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Quantum metrology with full and fast quantum control
Pavel Sekatski
1
, Michalis Skotiniotis
1 2
, Jan Kołodyński
3
, and Wolfgang Dür
1
1
Institut für Theoretische Physik, Universität Innsbruck, Technikerstr. 21a, A-6020 Innsbruck, Austria
2
Física Teòrica: Informació i Fenòmens Quàntics, Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellatera (Barcelona)
Spain
3
ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
September 6, 2017
We establish general limits on how precise
a parameter, e.g., frequency or the strength
of a magnetic ﬁeld, can be estimated with the
aid of full and fast quantum control. We con-
sider uncorrelated noisy evolutions of N qubits
and show that fast control allows to fully re-
store the Heisenberg scaling ( 1/N
2
) for all
rank-one Pauli noise except dephasing. For
all other types of noise the asymptotic quan-
tum enhancement is unavoidably limited to
a constant-factor improvement over the stan-
dard quantum limit (1/N) even when allow-
ing for the full power of fast control. The lat-
ter holds both in the single-shot and inﬁnitely-
many repetitions scenarios. However, even
in this case allowing for fast quantum control
helps to improve the asymptotic constant fac-
tor. Furthermore, for frequency estimation
with ﬁnite resource we show how a parallel
scheme utilizing any ﬁxed number of entangled
qubits but no fast quantum control can be out-
performed by a simple, easily implementable,
sequential scheme which only requires entan-
glement between one sensing and one auxiliary
qubit.
1 Introduction
Precision measurements play a fundamental role in
physics and beyond, as they constitute the main in-
gredient for many state-of-the-art applications and ex-
periments [1]. When investigating the limits of known
or speculative theories experimentally one quickly en-
ters into regimes where quantities and parameters
need to be measured with unprecedented precision.
In this context, it is of utmost importance to know
the ultimate limits nature sets on how precise any
given quantity can be determined and how to achieve
this.
These questions are at the focus of quantum metrol-
ogy [2]. Quantum mechanics is a probabilistic the-
ory and the intrinsically stochastic nature of mea-
surements ultimately limits the achievable precision.
When considering classical probes (or particles) inde-
pendently sensing a physical parameter, such as phase
or frequency, the maximum attainable precision (as
quantiﬁed by the Mean Squared Error—MSE) follows
the standard scaling, 1/N , where N is the number of
probes [3]. In turn it was shown that quantum entan-
glement allows one to achieve the so-called Heisen-
berg scaling (HS) in precision, 1/N
2
, a quadratic im-
provement as compared to classical approaches [4, 5].
These precision limits apply to both single-shot pro-
tocols [613] as well as protocols utilizing many rep-
etitions [3, 5, 14]. Still, it remains unclear to what
extent such an improvement can be harnessed in prac-
tice under non-idealized conditions, i.e., when taking
unavoidable noise and imperfections into account [15
17].
Due to the diﬃculty of obtaining exact precision
limits in the presence of noise, several asymptotic
lower bounds have been established for particular
noise models [1825]. These lower bounds are of-
ten not only cumbersome to evaluate but also hard
to optimize, relying on educated guesses, numerical
methods employing semi-deﬁnite programming or a
combination of the two. Nevertheless, these bounds
show that for typical uncorrelated noise processes the
possible gain due to the usage of quantum resources
is limited to a constant-factor improvement over the
standard scaling, as opposed to a diﬀerent scaling.
On the other hand, it was shown that for some types
of noise—namely noise perpendicular to the Hamilto-
nian that encodes the parameter of interest—the re-
striction to standard scaling can be circumvented, and
HS can be fully restored, by allowing for additional re-
sources, such as perfectly protected auxiliary particles
used to perform quantum error correction [2632].
In this work we develop a general framework that
provides us with analytic results for all types of noise
processes described by a time-homogeneous master
equation. In order to establish general limits, we ac-
count for the possibility of ancillary resources that do
not take part in the sensing process together with full
and fast quantum control (FFQC) of the system and
the ancillae
1
. Such FFQC allows one to eﬀectively
modify the Lindblad superoperator that describes the
noise process, a possibility which has hitherto not
1
We remark that we do not allow for additional control over
the environment as has been recently considered in [33, 34].
