All macroscopic quantum states are fragile and hard to prepare
Andrea López Incera
1
, Pavel Sekatski
2
, and Wolfgang Dür
1
1
Institut für Theoretische Physik, Universität Innsbruck, Technikerstraße 21a, 6020 Innsbruck, Austria
2
Departement Physik, Universität Basel, Klingelbergstraße 82, 4056 Basel, Switzerland
January 25, 2019
We study the effect of local decoherence
on arbitrary quantum states. Adapting tech-
niques developed in quantum metrology, we
show that the action of generic local noise
processes –though arbitrarily small– always
yields a state whose Quantum Fisher Infor-
mation (QFI) with respect to local observ-
ables is linear in system size N, indepen-
dent of the initial state. This implies that
all macroscopic quantum states, which are
characterized by a QFI that is quadratic in
N, are fragile under decoherence, and cannot
be maintained if the system is not perfectly
isolated. We also provide analytical bounds
on the effective system size, and show that
the effective system size scales as the inverse
of the noise parameter p for small p for all
the noise channels considered, making it in-
creasingly difficult to generate macroscopic
or even mesoscopic quantum states. In turn,
we also show that the preparation of a macro-
scopic quantum state, with respect to a con-
served quantity, requires a device whose QFI
is already at least as large as the one of the
desired state. Given that the preparation de-
vice itself is classical and not a perfectly iso-
lated macroscopic quantum state, the prepa-
ration device needs to be quadratically bigger
than the macroscopic target state.
1 Introduction
The last decade has seen increasing efforts to push
quantum mechanics to new regimes, and demon-
strate quantum features at ever larger scales. There
have been significant advances in the control and
manipulation of light and matter, and quantum ef-
fects have been demonstrated in a variety of systems
of up to mesoscopic size (see e.g. [17]). But will we
ever be able to prepare, maintain and confirm truly
Andrea Lóp ez Incera: andrea.lopez-incera@uibk.ac.at
Pavel Sekatski: pavel.sekatski@unibas.ch
Wolfgang Dür: wolfgang.duer@uibk.ac.at
macroscopic quantum states such as Schrödinger’s
cat? This is a question that is not only of philosoph-
ical interest, but may have impact on the planning
and the possibility of realizing future experiments.
It is sort of common knowledge that decoherence
destroys quantum features, and that macroscopic
quantum states are particularly susceptible. How-
ever, this statement is mainly restricted to a variety
of examples [816], and only a few general results
have been shown, see e.g. [17] for a review. Some
macroscopicity measures are intrinsically connected
with the stability of states under decoherence. Ref
[18] indeed takes the fragility under decoherence as
a measure for macroscopicity. More recent contribu-
tions take a different path, and define macroscopic-
ity by other means (see [17] for an overview). How-
ever, in some cases it turns out that the very criteria
that render a state macroscopic also imply that it is
susceptible to noise. One such example is given by
the measures of [19] and [20], which are based on
how good one can distinguish two (or more) states
that are in a quantum superposition with a classical
detector - or similarly how much information one
can extract. Intuitively, if a measurement appara-
tus can extract information to distinguish between
states, so can the environment. Hence there exists
a specific decoherence process that destroys quan-
tum coherence and thus macroscopicity for all such
states. Similar conclusions were obtained in [2123].
These statements are however restricted to specific
decoherence processes.
As argued in [17, 24], the variance of a state
w.r.t. a local observable A [25] –or more gener-
ally the Quantum Fisher Information (QFI), which
is the convex roof extension of the variance for mixed
states– can serve as a good measure for macroscopic-
ity. In fact many approaches to quantify macroscop-
icty make directly or indirectly use of the variance, or
the corresponding measures are closely related [17].
For a pure quantum state of N particles of the form
|ψi (|Ai + |Di) –where |Ai and |Di are classi-
cal states–, a large variance of O(N
2
) implies that
the expectation values for the two states w.r.t. A
are macroscopically distinct - i.e. the two states are
Accepted in Quantum 2019-01-10, click title to verify 1
arXiv:1805.09868v2 [quant-ph] 24 Jan 2019
very different. In fact all separable states have a
variance or QFI of O(N), while macroscopic quan-
tum states are characterized by a variance or QFI of
O(N
2
). The QFI also plays a central role in quan-
tum metrology, and in fact provides bounds like the
Cramér-Rao bound [26], for the achievable precision
in estimating a parameter.
Here we show that there are fundamental limi-
tations to maintain and prepare truly macroscopic
quantum states based on the QFI. In particular, we
find that any generic local decoherence process that
acts independently on the constituents of the multi-
particle quantum state renders a state microscopic
as the QFI after the noise process is O(N). This
result is obtained by adapting techniques developed
in the context of quantum metrology [27, 28], where
channel extension methods allow one to show that a
quantum scaling advantage in parameter estimation
disappears if one considers generic noise processes
[29, 30] by bounding the attainable QFI for any such
quantum channel. Similarly, we find that any initial
state is mapped to a state with QFI O(N) w.r.t. any
local observable under generic noise processes. The
only noise processes for which our methods do not
ensure linear scaling of the QFI are some restricted
rank 1 channels such as dephasing. Our methods
also allow us to obtain upper bounds on the effec-
tive size [18] of the system, where the effective size is
given by the minimal size of a quantum system that
can show the same features as the system in ques-
tion, i.e. has the same QFI. We consider different
types of noise processes and obtain analytic expres-
sions for the bound on the effective size, that scales
inversely proportional to the noise parameter p of
the single-qubit noise, N
eff
1/p, for small noise
(p 1). This implies that it becomes increasingly
difficult to maintain states of larger size.
Furthermore, we also obtain limitations on the
preparation process of macroscopic states. We find
that, in the typical case where the observable A is
the generator of the spatial rotation symmetry and
hence a conserved quantity for a closed system, the
preparation of a state with a certain QFI requires
an apparatus that already needs to have at least the
same QFI as the target state. This follows from the
equivalent of the processing inequality for the QFI
and the simple fact that, for a closed system, any
operation commutes with the symmetry. From this
one can conclude that the preparation of a macro-
scopic quantum state with QFI of O(N
2
) requires
an apparatus with a QFI that is quadratic in N. As
the preparation apparatus is usually considered to be
classical (and not itself in a quantum superposition
state), an apparatus of size M will only have a QFI of
O(M), and hence the apparatus needs to be quadrat-
ically bigger than the target state, M = O(N
2
).
