Generation of Large Number-Path Entanglement Using Linear Optics and
  Feed-Forward
Generation of Large Number-Path Entanglement Using Linear Optics and
Feed-Forward
Hugo Cable∗ and Jonathan P. Dowling
Horace C. Hearne Jr. Institute for Theoretical Physics,
Department of Physics and Astronomy, Louisiana State University, Baton Rouge LA70803.
(Dated: April 4, 2007)
We show how an idealised measurement procedure can condense photons from two modes into
one, and how, by feeding forward the results of the measurement, it is possible to generate efficiently
superpositions of components for which only one mode is populated, commonly called “N00N states”.
For the basic procedure, sources of number states leak onto a beam splitter, and the output ports
are monitored by photodetectors. We find that detecting a fixed fraction of the input at one output
port suffices to direct the remainder to the same port with high probability, however large the initial
state. When instead photons are detected at both ports, Schrödinger cat states are produced. We
describe a circuit for making the components of such a state orthogonal, and another for subsequent
conversion to a N00N state. Our approach scales exponentially better than existing proposals.
Important applications include quantum imaging and metrology.
The fundamental limits to optical detection for metrol-
ogy and imaging are quantum mechanical [1]. Of partic-
ular interest for reaching such quantum limits are path-
entangled states of photons of the form |N0〉+ eiφ|0N〉,
in a basis of photon-number states, commonly referred to
as “N00N” states. A variety of applications have been
suggested [2]. For lithography [3] and microscopy [4],
N00N state light would be used together with multi-
photon absorbers to achieve enhanced resolution. This is
because the de Broglie wavelength for an N -photon state
is a factor 1/N smaller than the wavelength associated
with the single photon, and the absorption rate scales
linearly with the incident intensity, rather than as the
N th power. Regarding applications to precision metrol-
ogy, whereby an interferometric setup is used to mea-
sure small phase shifts, N00N states achieve the Heisen-
berg limit, for which the phase uncertainty scales as 1/N
[5, 6, 7], and entanglement is a fundamental requirement
for achieving this limit. It has been rigorously demon-
strated that the cost of improving sensitivity (without
using entanglement) is higher intensities or longer coher-
ence times [8]. Classically the shot-noise limit applies,
attained for example by laser light, for which the uncer-
tainty scales as 1/
N , already a restraint in applications
such as magnetometry [9] and gyroscopy [10].
However, building a source of N00N states beyond two
photons is challenging. Three, four and six photon ex-
periments have been reported [11, 12, 13], but only in the
first two references were N00N states generated. In the-
ory a source could be made using a nonlinear crystal [14].
However, the required optical nonlinearity is not readily
available. An alternative is a non-deterministic approach
using linear optics, wherein the desired state is generated
on condition of a specific outcome at photodetectors. A
variety of schemes have been suggested which typically
rely on conditional destructive interference [15, 16, 17].
Electronic address: hcable@lsu.edu
However, so far none of these scales efficiently, that is
they all share the feature that exponentially decreasing
success probabilities outweigh the possible gains. Noting
that quantum algorithms, exhibiting polynomial and ex-
ponential speedups over their classical counterparts, may
be implemented scalably in a linear-optics approach [18],
we expect that it should be possible to do better. In this
Article we address this challenge by adapting a concep-
tually simple measurement procedure. Our method is as
follows. First, we aim to minimise the negative effects of
back-action in a sequence of detections, whereby earlier
measurements affect the outcomes of later ones. Next,
we engineer output states that closely approximate the
ideal case. Finally, we exploit feed-forward, for which cir-
cuits are actively switched in response to previous pho-
todetections. Feed-forward is an essential ingredient of
linear-optics based quantum computing, but is not used
in previous proposals for engineering N00N states.
