Quantum Joint Distributions \thanks{PACS No: 03.65.Ta}
School of Mathematics and Statistics
ACT 2601
Australia
In this paper we provide a method for constructing joint distributions for an arbitrary set of observables on finite dimensional Hilbert spaces irrespective of whether the observables commute or not. These distributions have a number of desirable properties: they agree with the standard quantum mechanical ones if the observables commute, they also depend continuously on the observables, and under unitary transformations they behave in a reasonable manner.
action.tex
\sect{action}{Action angle coordinates on $\PH$}

For the remainder of this article, we will assume
that $\dim \Hc = N < \infty$. Let $\{\psi_{1},\ldots,\psi_{N}\}$
be an orthonormal basis for $\Hc$. Define the projection operators
\leqn{eqA}{
P_{\psi_{k}} = |\psi_{k}\rangle\langle\psi_{k}| \, .
}
We can use the momentum map to define smooth functions
$p_{\psi_{k}}$ on $\PH$ by
\leqn{eqB}{
p_{\psi} := \E{P_{\psi_{k}}} \, .
}
Using \eqref{exp2} we get
\leqn{eqC}{
p_{\psi}([\psi]) = \frac{\ip{\psi_{k}}{\psi}}{\|\psi\|}^2
}
which is the transition probability from the state $\psi$ to
$\psi_{k}$. As the operators $P_{\psi_{k}}$ commute,
formula \eqref{poissonA} shows that the functions
$\{p_{\psi_{1}},\ldots,p_{\psi_{N}}\}$ are in involution, i.e.
\leqn{eqD}{
\{\psi_{j},\psi_{k}\} = 0 \quad 
\forall \; j,k =1,2,\ldots N \, .
}
It follows from $\id = \sum_{k=1}^{N} P_{\psi_{k}}$ that
\leqn{eqE}{
\sum_{j=1}^{N} p_{\psi_{k}} = 1\, ,
}
which shows that at most $(N-1)$ of the functions $p_{\psi_{k}}$
can be independent. It is not hard to show that the
set $\{p_{\psi_{2}},\ldots,p_{\psi_{N}}\}$ is independent.
That is the set of points in $\PH$ for which the
covectors $\{dp_{\psi_{2}},\ldots,dp_{\psi_{N}}\}$
are linearly dependent has measure zero. Consequently, we
can use these functions to construct action angle coordinates
of $\PH$ following the standard recipe, see \cite{k6081}
for details. This results in the following coordinate chart
\lalign{eqF}{
&\tau : T^{N-1}\times S \longrightarrow \PH \notag \\
&(\theta,\mathbf{I}) = ((\theta_{2},\ldots,
\theta_{N}),(I_{2},\ldots,I_{N}))\mapsto
\Bigl[ \bigl(1-\sum_{k=2}^{N}I_{k}\bigr)^{1/2}\psi_{1}
+ \sum_{j=2}^{N}e^{-i\theta_{j}}\sqrt{I_{j}}\psi_{j}
\Bigr] \label{eqF1}
}
where $T^{N-1}$ is the $(N-1)$ torus and
\leqn{eqG}{
S := \Bigl\{\, (I_{2},\ldots,I_{N})
\in \Rbb^{N-1} \, | \, 0 < I_{j}\,\; \sum_{j=2}^{N} I_{j} < 1 
\Bigr\} \, .
}
In this chart, the symplectic form $\omega$ is
given be
\leqn{eqH}{
\omega = \hbar \sum_{j=2}^{N} d\theta_{k} \wedge dI_{k} \, .
}
We also note that the functions $p_{\psi_{k}}$ have
the coordinate representations
\eqn{eqI}{
p_{\psi_{j}}(\theta,\mathbf{I}) = I_{j} \quad
j = 2,3,\ldots, N \, .
}
We can define a volume form $\mu$ on $\PH$ by
\leqn{eqJ}{
\mu := \left(\frac{-1}{2\pi\hbar}\right)^{N-1}\underset{\text{N-1 times}}{
\underbrace{
\omega\wedge \ldots \wedge \omega}} \; .
}
Locally this given by
\leqn{eqK}{
\mu = \frac{(N-1)!}{(2\pi)^{(N-1)}}
\, d\theta_{2}\wedge\ldots\wedge\theta_{N}\wedge
dI_{2}\wedge\ldots\wedge dI_{N} \, .
}
Then because the chart \eqref{eqF1} covers all of $\PH$
except for a set of measure zero, the volume of
$\PH$ is given by
\eqn{eqL}{
\Vol(\PH) = \frac{(N-1)!}{(2\pi)^{(N-1)}}
\int_{0}^{2\pi}\ldots\int_{0}^{2\pi}
d\theta_{2}\ldots d\theta_{N} \int_{S}
dI_{2}\ldots dI_{N} \, .
}
A straightforward calculation shows that
\eqn{eqM}{
\int_{S} dI_{2}\ldots dI_{N} = 1/(N-1)!
}
and hence $\Vol(\PH) = 1$.

conc.tex
\sect{conc}{Conclusion}


We have presented a method for constructing joint
distributions for an arbitrary number of observables
on finite dimensional Hilbert spaces.
In the case where the observables commute, the
joint distributions agree with the standard quantum
mechanical ones. Furthermore, we have shown that
that our joint distributions depend continuously on
the observables and transform correctly 
under unitary transformations.


cont.tex
\sect{cont}{Continuity of the joint distributions}

Let $\dH$ denote the Hausdorff distance, i.e. if $\Ac,\Bc \subset \PH$
then
\eqn{dHa}{
\dH(\Ac,\Bc) := \inf\{\: \epsilon \:|\: \Ac \subset \Bt(\Bc,\epsilon),
\Bc \subset \Bt(\Ac,\epsilon)\}
}
where for any set $\Cc \subset \PH$ we define
\eqn{dHb}{
\Bt(\Cc,\epsilon) := \{ x\in \PH \:|\: d(x,\Cc) < \epsilon \}\, .
}

\begin{prop} \label{contA} \mnote{[contA]}
Let $\beta = \{\psi_{1},\ldots,\psi_{N}\}$ and $ \gamma = 
\{\phi_{1},\ldots,\phi_{N}\}$
be two orthonormal bases for $\Hc$ and suppose $x\in \PH$. Then for
any $\epsilon > 0$ there exists a $\delta > 0$ such that  
\eqn{contA1}{
\dH(\Omega(x,\beta,\psi_{j}),\Omega(x,\gamma,\phi_{j})) \leq \epsilon
\quad j=1,2,\ldots, N\,
}
whenever $\|\phi_{j}-\psi_{j}\| < \delta$ for $j=1,2,\ldots,N$.
\end{prop}
\begin{proof}
For $j\neq k$,
\leqn{contA2}{
|\ip{\psi_{j}}{\phi_{k}}| = |\ip{\psi_{j}-\phi_{j}}{\phi_{k}}|
\leq \|\psi_{j}-\phi_{j}\|\,\|\phi_{k}\| < \delta\,1 = \delta \, .
}
Without loss of generality assume $\delta < 1$. Then
$\|\psi_{j}-\phi_{j}\| < \delta$ implies that
\leqn{contA3}{
|2-\ip{\psi_{j}}{\phi_{j}}-\ip{\phi_{j}}{\psi_{j}}| < \delta \, ,
}
and
\leqn{contA3b}{
|\ip{\psi_{j}}{\phi_{j}}-\ip{\phi_{j}}{\psi_{j}}|
= |\ip{\psi_{j}-\phi_{j}}{\phi_{j}}-\ip{\phi_{j}}{\psi_{j}-\phi_{j}}|
\leq 2\|\phi_{j}-\psi_{j}\|\,\|\phi_{j}\| < 2\delta \, .
}

Let $\psi\in \Hc$ satisfy $\|\psi\|=1$ and $x=[\psi]$. We will assume
that there exists a $\nu > 0$ such that 
\eqn{contA4}{
|\ip{\psi}{\psi_{j}}|\, ,\, |\ip{\psi}{\phi_{j}}| > \nu \quad
j=1,2,\ldots N\,.
}
The case where $|\ip{\psi}{\psi_{j}}|=0$ or $|\ip{\psi}{\phi_{j}}| =0$
for some $j$ can be taken care of by a slight modification of
the arguments below and will be left to the reader.

