\section{Introduction}

Based on ideas of Misra \cite{mis74}, it was essentially the late
S. Bugajski who recognized that the statistical (probabilistic) framework
of quantum mechanics can be understood as a reduced classical
probability theory on the projective Hilbert space $ \ph $
\cite{bug91;93a-d,bel95a;b,hol82,stu01,bus04}. In a forthcoming paper
\cite{bus07} a suggestive definition of a classical extension of
quantum mechanics is given and it is proved that every such classical
extension is essentially equivalent to the Misra-Bugajski scheme. Moreover,
it is known that $ \ph $, considered as a real differentiable manifold,
carries a Riemannian as well as a symplectic structure,
the latter enabling one to reformulate quantum dynamics
on the (in general infinite-dimensional) phase space $ \ph $
\cite{gue77,kib79,cir84,cir90a;b,bro01,bje05}. Thus, quantum mechanics
can be considered as some reduced classical statistical mechanics.

In order to take the projective Hilbert space $ \ph $ as a sample space
for classical probability theory, $ \ph $ must be equipped
with a measurable structure. To make $ \ph $ a differentiable manifold,
it should be equipped with a topology first. The elements of $ \ph $ can
be interpreted as equivalence classes of vectors $ \varphi \in \hi $,
$ \varphi \neq 0 $, as equivalence classes of unit vectors, as the
one-dimensional subspaces of $ \hi $, or as the one-dimensional
orthogonal projections acting in $ \hi $. So, on the one hand,
two qoutient topologies can be defined on $ \ph $ whereas,
on the other hand, all the different operator topologies induce
topologies on $ \ph $. In Section 2 we undertake a systematic review
and comparison, already sketched out in \cite{bug94}, of the various
topologies on $ \ph $ and show that all the topologies are the same,
thus giving a natural topology $ \mathcal{T} $ on $ \ph $. Furthermore,
in Section 3 we present a simple proof that the Borel structure
generated by the topology $ \mathcal{T} $ coincides with the measurable structure
generated by the transition probability functions of the pure
quantum states; our proof simplifies a proof of Misra from 1974 for
a corresponding statement \cite{mis74}.

\section{The Topology of the Projective Hilbert Space}\label{sec:top}

Let $ \hi \neq \{ 0 \} $ be a nontrivial separable complex
Hilbert space. Call two vectors of $ \hi^* := \hi \setminus \{0\} $
equivalent if they differ by a complex factor, and define the
{\it projective Hilbert space $ \ph $} to be the set of the corresponding
equivalence classes which are often called {\it rays}. Instead of
$ \hi^* $ one can consider only the unit sphere of $ \hi $,
$ S := \{\varphi \in \hi \, | \, \no{\varphi} = 1 \} $. Then two unit vectors
are called equivalent if they differ by a phase factor, and the set of
the corresponding equivalence classes, i.e., the set of the {\it
unit rays}, is denoted by $ S/S^1 $ (in this context, $ S^1 $ is
understood as the set of all phase factors, i.e., as the set of all
complex numbers of modulus $1$). Clearly, $ S/S^1 $ can be
identified with the projective Hilbert space $ \ph $. Furthermore,
we can consider the elements of $ \ph $ also as the one-dimensional
subspaces of $ \hi $ or, equivalently, as the one-dimensional
orthogonal projections $ P = P_{\varphi} = \kb{\varphi}{\varphi} $, $
\no{\varphi} = 1 $.

The set $ \hi^* $ and the unit sphere $S$ carry the topologies induced by
the metric topology of $ \hi $. Using the canonical projections
$ \mu \! : \hi^* \to \ph $, $ \mu(\varphi) := [\varphi] $, and
$ \nu \! : S \to S/S^1 $, $ \nu(\chi) := [\chi]_S $, where
$ [\varphi] $ is a ray and $ [\chi]_S $ a unit ray, we can equip
the quotient sets $ \ph $ and $ S/S^1 $ with their quotient topologies
$ \mathcal{T}_{\mu} $ and $ \mathcal{T}_{\nu} $. Considering $ \mathcal{T}_{\nu} $, a set
$ O \subseteq S/S^1 $ is called open if $ \nu^{-1}(O) $ is open.

\begin{theorem}\label{thm:ss1}
The set $ S/S^1 $, equipped with the quotient topology $ \mathcal{T}_{\nu} $,
is a second-countable Hausdorff space, and $ \nu $ is an open continuous
mapping.
\end{theorem}
\proof{
By definition of $ \mathcal{T}_{\nu} $, $ \nu $ is continuous. To show that
$ \nu $ is open, let $U$ be an open set of $S$. From
\[
\nu^{-1}(\nu(U)) = \nu^{-1}(\{ [\chi]_S \, | \, \chi \in U \})
                 = \bigcup_{\lambda \in S^1} \lambda U ,
\]
$ S^1 = \{ \lambda \in \C \, | \, |\lambda| = 1 \} $, it follows that
$ \nu^{-1}(\nu(U)) \subseteq S $ is open. So $ \nu(U) \subseteq S/S^1 $
is open; hence, $ \nu $ is open.

Next consider two different unit rays $ [\varphi]_S $ and $ [\psi]_S $ where
$ \varphi,\psi \in S $ and $ |\ip{\varphi}{\psi}| = 1 - \varepsilon $,
$ 0 < \varepsilon \leq 1 $. Since the mapping
$ \chi \mapsto |\ip{\varphi}{\chi}| $, $ \chi \in S $, is continuous,
the sets
\begin{equation}\label{u1}
U_1 := \left\{ \chi \in S \left| \,
              |\ip{\varphi}{\chi}| > 1 - \tfrac{\varepsilon}{2} \right.
                                                             \right\}
\end{equation}
and
\begin{equation}\label{u2}
U_2 := \left\{ \chi \in S \left| \,
              |\ip{\varphi}{\chi}| < 1 - \tfrac{\varepsilon}{2} \right.
                                                             \right\}
\end{equation}
are open neighborhoods of $ \varphi $ and $ \psi $, respectively. Consequently,
the sets $ O_1 := \nu(U_1) $ and $ O_2 := \nu(U_2) $ are open
neighborhoods of $ [\varphi]_S $ and $ [\psi]_S $, respectively. Assume
$ O_1 \cap O_2 \neq \emptyset $. Let $ [\xi]_S \in O_1 \cap O_2 $, then
$ [\xi]_S = \nu(\chi_1) = \nu(\chi_2) $ where $ \chi_1 \in U_1 $ and
$ \chi_2 \in U_2 $. It follows that $ \chi_1 $ and $ \chi_2 $ are equivalent,
so $ |\ip{\varphi}{\chi_1}| = |\ip{\varphi}{\chi_2}| $, in contradiction to
$ \chi_1 \in U_1 $ and $ \chi_2 \in U_2 $. Hence, $ O_1 $ and $ O_2 $
are disjoint, and $ \mathcal{T}_{\nu} $ is separating.

Finally, let $ \mB = \{ U_n \, | \, n \in \N \} $ be a countable base of
the topology of $S$ and define the open sets $ O_n := \nu(U_n) $. We show
that $ \{ O_n \, | \, n \in \N \} $ is a base of $ \mathcal{T}_{\nu} $. For
$ O \in \mathcal{T}_{\nu} $, we have that $ \nu^{-1}(O) $ is an open set of
$S$ and consequently $ \nu^{-1}(O) = \bigcup_{n \in M} U_n $ where
$ U_n \in \mB $ and $ M \subseteq \N $. Since $ \nu $ is surjective,
it follows that
\[
O = \nu(\nu^{-1}(O)) = \nu \left( \bigcup_{n \in M} U_n \right)
                     = \bigcup_{n \in M} \nu(U_n)
                     = \bigcup_{n \in M} O_n .
\]
Hence, $ \{ O_n \, | \, n \in \N \} $ is a countable base of
$ \mathcal{T}_{\nu} $. \ $\square$}

Analogously, it can be proved that the topology $ \mathcal{T}_{\mu} $ on $ \ph $
is separating and second-countable and that the canonical projection $ \mu $
is open (and continuous by the definition of $ \mathcal{T}_{\mu} $). Moreover,
one can show that the natural bijection $ \beta \! : \ph \to S/S^1 $,
$ \beta([\varphi]) := \left[ \frac{\varphi}{\no{\varphi}} \right]_S $,
$ \beta^{-1}([\chi]_S) = [\chi] $, is a homeomorphism. Thus, identifying
$ \ph $ and $ S/S^1 $ by $ \beta $, the topologies $ \mathcal{T}_{\mu} $ and
$ \mathcal{T}_{\nu} $ are the same.

We denote the real vector space of the self-adjoint trace-class operators by
$ \tsh $ and the real vector space of all bounded self-adjoint operators by
$ \bsh $; endowed with the trace norm and the usual operator norm,
respectively, these spaces are Banach spaces. As is well known,
$ \bsh $ can be considered as the dual space $ (\tsh)' $ where the duality
is given by the trace functional. Let $ \sh $ be the convex set of
all positive trace-class operators of trace $1$; the operators of
$ \sh $ are the density operators and describe the quantum states. We
recall that the extreme points of the convex set $ \sh $, i.e.,
the pure quantum states, are the one-dimensional orthogonal projections
$ P = P_{\varphi} $, $ \no{\varphi} = 1 $. We denote the set of these extreme
points, i.e., the extreme boundary, by $ \esh $.

The above definition of $ \ph $ and $ S/S^1 $ as well as of their
quotient topologies is related to a geometrical point of view. From
an operator-theoretical point of view, it is more obvious to identify
$ \ph $ with $ \esh $ and to restrict one of the various operator topologies
to $ \esh $. A further definition of a topology on $ \esh $ is suggested by
the interpretation of the one-dimensional projections $ P \in \esh $ as
the pure quantum states and by the requirement that
the transition probabilities between two pure states are
continuous functions. Next we consider, taking account of
$ \esh \subseteq \sh \subset \tsh \subseteq \bsh $, the metric topologies on
$ \esh $ induced by the trace-norm topology of $ \tsh $, resp.,
by the norm toplogy of $ \bsh$. After that we introduce the weak topology on
$ \esh $ defined by the transition-probability functions as well as
the restrictions of several weak operator topologies to $ \esh $. Finally,
we shall prove the surprising result that all the many toplogies on
$ \ph \cong S/S^1 \cong \esh $ are equivalent.

\begin{theorem}\label{thm:esh-dist}
Let $ P_{\varphi} = \kb{\varphi}{\varphi} \in \esh $ and
$ P_{\psi} = \kb{\psi}{\psi} \in \esh $ where
$ \no{\varphi} = \no{\psi} = 1 $. Then
\begin{enumerate}
\item[(a)]
\[ \rho_n(P_{\varphi},P_{\psi}) := \no{P_{\varphi} - P_{\psi}}
                              = \sqrt{1 - |\ip{\varphi}{\psi}|^2}
                              = \sqrt{1 - \tr{P_{\varphi} P_{\psi}}}
\]
where the norm $ \no{\cdot} $ is the usual operator norm
\item[(b)]
\[ \rho_{\mathrm{tr}}(P_{\varphi},P_{\psi})
               :=  \no{P_{\varphi} - P_{\psi}}_{\mathrm{tr}}
                = 2\no{P_{\varphi} - P_{\psi}},
\]
in particular, the metrics $ \rho_n $ and $ \rho_{\mathrm{tr}} $ on
$ \esh $ induced by the operator norm $ \no{\cdot} $ and the trace
norm $ \no{\cdot}_{\mathrm{tr}} $ are equivalent
\item[(c)]
\[
\no{P_{\varphi} - P_{\psi}} \leq \no{\varphi - \psi},
\]
in particular, the mapping $ \varphi \mapsto P_{\varphi} $ from $S$ into
$ \esh $ is continuous, $ \esh $ being equipped with $ \rho_n $ or
$ \rho_{\mathrm{tr}} $.
\end{enumerate}
\end{theorem}
\proof{
To prove (a) and (b), assume $ P_{\varphi} \neq P_{\psi} $, otherwise
the statements are trivial. Then the range of $ P_{\varphi} - P_{\psi} $
is a two-dimensional subspace of $ \hi $ and is spanned by the two linearly
independent unit vectors $ \varphi $ and $ \psi $. Since eigenvectors of
$P_{\varphi} - P_{\psi} $ belonging to eigenvalues $ \lambda \neq 0 $
must lie in the range of $ P_{\varphi} - P_{\psi} $, they can be written as
$ \chi = \alpha \varphi + \beta \psi $. Therefore, the eigenvalue problem
$ (P_{\varphi} - P_{\psi}) \chi = \lambda \chi $, $ \chi \neq 0 $,
is equivalent to the two linear equations
\begin{eqnarray*}
    (1- \lambda) \alpha + \ip{\varphi}{\psi} \beta & = & 0   \\
-\ip{\psi}{\varphi} \alpha -   (1 + \lambda) \beta & = & 0
\end{eqnarray*}
where $ \alpha \neq 0 $ or $ \beta \neq 0 $. It follows that
$ \lambda = \pm \sqrt{1 - |\ip{\varphi}{\psi}|^2} =: \lambda_{1,2} $. Hence,
$ P_{\varphi} - P_{\psi} $ has the eigenvalues
$ \lambda_1 $, $0$, and $ \lambda_2 $. Now, from
$ \no{P_{\varphi} - P_{\psi}} = \max \{ |\lambda_1|,|\lambda_2| \} $ and
$ \no{P_{\varphi} - P_{\psi}}_{\mathrm{tr}} = |\lambda_1| + |\lambda_2| $,
we obtain the statements (a) and (b).---From
\begin{eqnarray*}
\no{P_{\varphi} - P_{\psi}}^2
&  =   & 1 - |\ip{\varphi}{\psi}|^2
   =     \no{\varphi - \ip{\psi}{\varphi} \psi}^2
   =     \no{(I - P_{\psi}) \varphi}^2                                  \\
& \leq & \no{(I - P_{\psi}) \varphi}^2 + \no{\psi - P_{\psi} \varphi}^2    \\
&  =   & \no{(I - P_{\psi}) \varphi - (\psi - P_{\psi} \varphi)}^2         \\
&  =   & \no{\varphi - \psi}^2
\end{eqnarray*}
we conclude statement (c). \ $\square$}

