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\markboth{Quantum copying: A review }{ Mark Hillery }
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\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
Mathematical Physics and Quantum Field Theory, \newline
Electron. J. Diff. Eqns., Conf. 04, 2000, pp. 113--120\newline
http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)}
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 Quantum copying: A review 
\thanks{ {\em Mathematics Subject Classifications:} 81P68.
\hfil\break\indent
{\em Key words:} Quantum computing, qubits, no-cloning theorem, quantum copying.
\hfil\break\indent
\copyright 2000 Southwest Texas State University  and University of
North Texas. \hfil\break\indent
Published July 12, 2000. } }

\date{}
\author{ Mark Hillery }
\maketitle
\begin{abstract}
Quantum information is stored in two-level quantum systems
known as qubits.  The no-cloning theorem states that the
state of an unknown qubit cannot be copied.  This is in
contrast to classical information which can be copied.  If
one drops the requirement that the copies be perfect it is
possible to design quantum copiers.  This paper presents
a short review of the theory of quantum copying.
\end{abstract}

\section{Introduction}
The relatively recent confluence of quantum mechanics and
computer science has led to rapid and unexpected progress
in both fields.  In computer science it was shown that by
increasing the class of allowable algorithms to include
ones which operate according to quantum principles, it
is possible to perform certain tasks, such as factorization
and data base searching, much faster than had been previously
believed possible \cite{shor,grover}.  In quantum 
mechanics, it has led to the study of entirely new questions,
as well as some extremely interesting new applications.
For example, quantum error correcting codes provide novel
ways of protecting quantum systems from the effects of
decoherence \cite{code}.

This work grew out of the consideration of the limits which
physics places on computers.  Early work by Landauer and
Bennett focused on the limits due to thermodynamics, and
had the rather surprising outcome that thermodynamics
places no restrictions on what a computer can do \cite{max}. 
The next task was to see if quantum mechanics might place
any restrictions on what could be done, but Benioff and 
Feynman independently showed that it did not \cite{ben,feyn}.  
Deutsch, considering the matter further, showed that not 
only does quantum mechanics not hurt, it can, in fact, 
help \cite{deutsch}.  This conclusion was dramatically
confirmed by Shor, when he constructed a quantum
algorithm which can find the prime factors of an 
$N$-digit number in a time which is polynomial in $N$
\cite{shor}.

In addition to the work directly on quantum computation,
a second, but related, line of work emerged, that of
quantum information.  Classical information is carried
by bits which can have one of two values, $0$ or $1$.
The kind of information on which a quantum computer 
relies, quantum information, is carried by two-level
quantum systems, known as qubits, where one level 
corresponds to $0$ and the other to $1$.  Qubits,
unlike bits, can exist in a superposition of $0$
and $1$, and it is this fact that leads to their
unusual properties.  For example, a register 
consisting of qubits can be put into a superposition 
of all possible input values, e.g. $|00\rangle$, 
$|01\rangle$, $|10\rangle$, and $|11\rangle$ for a
two-qubit register, and any operation which acts
on the register acts on all of these values simultaneously.
This means that using quantum information instead
of classical leads automatically to a kind of 
parallel processing.  The study of the properties of
the information carried by qubits comprises the
field of quantum information.

Here we shall consider one of the more
unusual properties of quantum information - it cannot
be copied, or, to be more precise, it cannot be copied
perfectly.  This is not true of classical information,
which can, in principle, be copied without any difficulty
(though, as anyone who has struggled with a recalcitrant
copy machine can attest, the phrase ``in principle''
here covers up a lot).  The inability to copy quantum
information makes it useful in cryptography, and
prototype quantum cryptographic system have been 
constructed in several laboratories.  If one considers
approximate copies, however, or perfect copies which
can be produced with a probability less than one, of  
only a limited set of quantum states, then quantum copying
becomes possible.  Here we shall review both kinds of
copiers (more frequently known as quantum cloners).

