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Lower Bounds

on

Quantum Metrological Precision

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Fisika Teorikoa eta Zientziaren Historia SailaEuskal Herriko Unibertsitatea
(UPV/EHU)P. O. Box 644, 48080 Bilbao, Spain

Department of Theoretical PhysicsUniversity of the Basque Country (UPV/EHU)P.
O. Box 644, 48080 Bilbao, Spain

The LaTeX source of the thesis with figures is
athttps://gitlab.com/i-apellaniz/PhD-Thesis

A PDF copy of this work can be found athttps://gitlab.com/i-apellaniz/PhD-
Thesis/tree/master/out

This document was generated with the 2016 LaTeX distribution.The plots and
figures of this thesis were generated with Matplotlib,and then edited in
Inkscape and Blender.

2012-2017 Iagoba Apellaniz. This work is licensed under the Creative Commons
Attribution-ShareAlike 4.0 International License. To view a copy of this
license, visithttp://creativecommons.org/licenses/by-sa/4.0/deed.en_US

##  Prologue

prologueprologueProloguePrologue

This work is based on the doctoral project on quantum information which I
started in the summer of 2013. It collects part of the research I have done
during these years. In page Laburpena, I include a brief summary of the thesis
written in Basque. It gave me the opportunity to explain what I did these past
years in my tongue language. I find it useful for the people I love most, my
family and friends.

I will make an effort to be clear throughout the thesis in order to make it
useful for an audience as wide as possible. This way I hope it will be
readable by any person with a bachelor in science, particularly in physics or
at least with a partial knowledge in quantum mechanics. With that in mind, the
first and the second chapters will introduce the background to the reader as
well as the basic notions of quantum metrology, the field our work belongs to.
Even though I write this thesis for a broad audience in mind, a basic notions
in quantum physics and statistics is needed to follow it properly as I said
before. For instance, I will assume among other things that the reader knows
what probability is and which its properties are, or what a quantum state is
and what it represents. I will give references where to find complementary
material when necessary.

The thesis has been done within the research group in quantum information in
which Prof. Géza Tóth is the group leader and principal investigator. I have
to mention the rest of the members of the group Dr. Philipp Hyllus, Dr.
Giuseppe Vitagliano, Dr. Iñigo Urizar-Lanz, Dr. Zoltán Zinborás and Dr.
Matthias Kleinmann at the time I was working on the projects of this thesis.
Apart from the group of Géza Tóth based in Bilbao, Spain, this thesis also
collects some work done in collaboration with the Theoretical Quantum Optics
group lead by Prof. Otfried Gühne at the University of Siegen, Germany, and
the group of Prof. Carsten Klempt at the Leibniz University in Hannover, also
in Germany. The last one is an experimental group specialized on the creation
of exotic quantum states with very many particles with a variety of
applications in quantum technology.

In this thesis, first, we investigate the metrological usefulness of a family
of states known as unpolarized Dicke states, which turn to be very sensitive
to the magnetic field. Quantum mechanics plays a central role in achieving
such a high precision. Second, we investigate possible lower bounds on the
quantum Fisher information, a quantity that characterizes the usefulness of a
state for quantum metrology, using the theory of Legendre transforms such that
we obtain tight lower bounds based on few measurements of the initial quantum
state that will be used for metrology. And last but not least, we investigate
gradient magnetometry, i.e., we develop a theory to study the sensitivity of
some states of the change in space of the magnetic field.

Iagoba Apellaniz

Bilbao, March 02, 2017

##  Laburpena

laburpenalaburpenaLaburpenaLaburpena

Lan honek metrologia kuantikoaren baitan egindako hainbat ikerketa biltzen
ditu tesi moduan. Aurkezten dudan ikerlan hau, Zientzia eta Teknologia
Kuantikoko masterra bukatu ondoren, azkeneko lau urte hauetan Prof. Géza Tóth
irakaslearen lan-taldean burutua izan da. Tesi honetan agertzen ez diren beste
hainbat lan plazaratu ditugu nik eta elkarrekin lan egin dugun hainbat
ikertzailek. Publikatutako artikuluen lista tesi honen Publications.
orrialdean aurki daiteke.

Tesi honetan agertzen diren ikerketa lanak gauzatzeko ezinbestekoa izan da
nazioarteko elkarlana. Izandako elkarlanen artean Alemaniako Siegen hiriko
unibertsitatean dagoen Otfried Gühnek zuzentzen duen TQO taldea dago. Beste
kolaborazio garrantzitsu bat Italiako Florentzian dagoen unibertsitateko
Augusto Smerzik zuzentzen duen QSTAR taldea izan da. Azkenik, Alemaniako
Hannover hiriko unibertsitateko Carsten Klemptek zuzentzen duen ikerkuntza
talde esperimental batekin izandako elkarlana azpimarratu nahi da.

1. kapituluan teknologia kuantikoak eta metrologia kuantikoak duten garrantzia azpimarratzen da. Teknologia kuantikoa prozesu kuantikoez baliatzen baita klasikoki lortu ezin diren hainbat helburu lortzeko. Adibide gisa, ordenagailu kuantikoek hainbat posibilitate aldi berean aztertzeko izango luketen gaitasuna, edota simulazio kuantikoek modelo konplexu ezberdinak simulatzeko duten gaitasuna azpimarratzen dira.

Beste motatako teknologi kuantikoak alde batera utzita, metrologia kuantikoan
oinarritzen da lan hau. Metrologia kuantikoak zenbatetsi nahi diren
parametroen errorea txikiagotzeko aukera ematen du metrologia klasikoarekin
alderatuz gero. Klasikoki diseinatutako aparailu batek $N$ proba egin ondoren
errorea $\sqrt{N}$ aldiz txikitzea lortzen duen bitartean, metrologia
kuantikoa erabiliz errorea $N$ aldiz txikitu daitekeela aski ezaguna da [1,
2]. Muga gaindiezin hauei "shot-noise scaling" deritze aparatu klasikoen
kasurako eta "Heisenber scaling" egoera kuantiko orokorrek gainditzerik ez
duenarentzako.

Ezaguna da baita ere elkarlotura kuantikoak, hau da, mekanika kuantikoaren
propietatea eta klasikoki azalpenik ez duenak, zenbatespenean duen garrantzia.
Elkarlotura kuantikoa zenbatespena hobetzeko ezinbestekoa da, aldiz,
elkarlotura kuantiko mota guztiak ez dute balio errorea txikitzeko.
Elkarlotura kuantikoak eta metrologia kuantikoak duten erlazioaren hainbat
azterketa hurrengo erreferentzietan aurki daitezke [3, 4, 5, 6, 7, 8, 9, 10].

Honi guztiari azken urteotan metrologia kuantikoak piztu duen arreta gehitu
behar zaio. Kuantikak hobetutako metrologia erabiltzen da, adibidez, erloju
atomikoetan [4, 11, 12], zehaztasun handiko magnetometrian [13, 14, 15, 16,
17, 18, 19], edota uhin grabitazionalen detektagailuetan [20, 21, 22].

2. kapitulua metrologia kuantikoaren sarrera gisa uler daiteke. Bertan estatistikan oinarritutako hainbat kontzeptu azaltzen dira. Estatistika datuetatik ondorio ulerkorrak ateratzeko erabiltzen den zientzia matematikoa da. Adibidez, datu lagin baten batezbestekoa kalkulatzeko erabiltzen den prozedura, edota datu lagin baten bariantza kalkulatzeko erabilitako formulak azaltzen dira kapitulu honetan. Datu lagin batek zenbatetsi nahi den parametroari buruzko informazioa izan lezake bere baitan. Adibide bezala, pilota bat bosgarren pisutik jaurtitzerakoan lurra jotzeraino igarotako denbora neurtu da hainbat aldiz. Denbora ezberdin guzti hauek erabiliz, bosgarren pisuraino dagoen altuera kalkula daiteke grabitateak pilotarengan duen eragina aldez aurretik ezaguna denean.

Estatistikaren baitan kokatzen da datu lagin batetik zenbatespena egiterakoan
saihestezina den errorearen muga klasikoaren kalkulua. Muga hau Fisher
informazioan, hau da, laginaren probabilitate distribuzio funtzioa eta
zenbatetsi nahi den parametroaren aldaketaren arteko korrelazioa neurtzen duen
kantitatean, oinarrituta dago.

Kontzeptu hauek aztertu ondoren, mekanika kuantikoko tesi honetan erabilitako
hainbat tresna aurkezten dira. Tresna eta definizio hauek tesi hau hobeto
ulertzeko azaltzen dira tesiaren hasierako kapitulu honetan. Egoera
kuantikoaren definizio eta propietateak azaltzen dira, baita operadore
kuantikoenak ere. Egoera kuantikoa matrize baten bitartez irudikatu daiteke
gehienetan. Matrize honen karratuarekin egoera kuantiko bera lortzen bada
egoera purua dela esaten da. Egoera kuantiko nahasiak aldiz, egoera puruen
nahasketa baten bitartez adieraz daitezke.

Deskonposizio horien artean, egoera puruak beraien artean ortogonalak direnean
deskonposizio propioa dela esaten da. Egoera kuantiko bat beraz deskonposizio
propio baten bitartez adieraz daiteke
$\rho\equiv\sum_{\lambda}p_{\lambda}|{\lambda}\rangle{\\!}\langle{\lambda}|$,
non $p_{\lambda}$ probabilitate bat eta $|{\lambda}\rangle$ egoera puru bat
adierazten duten. Partikula multzo baten aurrean gaudenean, aurreko
propietateez gain, beste propietate interesante batzuk agertzen dira.
Elkarlotura kuantikoa, adibidez, partikula multzoetan definitzen da. Multzo
osoaren egoera kuantikoa banakorra ez denean elkarlotuta daudela esaten da.

Tesi honetako operadore erabilienak momentu angeluarraren osagaiak dira, bai
partikula bakarraren momentu angeluarrarenak, baita partikula guztien momentu
angeluar kolektiboenak ere. Momentu angeluarraren operadore hauek garrantzia
handia daukate magnetometrian. Partikula bakarraren spin operadoreak momentu
angeluar operadoreak dira, eta spin operadoreen bitartez deskribatzen da
partikulek eremu magnetikoekin daukaten interakzioa. Oinarrizkotzat hartu
daitekeen partikula bakarraren spin zenbakia gehienetan $\frac{1}{2}$ da. Honi
_qubit_ deitzen zaio kuantikoa.

Bestalde, momentu angeluarraren operadoreek deskonposizio propioan base berri
bat sortzen dute. Base honetan hainbat egoera kuantiko berezi topa daitezke,
esaterako, Dicke egoera simetrikoak edota singletea. Egoera hauek
aztertzerakoan ikusten da magnetometrian edota beste teknologia kuantikoetan
duten erabilgarritasuna.

Estatistikaren zenbatespen metodologia eta mekanika kuantikoa batzean
metrologia kuantikoa sortzen da. Metrologia kuantikoan zenbatespen prozesuaren
errorearen mugak aztertzen dira askotan. Tesi honetan aurkeztutako
ikerkuntzekin muga hauen bilaketan aurrera pausu garrantzitsuak eman dira.
Fisher informazio kuantikoa da normalean erabiltzen den tresnarik
esanguratsuena. Zenbatespena egiteko erabiltzen den egoera kuantikoan eta
interakzioak sortzen duen egoeraren eboluzioan oinarrituta dago Fisher
informazio kuantikoa. Beraz, ezinbestekoa da hasierako egoera kuantikoa
ezagutzea Fisher informazio kuantikoa kalkulatzeko. Hurrengo paragrafoetan,
aldez aurretik egoera kuantikoa zein den jakin gabe, muga hauek bilatzeko
garatu diren tresnak azaltzen dira.

3. kapituluan lehenbiziko ikerketa lana aurkezten da: Polarizatu gabeko Dicke egoeratik hurbil dauden egoera kuantiko nahasiek metrologian duten erabilgarritasuna. Egoera kuantiko puruak gauzatzea oso zaila da praktikoki, eta egoera nahasiak lortzen dira gehienbat laborategietan. Arrazoi honegatik, kapitulu honetan egoera nahasi hauek metrologian duten erabilgarritasunaren arabera sailkatzeko balio duen teknika aurkezten da.

Egoera ez polarizatuak egoera polarizatuak baino erabilgarriagoak izan
daitezke magnetometrian. Egoera polarizatuak erabiltzerakoan aldiz eremu
magnetikoaren magnitudea zenbatestea nahiko zuzena da. Egoerak denbora tarte
batean eremu magnetikoaren pean polarizazioan jasandako errotazioa neurtzen da
eta aldaketa honetatik eremu magnetikoaren zenbatespena egiten da. Bestalde,
egoera ez polarizatuak ezin dute teknika hau erabili, nahiz eta Fisher
informazio kuantikoa kalkulatzerakoan magnetometriarako erabilgarriagoak
direla argi dagoela ikusi.

Hau dela eta, Dicke egoera ez polarizatuek duten beste propietate bat
erabiltzen da, polarizazioaren sakabanaketa. Propietate hau polarizazioaren
neurketetan lortzen den datuen sakabanaketa da. Datuen sakabanaketa hau
Heisenbergen ziurgabetasun printzipioarekin lotuta dago. Dicke egoera ez
polarizatuetan sakabanaketa hau txiki izatetik $N^{2}$-ko proportzioetara
heltzen den magnitude bat da, beraz, $N^{2}$-ko proportzioetako aldaketa
neurtuko da. Polarizazioan oinarritutako zenbatespenak $N$-ko proportzioetara
heltzen diren bitartean, sakabanaketan oinarritutakoak zenbatespenak
"Heisenberg scaling" muga fisikotik hurbilago daude, ikusi 1.1. orrian dagoen
1.1. irudia. Polarizazioa gezi gorriak ematen duen bitartean, ziurgabetasuna
zirkulu urdinak ematen du.

Guzti honetan oinarrituta, polarizazioaren sakabanaketaren aldaketa
neurtzerakoan eremu magnetikoaren zenbatespena egin ahal da. 3. kapituluan
erroreen hedapenaren formula aplikatuz, zenbatespenean gertatuko den errorea
kalkulatzen dugu. Errore hau hasierako egoeraren itxarotako balioen funtzio
bezala idatzi ondoren, aski da lau behagarriren itxarotako balioak neurtzea.
Lau balore hauek neurtzearekin batera zenbatespenaren errorea lortuko dugu.
Errore hau Fisher informazio kuantikoaren gainetik egon arren, egoerak
sailkatzen laguntzen du beraz. Gainera, lau operadorereen neurketa partikula
asko duten egoeren tomografia egitea baino askoz errazagoa da esperimentalki.

Kapitulua bukatutzat emateko, errorearen formulan oinarrituta, are gehiago
sinplifikatzen dugu formula hau. Oraingoan, operadore biren itxarotako balioan
oinarritzen den beste ordezko ekuazio bat aurkezten dugu, honek dakarren
abantaila esperimentala azpimarratuz.

4. kapituluan, Fisher informazio kuantikoaren mugak aztertzen ditugu egoera kuantiko baten operadore ezberdinek daukaten itxarotako balioen funtzio bezala. Beraz, arazo berdinari egiten zaio aurre kapitulu honetan. Praktikoki egoera kuantikoa zehatz-mehatz jakitea ezinezkoa denez, eta are gutxiago partikula asko duten sistemetan, kapitulu hau behagarriek egoera kuantikoan duten itxarotako balioetan oinarritzen da Fisher informazio kuantikoa mugatzerako orduan.

Oraingoan aldiz, problema honi beste ikuspuntu batekin aztertzeari ekiten
zaio. Legendreren transformazioan oinarritutako elkarloturaren neurketak
egiteko metodo baten oinarrituta [23], Fisher informazio kuantikoaren
doitutako mugak topatzen dira.

Kapituluan zehar hainbat adibide garatzen dira. Metodo honek edozein behagarri
hartu eta beraren itxarotako balio bera duten egoera kuantiko guztien artean
Fisher informazio kuantiko baxuenekoa aukeratzea ahalbidetzen du. Metodo
honek, bat bakarra beharrean, hainbat behagarri hartu ditzake. Beraz, hainbat
behagarri sorta hartu eta beraien itxarotako balioak aldez aurretik
dakizkigula, metodoak emandako Fisher informazio kuantikoaren muga ezberdinak
aztertzen ditugu.

Azkenik, adibide konkretu batzuei jarraituz, gure metodoa partikula askotako
egoeretara nola luzatu daitekeen aztertzen dugu. Datu esperimentalak erabiliz,
aldez aurretik egindako esperimentuetarako zenbatespenaren mugak kalkulatzen
ditugu. Datu hauek [24] eta [7] erreferentziei jarraituz lortu ditugu.

Tesi honetan aurkezten dudan azkenengo ikerketa lana 5. kapituluan topa
daiteke: Eremu magnetikoaren gradientearen zenbatespenaren mugak atomo
multzoak erabiltzerakoan, izenekoa. Eremu magnetikoaren gradientea eremu
magnetikoak espazioan daukan aldaketa adierazten du. Aurreko kapituluetan ez
bezala, honetan, Fisher informazio kuantikoa kalkulatzen da.