According to statement (b) of Theorem \ref{thm:esh-dist}, the metrics $ \rho_n $ and
$ \rho_{\mathrm{tr}} $ give rise to the same topology
$ {\mathcal{T}}_n = {\mathcal{T}}_{\mathrm{tr}} $ as well as to the same uniform structures.

\begin{theorem}\label{thm:esh-sepc}
Equipped with either of the two metrics $ \rho_n $ and
$ \rho_{\mathrm{tr}} $, $ \esh $ is separable and complete.
\end{theorem}
\proof{
As a metric subspace of the separable Hilbert space $ \hi $, the unit sphere
$S$ is separable. Therefore, by statement (c) of Theorem \ref{thm:esh-dist}, the metric space
$ (\esh,\rho_n) $ is separable and so is $ (\esh,\rho_{\mathrm{tr}}) $
(the latter, moreover, implies the trace-norm separability of
$ \tsh $). Now let $ \{ P_n \}_{n \in \N} $ be a Cauchy sequence in
$ (\esh,\rho_{\mathrm{tr}}) $. Then there exists an operator
$ A \in \tsh $ such that $ \no{P_n - A}_{\mathrm{tr}} \to 0 $ as well as
$\no{P_n - A} \to 0 $ as $ n \to \infty $ (remember that, on $ \tsh $,
$ \no{\cdot}_{\mathrm{tr}} $ is stronger than $ \no{\cdot} $). From
\begin{eqnarray*}
\no{P_n - A^2}    =     \no{A^2 - P_n^2}
               & \leq & \no{A^2 - AP_n} + \no{AP_n - P_n^2}    \\
               & \leq & \no{A} \no{A - P_n} + \no{A - P_n}     \\
               & \to  &  0
\end{eqnarray*}
as $ n \to \infty $ we obtain $ A = \lim_{n \to \infty} P_n = A^2 $;
moreover,
\[
\tr{A} = \tr{AI} = \lim_{n \to \infty} \tr{P_nI} = 1.
\]
Hence, $A$ is a one-dimensional orthogonal projection, i.e.,
$ A \in \esh $. \ $\square$}

Next we equip $ \esh $ with the topology $ \mathcal{T}_0 $ generated by
the functions
\begin{equation}\label{p}
P \mapsto h_Q(P) := \tr{PQ} = |\ip{\varphi}{\psi}|^2
\end{equation}
where $ P = \kb{\psi}{\psi} \in \esh $, $ Q = \kb{\varphi}{\varphi} \in \esh $,
and $ \no{\psi} = \no{\varphi} = 1 $. That is, $ \mathcal{T}_0 $ is the coarsest
topology on $ \esh $ such that all the real-valued functions $ h_Q $
are continuous. Note that $ \tr{PQ} = |\ip{\varphi}{\psi}|^2 $ can be
interpreted as the transition probability between the two pure states
$P$ and $Q$.

\begin{lemma}\label{lem:esh}
The set $ \esh $, equipped with the topology $ \mathcal{T}_0 $, is
a second-countable Hausdorff space. A countable base of $ \mathcal{T}_0 $
is given by the finite intersections of the open sets
\begin{equation}\label{uklm}
\begin{array}{crl}
U_{klm} & := & h_{Q_k}^{-1} \left( \,
                            \left] q_l - \frac{1}{m},q_l + \frac{1}{m}
                                         \right[ \, \right)   \vspace{2mm}\\
        &  = & \left\{ P \in \esh \left| \,
               \left| \tr{PQ_k} - q_l \right| < \frac{1}{m}
                                                \right. \right\}
\end{array}
\end{equation}
where $ \{ Q_k \}_{k \in \N} $ is a sequence of one-dimensional orthogonal
projections being $ \rho_n $-dense in $ \esh $, $ \{ q_l \}_{l \in \N} $
is a sequence of numbers being dense in $ [0,1] \subseteq \R $, and
$ m \in \N $.
\end{lemma}
\proof{
Let $ P_1 $ and $ P_2 $ be any two different one-dimensional
projections. Choosing $ Q = P_1 $ in (\ref{p}), we obtain
$ h_{P_1}(P_1) = 1 \neq h_{P_1}(P_2) = 1 - \varepsilon $,
$ 0 < \varepsilon \leq 1 $. The sets
\[
U_1 := \left\{ P \in \esh \left| \,
                h_{P_1}(P) > 1 - \tfrac{\varepsilon}{2} \right. \right\}
\]
and
\[
U_2 := \left\{ P \in \esh \left| \,
                h_{P_1}(P) < 1 - \tfrac{\varepsilon}{2} \right. \right\}
\]
(cf.\ Eqs.~(\ref{u1}) and (\ref{u2})) are disjoint open neighborhoods of $ P_1 $ and
$ P_2 $, respectively. So $ \mathcal{T}_0 $ is separating.

For an open set $ O \subseteq \R $, $ h_Q^{-1}(O) $ is $ \mathcal{T}_0 $-open. We
next prove that
\begin{equation}\label{u}
U := h_Q^{-1}(O) = \bigcup_{U_{klm} \subseteq U} U_{klm}
\end{equation}
with $ U_{klm} $ according to (\ref{uklm}). Let $ P \in U $. Then
there exists an $ \varepsilon > 0 $ such that the interval
$ ]h_Q(P) - \varepsilon,h_Q(P) + \varepsilon[ $ is contained in $O$. Choose
$ m_0 \in \N $ such that $ \frac{1}{m_0} < \frac{\varepsilon}{2} $,
and choose a member $ q_{l_0} $ of the sequence $ \{ q_l \}_{l \in \N} $
and a member $ Q_{k_0} $ of $ \{ Q_k \}_{k \in \N} $ such that
$ |\tr{PQ} - q_{l_0}| < \frac{1}{2m_0} $ and
$ \no{Q_{k_0} - Q} < \frac{1}{2m_0} $. It follows that
\begin{eqnarray*}
|\tr{PQ_{k_0}} - q_{l_0}|
& \leq & |\tr{PQ_{k_0}} - \tr{PQ}| + |\tr{PQ} - q_{l_0}|    \\
& \leq &  \no{Q_{k_0} - Q} + |\tr{PQ} - q_{l_0}|            \\
&  <   &  \tfrac{1}{m_0}
\end{eqnarray*}
which, by (\ref{uklm}), means that $ P \in U_{k_0l_0m_0} $. We further have
to show that $ U_{k_0l_0m_0} \subseteq U $. To that end, let
$ \widetilde{P} \in U_{k_0l_0m_0} $. Then, from
\[
   \bigl| \tr{\widetilde{P}Q} - \tr{PQ} \bigr| \leq
   \bigl| \tr{\widetilde{P}Q} - \tr{\widetilde{P}Q_{k_0}} \bigr|
 + \bigl| \tr{\widetilde{P}Q_{k_0}} - q_{l_0} \bigr| + |q_{l_0} - \tr{PQ}|
\]
where the first term on the right-hand side is again smaller than
$ \no{Q - Q_{k_0}} $ and, by (\ref{uklm}), the second term is smaller than
$ \frac{1}{m_0} $, it follows that
\[
\bigl| h_Q(\widetilde{P}) - h_Q(P) \bigr|
 =   \bigl| \tr{\widetilde{P}Q} - \tr{PQ} \bigr|
\leq \tfrac{1}{2m_0} + \tfrac{1}{m_0} + \tfrac{1}{2m_0}
 = \tfrac{2}{m_0}
 < \varepsilon.
\]
This implies that $ h_Q(\widetilde{P}) \in
{]h_Q(P) - \varepsilon,h_Q(P) + \varepsilon[} \subseteq O $, i.e.,
$ \widetilde{P} \in h_Q^{-1}(O) = U $. Hence,
$ U_{k_0l_0m_0} \subseteq U $.

Summarizing, we have shown that, for $ P \in U $,
$ P \in U_{k_0l_0m_0} \subseteq U $. Hence,
$ U \subseteq \bigcup_{U_{klm} \subseteq U} U_{klm} \subseteq U $,
and assertion (\ref{u}) has been proved. The finite intersections of sets
of the form $ U = h_Q^{-1}(O) $ constitute a basis of the topology
$ \mathcal{T}_0 $. Since every set $ U = h_Q^{-1}(O) $ is the union of sets
$ U_{klm} $, the intersections of finitely many sets
$ U = h_Q^{-1}(O) $ is the union of finite intersections of sets
$ U_{klm} $. Thus, the finite intersections of the sets
$ U_{klm} $ constitute a countable base of $ \mathcal{T}_0 $. \ $\square$}

Later we shall see that the topological space $ (\esh,\mathcal{T}_0) $ is
homeomorphic to $ (\esh,\mathcal{T}_n) $ as well as to $ (S/S^1,\mathcal{T}_{\nu})
$. So it is also clear by Theorem \ref{thm:esh-sepc} or Theorem
\ref{thm:ss1} that $ (\esh,\mathcal{T}_0) $ is a second-countable Hausdorff
space. The reason for stating Lemma \ref{lem:esh} is that later we
shall make explicit use of the particular countable base given
there.

The weak operator topology on the space $ \bsh $ of the bounded
self-adjoint operators on $ \hi $ is the coarsest topology such that
the linear functionals
\[
A \mapsto \ip{\varphi}{A\psi}
\]
where $ A \in \bsh $ and $ \varphi,\psi \in \hi $, are continuous. It is
sufficient to consider only the functionals
\begin{equation}\label{afun}
A \mapsto \ip{\varphi}{A\varphi}
\end{equation}
where $ \varphi \in \hi $ and $ \no{\varphi} = 1 $. The topology $ \mathcal{T}_w $
induced on $ \esh \subset \bsh $ by the weak operator topology is the
coarsest topology on $ \esh $ such that the restrictions of the linear
functionals (\ref{afun}) to $ \esh $ are continuous. Since these restrictions
are given by
\[
P \mapsto \ip{\varphi}{P\varphi} = \tr{PQ} = h_Q(P)
\]
where $ P \in \esh $ and $ Q := \kb{\varphi}{\varphi} \in \esh $, the topology
$ \mathcal{T}_w $ on $ \esh $ is, according to (\ref{p}), just our topology $ \mathcal{T}_0 $.

Now we compare the weak topology $ \mathcal{T}_0 $ with the metric topology
$ \mathcal{T}_n $.

\begin{theorem}\label{thm:top-esh}
The weak topology $ \mathcal{T}_0 $ on $ \esh $ and the metric topology $ \mathcal{T}_n $ on
$ \esh $ are equal.
\end{theorem}
\proof{
According to (\ref{p}), a neighborhood base of $ P \in \esh $ w.r.t.\ $ \mathcal{T}_0 $
is given by the open sets
\begin{equation}\label{up}
\begin{array}{c}
U(P;Q_1,\ldots,Q_n;\varepsilon) \hspace{7.5cm}                \vspace{2mm}\\
\hspace{0.8cm}
\begin{array}{crl}
& := & \displaystyle{\bigcap_{i=1}^n h_{Q_i}^{-1}
        ( \, ]h_{Q_i}(P) - \varepsilon,h_{Q_i}(P) + \varepsilon[ \, )}
                                                              \vspace{2mm}\\
&  = & \bigl\{ \widetilde{P} \in \esh \, \bigl| \, \bigl|
        h_{Q_i}(\widetilde{P}) - h_{Q_i}(P) \bigr| < \varepsilon \
        {\rm for} \ i=1,\ldots,n \bigr\}                      \vspace{2mm}\\
&  = & \bigl\{ \widetilde{P} \in \esh \, \bigl| \, \bigl|
        \tr{\widetilde{P}Q_i} - \tr{PQ_i} \bigr| < \varepsilon   \
        {\rm for} \ i=1,\ldots,n \bigr\}
\end{array}
\end{array}
\end{equation}
where $ Q_1,\dots,Q_n \in \esh $ and $ \varepsilon > 0 $;
a neighborhood base of $P$ w.r.t.\ $ \mathcal{T}_n $ is given by the open balls
\begin{equation}\label{kep}
K_{\varepsilon}(P) := \bigl\{ \widetilde{P} \in \esh \, \bigl| \,
                      \bigl\| \widetilde{P} - P \bigr\| < \varepsilon
                                                        \bigr\}.
\end{equation}
If $ \bigl\| \widetilde{P} - P \bigl\| < \varepsilon $, then
\[
\bigl| \tr{\widetilde{P}Q_i} - \tr{PQ_i} \bigr|
  =   \bigl| \tr{Q_i(\widetilde{P} - P)} \bigr|
 \leq \no{Q_i}_{\mathrm{tr}} \bigl\| \widetilde{P} - P \bigr\|
  =   \bigl\| \widetilde{P} - P \bigr\|
  <   \varepsilon;
\]
hence, $ K_{\varepsilon}(P) \subseteq U(P;Q_1,\ldots,Q_n;\varepsilon) $. To
show some converse inclusion, take account of Theorem \ref{thm:esh-dist}, part (a), and note
that
\[
\bigl\| \widetilde{P} - P \bigr\|^2
  = 1 - \tr{\widetilde{P}P}
  = \bigl| \tr{\widetilde{P}P} - \tr{PP} \bigr|.
\]
In consequence, by (\ref{up}) and (\ref{kep}),
$ U(P;P;\varepsilon^2) = K_{\varepsilon}(P) $. Hence,
$ \mathcal{T}_0 = \mathcal{T}_n $. \ $\square$}

It looks surprising that the topolgies $ \mathcal{T}_0 $ and
$ \mathcal{T}_n $ coincide. In fact, consider the sequence
$ \{ P_{\varphi_n} \}_{n \in \N} $ where the vectors
$ \varphi_n \in \hi $ constitute an orthonormal system. Then,
w.r.t.\ the weak operator topology, $ P_{\varphi_n} \to 0 $ as
$ n \to \infty $ whereas $ \no{P_{\varphi_n} - P_{\varphi_{n+1}}} =1 $
for all $ n \in \N $. However, $ 0 \not\in \esh $; so
$ \{ P_{\varphi_n} \}_{n \in \N} $ is convergent neither w.r.t.\
$ \mathcal{T}_w = \mathcal{T}_0 $ nor w.r.t.\ $ \mathcal{T}_n $. Finally, like in the case
of the weak operator topology, there is a natural uniform structure
inducing $ \mathcal{T}_0 $. The uniform structures that are canonically related to
$ \mathcal{T}_0 $ and $ \mathcal{T}_n $ are different: $ \{ P_{\varphi_n} \}_{n \in \N} $ is
a Cauchy sequence w.r.t.\ the uniform structure belonging to $ \mathcal{T}_0 $
but not w.r.t.\ that belonging to $ \mathcal{T}_n $, i.e., w.r.t.\ the metric
$ \rho_n $.