One may also use our first result to argue that a
preparation apparatus will not be perfectly isolated
from the environment, and hence the resulting deco-
herence implies that the QFI of the apparatus is of
O(M). Notice that it was also found that a measure-
ment apparatus that is quadratically larger than the
system size is required to successfully confirm the
presence of a macroscopic quantum state [31]. Our
result extends this observation to the preparation
process.
The paper is organized as follows. In Sec. 2 we
introduce the QFI and discuss its role in quantum
metrology and macroscopicity. We also review the
channel extension method in quantum metrology. In
Sec. 3 we adapt the method to the macroscopicity
scenario and show that also in this case, one obtains
a linear QFI for all full rank and rank 2 channels. We
discuss rank 1 channels as well, and show that only
rank 1 Pauli channels may leave the QFI of states to
be of O(N
2
). Using a semidefinite program, we cal-
culate upper bounds on the effective size for finite
system sizes, and provide analytic results for sev-
eral noise channels such as local depolarizing noise
or amplitude damping in Sec. 4. Finally, we consider
the preparation process in Sec. 5 and show that for a
closed system the QFI with respect to the total spin
can not be increased, hence the effective size that
one can prepare is severely limited by the size of the
device used to prepare it.
2 Background and notation
2.1 QFI and metrology
Quantum Fisher Information (QFI) has been widely
used in the field of metrology, where some physical
parameter of interest ϕ is estimated with some pre-
cision ϕ (where
2
ϕ denotes the Mean Squared
Error of the unbiased estimator), bounded by the
so-called Cramér-Rao bound [26],
ϕ
1
r
F
ρ,
, (1)
where F
ρ,
denotes the QFI with respect to the
state ρ.
The QFI is mathematically defined as
F(ρ, ˙ρ) = 2
X
i,j:λ
i
+λ
j
6=0
|hi| ˙ρ |ji|
2
λ
i
+ λ
j
, (2)
Accepted in Quantum 2019-01-10, click title to verify 2
where ρ =
P
i
λ
i
|iihi| is the spectral decomposition of
the state ρ, and ˙ρ denotes the derivative with respect
to the parameter ϕ, which defines a one-dimensional
parametrization of the space of density matrices. In
the context of metrology and macroscopicity, the
unitary parametrization ρ(ϕ) = e
iHϕ
ρe
iHϕ
is nor-
mally used, where H denotes the time-independent
hamiltonian that generates the evolution of the state
ρ.
The general quantum metrology scenario consists
of N probes, e.g. atoms or photons, that sense the
parameter ϕ, so that by measuring the final state
one can estimate the value of the parameter. It has
been shown that, if the N probes sense the param-
eter independently, the achievable precision follows
the standard scaling (1/
N). This scaling can be
improved due to quantum properties such as entan-
glement, leading to the so-called Heisenberg scaling
(1/N) [32, 33], which corresponds to a quadratic
scaling (O(N
2
)) of the QFI with respect to the sys-
tem size N . However, this quadratic enhancement of
precision cannot be achieved if decoherence is taken
into account [3436]. In the context of metrology,
decoherence arises due to a noisy evolution of the
probes. In particular, we focus on the simplified
model analysed in [27], where N parallel, entan-
gled probes sense the parameter. However, the sens-
ing process may be subject to local noise on each
probe, so that the global evolution of the state of
N probes can be described by the tensor product
of the individual quantum channels Λ
N
ϕ
. In this
model, each probe undergoes a noisy evolution de-
scribed by a quantum operation that is of the form:
Λ
met
ϕ
(ρ) = E (u
ϕ
ρ u
ϕ
). The evolution u
ϕ
is applied
as a unitary rotation that encodes the parameter ϕ.
The decoherence process is introduced as a quantum
channel E characterized by the Kraus operators k
i
,
which do not depend on the parameter ϕ. Therefore,
the complete quantum channel reads
Λ
met
ϕ
(ρ) =
X
i
k
i
u
ϕ
ρ u
ϕ
k
i
=
X
i
K
met
i
(ϕ) ρ K
met
i
(ϕ).
(3)
2.2 Macroscopicity scenario
In the context of macroscopicity, we are interested
in the study of macroscopic quantum states them-
selves, and how the interaction with their surround-
ings eventually transforms them into classical sys-
tems. The main difference from the metrology sce-
nario is that the focus of study is not the system
evolution and how the noise interferes in the sens-
ing of a parameter. In fact, we optimize over all the
possible local observables. In our scenario, we use
the scaling of the QFI as a criteria to determine the
macroscopicity of a state after a given noise process
Λ(ρ). If the QFI of the state after the noise pro-
cess still scales as O(N
2
), the state has preserved its
macroscopicity.
In particular, we define macroscopic quantum
states in terms of the effective size [24]. A macro-
scopic state of N particles that has gone through
some noise channel, may perform as a quantum state
with N
eff
< N particles in a metrological task, i.e.
the QFI of the state after the noise may scale as a
state with N
eff
< N particles. The effective size is
defined as,
N
eff
(ρ) = max
H:local
F(ρ, H)
4N
, (4)
where F(ρ, H) denotes the QFI of the state ρ w.r.t a
local observable H. Therefore, we call a state macro-
scopic if N
eff
= O(N), that is, F(ρ, H) = O(N
2
).
