Table Of ContentQuantum Science and Technology
Abolfazl Bayat
Sougato Bose
Henrik Johannesson Editors
Entanglement
in Spin Chains
From Theory to Quantum Technology
Applications
Quantum Science and Technology
SeriesEditors
RaymondLaflamme,UniversityofWaterloo,Waterloo,ON,Canada
DanielLidar,UniversityofSouthernCalifornia,LosAngeles,CA,USA
ArnoRauschenbeutel,ViennaUniversityofTechnology,Vienna,Austria
RenatoRenner,InstitutfürTheoretischePhysik,ETHZürich,Zürich,Switzerland
Jingbo Wang, Department of Physics, University of Western Australia, Crawley,
WA,Australia
YaakovS.Weinstein,QuantumInformationScienceGroup,TheMITRECorpora-
tion,Princeton,NJ,USA
H.M.Wiseman,GriffithUniversity,Brisbane,QLD,Australia
SectionEditor
MaximilianSchlosshauer,DepartmentofPhysics,UniversityofPortland,Portland,
OR,USA
The book series Quantum Science and Technology is dedicated to one of today’s
most active and rapidly expanding fields of research and development. In par-
ticular, the series will be a showcase for the growing number of experimental
implementationsandpracticalapplicationsofquantumsystems.Thesewillinclude,
but are not restricted to: quantum information processing, quantum computing,
and quantum simulation; quantum communication and quantum cryptography;
entanglementandotherquantumresources;quantuminterfacesandhybridquantum
systems; quantum memories and quantum repeaters; measurement-based quantum
control and quantum feedback; quantum nanomechanics, quantum optomechanics
and quantum transducers; quantum sensing and quantum metrology; as well as
quantum effects in biology. Last but not least, the series will include books on
thetheoreticalandmathematicalquestionsrelevanttodesigningandunderstanding
these systems and devices, as well as foundational issues concerning the quantum
phenomena themselves. Written and edited by leading experts, the treatments will
be designed for graduate students and other researchers already working in, or
intendingtoenterthefieldofquantumscienceandtechnology.
Abolfazl Bayat • Sougato Bose •
Henrik Johannesson
Editors
Entanglement in Spin Chains
From Theory to Quantum Technology
Applications
Editors
AbolfazlBayat SougatoBose
InstituteofFundamentalandFrontier DepartmentofPhysicsandAstronomy
Sciences UniversityCollegeLondon
UniversityofElectronicScienceand London,UK
TechnologyofChina
Chengdu,China
HenrikJohannesson
DepartmentofPhysics
UniversityofGothenburg
Göteborg,Sweden
ISSN2364-9054 ISSN2364-9062 (electronic)
QuantumScienceandTechnology
ISBN978-3-031-03997-3 ISBN978-3-031-03998-0 (eBook)
https://doi.org/10.1007/978-3-031-03998-0
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Preface
The study of spin chains has played a pivotal role in furthering our understanding
of quantum many-body systems. Maybe most important, spin chain models have
servedasatheoreticallaboratoryfordevelopingnovelconceptsandmethodsinthis
endeavor.Awell-knownexampleistheearlyattemptbyHansBethetodiagonalize
the Hamiltonian of the Heisenberg chain, bringing forth the Bethe Ansatz [1] and,
as an offshoot, the notion of quantum integrability. Another example is the use
of the transverse-field Ising chain as a paradigm for systems exhibiting quantum
phasetransitions[2].Morerecently,workontheinteger-spinHeisenbergchains[3]
and their generalizations [4] has become a pillar of our conception of symmetry-
protectedtopologicalquantummatter,asacknowledgedbythe2016NobelPrizein
PhysicsawardedDuncanHaldane.
Evenmorerecently,thenotionsofquenchdynamicsandrelaxationinintegrable
models, ergodicity, thermalization, as well as the breaking of ergodicity in certain
circumstances have also been captured by spin chain models [5–8]. In short, by
allowing for multiple strongly anharmonic quantum systems (spins) with local
physicalinteractions,albeitwithinareducedspatialdimensionality,spinchainscan
beatoyuniverse:muchoftheemergentprinciplesofthephysicalworldaroundus
canbeseeninafullyquantumsettingthroughspinchains.