Eremu magnetikoa, beraz, parametro birekin zehaztuta dago, eremu magnetikoaren
parte homogeneoa eta gradientea. Ondorioz, parametro bat baino gehiago
zenbatetsi behar ditugu, nahiz eta gradiente parametroan soilik interesatuta
egon. Parametro bat baino gehiagoko metrologia kuantikoaren oinarrizko
problematzat hartu daiteke eremu magnetikoaren gradientearen zenbatespena.

Eremu magnetikoaren gradientea kalkulatzeko ezinbestekoa da egoera kuantikoak
espazioan daukan izaera aztertzea, hau da, egoera kuantikoak espazioa nola
betetzen duen jakitea. Egoera kuantikoaren espazioaren partea, partikula
puntualez osatutako egoera batera sinplifikatzen dugu, nahiz eta lortutako
emaitzak bestelako kasuetara ere egokitzen diren.

Adibidez, lehenengo kasuan atomoak espazioko puntu ezberdinetan jartzen dira
ilara zuzen bat sortuz. Atomo ezberdinek eremu magnetikoaren intentsitate
ezberdinak sumatuko dituzte. Spin egoeraren arabera beraz, Fisher informazio
ezberdinak kalkulatzen ditugu. Bigarren kasuan atomo guztiak espazioko bi
puntu ezberdinetan kokatuta daude, atomoen erdia puntu batean eta beste erdia
bestean. Kasu honetan topa daiteke eremu magnetikoaren zenbatespenerako spin
egoerarik onena, Heisenbergen printzipioez mugatutako zenbatespena ematen
duena.

Azkenengo kasuan, atomoak espazioan zehar era desordenatu baten sakabanatuta
daude. Esperimentu askotan topa daitekeen egoera da hau. Adibidez, atomoak
barrunbe batean daudenean. Spin egoera ezberdinak aztertzen ditugu eta kasu
bakoitzean beraien Fisher informazioa, zenbatespenean duten muga teorikoa,
kalkulatzen dugu.

Ondorio gisa, lan honetan aurkeztutako azterketek zenbatespen kuantikoa dute
jomuga. Lehenengo ikerkuntza bietan, esperimentuen konplexutasuna
sinplifikatzen da. Egoera kuantikoan oinarritu beharrean, behagarri batzuen
itxarotako balioetan oinarritzen baita zenbatespenaren errorearen muga. Metodo
hauen inplementazio praktikoak aztertu ditugu aldez aurretik egindako
esperimentuen datuak erabiliz. Honek guztiak, etorkizunean egingo diren
metrologia kuantikoko esperimentuetan, metodo hauek erabiltzea errazten du.
Bestalde, eremu magnetikoaren gradientearen azterketan topatu ditugun muga
teorikoak Heisenbergen proportzionaltasuna ahalbidetzen dute.
Proportzionaltasun hau bi partikula multzo erabiltzen direnean eta baita
multzo bakarra erabiltzen denean ere agertu daitekeela frogatu da. Partikula
multzo bakarra erabiltzerakoan beraz, partikula zenbakiarekin batera txikitzen
da errorea, esperimentua eta ondoren etor daitekeen inplementazio praktikoa
asko sinplifikatuz, eta Heisenbergen proportzionaltasuna oraindik ere
mantenduz.

##  Publications

publicationspublicationsPublicationsPublications

• Géza Tóth and Iagoba Apellaniz. Quantum metrology from a quantum information
science prespective. Journal of Physics A: Mathematical and Theoretical, 47
424006, (2014).

• Iagoba Apellaniz, Bernd Lücke, Jan Peise, Carsten Klempt and Géza Tóth.
Detecting metrologically useful entanglement in the vicinity of Dicke states.
New Journal of Physics, 17 083027, (2015).

• Iagoba Apellaniz, Matthias Kleinmann, Otfried Gühne and Géza Tóth. Optimal
detection of metrologically useful entanglement. Physical Review A, _in press_
, (2017).

In preparation

• Iagoba Apellaniz, Iñigo Urizar-Lanz, Zoltán Zimborás, Philipp Hyllus and
Géza Tóth. Precision bounds for gradient magnetometry with atomic ensembles.

Out of the scope of this thesis

• Giuseppe Vitagliano, Iagoba Apellaniz, Iñigo Luis Egusquiza and Géza Tóth.
Spin squeezing and entanglement for arbitrary spin. Physical Review A, 89
032307, (2014).

• Giuseppe Vitagliano, Iagoba Apellaniz, Matthias Kleinmann, Bernd Lücke,
Carsten Klempt and Géza Tóth. Entanglement and extreme spin squeezing of
unpolarized states. New Journal of Physics, 19 013027, (2017).

###### Contents

  1. Acknowledgments
  2. 1 Introduction
  3. 2 Background in quantum metrology
    1. 2.1 Background in statistics and theory of estimation
      1. 2.1.1 Probability, data samples, average and variance
      2. 2.1.2 Estimators and Fisher information
    2. 2.2 Quantum mechanics from metrology perspective
      1. 2.2.1 The quantum state, multiparticle state and entanglement
      2. 2.2.2 Angular momentum operators on multiparticle systems
      3. 2.2.3 Dynamics of quantum systems
    3. 2.3 Quantum metrology
      1. 2.3.1 Quantum magnetometry
      2. 2.3.2 Metrology with almost polarized states
      3. 2.3.3 The quantum Fisher information
  4. 3 Metrology in the vicinity of Dicke states
    1. 3.1 Unpolarized Dicke states for magnetometry
    2. 3.2 Precision based on the error propagation formula when $M=J_{x}^{2}$
      1. 3.2.1 The optimal precision
    3. 3.3 Testing the formula against some known states
    4. 3.4 Using our method with real experimental data
  5. 4 Witnessing metrologically useful entanglement
    1. 4.1 Lower bound on a convex function given some arbitrary expectation values
      1. 4.1.1 Lower bound based on a single observable
      2. 4.1.2 Measuring several observables
      3. 4.1.3 Lower bound on the QFI
      4. 4.1.4 Exploiting the symmetries
    2. 4.2 Examples
      1. 4.2.1 Fidelity measurements
      2. 4.2.2 Spin-squeezed states
      3. 4.2.3 Dicke states
    3. 4.3 Calculations for experimental data
      1. 4.3.1 Few-particle experiments
      2. 4.3.2 Many-particle experiments
    4. 4.4 Scaling of $\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{z}}]$ with $N$
  6. 5 Metrology of the gradient magnetic field
    1. 5.1 The setup
    2. 5.2 Cramér-Rao bound for gradient estimation
      1. 5.2.1 States insensitive to the homogeneous field: Single-parameter estimation
      2. 5.2.2 States sensitive to the homogeneous field: Two-parameter dependence
    3. 5.3 Testing our approach
      1. 5.3.1 Distinguishable atoms in a 1D lattice
      2. 5.3.2 Differential magnetometry with two ensembles
    4. 5.4 Magnetometry with a single atomic ensemble
      1. 5.4.1 Precision bound
      2. 5.4.2 Precision bounds for different spin-states
  7. 6 Conclusions
  8. Appendices
    1. A Multiparticle angular momentum subspaces
    2. B Husimi $\mathcal{Q}$-representation and the Bloch sphere
    3. C Calculation of $\langle{\\{J_{x}^{2},J_{y}^{2}\\}+\\{J_{x},J_{y}\\}^{2}}\rangle$
    4. D Properties of the spin-squeezing Hamiltonian
    5. E Legendre transform in 1-dimension
    6. F Calculation of $|\langle{\textnormal{D}_{N,m}}|{{}_{z}}|{\textnormal{D}_{N,N/2}}\rangle_{x}|^{2}$
    7. G Calculation of the QFI matrix elements

toctocTable of contentsTable of contents

##  Abbreviations, figures and tables

abbreviationsabbreviationsAbbreviations, figures and tablesAbbreviations,
figures and tables

### Abbreviations

BEC | - | Bose-Einstein condensate
---|---|---
GHZ | - | Greenberger-Horne-Zeilinger
PDF | - | Pobability distribution function
POVM | - | Positive-operator valued measure
QFI | - | Quantum Fisher information

### Figures

lof

### Tables

lot

thesisthesisThesisThesis

University of the Basque Country

PhD Thesis

Lower bounds on quantum metrological precision

Iagoba Apellaniz

_Supervised by_

Prof. Géza Tóth

March 02, 2017

dedicationdedicationDedicationDedication

_To my parents, my family_

_and to all the people_

_I have had around these years._

##  Acknowledgments

Iwould like to thank the people that has supported me in this endeavor.
Especially, I want to acknowledge my office and discussion mates, Giuseppe
Vitagliano and Iñigo Urizar-Lanz as well as Phillip Hyllus, Matthias Kleinmann
and Zoltán Zimborás. A special thank to my director Géza Tóth for all the
support, without whom my work at hand would not have been possible. I would
also like to thank more people from the department I belong to at the
University of the Basque Country, the Department of Theoretical Physics and
the History of Science. I appreciate the effort done by people like Iñigo L.
Egusquiza in promoting the Ph.D. program of the Department of Theoretical
Physics, which has successfully formed plenty of researchers now pursuing
cutting edge research projects world wide. I would also like to acknowledge
the Ph.D. students, named at the end, from different departments for the
friendly atmosphere I could enjoy working here at _our_ university.

I have been visiting several research groups during these years as well as
attending different conferences and workshops. First, I would like to thank
the TQO group at the University of Siegen as a whole, especially with whom I
have discussed or collaborated, Sanah Altenburg, Sabine Wölk and Otfried
Gühne. Second, I would also like to thank the QSTAR group at the University of
Florence, particularly Manuel Gessner, Luca Pezzè and Augusto Smerzi for
inspiring discussions. And finally, I would like to thank the people with whom
I also discussed or collaborated a lot but have not had the chance to visit
them, like Michał Oszmaniec, Jan Kolodynski, Morgan W. Mitchell and Antonio
Acín from ICFO in Barcelona, and Bernd Lücke, Jan Peise, and Carsten Klempt
from the University of Leibniz in Hannover.

I take this opportunity to cheer all the rest I have in mind right now:
Constantino, Roberta, Tobias, Frank, Roope, Nikolai, Mariami, Felix,
Christina, Ali, Daniela, Marius, Jannik and Tristan which I met them in
Siegen; Roberto, Fabian, Victor and Marco from Firenze; and last but not least
Mikel, Pablo, Julen, Iraultza, Joanes, Jon, Laura, Peio, Borja, Sofia,
Ariadna, Urtxi, Iñigo, Iker, João, Mattin, Olatz, Kepa, Carlos, Naiara, Diego,
Unai, Paul, Maria, Irene, Raul, Lluc, Montse and Santi, from Bilbao. I see us
like the components of an $N$-dimensional vector, each one with its own
direction, and at the same time pointing together as one! Thank you guys for
the nice and comfortable atmosphere!

##  1 Introduction Introduction

"The story I am about to tell, while it is not directly about John Bell,could
not have taken place without him."

Daniel M. Greenberger

Metrology plays an important role in many areas of physics and engineering
[25]. With the development of experimental techniques, it is now possible to
realize metrological tasks in physical systems that cannot be described well
by classical physics, instead quantum mechanics must be used for their
modeling. Quantum metrology [1, 26, 2, 27] is the novel field, which is
concerned with metrology using such quantum mechanical systems.

In quantum metrology, the quantumness of the system plays an essential role
[28, 29]. One can find bounds on the highest achievable precision of a
metrological setup. One of the usual methods to find such bounds is using the
theory of the quantum Fisher information (QFI) [30, 31, 32, 33]. There have
been efforts recently connecting quantum metrology to quantum information
science, in particular, to the theory of entanglement [34]. Entanglement, a
feature of quantum mechanics, lies at the heart of many problems has attracted
an increasing attention in recent years.

There are now efficient methods to detect entanglement with a moderate
experimental effort [35, 36]. However, in spite of intensive research, many of
the intriguing properties of entanglement are not fully understood. One of
such puzzling facts is that, while entanglement is a sought after resource,
not all entangled states are useful for some particular quantum information
processing task. For instance, it has been realized recently that entanglement
is needed in very general metrological tasks to achieve high precision [3].
Remarkably, this is true even in the case of millions of particles, which is
especially important for characterizing the entanglement properties of cold
atomic ensembles [4, 5, 6, 7, 8, 9]. However, there are highly entangled pure
states that are useless for metrology [10].

In the light of these results, beside verifying that a quantum state is
entangled, we should also show that it is useful for metrology. One of the
basic tasks of quantum metrology is magnetometry with an ensemble of spin-$j$
particles. Magnetometry with a state completely polarized works as follows.
The total spin of the ensemble is rotated by a homogeneous magnetic field
perpendicular to it. We would like to estimate the rotation angle or phase
$\theta$ based on some measurement. Then, the phase can be used to determine
the strength of the magnetic field, see Figure 1.1-(a).

\begin{overpic}[scale={1.15}]{img/IN_tpolarized_under_magnetic.pdf}
\put(5.0,10.0){\small(a)}
\end{overpic}\begin{overpic}[scale={1.15}]{img/IN_ssqueezed_under_magnetic.pdf}
\put(5.0,10.0){\small(b)} \end{overpic}

Figure 1.1:  (a) (red-arrow) Initial state $\rho_{\text{i}}$ pointing in the
$y$-direction. (blue-dashed-circle) Uncertainty ellipse of the polarization
perpendicular to the mean spin. The state is rotated with a speed proportional
to the strength of the magnetic field $B$ (green-arrow). Hence, the magnetic
field can be estimated from the final state $\rho_{\text{f}}$ (red-dashed-
arrow). (b) When the uncertainty ellipse is reduced in one direction
perpendicular to the polarization, the state is called a spin-squeezed state.
If the direction in which the uncertainty is reduced coincides with the
direction of the rotation, then it is easier to distinguish the final state
from the initial state. Which turns into a better precision for the estimation
of the magnetic field.

In recent years, quantum metrology has been applied in many scenarios, from
atomic clocks [4, 11, 12] and precision magnetometry [13, 14, 15, 16, 17, 18,
19] to gravitational wave detectors [20, 21, 22]. There have been many
experiments with fully polarized ensembles [27, 13, 15, 18, 37, 17, 38], in
which the collective spin of the particles is rotated as in Figure 1.1-(a) and
the angle of rotation is estimated by collective measurements. It has also
been verified experimentally that spin squeezing can result in a better
precision compared to fully polarized states [6, 27, 13, 38, 39, 40, 41, 42,
43] since spin-squeezed states are characterized by a reduced uncertainty in a
direction orthogonal to the mean spin [44, 45, 46, 47], see Figure 1.1-(b).

Besides almost fully polarized states, there are also unpolarized states
considered for quantum metrology. Prime examples of such states are
Greenberger-Horne-Zelinger (GHZ) states [48], which have already been realized
experimentally many times [49, 50, 51, 52, 53, 54, 55, 56]. Recently, new
types of unpolarized states have been considered for metrology, such as the
singlet state [57, 58] and the symmetric unpolarized Dicke states which have
been realized in cold gases [8, 59, 60].

In the present thesis we study how large precision can be achieved for
realistic noisy systems [61, 62]. We also study how such states can be
characterized with few measurements which reduce the experimental efforts
considerably. And finally, we discuss the multi-parameter estimation problem.
It turns out that some states not useful for homogeneous magnetometry may
become useful when we use them for differential magnetometry, which is one of
the most fundamental two-parameter estimation tasks.

##  2 Background in quantum metrology Background in quantum metrology

"In the real world, doing real experiments, statistics began to matter."

Roger J. Barlow

In this chapter, we will study the basics of quantum metrology, metrology
enhanced by quantum phenomena. First of all, metrology, as the science of
measuring, has played an essential role in the development of science and
technology. Metrology studies several aspects of the estimation process, for
example, which strategy to follow in order to improve the precision of the
estimation. Metrology also covers all intermediate processes, from the design
aspects of a precise measuring device, to the most basic mathematical concepts
that arise from the formulation of the different problems.

Historically, with the discovery of quantum theory and the subsequent
development of quantum mechanics, new opportunities emerged for advances in
metrology in the early decades of the twentieth century. Later with the
arrival of concepts like _qubit_ and _quantum cryptography_ , quantum theory
embraced the so-called quantum information theory, which merges the notions of
theory of information and computer science with quantum mechanics. Due to the
rapid development of quantum technologies, quantum information attracted a lot
of attention from the scientific community. Moreover, those emerging fields
rapidly became a very interesting interdisciplinary playground of science with
many scientists as well as resources involved.

Next, we will focus on quantum metrology, for which the role of entanglement,
an exclusive feature of quantum mechanics which cannot be described using
classical probabilistic theories, is essential. Entanglement connects quantum
metrology with concepts like local realism or quantum teleportation.
Nevertheless, throughout the thesis we will focus mainly on the achievable
precision for different systems and schemes.

On the other hand, we will also present some important tools and concepts of
statistics, without which many descriptions of important physical findings
would lack the rigorous interpretation needed. It basically helps to analyze
raw data to make it readable from the human perspective.

This chapter is organized as follows. In Section 2.1, we introduce the basic
concepts of statistics and estimation theory, focusing on what is needed for
the understanding of this thesis. In Section 2.2, we present the necessary
tools of quantum mechanics used throughout the text. Finally in Section 2.3,
we arrive at the main set of tools used in the quantum metrology framework and
by extension in the present work.

