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\hbox to 5.78in { {\bf PHYS 7895: Quantum Information Theory } \hfill #2 }
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\begin{document}
\lecture{9 --- September 23, 2015}{Fall 2015}{Prof.\ Mark M.\ Wilde}{Mark M.~Wilde}
This document is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
\section{Overview}
In the last lecture we discussed more general models of measurements, the POVM formalism, product states, separable states, and the partial trace operation.
The evolution of a quantum state is never perfect. In
this lecture, we discuss the most general approach to understanding quantum
evolutions:\ the \textit{axiomatic approach}. In this powerful approach, we
start with three physically reasonable axioms that should hold for any quantum
evolution and then deduce a set of mathematical constraints that any quantum evolution
should satisfy (this is known as the \textit{Choi-Kraus theorem}). For the rest of the course, we will refer to quantum evolutions satisfying these constraints as
\textit{quantum channels}.
\section{Axiomatic Approach to Noisy Quantum Evolutions}
\label{sec-nqt:axiomatic-approach}We now discuss a powerful approach to
understanding quantum physical evolutions called the \textit{axiomatic
approach}. Here we make three physically reasonable assumptions that any
quantum evolution should satisfy and then prove that these axioms imply
mathematical constraints on the form of any quantum physical evolution.
All of the constraints we impose are motivated by the reasonable requirement
for the output of the evolution to be a quantum state (density operator) if
the input to the evolution is a quantum state (density operator). An important
assumption to clarify at the outset is that we are viewing a quantum physical
evolution as a \textquotedblleft black box,\textquotedblright\ meaning that
Alice can prepare any state that she wishes before the evolution begins,
including pure states or mixed states. Critically, we even allow her to input
one share of an entangled state. This is a standard assumption in quantum
information theory, but one could certainly question whether this assumption
is reasonable. If we do accept this criterion as physically reasonable, then
the Choi-Kraus representation theorem for quantum evolutions follows as a consequence.
\begin{notation}
[Density Operators and Linear Operators]Let $\mathcal{D}(\mathcal{H})$ denote
the space of density operators acting on a Hilbert space $\mathcal{H}$, let
$\mathcal{B}(\mathcal{H})$ denote the space of square linear operators acting
on $\mathcal{H}$, and let $\mathcal{B}(\mathcal{H}_{A},\mathcal{H}_{B})$
denote the space of linear operators taking a Hilbert space $\mathcal{H}_{A}$
to a Hilbert space $\mathcal{H}_{B}$.
\end{notation}
Throughout this development, we let $\mathcal{N}$ denote a map which takes
density operators in $\mathcal{D}(\mathcal{H}_{A})$ to those in $\mathcal{D}%
(\mathcal{H}_{B})$. In general, the respective input and output Hilbert spaces
$\mathcal{H}_{A}$ and $\mathcal{H}_{B}$ need not be the same. Implicitly, we
have already stated a first physically reasonable requirement that we impose
on $\mathcal{N}$, namely, that\ $\mathcal{N}(\rho_{A})\in\mathcal{D}%
(\mathcal{H}_{B})$ if $\rho_{A}\in\mathcal{D}(\mathcal{H}_{A})$. Extending
this requirement, we demand that $\mathcal{N}$ should be \textit{convex
linear} when acting on $\mathcal{D}(\mathcal{H}_{A})$:%
\begin{equation}
\mathcal{N}(\lambda\rho_{A}+(1-\lambda)\sigma_{A})=\lambda\mathcal{N}(\rho
_{A})+(1-\lambda)\mathcal{N}(\sigma_{A}),
\label{eq-nqt:convex-linearity-channels}%
\end{equation}
where $\rho_{A},\sigma_{A}\in\mathcal{D}(\mathcal{H}_{A})$ and $\lambda
\in\left[ 0,1\right] $.
