Fault-tolerant Quantum Communication with Minimal Physical Requirements
L. Childress
We describe a novel protocol for a quantum repeater which enables long distance quantum communication through realistic, lossy photonic channels. Contrary to previous proposals, our protocol incorporates active purification of arbitrary errors at each step of the protocol using only two qubits at each repeater station. Because of these minimal physical requirements, the present protocol can be realized in simple physical systems such as solid-state single photon emitters. As an example, we show how nitrogen vacancy color centers in diamond can be used to implement the protocol, using the nuclear and electronic spin to form the two qubits.
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\begin{document}% You should use BibTeX and revtex.bst for references\bibliographystyle{apsrev}% Use the \preprint command to place your local institutional report% number on the title page in preprint mode.% Multiple \preprint commands are allowed.%\preprint{}%


\title{Fault-tolerant Quantum Communication with Minimal Physical Requirements}  

%\title{Fault-tolerant Quantum Communication Based on Solid-state Photon Emitters}  

\author{L. Childress}
\affiliation{Department of Physics, Harvard University, Cambridge,
%anders inserted space here (and in all places below)
Massachusetts, 02138}

\author{J. M. Taylor}
\affiliation{Department of Physics, Harvard University, Cambridge, Massachusetts, 02138}

\author{A. S. S\o rensen}
\affiliation{Department of Physics, Harvard University, Cambridge, Massachusetts, 02138}
\affiliation{ITAMP, Harvard-Smithsonian Center for Astrophysics,
Cambridge, Massachusetts, 02138}
\affiliation{The Niels Bohr Institute, University of
Copenhagen, DK-2100 Copenhagen \O, Denmark}
%\affiliation{$^\ast$To whom correspondence should be addressed; E-mail: anders.sorensen@nbi.dk.}
\author{M. D. Lukin}
\affiliation{Department of Physics, Harvard University, Cambridge, Massachusetts, 02138}
\affiliation{ITAMP, Harvard-Smithsonian Center for Astrophysics,
Cambridge, Massachusetts, 02138}

\date{\today}
\begin{abstract}
  We describe a novel protocol for a quantum repeater which enables long
  distance quantum communication through realistic, lossy photonic
  channels. Contrary to previous proposals, our protocol incorporates
  active purification of arbitrary errors at each step of the protocol
  using only two qubits at each repeater station.  Because of these minimal
  physical requirements, the present protocol can be realized in simple
  physical systems such as solid-state single photon emitters. As an example, we show how nitrogen vacancy color centers in diamond can be
  used to implement the protocol, using the nuclear and electronic
  spin to form the two qubits.

\end{abstract}


\pacs{03.67.Hk, 03.67.Mn, 78.67.Hc}\maketitle

Quantum communication holds promise for transmitting secure messages
via quantum cryptography, and for distributing quantum
information~\cite{gisin02}.  However, attenuation in optical fibers
fundamentally limits the range of direct quantum communication
techniques~\cite{brassard00}. If a photon is injected into an optical
fiber the probability to retrieve the photon after a distance $L$
decreases exponentially making transmission very impractical for long distances.
In principle, photon losses can be overcome by introducing
intermediate quantum nodes and utilizing a so-called quantum repeater
protocol~\cite{Briegel98}.  A repeater creates entanglement over long
distances by building a backbone of entangled pairs between
closely-spaced nodes.  Performing an entanglement swap at each
intermediate node~\cite{zukowski93} leaves the outer two nodes
entangled, and this long-distance entanglement can be used to teleport
quantum information~\cite{bennett93,bouwmeester97} or transmit secret
messages via quantum key distribution~\cite{ekert91}.  Even though
quantum operations are subject to errors, by incorporating
entanglement purification~\cite{Bennett96,Deutsch96} at each step, one
can create long-distance high-fidelity entangled pairs in a time that
scales polynomially with distance~\cite{Briegel98}.
 
In practice, current long-distance quantum communication schemes
remain challenging to implement in the laboratory.  For example, while
approaches based on photon storage in atomic ensembles~\cite{duan01}
are being explored, and these implementations are capable of
correcting errors caused by photon losses, they offer no protection
against more general errors such as those due to dynamical phase
fluctuations.  Other approaches attempt to create an interface between
light and single quantum bits (qubits)~\cite{blinov04,vanenk98}.  In
order for these schemes to be fully fault-tolerant, existing theories
\cite{Dur99} require that each node must contain a small quantum
computer whose size increases logarithmically with the communication
distance; the construction of such quantum computers represents a
considerable challenge.  In this Letter, we present a protocol for a
fully fault-tolerant quantum repeater in which each node is formed by
a two qubit quantum computer. We thereby avoid the increase in the
number of qubits required by previous protocols, which substantially
simplifies the experimental realization of quantum repeaters.