We remark that besides $ \mathcal{T}_0 $ and $ \mathcal{T}_w $ several further
weak topologies can be defined on $ \esh $. Let $ \csh $ be the Banach space of the compact
self-adjoint operators and remember that $ (\csh)' = \tsh $. So the weak
Banach-space topologies of $ \csh $, $ \tsh $, and $ \bsh $ as well as
the weak-* Banach-space topologies of $ \tsh $ and $ \bsh $ can
be restricted to $ \esh $, thus giving the topologies
$ \mathcal{T}_1 := \sigma(\csh,\tsh) \cap \esh $,
$ \mathcal{T}_2 := \sigma(\tsh,\csh) \cap \esh $,
$ \mathcal{T}_3 := \sigma(\tsh,\bsh) \cap \esh $,
$ \mathcal{T}_4 := \sigma(\bsh,\tsh) \cap \esh $, and
$ \mathcal{T}_5 := \sigma(\bsh,(\bsh)') \cap \esh $. Moreover,
the strong operator topology induces a topology $ \mathcal{T}_s $ on
$ \esh $. From the obvious inclusions
\[
\mathcal{T}_w \subseteq \mathcal{T}_1 \subseteq \mathcal{T}_2 \subseteq \mathcal{T}_3
      \subseteq \mathcal{T}_{\mathrm{tr}} ,
\]
\[
\mathcal{T}_1 = \mathcal{T}_4 \subseteq \mathcal{T}_5 = \mathcal{T}_1 ,
\]
and
\[
\mathcal{T}_w \subseteq \mathcal{T}_s \subseteq \mathcal{T}_n
\]
as well as from the shown equality
\[
\mathcal{T}_0 = \mathcal{T}_w = \mathcal{T}_n = \mathcal{T}_{\mathrm{tr}}
\]
it follows that the topologies $ \mathcal{T}_1,\ldots,\mathcal{T}_5 $ and $ \mathcal{T}_s $
also coincide with $ \mathcal{T}_0 $.

Finally, we show that all the topologies on $ \esh $ are equivalent to
the quotient topologies $ \mathcal{T}_{\mu} $ and $ \mathcal{T}_{\nu} $ on $ \ph $, resp.,
$ S/S^1 $.

\begin{theorem}\label{thm:ss1-esh}
The mapping $ F \! : S/S^1 \to \esh $, $ F([\varphi]_S := P_{\varphi} $ where
$ \varphi \in S $, is a homeomorphism between the topological spaces
$ (S/S^1,\mathcal{T}_{\nu}) $ and $ (\esh,\mathcal{T}_0) $.
\end{theorem}
\proof{
The mapping $F$ is bijective. The map
$ h_Q \circ F \circ \nu \! : S \to \R $
where $ h_Q $ is any of the functions given by Eq.~(\ref{p}) and
$ \nu $ is the canonical projection from $S$ onto $ S/S^1 $,
reads explicitly
\[
(h_Q \circ F \circ \nu)(\varphi) = h_Q(F([\varphi]_S)) = h_Q(P_{\varphi})
                              = \tr{P_{\varphi}Q}   = \ip{\varphi}{Q\varphi};
\]
therefore, $ h_Q \circ F \circ \nu $ is continuous. Consequently,
for an open set $ O \subseteq \R $,
\[
(h_Q \circ F \circ \nu)^{-1}(O) = \nu^{-1}(F^{-1}(h_Q^{-1}(O)))
\]
is an open set of $S$. By the definition of the quotient topology
$ \mathcal{T}_{\nu} $, it follows that $ F^{-1}(h_Q^{-1}(O)) $ is an open set of
$ S/S^1 $. Since the sets $ h_Q^{-1}(O) $, $ Q \in \esh $,
$ O \subseteq \R $ open, generate the weak topology $ \mathcal{T}_0 $,
$ F^{-1}(U) $ is open for any open set $ U \in \mathcal{T}_0 $. Hence,
$F$ is continuous.

To show that $F$ is an open mapping, let $ V \in \mathcal{T}_{\nu} $ be an open
subset of $ S/S^1 $ and let $ [\varphi_0]_S \in V $. Since the canonical
projection $ \nu $ is continuous, there exists an $ \varepsilon > 0 $
such that
\begin{equation}\label{nuke}
\nu(K_{\varepsilon}(\varphi_0) \cap S) \subseteq V
\end{equation}
where $ K_{\varepsilon}(\varphi_0) := \{ \varphi \in \hi \, | \,
\no{\varphi - \varphi_0} < \varepsilon \} $. Without loss of generality
we assume that $ \varepsilon < 1 $.

The topology $ \mathcal{T}_0 $ is generated by the functions $ h_Q $
according to (\ref{p}); $ \mathcal{T}_0 $ is also generated by the functions
$ P \mapsto g_Q(P) := \sqrt{h_Q(P)} = \sqrt{\tr{PQ}} $. In consequence,
the set
\[
U_{\varepsilon}
 :=  g_Q^{-1} \left( \, \left] 1 - \tfrac{\varepsilon}{2},
                               1 + \tfrac{\varepsilon}{2} \right[ \, \right)
\cap h_Q^{-1} \left( \, \left] 1 - \tfrac{\varepsilon^2}{4},
                               1 + \tfrac{\varepsilon^2}{4} \right[
                                                                  \, \right)
\]
where $ Q := P_{\varphi_0} $ and $ \varphi_0 $ and $ \varepsilon $ are specified
in the preceding paragraph, is $ \mathcal{T}_0 $-open. Using the identity
\[
1 - |\ip{\varphi_0}{\varphi}|^2 = \no{\varphi - \ip{\varphi_0}{\varphi} \varphi_0}^2
\]
where $ \varphi \in \hi $ is also a unit vector, we obtain
\begin{eqnarray*}
U_{\varepsilon}
& = & \left\{ P_{\varphi} \in \esh \left| \,
        |g_Q(P_{\varphi}) - 1| < \tfrac{\varepsilon}{2} \ {\rm and} \
        |h_Q(P_{\varphi}) - 1| < \tfrac{\varepsilon^2}{4} \right. \right\}   \\
& = & \left\{ P_{\varphi} \in \esh \left| \,
      \bigl| |\ip{\varphi_0}{\varphi}| - 1 \bigr| < \tfrac{\varepsilon}{2} \
                                                           {\rm and} \
      \bigl| |\ip{\varphi_0}{\varphi}|^2 - 1 \bigr| < \tfrac{\varepsilon^2}{4}
                                                       \right. \right\}   \\
& = & \left\{ P_{\varphi} \in \esh \left| \,
      \bigl| |\ip{\varphi_0}{\varphi}| - 1 \bigr| < \tfrac{\varepsilon}{2} \
                                                           {\rm and} \,
      \no{\varphi - \ip{\varphi_0}{\varphi} \varphi_0} < \tfrac{\varepsilon}{2}
                                                       \right. \right\}.
\end{eqnarray*}
Now let $ P_{\varphi} \in U_{\varepsilon} $. Since $ \varepsilon < 1 $, we have
that $ \ip{\varphi}{\varphi_0} \neq 0 $. Defining the phase factor
$ \lambda := \frac{\ip{\varphi}{\varphi_0}}{|\ip{\varphi}{\varphi_0}|} $,
it follows that
\begin{eqnarray*}
\no{\lambda \varphi - \varphi_0}
& = & \no{\lambda \varphi - \lambda \ip{\varphi_0}{\varphi} \varphi_0}
        + \no{\lambda \ip{\varphi_0}{\varphi} \varphi_0 - \varphi_0}         \\
& = & \no{\varphi - \ip{\varphi_0}{\varphi} \varphi_0}
        + \bigl\| |\ip{\varphi_0}{\varphi}| \varphi_0 - \varphi_0 \bigr\|    \\
& < & \tfrac{\varepsilon}{2} + \tfrac{\varepsilon}{2}            \\
& = & \varepsilon.
\end{eqnarray*}
That is, $ P_{\varphi} \in U_{\varepsilon} $ implies that
$ \lambda \varphi \in K_{\varepsilon}(\varphi_0) $; moreover,
$ \lambda \varphi \in K_{\varepsilon}(\varphi_0) \cap S $.

Taking the result (\ref{nuke}) into account, we conclude that,
for $ P_{\varphi} \in U_{\varepsilon} $,
$ [\varphi]_S = [\lambda \varphi]_S = \nu(\lambda \varphi) \in V $. Consequently,
$ P_{\varphi} = F([\varphi]_S) \in F(V) $. Hence,
$ U_{\varepsilon} \subseteq F(V) $. Since $ U_{\varepsilon} $
is an open neighborhood of $ P_{\varphi_0} $, $ P_{\varphi_0} $
is an interior point of $ F(V) $. So, for every $ [\varphi_0]_S \in V $,
$ F([\varphi_0]_S) = P_{\varphi_0} $ is an interior point of $ F(V) $, and
$ F(V) $ is a $ \mathcal{T}_0 $-open set. Hence, the continuous bijective map
$F$ is open and thus a homeomorphism. \ $\square$}

In the following, we identify the sets $ \ph $, $ S/S^1 $, and $ \esh $
and call the identified set the {\it projective Hilbert space
$ \ph $}. However, we preferably think about the elements of
$ \ph $ as the one-dimensional orthogonal projections $ P = P_{\varphi} $. On
$ \ph $ then the quotient topologies $ \mathcal{T}_{\mu} $, $ \mathcal{T}_{\nu} $, the
weak topologies $ \mathcal{T}_0 $, $ \mathcal{T}_w $, $ \mathcal{T}_1,\ldots,\mathcal{T}_5 $, $ \mathcal{T}_s $,
and the metric topologies $ \mathcal{T}_n $, $ \mathcal{T}_{\mathrm{tr}} $ coincide. So
we can say that $ \ph $ carries a natural topology $ \mathcal{T} $;
$ (\ph,\mathcal{T}) $ is a second-countable Hausdorff space.

For our purposes, it is suitable to represent this topology $ \mathcal{T} $ as
$ \mathcal{T}_0 $, $ \mathcal{T}_n $, or $ \mathcal{T}_{\mathrm{tr}} $. As already discussed,
the topologies $ \mathcal{T}_0 $, $ \mathcal{T}_n $, and $ \mathcal{T}_{\mathrm{tr}} $ are
canonically related to uniform structures. With respect to the uniform
structure inducing $ \mathcal{T}_0 $, $ \ph $ is not complete. The uniform
structures related to $ \mathcal{T}_n $ and $ \mathcal{T}_{\mathrm{tr}} $ are the same
since they are induced by the equivalent metrics $ \rho_n $ and
$ \rho_{\mathrm{tr}} $; $ (\ph,\rho_n) $ and $ (\ph,\rho_{\mathrm{tr}}) $
are separable complete metric spaces. So $ \mathcal{T} $ can be defined by
a complete separable metric, i.e., $ (\ph,\mathcal{T}) $ is a polish space.

