Article
Reference
Purification of single-photon entanglement with linear optics
SANGOUARD, Nicolas, et al.
Abstract
We show that single-photon entangled states of the form |0>|1>+|1>|0> can be purified with a simple linear-optics based protocol, which is eminently feasible with current technology.
Besides its conceptual interest, this result is relevant for attractive quantum repeater protocols.
SANGOUARD, Nicolas, et al . Purification of single-photon entanglement with linear optics.
Physical Review. A , 2008, vol. 78
DOI : 10.1103/PhysRevA.78.050301
Available at:
http://archive-ouverte.unige.ch/unige:1109
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Purification of single-photon entanglement with linear optics
Nicolas Sangouard,1,
*
Christoph Simon,2,*
Thomas Coudreau,1 and Nicolas Gisin21Laboratoire MPQ, UMR CNRS 7162, Université Paris 7, France
2Group of Applied Physics, University of Geneva, Switzerland 共Received 14 April 2008; published 13 November 2008兲
We show that single-photon entangled states of the form兩0典兩1典+兩1典兩0典can be purified with a simple linear- optics-based protocol, which is eminently feasible with current technology. Besides its conceptual interest, this result is relevant for attractive quantum repeater protocols.
DOI:10.1103/PhysRevA.78.050301 PACS number共s兲: 03.67.Pp, 03.65.Ud, 03.67.Hk Single-particle entanglement of the form 兩1典A兩0典B
+兩0典A兩1典B may be the simplest form of entanglement关1兴. It corresponds to a single particle that is in a superposition state of being in location共mode兲Aand of being in locationB. The particle can be a freely propagating photon as in proposals for linear-optics quantum computing 关2兴, but also, e.g., a single excitation in an atomic-ensemble-based quantum memory—i.e., a stored photon 关3兴. Single-photon entangle- ment has been teleported experimentally in a purely photonic experiment关4兴. Single-excitation entanglement in atomic en- sembles has also been used to implement the basic segment of a quantum repeater 关5兴. Note that in principle single- photon states of the form兩1典A兩0典B+兩0典A兩1典B can furthermore be converted into two-atom entangled states of the form 兩e典A兩g典B+兩g典A兩e典B共cf.关1兴兲, which are also often used in quan- tum communication schemes关6兴.
Entanglement purification is an important concept in quantum information. It was introduced in Ref. 关7兴, which showed that it is possible to convert two copies of a less entangled state into one copy of a more entangled state using only local operations and classical communication. The first entanglement purification protocols关7,8兴were formulated in terms of qubits and quantum gates; in particular, they re- quired controlled-NOT 共CNOT兲gates. Both for practical ap- plications and from a conceptual point of view, it is of great interest to look for implementations that are as simple as possible. For example, Ref. 关9兴 proposed a method for the purification of polarization-entangled photon pairs that could be realized with linear optical elements共without the need for
CNOT gates兲. This procedure was adapted to parametric down-conversion sources in Ref. 关10兴, leading to an experi- mental realization关11兴.
An important domain of application for entanglement pu- rification is in the context of long-distance quantum commu- nication. The direct distribution of quantum states is limited by the problem of photon loss in transmission. This can be overcome using quantum repeater protocols 关12兴 based on the creation and storage of entanglement for moderate- distance elementary links, followed by entanglement swap- ping. In practice, these operations cannot be performed with perfect fidelity, limiting the number of links that can be used.
This number and thus the achievable distance can be greatly increased if entanglement purification is used.
The linear-optics purification protocol of Ref.关9兴can be readily integrated into quantum repeater protocols that are based on photon-pair entanglement 关13,14兴. However, there are other very attractive protocols based on single-photon detections关15,16兴; see, e.g., Ref.关5兴for a recent related ex- periment. While being slower than the best known protocol 关14兴, they are rather simple and require significantly fewer resources to outperform the direct transmission of photons.