###  2.1 Background in statistics and theory of estimation

In this section, we will enumerate the basic concepts of statistics as well as
the estimation process. As we have said, the main mathematical tools used by
metrology belong to statistics. Moreover, we are especially interested in
estimation theory, which shows how to estimate properly some quantity based on
a data sample. The data can be of any kind. For instance, the data sample
might be a set of the heights of members of a basketball team, or the outcomes
of coin tosses, or even the wavelengths of photons coming out of some
radioactive sample*** For a more detailed introduction to statistics, see
Refs. [63, 64].

####  2.1.1 Probability, data samples, average and variance

The probability indicates the relative chance of an event to happen. For
instance, if there is a box with 10 red balls and 5 blue balls and assuming we
extract on of them randomly, the probability of obtaining a blue ball is given
by $\frac{n_{\text{blue}}}{n_{\text{total}}}=\frac{5}{10+5}=\frac{1}{3}$. In
the same way, the probability of obtaining a red ball is
$\frac{n_{\text{red}}}{n_{\text{total}}}=\frac{2}{3}$. Note that some
properties of the probabilities arise from this simple example. First, the
probability of any event is always given by a number in between zero and one.
Second, the sum of the probabilities of all possible events, in this case
either to extract a blue ball or a red ball, sum up to one.

Once we have a data sample, we normally face the task of analyzing the data to
extract the relevant properties from it. We assume that the data sample always
comes from a population sample which represents all the available data before
the measurements. Hence, the data sample might not be complete in the most
general case (one could lose some data in the measurement, or the data
population might be so large that we can only obtain part of it). The
measurement itself might also induce some error to the data (measuring a
continuous variable is always error prone, e.g., the height of people). Hence,
the data sample inherits a probability number for each of its data elements.
In the subsequent paragraphs, we will describe these relations between the
data sample and the corresponding probabilities, and we will enumerate the
most useful properties and formulas†††The notation used in this section
follows mainly the one used in Ref. [63]..

The magnitude we would like to measure from the data population is called a
random variable. When measuring some random variable $X$, a probability
distribution function (PDF) gives the probability of $x$ to be the outcome of
the measurement and it is denoted by $\text{Pr}(X{=}x)$. Second, due to the
random nature of the measurement, the $N$ elements of a data sample are
considered outputs of $N$ different random variables with their corresponding
values denoted by $\\{X_{i}{=}x_{i}\\}_{i=1}^{N}$, or
$\boldsymbol{X}{=}\boldsymbol{x}$ for short. The joint probability of those
random variables is in general not separable. This is due to the fact that the
data sample elements could depend on the rest of outcomes or some other more
complex relation that makes the most general case to be non-separable from the
probabilistic point of view. The PDF of the data sample is an $N$-variable
function denoted by

$\text{Pr}(\boldsymbol{X}{=}\boldsymbol{x})\equiv\text{Pr}(X_{1}{=}x_{1},X_{2}{=}x_{2},\dots,X_{N}{=}x_{N}).$
(2.1)

In the case of separable probabilities, Eq. (2.1) is written as

$\text{Pr}(\boldsymbol{X}{=}\boldsymbol{x})=\prod_{i=1}^{N}\text{Pr}(X_{i}{=}x_{i}),$
(2.2)

which is the case in many relevant situations.

When some indirect property of the system is defined as a function of the
measured random variables, the result is also a new random variable with
another assigned PDF. For example, we measure the position of a body at some
moment. If the system was at rest at the origin when $t=0$ and assuming that
the acceleration is constant, then from the measured final position one could
infer the value of the acceleration by using $A=2X/t^{2}$, where $X$ denotes
the final position at instant $t$ and $A$ the acceleration. If $X$ is a random
variable, which is the general case when measuring the position of some
physical system, then the probability assigned to the random variable $A$ is
computed by

$\text{Pr}(A{=}a)=\left.\frac{\text{d}X}{\text{d}A}\right|_{A=a}\text{Pr}(X{=}x)=\frac{2}{t^{2}}\text{Pr}(X{=}x).$
(2.3)

In general, for multiple random variables, we must require the following
identity

$|\text{Pr}(\boldsymbol{X}{=}\boldsymbol{x})\,\text{d}x_{1}\text{d}x_{2}\dots\text{d}x_{N}|=|\text{Pr}(\boldsymbol{Y}{=}\boldsymbol{y})\,\text{d}y_{1}\text{d}y_{2}\dots\text{d}y_{N}|,$
(2.4)

which leads to some interesting formulas we will discuss later.

We now stick to the simplest case in which the data consists of a collection
of values describing the same physical one-dimensional data population. We
assume that all the outcomes are independent from each other. Hence, the PDF
is of the form of Eq. (2.2). Next, we need some definitions used in this
thesis: namely the average, variance, and the statistical moments and central
moments. The arithmetic average is computed as

$\overline{x}=\frac{1}{N}\sum_{i=1}^{N}x_{i}.$ (2.5)

Note that there are other types of averages one can find in the literature.
The variance, which is related to the spread of the data, is computed as

$\sigma^{2}=\frac{1}{N}\sum_{i=1}^{N}(x_{i}-\overline{x})^{2},$ (2.6)

where $\sigma$ is the standard deviation. Different statistical moments are
computed by $\overline{x^{r}}=\frac{1}{N}\sum x_{i}^{r}$, while central
moments, i.e., statistical moments that are invariant under a global
translation of the parameter space, are of the form of
$c_{r}=\frac{1}{N}\sum(x_{i}-\overline{x})^{r}$. Note that the variance is
equivalent to the second central moment.

For completeness, when each element of the data consists of more than a single
magnitude the co-variance between two magnitudes is obtained as

$V_{X,Y}=\frac{1}{N}\sum_{i=1}^{N}(x_{i}-\overline{x})(y_{i}-\overline{y}),$
(2.7)

where it represents how both magnitudes influence each other. As an example,
we could measure two magnitudes simultaneously $(v_{i},a_{i})$, where $v_{i}$
is the velocity and $a_{i}$ the acceleration of a body, in a experiment to
estimate the friction coefficient of the air.

The data population is represented in most cases by the probability
distribution function. While the mean values over the data sample of any
function $g(x)$ are denoted with a bar, e.g., $\overline{x^{r}}$ for the
$r$-th moment over the data sample or $\overline{g(x)}$ for the average of
$g(x)$ itself, the mean values over the data population of any function $g(X)$
are denoted by $\text{E}[g(X)]$, e.g., the $r$-th moment over the data
population is $\text{E}[X^{r}]$. The data sample always consist of discrete
data elements. On the other hand, the data population can be represented by a
PDF for discrete values or by a PDF for continuous values. For completeness,
here are expressed the two definitions for the mean value over the data
population of a function $g(\boldsymbol{X})$ as

$\text{E}[g(\boldsymbol{X})]=\left\\{\begin{aligned} &\int
g(\boldsymbol{x})\text{Pr}(\boldsymbol{X}{=}\boldsymbol{x})\,\text{d}^{N}\boldsymbol{x},\\\
&\sum_{i,j,\dots}g(\boldsymbol{x})\text{Pr}(\boldsymbol{X}{=}\boldsymbol{x}).\end{aligned}\right.$
(2.8)

I also write the variance of $g(X)$ over the data population as
$\text{V}[g(X)]\equiv\text{E}[g(X)^{2}]-\text{E}[g(X)]^{2}$.

####  2.1.2 Estimators and Fisher information

Let us suppose that the data sample has encoded some parameters
$\boldsymbol{a}\equiv(a_{1},a_{2},\dots)$ we are interested in. Therefore, the
underlying probability, in general also unknown, might be conditioned by the
real values of the parameters $\boldsymbol{a}$. The probability of the data
sample is written as

$\text{Pr}(\boldsymbol{X}{=}\boldsymbol{x}|\boldsymbol{a}),$ (2.9)

where "$|\boldsymbol{a}$" indicates its dependency on these parameters. Note
that an estimate of one of the parameters must be based on the data sample
elements. A function of this type is called the _estimator_ and it is
connected to the PDF of the data population. Hence as mentioned before, the
estimator of $a$, one of the unknown parameters, is denoted by $\hat{a}$ and
the underlying PDF associated to it is computed from the PDF of the data
population as

$\text{Pr}(\hat{a}|\boldsymbol{a})\,\text{d}\hat{a}=\text{Pr}(\boldsymbol{x}|\boldsymbol{a})\,\text{d}x_{1}\text{d}x_{2}\dots\text{d}x_{N}.$
(2.10)

For short, we have omitted writing "$\boldsymbol{X}{=}$". The conditional
joint probability of $N$ random variables $\boldsymbol{X}$ is written simply
as $\text{Pr}(\boldsymbol{x}|\boldsymbol{a})$ then.

An estimator is a function of the data sample. For example, one of such
estimators can be the estimator of the mean value of the data population,
which is in general unknown. The mean value of the data population, which in
general we do not have access to, is denoted usually by $\mu$. Note that $\mu$
is in general different from the mean value of the data sample $\overline{x}$.
A valid estimator for the mean value $\mu$ would be the mean itself of the
data sample, i.e., $\hat{\mu}=\overline{x}$.

An important notion of an estimator is its _efficiency_. The smaller the
variance of the estimator the more efficient it is. Remember that an estimator
is considered a random variable so it must have a variance when our knowledge
about the population is incomplete.

Focusing now on what is more interesting for this thesis, an estimator of any
kind has a theoretical lower bound for its variance. For the proof of the
previous statement, which we will show only for continuous random variables
without loss of generality, we start with the normalization formula for any
given PDF

$\int\text{Pr}(\boldsymbol{x}|\boldsymbol{a})\,\text{d}^{N}\boldsymbol{x}=1.$
(2.11)

Next, we compute the partial derivative over $a$ such that

$\int{\partial_{a}}\text{Pr}(\boldsymbol{x}|\boldsymbol{a})\,\text{d}^{N}\boldsymbol{x}=\int\partial_{a}\big{[}\ln\text{Pr}(\boldsymbol{x}|\boldsymbol{a})\big{]}\text{Pr}(\boldsymbol{x}|\boldsymbol{a})\,\text{d}^{N}\boldsymbol{x}=0,$
(2.12)

where for the second equality we used the identity for logarithmic
derivatives. From Eq. (2.8), it turns out that Eq. (2.12) is the expectation
value of $\partial_{a}(\ln\text{Pr})$. Finally, if we have an _unbiased_
estimator, i.e., an estimator for which the expectation value
$\text{E}[\hat{a}]$ coincides with true value $a$, the partial differentiation
of $\text{E}[\hat{a}]$ over $a$ must be equal to one. Therefore, we apply
similar identities as in Eq. (2.12) to arrive at

$\begin{split}\partial_{a}\text{E}[\hat{a}]&=\partial_{a}a=\partial_{a}\int\hat{a}\text{Pr}(\boldsymbol{x}|\boldsymbol{a})\,\text{d}^{N}\boldsymbol{x}\\\
&=\int\hat{a}\,\partial_{a}\text{Pr}(\boldsymbol{x}|\boldsymbol{a})\,\text{d}^{N}\boldsymbol{x}=\int\hat{a}\,\partial_{a}\big{[}\ln\text{Pr}(\boldsymbol{x}|\boldsymbol{a})\big{]}\text{Pr}(\boldsymbol{x}|\boldsymbol{a})\,\text{d}^{N}\boldsymbol{x}=1,\end{split}$
(2.13)

where we have used the definition of the expectation value for continuous
variables Eq. (2.8) and we use the fact that the estimator is not a function
of the parameter $a$.

At this point, we invoke the Schwarz inequality for two real multidimensional
functions $g(\boldsymbol{x})$ and $h(\boldsymbol{x})$ such that $(\int
gh\,\text{d}^{N}\boldsymbol{x})^{2}\leqslant(\int
g^{2}\,\text{d}^{N}\boldsymbol{x})(\int h^{2}\,\text{d}^{N}\boldsymbol{x})$.
With this, we can obtain a lower bound for the variance of a general
estimator. First the definition of the variance over the data population of
$\hat{a}$ looks like

$V[\hat{a}]=E[(\hat{a}-a)^{2}]=\int(\hat{a}-a)^{2}\text{Pr}(\boldsymbol{x}|\boldsymbol{a})\,\text{d}^{N}\boldsymbol{x},$
(2.14)

for unbiased estimators. Second, since $a$ is not a function of
$\boldsymbol{x}$, subtracting $a$ times Eq. (2.12) to Eq. (2.13) we have that

$\int(\hat{a}-a)\,\partial_{a}\big{[}\ln\text{Pr}(\boldsymbol{x}|\boldsymbol{a})\big{]}\text{Pr}(\boldsymbol{x}|\boldsymbol{a})\,\text{d}^{N}\boldsymbol{x}=1.$
(2.15)

Hence, using the Schwarz inequality we can write

$\text{V}[\hat{a}]=\int(\hat{a}-a)^{2}\text{Pr}(\boldsymbol{x}|\boldsymbol{a})\,\text{d}^{N}\boldsymbol{x}\geqslant\frac{1}{\int\left(\partial_{a}\big{[}\ln\text{Pr}(\boldsymbol{x}|\boldsymbol{a})\big{]}\right)^{2}\text{Pr}(\boldsymbol{x}|\boldsymbol{a})\,\text{d}^{N}\boldsymbol{x}},$
(2.16)

which is also known as the Cramér-Rao bound or the Fisher inequality. The
denominator in the right hand-side is called generally the Fisher information
or simply information.

With this review of the most interesting properties of "classical" metrology,
from the point of view of this thesis, we conclude this section. In the next
section, we will discuss some properties of quantum mechanics and then we will
follow with another section which presents the basics of quantum metrology.

###  2.2 Quantum mechanics from metrology perspective

The ubiquitous probabilistic nature of quantum mechanics makes us to work with
probabilities on a regular basis. Moreover, if one studies fields connected to
experiments or some short of physical realizations, this probabilistic nature
of quantum mechanics becomes even more visible. On the other hand, exotic
features such as entanglement arise from quantum mechanics, which are directly
connected with the probabilistic properties of quantum system. The present
section is intended to describe quantum systems from the point of view of
metrology.

####  2.2.1 The quantum state, multiparticle state and entanglement

A formal mathematical description of the quantum state is given next. This
also allows us to introduce some notation used throughout the thesis. A
_state_ in quantum mechanics lives in a Hilbert space, $\mathcal{H}$. The
state $\rho$ has the following properties:

  1. i)

It is Hermitian, so it is invariant under the complex transposition,
$\rho=\rho^{\dagger}$ and all its eigenvalues are real.

  2. ii)

Its trace is equal to one, $\operatorname{tr}(\rho)=1$.

  3. iii)

It is positive semi-definite, i.e, all its eigenvalues are larger or equal to
zero, $\rho=\sum_{\lambda}p_{\lambda}\Pi_{\lambda}$ where
$p_{\lambda}\geqslant 0$ and
$\Pi_{\lambda}\equiv|{\lambda}\rangle{\\!}\langle{\lambda}|$ is the projector
to the eigenstate $|{\lambda}\rangle$, a vector state satisfying the following
eigenvalue equation $\rho|{\lambda}\rangle=p_{\lambda}|{\lambda}\rangle$. From
(ii), it follows that $\sum_{\lambda}p_{\lambda}=1$.

  4. iv)

If all $p_{\lambda}$ are zero except one, the state is called a pure state and
is equivalent to the projector to the corresponding eigenstate
$\rho=\Pi_{\lambda}=|{\lambda}\rangle{\\!}\langle{\lambda}|$.

  5. v)

Using the properties (iii) and (iv), it follows that the quantum states form a
convex set, where the extremal states are pure states.

  6. vi)

An expectation value of an observable $A$ is computed as
$\langle{A}\rangle=\operatorname{tr}(A\rho)$. To make a connection with the
previous section, the state might represent the so-called data population.
Hence, using the notation in Sec. 2.1, the following equivalence holds,
$\langle{\mathcal{O}}\rangle\equiv\text{E}[\mathcal{O}]$.

The composite system of $N$ different parties live in the Hilbert space
$\mathcal{H}=\mathcal{H}^{(1)}\otimes\mathcal{H}^{(2)}\otimes\cdots\otimes\mathcal{H}^{(N)}$
or $\mathcal{H}=\bigotimes_{i=1}^{N}\mathcal{H}^{(i)}$ for short, where
$\otimes$ stands for tensor product. For instance, this composite Hilbert
space could be used to represent a many-particle system, in this case $N$
particles. A _separable_ state in this Hilbert space can be written as [65]

$\rho_{\text{sep}}=\sum_{i}p_{i}\rho_{i}^{(1)}\otimes\rho_{i}^{(2)}\otimes\cdots\otimes\rho_{i}^{(N)},$
(2.17)

where $p_{i}$ are convex weights that add up to one and are equal to or larger
than zero. If a state cannot be written like Eq. (2.17), the state is
_entangled_. Note that Eq. (2.17) is a formal description of the separable
states, for more details see Refs. [36, 66]. One may note at this moment, that
relaxing the requirements of Eq. (2.17), one can lead to different
classifications of the states. Concepts like genuine multipartite
entanglement, $k$-producible states, or entanglement depth, among others,
arose from weaker constraints than Eq. (2.17) [36, 66].