Fix $\epsilon > 0$ and $k\in \{1,2,\ldots,N\}$. Suppose $\phi \in \Hc$
satisfies $\|\phi\|=1$ and $[\phi] \in \Omega(x,\gamma,\phi_{k})$.
Then by the definition of  $\Omega(x,\gamma,\phi_{k})$, 
$p_{\phi_{j}}([\phi])p_{\phi_{k}}([\psi]) \geq 
p_{\phi_{k}}([\phi])p_{\phi_{j}}([\psi])$ for $j=1,2,\ldots N$, or equivalently
\leqn{contA5}{
|\ip{\phi_{j}}{\phi}|^2|\ip{\phi_{k}}{\psi}|^2 \geq
|\ip{\phi_{k}}{\phi}|^2|\ip{\phi_{j}}{\psi}|^2 \quad j=1,2,\ldots N\, .
}
Define
\eqn{contA6}{
\Psi := \sum_{j=1}^{N} \ip{\phi_{j}}{\phi}\lambda_{j}\psi_{j}
}
where
\eqn{contA7}{
\lambda_{j} := \frac{|\ip{\psi_{j}}{\psi}|}{|\ip{\phi_{j}}{\psi}|}\, .
}
Then for $j=1,2,\ldots N$,
\alin{contA8}{
|\ip{\psi_{j}}{\Psi}|^2|\ip{\psi_{k}}{\psi}|^2 &= |\ip{\phi_{j}}{\phi}|^2
|\ip{\phi_{k}}{\psi}|^2|\lambda_{j}|^2|\lambda_{k}|^2 \\
& \geq |\ip{\phi_{k}}{\phi}|^2|\lambda_{k}|^2|\ip{\psi_{j}}{\psi}|^2
&&  \text{by \eqref{contA5}} \\
& = |\ip{\psi_{k}}{\Psi}|^2|\ip{\psi_{j}}{\psi}|^2\, ,
}
and hence $[\Psi]\in \Omega(x,\beta,\psi_{k})$.

Note that $\lambda_{j} = |\ip{\psi_{j}-\phi_{j}}{\psi}\ip{\phi_{j}}{\psi}^{-1}
+1|$ and so
\eqn{contA8}{
1-\frac{|\ip{\psi_{j}-\phi_{j}}{\psi}|}{|\ip{\phi_{j}}{\psi}|}
\leq \lambda_{j} \leq 1+
\frac{|\ip{\psi_{j}-\phi_{j}}{\psi}|}{|\ip{\phi_{j}}{\psi}|} \, .
}
But
\eqn{contA9}{
\frac{|\ip{\psi_{j}-\phi_{j}}{\psi}|}{|\ip{\phi_{j}}{\psi}|} \leq
\frac{\|\psi_{j}-\phi_{j}\|\,\|\psi\|}{\nu} < \frac{\delta}{\nu} \, ,
}
and hence
\leqn{contA10}{
1-\frac{\delta}{\nu} \leq \lambda_{j} \leq 1+\frac{\delta}{\nu} \quad 
j=1,2,\ldots,N \, .
}
Now, $\|\Psi\|^2 = \sum_{j=1}{N}|\ip{\phi_{j}}{\phi}|^2\lambda_{j}^2$
and $1=\|\phi\|^2 = \sum_{j=1}^{N}|\ip{\phi_{j}}{\phi}|^2$ implies
that 
\leqn{contA11}{
\left(1-\frac{\delta}{\nu}\right) \leq  \|\Psi\|^2 \leq 
\left(1+\frac{\delta}{\nu}\right)
}
by \eqref{contA10}. Also we have
\lalign{contA12}{
|&\ip{\Psi}{\phi}|  = \Bigl| \sum_{j,l=1}^{N}\ip{\phi}{\phi_{j}}
\lambda_{j}\ip{\psi_{j}}{\phi_{l}}\ip{\phi_{l}}{\phi}\Bigr| \notag \\
& = \Bigl| \sum_{j=1}^{N}|\ip{\phi}{\phi_{j}}|^2
\lambda_{j}\ip{\psi_{j}}{\phi_{j}} +
\sum_{j,l=1 \;  j\neq l}^{N} \ip{\phi}{\phi_{j}}
\lambda_{j}\ip{\psi_{j}}{\phi_{l}}\ip{\phi_{l}}{\phi}\Bigr| \notag \\
& \geq \Bigl|\sum_{j=1}^{N}|\ip{\phi}{\phi_{j}}|^2
\lambda_{j}\ip{\psi_{j}}{\phi_{j}}\Bigr| - 
\sum_{j,l=1 \;  j\neq l}^{N} |\ip{\phi}{\phi_{j}}
\lambda_{j}\ip{\psi_{j}}{\phi_{l}}\ip{\phi_{l}}{\phi}| \notag \\
& = \Bigl|\sum_{j=1}^{N}|\ip{\phi}{\phi_{j}}|^2 -
\sum_{j=1}^{N}|\ip{\phi}{\phi_{j}}|^2(1-
\lambda_{j}\ip{\psi_{j}}{\phi_{j}})\Bigr| - 
\sum_{j,l=1 \;  j\neq l}^{N} |\ip{\phi}{\phi_{j}}
\lambda_{j}\ip{\psi_{j}}{\phi_{l}}\ip{\phi_{l}}{\phi}| \notag \\
& \geq \Bigl|\sum_{j=1}^{N}|\ip{\phi}{\phi_{j}}|^2\Bigr| -
\sum_{j=1}^{N}|\ip{\phi}{\phi_{j}}|^2|(1-
\lambda_{j}\ip{\psi_{j}}{\phi_{j}})| - 
\sum_{j,l=1 \;  j\neq l}^{N} |\ip{\phi}{\phi_{j}}
\lambda_{j}\ip{\psi_{j}}{\phi_{l}}\ip{\phi_{l}}{\phi}|\, . \label{contA12a}
}
But note that
\leqn{contA13}{
\Bigl|\sum_{j=1}^{N}|\ip{\phi}{\phi_{j}}|^2\Bigr| = \|\phi\|^2=1\, ,
\quad |\ip{\phi}{\phi_{j}}|\leq \|\phi\|^2\|\phi_{j}\|^2 =1 \, ,
}
and
\lalign{contA14}{
|\lambda_{j}&\ip{\psi_{j}}{\phi_{j}}-1|  =
|(\lambda_{j}-1+1)\ip{\psi_{j}}{\phi_{j}}-1| 
 \leq |\lambda_{j}-1||\ip{\psi_{j}}{\phi_{j}}| + |\ip{\psi_{j}}{\phi_{j}}-1|
\notag \\
& \leq |\lambda_{j}-1|\|\psi_{j}\|\|\phi_{j}\| + 
\frac{1}{2}|\ip{\psi_{j}}{\phi_{j}}-1+\ip{\phi_{j}}{\psi_{j}}-1|
+ \frac{1}{2}|\ip{\psi_{j}}{\phi_{j}}-\ip{\phi_{j}}{\psi_{j}}| \notag \\
& \leq \frac{\delta}{\nu} + \frac{\delta}{2}+\delta = \left(
\frac{1}{\nu} + \frac{3}{2}\right)\delta \label{contA14a}
}
where in deriving the last inequality we used \eqref{contA3}, \eqref{contA3b},
and \eqref{contA10}. Using \eqref{contA2}, \eqref{contA10}, \eqref{contA13},
and \eqref{contA14a}, we can write \eqref{contA12a} as 
\leqn{contA15}{
|\ip{\Psi}{\phi}| \geq 1 - N \left(
\frac{1}{\nu} + \frac{3}{2}\right)\delta-(N-1)^2 \left(
\frac{\delta}{\nu} + 1\right)\delta \, .
}
Therefore \eqref{contA11} and \eqref{contA15} imply that
\eqn{contA16}{
1 \leq \frac{|\ip{\Psi}{\phi}|^2}{\|\Psi\|^2\|\phi\|^2}
\geq \frac{(1 - N(1/\nu + 3/2)\delta-(N-1)^2(
\delta/\nu + 1)\delta)^2 }{(\delta/\nu + 1)^2} \, ,
}
or by \eqref{transA} and \eqref{transC}
\eqn{contA17}{
1 \geq \cos^2\left(\frac{d([\Psi],[\phi])}{\sqrt{2\hbar}}\right)
\geq \frac{(1 - N(1/\nu + 3/2)\delta-(N-1)^2(
\delta/\nu + 1)\delta)^2 }{(\delta/\nu + 1)^2} \, .
}
This shows that by choosing $\delta > 0$ small enough we get
$d([\Psi],[\phi]) \leq \epsilon$. Since $[\Psi]\in \Omega(x,\beta,\psi_{k})$
and $[\phi]\in  \Omega(x,\gamma,\phi_{k})$ was chosen arbitrarily, we
have that $d(y, \Omega(x,\beta,\psi_{k})\leq \epsilon$ for all
$x \in \Omega(x,\gamma,\phi_{k})$. The same arguments show that
$\delta$ can be chosen small enough so that 
$d(y, \Omega(x,\gamma,\phi_{k})\leq \epsilon$ for all
$x \in \Omega(x,\beta,\psi_{k})$. Therefore it follows that \\
$\dH(\Omega(x,\beta,\psi_{k}),\Omega(x,\gamma,\phi_{k})) \leq \epsilon$. 
\end{proof}

