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arXiv:1212.5115v1 [quant-ph] 20 Dec 2012
Qudit-Teleportation for photons with linear optics
Sandeep K. Goyal,1, 2, ∗Patricia E. Boukama-Dzoussi,1Sibasish Ghosh,2Filippus S. Roux,3and Thomas Konrad1, 4
1School of Chemistry and Physics, University of KwaZulu-Natal, Durban, South Africa
2Optics and Quantum Information Group, The Institute of Mathematical Sciences, CIT campus, Chennai 600 113, India
3CSIR National Laser Centre, PO Box 395, Pretoria 0001, South Africa
4National Institute of Theoretical Physics (Durban Node), South Africa
Quantum Teleportation, the transfer of the state of one quantum system to another without
direct interaction between both systems, is an important way to transmit information encoded in
quantum states and to generate quantum correlations (entanglement) between remote quantum
systems. So far, for photons only superpositions of two distinguishable states (one ”qubit”) could
be teleported . We here show how to teleport a ”qudit”, i.e., a superposition of an arbitrary number
dof distinguishable states present in the orbital angular momentum of a single photon using d
beam splitters and dadditional entangled photons. The same entanglement resource might also be
employed to collectively teleport the state of d/2 photons at the cost of one additional entangled
photon per qubit. This is superior to existing schemes for qubits which require an additional pair
of entangled photons per qubit.
In classical physics it is possible, in principle, to de-
tect the state of a single system, for example the po-
sition and momentum of a point particle, transmit the
information about that state to a remote location and
then reconstruct it within a second system. This con-
cept of “classical teleportation” underlies telecommuni-
cation techniques such as the transfer of documents via
facsimile. Quantum physics, however, excludes the pos-
sibility to detect or duplicate the state of a single micro-
scopic system1and therefore rules out all forms of classi-
cal teleportation with atoms, photons or other quantum
systems. It is thus surprising2, that the state transfer
between quantum systems can nevertheless be realized
according to the rules of quantum physics by means of
“quantum teleportation”3. This procedure makes use of
correlations between quantum systems - entanglement -
which cannot be described by local-realistic theories4,
such as classical mechanics or electrodynamics nor any
other theory within classical physics.
Quantum teleportation lies at the core of quantum
communication which is the quantum analog of telecom-
munication, and can also be employed to enhance the suc-
cess probability in quantum computing with photons5–8 .
Moreover, it is one of the crucial ingredients9,10 for en-
abling long-distance quantum cryptography - a technique
to transmit information secured against eavesdropping.
The importance of quantum teleportation for quan-
tum information processing and communication can be
seen from the long list of experimental realizations of
teleportation of the state of a two-level system corre-
sponding to the smallest unit of quantum information
- one quantum bit (qubit),11–23. In these realizations
single qubits were encoded in the polarization of pho-
tons or in the superposition of vacuum and one photon
states21. Quantum teleportation with two-level atoms
has been demonstrated24–26 and it has also been designed
for three- and four-level atomic systems27,28.
At present, light is the only candidate for quantum
communication and quantum cryptography over large
distances because of its small interaction with its envi-
ronment as compared to matter29 . At the same time
its small interaction makes it difficult to manipulate the
states of light in order to achieve teleportation with pho-
tons. There are two main challenges in realizing tele-
portation: (i) photons sharing maximal quantum cor-
relations (entanglement) have to be generated and dis-
tributed between the sender and the receiver of quantum
information and (ii) the input photons and the photons
of the sender have to be projected into a maximally en-
tangled state by a joint measurement of both (a so-called
Bell measurement) in order to transfer the state of the
input photons to the photons held by the receiver. Both
challenges can in principle be overcome using non-linear
optical media, which manipulate the light depending on
its intensity. The corresponding processes, however, have
a very small efficiency on the single photon level. For ex-
ample a non-linear Beta Barium Borate (BBO) crystal is
used to generate, from one photon, two photons of half
the input-frequency in an entangled state (challenge (i))
with a success probability of approximately 10−6per in-
coming photon30. The efficiency of a Bell measurement
by means of non-linear optics in order to meet challenge
(ii) is even lower17, at about 10−10. Therefore it is desir-
able to design a more efficient solution to both challenges
based on linear optics. Here we focus on challenge (ii)
and briefly discuss a solution of challenge (i) which will
be given explicitly in a subsequent article.