It is important to describe one of such classifications in order to
characterize the different levels or multipartite entanglement followed in
this work. We call a state $k$-producible [67], if it can be written as a
mixture of the tensor product of different multipartite states with at most
$k$ parties in each,

$\rho_{k\text{-pro}}=\sum_{i}p_{i}\rho^{(\alpha,\dots,\beta)}_{i}\otimes\rho^{(\gamma,\dots,\delta)}_{i}\otimes\dots,$
(2.18)

where superscript indexes between parenthesis go from 1 to $N$ and denote to
which parties belong to the state, and where each index appears once in each
sum element. For instance, a separable state like Eq. (2.17) is 1-producible.
If a state cannot be written as $k$-producible, then it must be
$(k{+}1)$-entangled. This defines the entanglement depth, see Figure 2.1.
Later on, these concepts of entanglement and entanglement depth will arise
naturally on the metrological framework [68].

Figure 2.1:  Hierarchic sets for $k$-producible states, were $k$-producible
set contains $(k{-}1)$-producible states. Based on the Eq. (2.18), one can see
that a state that cannot be written as $k$-producible must be
$(k{+}1)$-entangled or equivalently it must have $k{+}1$ entanglement depth. A
separable state can always be written as 1-producible state by definition.

####  2.2.2 Angular momentum operators on multiparticle systems

We now present a set of operators that will appear many times in all chapters,
namely the angular momentum operators. Again these definitions allow us to
introduce much of the notation used on this thesis. For a single-particle with
$d$ discrete levels, and therefore with spin $j=(d-1)/2$, the eigenvalue
equation for the angular momentum projection operators are

$j_{l}^{(n)}|{m}\rangle_{l}^{(n)}=m|{m}\rangle_{l}^{(n)}$ (2.19)

for $m=-j,\dots,+j$ and where $l=x,y,z$. It is usual to omit the subscript of
$|{m}\rangle_{l}^{(n)}$ when $l=z$ because it is typically the preferred
direction and the superscript ($n$) can also be omitted in some cases, e.g.,
$|{\psi}\rangle=|{m_{1}}\rangle\otimes|{m_{2}}\rangle\otimes\dots\otimes|{m_{N}}\rangle$.
In many cases, it is also usual to merge all single-particle states into a
single ket such that $|{\psi}\rangle=|{m_{1},m_{2},\dots,m_{N}}\rangle$‡‡‡
This notation is normally used in many-body quantum mechanics, see Refs. [69,
70]..

The square of the total angular momentum,
$\boldsymbol{j}^{2}=j_{x}^{2}+j_{y}^{2}+j_{z}^{2}$, for a single-party $(n)$
acts on any state $\rho$ simply as

$(\boldsymbol{j}^{2})^{(n)}\rho=j_{n}(j_{n}+1)\rho,$ (2.20)

where the Hilbert space where $\rho$ is defined must contain
$\mathcal{H}^{(n)}$ and where $j_{n}$ is the spin number of such a particle.
Note that in order to distinguish the spin number $j_{n}$ and the operators
$j_{l}^{(n)}$ we may use the fact that the operators are attached to a Hilbert
space with a superscript or even we can use a "hat" to denote which of them is
an operator like $\hat{j_{l}}$ and which is not.

The collective angular momentum projection operators $J_{l}$ are defined as
the sum of their respective single-party spin operators $j_{l}^{(n)}$ such
that they are extended to the remaining of the Hilbert spaces by tensor
products of the identity operators defined in the rest of subspaces,

$J_{l}=\sum_{i=1}^{N}\mathbbm{1}^{(1,\dots,i-1)}\otimes
j_{l}^{(i)}\otimes\mathbbm{1}^{(i+1,\dots,N)}\equiv\sum_{i=1}^{N}j_{l}^{(i)},$
(2.21)

where $\mathbbm{1}$ stands for the identity operator and where in the last
equality the identity operators are omitted for simplicity. On the other hand,
note that the squares of the different projections of total angular momentum
are not equal to the sum of the square angular momentum projections of each of
the parties. The square angular momentum components are

$J_{l}^{2}=\sum_{i,j}^{N}j_{l}^{(i)}j_{l}^{(j)}=\sum_{i=1}^{N}(j_{l}^{2})^{(i)}+\sum_{i\neq
j}^{N}j_{l}^{(i)}j_{l}^{(j)},$ (2.22)

for $l=x,y,z$. Therefore, neither the square of the total angular momentum is
the sum of the square of all single-party angular momenta but

$\boldsymbol{J}^{2}=\sum_{i}^{N}\boldsymbol{j}^{(i)}+\sum_{l=x,y,z}\sum_{i\neq
j}^{N}j_{l}^{(i)}j_{l}^{(j)},$ (2.23)

where we separated the sum into two parts. The first one corresponds to the
sum of all single-party square angular momentum and the second corresponds to
the product of angular momentum projection operators of two distinct parties
summed for all $l=x,y,z$. Many more combinations of these single-party
operators may arise in different contexts. In the Appendix A, we discuss in
more detail the addition of the angular momenta and the block structure that
arises in this new basis, e.g., the symmetric subspace or the singlet
subspace.

####  2.2.3 Dynamics of quantum systems

The most basic evolution of the state is represented by unitary evolution
operators denoted by $U$ and those are the only ones appearing throughout the
thesis. On the other hand, there are other types of dynamics involving
particle losses, entropy changes in the system and open quantum systems in
general. These transformations are governed by master-equations such as the
Lindblad equation [71, 72, 73].

For the understanding of this thesis it is enough to present the unitary
evolution operators. We also restrict ourselves to the case in which the
generator $G$ is constant, so are in general the Hamiltonians of the
metrological setups. The unitary evolution operator is defined as

$U=\exp(-i\alpha G)=\sum_{n=0}^{\infty}\frac{(-i\alpha G)^{n}}{n!},$ (2.24)

for the amount of change $\alpha$ and where we use the matrix exponentiation
in the last equality. When a constant Hamiltonian $H$ acts on an initial state
$\rho$, the state evolves in time as

$\rho(t)=U\rho U^{\dagger}=e^{-itH/\hbar}\rho e^{+itH/\hbar}.$ (2.25)

Note that all information we can extract from the system comes in the form of
expectation values of different operators at different times,
$\langle{A}\rangle(t)=\operatorname{tr}(A\rho(t))$. When the state evolves in
time but the operators are constant the picture of the system is called the
Schrödinger-picture. Using the cyclic property of the trace,
$\operatorname{tr}(ABC)=\operatorname{tr}(CBA)$, the Heisenberg-picture, a
dual interpretation of the same physical system emerges, in which the state
remains the same while the operators evolve in time. It is well known that the
operators in this picture evolve as $A(t)=U^{\dagger}AU$, where $A$ is the
initial operator, see Refs. [69, 70].

###  2.3 Quantum metrology

In this section, we summarize important recent advances in quantum metrology.
Quantum metrological setups encode some unknown parameter into the quantum
state. Then, from the readout of the final state one could in principal
estimate this parameter. The basic ideas of quantum metrology emerge when one
applies the notions of estimation theory to the intrinsic probabilistic nature
of quantum mechanics. Merging the probabilistic features of quantum mechanics
and the estimation theory is not trivial. Nevertheless, with the initial
pioneering works of C. W. Helstrom, W. K. Wootters, and S. L. Braunstein and
C. M. Caves, in 1969, 1981 and 1992-1994 respectively [30, 74, 75, 76], until
the works of V. Giovannetti et al, and M. G. Paris, roughly two decades later
[1, 2], quantum metrology got firm foundations. Later, advanced works in
quantum metrology appeared [10, 77, 78, 79, 80] together with experimental
realizations [37, 17, 8], which raised the interest in this topic. In this
section, we will highlight the most important aspects of this field and with
this we will conclude this introductory chapter.

The most basic scheme for a metrological setup in the present context is the
following. First, a state $\rho$ is prepared followed by a general evolution
represented by a mapping $\Lambda_{\theta}$ in which the unknown parameter
$\theta$ is imprinted on the state. Finally, the outgoing state is
characterized by some measured quantity $\langle{M}\rangle$, which allows us
to infer the value of the parameter $\theta$. Figure 2.2 illustrates the main
steps of quantum metrology.

Figure 2.2:  Sequence of the different steps of the estimation process in
quantum metrology. First, an input state $\rho$ enters the region in which the
unknown parameter $\theta$ is imprinted on it, which is represented with
$\Lambda_{\theta}$ for the most general case. The encoded parameter $\theta$
is inferred from the measured quantity $\langle{M}\rangle$ over the final
state $\rho_{\theta}$.

In the many-particle case, most of the metrology experiments have been done
for systems with simple Hamiltonians that do not contain interaction terms.
Such Hamiltonians cannot create entanglement between the particles. A typical
situation is that we rotate our many particle state by some angle and we want
to estimate the rotation angle $\theta$. It has been shown that particles
exhibiting quantum correlations, or more precisely, quantum entanglement [36,
66], provide a higher precision than an ensemble with non-entangled particles.
The most important question is how the achievable precision of the angle
estimation $(\Delta\theta)^{2}$ scales with the number of particles $N$. Very
general derivations lead to, at best,

$(\Delta\theta)^{2}\sim\frac{1}{N}$ (2.26)

for non-entangled particles. The equation above is called _shot-noise_ or
standard scaling, the term originating from the shot-noise in electronic
circuits, which is due to the discrete nature of the electric charge. On the
other hand, quantum entanglement makes it possible to reach

$(\Delta\theta)^{2}\sim\frac{1}{N^{2}}$ (2.27)

which is called the _Heisenberg_ scaling. Note that if the Hamiltonian of the
dynamics has interaction terms, then these bounds can be surpassed [66, 81,
82, 83, 84, 85, 86].

It is time to mention that the calculations above have been carried out for an
ideal situation. When an uncorrelated noise is present in the system, it turns
out that for large enough particle number the scaling becomes shot-noise
scaling [62]. The possible survival of a better scaling under correlated
noise, under particular circumstances, or depending on some interpretation of
the metrological task, is at the center of attention currently. All these are
strongly connected to the question of whether strong multipartite entanglement
can survive in a noisy environment.

Finally, note that often, instead of $(\Delta\theta)^{2}$ one calculates its
inverse, which is large for high precision. It scales as
$(\Delta\theta)^{-2}\sim N$ for shot-noise scaling and as
$(\Delta\theta)^{-2}\sim N^{2}$ for the Heisenberg scaling.

####  2.3.1 Quantum magnetometry

Without loss of generality, we present in this section the characterization of
the precision of one of the simplest metrological tasks, namely the estimation
of a homogeneous magnetic field based on the interaction between the system
and the field. In this section, we will mostly study the interaction of a
system with a homogeneous magnetic field in the $z$-direction. In the Chapter
5, we will show a different situation in which the magnetic field changes
linearly with the position of the system. Coupling the magnetic moment of the
state and the magnetic field, we imprint the magnetic field strength on the
evolved state. Finally, measuring how the state has changed one could in
principle infer on the strength of the magnetic field.

In general, we will say that the magnetic moment of the state comes
exclusively from the spin angular momentum, neglecting any possible
contribution from the orbital angular momentum. This way the physics is
simpler. This is justified in the sense that most of the recent experiments in
this context have been carried out with ion-traps, Bose-Einstein condensates
(BEC) or at most cold atomic ensembles, which have indeed a negligible orbital
angular momentum.

Beside this considerations, the interaction Hamiltonian can be written as

$H=-\boldsymbol{\mu}\cdot\boldsymbol{B},$ (2.28)

Now in the simplest case we will choose the magnetic field to be pointing in
the $z$-direction as ${B}=B\boldsymbol{k}$, where $\boldsymbol{k}$ is the
unitary vector pointing to the $z$-direction. This way one does not need to
determine the direction of the magnetic field.

The magnetic moment of the system is proportional to the total angular
momentum, $\boldsymbol{\mu}=-\mu_{\text{B}}g_{\text{s}}\hbar\boldsymbol{J}$,
where $\mu_{\text{B}}$ and $g_{\text{s}}$ are the Bohr magneton and the
anomalous gyromagnetic factor respectively, and where we multiply it by the
Plank reduced constant $\hbar$. Note that throughout all the thesis the
angular momentum component were defined with $\hbar=1$ for simplicity, e.g.,
see Eq. (2.19). Hence, one can rewrite the interaction Hamiltonian as

$H=\gamma BJ_{z}.$ (2.29)

where $\gamma=\mu_{\text{B}}g_{\text{s}}\hbar$ and we have used that
$\boldsymbol{J}\cdot\boldsymbol{k}=J_{z}$. Finally, the unitary operator
leading the evolution of the system can be written as

$U=\exp(-i\theta J_{z}),$ (2.30)

where the magnetic field strength is encoded into the phase-shift
$\theta=-\mu_{\text{B}}g_{\text{s}}tB$ and $t$ is the evolution time.

We have to mention that for large particle ensembles, typically only
collective quantities can be measured in order to characterize the state in
different phases of the metrological sequence, see Figure 2.2. Such collective
quantities are in this case the total angular momentum components defined in
Eq. (2.21) and their linear combinations. More concretely, we can measure the
expectation values of any direction

$J_{\boldsymbol{n}}:=\sum_{l=x,y,z}n_{l}J_{l},$ (2.31)

where $\boldsymbol{n}=(n_{x},n_{y},n_{z})$ is a unit vector describing the
direction of the component.

####  2.3.2 Metrology with almost polarized states, including spin-squeezed
states

Let us present one of the basic approaches to calculate the metrological
precision of a quantum setup. In order to estimate the phase-shift $\theta$ we
measure the expectation value of a Hermitian operator, which we will denote by
$M$ in the following. If the evolution time is a constant, then estimating
$\theta$ is equivalent to estimating the magnetic field strength $B$ appearing
in Eq. (2.29). The precision of the estimation can be characterized with the
error propagation formula as

$(\Delta\theta)^{2}=\frac{(\Delta
M)^{2}}{|\partial_{\theta}\langle{M}\rangle|^{2}},$ (2.32)

where $\langle{M}\rangle$ is the expectation value of the operator $M$. We
used that $\langle{M}\rangle$ is a random variable with probability
$\text{Pr}(\langle{M}\rangle=m|\theta)$ for the outcomes of the measurements,
see the Section 2.1.2. For a recent review discussing this approach in detail,
see Ref. [79]. Based on the formula Eq. (2.32), one can see that the larger
the slope $|\partial_{\theta}\langle{M}\rangle|$, the higher the precision. On
the other hand, the larger the variance $(\Delta M)^{2}$, the lower the
precision. Figure 2.3 helps to interpret the quantities appearing in Eq.
(2.32).

Figure 2.3:  (Blue solid) Functional relation between the expectation value
$\langle{M}\rangle$ and the estimator of the wanted parameter $\hat{\theta}$.
(Green dashed) One to one correspondence when the estimator $\hat{\theta}$ is
based on $\langle{M}\rangle$. (Red dotted) Obtaining the error of the estimate
is based on Eq. (2.32). The slope of the curve at that point, denoted with
$\tan\alpha$, directly relates the uncertainty $\sigma_{M}$ on the measured
quantity $\langle{M}\rangle$ and the error on the estimation
$\sigma_{\theta}$.

We focus on multi-partite systems of spin-$\frac{1}{2}$ particles, widely
known as qubits. Let us start with an almost polarized state. The state
totally polarized along the $l$-direction is written as

$|{+j}\rangle_{l}^{\otimes
N}\equiv|{+j}\rangle_{l}^{(1)}|{+j}\rangle_{l}^{(2)}\cdots|{+j}\rangle_{l}^{(N)},$
(2.33)

For $l=y$, the state Eq. (2.33) is almost all the times one of the states that
saturates the shot-noise limit, which will be defined later in this section.
In the present case, the spin-vector of the ensemble, originally pointing into
the $y$-direction, will be rotated. The rotation after the evolution is used
to estimate the strength of the magnetic field. A way to measure the rotation
of the system is to measure the expectation value $\langle{J_{x}}\rangle$,
which is zero at the beginning.

For small angles of $\theta$ and using the Eq. (2.32) after substituting $M$
by $J_{x}$, we have that

$(\Delta\theta)^{-2}=\frac{|\partial_{\theta}\langle{J_{x}}\rangle|^{2}}{(\Delta
J_{x})^{2}},$ (2.34)

for a state almost completely polarized along the $y$-axis. Next, we have to
compute these values for small angles of $\theta$.

