Fermat's Library | All macroscopic quantum states are fragile and hard to prepare annotated/explained version.

All macroscopic quantum states are fragile and hard to prepare

Andrea López Incera

1

, Pavel Sekatski

2

, and Wolfgang Dür

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. [1–7]). 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 [8–16], 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 [21–23].

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

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,

dρ

dϕ



, (1)

where F



ρ,

dρ

dϕ



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 [34–36]. 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 (Λ

ϕ

(ρ)) ≤ max

ρ

ext

F (Λ

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 (Λ

⊗N

ϕ

(ρ

N

)) ≤ max

ρ

N

ext

F (Λ

⊗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 (Λ

⊗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

(Λ

⊗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

+ 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

+ 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

+ 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

, which

in turn gives n

2

1

+ n

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

=

±

(1−p)

√

1+p

√

p(p

2

+p−1)

. 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

X

i=1

σ

i

ρσ

i

, (33)

where 0 ≤ p < 0.75 and the Kraus operators are

k

0

=

√

1 − pσ

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

ϕ→0

1−F

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

+ iλ

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

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

+

p

A

2

− B

2

) + 2n

2

Im(λ

1

)λ

2

= 0; (60a)

h

Re(λ

1

) (

√

A + B −

√

A − B) + λ

2

(

√

A + B +

√

A − B)

i

Im(h

01

)+

Im(λ

1

) (

√

A + B −

√

A − B) Re(h

01

) + n

2

(|λ

1

|

2

− λ

2

−

p

A

2

− B

2

) − 2n

1

Im(λ

1

)λ

2

= 0, (60b)

where the variables are: h

00

, h

11

, Re(h

01

), Im(h

01

). The set of equations (59) refers only to variables h

00

and h

11

, whereas the set (60) refers to variables Re(h

01

) and Im(h

01

). We denote them as set1 and set2

respectively. In order to show that β = 0, both set1 and set2 need to have solutions.

According to Rouché-Capelli theorem [38], a system of linear equations Sx = c is consistent (has at least

one solution) if and only if the rank of its coeﬃcient matrix (S) is equal to the rank of its augmented matrix

(S|c).

Let us ﬁrst consider set1, with coeﬃcient and augmented matrices,

(S

1

|c

1

) =

A 1 − A −2n

3

B

B −B n

3

(1 − 2A)

!

, (61)

where det S

1

= −B. If det S

1

6= 0, rank(S

1

) = rank(S

1

|c

1

) = 2 so set1 is consistent. If det S

1

= 0, i.e.

B = 0, rank(S

1

) = 1 so the condition rank(S

1

|c

1

) = 1 needs to be fulﬁlled. This condition is fulﬁlled if

n

3

(1 − 2A) = 0, that is, either n

3

= 0 or A = 1/2.

Similarly, we study set2, that has det S

2

= 2B. As before, if B 6= 0, set2 is consistent. Otherwise,

condition rank(S

2

|c

2

) = 1 needs to be fulﬁlled, together with condition n

3

(1 −2A) = 0, since both set1 and

set2 need to be consistent to get β = 0.

Altogether, the options for setting β = 0 are:

Accepted in Quantum 2019-01-10, click title to verify 13

· Re(λ

1

) 6= 0, λ

2

6= 0 Both set1 and set2 are consistent.

· (i) Re(λ

1

) = 0, λ

2

6= 0 or (ii) λ

2

= 0, Re(λ

1

) 6= 0 In both cases (i) and (ii), rank 1 Pauli channel is re-

covered, since the Kraus operator k

1

is of the form

(i) k

1

= i|m|σ

~m

, (62)

where σ

~m

= ~m ·~σ, with ~m = (

Im(λ

1

)

|m|

,

λ

2

|m|

, 0) and |m| =

q

Im(λ

1

)

2

+ λ

2

. Or,

(ii) k

1

= λ

1

σ

1

. (63)

In this case, the only possibilities for rank 1 Pauli channels to fulﬁll β = 0 are that, either the hamiltonian

direction is the same as the noise direction (H ∝ k

1

), or the noise parameters must fulﬁll the condition

|λ

1

|

2

+ λ

2

= 1/2 (which is equivalent to a measurement in the eigenbasis of (i)σ

~m

or (ii)σ

1

). If this is

so, the system is consistent (in fact, it has inﬁnite solutions).

These cases, where β = 0, correspond to types of noise channels that lead to a linear scaling (O(N )) of

the QFI bound. However, in this context of macroscopicity, we are interested in the eﬀective size, that is,

we optimize over all the possible "virtual" hamiltonians H. Therefore, the hamiltonian direction can always

be chosen to not be parallel to the noise direction.