Considering this specific macroscopicity scenario,
in which some noise process is applied to the initially
macroscopic quantum state, the one-probe quantum
channel has the form Λ
mac
ϕ
= u
ϕ
E(ρ) u
ϕ
; where a
perfect unitary evolution u
ϕ
is applied after the noise
process E(ρ) =
P
i
k
i
ρk
i
, described by the Kraus op-
erators k
i
. The noise process is applied before since
we are interested in computing the QFI of the noisy
state to analyse if the state is still macroscopic or
not. Note that a perfect evolution is considered, as
it does not represent a real physical process, but the
"virtual" evolution on which the QFI of the noisy
state E(ρ) depends. This evolution is just used to
compute the QFI but, for the purpose of macro-
scopicity, it does not have any further meaning. The
total channel has the form,
Λ
mac
ϕ
(ρ) =
X
i
u
ϕ
k
i
ρ k
i
u
ϕ
=
X
i
K
mac
i
(ϕ) ρ K
mac
i
(ϕ),
(5)
where K
mac
i
(ϕ) = u
ϕ
k
i
are the Kraus
operators of the total channel, satisfying
P
i
K
mac
i
(ϕ)K
mac
i
(ϕ) = . For simplicity, we
denote K
mac
i
(ϕ) as K
i
(ϕ) from now on. This Kraus
representation is not unique, as there exist unitary
equivalent representations,
˜
K
i
(ϕ) =
X
j
U
ϕ(ij)
K
j
(ϕ), (6)
where U
ϕ
= e
iϕh
denotes a unitary matrix gener-
ated by the hermitian matrix h.
2.3 Channel extension method
Despite the numerous advantages of the QFI, it is
not always possible to analytically compute it, es-
Accepted in Quantum 2019-01-10, click title to verify 3
pecially when considering some scenarios in metrol-
ogy, where it is also necessary to maximize the QFI
over all possible input states to obtain the maxi-
mum achievable precision. However, the channel ex-
tension method [35] provides an upper bound (CE
bound, where CE stands for channel extension) on
the QFI of a N -probe state, based solely on the
Kraus operators that describe the noise channel for
one probe. Intuitively, by extending the system
space -e.g. by adding an ancilla- and making the
channel act trivially on this extension, the precision
of an estimator can only increase, i.e.
F
ϕ
(ρ)) max
ρ
ext
F
ϕ
A
(ρ
ext
)), (7)
where ρ
ext
B(H
S
H
A
) denotes the joint state
system+ancilla.
Based on this idea, the CE bound can be derived
for both one-probe and N -probe channels. Consid-
ering a setting of N parallel, entangled probes that
undergo some local, uncorrelated noise process, the
CE bound has the form [35],
F
N
ϕ
(ρ
N
)) max
ρ
N
ext
F
N
ϕ
A
(ρ
N
ext
)) (8)
4 min
˜
K
{N ||α
˜
K
|| + N (N 1) ||β
˜
K
||
2
},
(9)
where ρ
N
ext
B(H
N
S
H
A
) and || || denotes the
operator norm. The parameters α
˜
K
and β
˜
K
are de-
fined in terms of the Kraus operators of the one-
probe channel
˜
K
i
as,
α
˜
K
=
˙
˜
K
˙
˜
K =
X
i
˙
˜
K
i
˙
˜
K
i
, (10)
β
˜
K
= i
˙
˜
K
˜
K = i
X
i
˙
˜
K
i
˜
K
i
, (11)
where
˙
˜
K
i
(ϕ) =
ϕ
˜
K
i
(ϕ). Throughout this work, we
make use of bold letters to designate the vector char-
acter of the Kraus operators, e.g.
˜
K denotes a col-
umn vector with
˜
K
i
as its elements. We also remark
that the optimization over all possible Kraus repre-
sentations in eq. (9) is equivalent to a minimization
over the possible h matrices (see eq. (6)).
This CE bound can be directly applied to the
macroscopicity scenario, the only difference being
the definition of the Kraus operators, that in this
case have the form K
i
(ϕ) = u
ϕ
k
i
(note that the evo-
lution is applied after the noise process).
3 Linear scaling of QFI
In this section, we apply the methods used in the
context of metrology [27, 28] to macroscopicity. In
particular, we study the scaling of the QFI bound
(9) as a way to quantify the macroscopicity of states
that have gone through a decoherence process. We
show that all noise processes, except some restricted
rank 1 Pauli channels, destroy the macroscopicity
of the initial quantum state, as the QFI bound of
the decohered state scales at most linearly in N . A
sufficient condition for the bound (9) to scale linearly
in N is β
˜
K
= 0, i.e. any noise process that satisfies
this condition yields a non-macroscopic state.
In order to study the bound (9) analytically, the
expressions for α
˜
K
and β
˜
K
in eqs. (10) and (11) need
to be written in terms of the variables of interest, i.e.
the h matrix and the Kraus operators that describe
the given noise process k
i
.
Considering N-qubit states, the one-qubit evolu-
tion u
ϕ
= e
iϕH
can be described as a generic ro-
tation in the Bloch sphere, i.e. H = σ
~n
= ~n · ~σ =
n
1
σ
1
+ n
2
σ
2
+ n
3
σ
3
. Given that
˜
K = U
ϕ
u
ϕ
k (see
eq. (5) and (6)) -where U
ϕ
acts on the space of
Kraus operators and u
ϕ
acts on the Hilbert space
of the one-qubit system-, its derivative is
˙
˜
K =
i(h + H)U
ϕ
u
ϕ
k. Hence, the expressions (10), (11)
for α
˜
K
and β
˜
K
have the form,
α = k
h
2
k + k
H
2
k + 2 k
hHk, (12)
β = k
(h + H)k. (13)
Note that we denote α
˜
K
, β
˜
K
as α, β for simplic-
ity. We remark that, for the purpose of this work,
the free variables –over which we optimize– are the
Gauge hamiltonian h, that comes from the unitary
equivalence of Kraus representations; and the phys-
ical hamiltonian H, that is also optimized in the
context of macroscopicity, where we do not deal with
any specific physical evolution. The Kraus operators
k are fixed depending on the noise process we study.
In the following subsections, we study how ini-
tially macroscopic quantum states are affected by
different noise processes. Thus, we start by
analysing if such processes fulfill the condition β = 0
(QF I = O(N)), which, in turn, means that they de-
stroy the macroscopicity of the initial states.
3.1 Linear scaling of the QFI bound for general
channels
We consider general full rank, rank 2 and rank 1
channels separately to show that all noise processes
Accepted in Quantum 2019-01-10, click title to verify 4
satisfy the condition β = 0 in eq. (13), except for
rank 1 Pauli channels. Thus, the macroscopicity
of the initial state cannot be maintained after such
noise processes.
For the purpose of this section, it is enough to
show that the different channels can fulfill β =
0. Thus, we only need to demonstrate that the
term k
hk in eq. (13) spans the whole vector
space of Hermitian operators S span
h
k
hk =
span{ , σ
x
, σ
y
, σ
z
}; so that the hermitian term
k
Hk can always be cancelled out (for more details
see Appendix B).
