Coherence-based operational nonclassicality criteria Luca Innocenti1 2Lukaˇs Lachman1and Radim Filip1 1Department of Optics Palack y University 17. Listopadu 12 771 46 Olomouc Czech Republic

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Coherence-based operational nonclassicality criteria

Luca Innocenti,

1, 2

Luk

s Lachman,

and Radim Filip

Department of Optics, Palack

y University, 17. Listopadu 12, 771 46 Olomouc, Czech Republic

Universit

a degli Studi di Palermo, Dipartimento di Fisica e Chimica – Emilio Segr

e, via Archira 36, I-90123 Palermo, Italy

e nonclassicality of quantum states is a fundamental resource for quantum technologies and quantum

information tasks in general. In particular, a pivotal aspect of quantum states lies in their coherence properties,

encoded in the nondiagonal terms of their density matrix in the Fock-state bosonic basis. We present opera-

tional criteria to detect the nonclassicality of individual quantum coherences that only use data obtainable

in experimentally realistic scenarios. We analyze and compare the robustness of the nonclassical coherence

aspects when the states pass through lossy and noisy channels. e criteria can be immediately applied to

experiments with light, atoms, solid-state system and mechanical oscillators, thus providing a toolbox allowing

practical experiments to more easily detect the nonclassicality of generated states.

I. INTRODUCTION

e nonclassicality of quantum states is of utmost importance

for quantum information tasks [

], ranging from quantum

communication and computation [

–

], quantum sensing [

and thermodynamics [

]. Several notions of nonclassicality

have been explored in dierent contexts. For bosonic systems,

the indivisibility of single bosons has for a long time been

considered a direct experimental manifestation of nonclassi-

cality [

–

]. Another type of nonclassicality is the impossi-

bility of a state to be writable as a convex decomposition of

coherent states [

–

]. is can be formalised as the failure

of a state ρto be decomposable as

ρ=Zd2αP (α)|αihα|(1)

for some probability distribution

[

]. Operationally,

coherent states

|αi

are ideal states of a linear oscillator driven

by external coherent force. However, reconstructing the

function experimentally is highly nontrivial [

], and cri-

teria to detect

-nonclassicality include witness-based ones,

relying on bounds on expectation values with respect to the

function [

]; hierarchies of necessary and sucient non-

classicality criteria based on the moments of distribution [

–

]; and criteria based on dierent approaches [

–

]. e

above methods share the shortcoming of relying on global

properties of the state, such as statistical moments, rather

than being tailored to the specic information acquired in

a given experimental scenario. Other nonclassicality crite-

ria, based on photon-click statistics [

–

], are based on

operationally measurable quantities, but are tied to specic

detection schemes.