This paper is dedicated to Professor Eyvind Wichmann on
his 70th birthday.  While I had taken previous courses
in quantum mechanics, I really learned the subject from a
one-year graduate course which Eyvind taught, and this
knowledge was deepened during my Ph.D.\ studies with him.  
It was one of the most challenging and one of the best 
courses which I have ever taken (the only other 
candidate was the quantum field theory course I took
from him the following year), and I still refer rather 
often to the set of notes which he distributed during
this course.  I hope what is in this paper will
demonstrate that what he taught has been put to good 
use. 

\section{No-cloning theorem}
The original realization that quantum information 
cannot be cloned is due to Wooters and Zurek \cite{wooters}.
The argument is very short and depends only on the 
linearity of quantum mechanics.  We assume that the
cloner has degrees of freedom of its own and operates
by means of a unitary transformation acting on 
a Hilbert space which is a tensor product of the
space of the cloner itself, $\mathcal{H}_{x}$, that of 
the input qubit, $\mathcal{H}_{a}$, and the space of a 
``blank'' qubit, $\mathcal{H}_{b}$ which is
to become the copy.  The input qubit is in an
arbitrary input state, $|\psi\rangle_{a} =\alpha
|0\rangle_{a} +\beta |1\rangle_{a}$, while we assume that
the ``blank'' qubit is always initially in the state
$|0\rangle_{b}$.  The cloner is assumed to be initially
in the state $|Q\rangle_{x}$, no matter what the state
of the input qubit.  An ideal cloner would have the
following action:
\begin{equation}
\label{ideal}
|\psi\rangle_{a}|0\rangle_{b}|Q\rangle_{x}\rightarrow
|\psi\rangle_{a}|\psi\rangle_{b}|Q^{\prime}
\rangle_{x} .
\end{equation}

If the cloner is to duplicate any initial state, then
clearly it must duplicate the basis states properly.  This
means that
\begin{eqnarray}
|0\rangle_{a}|0\rangle_{b}|Q\rangle_{x}&\rightarrow &
|0\rangle_{a}|0\rangle_{b}|Q_{0}\rangle_{x} \nonumber \\
|1\rangle_{a}|0\rangle_{b}|Q\rangle_{x}&\rightarrow &
|1\rangle_{a}|1\rangle_{b}|Q_{1}\rangle_{x} .
\end{eqnarray}
These equations, however, determine the transformation on
any input state, because it is realized by a linear 
transformation.  In particular
\begin{equation}
\label{actual}
|\psi\rangle_{a}|0\rangle_{b}|Q\rangle_{x}\rightarrow
\alpha |0\rangle_{a}|0\rangle_{b}|Q_{0}\rangle_{x}+\beta
|1\rangle_{a}|1\rangle_{b}|Q_{1}\rangle_{x} .
\end{equation}
A comparison of Eqs. (\ref{ideal}) and (\ref{actual})
shows that the right-hand sides cannot be equal for 
all values of $\alpha$ and $\beta$; the dependence
of one is quadratic while that of the other is linear.
Therefore, a perfect cloner cannot exist.

\section{Universal quantum copiers}
Now let us assume we want to copy
the state $|\psi\rangle$ given above, but we shall
drop the requirement that the copies be perfect.  It
will be useful in what follows to parameterize 
$\alpha$ and $\beta$ as
\begin{equation}
\alpha=\sin\vartheta e^{i\varphi},\hspace{1cm}
\beta=\cos\vartheta.
\label{1.1}
\end{equation}
We now need to ask ourselves what requirement we would
like our imperfect copy machine to satisfy.  One possibility
is the Universal Quantum Copy Machine (UQCM) \cite{buzek1},
which is specified by the following conditions.

{\bf (i)}The  state of the original system and its quantum 
copy at the output of the quantum copier, described by the
reduced density operators
$\hat{\rho}^{(out)}_{a}$ and $\hat{\rho}^{(out)}_{b}$, 
respectively, are identical, i.e.,
\begin{eqnarray}
\hat{\rho}^{(out)}_{a}   = \hat{\rho}^{(out)}_{b}
\label{1.3}
\end{eqnarray}