\begin{prop} \label{contB} \mnote{[contB]}
Let $\beta = \{\psi_{1},\ldots,\psi_{N}\}$, 
$\beta' = \{\psi_{1}',\ldots,\psi_{N}'\}$, 
$\gamma = \{\phi_{1},\ldots,\phi_{N}\}$, and 
$\gamma' = \{\phi_{1}',\ldots,\phi_{N}'\}$  be orthonormal bases  and
suppose $x\in \PH$. Then for any $\epsilon > 0$ there exists a
$\delta > 0$ such that  
\eqn{contB1}{
|\emph{\Vol}(\Omega(x,\beta,\psi_{j})\cap\Omega(x,\gamma,\phi_{k})) -
\emph{\Vol}(\Omega(x,\beta',\psi_{j}')\cap\Omega(x,\gamma',\phi_{k}'))| 
\leq \epsilon
}
for $j,k=1,2,\ldots,N$ whenever $\|\psi_{j}-\psi_{j}'\| < \delta$ and $\|\phi_{j}-\phi_{j}'\|<\delta$
for $j=1,2,\ldots,N$.
\end{prop} 

\begin{proof}
Fix $\eta > 0$. Given an $\epsilon > 0$, there exists,
by proposition \ref{contA}, a $\delta > 0$
such that $\|\psi_{j}-\psi_{j}'\| < \delta$ and $\|\phi_{j}-\phi_{j}'\|<\delta$
for $j=1,2,\ldots,N$ implies that $\dH(\Omega(x,\beta,\psi_{j}),\Omega(x,\beta',\psi_{j}'))
< \eta$ and $\dH(\Omega(x,\gamma,\phi_{j}),\Omega(x,\gamma',\phi_{j}'))
< \eta$ . So
\alin{contB2}{
\Omega(x,\beta,\psi_{j}) \subset \Bt(\Omega(x,\beta',\psi_{j}'),\epsilon)
&= \Omega(x,\beta',\psi_{j}')\cup T^{1}_{\epsilon} \\
\Omega(x,\gamma,\phi_{k}) \subset \Bt(\Omega(x,\gamma',\phi_{k}'),\epsilon)
&= \Omega(x,\gamma',\phi_{k}')\cup T^{2}_{\epsilon}
}
where
\eqn{contB3}{
T^{1}_{\epsilon} := \Bt(\Omega(x,\beta',\psi_{j}'),\epsilon) \setminus 
\Omega(x,\beta',\psi_{j}') \quad \text{and}\quad
T^{2}_{\epsilon} := \Bt(\Omega(x,\gamma',\phi_{k}'),\epsilon) \setminus 
\Omega(x,\gamma',\phi_{k}') \, .
}
Therefore 
\eqn{contB4}{
\Omega(x,\beta,\psi_{j})\cap \Omega(x,\gamma,\phi_{k}) \subset
\Omega(x,\beta',\psi_{j}')\cap \Omega(x,\gamma',\phi_{k}')
\cup T^{3}_{\epsilon}\cup T^{4}_{\epsilon}\cup T^{5}_{\epsilon}
}
where $T^{3}_{\epsilon} := \Omega(x,\beta',\psi_{j}')\cap T^{2}_{\epsilon}$,
$T^{3}_{\epsilon} := \Omega(x,\gamma',\phi_{k}')\cap T^{1}_{\epsilon}$, and
$T^{5}_{\epsilon}:= T^{1}_{\epsilon}\cap T^{2}_{\epsilon}$.
Thus
\eqn{contB5}{
\Vol(\Omega(x,\beta,\psi_{j})\cap \Omega(x,\gamma,\phi_{k}) ) \leq
\Omega(x,\beta',\psi_{j}')\cap \Omega(x,\gamma',\phi_{k}') +
\sum_{s=3}^{5}\Vol(T^{s}_{\epsilon})\, . 
}
It is clear that $\Vol(T^{s}_{\epsilon})\rightarrow 0$ as $\epsilon \rightarrow 0$
which shows that by choosing $\delta$ small enough we get 
\eqn{contB6}{
\Vol(\Omega(x,\beta,\psi_{j})\cap \Vol(\Omega(x,\gamma,\phi_{k}) ) \leq
\Omega(x,\beta',\psi_{j}')\cap \Omega(x,\gamma',\phi_{k}')) + \eta \, .
}
Similar arguments show that by choosing $\delta$ small enough we also
have 
\eqn{contB7}{
\Vol(\Omega(x,\beta',\psi_{j}')\cap \Vol(\Omega(x,\gamma',\phi_{k}') ) \leq
\Omega(x,\beta,\psi_{j})\cap \Omega(x,\gamma,\phi_{k})) + \eta \, .
}
Therefore we have established that
\eqn{contB8}{
|\Vol(\Omega(x,\beta',\psi_{j}')\cap \Omega(x,\gamma',\phi_{k}') ) -
\Vol(\Omega(x,\beta,\psi_{j})\cap \Omega(x,\gamma,\phi_{k})) |\leq \eta\, .
}
\end{proof}

As before, let $\Sa := \{\, A\in \mathcal{L}(\Hc)\;|\; A^{\dagger}=A\;\}$ be the set of
self-adjoint operators and $\Dc\subset \Sa$ be the set of all self-adjoint 
operators with distinct eigenvalues. Note that $\Dc$ is open in $\Sa$.

\begin{lem} \label{contC} \mnote{[contC]}
For each $A\in\Dc$ there exist a neighborhood $\Nc_{A}\in \Dc$ and
smooth maps
\eqn{contC1}{
\alpha_{j} :\Hc_{A}\rightarrow \Rbb \quad \text{and}\quad
\phi_{j} : \Nc_{A}\rightarrow S(\Hc)  \quad j=1,2,\ldots,N
}
such that for all $B\in \Nc_{A}$ 
\eqn{contC2}{
B\phi_{j}(B)=\alpha_{j}(B)\phi_{j}(B) \quad j=1,2,\ldots,N\, .
}
\end{lem}
\begin{proof}
Let $a_{1} < a_{2} < \cdots < a_{N}$ be the distinct eigenvalues for $A$ and
$\psi_{1},\ldots,\psi_{N}$ the corresponding unit eigenvectors. Define
a map 
\eqn{contC3}{
F^{j} : \Sa\times \Hc \rightarrow \Hc
}
by
\eqn{contC4}{
F^{j}(B,\psi) := (B- a_{j}\ip{\psi_{j}}{\psi}\id)\psi \, .
}
This map is smooth and also satisfies $F^{j}(A,\psi_{j})=0$.Now
\eqn{contC5}{
\text{D}_{2} F^{j}(B,\psi)\cdot\delta\psi =
 (B - a_{j}\ip{\psi_{j}}{\psi}\id)\delta\psi  -  a_{j}\ip{\psi_{j}}{\delta\psi}\psi \, .
}
Without loss of generality we can assume that $a_{j}\neq 0$ because if it were then
we could replace $A$ be a nearby operator in $\Dc$ with $a_{j}\neq 0$. Expanding
$\delta\psi$ as
\eqn{contC6}{
\delta\psi = \sum_{j=1}^{N} c_{j}\psi_{j}
}
we get
\eqn{contC7}{
\text{D}_{2}F^{j}(B,\psi_{j})\cdot\delta\psi = 
\sum_{k=1,k\neq j}^{N} c_{k}(a_{k}-a_{j})\psi_{k} - a_{j}c_{j}\psi_{j} \; .
}
Since $a_{j}\neq 0$ and $a_{k}-a_{j}\neq 0$ for $k\neq j$ it is clear from
the above result that $\text{D}_{2}F^{j}(B,\psi_{j})\Hc \rightarrow \Hc$ is
an isomorphism. Therefore by the implicit function theorem there exists a
neighborhood $\Nc_{A}$ of $A$ and a map $\tilde{\phi}_{j} :\Nc_{A} \rightarrow \Hc$
such that 
\leqn{contC8}{
F^{j}(B,\tilde{\phi}_{j} (B))=0   \quad \text{for all $B\in \Nc_{A}$}
\quad \text{and} \quad \tilde{\phi}_{j}(A) = \psi(j)\; . 
}
Since $\|\psi_{j}\|=1$ we can by shrinking $\Nc_{A}$ if necessary define
\eqn{contC9}{
\phi_{j} : \Nc_{A} \rightarrow S(\Hc)\; :\; B \rightarrow
\frac{\tilde{\phi}_{j}(B)}{\|\tilde{\phi}_{j} (B)\|} \; .
}
It follows from \eqref{contC8} that
\eqn{contC10)}{
(B-\alpha_{j}(B)\id)\phi_{j}(B) = 0 \quad \text{for all $B\in \Nc_{A}$}
}
where
\eqn{contC11)}{
\alpha_{j}(B) := a_{j}\ip{\phi_{j}}{\tilde{\phi}_{j}(B)} \, .
}
This shows that $\alpha_{j} :\Nc_{A} \rightarrow \mathbb{C}$ is smooth.
However, since $B$ is self-adjoint, it has only real eigenvalues and we
must have $\alpha_{j}(\Nc_{A}) \subset \Rbb$.
\end{proof}

