Quantum optical coherence can survive photon losses: a continuous-variable quantum erasure correcting code

Abstract

A fundamental requirement for enabling fault-tolerant quantum information
processing is an efficient quantum error-correcting code (QECC) that robustly
protects the involved fragile quantum states from their environment. Just as
classical error-correcting codes are indispensible in today's information
technologies, it is believed that QECC will play a similarly crucial role in
tomorrow's quantum information systems. Here, we report on the first
experimental demonstration of a quantum erasure-correcting code that overcomes
the devastating effect of photon losses. Whereas {\it errors} translate, in an
information theoretic language, the noise affecting a transmission line, {\it
erasures} correspond to the in-line probabilistic loss of photons. Our quantum
code protects a four-mode entangled mesoscopic state of light against erasures,
and its associated encoding and decoding operations only require linear optics
and Gaussian resources. Since in-line attenuation is generally the strongest
limitation to quantum communication, much more than noise, such an
erasure-correcting code provides a new tool for establishing quantum optical
coherence over longer distances. We investigate two approaches for
circumventing in-line losses using this code, and demonstrate that both
approaches exhibit transmission fidelities beyond what is possible by classical
means.

Full text

Page 1

Quantum optical coherence can survive photon
losses using a continuous-variable quantum
erasure-correcting code
Mikael Lassen1,2*, Metin Sabuncu1,2, Alexander Huck1, Julien Niset3,4, Gerd Leuchs2,5, Nicolas J. Cerf3
and Ulrik L. Andersen1*
A
quantum information processing is an efficient quantum
error-correcting code that robustly protects the involved
fragile quantum states from their environment1–10. Just as clas-
sical error-correcting codes are indispensible in today’s infor-
mation technologies, it is believed that quantum error-
correcting code will play a similarly crucial role in tomorrow’s
quantum information systems. Here, we report on the exper-
imental demonstration of a quantum erasure-correcting code
that overcomes the devastating effect of photon losses. Our
quantum code is based on linear optics, and it protects a
four-mode entangled mesoscopic state of light against era-
sures. We investigate two approaches for circumventing in-
line losses, and demonstrate that both approaches exhibit
transmission fidelities beyond what is possible by classical
means. Because in-line attenuation is generally the strongest
limitation to quantum communication, such an erasure-correct-
ing code provides a new tool for establishing quantum optical
coherence over longer distances.
Quantum information protocols are inevitably affected by noise,
which in turn produces errors in the extremely sensitive processed
quantum information1. To take full advantage of quantum infor-
mation processing, including long-distance quantum communi-
cation and fault-tolerant quantum computing11–13, these errors
must be corrected efficiently. This can be done by encoding the
information in special quantum error-correcting codes, which
introduce redundancy, protecting the fragile quantum information
from environment-induced decoherence. Using such codes, trans-
mission errors can be diagnosed through so-called syndrome
measurements, the results of which are used to correct the corrupted
quantum information.
Quantum error correcting codes (QECC) were first discovered
for discrete variable qubit systems2–7and later extended to systems
in which information is encoded into observables with a continuous
spectrum8–10,14,15. Only a few experimental implementations
demonstrating quantum error correction have been carried out to
date, such as those in nuclear magnetic resonance systems16, in
an ion-trap system17, and in a pure optical system18,19. All these
works have reported on the correction of errors that are the mani-
festation of line noise. However, it is very often the loss of
photons in a transmission line (corresponding to erasures in an
information theoretic language) that is the main obstacle to the sur-
vival of quantum coherence. Here, we report on the first experimen-
tal realization of a quantum erasure-correcting code that
fundamentalrequirementforenabling fault-tolerant
simultaneously protects two independent continuous-variable
(CV)quantumsystemsagainst
transmission channel.
