Manipulating biphotonic qutrits
B. P. Lanyon, T. J. Weinhold, N. K. Langford, J. L. O'Brieny, K. J. Reschz, A. Gilchrist, and A. G. White
Department of Physics and Centre for Quantum Computer Technology,
University of Queensland, QLD 4072, Brisbane, Australia.
yCentre for Quantum Photonics, H. H. Wills Physics Laboratory & Department of Electrical and Electronic Engineering,
University of Bristol, Merchant Venturers Building, Woodland Road, Bristol, BS8 1UB, UK
z Institute for Quantum Computing and Department of Physics & Astronomy,
University of Waterloo, Waterloo, N2L 3G1, Canada.
Quantum information carriers with higher dimension than the canonical qubit oer signicant ad-
vantages. However, manipulating such systems is extremely dicult. We show how measurement
induced non-linearities can be employed to dramatically extend the range of possible transforms on
biphotonic qutrits; the three level quantum systems formed by the polarisation of two photons in
the same spatio-temporal mode. We fully characterise the biphoton-photon entanglement that un-
derpins our technique, thereby realising the rst instance of qubit-qutrit entanglement. We discuss
an extension of our technique to generate qutrit-qutrit entanglement and to manipulate any bosonic
encoding of quantum information.
Higher dimensional systems oer advantages such as
increased-security in a range of quantum information pro-
tocols [1{7], greater channel capacity for quantum com-
munication [8], novel fundamental tests of quantum me-
chanics [9, 10] and more ecient quantum gates [11]. Op-
tically such systems have been realised using polarisation
[12] and transverse spatial modes [1, 13]. However in each
case state transformation techniques have proved dicult
to realise. In fact, performing such transformations is a
signicant problem in a range of physical architectures.
The polarisation of two photons in the same spatio-
temporal mode represents a three-level bosonic quantum
system, a biphotonic qutrit, with symmetric logical ba-
sis states: j03ij2H ; 0V i, j13i(j1H ; 1V i+j1V ; 1Hi)=
p
2,
and j23ij0H ; 2V i [14]. The simple optical tools which al-
low full control over the polarisation of a photonic qubit
are insucient for full control over a biphotonic qutrit
[15]. Consequently even simple state transformations
required in qutrit generation, processing and measure-
ment are extremely limited. Signicant progress has been
made in biphoton state generation. For example, com-
plex arbitrary state preparation techniques that employ
multiple nonlinear crystals [16] and non-maximally en-
tangled states [17] have been developed.
Here we present and demonstrate a technique that dra-
matically extends the range of biphotonic qutrit trans-
forms, for use in all stages of qutrit manipulation. The
technique is based on a Fock-state lter which employs
a measurement-induced nonlinearity to conditionally re-
move photon-number (Fock) states from superpositions
[18{23]. We rst demonstrate the action of the lter as a
qutrit polariser, which can conditionally remove a single
logical qutrit state from a superposition. We then com-
bine this nonlinear operation with standard waveplate ro-
tations to demonstrate the dramatically increased range
of qutrit transforms it enables. Finally we present the
rst instance and full characterisation of a polarisation
entangled photon-biphoton state, which underpins the
P3 qutrit tomographyqutrit-qubit source
CPBS λ/4

ancillaqubitλ/2
qutrit
D1 D2 D3
D4PDC
H (θ )
3 H (φ ) 3
a
b
c
d
1
2
50% BS
FIG. 1: Experimental schematic. Emission from a parametric
down conversion (PDC) crystal is coupled into single mode -
bre and injected into modes 1 and 2. Coincident (C) detection
of photons at D1-4 selects, with high probability, the cases of
double photon-pair emission from the PDC source.
power of our technique. Such qubit-qutrit states have
been studied extensively [24{30] and we suggest an ex-
tension to generate qutrit-qutrit entanglement.
We generate our qutrits through double-pair emission
from a spontaneous parametric down conversion source
(Fig. 1). Measurement of a four-fold coincidence between
detectors D1-D4 selects, with high probability, the cases
of double-pair emission into inputs 1 and 2. The bipho-
ton state in mode 1 is passed through a horizontal po-
lariser to prepare the logical qutrit state j03i. Input 2 is
passed through a 50% beam splitter; detection of a sin-
gle photon at D1 signals, with high probability, a single
photon in mode b; which is passed through a polaris-
ing beam splitter to prepare a polarisation ancilla qubit
(j02ij1Hi, j12ij1V i) in the logical state j02i. Thus a
qubit and qutrit arrive simultaneously at the central 50%
beam splitter.