The above lemma allows us to compare the joint distributions for the
self-adjoint operators $A,B$ and $A',B'$ provided that $A$ and $A'$,
and $B$ and $B'$ are close enough. To see this fix $A,B\in \Dc$.
Let $\Nc_{A},\Nc_{B}\subset \Dc$, $\phi_{j}^{A} : \Nc_{A} \rightarrow
S(\Hc)$, $\phi_{j}^{B} : \Nc_{B} \rightarrow S(\Hc)$,
$\alpha^{A}_{j} : \Nc_{A} \rightarrow S(\Hc)$, and
$\alpha^{B}_{j} : \Nc_{B} \rightarrow S(\Hc)$ be as in lemma \ref{contC}.
Suppose $A'\in \Nc_{A}$ and $B'\in \Nc_{B}$, then we can define
\lgath{contD}{
\psi^{A'}_{j} := \phi^{A}_{j}(A') \, , \quad \psi^{B'}_{j} := \phi^{B}_{j}(B') \, , \quad 
\psi^{A}_{j} := \phi^{A}_{j}(A') \, , \quad \psi^{B}_{j} := \phi^{B}_{j}(B')\, , \label{contD1}\\ 
a_{j}' := \alpha^{A}_{j}(A') \, ,\quad b_{j}' := \alpha^{B}(B') \, , \quad
a_{j} := \alpha^{A}_{j}(A) \, ,\quad b_{j} := \alpha^{B}(B) \, . \label{contD2}
}
For any $x\in \PH$, we can compare the joint distributions
$\Prob(A=a_{j},B=b_{k}\,|\,x)$ and $\Prob(A'=a_{j}',B'=b_{k}'\,|\,x)$ by
computing
\lalign{contE}{
\|\Prob(A,B&\,|\,x)- \Prob(A',B'\,|\,x)\| \notag \\ &:= 
\sum_{j,k=1}^{N} |\,\Prob(A=a_{j},B=b_{k}\,|\,x)-\Prob(A'=a_{j}',B'=b_{k}'\,|\,x)\,| \, . \label{contE.1}
}

\begin{thm} \label{contF}\mnote{[contF]}
For any $x\in \PH$ and $\epsilon > 0$ there exists a $\delta > 0$ such that if
$A,A',B,B'\in \Dc$ and $\|A-A'\|$, $\|B-B'\|< \delta$ then
\eqn{contF1a}{
\|\emph{\Prob}(A,B\,|\,x)- \emph{\Prob}(A',B'\,|\,x)\| < \epsilon\, .
}
\end{thm}

\begin{proof}
We  know from the discussion following lemma \ref{contC}, that for
any $\eta > 0$ there exists a $\delta > 0$ such that if
$\|A -A'\| < \delta$ and  $\|B-B'\|< \delta$ then 
$\|\psi_{j}^{A}-\psi_{j}^{A'}\|, \|\psi_{j}^{B}-\psi_{j}^{B'}\| < \eta $
for $j=1,2,\ldots,N$ where $\psi_{j}^{A}$,$\psi_{j}^{A'}$, $\psi_{j}^{B}$,
and $\psi_{j}^{B'}$ are as defined by \eqref{contD1}. Now,
\alin{contF1}{
 |\,\Prob(&A=a_{j},B=b_{k}\,|\,x)-\Prob(A'=a_{j}',B'=b_{k}'\,|\,x)\,| \\
& = |\, \Vol\bigl(\Omega(x,\beta^{A},\psi_{j}^{A})\cap\Omega(x,\beta^{B},\psi_{k}^{B})\bigr)
-\Vol\bigl(\Omega(x,\beta^{A'},\psi_{j}^{A'})\cap\Omega(x,\beta^{B'},\psi_{k}^{B'})\bigr)\, | \, .
}
Proposition \ref{contB} then shows that if we choose $\eta$ small enough
then 
\eqn{contF2}{
|\, \Vol\bigl(\Omega(x,\beta^{A},\psi_{j}^{A})\cap\Omega(x,\beta^{B},\psi_{k}^{B})\bigr)
-\Vol\bigl(\Omega(x,\beta^{A'},\psi_{j}^{A'})\cap\Omega(x,\beta^{B'},\psi_{k}^{B'})\bigr)\, |
< \frac{\epsilon}{N^2}\, 
}
and the proof follows immediately from \eqref{contE.1}.  
\end{proof}

intro.tex
\sect{intro}{Introduction}

As discussed by Ballentine \cite{k6679}, the
assumption that a particle has a definite
position and momentum  
is not incompatible with the uncertainty 
principle.  
More generally, the assumption
that a quantum state has definite
values for certain observables even if
the observables do not commute is compatible
with the statistical interpretation of
quantum mechanics.
Therefore, as indicated by Ballentine \cite{k6679}, there is, 
in principle, no problem
with admitting joint distributions for any set of
observables commutative or otherwise. 

As the standard quantum formalism does
not provide a recipie for constructing
joint distributions associated with
non-commuting observables, the formalism
must be extended.
The problem can be thought of as the
problem of constructing joint distributions with
given marginals. In \cite{CohZap80} (see also 
\cite{Coh84,Coh88}), 
Cohen and Zaparovanny provided a method for 
constructing joint distributions with given 
marginals. It was later shown in \cite{FinGro84,SchSkl86}  
that the Cohen-Zaparovanny construction actually generated
all possible joint distribtuions. Consequently, it is
possible to construct joint distributions for any set
of observables, commuting or not. The difficulty 
is that the Cohen-Zparovanny construction does not single 
out a particlular distribution.

In this paper we provide a method for constructing 
joint distributions
of an arbitrary set of
observables on finite dimensional Hilbert spaces 
irrespective of whether the observables
commute or not. We restrict ourselves to
observables with distinct eigenvalues. This is not
entirely necessary because if there is some degeneracy then
the construction still goes through but does
not result in a unique distribution. Instead, a number
of distributions are possible and we do not
have a criterion for selecting out a particular
one.

The method we employ is technically simple. 
We first represent the usual
transition probabilities as the volume of certain regions
in projective Hilbert space. The joint distributions are
then constructed naturally via the volume of the intersected
regions. This is the strength of our method as compared to
\cite{CohZap80,Coh84,Coh88} because once we choose to represent the transition
probablities as volumes then then the joint distributions arise
naturally. 
For all the technical simplicity of our method, the 
drawback is that the joint distributions are difficult
compute as they require the evaluation of complicated
integrals on projective Hilbert space.
However, we do show that these distributions have a 
number of desirable properties. For example, they
agree with the standard ones if the observables
commute. They also depend continuously on the
observables. Finally, under unitary transformations
they behave in a reasonable manner.