\section{The Measurable Structure of $\ph$}\label{sec:meas}

It is almost natural to define a measurable structure on the
projective Hilbert space $ \ph $ by the $ \sigma $-algebra $ \Xi =
\Xi(\mathcal{T}) $ generated by the $ \mathcal{T} $-open sets, i.e., $ \Xi $ is the
smallest $ \sigma $-algebra containing the open sets of the natural
topology $ \mathcal{T} $. In this way $ (\ph,\Xi) $ becomes a measurable
space where the elements $ B \in \Xi $ are the Borel sets of $ \ph
$. However, since the topology $ \mathcal{T} $ is generated by the
transition-probability functions $ h_Q $ according to Eq.\
(\ref{p}), it is also obvious to define the measurable structure of
$ \ph $ by the $ \sigma $-algebra $ \Sigma $ generated by the
functions $ h_Q $, i.e., $ \Sigma $ is the smallest $ \sigma
$-algebra such that all the functions $ h_Q $ are measurable. A
result due to Misra (1974) \cite[Lemma 3]{mis74} clarifies the
relation between $ \Xi $ and $ \Sigma $. Before stating that result,
we recall the following simple lemma which we shall also use later.

\begin{lemma}\label{lem:sigt-sigb}
Let $ (M,\mathcal{T}) $ be any second-countable topological space,
$ \mB \subseteq \mathcal{T} $ a countable base, and $ \Xi = \Xi(\mathcal{T}) $ the
$ \sigma $-algebra of the Borel sets of $M$. Then
$ \Xi = \Xi(\mathcal{T}) = \Xi(\mB) $ where $ \Xi(\mB) $ is the
$ \sigma $-algebra generated by $ \mB $; $ \mB $ is
a countable generator of $ \Xi $.
\end{lemma}
\proof{
Clearly, $ \Xi(\mB) \subseteq \Xi(\mathcal{T}) $. Since every open set
$ U \in \mathcal{T} $ is the countable union of sets of $ \mB $, it follows that
$ U \in \Xi(\mB) $. Therefore, $ \mathcal{T} \subseteq \Xi(\mB) $ and consequently
$ \Xi(\mathcal{T}) = \Xi(\mB) $. \ $\square$}

\begin{theorem}[{\rm Misra}]\label{thm:misra}
The $ \sigma $-algebra $ \Xi = \Xi(\mathcal{T}) $ of the Borel sets of the
projective Hilbert space $ \ph $ and the $ \sigma $-algbra $ \Sigma $
generated by the transition-probability functions $ h_Q $, $ Q \in \ph $,
are equal.
\end{theorem}
\proof{
Since $ \mathcal{T} $ is generated by the functions $ h_Q $, the latter are
continuous and consequently $ \Xi $-measurable. Since $ \Sigma $ is
the smallest $ \sigma $-algebra such that the functions $ h_Q $ are
measurable, it follows that $ \Sigma \subseteq \Xi $.

Now, by Lemma \ref{lem:esh}, $ \mathcal{T} $ is second-countable, and a countable base
$ \mB $ of $ \mathcal{T} $ is given by the finite intersections of the sets
$ U_{klm} $ according to Eq.\ (\ref{uklm}). Since $ U_{klm} \in \Sigma $,
it follows that $ \mB \subseteq \Sigma $. By Lemma \ref{lem:sigt-sigb}, we conclude that
$ \Xi = \Xi(\mB) \subseteq \Sigma $. Hence, $ \Xi = \Sigma $. \ $\square$}

We remark that our proof of Misra's theorem is much easier than
Misra's proof from 1974. The reason is that we explicitly used the
countable base $ \mB $ of $ \mathcal{T} $ consisting of $ \Sigma $-measurable sets.

Finally, consider the $ \sigma $-algebra $ \Xi_0 $ in $ \ph $ that is
generated by all $ \mathcal{T} $-continuous real-valued functions on $ \ph $, i.e.,
$ \Xi_0 $ is the $ \sigma $-algebra of the Baire sets of $ \ph $. Obviously,
$ \Sigma \subseteq \Xi_0 \subseteq \Xi $; so Theorem \ref{thm:misra} implies that
$ \Xi_0 = \Xi $. This result is, according to a general theorem,
also a consequence of the fact that the topology $ \mathcal{T} $ of
$ \ph $ is metrizable.

Summarizing, our result $ \Sigma = \Xi_0 = \Xi $ manifests that
the projective Hilbert space carries, besides its natural topology
$ \mathcal{T} $, also a very natural measurable structure $ \Xi $.


\section{Supplemental Material}
\setcounter{page}{1}
\subsection{Determination of State Populations}
Our photon-count histograms are well approximated as a weighted sum of three Poissonians (a two-parameter probability mass function, because the three weights $P_2$, $P_1$, and $P_0$ must add up to unity). These three Poissonians correspond to two bright ions ($\qubitdowndown$), a single bright ion ($\qubitdownup$ and $\qubitupdown$), and two dark ions ($\qubitupup$). If we define $k_2$, $k_1$, and $k_0$ as the mean number of counts in each Poissonian (determined via independent measurements and maintained at fixed values during the following fitting procedure), we can write the probability mass function (PMF) as
\begin{equation*}
\begin{split}
    \mathrm{PMF}(n;P_0,P_1)&=P_0\frac{k_0^n \exp{(-k_0)}}{n!}+P_1\frac{k_1^n \exp{(-k_1)}}{n!} \\
    &+(1-P_1-P_0)\frac{k_2^n \exp{(-k_2)}}{n!}.
\end{split}
\end{equation*}
For a given dataset of identical experiments, with each repetition in the dataset labeled by $i$, we have a set of measured photon counts $\{n_i\}$. With this we define the usual log-likelihood as
\begin{equation*}
\begin{split}
    l(P_0,P_1)=\sum_{i} \ln(\mathrm{PMF}(n_i;P_0,P_1)).
\end{split}
\end{equation*}
We then numerically maximize this function to find the most likely values for $P_0$, $P_1$, and $P_2=1-P_1-P_0$.

\subsection{Fidelity Distribution}
\begin{figure}[htp]
	\includegraphics{fidelity_histogram.pdf}
	\caption{\label{fig:fidelity_histogram} Histogram of bootstrapped Bell-state fidelities.}
\end{figure}
Figure~\ref{fig:fidelity_histogram} gives a histogram of the Bell-state fidelities obtained from the bootstrapping procedure defined in the main text.

\subsection{Gate Errors}
1-qubit rotation fidelity is limited by qubit frequency noise, laser phase noise, and laser intensity fluctuations. Qubit frequency noise results from variations in the background magnetic field; our qubit is first-order sensitive to these variations which arise in our lab from changes such as the movement of a chair or the motion of a nearby elevator. Although the incorporation of a spin echo into our gate should alleviate the influence of these frequency variations, 1-qubit rotations (of finite duration) still suffer to some extent. To further reduce these errors, we interleave calibrations of the qubit frequency within our gate experiments, performing a Ramsey frequency calibration experiment between each repetition of the gate experiment. In this way the frequency is stabilized to approximately $\pm$250~Hz, a level at which it is expected to contribute an error of only $4\times 10^{-6}$ to the 1-qubit pulses. We use a NdFeB permanent magnet to minimize fast fluctuations of the $1.07~\text{mT}$ experiment bias field at the expense of slow (correctable) thermal drifts. The remaining error is likely caused by faster phase noise, intensity noise, or fast ambient magnetic field fluctuations.

Additional errors arise due to variations in the phase of the 729 nm laser between the various 1-qubit pulses. Such phase fluctuations can occur due to instability in the laser itself, vibrations of the mirrors along the beampath, and motion of the ion or vacuum chamber. Ramsey coherence measurements performed by repeating the gate experiment [pulses (1), (3), and (5) of Fig.~\ref{fig:pulse_profiles}(b)] while extinguishing the ODF beams indicate that this error contribution is $<2\times 10^{-4}$. Here a marked improvement was observed after the installation of an electro-optic modulator in the output path of the 729 nm laser for phase-noise stabilization.

Fundamentally the fidelity is limited by metastable decay from the $D_{5/2}$ level as well as by spontaneous photon scattering from the ODF beams. These contributions are discussed in detail in Ref.~\cite{sawyer_2021}. We estimate a photon scattering error of $1.1\times 10^{-5}$ at our c.m. frequency of 2.5~MHz. We estimate the metastable decay error as follows. Assuming on average that 50\% of the population lies in the $D_{5/2}$ level until the detection laser is turned on 50~$\mu$s later (this includes the time required for the parity analysis pulse as well as some additional short delays), there is a metastable decay error of $2\times 0.5\times 50\:\mu\mathrm{s}/1.174\:\mathrm{s}=4.3\times 10^{-5}$. In addition, the ions are illuminated with the detection laser for 50~$\mu$s before we begin counting photons (this allows the intensity of the laser to stabilize). However, due to the Zeno effect, the metastable decay rate during this second interval is roughly half its unperturbed value, leading to an error of $2.1\times 10^{-5}$ (we have confirmed this reduction in decay rate via a full simulation of the Schr{\"o}dinger equation including all the relevant levels in $^{40}$Ca$^{+}$). Therefore, the total metastable decay error is approximately $6\times 10^{-5}$. A closely related source of error is metastable decay during the measurement itself. In the case of 100~$\mu$s (200~$\mu$s) measurement duration used for the data in Fig.~\ref{fig:parity_amplitude} (Fig.~\ref{fig:population_scan}), we calculate a lifetime-limited detection error of $4.3\times 10^{-5}$ ($8.6\times 10^{-5}$), again considering the reduced decay rate during illumination. These are likely overestimates, because decays that occur later during the measurement interval should only minimally impact the photon count histograms. We have chosen not to SPAM-correct our reported fidelity in order to reduce statistical complexity.

To initialize to $\qubitdowndown$, we employ optical pumping with a circularly polarized 397 nm laser beam followed by seven cycles of two-step optical pumping with the 729~nm and 854~nm laser beams exciting along $S_{1/2}$~$(m_j=-1/2)\rightarrow D_{5/2}$~$(m_j= 1/2)\rightarrow P_{3/2}$. The initial 397~nm pumping step yields an error of $\sim1\times10^{-3}$, and the following steps further reduce this error to $<10^{-6}$, which we exclude from the error budget table in the Letter.

In the Lamb-Dicke limit there is no dependence of the gate on ion temperature~\cite{sorensen_2000}. A derivation of the higher-order deviations from perfect Lamb-Dicke assumptions reveals a gate error of $\epsilon_g=(\pi^2/4) \eta^4 \bar{n}(2\bar{n}+1)$, where $\bar{n}$ represents the thermal mean number of quanta in and $\eta$ is the Lamb-Dicke parameter of the mode \cite{sorensen_2000,ballance_2016}. This dependence, which applies independently for both the BM and c.m. finite-temperature contributions, is confirmed via numerical simulations. We estimate $\bar{n}_\mathrm{BM}<0.1$, $\bar{n}_\mathrm{c.m.}<0.1$, $\eta_\mathrm{BM}=0.063$, $\eta_\mathrm{c.m.}=0.083$ so that $\epsilon_g<2\times 10^{-5}$.

Heating of the gate mode during operation is unavoidable. We operate the gate near the BM mode instead of near the c.m. mode because of the former's significantly lower $<1.4$~quanta/s heating rate (the c.m. heating rate is $33(14)$~quanta/s). This translates into a gate error of $\epsilon_g=\dot{\bar{n}} \tau_g/(2 K)=7\times 10^{-6}$ ($\tau_g$ here represents the sum of the two ODF pulse durations, and $K=2$ is the number of loops in phase space) \cite{sorensen_2000}.

Variations in the trapping potential can lead to errors via two possible mechanisms. Fast electric field variations can change the position of the trap potential within the Ramsey sequence [Fig~\ref{fig:pulse_profiles}(b)]. This manifests as phase noise both on the optical-dipole force interaction and on the 1-qubit rotations \cite{pino_2020}. For the 2~Hz low-pass filters in use on our dc electrodes, we expect this error contribution is negligible. However, it may become relevant if filters with a higher cutoff frequency are used in the future.

Slower variations in the trapping potential (for example due to thermal drift in the DAC electronics or due to slow charging or discharging of the trap surfaces) can cause variations in the trap frequency from shot to shot of the Ramsey sequence. We have characterized the trap frequency by driving ion motion with a radio-frequency electric field and detecting the ions’ response. For an 8~ms pulse, the resulting frequency response is well fit to a Gaussian with a standard deviation of $\Delta \omega/2\pi=63(11)$~Hz. This predicts an error $\epsilon_g=(\pi^2/4) (\Delta \omega/\delta)^2\approx1\times 10^{-6}$ for our gate detuning $\delta/2\pi=114$~kHz. It is likely that this frequency uncertainty is dominated by Kerr cross-coupling between the (Doppler-cooled) radial rocking modes and the BM mode \cite{nie_2009}

To lowest order, fluctuations in laser intensity between the two ODF pulses lead to errors because of the associated uncompensated phase accumulation. To estimate the effect of gate laser beam power fluctuations, we measure the power stability of the two 532~nm laser beams separately with the photodetectors used for power stabilization. From this, we calculate an Allan variance of power fluctuations at $\tau_\mathrm{ODF}=12$~$\mu$s that is consistent with the photon shot noise at each detector. The resulting error contribution is bounded from above by $1\times10^{-5}$~\cite{sawyer_2021}, which is included with the first entry in Table~\ref{table:error_budget}.

In addition to power instability, beam pointing or ion position variations will also cause intensity to fluctuate. We measure this experimentally by performing pulses (1)-(5) while illuminating the ions with only one of the two ODF beams. Performing these measurements with sufficiently many repetitions to accumulate good statistics is time consuming, but we routinely observe no errors in $1\,000$ repetitions, bounding this error source to $<10^{-3}$. Extending this characterization to include a similar number of experiments ($\sim10\,000$) as are used for Bell-state verification would provide a more restrictive bound. Slow (shot-to-shot) variations in ODF laser-beam intensity will also contribute to the Bell-state infidelity. This error source is difficult to quantify independently of the entangling gate but is reflected in the first entry of Table~\ref{table:error_budget}.