We begin by considering the thought experiment de-
picted in Fig. 1(a). Here, cavity modes labeled A and
B are assumed to start with a well-defined photon num-
ber N . They are each coupled to an external mode by
a weakly transmissive mirror, and these modes are com-
bined at a 50:50 beam splitter, and then subject to par-
tial photodetection. The beam splitter acts to make the
origin of the photons indistinguishable. When a photon
is registered at the left or right photodetector labeled
DL or DR, the transformation is given by the Kraus
operators L̂ = (â − b̂)/
2 or R̂ = (â + b̂)/
2 respec-
tively (where â and b̂ are the annihilation operators for
modes A and B). To obtain the corresponding prob-
abilities it is necessary to normalise by the total pho-
ton number prior to detection. We suppose now that a
string of detections occur only at Dr, say (by adjusting
the path length difference of the cavities between detec-
tions, with a phase shifter, the same state for the cav-
ity modes can be obtained in every case). After this,
the detectors are removed and the system evolves to a
final state with all the remaining population at the out-
put ports. Denoting by |ψAB〉 the state of modes A
http://arxiv.org/abs/0704.0678v1
mailto:hcable@lsu.edu
FIG. 1: Two measurement-based procedures, inducing rela-
tive phase correlations between the principal modes, each one
with N photons at the start. In (a), population leaks from
cavity modes A and B into external modes, which are com-
bined at a beam splitter. Photons are detected one at a time
by photodetectors Dl and Dr. In (b), all modes are propagat-
ing. Beam splitters couple some fraction f from the principal
modes one and two into ancillae modes three and four, which
are initially the vacuum. The ancillae are subsequently com-
bined at a beam splitter, and subjected to number-resolving
photodetection at Dl and Dr.
and B after r initial detections, we find that |ψAB〉 =
R̂r |N〉 |N〉 /
〈N | 〈N |
|N〉 |N〉, normalising
to unity. The probability Pcond of finding all the remain-
ing photons “condensed” at the right output port (and
none at the left output port) is as follows,
Pcond = 〈ψAB|
|ψAB〉 /S!
(rCk)
SCN−k
where S ≡ 2N − r denotes the total remaining photon
number, C denotes a binomial coefficient, and we assume
that r < N . Evaluating the value of Pcond numerically
for initial states of increasing size, we find that its value
is determined asymptotically by the proportion of the
input that is measured. For example, setting r either as
one quarter or one third of 2N suffices for Pcond > 0.6 or
Pcond > 0.7, respectively.
We have found for our thought experiment, that later
detections tend to strongly reinforce earlier ones. Hence,
the effect of measurement back-action here is useful for
state engineering, and in what follows we adapt the mea-
surement process for N00N state generation. We di-
vide our analysis into three stages. First, we translate
our thought experiment into a mathematically equivalent
procedure based purely on linear optics, and consider the
general case for which photons are detected at both pho-
todetectors. The localisation phenomena resulting from
this measurement process, have been studied extensively
in the context of the debate over the existence of abso-
lute optical coherence in common quantum-optical exper-
iments [19]. It has been demonstrated that well-defined
correlations in the relative optical phase evolve, for the
remaining population, and play a central role in the ongo-
ing dynamics. Hence, in the second stage of our analysis,
we investigate simple procedures for manipulating phase
correlations, and relate states with well-defined correla-
tions in the relative phase and N00N states. Finally, we
identify a method based on feed-forward to enable N00N
states to be generated efficiently.
