Scattering theory of plasmon-assisted entanglement transfer and
distillation
Velsen, J.L. van; Tworzydlo, J.; Beenakker, C.W.J.
Citation
Velsen, J. L. van, Tworzydlo, J., & Beenakker, C. W. J. (2003). Scattering theory of
plasmon-assisted entanglement transfer and distillation. Physical Review A, 68, 043807.
doi:10.1103/PhysRevA.68.043807
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Scattering theory of plasmon-assisted entanglement transfer and distillation
J. L. van Velsen,1 J. Tworzydło,1,2and C. W. J. Beenakker1
1
Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands 2Institute of Theoretical Physics, Warsaw University, Hoz˙a 69, 00-681 Warszawa, Poland
共Received 18 November 2002; published 8 October 2003兲
We analyze the quantum-mechanical limits to the plasmon-assisted entanglement transfer observed by Al-tewischer, van Exter, and Woerdman关Nature 418, 304 共2002兲兴. The maximal violation S of Bell’s inequality at the photodetectors behind two linear media 共such as the perforated metal films in the experiment兲 can be described by two ratio’s1,2of polarization-dependent transmission probabilities. A fully entangled incident
state is transferred without degradation for 1⫽2, but a relatively large mismatch of 1 and 2 can be
tolerated with a small reduction of S. We predict that fully entangled Bell pairs can be distilled out of partially entangled radiation if1and2satisfy a pair of inequalities.
DOI: 10.1103/PhysRevA.68.043807 PACS number共s兲: 42.50.Dv, 03.67.Mn, 03.65.Ud, 73.20.Mf
The motivation for this work came from the recent re-markable demonstration by Altewischer, van Exter, and Woerdman of the transfer of quantum-mechanical entangle-ment from photons to surface plasmons and back to photons 关1兴. Since entanglement is a highly fragile property of a two-photon state, it came as a surprise that this property could survive, with little degradation, the conversion to and from the macroscopic degrees of freedom in a metal 关2兴.
We present a quantitative description of the finding of Ref.关1兴 that the entanglement is lost if it is measured during transfer, that is to say, if the medium through which the pair of polarization-entangled photons is passed acts as a ‘‘which-way’’ detector for polarization. Our analysis explains why a few percent degradation of entanglement could be realized without requiring a highly symmetric medium. We predict that the experimental setup of Ref.关1兴 could be used to ‘‘dis-till’’关3,4兴 fully entangled Bell pairs out of partially entangled incident radiation, and we identify the region in parameter space where this distillation is possible.
We assume that the medium is linear, so that its effect on the radiation can be described by a scattering matrix. The assumption of linearity of the interaction of radiation with surface plasmons is central to the literature on this topic 关5–9兴. We will not make any specific assumptions on the mode and frequency dependence of the scattering matrix, but extract the smallest number of independently measurable pa-rameters needed to describe the experiment. By concentrat-ing on model-independent results we can isolate the funda-mental quantum-mechanical limitations on the entanglement transfer, from the limitations specific to any particular trans-fer mechanism.
The system considered is shown schematically in Fig. 1. Polarization-entangled radiation is scattered by two objects and detected by a pair of detectors behind the objects in the far field. The objects used in Ref. 关1兴 are metal films perfo-rated by a square array of subwavelength holes. The trans-mission amplitude t⬘,iof object i⫽1,2 relates the
transmit-ted radiation 共with polarization ⫽H,V) to the incident radiation共polarization
⬘
⫽H,V). We assume a single-mode incident beam and a single-mode detector共smaller than the coherence area兲 so that we require a set of eight transmission amplitudes t⬘,i out of the entire scattering matrix 共whichalso contains reflection amplitudes and transmission
ampli-tudes to other modes兲. The extension to a multimode theory 共needed to describe some aspects of the experiment 关1兴兲 is left for a future investigation关10兴. We do not require that the scattering matrix be unitary, so our results remain valid if the objects absorb part of the incident radiation.
The radiation incident on the two objects is in a known, partially entangled state and we wish to determine the degree of entanglement of the detected radiation. It is convenient to use a matrix notation. The incident two-photon state has the general form 兩⌿in
典
⫽aHH in 兩HH典
⫹a HV in 兩HV典
⫹a VH in 兩VH典
⫹a VV in 兩VV典
. 共1兲 The four complex numbers a⬘ in form a matrix Ain⫽
冉
aHHin aHVin aVHin aVVin冊
. 共2兲FIG. 1. Main plot: efficiency of the entanglement transfer for a fully entangled incident state, as given by Eq. 共14兲. The maximal violation Smaxof Bell’s inequality at the photodetectors is plotted as
a function of the ratio1/2⫽T1⫹T2⫺/T1⫺T2⫹of the
Normalization of兩⌿in
典
requires Tr AinAin†⫽1, with ‘‘Tr’’be-ing the trace of a matrix.
