mechanism of Kondo-stabilized spin liquid  \cite{Coleman-Andrei, sm-phase-PNAS}.  The development of superconductivity in this co-existing phase requires higher-order processes involving both the Kondo and $t$-RVB terms: the mean-field $t$-RVB pairings of the local $f$ fermions provide preformed Cooper pairs. When the Kondo hybridization field $\chi$ gets Bose-condensed ($x\neq 0$), the local fermions delocalize into the conduction band and make the preformed $t$-RVB Cooper pairs   superconduct \cite{Nevidomskyy}. These  processes can be   described by the effective mean-field Hamiltonian $H_{sc} = \sum_{\bm{k}}\left(\bar{\Delta}_{\bm{k}}^{\downarrow *}c_{-\bm{k}\downarrow
}c_{\bm{k}\downarrow} + \bar{\Delta}_{\bm{k}}^{\uparrow *}c_{-\bm{k}\uparrow
}c_{\bm{k}\uparrow} +H.c.\right)$, where the effective gap functions  take the form $\bar{\Delta}_{\bm{k}}^{\downarrow *} =V_{1\bm{k}} V_{1,-\bm{k}}\Delta_{\bm{k}}^{\uparrow*} \sim x^2 \Delta_t (\sin^2 k_x +\sin^2 k_y)(\sin k_x -i\sin k_y) $ and $\bar{\Delta}_{\bm{k}}^{\uparrow *} = V_{2\bm{k}} V_{2,-\bm{k}}\Delta_{\bm{k}}^{\downarrow*} \sim x^2 \Delta_t (\sin^2 k_x +\sin^2 k_y)(\sin k_x + i\sin k_y)$ with the size of the superconducting gap being proportional to $x^2 \Delta_t$. The superconducting gap function  $\bar{\Delta}_{\bm{k}}^{\uparrow} $ we obtained here shows a $f$-wave-like pairing symmetry on a generic anisotropic (non-circular) 2D Fermi surface.  Nevertheless, as we are taking the continuous limit of the conduction band here,  $\bar{\Delta}_{\bm{k}}^{\uparrow} $ can be expressed as a product of $s$ and $p\pm i p^\prime$ pairing orders, i.e., $\bar{\Delta}_{\bm{k}}^{\uparrow/\downarrow *} \sim k^2 (k_x \pm i k_y)$ with $k^2 \equiv k_x^2 + k_y^2$ on a circular Fermi surface but only the $p\pm i p^\prime$ component plays a role here. \added{Note that we find the co-existing superconducting state persists for an arbitrary small value of $J_H/J_K \to 0^+$. This is likely due to the overestimation of the co-existing phase at the mean-field level. Upon including fluctuations of the Kondo and  $t$-RVB order parameters beyond the mean-field level, we expect a narrower co-existing superconducting phase.}  A first-order transition similar to the results found in Refs. \cite{sm-phase-PNAS,Senthil-PRL-2003-fractionalized} is observed at the transition of the  $t$-RVB and the co-existing superconducting phases (see Fig. \ref{GS})}. \added{The bulk band structure in the co-existing superconducting state is shown in Fig. \ref{fig:3d-disp}.}
\subsection{Topological invariance}

\begin{figure}[t]
    \centering
    \includegraphics[width=0.45\textwidth]{ribbon-band-zoomin.pdf}
    \caption{The left figure displays the electronic band structure of the coexisting superconductor state for a strip with $N_y = 81$ described by $\mathcal{H}_A$ at $J_K/t = 0.3$ and $J_H/t = 1.0$. Three pairs of edge states with Dirac spectra are observed near $k_x = 0$ (the pink curves). The edge states at zero energy correspond to the Majorana zero modes. Due to the time-reversal symmetry of the model, the band structure for a strip for $\mathcal{H}_B$ is identical to that of $\mathcal{H}_A$. The close-up band structures near three pairs of edge states (pink curves) on the top, middle and bottom bounded by the red squares are shown on the right figures.  }
    \label{JH_3.0_EBS_AB}
\end{figure}