Although linear optics, that is, the use of beam split-
ters, phase shifters and mirrors, does not allow the re-
alization of a complete Bell measurement31, a simple
50 : 50 beam splitter can be used as a filter to pro ject
two incoming photons onto a particular entangled state
in a certain percentage of the cases. Two photons inci-
dent on the input ports of a beam splitter do not pro-
duce a coincidence count in two detectors in the output
ports (Hong-Ou-Mandel effect32,33) unless they possess
an anti-symmetric component with respect to their in-
ternal degree of freedom, e.g. their polarization. A coin-
cidence count thus effectively projects onto an antisym-
metric state. For two polarized photons entering in dif-
2
Propagation
Direction
FIG. 1: Schematic diagram of the helical wavefront of a light
beam.
ferent input ports of the beam splitter there is only one
such state:
|ψi=1
√2(|HV i − |V Hi),(1)
i.e, elementary excitations of the first and second spatial
mode (represented by the first and second slot in the state
symbol) which carry horizontal and vertical polarization,
respectively, superposed with excitations of these modes
with swapped polarizations. This state is antisymmet-
ric, because it changes sign under a permutation of the
first and second mode (slot), and maximally entangled,
a condition that allows the realization of the teleporta-
tion of a qubit11 encoded in the polarization of a single
photon. Moreover, as we shall see in the following, this
phenomenon can also be employed for the simultaneous
teleportation of multiple qubits encoded in the orbital
angular momentum (OAM) of photons.
It was noticed by Allen et al.34 in 1992 that light with
a phase distribution exp(ilφ) depending on the azimuthal
angle φin the plane orthogonal to its direction of prop-
agation carries an orbital angular momentum of an inte-
ger ltimes Planck’s constant ~per photon. Such light is
characterized by helical (screw-like) wavefronts (cp. Fig.
1) and can for example be produced by means of spa-
tial light modulators - thin liquid crystal displays (LCDs)
which imprint the helical phase pattern or superpositions
of such patterns. The orbital angular momentum of a
photon can thus be used to carry information and repre-
sents a quantum system with an unrestricted number of
levels.
Quantum teleportation using an incomplete Bell mea-
surement ( a “Bell filter”) can be applied to systems with
an arbitrary number of levels, not only simple two-level
systems (such as polarized photons). Let us review how.
Teleportation involves three parties Alice, Bob and Char-
lie, cp. Fig 2. Alice and Bob are far apart and share a
|χ>|χ>
Sψ
Fψ
A BC
FIG. 2: Teleportation using a filter: source Sψproduces a pair
of systems Aand Bin state |Ψi. System Aand system Care
to be projected by a filter Fψinto state |Ψi. If successful, the
filtering transfers the initial state of Cto system B.
pair of systems in a maximally entangled state
|ΨiAB =1
√D
D−1
X
i=0 |AiiA⊗ |BiiB,(2)
which is a superposition of products of orthogonal basis
states of their systems Aand B. Charlie provides Alice
with an unknown state |χi, which has to be transferred
from Charlie’s system Cto Bob’s system B. For this
purpose systems Cand Bmust be similar - Bhas to
support the same states as C. The state to be teleported
can thus be expressed as a superposition of Bob’s basis
states: |χi=PD−1
k=0 αk|Bki. Alice successfully teleports
the state |χiif she carries out a measurement which acts
like a filter and projects systems Cand Aonto the en-
tangled state |Ψi:
|χiC⊗ |ΨiAB →([|ΨihΨ|]CA ⊗11B)|χiC⊗ |ΨiAB
=1
D3/2
D−1
X
k,l=0
αk|BliC⊗ |AliA⊗ |BkiB
=1
D|ΨiCA ⊗ |χiB.(3)
According to the rules of quantum mechanics the likeli-
hood for such a projection to occur is given by the square
of the length of the resulting state vector, p= 1/D2.