The four transmission amplitudes t⬘,i of object i⫽1,2
form the matrix
Ti⫽
冉
tHH,i tHV,i tVH,i tVV,i
冊
. 共3兲
The transmitted two-photon state兩⌿out
典
has matrix ofcoef-ficients
Aout⫽Z⫺1/2T1AinT2
t
, 共4兲
with normalization factor
Z⫽Tr 共T1AinT2
t兲共T
1AinT2
t兲†. 共5兲
共The superscript ‘‘t’’ denotes the transpose of a matrix.兲 We quantify the degree of entanglement in terms of the Clauser-Horne-Shimony-Holt parameter S 关11兴, which mea-sures the maximum violation of Bell’s inequality and was used in the experiment of Ref. 关1兴. This parameter can be obtained from a decomposition of兩⌿
典
into a superposition of a fully entangled state共with weight冑
P) and a factorizedstate orthogonal to it 关12,13兴. The relation is
S⫽2
冑
1⫹P2, P2⫽4DetAA†, 共6兲 with ‘‘Det’’ being the determinant and 0⭐P⭐1. 共The con-currence 关14兴 is identical to P.兲 A fully entangled state hasP⫽1, S⫽2
冑
2, while a factorized state has P⫽0, S⫽2. The fully entangled state could be the Bell pair (兩HV典
⫺兩VH典
)/冑
2, or any state derived from it by a local unitary transformation (A→UAV with U,V arbitrary unitary matri-ces兲. The degree of entanglement Pin⫽2兩Det Ain兩 of thein-cident state is given and we seek the degree of entanglement
Pout⫽2兩Det Aout兩 of the transmitted state. We are particularly
interested in the largest Poutthat can be reached by applying
local unitary transformations to the incident state. This would correspond to the experimental situation that the po-larizations of the two incoming photons are rotated indepen-dently, in order to maximize the violation of Bell’s inequality of the detected photon pair.
Before proceeding with the calculation we introduce some parametrizations. The Hermitian matrix product TiTi
† has the eigenvalue-eigenvector decomposition T1T1 †⫽U†
冉
T1⫹ 0 0 T1⫺冊
U, T2T2 †⫽V†冉
T2⫹ 0 0 T2⫺冊
V. 共7兲 The matrices of eigenvectors U,V are unitary and the trans-mission eigenvalues Ti⫾ are real numbers between 0 and 1.We order them such that 0⭐Ti⫺⭐Ti⫹⭐1 for each i⫽1,2.
We will see that the maximal entanglement transfer depends only on the ratiosi⫽Ti⫹/Ti⫺. This parametrization there-fore extracts the two significant real numbers 1,2 out of
eight complex transmission amplitudes. The Hermitian ma-trix product AinAin † has eigenvalues ⫾⫽12⫾ 1 2(1⫺Pin 2)1/2.
These appear in the polar decomposition
UAinV⫽ei
冉
u⫹ u⫺ ⫺u⫺* u⫹*冊
冉
冑
⫹ 0 0冑
⫺冊
冉
v⫹ v⫺ ⫺v⫺* v⫹*冊
. 共8兲 The phase is real and u⫾,v⫾ are complex numbers con-strained by 兩u⫾兩⫽(1 2⫾u)1/2, 兩v⫾兩⫽( 1 2⫾v)1/2, with real u,v苸(⫺12, 12). These numbers can be varied by local unitary
transformations, so later on we will want to choose values which maximize the detected entanglement.
With these parametrizations a calculation of the determi-nant of Aout leads to the following relation between Pinand
Pout: Pout⫽ Pin
冑
12 共1⫺1兲共2⫺1兲冋
⫹Q⫹⫹⫺Q⫺⫺2冑
⫹⫺ ⫻冉
1 4⫺u 2冊
1/2冉
1 4⫺v 2冊
1/2 cos⌽册
⫺1 , 共9兲 Q⫾⫽冉
u⫾1 2 1⫹1 1⫺1冊冉
v⫾1 2 2⫹1 2⫺1冊
. 共10兲The phase⌽ equals the argument of u⫹u⫺*v⫹v⫺. To maxi-mize Pout we should choose⌽⫽0.