Entanglement is perhaps the most striking feature of general states of multiple
interactingquantumparticles.Thus,spinchainsofferamostnaturalarenatostudy
entanglement, as it provides states that should be physically accessible in nature,
in equilibrium or through dynamics. Motivated by that, almost as soon as one of
theearliestmeasuresofquantumentanglementbetweentwoqubitsinmixedstates,
namely the concurrence, was formulated by Wootters [9], he, and others [10, 11],
applied it to ground and thermal states of spin chains. Using this measure, the
behaviorofentanglementatquantumphasetransitionswasfirstcharacterized[12,
13].Apracticalapplicationofspinchainsasaquantumcommunicationchannelwas
also quantified using concurrence [14]. It was understood that to characterize the
moreglobalaspectsofentanglement,onewouldneedtocalculatethevonNeumann
entropy[15]betweenthecomplementarypartsofamany-bodysystem[16,17],and
fieldtheoreticalmethodstocomputethesamewereformulated[18].Entanglement
v
vi Preface
between non-complementary parts of a spin chain, as quantified by the negativity
[19–21],werealsostudied[22–24].
Another reason for the renewed interest in spin chains in the last two decades
has been their realization using experimental quantum simulators, based on, e.g.,
ultracoldatoms[25],trappedions[26,27],superconductingcircuits[28],electrons
in solid state spin arrays [29, 30], and NMR systems [31]. This interest has been
furtherfueledbyaremarkablecross-fertilizationbetweencondensedmatterphysics
and quantum information theory, some of which has been noted in the preceding
paragraph. Work has flowed in two directions, mostly centered around the very
concept of quantum entanglement: “How to better understand quantum many-
bodysystemsbyexploitingvariousentanglementmeasures(withspinchainsbeing
arguablythesimplestsuchsystems)?”and,conversely,“Howtousespinchainsfor
practical purposes; analyzing, drafting, and eventually building working quantum
informationdevicesdrawingonentanglementresources?”Thesearethetwocircles
ofproblemsaddressedbythecontributorstothisvolume.
We envision two categories of readers. First, the volume will be useful for
graduate students at the start of their PhDs as an up to date and comprehensive
review on the rich and highly diverse area that the field of entanglement in spin
chains has now become. Secondly, active researchers knowledgeable in some
aspects of the field can acquire a quick insight into other topics of interest to
which they can potentially contribute. In this way, our volume is intended to
serve as a bridge between condensed matter theorists, quantum information and
quantumcomputationtheorists,andexperimentalistsworkingonspinchainsystems
invariousplatforms.
Thefirstfourchaptersareonthesimplestscenario—thatofstatic(equilibrium)
phases of spin chains and the various measures of quantum entanglement one can
use to understand their strongly correlated character. We begin with Chap.1 by
Thomale, which, after a pedagogic introduction to entanglement in spin chains,
introducesoneofthemostfundamentalquantitiesthatcharacterizethemany-body
quantumcorrelatedaspectsofaspinchain,namelyitsentanglementspectrum.Itis
describedhowtheentanglementspectrum,whencomputedforvariouscutsofaspin
chain,inbothpositionandmomentumspace,revealsaspectsofaspinchainsuchas
itstopologicalfeatures,aswellasthenatureofitslowenergyexcitations.Next,in
Chap.2,wepresentacontributionbyLi,Yu,andLin,whichagain,startspedagog-
ically by introducing the concept of quantum phase transitions in spin chains, and
thesimplestquantitiessuchasconcurrence,entanglemententropy,quantumdiscord,
and measures of quantum coherence, which can be used to characterize them and
theirquantumness.InChap.3,weincludeacontributionbyRoyandSaleurwhich
bringsabouttheareaofcomputation ofentanglement entropies inconformalfield
theories(CFTs)throughwhichspinchainsaregenericallydescribednearthecritical
point.Inparticular,spinchainsmayhaveboundariesanddefects;theauthorsdiscuss
thecontributionofthesetotheentanglemententropy.Next,inChap.4,Laflorencie
provides a contribution on entanglement measures for phases of disordered spin
chains,startingwithfreefermionmodelsandthenadvancingtointeractingfermions
andmodelsofmany-bodylocalization.