Thus the success probability of this teleportation scheme,
which uses only one of the outcomes of a Bell measure-
ment, decreases with the number Dof basis states in
which quantum information is encoded. The advantage
lies in the fact that this concept of teleportation, which
is used for photonic qubits (D= 2) can be generalised to
teleport states with arbitrary Dusing linear optics. In
order to do so, we have to identify a photonic system with
a unique antisymmetric state and a linear optical device
that plays the role of the beam splitter in the qubit case,
i.e., a filter for antisymmetric states. The uniqueness is
required to guarantee that the filter yields the same state
|ΨiCA which is initially shared by Alice and Bob, i.e, the
state |ΨiAB. The dimension of the space spanned by the
antisymmetric states of composite systems (only if the
dimension equals one do we have a unique antisymmet-
ric state!) can be easily determined by means of Young
tableaux (see Supplementary Information). It turns out
that only dsystems each with dlevels posses a unique
antisymmetric state. As a consequence, a generalization
of the teleportation scheme for photonic qubits by means
3
C
A A A
1 2
(d−1)/d
d/(d+1) 1/2
d−1
FIG. 3: Bell filter: array of dbeam splitters which projects
Charlie’s incoming photon and Alice’s d−1 photons into the
antisymmetric state |Ψi. Below each beamsplitter it’s reflec-
tivity is indicated.
of a Bell filter for antisymmetric states requires dpho-
tons propagating on different paths each with a quantized
degree of freedom, e.g OAM, with d-levels, i.e., dqudits.
Photonic states can be conveniently expressed by
means of creation operators a†acting on the vacuum
state |0i. In our case these operators carry two indices
— the first one, j, specifies one of dpossible propagation
paths whereas the second one, l= 1 ...d, denotes the
orbital angular momentum value l~of the photon. For
example the state |ψi=|12i−|21iof two photons prop-
agating on different paths with two OAM values l= 1,2
– which is the OAM analog of the polarization state in
Eq. (1) – can be written by means of a determinant of
creation operators:
|ψi=1
√2a†
11a†
22 −a†
12a†
21|0i=1
√2det a†
11 a†
12
a†
21 a†
22 |0i
(4)
It is obvious that the state |ψiis antisymmetric, since
under permutation of the propagation paths it is trans-
fered to −|ψi. The antisymmetry is represented by the
determinant: a swap of rows corresponding to the permu-
tation results in a minus sign of the determinant. Using
the same logic the antisymmetric state of dphotons with
dOAM values can be expressed by the determinant of a
d×dMatrix Λ
|Ψi=1
√d!det(Λ)|0i(5)
where
Λ =
a†
11 a†
12 ··· a†
1d
a†
21 a†
22 ··· a†
2d
.
.
..
.
.....
.
.
a†
d1a†
d2··· a†
dd
(6)
Here |Ψiis the antisymmetric state which was sought,
since a permutation of any two propagation directions
corresponding to a swap of two rows of the determinant
in Eq. (5) leads to a change of sign of the state |Ψi. It
turns out that a certain array of dbeam splitters (cp.
Fig. 3)– acts as a Bell filter and projects onto |Ψiin case
of a coincidence count at all output ports, which can be
checked for any finite dimension dby direct calculation.
Our notation makes it a simple matter to understand the
reason. Since the beam splitters only superimpose the in-
coming light fields (first index of the creation operators)
independent of their orbital angular momenta (second in-
dex), the resulting linear transformation can be expressed
by a rotation matrix Uacting on the creation operators
˜a†
kj =XUkia†
ij ,(7)
which leads to a linear transformation of the matrix Λ
˜
Λ = UΛ (8)
and the invariance of the antisymmetric state: |Ψi →
det(˜
Λ)|0i= det(U) det(Λ)|0i= det(Λ)|0i=|Ψi. Hence
one photon will leave from each of the doutput ports
of the multiport thus leading to a possible coincidence
count.
In order to use the antisymmetric state given in Eq.
(5), the dphotons must be divided between Alice (npho-
tons) and Bob (d−nphotons) such that both share a
bipartite maximally entangled state. If Charlie now pro-
vides Alice with d−nphotons, she can teleport the state
of these photons by projecting them onto an antisym-
metric state by means of a balanced multiport. In order
to check for maximal entanglement we once more take
advantage of the representation of |Ψiin terms of a de-
terminant. Expanding the determinant with respect to
the first row, we obtain the state:
|Ψi=1
√d!det(Λ)|0i=1
√d!
d
X
i=1
(−1)i+1a†
1idet(Λ1i)|0i
=1
√d
d
X
i=1 |Aii|ii,(9)
where |Aii= 1/p(d−1)! det(Λ1i)|0iand Λi1is the
(d−1) ×(d−1) submatrix obtained by omitting the
i-th column and the first row (a so-called minor of Λ)
and (−1)i+1a†
1i|0i=|ii. It is remarkable that the ex-
pansion of the determinant results in a maximally en-
tangled bipartite state of the form (2) with D=dand
|Bii=|ii, implying that Alice and Bob obtain d−1 and
one photon, respectively. This partition of photons al-
lows Alice to teleport any state of a single d-level photon
from Charlie by sending it along with her d−1 photons
into a balanced multiport and subsequently obtaining a
coincidence count in all its output ports.
