High-bandwidth hybrid quantum repeater.

Abstract

We present a physical- and link-level design for the creation of entangled pairs to be used in quantum repeater applications where one can control the noise level of the initially distributed pairs. The system can tune dynamically, trading initial fidelity for success probability, from high fidelity pairs (F=0.98 or above) to moderate fidelity pairs. The same physical resources that create the long-distance entanglement are used to implement the local gates required for entanglement purification and swapping, creating a homogeneous repeater architecture. Optimizing the noise properties of the initially distributed pairs significantly improves the rate of generating long-distance Bell pairs. Finally, we discuss the performance trade-off between spatial and temporal resources.

Full text

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arXiv:0808.0307v1 [quant-ph] 3 Aug 2008
A high bandwidth quantum repeater
W. J. Munro,1,2, ∗R. Van Meter,2,3Sebastien G.R. Louis,2,4and Kae Nemoto2
1Hewlett-Packard Laboratories, Filton Road, Stoke Gifford, Bristol BS34 8QZ, United Kingdom
2National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan
3Keio University, 5322 Endo, Fujisawa, Kanagawa, 252-8520, Japan
4Department of Informatics, School of Multidisciplinary Sciences,
The Graduate University for Advanced Studies, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430 Japan
We present a physical- and link-level design for the creation of entangled pairs to be used in
quantum repeater applications where one can control the noise level of the initially distributed pairs.
The system can tune dynamically, trading initial fidelity for success probability, from high fidelity
pairs (F=0.98 or above) to moderate fidelity pairs. The same physical resources that create the long-
distance entanglement are used to implement the local gates required for entanglement purification
and swapping, creating a homogeneous repeater architecture. Optimizing the noise properties of
the initially distributed pairs significantly improves the rate of generating long-distance Bell pairs.
Finally, we discuss the performance trade-off between spatial and temporal resources.
PACS numbers: 03.67.Hk, 03.67.Mn, 42.50.Pq
Quantum information has reached a very interesting
stage in its development, where we have seen many fun-
damental experiments laying the foundation for practi-
cal systems[1]. Now certain applications, such as quan-
tum key distribution (QKD), are being readied for com-
mercial use[2, 3], where practical distances hover around
the 150km mark. Any quantum communication longer
than this limit suffers severely from noise and exponential
loss in the quantum communication channel. Hence, the
quantum communication for either QKD over a distance
beyond the limit or, more generally, all distributed quan-
tum information processing, requires the development of
a high-bandwidth repeater which can distribute and po-
tentially process quantum information given these con-
straints. In a quantum repeater system, initial imperfect
Bell pairs (which we call base-level pairs) are distributed
over channel segments. These base pairs are then purified
to high fidelity Bell pairs and connected via entanglement
swapping, resulting in entanglement between the qubits
at distant stations. Iterating this procedure creates Bell
pairs at even larger distances. These pairs can be used
in many different applications, including QKD, quantum
communication, distributed quantum computation, and
quantum metrology and related uses[4, 5, 6, 7, 8].
The many recently-proposed schemes for the design of
a quantum repeater fall into two categories. The majority
of the schemes focus on the heralded creation of very high
fidelity base-level pairs[9, 10, 11, 12, 13, 14, 15, 16, 17].
For longer segment lengths, the generation of these high
fidelity pairs comes at the expense of a very low prob-
ability of success, which becomes one of the major bot-
tlenecks in the overall performance of a repeater system.