Accepted in Quantum 2017-09-04, click title to verify 1
arXiv:1603.08944v5 [quant-ph] 5 Sep 2017
been considered
2
. Our approach allows us to identify
all types of qubit noise that can be fully corrected, and
thus restore HS in precision. In addition, for all other
types of uncorrelated noise we are able to provide fully
analytic bounds applicable to the entire hierarchy of
metrology schemes. In contrast to the available meth-
ods [19, 21], our bounds can be generally determined
without need to explicitly solve the system dynamics.
Our main results can be summarized as follows:
(i) FFQC allows to restore HS by completely elimi-
nating any Pauli rank-one noise (that is not par-
allel to the Hamiltonian) at the cost of slowing
down the unitary evolution by a constant factor;
(ii) All other noise processes unavoidably limit the
quantum gain to a constant factor improvement
over standard scaling despite full quantum con-
trol. We obtain analytic bounds for the achiev-
able improvement factor.
(iii) For standard scaling-limited noise processes
FFQC may yet allow for signiﬁcant improvement
of precision in case of limited resources. We pro-
vide explicit examples demonstrating the advan-
tage of FFQC-assisted schemes over ones without
intermediate control for both continuous and dis-
crete processes.
Let us already stress that our results do not per-
tain only to the frequentist approach to parameter
estimation, which assumes an inﬁnite number of pro-
tocol repetitions (i.e., suﬃciently large statistics), but
also apply to the Bayesian approach within which one
considers a ﬁnite statistical data or, in the extreme
case, even a single experimental run. Moreover, the
most general protocol involving FFQC lends itself nat-
urally to adaptive strategies in which one is allowed
to modify the protocol (measurements, control oper-
ations etc.) "on-the-ﬂy"—basing on the record of the
The paper is organized as follows. In Sec. 2 we recall
some recent results in quantum metrology, and com-
pare them through a hierarchy of quantum metrol-
ogy schemes accounting for various levels of control.
We show that the FFQC scheme, that tops the hi-
erarchy, is the most powerful one allowed by quan-
tum mechanics. Sec. 3 introduces the model of noisy
quantum processes described by a time-homogeneous
master equation, and reviews the concept of quantum
Fisher information (QFI). In Sec. 4, we review upper
bounds on the QFI for metrology schemes, account-
ing for intermediate control, and generalize them to
incorporate FFQC (Sec. 4.2.1) demonstrating (i) and
(ii). Our extension of the bounds to the limit of short
evolution times is crucial to show (ii), and is the key
to obtain analytic bounds (as it removes the necessity
2
To the best of our knowledge, quantum control has been
used to modify the overall system-environment dynamics but
not the dissipative evolution itself [29, 35].
Preparation
...
...
...
...
Control #1
...
...
(a)
(c)
(d)
(b)
(c)
(d)
Preparation
...
...
ancillae
(b)
(a)
ancillae
ancillae
Control #2
Control #k
Control #1
Control #2
Control #m
Figure 1: Hierarchy of quantum metrology protocols for sens-
ing a frequency-like parameter ω that is encoded on each
probe during its noisy evolution E
ω,t
. Within the standard
schemes of type (a), N probes independently sense the pa-
rameter for a time t. In (b), additionally an unlimited number
of ancillae is allowed to perform a more general measurement
strategy that may include a single error-correcting step at the
end of the protocol. In (c), the sensing process is interspersed
at time intervals, δt, with quantum control operations act-
ing on both the sensing probes and ancillae. Finally in (d),
control operations are further allowed to be of inﬁnitesimally
small duration, i.e, δt dt, so that it is enough to con-
sider a single probe that sequentially senses the parameter
for the elongated total time t
0
= Nt. As full and fast quan-
tum control (FFQC), applied frequently on the global state,
may constitute swap operations any protocol of type (c) can
be simulated. Hence, the schemes form the following hi-
erarchy: (a)(b)(c)(d), when ordered in terms of their
ultimate power.
to solve system dynamics). In Sec. 5 we discuss the
performance of FFQC-assisted metrology both in the
single shot regime as well as the regime of asymptoti-
cally many repetitions. Sec. 6 considers protocols with
limited resources, where we show that FFQC-assisted
schemes outperform parallel schemes without inter-
mediate control (see (iii)). This is demonstrated for
noisy frequency estimation in Sec. 6.1, and for noisy
phase estimation in Sec. 6.2. We summarize and con-
clude our results in Sec. 7. A reader who is familiar
with quantum metrology, or who is primarily inter-
ested in the results, can directly proceed to sections
4.1 and 4.2.4 where the ﬁrst two of our main results
((i) and (ii)) are presented.