{\bf (ii)} If no {\em a priori} information about the 
input state of the original system is available, then 
it is reasonable to require that all pure states should be 
copied equally well. One way to implement this condition 
is to design a quantum copier so that the distances between
density operators of each system at the output 
($\hat{\rho}^{(out)}_{a}$ and $\hat{\rho}^{(out)}_{b}$)
and the ideal density operator $\hat{\rho}^{(id)}$
which describes the input state of the original qubit,
are input state independent.  Quantitatively this means that 
if we employ the Hilbert-Schmidt norm to define our 
distance between two density matrices,
\begin{equation}
d(\hat{\rho}_{1};\hat{\rho}_{2}):= \left( {\rm Tr}
\left[\left(
\hat{\rho}_{1}-\hat{\rho}_{2}\right)^2\right] \right)^{1/2},
\label{1.4}
\end{equation}  
then the quantum copier should be such that
\begin{eqnarray}
d(\hat{\rho}_{a}^{(out)};\hat{\rho}^{(id)})=
{\rm const.} 
\label{1.5}
\end{eqnarray}

{\bf (iii)}
Finally, we would also like to require that
the copies are as close as possible to the ideal output
state, which is, of course, just the input state.
This means that we want the constant in Eq. (\ref{1.5}) 
to be as small as possible.

Originally, the UQCM was found by guessing a transformation 
which contained two free parameters, and then determining 
them by demanding that condition (ii) be satisfied, and that 
the distance between the two-qubit output density matrix and 
the ideal two-qubit output be input state independent.  That 
the UQCM machine satisfies condition (iii) has been shown  
by Gisin and Massar, and by Bruss, et.\ al.\ 
\cite{gisin,bruss}.

The unitary transformation which implements the
UQCM \cite{buzek1} is given by
\begin{eqnarray}
|0\rangle_{a} |0\rangle_{b}|Q\rangle_{x} &\rightarrow & 
\sqrt{\frac{2}{3}}|00
\rangle_{ab}|\uparrow\rangle_{x}+\sqrt{\frac{1}{3}}
|+\rangle_{ab}
|\downarrow\rangle_{x} \nonumber \\
|1\rangle_{a} |0\rangle_{b}|Q\rangle_{x} &\rightarrow & 
\sqrt{\frac{2}{3}}|11
\rangle_{ab}|\downarrow\rangle_{x}+\sqrt{\frac{1}{3}}
|+\rangle_{ab}|\uparrow\rangle_{x},
\label{1.7}
\end{eqnarray}
where
\begin{equation}
|+\rangle_{ab} = \frac{1}{\sqrt{2}}(|10\rangle_{ab}+
|01\rangle_{ab}),
\label{1.8}
\end{equation}
and satisfies conditions (i), (ii), and (iii).
The system labeled by $a$ is the original (input) qubit, 
while the other system $b$ represents the qubit onto which 
the information is copied, and is analogous to ``blank 
paper'' in a copier.  The states of the copy machine
are labeled by $x$.  The state space of the copy machine 
is two dimensional, and it is spanned by the vectors
$|\uparrow\rangle_{x}$ and $|\downarrow\rangle_{x}$.
We assume that copy machine is always 
in the same state, $|Q\rangle_{x}$, initially.  
The reduced density operators of both copies
at the output are equal and they can be expressed as
(we give the expressions for qubit $a$, the ones for qubit 
$b$ are similar)
\begin{equation}
\hat{\rho}_{a}^{(out)}
=\frac{5}{6}|\psi\rangle_{a}\langle\psi|+
\frac{1}{6}|\psi_{\perp}\rangle_{a}\langle\psi_{\perp}|,
\label{1.9}
\end{equation}
where
\begin{equation}
|\psi_{\perp}\rangle_{a}=\beta^{\ast} |0\rangle_{a}-
\alpha^{\ast} |1\rangle_{a} ,
\label{1.10}
\end{equation}
is the state orthogonal to $|\psi\rangle_{a}$.  This
implies that the copy contains $5/6$ of the state we 
want and $1/6$ of the one we do not.

The density operator $\rho_{a}^{(out)}$ given by
Eq.(\ref{1.9})
can be rewritten in a ``scaled'' form:
\begin{equation}
\hat{\rho}_{a}^{(out)}
= s \hat{\rho}_{a}^{(id)}  + \frac{1-s}{2} \hat{1} ,
\label{1.11}
\end{equation}
which guarantees that the distance (\ref{1.4}) is 
input-state independent,
i.e. the condition specified by Eq. (\ref{1.5}) is 
automatically fulfilled. The scaling
factor in Eq.(\ref{1.11}) is $s=2/3$.