We also note that the volume representation
of transition probablities introduced here can be used
to simplify the constructions used in the hidden
measurment approach to quantum probabilites
\cite{Aert86,Aert98,Coec95a,Coec95b}. 
Moreover, the present approach to transition
probabilites can also be used to easily construct
new hidden measurement systems. We will report on this work elsewhere. 


joint.tex
\sect{joint}{Joint distributions}

To generalize the construction in the previous section to arbitrary but
finite dimensions we must first find a method for generalizing the
decomposition $\PH = \Omega_{1}\cup \Omega_{2}$. So for the moment,
let us still assume that $\dim \Hc = 2$ and that $\{\psi_{1},\psi_{2}\}$
is an orthonormal basis. Now, a short calculation shows that
\eqn{jointA}{
x \in \Omega_{k} \quad \text{if and only if} \quad p_{\psi_{j}}(x)
p_{\psi_{k}}(y) \geq p_{\psi_{k}}(x)p_{\psi_{j}}(y) \quad \text{for} \: 
j=1,2 \, ,
}
where
\eqn{jointB}{
y=[\psi] \, .
}
This motivates us to make the following definition. Let $\Hc$ be
an $N$ dimensional Hilbert space. Suppose $\beta = \{\psi_{1},\ldots,
\psi_{N}\}$ is an orthonormal basis for $\Hc$. Then define a region 
$\Omega(y,\beta,\psi_{j})$ of
$\PH$ that depends on a point $y\in \PH$, the basis $\beta$, and
a particular basis vector $\psi_{j}$ by
\eqn{Omega}{
\Omega(y,\beta,\psi_{k}) := \{\, x\in \PH\:|\: p_{\psi_{j}}(x)
p_{\psi_{k}}(y) \geq p_{\psi_{k}}(x)p_{\psi_{j}}(y) \quad 
\text{for} \: j=1,2,\ldots,N\,\}\, .
}

It is useful to introduce an alternate characterization for 
$\Omega(y,\beta,\psi_{k})$ which although it looks more
complicated is actually easier to work with. To start, consider
the following vectors in $\Rbb^{N-1}$ 
\eqn{xivects}{
\xi_{1} := 0 \quad \text{and}\quad \xi_{j+1}:=(0,\ldots,\overset{\text{jth}}{1}
,\ldots,0) \quad j=1,2,\ldots,N-1\, .
}
Let
\leqn{xit}{
\xit := \sum_{j=1}^{N} p_{\psi_{j}}(y)\xi_{j} 
}
and define
\leqn{Sk}{
S(y,\beta,\psi_{k}) := \Bigl\{ \sum_{j=1,j\neq k}^{N} 
I_{j}\xi_{j} +
I_{k}\xit \in \Rbb^{N-1} \: \Bigl| \: I_{k}\geq 0 \quad \text{and} \quad \sum_{j=1}^{N} I_{j}=1
\Bigr\} \, .
}

\begin{prop} \label{jointC} \mnote{[jointC]}
Suppose $\beta = \{\psi_{1},\ldots,
\psi_{N}\}$ is an orthonormal basis for $\Hc$ and $y\in \PH$. Then
\eqn{jointC1}{
\Omega(y,\beta,\psi_{k}) = \Bigl\{ x \in \PH \:\Bigl| \:
\sum_{j=1}^{N} p_{\psi_{j}}(x)\xi_{j} \in S(y,\beta,\psi_{k}) \: \Bigr\} \, .
} 
\end{prop}
\begin{proof}
Assume that $p_{\psi_{k}}(y)\neq 0$. The case  $p_{\psi_{k}}(y) = 0$
will be left to the reader. Then
using \eqref{xit}, we can write $\sum_{j=1}^{N-1}p_{\psi_{j}}(x)\xi_{j}$  as
\eqn{jointC2}{
\sum_{j=1}^{N-1}p_{\psi_{j}}(x) = \sum_{j=1,j\neq k}^{N} \left( p_{\psi_{j}}(x)
-\frac{p_{\psi_{j}}(y)}{p_{\psi_{k}}(y)}p_{\psi_{k}}(x)\right)\xi_{j}
+ \frac{p_{\psi_{k}}(x)}{p_{\psi_{k}}(y)}\xit \, .
}
From \eqref{eqE} it is easy to see that
\lgath{jointC2}{
\sum_{j=1,j\neq k}^{N}\left( p_{\psi_{j}}(x)
-\frac{p_{\psi_{j}}(y)}{p_{\psi_{k}}(y)}p_{\psi_{k}}(x)\right) +
\frac{p_{\psi_{k}}(x)}{p_{\psi_{k}}(y)} = 1
}
These two results along with the fact that $p_{\psi_{j}}\geq 0$ 
show by the definition of $S(y,\beta,\psi_{k})$ that
\eqn{jointC3}{
\sum_{j=1}^{N-1}p_{\psi_{j}}(x)\xi_{j} \in S(y,\beta,\psi_{k})\quad \Longleftrightarrow \quad
p_{\psi_{j}}(x)p_{\psi_{k}}(y)
\geq p_{\psi_{j}}(y)p_{\psi_{k}}(x) \quad j=1,2,3,\ldots,N \, .
}
\end{proof}

The next two propositions show that the sets $\Omega(y,\beta,\psi_{j})$
have the required properties to be considered a generalization of the sets 
$\Omega_{1}$ and $\Omega_{2}$ from the previous section.

\begin{prop} \label{jointD} \mnote{[jointD]}
Suppose $\beta = \{\psi_{1},\ldots, 
\psi_{N}\}$ is an orthonormal basis for $\Hc$ and $y\in \PH$.
Then 
\eqn{jointD1}{
\PH = \bigcup_{k=1}^{N} \Omega(y,\beta,\psi_{k})
}
and
\eqn{jointD2}{
\emph{\Vol}\bigl(\Omega(y,\beta,\psi_{j})\cap \Omega(y,\beta,\psi_{k})\bigr)
 = 0 \quad
\text{for}\; j\neq k\, .
}
\end{prop}
\begin{proof}
Let $\Sbar$ denote the closure of $S$ defined by \eqref{eqG}, i.e.
\eqn{jointD3}{
\Sbar = \Bigl\{ (a_{2},\ldots,a_{N})\in 
\Rbb^{N-1}\: |\: a_{j}\geq 0 \; \text{and}\;
\sum_{j=2}^{N}a_{j} \leq 1 \; \Bigr\}
}
and define a map $\Jt :\PH \rightarrow \Rbb^{N-1}$ by
\eqn{jointD4}{
\Jt(x) = \sum_{j=2}^{N} p_{\psi_{j}}(x)\xi_{j} \, .
}
Since $p_{\psi_{j}}\geq 0$ and $\sum_{j=1}^{N}p_{\psi_{j}}=1$, we
have that $\Jt(\PH) \subset \Sbar$. To see that this inclusion is
actually an equality, consider any state vector 
\eqn{jointD5}{
\phi = \sum_{j=1}^{N} c_{j}\psi_{j}
} 
where at least one of the coefficients $c_{j}$ is non-zero. Then
$\phi \in \Hcx$ and
\eqn{jointD6}{
\Jt([\phi]) := \frac{1}{\sum_{k=1}^{N}|c_{k}|^2}\sum_{j=2}^{N}|c_{j}|^2
\xi_{j}\, .
} 
It follows directly from this formula that $\Jt(\PH)=\Sbar$.
Also, it is not hard to verify that
\eqn{jointD7}{
\Sbar = \bigcup_{j=1}^{N} S(y,\beta,\psi_{j}) \, .
}
The above two results and proposition \ref{jointC} then imply that
$\PH = \bigcup_{j=1}^{N} \Omega(y,\beta,\psi_{j})$. 