We consider the transmission of a quantum state of light through
a channel that either transmits the information perfectly or com-
pletely erases it with an error probability PE. The density matrix
of the transmitted state is given by
photon lossesinthe
r = (1 − PE)|alka| + PE|0lk0|(1)
where the state |al comprises the transmitted quantum information
and |0l is the vacuum state arising due to channel erasure. Such an
error model, which can be viewed as random fading, is likely to
occur as a result of time jitter noise or beam pointing noise in an
atmospheric transmission channel, for example20–22. The CV code
for protecting quantum information from erasures is a four-mode
entangled mesoscopic state in which two (information-carrying)
quantum states are encoded with the help of a two-mode entangled
vacuum state15. As a result of the redundancy, the damage that the
erasure has made regarding the quantum states can be reversed by a
simple decoding procedure followed by measurements that deter-
mine the damage and a feedforward step that corrects the output
states. It is remarkable that the quantum coding, decoding and cor-
rection is based on simple linear optical components and Gaussian
resources. This is not a violation of the recent No-Go theorem for
QECC of Gaussian states with Gaussian transformation23because
the stochastic erasure noise occurring in the transmission channel
produces a non-Gaussian state.
We investigate two different detection strategies for correcting
the corrupted quantum states. The first strategy actively corrects
the errors of the transmitted state according to the outcome of the
measurement, whereas the second strategy filters out the corrupted
state when an error is detected. A fundamental difference between
the two approaches is that the former is deterministic and therefore
always recovers a corrected output state, whereas the latter is prob-
abilistic, which means that the corrected output state is only some-
times recovered. As for most QECC schemes, the deterministic
approach comes with a price: for the scheme to work one may
only allow for the occurrence of a given number of errors
(a single error in this case) and one needs to know in which
mode the error occurred after the transmission of the particular
quantum state. As was shown in ref. 15, these requirements may,
however, be relaxed by using a probabilistic approach. We note
1Department of Physics, Technical University of Denmark, Fysikvej, 2800 Kongens Lyngby, Denmark,
Scharowskystrasse 1, 91058 Erlangen, Germany,
1050 Brussels, Belgium,
Nu ¨rnberg, Staudtstrasse 7/B2, 91058 Erlangen, Germany. *e-mail: mlassen@fysik.dtu.dk; ulrik.andersen@fysik.dtu.dk
2Max Planck Institute for the Science of Light, Gu ¨nther
3Quantum Information and Communication, Ecole Polytechnique, CP 165, Universite Libre de Bruxelles,
4Department of Physics, Hunter College of CUNY, 695 Park Avenue, New York, New York 10065, USA,
5University Erlangen–
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that a subpart of the deterministic circuit has been implemented in
the context of CV quantum secret sharing24.
Schematicsoftheset-uparedepictedinFig.1,showinganencod-
ingstationwherethefour-modecodeisprepared,anerasurechannel
whereinformationisrandomlyerased,andadecodingstation,where
measurement outcomes either correct or filter the corrupted state.
The key element in the preparation stage is a two-mode Gaussian
entangled source, which exhibits quantum correlations between
pairs of conjugate quadrature amplitudes (see Methods). This state
interferes with two signal states to form the final four-mode code
comprising four optical beams. A vacuum state is chosen as one of
the input signals, and the other input is prepared in a coherent
state. This choice simplifies the experimental realization, but is not
an intrinsic limitation of the scheme. The four resulting beams are
then dispersed into four free-space transmission channels that can
be independently blocked to simulate any combination of erasures.
Atthedecodingstationtheinterferencesarereversedintwobalanced
beamsplitters, and two of the resulting outputs are jointly measured
inanentangledmeasurementstrategy25inwhichthemodesinterfere
on a balanced beamsplitter and conjugate quadratures are measured
at the two outputs (see Fig. 1). The resulting outcomes are now used
either to deterministically correct the errors through conditional
linear displacements or to probabilistically filter out the loss-
infected states.