The operation of a Fock-lter relies on nonclassical in-
terference eects [31]. When two indistinguishable pho-
tons are injected into modes a and b (Fig. 1), the prob-
ability of detecting a single photon in mode d is zero; if
two or more photons are injected into mode a then this
probability is non-zero. By injecting a single photon into
mode b and detecting a single photon in mode d, single
photon terms can therefore be removed from any pho-
ar
X
iv
:0
70
7.
28
80
v2
[
qu
an
t-p
h]
1
6 O
ct
20
07

2ton number superposition states arriving in mode a. In
fact, by varying the re
ectivity of the beam splitter it is
possible to conditionally remove any number-state from
a superposition [22]. This Fock-state lter acts only on
light with the same polarisation as the ancilla (in our
case, horizontal), so by detecting a single horizontal pho-
ton in mode d, the logical qutrit state j13i is blocked,
since it contains a single photon with the same polarisa-
tion as the ancilla. The remaining logical qutrit states
are only attenuated.
Thus for a beam splitter of re
ectivity 50% the l-
ter acts, on an initial qutrit state of j03i+j13i+
j23i,
as a qutrit polariser described by the operator
P3=j03ih03jj23ih23j. Note that a standard horizontal
or vertical linear polariser would act as a lossless qutrit
polariser, but one restricted to simultaneously removing
either the logical j03i and j13i, or the j13i and j23i states.
Such a polariser could therefore not leave just the j13i
state. By varying the polarisation of the ancilla, and the
re
ectivity of the central beam splitter, the operation of
our lossy qutrit polariser can be tuned to preferentially
remove the j03i, j13i or j23i states. We choose to demon-
strate removal of the j13i state and include the general
operation of the lter for an arbitrary beam splitter re-
ectivity [32].
The qutrit polariser oers a powerful tool for trans-
forming between qutrit states. For example, consider the
initial qutrit state j03i injected into input 1, the red dot
of Fig. 2a). The black ring shows the limited range of
qutrit states, with real coecients, that are accessible
using waveplates [33]. By including the qutrit polariser
the range is dramatically extended to the closed sphere
in Fig 2; the transformation to any real state is possible.
We measure our qutrits by passing mode c through
a 50% beam splitter and performing polarisation analy-
sis of the two outputs in coincidence, as shown in Fig. 1.
This analysis non-deterministically discriminates the log-
ical states j03i, j13i, and j23i with probabilities p(03)= 12 ,
p(13)= 14 and p(23)=
1
2 . Combining it with single qubit
rotations after the beam splitter allows us to perform
full qutrit state tomography of mode c. Complete qutrit
tomography requires nine independent measurements,
which we construct from logical basis states and two-
part superpositions [1]. Our method diers from that of
Refs [14, 15] and is described in additional online ma-
terial. We use convex optimisation to reconstruct the
qutrit density matrix and Monte-Carlo simulations for
error analysis [34, 35].
Ideally both the central and tomography beam split-
ters re
ect 50% of both polarisations. In practice, we
found that they deviated by a few percent and impart
undesired unitary rotations on the re
ected and trans-
mitted modes. For the tomography beam splitter, these
imperfections modied the nine measured qutrit states;
we characterised this eect and incorporated it into the
tomographic reconstruction. We also found that the ef-
|1〉 3
|0〉3
|2〉3
FIG. 2: Comparison of the range of linearly polarised qutrit
states achievable by transforming the state j03i (red dot);
when using only waveplate operations (black ring); by incor-
porating our qutrit polariser,Q3()H3()P3(
p
0:5)H3()j03i
(sphere) [32, 33].
fect of the imperfect central beam splitter on the perfor-
mance of the qutrit polariser was negligible.