The specific cases are analysed in detail in Ap-
pendix A (general rank 1 channels), App. C (gen-
eral rank 2 channels) and App. D (general full rank
channels). We obtain that all the noise processes
can satisfy condition β = 0 except for rank 1 Pauli
channels, which implies that their QFI bound is lin-
ear in N. Hence, the macroscopicity of the initial
state can survive, in principle, only if it undergoes a
noise process described by a rank 1 Pauli channel.
One possible physical interpretation of this result
is the following: if one considers the channel exten-
sion method, in which the channels acts trivially on
some ancillas, then it is shown in [37] that there ex-
ists an error correction code that can remove rank 1
Pauli noise. Thus, the macroscopicity of the initial
state can be maintained after this type of channel
because the noise process can be corrected without
destroying the state. In the case where no ancillas
are allowed, this more physical interpretation is still
an open question since, to our best knowledge, there
is not an error correction in such scenarios.
For the rest of channels, we can further analyse
bound (9) to get a bound on the effective size (see
Sec. 4).
3.2 Full rank and rank 2 Pauli channels
In this section, we explicitly compute the conditions
h should satisfy to fulfill β = 0 for some useful ex-
amples such as full rank and rank 2 Pauli channels.
Pauli channels are described as,
E(ρ) =
r
X
i=0
p
i
σ
i
ρσ
i
, (14)
with Kraus operators {k
i
=
p
i
σ
i
}
i=0,..,r
, that is,
each transformation described by σ
i
is applied with
probability p
i
> 0 (thus,
P
i
p
i
= 1).
Let us first consider Full rank Pauli channels
(r = 3). We introduce the following notation for
the vector k and the h matrix,
k =
k
0
k
1
k
2
k
3
=
p
0
σ
0
p
1
σ
1
p
2
σ
2
p
3
σ
3
=
p
0
~σ
p
, (15)
h =
h
00
~
h
~
h H
. (16)
With this notation, the condition β = 0 (eq. (13))
reads,
p
0
h
00
+p
0
(
~
h
+
~
h) ~σ
p
+~σ
p
H~σ
p
+p
0
σ
~n
+~σ
p
σ
~n
~σ
p
= 0.
(17)
Specifically, the conditions for β = 0 that restrict
the h components are:
3
X
i=0
p
i
h
ii
= 0; (18)
2
p
0
p
i
Re(h
0i
) 2
p
j
p
k
Im(h
jk
)+
+(p
0
+ p
i
p
j
p
k
)n
i
= 0, (19)
for i, j, k = 1, 2, 3, where i 6= j 6= k and j < k. Re()
and Im() denote the real and the imaginary part re-
spectively. These conditions can always be fulfilled
with the proper choice of h, since all equations are
linearly independent and the number of variables ex-
ceeds the number of equations.
Similarly, we can study rank 2 Pauli channels and
get,
2
X
i=0
p
i
h
ii
= 0; (20)
2
p
0
p
1
Re(h
01
) + (p
0
+ p
1
p
2
)n
1
= 0; (21)
2
p
0
p
2
Re(h
02
) + (p
0
+ p
2
p
1
)n
2
= 0; (22)
2
p
1
p
2
Im(h
12
) + (p
0
p
1
p
2
)n
3
= 0. (23)
These conditions will be useful in the following
sections (in particular, in Sec. 4.1).
4 Upper bounds on effective size
Provided that β = 0, the linear bound on the QFI
has the form,
F
N
ϕ
(ρ
N
)) 4N min
˜
K
||α
˜
K
||, (24)
where the minimization over the possible Kraus rep-
resentations is, indeed, a minimization over the pos-
sible h matrices
1
that satisfy β = 0.
1
Although the minimization over h gives a tighter bound,
we remark that any choice of h that satisfies β = 0 is also a
Accepted in Quantum 2019-01-10, click title to verify 5
In the context of macroscopicity, such a linear
bound means that the macroscopicity of the ini-
tial N-qubit state cannot be maintained whenever
the system is subjected to a generic local noise pro-
cess. However, we can extract more information
from bound (24). By optimizing over the possible
local hamiltonians, a bound on the effective size N
eff
(Eq. (4)) can be derived,
N
eff
N
ϕ
(ρ
N
)) max
H:local
||α||
min
, (25)
where ||α||
min
= min
˜
K
||α
˜
K
||. The bound N
max
eff
=
max
H
||α||
min
represents the maximum number of
particles among which quantum correlations can re-
sist the decoherence action. As a simple example,
consider an initial state of N = 1000 qubits with
N
max
eff
= 20 after some noise process that fulfills
β = 0. The final state is no longer macroscopic,
but quantum coherence can be maintained among
groups of maximally 20 qubits.
4.1 Analytical results
In this section, we derive analytical expressions for
the bound on the effective size, and in all cases we
find that N
max
eff
1/p when the noise parameter p is
very low. In addition, we derive analytical bounds
for amplitude damping and depolarizing channels
p.
Throughout this section, we write α in terms of
the Pauli basis as α =
P
3
i=0
a
i
σ
i
, with coefficients
a
i
obtained by eq. (12). With this nota-
tion, α = a
0
σ
0
+ Cσ
~c
, where C =
q
a
2
1
+ a
2
2
+ a
2
3
and ~c =
a
1
C
,
a
2
C
,
a
3
C
. Hence, its singular val-
ues are {|a
0
+ C|, |a
0
C|}, i.e. ||α|| = |a
0
| +
q
a
2
1
+ a
2
2
+ a
2
3
.
4.1.1 Amplitude damping
The amplitude damping channel is described by the
Kraus operators,
k
0
=
1 0
0
1 p
!
, k
1
=
0
p
0 0
!
, (26)
where 0 p < 1.
Condition β = 0 leads in this case to a fixed h
matrix (no optimization over h can be done) of the
form,
h =
n
3
(n
1
+ in
2
)
q
1
p
1
(n
1
in
2
)
q
1
p
1 (
2
p
3)n
3
,
(27)
valid upper bound. In any case, it is not known if the upper
bound (24) is saturable for non-extended channels
where ~n = (n
1
, n
2
, n
3
) denotes the hamiltonian di-
rection.