The physical interpretation of this convex-linearity requirement is in terms
of repeated experiments. Suppose a large number of experiments are conducted
in which identical quantum systems are prepared in the state $\rho_{A}$ for a
fraction $\lambda$ of the experiments and in the state $\sigma_{A}$ for the
other fraction $1-\lambda$ of the experiments. Suppose further that it is not
revealed which states are prepared for which experiments. Before you are
allowed to perform measurements on each system, the evolution $\mathcal{N}$ is
applied to each of the systems. The density operator characterizing the state
of each system for these experiments is then $\mathcal{N}(\lambda\rho
_{A}+(1-\lambda)\sigma_{A})$. You are then allowed to perform a measurement on
each system, which after a large number of experiments allow you to infer that
the density operator is $\mathcal{N}(\lambda\rho_{A}+(1-\lambda)\sigma_{A})$.
Now, in principle, it could have been revealed which fraction of the
experiments had the state $\rho_{A}$ prepared and which fraction had
$\sigma_{A}$ prepared. In this case, the density operator describing the
$\rho_{A}$ experiments would be $\mathcal{N}(\rho_{A})$ and that describing
the $\sigma_{A}$ experiments would be $\mathcal{N}(\sigma_{A})$. So, it is
reasonable to expect that the statistics observed in your measurement outcomes
in the first scenario would be consistent with those observed in the second
scenario, and this is the physical statement that the requirement \eqref{eq-nqt:convex-linearity-channels}\ makes.
Now, it is mathematically convenient to extend the domain and range of the
quantum channel to apply not only to density operators but to all linear
operators. To this end, it is possible to find a unique linear extension
$\widetilde{\mathcal{N}}$ of any quantum evolution $\mathcal{N}$ defined as
above (originally defined exclusively by its action on density operators and
satisfying convex linearity). See Section~\ref{app:unique-linear-ext}\ for a
full development of this idea. Thus, it is reasonable to associate this unique
linear extension $\widetilde{\mathcal{N}}$\ to the quantum physical evolution
$\mathcal{N}$ mathematically, and in what follows (and for the rest of the
book), we simply identify a physical evolution $\mathcal{N}$\ with its unique
linear extension $\widetilde{\mathcal{N}}$, and this is what we call a
\textit{quantum channel}. For these reasons, we now impose that any quantum
channel $\mathcal{N}$ is linear:
\begin{criterion}
[Linearity]\label{crit-nqt:linearity}A quantum channel $\mathcal{N}$ is a
linear map:%
\begin{equation}
\mathcal{N}(\alpha X_{A}+\beta Y_{A})=\alpha\mathcal{N}(X_{A})+\beta
\mathcal{N}(Y_{A}),
\end{equation}
where $X_{A},Y_{A}\in\mathcal{B}(\mathcal{H}_{A})$ and $\alpha,\beta
\in\mathbb{C}$.
\end{criterion}
We have already demanded that quantum physical evolutions should take density
operators to density operators. Combining with linearity (in particular, scale
invariance) implies that quantum channels should preserve the class of
positive semi-definite operators. That is, they should be positive maps, as
defined below:
\begin{definition}
[Positive Map]A linear map $\mathcal{M}:\mathcal{B}(\mathcal{H}_{A}%
)\rightarrow\mathcal{B}(\mathcal{H}_{B})$ is positive if $\mathcal{M}(X_{A})$
is positive semi-definite for all positive semi-definite $X_{A}\in
\mathcal{B}(\mathcal{H}_{A})$.
\end{definition}
If we were dealing with classical systems, then positivity would be sufficient
to describe the class of physical evolutions. However, above we argued that we
are working in the \textquotedblleft black box\textquotedblright\ picture of
quantum physical evolutions, and here, in principle, we allow for Alice to
prepare the input system $A$ to be one share of an arbitrary two-party state
$\rho_{RA}\in\mathcal{D}(\mathcal{H}_{R}\otimes\mathcal{H}_{A})$, where $R$ is
a reference system of arbitrary size. So this means that the evolution
consisting of the identity acting on the reference system $R$ and the map
$\mathcal{N}$ acting on system $A$ should take $\rho_{RA}$ to a density
operator on systems $R$ and $B$. Let $\operatorname{id}_{R}\otimes
\mathcal{N}_{A\rightarrow B}$ denote this evolution, where $\operatorname{id}%
_{R}$ denotes the identity superoperator acting on the system $R$.