In addition to presenting an algorithm for resource-efficient
entanglement propagation and purification, we also present a physical
system in which it can be implemented.  The general protocol
is relevant to a variety of systems, including trapped atoms in a
cavity~\cite{mckeever04} or trapped ions~\cite{blinov04}.  The reduced physical requirements facilitate  development of a scheme which is,
to our knowledge, the first realistic proposal for the construction of
a quantum repeater in a solid state environment. 
 %Because our protocol only requires two qubits at each node, the nodes can be constructed in
%simple physical systems, which need not be scalable. 
 In particular,
we describe how the repeater nodes may constructed from single photon
emitters in solid state systems by using the nuclear spin degree of
freedom to store quantum information while the electron spin is used
for communication with neighboring nodes. This may be accomplished,
e.g., in nitrogen-vacancy (NV) centers in diamond
~\cite{weinfurter00,beveratos02}, single quantum dots \cite{santori02,
  bracker04, taylor03}, etc.  We start with the details of our
protocol for a fault-tolerant quantum repeater.  Then, a specific
implementation using NV centers in diamond is developed to demonstrate
the scheme.

 

  \begin{figure}\vspace{.0in}
\centerline {
\includegraphics[width=3in]{Figure1.eps}
}
\vspace{.0in}\caption{Protocol for fault-tolerant quantum
communication.   Each node is represented by an oval containing two qubits (circles).  Entangled states are represented by a solid or dashed line between the entangled qubits.
% Solid (black) lines are entanglement between nuclear spins, while dotted lines (gray) are entanglement between electron spins.  Dashed lines of either color indicate lower fidelity entangled pairs.
%LILY: I think part (a) should be moved to fig 3. OK Sounds good.
 (a) Entanglement connection.  %propagation by swapping. 
 (b) Nested entanglement purification.   Dots indicate an arbitrary number of
nodes.  %(i) We split the set of repeaters in half, create two purified A pairs, and connect them via adjacent central nodes.  (ii) This yields an entangled B pair. (iii) While storing the entangled pair in the endpoint stations, we create and purify entangled A pairs using the nearest neighbor and central nodes.  (iv) Bell state measurement of the central and nearest neighbor nodes creates a C pair which is used to purify the B pair. 
} 
\label{repeater}\end{figure}

A protocol for a simple repeater is presented in Fig. \ref{repeater}
(a). The total communication channel is divided into small segments of
length $L_0$ by a set of quantum nodes, each containing a two qubit
quantum computer.  Initially, two qubits in neighboring nodes are
prepared in an entangled state
$|\Psi_-\rangle=(|01\ra-|10\ra)/\sqrt{2}$, where $|0\ra$ and $|1\ra$
are the two states of the qubits [solid line between upper qubits in
the first and second node in Fig. \ref{repeater} (a)].  As detailed
below, we envision that such entangled states can be prepared
probabilistically between distant nodes with state-selective light
scattering.  Because the optical fibers connecting the two nodes are
lossy, this step is necessarily probabilistic, and has to be repeated
until successful.  Simultaneously, an entangled state is prepared
between two nearby nodes [solid line between upper qubits in the third
and fourth node in Fig.~\ref{repeater}(a)].  Once all entanglement
steps succeed the entangled pairs are mapped into storage states
(i.e., nuclear spins) and the remaining qubits in the central nodes
are entangled in a similar fashion [dotted line between lower qubits
in Fig. \ref{repeater} (a)].  Finally, an entanglement swap
\cite{zukowski93} is performed at each node, which teleports the
entanglement between the nodes so that eventually the outer two qubits
are entangled [Figure~\ref{repeater}(a), right].  Since entanglement
is generated only over a fixed distance $L_0$, this procedure avoids
the exponentially small probability for a photon to travel the full
length of the channel and thus allows the construction of long
distance entanglement from short range entanglement---provided that
there are no errors.

A single error in the chain will destroy the final state, making the
fidelity decay exponentially with distance.  To extend entanglement to
long distances in the presence of errors, active purification is
required at each level of the repeater scheme.  According to a standard
purification protocol ~\cite{Deutsch96},  we prepare two entangled pairs between two nodes.  Local two-qubit operations at each node are followed by measurement of one qubit at each node.   Conditioned on a successful outcome
of the measurement, this procedure yields an entangled pair of higher
fidelity between the remaining entangled qubits in the two nodes.  In Fig.~1(b) we present a protocol
for incorporating entanglement purification into a two-qubit repeater
scheme.  For clarity, we distinguish three types of entangled pairs A,
B, and C, labeled according to their purity.  A-pairs are fully
purified high fidelity pairs ready to be used in the next step of the
protocol, B-pairs are intermediate pairs, which are being purified to
A-pairs, and C-pairs are the lowest quality pairs which are used to
purify the B pairs.