To compute them we use the Heisenberg picture of the operator $J_{x}$, which
after applying the unitary operator Eq. (2.30), is written as

$J_{x}(\theta)=U^{i\theta J_{z}}J_{x}U^{-i\theta
J_{z}}=\text{c}_{\theta}J_{x}-\text{s}_{\theta}J_{y},$ (2.35)

where we introduce a notation for trigonometric functions such that
$\text{c}_{x}=\cos(x)$, $\text{s}_{x}=\sin(x)$ and $\text{t}_{x}=\tan(x)$. The
square of $J_{x}$ is simply

$J_{x}^{2}(\theta)=\text{c}_{\theta}^{2}J_{x}^{2}-2\text{c}_{\theta}\text{s}_{\theta}\\{J_{x},J_{y}\\}+\text{s}_{\theta}^{2}J_{y}^{2},$
(2.36)

where $\\{A,B\\}=AB+BA$ is the anti-commutator. We also compute the derivative
with respect to $\theta$ of $\langle{J_{x}}\rangle$ as

$\partial_{\theta}\langle{J_{x}}\rangle(\theta)=\partial_{\theta}\text{c}_{\theta}\langle{J_{x}}\rangle-\partial_{\theta}\text{s}_{\theta}\langle{J_{y}}\rangle=-\text{s}_{\theta}\langle{J_{x}}\rangle-\text{c}_{\theta}\langle{J_{y}}\rangle.$
(2.37)

Hence, assuming that we are in the small angle limit we can neglect the terms
proportional to $\sin(\theta)$ when computing the Eq. (2.34). Finally, the
precision of the estimation is given by

$(\Delta\theta)^{-2}=\frac{|\text{c}_{\theta}\langle{J_{y}}\rangle|^{2}}{\text{c}_{\theta}^{2}\langle{J_{x}^{2}}\rangle-(\text{c}_{\theta}\langle{J_{x}}\rangle)^{2}}=\frac{\langle{J_{y}}\rangle^{2}}{(\Delta
J_{x})^{2}},$ (2.38)

when we measure $\langle{J_{x}}\rangle$ to estimate the angle $\theta$.

For the state totally polarized along the $y$-axis, the initial expectation
values $\langle{J_{x}}\rangle$, $\langle{J_{x}^{2}}\rangle$ and
$\langle{J_{y}}\rangle$ needed to compute the precision are

$\displaystyle\langle{J_{x}}\rangle_{\text{tp}}$ $\displaystyle=0,$ (2.39a)
$\displaystyle\langle{J_{x}^{2}}\rangle_{\text{tp}}$
$\displaystyle=\tfrac{N}{4},$ (2.39b)
$\displaystyle\langle{J_{y}}\rangle_{\text{tp}}$ $\displaystyle=\tfrac{N}{2}.$
(2.39c)

Thus, we obtain a precision that scales linearly with $N$,

$(\Delta\theta)^{-2}_{\text{tp}}=\frac{N^{2}/4}{N/4}=N.$ (2.40)

Note that the totally polarized state Eq. (2.33) is a separable pure state
with all particles pointing into the $y$-direction. Hence, we obtained the
shot-noise scaling, even with very simple qualitative arguments.

A way of improving the precision is considering that the variances of the
angular momentum components are bounded by the Heisenberg uncertainty relation
as [44]

$(\Delta J_{x})^{2}(\Delta
J_{z})^{2}\geqslant\frac{1}{4}|\langle{J_{y}}\rangle|^{2}.$ (2.41)

When decreasing the variance $(\Delta J_{x})^{2}$ to obtain a better
precision, our state fulfills

$(\Delta J_{x})^{2}<\frac{1}{2}|\langle{J_{y}}\rangle|,$ (2.42)

where the main spin points along the $y$-axis and where we would obtain
$(\Delta J_{x})^{2}=\frac{1}{2}|\langle{J_{y}}\rangle|$ for totally polarized
states. Such states are called _spin-squeezed_ states [44, 45, 46, 47] and
they can in principal overcome the shot-noise scaling.

Next, we can ask, what the best possible phase estimation precision is for the
metrological task considered in this section. For that, we have to use the
following inequality based on general principles of angular momentum theory

$\langle{J_{x}^{2}+J_{y}^{2}+J_{z}^{2}}\rangle\leqslant\frac{N(N+2)}{4}.$
(2.43)

Note that Eq. (2.43) is saturated only by symmetric multiparticle states, see
Appendix A. Together with the identity connecting the second moments,
variances and expectation values

$(\Delta J_{l})^{2}+\langle{J_{l}}\rangle^{2}=\langle{J_{l}^{2}}\rangle,$
(2.44)

Eq. (2.43) leads to a bound on the uncertainty for the angular momentum
component in the anti-squeezed direction, in this case the $z$-direction, as

$(\Delta
J_{z})^{2}\leqslant\frac{N(N+2)}{4}-\langle{J_{y}}\rangle^{2}=\frac{N}{2}+\frac{N^{2}}{4}\left(1-\frac{\langle{J_{y}}\rangle^{2}}{J_{\max}^{2}}\right),$
(2.45)

where $J_{\max}=N/2$ is the maximum value an angular momentum component can
take. This leads to a bound in the precision when measuring
$\langle{J_{x}}\rangle$ as

$(\Delta\theta)^{-2}=\frac{\langle{J_{y}}\rangle^{2}}{(\Delta
J_{x})^{2}}\leqslant 4(\Delta J_{z})^{2}\leqslant
2N+N^{2}\left(1-\frac{\langle{J_{y}}\rangle^{2}}{j_{\max}}\right),$ (2.46)

which indicates that the precision is limited for almost completely polarized
states to the shot-noise scaling and that less polarized states can overcome
the shot-noise limit. The bound above is not optimal, as for the fully
polarized states we would expect $N$ and we obtain $2N$.

####  2.3.3 The quantum Fisher information

In this section, we review the theoretical background of the Fisher
information, the Cramér-Rao bound and we introduce the quantum Fisher
information.

First of all, we have already introduced the Fisher information and the
Cramér-Rao bound in Section 2.1.2. The formula Eq. (2.16) tells us that if we
measure an operator, say $M$, and we know its probability distribution
function $\text{Pr}(\langle{M}\rangle=m|\theta)$, we can bound the achievable
precision. In quantum mechanics the PDF of a state $\rho_{\theta}$ when
measuring the operator $M$ is given by

$\text{Pr}(m|\theta)=\operatorname{tr}({\Pi_{m}\rho_{\theta}}),$ (2.47)

where $\Pi_{m}$ are the projector operators or the eigenstates on which $M$ is
expanded, $M=\sum m\Pi_{m}$. Following arguments found on Refs. [1, 2], one
can arrive at a universal bound valid for any measurement. This bound is
called the quantum Cramér-Rao bound given as

$(\Delta\theta)^{-2}\leqslant\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{z}}],$
(2.48)

where $\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{Z}}]$ denotes the
quantum Fisher information for the initial state $\rho$ and the unitary
evolution generated by $J_{z}$. In principle, it might be difficult to find
the operator that leads to the best estimation precision just by trying
several operators. Fortunately, the theory tells us that the bound Eq. (2.48)
can be saturated [87, 31].

As a direct consequence, based on Eq. (2.32) for any given $M$, we have that

$\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{z}}]\geqslant\frac{|\partial_{\theta}\langle{M}\rangle|^{2}}{(\Delta
M)^{2}}$ (2.49)

is an upper bound for the precision obtained by measuring $\langle{M}\rangle$.
For example, when the rotation of the system is estimated by measuring
$\langle{J_{x}}\rangle$, based on the Eq. (2.38) the quantum Fisher
information is bounded from below by [3]

$\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{z}}]\geqslant\frac{\langle{J_{y}}\rangle^{2}}{(\Delta
J_{x})^{2}}.$ (2.50)

The quantum Fisher information can be computed by closed formulas such as

$\displaystyle\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{z}}]$
$\displaystyle=2\sum_{\lambda,\nu}\frac{(p_{\lambda}-p_{\nu})^{2}}{p_{\lambda}+p_{\nu}}|\langle{\lambda}|{J_{z}}|{\nu}\rangle|^{2},$
(2.51) $\displaystyle\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{z}}]$
$\displaystyle=\inf_{p_{k},|{\psi_{k}}\rangle}4\sum_{k}p_{k}(\Delta
J_{z})^{2}_{|{\psi_{k}}\rangle},$ (2.52)

for the case of unitary evolution Eq. (2.30). In the Eq. (2.51), the state is
decomposed in its eigenbasis as
$\rho=\sum_{\lambda}p_{\lambda}|{\lambda}\rangle{\\!}\langle{\lambda}|$. On
the other hand, in the Eq. (2.52), the state is arbitrarily decomposed as
$\rho=\sum_{k}p_{k}|{\psi_{k}}\rangle{\\!}\langle{\psi_{k}}|$. Hence, the
first equation is based on the eigen-decomposition of the state whereas the
second is the convex-roof of $4(\Delta J_{z})^{2}$ [2, 88, 89]. Both formulas
yield the same expression for pure states

$\mathcal{F}_{\textnormal{Q}}[{\textstyle|{\psi}\rangle,J_{z}}]=4(\Delta
J_{z})^{2}.$ (2.53)

Using the identity $(a-b)^{2}=(a+b)^{2}-4ab$ and that
$p_{\lambda}=\delta_{\lambda,1}$ (we choose the pure state to be $|{1}\rangle$
without loss of generality) in Eq. (2.51), we arrive at the result

$\begin{split}\mathcal{F}_{\textnormal{Q}}[{\textstyle|{\psi}\rangle,J_{z}}]&=2\sum_{\lambda,\nu}\left(p_{\lambda}+p_{\nu}-\frac{4p_{\lambda}p_{\nu}}{p_{\lambda}+p_{\nu}}\right)|\langle{\lambda}|{J_{z}}|{\nu}\rangle|^{2}\\\
&=4\operatorname{tr}(J_{z}\rho)-8\sum_{\lambda,\nu}\frac{p_{\lambda}p_{\nu}}{p_{\lambda}+p_{\nu}}|\langle{\lambda}|{J_{z}}|{\nu}\rangle|^{2}\\\
&=4(\Delta J_{z})^{2}.\end{split}$ (2.54)

From the Eq. (2.52), we also obtain the same result since the convex-roof over
a pure state must be computed in the state itself.

Some interesting properties of the quantum Fisher information emerge from
these expressions and we summarize them in the following list:

  1. i)

The QFI is convex in states, as it is directly shown in Eq. (2.52). It can
also be proven if one starts from Eq. (2.51),

$\mathcal{F}_{\textnormal{Q}}[{\textstyle
p\rho_{1}+(1-p)\rho_{2},J_{z}}]\leqslant
p\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho_{1},J_{z}}]+(1-p)\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho_{2},J_{z}}].$
(2.55)

  2. ii)

Based on Eq. (2.52), it has recently been shown, that the quantum Fisher
information is the largest convex function that fulfills Eq. (2.53) [88, 89].

  3. iii)

For pure states
$\mathcal{F}_{\textnormal{Q}}[{\textstyle|{\psi}\rangle,J_{z}}]=4(\Delta
J_{z})^{2}$, as it has been already shown.

  4. iv)

For all states, the QFI is smaller or equal to four times the variance

$\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{z}}]\leqslant 4(\Delta
J_{z})^{2}_{\rho}.$ (2.56)

It can be proven that the variance is the concave-roof of the variance itself
[88]. Hence, the main relation between the quantum Fisher information and the
variance can be summarized as follows. For any decomposition
$\\{p_{k},|{\psi_{k}}\rangle\\}$ of a state $\rho$ we have

$\frac{1}{4}\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{z}}]\leqslant\sum_{k}p_{k}(\Delta
J_{z})^{2}_{|{\psi_{k}}\rangle}\leqslant(\Delta J_{z})^{2}_{\rho},$ (2.57)

where the upper bound and the lower bound are both tight in the sense that
there are decompositions that saturate the first inequality, and there are
others that saturate the second one.

After the discussion relating the quantum Fisher information to the variance,
and examining its convexity properties, we list some further useful relations
for the QFI. In this case, we substitute the generator of the phase shift
$J_{z}$ by some more general Hermitian operators. From Eq. (2.51), we can
obtain directly the following identities:

  1. i)

The formula Eq. (2.51) does not depend on the diagonal elements of the
generator written in the eigenbasis of the state. Hence,

$\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,A}]=\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,A+D}],$
(2.58)

where $D$ is an arbitrary matrix that is diagonal in the eigenbasis of the
state, i.e., $[\rho,D]=0$.

  2. ii)

The following identity holds for all unitary dynamics $U$ as it could be
expected from the Schrodinger vs Heisenberg pictures,

$\mathcal{F}_{\textnormal{Q}}[{\textstyle U\rho
U^{\dagger},A}]=\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,U^{\dagger}AU}].$
(2.59)

In particular the QFI does not change for unitary dynamics of the type
$U=e^{-iB}$ when $[A,B]=0$.

  3. iii)

The quantum Fisher information is additive under tensor product as

$\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho^{(1)}\otimes\rho^{(2)},A^{(1)}\otimes\mathbbm{1}^{(2)}+\mathbbm{1}^{(1)}\otimes
A^{(2)}}]=\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho^{(1)},A^{(1)}}]+\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho^{(2)},A^{(2)}}].$
(2.60)

For $N$-fold tensor product of the system, we obtain an $N$-fold increase in
the quantum Fisher information as

$\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho^{\otimes
N},\textstyle\sum_{n=1}^{N}A^{(n)}}]=N\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,A}],$
(2.61)

where $A^{(n)}\equiv A$ for all $n$.

  4. iv)

The quantum Fisher information is additive under a direct sum [90]

$\mathcal{F}_{\textnormal{Q}}[{\textstyle\bigoplus_{k}p_{k}\rho_{k},\bigoplus_{k}A_{k}}]=\sum_{k}p_{k}\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho_{k},A_{k}}],$
(2.62)

where $\sum_{k}p_{k}=1$. The above equation is relevant, for instance, for
experiments where the particle number variance is not zero, and the $\rho_{k}$
corresponds to density matrices with a fixed particle number [10, 77].

Entanglement and the quantum Fisher information

The quantum Fisher information is strongly related to entanglement. In this
section, we discuss now this relation and we review some important facts
concerning it. We will show that entanglement is needed to overcome the shot-
noise sensitivity in very general metrological tasks. Moreover, not only
entanglement but multipartite entanglement is necessary for a maximal
sensitivity. All these statements will be derived in a very general framework,
based on the quantum Fisher information. We will also briefly discuss the
question whether inter-particle entanglement is an appropriate notion for our
systems.

Let us first examine the upper bounds on the QFI for general quantum states
and for separable states. Due to the Cramér-Rao bound (2.48), they are also
bounds for the sensitivity of the phase estimation. Entanglement has been
recognized as an advantage for several metrological tasks (see, e.g., Refs.
[46, 91]). For a general relationship for linear interferometers, we can take
advantage of the properties of the quantum Fisher information discussed
before. Since for pure states the quantum Fisher information equals four times
the variance, for pure product states we can have at most

$\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{z}}]=4(\Delta
J_{z})^{2}=4\sum_{n=1}^{N}(\Delta j_{z}^{(n)})^{2}\leqslant 4Nj^{2},$ (2.63)

where $j$ is the spin of the particles. Note that for spin-$\frac{1}{2}$ we
recover the usual threshold one can find for instance in Ref. [26]. For the
second equality, we used the fact that for a product state the variance of a
collective observable is the sum of the single-particle variances. Due to the
convexity of the QFI, this upper bound is still valid for all separable states
of the form Eq. (2.17) and we obtain [3]

$\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{z}}]\leqslant 4Nj^{2},$
(2.64)

a bound for not entangled states. All states violating Eq. (2.64) are
entangled. The entangled states make it possible to surpass this bound, the
shot-noise limit, and some might be more useful than separable states for the
metrological tasks at hand.

The maximum achievable precision for general states, called the Heisenberg
limit, can be obtained evaluating the Eqs. (2.51) or (2.52) for pure states
only. Therefore, similarly we have

$\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{z}}]=4(\Delta
J_{z})^{2}\leqslant 4N^{2}j^{2},$ (2.65)

which is a valid bound still for mixed states due to the convexity of the QFI.
Note that such a bound has the Heisenberg scaling Eq. (2.27). Note also that
our derivation is very simple, and does not require any information about what
operator we measure to estimate $\theta$. Equation (2.64) has been used
already to detect entanglement based on the metrological performance of the
quantum states in Refs. [60, 8].

At this point one might ask whether all entangled states can provide a
sensitivity larger than the shot-noise limit. This would show that
entanglement is equivalent to metrological usefulness. Concerning linear
interferometers, it has been proven that not all entangled states violate the
shot-noise limit Eq. (2.64), even allowing local unitary transformations. Thus
not all entangled states are useful for phase estimation [10]. It has been
shown that there are even highly entangled pure states that are not useful.
Hence, the presence of entanglement is a necessary but rather than a
sufficient condition.

The quantum Fisher information can be used to define an entanglement parameter
that characterizes the metrological usefulness as

$\chi=\frac{\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{z}}]}{N},$ (2.66)

which is not larger than one for separable states and it can take at most the
value of $N$ as is deduced from Eq. (2.65).

Based on methods similar to the ones used to find the bound for separable
states in Eqs. (2.64) and (2.65), for $k$-producible states the QFI is bounded
from above as [77, 92]

$\begin{split}\chi&\stackrel{{\scriptstyle\phantom{k\ll
N}}}{{\leqslant}}\frac{sk^{2}+(N-sk)^{2}}{N}\stackrel{{\scriptstyle k\ll
N}}{{\approx}}k,\end{split}$ (2.67)

where $s$ is the integer part of $\frac{N}{k},$ and we write out explicitly
the bound for $k\ll N$. It is instructive to write the equation above for the
case in which $N$ is exactly divisible by $k$ as

$\chi\leqslant{k}.$ (2.68)

Thus, the bounds reachable by $k$-producible states are distributed linearly
in $k$.