We first analyze this expression for the case of a fully entangled incident state, as in the experiment of Ref.关1兴. For
Pin⫽1 one has ⫹⫽⫺⫽1/2, and Eq. 共9兲 simplifies to
Pout⫽ 4
冑
12 共1⫹1兲共2⫹1兲⫹4a共1⫺1兲共2⫺1兲 , 共11兲 a⫽uv⫺冉
1 4⫺u 2冊
1/2冉
1 4⫺v 2冊
1/2 cos⌽. 共12兲Since i⭓1 and 兩a兩⭐1
4 we conclude that the degree of
en-tanglement is bounded by Pmin⭐Pout⭐Pmax, with
Pmin⫽ 2
冑
12 1⫹12 , Pmax⫽ 2冑
1/2 1⫹1/2 . 共13兲The maximum Pmax can always be reached by a proper
choice of the共fully entangled兲 incident state, so the maximal violation of Bell’s inequality is given by
Smax⫽2
冑
1⫹41/2
共1⫹1/2兲2
. 共14兲
The dependence of Smaxon 1/2 is plotted in Fig. 1. Full entanglement is obtained for 1⫽2, hence for T1⫹T2⫺
⫽T1⫺T2⫹. Generically, this requires either identical objects
(T1⫾⫽T2⫾) or nonidentical objects with Ti⫹⫽Ti⫺. If 1
⫽2 there are no which-way labels and entanglement fully
survives with no degradation.
Small deviations of1/2 from unity only reduce the
en-tanglement to second order:
Smax⫽2
冑
2关1⫺1
16共1/2⫺1兲2⫹O共1/2⫺1兲3兴. 共15兲
van VELSEN, TWORZYDŁO, AND BEENAKKER PHYSICAL REVIEW A 68, 043807 共2003兲
So for a small reduction of the entanglement one can tolerate a large mismatch of the transmission probabilities. In par-ticular, the experimental result S⫽2.71 for plasmon-assisted entanglement transfer 关1兴 can be reached with more than a factor two of mismatch (S⫽2.71 for1/2⫽2.4).
As a simple example we calculate the symmetry param-eter 1/2 for a Lorentzian transmission probability, appro-priate for plasmon-assisted entanglement transfer关5–9兴. We take
Ti⫾⫽
T ⌫2
共0⫺i⫾兲2⫹⌫2
, 共16兲
where0 is the frequency of the incident radiation,⌫ is the
linewidth, and T is the transmission probability at the reso-nance frequency i⫾. 共For simplicity we take
polarization-independent ⌫ and T.兲 The transmission is through an
opti-cally thick metal film with a rectangular array of
subwavelength holes共lattice constants Li⫾). The dispersion
relation of the surface plasmons is i⫾⫽(1
⫹1/⑀)1/22nc/Li⫾关9兴, where⑀is the real part of the
dielec-tric constant and n is the order of the resonance, equal to the number of plasmon-field oscillations in a lattice constant. We break the symmetry by taking one square array of holes and one rectangular array 共lattice constants L0⫽L1⫹⫽L2⫹
⫽L2⫺and L1⫽L1⫺). The lattice constant L0is chosen such
that the incident radiation is at resonance. The symmetry parameter becomes 1 2⫽1⫹共2兲 2
冉
nl L0 ⫺nl L1冊
2 , l⫽c ⌫冑
⑀⫹1 ⑀ . 共17兲The length l is the propagation length of the surface plasmon. 关We have taken c(1⫹1/⑀)1/2for the plasmon group velocity, valid if 0 is not close to the plasma frequency关9兴.兴
Com-bining Eqs. 共15兲 and 共17兲 we see that the deviation of Smax
from 2
冑
2 共the degradation of the entanglement兲 is propor-tional to the fourth power of the difference between the num-ber of oscillations of the plasmon field along the two lattice vectors.Turning now to the more general case of a partially en-tangled incident state, we ask the following question: Is it possible to achieve Pout⫽1 even if Pin⬍1? In other words, can one detect a 2
冑
2 violation of Bell’s inequality after transmission even if the original state was only partially en-tangled? Examination of Eq. 共9兲 shows that the answer to this question is: Yes, provided1 and2 satisfy冏
ln12
冏
⭐2 arcosh共Pin⫺1兲 and ln
12⭓2 arcosh共Pin⫺1兲.
共18兲 The allowed values of1 and2 lie in a strip that is open at
one end, see Fig. 2. The boundaries are reached at兩u兩⫽兩v兩 ⫽1
2. The region inside the strip is reached by choosing both
兩u兩 and 兩v兩⬍1/2. For Pin⫽1 the strip collapses to the single
line 1⫽2, in agreement with Eq.共13兲.
The possibility to achieve Pout⫽1 for Pin⬍1 is an
ex-ample of distillation of entanglement 关4兴. The distillation method used here is the Procrustean method of Bennett et al. 关3兴. It requires only local linear filters 共the metal films in our case兲 and classical communication 共the coincidence counter兲. See Ref.关15兴 for an experimental realization and Refs. 关16– 20兴 for other distillation schemes. As it should, no entangle-ment is created in this operation. Out of N incoming photon-pairs with entanglement Pin one detects NZ pairs with
entanglement Pout⫽PinZ⫺1
冑
T1⫹T1⫺T2⫹T2⫺, so thatNZ Pout⭐NPin.