The argument proceeds inductively: We assume that we have a method to
create and purify A pairs over distances up to $L_n = n L_0$ using
only two qubits per node and show that we can use these to generate
and purify A pairs over a distance $L_{2n+1} = (2n+1)L_0$.  It is
fairly straightforward to construct a B-pair over distance $L_{2n+1}$
by creating two purified A-pairs over the distance $L_n$, and
connecting them via an entangled pair between the central nodes (see
Fig 1(b)(i-ii)).  Previous schemes \cite{Briegel98, Dur99} have
constructed a C-pair in the same manner---at the cost of requiring an
extra qubit in the outermost nodes.  Instead, we employ the unused
nearest-neighbor nodes, creating two A-pairs and three short range
pairs, as shown in Fig.~\ref{repeater} (b)(iii).  Performing an
entanglement swap at the central and nearest neighbor nodes creates a
C pair over the distance $L_{2n+1}$ which can be used to purify the B
pair, see Fig.~\ref{repeater} (b)(iv).  We then perform the
purification protocol \cite{Deutsch96}, which, if successful, results
in a pair B-pair with higher fidelity.  The generation of C-pairs and
purification may then be repeated.  After generating C pairs for $m$
consecutive successful purification steps, (a technique sometimes
referred to as ``entanglement pumping"), the stored pair becomes a
purified A-pair over the full distance $L_{2n+1}$.  We remark that
this procedure is most efficient when $n \sim 2^k$ for integer $k$.


The fidelity obtained at the end of this nested purification
procedure, $F(m, L/L_0, F_0, p, \eta)$, depends on the number of
purification steps $m$, the total number of nodes $L/L_0$, the
initial fidelity $F_0$ between adjacent nodes, and the reliability of
measurements $\eta\leq1$ and local two-qubit operations $p\leq1$
required for entanglement purification and
connection.  There is little theoretical insight gained from the mathematical form of $F$ (see Refs.~\cite{Dur99,andersXX}) but it is easily evaluated numerically. In Fig. 2 we show the result of a numerical investigation of the
protocol in the presence of errors. In this analysis we assume that
the qubits do not decohere 
significantly over the communication time.
In Fig.~2 (a)  we show the fidelity as a function of distance. The curves show
the fidelity obtained by using three purification steps $m=3$. As seen
in the figure the fidelity saturates and only shows a very limited
decrease in fidelity with distance, demonstrating the ability of the protocol to
correct  error and the applicability of the protocol for long distance quantum
communication. As shown
in Fig. 2 (b),  the required time scales polynomially with distance.
%\begin{equation}
%  \label{eq:1}
%  T\sim T_0 {\left(\frac{L}{L_0}\right)}^\beta,
%\end{equation}
% where $T_0$ is the time to generate entanglement between nearest
% neighbors. Because the success probability depends on the Fidelity,
% the exponent $\beta$ depends on the initial Fidelity and reliability
% of gates and measurements, but for the parameters of
% Fig. \ref{asymp} (a) we find $\beta\sim ???$ LILY FILL IN HERE.
%Compared to the protocol \cite{briegel98} which yields $\beta $ this is
%slightly  slower but not significantly different.
%lily added: 



To determine the tolerance to the initial fidelity $F_0$ and to errors
in gates  $1-p$ and measurements
$1-\eta$, it is useful to consider the limiting case of many
purification steps and large distances.   
As the number of purification steps increases  $m\rightarrow \infty$,
the fidelity at a given distance $L$ grows, eventually saturating at 
 a fixed point 
 \beq
 F\rightarrow F _{FP}(L, F_0, p, \eta)\ .
 \eeq %at which
Additional purification steps yield no further benefit at the fixed point, because
the increase in fidelity they offer is cancelled by the likelihood of errors in the purification
procedure~\cite{Dur99}.   Moreover, as $L$ increases,  the fidelity
may 
approach an asymptotic value 
\beq
F_{FP}\rightarrow F_{\infty}(F_0, p,
\eta)\ ,
\eeq
 which is independent of distance~\cite{andersXX}.  For
comparison these quantities are also shown in Fig. \ref{asymp} (a). 
%anders Changed to FIg.
%lily added the following sentence:
%In Fig. 2 we show the result of a numerical investigation of the
%protocol in the presence of errors. In this analysis we assume that
%the qubits do not decohere 
%significantly over the communication time.
%In Fig.~2 (a)  we show the fidelity as a function of distance. The curves show
%the fidelity obtained by using three purification steps $m=3$,
%as well as $F_{FP}$ and $F_{\infty}$. % for various initial fidelities.  % The results illustrate the efficiency of the purification protocol: for initial fidelities $F_0\gtrsim 97\%$, three purification steps ($m=3$) suffice to produce entanglement at large distances.  
%As shown
%in Fig. 2 (b) the required time scales polynomially with distance. % Th reader
%should be able to figure this out. (note, that it is a log-log
%scale). 
The asymptotic fidelity $F_\infty$ in Fig.~2 (c)  shows  
that our scheme will operate in the presence of $1-p \lsim 1\%$ errors
in local operations and percent-level phase errors in initial
entanglement fidelity.  For the specific physical system presented
below, the most likely error in entanglement generation between neighboring nodes results in an incoherent %s resulting in an incoherent
admixture of the state $|\Psi_+\ra=(01\ra+|10\ra)$, which we refer to as a phase error. 
Above we have assumed that only this type of error matters, but other types are
in principle possible.  In Fig.~2d we account for arbitrary errors in the initial entanglement by allowing  incoherent admixtures of the other two Bell states $\ket{\Phi_{\pm}} = (\ket{00}\pm
\ket{11})/\sqrt{2}$, each with weight $\upsilon(1-F_0)$~\cite{Deutsch96} (the weight of phase errors is thus $(1-2\upsilon)(1-F_0)$).     
%On average, the final 
% fidelity obtained by the nested entanglement purification procedure
 % depends only on the diagonal elements of the density matrix in the
% Bell state basis \cite{Deutsch96} %anders changed to \Psi_+
 % $\{\ket{\Psi_-}, \ket{\Phi_-}, \ket{\Phi_+},
%\ket{\Psi_+}\}$, where %anders made two kets to one
%$\ket{\Phi_{\pm}} = (\ket{00}\pm
%\ket{11})/\sqrt{2}$.   We parameterize the
%initial fidelity by $\{F_0, (1-F_0)\upsilon, (1-F_0)\upsilon,
%(1-F_0)(1-2\upsilon)\}$ in this basis, so that the shape parameter
%$\upsilon$ quantifies the weight of non-phase errors.    %Lily removed
%The asymptotic fidelity shown in 
Although the protocol we use is most effective for purifying phase
errors, Fig.~2d indicates that  it also tolerates arbitrary errors. 