First, we translate our thought experiment into a
mathematically-equivalent procedure, based on linear op-
tics, as depicted in Fig. 1(b). We label this optical cir-
cuit, Circuit I. Here all modes are propagating, and a
source is assumed to supply dual Fock states |N〉|N〉 to
the principal modes one and two. Beam splitters of re-
flectance f couple modes one and two to ancillae modes
three and four, which are combined at a 50:50 beam
splitter. They are then measured by number-resolving
photodetectors labeled Dl and Dr, where on average a
fraction f of the input photons are registered. We now
consider the state |ψl,r〉 generated in modes one and two
after l photons are registered at Dl and r at Dr. Fol-
lowing Ref. [20], it is convenient to adopt a representa-
tion in terms of coherent states, which are of the form,
|α〉 ≡ ||α| exp(iθ)〉 ∝
|α|2k/k! exp (ikθ) |k〉 in a
basis of Fock states. It has been shown that,
|ψl,r〉∝
dθavd∆exp (−iSθav)×
G(∆−∆0)+exp (iσ)G(∆+∆0)
|α1〉|α2〉,(1)
where S ≡ 2N − l − r, αj = |αj | exp(iθj), θav =
(θ1 + θ2)/2 and ∆ ≡ θ2 − θ1. The superposition phase
σ takes the value lπ, and hence the measurement record
must be known exactly. The scalar function G(X) is
given to good approximation by the Gaussian expression
−(l + r)X2/4
. The total photon number is equal
to S. There are well-defined correlations in the relative-
optical phase parameter ∆ at values plus and minus ∆0,
determined only by the ratio of l to r. These correla-
tions are multi-valued whenever photons are registered
at both photodetectors, and a Schrödinger cat state is
generated. We can see that cats are generated as a re-
sult of the symmetry of the setup. Specifically, L̂ and
R̂ are invariant under an exchange of the labeling of the
modes, a transformation which reverses the sign of the
relative phase. The generation of cat states therefore
also requires precise phase stability between the modes.
Turning to the source, we see that the state of the input
can be a mixture of the form
N PN |N〉|N〉〈N |〈N |, since
∆0 is independent of N , and standard linear optical ele-
ments obey a superselection rule for the photon number.
Several two-mode squeezing processes strongly suppress
relative number fluctuations, and hence might serve as
practical sources of light described by these mixed states.
For the second stage in our analysis, we identify the
outputs of Circuit I as examples of quantum reference
frames — reference frames for a classically defined pa-
rameter composed of finite quantum resources. Quantum
reference frames are subject to depletion and degradation
as they are used, and are currently of interest for proto-
cols in the field of quantum information, in which they
are regarded as a resource [21]. By making an analogy
to classical phase references we can now identify simple
ways in which states of the form Eq. (1) can be manip-
ulated. For the current purposes we can assume that
a large number of detections have been performed and
define,
|ψ∞ (∆0)〉 ∝
dθ exp (−iSθ) |α〉|α exp(i∆0)〉, (2)
where α = |α|eiθ, for a state with a total photon num-
ber S and a relative phase of ∆0 (assumed to be nor-
malized). Relative phase correlations between more than
two modes are transitive and are transformed additively
by phase shifters. A phase reference can be extended to
additional modes by combining it with the vacuum at
a beam splitter. As an example, a 50:50 beam split-
ter, which we denote here by Ubs, beating light in a
Fock state with S photons against the vacuum yields,
Ubs|S〉|0〉 ∝ |ψ∞(0)〉, and Ubs|0〉|S〉 ∝ |ψ∞(π)〉. There-
fore, we see that a simple circuit, consisting only of a
beam splitter and a phase shifter, can convert a cat state
generated by Circuit I to aN00N state, whenever the rel-
ative phase correlations differ by π. This happens when
l = r and ∆0 = π/2. We label this circuit, Circuit III
(anticipating an intermediate process modifying the cat
states for the general case).
Before proceeding to the final stage of our analy-
sis, we consider a simple N00N -state generator, that
attempts to convert every cat state generated by Cir-
cuit I using Circuit III. This method might be expected
to yield close approximations to N00N states, when-
ever the the relative phases of the cat state are close
to plus and minus π/2. The situation is summarized
in Fig. 2(a). To measure the quality of the output
state we adopt the fidelity, denoting it by F . For
the measurement-induced condensation, considered at
the start, F takes the same same value as Pcond. For
schemes generating N00N states, it is necessary to ac-
count for the phase of the superposition, and we define
F = maxφ
〈0S| + exp (−iφ) 〈S0|
|ψoutput〉
/2, where
S is the total photon number of the state |ψoutput〉.