How do we describe the evolution $\operatorname{id}_{R}\otimes\mathcal{N}%
_{A\rightarrow B}$ mathematically? Let $X_{RA}$ be an arbitrary operator
acting on $\mathcal{H}_{R}\otimes\mathcal{H}_{A}$, and let $\{|i\rangle_{R}\}$
be an orthonormal basis for $\mathcal{H}_{R}$. Then we can expand $X_{RA}$
with respect to this basis as follows:%
\begin{equation}
X_{RA}=\sum_{i,j}|i\rangle\langle j|_{R}\otimes X_{A}^{i,j},
\end{equation}
and the action of $\operatorname{id}_{R}\otimes\mathcal{N}_{A\rightarrow B}$
on $X_{RA}$ (for linear $\mathcal{N}$) is defined as follows:%
\begin{align}
\left( \operatorname{id}_{R}\otimes\mathcal{N}_{A\rightarrow B}\right)
\left( X_{RA}\right) & =\left( \operatorname{id}_{R}\otimes
\mathcal{N}_{A\rightarrow B}\right) \left( \sum_{i,j}|i\rangle\langle
j|_{R}\otimes X_{A}^{i,j}\right) \\
& =\sum_{i,j}\left( \operatorname{id}_{R}\otimes\mathcal{N}_{A\rightarrow
B}\right) \left( |i\rangle\langle j|_{R}\otimes X_{A}^{i,j}\right) \\
& =\sum_{i,j}\operatorname{id}_{R}\left( |i\rangle\langle j|_{R}\right)
\otimes\mathcal{N}_{A\rightarrow B}\left( X_{A}^{i,j}\right) \\
& =\sum_{i,j}|i\rangle\langle j|_{R}\otimes\mathcal{N}_{A\rightarrow
B}\left( X_{A}^{i,j}\right) . \label{eq-nqt:action-id-N}%
\end{align}
That is, the identity superoperator $\operatorname{id}_{R}$ has no effect on
the $R$ system. The above development leads to the notion of a linear map
being \textit{completely positive} and our next criterion for any quantum
physical evolution:
\begin{definition}
[Completely Positive Map]\label{def-nqt:completely-positive}A linear map
$\mathcal{M}:\mathcal{B}(\mathcal{H}_{A})\rightarrow\mathcal{B}(\mathcal{H}%
_{B})$ is completely positive if $\operatorname{id}_{R}\otimes\mathcal{M}$ is
a positive map for a reference system $R$ of arbitrary size.
\end{definition}
\begin{criterion}
[Complete Positivity]\label{crit-nqt:CP}A quantum channel is a completely
positive map.
\end{criterion}
There is one last requirement that we impose for quantum physical evolutions,
known as \textit{trace preservation}. This requirement again stems from the
reasonable constraint that $\mathcal{N}$ should map density operators to
density operators. That is, it should be the case that $\operatorname{Tr}%
\{\rho_{A}\}=\operatorname{Tr}\{\mathcal{N}(\rho_{A})\}=1$ for all input
density operators $\rho_{A}$. However, now that have argued for linearity of
every quantum physical evolution, trace preservation on density operators
combined with linearity implies that quantum channels are trace preserving on
the set of all operators. This is due to the fact that there are sets of
density operators that form a basis for $\mathcal{B}(\mathcal{H}_{A})$.