They will thus be easier to implement in the short term and may well be the first repeater protocols achieving a genuine advantage compared to direct transmission. 共More detailed resource counts are given in Ref.关14兴.兲As a consequence of their relative simplicity, protocols based on single-photon de- tections are also significantly less sensitive to imperfections, such as non-unit-memory readout or photon detection effi- ciencies, as compared with protocols based on two-photon detections. The main drawback of protocols based on single- photon detections is that, unlike protocols based on two- photon detections, they are interferometrically sensitive to path length fluctuations 关17兴. Purification of single-photon entanglement is thus particularly important in this context.
However, to our knowledge, no purification procedure for single-photon entanglement has been proposed so far.
Here we show that single-photon entanglement purifica- tion can be realized in a very simple way using only linear- optical elements and photon detectors. Suppose that Alice and Bob ideally want to share a maximally entangled state
+
ab=冑12共兩1典A兩0典B+兩0典A兩1典B兲=冑12共a†+b†兲兩0典, but that there are phase errors due, e.g., to channel length fluctuations, such that without entanglement purification they can only distrib- ute copies of the state
ab=F兩+ ab典具+
ab兩+共1 −F兲兩− ab典具−
ab兩, 共1兲
with 12⬍F⬍1, where −ab=冑12共a†−b†兲兩0典 is the state after a phase error 关18兴; F=12 corresponds to the case where all phase information has been lost and no entanglement is left.
Note that losses and phase errors are the most significant practical limitations in the present context. In the context of quantum repeaters, vacuum 共兩0A典兩0B典兲 and multiphoton 共兩1A典兩1B典兲 components are also relevant errors. However, vacuum components do not decrease the fidelity of the dis- tributed state since the final measurement in schemes based on single-photon detections post-selects the cases where there was a photon in the output. For multiphoton compo- nents, one can use a specific architecture based on single-
*These authors contributed equally to this work.
photon sources which does not create multiphoton compo- nents in the elementary link, making it very efficient关16兴.
We now show how, starting from two copies of the state, Eq.共1兲, one can create one copy of a state of the same form with fidelityF˜⬎F. We look for a purification protocol that has the simple form shown in Fig. 1. Alice and Bob both perform a linear unitary transformation on their two modes 共a1, a2 and b1, b2, respectively兲, and then they each detect one of the output modes共da anddb兲. The goal is to have a higher-fidelity single-photon entangled state in modes˜a and
˜b; cf. Fig.1. The events of interest are thus those where one photon is detected in eitherdaordb, leaving the other photon in the modes˜aand˜b. The most general form for the matrix describing Alice’s transformation is
U共,,兲=
冋
cossineeii − sincosee−i−i,册
, 共2兲with兵a1†兩0典,a2†兩0典其=U兵a˜†兩0典,da†兩0典其. Bob can choose a differ- ent transformation, denoted V共⬘,⬘,⬘兲, which is obtained fromUby replacing the arguments by共⬘,⬘,⬘兲.
We now determine the state of the output modesa˜ and˜b conditional on the detection of a single photon in mode da
and of zero photons in mode db. The initial state
a1b1丢a2b2can be seen as a probabilistic mixture of the four pure states +a1b1丢+a2b2, −a1b1丢+a2b2, +a1b1丢−a2b2, and
− a1b1
丢− a2b2
with probabilities F2, F共1 −F兲, F共1 −F兲, and 共1 −F兲2 respectively. These states give rise to conditional states 21共Aa˜†+B−˜b†兲兩0典, 12共Aa˜†−B+˜b†兲兩0典, 12共Aa˜†+B+b˜†兲兩0典, and 21共Aa˜†−B−˜b†兲兩0典, respectively, where A= cos 2 and B⫾= cose−icos⬘ei⬘⫾sine−isin⬘ei⬘are functions of the matrix elements ofUandV. Based on these formulas one can calculate the trace of the conditional state˜ab,
tr˜ab=A2
4 +F2+共1 −F兲2
4 兩B−兩2+F共1 −F兲
2 兩B+兩2, 共3兲
and its fidelity with respect to the desired state+ab,
F˜=1
2+ A关F2−共1 −F兲2兴Re共B−兲
A2+关F2+共1 −F兲2兴兩B−兩2+ 2F共1 −F兲兩B+兩2. 共4兲 The case of one photon detected in db and no photon inda
leads to analogous expressions; it is sufficient to exchange the roles of the matrix elements ofUandV共i.e., primed and unprimed angles兲.