We now address the topological properties of the coexisting superconducting state.  Since this system is invariant under time-reversal transformation, the bulk topological properties of the coexisting Kondo-RVB superconducting state with $p\pm i p^\prime$ spin-triplet RVB pairing can be thus characterized by  the $Z_2$ Chern number $c_T$ (or time-reversal polarization) \cite{Fu-kane-PRB-2005,Fu-kane-PRL-2007,DNSheng-PRL-2006}, given by
\begin{align}
    c_{T}=\frac{c_{A}-c_{B}}{2}
\end{align}
with $c_{I}$ $(I \in A,B)$ being the Thouless-Kohmoto-Nightingal-den Nijs (TKNN) number \cite{TKNN-PRL} of $H_I$, defined as

\begin{align}
    c_I = \frac{1}{2\pi}\int_{\bm{k}\in\text{FBZ}} d\bm{S}_{\bm{k}}\cdot  \left( \bm{\nabla}_{\bm{k}} \times \bm{\mathcal{A}}^I_{\bm{k}}\right).
\end{align}

The Berry's connection  $\bm{\mathcal{A}}^I_{\bm{k}}$ for $H_I$ is given by $ \bm{\mathcal{A}}^I_{\bm{k}} \equiv  i\sum_{n\in I} \langle u_{n\bm{k}}^{I}|\bm{\nabla_{\bm{k}}}|u_{n\bm{k}}^{I}\rangle$ with $|u_{n\bm{k}}^{I}\rangle$ being the  normalized Bloch state of the $n$-th filled band for $H^I_{\bm{k}}$. We numerically calculate the TKNN numbers \cite{Hatsugui-chern-numb}, $c_A$ and $c_B$, and find that $c_A = -c_B = 1$ in the co-existing phase, indicating a topologically non-trivial $Z_2$ Chern number $c_T = 1$. By the bulk-edge correspondence, we expect this co-existing superconducting state to support a pair of counter-propagating Majorana zero modes at the edges of a finite-sized strip. Further band structure calculations of our model on a strip in the following subsection  confirm our expectation.
\subsection{Edge states of the coexisting Kondo-RVB spin-triplet $p\pm ip^\prime$-wave superconducting state}

\begin{figure*}[ht]
    \centering
    \includegraphics[width=0.9\textwidth]{MF-edgestate.pdf}
    \caption{\added{Figures (a) and (d) show the Bogoliubov excitation spectra of $\mathcal{H}_A$ and $\mathcal{H}_B$,  respectively, near the chemical potential on a nano-strip with $N_y = 81$ chains. Figures (b), (c) and (e), (f) demonstrate the probability density of the Majorana edge state wave functions of $\mathcal{H}_A$ and $\mathcal{H}_B$ as a function of atom position $y_i$, $\left| \gamma^\Gamma_{I, k_x}(y_i)  \right|^2$  with $I = A,B$ and $\Gamma = R,L$ (pink curves in (a) and (d)), at a fixed energy $E \equiv E(k_x = \pm 0.03)$. The probability density is described by $\left| \gamma^\Gamma_{I, k_x} (y_i) \right|^2 = \left( \left| u^\Gamma_{I,k_x} \right|^2,\, \left| \bar{u}^\Gamma_{I,k_x} \right|^2,\, \left| v^\Gamma_{I,k_x} \right|^2, \, \left| \bar{v}^\Gamma_{I,k_x} \right|^2   \right) (y_i) $. The parameters are $J_K/t = 0.3$, $J_H/t = 1.0$, and doping $\delta = -0.3$. The edge states are of the helical type, as schematically represented in (g). }\deleted{The probability densities are calculated at finite momenta at $J_H/t = 0.3$. For $\mathcal{H}_{\mathbold{k}}^{A}$ (top left in blue), $|\Psi_{A,k}|^2$ (top middle) dominates at one edge and $|\Psi_{A,-k}|^2$ (bottom middle) dominates at the other edge. And they reverse for $\mathcal{H}_{\mathbold{k}}^{B}$ (bottom left in red and right panels). Clearly, $|\Psi_{A,k}|^2$ and $|\Psi_{B,-k}|^2$ have the same behavior and $|\Psi_{A,-k}|^2$ and $|\Psi_{B,k}|^2$ share the same behavior.}}
    \label{JH_3.0_HMF_AB}
\end{figure*}
\begin{figure*}[ht]
    \centering
    \includegraphics[width=0.9\textwidth]{up-edgestate.pdf}
    \caption{\added{The finite-energy ($E(k_x)>0$) Bogoliubov excitation spectra of (a) $\mathcal{H}_A$ (shown on top right in Fig. \ref{JH_3.0_HMF_AB}) and (d) $\mathcal{H}_B$. A pair of ``helical" edge states is found to exist at finite energy [pink curve in (a) and (d)], and their probability densities are shown in (b) and (c), (e) and (f), respectively, at a fixed energy $E(k_x = \pm 0.22)$.}}
    \label{fig:ribbon-upedgestate-Ny61}
\end{figure*}