Indeed, one such basis of density operators is as follows:%
\begin{equation}
\rho_{A}^{x,y}=\left\{
\begin{array}
[c]{cc}%
|x\rangle\langle x|_{A} & \text{if }x=y\\
\frac{1}{2}\left( |x\rangle_{A}+|y\rangle_{A}\right) \left( \langle
x|_{A}+\langle y|_{A}\right) & \text{if }xy
\end{array}
\right. . \label{eq-nqt:density-op-basis}%
\end{equation}
Consider that for all $x,y$ such that $x0$ that we have scale invariance:%
\begin{align}
\widetilde{\mathcal{N}}(sP_{A}) & =\operatorname{Tr}\{sP_{A}\}\mathcal{N}%
(\left[ \operatorname{Tr}\{sP_{A}\}\right] ^{-1}sP_{A})\\
& =s\operatorname{Tr}\{P_{A}\}\mathcal{N}(\left[ \operatorname{Tr}%
\{P_{A}\}\right] ^{-1}P_{A})\\
& =s\widetilde{\mathcal{N}}(P_{A}).
\end{align}
Furthermore, for two non-zero positive semi-definite operators $P_{A}$ and
$Q_{A}$, we have the following additivity relation:%
\begin{equation}
\widetilde{\mathcal{N}}(P_{A}+Q_{A})=\widetilde{\mathcal{N}}(P_{A}%
)+\widetilde{\mathcal{N}}(Q_{A}), \label{eq-app-linear:additive-PSD}%
\end{equation}
which follows because%
\begin{align}
& \widetilde{\mathcal{N}}(P_{A}+Q_{A})\nonumber\\
& =\operatorname{Tr}\{P_{A}+Q_{A}\}\mathcal{N}(\left[ \operatorname{Tr}%
\{P_{A}+Q_{A}\}\right] ^{-1}(P_{A}+Q_{A}))\\
& =\operatorname{Tr}\{P_{A}+Q_{A}\}\mathcal{N}\left( \frac{1}%
{\operatorname{Tr}\{P_{A}+Q_{A}\}}P_{A}+\frac{1}{\operatorname{Tr}%
\{P_{A}+Q_{A}\}}Q_{A}\right) \\
& =\operatorname{Tr}\{P_{A}+Q_{A}\}\mathcal{N}\left( \frac{\operatorname{Tr}%
\{P_{A}\}}{\operatorname{Tr}\{P_{A}+Q_{A}\}}\frac{P_{A}}{\operatorname{Tr}%
\{P_{A}\}}+\frac{\operatorname{Tr}\{Q_{A}\}}{\operatorname{Tr}\{P_{A}+Q_{A}%
\}}\frac{Q_{A}}{\operatorname{Tr}\{Q_{A}\}}\right) \\
& =\operatorname{Tr}\{P_{A}\}\mathcal{N}\left( \frac{P_{A}}%
{\operatorname{Tr}\{P_{A}\}}\right) +\operatorname{Tr}\{Q_{A}\}\mathcal{N}%
\left( \frac{Q_{A}}{\operatorname{Tr}\{Q_{A}\}}\right) \\
& =\widetilde{\mathcal{N}}(P_{A})+\widetilde{\mathcal{N}}(Q_{A}),
\end{align}
where in the fourth equality, we exploited convex linearity of the quantum
physical evolution $\mathcal{N}$.
For the next step, recall that any Hermitian operator $T_{A}$\ can be written
as a linear combination of a positive part and a negative part:\ $T_{A}%
=T_{A}^{+}-T_{A}^{-}$, where both $T_{A}^{+}$ and $T_{A}^{-}$ are positive
semi-definite operators. So we define the action of $\widetilde{\mathcal{N}}$
on any Hermitian operator $T_{A}$ as follows:%
\begin{equation}
\widetilde{\mathcal{N}}(T_{A})\equiv\widetilde{\mathcal{N}}(T_{A}%
^{+})-\widetilde{\mathcal{N}}(T_{A}^{-}).
