improved to 1/n: the “Heisenberg limit” (Dunning-
ham and Kim, 2006).
φ
D1
D2
Beam
Splitter
Beam
Splitter
Input 1
Input 2
Figure 1: A Mach-Zehnder interferometer which can be
used to measure a phase φ. Photons are send through the
beam splitters and phase shift, then measured at the detec-
tors D1 and D2.
In order to create an entangled number state we
replace the ﬁrst beam splitter in the interferometer in
Fig. 1 by a “quantum beam splitter” (QBS). A quan-
tum beam splitter is like an ordinary interferometer,
but with a nonlinearity in one arm (Dunningham and
Kim, 2006) instead of the phase, and has the follow-
ing effect on state |n,0i:
|n,0i
QBS
−−→
1
√
2
[|n,0i+i|0,ni] (1)
The state on the right hand side is known as a
NOON state (Dowling, 2008). After creating the
NOON state, we then apply the phase shift to one
of the paths in the interferometer, as we did with the
SP state. We can then send this state through a sec-
ond QBS, and measure the number of particles at the
detectors. If we send a NOON state of n particles
through through this scheme, then the precision of
phase measurement can be seen to vary as δφ = 1/n,
the Heisenberg limit (Dunningham and Kim, 2006).
However, there is a problem with such an ap-
proach because NOON states are highly fragile to par-
ticle loss. Losing just one particle from a NOON
state (Eq. 1) will project the state onto either
e
inφ
|n,0i or i|0,ni. The global phase in each of
these cases is not physical and cannot be measured,
so all the phase information is lost when we lose
just one particle. Despite this, a number of clever
schemes have been devised with robustness to loss
which still capture sub shot noise limit precision, al-
beit not quite at the Heisenberg limit. An example of
one of these schemes is a NOON “chopping” strat-
egy (Dorner et al., 2009), in which multiple smaller
NOONs are sent through an interferometer instead
of one big one. Other examples include unbalanced
NOON states (Demkowicz-Dobrzanski et al., 2009),
BAT states (Gerrits et al., 2010) and mixtures of SP
and NOON states (Gkortsilas et al., 2012). While
these states can beat the shot noise limit when loss is
included, they are still fragile, and with large amounts
of loss they are outperformed by classical strategies.
We will now turn to a state which shows huge poten-
tial, as it is intrinsically robust to the effects of loss:
coherent states.
2.2 Entangled Coherent States
A coherent state is deﬁned as:
|αi = e
−
|α|
2
2
∞
∑
n=0
α
n
√
n!
|ni, (2)
where α is a complex amplitude and n is the particle
number. In order to achieve quantum enhanced mea-
surement, we still need to create an entangled coher-
ent state (ECS) (Sanders, 2012; Hirota et al., 2011):
N
|α,0i+e
iθ
|0,αi
(3)
where N = 1/
p
2 +2e
−|α|
2
cosθ. We could create
the ECS with θ = π/2 by sending input state |α,0i
through the QBS. Alternatively, the ECS with θ = 0
can be created by interacting a “cat state” N
α
(|+αi+
|−αi) with a coherent state |αiat a beam splitter (Joo
et al., 2011). This latter scheme is likely to be more
experimentally feasible. However, the issue of exper-
imental implementations will be left to later work.
In order to investigate the potential of the ECS in
quantum metrology, Joo et. al. (Joo et al., 2011)
calculated its quantum Fisher information (QFI). The
QFI for a general state ρ (Braunstein and Caves, 1994;
Boixo et al., 2009) is given by:
F
Q
= Tr
ρA
2
(4)
where A is found from solving the symmetric loga-
rithmic derivative ∂ρ/∂φ = 1/2 [Aρ + ρA]. The preci-
sion in the phase measurement (more speciﬁcally the
lower bound on the standard deviation) is given by the
quantum Cram
´
er-Rao bound (Braunstein and Caves,
1994):
δφ ≥
1
p
µF
Q
, (5)
where µ is the number of copies of the state (i.e. times
that the measurement is independently repeated).
This gives the best possible precision with which a
state can measure a phase. For NOON states and SP
states the quantum Cram
´
er-Rao bound gives us the
Heisenberg and shot noise limits respectively.
Joo et. al. used the QFI to show that with and
without loss the ECS can achieve better precisions
than SP, NOON, and some other candidate states.
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