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 [27–30]. 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 [27–30], dynamical-decoupling pulses [39–44] 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 [9–12], 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