\deleted{Via bulk-edge correspondence, the non-trivial topological $Z_2$ number $c_T$ of the bulk coexisting  superconducting state indicates a pair of gapless counter-propagating edge states to exist at edges. Accompanying with the particle-hole symmetry of our model, the edge state can therefore be regarded as helical Majorana fermion.  We confirm this expectation by examining the band structure and the edge-state wave function of the Kondo-RVB coexisting superconducting phase on a finite-sized strip}

We now check whether our model would support helical Majorana zero modes at the edge of a finite-sized system. We shall examine our model's band structures and edge-state wave functions on a finite-sized strip  that extends infinitely along the $x$ direction but contains a finite number of lattice sites in $y$.  The results are shown in Figs. \ref{JH_3.0_EBS_AB} to \ref{fig:ribbon-upedgestate-Ny61}. As shown in Fig. \ref{JH_3.0_EBS_AB},  gapless Dirac spectra of the Bogoliubov excitations around $k_x = 0$ near zero energy are observed, exhibiting one of the typical features of topological edge states. The excitations can be effectively described  by the linear-dispersed Hamiltonian $ \tilde{H}_{I}=\sum_{k_x}v_x|k_x| \left( \gamma^{R\,\dagger}_{I,k_x} \gamma^{R}_{I,k_x}-\gamma^{L\,\dagger}_{I,k_x}\gamma^L_{I,k_x}\right)$ with

\begin{align}
    \gamma_{I,k_x}^\Gamma =&\sum_{y_i}\left[u^\Gamma_{I,k_x}(y_i)c_{k_x,y_i,\uparrow}+\bar{u}^\Gamma_{I,k_x}(y_i) c_{-k_x,y_i,\uparrow}^{\dagger} \right.\nonumber \\
    &   \left.+v^\Gamma_{I,k_x}(y_i)f_{k_x,y_i,\downarrow}+\bar{v}^\Gamma_{I,k _x}(y_i)f_{-k_x,y_i,\downarrow}^{\dagger}\right]

    \label{eq:low-E-HA-gamma}
\end{align}

with $u,\bar{u}$ and $v,\bar{v}$ being the coherent factors. In Eq. (\ref{eq:low-E-HA-gamma}), $I \in A,B$, $\Gamma \in R,L $, and $\gamma^{R/L}_{A/B,k_x}$  represents the right/left-moving Bogoliubov quasiparticle  of $\tilde{H}_{A/B}$.  Here, $v_x$ in $\tilde{H}_I$ denotes the velocity. Due to time-reversal symmetry, $H_A$ is the time-reversal partner of $H_B$, and thus their   spectra are identical. \deleted{and the effective Hamiltonian of the low-energy excitations of $H_B$ takes a similar form to Eq. (\ref{eq:low-E-HA-gamma}).} The low-energy eigenstates with Dirac spectra near $k_x = 0$ for both $H_A$ and $H_B$ exhibit the typical property of edge states, as their probability densities accumulate mostly at the edges of strip, as shown in Fig. \ref{JH_3.0_HMF_AB}. Combining the directions of propagation inferred from the velocity $v_x \sim \partial E(k_x)/\partial k_x$, we can classify these edge states  into two groups, each of them constitutes a pair of counter-propagating  edge states (see Fig. \ref{JH_3.0_HMF_AB}), revealing the nature of helical Majorana zero modes. The helical type of the Majorana zero modes is the consequence of time-reversal symmetry of our model, reminiscent of the well-known Kane-Mele model on a single-layered graphene \cite{2005-PRL-KM-Z2,2005-PRL-KM}. Remarkably, in addition to the Majorana fermions at zero energy,  two pairs of  counter-propagating edge-states are observed at finite energy, see Fig. \ref{fig:ribbon-upedgestate-Ny61}. The two pairs of  edge states correspond to the edge states of the topological Kondo insulator, where the spin-triplet RVB order parameter is absent ($\Delta_t = 0 $) \cite{dzero-TKI-PRL,dzero-TKI-PRB,dzero-Ann-TKI}.

\begin{figure}
    \centering
    \includegraphics[width=0.45\textwidth]{T-x-Dt.pdf}
    \caption{Plot of the temperature-dependent mean-field order parameters $x(T)/t$ and $\Delta_t (T)/t$ with $k_B = 1$, $J_K/t = 0.3$ and $J_H/t = 1.0$ fixed. Inset shows the enlarged plot of $\Delta_t (T)$. The single-impurity Kondo temperature occurs at $T_{\text{onset}}/t \approx 0.16$ while the transition of superconductivity takes places at temperature $T_c /t \approx 0.015$.}
    \label{fig:T-MF}
\end{figure}