But this is not the only possible partitioning of pho-
tons which leads to a maximally entangled state between
Alice and Bob. Strickingly, any partition (n, d −n) with
0< n < d of an antisymmetric state of dparticles pos-
sesses this property, and this can be easily understood by
virtue of the rules to calculate determinants (see Supple-
mentary Information). For the partition (n, d −n) one
obtains a bipartite state as given in Eq. (2) with D=d
n
and |Aiias well as |Biigiven in terms of minors of Λ. The
4
|χ>
|χ>nd−n
d−n FψSψ
C A B
FIG. 4: Teleportation of any antisymmetric state |χiof d−n
photons: source Sψproduces a pair of systems Aand Bin
state |Ψiwith nand d−nphotons respectively. System
Awith nphotons and system Cwith d−nphotons are to
be projected by a filter Fψinto state |Ψi. If successful, the
filtering transfers the initial state |χiof Cto system B.
0 20 40 60 80 100 120 140 160
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Number of photons
Number of qubits per photon
FIG. 5: The graphs show the number of quantum informa-
tion units (qubits) teleported per additional photon versus the
number of additional photons required for individual-qubit
teleportation (red) and the optimal new technique (blue).
maximum amount of quantum information can be tele-
ported with a (d/2, d/2) partiton of an even number dof
photons prepared in state |Ψi. In this case Charlie has
to provide d/2 photons, cp. Fig. 4, and can send for large
dsimultaneously approximately dqubits encoded in the
D=d
d/2= 2d+O(d) dimensional, anti-symmetric sub-
space spanned by the |Bii. This is an exponential gain
compared to the teleportation of a single qudit (corre-
sponding to log2(d) qubits) carried by one photon, which
requires the same entanglement resource – an antisym-
metric state of dphotons |Ψi.
In comparison, teleporting dqubits individually re-
quires the same number of maximally entangled photon
pairs, i.e., a total of 2dadditional photons, resulting in
an efficiency of sending half a qubit per additional pho-
ton or 1/3 of a qubit per photon generated. Our scheme
yields double these rates: it thus requires half the num-
ber photons (cp Fig. (5)) to teleport the same amount
of quantum information. This is an important improve-
ment since the number of photons, that can be generated
per time-unit is the limiting factor for the bandwith of
photonic quantum communication.
In the foregoing discussion we assumed the existence of
a source SΨthat produces the state |Ψiinitially shared
by Alice and Bob (cp. challenge (i)). Indeed for qutrits,
such a source can be realized by yet another balanced
multiport which acts as a Bell-filter for the antisymmet-
ric state |Ψi, provided the input consist of a single photon
and a antisymmetric state of two photons and one pho-
ton leaves each of its output ports. In the general case
nondestructive heralding techniques based on non-linear
optical effects38,39 might be employed, which however re-
duce the efficiency of the teleportation scheme and will
be discussed together with other preparation methods
elsewhere.
Supplementary Information
Dimension of the antisymmetric subspace
The dimension of an antisymmetric subspace can be
calculated using combinatorial objects called Young
tableaux, which provide a technique of keeping track of
the constraints imposed by the permutation symmetry
of the system. Here we represent a basis state of a sys-
tem by a box, a, where anumbers the basis state. A
basis of the symmetric combinations of two systems can
be depicted by a row of two boxes, . Similarly, a
basis of anti-symmetric states is represented by a col-
umn of boxes, . Since we are interested only in the
antisymmetric part, we focus on columns only. The di-
mension of the corresponding subspace, i.e., the number
of basis states, is obtained by counting the possibilities
for filling the boxes with numbers according to certain
rules. For the antisymmetric subspace we start filling
the numbers in descending order, from top to bottom.
For a system which consists of two subsystems, a Young
tableaux reads:
a
b≡ |a bi − |b ai,(10)
where ais always greater than b. Therefore, if the total
number of basis states available for each subsystem is
two, i.e., we are dealing with two qubits, there is only
one possibility, namely a= 2 and b= 1, therefore we
obtain an antisymmetric subspace of dimension one. If
the available states are more than two, say d, then we
have d−1 options for aand given a,a−1 options for b.