After substituting in eq. (12), coefficients a
i
for
α =
P
i
a
i
σ
i
read,
a
0
=
1
p
2p
n
2
3
+
1
p
+ 2p 2, (28)
a
1(2)
= 2
1 p
p
n
1(2)
n
3
, (29)
a
3
=
2p
2
p
n
2
3
2p + 2. (30)
As explained above, the analytic expression for
||α|| is given by ||α|| = |a
0
| +
q
a
2
1
+ a
2
2
+ a
2
3
. Note
that, in this case, the h matrix is fixed, so ||α|| is
directly computed (without minimization).
In order to compute the bound on the effective size
N
eff
max
~n
||α||, we only need to optimize over the
hamiltonian directions. Considering that |~n| = 1,
the optimal ~n is obtained by solving
||α||
n
3
= 0. Note
that, in this case, the norm depends on a
2
1
+a
2
2
, which
in turn gives n
2
1
+ n
2
2
that can be written in terms of
n
3
as 1 n
2
3
. Since we study the function ||α|| in the
region n
3
[1, 1], we can distinguish two regimes,
1/
2 < p < 1 There are two maxima at n
3
=
±
(1p)
1+p
p(p
2
+p1)
. The bound on the effective size
is,
N
eff
3
p
1
p
2
+
2 3p
p
2
+ p 1
. (31)
0 p 1/
2 There are only local maxima at
n
3
= ±1, that give the bound,
N
eff
4
p
4. (32)
4.1.2 Local depolarizing noise
Local depolarizing noise is a full rank Pauli channel
of the form,
E(ρ) = (1 p)ρ +
p
3
3
X
i=1
σ
i
ρσ
i
, (33)
where 0 p < 0.75 and the Kraus operators are
k
0
=
1
0
, {k
i
=
q
p
3
σ
i
}
i=1,2,3
. Due to the
symmetry of the channel, ||α|| does not depend on
the direction of the hamiltonian H, so w.l.o.g. we
can choose ~n = (0, 0, 1). Thus, the bound on the
effective size is directly given by N
eff
||α||
min
.
Making use of the numerical methods described in
Sec. 4.2, we obtain that the optimal h reads,
h
03
= h
30
= Re(h
03
),
h
12
= h
21
= iIm(h
12
),
h
ij
= 0 otherwise, (34)
Accepted in Quantum 2019-01-10, click title to verify 6
whose terms are related by the condi-
tion β = 0 (eq. (19)): Im(h
12
) =
1
p
p
3p(1 p) Re(h
03
) + (
3
2
2p)
. Consider-
ing also that ~n = (0, 0, 1), coefficients a
i
now
read,
a
0
= (3
8
3
p) Re(h
03
)
2
+
6
3
s
1
p
1 Re(h
03
)
+ (
3
2p
1),
a
1
= a
2
= a
3
= 0. (35)
Optimizing the analytical expression ||α|| = |a
0
|
with respect to Re(h
03
) is straightforward and it
leads to the bound
N
eff
1
2p
+
1
9 8p
1. (36)
It can easily be seen that for p 1, N
eff
1
2p
8
9
.
4.1.3 Pauli channels with p 1
In this section, we obtain analytical bounds for full
rank and rank 2 Pauli channels considering that the
amount of noise applied to the macroscopic state
is infinitesimal, i.e. the noise parameters p
i
1
i 6= 0.
Full rank Pauli channels. E(ρ) =
P
3
i=0
p
i
σ
i
ρσ
i
. For this analysis, p
0
p
i
and p
i
1
for i = 1, 2, 3.
First, we consider the expression for β = 0 in
eq. (19). The last term (p
0
+ p
i
p
j
p
k
)n
i
' p
0
n
i
is of order O(1), so that Re(h
0i
) O(1/
p) and
Im(h
jk
) O(1/p) (where p generically denotes the
noise parameters, since p
1
p
2
p
3
) in order to get
β = 0. We remark that we use the index notation
i, j, k = 1, 2, 3, i 6= j 6= k, j < k throughout this sec-
tion. Furthermore, instead of minimizing over the
possible Kraus representations, we make an ansatz
for the Gauge hamiltonian h to simplify the analysis.
In this case, we consider,
h =
0 h
01
h
02
h
03
h
01
0 ih
12
ih
13
h
02
ih
12
0 ih
23
h
03
ih
13
ih
23
0
, (37)
where h
ij
i, j and h
0i
O(1/
p), h
jk
O(1/p) as argued above. Considering this, the terms
in h
2
scale as,
h
2
O(
1
p
) O(
1
p
p
)
O(
1
p
p
) O(
1
p
2
)
. (38)
Once we have settled the h matrix, we analyse the
resulting α term by term (eq. (12)) in order to get
the scaling of the bound N
eff
max
H
||α||
min
. The
first term k
h
2
k scales as O(1/p), whereas the other
two terms 2 k
hHk + scale as O(1). Thus, we can
approximate α k
h
2
k.
The corresponding a
i
coefficients are then,
a
0
'
3
X
i=1
h
2
0i
+
3
X
j<k=1
(p
j
+ p
k
)h
2
jk
+ o(1/p),
a
1
= a
2
= a
3
' 0, (39)
which give ||α|| = a
0
.
Taking into account condition (19), i.e. h
0i
=
2
p
j
p
k
h
jk
n
i
2
p
i
, we analytically minimize a
0
over the
variables h
jk
, and we get,
a
0
min
'
3
X
i=1
p
j
+ p
k
4p
j
p
k
+ 4p
i
(p
j
+ p
k
)
n
2
i
, i 6= j 6= k j < k,
(40)
for the optimal value h
jk
=
n
i
p
j
p
k
2(p
i
p
j
+p
i
p
k
+p
j
p
k
)
.
Finally, we analytically maximize over the hamil-
tonian direction ~n (subject to |~n| = 1) and we get,
N
eff
p
1
+ p
2
4p
1
p
2
+ 4p
3
(p
1
+ p
1
)
+ o(1/p), (41)
where we have assumed w.l.o.g. that p
1,2
> p
3
2
.
The optimal hamiltonian direction is (0, 0, 1).