A frequency-doubled mode-locked Ti:Sapphire laser
(820 nm!410 nm,
=80fs at 82 MHz repetition rate)
is used to produce photon pairs via parametric down con-
version from a Type I phase-matched 2mm Bismuth Bo-
rate (BiBO) crystal, ltered by blocked interference l-
ters (8201.5 nm). We collect the down-conversion into
single-mode optical bers. Photons are detected using
ber-coupled single photon counting modules and coinci-
dences measured using a Labview (National Instruments)
interfaced quad-logic card (ORTEC CO4020). When di-
rectly coupled into detectors the source yielded two-folds
at 60 kHz and singles rates at 220 kHz. At the output
of the complete circuit we observed four-fold coincidence
rates at approximately 1 Hz.
The quality of the non-classical interference underpin-
ning the qutrit polariser can be measured directly [22].
Ref. [23] relates non-classical visibilities to a Fock-state
lter's ability to block single photon terms. Using this
technique, and measured visibilities of V1=97  1% &
V2=684%, we predict an extinction ratio of 5(2):1; our
qutrit polariser will pass the logical j03i and j23i states
at ve times the rate it passes the logical j13i state.
To demonstrate the qutrit polariser we include a half-
waveplate in mode a set to =8 to generate a superposi-
tion qutrit state with all logical states populated, of the
form [33]:
H3()j03i= cos2 2j03i+ sin
2 2j23i+ sin 4j13i=
p
2 (1)
We measure the output state in mode c without apply-
ing the qutrit polariser. This is achieved by blocking the
ancilla photon in mode b, and performing qutrit tomog-
raphy of mode c in two-fold coincidence between D3 and
D4. The experimentally reconstructed density matrix is
shown in Fig. 3a) and has a near perfect delity between
the measured and ideal states, F=971%, and a low lin-
ear entropy, SL=67%, [36, 37]. We then prepared the

3 0
1
2 012
−0.50
0.5
−0.50
0.5
−0.50
0.5
−0.50
0.5
−0.5
0
0.5
−0.50
0.51
−0.5
0
0.5
−0.50
0.51
a) i. b) i. c) i. d) i.
ii. ii. ii. ii.
FIG. 3: Comparison of real parts of (i) ideal and (ii) measured
qutrit density matrices. a) The measured output state with
the qutrit polariser `o' (Eq. 1 for =8 ). b) The output
state with the qutrit polariser `on' showing the removal of the
logical j13i qutrit state. c)-d) Newly accessible qutrit states
j13i and (j03ij13ij23i)/
p
3, respectively. States b-d) all lie
on the surface of the sphere of Fig. 2, but not on the ring.
output state by unblocking the ancilla and, as in all fur-
ther cases, perform tomography of mode c in four-fold co-
incidence between D1-4. The qutrit polariser is now `on'
and we expect the absorption of the logical j13i state.
The reconstructed density matrix is shown in Fig. 3b)
and has a lower delity with the ideal, F=788%, and
linear entropy SL=4714%. The relative reduction in
the logical j13i state probability, when the lter is turned
on, yields an extinction ratio of 6:80(0:07):1, consistent
with that predicted above.
Measured non-classical visibilities are signicantly lim-
ited by higher-order parametric down conversion photon
number terms [38, 39]. After removing these eects, as
described in reference [23], we nd a corrected two-fold
visibility of V 01=100 1%, that would be measured given
an ideal two-photon source (higher-order eects cannot
be distinguished from experimental uncertainty in the
four-fold visibility). This corrected visibility can then be
used to predict the potential performance of our circuit
given an ideal photon source [23]; in this case we predict
that the lter would pass the logical j03i and j23i states
at least 24 times the rate it passes the logical j13i state.
Clearly the performance of our qutrit polariser is signi-
cantly limited by higher-order emissions from our optical
source.
Figures 3c-d) show experimentally reconstructed den-
sity matrices of newly accessible states achieved by in-
corporating the qutrit polariser with half-waveplate op-
erations applied to the initial state of j03i; j13i and
(j03ij13ij23i)/
p
3. The delities with the ideal are
773%, and 837% with linear entropies 517% and
3815%, respectively. These delities exceed the max-
imum achievable using only linear waveplates (50%) by
91 and 51 standard deviations, respectively.
The qutrit polariser employs a measurement-induced
non-linearity whereby the biphoton becomes entangled
with the ancilla photon. Instead of detecting the ancilla
in a single, xed polarisation state, we can also use to-
mographic measurements to directly investigate this re-
sultant entangled qubit-qutrit system. Without emphasis
to the physical systems involved, such states where rst
studied by Peres as a special case of his negativity cri-
terion for entanglement; a negativity of 0 (>0) is con-
clusive of a separable (entangled) state [24, 40]. More
recently these states have received a signicant amount
of attention [24{29] and have been predicted to exhibit
novel entanglement sudden death phenomena [30] .