Evaluating F for our N00N -state generator, when Cir-
cuit I generates a cat state with relative phase compo-
nents at ±∆0 and total photon number S, we find to
first approximation that F ∼ cos2S [(∆0 − π/2) /2]. As
with other proposals, this scheme in fact scales exponen-
tially poorly whenever the relative phase correlations are
less than π apart, as is typically the case. Inspecting
the overlap for different relative phase components, as
in Eq. (2) with total photon number S, we find that
〈ψ∞(∆1)|ψ∞(∆2)〉
cos [(∆2 −∆1) /2]
. The poor
scaling can be attributed to the non-orthogonality of the
cat state components.
We now proceed to the final stage of our analysis. Our
previousN00N -state generator is effective when Circuit I
FIG. 2: (a) and (b) illustrate complete N00N state generators
in outline. Circuit I produces Schrödinger cat states with two
relative phase components non-deterministically (green). The
generators are terminated by a fixed unitary (yellow) circuit
consisting of beam splitters and phase shifters. In (b) the cat
states are corrected, using an additional measurement process
conditioned on the previous detection outcomes.
generates cat states with components which are orthog-
onal. However, this occurs with low probability. It is
not possible to improve the situation with any combi-
nation of (idealised) beam splitters and phase shifters,
since these implement unitary transformations. Hence,
we now devise a circuit, labeled Circuit II, to make input
cat components orthogonal, using additional processes of
measurement and feed-forward. This more sophisticated
scenario is depicted in Fig. 2(b). To identify a suitable
circuit, we investigate how a beam splitter transforms
phase references, starting with two classical fields. Here
the field in each mode is represented by a complex num-
ber, with the square amplitude corresponding to the in-
tensity, and the phase to the optical one. A 50:50 beam
splitter, configured so as not to impart additional phase
shifts to the modes, outputs two classical fields described
by the sum and difference of the values for the inputs (al-
tering both the square amplitudes and the phases). If the
input has a relative phase of 0 or π, and equal intensities
for each mode, the population is transferred entirely into
one mode. On the other hand, if the input has a relative
phase of plus or minus π/2, and equal intensities in each
mode, the relative phase and intensities are preserved.
Moving to the quantum case, we consider the action of
the beam splitter for a state, defined as in Eq. (2), with
a relative phase of ∆0, a total photon number S, and an
intensity S/2 in each mode. Computing the final state
explicitly, we find a scenario similar to the classical case,
Ubs|ψ∞〉 ∝
dθ exp (−iSθ)×
I1 exp (iθ)〉|
I2 exp [i (θ ± π/2)]〉 . (3)
This has an intensity SI1/ (I1 + I2) = [1− cos(∆0)] /2
in mode one and SI2/ (I1 + I2) = S [1 + cos(∆0)] /2 in
mode two, and a relative phase of plus π/2 when 0 <
∆0 ≤ π/2 and of minus π/2 when −π/2 ≤ ∆0 < 0
(we consider cases for which the intensity is increased in
favour of mode two).
The symmetry of the beam splitter transformation
makes it useful for altering the cats generated by Circuit
I, so that the relative phases are different by π. How-
ever, it creates a difference in the intensities between the
modes. To correct this, we propose beating mode two
against the vacuum, so as to to move the difference of
the intensities to an ancillary mode, which can be re-
moved by a photodetection. This method depends criti-
cally on feeding forward the result of the detections per-
formed by Circuit I, so that a variable beam splitter can
be set according to the value of ∆0. A variable beam
splitter can be implemented with 50:50 beam splitters
and variable phase shifters. The cost of correction is a
decrease in the total photon number, which varies non-
deterministically. As can be seen from Eq. (3), a fraction
of cos (∆0) of the photons are lost on average. Overall,
Circuits I through III constitute a complete N00N -state
generator. Additional mathematical analysis is given in
the supplementary online text. Runs for which Circuit
I fails to generate a Schrödinger cat state, or too many
photons are lost in the detection process are discarded.