First, we describe the deterministic correction strategy. Figure 2a
shows a scan of the quantum-mechanical oscillator comprising the
coherent state quantum information of the input state. The infor-
mation encoded in a coherent state can be concisely described by
the conjugate quadrature operators: the amplitude ˆ x and the
phase ˆ p such that |al = |kˆ xl + ikˆ pll. For the specific measurement
run shown in Fig. 2a, |al ≈|3þ 3il. Figure 2b illustrates the
measurements at the homodyne detector HD1 after the four-
mode state has been transmitted through the channel with erasure
on channel 2. The state is clearly seen to be corrupted as the first
and second moments of the quantum oscillator are significantly
changed. However, by using the measurement outcomes (xm, pm)
of the homodyne detectors (Fig. 2c) to linearly displace the ampli-
tude and phase quadratures of the transmitted state (xo, po) with the
gain,G
(xo? xo+
quantum state is partially recovered, as shown qualitatively in
Fig. 2d for G ¼ 1.97. The accuracy in the estimation of the error
(and thus the precision of the displacement) depends crucially on
the degree of squeezing: by using infinite squeezing, the transmitted
states can, in principle, be perfectly corrected15. With finite squeez-
ing the protocol can be quantified by the fidelity between the input
state and the corrected output state. Based on the measurements
presented above, the fidelities are computed for various gains and
the results are depicted by the blue squares in Fig. 2e. A
maximum fidelity of 0.57+0.02 is obtained, which clearly surpasses
the classical benchmark of 0.50. Similar fidelities are achieved for
the erasure of channel 1, whereas fidelities close to unity are
obtained when channels 3 or 4 are blocked. Measurements for
which the two-mode squeezed state was replaced by vacua were
also carried out for different displacement gains. The resulting fide-
lities are depicted in Fig. 2e by the red circles, and they nicely illus-
trate the need for entanglement. Although the protocol has been
implemented only for a specific pair of input coherent states, it
will work equally well for any coherent state as the protocol is invar-
iant under displacements. Thus any input alphabet of coherent
states can be corrected.
??
G
√
xm,po? po+
??
G
√
pm),theoriginal
A/D card
Amplifier
Decoding and verification
+
Channel 1
Channel 2
Channel 3
Channel 4
BBS
LO
LO
AM
HD1
Digital
data
processing
HD2
Down mixer
Information
channels
Encoding and state preparation
OPO 2
PM
OPO 1
P
Coherent state
Pump
Pump
Seed
Seed
X
xm
pm
BBS
BBS
BBS
BBS
BBS
BBS
Low pass
BBS
SM
X
P
Vacuum state
P
X
θ
b
a
c
|0
|0
|α
|α
Squeezed state
P
X
Squeezed state
−
−
−
θ
Figure 1 | Schematics of the experimental QECC set-up. a, The four-mode code is prepared through linear interference at three balanced beamsplitters
(BBS) between the two input states, |al and |0l, and two ancillary squeezed vacuum states. The latter states are produced in two optical parametric
oscillators, OPO 1 and OPO 2, and the coherent state is prepared via a coherent modulation at 5.5 MHz produced by an amplitude (AM) and a phase
modulator (PM). b, The encoded state is injected into four free-space channels that can be independently blocked, thereby mimicking erasures. c, The
corrupted state is decoded, the error is detected by the syndrome measurement (SM) and the state is deterministically corrected or probabilistically selected.
The measurement is an entangled measurement in which the phase and amplitude quadratures of the two emerging states are jointly measured (for example,
see ref. 25). The error correcting displacement or post-selection operation is carried out electronically after the measurement of the transmitted quantum
states. These states are measured with two independent homodyne detectors that allow for full quantum state characterization, by scanning the phases (u)
of the local oscillators (LOs) with respect to the phases of the signals. All erasure events are obtained by blocking the beam paths.
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We now proceed by discussing the results of probabilistic
recovery of quantum information, where the objective is to use a
probabilistic QECC to surpass the fidelity associated with a single
channel. In contrast to the deterministic approach, for which strin-
gent conditions are put on the channels, the probabilistic approach
is much less stringent. One may allow for multiple erasures and, in
addition, there are no requirements regarding knowledge of the
occurrence and location of the erasures. In this case, the density
matrix of the transmitted state is given by
rtrans=
?