Preface vii
Wenextproceedtoincluderesearchonspinchainsattheinterfaceofquantum
information processing and quantum computation. We start by Chap.5, which
contains a contribution by Wei, Raussendorf, and Affleck. This chapter first
introduces the background on the AKLT models of gapped integer spin chains
and important topics such as tensor networks and matrix product states in this
context. Subsequently, their applications in the study of topological order and in
measurement-based quantum computation are discussed among other topics. In
Chap.6, Banchi discusses the application of machine learning in measuring the
entanglementnegativityinspinchains.Inthischapter,firstanefficientmethodfor
estimatingthemomentsofpartiallytransposeddensitymatricesisintroduced,then,
by feeding these moments into a neural network, the entanglement negativity is
measured. Next, in Chap.7, Amico, Korepin, Hamma, Giampaolo, and Franchini
discuss the local convertibility in spin systems just by means of local operations
and classical communication with respect to a given bipartition of the system.
They show that the absence of differential local convertibility is an indicator of a
higher computational power of the system’s quantum phase, which is also usually
connectedwiththeexistenceoflong-rangeentanglement,topologicalorder,oredge
states. In Chap.8, Rattacaso, Passarelli, Lucignano, and Fazio address the inverse
problemoffindingaparentHamiltonianforagivenquantumstate.Inparticular,the
scalingofcomplexityforreconstructingtheparentHamiltonianwithrespecttothe
systemsizeisinvestigated.
The emergence of quantum simulators in various physical platforms have put
spin chains at the forefront of quantum simulation. Two chapters of the book
are devoted to this important subject. In Chap.9, Potter and Vasseur review the
dynamics of quantum information in ensembles of random quantum circuits from
a statistical mechanics perspective. The chapter first explores universal features
for entanglement growth, operator spreading, thermalization, and chaos in unitary
random quantum circuits. The text then goes through the dynamics of monitored
random circuits and exhibits new types of measurement-induced phases and
criticality. Proceeding to Chap.10, Lunt, Richter, and Pal investigate the quantum
simulationinnoisyquantumcircuits.Twoclassesofdynamicsandtheirphasetran-
sitionsarestudiedusingrandom-circuitmodels.First,hybridcircuitsconsistingof
unitarygatesinterspersedwithnonunitaryprojectivemeasurementsareinvestigated.
Second,random-circuitsamplingexperimentsareconsideredandtheusefulnessof
randomquantumstatesforsimulatingquantummany-bodydynamicsarediscussed.
The dynamics of spin chains have been the subject of many studies, and thus
fourchaptersofthisbookhavebeendedicated tothistopic.InChap.11,Bentsen,
Daley,andSchachenmayerinvestigatethedynamicsofentanglementinspinchains
with long-range interaction. Several types of long-range interacting systems, such
assparsecouplinggraphsandspinchainswithalgebraicallydecayinginteractions,
are discussed. In Chap.12, Lorenzo, Plastina, Consiglio, and Apollaro provide
a formalism for describing quantum state transfer and multi-qubit entanglement
distribution and generation in spin networks. Next, in Chap.13, Papic presents a
pedagogicalintroductiontoweakergodicitybreakingphenomena,includingHilbert
space fragmentation and quantum many-body scars, through the lens of quantum
viii Preface
entanglement.Insuchsystems,whilemostinitialstatesquicklyequilibrate,certain
initial conditions may lead to slow relaxation. In Chap.14, Murciano, Alba, and
Calabrese provide a conformal field theory perspective to the quench dynamics of
the moments of the partially transposed density matrices and their corresponding
Renyinegativities.Whiletheseresultsaregenerictointeractingintegrablesystems,
numericalsimulationsareprovidedforfree-fermionicandbosoniclattices.
Thanks to the advancement of quantum technologies, spin chains can now be
tested in several experimental platforms. The first such system discussed in the
book is cold atoms. In Chap.15, Sengupta provides a review on accessible phases
and the dynamics of ultracold bosons in tilted optical lattices. Quenches, ramps,
andperiodicdrivesforinducingFloquetdynamicsareinvestigatedinthischapter.
Next, in Chap.16, Cappellaro, Peng, and Ramanathan focus on NMR systems
for realization of spin chains. In particular, Hamiltonian engineering tools and
metricsofcorrelationandentanglementarediscussedthroughseveralparadigmatic
examplesofintegrableandnon-integrabledynamicsinlargenuclearspinsystems.