It can easily be seen that the previous result for
the depolarizing channel (N
max
eff
= 1/2p + o(1/p)) is
recovered by setting p
1
= p
2
= p
3
= p/3.
Rank 2 Pauli channels. E(ρ) =
P
2
i=0
p
i
σ
i
ρσ
i
.
For this analysis, p
0
p
i
and p
i
1 for i = 1, 2.
Equivalently, other combinations of two Pauli ma-
trices can also be considered.
Analogously, we can apply the previous procedure
to rank 2 Pauli channels. As before, we can approx-
imate α k
h
2
k (O(1/p)) when we consider the
ansatz,
h =
0 h
01
h
02
h
01
0 ih
12
h
02
ih
12
0
, (42)
2
One can always unitarily rotate the Kraus operators k
such that the noise parameters are ordered like this.
Accepted in Quantum 2019-01-10, click title to verify 7
where h
0i
O(1/
p) for i = 1, 2 and h
12
O(1/p).
Furthermore, the β = 0 conditions (see Sec. 3) com-
pletely determine h
01(2)
'
n
1(2)
2
p
1(2)
and h
12
'
n
3
2
p
1
p
2
.
Optimizing over the hamiltonian direction, we get
N
eff
1
4
1
p
1
+
1
p
2
+ o(1/p), (43)
for the optimal direction ~n = (0, 0, 1).
We have shown that for all the analytical cases
studied (full rank and rank 2 Pauli channels and am-
plitude damping), the effective size bound scales as
N
eff
1/p when low values of the noise parameter p
are considered. This implies that macroscopic quan-
tum states are extremely fragile under decoherence,
since, even if the macroscopic state is subjected to
a very small amount of noise, a little increase in the
noise parameter leads to a decay in the macroscop-
icity of the system.
4.2 Numerical results
In order to obtain the upper bound (25), we first
optimize over the Kraus representations numerically.
Eq. (24) can be transformed into a semi-definite pro-
gram (SDP) [27, 28] of the form (see Appendix E),
minimize t,
subject to
t
˙
˜
K
˙
˜
K t
0, (44)
β
˜
K
= 0,
where ||α||
min
= t
2
min
and
˙
˜
K = i(h + H)k (note
that we have chosen U
ϕ=0
= and u
ϕ=0
=
w.l.o.g).
Once we get ||α||
min
via SDP, we optimize over the
possible hamiltonians (H = σ
~n
for qubits) by plot-
ting the parameter ||α||
min
for each hamiltonian di-
rection ~n, so that the optimal direction can be easily
visualized. We make use of the stereographic coordi-
nates to map each three-dimensional vector ~n inside
the Bloch sphere onto a point P inside a circle of
radius one. Considering that the QFI is symmetric,
we only have to take into account the lower half of
the Bloch sphere. Thus, the map is a one-to-one
correspondence between ~n and P ,
P =
n
1
1 n
3
,
n
2
1 n
3
, (45)
where ~n = (n
1,
n
2
, n
3
).
We apply these methods to different types of noise
processes. The results are presented below.
4.2.1 Full rank Pauli channels
In this section, we present the results obtained for
different choices of full rank Pauli channels,
E(ρ) =
3
X
i=0
p
i
σ
i
ρσ
i
, (46)
where
P
i
p
i
= 1.
The effective size bound N
max
eff
is given by the
maximum value of ||α||
min
(yellow points in the fig-
ures).
We consider a Pauli channel with p
1
6= p
2
6= p
3
(Fig. 1), where p
3
> p
1
> p
2
. Fig. 1(a) shows
the case where a global 0.07% of noise is applied
(p
0
= 0.9993), leading to an effective size bound
N
max
eff
= 1070. In Fig. 1(b), all the noise parameters
are increased, giving a bound on the effective size
of N
max
eff
= 105 particles, which means that, with
a global 0.7% of noise, quantum coherence is main-
tained among groups of maximum 105 particles in-
side the initial state of N particles. In Fig. 1(c), a
global 7% of noise gives N
max
eff
= 9. The optimal
hamiltonian direction is close to the y axis in these
cases. We show in Fig. 1(d) how the optimal hamil-
tonian direction changes to the x axis when the noise
parameters are not infinitesimal.
4.2.2 Rank 2 Pauli channels
We now continue with the study of rank 2 Pauli
channels, i.e. E(ρ) =
P
2
i=0
p
i
σ
i
ρσ
i
, with
P
i
p
i
= 1.
As in the previous section, the values of ||α||
min
for each hamiltonian direction ~n are plotted. The
maximum value of ||α||
min
(yellow points) is the up-
per bound on the effective size, N
max
eff
.
We consider Pauli channels with p
2
= 0. The ef-
fective size bounds are the same (for the same values
of the noise parameters) if p
3
= 0 or p
1
= 0 instead.
The only difference is the optimal hamiltonian di-
rection (see Fig. 2).
In particular, we consider the case where the noise
introduced by σ
z
is much bigger than the σ
x
(p
3
>
p
1
). It can be seen (Fig. 2(a)) that, for very low
values of the noise parameters (p
1
= 2 · 10
5
, p
3
=
1.5 · 10
4
), the effective size bound is macroscopic
(N
max
eff
= 14160), corresponding to hamiltonian di-
rections close to the y axis (0, ±1, 0). If noise pa-
rameters increase, the effective size bound rapidly
decreases to N
max
eff
= 135 (Fig. 2(b)) and N
max
eff
= 8
(Fig. 2(c)). We show in Fig. 2(d) how the effective
size bound is analogous for rank 2 Pauli channels
with p
3
= 0, the only difference being the optimal
hamiltonian direction.