Accepted in Quantum 2017-09-04, click title to verify 2
2 Quantum metrology protocols
In a standard quantum metrology protocol, a system
consisting of N probes (photons, atoms etc.) is care-
fully engineered, so that each probe may be used to
independently sense a parameter of interest ω. As de-
picted in Fig. 1(a), the experimentalist prepares then
a suitably entangled state of the N probes, which un-
dergo the sensing process in parallel for a time t, dur-
ing which the unknown parameter ω is imprinted on
the state of the probes independently. The ﬁnal state
of the system is measured in order to most precisely
retrieve information regarding ω.
If the sensing process is noiseless, the strategy (a)
of Fig. 1 is known to optimally achieve the HS [5].
However, various uncorrelated noise types have been
shown to constrain the precision to follow the stan-
dard scaling in the asymptotic N limit, even if one op-
timally prepares the probes in an entangled but noise
robust state [15, 18, 19, 21].
A more powerful metrological protocol is depicted
in Fig. 1(b). Here, in addition to the N sensing
probes, the experimenter is equipped with ancillary
particles that do not take part in the sensing pro-
cess, and may be used to implement single-step error
correction at the ﬁnal measurement stage [22]. No-
tice that the protocol of Fig. 1(b) is more general
than that of Fig. 1(a); indeed one recovers the lat-
ter by choosing not to entangle the sensing probes
with the ancillae. Consequently any bound on pre-
cision valid for scheme (b) also applies to scenario
(a). Methods have been recently proposed that al-
low to derive bounds on the precision for protocols
(b) from a Kraus representation of a single chan-
nel E
ω,t
[21, 22]. Although these methods allowed
to prove the asymptotic standard scaling for various
noise types, the derived bounds cannot be guaranteed
to be tight (even in the asymptotic N limit). Never-
theless, in case of dephasing and particle-loss noise
types, the corresponding bounds have been shown to
be asymptotically achievable already within scheme
(a) [24, 25, 36]. For amplitude-damping noise, how-
ever, scheme (b) has been shown to give a strictly
better scaling than scheme Fig. 1(a) [37].
An interesting situation is the case of the X-noise: a
noise similar to dephasing but with the generator per-
fectly transversal to the Hamiltonian encoding the pa-
rameter. For a ﬁxed sensing time t the bounds [19, 21]
impose asymptotic standard scaling. However, in fre-
quency estimation one strongly beneﬁts from decreas-
ing the sensing time while increasing the number of
probes N. Indeed, by optimizing the sensing time
of the protocols (a) and (b) for each given N, it was
shown that one may asymptotically beat the standard
scaling and achieve the 1/N
5/3
precision scaling [38].
Moreover, it has been shown that one can even re-
store the HS of precision for such strictly transversal
noise after allowing for fast possibly multi-step error
correction [2730]. This is a particular case of the
metrological scenario depicted in Fig. 1(c). Here, in
addition to employing ancillary particles, the exper-
imenter is capable of freely interjecting the sensing
process with m control pulses that may act on both
probes and ancillae representing, e.g. error correction
steps [2730], dynamical-decoupling pulses [3944] or
any general adaptive feedback scheme [45, 46]. No-
tice that protocol (c) is more general than Fig. 1(b)
and one can obtain the latter from the former sim-
ply by allowing the intermediate operations to be the
identity.