\section{Discussions and Conclusions}

\added{We now discuss the application of our results for heavy-electron superconductors, particularly the Kondo lattice compound UTe$_2$. Experimental evidence indicates that this compound does not show  long-range magnetic order and is in the vicinity of the ferromagnetic quantum critical point, exhibiting both strong ferromagnetic fluctuations, possibly due to magnetic frustrations induced by sub-leading antiferromagnetic fluctuations \cite{YFYang-UTe, QM-UTe2-nature}, and Kondo screening \cite{Jiao-2020-UTe2,Ran-FM-UTe,Paglione-PRL-ARPES}. The DFT+$U$ calculations indicate that the dynamics of electron bands and the physical properties of UTe$_2$ are dominated by the electrons near the quasi-two-dimensional (cylindrical) Fermi surface with weak $k_z$ dependence despite its 3D crystal structure \cite{YFYang-UTe}. Superconductivity is reached at $T_c = 1.6$K, while the resistivity maximum observed at $T^\star \approx 15\sim 75$ $\text{K}$ reveals signature of coherent Kondo scattering \cite{rho-c-axis-PRB,Ran-FM-UTe}, indicating $T^\star/T_c \approx 10 \sim 50$. The superconductivity can, in general, co-exist and compete with the Kondo effect \cite{Jiao-2020-UTe2}. When a magnetic field is applied along the hard-magnetic axis $b$ of UTe$_2$ and before entering the superconducting phase,  a correlated paramagnetic phase is observed  below the temperature at which the magnetic susceptibility shows a broad maximum \cite{Braithwaite-CommPhys-PBphaseDiag}. Similar spin-liquid behavior has been observed in the magnetic susceptibility of another heavy fermion compound CePdAl  \cite{2019-Sun-CePdAl}. This similarity  suggests this correlated paramagnetic phase may feature  short-range magnetic order. Our theoretical framework based on  competition and collaboration between a Kondo-screened and a ferromagnetic $t$-RVB  spin-liquid states on a two-dimensional Kondo lattice is consistent with the above observations in UTe$_2$. It, therefore, constitutes a promising approach to account for its exotic phenomena. On the other hand, the chiral in-gap state, a signature of chiral topological superconductor, has been observed by scanning tunneling spectroscopy in the superconducting phase of UTe$_2$  \cite{Jiao-2020-UTe2}. Combining with the ferromagnetic fluctuations that are known to induce spin-triplet pairing, people believe UTe$_2$ is a promising candidate for the spin-triplet chiral  topological superconductor \cite{Jiao-2020-UTe2,Ran-FM-UTe}.  Furthermore, the superconducting phase co-existing with Kondo coherence in this material strongly suggests the role played by the Kondo effect in this possible topological superconductor. The topological Kondo superconducting state with equal-spin spin-triplet  $p$-wave   pairings we proposed here bears striking similarities to and strong relevance for the experimental observations on UTe$_2$: (i) the $d$- and $f$-orbitals electrons with their angular momentum quantum number  differing by $1$ in the uranium atoms of UTe$_2$ likely give rise to the odd-parity Kondo effect \cite{dzero-TKI-PRL,dzero-TKI-PRB,dzero-Ann-TKI}, (ii) the $t$-RVB state in our theory may be considered as one possible realization of the short-ranged ferromagnetic fluctuations in UTe$_2$, (iii) the Kondo-$t$-RVB co-existing superconducting state we find here qualitatively agrees with the co-existence between superconductivity and Kondo effect observed in UTe$_2$,  (iv) the high upper critical field exceeding the Pauli limit \cite{Ran-FM-UTe,Aoki-JPSJ-2020-FM} implies that the superconducting state of UTe$_2$ may have equal-spin Cooper pairs, and (v) the  effective pairing $\overline{\Delta}_{\bm{k}}^\sigma$ formed in the conduction band mentioned in Section \ref{sec:MFPD} shows  characteristics of  spin-triplet point-node gap structure \cite{point-node-Paglione}. Various characteristic temperature scales estimated from our mean-field calculations with $J_H/t = 1.0$ and $J_K/t = 0.3$ at finite temperatures agree reasonably well  with experimental observations (see Fig. \ref{fig:T-MF}): The superconducting transition temperature  $T_c$, theoretically determined from our mean-field analysis $T_c = \text{Min}[T(x = 0), T(\Delta_t = 0)]$, shows $T_c \approx 0.015 t \approx 2.3\text{\,K}$ by taking  estimated values of $t = 150\, \text{K}$ and  half-bandwidth  $D = 1.25t$ \cite{Paglione-PRL-ARPES}. The Kondo coherent scale can be obtained by $T^\star = x^2(T=0)/D \approx 17.4\,\text{K}$ \cite{Burdin-PRL}. The  ratio $T^\star/T_c \approx 8$ is in reasonable agreement with experimental observations.  The onset temperature $T_{\text{onset}}$ of  Kondo hybridization, which occurs at $x(T=T_{\text{onset}}) = 0$, displays $T_{\text{onset}} \approx 0.16t \approx 24\,\text{K}$, within the theoretically estimated range $10\text{K}< T_{\text{onset}}<100 \text{K}$ by DMFT calculation \cite{Paglione-PRL-ARPES}. Meanwhile, there have been evidences of TRS breaking in UTe$_2$ from the observed two superconducting transitions and a finite polar Kerr effect at $T<T_c$ \cite{Hayes-KerrEff-Science}, likely due to proximity to the ferromagnetic ordered phase.  A number of theoretical attempts were proposed based on these observations \cite{Agterberg-topo-band-PRB,Coleman-2022-UTe}. However, the observed single superconducting transition near ambient pressure and zero field \cite{Braithwaite-CommPhys-PBphaseDiag,  Rosa-SingleTran-CommMat, Rousel-arxiv-singletransition} as well as the theoretically proposed unitary triplet pairing \cite{YFYang-UTe} suggest TRS may be preserved in UTe$_2$. Though our results shown above are obtained in the presence of TRS, the chiral $p$-wave superconducting state with chiral Majorana zero mode at edges  is expected to occur here once a  time-reversal breaking magnetic field is applied \cite{sato-PRB-NSC}. Our distinct predictions with and without fields serve as theoretical guidance for future experiments to distinguish the time-reversal breaking from time-reversal preserving triplet pairing states  in UTe$_2$.  Since the Kondo correlations stabilize the $t$-RVB spin liquid in the co-existing superconducting phase, it is expected to be robust against gauge-field fluctuations beyond the mean field.  Our approach and results are distinct from the spin-triplet non-topological superconducting state recently proposed based on the Hund's-Kondo coupling and $S_z=0$ $t$-RVB state to account for UTe$_2$ \cite{Coleman-2022-UTe}.}