\begin{figure}\vspace{.0in}
\centerline {
\includegraphics[width=3.5in]{Figure2.eps}
}
\vspace{.0in}\caption{ 
(a) Fidelity scaling with distance.   Points show results using 3
    purification steps at each nesting level;  dashed lines show the
    fixed point $F_{FP}$ at each distance; dotted lines indicate the 
asymptotic fidelity $F_{\infty}$.   For (a) and (b), measurements and
local two-qubit operations $\eta=p$ contain $0.5\%$ errors.  For (a),
(b), and 
(c), the initial fidelity $F_0$ is (i) 100\% (ii) 99\% (iii) 98\% (iv)
97\% (v) 96\% with phase errors only.   (b) Time scaling with distance
for m=3, given in units of $T_0 \gg L_0/c$, %anders removed  = (t_0 + t_c)/P$,
the time required to
generate entanglement between nearest neighbors,  and $L_0$, the
distance between nearest neighbors. 
%anders removed   Measurement and local operation
%times are neglected.
%Note that the axes are logarithmic, so time scales polynomially with distance.
(c) Long-distance asymptote dependence on operation and measurement errors.  (d) Long-distance asymptote dependence on error
    type %anders chnged here  For the calculations shown,  
    ($F_0 = 0.99$,  %and the shape    parameter ranges from 
    $\upsilon = 0, 0.1, 0.2, 0.3$).}  
\label{asymp}\end{figure}

%anders changed to Fig.
%Fig.~2b demonstrates that our scheme %lily removed: permits generation of high-fidelity, long distance entangled pairs in the presence of percent-level errors lily added: 
%operates 
%in polynomial time. 
For the implementations discussed below, the overall time scale is set
by the classical communication time between nodes, and the fidelity is
limited by the photon emission probability $P_{\rm{em}}$ from each
node. 
As an example,
using a high photon %\lily added 
collection efficiency, a photon loss rate of $\sim 0.2$ dB/km,
spacing $L_0 \sim 20$ km, %lily addded 
an initial fidelity $F_0$ set by 
an emission probability $P_{\rm{em}} \sim 8\%$ (see Eq.~\ref{fidelity} below), %lily added
local errors $\eta = p = 0.5\%$, and just one purification step at
each nesting level, our scheme could potentially produce entangled
pairs with fidelity $F\sim 0.8$ sufficient to violate Bell's
inequalities over 1000 km in a few seconds.    %lily removed 
%For comparison, under the same set of assumptions direct entanglement
%schemes would require $\sim10^{10}$ years.    
Moreover, %it is likely that our scheme's  
the bit-rate could likely be significantly improved by employing
optimal control theory to tailor the details of the repeater protocol
to the parameters of a desired implementation.  


The above analysis demonstrates that two qubits per repeater node are sufficient for fully fault tolerant long distance quantum communication.  To illustrate the possibilities opened up by reduced physical requirements, 
we now turn to a specific example for implementation of
the %presented 
protocol in a solid state system: the nitrogen vacancy (NV) center in diamond.  The qubits required for entanglement connection and purification are realized in the electronic triplet ground state and the nuclear spin of a nearby $^{13}$C impurity.   Each spin can be manipulated by magnetic resonance techniques, and strong hyperfine interactions %can be realized in pair of electronic spin sub-levels  that allows for spin-selective optical excitation, and a proximal nuclear spin that provides an auxiliary memory.  
 % The nitrogen vacancy (NV) center in diamond has   a triplet electron spin ground state is strongly coupled to a nearby $^{13}$C impurity nuclear spin, which can have a very long coherence time~\cite{Ramanathan04}. 
couple the two qubits, allowing experimental demonstration of two-qubit gates~\cite{jelezko04b}.   %Electron spin resonance (ESR) and nuclear magnetic  resonance (NMR) have already been employed to manipulate coupled electron and nuclear spins~\cite{jelezko04b}, and a
A single NV-center
with a nearby $^{13}$C can therefore effectively 
constitute a two qubit quantum computer. %Furthermore the possibility of optically accessing the electron spin allows this system to be used for a quantum repeater.    

