CHAPTER 4 Obtainable Heisenberg limit in single parameter unitary esti-
4.3 Interferometer with a fixed total number of photons
The most canonical example of applying the above theorem is the problem of estimating the unknown phase in the interferometer. In such case the boundEq. (2.5) gives:
∆2θ˜≥ 1
N2, (4.12)
without guarantee of saturability. It was shown [Luis and Peřina [1996],Bužek et al. [1999], Berry and Wiseman [2000]], that for estimation of a completely unknown phase the additionalπ2 factor appears.
First, note that the valueθ should be identified withθ+ 2π; therefore, we need to choose a proper cost function considering this fact. The common choice is
C(θ,θ) = 4 sin˜ 2 ˜
θ−θ 2
, (4.13)
which for small difference θ˜−θ may be well approximated by (˜θ−θ)2. The average cost for the covariant measurement is therefore equal:
C= 1 2π
Z
dθ˜Tr(ρ0Uθ˜M0U˜†
θ)4 sin2˜
θ 2
, (4.14)
(so in notation fromEq. (4.11)g= 2πθ ). Without loss, we may restrict ourselves to fully symmetric space and then use the eigenbasis of Uθ˜. Since the mean cost is linear in ρ, we may restrict to a pure input state|ψN⟩, for which
Uθ|ψN⟩=Uθ
N
X
m=0
cm|m⟩
!
=
N
X
m=0
eimθcm|m⟩, (4.15)
where |m⟩ is the state with m photons in the upper (sensing) arm. After applying identity 4 sin2˜
θ 2
= 2−eiθ−e−iθ we obtain
C= Z dθ
2π
X
jk
c∗jckeikθMkje−ijθ
(2−eiθ−e−iθ) =X
jk
c∗jckMkj(2δj,k−δj+1,k−δj−1,k). (4.16) From conditionEq. (4.10) ∀mMmm = 1; moreover, without loss of generality, we may assume that
∀mcm ≥ 0 (otherwise, one can always redefine the vectors of the basis, by multiplying them by
proper phase factors). Therefore, the minimum value will be obtained for the maximal possible Mm,m+1. From positivity condition for M, Mm,m+1 ≤ 1, where the equality is satisfied if one chooses asM the projection onto vector PN
m=0|m⟩, such that:
1
2πMθ˜=|θ⟩⟨˜ θ|,˜ with |θ⟩˜ = 1
√2π
N
X
m=0
eimθ˜|m⟩. (4.17)
After that, the problem simplifies to:
min
|ψN⟩,{Mθ˜}
C= min
{cm} N−1
X
m=1
(2cmcm−cmcm+1−cmcm−1), (4.18) which is equivalent to finding the lowest eigenvector of the following tri-diagonal matrix:
2 −1 0 0
−1 2 −1 0 . . .
0 −1 2 −1
0 0 −1 2
... ... −1
−1 2
. (4.19)
The exact solution is known:
|ψN⟩= r 1
N + 2
N
X
m=0
sin
(m+ 1)π N+ 2
|m⟩ (4.20)
for which
C= 2
1−cos π
N+ 2
= π2 N2 +O
1 N3
, (4.21)
so indeed additionalπ2 factor appears when compared withEq. (4.12).
While the above provides the exact and complete solution to the problem, some comments are worth mentioning here.
First, while above we have used covariant measurement, the same cost may also be obtained by using the projective measurement, which may be seen as the discrete version of the previously discussed:
{Mj}={|χj⟩⟨χj|}Nj=0, |χj⟩= 1
√N + 1
N
X
m=0
eiN+12πjm|m⟩, ⟨χj|χk⟩=δjk. (4.22) (note that this is a specific feature of this peculiar model and does not need to be satisfied for
arbitrary covariant problems). This choice also has an intuitive interpretation, as this basis may be obtained by applying discrete Fourier transform to the basis with well-defined photons number in upper arm {|m⟩}Nm=0. Therefore, it stays in the analog to the problem of the position shift estimation in continuous space, where the shift generator is momentum operator pˆ= 1idxd, while the optimal measurement is{|x⟩}(which may be obtained by applying Fourier transform to{|p⟩}).
Nevertheless, adequacy is not exact, as for the phase θ (unlike for position x), no corresponding observable exists.
Second, even if the problem is analytically solvable for finite N, it will be useful to see what approximation may be used in the limit N → ∞ (that will allow us to understand better a more general case). Let us introduce m/N → µ and consider fˆ(µ) and the stateEq. (4.15) withcm =
√1
Nf(m/N)ˆ . ThenEq. (4.18)takes form:
|ψNmin⟩,{Mθ˜}C= min
fˆ
1 N2 · 1
N
N
X
m=1
f(m/N)ˆ 2 ˆf(m/N)−f(m/Nˆ + 1/N)−fˆ(m/N−1/N)
(1/N)2 , (4.23)
which, assuming that fˆ(µ) does not change very rapidly at distance of order 1/N, may be ap- proximated by (see [Imai and Hayashi [2009], Hayashi [2016]] for more discussion about exact correspondence in the limitN → ∞):
min
|ψN⟩,{Mk}
C ≈ 1 N2 min
f
Z 1 0
dµfˆ∗(µ)
− d2 dµ2
fˆ(µ), fˆ(0) = ˆf(1) = 0, (4.24) so the problem is equivalent to minimizing the kinetic energy of a single particle in the infinite well.
The solution is thenfˆ(µ) =√
2 sin(µπ), for which the cost is Nπ22.
Third, the vital difference between minimizing variance and minimizing the risk of coarse error in this context is worth noting. In the many repetitions scenario, in the limit of large k, for the ML estimator, the distribution of difference θ˜ML−θ converges to the gaussian one with the variance k, so the probability of making gross error decreases exponentially with the number of resources.
Contrary, in the case discussed in this section, the probability of getting outcomeθ˜where the actual value of the parameter is θis given as (for optimal covariant measurement Eq. (4.17)):
p(˜θ|θ) = 1 2π
N
X
k=0
eimθ˜e−imθcm
!2
≈ 1 2π
Z 1 0
dµeiN µ(˜θ−θ)f(µ)ˆ 2
=
=|f(N(˜θ−θ))|2 = 2π(1 + cos(N(˜θ−θ))
(N2(˜θ−θ)2−π2))2 , (4.25) so it decreases only like forth power of N. That is – the strategy using all resources to minimize the quadratic cost (or the one close to quadratic) turns out to be extremely poor for the problem
of discriminating the final sized region to which θ belongs. It was discussed in [Imai and Hayashi [2009]] that for optimal usage of all resources, these two problems need to be discussed independently, and completely different input states turn out to be optimal in both of them (while in the many repetitions scenario, the same strategy optimized both problems).
At last, the broad literature is dedicated to analyzing the relation between the results obtained for the phase estimation in both QFI and Bayesian formalism. The complex analysis of plenty of measurement strategies may be found in [Berry et al. [2009]]. As mentioned in Ch. 3, in [Higgins et al. [2009]], it was analytically proven that the Heisenberg scaling with all resources N may be obtained by a proper sequence of the measurement for n00n states with different n in a different iteration, which was also demonstrated in the experiment [Higgins et al.[2007,2009]]. In [Kaftal and Demkowicz-Dobrzański [2014]], the numerical analysis suggests that the factor π2 may be exactly obtained for such strategy if one allows for collective measurement on the quantum superposition of all these n00nstates.