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
ˆ
H
tot
=
ˆ
H
det
+
ˆ
H
int
,
(S1)
where
ˆ
H
det
is the free Hamiltonian for the detector, and
ˆ
H
int
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
ˆ
A
of the detector
at time
t
in the Heisenberg picture is given by
ˆ
A
(
t
)
=
ˆ
U
†
I
(
−∞
,
t
)
ˆ
A
(0)
(
t
)
ˆ
U
I
(
−∞
,
t
)
(S2)
with
ˆ
A
(0)
(
t
) denoting the operator under the free evolution:
ˆ
A
(0)
(
t
)
≡
ˆ
U
†
0
(
−∞
,
t
)
ˆ
A
ˆ
U
0
(
−∞
,
t
)
.
(S3)
The unitary operator for the free-evolution part is defined as
ˆ
U
0
(
−∞
,
t
)
≡ T
exp
{−
(
i
/
~
)
∫
t
−∞
d
t
′
ˆ
H
det
(
t
′
)
}
with
T
being the
time-ordering, and, for the interaction part, we have defined
ˆ
U
I
(
−∞
,
t
)
≡T
exp
{−
(
i
/
~
)
∫
t
−∞
d
t
′
ˆ
H
(0)
int
(
t
′
)
}
.
For the measurement to be linear,
ˆ
H
det
only involves linear
or quadratic functions of canonical coordinates, among which
their commutators are classical numbers, i.e., not operators;
the interaction
ˆ
H
int
is in the bilinear form:
ˆ
H
int
=
−
ˆ
F x
(
t
)
.
(S4)
As a result, Eq. (S2) leads to the following exact solution to
the input-port observable
ˆ
F
and output-port observable
ˆ
Z
:
ˆ
Z
(
t
)
=
ˆ
Z
(0)
(
t
)
+
∫
+
∞
−∞
d
t
′
χ
ZF
(
t
,
t
′
)
x
(
t
′
)
,
(S5)
ˆ
F
(
t
)
=
ˆ
F
(0)
(
t
)
+
∫
+
∞
−∞
d
t
′
χ
F F
(
t
,
t
′
)
x
(
t
′
)
.
(S6)
The susceptibility
χ
AB
(
A
,
B
=
Z
,
F
), which describes the de-
tector response to the signal, is defined as
χ
AB
(
t
,
t
′
)
≡
i
~
[
ˆ
A
(0)
(
t
)
,
ˆ
B
(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 di
ff
erent
times precisely without introducing additional noise. This can
happen only if
ˆ
Z
(
t
) commutes with itself at di
ff
erent times,
namely,
[
ˆ
Z
(
t
)
,
ˆ
Z
(
t
′
)]
=
0
∀
t
,
t
′
.
(S8)
It is called the condition of simultaneous measurability in
Ref. [S3] which also shows that it implies
[
ˆ
Z
(0)
(
t
)
,
ˆ
Z
(0)
(
t
′
)]
=
[
ˆ
F
(0)
(
t
)
,
ˆ
Z
(0)
(
t
′
)]
Θ
(
t
−
t
′
)
=
0
,
(S9)
or equivalently,
χ
ZZ
(
t
,
t
′
)
=
χ
FZ
(
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
di
ff
erence, i.e.,
χ
AB
(
t
,
t
′
)
=
χ
AB
(
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
ˆ
Z
(
ω
)
=
ˆ
Z
(0)
(
ω
)
+
χ
ZF
(
ω
)
x
(
ω
)
,
(S12)
ˆ
F
(
ω
)
=
ˆ
F
(0)
(
ω
)
+
χ
F F
(
ω
)
x
(
ω
)
.
(S13)
in which the Fourier transform
ˆ
A
(
ω
)
≡
∫
+
∞
−∞
d
t e
i
ω
t
ˆ
A
(
t
). Fur-
thermore, we consider the detector being in a stationary state,
i.e., its density matrix ˆ
ρ
det
commuting with
ˆ
H
det
. 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
)
≡
∫
+
∞
−∞
d
t e
i
ω
t
Tr[ ˆ
ρ
det
ˆ
A
(0)
(
t
+
τ
)
ˆ
B
(0)
(
τ
)]
,
(S14)
where
τ
can be arbitrary due to the stationarity, and we have
assumed Tr[ ˆ
ρ
det
ˆ
A
]
=
Tr[ ˆ
ρ
det
ˆ
B
]
=
0 without loss of general-
ity. Or equivalently, the spectral density can also be defined
through
Tr[ ˆ
ρ
det
ˆ
A
(0)
(
ω
)
ˆ
B
(0)
†
(
ω
′
)]
≡
2
π
S
AB
(
ω
)
δ
(
ω
−
ω
′
)
.
