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prop_classical_quantum.lyx
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\begin_layout Title
Note about the Relation Between Classical and Quantum
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\begin_layout Author
Yu-Tsung Tai
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\begin_layout Affiliation
Department of Mathematics, Indiana University, Bloomington, Indiana 47405,
USA
\end_layout
\begin_layout Affiliation
Department of Computer Science, Indiana University, Bloomington, Indiana
47405, USA
\end_layout
\begin_layout Author
Andrew J.
Hanson
\end_layout
\begin_layout Affiliation
Department of Informatics, Indiana University, Bloomington, Indiana 47405,
USA
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\begin_layout Author
Gerardo Ortiz
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\begin_layout Affiliation
Department of Physics, Indiana University, Bloomington, Indiana 47405, USA
\end_layout
\begin_layout Author
Amr Sabry
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\begin_layout Affiliation
Department of Computer Science, Indiana University, Bloomington, Indiana
47405, USA
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today
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%
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keywords{Finite precision measurement, the Born rule, Gleason's
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% theorem, the Kochen-Specker theorem, interval-valued probability measure,
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status collapsed
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% discrete quantum theory}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin_layout Standard
This note is to prove that quantum probability becomes classical when restrictin
g on a commuting subspace of events.
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\begin_layout Section
Basic Definitions
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Before diving into the theorems and proofs, we summarize the definitions
of classical and quantum probability in the following table.
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\noindent
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Table I.
Compare the classical probability and quantum probability.
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Classical
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Quantum
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\end_inset
</cell>
<cell alignment="block" valignment="top" topline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Projectors
\begin_inset space ~
\end_inset
\begin_inset Formula $E_{0}$
\end_inset
and
\begin_inset space ~
\end_inset
\begin_inset Formula $E_{1}$
\end_inset
is called
\emph on
disjoint
\emph default
if
\begin_inset Formula $E_{0}\cap E_{1}=\emptyset$
\end_inset
\emph on
.
\end_layout
\end_inset
</cell>
<cell alignment="block" valignment="top" topline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="block" valignment="top" topline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Projectors
\begin_inset space ~
\end_inset
\begin_inset Formula $P_{0}$
\end_inset
and
\begin_inset space ~
\end_inset
\begin_inset Formula $P_{1}$
\end_inset
is called
\emph on
orthogonal
\emph default
if
\begin_inset Formula $P_{0}P_{1}=\mathbb{0}$
\end_inset
\emph on
.
\end_layout
\end_inset
</cell>
</row>
<row topspace="default">
<cell alignment="block" valignment="top" topline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Set of events,
\begin_inset Formula $\events$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="block" valignment="top" topline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="block" valignment="top" topline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Set of all subsets of a given set
\begin_inset space ~
\end_inset
\begin_inset Formula $\Omega$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="block" valignment="top" topline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="block" valignment="top" topline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Set of all projectors on a given Hilbert space
\begin_inset space ~
\end_inset
\begin_inset Formula ${\cal H}$
\end_inset
\end_layout
\end_inset
</cell>
</row>
<row topspace="default">
<cell alignment="block" valignment="top" topline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\emph on
Subspace
\emph default
of events,
\begin_inset Formula $\events'\subseteq\events$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="block" valignment="top" topline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="block" valignment="top" topline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\events'$
\end_inset
is subject to the following constraints:
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
begin{subequations}
\end_layout
\end_inset
\begin_inset Formula
\begin{eqnarray}
& & \emptyset\in\events'\,,\\
& & \Omega\in\events'\,,\\
& & \overline{E}\in\events'\,,
\end{eqnarray}
\end_inset
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
end{subequations}
\end_layout
\end_inset
and satisfying for each pair of sets
\begin_inset Formula $E_{0}\in\events'$
\end_inset
and
\begin_inset Formula $E_{1}\in\events'$
\end_inset
,
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
begin{subequations}
\end_layout
\end_inset
\begin_inset Formula
\begin{eqnarray}
& & E_{0}\cap E_{1}\in\events'\,,\\
& & E_{0}\cup E_{1}\in\events'\,.
\end{eqnarray}
\end_inset
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
end{subequations}
\end_layout
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="block" valignment="top" topline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="block" valignment="top" topline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\events'$