The most general and powerful metrology protocol,
and the main focus of the current work, is the one de-
picted in Fig. 1(d). Here, the experimenter prepares
a suitably entangled state between a single probe and
many ancillae, and is capable of frequently interject-
ing the evolution with FFQC—an arbitrary number,
k, of most general intermediate control operations act-
ing on the overall state of probe-plus-ancillae. More-
over, by choosing k suﬃciently large one can ensure
the sensing time spent by the probe in between suc-
cessive FFQC steps to be inﬁnitesimal. It is this ad-
ditional power of frequently interjecting the sensing
process with fast quantum control that we exploit
throughout the remainder of this work to establish
the ultimate bounds on precision. This intermediate
control allows one in fact to modify the noise process,
a possibility that has not been considered in previous
approaches.
To see that this is indeed the most general strategy,
we note that for any protocol (c), with N parallel
probes and total sensing time t, there exists a protocol
of type (d) with total sensing time t
0
= Nt whose
metrological performance is just as good as that of
(c) or better. We note in passing that protocols (c)
and (d) are not equivalent; not only are the timesteps
in protocol (d) assumed to be inﬁnitesimally small,
but protocol (d) allows us to use the same resources
(namely N and t) in a "sequentialized" fashion.
The above metrological schemes apply equally to
the case of Bayesian parameter estimation, i.e., when
considering only a ﬁnite number of experimental rep-
etitions and a prior probability p(ω) representing our
knowledge about the parameter ω. Examples of
such Bayesian scenarios include reference frame align-
ment [912], adaptive Bayesian estimation [7, 13, 47],
and single-shot estimation [48, 49].
2.0.1 A remark on error correction
Let us brieﬂy remark that the most general
error correction step (encoding+syndrome read-
out+correction) only requires the dimension of the
space describing the ancillae to be the same as the
system one. Indeed, given that the system containing
the sensing probes has an overall dimension d that is
smaller than the dimension of the ancillary space, any
bipartite pure state of the two can be written using the
Accepted in Quantum 2017-09-04, click title to verify 3
Schmidt decomposition as
P
d
i=1
λ
i
|ii
S
|ψ
i
i
A
. Here,
{|ii}
d
i=1
denotes a given ﬁxed basis for the system,
while {|ψ
i
i}
d
i=1
is a set of orthonormal states of ancil-
lae that spans a d-dimensional subspace. Thus, since
any error correction protocol may be implemented by
restricting to states of the above form, it suﬃces to
choose the overall space describing the ancillae to be
of same dimension as the system.
3 Background
In order to assess the performance of any of the proto-
cols introduced in Sec. 2, we review below the crucial
tool of quantum metrology—the QFI. In addition, in
Sec. 3.2 we describe the general time-homogeneous
qubit noise processes we consider throughout this
work.
Notation. Before doing so, let us ﬁrst introduce the
notation we utilize. In what follows, we denote the
derivative with respect to the estimated parameter ω
(whenever applied to operators, vectors or matrices;
diﬀerentiating adequately the entries) as
˙
:=
d
dω
; and
shorten the trigonometric functions to c
θ
:=cos(θ) and
s
θ
:= sin(θ). ||A|| := max
ψ
|hψ|A |ψi| stands for the
operator norm, whereas and I for the identity oper-
ator and identity linear map respectively. We use bold
face to denote vectors, so that r := (x, y, z)
T
repre-
sents a 3-dimensional complex vector, while x, y and
z the Cartesian unit vectors. We denote the standard
Pauli 4- and 3-vectors as ς := (σ
0
, σ
1
, σ
2
, σ
3
)
T
and
σ := (σ
1
, σ
2
, σ
3
)
T
respectively (σ
0
:= ). The scalar
and outer products of vectors are then deﬁned in the
usual way irrespectively of the type of vector entries,
i.e., v
v
0
:=
P
i
v
i
v
0
i
and [v
0
v
]
ij
:= v
0
i
v
j
. We write
σ
n
:= n
T
σ to denote a Pauli operator in the spa-
tial direction n = (s
θ
c
φ
, s
θ
s
φ
, c
θ
)
T
. We reserve the
special font, M, for matrices whose entries, M
ij
, are
(complex) numbers, so that the multiplication of all
types of vectors can then be deﬁned in the standard
way: Mv =(
P
j
M
1j
v
j
,
P
j
M
2j
v
j
, . . . )
T
. For example,
any linear qubit map, L, can be unambiguously repre-
sented by constructing its corresponding matrix L (in
the Pauli basis) such that L =ς
Lς =
P
3
µ,ν=0
σ
µ
L
µν
σ
ν
.