(S15)
The corresponding symmetrized version of the previously de-
fined spectral density is
̄
S
AB
(
ω
)
≡
1
2
[
S
AB
(
ω
)
+
S
BA
(
−
ω
)]
,
(S16)
which is a summation of both the positive-frequency and
negative-frequency spectra.
2
From the definitions of the susceptibility and spectral den-
sity, we have a general equality relating them to each other:
χ
AB
(
ω
)
−
χ
∗
BA
(
ω
)
=
i
~
[
S
AB
(
ω
)
−
S
BA
(
−
ω
)]
.
(S17)
When applying this to the case with
ˆ
A
=
ˆ
B
, it leads to the
famous Kubo’s formula:
Im[
χ
AA
(
ω
)]
=
1
2
~
[
S
AA
(
ω
)
−
S
AA
(
−
ω
)]
.
(S18)
Such an imaginary part of the susceptibility Im[
χ
AA
(
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
ˆ
Z
1
,
2
and
ˆ
F
can be generally represented in terms of
the ingoing field:
ˆ
Z
(0)
1
,
2
(
t
)
=
∫
∞
−∞
d
t
′
Z
1
,
2
(
t
−
t
′
)
ˆ
d
(
t
′
)
+
h
.
c
.,
(S19)
ˆ
F
(0)
(
t
)
=
∫
∞
−∞
d
t
′
F
(
t
−
t
′
)
ˆ
d
(
t
′
)
+
h
.
c
..
(S20)
Here
Z
and
F
are some complex-valued functions; h
.
c
.
de-
notes Hermitian conjugate;
ˆ
d
(
t
) is annihilation operator of the
ingoing field that satisfies the following commutator relation:
[
ˆ
d
(
t
)
,
ˆ
d
†
(
t
′
)]
=
δ
(
t
−
t
′
)
.
(S21)
In the frequency domain, Eqs. (S19) and (S20) can be rewrit-
ten as
ˆ
Z
(0)
1
,
2
(
ω
)
=
Z
1
,
2
(
ω
)
ˆ
d
(
ω
)
+
Z
∗
1
,
2
(
−
ω
)
ˆ
d
†
(
−
ω
)
,
(S22)
ˆ
F
(0)
(
ω
)
=
F
(
ω
)
ˆ
d
(
ω
)
+
F
∗
(
−
ω
)
ˆ
d
†
(
−
ω
)
,
(S23)
and the commutator for the ingoing field is
[
ˆ
d
(
ω
)
,
ˆ
d
†
(
ω
′
)]
=
2
πδ
(
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
ˆ
Z
1
,
2
satisfies
[
ˆ
Z
k
(
t
)
,
ˆ
Z
l
(
t
′
)]
=
−
σ
kl
y
δ
(
t
−
t
′
)
,
(S25)
where
k
,
l
=
1
,
2 and
σ
y
is the Pauli matrix. In the frequency
domain, the above commutator reads
[
ˆ
Z
k
(
ω
)
,
ˆ
Z
†
l
(
ω
′
)]
=
−
2
πσ
kl
y
δ
(
ω
−
ω
′
)
.
(S26)
Together with Eq. (S24), this implies the following constraint
on those functions in Eq. (S22):
Z
k
(
ω
)
Z
∗
l
(
ω
)
−
Z
∗
k
(
−
ω
)
Z
l
(
−
ω
)
=
−
σ
kl
y
,
(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.,
ˆ
ρ
det
=
|
0
〉〈
0
|
.
(S28)
Correspondingly, we have Tr[ ˆ
ρ
det
ˆ
d
(
ω
)
ˆ
d
†
(
ω
′
)]
=
2
πδ
(
ω
−
ω
′
)
and Tr[ ˆ
ρ
det
ˆ
d
†
(
ω
)
ˆ
d
(
ω
′
)]
=
0, which are equivalent to
S
ˆ
d
ˆ
d
†
(
ω
)
=
1
,
S
ˆ
d
†
ˆ
d
(
ω
)
=
0
.
(S29)
From Eqs. (S22) and (S23), the above spectral density for
ˆ
d
leads to
S
Z
1
,
2
F
(
ω
)
=
Z
1
,
2
(
ω
)
F
∗
(
ω
)
,
(S30)
S
F F
(
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
Z
1
F
(
ω
)
̄
S
∗
Z
2
F
(
ω
)]
=
1
8
[
S
F F
(
ω
)
−
S
F F
(
−
ω
)]
.