\end{document}

\section{Supplemental Material}
\setcounter{page}{1}
\subsection{Determination of State Populations}
Our photon-count histograms are well approximated as a weighted sum of three Poissonians (a two-parameter probability mass function, because the three weights $P_2$, $P_1$, and $P_0$ must add up to unity). These three Poissonians correspond to two bright ions ($\qubitdowndown$), a single bright ion ($\qubitdownup$ and $\qubitupdown$), and two dark ions ($\qubitupup$). If we define $k_2$, $k_1$, and $k_0$ as the mean number of counts in each Poissonian (determined via independent measurements and maintained at fixed values during the following fitting procedure), we can write the probability mass function (PMF) as
\begin{equation*}
\begin{split}
    \mathrm{PMF}(n;P_0,P_1)&=P_0\frac{k_0^n \exp{(-k_0)}}{n!}+P_1\frac{k_1^n \exp{(-k_1)}}{n!} \\
    &+(1-P_1-P_0)\frac{k_2^n \exp{(-k_2)}}{n!}.
\end{split}
\end{equation*}
For a given dataset of identical experiments, with each repetition in the dataset labeled by $i$, we have a set of measured photon counts $\{n_i\}$. With this we define the usual log-likelihood as
\begin{equation*}
\begin{split}
    l(P_0,P_1)=\sum_{i} \ln(\mathrm{PMF}(n_i;P_0,P_1)).
\end{split}
\end{equation*}
We then numerically maximize this function to find the most likely values for $P_0$, $P_1$, and $P_2=1-P_1-P_0$.

\subsection{Fidelity Distribution}
\begin{figure}[htp]
	\includegraphics{fidelity_histogram.pdf}
	\caption{\label{fig:fidelity_histogram} Histogram of bootstrapped Bell-state fidelities.}
\end{figure}
Figure~\ref{fig:fidelity_histogram} gives a histogram of the Bell-state fidelities obtained from the bootstrapping procedure defined in the main text.

\subsection{Gate Errors}
1-qubit rotation fidelity is limited by qubit frequency noise, laser phase noise, and laser intensity fluctuations. Qubit frequency noise results from variations in the background magnetic field; our qubit is first-order sensitive to these variations which arise in our lab from changes such as the movement of a chair or the motion of a nearby elevator. Although the incorporation of a spin echo into our gate should alleviate the influence of these frequency variations, 1-qubit rotations (of finite duration) still suffer to some extent. To further reduce these errors, we interleave calibrations of the qubit frequency within our gate experiments, performing a Ramsey frequency calibration experiment between each repetition of the gate experiment. In this way the frequency is stabilized to approximately $\pm$250~Hz, a level at which it is expected to contribute an error of only $4\times 10^{-6}$ to the 1-qubit pulses. We use a NdFeB permanent magnet to minimize fast fluctuations of the $1.07~\text{mT}$ experiment bias field at the expense of slow (correctable) thermal drifts. The remaining error is likely caused by faster phase noise, intensity noise, or fast ambient magnetic field fluctuations.

Additional errors arise due to variations in the phase of the 729 nm laser between the various 1-qubit pulses. Such phase fluctuations can occur due to instability in the laser itself, vibrations of the mirrors along the beampath, and motion of the ion or vacuum chamber. Ramsey coherence measurements performed by repeating the gate experiment [pulses (1), (3), and (5) of Fig.~\ref{fig:pulse_profiles}(b)] while extinguishing the ODF beams indicate that this error contribution is $<2\times 10^{-4}$. Here a marked improvement was observed after the installation of an electro-optic modulator in the output path of the 729 nm laser for phase-noise stabilization.

Fundamentally the fidelity is limited by metastable decay from the $D_{5/2}$ level as well as by spontaneous photon scattering from the ODF beams. These contributions are discussed in detail in Ref.~\cite{sawyer_2021}. We estimate a photon scattering error of $1.1\times 10^{-5}$ at our c.m. frequency of 2.5~MHz. We estimate the metastable decay error as follows. Assuming on average that 50\% of the population lies in the $D_{5/2}$ level until the detection laser is turned on 50~$\mu$s later (this includes the time required for the parity analysis pulse as well as some additional short delays), there is a metastable decay error of $2\times 0.5\times 50\:\mu\mathrm{s}/1.174\:\mathrm{s}=4.3\times 10^{-5}$. In addition, the ions are illuminated with the detection laser for 50~$\mu$s before we begin counting photons (this allows the intensity of the laser to stabilize). However, due to the Zeno effect, the metastable decay rate during this second interval is roughly half its unperturbed value, leading to an error of $2.1\times 10^{-5}$ (we have confirmed this reduction in decay rate via a full simulation of the Schr{\"o}dinger equation including all the relevant levels in $^{40}$Ca$^{+}$). Therefore, the total metastable decay error is approximately $6\times 10^{-5}$. A closely related source of error is metastable decay during the measurement itself. In the case of 100~$\mu$s (200~$\mu$s) measurement duration used for the data in Fig.~\ref{fig:parity_amplitude} (Fig.~\ref{fig:population_scan}), we calculate a lifetime-limited detection error of $4.3\times 10^{-5}$ ($8.6\times 10^{-5}$), again considering the reduced decay rate during illumination. These are likely overestimates, because decays that occur later during the measurement interval should only minimally impact the photon count histograms. We have chosen not to SPAM-correct our reported fidelity in order to reduce statistical complexity.

To initialize to $\qubitdowndown$, we employ optical pumping with a circularly polarized 397 nm laser beam followed by seven cycles of two-step optical pumping with the 729~nm and 854~nm laser beams exciting along $S_{1/2}$~$(m_j=-1/2)\rightarrow D_{5/2}$~$(m_j= 1/2)\rightarrow P_{3/2}$. The initial 397~nm pumping step yields an error of $\sim1\times10^{-3}$, and the following steps further reduce this error to $<10^{-6}$, which we exclude from the error budget table in the Letter.

In the Lamb-Dicke limit there is no dependence of the gate on ion temperature~\cite{sorensen_2000}. A derivation of the higher-order deviations from perfect Lamb-Dicke assumptions reveals a gate error of $\epsilon_g=(\pi^2/4) \eta^4 \bar{n}(2\bar{n}+1)$, where $\bar{n}$ represents the thermal mean number of quanta in and $\eta$ is the Lamb-Dicke parameter of the mode \cite{sorensen_2000,ballance_2016}. This dependence, which applies independently for both the BM and c.m. finite-temperature contributions, is confirmed via numerical simulations. We estimate $\bar{n}_\mathrm{BM}<0.1$, $\bar{n}_\mathrm{c.m.}<0.1$, $\eta_\mathrm{BM}=0.063$, $\eta_\mathrm{c.m.}=0.083$ so that $\epsilon_g<2\times 10^{-5}$.

Heating of the gate mode during operation is unavoidable. We operate the gate near the BM mode instead of near the c.m. mode because of the former's significantly lower $<1.4$~quanta/s heating rate (the c.m. heating rate is $33(14)$~quanta/s). This translates into a gate error of $\epsilon_g=\dot{\bar{n}} \tau_g/(2 K)=7\times 10^{-6}$ ($\tau_g$ here represents the sum of the two ODF pulse durations, and $K=2$ is the number of loops in phase space) \cite{sorensen_2000}.

Variations in the trapping potential can lead to errors via two possible mechanisms. Fast electric field variations can change the position of the trap potential within the Ramsey sequence [Fig~\ref{fig:pulse_profiles}(b)]. This manifests as phase noise both on the optical-dipole force interaction and on the 1-qubit rotations \cite{pino_2020}. For the 2~Hz low-pass filters in use on our dc electrodes, we expect this error contribution is negligible. However, it may become relevant if filters with a higher cutoff frequency are used in the future.

Slower variations in the trapping potential (for example due to thermal drift in the DAC electronics or due to slow charging or discharging of the trap surfaces) can cause variations in the trap frequency from shot to shot of the Ramsey sequence. We have characterized the trap frequency by driving ion motion with a radio-frequency electric field and detecting the ions’ response. For an 8~ms pulse, the resulting frequency response is well fit to a Gaussian with a standard deviation of $\Delta \omega/2\pi=63(11)$~Hz. This predicts an error $\epsilon_g=(\pi^2/4) (\Delta \omega/\delta)^2\approx1\times 10^{-6}$ for our gate detuning $\delta/2\pi=114$~kHz. It is likely that this frequency uncertainty is dominated by Kerr cross-coupling between the (Doppler-cooled) radial rocking modes and the BM mode \cite{nie_2009}

To lowest order, fluctuations in laser intensity between the two ODF pulses lead to errors because of the associated uncompensated phase accumulation. To estimate the effect of gate laser beam power fluctuations, we measure the power stability of the two 532~nm laser beams separately with the photodetectors used for power stabilization. From this, we calculate an Allan variance of power fluctuations at $\tau_\mathrm{ODF}=12$~$\mu$s that is consistent with the photon shot noise at each detector. The resulting error contribution is bounded from above by $1\times10^{-5}$~\cite{sawyer_2021}, which is included with the first entry in Table~\ref{table:error_budget}.

In addition to power instability, beam pointing or ion position variations will also cause intensity to fluctuate. We measure this experimentally by performing pulses (1)-(5) while illuminating the ions with only one of the two ODF beams. Performing these measurements with sufficiently many repetitions to accumulate good statistics is time consuming, but we routinely observe no errors in $1\,000$ repetitions, bounding this error source to $<10^{-3}$. Extending this characterization to include a similar number of experiments ($\sim10\,000$) as are used for Bell-state verification would provide a more restrictive bound. Slow (shot-to-shot) variations in ODF laser-beam intensity will also contribute to the Bell-state infidelity. This error source is difficult to quantify independently of the entangling gate but is reflected in the first entry of Table~\ref{table:error_budget}.

\end{document}


\section{Introduction}

The prospects for new physics searches at the LHC and other future
colliders are already constrained by rare processes that are sensitive to
small deviations from the Standard Model. A prime example of such a
low-energy constraint is $b \to s \gamma$ decay \cite{bsg,bsgth}. This, the anomalous
magnetic moment of the muon, $g_\mu - 2$ \cite{g-2}, and the Higgs mass, $m_h$ \cite{mh},
are among the most important indirect constraints on extensions of the
Standard Model, such as the minimal supersymmetric extension of the
Standard Model (MSSM).

The decay $B_s \to \mu^+ \mu^-$ has been emerging as another interesting
potential constraint on the parameter space of models for physics beyond
the Standard Model, such as the MSSM \cite{Dedes,Arnowitt,ko,baer}.
The Fermilab Tevatron collider
already has an interesting upper limit on the branching ratio for $B_s \to
\mu^+ \mu^-$ \cite{cdf}, and is expected soon to increase significantly its
sensitivity to $B_s \to \mu^+ \mu^-$ decay, whilst the LHC sensitivity
will reach down to the Standard Model rate \cite{Buras}. Since its branching ratio is
known to increase rapidly for large values of the ratio of Higgs v.e.v.'s,
$\tan \beta$ \cite{calcs,Babu,Bobeth}, increasing like its sixth power, these present and future
sensitivities are particularly important for models with large $\tan
\beta$.

In view of its potential importance for the MSSM, it is important to treat
the $B_s \to \mu^+ \mu^-$ decay constraint with some care, as has already
been done for the $b \to s \gamma$, $g_\mu - 2$ and $m_h$ constraints. In
each of these cases, the interpretation depends on uncertainties in
theoretical calculations, which should be propagated carefully and
combined with the experimental errors if limits are to be calculated at
some well-defined confidence level, or if a global fit to MSSM parameters
is to be attempted.

Both of these issues are important also in the case of $B_s \to \mu^+
\mu^-$ decay. As concerns the theoretical uncertainties, it is important
to include consistently all the available one-loop MSSM contributions, and
avoid any approximate treatments of any individual terms, since
cancellations are potentially important, and also to include knowledge of
the higher-order QCD corrections to the lowest-order MSSM loop diagrams.
As we discuss in this paper, other sources of error and uncertainty are
also important. These include, for example, the uncertainties in the $B_s$
meson parameters, principally the decay constant $f_{B_s}$. Since the $B_s
\to \mu^+ \mu^-$ decay rate is dominated by the exchange of the
pseudoscalar Higgs boson $A$, the value of $m_A$ is also an important
uncertainty.

Bounds on supersymmetry are often explored in a constrained model with
universal soft supersymmetry-breaking scalar masses $m_0$, gaugino masses
$m_{1/2}$ and trilinear couplings $A_0$ at some GUT input scale. In this
CMSSM, the values of $m_A$ and magnitude of the Higgs mixing parameter
$\mu$ (but not its sign) are in principle fixed by the electroweak
vacuum conditions. However, these predictions are necessarily approximate.
For example, the value of $m_A$ predicted as a function of the independent
parameters $m_{1/2}, m_0, A_0$ and $\tan \beta$ has significant
uncertainties associated with the lack of precision with which the heavy
quark masses $m_t$ and $m_b$ are known, as we discuss extensively later in
this paper. Moreover, rather different values of $m_A$ would be predicted
in models where the universality assumptions of the CMSSM are relaxed. For
example, much smaller values of $m_A$ are attainable in models with
non-universal Higgs masses (NUHM).

When interpreting experimental upper bounds (or measurements) within any
specific model such as the CMSSM, care must be taken to incorporate the
uncertainties in auxiliary parameters such as $f_{B_s}$, $m_t$ and $m_b$.
These should be propagated and combined with the experimental likelihood
function when quoting sensitivities in, e.g., the $(m_{1/2}, m_0)$ plane
at some fixed level of confidence. Moreover, one must also be aware of
model dependences within the assumed framework such as the value of $A_0$
in the CMSSM, as well as the effects of possible deviations from the model
framework such as non-universal Higgs masses.