It is also instructive to define a QFI averaged over all possible directions.
Simple calculations show that

$\underset{\boldsymbol{n}}{\text{avg}}\,\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{\boldsymbol{n}}}]\equiv\int_{\boldsymbol{n}=(\text{c}_{\varphi}\text{s}_{\vartheta},\text{s}_{\varphi}\text{s}_{\vartheta},\text{c}_{\vartheta})}\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{\boldsymbol{n}}}]\sin(\vartheta)\,\text{d}\varphi\text{d}{}\vartheta=\frac{1}{3}\sum_{l=x,y,z}\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{l}}],$
(2.69)

where we used the spherical coordinates such that $\varphi$ and $\vartheta$
are the azimuthal and polar angles respectively. Therefore, bounds similar to
Eqs. (2.64) and (2.65) for separable and general states can be obtained for
the average quantum Fisher information as

$\displaystyle\underset{\boldsymbol{n}}{\text{avg}}\,\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{\boldsymbol{n}}}]$
$\displaystyle\leqslant\frac{2}{3}N,$ (2.70a)
$\displaystyle\underset{\boldsymbol{n}}{\text{avg}}\,\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{\boldsymbol{n}}}]$
$\displaystyle\leqslant\frac{1}{3}N(N+2).$ (2.70b)

Similarly to Eq. (2.66), an entanglement criterion can be constructed for the
averaged QFI. Similar to Eq. (2.66) an entanglement criterion can be
constructed for the averaged QFI.

Finally, we also mention that bound entangled states can also be detected with
the entanglement criteria based on the quantum Fisher information. Bound
entanglement is a weak type of entanglement, which is not distillable with
local operations and classical communication [35, 36]. Ref. [77] presented
states that were detected as bound entangled based on the criterion for the
average QFI Eq. (2.69). Ref. [93] presented states that violate the criterion
based on a bound for the quantum Fisher information similar to Eq. (2.66).

##  3 Metrology in the vicinity of Dicke states Metrology in the vicinity of
Dicke states

"An experimentalist should not be unduely inhibited by theoretical
untidyness."

Robert H. Dicke

In this chapter we will present recent results regarding the metrological
usefulness of a family of unpolarized states. Such states can be used as trial
states to estimate the homogeneous magnetic field strength, see Section 2.3.1
for references about magnetometry. It turns out that unpolarized states are
the most adequate states to reach the Heisenberg limit, as it was shown in the
Section 2.3. Hence, these states have attracted considerable interest.

One of the figures of merit of these states is the so-called unpolarized Dicke
state [94] given in the $l$-basis, which consists of an equal number of qubits
pointing in the $l$-direction and pointing in the opposite direction while the
whole state is symmetrized, and where $l=x,y,z$. It can be written as

$|{\textnormal{D}_{N}}\rangle_{l}\equiv|{\textnormal{D}_{N,N/2}}\rangle_{l}:=\binom{N}{N/2}^{-\frac{1}{2}}\sum_{k\in\sigma_{\text{s}}}\mathcal{P}_{k}(|{0}\rangle_{l}^{\otimes
N/2}|{1}\rangle_{l}^{\otimes N/2}),$ (3.1)

where $\mathcal{P}_{k}$ denote all the distinct permutations in
$\sigma_{\text{s}}$. $|{0}\rangle_{l}$ and $|{1}\rangle_{l}$ are single-
particle quantum states of a spin-$\frac{1}{2}$ system in the $l$-basis, see
Appendix A for more details. Note that in Eq. (3.1) we omit the subscript
giving the number of $|{1}\rangle$’s which is the notation we will follow in
this chapter. Such a state is known to be highly entangled [95, 96] and can
reach Heisenberg scaling when used for magnetometry [97].

One of the characteristics of state Eq. (3.1) is that it is an eigenstate of
the collective operator $J_{l}$ with corresponding eigenvalue equal to zero.
At the same time, it lives in the subspace where the collective total spin is
maximum, i.e, $\langle{\boldsymbol{J}^{2}}\rangle=N(N+2)/4$. Based on these
and the fact that the state is unpolarized, we can see that has a very large
uncertainty for the collective spin operators perpendicular to $J_{l}$.

For metrology, we chose the magnetic field to be pointing in the $z$-axis.
Hence, the Dicke state we choose must be an eigenstate of a perpendicular
component of the angular momentum operator such as $J_{x}$. Hence, we will
consider a scheme in which the state is rotated around the $z$-direction and
the rotation angle must be estimated based on collective measurements. A
criterion to detect the metrological usefulness of states of this type has
been derived in Ref. [98].

In this chapter, we present a condition for metrological usefulness for the
case when the second moment of a total angular momentum component is measured
to obtain an estimate for the rotation angle. Our method is expected to
simplify the experimental determination of metrological sensitivity since it
is much easier to measure the collective operators of the state than carrying
out the metrological procedure and measure directly the sensitivity. We also
test our approach using the experimental results of Refs. [8, 60], which
realize parameter estimation with a Dicke state. Thus, our work is expected to
be useful for similar experiments in the future.

The chapter is organized as follows. In Section 3.1, we review some important
concepts behind the theory of metrology with unpolarized states. In Section
3.2, we present our criterion. In Section 3.3, we compare our findings to
sensitivity bounds obtained from the quantum Fisher information. In Section
3.4, we show how to apply our criterion to experimental results.

###  3.1 Unpolarized Dicke states for magnetometry

In Eq. (2.46) we have shown that unpolarized states may overcome the shot-
noise limit, i.e., the precision achieved by completely polarized states.
While the quantum Fisher information would give us directly the performance of
the state, we typically cannot compute it because a complete knowledge of the
state would be necessary, see Eq. (2.51). On the other hand, we can use the
error propagation formula Eq. (2.32) to obtain a bound on the achievable
precision which at the same time bounds the QFI.

As one can see in Figure 3.1, a pure unpolarized Dicke state
$|{\textnormal{D}_{N}}\rangle_{x}$ of 16 qubits, an eigenstate of the $J_{x}$
operator, is rotated around the $z$-axis. The state is unpolarized so the
expectation value of any component of the total angular momentum remains zero.
It turns out that measuring the evolution of the second moment of $J_{x}$
allows the estimation of rotation angle $\theta$, and therefore, the magnetic
field. The expectation value $\langle{J_{x}^{2}}\rangle$ is initially zero for
a pure unpolarized Dicke state, and it increases rapidly as it can be seen in
the Figure 3.1. Another observation is that for $\theta=\pi/2$ the value of
$\langle{J_{x}^{2}}\rangle$ will be at its maximum proportional to
$\langle{\boldsymbol{J}^{2}}\rangle$ or equivalently to $\mathcal{J}_{N/2}$,
see Eq. (A.3). Hence, the change in the second moment over the phase shift
must be in this case proportional to $N^{2}$. We are lead to the conclusion
that one only needs to measure the second moment of the collective spin
$J_{x}$ to achieve Heisenberg scaling for the estimation.

Figure 3.1:  Sequence of the evolution of an unpolarized Dicke state of 16
qubits for $\theta=\\{i\pi/6\\}_{i=0}^{4}$. (sequence-above) Bloch spheres
representing the Husimi quasi-probabilistic distribution $\mathcal{Q}$ of the
state, see Appendix B for details. (sequence-below) PDF of the $J_{x}$
positive-operator valued measure (POVM) for each step of the sequence.

In certain situations, it is better to use Eq. (2.32) rather than Eq. (2.48)
for calculating the achievable precision, since it gives the precision for a
particular operator to be measured in an experimental setup. This is
reasonable, since in a typical experiment, only a restricted set of operators
can be measured. In this work, we will consider many-particle systems in which
the particles cannot be accessed individually, and only collective quantities
can be measured. As we said, measuring the second moment of $J_{x}$ is a valid
choice to estimate the rotation angle. In the following equation, we show the
error propagation formula when measuring the second moment of the $J_{x}$
total angular momentum component,

$(\Delta\theta)^{-2}=\frac{|\partial_{\theta}\langle{J_{x}}\rangle|^{2}}{(\Delta
J_{x}^{2})^{2}}.$ (3.2)

Since Eq. (3.2) is always smaller than $\mathcal{F}_{\textnormal{Q}}$, and as
a consequence of Eq. (2.66), if

$\frac{|\partial_{\theta}\langle{J_{x}}\rangle|^{2}}{(\Delta
J_{x}^{2})^{2}}\geqslant N$ (3.3)

holds, then the system is entangled. Hence again, entanglement is required for
a large metrological precision. Based on Eq. (2.67), we can bound the
entanglement depth from below of the systems as follows. Similarly to the
previous paragraph, if for a quantum state

$\frac{|\partial_{\theta}\langle{J_{x}}\rangle|^{2}}{(\Delta
J_{x}^{2})^{2}}\geqslant kN$ (3.4)

holds, then it is at least $(k+1)$-entangled.

###  3.2 Precision based on the error propagation formula when $M=J_{x}^{2}$

With the aim of obtaining the precision, Eq. (3.2), we will compute the
dependence on $\theta$ of the expectation value of the operator $J_{x}$ and
higher order moments. We will use the Heisenberg picture, where the operators
evolve in time while the state remains the same. The operator $J_{x}$ can be
written as a function of $\theta$ in the following way

$J_{x}(\theta)=e^{i\theta J_{z}}J_{x}(0)e^{-i\theta
J_{z}}=J_{x}(0)\text{c}_{\theta}-J_{y}(0)\text{s}_{\theta},$ (3.5)

where $J_{l}(0)$ for $l=x,y,z$ are the collective angular momentum operators
at time equal zero. We will denote them by $J_{l}$ from now on. The notation
for the trigonometric functions $\text{c}_{\theta}$ and $\text{s}_{\theta}$
was introduced in Section 2.3.2.

We need to compute the second and the fourth moments of $J_{x}$ as it is
required by the Eq. (3.2). But before any calculation we will make a
simplifying assumption which turns out to be true in the most common
situations. The assumption is that both expectation values are even functions
of $\theta$, so

$\begin{split}\langle{J_{x}^{2}(\theta)}\rangle&=\langle{J_{x}^{2}(-\theta)}\rangle,\\\
\langle{J_{x}^{4}(\theta)}\rangle&=\langle{J_{x}^{4}(-\theta)}\rangle\end{split}$
(3.6)

holds. This way we can omit the terms that are odd in $\theta$. In Section
3.4, we will see that unitary dynamics of some experimentally prepared states
have this property. The assumption Eq. (3.6) is needed to obtain a closed
formula for the precision of the phase estimation.

The square of $J_{x}$ in the Heisenberg picture is written as

$J_{x}^{2}(\theta)=J_{x}^{2}\text{c}_{\theta}^{2}+J_{y}^{2}\text{s}_{\theta}^{2}-(J_{x}J_{y}+J_{y}J_{x})\text{c}_{\theta}\text{s}_{\theta}.$
(3.7)

Hence, due to the first constraint of Eq. (3.6) and Eq. (3.7), we require that

$\langle{\\{J_{x},J_{y}\\}}\rangle=0.$ (3.8)

Eq. (3.8) is based on expectation values of the initial state state. The
condition Eq. (3.8) is fulfilled in a typical experiment.

As we have done with the expectation value of the square of $J_{x}$, now we do
the same for $J_{x}^{4}$. This way one will be able to distinguish which other
expectation value of combination of operators must vanish in order to have Eq.
(3.6) guarantied. The fourth power of $J_{x}$ can be written in the Heisenberg
picture as

$J_{x}^{4}(\theta)=J_{x}^{4}\text{c}_{\theta}^{4}+J_{y}^{4}\text{s}_{\theta}^{4}+(J_{x}^{2}J_{y}^{2}+J_{x}J_{y}J_{x}J_{y}+J_{x}J_{y}^{2}J_{x}+J_{y}J_{x}J_{y}J_{x}+J_{y}J_{x}^{2}J_{y}+J_{y}^{2}J_{x}^{2})\text{c}_{\theta}^{2}\text{s}_{\theta}^{2}\\\
-(J_{x}^{3}J_{y}+J_{x}^{2}J_{y}J_{x}+J_{x}J_{y}J_{x}^{2}+J_{y}J_{x}^{3})\text{c}_{\theta}^{3}\text{s}_{\theta}-(J_{x}J_{y}^{3}+J_{y}J_{x}J_{y}^{2}+J_{y}^{2}J_{x}J_{y}+J_{y}^{3}J_{x})\text{c}_{\theta}\text{s}_{\theta}^{3}.$
(3.9)

And again assuming that the expectation value of $J_{x}^{4}(\theta)$ must be
an even function of $\theta$, we see that the terms multiplied by
$\text{c}_{\theta}^{3}\text{s}_{\theta}$ and
$\text{c}_{\theta}\text{s}_{\theta}^{3}$, respectively, must be zero. So, the
expectation value of
$(J_{x}^{3}J_{y}+J_{x}^{2}J_{y}J_{x}+J_{x}J_{y}J_{x}^{2}+J_{y}J_{x}^{3})$ and
$(J_{x}J_{y}^{3}+J_{y}J_{x}J_{y}^{2}+J_{y}^{2}J_{x}J_{y}+J_{y}^{3}J_{x})$ must
vanish. Hence, the second constraint of the Eq. (3.6) can be fulfilled if

$\begin{split}\langle{\big{\\{}J_{x}^{2},\\{J_{x},J_{y}\\}\big{\\}}}\rangle=0,\\\
\langle{\big{\\{}J_{y}^{2},\\{J_{x},J_{y}\\}\big{\\}}}\rangle=0.\end{split}$
(3.10)

Finally, we can write the evolution of second and fourth moments of the
$J_{x}$ operator as

$\displaystyle\langle{J_{x}^{2}(\theta)}\rangle=\;$
$\displaystyle\langle{J_{x}^{2}}\rangle\text{c}_{\theta}^{2}+\langle{J_{y}^{2}}\rangle\text{s}_{\theta}^{2}$
(3.11a)
$\displaystyle\begin{split}\langle{J_{x}^{4}(\theta)}\rangle=\;&\langle{J_{x}^{4}}\rangle\text{c}_{\theta}^{4}+\langle{J_{y}^{4}}\rangle\text{s}_{\theta}^{4}\\\
&+\langle{\\{J_{x}^{2},J_{y}^{2}\\}+\\{J_{x},J_{y}\\}^{2}}\rangle\text{c}_{\theta}^{2}\text{s}_{\theta}^{2}.\end{split}$
(3.11b)

From here, we are able to write the evolution of the variance of the second
moment when Eq. (3.6) is fulfilled. We obtain

$\begin{split}(\Delta
J_{x}^{2}(\theta))^{2}&=\langle{J_{x}^{4}(\theta)}\rangle-\langle{J_{x}^{2}(\theta)}\rangle^{2}\\\
&=\langle{J_{x}^{4}}\rangle\text{c}_{\theta}^{4}+\langle{J_{y}^{4}}\rangle\text{s}_{\theta}^{4}+\langle{\\{J_{x}^{2},J_{y}^{2}\\}+\\{J_{x},J_{y}\\}^{2}}\rangle\text{c}_{\theta}^{2}\text{s}_{\theta}^{2}-\big{(}\langle{J_{x}^{2}}\rangle\text{c}_{\theta}^{2}+\langle{J_{y}^{2}}\rangle\text{s}_{\theta}^{2}\big{)}^{2}\\\
&=\big{(}\langle{J_{x}^{4}}\rangle-\langle{J_{x}^{2}}\rangle^{2}\big{)}\text{c}_{\theta}^{4}+\big{(}\langle{J_{y}^{4}}\rangle-\langle{J_{y}^{2}}\rangle^{2}\big{)}\text{s}_{\theta}^{4}+\big{(}\langle{\\{J_{x}^{2},J_{y}^{2}\\}+\\{J_{x},J_{y}\\}^{2}}\rangle-2\langle{J_{x}^{2}}\rangle\langle{J_{y}^{2}}\rangle\big{)}\text{c}_{\theta}^{2}\text{s}_{\theta}^{2}\\\
&=(\Delta J_{x}^{2})^{2}\text{c}_{\theta}^{4}+(\Delta
J_{y}^{2})^{2}\text{s}_{\theta}^{4}+\big{(}\langle{\\{J_{x}^{2},J_{y}^{2}\\}+\\{J_{x},J_{y}\\}^{2}}\rangle-2\langle{J_{x}^{2}}\rangle\langle{J_{y}^{2}}\rangle\big{)}\text{c}_{\theta}^{2}\text{s}_{\theta}^{2}.\end{split}$
(3.12)

In order to compute the Eq. (3.2), we also need the modulus square of the
derivative of the second moment of the $J_{x}$ operator. Using Eq. (3.11a) for
the expression of the evolution of the second moment, the numerator of Eq.
(3.2) follows