The fidelities at the output are, on average, 0.87, 0.94 or
0.98, when a fraction of one third, one half, or two third
respectively of the input photons are detected by Circuit
I. Higher fidelities are possible when the photon number
at the input is small. If allowance is made for sufficient
input photons to be detected by Circuit I, and a further
half to be detected in Circuit II, the probability of failure
is not too large.
Finally, we suggest some possibilities for experimen-
tal implementation. For the source, we propose an op-
tical parametric oscillator setup for which the two mode
squeezed output of an optical parametric amplifier is en-
hanced by a cavity [22]. Note, however, that the cur-
rent purposes require twin beams of a much lower inten-
sity than is typical in many experiments, and that the
beams must be rendered frequency degenerate. Tech-
niques of feed-forward and photodetection are being de-
veloped with a view to quantum information technolo-
gies [23, 24]. For the source, an important problem is
imperfect correlation between the modes. If, for exam-
ple, two independent lasers of equal intensity provide the
input, the scheme generates the intended relative phase
correlations, but no entanglement [25]. Photodetectors
are subject to loss and dark counts. Losses will act to
degrade the source, reducing the relative number corre-
lation and increasing the uncertainty in the total photon
number. Dark counts are more problematic, mixing over
the phase for the superposition in Eq. (1). An alter-
native suggestion is using trapped bosonic atoms. One
possibility might be to work in a regime for which the
atomic wave-packets are much longer than the typical
scattering length, as proposed in Ref. [26]. Another is
to use Bose-Einstein condensates, for which a variety of
coherent operations have been demonstrated. Number-
resolved condensates might be obtained from the Mott
Insulator phase, while relative-number squeezing can be
achieved by different techniques.
In conclusion, we have proposed for the first time a
linear-optics based scheme that generates large N00N
states efficiently, the photon number at the output scal-
ing with that of the source — all the while maintaining
high fidelities, high success probabilities and a fixed num-
ber of circuit components. As well as being of immediate
interest for a range of applications, our results have con-
nections with other topics. For example, the scaling we
derive for our measurement-induced condensation pro-
cedure is of relevance to the study of the interference of
light from independent sources and localizing relative op-
tical phase, phenomena with analogs in different physical
systems [27]. We have left as an open question the extent
to which the scaling can be attributed to Bose statistics.
Regarding our N00N -state generators, the creation of
macroscopic entangled states is of interest for exploring
the quantum-classical transition. Finally, our study of
Schrödinger cat states may have application to quantum
computing, where Schrödinger cat states, defined for one
mode only, have been proposed to encode qubits, which
may be manipulated using standard experimental tech-
niques [28].
Acknowledgements
The authors would like to acknowledge support from the
Hearne Institute, the Army Research Office, and the Dis-
ruptive Technologies Office. H. C. would like to thank
Terry Rudolph, Ryan Glasser, Sonja Daffer and Yuan
Liang Lim for helpful discussions.
[1] Giovannetti, V., Lloyd, S. & Maccone, L. Quantum-
enhanced measurements: beating the standard quantum
limit. Science 306, 1330 (2004).
[2] Kapale, K. T., Didomenico, L. D., Lee, H., Kok, P. &
Dowling, J. P. Quantum interferometric sensors. Con-
cepts of Physics II, 225 (2005).
[3] Boto, A. N. et al. Quantum interferometric optical lithog-
raphy: exploiting entanglement to beat the diffraction
limit. Phys. Rev. Lett. 85, 2733 (2000).
[4] Teich, M. C. & Saleh, B. E. A. Microscopy with quantum-
entangled photons. Českloslovenský časopis pro fyziku
47, 3 (1997). English translation.
[5] Bollinger, J. J., Itano, W. M., Wineland, D. J. & Heinzen,
D. J. Optimal frequency measurements with maximally
correlated states. Phys. Rev. A 54, R4649 (1996).
[6] Ou, Z. Y. Fundamental quantum limit in precision phase
measurement. Phys. Rev. A 55, 2598 (1997).