Another significant issue is that in the majority of these
schemes, local gates between multiple qubits within a
single repeater station are difficult. An alternative ap-
proach has recently been proposed, instead creating base-
level pairs of moderate fidelity and high heralded success
probability[18, 19]. In this second approach, the phys-
ical resources used for long-distance entanglement also
efficiently implement local gates, facilitating the purifi-
cation of moderate fidelity pairs back to high fidelity
pairs. For instance with 16 qubits/node with a 10km
spacing rate 15 pairs of fidelity F=0.98 can be achieved
over a 1280km repeater network, however at longer re-
peater node spacing distanes (> 40km) the rate fails to
zero. This node spacing isssue is one of the key limita-
tions for this second approach. These two approaches are
radically different in the use of physical resources and in
technological requirements. It is hence not trivial to di-
rectly compare the feasibility and efficiency of schemes in
different approaches. However, it has been thought that
these two approaches are complementary to each other,
trading high fidelity for high success probability or vice
versa. Quantum repeaters of the latter kind typically use
coherent light instead of the single photons common in
the former category. It has been believed that a coherent-
light quantum repeater is fundamentally unable to gen-
erate high-fidelity Bell pairs and hence is unable to cope
with severe loss in the quantum channel. In this letter
we address this shortcoming by presenting the design of
a new scheme for entanglement distribution utilizing co-
herent light and demonstrate that in fact such a system is
more flexible over a wide range of losses without serious
overhead in physical resources. This advance will have a
significant impact on the overall repeater performance.
In this letter we consider the design of a repeater seg-
ment where one can dynamically vary the quality of the
base-level entangled pairs from very high to moderate fi-
delity. This tuneability will allow one to trade the fidelity
of the entangled pairs against the probability of their suc-
cessful distribution for a given segment length. We will
also ensure that the same physical resources can be used
to implement the local gates necessary for the entangle-
ment purification and entanglement swapping[20, 21]. In

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this case, both requirements can be met via a controllable
interaction between our qubit and light field.
The core interaction between our qubit and field in
our cavity quantum electrodynamics (CQED) system
is the Jaynes-Cummings Hamiltonian given by HJC =
?g?a†σ−+ aσ+
for the dipole transition. a (a†) refers to the annihilation
(creation) operators of the electromagnetic field mode in
a cavity and σ+(σ−) the raising (lower) operators of the
qubit with ground state |0? and excited state |1?. Our
qubit is a solid-state electronic spin which is also coupled
to a nuclear spin qubit, allowing for a coherent transfer of
quantum information to the long-lived nuclear spin qubit.
Physically, the electronic and nuclear-spin systems may
be achieved, for example, by single electrons trapped in
quantum dots, NV centers in diamond or neutral donor
impurities in semiconductors.
Our basic J-C Hamiltonian can be used to imple-
ment a controlled displacement D(βσz) ≡ eσz(βa†−β∗a)
operation[21, 22] between the qubit and field, where
σz = |0??0| − |1??1|. This operation displaces the field
mode by an amount β conditioned on the state of the
qubit. There are a number of ways this interaction can be
achieved ranging from shaped pulse sequences modulat-
ing the qubit or field[24] to sequences of controlled rota-
tions and unconditional displacement operations[22, 23].
?where 2g is the vacuum Rabi splitting
Phase Reference
probe
channel
qubit
cavity
detector
optical circulator
FIG. 1: Schematic of an entanglement distribution scheme
based on two qubits in individual cavities interacting indi-
rectly via a shared probe beam and controlled displacement
operations.An optical circulator before the second qubit
routes the probe field into the cavity and then the probe beam
leaking out of the cavity to the detector. A phase reference is
sent along the same lossy channel.
These controlled displacement operations now form the
basis of an efficient and tunable entanglement distribu-
tion scheme, as depicted in Fig (1). The scheme works as
follows: our first qubit is prepared in an equal superpo-
sition of both basis states (|0? + |1?)/√2 with the probe
mode in the cavity initially being the vacuum. The qubit
interacts with the probe beam via the controlled displace-
ment operation D(βσz) resulting in the combined qubit-
probe state (|0?|β? + |1?| − β?)/√2. The probe beam is
either switched out or leaks out of the cavity and is trans-
mitted over the noisy loss channel to the second cavity.
Here the second qubit and probe mode interact via a
controlled displacement operation. The probe beam then
leaks out of this second cavity and is measured, project-
ing our qubits into an appropriate entangled state. In
more detail, if our two distributed qubits are prepared
in a state (|00? + |01? + |10? + |11?)/2 then after both
controlled displacements and the noisy channel our two
qubit-light field state can be represented by
ρ =1 + e−¯ γ/2
2
|Z+??Z+| +1 − e−¯ γ/2
√
N±
2
|Φ+?|c±? +
√2?, |Ψ±? = |01±10
2
|Z−??Z−|
√
N∓
2
|Φ−?|c∓? +
√2?, |c±? =
(1)
where
1
√2|Ψ±?|0?, |Φ±? = |00±11
(|2βe−l/2l0? ± | − 2βe−l/2l0?)/?2N± with N± = 1 ±
exp?−8|β|2e−l/l0?
