The quantum Cram\'er-Rao bound (QCRB) sets a fundamental limit for the
measurement of classical signals with detectors operating in the quantum
regime. Using linear-response theory and the Heisenberg uncertainty relation,
we derive a general condition for achieving such a fundamental limit. When
applied to classical displacement measurements with a test mass, this condition
leads to an explicit connection between the QCRB and the Standard Quantum Limit
which arises from a tradeoff between the measurement imprecision and quantum
backaction; the QCRB can be viewed as an outcome of a quantum non-demolition
measurement with the backaction evaded. Additionally, we show that the test
mass is more a resource for improving measurement sensitivity than a victim of
the quantum backaction, which suggests a new approach to enhancing the
sensitivity of a broad class of sensors. We illustrate these points with laser
interferometric gravitational wave detectors.Comment: revised version with supplemental materials adde

Adhikari, Rana X

Chen, Yanbei

Ma, Yiqiu

Miao, Haixing

Pang, Belinda

English

arXiv

The quantum Cramér-Rao bound (QCRB) sets a fundamental limit for the measurement of classical signals with detectors operating in the quantum regime. Using linear-response theory and the Heisenberg uncertainty relation, we derive a general condition for achieving such a fundamental limit. When applied to classical displacement measurements with a test mass, this condition leads to an explicit connection between the QCRB and the standard quantum limit that arises from a tradeoff between the measurement imprecision and quantum backaction; the QCRB can be viewed as an outcome of a quantum nondemolition measurement with the backaction evaded. Additionally, we show that the test mass is more a resource for improving measurement sensitivity than a victim of the quantum backaction, which suggests a new approach to enhancing the sensitivity of a broad class of sensors. We illustrate these points with laser interferometric gravitational-wave detectors

Adhikari, Rana X.

Caltech Authors - Main

Towards the Fundamental Quantum Limit of Linear Measurements of Classical Signals

The quantum Cram\'er-Rao bound (QCRB) sets a fundamental limit for the measurement of classical signals with detectors operating in the quantum regime. Using linear-response theory and the Heisenberg uncertainty relation, we derive a general condition for achieving such a fundamental limit. When applied to classical displacement measurements with a test mass, this condition leads to an explicit connection between the QCRB and the Standard Quantum Limit which arises from a tradeoff between the measurement imprecision and quantum backaction; the QCRB can be viewed as an outcome of a quantum non-demolition measurement with the backaction evaded. Additionally, we show that the test mass is more a resource for improving measurement sensitivity than a victim of the quantum backaction, which suggests a new approach to enhancing the sensitivity of a broad class of sensors. We illustrate these points with laser interferometric gravitational wave detectors