On injection of the qutrit state given by Eq. 1 into the
Fock-lter, we nd the following qubit-qutrit joint state
of modes c and d:
fcos2 2j02; 03i+ sin 4j12; 03i (2)
+ sin2 2(
p
2j12; 13i j02; 23i)g=N;
where N=
p
2 cos 4. By varying  we can tune the level
of entanglement from zero (=0) to near-maximal (=4 ),
with corresponding negativities of 0 to
p
8=9  0:94, re-
spectively. To perform qubit-qutrit state tomography we
use 36 independent measurements constructed from all
of the combinations of the aforementioned nine qutrit
states and four qubit states [H,V,D,R]. Fig. 4 shows the
measured density matrix for the near-maximally entan-
gled case, which corresponds to the preparation of two
vertically polarised photons in mode a. There is a high
delity with the ideal of 813%, low linear entropy of
175% and the state is highly entangled with a negativ-
ity of 0:770:05. We note that a maximally entangled
state is predicted for =4 and a central beam splitter
re
ectivity of R=
p
2=(
p
2+1)  58:6%.
Entangling information carriers to ancilla qubits is an
extremely powerful technique [41]: such correlations play
a central role in the power of the Fock lter to trans-
form biphotonic qutrits. However, the application of our
technique is not limited to extending transforms on sin-
gle qutrits. We propose that the generation of qubit-
qutrit entanglement oers a path to realise multi-qutrit
operations. For example, a pair of entangled qubit-qutrit
states could be used to create qutrit-qutrit entanglement
by projecting the qubits into an entangled state using
well-known techniques. The much anticipated develop-
ment of high-brightness single-photon sources will make
such experiments feasible in the near future. We wish to
emphasize that our technique is not limited to manipu-
lating biphotons. The Fock-lter can be applied to any
system where measurement can induce non-linear eects;
that is any bosonic encoding of quantum information, in-
cluding bosonic atoms [42] and time-bin, frequency and
orbital angular momentum encoding of photons.
We have shown that measurement induced nonlinear-
ities oer signicant advantages for the manipulation of
higher dimensional bosonic information carriers, specif-
ically biphotonic qutrits. We demonstrated a nonlinear
qutrit-polariser, capable of conditionally removing a sin-
gle logical qutrit state from a superposition, and greatly
extending the range of possible qutrit transforms. Such
tools could nd immediate application to quickly gener-

4

0,0 0,1 0,2 1,0 1,1 1,2
−0.5
0
0.5
−0.5
0
0.5
−0.5
0
0.5
a)
b)
0,00,10,21,01,11,2i. ii.
FIG. 4: Comparison of entangled qubit-qutrit density matri-
ces. a) ideal, b) & c) measured real & imaginary parts. There
is high delity with the ideal (81  3%), low linear entropy
(17 5%) and the state is highly entangled with a negativity
of 0:77  0:05. The ideal state is given by Eq. 2 for ==4.
Note the axis label: x; j represents the qubit logical state x
and the qutrit logical state j i.e. jx2; j3i.
ate the mutually unbiased basis states required for opti-
mum security in quantum key distribution protocols in-
volving qutrits [5{7] or as a ltering technique to manipu-
late entanglement in qutrit-qutrit states. Finally we fully
characterised the entangled photon-biphoton state that
underpins the power of our technique. This is the rst in-
stance of the generation and characterisation of entangle-
ment between these distinct physical systems and makes
recent theoretical proposals experimentally testable [30].
Besides oering a path to implement novel multi-qutrit
operations we propose that our technique can be ex-
tended to manipulate any bosonic encoding of quantum
information.
NOTE: After completion of this work , several authors
presented proposals for which our technique is directly
relevant [43{45].
This work was supported in part by the Australian Re-
search Oce and the US Disruptive Technologies Oce.
BPL and AGW wish to acknowledge funding by the En-
deavor Europe Award and Federation Fellow programs,
respectively.
[1] N. K. Langford et al, Phys. Rev. Lett. 93, 053601 (2004).
[2] R. W. Spekkens and T. Rudolph, Phys. Rev. A 65,
012310 (2001).