[7] Boixo, S., Flammia, S. T., Caves, C. M. & Geremia,
J. M. Generalized limits for single-parameter quantum
estimation. Phys. Rev. Lett. 98, 090401 (2007).
[8] Giovannetti, V., Lloyd, S. & Maccone, L. Quantum
metrology. Phys. Rev. Lett. 96, 010401 (2006).
[9] Kominis, I. K., Kornack, T. W., Allred, J. C. & Romalis,
M. V. A subfemtotesla multichannel atomic magnetome-
ter. Nature 422, 596 (2003).
[10] Dowling, J. P. Correlated input-port, matter-wave inter-
ferometer: quantum-noise limits to the atom-laser gyro-
scope. Phys. Rev. A 57, 4736 (1998).
[11] Mitchell, M. W., Lundeen, J. S. & Steinberg, A. M.
Super-resolving phase measurements with a multi-photon
entangled state. Nature 429, 161 (2004).
[12] Walther, P. et al. De broglie wavelength of a non-local
four-photon state. Nature 429, 158 (2004).
[13] Resch, K. J. et al. Time-reversal and super-resolving
phase measurements. quant-ph/0511214 (2005).
[14] Sanders, B. C. Quantum dynamics of the nonlinear rota-
tor and the effects of continual spin measurement. Phys.
Rev. A 40, 2417 (1989).
[15] Fiurášek, J. Conditional generation of n-photon entan-
gled states of light. Phys. Rev. A 65, 053818 (2002).
[16] Zou, X., Pahlke, K. & Mathis, W. Generation of entan-
gled photon states by using linear optical elements. Phys.
Rev. A 66, 014102 (2002).
[17] Kok, P., Lee, H. & Dowling, J. P. Creation of large-
photon-number path entanglement conditioned on pho-
todetection. Phys. Rev. A 65, 052104 (2002).
[18] Knill, E., Laflamme, R. & Milburn, G. J. A scheme for
efficient quantum computation with linear optics. Nature
409, 46 (2001).
[19] Mølmer, K. Optical coherence: A convenient fiction.
Phys. Rev. A 55, 3195 (1997).
[20] Sanders, B. C., Bartlett, S. D., Rudolph, T. & Knight,
P. L. Photon-number superselection and the entangled
coherent-state representation. Phys. Rev. A 68, 042329
(2003).
[21] Bartlett, S. D., Rudolph, T. & Spekkens, R. W. Refer-
ence frames, superselection rules, and quantum informa-
tion. quant-ph/0610030v2 (2006).
[22] Zhang, Y., Kasai, K. & Watanabe, M. Investigation of
the photon-number statistics of twin beams by direct de-
tection. Opt. Lett. 27, 1244 (2002).
[23] Kok, P. et al. Linear optical quantum computing with
photonic qubits. Rev. Mod. Phys. 79, 135 (2007).
[24] Prevedel, R. et al. High-speed linear optics quantum
computing using active feed-forward. Nature 445, 65
(2007).
[25] Cable, H., Knight, P. L. & Rudolph, T. Measurement-
induced localization of relative degrees of freedom. Phys.
Rev. A 71, 042107 (2005).
[26] Popescu, S. KLM quantum computation with bosonic
atoms. quant-ph/0610043 (2006).
[27] Rau, A. V., Dunningham, J. A. & Burnett, K.
Measurement-induced relative-position localization
through entanglement. Science 301, 1081 (2003).
[28] Jeong, H. & Ralph, T. C. Schrodinger cat states for quan-
tum information processing. quant-ph/0509137 (2005).
Supplementary Material: Methods
In these supplementary notes, we provide further anal-
ysis of our N00N -state generator, consisting of Circuits
I, II and III, as depicted in outline in Fig. 2(b). First,
we specify notation for beam splitters, phase shifters and
states with well-defined relative phase correlations. For
the lossless beam splitter, we choose a notation which
makes explicit the “rotation” performed by such a device.