3.1 QFI and its role in quantum metrology
In all metrology schemes discussed in Sec. 2 the in-
formation about the estimated parameter of interest
is encoded in the ﬁnal state of the system consisting
generally of both probes and ancillae. Thus, we de-
note the ﬁnal state of the system in any of the schemes
(a,b,c,d) of Fig. 1 as %
ω
. It is then %
ω
that is mea-
sured, in order to most precisely construct an estimate
of the parameter of interest, ω.
How best to quantify the precision of this estimate
depends crucially on how the information about ω has
been collected. If we are freely allowed to repeat the
protocol suﬃciently many times and collect statistics,
then the best precision of estimation, as quantiﬁed by
the mean squared error (MSE), is determined by the
well-known Cramér-Rao bound (CRB) [50, 51]
δ
2
ω
1
νF(%
ω
)
, (1)
where F(%
ω
) is the QFI of the state and ν is the num-
ber of repetitions. On the other hand, if the informa-
tion about ω is inferred from the outcome of just a
single repetition of the protocol, then the single-shot
MSE has to be averaged over the prior distribution
representing the knowledge we possess about ω prior
to its estimation. Then, the average mean squared
error (AvMSE ) can be lower-bounded by either the
Bayesian Cramér-Rao bound (BCRB) [49] (for well
behaved prior distributions) or the Ziv-Zakai bound
(ZZB) [48].
Whether considering the CRB, BCRB, or
ZZB, the crucial quantity of interest is the
QFI: F(%
ω
). Given the spectral decomposition
%
ω
=
P
i
p
i
(ω) |ψ
i
(ω)ihψ
i
(ω)|, the QFI of the state %
ω
evaluated with respect to the estimated parameter ω
F(%
ω
) := 2
X
i,j
p
j
+p
j
6=0
1
p
i
(ω) + p
j
(ω)
|hψ
i
(ω)| ˙%
ω
|ψ
j
(ω)i|
2
.
(2)
On the other hand, the QFI may be equivalently de-
ﬁned by using its relation with the ﬁdelity between
quantum states [52], F (ρ, σ):=tr
p
σρ
σ:
F(%
ω
) := 8 lim
dω0
1 F (%
ω
, %
ω+dω
)
dω
2
. (3)
The QFI satisﬁes the following important proper-
ties. It is additive, so that if the ﬁnal state of the
system is a product state, i.e., %
ω
= ρ
N
ω
, then its
QFI satisﬁes F(ρ
N
ω
)= NF(ρ
ω
) and thus gives rise to
standard scaling [3]. If the ﬁnal state of the probes is
pure, %
ω
=|ψ
ω
ihψ
ω
|, then
F(%
ω
) = 4
˙
ψ
ω
|
˙
ψ
ω
˙
ψ
ω
|ψ
ω
2
. (4)
Moreover, if the parameter is also encoded via
a unitary, so that |ψ
ω
i = e
iωH
|ψi where H is
some Hamiltonian, then F(%
ω
) = 4(hψ|H
2
|ψi
hψ|H |ψi
2
) =: 4 Var
|ψi
(H). In this case the QFI
is maximized by preparing the probes in the state
|ψi =
1
2
(|λ
max
i+|λ
min
i), where
λ
max/min
are the
eigenstates corresponding to the maximum/minimum
eigenvalues of H. Furthermore, if the N probes are
subjected to a unitary evolution generated by a lo-
cal Hamiltonian, e.g., H =
1
2
P
N
n=0
σ
(n)
3
, then such a
state corresponds to the GHZ state and attains the
HS by virtue of the CRB (1) [3]. However, note that
the QFI is important not only for metrology but also
for entanglement detection [5355]; deﬁnes a natural
Accepted in Quantum 2017-09-04, click title to verify 4
geometric distance between quantum states [56], thus
serving as a tool to derive speed limits on quantum
evolution [56, 57], as well as constitutes a measure of
macroscopicity [58]. The QFI is the primary focus of
this work and we further discuss its role in quantify-
ing the precision of estimation in metrology protocols
in more detail in Sec. 5.