\label{eq-app-linear:define-hermitian}%
\end{equation}
To see that the following additivity relation holds for all Hermitian $S_{A}$
and $T_{A}$%
\begin{equation}
\widetilde{\mathcal{N}}(S_{A}+T_{A})=\widetilde{\mathcal{N}}(S_{A}%
)+\widetilde{\mathcal{N}}(T_{A}), \label{eq-app-linear:additive-hermitian}%
\end{equation}
consider that%
\begin{equation}
S_{A}+T_{A}=(S_{A}+T_{A})^{+}-(S_{A}+T_{A})^{-},
\end{equation}
while also%
\begin{equation}
S_{A}+T_{A}=S_{A}^{+}+T_{A}^{+}-S_{A}^{-}-T_{A}^{-}.
\end{equation}
Equating both sides, we find that%
\begin{equation}
(S_{A}+T_{A})^{+}+S_{A}^{-}+T_{A}^{-}=(S_{A}+T_{A})^{-}+S_{A}^{+}+T_{A}^{+}.
\end{equation}
Now we exploit this equality, \eqref{eq-app-linear:additive-PSD}, and the
definition in \eqref{eq-app-linear:define-hermitian} to establish \eqref{eq-app-linear:additive-hermitian}.
The final step is to extend the action of $\widetilde{\mathcal{N}}$ to all
operators $X_{A}\in\mathcal{B}(\mathcal{H}_{A})$. Here, we recall that any
linear operator can be written in terms of a real and imaginary part as
follows:%
\begin{equation}
X_{A}^{R}\equiv\frac{1}{2}\left( X_{A}+X_{A}^{\dag}\right)
,\ \ \ \ \ \ \ \ X_{A}^{I}\equiv\frac{1}{2i}\left( X_{A}-X_{A}^{\dag}\right)
,
\end{equation}
where by inspection, $X_{A}^{R}$ and $X_{A}^{I}$ are Hermitian operators. So
we define%
\begin{equation}
\widetilde{\mathcal{N}}(X_{A})\equiv\widetilde{\mathcal{N}}(X_{A}%
^{R})+i\widetilde{\mathcal{N}}(X_{A}^{I}).
\end{equation}
This completes the development of a well defined linear extension
$\widetilde{\mathcal{N}}$\ of the quantum physical evolution $\mathcal{N}$.
To show that it is unique, recall that any operator $X_{A}$ can be expanded as
a linear combination of density operators from the basis $\{\rho_{A}^{x,y}\}$,
defined in \eqref{eq-nqt:density-op-basis}, as follows:%
\begin{equation}
X_{A}=\sum_{x,y}\alpha_{x,y}\rho_{A}^{x,y},
\end{equation}
where $\alpha_{x,y}\in\mathbb{C}$ for all $x$ and $y$. It is straightforward
to show from the above development that%
\begin{equation}
\widetilde{\mathcal{N}}(X_{A})=\sum_{x,y}\alpha_{x,y}\mathcal{N}(\rho
_{A}^{x,y}).
\end{equation}
Now suppose that $\mathcal{N}^{\prime}$ is some other linear map for which
$\mathcal{N}^{\prime}(\rho_{A})=\mathcal{N}(\rho_{A})$ for all $\rho_{A}%
\in\mathcal{D}(\mathcal{H}_{A})$. Then the following equality holds for all
$X_{A}\in\mathcal{B}(\mathcal{H}_{A})$:%
\begin{equation}
\mathcal{N}^{\prime}(X_{A})=\sum_{x,y}\alpha_{x,y}\mathcal{N}^{\prime}%
(\rho_{A}^{x,y})=\sum_{x,y}\alpha_{x,y}\mathcal{N}(\rho_{A}^{x,y}%
)=\widetilde{\mathcal{N}}(X_{A}).
\end{equation}
As a result, $\mathcal{N}^{\prime}=\widetilde{\mathcal{N}}$, given that they
have the same action on every operator $X_{A}\in\mathcal{B}(\mathcal{H}_{A})$.
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