16
i=1
Piri
(2)
where riis the output density matrix corresponding to one of the
16 different erasure patterns that may occur during transmission
and Piindicates the associated probability running from PE
(1 2PE)4corresponding to complete erasure and complete trans-
mission, respectively. Using the tomographic maximum likelihood
algorithm for reconstructing the density matrices via homodyne
detection, we fully characterized the input and output states for
various cases. Figure 3a shows the density matrix of the coherent
input state in the 30 × 30 Fock state basis. This state was then
mixed with the entangled state and sent through the four channels.
4to
Subsequently, we performed 16 different full measurement runs by
interchangeably blocking the four channels corresponding to the
16 different transmission patterns. The measurement outcomes are
then weighted by the probabilities Pito create the density matrix of
thetransmittedmixedstate.Thecorruptedstateswerethenprobabil-
istically corrected by conditioning on the outcomes of the syndrome
measurements (SMs). For the realization in Fig. 3 we used the con-
dition that if the measurement outcomes obeyed |xm| .0.8 and
|pm| . 0.8 (found from optimization15), an error was detected and
the resulting transmitted state was discarded. After this filtering
operation, we reconstructed the density matrix based on the
reduceddataset;theresultisshowninFig.3cforPE¼ 0.25.Byrepla-
cing the entangled states with vacua, the resulting density matrix is
largely changed, as illustrated in Fig. 3d. To determine the fidelity
F between the input state (with density matrix rin) and the filtered
output state (with density matrix rout), we used the general
expression F = [Tr(
we computed a fidelity of F¼ 0.82+0.02, which clearly surpasses
the transmission fidelity of F≃ 0.75 obtained by transmission in a
singlechannelwithsimilarerasureprobabilityandwithnoerrorcor-
rection applied. It is interesting to note that even without entangle-
ment, the protocol also outperforms the single channel approach15.
For the corresponding entanglement-free set-up, in which the two-
mode squeezed state is replaced by vacua, the corrected density
??????????????? ?
????
rout
√
rin
????
rout
√
?
)]2. For the example in Fig. 3c
Amplitude (SNU)
Amplitude (SNU)
d
e
−3 −2 −101234
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Phase (SNU)Amplitude (SNU)
Probability
Probability
i Without squeezing
0.0
0.1
0.2
0.3
0.4
0.5
0.6
−3 −2 −101234
Amplitude (SNU)
Probability
0.0
0.1
0.2
0.3
0.4
0.5
0.6ii With squeezing
−3 −2 −101234−3 −2 −101234
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Phase (SNU)
Probability
SNL
a
6
5
4
3
2
1
0
−1
−2
−3
−4
−5
−6
SNL
Fidelity
Displacement gain
1.61.71.8 1.92.02.12.22.3 2.4
0.30
0.35
0.40
0.45
0.50
0.55
0.60
G = 1.97
LO phase
02π
3π/2
π/2
π
LO phase
02π
3π/2
π/2
π
LO phase
0
2π
3π/2
π/2
π
Amplitude (SNU)
b
c
6
5
4
3
2
1
0
–1
–2
–3
–4
–5
–6
6
5
4
3
2
1
0
–1
–2
–3
–4
–5
–6
Figure 2 | Results of the deterministic QECC protocol. a, Phase scan of the input coherent state with the excitation |al≈ |3þ3il. b,c, Phase scans of the
output state measured at HD1 before correction (b) and of the corrected output state (c). d, Histograms of the marginal distributions of the amplitude and
phase quadratures of the joint syndrome measurement (in shot noise units, SNU). Red and blue curves correspond to the marginal distributions for a shot-
noise-limited (SNL) state, whereas the black curves are the best Gaussian fits to the histograms. e, Fidelity is plotted as a function of the displacement gain
with (blue squares) and without (red circles) the use of entanglement. The dashed and solid lines are the theoretically predicted fidelities for 0 dB and 2 dB
of two-mode squeezing, respectively. The error bars depend on the measurement error, which is mainly associated with the stability of the system over time
and the finite resolution of the analog-to-digital converter. This amounts to an error of+3% for all fidelities.