3.2 Time-homogeneous qubit evolution
The interaction dynamics via which a probe senses
the parameter is generally described by a quantum
channel [59] (see Fig. 1), E
ω,t
, that encodes the pa-
rameter ω onto the state of the probe over the inter-
rogation time t: ρ
ω
(t)=E
ω,t
(ρ). The channel consists
of a unitary evolution part, encoding the parameter
of interest, and a non-unitary (noise) part describ-
ing additional system-environment interactions. In
this work we shall consider that the non-unitary part
arises due to uncorrelated noise processes described
by a time-homogeneous master equation of Lindblad
form—often referred to as the semigroup dynamics
[60]—which provides an appropriate description for
most physically relevant noise processes. In particu-
lar, in such a typical setting the environment is as-
sumed to be suﬃciently large such that it disturbs
the system in the same fashion independently of the
system state and the time instance [61]. Thus, the
time-homogeneity of evolution allows us to unambigu-
ously apply the FFQC techniques as any operation
applied on the system does not aﬀect the environ-
ment. Moreover, one may then freely swap the sensing
particles at any time with the ancillary ones, which
experience afterwards the same ﬁxed noisy dynam-
ics. In contrast, noise processes described by time-
inhomogeneous master equations have been recently
considered within the context of quantum metrology
in [6265], where time-inhomogeneity has been shown
to be beneﬁcial at short time-scales— the so-called
Zeno regime—where the bath can no longer be as-
sumed to be uncorrelated from the system. Let us
stress that such a regime, however, does not allow for
the general control operations considered here to be
unambiguously applied without explicitly modelling
the environment [35, 66].
We describe the evolution of a single qubit probe
by following time-homogeneous master equation
dρ
ω
(t)
dt
= i
ω
2
[σ
3
, ρ
ω
(t)] + L(ρ
ω
(t)), (5)
where ω is the parameter to be estimated and the
noise is given by the Lindblad super-operator, L. The
Liouvillian is then generally deﬁned as
L(ρ) :=
1
2
3
X
µ,ν=0
L
µν
([σ
µ
ρ, σ
ν
] + [σ
µ
, ρσ
ν
]) , (6)
where L is the matrix representation of the Lindblad
superoperator – an Hermitian positive semi-deﬁnite
matrix whose entries are independent of t and ω.
We note that all the noise terms L
0i
and L
i0
can be
straightforwardly corrected within FFQC as they lead
to an additional Hamiltonian evolution term in the
Eq. (5), sometimes referred to as Lamb-shift. This
ω-independent Hamiltonian term can be cancelled by
continuously applying the inverse unitary rotation. In
what follows we assume that the control operations
already incorporate such unitary and will, thus, be
concerned with the restriction of L to the subspace
spanned by {σ
1
, σ
2
, σ
3
}, which we denote as
¯
L.
We can group all the relevant noise processes into
three important families:
Rank-one Pauli noise of strength γ
L
1P
n
(ρ) :=
γ
2
(σ
n
ρσ
n
ρ), (7)
with the particular case of dephasing noise L
1P
z
for
which σ
z
= σ
3
.
General rank-one noise L
1G
r
deﬁned by the matrix
¯
L
1G
r
:= r r
, (8)
where r = (x, y, z)
T
and the strength of the noise
is denoted γ/2 = |r|
2
. The special case Re(r) ×
Im(r) = 0 corresponds to the rank-one Pauli noise
of Eq. (7), whereas whenever |Re(r)|= |Im(r)| and
Re(r)
T
Im(r) = 0 an amplitude damping channel is
recovered, which represents spontaneous emission
along some particular direction.
Rank-two Pauli noise L
2P
deﬁned by the matrix
¯
L
2P
:=
1
2
R
T
γ
1
γ
2
0
!