The remaining requirement for a quantum repeater is an entanglement generation scheme. % that does not disturb the quantum state of the other qubit in the node (which may be entangled with another node).
%For the NV-centers we present a novel entanglement scheme which is specifically designed to allow the nuclear spin to store quantum information while the electron spin is entangled with an outgoing photon. 
%To generate entanglement we propose to use the strong, state-selective optical transition %anders changed to Fig. (Fig.~3a) near 637nm which has been  used for robust generation of  single photons on demand ~\cite{weinfurter00,beveratos02} and single spin measurement \cite{jelezko04}.
%For a two-qubit system, we detail entanglement
%generation between nearest-neighbor nodes in a solid-state
%environment.   
In atoms and ions, %separated by a large distance $L_0$, 
entanglement between spatially separated systems can be 
generated probabilistically by, %a variety of means
 e.g.,  Raman
scattering~\cite{cabrillo99} or polarization-dependent
fluorescence~\cite{blinov04} %lily added
followed by photon interference.   In the spirit of reducing the physical requirements on the system, we will show that probabilistic entanglement generation can succeed even in the absence of polarization selection rules, allowed Raman transitions, or even radiatively broadened optical transitions.    Our scheme requires only state-selective light scattering (see Fig.~\ref{levels}a, inset), and is thus applicable to a variety of solid-state emitters including the NV center~\cite{weinfurter00,beveratos02, jelezko04}.   %For the present purpose, however, it is essential that one of the qubits (the electron) can be entangled with another node, while the other qubit (the nuclear spin) %, which may be entangled with other nodes,   %remains %completely 
 %undisturbed (as it may be entangled with yet another node).   
 Furthermore, this simple level structure facilitates entanglement of one qubit (the electron spin) while leaving the other qubit (the nuclear spin) undisturbed.   In particular, by choosing a scattering transition between $M_s = 0$ electron spin states, we can eliminate sensitivity to the nuclear spin state during entanglement generation (see Fig.~\ref{levels}b).  The desire for a simple requirement for level structure combined with the necessity of preserving the nuclear spin state motivates discussion of a new entanglement generation scheme based on state-selective elastic light scattering and photon interference.
%Here we propose a setup which satisfies this stringent requirement.  %This puts stringent requirements on the entangling scheme.  
%The ground state of a coupled $^{13}$C-NV center system exhibits an electron spin energy splitting which is an order of magnitude larger than the hyperfine interaction, effectively decoupling the nuclear and electronic spin states.  Furthermore, the $M_s = 0$ optical transition frequency is independent of the nuclear spin state, and spectrally isolated from the $M_s = 1$ optical transition.  We can thus use a state-selective $M_s = 0$ optical transition to both generate entanglement between neighboring nodes and measure or reinitialize the electron spin without affecting the nuclear spin state.   
%but here we propose a setup which achieves this: 
%Spin selective
% fluorescence allows electron spin initialization, 
%measurement~\cite{jelezko04} and entanglement with outgoing photons;
% and this step depends on
%the physical system used.         
%However, solid-state emitters often do not exhibit appropriate selection rules, and for our repeater protocol one must carefully choose an optical transition that is independent of the nuclear spin state. 
%in the ground state,
%the energy splitting between electron spin states $M_s = 0$ and $M_s =
%\pm 1$ is an order of 
%magnitude larger than the hyperfine interaction, effectively
%decoupling the nuclear and electronic spin states. %With a laser field
%The $M_s = 1$ and $M_s = 0$ optical transitions have different frequencies, so an applied laser field can selectively excite the $M_s=0$ state. %can be selectively excited to an excited state with $M_s=0$ because the splitting between  $M_s = 0$ and $M_s = \pm 1$ is different for the excited and ground states. 
  %Since  the
%optical transition frequency between states with $M_s=0$ is independent of
%nuclear spin state,  the nuclear spin state remains undisturbed
%during light scattering. We can thus use the optical transition to 
%generate entanglement between neighboring nodes, in a probabilistic
%entanglement protocol, and in the event of a failed entanglement
%generation, detect the electronic spin state and reinitialize the
%entanglement protocol. 
% For  the NV center we propose to use the closed  optical transition between states with $M_s=0$. 
%The frequency of this
%transition is independent of 
%nuclear spin state, leaving the nuclear spin %state 
%undisturbed during entanglement generation.
%The use of this transition, %of nuclear-spin-insensitive entanglement
%generation, 
%combined with restrictions imposed by homogeneous optical broadening, %discussed below
%necessitates a novel entanglement mechanism.  We therefore present a scheme %for  probabilistic entanglement generation 
%applicable to the NV-center which is
%based on 
%state-selective elastic light scattering and photon interference.  %(see Fig.~3 (a)).  