We exemplify these points in a discussion of uncertainties in the
interpretation of the present and likely future sensitivities of the
Fermilab Tevatron collider and the LHC to $B_s \to \mu^+ \mu^-$ decay,
assuming $\mu > 0$ as preferred by $g_\mu - 2$. In particular, we show
that the uncertainties in $f_{B_s}$, $m_t$ and $m_b$ each shrink
significantly the regions that might otherwise appear to be excluded by
the present limit, or might appear to be if the decay is not discovered at
the likely future sensitivity. We compare the resulting $B_s \to \mu^+
\mu^-$ constraints with other existing constraints such as $b \to s
\gamma$, discussing how they vary with $A_0$ and commenting on the
situation within the NUHM.

\section{Calculation of {\boldmath $B_s \goto \mu^+ \, \mu^-$} Decay}

The branching ratio for the decay $B_s \goto \mu^+ \, \mu^-$ is given by
\begin{eqnarray}
\mathcal{B}(B_s \goto \mu^+ \, \mu^-) &=& \frac{G_F^2 \alpha^2}{16 \pi^3} \frac{M_{B_s}^5 f_{B_s}^2
\tau_B }{4} |V_{tb}V_{ts}^*|^2 \sqrt{1-\frac{4 m_\mu^2}{M_{B_s}^2}} \nonumber \\
  &\times& \left\{    \left(1-\frac{4 m_\mu^2}{M_{B_s}^2}\right) | C_S |^2
  + \left |C_P-2 \, C_A \frac{m_\mu}{M_{B_s}^2} \right |^2   \right\} \, ,
\label{eq:braratio}
\end{eqnarray}
where the one-loop corrected Wilson coefficients $C_{S,P}$ are taken
from~\cite{Bobeth} and $C_A$ is defined in terms of $Y(x_t)$,
following~\cite{Logan}, as $C_A=Y(x_t)/\sin^2 \theta_W$ where
\begin{equation}
Y(x_t)=1.033 \left( \frac{m_t(m_t)}{170 {\rm \, Ge\kern-0.125em V}} \right)^{1.55} \, .
\end{equation}
The function $Y(x_t)$ incorporates both leading \cite{Inami} and
next-to-leading order~\cite{Buras} QCD corrections, and $m_t(m_t)$ is the
running top-quark mass in the $\overline{MS}$ scheme.
The precise value of $m_t(m_t)$ depends somewhat
on the set of supersymmetric parameters and our choice of the physical top
quark mass $m_t=178 \pm 4$~GeV~\cite{D0} that we use in this paper.
The Wilson coefficients $C_{S,P}$ have been multiplied by
$1/(1+\epsilon_b)^2$, where $\epsilon_b$ incorporates the full
supersymmetric one-loop correction to the bottom-quark Yukawa
coupling~\cite{mbcor,Carena, Pierce}. It is known that, since $\epsilon_b$ is
proportional to $\tan \beta$, this correction may be significant in the
large-$\tan \beta$ regime we study here \cite{Dedes,Arnowitt}.
These corrections to $\epsilon_b$ are flavour independent.
In addition, it is important to include the flavour-violating
contributions arising from the Higgs and chargino couplings
at the one-loop level. These corrections result in effective
one-loop corrected values for the  Kobayashi-Maskawa (KM)
matrix elements~\cite{Babu,Isidori}, which we include as described
in~\cite{Buras1,Tata}. In particular, these corrections modify the Wilson
coefficients involved in Eq.~(\ref{eq:braratio}), as can be seen
in Eqs. (6.35) and (6.36) in ~\cite{Buras1} or in Eq. (14) in ~\cite{Tata}.
In addition, the latter study includes the effects of squark mixing, which are included here as well.
Below, we discuss the behaviour of the dominant
contribution to the Wilson coefficients, mainly  to illustrate their
dependence on the pseudo-scalar Higgs boson mass.

The Wilson coefficients $C_{S,P}$ receive four contributions in the
context of MSSM, due to Higgs bosons, counter-terms, box and penguin
diagrams.  The Higgs-boson corrections were calculated in~\cite{Logan},
and the rest of the supersymmetric corrections in~\cite{calcs,Babu}.
The full one-loop corrections have been studied and presented
comprehensively in~\cite{Bobeth}.
In our numerical analysis, we implement the full
one-loop corrections taken from this work
as well as the dominant NLO QCD corrections
studied in~\cite{Buras2}, as well as
the flavour-changing gluino contribution~\cite{Tata,Bobeth2} .
The Higgs-boson, box and
penguin corrections to $C_{S,P}$ are proportional to $\tan^2\beta$, while
the counter-term corrections dominate in the large-$\tan \beta$ limit, as
they are proportional to $\tan^3\beta$.

In order to understand the behaviour of the branching ratio in the
$(m_{1/2},m_0)$ plane in the context of the CMSSM, we focus attention on
the counter-terms which are mediated by $A,H,h$ exchange as
seen in Eqs. (5.1) and (5.2) of~\cite{Bobeth}. As seen in Eq. (5.13) of~\cite{Bobeth},
the $B_s \goto \mu^+ \, \mu^-$ decay amplitude $\propto 1/m_{A}^2$ in the large-$\tan \beta$
limit, and the decay rate $\propto 1/m_{A}^4$.  This underlines the
importance of knowing or calculating $m_{A}$ as accurately as
possible.

The counter-term contribution to $C_{S,P}$ is given by \cite{Bobeth}
\begin{eqnarray}\label{susy:result:count}
C_{S,P}^{\rm CT} &=&\mp
\frac{m_\mu\tan^3\beta}{\sqrt{2}M_W^2
m_{A}^2}\sum_{i=1}^{2}\sum_{a=1}^{6}\sum_{m,n=1}^{3}
 [m_{\tilde{\chi}_i^{\pm}} D_3(y_{ai})U_{i2}(\Gamma^{U_L})_{am}\Gamma^a_{imn}],
\end{eqnarray}
where
\begin{equation}\label{susy:result:gamma}
\Gamma^a_{imn}= \frac{1}{2\sqrt{2}\sin^2\theta_W}
[\sqrt{2}M_W V_{i1}(\Gamma^{U_L\dagger})_{na}-(M_U)_{nn}V_{i2}
     (\Gamma^{U_R\dagger})_{na}]\lambda_{mn},
\end{equation}
and $M_U \equiv  {\rm diag}(m_u, m_c, m_t)$. $U$ and $V$ are the chargino
diagonalization matrices, $\Gamma^{U_L}$ and  $\Gamma^{U_R}$ are $6 \times 3$
squark diagonalization matrices, and $D_3(x) \equiv x\ln x/(1-x)$.
Additionally, $y_{ai}$ is defined in
Eq. (5.10) of~\cite{Bobeth} as $y_{ai} \equiv \msup{a}^2/\mchr{i}^2$,
where $\msup{a}^2 \equiv \{ m_{{\tilde{u}_L}}^2, m_{{\tilde{c}_L}}^2,
m_{{\tilde{t}_1}}^2,
m_{{\tilde{u}_R}}^2, m_{{\tilde{c}_R}}^2,  m_{{\tilde{t}_2}}^2 \}$.
Finally, $\lambda_{mn} \equiv V_{mb}V^*_{ns}/V_{tb}V^*_{ts}$.

We can split the counter-term contribution  into two terms:
one that is proportional to $M_W$ and another that is
proportional to $m_t$.
Starting with the term whose numerator depends on $M_W$,
it is easy to see that the non-vanishing terms
stem from the following combinations of
indices:
$\{a,n,m\}=\{111,222,333,633\}$ and $i=1,2$. However, the first term
$\{111\}$ is
suppressed, as it is  proportional to
$\lambda_{11}=V_{ub}\, V_{us}/V_{tb}\, V_{ts} \simeq - 0.022$,
whereas the
second is not suppressed, because it is
proportional to
$\lambda_{22}=V_{cb}\, V_{cs}/V_{tb}\, V_{ts} \simeq - 1$.
Nor are the third and fourth terms suppressed, as they are multiplied by
$\lambda_{33}=1$.
Thus, the part of the counter-term contribution to the Wilson coefficient
that is $\propto M_W$ is
\begin{eqnarray}
C_S^{CT,M_W}=&-& \sqrt{2} M_W \, f  \left\{  \mchr{1}\, V_{11}\,U_{12}
\left[ \lambda_{22}\, D_3(y_{21})+ \lambda_{33}\, \left( \cos^2 \theta_{\tilde t} \, D_3(y_{31}) + \sin^2 \theta_{\tilde t} \,
D_3(y_{61} ) \right)  \right] \right. \nonumber \\
&+& \left.   \mchr{2}\, V_{21}\,U_{22}
\left[ \lambda_{22}\,D_3(y_{22})+ \lambda_{33}\, \left(\cos^2 \theta_{\tilde t} \, D_3(y_{32}) + \sin^2 \theta_{\tilde t}
\, D_3(y_{62}) \right) \right]    \right\} ,
\label{eq:ctmw}
\end{eqnarray}
where
\begin{equation}
f \equiv \frac{m_\mu \tan^3\beta }{4M_W^2 \sin^2 \theta_W m_A^2} \, ,
\label{eq:ffact}
\end{equation}
and we have ignored in (\ref{eq:ctmw}) terms that are proportional
to $\lambda_{11}$. The unitarity
of the KM
matrix implies that $\lambda_{11}+\lambda_{22}+\lambda_{33}=0$, which
for small $\lambda_{11}$ means $\lambda_{22}=-\lambda_{33}$, resulting
in the suppression of $C_S^{CT,M_W}$.

Turning now to the terms that increase with the charge-2/3 quark masses,
we see that the terms with $n=3$ (the top-quark contributions) dominate
the first- and second-generation terms in $\Gamma_{imn}^a$.
Specifically, the dominant terms have $\{a,n,m\}=\{333,633\}$. In
addition, we notice that the $i=2$ part is important, since it is
proportional to $V_{22} \, U_{22}$, while the $i=1$ term is proportional
to $V_{12} \, U_{12}$. Hence it is sufficient to take
\begin{equation}
C_S^{CT,m_t} = m_t  \, f \,\, \left(  \frac{\sin2\theta_{\tilde t}}{2}\right)
                     \mchr{2}\,      [  D(y_{32})-D(y_{62}) ] \, ,
\label{eq:ctmt}
\end{equation}
where we set $V_{22}\simeq U_{22} \simeq 1$.
The expression
(\ref{eq:ctmt}) is the approximation derived in~\cite{Arnowitt}.
The GIM cancellation in  (\ref{eq:ctmw}) means that the counterterm
contribution to the Wilson coefficients, which is the dominant one, can be
approximated relatively well by (\ref{eq:ctmt}). However,
in our analysis we use the full expressions given in~\cite{Bobeth}, as
well as the gluino correction and the flavour-violating corrections to
Kobayashi-Maskawa matrix elements as described above.

\section{Dependence of {\boldmath $m_A$}  on {\boldmath $m_t$} and on
{\boldmath $m_b$}}

It is clear from the discussion above that the mass of the
pseudoscalar Higgs boson, $m_{A}$, is an important ingredient in
calculating the branching ratio for the decay $B_s \goto \mu^+ \, \mu^-$, since it enters in
the fourth power in the denominators of the Wilson coefficients $C_S$ and
$C_P$ in (\ref{eq:braratio}). Therefore, to further our discussion of the
uncertainties in the $B_s \goto \mu^+ \, \mu^-$ branching ratio, we now discuss the
uncertainties in the calculated value of $m_A$.

The electroweak symmetry breaking conditions may be written in the form:
\begin{equation}
m_A^2  = m_{H_1}^2 + m_{H_2}^2 + 2 \mu^2 + \Delta_A
\label{eq:mamass}
\end{equation}
and
\begin{equation}
\mu^2 = \frac{m_{H_1}^2 - m_{H_2}^2 \tan^2 \beta + \frac{1}{2} m_{\ss Z}^2 (1 - \tan^2 \beta)
+ \Delta_\mu^{(1)}}{\tan^2 \beta - 1 + \Delta_\mu^{(2)}},
\label{eq:minmu}
\end{equation}
where $\Delta_A$ and $\Delta_\mu^{(1,2)}$ are loop
corrections~\cite{Pierce,Barger:1993gh,deBoer:1994he,Carena:2001fw,erz}.
The exact forms of the radiative corrections to $\mu$ and $m_A$
are not needed here, but it is important to note that, at large $\tan
\beta$, the dominant contribution to $\Delta_\mu^{(1)}$ contains a term
which is proportional to $h_t^2 \tan \beta^2$, whereas the dominant
contribution to $m_A^2$ contains terms proportional to
$h_t^2 \tan \beta$ and $h_b^2 \tan \beta$.
Therefore, the $m_{H_2}^2$ term along with a piece proportional
to $h_t^2$ in $\Delta_\mu^{(1)}$ are dominant in (\ref{eq:minmu}) in the
large-$\tan \beta$ regime, so $\mu$ depends rather mildly on $m_b$.