[3] G. Molina-Terriza et al, Phys. Rev. Lett. 92, 167903
(2004).
[4] S. Groblacher et al, New J. Phys. 8, 75 (2006).
[5] D. Bru and C. Macchiavello, Phys. Rev. Lett. 88,
127901 (2002).
[6] N. J. Cerf et al, Phys. Rev. Lett. 88, 127902 (2002).
[7] T. Durt et al, Phys. Rev. A 67, 012311 (2003).
[8] M. Fujiwara et al, Phys. Rev. Lett. 90, 167906 (2003).
[9] D. Collins et al, Phys. Rev. Lett. 88, 040404 (2002).
[10] D. Kaszlikowski et al, Phys. Rev. A 65, 032118, (2002).
[11] T. C. Ralph et al, Phys Rev A 75, 022313 (2007).
[12] A. Aspect et al, Phys. Rev. Lett. 47, 460 (1981).
[13] A. Mair et al, Nature 412, 313 (2001).
[14] Y. Bogdanov et al , Quantum Informatics 2004, pp. 202{
212, (2005).
[15] Y. I. Bogdanov, Phys. Rev. A, 70, 042303, (2004).
[16] Y. I. Bogdanov, Phy. Rev. Lett. 93, (2004).
[17] G. Vallone et al, Phys. Rev. A 76, 012319 (2007).
[18] A. Grudka and A. Wojcik, Phys. Rev. A 66, 064303
(2002).
[19] H. F. Hofmann and S. Takeuchi, Phys. Rev. Lett. 88,
147901 (2002).
[20] X. Zou et al, Phys. Rev. A 66, 064302 (2002).
[21] K. Sanaka et al, Phys. Rev. Lett. 92, 017902 (2004).
[22] K. Sanaka et al, Phys. Rev. Lett. 96, 083601 (2006).
[23] K. J. Resch et al, Phys. Rev. Lett. 98, 203602 (2007).
[24] A. Peres, Phys. Rev. Lett. 77, 1413 (1996).
[25] P. B. Slater et al, Phys. Rev. A 71, 052319 (2005).
[26] A. Cabello et al, Phys. Rev. A 72, 052112 (2005).
[27] O. Osenda and G. A. Raggio, Phys. Rev. A 72, 064102
(2005).
[28] S. Jami et al quant-ph/0606039, (2006).
[29] P. B. Slater, quant-ph/0702134, (2007).
[30] K. Ann and G. Jaeger, quant-ph/0707.4485, (2007).
[31] C. K. Hong , Z. Y. Ou, and L. Mandel, Phys. Rev. Lett.
59, 2044 (1987).
[32] For our Fock-lter with re
ectivity R=r2, P3(r)=
2
4
r(2 3r2) 0 0
0 r(1 2r2) 0
0 0 r3
3
5 :
[33] From Ref. [15], the waveplate action on a qutrit is:
2
4
t2
p
2tr r2

p
2tr jtj2jrj2
p
2tr
r2
p
2tr t2
3
5 ;
where t= cos +i sin  cos 2, r=i sin  sin 2, and  is the
waveplate angle. For a half-wave plate, H3(), ==2;
for a quarter-wave plate, Q3(), ==4.
[34] A. Doherty and A. Gilchrist, in preparation, (2007).
[35] J. L. O'Brien et al, Phys. Rev. Lett. 93, 080502 (2004).
[36] Fidelity is F (; )fTr[
pp

p
]g2; linear entropy is
SL d(1Tr[2])=(d1), where d is the state dimension.
[37] A. G. White et al, JOSA B 24, 172 (2007).
[38] J. Fulconis et al, arXiv:quant-ph/0611232 (2007).
[39] T. J. Weinhold et al, in preparation, (2007).
[40] Negativity is dened as N=maxf0;
P
i ig where i are
the negative eigenvalues of the partial transpose of the
target density matrix.
[41] E. Knill, R. La
amme, and G. J. Milburn, Nature 409
(2001).
[42] S. Popescu, quant-ph/0610043, (2006).
[43] I. Bregman et al, quant-ph/0709.3804, (2007).
[44] C. Bishop and M. S. Byrd quant-ph/0709.0021, (2007).
[45] M. Ali, et al., arxiv:0710.2238, (2007).