From proposition \ref{jointC} and the definition of $\Jt$, we
have that $\Omega(y,\beta,\psi_{j}) = \Jt^{-1}(S(y,\beta,\psi_{j}))$.
Consequently 
\eqn{jointD8}{
\Omega(y,\beta,\psi_{j})\cap \Omega(y,\beta,\psi_{k}) 
= \Jt^{-1}(S(y,\beta,\psi_{j})) \cap \Jt^{-1}(S(y,\beta,\psi_{k}))
= \Jt^{-1}(S(y,\beta,\psi_{j})\cap S(y,\beta,\psi_{k})) \, .
}
But for $j\neq k$, the set 
$S(y,\beta,\psi_{j})\cap S(y,\beta,\psi_{k})$ lies inside an $N-2$
dimensional subset of $\Rbb^{N-1}$ and hence
$\Jt^{-1}(S(y,\beta,\psi_{j})\cap S(y,\beta,\psi_{k}))$ must
have measure zero. Therefore for $j\neq k$ the formula 
$\Vol\bigl(\Omega(y,\beta,\psi_{j})\cap \Omega(y,\beta,\psi_{k})\bigr)=0$
follows.
\end{proof}

\begin{prop} \label{jointE} \mnote{[jointE]}
Suppose $\{\psi_{1},\ldots,\psi_{N}\}$ is an orthonormal basis for $\Hc$ and $y\in \PH$.
Then
\eqn{jointE1}{
\emph{\Vol}(\Omega(y,\beta,\psi_{j})) = p_{\psi_{j}}(y) \qquad  j=1,2,\ldots,N \, .
}
\end{prop}
\begin{proof}
It is enough to prove it for $j=N$. From proposition \ref{jointC}, and
equations \eqref{eqH}-\eqref{eqK} it is clear that
\alin{jointE2}{
\Vol(\Omega(y,\beta,\psi_{N})) &= \frac{(N-1)!}{(2\pi)^(N-1)}
\int_{0}^{2\pi}\ldots\int_{0}^{2\pi} d\theta_{2}\ldots
d\theta_{N} \int_{S(y,\beta,\psi_{N})}dI_{2}\ldots dI_{N} \\
& = (N-1)! \int_{S(y,\beta,\psi_{N})}dI_{2}\ldots dI_{N} 
}
But from \eqref{xit} and \eqref{Sk} it is easy to verify that
\eqn{jointE3}{
\int_{S(y,\beta,\psi_{N})}dI_{2}\ldots dI_{N} = \frac{1}{(N-1)!}
p_{\psi_{N}}(y) \, ,
}
which completes the proof.
\end{proof}

We are now ready to construct the joint distribution. We only handle
the case of two observables with the generalization to more
observables being obvious.

\begin{thm} \label{jointF} \mnote{[jointF]}
Suppose $A$ and $B$ are two self adjoint operators with distinct eigenvalues.
Let  $A = \sum_{j=1}^{N} a_{j} \proj{\psi^{A}_{j}}$ and
$B = \sum_{j=1}^{N} b_{j} \proj{\psi^{B}_{j}}$ be the unique spectral
representations of $A$ and $B$. Then for each $y\in \PH$,
\eqn{jointF1}{
\emph{\Prob}(A=a_{j},B=b_{k} | y) := \emph{\Vol}\bigl(\Omega(y,\beta^{A},\psi^{A}_{j})
\cap \Omega(y,\beta^{B},\psi^{B}_{j})
\bigr)
}
is a well defined probability distribution. Moreover, the marginal probability distributions 
\eqn{jointF2}{
\emph{\Prob}(A=a_{j} | y) := \sum_{k=1}^{N} \emph{\Prob}(A=a_{j},B=b_{k} | y)
}
and
\eqn{jointF3}{
\emph{\Prob}(B=b_{k} | y ) := \sum_{j=1}^{N} \emph{\Prob}(A=a_{j},B=b_{k} | y)
}
satisfy
\eqn{jointF4}{
\emph{\Prob}(A=a_{j} | y) = p_{\psi^{A}_{j}}(y) \quad \text{and} \quad
\emph{\Prob}(B=b_{k} | y ) = p_{\psi^{B}_{j}}(y) \, ,
}
and hence agree with the usual quantum probability distributions.
\end{thm}
\begin{proof}
From proposition \ref{jointD} we get
\alin{jointF5}{
\PH = \PH \cap \PH & = \Bigl(\bigcup_{j=1}^{N}
\Omega(y,\beta^{A},\psi^{A}_{j})\Bigr)
\cap\Bigl(\bigcup_{k=1}^{N}\Omega(y,\beta^{B},\psi^{B}_{k})\Bigr) \\
 &= \bigcup_{j,k=1}^{N} \Omega(y,\beta^{A},\psi^{A}_{j}) \cap
\Omega(y,\beta^{B},\psi^{B}_{k})\, .  
}
Also from proposition \ref{jointD} we get that 

\lgath{jointF6}{
\Vol\Bigl( \bigl(\Omega(y,\beta^{A},\psi^{A}_{j}) \cap 
\Omega(y,\beta^{B},\psi^{B}_{k})\bigr) \cap
\bigl(\Omega(y,\beta^{A},\psi^{A}_{l}) \cap 
\Omega(y,\beta^{B},\psi^{B}_{m})\bigr) \Bigr) \notag \\
\leq \left\{
\begin{array}{ll}
\Vol(\Omega(y,\beta^{A},\psi^{A}_{j})\cap\Omega(y,\beta^{A},\psi^{A}_{l}))=0&
\text{if $j\neq l$} \\
\Vol(\Omega(y,\beta^{B},\psi^{B}_{k}) 
\cap\Omega(y,\beta^{B},\psi^{B}_{m})) = 0
& \text{if $k\neq m$}
\end{array}
\right. \, . \label{jointF6b}
}
Combining these two results shows that
\eqn{jointF7}{
1 = \Vol(\PH) = \sum_{j,k=1}^{N} \Vol\bigl(\Omega(y,\beta^{A},\psi^{A}_{j}) 
\cap \Omega(y,\beta^{B},\psi^{B}_{k})\bigr) \, , 
}
and hence
$1 = \sum_{j,k=1}^{N} \Prob(A=a_{j}, B=b_{k} | y)$ .
This along with the fact that $\Prob(A=a_{j}, B=b_{k} | y) =
 \Vol\bigl(\Omega(y,\beta^{A},\psi^{A}_{j}) 
\cap \Omega(y,\beta^{B},\psi^{B}_{k})\bigr) \geq 0$ shows that
$\Prob(A=a_{j}, B=b_{k} | y)$ is a well defined probability distribution.

Now,
\alin{jointF8}{
\Omega(y,\beta^{A},\psi^{A}_{j}) &= \Omega(y,\beta^{A},\psi^{A}_{j}) \cap \PH 
 = \Omega(y,\beta^{A},\psi^{A}_{j}) \cap \bigcup_{k=1}^{N}
\Omega(y,\beta^{B},\psi^{B}_{k})\\ & =  \bigcup_{k=1}^{N} 
\Omega(y,\beta^{A},\psi^{A}_{j}) \cap \Omega(y,\beta^{B},\psi^{B}_{k})\, .
}
This result along with \eqref{jointF6b} shows that
\eqn{jointF9}{
\Vol\bigl(\Omega(y,\beta^{A},\psi^{A}_{j})\bigr)
= \sum_{k=1}^{N} \Vol\bigl(\Omega(y,\beta^{A},\psi^{A}_{j}) \cap \Omega(y,\beta^{B},\psi^{B}_{k})\bigr) \, . 
}
Using this and proposition \ref{jointE}, we find that
\eqn{jointF10}{
{p_{\psi^{A}_{j}}(y)} = \sum_{k=1}^{N} {\Prob( A_{j}=a_{j}, B_{k}=b_{k}|y)}\, ,
}
as required. The same arguments can be used to show that \\
${p_{\psi^{B}_{k}}(y)} = \sum_{j=1}^{N} {\Prob( A_{j}=a_{j}, B_{k}=b_{k}|y)} $.
\end{proof}

It is important to realize that the above proposition is also valid for operators
without distinct eigenvalues once a basis of orthonormal eigenvectors
is fixed. The problem is that unlike the case where the eigenvalues are
distinct there is no unique choice of basis and it is not hard to see
that the joint distribution will depend on the basis that is chosen.
Therefore, two different bases will, in general, lead to two different
joint distributions. What is needed is a reasonable criteria that will select
a particular basis.   