We note once again that the UQCM copies all input states 
equally well, and, therefore, it is suitable for copying 
when no {\it a priori} information about the state of the
original qubit is available. This corresponds to a 
uniform prior probability distribution
on the state space of a qubit (Poincare sphere). 
This suggests that another measure of the quality of 
the copies is the average fidelity ${\cal F}$
which is equal to the mean overlap between a copy 
and the input state
\cite{gisin}
\begin{equation}
{\cal F}=\int\, d\Omega  _{a}\langle \psi|
\hat{\rho}_{a}^{(out)}| \psi \rangle_{a},
\label{1.12}
\end{equation}
where $\int d\Omega=\int_0^{2\pi} d\varphi\,\int_0^{\pi} 
d\vartheta
\sin\vartheta/4\pi$. It is easy to see that 
the fidelity ${\cal F}$
and the scaling factor $s$ are related as
\begin{equation}
s= 2{\cal F}-1.
\label{1.13}
\end{equation}
The UQCM transformation is the one which minimizes 
${\cal F}$ \cite{gisin}.

It is possible to generalize universal, that is, 
input-state-independent, quantum copiers in several
ways.  One is to suppose that you wish to produce
more than two copies (by the number of copies we mean
the number of output qubits which are similar to the
input qubit).  One would expect that the
more copies one produces, the lower the quality
of each copy.  It is also possible to imagine that
instead of feeding in just one original qubit into
the copier, if one had several identical qubits
one could feed all of them into the copier to produce
better copies. For example, starting with
two identical originals, one could produce 3 copies.
The quality of these copies would presumably be better 
than that of 3 copies produced from only one original.
Specific transformations have been found for a
copier which starts with $N$ originals and produces 
$M\geq N$
copies \cite{gisin}, and these have been shown to be
optimal \cite{gisin}-\cite{werner2}.  The reduced density
matrixes of the output qubits are in the scaled form
(see Eq. (\ref{1.11})) with a scaling factor
\begin{equation}
s=\frac{N(M+2)}{M(N+2)} .
\end{equation} 
Note that the scaling factor is an increasing function
of the number of originals and a decreasing function
of the number of copies, as expected.

Another possible generalization is to consider copying
systems of dimension higher than two.  In this case,
because a system with $d>2$ dimensions contains more
quantum information than a two-dimensional one, we 
would expect that the quality of the copies will
be an decreasing function of $d$.  An explicit
copying transformation is known for general $d$ in
the case in which we have one original and 
produce two copies \cite{buzek2}.  In the case 
in which we have $N$ identical originals and produce
$M\geq N$ copies, it is known that the optimal
transformation again leads to reduced density
matrixes for the individual copies of scaled form
with the scaling factor \cite{werner1,werner2}
\begin{equation}
s=\frac{N(M+d)}{M(N+d)} .
\end{equation}
As expected, it is a decreasing function of $d$.

\section{Probabilistic copiers}
Instead of trying to construct a device which will
copy all possible input states, one could design one
which will copy only states from a particular set
of allowed input states.  Presumably a copier of
this type can achieve higher fidelities for its
copies than one which is required to accept all 
possible states as inputs.  In fact, it might even
be possible for the copier to be perfect for certain
input sets.  This suspicion is correct, but the size
of the input sets for which it is true is small.  If
the input set contains any two states which are not
orthogonal, a perfect copier is impossible \cite{hill},
and if we are copying qubits, then this means that it
is not possible to build a perfect copier for input
sets of more than two states.

There is a way around this problem, however.  What we
have been discussing in the previous paragraph is a copier
which produces perfect copies and works every time.
What happens if we relax this latter requirement?  We
still demand that the copies for our allowed input set be
perfect, and we permit the copier to have a certain
probability of failure, but impose the requirement that
the copier lets us know when it has failed and when it
has succeeded.  One can think of the copier as
having a red light on top.  We put in one of our allowed
input states, and the copier produces an output.  If the
red light stays off, the copying process has succeeded,
and our output consists of perfect copies.  If it goes
on, which it does with a certain nonzero probability
if the input set contains nonorthogonal states,  then
the copying process has failed, and we discard the 
output.