We illustrate in Fig.~\ref{fig:errors} the logarithmic sensitivity of
$m_A$, namely $\Delta m_A/m_A$, to $m_b$ and $m_t$ along slices through
the $(m_{1/2}, m_0)$ plane for $\tan \beta = 57$, $A_0 = 0$ and $\mu > 0$.
We use as representative errors $\Delta m_t = 1$~GeV and $\Delta
m_b = 0.1$~GeV.
Panel (a) shows the effect of varying $m_0$ for fixed $m_{1/2} =
300$~GeV, and panel (b) shows the effect of varying $m_{1/2}$ for fixed
$m_0 = 400$~GeV.

\begin{figure}
\vskip 0.5in
\vspace*{-0.75in}
\begin{minipage}{8in}
\epsfig{file=error2As.eps,height=3in}
\hspace*{-0.17in}
\epsfig{file=error1As.eps,height=3in}
\hfill
\end{minipage}
\caption{
{\it
The sensitivities of $m_A$ to $m_t$ and $m_b$, assuming $\Delta m_t =
1$~GeV
and $\Delta m_b = 0.1$~GeV, along slices through the $(m_{1/2}, m_0)$
CMSSM plane for $A_0 = 0$ and $\tan \beta = 57$. Panel (a) fixes $m_{1/2}
= 300$~GeV and varies $m_0$, while panel (b) fixes $m_0
= 400$~GeV and varies $m_{1/2}$.}
}
\label{fig:errors} \end{figure}

One can understand the behaviour depicted by employing (\ref{eq:mamass}),
(\ref{eq:minmu}) and the renormalization-group equations (RGEs) that
govern the evolution of the $m_{H_i}$ from $M_{GUT}$ to $M_Z$. The
one-loop RGE for $m_{H_1}^2$ depends on $h_b$:
\begin{equation}
\frac{\partial m_{H_1}^2}{\partial \ln Q}=\frac{1}{16 \pi^2}
\left\{  (-6g_2^2 \, M_2^2-\frac{6}{3}g_1^2 \,M_1^2)\,
+\,6|h_b|^2(m_{Q_3}^2 +  m_{D_3}^2 + m_{H_1}^2 +A_b^2 )     \right\} \, ,
\label{eq:rgeh1}
\end{equation}
where we ignore, for simplicity, the $h_\tau$ contribution, while the RGE
for $m_{H_2}^2$ depends on $h_t$:
\begin{equation}
\frac{\partial m_{H_2}^2}{\partial \ln Q}=\frac{1}{16 \pi^2}
\left\{  (-6g_2^2 \, M_2^2-\frac{6}{3}g_1^2 \,M_1^2)\,
+\,6|h_t|^2(m_{Q_3}^2 +  m_{U_3}^2 + m_{H_2}^2 +A_t^2 )     \right\} \, .
\label{eq:rgeh2}
\end{equation}

We see from (\ref{eq:rgeh1}) that, as $m_b$ increases, the value of
$m_{H_1}^2$ at $M_Z$ decreases, tending to decrease $m_A$
(\ref{eq:mamass}). More importantly, the term proportional to $h_b^2$ in
$\Delta_A$ in (\ref{eq:mamass}) is proportional to $m_{{\tilde
b}_1}^2/(m_{{\tilde b}_1}^2 - m_{{\tilde b}_2}^2)\log(m_{{\tilde b}_1}^2/
m_{{\tilde b}_2}^2) - 1$, which is negative across the $(m_{1/2}, m_0)$
plane for $\tan \beta = 57$ and $A_0 = 0$, with a magnitude that decreases
with $m_{1/2}$. Thus, as $m_b$ increases, we obtain a further decrease in
$m_A$. This term in fact provides most of the numerical dependence of
$m_A$ on $m_b$. Since it is enhanced by $\tan\beta$, the $m_b$ dependence
becomes milder for smaller values of $\tan\beta$.

In addition, the effect on $m_A$ is augmented if the role of the bottom
Yukawa coupling in the RGE (\ref{eq:rgeh1}) is enhanced, which occurs at
large $m_0$. This increases the sensitivity
of $m_A$ to $m_b$, as seen in Fig. \ref{fig:errors}(a). In contrast,
when $m_{1/2}$ is increased, the sensitivity to $\Delta_A$ is diminished
and, at the same time, the gaugino part of the RGE is enhanced. Both
effects lead to a reduced change in $m_A$ at large $m_{1/2}$, as can be
seen in Fig. \ref{fig:errors}(b).

Turning now to the dependence of $m_A$ on $m_t$, we see that the evolution
of $m_{H_2}^2$ shown in (\ref{eq:rgeh2}) is similar to (\ref{eq:rgeh1}),
apart from the substitution of $h_t$ and analogous mass substitutions. As
$m_t$ increases, the value of $m_{H_2}^2$ at $M_Z$ is driven to larger
negative values. However, the change in $m_A$ is dominated by the change
in $\mu$, which grows with an increase in $m_t$. The net effect is an
increase in $m_A$, as seen in Fig. \ref{fig:errors}, which increases with
$m_0$ and decreases with $m_{1/2}$, as seen in panels (a) and (b),
respectively.

Since the $B_s \goto \mu^+ \, \mu^-$ decay rate depends on the fourth power of $m_A$, the
sensitivity of $m_A$ to both $m_b$ and $m_t$ displayed in
Fig.~\ref{fig:errors} can lead to a large uncertainty in $B_s \goto \mu^+ \, \mu^-$ for $\tan
\beta = 57$. We have also evaluated the sensitivities to $m_t$ and $m_b$
for $\tan \beta = 40$. These sensitivities do not vary significantly with
$m_{1/2}$ nor with $m_0$. They are always smaller
than for $\tan \beta = 57$, and the difference is rather substantial for
large $m_0$ and small $m_{1/2}$.

Numerically, in the following analysis we assume
\begin{equation}
\Delta m_t \, = \, 4~{\rm GeV}, \; \Delta m_b \, = \, 0.11~{\rm GeV},
\label{deltamtmb}
\end{equation}
with central values for the physical pole mass $m_t = 178$~GeV and the
running mass $m_b^{\overline {MS}}(m_b) = 4.25$~GeV.
The first of the uncertainties in (\ref{deltamtmb}) is taken directly from
measurements at
the Fermilab Tevatron collider, and may be reduced by a factor of 2 to 4
by future measurements there and at the LHC. The following analysis shows
that such reductions would be most welcome also in the analysis of $B_s \goto \mu^+ \, \mu^-$
decay. The uncertainty in $m_b$ is taken from a recent review~\cite{E-KL}
that combines determinations from ${\bar b}b$ systems, $b$-flavoured
hadrons and high-energy processes. Our 1-$\sigma$ range given
in~\cite{E-KL} is contained
within the preferred range quoted by the Particle Data Group~\cite{pdg},
and is very
similar to the ranges quoted recently by the UKQCD group~\cite{UKQCD}
in the unquenched approximation and
in the review
given by Rakow at the Lattice 2004 conference~\cite{Rakow}.

It is easy to see how important these uncertainties could be. For example,
when $\Delta m_A/m_A = 0.05$ for $\Delta m_t = 1$~GeV, which occurs when
$\tan \beta = 57$ for $(m_{1/2}, m_0)  = $ (300, 900) or (100, 400)~GeV,
the change in $m_A$ for $|\Delta m_t| = 4$~GeV is $\pm 0.2 m_A$,
corresponding to a change in the $A$ contribution to the $B_s \goto \mu^+ \, \mu^-$ decay rate
by a factor 2.07 or 0.41, depending on the sign of $\Delta m_t$.

We note in passing that both the CDF~\cite{CDFH} and D0~\cite{D0H}
collaborations have recently published new upper limits on Higgs
production at the Fermilab Tevatron collider. In particular, the D0
limit~\cite{D0H} is relevant to the MSSM at very large $\tan \beta$ and
small $m_A$. However, such small values of $m_A$ are already excluded at
large $\tan \beta$ in the CMSSM by other constraints such as $b \to s
\gamma$ and the lower limit on $m_h$, so the present D0 limit does not
further restrict the part of the CMSSM parameter space of interest here.

\section{The Effects of Uncertainties on {\boldmath $B_s \goto \mu^+ \, \mu^-$} Limits in the
CMSSM}

In order to assess how important these auxiliary uncertainties may be in
the interpretation of $B_s \goto \mu^+ \, \mu^-$ experiments, we display in
Fig.~\ref{fig:CMSSMerrors} their individual effects on the present $B_s \goto \mu^+ \, \mu^-$
constraint in the $(m_{1/2}, m_0)$ plane of the CMSSM for $A_0 = 0$, $\mu
> 0$ and $\tan \beta = 57$. This is close to the largest value of $\tan
\beta$ for which we find suitable electroweak vacua in generic domains of
the $(m_{1/2}, m_0)$ plane, and so maximizes the potential impact of the
$B_s \goto \mu^+ \, \mu^-$ constraint, which increases asymptotically as the sixth power of
$\tan \beta$. The dark (brick) shaded regions in the bottom-right corners
of each panel are excluded because there the lightest supersymmetric
particle (LSP) would be the charged ${\tilde \tau_1}$. The pale (blue)
shaded strips are those favoured by WMAP, if all the cold dark matter is
composed of LSPs. The supersymmetric spectrum
and relic density calculations have been descibed elsewhere
(see e.g. \cite{else}).  The near-vertical dashed (black) lines at small
$m_{1/2}$ are the constraint imposed by the non-observation of charginos
at LEP, and the near-vertical dash-dotted (red) lines are those imposed by
the non-observation of the lightest MSSM Higgs boson, as calculated using
the {\tt FeynHiggs} code \cite{FeynHiggs}.

The medium (green) shaded regions are excluded
by the rare decay $b \to s \gamma$.
The branching ratio for this  has been measured
by the CLEO, BELLE and BaBar collaborations~\cite{bsg}.
The theoretical prediction of $b \to s
\gamma$~\cite{bsgth,Ambrosio,Buras1} contains uncertainties which stem from
the uncertainties
in $m_b$, $\alpha_s$, the measurement of the semileptonic branching ratio
of the $B$ meson, and the effect of the scale dependence.
In particular, the scale dependence of the theoretical prediction arises from
the dependence on three scales: the scale where the QCD corrections to the
semileptonic decay are calculated and the high and low energy scales
relevant to $b \to s \gamma$ decay~\cite{scale}.
These sources of uncertainty can be
combined to determine a total
theoretical uncertainty~\footnote{According to a recent
analysis~\cite{Neubert},
these theoretical uncertainties may be significantly larger, resulting to
a weaker bound on the masses of supersymmetric particles.}.
The experimental measurement is converted into a Gaussian likelihood and
convolved with a theoretical likelihood to determine the total
likelihood~\cite{efgo},
which is used to calculate the excluded region at 95\% CL.
It is important to note that the dependence of this excluded
region on $m_A, m_b,$ and $m_t$ is quite weak in comparison,
as we have checked numerically.

Finally, the ellipsoidal contours represent the nominal $B_s \goto \mu^+ \, \mu^-$ branching
ratio, calculated (like all the others) using the current central values
$m_t = 178$~GeV and $m_b^{\overline {MS}}(m_b) = 4.25$~GeV. The numerical
labels for the two outer solid lines are exponents in the branching ratio:
$10^{-7}, 10^{-8}$, the thinner-dashed line is for $2 \times 10^{-8}$, the
thicker-dashed line for $5 \times 10^{-8}$.

The most stringent experimental upper bound on the $B_s \goto \mu^+ \, \mu^-$ branching
ratio is that given by an updated CDF measurement:
$1.5 \times 10^{-7}$ ($2.0 \times 10^{-7})$ at the 90\% (95\%)
CL~\cite{cdf}.
The innermost thick solid line of  Fig.~\ref{fig:CMSSMerrors} is
the contour for the present nominal 95\% CL experimental upper limit of
$2.0 \times
10^{-7}$.

\begin{figure}
\vskip 0.5in
\vspace*{-0.75in}
\begin{minipage}{8in}
\epsfig{file=currnotnobfs.eps,height=3.3in}
\hspace*{-0.17in}
\epsfig{file=currnobfs.eps,height=3.3in}
\hfill
\end{minipage}
\begin{minipage}{8in}
\epsfig{file=currnotfs.eps,height=3.3in}
\hspace*{-0.2in}
\epsfig{file=currfs.eps,height=3.3in} \hfill
\end{minipage}
\caption{
{\it
The effects of auxiliary uncertainties on the region of the $(m_{1/2},
m_0)$ plane for $A_0 = 0$, $\mu > 0$ and $\tan \beta = 57$ currently
excluded by the
Fermilab Tevatron collider. (a) The effect of $B_s$ meson uncertainties
alone, principally that in $f_{B_s}$. (b) These uncertainties combined
with the uncertainty $\Delta m_t = 4$~GeV. (c) The $B_s$ meson
uncertainties combined with the uncertainty $\Delta m_b = 0.11$~GeV. (d)
All uncertainties combined. The various contours and shadings in the
$(m_{1/2}, m_0)$ plane are explained in the text.}
}
\label{fig:CMSSMerrors}
\end{figure}

Panel (a) of Fig.~\ref{fig:CMSSMerrors} displays as a blue shaded region
the effect on the interpretation of the present experimental limit of
including the uncertainties in the $B_s$ meson parameters. The most
important uncertainty is that in the decay constant $f_{B_s}$, for which
we assume~\cite{bernard}
\begin{equation}
f_{B_s} = 230 \pm 30~{\rm MeV}.
\end{equation}
In addition, we use~\cite{pdg}
\begin{equation}
m_{B_s}=5369.6 \pm 2.4~{\rm MeV}, \qquad \tau_{B_s}=(1.461 \pm 0.057) \times
10^{-12}~{\rm s}.
\end{equation}
In calculating the uncertainty we add
quadratically the uncertainties that result from these errors as well as
those in the KM elements. We see that incorporating these
uncertainties does not change the overall shape of the excluded region,
but does shrink it slightly. There may be some possibility to reduce
the uncertainty in $f_{B_s}$ in the foreseeable future, but in this
analysis we retain it fixed in the following panels and other figures.