Finally, in Chap.17, Nichol provides a review on the fundamental aspects of
semiconductor quantum dots and the Heisenberg exchange coupling that occurs
betweenneighboringquantumdots.Suchcouplingsallowforquantuminformation
processing,quantumstatetransfer,andthesimulationofspin-chaindynamics.
Chengdu,China AbolfazlBayat
London,UK SougatoBose
Göteborg,Sweden HenrikJohannesson
References
1. H.Bethe,Z.Physik71,205(1931)
2. P.Pfeuty,Ann.Phys.57,79(1970)
3. F.D.M.Haldane,Phys.Lett.93A,464(1983);Phys.Rev.Lett.50,1153(1983)
4. I. Affleck, T. Kennedy, E. Lieb, H. Tasaki, Phys. Rev. Lett. 59, 799 (1987);
Commun.Math.Phys.115(1988)
5. K.Sengupta,S.Powell,S.Sachdev,Phys.Rev.A69,053616(2004)
6. P.Calabrese,F.H.L.Essler,M.Fagotti,Phys.Rev.Lett.106,227203(2011)
7. M.Žnidariè,T.Prosen,P.Prelovšek,Phys.Rev.B77,064426(2008)
8. A.Pal,D.A.Huse,Phys.Rev.B82,174411(2010)
9. W.K.Wootters,Phys.Rev.Lett.80,2245(1998)
10. K.M.O’Connor,W.K.Wootters,Phys.Rev.A63,052302(2001)
11. M.C.Arnesen,S.Bose,V.Vedral,Phys.Rev.Lett.87,017901(2001)
12. A.Osterloh,L.Amico,G.Falci,R.Fazio,Nature416,608(2002)
13. T.J.Osborne,M.A.Nielsen,Phys.Rev.A66,032110(2002)
14. S.Bose,Phys.Rev.Lett.91,207901(2003)
15. S.Popescu,D.Rohrlich,Phys.Rev.A56,R3319(R)(1997)
16. G.Vidal,J.I.Latorre,E.Rico,A.Kitaev,Phys.Rev.Lett.90,227902(2003)
Preface ix
17. B.-Q.Jin,V.E.Korepin,J.Stat.Phys.116,79(2004)
18. P.Calabrese,J.Cardy,J.Stat.Mech.P06002(2004)
19. K. Zyczkowski, P. Horodecki, A. Sanpera, M. Lewenstein, Phys. Rev. A 58,
883(1998)
20. J.Lee,M.Kim,Y.Park,S.Lee,J.Mod.Opt.47,2151(2000)
21. G.Vidal,R.F.Werner,Phys.Rev.A65,032314(2002)
22. H. Wichterich, J. Molina-Vilaplana, S. Bose, Phys. Rev. A 80, 010304(R)
(2009)
23. A.Bayat,P.Sodano,S.Bose,Phys.Rev.B81,064429(2010)
24. P.Calabrese,J.Cardy,E.Tonni,Phys.Rev.Lett.109,130502(2012)
25. T. Fukuhara, P. Schauß, M. Endres, S. Hild, M. Cheneau, I. Bloch, C. Gross,
Nature502,76(2013)
26. P. Richerme, Z.-X. Gong, A. Lee, C. Senko, J. Smith, M. Foss-Feig, S.
Michalakis,A.V.Gorshkov,C.Monroe,Nature511,198(2014)
27. P. Jurcevic, B.P. Lanyon, P. Hauke, C. Hempel, P. Zoller, R. Blatt, C.F. Roos
Nature511,202(2014)
28. R.Barends,et.al.,Nature534,222(2016)
29. H. Qiao, Y.P. Kandel, K. Deng, S. Fallahi, G.C. Gardner, M.J. Manfra, E.
Barnes,J.M.Nichol,Phys.Rev.X10,031006(2020)
30. C.J. van Diepen, T.-K. Hsiao, U. Mukhopadhyay, C. Reichl, W. Wegscheider,
L.M.K.Vandersypen,Phys.Rev.X11,041025(2021)
31. M. Steffen, W. van Dam, T. Hogg, G. Breyta, I. Chuang, Phys. Rev. Lett. 90,
067903(2003)