represents the attenuation of the probe in the chan-
nel. Now the probe beam in Eqn (1) is in one of three
possible states, |c±? or the vacuum |0?.
state |c−? is orthogonal to both |c+? and |0?.
ever, |c+? and |0? are non-orthogonal with an overlap
2exp?−4|β|2e−l/l0?/(1+exp?−8|β|2e−l/l0?). Of course,
if βe−l/2l0≫ 1, these two states are effectively orthogo-
nal, and so one could distinguish all probe beam states.
However, the channel loss has had two significant effects:
first, it has mixed |Z+? with |Z−? with a mixing param-
eter1+e−¯ γ/2
2
. This mixing parameter is small only when
|β|2(1−e−l/l0) ≪ 1, which is in conflict with the desire to
have all probe beam states nearly orthogonal for chan-
nels of moderate length. Second, the channel has also
attenuated the probe beam’s amplitude and so the sec-
ond controlled displacement must be by D?βe−l/2l0σz2
rather than D(βσz2) to minimize the effect of the loss.
We now turn our attention to the measurement of the
probe and the resulting conditioning it causes on the dis-
tributed qubits. There are various measurement strate-
gies, ranging from highly idealized cat state projectors
(CSP) |c−??c−|, to single photon detection (SPD). Using
such detection strategies, our qubits are conditioned to
|Z±?
=
and ¯ γ = 2|β|2(1 − e−l/l0). Here l/l0
The odd cat
How-
?
ρ(F) = F|Φ−??Φ−| + (1 − F)|Φ+??Φ+|
(2)
with heralded success probabilities
PCSP[F,l/l0] =
1
4
?
1 − [2F − 1]
8e−l/l0
1−e−l/l0
?
(3)
PSPD[F,l/l0,η2] =
1
2
d
dλ[2F − 1]
4η2e−l/l0
1+(7−8η2)e−l/l0λ
?????
λ=1
(4)
respectively. The single photon detector is assumed to
have a non-unit quantum efficiency η2. Eqns (2 - 4) have
been expressed in terms of the fidelity F of the |Φ−? state
generated, the attenuation parameters l/l0 and the de-
tection efficiency rather than β. The initial displacement
β can be expressed in terms of F, l/l0, P and η2. We
also need to point out that Eqn (2) is a mixture of only
two Bell states, |Φ±?, which has important advantages
in entanglement purification. Such states are much more
efficient to purify.

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In Fig (2), we plot these probabilities P[F,l/l0] ver-
sus fidelity for both measurement strategies (with η2=
0.9,1) for an attenuation length l/l0 = 0.8. Our ideal-
ized cat projector (CSP) allows a nice range of fidelities
to be achieved, ranging from F = 1/2 to F = 1. For
higher fidelities, lower success probabilities are achieved.
The major issue with the cat projector is that it is dif-
ficult to implement in practice, but for the moderate to
high fidelity regimes, the single photon detection scheme
closely follows the cat projector results and so is an ex-
cellent compromise. It also highlights that single photons
(and hence single-photon sources) are not needed for cre-
ating high fidelity distributed Bell pairs. The sending
of coherent states (whether weak or strong) can achieve
the same goal. High fidelity pairs can be generated over
long distances but at the expense of the success proba-
bility. Other measurements techniques (such as homo-
dyne, bucket or vacuum detection) also result in entan-
gled states, but these states tend be composed of more
than two Bell states, and so are more difficult to purify.
However, they can provide success probabilities greater
than one half but result in low fidelity final states.
0.6 0.70.8 0.9
1
F
0.05
0.1
0.15
0.2
0.25
P
a)
b)
c)
β=0.2
β=0.4
FIG. 2: Plot of the probability P[F,l/l0] of successfully estab-
lishing an entangled pair at an attenuation length of l/l0 = 0.8
(20 km in commercial fiber) versus fidelity for the different
measurement strategies: a) odd state cat projector CSP, b)
ideal SPD and c) SPD with η2= 0.9.