B General considerations on the existence of linear QFI bound

The canonical Kraus representation of a general rank-r channel E is given by k = (k

0

. . . k

r

), with tr k

†

i

k

j

= 0

for i 6= j. In order to derive a linear bound on the QFI of state after the noise, we show that there exist

choices of Gauge Hamiltonians h which set β = 0 in Eq. (13) for all directions of the physical Hamiltonian

H = σ

n

. As k

†

Hk is Hermitian, it is suﬃcient to show that the Gauge term spans the whole vector space

of Hermitian operators

S ≡ span

h

k

†

hk = span{ , σ

x

, σ

y

, σ

z

}. (64)

Note that, for our goal, two channels E and E

0

related by a posterior unitary transformation E

0

= U ◦E are

equivalent. This follows from the fact that U can be absorbed in the parametrization of the physical Hamil-

tonians H

0

= UHU

†

, and we anyway optimize over all directions of H. On the other hand, such a unitary

modiﬁes the Kraus operators k

i

→ Uk

i

. It follows that we can always choose the unitary transformation

such that k

0

= k

†

0

is Hermitian. Hence, we assume this in the following.

Let us further rewrite the problem in a more convenient form by a reparametrization of h. To this end we

introduce a positive diagonal matrix

d =







√

d

0

.

√

d

r







(65)

with d

i

= tr k

†

i

k

i

> 0. We have that

k

†

hk = k

†

d

−1

d h d d

−1

k, (66)

such that a change of variable h → d h d and k → d

−1

k does not aﬀect S. The advantage of this parametriza-

tion is that the new operators are orthonormal tr k

†

i

k

j

= δ

ij

with respect to the Hilbert-Schmidt inner product.

Note that the new operators do not satisfy the trace preserving condition

P

k

†

i

k

i

6= anymore, instead one

has

X

d

i

k

†

i

k

i

= (67)

C General Rank 2 noise channels

With the notations of App. B, the problem for the general rank-2 noise is to show that S spans all Hermitian

operators for any choice of orthonormal operators k

0

= k

†

0

, k

1

and k

2

.

Accepted in Quantum 2019-01-10, click title to verify 14

With the operator k

0

being Hermitian, we can choose a basis in which it is diagonal and reads

k

0

= c

θ

+ s

θ

σ

z

(68)

with c

θ

> 0

3

. The ﬁrst observation is that the diagonal term of the Gauge Hamiltonian h =

diag(d

0

a, d

1

b, d

2

b) yields

k

†

hk = b + d

0

(a − b)( + 2c

θ

s

θ

σ

z

), (69)

where we used (67). If s

θ

6= 0 these terms span all diagonal Hamiltoninas, while for s

θ

= 0 it only spans

. We will treat the two cases separately. But before, note that the other two operators k

1

and k

2

are

orthogonal to k

0

and hence can be expressed as

k

i

= γ

i

(s

θ

− c

θ

σ

z

) + ξ

i

σ

x

+ ζ

i

σ

y

(70)

with |γ

i

|

2

+ |ξ

i

|

2

+ |ζ

i

|

2

= 1. In addition, we can choose the free phases k

i

→ e

iϕ

i

k

i

such that γ

i

∈ .

C.1 s

θ

6= 0 case

It remains to show that the remaining terms in h allow to span σ

x

and σ

y

. To do so, ﬁrst note that there exists

a Kraus representation of the channel

˜

k = u

(12)

(ν)k, one can also think of it as a unitary transformation of

the Gauge Hamiltonian

˜

h = u

(1)

(ν) h u

(12)†

(ν),

u

(12)

(ν) =







1 0 0

0 c

ν

s

ν

0 −s

ν

c

ν







(71)

after which ˜γ

2

= 0.

• Furthermore, assuming that ˜γ

1

6= 0, there exists a subsequent unitary transformation

¯

k = u

(01)

(ν)

˜

k

u

(01)

(ν) =







c

ν

s

ν

0

−s

ν

c

ν

0

0 0 1







, (72)

which does not change

¯

k

2

=

˜

k

2

and after which

¯

k

1

= λ + µ

x

σ

x

+ µ

y

σ

y

has no overlap with σ

z

and

λ ∈ . For this new h let us consider the contribution to S stemming from

¯

h

12

= r

12

+ i i

12

, one gets

r

12

(

¯

k

†

1

¯

k

2

+

¯

k

†

2

¯

k

1

)+ (73)

i

12

i(

¯

k

†

1

¯

k

2

−

¯

k

†

2

¯

k

1

). (74)

Pluging in the expressions for the operators give

(

¯

k

†

1

¯

k

2

+

¯

k

†

2

¯

k

1

) = 2λ(ξ

0

2

σ

x

+ ζ

0

2

σ

y

) + O

r

|{z}

span( ,σ

z

)

(75)

i(

¯

k

†

1

¯

k

2

−

¯

k

†

2

¯

k

1

) = −2λ(ξ

00

2

σ

x

+ ζ

00

2

σ

y

) + O

i

|{z}

span( ,σ

z

)

, (76)

and the contributions O

r

and O

i

can be ignored. The operators (ξ

0

2

σ

x

+ ζ

0

2

σ

y

) and (ξ

00

2

σ

x

+ ζ

00

2

σ

y

) span

all oﬀ diagonal Hamiltonians (which completes the proof), unless they are linearly dependent. In the

latter case we have

¯

k

2

=

˜

k

2

= c



σ

x

+ s



σ

y

∝ ξ

0

2

σ

x

+ ζ

0

2

σ

y

, (77)

after we get rid of the phase factor. Considering the terms stemming from

˜

h

02

with

˜

k

0

= k

0

gives the

following contribution

i(

˜

k

†

0

˜

k

2

− k

†

2

k

0

) = s

θ

[σ

z

, c



σ

x

+ s



σ

y

], (78)

this operator is oﬀ-diagonal and orthogonal to ξ

0

2

σ

x

+ ζ

0

2

σ

y

, hence recover the span of all oﬀ-diagonal

Hamiltonians.