As a result the total number of combinations is given by
1+2+...+d−1 = d(d−1)/2, which is the dimension of the
antisymmetric subspace for a pair of d-level systems each
of which can carry one ”qudit” of quantum information.
This can be generalized for systems with nsubsys-
tems. Now we have nnumbers {a1> a2>···> an}in
a column of boxes. The dimension of this antisymmetric
subspace is given by the binomial coefficient d
n,
i.e, dchoose n, where nis the number of subsystems.
This equals one only when n=d, giving us a unique
antisymmetric state for dqudits.
Laplace expansion for the determinant of a matrix
The determinant of an n×nmatrix Awith elements aij
can be calculated by an expansion with respect to the
5
first row of Aas follows:
det(A) =
n
X
i=1
(−1)1+ia1idet(A1i).(11)
Here det(A1i) is the determinant of the (n−1) ×(n−1)
submatrix of Aobtained from eliminating the first row
and i-th column. In fact this is just a special case of a
simultaneous expansion of the determinant with respect
to several rows. For example, expanding with respect to
the first two rows of a 4 ×4 matrix A, we obtain:
++
−
−
=
det(A) +
,
where each block on the right-hand side represents the
product of the determinant of the submatrix (minor)
with blue dot and the minor with the red dot. In
general, any such Laplace expansion assumes the form
det(A) = Picidet(Ai) det(Bi), where ci=±1 and the
Ai(Bi) are minors of Awhich differ at least in one
column40. The possible Laplace expansions of det Λ in
(5) correspond to the different distributions of the dpho-
tons between Alice and Bob. Each distribution leads to
orthogonal states |Aii ∝ det(Ai)|0ion Alice’s side, and
on Bob’s side accordingly, and therefore to a maximally
entangled state shared between both parties.
Acknowledgements
We thank A Forbes, P Krumm and K Garapo for useful
discussions.
∗Electronic address: goyal@ukzn.ac.za
1Wootters, W. K. & Zurek, W. H. A single quantum cannot
be cloned. Nature 299, 802–803 (1982).
2Werner, R. F. Optimal cloning of pure states. Phys. Rev.
A58, 1827–1832 (1998).
3Bennett, C. H. et al. Teleporting an unknown quantum
state via dual classical and einstein-podolsky-rosen chan-
nels. Phys. Rev. Lett. 70, 1895–1899 (1993).
4Bell, J. S. On the einstein podolsky rosen paradox. Physics
1, 195 (1964).
5Brassard, G., Braunstein, S. L. & Cleve, R. Teleportation
as a quantum computation. Physica D: Nonlinear Phe-
nomena 120, 43 – 47 (1998).
6Gottesman, D. & Chuang, I. L. Demonstrating the viabil-
ity of universal quantum computation using teleportation
and single-qubit operations. Nature 402, 390–393 (1999).
7Knill, E., Laflamme, R. & Milbum, G. J. A scheme for
efficient quantum computation with linear optics. Nature
409, 46 (2001).
8Duan, 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).
9Briegel, H.-J., D¨ur, W., Cirac, J. I. & Zoller, P. Quan-
tum repeaters: The role of imperfect local operations in
quantum communication. Phys. Rev. Lett. 81, 5932–5935
(1998).
10 A. I. Lvovsky, W. Tittel, B. C. Sanders. Optical quantum
memory. Nature Photonics 3, 706–714 (2009).
11 Bouwmeester, D. et al. Experimental quantum teleporta-
tion. Nature 390, 575–579 (1997).
12 Boschi, D., Branca, S., De Martini, F., Hardy, L. &
Popescu, S. Experimental realization of teleporting an un-
known pure quantum state via dual classical and einstein-
podolsky-rosen channels. Phys. Rev. Lett. 80, 1121–1125
(1998).
13 Furusawa, A. et al. Unconditional quantum teleportation.
Science 282, 706–709 (1998).
14 Nielsen, M. A., Knill, E. & Laflamme, R. Complete quan-
tum teleportation using nuclear magnetic resonance. Na-
ture 396, 52–55 (1998).
15 Pan, J.-W., Daniell, M., Gasparoni, S., Weihs, G. &
Zeilinger, A. Experimental demonstration of four-photon
entanglement and high-fidelity teleportation. Phys. Rev.
Lett. 86, 4435–4438 (2001).