We now have a highly tunable segment where one can
dynamically vary the quality of the base-level entangled
pair from low to high fidelity utilizing non-unit efficiency
single-photon detection. These pairs can be used in a re-
peater protocol to create long-distance entangled pairs.
The basic repeater protocol works as follows: multiple
copies of lower fidelity base-level pairs are purified to cre-
ate high fidelity pairs, then entanglement swapping of the
high fidelity pairs between adjacent repeater nodes cre-
ates longer distance but lower fidelity pairs. These result-
ing pairs can then purified to high fidelity pairs and en-
tanglement swapping gives even longer range pairs. The
procedure is iterated until pairs over the desired length
are obtained. The purification and entanglement swap-
ping protocols require efficient local two-qubit C-Z (or
CNOT) gates. In our architecture these can be achieved
using another sequence of controlled displacement op-
erations D(iβ2σz2)D(β1σz1)D(−iβ2σz2)D(−β1σz1)
exp[2i β1β2σz1⊗ σz2] with β1and β2satisfying β1β2=
π/8 [21, 22]. These quantum bus (qubus) based local
gates are needed throughout the protocol, and so the
more efficient and robust they are the better our overall
performance.
One of the major issues for performance becomes
the chosen quality of base-level entangled pairs for our
lowest-level segment.Conventional quantum repeater
wisdom generally suggests that before one performs en-
tanglement swapping to create longer pairs, one should
purify the noisy base-level pairs as best as possible. For
our discussions here we will set a working fidelity of
F = 0.98 before attempting entanglement swapping.
There are now a number of ways we can use our tun-
able segment to achieve this required fidelity. We could
just directly create a pair of that fidelity (see Table I), or
we could create lower-fidelity pairs and purify them us-
ing standard protocols[25, 26, 27, 28]. Which approach
is best will depend on both the probability of generat-
ing the entangled pairs Pgand the purification probabil-
ity Ppur. For a one-round purification protocol with no
limitation on the physical resources, the effective prob-
ability of generating the final fidelity pair (per channel)
is given by Peff = PgPpur/2. The purification prob-
ability depends critically on the form of the initial en-
tangled pairs, and so our engineering of Eqn (2) above
is very advantageous. A mixture of two Bell states is
much easier to purify than more general mixed states.
In this situation, two copies of ρ(F) can be purified to
a new entangled ρ(F′= F2/(1 − 2F + 2F2)) with a
Ppur(F) = F2+(1−F)2[28]. This gives an overall prob-
ability of success for generating a F = 0.98 pair from
two F = 0.9 pairs of Peff = 0.0547. In comparison, we
have generation probabilities of 0.03374 for directly man-
ufacturing the F = 0.98 pair. Thus, using lower fidelity
pairs and purification improves our overall probability of
generating the final pair, assuming efficient local gates.
≡
F=0.98 F=0.9 F=0.75
PSPD[F,0.4,0.9] 0.05076 0.16693 0.13996
PSPD[F,0.8,0.9] 0.03374 0.13338 0.17970
PSPD[F,1.6,0.9] 0.01494 0.07157 0.15528
TABLE I: Success probability of generating a Bell state of
fidelity F conditioned on single photon detection (η2= 0.9).
Does a multiple-round purification protocol improve
our results? Moving to multiple rounds, there are a num-
ber of choices for how to implement the protocol[25, 27].
As an illustration, consider a symmetric purification
protocol[27] where four F = 0.75 pairs are purified to the
F = 0.98 pair. In this case, we have an effective genera-
tion probability of Pg = 0.0236 which is lower than the
probability achieved by the single-round protocol. The
difference is primarily due to the lower probability of suc-

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cessfully purifying lower fidelity pairs.