Adhikari, Rana

University of Birmingham Research Portal

Supplemental Material for “Towards the Fundamental Quantum Limit of Linear Measurements of
Classical Signals”
Haixing Miao, Rana X Adhikari, Yiqiu Ma, Belinda Pang, and Yanbei Chen
I. LINEAR-RESPONSE THEORY
Here we briefly introduce the linear-response theory that
has been applied in our analysis. One can refer to Refs. [S1–
S4] for more details. Given the model illustrated in Fig. 1 of
the main paper, the Hamiltonian for the measurement setup is
Hˆtot = Hˆdet + Hˆint , (S1)
where Hˆdet is the free Hamiltonian for the detector, and Hˆint
describes the coupling between the classical signal and the
detector. We consider the steady state with the coupling turned
on at t = −∞. The solution to any operator Aˆ of the detector
at time t in the Heisenberg picture is given by
Aˆ(t) = Uˆ†I (−∞, t)Aˆ(0)(t)UˆI(−∞, t) (S2)
with Aˆ(0)(t) denoting the operator under the free evolution:
Aˆ(0)(t) ≡ Uˆ†0(−∞, t)Aˆ Uˆ0(−∞, t) . (S3)
The unitary operator for the free-evolution part is defined as
Uˆ0(−∞, t) ≡ T exp{−(i/~)
∫ t
−∞ dt
′Hˆdet(t′)} with T being the
time-ordering, and, for the interaction part, we have defined
UˆI(−∞, t) ≡ T exp{−(i/~)
∫ t
−∞ dt
′Hˆ(0)int (t
′)}.
For the measurement to be linear, Hˆdet only involves linear
or quadratic functions of canonical coordinates, among which
their commutators are classical numbers, i.e., not operators;
the interaction Hˆint is in the bilinear form:
Hˆint = −Fˆx(t) . (S4)
As a result, Eq. (S2) leads to the following exact solution to
the input-port observable Fˆ and output-port observable Zˆ:
Zˆ(t) = Zˆ(0)(t) +
∫ +∞
−∞
dt′χZF(t, t′) x(t′) , (S5)
Fˆ(t) = Fˆ(0)(t) +
∫ +∞
−∞
dt′χFF(t, t′) x(t′) . (S6)
The susceptibility χAB (A, B = Z, F), which describes the de-
tector response to the signal, is defined as
χAB(t, t′) ≡ i
~
[Aˆ(0)(t), Bˆ(0)(t′)]Θ(t − t′) (S7)
with Θ(t) being the Heaviside function. Notice that the sus-
ceptibilities are classical numbers and only involve operators
under the free evolution, which are consequences of the de-
tector being linear.
For the measurement to be continuous, we need to be able
to projectively measure the output-port observable at different
times precisely without introducing additional noise. This can
happen only if Zˆ(t) commutes with itself at different times,
namely,
[Zˆ(t), Zˆ(t′)] = 0 ∀t, t′ . (S8)
It is called the condition of simultaneous measurability in
Ref. [S3] which also shows that it implies
[Zˆ(0)(t), Zˆ(0)(t′)] = [Fˆ(0)(t), Zˆ(0)(t′)]Θ(t − t′) = 0 , (S9)
or equivalently,
χZZ(t, t′) = χFZ(t, t′) = 0 , (S10)
which is central to the discussion of continuous, linear quan-
tum measurements.
When the free Hamiltonian for the detector is time-
independent, the susceptibility will only depend on the time
difference, i.e.,
χAB(t, t′) = χAB(t − t′) , (S11)
which is the case considered in the main paper. This allows us
to move into the frequency domain, and rewrite Eqs. (S5) and
(S6) as
Zˆ(ω) = Zˆ(0)(ω) + χZF(ω) x(ω) , (S12)
Fˆ(ω) = Fˆ(0)(ω) + χFF(ω) x(ω) . (S13)
in which the Fourier transform Aˆ(ω) ≡ ∫ +∞−∞ dt eiωtAˆ(t). Fur-
thermore, we consider the detector being in a stationary state,
i.e., its density matrix ρˆdet commuting with Hˆdet. The statisti-
cal property of the relevant operators, which defines the quan-
tum noise of the detector, can then be quantified by using the
frequency-domain spectral density, which is given by
S AB(ω) ≡
∫ +∞
−∞
dt eiωtTr[ρˆdet Aˆ(0)(t + τ)Bˆ(0)(τ)] , (S14)
where τ can be arbitrary due to the stationarity, and we have
assumed Tr[ρˆdetAˆ] = Tr[ρˆdetBˆ] = 0 without loss of general-
ity. Or equivalently, the spectral density can also be defined
through