In conclusion, we propose \replaced{a first  realization of}{a microscopic mechanism} the topological superconductivity in the Kondo lattice model, \added{a distinct class of topological superconductors due to purely  strong electron correlations without employing spin-orbit coupling or  proximity effect. } \deleted{which is relevant for possible topological heavy-fermion superconducting states in heavy-fermion compounds.} \added{A topological Kondo superconductor essentially constitutes of 1)  itinerant $c$ and  localized $f$ bands with different orbital quantum numbers, 2) strong Hubbard interaction of the $f$ electrons, 3) odd-parity Kondo hybridization of the $c$ and $f$ bands, and 4) the attractive exchange interaction of the $f$ electrons with spin-triplet correlations.} Starting from the odd-parity Anderson lattice model, we obtain the unconventional type of Kondo hybridization and \added{ferromagnetic RKKY-like} interaction via perturbation theory, \added{leading} \deleted{which lead} to spin-triplet resonating-valence-bond (RVB) pairing between $f$-electrons with \added{time-reversal invariant} $p \pm ip^\prime$-wave gap symmetry. Via the mean-field approach, we find a Kondo triplet-RVB coexisting phase in the intermediate range of the Kondo to RKKY coupling ratio. This phase is shown as a time-reversal invariant topological superconducting state with a spin-triplet $p \pm ip^\prime$-wave RVB pairing gap. It exhibits non-trivial topology in the bulk band structure, and supports helical Majorana zero modes at edges. \added{Our prediction in the presence of a time-reversal breaking field leads to chiral $p$-wave spin-triplet topological  Kondo superconductor.  Our results on the superconducting transition temperature, Kondo coherent scale, and onset temperature of Kondo hybridization  not only qualitatively but also  quantitatively agree with the observations for UTe$_2$. The theoretical framework we propose here opens up the search for topological superconductors induced by strongly electronic correlations on the Kondo lattice compounds.
}

\section{Acknowledgements} This work is supported by the Ministry of Science and Technology Grants 104-2112-M-009-004-MY3 and 107-2112-M-009-010-MY3, the National Center for Theoretical Sciences of Taiwan, Republic of China (to C.-H. C.).

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