The blue shaded region in panel (b) of Fig.~\ref{fig:CMSSMerrors}
incorporates the present uncertainty in $m_t$, assumed to be $\Delta m_t =
4$~GeV, which is propagated through the CMSSM calculation of $m_A$ as
discussed in the previous Section. We see that this uncertainty is more
important for larger $m_0$, truncating the upper part of the exclusion
domain. This effect can readily be understood from panel (a) of
Fig.~\ref{fig:errors}, where we saw that $\Delta m_t$ has a particularly
important effect on $m_A$ at large $m_0$.

The blue shaded region in panel (c) of Fig.~\ref{fig:CMSSMerrors} shows
the parallel effect of the uncertainty in $m_b$, assumed to be $\Delta m_b
= 0.11$~GeV, which is also propagated through the CMSSM calculation of
$m_A$ as discussed in the previous Section. This uncertainty is also more
important for larger $m_0$, providing an independent mechanism for
truncating the upper part of the exclusion domain. This can also readily
be understood from panel (a) of Fig.~\ref{fig:errors}, where we saw that
$\Delta m_b$ also has a particularly important effect on $m_A$ at large
$m_0$.

There is also some tendency in both panels (b) and (c) of
Fig.~\ref{fig:CMSSMerrors} for the exclusion domain to separate from the
axis $m_{1/2} \sim 100$ GeV, particularly at large $m_0$. This can be understood
from panel (b) of Fig.~\ref{fig:errors}, where we see that the effects of
both $\Delta m_t$ and $\Delta m_b$ on $m_A$ are enhanced when $m_{1/2} \mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}}
200$~GeV. The uncertainties in each of $m_t$ and $m_b$ become
particularly important when $m_{1/2}$ is small and $m_0$ large, as seen
separately in panels (b) and (c) of Fig.~\ref{fig:CMSSMerrors}.

The similar tendencies in panels (b) and (c) of Fig.~\ref{fig:CMSSMerrors}
are reinforced when we combine the uncertainties in $m_t$ and $m_b$, as
shown by the blue shaded region in panel (d). We find that $B_s \goto \mu^+ \, \mu^-$ decay
is currently unable to exclude any value of $m_0$ above about 350~GeV,
whereas the exclusion region would have extended up to $m_0 \sim 450$~GeV
if the auxiliary uncertainties had not been taken into account, and $\sim
400$~GeV if either of the $m_t$ or $m_b$ uncertainties had been ignored.
On the other hand, the reduction in the excluded range of $m_{1/2}$ at
lower $m_0$ is less important, typically $\mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}} 30$~GeV.

We observe that the region currently excluded by $B_s \goto \mu^+ \, \mu^-$ is always included
within the region already excluded by $b \to s \gamma$ and/or $m_h$, even
without including the auxiliary uncertainties. The same is even more true
for smaller values of $\tan \beta$: in the case $\tan \beta = 40$ (not
shown), the region currently excluded by $B_s \goto \mu^+ \, \mu^-$ has $m_{1/2} \mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}} 180$~GeV
and $m_0 \mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}} 170$~GeV, within the strips excluded by $m_h$ and
$m_{\chi^\pm}$ but allowed by $b \to s \gamma$.  We recall that the $b \to
s \gamma$ limit is very dependent on the assumption of universal scalar
masses for the squarks, which does not play a role elsewhere in the
analysis of constraints on CMSSM parameters, and is of course untested.
Clearly $B_s \goto \mu^+ \, \mu^-$ has the potential to complement or even, in the future,
supplant the $b \to s \gamma$ constraint, though it also relies on
squark-mass universality.

The input value of the trilinear soft supersymmetry-breaking parameter
$A_0$ has a significant effect on the allowed CMSSM parameter space, and
is also important for $m_h$ as well as the $b \to s \gamma$ and $B_s \goto \mu^+ \, \mu^-$
decays. Therefore, we display in Fig.~\ref{fig:varyA} the interplays of
these constraints for $\tan \beta = 57$, $\mu > 0$, and (a) $A_0 = 2
m_{1/2}$ and (b) $A_0 = - 2 m_{1/2}$. The qualitative conclusions are
similar to the $A_0 = 0$ case discussed previously: the region currently
disallowed by $B_s \goto \mu^+ \, \mu^-$ decay largely overlaps with the regions previously
disfavoured by $m_h$ and $b \to s \gamma$, and decreases in extent as
$A_0$ is reduced from positive to negative values.

\begin{figure}
\vskip 0.5in
\vspace*{-0.75in}
\begin{minipage}{8in}
\epsfig{file=currApfs.eps,height=3.3in}
\hspace*{-0.17in}
\epsfig{file=currAnfs.eps,height=3.3in}
\hfill
\end{minipage}
\caption{
{\it
The disallowed regions in the $(m_{1/2}, m_0)$ plane for
$\mu > 0$ and $\tan \beta = 57$, for (a) $A_0 = 2 m_{1/2}$ and (b) $A_0 =
- 2 m_{1/2}$. The various contours and shadings in the $(m_{1/2}, m_0)$
plane are as explained in the text describing Fig.~\ref{fig:CMSSMerrors}.
}}
\label{fig:varyA}
\end{figure}

\section{Treatment of Errors}

There are several ways to treat the auxiliary errors in the $B_s \goto \mu^+ \, \mu^-$
analysis. In the above, we have implicitly assumed one of the most
conservative treatments, in the sense that it excludes the smallest region
of the $(m_{1/2}, m_0)$ plane for given fixed values of the uncertainties.
However, other treatments are possible, and here we compare their results.
For this comparison, we include all the uncertainties in $f_{B_s}$, $m_t$
and $m_b$ discussed in the previous Section.

In our previous treatment, see panel (d) of Fig.~\ref{fig:CMSSMerrors}, we
assumed that all the uncertainties have Gaussian error distributions, and
defined the allowed region by discarding the upper 2.5\% tail of the
likelihood distribution obtained by combining the experimental and
auxiliary errors.  This would be correct if the central value of the
experimental measurement, after subtracting any backgrounds, was strictly
zero. Alternatively, one might discard the upper 5 \% of the combined
likelihood distribution.
This would give the correct experimental upper limit if the central
experimental value were far enough above zero that no significant part of
the lower tail of the likelihood distribution extends below zero $B_s \goto \mu^+ \, \mu^-$
decay rate, but is otherwise clearly more conservative than the previous
treatment.  Finally, experimentalists sometimes subtract one theoretical
(systematic) error from the measured result and then plot the 95 \%
confidence-level contour given by the experimental error.

The two alternative prescriptions
yield similar upper limits on $m_0$, though the shapes of the allowed regions
are different. They both reach up to $m_0 \sim 400$~GeV,
as compared the range $m_0 \mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}} 350$~GeV found in the previous analysis,
shown in panel (d) of Fig.~\ref{fig:CMSSMerrors}. We prefer to use the
more conservative prescription used in drawing the previous figures, also
because it is demonstrably appropriate if the central experimental value
is negligible, as is currently the case.

\section{Possible Future Developments}

It is expected that the Fermilab Tevatron collider experiments will continue
to improve the present sensitivity to $B_s \goto \mu^+ \, \mu^-$ decay.
To assess the likely impact of this improved sensitivity, we
exhibit in Fig.~\ref{fig:future} the potential $(m_{1/2}, m_0)$ planes for
$A_0 = 0$ and $\tan \beta = 57, 40$, obtained using the conservative error
prescription and neglecting possible improvements in the determinations of
$f_{B_s}$, $m_t$ and $m_b$, in the pessimistic case that no signal is
seen.

\begin{figure}
\vskip 0.5in
\vspace*{-0.75in}
\begin{minipage}{8in}
\epsfig{file=futfs.eps,height=3.3in}
\hspace*{-0.17in}
\epsfig{file=fut40fs.eps,height=3.3in}
\hfill
\end{minipage}
\begin{minipage}{8in}
\epsfig{file=lhcfs.eps,height=3.3in}
\hspace*{-0.17in}
\epsfig{file=lhc40fs.eps,height=3.3in}
\hfill
\end{minipage}
\caption{
{\it
The potential disallowed regions in the $(m_{1/2}, m_0)$ plane for $A_0 =
0$, $\mu > 0$ and (a) $\tan \beta = 57$, (b) $\tan \beta = 40$, obtained
assuming a Fermilab Tevatron upper limit on $B_s \goto \mu^+ \, \mu^-$ that is improved
to a 95\% CL upper limit of $5 \times 10^{-8}$,
with no parallel reductions in the uncertainties in $f_{B_s}$,
$m_t$ and $m_b$. Panels (c) and (d) show the corresponding disallowed
domains assuming a conjectural LHC measurement $BR(B_s \goto \mu^+ \, \mu^-) = (3.9 \pm
1.3) \times 10^{-9}$. The various contours and shadings in the $(m_{1/2},
m_0)$
plane are as explained in the text describing Fig.~\ref{fig:CMSSMerrors}.
}}
\label{fig:future}
\end{figure}

In panel  (a) of Fig.~\ref{fig:future} for $\tan \beta = 57$, we show  that
the $B_s \goto \mu^+ \, \mu^-$ constraint would, under these pessimistic assumptions, begin to
exclude a region in the neighbourhood of $(m_{1/2}, m_0) = (400, 400)$~GeV
that is favoured by WMAP and allowed by all the other present constraints,
assuming a future 95\% CL upper limit of $5 \times 10^{-8}$.
The region at small $m_{1/2}$ is allowed because of the high sensitivity to
$m_t$ and $m_b$ seen in panel (b) of Fig.~\ref{fig:errors}.  For $\tan
\beta = 40$, as seen in panel (b) of Fig.~\ref{fig:future}, the region
disallowed by $B_s \goto \mu^+ \, \mu^-$ would still lie within the region already disallowed
by $b \to s \gamma$. The much reduced sensitivity of $B_s \goto \mu^+ \, \mu^-$ decay to $m_t$
and $m_b$ at small $m_{1/2}$ for $\tan \beta = 40$ implies that there is
no allowed `gap' at small $m_{1/2}$.

Panels (c) and (d) of Fig.~\ref{fig:future} show the corresponding
sensitivities at the LHC, assuming a conjectural 3-$\sigma$ measurement
whose central value coincides with the central value predicted by the
Standard Model, i.e., $BR(B_s \goto \mu^+ \, \mu^-) = (3.9 \pm 1.3) \times 10^{-9}$. In
panel (c) for $\tan \beta = 57$ we see that such a measurement would cover
a very large fraction (but not all) of the CMSSM parameter space allowed
by WMAP. On the other hand, the fraction covered in panel (d) for $\tan
\beta = 40$ is somewhat smaller. Of course, it is quite likely that the
present errors in $f_{B_s}$, $m_t$ and $m_b$ will be substantially reduced
by the time of this ultimate LHC measurement. Reducing the $f_{B_s}$
error, in particular, would reduce the scope available for a CMSSM
contribution, and extend the $B_s \goto \mu^+ \, \mu^-$ exclusion region to larger $m_{1/2}$.

\section{Conclusions}

We have seen that the interpretation of the present and prospective
experimental limits on $B_s \goto \mu^+ \, \mu^-$ decay are very sensitive to auxiliary
uncertainties, principally those in $f_{B_s}$, $m_t$ and $m_b$.  At the
present time, these restrict significantly the regions of the CMSSM
parameter space that can be excluded by the current upper limit on this
decay, and their uncertainties may not be reduced significantly during the
remaining operation of the Fermilab Tevatron collider, with the likely
exception of $m_t$. However, one might hope that the uncertainties in each
of $f_{B_s}$, $m_t$ and $m_b$ could be reduced by the time the LHC
achieves its ultimate sensitivity to $B_s \goto \mu^+ \, \mu^-$ decay. As an exercise, we have
considered the possibility that their uncertainties might be reduced to
$\Delta f_{B_s} = 10$~MeV, $\Delta m_t = 1$~GeV and $\Delta m_b =
0.05$~GeV. In this case, the LHC reach in the $(m_{1/2}, m_0)$ plane for
$\tan \beta = 57$ would extend to $m_{1/2} \mathrel{\raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}}} 1400$~GeV along the WMAP
strip.

Beyond the framework of the CMSSM, it would be interesting to study the
interpretation of the $B_s \goto \mu^+ \, \mu^-$ constraint also in the frameworks of more
general models, such as those with non-universal Higgs masses
(NUHM)~\footnote{We expect the situation in the general low-energy
effective supersymmetric theory (LEEST) to be similar.}~\cite{nuhm}. A
complete study of the situation within the NUHM would take us beyond the
scope of this paper, so we restrict ourselves to a few remarks. The models
likely to be disfavoured by $B_s \goto \mu^+ \, \mu^-$ are those with a low value of $m_A$ and
large $\tan \beta$. Such models also tend to predict large
neutralino-nucleus scattering cross sections~\cite{ko}. We have examined a
sample of NUHM scenarios that are apparently excluded by the recent
CDMS~II upper limit on the direct scattering of supersymmetric dark
matter, and found that about half of them are excluded by $B_s \goto \mu^+ \, \mu^-$. The next
step would be to examine models apparently allowed by CDMS~II, to see how
many of them are also excluded by $B_s \goto \mu^+ \, \mu^-$. However, for consistency, one
should also study the effects on the dark matter scattering cross section
of auxiliary uncertainties such as those in $m_t$ and $m_b$, which has not
yet been done in the manner described here for $B_s \goto \mu^+ \, \mu^-$. We plan to present
elsewhere such a unified treatment of the uncertainties.

\section*{Acknowledgments}
\noindent
The work of K.A.O. and V.C.S. was supported in part
by DOE grant DE--FG02--94ER--40823. We would like to thank
D. Cronin-Hennessy, C.~Sachrajda
and M.~Voloshin for helpful discussions.