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matrix is shown in Fig. 3c and a fidelity of F¼ 0.77+0.02
is computed.
The fidelity as a function of the error probability PEand the
threshold value |xth| ¼ |pth| are illustrated in Fig. 4a and b, respect-
ively. It can be seen that the entanglement-free QECC performs
better than the single channel approach (that is, without error cor-
rection) forerror probabilities up to ?28%. It is also evident that the
use of the entanglement-based code further increases the fidelity for
all evaluated error probabilities. As expected, we see in Fig. 4b that
the fidelity increases asthe threshold value is lowered. This is associ-
ated, however, with a lower probability of success.
We have successfully demonstrated a continuous-variable
quantum erasure-correcting code that protects quantum optical
coherence from erasures (or probabilistic photon losses). The deter-
ministic version of our scheme has the additional advantage that it
enables a direct processing of the error-corrected output state in
downstream applications, circumventing the need for a quantum
repeater configuration that requires quantum entanglement distilla-
tionandquantummemory.Moreover,thedeterministicversionmay
berelevantintheperspectiveoffault-tolerantimplementationofCV
quantum information computing; probabilistic gates are applied at
the level of the encoded states and gate success (or failure) translates
into transmission (or erasure) of the corresponding mode.
Although the protocol has only been experimentally accessed for
quantum information encoded in coherent states of light, it enables
the faithful transmission of other quantum states such as squeezed
states, qubit states or even bipartite entangled states. Furthermore,
the error model can be extended to also include partial erasure
and random phase noise, which often occurs in free-space trans-
mission in a turbulent atmosphere21. As an outlook, it is intriguing
to address the universality of our protocol with respect to arbitrary
input states and arbitrary non-Gaussian error models.
Methods
Generation of two-mode squeezing. Two-mode squeezing (or CV entanglement)
was produced through the interference of two single-mode squeezed statesgenerated
in optical parametric oscillators (OPOs)operating below threshold. The main source
for the OPOs was a Diabolo laser (Innolight) delivering 400 mW of infrared light
(1,064 nm) and 600 mW of green light (532 nm). These light beams were both sent
through empty, high-finesse triangular cavities to clean their spatial modes as well as
to filter out classical amplitude and phase noise. The resulting beams then served as
pump beams, seed beams and locking beams for the OPOs. The OPOs were bowtie-
shaped cavities each with a type I periodically poled potassium titanyl phosphate
(KTP) crystal (1 × 2 ×10 mm3) for nonlinear downconversion. The OPO cavities
each consisted of two curved mirrors (radius of curvature, 25 mm) and two plane
mirrors. Three of the mirrors in each cavity were highly reflective at 1,064 nm
(R . 99.95%), and the output couplers had a transmission of 8%. The transmittance
of the mirrors at the pump wavelength (532 nm) was more than 95%. Together with
0
10
20
30
0
10
20
30
0.00
0.02
0.04
0.06
0.08
0
10
20
30
0
10
20
m
30
0.00
0.02
0.04
0.06
0
10
20
30
0
10
20
30
0.0
0.1
0.2
|p(n,m)|
|p(n,m)|
|p(n,m)|
|p(n,m)|
n
n
n
m
m
F = 0.82
F = 0.77
F = 0.75
0
10
20
30
0
10
20
30
0.00
0.05
0.10
n
m
a
c
d
Input state
Output—single channel
b
Output—entanglement-free QECC
Output—entanglement-based QECC
Figure 3 | Results of the probabilistic QECC protocol. a, Density matrix in the 30×30 Fock state basis of the input coherent state with the excitation
|al≈|3þ3il. b, Density matrix of the output state using a single erasure channel as described in equation(1). The fidelity to the input state is F¼0.75.
c, Density matrix of the corrected output state using the entangled four-mode code. The fidelity to the input state is F¼0.82. d, Density matrix for the
corrected output state when the entangled states are blocked, and thus replaced by vacua. The fidelity is computed to be F¼0.77. The erasure probability
for all realizations is PE¼0.25.