In conclusion, we have shown that optical entanglement transfer and distillation through a pair of linear media can be described by two ratios 1 and2 of polarization-dependent
transmission probabilities. For fully entangled incident radia-tion, the maximal violation of Bell’s inequality at the detec-tors is given by function 共14兲 of 1/2 which decays only
slowly around the optimal value 1/2⫽1. Distillation of a
fully entangled Bell pair out of partially entangled incident radiation is possible no matter how low the initial entangle-ment, provided that 1 and 2 satisfy the two inequalities 共18兲.
Our results provide a simple way to describe the experi-ment of Ref. 关1兴 on plasmon-assisted entanglement transfer, in terms of two separately measurable parameters. By chang-ing the square array of holes used in Ref. 关1兴 into a rectan-gular array 共or, equivalently, by tilting the square array rela-tive to the incident beam兲, one can move away from the point1⫽2⫽1 and search for the entanglement distillation predicted here. The possibility of extracting Bell pairs by manipulating surface plasmons may have interesting applica-tions in quantum information processing.
This work was supported by the Dutch Science Founda-tion NWO/FOM and by the U.S. Army Research Office 共Grant No. DAAD 19-02-0086兲. J.T. acknowledges the fi-nancial support provided through the European Community’s Human Potential Program, under Contract No. HPRN-CT-2000-00144, Nanoscale Dynamics. We have benefitted from discussions with J. Preskill and J.P. Woerdman.
FIG. 2. The shaded strips indicate the values of ln1and ln2 for which Pout⫽1 can be reached with Pin⫽0.5 共horizontally shaded兲 and Pin⫽0.9 共vertically shaded兲, in accordance with Eq.
关1兴 E. Altewischer, M.P. van Exter, and J.P. Woerdman, Nature 共London兲 418, 304 共2002兲.
关2兴 W. Barnes, Nature 共London兲 418, 281 共2002兲.
关3兴 C.H. Bennett, H.J. Bernstein, S. Popescu, and B. Schumacher,
Phys. Rev. A 53, 2046共1996兲.
关4兴 M.A. Nielsen and I.L. Chuang, Quantum Computation and Quantum Information 共Cambridge University Press,
Cam-bridge, 2000兲.
关5兴 U. Schro¨ter and D. Heitmann, Phys. Rev. B 58, 15419 共1998兲. 关6兴 M.M.J. Treacy, Appl. Phys. Lett. 75, 606 共1999兲.
关7兴 J.A. Porto, F.J. Garcı´a-Vidal, and J.B. Pendry, Phys. Rev. Lett. 83, 2845共1999兲.
关8兴 L. Martı´n-Moreno, F.J. Garcı´a-Vidal, H.J. Lezec, K.M.
Pel-lerin, T. Thio, J.B. Pendry, and T.W. Ebbesen, Phys. Rev. Lett.
86, 1114共2001兲.
关9兴 H. Raether, Surface Plasmons 共Springer, Berlin, 1988兲. 关10兴 An essential complication which appears in the case of
multi-mode detection is that the detected polarization state is no longer a pure state, but is a mixed state that has to be described by a density matrix. In the case of single-mode detection con-sidered here, the only way in which a linear medium can lead
to loss of purity is by thermal fluctuations共either from the two objects or from the electromagnetic environment of the detec-tors兲. This thermal noise is insignificant at room temperature and optical frequencies.
关11兴 J.F. Clauser, M.A. Horne, A. Shimony, and R.A. Holt, Phys.
Rev. Lett. 23, 880共1969兲.
关12兴 A.F. Abouraddy, B.E.A. Saleh, A.V. Sergienko, and M.C.
Teich, Phys. Rev. A 64, 050101共2001兲.
关13兴 N. Gisin, Phys. Lett. A 154, 201 共1991兲.
关14兴 S. Hill and W.K. Wootters, Phys. Rev. Lett. 78, 5022 共1997兲;
W.K. Wootters, ibid. 80, 2245共1998兲.
关15兴 P.G. Kwiat, S. Barraza-Lopez, A. Stefanov, and N. Gisin,
Na-ture共London兲 409, 1014 共2001兲.
关16兴 C.H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J.A.
Smolin, and W.K. Wootters, Phys. Rev. Lett. 76, 722共1996兲.
关17兴 H.-W. Lee and J. Kim, Phys. Rev. A 63, 012305 共2000兲. 关18兴 D. Gottesman and J. Preskill, Phys. Rev. A 63, 022309 共2001兲. 关19兴 T. Yamamoto, M. Koashi, and N. Imoto, Phys. Rev. A 64,
012304共2001兲.
关20兴 J.-W. Pan, C. Simon, C. Brukner, and A. Zeilinger, Nature 共London兲 410, 1067 共2001兲.
van VELSEN, TWORZYDŁO, AND BEENAKKER PHYSICAL REVIEW A 68, 043807 共2003兲