Accepted in Quantum 2019-01-10, click title to verify 8
||α ||
min
x sphere axis
-1 0 1
y sphere axis
-1
-0.5
0
0.5
1
0
200
400
600
800
1000
(a) p
1
= 2 · 10
4
p
2
= 10
4
p
3
= 4 · 10
4
||α ||
min
x sphere axis
-1 0 1
y sphere axis
-1
-0.5
0
0.5
1
0
20
40
60
80
100
(b) p
1
= 2 · 10
3
p
2
= 10
3
p
3
= 4 · 10
3
||α ||
min
x sphere axis
-1 0 1
y sphere axis
-1
-0.5
0
0.5
1
0
2
4
6
8
(c) p
1
= 0.02 p
2
= 0.01
p
3
= 0.04
||α ||
min
x sphere axis
-1 0 1
y sphere axis
-1
-0.5
0
0.5
1
0
0.05
0.1
0.15
0.2
(d) p
1
= 0.2 p
2
= 0.1
p
3
= 0.4
Figure 1: Asymmetric Pauli channel p
2
< p
1
< p
3
. ||α||
min
values for the possible hamiltonian directions (points inside
the circle) are represented through a color code. The bound
on the effective size N
max
eff
is given by the maximum value
of ||α||
min
(yellow points). ~n
optimal
is close to the y axis in
all the cases considered except for fig. 1(d), where the noise
parameters are not infinitesimal, and ~n
optimal
gets close to
the x axis. It can be seen how N
max
eff
rapidly decays as the
noise parameters have higher values. (a) N
max
eff
= 1070 (b)
N
max
eff
= 105 (c) N
max
eff
= 9 (d) N
max
eff
= 0.
5 Preparation of macroscopic quantum
states
We now step back from the effect of noise on the ef-
fective size of quantum states, and consider the lim-
itations on the preparation of macroscopic states in
the first place, arising from the finite size of prepara-
tion device and fundamental symmetries of nature.
Then we come back to the noise, and interpret the
results of the previous section in the new light.
Consider a closed system composed of a "refer-
ence frame" |RFi and a target state |ψ
0
i, we call
the global state ρ = |ψ
0
ihψ
0
| |RFihRF|. Both the
RF and the target subsystems are composite of M
and N particles respectively, that we assume to be
qubits for simplicity (this is not crucial for the fol-
lowing argument). Define a local observable A
tot
of
the global system as the sum of local observables for
the subsystems
A
tot
= A
sys
+ A
RF
=
N+M
X
i=1
σ
(i)
n
, (47)
where each σ
(i)
n
acts on the i-th qubit.
||α ||
min
x sphere axis
-1 0 1
y sphere axis
-1
-0.5
0
0.5
1
0
5000
10000
(a) p
1
= 2 · 10
5
p
2
= 0
p
3
= 1.5 · 10
4
||α ||
min
x sphere axis
-1 0 1
y sphere axis
-1
-0.5
0
0.5
1
0
50
100
(b) p
1
= 0.002 p
2
= 0
p
3
= 0.015
||α ||
min
x sphere axis
-1 0 1
y sphere axis
-1
-0.5
0
0.5
1
0
2
4
6
8
(c) p
1
= 0.02 p
2
= 0
p
3
= 0.15
||α ||
min
x sphere axis
-1 0 1
y sphere axis
-1
-0.5
0
0.5
1
0
5000
10000
(d) p
1
= 2 · 10
5
p
2
= 1.5 · 10
4
p
3
= 0
Figure 2: Rank 2 Pauli channel. ||α||
min
values for the
possible hamiltonian directions ~n are represented. The max-
imum value of ||α||
min
(yellow points) corresponds to the
bound on the effective size N
max
eff
. ~n
optimal
is close to the y
axis when we consider p
2
= 0 (a,b,c) and it changes to be-
ing close to the z axis when p
3
= 0 (d). It can be seen that
N
max
eff
is still high for infinitesimal values of the noise param-
eters (a,d), but it rapidly decays when the noise parameters
are increased (b,c). (a) N
max
eff
= 14160 (b) N
max
eff
= 135
(c) N
max
eff
= 8 (d) Note that this case is analogous to (a),
i.e. N
max
eff
= 14160, the only difference being the optimal
hamiltonian direction.
As the total system is closed it obeys the usual
symmetries and conservation laws. In particular, its
time evolution commutes with spatial rotations, as
the isotropy of space requires the physics of a closed
system to be independent of the choice of orientation
of the coordinate system. Hence, the operations that
can be performed on the system also commute with
spatial rotations. The most general operation that
can be performed on a quantum system is captured
by the notion of quantum instruments: a quantum
instrument E acting on N + M qubits is a collection
of CP maps E
k
such that
P
k
E
k
is trace preserving
and
E (ρ) =
X
k
E
k
(ρ) |kihk|, (48)
where |ki is a classical label of the outcome of the
instrument. As mentioned above, because the sys-
tem is closed the isometry of space implies that the
rotations around the axis n whose unitary represen-
tation is U
ϕ
= e
iϕA
tot
commute with the operation
E
E U
ϕ
= U
ϕ
E , (49)
Accepted in Quantum 2019-01-10, click title to verify 9
(where U
ϕ
is formally extended to act trivially on
the outcome label |ki). Now consider the average
Fisher information of the state after the action of
the instrument
hFi =
X
k
p
k
F
E
k
(ρ)
p
k
, A
tot
, (50)
with p
k
= tr E
k
(ρ) the probability of the outcome k.
Because all |ki are orthogonal we have
hFi = F(E (ρ), A
tot
label
). (51)
This follows from the block diagonal structure of the
state E (ρ) with respect to k and the fact that A
tot
label
only acts non-trivially within each block. The
QFI in Eq. (51) is the susceptibility of the fidelity
F
ϕ
= F (U
ϕ
E (ρ) U
ϕ
, E (ρ)) (52)
with respect to small deviations of ϕ from zero F
lim
ϕ0
1F
ϕ
ϕ
2
. From the symmetry (49) we get
F
ϕ
= F (E (U
ϕ
ρ U
ϕ
) , E (ρ)) F (U
ϕ
ρ U
ϕ
, ρ), (53)
where for the last step we use the processing inequal-
ity for fidelity (the fidelity between two states can
not decrease under posprocessing E ). From the last
inequality we get that the QFI of the initial state
can only decrease under manipulations that com-
mute with the observable A
tot
hFi F(ρ, A
tot
) = F(|ψ
0
i, A
sys
) + F(|RFi, A
RF
).
(54)
Note that this relation holds if one traces out some
qubits for some outcomes k, this follows from the
fact that the tracing of subsystems also commutes
with U
ϕ
which acts locally on each qubit.
To summarise, we have just shown that the QFI of
a closed system with respect to an observable A
tot
,
that commutes with the symmetries e.g. is a gen-
erator of the symmetries, can not be increased by
any kind of manipulations. This is a direct conse-
quence of the "processing inequality for QFI" a
direct consequence of the processing inequality for
fidelity.