The next proposition shows that the joint distributions transform correctly
under unitary transformations.

\begin{prop} \label{jointG} \mnote{[jointG]}
Suppose $U \in \U$, $y\in \PH$, and $A = \sum_{j=1}^{N} a_{j} \proj{\psi^{A}_{j}}$ and
$B = \sum_{j=1}^{N} b_{j} \proj{\psi^{B}_{j}}$ are two self-adjoint operators
with distinct eigenvalues. Let
\alin{jointG1}{
A' := UAU^{-1} & = \sum_{j=1}^{N} a_{j} \proj{\psi^{A'}_{j}}  \qquad
\psi^{A'}_{j} := U\psi^{A}_{j}\, , \\
B' := UBU^{-1} & = \sum_{j=1}^{N} b_{j} \proj{\psi^{B'}_{j}}  \qquad
\psi^{B'}_{j} := U\psi^{B}_{j}\, . \\
}
Then
\eqn{jointG2}{
\emph{\Prob}(A'=a_{j}, B'=b_{k}|y)
= \emph{\Prob}(A=a_{j}, B=b_{k}|\rho_{U}(y))\, .
}
\end{prop}
\begin{proof}
To start, suppose $\alpha := \{\psi_{1},\ldots,\psi_{N}\}$ is an orthonormal
basis. From the definition of the functions $p_{\psi}$ it is easy to
see that
\eqn{jointG3}{
p_{U\psi_{j}}(y) = p_{\psi_{j}}(\rho_{U}(y)) \quad j=1,2,\ldots,N \; .
}
It then follows directly from the definition of the 
set $\Omega(y,\alpha,\psi_{j})$ that
\eqn{jointG4}{
\Omega(\rho_{U}(y),\alpha,\psi_{j}) = \rho_{U}(\Omega(y,\alpha',\psi_{j}'))
}
where $\alpha' := \{\psi_{1}':= U\psi_{1},\ldots,\psi_{N}' := U\psi_{N}\}$.
Therefore,
\leqn{jointG5}{
\Omega(\rho_{U}(y),\beta^{A},\psi^{A}_{j})\cap
\Omega(\rho_{U}(y),\beta^{B},\psi^{B}_{k})
= \rho_{U}\bigl(\Omega(y,\beta^{A'},\psi^{A}_{j})\cap
\Omega(y,\beta^{B'},\psi^{B'}_{k})
 \bigr)
} 
for all $j,k=1,2,\ldots,N$.
Since $\rho_{U}^{*}\omega = \omega$ it follows that the volume form
$\mu$ is invariant under the action of $U$, i.e. $\rho_{U}^{*}\mu = \mu$.
So \eqref{jointG5} and the change of variables theorem
implies that
\eqn{jointG6}{
\Vol\bigl(\Omega(\rho_{U}(y),\beta^{A},\psi^{A}_{j})\cap
\Omega(\rho_{U}(y),\beta^{B},\psi^{B}_{k})\bigr)
= \Vol\bigl(\Omega(y,\beta^{A'},\psi^{A}_{j})\cap
\Omega(y,\beta^{B'},\psi^{B'}_{k})
 \bigr) 
}
for all $ j,k=1,2,\ldots,N$,
which completes the proof.
\end{proof}

proj.tex
\sect{PHS}{Projective Hilbert Space}

In this section we review some basic results about projective Hilbert
space. We use the  book \cite{k6081} as our standard reference.
Let $(\Hc,\ip{\cdot}{\cdot})$ be a complex Hilbert space where the inner product
$\ip{\cdot}{\cdot}$ is taken to be linear in the second variable. Define 
\eqn{Hcross}{
\Hcx := \Hc\setminus\{0\} \quad \text{and} \quad \Cbbx := \Cbb\setminus\{0\}.
} 
On $\Hcx$ we can define an equivalence relation $\sim$ by 
\eqn{equiv}{
\text{$\psi \sim \phi$ if and only if there exists a $\lambda \in \Cbbx$ such that 
$\psi = \lambda \phi$.}
}
Letting $[\psi]$ denote the equivalence class for $\psi \in \Hcx$, 
we have
\eqn{eclass}{
[\psi] := \{\, \lambda \psi \, |\, \lambda \in \Cbbx\,\} \; .
}
Projective Hilbert $\PH$ space is defined as
\eqn{PH}{
\PH := \Hcx/\sim = \{\, [\psi] \,|\, \psi \in \Hcx\, \} \, .
}
It is well known that $\PH$ carries a 
Hilbert manifold structure for which the canonical projection
\eqn{proj}{
\pi : \Hcx \rightarrow \PH\; ; \;\psi \mapsto [\psi]
}
is a $C^{\infty}$ submersion. As a consequence for any $q\in \PH$ and
$v_{q} \in \T_{q} \PH$ there exists a $\psi \in \Hcx$ and $\phi \in \Hc$
such that
\eqn{projtan}{
q = [\psi] \quad \text{and} \quad v_{q} = \T_{\psi} \pi\cdot \phi \, .
}
This can be used to define a complex structure $\Jbb$ and a strongly
non-degenerate symplectic form $\omega$
on $\PH$ via the formulas
\leqn{cstruc}{
\Jbb(\T_{{\psi}} \pi \cdot \phi) := \T_{\psi}\cdot i\phi 
}
and
\leqn{sympl}{
\omega_{[\psi]}(\T_{\psi}\pi\cdot\phi_{1},\T_{\psi}\pi\cdot\phi_{2}) := 
2\hbar\|\psi\|^{-4} \text{Im}\left( \ip{\phi_{1}}{\phi_{2}}\|\psi\|^{2}
-\ip{\phi_{1}}{\psi}\ip{\psi}{\phi_{2}}\right) 
}
for every $\psi \in \Hcx$ and $\phi,\phi_{1},\phi_{2}\in \Hc$. 
It should be noted that 
\leqn{metric}{
g(v,w) = \omega(v,\Jbb w)
} 
defines a Riemannian metric on $\PH$ and hence establishes that $\PH$ is a K\"{a}hler manifold.

Given a function $f \in  C^{\infty}(\PH)$, the
fact that the symplectic form is strongly non-degenerate implies
that the following equation
\leqn{Hamilton}{
\ti_{X_{f}} \omega  = \td f
}
uniquely defines the vector field $X_{f}$.
The Poisson bracket $\{\cdot,\cdot\}$ is then defined via
\leqn{poissonB}{
\{f,g\} := \omega(X_{f},X_{f}) \quad \forall \; f,g \in  C^{\infty}(\PH) \; .
}

Let $\Lc(\Hc)$ denote the set of bounded linear operators on $\Hc$.
Then the unitary group $\U$ is defined by
\eqn{group}{
\U := \{ U \in \Lc(\Hc) \, | \, \ip{U\psi}{U\phi} = \ip{\psi}{\phi} \quad
\forall \psi,\phi \in \Hc \, \} \; .
}
Its Lie algebra $\uf$ is the set of skew-adjoint operators, i.e.
\eqn{liealg}{
\uf := \{ A \in \Lc(\Hc) \, | \, A^{\dagger} = - 
A \, \} \, .
}
Here we are using $\dagger$ to denote the adjoint of an operator.
The following map
\eqn{gaction}{
\rho \; : \; \U \times \PH \rightarrow \PH \; : \; (U,[\psi]) \rightarrow
[U\psi] 
}
defines an action of $\U$ on $\PH$ by symplectomorphism (i.e. 
$\rho_{U}^{*}\omega = \omega$ for all $U\in \U$). There exists
an equivariant momentum mapping $\J\, :\, \PH \longrightarrow \uf^{*}$ for this action defined by
\leqn{mmap}{
\langle \J([\psi]),A\rangle := 
i\hbar\frac{\ip{\psi}{A\psi}}{\|\psi\|^{2}} \quad
\forall \; \psi \in \Hc,\, A\in \uf \, , 
}
where $ \langle \cdot, \cdot \rangle $ denotes the canonical pairing between
$\uf^{*}$ and $\uf$.
Letting $\mathcal{C}^{\infty}(\PH)$ denote
the set of smooth functions on $\PH$, the momentum map can be viewed
as a map
$\J\, :\, \uf \longrightarrow \mathcal{C}^{\infty}(\PH)$ by
defining
\leqn{mmapA}{
\J(A)(x) := \langle \J(x),A\rangle \quad \forall \; x \in \PH\; .
}
Because of the equivariance of the momentum map $\J$, we have the
useful property
\leqn{poisson}{
\{\J(A),\J(B)\} = \J([A,B]) 
\quad \forall \; A,B \in \uf\, .
}