We now need to turn our attention to a more detailed
discussion of the physical resources required for creat-
ing our long distance pairs. It is important to consider
both the spatial and temporal resources necessary. The
tunability of our source allows us significant flexibility in
how we use resources. We can implement the minimum
physical resource strategies (two qubits per station) de-
veloped by Harvard[11], as well as the modest physical
resource sixteen qubit per station schemes of van Loock
et al.[18, 19]. Due to our low to moderate probabilities
of successfully creating the base pairs between stations,
the minimum physical resource approach will have sig-
nificant performance issues due to this distribution bot-
tleneck. Significant time will be spent idle waiting for
the base pairs to be successfully created between all sta-
tions. However, by allowing moderate physical resources,
we can simultaneously attempt to create multiple base
pairs per segment, and so reduce the time waiting for the
necessary resources to become available. This approach
will dramatically increase the throughput of the entire
repeater chain. To quantify this degree of the improve-
ment, we have performed a Monte Carlo simulation of
a nested entanglement protocol over 51.2 l/l0(1280 km)
for varying numbers of qubit per station with dynam-
ical resource allocation[29].
in Table(II). First, they show that, of the three seg-
ment lengths considered, the best results were obtained
from the 0.8 l/l0(20km) situation. For longer distances,
the initial success probability drops dramatically, and for
shorter segment lengths, errors in local gates have a sig-
nificant impact. We also found that the protocol could
run with local gate error rates exceeding 1%, though in
this situation the generation rate falls to only a few pairs
per second. Raising the number of qubits in each half sta-
tion also gives a slight improvement compared to stacking
smaller repeater nodes in parallel. Still, our results show
a good generation rate with 8-16 qubits per half node.
In the scheme of van Loock, with 16 qubits per half node
with 0.4 l/l0segment spacing (4096 total qubits), a rate
of 15 F=0.98 Bell pairs/second was achieved. Using our
new scheme (for the same total number of qubits), we
achieve a rate of 3190 pairs/second, an improvement of
over two orders of magnitude. Our new scheme also pro-
duces relatively high throughput of 437 pairs at a link
distance of 40km compared with zero for the orginal van
Loock case. In the new scheme, the fidelity remains high
over long distances, but the probability of success de-
clines. Finally, some of our improvements in the protocol
are obtained by tuning the base-level fidelity to optimize
the number of purification rounds before entanglement
swapping.
To summarize we have shown how to implement a
high bandwidth quantum repeater using the fundamen-
tal atom-light interactions in quantum optics, through
a qubus-mediated entangling operation. This new ap-
The results are presented
qubits
half station
8
16
32
0.4 l/l0
0.8 l/l0
1.6 l/l0
520 (2048)
1097 (4096) 1528 (2048) 437 (1024))
2297 (8192) 3190 (4096) 987 (2048))
693 (1024)193 (512)
TABLE II: Rate of final generation of distant 51.2 l/l0 (1280
km) entangled pairs of minimum fidelity F = 0.98 result-
ing from a nested entanglement protocol with 8, 16 and 32
qubits per half-station. The stations are separated by either
0.4 l/l0, 0.8 l/l0 or 1.6 l/l0 attenuation lengths.The number
in the brackets indicate the total numbers of qubits over all
repeater stations. We have assumed an initial fidelity reduc-
tion in the base pairs due to loss in the fiber (assumed to be
0.17dB/km) and distortion of 0.1% due to local losses during
the measurement-free displacement-based C-Z gate.
proach refutes the criticism that hybrid repeaters can-
not adapt to a wide range of losses and offers flexi-
bility on fidelity and entanglement success probability.
In this hybrid scheme, the only required interactions
are controlled displacement operations between the light
field and qubit. These controlled displacement opera-
tions result in the distribution of moderate- to high-
fidelity entanglement between the qubits in the repeater
stations, conditioned on single-photon detection of the
qubus mode.They also can implement a determinis-
tic C-Z gate for use in all the purification and entan-
glement swapping steps. Such tools allow for the natu-
ral design of a scalable and homogeneous quantum re-
peater network and thus a distributed computation with
many concurrent steps happening in different locations.
Using high-efficiency single photon detection, we have
shown that long-distance communication rates over 1000
pairs/second with a fidelity above 98% are possible with
only modest resources.
Acknowledgments: We thank T. Ladd for the use of the
base simulation code upon which our simulations are
based and also thank him, L. Jiang, P. van Loock, T. P.
Spiller and J. M. Taylor for valuable discussions. This
work was supported in part by MEXT and NICT in
Japan and the EU project QAP.
∗Electronic address: bill.munro@hp.com
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