As of yet, no nonclassicality criterion specically tailored at

individual quantum coherences — as opposed to requiring a

more complete (oen tomographically complete) knowledge

of the state — is known. A possible reason for this is that while

the shape of the set of classical states when only diagonal ma-

trix elements are being observed is relatively manageable via

generalised Klyshko-like inequalities [

], nding similar

inequalities when also coherences are involved is highly non-

trivial. However, being quantum coherences a useful resource

for a variety of quantum information tasks [

], understand-

ing the nonclassicality involving individual coherences would

be a valuable from both experimental and fundamental view-

points. In this Leer we lay out a framework to characterise

the nonclassicality with Fock-state quantum coherences, by

devising operational criteria to certify the nonclassicality of

states leveraging their coherences. We can thus discuss the

role of coherence-based observables on certifying incompat-

ibility with classical states of the form (1) Opposite to what

was the case when characterising nonclassicality using only

Fock state probabilities [

], we nd that when coherences are

involved it is also pivotal to consider the boundary of the set

of all states in the considered spaces, as in some situations the

two can partially overlap, resulting in more care being needed

when devising nonclassicality criteria. To ensure seamless ap-

plicability to experimental scenarios, our criteria only exploit

knowledge of the expectation values of few observables, as

one would have access to in realistic cirumstances. To achieve

this, we devise an approach to nonclassicality detection based

on incomplete knowledge of the density matrix [

], ex-

tending the current state of the art by analysing the infor-

mation hidden in o-diagonal terms. ese elements are di-

rectly measurable by Ramsey-like interferometry of trapped

ion [

], superconducting circuit experiments [

], and elec-

tromechanical oscillators [

]. For light, atomic ensembles,

and optomechanical oscillators, they can be reconstructed

using homodyne tomography. We compare our criteria to

those relying only on Fock-state probabilities [

], and

analyze the nonclassical depth of various quantum coherences

represented by dierent o-diagonal elements.

We nd that observing coherence terms can provide enhanced

predictive power in terms of nonclassicality detection, and

showcase this in several instances of nonclassicality in one-,

two-, and three-dimensional spaces. More precisely, we nd

that, remarkably, in some situations the Fock state probabili-

ties alone are sucient to detect all of the existing nonclassi-

cality, whereas in other situations adding knowledge about

coherence terms provides enhanced predictive power. More-

over, we show how each set of dierent measured observables

provides a distinct boundary of nonclassicality, and study the

behaviour in these spaces of superposition states subject to

aenuation and thermal noise. is further highlights how

arXiv:2210.04390v1 [quant-ph] 10 Oct 2022

dierent types of noise aect the observable nonclassicality

in nontrivial ways, even in relatively low-dimensional spaces.

II. GENERAL FRAMEWORK

Support function and support hyperplanes — Suppose we are

given the expectation values

hOii

for some set of observ-

ables

, and want to gure out whether these measure-

ments are compatible with some classical state. Given the

relevant Hilbert space

, we will denote with

the set of

density matrices in this space, and with

C ⊂ Q

the con-

vex hull of the coherent states. Let us also denote with

O(ρ)≡(Tr(Okρ))n

k=1

the set of expectation values resulting

from measuring

. We seek a method to determine whether,

given an unknown state

, whether there is some

σ∈ C

com-

patible with the observed measurements, that is, to determine

whether O(ρ)∈ {O(σ) : σ∈ C}.

e convexity of

and

allows to characterize them via sup-

porting hyperplanes, using the tools of convex geometry [

Any closed convex set

A⊂Rn

is characterized be its support

function

hA:Rn→R

, dened as

hA(n) = supx∈Ahn,xi

Geometrically,

hA(n)

represents the distance from the ori-

gin to the hyperplane tangent to

orthogonal to

. Denote

with

hC(n)

and

hQ(n)

the support functions of

and

respectively, in the space of interest. More explicitly, we

consider the structure of

and

when projected onto the

nite-dimensional subspaces spanned by the observables mea-

sured in a given context. is allows to devise criteria with

a direct operational signicance. We can then translate the

task of nonclassicality detection into nding whether there

such that

hQ(n)> hC(n)

. Whenever this is the case, it

is possible to nd a set of measurement results

O∈Rn

such

that

n·O> hC(n)

, which certies that these measurement

results are not compatible with any classical state. By study-

ing the structure of

hC(n)

and

hQ(n)

for all

, we can fully

characterize the geometry of the classical set, and oen end

up with Klyshko-like nonclassicality criteria [

]. Notably,

in many of the scenarios considered here, we will be able to

derive the relevant criteria without explicitly involving the

corresponding support function. is is possible in suciently

simple situations where we can devise ad-hoc procedures to

reach the conclusion. ese criteria are equivalent to the full

set of criteria of the form

n·O> hC(n)

, for all

. In a sense,

we can understand these ad-hoc derivations as corresponding

to a full characterization of the support functions

hC(n)

for

all values of

. Directly using the support function remains

nonetheless very useful, as we will show in some explicit

cases.