Let $\Sa$ denote the set of bounded self-adjoint operators
on $\Hc$. For each operator $H \in \Sa$ we define
a smooth function $\E{H}$ on $\PH$ via
\leqn{exp1}{
\E{H} := \J(-\ioh H) \, .
}
Thus $\E{H}$ is just the usual expectation of the
observable $H$, i.e.
\leqn{exp2}{
\E{H}([\psi]) = \frac{\ip{\psi}{H\psi}}{\|\psi\|^2}\, . 
}
With this notation \eqref{poisson} can be written
in the more familiar form 
\leqn{poissonA}{
\{\E{A},\E{B}\} = \E{\ioh[B,A]} \; .
}

qjd.tex
\documentclass{article}


\usepackage{graphicx,amsmath,amsthm,amssymb,array,latexsym,cite,epsfig,finalT}
%\usepackage{doublespace}

\numberwithin{equation}{section}

\newcommand{\sauthor}{T. A. Oliynyk}
\newcommand{\stitle}{Quantum Joint Distributions}


\input qjd.def


\begin{document}

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\input qjd_tit.tex
\input intro.tex
\input proj.tex
\input action.tex
\input vrep.tex
\input joint.tex
\input cont.tex
\input conc.tex


\section*{Acknowledgements} 
\noindent This work was partially supported
by the ARC grant A00105048 at the University of Canberra.

%\end{spacing}

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\bibliographystyle{amsplain}
\bibliography{qjd}

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qjd_tit.tex
\title{Quantum Joint Distributions
\thanks{\emph{PACS No:} 03.65.Ta}
}
\author{
Todd A. Oliynyk \thanks{todd.oliynyk@ise.canberra.edu.au} \\
School of Mathematics and Statistics\\
University of Canberra\\ ACT 2601, Australia }

\maketitle

\begin{abstract}

In this paper we provide a method for constructing
joint distributions for an arbitrary set of
observables on finite dimensional Hilbert spaces
irrespective of whether the observables
commute or not.  These distributions have a
number of desirable properties: they
agree with the standard quantum mechanical ones if the observables
commute, they also depend continuously on the
observables, and  under unitary transformations
they behave in a reasonable manner. 
\end{abstract}

vrep.tex
\sect{vreps}{Volume representation of transition probabilities}

Suppose $\psi,\phi \in \Hcx$. Then the transition probability from
the state $\psi$ to $\phi$, or vice versa, is given by
\leqn{trans}{
T(\psi,\phi) := \frac{|\ip{\psi}{\phi}|^{2}}{\|\psi\|^{2}\|\phi\|^{2}}\;.
}
Because this formula is invariant under scaling of $\phi$ or $\psi$ by
non-zero complex numbers, it passes to a well defined function on 
$\PH\times \PH$ given by
\leqn{transA}{
T([\psi],[\phi]) = \frac{|\ip{\psi}{\phi}|^{2}}{\|\psi\|^{2}\|\phi\|^{2}}\;
\quad \forall\; \psi,\phi \in \Hcx\, .
}
\begin{figure}
\begin{center}
\epsfig{file=qjdfig1.eps,height=7cm}
\end{center}
\caption{Volume representation of transition probabilities} \label{fig1}
\end{figure}
It was shown in \cite{k6678} that if we let $d(x,y)$ denote the geodesic distance between points $x,y \in \PH$ then the distance $d(x,y)$ is related to
the transition probability $T(x,y)$ via the formula
\leqn{transC}{
T(x,y) = \cos^{2}\Bigl(\frac{d(x,y)}{\sqrt{2\hbar}}\Bigr) \, .
} 
This shows that there exists a representation of the transition probability
in terms of the geodesic distance. The question now is, are there other 
representations for the transition probability in terms of geometrical objects
on $\PH$? We will show that the transition probability, at least for finite
dimensional Hilbert spaces, can be related the volume of certain regions
in $\PH$. To motivate this, we will first look at $\PH$ where $\Hc$ is
a 2 dimensional Hilbert space. Then $\PH \cong S^{2}$ where $S^{2}$ is
the ordinary two sphere in $\Rbb^{3}$. Suppose $\{\psi_{1},\psi_{2}\}$ is
an orthonormal basis for $\Hc$ and $\psi \in \Hcx$ is an  arbitrary state
vector. Since $\{\psi_{1},\psi_{2}\}$ is orthonormal, we can choose
them to be the north and south poles of $S^{2}$ as in Figure \ref{fig1}.
The symplectic form on $S^2$ is
\eqn{S2symp}{
\omega = \frac{\hbar \sin\theta}{2}d\phi \wedge d\theta
} 
while the volume form $\mu$ is given by
\eqn{S2vol}{
\mu = \frac{1}{2\hbar\pi}\omega \; .
}
The normalization on the volume form is chosen so that $\int_{\PH}\mu = 1$.
 
Referring to Figure \ref{fig1}, let $\Omega$ be the shaded region between
the points $[\psi_{1}]$ and $[\psi]$. Then a straightforward calculation shows
that
\eqn{volrepA}{
T([\psi_{2}],[\psi]) = \Vol(\Omega) := \int_{\Omega} \mu\, . 
}
Of course if we let $\gamma$ be the geodesic between $[\psi]$ and
$[\psi_{2}]$ represented by the dashed line in Figure \ref{fig1} then
we also have
\eqn{volrepB}{
T([\psi_{2}],[\psi]) = \cos^{2}\bigl((2\hbar)^{-1}
(\text{geodesic lenth of $\gamma$})\bigr)\, .
}
Letting $\Omega^{c}$ denote the complement of $\Omega$ we also have
\eqn{volrepC}{
T([\psi_{1}],[\psi]) = \Vol(\Omega^{c}) := \int_{\Omega^{c}} \mu\, . 
}
It is interesting to note that the conservation of probability
\eqn{volrepD}{
1 = T([\psi_{1}],[\psi]) + T([\psi_{2}],[\psi])
}
has the simple geometric representation
\eqn{volrepE}{
\PH = \Omega \cup \Omega^{c} \, .
}
Now, consider the self-adjoint operator
\eqn{volrepF}{
A = a_{1}\proj{\psi_{1}} + a_{2}\proj{\psi_{2}} 
}
where the eigenvalues $a_{1}$ and $a_{2}$ are distinct.
Then upon measurement, the probability
of a measurement of the observable $A$ yielding a value $a_{j}$ given
that it is in the state $[\psi]$ is
\eqn{measur}{
\Prob(A=a_{j}\,|\,[\psi]) = \Vol(\Omega_{j}) \quad j=1,2 \, ,
}
where we let
\eqn{measurA}{
\Omega_{2} := \Omega \quad \text{and} \quad \Omega_{1} := \Omega^{c}  \; .
}
Now suppose $\{\phi_{1},\phi_{2}\}$ is another orthonormal basis and
$B$ is the operator,
\eqn{measurB}{
B = b_{1}\proj{\phi_{1}} + b_{2}\proj{\phi_{2}}\, .
} 
Then as above, we find two sets $\Sigma_{1}$ and $\Sigma_{2}$
such that
\eqn{measurC}{
\PH = \Sigma_{1} \cup \Sigma_{2}
}
and
\eqn{measurD}{
\Prob(B=b_{j}\,|\,[\psi]) = \Vol(\Sigma_{j}) \quad j=1,2 \, .
}

So far we have not done anything new, except express the usual
transition probabilities in terms of the volume of certain regions
in $\PH$. However, we observe that there is a natural way to
construct a joint distribution for the observables $A$ and $B$
which is given by
\eqn{jointAA}{
\Prob(A=a_{j},B=b_{k}\,|\,[\psi]) := \Vol(\Omega_{j}\cap\Sigma_{k})
\quad j,k=1,2 \, .
}
It is important to realize that this formula is valid even if
$A$ and $B$ do not commute. It is also easy to see that it gives
the correct joint distribution if $A$ and $B$ do commute.
\begin{figure}
\begin{center}
\epsfig{file=qjdfig2.eps,height=7cm}
\end{center}
\caption{Volume representation of joint probabilities}\label{fig2}
\end{figure}
See figure \ref{fig2} for an illustration of the region $\Omega_{2}\cap
\Sigma_{2}$, the volume of which would yield the transition probability 
$\Prob(A=a_{2},B=b_{2}\,|\,[\psi])$. 

qjd.bbl
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
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