Tr[ρˆdet Aˆ(0)(ω)Bˆ(0)†(ω′)] ≡ 2pi S AB(ω)δ(ω − ω′) . (S15)
The corresponding symmetrized version of the previously de-
fined spectral density is
S¯ AB(ω) ≡ 12[S AB(ω) + S BA(−ω)] , (S16)
which is a summation of both the positive-frequency and
negative-frequency spectra.
2From the definitions of the susceptibility and spectral den-
sity, we have a general equality relating them to each other:
χAB(ω) − χ∗BA(ω) =
i
~
[S AB(ω) − S BA(−ω)] . (S17)
When applying this to the case with Aˆ = Bˆ, it leads to the
famous Kubo’s formula:
Im[χAA(ω)] =
1
2~
[S AA(ω) − S AA(−ω)] . (S18)
Such an imaginary part of the susceptibility Im[χAA(ω)] quan-
tifies the dissipation, and, in the thermal equilibrium, it is
related to the symmetrized spectral density S¯ AA(ω) through
the fluctuation-dissipation theorem. The measurement pro-
cess is far from the thermal equilibrium, and therefore the
usual fluctuation-dissipation theorem cannot be applied. Nev-
ertheless, when the detector is ideal at the quantum limit with
minimum uncertainty, we can also find some general relations
between the susceptibility and the symmetrized spectral den-
sity, e.g., Eq. (18) and Eq. (21) in the main paper, the later of
which will be proven in the next section.
II. PROOF OF EQ. (21)
Here we show the proof of Eq. (21) in the main paper. In
the continuous, linear measurements, the detector is a contin-
uum field that contains many degrees of freedom which are
coupled to each other through the free evolution. The degrees
of freedom for the input and output port that we pick are con-
tinuously driven by the ingoing part of the continuum field,
which is similar to the in field introduced in Ref. [S5]. In the
steady state with the initial condition decaying away, their ob-
servables Zˆ1,2 and Fˆ can be generally represented in terms of
the ingoing field:
Zˆ(0)1,2(t) =
∫ ∞
−∞
dt′Z1,2(t − t′)dˆ(t′) + h.c. , (S19)
Fˆ(0)(t) =
∫ ∞
−∞
dt′F (t − t′)dˆ(t′) + h.c. . (S20)
Here Z and F are some complex-valued functions; h.c. de-
notes Hermitian conjugate; dˆ(t) is annihilation operator of the
ingoing field that satisfies the following commutator relation:
[dˆ(t), dˆ†(t′)] = δ(t − t′) . (S21)
In the frequency domain, Eqs. (S19) and (S20) can be rewrit-
ten as
Zˆ(0)1,2(ω) = Z1,2(ω)dˆ(ω) +Z
∗
1,2(−ω)dˆ†(−ω) , (S22)
Fˆ(0)(ω) = F (ω)dˆ(ω) +F ∗(−ω)dˆ†(−ω) , (S23)
and the commutator for the ingoing field is
[dˆ(ω), dˆ†(ω′)] = 2pi δ(ω − ω′) . (S24)
A natural choice for the output port is the outgoing part of
the continuum field, similar to the out field in Ref. [S5], which
guarantees that the condition in Eq. (S8) can be fulfilled due
to causality. Its two conjugate variables Zˆ1,2 satisfies
[Zˆk(t), Zˆl(t′)] = −σkly δ(t − t′) , (S25)
where k, l = 1, 2 and σy is the Pauli matrix. In the frequency
domain, the above commutator reads
[Zˆk(ω), Zˆ
†
l (ω
′)] = −2piσkly δ(ω − ω′) . (S26)
Together with Eq. (S24), this implies the following constraint
on those functions in Eq. (S22):
Zk(ω)Z ∗l (ω) −Z ∗k (−ω)Zl(−ω) = −σkly , (S27)
which is an important equality for the proof.
We first prove Eq. (21) in the case when the detector is in
the vacuum state, i.e.,
ρˆdet = |0〉〈0| . (S28)
Correspondingly, we have Tr[ρˆdet dˆ(ω)dˆ†(ω′)] = 2pi δ(ω − ω′)
and Tr[ρˆdet dˆ†(ω)dˆ(ω′)] = 0, which are equivalent to
S dˆdˆ† (ω) = 1 , S dˆ†dˆ(ω) = 0 . (S29)
From Eqs. (S22) and (S23), the above spectral density for dˆ
leads to
S Z1,2F(ω) = Z1,2(ω)F
∗(ω) , (S30)
S FF(ω) = |F (ω)|2 . (S31)
Using the constraint in Eq. (S27) and the definition of sym-
metrized spectral density Eq. (S16), we find
Im[S¯ Z1F(ω)S¯
∗
Z2F(ω)] =
1
8
[S FF(ω) − S FF(−ω)] . (S32)
With the Kubo’s formula Eq. (S18):
Im[χFF(ω)] =
1
2~