While

hQ(n)

is generally easier, as it amounts to nding the

largest eigenvalue of

n·O≡PiniOi

, that is, computing

the operator norm

kn·Okop

. On the other hand, computing

hC(n)

is in general more dicult, as it involves maximising

PiniTr(Oiρ)

over the set of

ρ∈ C

. Nonetheless, even

though characterising

hQ(n)

is relatively straightforward for

any xed value of

, this does not trivially translate into an

algebraic characterisation of the boundary of

itself. We

show here how to tackle this task in several cases of interest.

Coherence terms — To focus on the nonclassicality of coher-

ences, we will consider as basic observables

Xjk ≡ |jihk|+

|kihj|

and

Yjk ≡i(|kihj|−|jihk|)

, which are a straightfor-

ward generalisation of non-diagonal Pauli matrices in higher

dimensions. ese naturally capture information hidden in

coherence terms, that is not directly accessible via projections

of the form

Pj≡ |jihj|

. e expectation value of

Xjk, Yjk

on a coherent state

|αi

with

α=√µeiφ

are related to the

Fock-state number probabilities Pi≡ |iihi|as

Xij = 2pPiPjcos(φ(i−j)), Yij = 2pPiPjsin(φ(i−j)).(2)

For ease of notation, here and in the rest of the paper, we

will with some abuse of notation conate the operators

Xjk

with their expectation values on a given state

hXjkiρ≡

Tr(Xjkρ)

. For example, eq. (2) would be more precisely

wrien as

hXjkiα= 2phPjiµhPkiµcos(φ(j−k)).

generally, we can consider the rotated operators

Rjk(θ)≡

cos(θ)Xjk + sin(θ)Yjk

, whose expectation value on coherent

states reads Rjk(θ) = 2pPjPkcos(θ−φ(j−k)).

antum boundary — When only dealing with Fock-state

probabilities, any probability distributions is compatible with

some quantum state, and thus the boundary of

is simply

dened by the relations

PjPj≤1

and

0≤Pj≤1

. e

situation changes signicantly when coherence terms are be-

ing considered. Finding the boundary of

then amounts to

guring out the conditions under which the observed expec-

tation values t into a positive semidenite matrix. Further

details on how this process results in dierent inequalities are

given on a per-case basis in the text, and we also include for

completeness a more general discussion in the SM.

III. NONCLASSICALITY CRITERIA

We will discuss here the nonclassicality certiable via non-

diagonal elements of the density matrix in the Fock state basis,

as well as the nonclassicality encoded in nontrivial combina-

tions of dierent coherence terms, or in nontrivial combina-

tions of both coherence terms and Fock-state probabilities.

One-dimensional criteria — We rst study the class of nonclas-

sicality criteria associated to an individual coherence term

Rjk(θ)

. ese are the easiest to apply in any experimental

scenario where coherences are measured. In these spaces,

the set of all states is bounded by

|Rjk(θ)| ≤ 1

, with bound

saturated by the state

√2(|ji+eiθ |ki)

. On the other hand,

the corresponding classical bound is

|Rjk(θ)| ≤ max

ρ∈C 2√ρjj ρkk = 2e−j+k

2[(j+k)/2]j+k

√j!k!,(3)

where the maximisation can be restricted to the set of coherent

states. e corresponding bound for the set of all states is

instead

|Rjk(θ)| ≤ 1

, saturated by the state

√2(|ji+eiθ |ki)

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Coherence-basedoperationalnonclassicalitycriteriaLucaInnocenti,1,2LukasLachman,1andRadimFilip11DepartmentofOptics,PalackyUniversity,17.Listopadu12,77146Olomouc,CzechRepublic2UniversitadegliStudidiPalermo,DipartimentodiFisicaeChimica{EmilioSegre,viaArchira36,I-90123Palermo,Italyenonclassicalit...

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Coherence-based operational nonclassicality criteria Luca Innocenti1 2Lukaˇs Lachman1and Radim Filip1 1Department of Optics Palack y University 17. Listopadu 12 771 46 Olomouc Czech Republic

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