$\begin{split}|\partial_{\theta}\langle{J_{x}^{2}(\theta)}\rangle|^{2}&=|-2\langle{J_{x}^{2}}\rangle\text{c}_{\theta}\text{s}_{\theta}+2\langle{J_{y}^{2}}\rangle\text{c}_{\theta}\text{s}_{\theta}|^{2}\\\
&=4\langle{J_{y}^{2}-J_{x}^{2}}\rangle^{2}\text{c}_{\theta}^{2}\text{s}_{\theta}^{2}.\end{split}$
(3.13)

From the equations above directly follows expression for the precision of
$\theta$,

$\begin{split}(\Delta\theta)^{2}&=\frac{(\Delta
J_{x}^{2})^{2}\text{c}_{\theta}^{4}+(\Delta
J_{y}^{2})^{2}\text{s}_{\theta}^{4}+\big{(}\langle{\\{J_{x}^{2},J_{y}^{2}\\}+\\{J_{x},J_{y}\\}^{2}}\rangle-2\langle{J_{x}^{2}}\rangle\langle{J_{y}^{2}}\rangle\big{)}\text{c}_{\theta}^{2}\text{s}_{\theta}^{2}}{4\langle{J_{y}^{2}-J_{x}^{2}}\rangle^{2}\text{c}_{\theta}^{2}\text{s}_{\theta}^{2}}\\\
&=\frac{(\Delta J_{x}^{2})^{2}\text{t}_{\theta}^{-2}+(\Delta
J_{y}^{2})^{2}\text{t}_{\theta}^{2}+\langle{\\{J_{x}^{2},J_{y}^{2}\\}+\\{J_{x},J_{y}\\}^{2}}\rangle-2\langle{J_{x}^{2}}\rangle\langle{J_{y}^{2}}\rangle}{4\langle{J_{y}^{2}-J_{x}^{2}}\rangle^{2}}.\end{split}$
(3.14)

To this calculations further computations follow mainly regarding to the
following expectation value
$\langle{\\{J_{x}^{2},J_{y}^{2}\\}+\\{J_{x},J_{y}\\}^{2}}\rangle$. This
calculus is left for the Appendix C. Finally, the expression Eq. (3.14) can be
written as

$(\Delta\theta)^{2}=\frac{(\Delta J_{x}^{2})^{2}\text{t}_{\theta}^{-2}+(\Delta
J_{y}^{2})^{2}\text{t}_{\theta}^{2}+4\langle{J_{y}^{2}}\rangle-3\langle{J_{z}^{2}}\rangle-2\langle{J_{x}^{2}}\rangle(1+\langle{J_{y}^{2}}\rangle)+6\langle{J_{x}J_{y}^{2}J_{x}}\rangle}{4\langle{J_{y}^{2}-J_{x}^{2}}\rangle^{2}}.$
(3.15)

We have verified the correctness of our analytic formula Eq. (3.15) comparing
it with a numerical simulation of the Eq. (3.2) for the ground-state of
$H=J_{x}^{2}+J_{y}$ for 6 qubits, $|\text{GS}\rangle$. We computed the
evolution of the expectation values of the second and the fourth moments of
the operator $J_{x}$ for $\theta\in[0,\pi]$, for thousand of equidistant
points, from which we obtained the bound, see Figure 3.2-(a). Finally, we have
also checked that the constraints assumed at the beginning of this section are
fulfilled. For that, we considered the range $\theta\in[-\pi,\pi]$ and we have
computed the expectation values, see Figure 3.2-(b). We can conclude saying
that our formula Eq. (3.15) reproduces exactly the evolution of the error
propagation formula, Eq. (3.2).

\begin{overpic}[scale={.65}]{img/VD_simulation.pdf} \put(5.0,10.0){\small(a)}
\end{overpic}\begin{overpic}[scale={.65}]{img/VD_parity_simulation.pdf}
\put(5.0,10.0){\small(b)} \end{overpic}

Figure 3.2:  (a) Evolution of the precision $(\Delta\theta)^{-2}/N$ for 6
qubits based on the simulation of the system $|{\text{GS}}\rangle$ and its
expectation values. The agreement with the Eq. (3.15) is shown in the inset
plot where the square of the difference between two approaches are plotted,
the analytically obtained result and the simulation. The difference is more or
less two orders of magnitude below the actual value for the relevant points,
which is mainly because of computing the derivative near the points at which
the expectation value $\langle{J_{x}^{2}}\rangle$ and $(\Delta J_{x}^{2})^{2}$
are both close to zero. (b) With the system at hand, we verified the parity
with respect to $\theta$ of the expectation values of the second and the
fourth moment, so to fulfill the constraint Eq. (3.6).

####  3.2.1 The optimal precision

First of all, note that all the dependence on the phase shift $\theta$ is in
the first two terms of the numerator of Eq. (3.15). Hence, one can minimize
the sum on the first two terms in order to find where the precision is best.
So it follows that for the optimal angle

$\tan^{2}(\theta_{\text{opt}})=\sqrt{\frac{(\Delta J_{x}^{2})^{2}}{(\Delta
J_{y}^{2})^{2}}}$ (3.16)

holds. By substituting Eq. (3.16) into Eq. (3.15), we obtain the optimal bound
as

$(\Delta\theta)^{2}_{\text{opt}}=\frac{\sqrt{(\Delta J_{x}^{2})^{2}(\Delta
J_{y}^{2})^{2}}+4\langle{J_{y}^{2}}\rangle-3\langle{J_{z}^{2}}\rangle-2\langle{J_{x}^{2}}\rangle(1+\langle{J_{y}^{2}}\rangle)+6\langle{J_{x}J_{y}^{2}J_{x}}\rangle}{4\langle{J_{y}^{2}-J_{x}^{2}}\rangle^{2}}.$
(3.17)

.

We conclude this section checking our bound for pure unpolarized Dicke state
aligned with the $x$-axis, $|{\textnormal{D}_{N}}\rangle_{x}$, whose precision
bound is well known using the QFI for pure states Eq. (2.53),

$\mathcal{F}_{\textnormal{Q}}[{\textstyle|{\textnormal{D}_{N}}\rangle_{x},J_{z}}]=4(\Delta
J_{z})^{2}_{|{\textnormal{D}_{N}}\rangle_{x}}=\frac{N(N+2)}{2}.$ (3.18)

With this aim we compute all the expectation values needed for the Eq. (3.17)
which almost all of them are trivial,
$\langle{J_{x}J_{y}^{2}J_{x}}\rangle=\langle{J_{x}^{4}}\rangle=\langle{J_{x}^{2}}\rangle=0$
since the state is an eigenstate of $J_{x}$ with an eigenvalue zero. The last
expectation value is obtained as

$\langle{J_{y}^{2}}\rangle=\langle{J_{z}^{2}}\rangle=\frac{N(N+2)}{8}.$ (3.19)

Note that for the Eq. (3.19) we use that the state is invariant under
rotations over the $x$-axis, the sum of all the second moments must give
$\langle{\boldsymbol{J}^{2}}\rangle=\frac{N(N+2)}{4}$, and
$\langle{J_{x}^{2}}\rangle=0$. Hence, the Eq. (3.19) holds.

From the Eq. (3.19) using the expression for the optimal precision Eq. (3.17)
and substituting all the terms that are zero, one arrives at the following
formula for the precision of the phase shift for a pure unpolarized Dicke
state,

$(\Delta\theta)^{2}_{\text{opt}}=\frac{2}{N(N+2)},$ (3.20)

which coincides exactly with the inverse of the quantum Fisher information for
such state Eq. (3.18) [8]. Hence for the ideal Dicke state, the Cramér-Rao
bound Eq. (2.48) is saturated, which means that estimating the phase shift
$\theta$ using the measurement of $\langle{J_{x}^{2}}\rangle$ is optimal.
Based on Eq. (3.16), we add that the optimal angle for the ideal Dicke state
is $\theta_{\text{opt}}=0$.

###  3.3 Testing the formula against some known states

In this section, we will compare our criteria based on few expectation values
against the corresponding quantum Fisher information obtained for some known
states. We find that our formula gives a good lower bound on the quantum
Fisher information, which is the best achievable precision when any
measurement is allowed. However note that the Cramér-Rao bound might be
impractical.

Let us consider first the spin-squeezed states. Those states will be defined
as the ground states $|{\text{GS}}\rangle_{\lambda}$ of the spin-squeezing
Hamiltonian given as

$H_{\lambda}=J_{x}^{2}-\lambda J_{y},$ (3.21)

see Appendix D. For $\lambda>0$, the ground state is unique, and it is in the
symmetric subspace. Hence, we can restrict our attention to this subspace for
our computations, and hence we can model larger systems. For
$\lambda\rightarrow\infty$, the ground state is the state totally polarized in
the $y$-direction Eq. (2.33). And for $\lambda\rightarrow 0^{+}$, it is the
Dicke state Eq. (3.1). Note that for $\lambda=0$ the eigenvalue is degenerate,
so there are more than one ground states. On the other hand, we still can use
a limit in which $\lambda$ tends to zero from the positive axis to arrive at
the Dicke state. Figure 3.3-(a) shows the sensitivity we obtained together
with the QFI for the same states. Our bound is close to the QFI when the state
is well polarized. It also coincides with the bound in the $\lambda\rightarrow
0^{+}$ limit, when the ground state is close to the unpolarized Dicke state.

\begin{overpic}[scale={.65}]{img/VD_against_spsq.pdf}
\put(5.0,10.0){\small(a)}
\end{overpic}\begin{overpic}[scale={.65}]{img/VD_against_therm.pdf}
\put(5.0,10.0){\small(b)} \end{overpic}

Figure 3.3:  Comparison between our formula for the precision and the QFI for
different states. (a) Comparison with ground states of $H_{\lambda}$. (b)
Comparison with thermal mixture of Dicke states.

The second family of states we use to test our formula are the Gaussian
mixture of Dicke states around the unpolarized Dicke state, which have the
following form as function of $T$ as

$\rho_{T}\propto\sum_{n=0}^{N}e^{-\frac{(n+N/2)^{2}}{T}}|{\textnormal{D}_{N,n}}\rangle_{x}\langle{\textnormal{D}_{N,n}}|_{x}$
(3.22)

for even $N$, where $|{\textnormal{D}_{N,n}}\rangle$ is defined in Eq. (A.5).
It can be used to model a noisy or thermal unpolarized Dicke state. For $T=0$,
we obtain the pure unpolarized Dicke state. For $T>0$, other symmetric Dicke
states in the vicinity of the unpolarized one are also populated. The result
can be seen in Figure 3.3-(b). Again, our bound seems to be quite close to the
corresponding QFI.

Note also that in Figure 3.3, based on Eq. (2.67), if the bound turns to be
greater than $k$ integer, then a metrologically useful $(k+1)$-particle
entanglement is detected in the system. Note that this is true whenever $k$ is
a divisor of $N$, or $k\ll N$.

Although showing how the optimal precision formula behaves compared with the
quantum Fisher information for those two families of states, we will now prove
that they indeed fulfill the constraints appearing in Eq. (3.6). Hence, we
compute the Eq. (3.6) for the spin-squeezed states
$|{\text{GS}}\rangle_{\lambda}$. For that it is enough to know that since
those states are non-degenerate ground states, they must have the same
symmetries as the Hamiltonian. In this case, the Hamiltonian Eq. (3.21) is
invariant under $x\leftrightarrow-x$ and $z\leftrightarrow-z$, thus it must be
invariant under $r\pi$ angle rotations under the $y$-axis for $r$ integer, for
more references about symmetries in quantum mechanics see Refs. [70, 69].
Hence, we can write for the evolution of the expectation value of any power
$m$ of $J_{x}$ such that

$\operatorname{tr}(e^{+i\theta J_{z}}J_{x}^{m}e^{-i\theta
J_{z}}\rho_{\lambda})=\operatorname{tr}(e^{+i\theta J_{z}}J_{x}^{m}e^{-i\theta
J_{z}}e^{-ir\pi J_{y}}\rho_{\lambda}e^{+ir\pi J_{y}}),$ (3.23)

where $\rho_{\lambda}=|{\text{GS}}\rangle{\\!}\langle{\text{GS}}|_{\lambda}$.
Then for the case $r=1$, we can use the cyclic property of the trace to arrive
at

$\begin{split}\operatorname{tr}(e^{+i\pi J_{y}}e^{+i\theta
J_{z}}J_{x}^{m}e^{-i\theta J_{z}}e^{-i\pi
J_{y}}\rho_{\lambda})&=\operatorname{tr}(e^{+i\theta(-1)J_{z}}(-1)^{m}J_{x}^{m}e^{-i\theta(-1)J_{z}}\rho_{\lambda})\\\
&=\operatorname{tr}(e^{-i\theta J_{z}}(-1)^{m}J_{x}^{m}e^{+i\theta
J_{z}}\rho_{\lambda}),\end{split}$ (3.24)

or equivalently

$\langle{J_{x}^{m}(\theta)}\rangle_{\rho_{\lambda}}=\langle{(-1)^{m}J_{x}^{m}(-\theta)}\rangle_{\rho_{\lambda}},$
(3.25)

which implies that for even $m$, and specially for $m=2,4$, the expectation
values are an even function of $\theta$, and that for odd $m$ the expectation
values are odd functions of $\theta$, which proves the Eq. (3.6) for the
present case.

On the other hand, the eigenstates of the thermal state $\rho_{T}$ are also
eigenstates of $J_{x}$, and hence the state invariant under rotations around
the $x$-axis. This still holds for the entire state, since it is a statistical
mixture of states invariant under rotations around the $x$-axis. Moreover, it
is also invariant if the state is rotated around the $x$-axis by an angle
$\pi.$ Hence, we have for the evolution of the expectation values of
$J_{x}^{m}$ that

$\operatorname{tr}(e^{+i\theta J_{z}}J_{x}^{m}e^{-i\theta
J_{z}}\rho_{T})=\operatorname{tr}(e^{+i\theta J_{z}}J_{x}^{m}e^{-i\theta
J_{z}}e^{-i\pi J_{x}}\rho_{T}e^{+i\pi J_{x}})$ (3.26)

holds for any $m$. Finally, using again the cyclic properties of the trace, we
flip the signs of the of angular momentum components orthogonal to $J_{x}$, so
in this case $J_{y}\rightarrow-J_{x}$ and $J_{z}\rightarrow-J_{z}$, and we
arrive at

$\operatorname{tr}(e^{+i\pi J_{x}}e^{+i\theta J_{z}}J_{x}^{m}e^{-i\theta
J_{z}}e^{-i\pi J_{x}}\rho_{T})=\operatorname{tr}(e^{-i\theta
J_{z}}J_{x}^{m}e^{+i\theta J_{z}}\rho_{T}).$ (3.27)

We conclude that for the thermal state this case all the moments of the
$J_{x}$ operator are even functions of $\theta$ for the thermal state, i.e.,
$\langle{J_{x}^{m}(\theta)}\rangle_{\rho_{T}}=\langle{J_{x}^{m}(-\theta)}\rangle_{\rho_{T}}$,
which proves that the Eq. (3.6) holds for this case too.

###  3.4 Using our method with real experimental data

In reference [24], a state is produced in the laboratory with the proper
characteristics of an unpolarized Dicke state, small variance in one
direction, say $x$, and a very large variance in the directions perpendicular
to the $x$-axis. In the cited experiment with $N$ qubits, it is possible to
determine the operator $J_{x}$ as the population imbalance of the two levels
as

$J_{x}=\frac{1}{2}(N_{1,x}-N_{0,x}),$ (3.28)

where $N_{m,x}$ is the number of particles in the state $|{m}\rangle_{x}$.
Hence, measuring the population imbalance and collecting the statistics of the
measurements, the expectation values of all moments of $J_{x}$ can be
obtained. In practice, it is possible to measure the lower order moments like
$\langle{J_{x}^{2}}\rangle$ and $\langle{J_{x}^{4}}\rangle$, while higher
order moments need too many repetitions of the experiment to collect enough
statistics.

The other two global operators $J_{y}$ and $J_{z}$ are obtained by rotating
the system using a $\frac{\pi}{2}$ microwave coupling pulse before the
measurement of the population imbalance. Whether $J_{y}$ or $J_{z}$ is
obtained depends on the relation between the microwave phase and the phase of
the initial BEC. The condensate phase represents the only possible phase
reference in analogy to the local oscillator in optics. Intrinsically, it has
no relation to the microwave phase, such that it homogeneously average over
all possible phase relations. From another point of view, one can also say
that the fluctuation of the magnetic field results in a random rotation of the
spin around the $z$-axis. Hence, what is obtained in this case is

$J_{\alpha}=\sin(\alpha)J_{y}+\cos(\alpha)J_{z},$ (3.29)

where $\alpha$ is an angle, and we need to consider the average over all
possible angles. Effectively, the state has the following form

$\rho=\frac{1}{2\pi}\int e^{-i\alpha J_{x}}\rho_{0}e^{i\alpha
J_{x}}\,\text{d}\alpha,$ (3.30)

where $\rho_{0}$ is what we would obtain if we would have access to the phase
reference. Note that the integration over the rotation angle $\alpha$ does not
create entanglement. If the state $\rho$ is entangled then $\rho_{0}$ has to
be also entangled.