Our entanglement generation scheme proceeds as follows:  We consider
two NV centers separated by a distance $L_0$, such that %We consider
%the situation in  which  
%Each emitter is placed inside a photonic cavity, whose output  is coupled to a photonic fiber. %.    The cavity is not essential for our scheme, but it enhances collection efficiency and frequency selection.   We also assume that 
%The optical transition frequencies in each
%node are matched by careful selection or tuning of the emitters.   The emitter level structure is simple:
each node scatters light only if its  electron spin is in state
$\ket{0}$. 
 Two adjacent nodes thus form state-selective mirrors in
an interferometer  (see Fig.~3(a)).    Our scheme relies on balancing
this interferometer so that when both nodes are in the 
scattering state $\ket{0}$, the outgoing photons will always exit one
detector arm $D_+$.   A detection event in the other arm
$D_-$ %must then correspond to scattering by one of the two nodes;
%since the photon could have scattered off either node, a 
%in particular, the $D_-$
%measurement 
can then project the spins onto an entangled state. 

We now address this scheme quantitatively, and determine the entanglement fidelity it can produce.  The scheme starts with each node %is prepared 
in 
%a superposition 
the state $(|0\ra +|1\ra)/\sqrt{2}$; state $|0\ra$ is then coupled to an excited level that decays radiatively at a rate $\gamma$.
In the weak excitation limit, we can adiabatically eliminate the
excited state, resulting in coherent scattered light.  %and the scattered light is in a coherent state, %scattered off
%state $|0\ra$ is well described as a coherent state,
 %so that
The combined state of node $i$ and the scattered light field is given by  
$\ket{\psi}_i \approx ( \ket{1} + T_i \ket{0} ) / \sqrt{2}$ with  
$ T_i = e^{-\sqrt{P_{\rm{em}}} (\sqrt{1-\epsilon}\hat b^{\dag}_i +
\sqrt{\epsilon} \hat a^{\dag}_i)-P_{\rm{em}}/2}, 
\label{coherentstate}$  
where $P_{\rm{em}}$ is the total emission probability, $\epsilon$ is
the total collection, propagation, and detection efficiency, and $\hat
a_i, \hat b_i$ 
are the annihilation operators for the 
field reaching the beamsplitter and other (loss) fields, respectively.   
In the limit $P_{\rm{em}}\rightarrow 0 $,  a detection event in %detector 
$D_-$ (mode $\hat d_-
\propto \hat a_1 - \hat a_2$) projects the system onto a maximally
entangled state
$\hat d_- (T_1 \ket{01} + T_2 \ket{10})/2 \propto
% (|01\ra-|10\ra)/\sqrt{2}=
\ket{\Psi_-}$.
For finite $P_{\rm{em}}$,  there is a chance $\sim P_{\rm{em}}$ 
that %,during a successful $D_-$ detection event, 
an undetected photon was emitted into the environment. After a
detection in $D_-$, the nodes cannot be in the $|00\ra$ or $|11\ra$
state; the additional photon emission thus introduces some admixture of the state $\ket{\Psi_+}$ % = (\ket{01} + \ket{10})/\sqrt{2}$ 
(a phase error).   %Since $P_{\rm{em}}$ determines how long it takes for successful entanglement generation, it cannot be too small, so we expect phase errors to dominate infidelity in the entangled pair. 
We find that the 
scheme succeeds with probability 
$P = (1/2)\left(1-e^{-P_{\rm{em}}\epsilon/2}\right)\approx \epsilon P_{\rm{em}}/4$,
producing the state $|\Psi_-\rangle$ with fidelity
\beq
F_0 = \frac{1}{2}\left(1+e^{-P_{\rm{em}}\left(1-\epsilon\right)}\right) \approx 1 - \frac{P_{\rm em} (1- \epsilon)}{2}
\label{fidelity}
\eeq
in time $T_0 \approx (t_0+t_c) /P$.
 