Our goal is to find the transformationsUandVthat maxi- mize the output fidelity, Eq.共4兲. One can see that the optimal fidelity is achieved forB+= 0 and Im共B−兲= 0. Both these con- ditions can be satisfied choosing =⬘==⬘= 0 and
⬘=−2, giving B−= sin 2. Due to the above-mentioned symmetry, this choice also maximizes the output fidelity for a detection in db. Note that the optimum choice of transfor- mations U and V is asymmetric—i.e., U⫽V. The fidelity now depends on the single parameter:
F˜ =1
2+ 关F2−共1 −F兲2兴tan 2
1 +关F2+共1 −F兲2兴tan22. 共5兲 For situations where the fidelity of the initial states is known, one can use Eq.共5兲to optimize the parameteras a function of F. One finds
tan 2opt= 1
冑
F2+共1 −F兲2, 共6兲 giving the optimized output fidelityF˜
opt=1
2 + F2−共1 −F兲2
2
冑
F2+共1 −F兲2; 共7兲 cf. Fig.2.One can show that this is the optimal fidelity that can be achieved even if one admits auxiliary input modes that are initially empty. In this more general case the fidelity is still given by Eq. 共4兲, but A, B+, and B− now depend on the relevant coefficients of a larger unitary matrix. However, op- timizing Eq.共4兲as ifA,B+, andB−were independent vari- ables, one finds the same optimum expression, Eq.共7兲. This FIG. 1. 共Color online兲Scheme for entanglement purification of
single-photon entanglement. Alice and Bob share two entangled single-photon states a1b1 and a2b2 with fidelity F. Both parties apply local unitary linear-optical transformationsUand Vto their respective modes. For appropriately chosenUandV, the detection of one photon in either da or db projects modes˜a and b˜ into a single-photon entangled state with higher fidelity.
FIG. 2. 共Color online兲 Output fidelity as a function of input fidelity for the optimized protocol, where the parameteris adapted to the input fidelity共F˜
opt, solid line兲, and for a simplified protocol where=8 for all input fidelities共F˜
1, dashed line兲. As a reference, the straight lineF˜=Fis also shown共dotted line兲.
SANGOUARDet al. PHYSICAL REVIEW A78, 050301共R兲 共2008兲
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implies that Eq. 共7兲 cannot be improved for any number of auxiliary empty modes. The situation may be different for auxiliary photons.
The success probability for the optimized protocol as de- scribed above ispopt= 2⫻tr˜absince both cases共detection in da, no detection in db and vice versa兲contribute. One finds
popt= F2+共1 −F兲2
1 +F2+共1 −F兲2. 共8兲 For situations where the input fidelityFis not known a pri- ori, one has to choose a value ofthat is independent ofF.
For example, ifFis unknown, but expected to be close to 1, one could choose the value of optfor F= 1, which gives
=8. For this simplified protocol one finds an output fidelity F˜
1= F2+F
1 +F2+共1 −F兲2 共9兲 and a success probability
p1=1 +F2+共1 −F兲2
4 , 共10兲
which is close to 1/2 forF close to 1. Figure 2 shows the output fidelitiesF˜
optandF˜
1both for the optimal protocol and for the simplified protocol. Both protocols achieve substan- tial purification, and the simplified protocol is almost as ef- fective as the optimal one. It is interesting to consider the regime of high fidelities. For F= 1 −⑀ both F˜opt andF˜
1 are equal to 1 −⑀2+O共⑀2兲; i.e., the purification protocol divides the error by a factor of 2. Note that in quantum repeater protocols the error is approximately doubled with every level of entanglement swapping. This means that the present pro- tocol has the potential of significantly increasing the number of possible levels and, thus, the achievable total distance.
The proposed protocols are very feasible with current technology. For example, for the simplified protocol, Alice’s linear operationUsimply corresponds to a beam splitter with 共amplitude兲transmission coefficient cos8, corresponding to an intensity transmission of 85%, and Bob’s operationVto a beam splitter with amplitude transmission coefficient cos−38 , corresponding to an intensity transmission of 15%.