(S32)
With the Kubo’s formula Eq. (S18):
Im[
χ
F F
(
ω
)]
=
1
2
~
[
S
F F
(
ω
)
−
S
F F
(
−
ω
)]
,
(S33)
finally it gives rise to Eq. (21) in the main paper, i.e.,
Im[
̄
S
Z
1
F
(
ω
)
̄
S
∗
Z
2
F
(
ω
)]
=
~
4
Im[
χ
F F
(
ω
)]
.
(S34)
We can further show that Eq. (S34) also holds for the gen-
eral, stationary, pure Gaussian state—multi-mode squeezed
state ˆ
ρ
det
=
ˆ
S|
0
〉〈
0
|
ˆ
S
†
, in which the squeezing operator
ˆ
S
is
defined as [S6]
ˆ
S≡
exp
{
∫
∞
−∞
d
ω
2
π
[
ξ
(
ω
)
ˆ
d
†
(
ω
)
ˆ
d
†
(
−
ω
)
−
h
.
c
.
]
}
(S35)
with
ξ
(
ω
)
=
ξ
(
−
ω
). This is because
ˆ
S
only makes a Bogoli-
ubov transformation of
ˆ
d
. The spectral densities in Eqs. (S30)
and (S31) are in the same form as in the case of vacuum state,
after replacing
Z
1
,
2
by
Z
′
1
,
2
and
F
by
F
′
:
Z
′
1
,
2
(
ω
)
≡
Z
1
,
2
(
ω
) cosh
r
s
+
e
−
i
φ
s
Z
∗
1
,
2
(
−
ω
) sinh
r
s
,
(S36)
F
′
(
ω
)
≡
F
(
ω
) cosh
r
s
+
e
−
i
φ
s
F
∗
(
−
ω
) sinh
r
s
,
(S37)
where the real-valued functions
r
s
and
φ
s
are defined through
ξ
(
ω
)
≡
r
s
(
ω
)
e
i
φ
(
ω
)
. Such a transform will leave Eq. (S34) un-
changed.
3
III. MINIMUM OF
|
̄
S
ZF
/χ
ZF
|
Here we prove Eq. (22) of the main paper. Given the output-
port observable
ˆ
Z
=
ˆ
Z
1
sin
θ
+
ˆ
Z
2
cos
θ
, we have
̄
S
ZF
(
ω
)
=
̄
S
Z
1
F
(
ω
) sin
θ
+
̄
S
Z
2
F
(
ω
) cos
θ,
(S38)
χ
ZF
(
ω
)
=
χ
Z
1
F
(
ω
) sin
θ
+
χ
Z
2
F
(
ω
) cos
θ.
(S39)
The absolute value of their ratio is simply, for
θ
,
0,
R≡
∣
∣
∣
∣
∣
∣
̄
S
ZF
(
ω
)
χ
ZF
(
ω
)
∣
∣
∣
∣
∣
∣
=
∣
∣
∣
∣
∣
∣
̄
S
Z
1
F
(
ω
)
+
̄
S
Z
2
F
(
ω
) cot
θ
χ
Z
1
F
(
ω
)
+
χ
Z
2
F
(
ω
) cot
θ
∣
∣
∣
∣
∣
∣
.
(S40)
Using Eqs. (S10) and (S17), we can express the susceptibil-
ity
χ
Z
1
,
2
F
in terms of the unsymmetrized spectral density:
χ
Z
1
,
2
F
(
ω
)
=
i
~
[
S
Z
1
,
2
F
(
ω
)
−
S
FZ
1
,
2
(
−
ω
)]
.
(S41)
Form the expressions for
S
Z
1
,
2
F
shown in Eq. (S30), the above
ratio can be rewritten as
R
=
~
2
∣
∣
∣
∣
∣
1
+
αβ
1
−
αβ
∣
∣
∣
∣
∣
,
(S42)
where we have defined
α
≡
Z
∗
1
(
−
ω
)
+
Z
∗
2
(
−
ω
) cot
θ
Z
1
(
ω
)
+
Z
2
(
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
e
i
φ
α
with
φ
α
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
≡
φ
α
+
arctan[Re(
β
)
/
Im(
β
)]
.
(S47)
Due to the one-to-one mapping between
θ
and
φ
′
α
, minimizing
R
over
θ
is therefore equivalent to that over
φ
′
α
. The minimum
of
R
is achieved when
φ
′
α
=
π/
2 and
R
min
=
~
2
∣
∣
∣
∣
∣
1
−|
β
|
1
+
|
β
|
∣
∣
∣
∣
∣
.
(S48)
It is always smaller than
~
/
2, i.e.,
R
min
≤
~
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 or
F
(
ω
)
=
0, which is equivalent to
S
F F
(
−
ω
)
=
0 or
S
F F
(
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.
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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).