A beam splitter with transmittance τ and reflectance
(1− τ) acts to transform the annihilation operators ôj
for modes labeled j, according to the relations,
cos(γ) − sin(γ)
sin(γ) cos(γ)
with angular parameter γ, where τ = cos2 (γ) and 0 ≤
γ ≤ π/2. We denote this transformation Ubs (γ), and we
include, where necessary, phase shifts of χ at the input
port and −χ at the output port of the first mode, so that
Ubs (γ, χ) ≡ exp
γ exp(iχ)ô1ô
2 − γ exp(−iχ)ô
. For
example, Ubs (arccos (
τ ), π/2) corresponds to a sym-
metric beam splitter. We denote a phase shift trans-
formation on mode j, exp
†ôjχ
, by Ups (χ). For a
state defined, as in Eq. (2), with a total photon number S
and relative phase ∆0, it is helpful to incorporate a phase
factor exp (−iS∆0/2) into the normalisation (making the
definition symmetric between the modes). We then adopt
the following notation for a normalised Schrödinger cat
state,
|ψcat (∆0,Λ)〉 ∝ |ψ∞ (∆0)〉+ exp (iΛ) |ψ∞ (−∆0)〉 ,
having components with relative phases plus and minus
∆0, a phase for the superposition Λ (with the overall
normalisation constant assumed positive).
Next, we elaborate on the sequence of operations per-
formed by our N00N -state generator. We assume the
final state should have at least P photons, and that the
correlations in the relative phase are ideal. For the first
step, Circuit I, depicted in Fig. 1(b), implements the
transformation,
|l, r〉〈l, r|
3,4 Ubs
conditioned on the detection of l photons in mode 3 and
r photons in mode 4. A dual Fock state from the source
evolves to a cat state according to,
Circuit I :|N〉1|N〉2|0〉3|0〉4 −→ |ψcat (∆0, lπ)〉1,2 .
This cat state has relative phase components with values
plus and minus ∆0 ≡ 2 arccos
r/(l + r)
, and total
photon number 2N − l − r. Runs for which l or r are
zero must be discarded. It has been shown that values
for the relative phase are generated with approximately
equal frequency across the range [25], and hence these
failure events do not affect the scaling of the generator.
Next, Circuit II acts to transform the relative phase
correlations, to plus and minus π/2, in every case. When
r ≥ l, the relative phase correlations lie in the range
[−π/2, π/2], and a 50:50 beam splitter acting on the prin-
cipal modes corrects the relative phase correlations, while
increasing the intensity in mode two (and decreasing it in
mode one). To achieve the same outcome when l < r, we
suppose that a phase shift of π is applied in advance (on
either mode). This transforms the cat state generated by
Circuit I as,
Ups (π) |ψcat (∆0(l, r), lπ)〉 ∝ |ψcat (∆0(r, l), rπ)〉 .
Next, a beam splitter, with transmittance
[1− cos (∆0)] / [1 + cos (∆0)], transfers the differ-
ence of the intensities to the ancillary mode five. A
circuit for implementing the variable beam splitter is
given by the relation,
Ubs (γ)2,5 ≡
Ubs (π/4, π/2)2,5 Ups (γ)5 Ups (−γ)2 Ubs (π/4,−π/2)2,5 .
A photodetector measures Q photons in mode 5. Overall,
Circuit II implements the transformation,
|Q〉 〈Q|
Ubs {arccos [tan (∆0/2)]}2,5 Ubs (π/4)1,2 .
The cat state evolves as,
Circuit II :
|ψcat (∆0, lπ)〉−→|ψcat [π/2, lπ + (2N − l− r −Q)π/2]〉 .