16 Jennewein, T., Weihs, G., Pan, J.-W. & Zeilinger, A.
Experimental nonlocality proof of quantum teleportation
and entanglement swapping. Phys. Rev. Lett. 88, 017903
(2001).
17 Kim, Y., Kulik, S. & Shih, Y. Quantum teleportation of a
polarization state with a complete bell state measurement.
Physical Review Letters 86, 1370–1373 (2001).
18 Lombardi, E., Sciarrino, F., Popescu, S. & De Martini,
F. Teleportation of a vacuum–one-photon qubit. Physical
review letters 88, 70402 (2002).
19 Marcikic, I., de Riedmatten, H., Tittel, W., Zbinden, H. &
Gisin, N. Long-distance teleportation of qubits at telecom-
munication wavelengths. Nature 421, 509–513 (2003).
20 Bowen, W. P. et al. Experimental investigation of
continuous-variable quantum teleportation. Phys. Rev. A
67, 032302 (2003).
21 Babichev, S. A., Ries, J. & Lvovsky, A. I. Quantum scis-
sors: Teleportation of single-mode optical states by means
of a nonlocal single photon. Europhys. Lett: 64, 1 (2003).
22 Ursin, R. et al. Quantum teleportation across the danube.
Nature 430, 849 (2004).
23 Fattal, D., Diamanti, E., Inoue, K. & Yamamoto, Y.
Quantum teleportation with a quantum dot single photon
source. Phys. Rev. Lett. 92, 037904 (2004).
24 Riebe, M. et al. Deterministic quantum teleportation with
atoms. Nature 429, 734 (2004).
25 Barrett, M. D. et al. Deterministic quantum teleportation
of atomic qubits. Nature 429, 737 (2004).
26 Olmschenk, S. et al. Quantum teleportation between dis-
tant matter qubits. Science 323, 486 (2009).
27 Al-Amri, M., Evers, J. & Zubairy, M. S. Quantum telepor-
tation of four-dimensional qudits. Phys. Rev. A 82, 022329
(2010).
28 Ritter, S. et al. An elementary quantum network of single
6
atoms in optical cavities. Nature 484, 195 (2012).
29 The maximal distance for quantum communication
achieved with photons so far was 144 km through the at-
mosphere limited mainly by absorption. Much further dis-
tances seem only possible using teleportation of entangled
photons in conjunction with so-called quantum repeaters9.
30 Boyd, R. Nonlinear optics. Electronics & Electrical (Aca-
demic Press, 2003).
31 L¨utkenhaus, N., Calsamiglia, J. & Suominen, K.-A. Bell
measurements for teleportation. Phys. Rev. A 59, 3295–
3300 (1999).
32 Hong, C. K., Ou, Z. Y. & Mandel, L. Measurement of
subpicosecond time intervals between two photons by in-
terference. Phys. Rev. Lett. 59, 2044–2046 (1987).
33 Bose, S. & Home, D. Generic entanglement generation,
quantum statistics, and complementarity. Phys. Rev. Lett.
88, 050401 (2002).
34 Allen, L., Beijersbergen, M. W., Spreeuw, R. J. C. & Wo-
erdman, J. P. Orbital angular momentum of light and
the transformation of laguerre-gaussian laser modes. Phys.
Rev. A 45, 8185–8189 (1992).
35 Reck, M., Zeilinger, A., Bernstein, H. J. & Bertani, P.
Experimental realization of any discrete unitary operator.
Phys. Rev. Lett. 73, 58–61 (1994).
36 Lim, Y. L. & Beige, A. Generalized hongoumandel exper-
iments with bosons and fermions. New Journal of Physics
7, 155 (2005).
37 Lim, Y. L. & Beige, A. Multiphoton entanglement through
a bell-multiport beam splitter. Phys. Rev. A 71, 062311
(2005).
38 Konrad, T., Nock, M., Scherer, A. & Audretsch, J. Pro-
duction of heralded pure single photons from imperfect
sources using cross-phase-modulation. Phys. Rev. A 74,
032331 (2006).
39 Invernizzi, C., Olivares, S., Paris, M. G. A. & Banaszek,
K. Effect of noise and enhancement of nonlocality in on/off
photodetection. Phys. Rev. A 72, 042105 (2005).
40 Lamcaster, P. & Tismenetsky, M. The theory of matrices
(Harcourt Brace Jovanovich Publishers, 1985).