R
, (9)
where SO(3) and R
= R
z
(ϕ)R
y
(θ)R
z
(ξ) is
its matrix representation written in the Euler form
with angles (ϕ, θ, ξ). The special case of R
= cor-
responds to asymmetric X-Y noise, which we con-
veniently parametrize with γ
1
=γ p and γ
2
=γ(1p)
for 0p1 to deﬁne
L
X-Y
(ρ) :=
γ
2
(p σ
1
ρ σ
1
+ (1 p)σ
2
ρ σ
2
ρ).(10)
In order to explicitly determine the form of the
quantum channel describing the qubit evolution, one
has to integrate Eq. (5) [67]. The resulting dynamics
can then be expressed, e.g, with help of the corre-
sponding dynamical matrix, S (speciﬁed in the Pauli
operator basis), or via a Kraus representation as fol-
lows:
E
ω,t
(ρ) =
3
X
µ,ν=0
S
µν
(ω, t) σ
µ
ρ σ
ν
(11)
=
X
i
K
i
(ω, t) ρ K
i
(ω, t)
. (12)
Note that the Kraus representation of Eq. (12) is not
unique as starting from a set of r linearly independent
Accepted in Quantum 2017-09-04, click title to verify 5
Kraus operators, K = (K
1
, . . . , K
r
)
T
, we may simply
construct another valid set K
0
=uK choosing u to be
any (potentially ω-,t- dependent) r×r unitary matrix.
4 Noisy metrology with full and fast
control
In this section we derive the main results of our work.
In Sec. 4.1 we outline the optimal FFQC proto-
col suitable for correcting rank-one Pauli noise that
is not parallel to the Hamiltonian (7). In Sec. 4.2
we introduce the necessary tools required to bound
the QFI for FFQC (Fig. 1(d)), and show that any
other noise unavoidably leads to a linear scaling of the
QFI. We exemplify our ﬁndings by deriving analyti-
cal bounds for the QFI for several physically relevant
noise-types. Finally, in Sec. 4.3 we discuss diﬀerent
aspects in which FFQC allows to outperform scenar-
ios without fast control, using the cases of X-Y noise
(7) and transversal rank-one Pauli noise (40c) as ex-
amples.
4.1 Removing rank-one Pauli noise
Here we construct a general FFQC strategy that al-
lows one to correct for any rank-one Pauli noise L
1P
n
introduced in Eq. (7) with n 6= z. For convenience,
we perform a change of basis so that the noise is gen-
erated by σ
1
(n=x), whereas ω is encoded via a ro-
tated Hamiltonian σ
θ
:= s
θ
σ
3
+ c
θ
σ
1
. We consider
the qubit probe to be aided by an ancillary qubit
and deﬁne the two-qubit code space as a subset of
H
S
H
A
H
C
:= span{|00i, |11i}, with correspond-
ing projector Π
C
:= |00ih00| + |11ih11|, and the error
space, H
E
:=span{|01i, |10i}, with Π
E
:= Π
C
.
Let the probe-plus-ancilla be prepared in a pure
state % = |ψihψ| with |ψi= α |00i + β |11i H
C
. The
dynamics on the probe-plus-ancilla is then described
by Eq. (7) with only the probe system evolving
d%
dt
= i
ω
2
[σ
θ
, %] + L
1P
x
I
A
(%). (13)
Integrating Eq. (13) over an elementary timestep dt,
we may write the probe-plus-ancilla state, up to ﬁrst
order in dt, as
%(dt) = %(1
γ
2
dt) i
ω
2
[σ
θ
, %]dt (14)
+
γ
2
(σ
1
)%(σ
1
)dt + O(dt
2
).
Projecting %(dt) onto the code and error subspaces
yields the unnormalized states %
C
(dt) = Π
C
%(dt
C
and %
E
(dt)= Π
E
%(dt
E
given by
%
C
(dt) = %(1
γ
2
dt) i s
θ
ω
2
[σ
3
, %]dt + O(dt
2
),
%
E
(dt) =
γ
2
(σ
1
)%(σ
1
) dt + O(dt
2
) (15)
respectively.