3

In the case where k

0

= σ

z

one can always go to the channel σ

z

◦ E, for which k

0

= . In addition, if c

θ

< 0 the Kraus

operator can be simply multiplied by −1, without aﬀecting the channel E .

Accepted in Quantum 2019-01-10, click title to verify 15

• if ˜γ

1

= ˜γ

2

= 0, we get

k

1

= c

ω

σ

x

+ e

iφ

s

ω

σ

y

(79)

k

2

= s

ω

σ

x

− e

iφ

c

ω

σ

y

. (80)

Applying the unitary

u =







1 0 0

0 c

ω

s

ω

0 e

−iφ

s

ω

−e

−iφ

c

ω







, (81)

gives



ˆ

k

1

ˆ

k

2



=



σ

x

σ

y



. Hence, the following terms

i

2

(

ˆ

k

†

0

ˆ

k

1

−

ˆ

k

†

1

ˆ

k

0

) = −s

θ

σ

y

(82)

i

2

(

ˆ

k

†

0

ˆ

k

2

−

ˆ

k

†

1

ˆ

k

2

) = s

θ

σ

x

(83)

span the oﬀ-diagonal Hamiltonians. This completes the proof for the case s

θ

6= 0.

C.2 s

θ

= 0 case

This case corresponds to k

0

= . Without loss of generality we can set the basis of the Pauli operators such

that

k

1

= c

δ

σ

x

+ is

δ

(c

λ

σ

x

+ s

λ

σ

y

) (84)

with c

δ

> 0 and s

λ

6= 0. k

0

and k

1

yield the following contributions to S

k

†

0

k

0

= (85)

1

2

(k

†

0

k

1

+ k

†

1

k

0

) = c

δ

σ

x

(86)

i

2

(k

†

0

k

1

− k

†

1

k

0

) = −s

δ

(c

λ

σ

x

+ s

λ

σ

y

) (87)

k

†

1

k

1

= + c

δ

s

δ

s

λ

σ

z

, (88)

which span the whole vector space unless c

δ

= 1 and k

1

= σ

x

. We can repeat the same argument for k

2

,

ﬁnishing the proof unless it is also a Pauli operator. In consequence, up to a basis change the only remaining

case for which the existence of β = 0 has not been demonstrated is (k

0

, k

1

, k

2

) = ( , σ

x

, σ

y

). But for this

channel we obtain

1

2

(k

†

0

k

1

+ k

†

1

k

0

) = σ

x

(89)

1

2

(k

†

0

k

2

+ k

†

2

k

0

) = σ

y

(90)

i

2

(k

†

1

k

2

− k

†

2

k

1

) = −σ

z

, (91)

which completes the proof.

D Full-rank noise

With the notations of App. B the full rank noise is speciﬁed by 4 operators

k

i

= ξ

k

i

σ

k

, (92)

with orthonormal vectors

P

k

ξ

k∗

i

ξ

k

j

= δ

ij

. So the Kraus operators form an (orthonormal) basis of the complex

Hilbert space represented by complex 2 × 2 matrices with the Hilbert-Schmidt inner product. As any two

Accepted in Quantum 2019-01-10, click title to verify 16

bases are related by a unitary transformation, and full-rank channel in our notation can be related to

˜

k

0

= (93)

˜

k

1

= σ

x

(94)

˜

k

2

= σ

y

(95)

˜

k

3

= σ

z

(96)

by some unitary transformation. In this form, it is direct to see that

˜

k

†

h

˜

k spans all hermitian matrices.

E QFI bound as a semi-deﬁnite programming

Given the matrix α, the minimization of its operator norm (maximum singular value) can be expressed as,

minimize t,

subject to t

2

−

˙

˜

K

†

˙

˜

K  0,

β

˜

K

= 0, (97)

from where it can be seen that ||α|| = t

2

.

Due to the Schur complement condition for positive semideﬁniteness, eq. (97) is equivalent to,

minimize t,

subject to





t

d

1

˙

˜

K

†

˙

˜

K t

(r+1)·d

2





 0,

β

˜

K

= 0, (98)

where the Kraus operators {

˙

˜

K

i

}

r

i=0

are d

2

× d

1

matrices and

d

is the d × d identity matrix. Problem (98)

is a semi-deﬁnite program since the Kraus operators

˙

˜

K

i

= −i

P

j

h

ij

k

j

− iσ

~n

k

i

are linear in h. Explicitly,

condition β

˜

K

= 0 is in our case

P

ij

h

ij

k

†

i

k

j

+

P

i

k

†

i

σ

~n

k

i

= 0.

To solve problem (98) we used CVX, a package for specifying and solving convex programs [39, 40].

Accepted in Quantum 2019-01-10, click title to verify 17

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