The optimized protocol requires beam splitters with varying transmission depending on the input fidelity F. Note that if the modes a1 anda2 共and analogously b1 andb2兲 are con- verted to the polarization states of a single spatial mode关5兴, then the required generalized beam splitters are very easy to realize combining wave plates and polarizing beam splitters.
The protocol relies on single-photon interference and on the bosonic character of indistinguishable photons. Single- photon interference is equivalent to classical interference and can be performed with extremely high precision. The most significant experimental challenge is likely to be the genera- tion of highly indistinguishable photons. Their degree of in- distinguishability can be quantified by their mode overlap, which corresponds experimentally to the visibility V of the
“Hong-Ou-Mandel dip” 关19兴—i.e., the extent to which the two photons “bunch” after a 50-50 beam splitter 共V= 1 cor- responds to perfect bunching兲. Reference关20兴has reported a
very impressive visibility V= 0.994, which is largely suffi- cient for high-fidelity purification; cf. Fig. 3. The result of Ref.关20兴was achieved for photons from the same pair for a parametric down-conversion source, which is likely to be the most promising system for initial demonstrations of the pro- posed protocol. In general, Fig. 3 shows that purification is possible for dip visibilities that should be achievable in a wide range of systems.
The protocol in its ideal form requires highly efficient photon-number-resolving single-photon detectors. We now discuss the impact of nonperfect detection efficiency. The nondetection of a second photon that is actually present in modedaor db will lead to a vacuum component in modes˜a andb˜. However, it will not reduce the fidelity for the single- photon component of the output. If the purification protocol is applied in the context of the quantum repeater schemes of Refs. 关15兴, the vacuum component leads to a lower success probability for the subsequent entanglement swapping steps and thus to a lower entanglement generation rate. On the other hand, the fidelity of the generated long-distance en- tanglement only depends on the fidelity of the single-photon component 共and thus is independent of the detection effi- ciency兲, since the final measurement in these schemes post- selects the cases where there was a photon in the output.
Note that highly efficient detectors are being developed; e.g., Ref.关21兴has recently reported photon-number-resolving de- tectors with 95% efficiency at 1556 nm, a wavelength ideally suited for long-distance transmission in optical fibers.
The proposed scheme could be demonstrated both in purely photonic experiments and in experiments involving quantum memories for photons as an important further step towards the realization of a quantum repeater. The first type of experiment, in addition to the components discussed above, just requires two sources of single photons, which could be realized conditionally based on parametric down- conversion关22兴or directly using quantum dots关23兴or single atoms关24兴.
The second type of experiment would be very similar to the experiment of Ref. 关5兴. Single-particle entanglement would first be created between two pairs of atomic en- sembles. The stored excitations are then reconverted into photons, combined on linear optical elements as described above and detected. The conditions for the successful real-
0.6 0.7 0.8 0.9 1.0
0.01 0.01 0.02 0.03 0.04 0.05
FIG. 3.共Color online兲Fidelity improvementF˜
1−Fas a function ofFfor Hong-Ou-Mandel dip visibilitiesV= 1, 0.99, and 0.98共top to bottom兲.
ization of the purification scheme proposed here, in particu- lar the indistinguishability of the photons emitted by differ- ent ensembles, are similar as for the experiment of Ref.关5兴.
Interesting further experimental steps would be to reabsorb the purified delocalized photon in a memory and, of course, to increase the distance between the two parties.
In conclusion, we have presented a very simple scheme for the purification of single-photon entanglement which is realizable with current technology. We have furthermore shown that the scheme achieves the optimal fidelity for any number of auxiliary vacuum modes. We find the simplicity
of the scheme remarkable from a conceptual point of view. It also constitutes important progress for the implementation of quantum repeaters based on single-particle entanglement 关15兴.
We thank J.-D. Bancal, C. Branciard, N. Brunner, R.
Dubessy, H. de Riedmatten, and H. Zbinden for helpful com- ments. This work was supported by the EU Integrated Project “Qubit Applications,” the Swiss NCCR “Quantum Photonics,” and the French National Research Agency 共ANR兲Project No. ANR-JC05គ61454.
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