We derived a full probability distribution for the out-
comes of the photodetection performed by Circuit II,
Prob(Q = 0, · · · , S − 1) = 1
1 + (−1)l cosS (∆0)
SCQ [1−cos (∆0)]S−Q cosQ (∆0)
Prob(Q = S) =
1 + (−1)l cosS (∆0)
1 + (−1)l
cosS (∆0) ,
where S = 2N − l − r is the total photon number prior
to detection, and C denotes a binomial coefficient. This
probability distribution is approximately binomial, and
the expected number of detections is S cos (∆0). If too
many photons are lost in Circuits I and II the run must
be aborted. The probability of this can be made small
by taking P ≃ N(1−f). In principle, excess photons can
be removed by an additional process, similar to Circuit
Finally, Circuit III implements the unitary transfor-
mation,
Ubs (π/4, π)1,2 Ups (π/2)2 .
The corrected cat state evolves as,
Circuit III :
|ψcat (π/2, lπ+(2N−l−r−Q)π/2)〉−→|P, 0〉+(−1)l |0, P 〉 ,
yielding the desired N00N state, with P = 2N−l−r−Q.
It may be noted that the superposition phase for the
N00N -state at the output depends on the measurement
record at the photodetectors. When l < r, the additional
phase shift in Circuit II causes this phase to be rπ rather
than lπ.
Next, we estimate the fidelities of the states produced
by our N00N state generator, and clarify its behaviour
for large photon number. To do this, we first compute
the fidelity for one component of a cat state generated
by Circuit I, which we denote by |ψG(∆0)〉. We assume,
as in Eq. (1), that the function G(X), describing the
localisation of the relative phase, assumes its Gaussian
asymptotic form. Note that the rate of localisation is
faster when phase correlations evolve at more than one
value. We assume that the state at the input is the dual
Fock state |N〉1|N〉2, and that a total of D = l + r de-
tections have occurred. Then,
〈ψG(∆0)|ψ∞ (∆0)〉
∫ π/2
d∆cosS
∫ π/2
∫ π/2
d∆d∆′ cosS
∆′2 +∆2
where S = 2N − D is the total photon number. This
result was derived assuming that S and D are not small.
The value for the fidelity depends only on the ratio of
detections to input photons. For example, when D/2N
is one half, F = 0.94, and when D/2N is two thirds,
F = 0.98. To verify this result, we computed numerically
exact values for the fidelity,
〈ψl,r|ψcat (∆0(l, r), lπ)〉
for a range of states |ψl,r〉 generated by Circuit I. For
input state |3〉1|3〉2, the fidelity is 0.94 for (l, r) = (1, 2)
and (2, 1), and anomalously it is 1 for (l, r) = (1, 1). For
input state |5〉1|5〉2 the values are 0.94 and 0.96 when
l + r = 5, and range from 0.96 to 1 when l + r = 6,
while for input state |15〉1|15〉2 the values range from
0.92 to 0.96 when l+ r = 15, and from 0.96 to 0.99 when
l + r = 20.
Finally, we performed a complete numerical simulation
of the N00N -state generator, to verify that Circuits I, II
and III work together as predicted. In particular, it was
necessary to check that Circuits II and Circuit III func-
tion as expected when the relative phase correlations for
the cat states are not perfectly well-defined. The results
are shown in Fig. 3. Each point in the plots corresponds
to a particular choice of input state, and measurement by
Circuit I. The height corresponds to the expected pho-
ton number for the output N00N state, and the color
to its fidelity. Averages are taken over all possible out-
comes to the third photodetection performed by Circuit
II. For comparison, the mesh shows the predictions of
the preceding analysis. Good agreement is seen between
these analytical predictions and the numerical results.
However, inspection of individual outcomes in Circuit II
reveals that the high fidelities are not maintained in ev-
ery case. Roughly speaking, improbable outcomes were
often found to have low fidelity.
0.88 0.9 0.92 0.94 0.96 0.98 1
Fidelity
FIG. 3: Fidelities (color) and photon number (vertical axis)
are displayed for outputs of our N00N states generator. Each
point corresponds to a possible outcome to Circuit I, for which
D photons are detected. Input states |N〉|N〉2 are considered
for N up to 15. Going from left to right, D/2N is one third,
one half and two thirds.