This limitation on the preparation of macroscopic
states allows to shed new light on the results of previ-
ous sections. Typically, in experiments the reference
frame system (e.g. a laser beam, a magnet, a double
slit...) used for the preparation is not well isolated,
but rather an open quantum system subject to all
kinds of interactions with the environment and de-
coherence. Consequently, we know from the results
of previous sections that its QFI O(M) can only
scale linearly with its size M . Hence, in order to
prepare a macroscopic state of size N, one requires a
reference frame system (a preparation device) which
is quadratically bigger M N
2
. We note that sim-
ilar results have been obtained for the detection of
macroscopic states in [31], where one can also find
far-reaching quantitative interpretation for this rela-
tion.
6 Summary and conclusion
Quantum Fisher Information (QFI) has been used
in the context of metrology to derive the Cramér-
Rao bound for the achievable precision of an es-
timation protocol. Large scale systems exhibiting
quantum properties such as entanglement have been
proven to achieve a quadratic scaling of the QFI with
the system size N, thus increasing the sensitivity of
the parameter estimation over the standard scaling
(QFI O(N )) achieved by separable states. As ar-
gued in [24], these different scalings of the QFI can
be used to identify genuinely macroscopic quantum
states as resource states that improve a metrological
protocol.
In this work, we adapt methods used in the field
of metrology to study the macroscopicity of N-qubit
states that have undergone a given noise process. In
particular, we analyse the Channel Extension bound
(CE bound), that gives an upper bound on the QFI
based solely on the Kraus operators of the local noise
channel acting on a single qubit.
We show that general full rank and rank 2 chan-
nels (including Pauli channels) present a linear scal-
ing of the QFI bound. With respect to rank 1 chan-
nels, only rank 1 Pauli channels such as dephasing
maintain a quadratic scaling of the QFI bound. This
means that initially macroscopic quantum states are
fragile under all local, uncorrelated noise channels
except for rank 1 Pauli channels.
In addition, we have studied the bound on the
effective size, that is, the minimal system size of a
quantum system that performs equivalently to the
system of study, i.e. the QFI of the state after the
noise may scale as a quantum state with N
eff
< N
particles. Hence, we consider that the state after the
noise process is macroscopic if N
eff
= O(N).
For all the noise channels considered, the effective
size rapidly decays with increasing noise parameter
p. Only for very low values of the noise parameter
(p 0), N
eff
bound can still be of the order of N .
We obtain analytic expressions for the N
eff
bound
for depolarizing noise, amplitude damping and full
rank and rank 2 Pauli channels with p 0. In all
cases, N
eff
1
p
in the limit p 1.
Accepted in Quantum 2019-01-10, click title to verify 10
Finally, we have shown that in a closed system
the average QFI of a subsystem with respect to a
conserved quantity and after any kind of manipula-
tions can not exceed the original QFI of the total
state. Hence the effective size of target state is lim-
ited by the QFI of the reference frame state used for
the preparation. A preparation device is typically an
open system subject to noise, and hence, in lights of
the results above, has a QFI scaling linearly with its
size. This allows us to conclude that the preparation
of a macroscopic state of some size requires a device
which is quadratically bigger.
Acknowledgements
We would like to thank Florian Fröwis, Michalis Sko-
tiniotis and Jan Kołodyński for helpful discussions
on parts of this project. This work was supported
by the Austrian Science Fund (FWF): P28000-N27,
P30937-N27 and SFB F40-FoQus F4012, by the
Swiss National Science Foundation (SNSF) through
Grant number P300P2_167749, the Army Research
Laboratory Center for Distributed Quantum Infor-
mation via the project SciNet and the EU via the
integrated project SIQS.
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Accepted in Quantum 2019-01-10, click title to verify 12
A General Rank 1 noise channels
General rank 1 noise channels can be generally expressed in terms of the basis {σ
i
}
i=0...3
as,
k
0
=
3
X
i=0
µ
i
σ
i
, (55)
k
1
=
3
X
i=0
ν
i
σ
i
, (56)
with µ
i
, ν
i
i. Since
P
i
k
i
k
i
= , these Kraus operators are related by k
0
=
q
k
1
k
1
.
In order to simplify eq. (56), unitarily rotated Kraus operators k
0
= Uk can be chosen (e.g. U = e
iθσ
1
, with
tan θ = i
ν
0
µ
0
) so that k
0
1
does not contain σ
0
. Thus, operator k
0
1
(that we rename as k
1
again for simplicity)
can be written as k
1
=
~
λ·~σ, where
~
λ
0
denotes the real part and
~
λ
00
denotes the imaginary part of the complex
vector
~
λ. Since we are considering rank 1 channels -k
0
is completely determined by k
1
-, and the hamiltonian
direction is optimized over all possible directions (see eq. (4)), direction
~
λ
0
can be arbitrarily chosen and
~
λ
00
can be decomposed as
~
λ
00
=
~
λ
00
k
+
~
λ
00
, where
~
λ
00
k
is parallel to
~
λ
0
and
~
λ
00
perpendicular. Thus, the expression
for k
1
reads,
k
1
= Re(λ
1
)σ
1
+ i (Im(λ
1
)σ
1
+ λ
2
σ
2
) = λ
1
σ
1
+
2
σ
2
, (57)
where λ
1
and λ
2
. Since k
0
=
q
k
1
k
1
,
k
0
=
A + B 0
0
A B
!
, (58)
where A + B > 0, A B > 0; with A = 1 |λ
1
|
2
λ
2
2
and B = 2 Re(λ
1
)λ
2
.
Considering β = k
(h + H)k (eq. (13)), the set of equations for β = 0 is,
Ah
00
+ (1 A)h
11
+ 2n
3
B = 0; (59a)
Bh
00
Bh
11
+ n
3
(2A 1) = 0; (59b)
h
Re(λ
1
) (
A + B +
A B) + λ
2
(
A + B
A B)
i
Re(h
01
)
Im(λ
1
) (
A + B +
A B) Im(h
01
) + n
1
(|λ
1
|
2
λ
2
2
+
p
A
2
B
2
) + 2n
2
Im(λ
1
)λ