%Combined with the considerations above,  we find that the 
%scheme succeeds with probability 
%$P = (1/2)\left(1-e^{-P_{\rm{em}}\epsilon/2}\right)\approx \epsilon P_{\rm{em}}/4$,
%producing the state $|\Psi_-\rangle$ in time $T_0 \approx (t_0+t_c) /P$ with fidelity
%%lily removed dark  count references
%\ba
%F_0 & \approx & \frac{1}{2}\left(1+e^{-P_{\rm{em}}\left(1-\epsilon\right)}\right)
%-  \gamma_e(t_0 +  t_c)\nonumber \\ &&% -  \gamma_{dc}\frac{t_0}{P} 
%-
%\frac{3}{2}\frac{\Gamma}{\Gamma + \gamma}\frac{\kappa}{\kappa +
%\gamma}\frac{1}{1+4g^2/\kappa(\gamma + \Gamma)}. 
%\label{fidelity}
%\ea
%The first term arises from multiple photon emission;  the
% second term accounts for %lily added
% phase errors caused by electron spin dephasing (at rate $\gamma_e$)
% during the excitation time $t_0$ and classical
% communication time. $t_c$; % the third term accounts for detector dark  counts at rate $\gamma_{dc}$; 
%The last term arises from homogeneous broadening. here we have assumed
%that the line is broadened form the natural linewidth $\gamma$ to
%$\gamma+\Gamma$ by a white noise process, and that the center is
%placed in a cavity with vacuum Rabi frequency $g$ and cavity decay
%rate $\kappa$.  
%For realistic emitters  placed into a cavity with %lily removed either 
%a narrow
%linewidth 
%$\kappa\ll\gamma$ or a large Purcell factor  $4g^2/(\kappa(\gamma + 
%\Gamma))\gg 1$, % reduces the error introduced by the  broadening
% the first two terms should dominate the error. % Both of these %lily changed contributions terms introduce phase errors. 

%Using this procedure, electron spin entanglement can be generated
%between pairs of nodes.  The electron spin state is then mapped onto
%the nuclear spin qubit for long-term storage  
%using the hyperfine interaction, leaving the electronic degree of
%freedom 
%available to generate entanglement between unconnected nodes.%, as required for the repeater protocol.   

%illustrated in %anders changed to Fig. 
%Fig.~1b.
%%   Because we have access to optical detection of individual electron spin states~\cite{jelezko04} and effective two-qubit operations based on the hyperfine interaction~\cite{taylor04,jelezko04b}, we may projectively measure all four Bell states in the electronic/nuclear manifold associated with each emitter.  
%The outcomes of the Bell state measurements reveal the appropriate
%local rotations to obtain a singlet state in the remaining pair of
%nuclear spins, implementing a deterministic entanglement
%This implements a deterministic entanglement
%swap~\cite{bennett93,zukowski93} and extends the distance between the
%entangled pairs.   %By performing this procedure in parallel, and
%iterating the process for $N \propto \log_2{(L/L_0)}$ layers, we
%obtain the desired nuclear spin entanglement over distance $L$ in a
%time $\propto L\log_2{(L/L_0)}$.  
\begin{figure}\vspace{.0in}
\centerline {
\includegraphics[width=3.5in]{Figure3.eps}
}
\vspace{.0in}\caption{  (a) Interferometric arrangement for entanglement
generation.  Inset shows relevant level scheme.  (b) Implementation with NV centers.  Electronic spin ($\ket{0}, \ket{1}$) and $^{13}$C nuclear spin
($\ket{\uparrow}, \ket{\downarrow}$) states %are %of the NV center 
%coupled to a $^{13}$C  nuclear spin; 
are coupled by  optical, microwave, and RF transitions.  The $M_s = -1$ electronic state can be shifted out of resonance by a small magnetic field.  %(b) Implementation with quantum dots.  Electronic ($\ket{\uparrow}, \ket{\downarrow}$) and collective nuclear ($\ket{0}, \ket{1}$) states and their transitions for     singly doped quantum dots in a polarized nuclear spin lattice.  To suppress background light scattering, two-photon excitation can be used in both cases.
} 
\label{levels}\end{figure}