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the pump, a seed beam was injected at 1,064 nm, the brightness of which facilitated
the construction of the various phase locks in the set-up. To lock the phase of the
cavities we used auxiliary counterpropagating beams at 1,064 nm. Amplitude
quadrature squeezed beams were then produced by deamplification of the seed
beams. Using a spectrum analyser we measured squeezing of 23.4+0.2 dB and
22.7+0.2 dB below the shot noise level (at 5.5 MHz, a resolution bandwidth of
300 kHz and video bandwidth of 300 Hz) for the two OPOs. The two amplitude
squeezed beams with a relative phase of p/2 were then brought to interference at a
50/50 beamsplitter (visibility greater than 98%) to produce the two-mode squeezing,
which was then launched into the quantum error correction coding set-up. The
average measured two-mode squeezing was approximately 22.0+0.5 dB below the
shot noise level.
Data acquisition. All measurements were performed at a sideband frequency of
5.5 MHz. The electronic outputs of all detectors were mixed down with an electronic
local oscillator at 5.5 MHz, low-pass filtered (300 kHz), amplified (FEMTO
DHPVA-100) and finally digitized by a 14-bit analog-to-digital converter at
10 Msamples/s. The signal was sampled around the 5.5 MHz sideband to avoid low-
frequency classical noise. The time trace of the coherent state is shown in Fig. 2
before (Fig. 2a) and after (Fig. 2b) total erasure of channel 2 (see Fig. 1). These are
typical traces of a homodyne measurement of a coherent state, where the local
oscillator is scanned from 0 to 2p. The traces consist of ?220,000 data points. From
the down-mixed time traces we reconstructed the density matrices and the Wigner
functions using a maximum likelihood algorithm.
Received 10 March 2010; accepted 10 June 2010;
published online 25 July 2010
References
1. Nielsen, M. A. & Chuang, I. L. Quantum Computation and Quantum
Information (Cambridge Univ. Press, 2000).
2. Shor, P. W. Scheme for reducing decoherence in quantum computer memory.
Phys. Rev. A 52, R2493–R2496 (1995).
3. Steane, A. M. Error correcting codes in quantum theory. Phys. Rev. Lett. 77,
793–797 (1996).
4. Calderbank, A. R. & Shor, P. W. Good quantum error-correcting codes exist.
Phys. Rev. A. 54, 1098–1105 (1996).
5. Laflamme, R., Miquel, C., Paz, J. P. & Zurek, W. H. Perfect quantum error
correcting code. Phys. Rev. Lett. 77, 198–201 (1996).
6. Bennett, C. H., DiVincenzo, D. P., Smolin, J. A. & Wootters, W. K. Mixed-state
entanglement and quantum errorcorrection. Phys. Rev. A 54, 3824–3851 (1996).
7. Grassl, M., Beth, Th. & Pellizzari, T. Codes for the quantum erasure channel.
Phys. Rev. A 56, 33–38 (1997).
8. Braunstein, S. Error correction for continuous quantum variables. Phys. Rev.
Lett. 80, 4084–4087 (1998).
9. Lloyd, S. & Slotine, J.-J. E. Analog quantum error correction. Phys. Rev. Lett. 80,
4088–4091 (1998).
10. Braunstein, S. L. Quantum error correction for communication with linear
optics. Nature 394, 47–49 (1998).
11. Duan, L. M., Lukin, M. D., Cirac, J. I. & Zoller, P. Long-distance quantum
communication with atomic ensembles and linear optics. Nature 414,
413–418 (2001).
12. Yuan, Z.-S. et al. Experimental demonstration of a BDCZ quantum repeater
node. Nature 454, 1098–1101 (2008).
13. Knill, E. Quantum computing with realistically noisy devices. Nature 434,
39–44 (2005).
14. Wilde, M. M., Krovi, H. & Brun, T. A. Entanglement-assisted quantum error
correction with linear optics. Phys. Rev. A 76, 052308 (2007).