This type of copier was proposed and the transformation
for it constructed by Duan and Guo \cite{duan1, duan2}. 
Let us suppose that we want to copy one of two states,
$|\psi_{1}\rangle$ and $|\psi_{2}\rangle$, where, for
simplicity, we shall assume that their inner product is
real and nonnegative, $\langle\psi_{1}|\psi_{2}\rangle
= r \geq 0$.  The copying transformation is given by
\begin{equation}
|\psi_{j}\rangle_{a}|0\rangle_{b}|Q\rangle_{x}
\rightarrow \frac{1}{\sqrt{1+r}}(|\psi_{j}\rangle_{a}
|\psi_{j}\rangle_{b}|X_{0}\rangle_{x}+\sqrt{r}|\Phi
\rangle_{ab}|X_{1}\rangle_{x}) ,
\end{equation}
for $j=1,2$.  Here we have that $\,_{x}\langle X_{j}|
X_{k}\rangle_{x} =\delta_{jk}$ and $|\Phi\rangle_{ab}$
is an arbitrary state of the $a-b$ system with norm one.  
This transformation is consistent with unitarity,
has the same failure probability for each input state,
and gives
the lowest possible failure probability.  The way it
works is that we take a qubit, which is either in the
state $|\psi_{1}\rangle_{a}$ or $|\psi_{2}\rangle_{a}$,
but we do not know which, and send it through a device
which implements the above transformation.  We then
measure the device itself in order to determine whether 
it is in the state $|X_{0}\rangle_{x}$ or $|X_{1}
\rangle_{x}$.  If it is in the former, the transformation
has succeeded, and we have two copies of the input state
as our output.  The probability of this happening is
$1/(1+r)$.  If it is the latter the transformation has
failed, and we discard the output.

A number of generalizations of this type of copier are
possible.  Larger sets of allowed states can be 
considered \cite{duan2} or one might have multiple
copies of the input state and wish to produce more
than two copies at the output \cite{chefles1}.  It
is also possible to design copiers which lie between
the universal and probabilistic ones.  In particular,
they are designed to copy a finite set of allowed states,
the copies they produce are not perfect, and these
are produced with only a certain probability 
\cite{chefles2}.  In these one finds that there is 
a trade-off between the quality of the copies and
the probability of successfully producing them.
The success probability for producing perfect copies 
is the smallest, and this probability increases as the
quality of the copies decreases.

\section{Conclusion}
The study of quantum copying has increased our 
understanding of the properties of quantum
information.  What its role in devices which
process quantum will be is not yet clear.  It can
perhaps best be used to split the quantum information
in a qubit, allowing some of that information to be
processed in one way and some in another.  The
development of quantum algorithms is a very active
field of research, and quantum copying should prove
to be a useful part of the quantum programmer's
tool kit.

A final question is how can one actually build a
quantum copier.  Designs in terms of quantum logic
networks exist \cite{chefles2}-\cite{buzek4}, and
the gates from which these are constructed have
actually been realized in several laboratories.
However, putting together enough of them to build
a quantum copier is still a major problem.  
Nevertheless, it may soon be possible to actually
construct a quantum copier due to a clever proposal 
by Simon, Weihs, and Zeilinger \cite{simon}.
They showed how two-photon down conversion can be 
utilized to realize the UQCM on a single qubit .
This proposal can be realized with present technology,
and the hope is that a working quantum copier will
be a reality in a few years.

\subsection*{Acknowledgments}
I would like to thank Vladimir Bu\v{z}ek for what has 
been, and continues to be, a very fruitful collaboration.  
In addition, I would like to thank the National Science
Foundation for their support of this effort, most
recently under grant PHY-9970507.

 
\bibliographystyle{unsrt}
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\end{thebibliography}

\noindent{\sc Mark Hillery} \\ 
Department of Physics and Astronomy \\
Hunter College of the City University of New York \\ 
695 Park Avenue \\
 New York, NY 10021, USA \\
e-mail:  mhillery@shiva.hunter.cuny.edu

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