Finally, we mention %discuss %several 
some technical aspects of the proposed implementation. 
First, % of all,
interferometer stabilization poses a challenge, but has been achieved over $\sim10$ km  distances \cite{holman05}.  
%In  addition, slow drifts in the fibers may be compensated for by using a double pass system where all pulses traverse the same path %, cancelling changes in optical path length 
%\cite{andersXX,muller97}).  
Alternatively the interferometric setup
may be replaced by a photon coincidence detection, which is less
susceptible to path length fluctuations \cite{barrett04,simon}.
 %Moreover, the effect of small imperfections can be corrected by subsequent entanglement purification
% our scheme incorporates entanglement purification the interferometer
%need not be perfect. %: small phase instabilities will simply decrease
%the fidelity of entanglement between adjacent pairs. 
%Second, inhomogeneous broadening of solid-state emitters will require careful selection and/or tuning with external fields to ensure that the emitters have the same optical frequency. %to the homogeneous broadening considered above, solid
%state emitters have an inhomogeneous broadening because the resonance frequency of different emitters are different. This broadening can be avoided by carefully selecting emitters of similar frequency combined with external fields for fine tuning. 
Another important source of error is the homogeneous broadening 
typically found in solid-state emitters. 
%% Here we present a method to
%% remove the effect of this broadening on the entanglement preparation.
%% We model the dephasing  by  a random  energy  shift of the excited state 
%% %$\Delta$ 
%% with white-noise characteristics
%% ($\mean{\Delta(t)\Delta(t')} = \Gamma\delta(t-t')$).  
%% Solving the Heisenberg equations for this system under weak driving
%% \cite{andersXX} we 
%% find that the spectrum of the emitted light consists of a narrow
%% coherent peak at the frequency of the driving laser and a broad
%% incoherent background of width $\Gamma$. By using a narrow frequency
%% filter we can block %thus filter out 
%% the incoherently scattered photons, and
%% the broadening mainly results in a reduction of the success
%% probability, not the fidelity. 
%% Specifically we consider atoms coupled to a cavity with vacuum Rabi
%% coupling $g$ and linewidth $\kappa$. %, and a
%% Averaging over the noise,
%% we obtain the fidelity loss associated with homogeneous broadening~\cite{andersXX}.  
%Putting these considerations together,
The effect of this broadening can, however, be reduced by sending the
light through a narrow frequency filter or a using a cavity \cite{andersXX}.
For NV centers coupled to cavities with Purcell factors  $\sim
10$~\cite{santori02}, we find that the dominant source of error is
electron spin 
decoherence. %during the classical communication period.  
Using an emission probability $P_{\rm{em}}\sim 5\%$, a collection efficiency
$\epsilon \sim 0.2$, %and a  classical communication time of $t_c \sim
and $t_c \sim70\mu$s over $L_0 \sim 20$ km, we find %the fidelity of directly entangled pairs can reach
$F_0 \sim 97\%$ for electron spin coherence
times in the range of a few milliseconds.  According to our numerical calculations, this fidelity is sufficient for long distance quantum communication.

% Electron spin coherence
%times in the range of $100\mu$s have been observed at room temperature
% and significant improvements are expected for
% high purity samples at low temperatures~\cite{Kennedy03}.  The large
% hyperfine splitting allows fast local operations between electron and
% nuclear spin degrees of freedom on a timescale $\sim
% 100$ns~\cite{jelezko04b},  much  
%shorter than the decoherence time, allowing $1-p < 1\%$. Finally,  cavity
%enhanced collection should significantly improve observed measurement
%efficiencies of $\eta \sim 80\%$~\cite{jelezko04b}.  



%Further speed-up may %also 
%be possible %in the case 
%when collection efficiency
%is very high, e.g., by using coincidence detection in combination with
%time-bin encoding~\cite{gisin02}. This approach also has
%advantages in terms of interferometric stability \cite{barrett04,simon}.



%% Semiconductor quantum dots~\cite{santori02} represent
%% another promising physical implementation (Fig.~\ref{levels} (b)). 
%%   By doping a neutral dot
%% with a single electron, the ground state of the dot gains an electron
%% spin degree of freedom.  This spin state has 
%% been prepared and measured by state-selective optical transitions and 
%% Raman-based ESR in GaAs dots~\cite{bracker04}.   Hyperfine coupling of the electron spin to
%% the surrounding lattice of nuclear spins  allows nuclear
%% polarization 
%% and use of  collective nuclear spin
%% excitations as a quantum memory~\cite{taylor03} 
%% with potential coherence times on the order of
%% seconds~\cite{Ramanathan04}.  Although optical transitions in doped quantum dots can exhibit  homogeneous broadening $\Gamma  \sim 100$ GHz $\sim 10-100\gamma$~\cite{kiraz02},  the corresponding error  can be made negligible by sending the output from the cavity 
%% through a frequency filter with a linewidth of a few hundred MHz (with
%% weak driving this will have negligible effect on the collection
%% efficiency \cite{andersXX}). 
%% Assuming a high degree of nuclear spin
%% polarization ($P_{\rm n} \gtrsim 0.95$)~\cite{bracker04} and active ESR
%% pulse correction,  the electron spin dephasing time is expected to be $1$
%% ms~\cite{golovach03}.  %lily added
%% Both implementations in practice may require frequency conversion to telecom wavelengths.  

%anders changed this paragraph
%In conclusion, we have shown that by combining state-of-the-art
%solid state quantum optical emitters with techniques for
%electron and  nuclear spin manipulation, quantum communication over
%long distances can be achieved. 

In conclusion, we have shown that a fully fault tolerant quantum repeater can be constructed using only two qubits per node.  This opens up the possibility to build repeaters in simple systems with only two degrees of freedom, such as coupled nuclear and electronic spins.  We have exemplified this with a particular implementation in NV-centers, but the concept can be applied to a variety of physical systems \cite{andersXX}. 
%The authors wish to thank Phillip Hemmer, Aryesh Mukherjee,      Alexander Zibrov, and Gurudev Dutt. 

     The authors wish to thank P. R. Hemmer, A. Mukherjee,      A. Zibrov, and G. Dutt.  This work is supported by DARPA, NSF, ARO-MURI, and the Packard, Sloan and Hertz Foundations and  the Danish Natural Science Research Council.  

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quant-ph0410123.extracted_bib
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