[S FF(ω) − S FF(−ω)] , (S33)
finally it gives rise to Eq. (21) in the main paper, i.e.,
Im[S¯ Z1F(ω)S¯
∗
Z2F(ω)] =
~
4
Im[χFF(ω)] . (S34)
We can further show that Eq. (S34) also holds for the gen-
eral, stationary, pure Gaussian state—multi-mode squeezed
state ρˆdet = Sˆ|0〉〈0|Sˆ†, in which the squeezing operator Sˆ is
defined as [S6]
Sˆ ≡ exp
{∫ ∞
−∞
dω
2pi
[ξ(ω)dˆ†(ω)dˆ†(−ω) − h.c.]
}
(S35)
with ξ(ω) = ξ(−ω). This is because Sˆ only makes a Bogoli-
ubov transformation of dˆ. The spectral densities in Eqs. (S30)
and (S31) are in the same form as in the case of vacuum state,
after replacing Z1,2 by Z ′1,2 andF byF
′:
Z ′1,2(ω) ≡ Z1,2(ω) cosh rs + e−iφsZ ∗1,2(−ω) sinh rs , (S36)
F ′(ω) ≡ F (ω) cosh rs + e−iφsF ∗(−ω) sinh rs , (S37)
where the real-valued functions rs and φs are defined through
ξ(ω) ≡ rs(ω)eiφ(ω). Such a transform will leave Eq. (S34) un-
changed.
3III. MINIMUM OF |S¯ ZF/χZF |
Here we prove Eq. (22) of the main paper. Given the output-
port observable Zˆ = Zˆ1 sin θ + Zˆ2 cos θ, we have
S¯ ZF(ω) = S¯ Z1F(ω) sin θ + S¯ Z2F(ω) cos θ , (S38)
χZF(ω) = χZ1F(ω) sin θ + χZ2F(ω) cos θ . (S39)
The absolute value of their ratio is simply, for θ , 0,
R ≡
∣∣∣∣∣∣ S¯ ZF(ω)χZF(ω)
∣∣∣∣∣∣ =
∣∣∣∣∣∣ S¯ Z1F(ω) + S¯ Z2F(ω) cot θχZ1F(ω) + χZ2F(ω) cot θ
∣∣∣∣∣∣ . (S40)
Using Eqs. (S10) and (S17), we can express the susceptibil-
ity χZ1,2F in terms of the unsymmetrized spectral density:
χZ1,2F(ω) =
i
~
[S Z1,2F(ω) − S FZ1,2 (−ω)] . (S41)
Form the expressions for S Z1,2F shown in Eq. (S30), the above
ratio can be rewritten as
R = ~
2
∣∣∣∣∣1 + αβ1 − αβ
∣∣∣∣∣ , (S42)
where we have defined
α ≡ Z
∗
1 (−ω) +Z ∗2 (−ω) cot θ
Z1(ω) +Z2(ω) cot θ
, (S43)
β ≡ F (−ω)
F ∗(ω)
. (S44)
With the constraint Eq. (S27), one can show that
|α| = 1 . (S45)
We can therefore write α as eiφα with φα being real, and obtain
R = ~
2
[
1 + |β|2 − 2|β| sin φ′α
1 + |β|2 + 2|β| sin φ′α
]1/2
, (S46)
in which we have introduced
φ′α ≡ φα + arctan[Re(β)/Im(β)] . (S47)
Due to the one-to-one mapping between θ and φ′α, minimizingR over θ is therefore equivalent to that over φ′α. The minimum
of R is achieved when φ′α = pi/2 and
Rmin = ~2
∣∣∣∣∣1 − |β|1 + |β|
∣∣∣∣∣ . (S48)
It is always smaller than ~/2, i.e.,
Rmin ≤ ~2 , (S49)
and reaches the equal sign when either
|β| = 0 or |β| → ∞ . (S50)
From the definition of β Eq. (S44), this corresponds to either
F (−ω) = 0 orF (ω) = 0, which is equivalent to
S FF(−ω) = 0 or S FF(ω) = 0 , (S51)
according to Eq. (S31). With the same argument as the one
presented in the previous section, the above conclusion is not
conditional on whether the detector is in the vacuum state or
in the general, stationary, pure Gaussian state.
Q.E.D.
[S1] R. Kubo, Reports on Progress in Physics 29, 255 (1966).
[S2] V. B. Braginsky and F. Khalilli, Quantum Measurement (Cam-
bridge University Press, 1992).
[S3] A. Buonanno and Y. Chen, Phys. Rev. D 65, 042001 (2002).
[S4] A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and
R. J. Schoelkopf, Rev. Mod. Phys. 82, 1155 (2010).
[S5] C. W. Gardiner and M. J. Collett, Phys. Rev. A 31, 3761 (1985).
[S6] K. J. Blow, R. Loudon, S. J. D. Phoenix, and T. J. Shepherd,
Phys. Rev. A 42, 4102 (1990).


Haixing Miao

Rana X Adhikari

Yiqiu Ma

Belinda Pang

Yanbei Chen

Crossref

Physical Review Letters

https://authors.library.caltech.edu/79813/2/1608.00766.pdf

Towards the Fundamental Quantum Limit of Linear Measurements of
  Classical Signals

Towards the Fundamental Quantum Limit of Linear Measurements of Classical Signals

Abstract

Similar works

Full text

Available Versions

Caltech Authors - Main

University of Birmingham Research Portal

Caltech Authors - Main

University of Birmingham Research Portal

Crossref