Let us see the consequences of our state having the form Eq. (3.30). For the
density matrix $\rho$, since it is invariant under rotations around the
$x$-axis, we have

$\langle{J_{\alpha}^{m}}\rangle=\langle{J_{y}^{m}}\rangle=\langle{J_{z}^{m}}\rangle$
(3.31)

for all $m$. Hence the expectation values of $\langle{J_{y}^{m}}\rangle$ and
$\langle{J_{z}^{m}}\rangle$ can be obtained from the statistics of measuring
$J_{\alpha}$. Moreover, for states of the form Eq. (3.30) the unitary dynamics
will fulfill the condition Eq. (3.6).

There is a single remaining term in the expression for the achievable
precision Eq. (3.17), the expectation value for
$\langle{J_{x}J_{y}^{2}J_{x}}\rangle$, which can be bounded as

$\begin{split}\langle{J_{x}J_{y}^{2}J_{x}}\rangle&=\frac{\langle{J_{x}(J_{y}^{2}+J_{z}^{2})J_{x}}\rangle}{2}=\frac{\langle{J_{x}(J_{x}^{2}+J_{y}^{2}+J_{z}^{2})J_{x}}\rangle-\langle{J_{x}^{4}}\rangle}{2}\\\
&\leqslant\frac{N(N+2)}{8}\langle{J_{x}^{2}}\rangle-\frac{\langle{J_{x}^{4}}\rangle}{2},\end{split}$
(3.32)

where the last inequality is due to that for all states
$\langle{\boldsymbol{J}^{2}}\rangle\leqslant\mathcal{J}_{N/2}$, see Eq. (A.3),
while symmetric states saturate the inequality in Eq. (3.32). Note that
obtaining $\langle{J_{x}J_{z}^{2}J_{x}}\rangle$ can be hard experimentally. In
any case, this simplification can only make our estimation of the precision
worse while for symmetric states the equality holds. Hence, the lower bound
for the achievable precision can be written as

$(\Delta\theta)^{2}_{\text{opt}}\leqslant\frac{\sqrt{(\Delta
J_{x}^{2})^{2}(\Delta
J_{y}^{2})^{2}}+\langle{J_{y}^{2}}\rangle+\frac{3N(N+2)-8}{4}\langle{J_{x}^{2}}\rangle-2\langle{J_{x}^{2}}\rangle\langle{J_{y}^{2}}\rangle-3\langle{J_{x}^{4}}\rangle}{4\langle{J_{y}^{2}-J_{x}^{2}}\rangle^{2}},$
(3.33)

where some terms were reordered and further simplified.

It is worth to study the case appearing in Ref. [60] and apply our methods
such that we obtain conclusions about the metrological usefulness of the
state. The system under consideration has around $N=7900$. Note that using the
expectation value of the particle number, in our case
$\langle{N}\rangle=7900$, cannot overestimate any lower bound on the
precision. For a discussion about entanglement criteria in systems with
particle number fluctuations see Ref. [99]. The measured data for the system
yields

$\begin{aligned} \langle{J_{x}^{2}}\rangle&=112\pm 31,\\\
\langle{J_{x}^{4}}\rangle&=40\times 10^{3}\pm 22\times
10^{3},\end{aligned}\quad\begin{aligned} \langle{J_{y}^{2}}\rangle&=6\times
10^{6}\pm 0.6\times 10^{6},\\\ \langle{J_{y}^{4}}\rangle&=6.2\times 10^{13}\pm
0.8\times 10^{13}.\end{aligned}$ (3.34)

Hence, we obtain the maximum precision as

$\frac{(\Delta\theta)^{-2}_{\text{opt}}}{N}\geqslant 3.7\pm 1.5.$ (3.35)

The statistical uncertainties of Eqs. (3.34) and (3.35) have been obtained
through bootstraping, while the direct substitutions of expectation values
would yield to 3.3 of gain over the shot-noise limit $(\Delta\theta)^{-2}=N$.
Based on Eq. (2.66), this proves the presence of metrologically useful
entanglement [3]. Based on Eq. (2.67), it even demonstrated that the quantum
state had metrologically useful 4-particle entanglement. Assuming an error of
a standard deviation, Eq. (3.35) still proves 3-particle entanglement.

Next we plot the value for the precision substituting directly the
experimental data into Eq. (3.15), see Figure 3.4. Since we cannot obtain the
expectation value $\langle{J_{x}J_{y}^{2}J_{x}}\rangle$ we approximate it with
the right-hand side of Eq. (3.32). With that we underestimate
$(\Delta\theta)^{-2}$.

Figure 3.4:  (solid) The precision as a function of the parameter $\theta$
given by Eq. (3.15) varies through the evolution. Note that for the initial
moment the precision is zero. (dashed) We highlight where the precision
reaches its maximum at $\theta\approx 0.0057$. (gray-area) It represent the
region where the precision is below the shot-noise limit.

Thus, we could detect metrological usefulness by measuring the second and
fourth moments of the collective angular momentum components. For future
applications of our scheme, it is important to reduce further the number of
quantities we need to obtain a lower bound for the precision. In practice, one
can easily avoid the need for determining $\langle{J_{y}^{4}}\rangle$. Note
that if we measure $J_{y}$ then the distribution of the values obtained is
strongly non-Gaussian. The values $\pm N/2$ appear most frequently, and the
value 0 appears least frequently [8]. See also the distribution of a pure
Dicke state when $\theta=\pi/2$ in the Figure 3.1. The state has more overlap
with the eigenstates of the edges than in the middle. Based on
$\langle{AB}\rangle\leqslant\lambda_{\max}(A)\langle{B}\rangle$, where
$\lambda_{\max}(A)$ is the largest eigenvalue of $A$, for two commuting
positive-semidefinite observables,

$\langle{J_{y}^{4}}\rangle\leqslant\frac{N^{2}}{4}\langle{J_{x}^{2}}\rangle.$
(3.36)

Since even for a noisy Dicke state $\langle{J_{y}^{2}}\rangle$ is very large,
the above equation is a very good upper bound. Substituting it into the Eq.
(3.17), we will underestimate $(\Delta\theta)^{-2}$.

It is also possible to approximate $\langle{J_{x}^{4}}\rangle$ with
$\langle{J_{x}^{2}}\rangle$ in the sense that it is small and that mainly its
value comes from technical noise,

$\langle{J_{x}^{4}}\rangle\approx\beta\langle{J_{x}^{2}}\rangle^{2}.$ (3.37)

This approximation, even if it is not a strict bound on the precision, can be
very useful in order to characterize the metrological usefulness of the state
based only on second statistical moments of only two angular momentum
components, namely $\langle{J_{y}^{2}}\rangle$ and
$\langle{J_{x}^{2}}\rangle$. Those two expectation values are related with how
thin is the state in one direction and how wide in the perpendicular ones. So
in this case we use $\beta=3$ assuming that the distribution function has a
Gaussian shape.

From these considerations we are able to write a second bound with fewer
expectation values for the optimal precision such that

$(\Delta\theta)^{2}_{\text{opt}}\leqslant\frac{\langle{J_{y}^{2}}\rangle+\frac{3N(N+2)-8}{4}\langle{J_{x}^{2}}\rangle+\Big{(}\sqrt{\frac{N^{2}}{2\langle{J_{y}^{2}}\rangle}-2}-2\Big{)}\langle{J_{y}^{2}}\rangle\langle{J_{x}^{2}}\rangle-9\langle{J_{x}^{2}}\rangle^{2}}{4\langle{J_{y}^{2}-J_{x}^{2}}\rangle^{2}}.$
(3.38)

We have used this formula to compute the bound for the optimal precision with
the measured data shown on Eq. (3.34),
$(\Delta\theta)^{-2}_{\text{opt}}\geqslant 2.9N$, see Figure 3.5. It turns out
that even this way, 3-particle metrologically useful entanglement is detected.

\begin{overpic}[scale={.65}]{img/VD_exper_contour.pdf}
\put(5.0,10.0){\small(a)}
\end{overpic}\begin{overpic}[scale={.65}]{img/VD_exper_slice.pdf}
\put(5.0,10.0){\small(b)} \end{overpic}

Figure 3.5:  (a) Precision bound as a function of $\langle{J_{y}^{2}}\rangle$
and $\langle{J_{x}^{2}}\rangle$. The expectation value
$\langle{J_{y}^{2}}\rangle$ is normalized with $\mathcal{J}_{N/2}$. (solid)
Different boundaries for metrologically useful entanglement depths. (cross)
Experimental data. (blue-ellipse) Region with one $\sigma$ confidence from the
experimental data. (vertical-dashed) Constant $\langle{J_{y}^{2}}\rangle$
cross section plotted in (b). (b) Constant $\langle{J_{y}^{2}}\rangle$ cross
section for the precision bound. (solid) Precision bound based on the Eq.
(3.38). One can see that decreasing further the uncertainty in
$\langle{J_{x}^{2}}\rangle$ can improve the bound significantly. (white-point)
Experimental data with the corresponding errors bars. (gray-area) Shot-noise
limit. The point is slightly below the 4-particle entanglement level.

In Figure 3.5-(a), we show the two-dimensional plot that is obtained based on
these considerations. The regions with various levels of multipartite
entanglement can clearly be identified. The ideal Dicke state corresponds to
the bottom-right corner, where $\langle{J_{x}^{2}}\rangle=0$ and
$\langle{J_{y}^{2}}\rangle$ is largest. In Figure 3.5-(b), the cross section
of the two-dimensional plot is shown.

Summarizing, we have shown that characterizing the metrological usefulness of
noisy Dicke states can be done using few experimental data. Furthermore, the
precision bound is computed with the expectation values of the second and
fourth moments of the angular momentum in the directions perpendicular to the
magnetic field. We have shown some simplifying techniques that reduce the
experimental efforts needed to characterize the state.

##  4 Witnessing metrologically useful  X entanglement Witnessing
metrologically useful  X entanglement

"All the truths of mathematics are linked to each other,and all means of
discovering them are equally admissible."

Adrien-Marie Legendre

Typically, one has no access to the density matrix of the system which is used
for metrology or for some other quantum information processing task. Moreover,
for systems in which the particle number is very large, which is the case when
one would like to do metrology with quantum states, the details of the density
matrix cannot be known due to practical reasons. Since the quantum Fisher
information is based on the complete knowledge of the density matrix, methods
to avoid the complete tomography must be developed as we have shown a
practical case in the previous chapter. In this chapter, we obtain a general
procedure to get an optimal bound for the quantum Fisher information based on
as many expectation values of the initial state as one is ready to measure.
Two main features are worth to mention again. First, in general this method
gives us a tight bound. Second, the bound is based on the expectation values
of the initial state only, so it is not necessary to perform an evolution of
the state. This is in contrast to other approaches one can find in the
literature, ours needs a significantly smaller experimental effort.

From Eq. (2.50), a lower bound on the quantum Fisher information based on
expectation values of the initial state is

$\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{z}}]\geqslant\frac{\langle{J_{x}}\rangle^{2}}{(\Delta
J_{y})^{2}}.$ (4.1)

where the state is polarized along the $x$-axis [3]. In the previous chapter
we have also shown one of these bounds specifically designed for unpolarized
Dicke states (3.1).

The setup is described in Section 2.3.1 and consists of in estimating the
homogeneous magnetic field strength, in this case, which points towards the
$z$-axis. Therefore, we will base our calculations on the quantum Fisher
information $\mathcal{F}_{\textnormal{Q}}[{\textstyle\rho,J_{z}}]$, defined in
Eqs. (2.51) and (2.52), see Section 2.3.3 for more information about the
properties of the QFI.

###  4.1 Lower bound on a convex function given some arbitrary expectation
values

Our problem could be solved with Lagrange multipliers or Legendre transforms.
We follow the latter method since obtaining a lower bound on a convex function
for states with some given operator expectation values has already been
studied by O. Gühne et al. and J. Eisert et al. in Refs. [23, 100]
respectively, mainly from the perspective of entanglement measures. The
illustrated techniques are based on the well known Legendre transform for
differentiable functions, see Appendix E for more details. We first review in
this section the state-of-the-art solution to this problem. Later on, we
extend it to the quantum Fisher information. For simplicity in the next
section, Section 4.1.1, we assume that a single expectation value is given. An
extension to the case in which more expectation values are given will follow
in the Section 4.1.2. Finally, we will summarize our results with an explicit
formula which will be used to compute the bounds in the most general case.

####  4.1.1 Lower bound based on a single observable

When a convex function $g(\rho)$ is given together with an expectation value
of some operator $w=\operatorname{tr}(\rho W)$, a tight lower bound,
$\mathcal{B}_{\,g}(w)$, can be obtained as [101, 23, 100]

$\begin{split}g(\rho)\geqslant\mathcal{B}_{\,g}(w):=\,&\sup_{r}\\{rw-\hat{g}(rW)\\}\\\
=\,&\\{\inf_{\rho}g(\rho)\,|\,w=\operatorname{tr}(\rho W)\\},\end{split}$
(4.2)

where $\hat{g}(rW)$ is the Legendre transform of $g(\rho)$ and the second
equality expresses the tightness of the bound. The Legendre transform in this
context is defined as

$\hat{g}(rW)=\sup_{\rho}\\{\langle{rW}\rangle_{\rho}-g(\rho)\\},$ (4.3)

where the maximization is over _all_ possible states. This method to obtain
the lower bound has been used to compute entanglement measures [23, 100].

Following the theory one can find that if the convex function $g(\rho)$ is
defined as a convex roof over all possible convex decompositions of the state,
the optimisation of Eq. (4.3) can be reduced to an optimization over pure
states only, thus simplifying the calculation [23, 100]

$\begin{split}\hat{g}(rW)&=\sup_{\rho}\\{\langle{rW}\rangle_{\rho}-g(\rho)\\}\\\
&=\sup_{\rho}\Big{\\{}r\langle{W}\rangle_{\rho}-\inf_{\\{p_{k},|{\phi_{k}}\rangle\\}}\big{\\{}\sum_{k}p_{k}g(|{\phi_{k}}\rangle)\big{\\}}\Big{\\}}\\\
&=\sup_{\\{p_{k},|{\phi_{k}}\rangle\\}}\Big{\\{}\sum_{k}p_{k}\langle{rW}\rangle_{|{\phi_{k}}\rangle}-\inf_{\\{p_{k},|{\phi_{k}}\rangle\\}}\big{\\{}\sum_{k}p_{k}g(|{\phi_{k}}\rangle)\big{\\}}\Big{\\}}\\\
&=\sup_{\\{p_{k},|{\phi_{k}}\rangle\\}}\Big{\\{}\sum_{k}p_{k}\big{\\{}\langle{rW}\rangle_{|{\phi_{k}}\rangle}-g(|{\phi_{k}}\rangle)\big{\\}}\Big{\\}}\\\
&=\sup_{|{\psi}\rangle}\big{\\{}\langle{rW}\rangle_{|{\psi}\rangle}-g(|{\psi}\rangle)\big{\\}}.\end{split}$
(4.4)

However, even an optimization over all pure states is feasible numerically
only for small systems. We will show later in this section how to circumvent
this problem in the case of the QFI. The convex roof construction has the
following form

$g(\rho)=\inf_{\\{p_{k},\psi_{k}\\}}\sum_{k}p_{k}g(|{\psi_{k}}\rangle),$ (4.5)

where the mixed state is decomposed into
$\rho=\sum_{k}p_{k}|{\psi_{k}}\rangle{\\!}\langle{\psi_{k}}|$. Among other
definitions of the QFI in the literature, there is one that defines it as the
convex roof of $4(\Delta J_{z})^{2}$, the variance of the generator, as it has
been shown in Ref. [95, 89], we can compute the Legendre transform optimizing
for pure states only. Hence, we will be able to use this simplification to
apply this method to obtain the lower bound on the QFI. Note that in this
context, the QFI is the convex roof of four times the variance of the
generator, Eq. (2.52).

####  4.1.2 Measuring several observables

For some cases, it is interesting to characterize the quantum state with
several measurements rather than with a single one. For instance, we might
want to use, as it is done with the spin-squeezed states the absolute
polarization and the variance of one of the orthogonal components of the
angular momentum to detect entanglement and metrological usefulness [3]. So
far, we studied the case in which a single measurement is used. Its extension
to several expectation values is indeed straight-forward. We can generalize
Eqs. (4.2) and (4.3) for several observables $\\{W_{i}\\}_{i=1}^{M}$ as
follows [23]

$\mathcal{B}_{\,g}(w_{1},w_{2},\dots):=\sup_{\boldsymbol{r}}\big{\\{}\boldsymbol{rw}-\sup_{\rho}\\{\langle{\boldsymbol{rW}}\rangle-g(\rho)\\}\big{\\}},$
(4.6)

where $\boldsymbol{ab}=\sum_{k=1}^{M}a_{k}b_{k}$, the usual notation for
scalar products of two vectors.

####  4.1.3 Lower bound on the quantum Fisher information

After we have shown how to find a lower bound for a general convex function of
the state based on its expectation values and how to simplify that method for
the case in which the function is defined as a convex roof, now we are in the
position to achieve the main goal of this chapter. First of all, we note that
for the quantum Fisher Information the inner maximization, the Legendre
transform, is obtained optimizing a function quadratic in expectation values,