If an error is detected, we simply apply σ
1
on the
sensing probe (so after correction ¯%
E
(dt) := (σ
1
)%
E
(dt)(σ
1
)), and otherwise do nothing. Hence,
the state of probe-plus-ancilla after an inﬁnitesimal-
timestep followed by fast error correction is the mix-
ture
¯%(dt) := %
C
(dt) + ¯%
E
(dt)
= % i s
θ
ω
2
[σ
3
, %] dt + O(dt
2
), (16)
which, to ﬁrst order in dt, is equivalent to a unitary
evolution under the projected Hamiltonian
ω
2
s
θ
σ
3
. As
the measurement may always be adjusted to compen-
sate for a known rotation of the Hamiltonian encoding
the parameter, the above strategy perfectly corrects
rank-one Pauli noise (7) at the price of slowing the
evolution down by a factor s
θ
=
p
1 (n
T
z)
2
.
We summarize the above result in the following ob-
servation:
Result 1. A general FFQC strategy allows one to
completely eliminate the impact of any rank-one Pauli
noise L
1P
n
(7) that is not exactly parallel to the
parameter-encoding Hamiltonian. The resulting dy-
namics of the system are then described by a noise-
less evolution with a rotated Hamiltonian and the es-
timated parameter being rescaled to
p
1 (n
T
z)
2
ω,
which still yields HS in precision.
Note that in the above derivation nothing forbids us
from replacing the scalar parameter ω with any opera-
tor B acting on an additional system described by the
Hilbert space H
B
. Consequently, one can eﬀectively
modify the noisy dynamics of any %
BS
B(H
B
H
S
):
d%
BS
dt
= i
1
2
[B σ
θ
, %
BS
] + I
B
L
1P
x
(%
BS
), (17)
by adding an ancillary qubit and implementing the
FFQC strategy described above. Then, the dynamics
of the error-corrected ¯%
BS
= tr
A
{¯%
BSA
} is governed
by
d¯%
BS
dt
= i
p
1 (n
T
z)
2
2
[B σ
3
, ¯%
BS
], (18)
with the rank-one Pauli noise removed. Hence, our
strategy can be directly used for the implementation
of unitary gates U = e
ig Bσ
3
in the presence of rank-
one Pauli noise on the qubit.
4.2 Other noise-types: unavoidable linear scal-
ing
In this subsection we prove our main no-go result,
F
L
4 α
L
t
0
, showing that even with FFQC the QFI
is bound to a linear scaling for all uncorrelated noise-
types other than rank-one Pauli noise, and to min-
imize the constant α
L
appearing in this bound for
several practically relevant noise-types introduced in
Accepted in Quantum 2017-09-04, click title to verify 6
Sec. 3.2. These results are presented in Sec. 4.2.4
and Sec. 4.2.5 respectively. The preceding sections
are devoted to elaborate the method that allows us to
derive these results. In Sec. 4.2.1 and Sec. 4.2.2 we
describe the channel extension (CE) method used to
upper bound the QFI [18, 21, 22, 37] and then review
the bounds it yields for strategies (a), (b) and (c) of
Fig. 1. However, the CE method fails to yield a mean-
ingful bound when considering the limit in which the
sensing process lasts for an inﬁnitesimal time-step. In
Sec. 4.2.3 we resolve this issue by amending the CE
method in order to obtain an upper bound on the QFI
for strategies that employ FFQC.
We stress that we are able to derive analytic bounds
by extending the CE method to the limit dt 0,
in which the problem actually simpliﬁes: one does
not have to solve the master equation and determine
the full dynamics! This is even more remarkable as
these bounds apply to any metrological scheme (see
the hierarchy in Fig. 1)
4.2.1 Upper bounds on the QFI
We would like to compare the maximum achievable
QFI in the various metrology schemes introduced in
Sec. 2 and, in particular, quantify the improvement (if
any) between protocols (b), (c) and (d). To this end
we use variations of the CE method [18, 21, 22, 37]
that allow one to upper-bound the QFI depending
on the scenario considered. Crucially, they constrain
the QFI using solely the properties of the quantum
channel responsible for encoding the parameter.
CE bound. The QFI of a state representing the
output of a quantum channel encoding the parameter
may always be upper-bounded purely by the channel
properties after performing a channel extension (CE).