15. Niset, J., Andersen, U. L. & Cerf, N. J. Experimentally feasible quantum erasure-
correcting code for continuous variables. Phys. Rev. Lett. 101, 130503 (2008).
16. Cory, D. G. et al. Experimental quantum error correction. Phys. Rev. Lett. 81,
2152–2155 (1998).
17. Chiaverini, J. et al. Realization of quantum error correction. Nature 432,
602–605 (2004).
18. Pittmann, T. B., Jacobs, B. C. & Franson, J. D. Demonstration of quantum error
correction using linear optics. Phys. Rev. A 71, 052332 (2005).
19. Aoki, T. et al. Quantum error correction beyond qubits, Nature Phys. 5,
541–546 (2009).
20. Wittmann, C. et al. Quantum filtering of optical coherent states. Phys. Rev. A 78,
032315 (2008).
21. Majumdar, A. K. & Ricklin, J. C. Free-Space Laser Communications
(Springer, 2008).
22. Elser, D. et al. Feasibility of free space quantum key distribution with coherent
polarization states, N. J. Phys. 11, 045014 (2009).
23. Niset, J., Fiurasek, J. & Cerf, N. J. No-go theorem for Gaussian quantum error
correction. Phys. Rev. Lett. 102, 120501 (2009).
24. Lance, A. M. et al. Tripartite quantum state sharing. Phys. Rev. Lett. 92,
177903 (2004).
25. Niset, J. et al. Superiority of entangled measurements over all local strategies for
the estimation of product coherent states. Phys. Rev. Lett. 98, 260404 (2007).
26. Van Loock, P. et al. Broadband teleportation. Phys. Rev. A 62, 022309 (2000).
0.00.10.2
PE
0.30.4
0.5
0.6
0.7
0.8
0.9
1.0
Fidelity
PE = 0.25
0.20.40.6
Threshold (SNU)
0.81.01.2
0.65
0.70
0.75
0.80
0.85
0.90
Fidelity
a
b
Figure 4 | Performance of the probabilistic QECC protocol. a,b, Transmission fidelity plotted as a function of the erasure probability (a) and the post-
selection threshold value (b). In both plots the dashed line represents the single channel (code-free) fidelity. The circles correspond to the entanglement-free
error correction code, and the squares correspond to the entanglement-based code. The dashed and solid lines are the theoretically predicted fidelities for
the two cases. The directly measured squeezing values at the entrance to the protocol were ?3 dB (see Methods). The black dotted lines are the fidelity for
the transmission in a single channel. For the theoretical model, we used a single-mode picture15which is justified by the small bandwidth of the
measurements (300 kHz); otherwise a broadband description should be used26. The error bars are estimated as explained in the caption of Fig. 2.
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Acknowledgements
This work was supported by the Future and Emerging Technologies programme of the
European Commission under the FP7 FET-Open grant no. 212008 (COMPAS), by the
Danish Agency for Science Technology and Innovation under grant no. 274-07-0509, by
the Deutsche Forschungsgesellschaft, and by the Interuniversity Attraction Poles program
of the Belgian Science Policy Office under grant IAP P6-10 (photonics@be).
Author contributions
M.L. and M.S. performed the experiments. J.N. and N.J.C. performed the theoretical
calculations. M.L., M.S. and A.H. performed the data analysis. M.L., N.J.C. and U.L.A.
wrote the manuscript. M.L., M.S., U.L.A. and G.L. performed the project planing. All
authors discussed the results and implications and commented on the manuscript at all
stages. Correspondence and requests for additional materials should be addressed to
M.L. or U.L.A.
Additional information
Theauthorsdeclarenocompetingfinancialinterests.Reprintsandpermissioninformationis
available online at http://npg.nature.com/reprintsandpermissions/. Correspondence and
requests for materials should be addressed to M.L. and U.L.A.
NATURE PHOTONICS DOI: 10.1038/NPHOTON.2010.168
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NATURE PHOTONICS | VOL 4 | OCTOBER 2010 | www.nature.com/naturephotonics
705
© 2010 Macmillan Publishers Limited. All rights reserved.