Table Of ContentQuantum Science and Technology
Bei Zeng
Xie Chen
Duan-Lu Zhou
Xiao-Gang Wen
Quantum
Information
Meets Quantum
Matter
From Quantum Entanglement to
Topological Phases of Many-Body
Systems
Quantum Science and Technology
Series Editors
Raymond Laflamme, Waterloo, ON, Canada
Gaby Lenhart, Sophia Antipolis, France
Daniel Lidar, Los Angeles, CA, USA
Arno Rauschenbeutel, Vienna University of Technology, Vienna, Austria
Renato Renner, Institut für Theoretische Physik, ETH Zürich, Zürich, Switzerland
MaximilianSchlosshauer,DepartmentofPhysics,UniversityofPortland,Portland,
OR, USA
Yaakov S. Weinstein, Quantum Information Science Group, The MITRE
Corporation, Princeton, NJ, USA
H. M. Wiseman, Brisbane, QLD, Australia
Aims and Scope
The book series Quantum Science and Technology is dedicated to one of today’s
mostactiveandrapidlyexpandingfieldsofresearchanddevelopment.Inparticular,
the series will be a showcase for the growing number of experimental implemen-
tations and practical applications of quantum systems. These will include, but are
not restricted to: quantum information processing, quantum computing, and
quantum simulation; quantum communication and quantum cryptography; entan-
glement and other quantum resources; quantum interfaces and hybrid quantum
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 the
theoretical and mathematical questions relevant to designing and understanding
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
intending to enter the field of quantum science and technology.
More information about this series at http://www.springer.com/series/10039
Bei Zeng Xie Chen Duan-Lu Zhou
(cid:129) (cid:129) (cid:129)
Xiao-Gang Wen
Quantum Information Meets
Quantum Matter
From Quantum Entanglement to Topological
Phases of Many-Body Systems
123
BeiZeng XieChen
University of Guelph Physics
Guelph,ON, Canada California Institute ofTechnology
Pasadena,CA, USA
Duan-LuZhou
Institute of Physics Xiao-GangWen
ChineseAcademy of Sciences Massachusetts Institute of Technology
Beijing,China Cambridge, MA, USA
ISSN 2364-9054 ISSN 2364-9062 (electronic)
QuantumScience andTechnology
ISBN978-1-4939-9082-5 ISBN978-1-4939-9084-9 (eBook)
https://doi.org/10.1007/978-1-4939-9084-9
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Foreword
In1989,IattendedaworkshopattheUniversityofMinnesota.Theorganizershad
hoped the workshop would spawn new ideas about the origin of high-temperature
superconductivity, which had recently been discovered. But I was especially
impressed by a talk about the fractional quantum Hall effect by a young physicist
named Xiao-Gang Wen.
FromWen,Iheardforthefirsttimeaboutaconceptcalledtopologicalorder.He
explainedthatforsomequantumphasesoftwo-dimensionalmatterthegroundstate
becomes degenerate when the system resides on a surface of nontrivial topology
such as a torus, and that the degree of degeneracy provides a useful signature for
distinguishing different phases. I was fascinated.
Up until then, studies of phases of matter and the transitions between them
usuallybuiltonprinciplesannunciateddecadesearlierbyLevLandau.Landauhad
emphasized the crucial role of symmetry, and of local-order parameters that dis-
tinguishdifferentsymmetryrealizations.ThoughmuchofwhatWensaidwentover
myhead,Ididmanagetogleanthathewasproposingawaytodistinguishquantum
phasesfoundedonmuchdifferentprinciplesthanLandau’s.Asaparticlephysicist,
I deeply appreciated the power of Landau theory, but I was also keenly aware that
the interface of topology and physics had already yielded many novel and fruitful
insights.
Mullingovertheseideasontheplaneridehome,Iscribbledafewlinesofverse:
Nowweareallowed
TodisavowLandau.
Wow…
Without knowing where it might lead, one could sense the opening of a new
chapter.
At around that same time, another new research direction was beginning to
gathersteam,thestudyofquantuminformation.RichardFeynmanandYuriManin
hadsuggestedthatacomputerprocessingquantuminformationmightperformtasks
beyondthereachofordinarydigitalcomputers.DavidDeutschformalizedtheidea,
which attracted the attention of computer scientists, and eventually led to Peter
v
vi Foreword
Shor’s discoverythat aquantumcomputer can factor large numbersinpolynomial
time. Meanwhile, Alexander Holevo, Charles Bennett, and others seized the
opportunity to unify Claude Shannon’s information theory with quantum physics,
erecting new schemes for quantifying quantum entanglement and characterizing
processes in which quantum information is acquired, transmitted, and processed.
ThediscoveryofShor’salgorithmcausedaburstofexcitementandactivity,but
quantuminformationscienceremainedoutsidethemainstreamofphysics,andfew
scientists at that time glimpsed the rich connections between quantum information
and the study of quantum matter. One notable exception was Alexei Kitaev, who
had two remarkable insights in the 1990s. He pointed out that finding the
ground-state energy of a quantum system defined by a “local” Hamiltonian, when
suitablyformalized,isashardasanyproblemwhosesolutioncanbeverifiedwitha
quantumcomputer.ThisidealaunchedthestudyofHamiltoniancomplexity.Kitaev
alsodiscernedtherelationshipbetweenWen’sconceptoftopologicalorderandthe
quantum error-correcting codes that can protect delicate quantum superpositions
from the ravages of environmental decoherence. Kitaev’s notion of a topological
quantum computer, a mere theorist’s fantasy when proposed in 1997, is by now
pursued in experimental laboratories around the world (though the technology still
hasfartogobeforetrulyscalablequantumcomputerswillbecapableofaddressing
hard problems).
Thereafter progress accelerated, led by a burgeoning community of scientists
workingattheinterfaceofquantuminformationandquantummatter.GuifreVidal
realizedthatmany-particlequantumsystemsthatareonlyslightlyentangledcanbe
succinctlydescribedusingtensornetworks.Thisnewmethodextendedthereachof
mean-field theory and provided an illuminating new perspective on the successes
oftheDensityMatrixRenormalizationGroup(DMRG).Byprovingthattheground
stateofalocalHamiltonianwithanenergygaphaslimitedentanglement(thearea
law), Matthew Hastings showed that tensor network tools are widely applicable.
These tools eventually led to a complete understanding of gapped quantum phases
in one spatial dimension.
The experimental discovery of topological insulators focused attention on the
interplay of symmetry and topology. The more general notion of a symmetry-
protectedtopological(SPT)phasearose,inwhichaquantumsystemhasanenergy
gapinthebulkbutsupportsgaplessexcitationsconfinedtoitsboundarywhichare
protected by specified symmetries. (For topological insulators, the symmetries are
particle-number conservation and time reversal invariance.) Again, tensor network
methods proved to be well suited for establishing a complete classification of
one-dimensional SPT phases, and guided progress toward understanding higher
dimensions, though many open questions remain.
WenowhaveamuchdeeperunderstandingoftopologicalorderthanwhenIfirst
heard about it from Wen nearly 30 years ago. A central new insight is that topo-
logicallyorderedsystemshavelong-rangeentanglement,andthattheentanglement
has universal properties, like topological entanglement entropy, which are insen-
sitivetothemicroscopicdetailsoftheHamiltonian.Indeed,topologicalorderisan
intrinsicpropertyofaquantumstateandcanbeidentifiedwithoutreferencetoany
Foreword vii
particular Hamiltonian at all. To understand the meaning of long-range entangle-
ment,imagineaquantumcomputerwhichappliesasequenceofgeometricallylocal
operations to an input quantum state, producing an output product state which is
completely disentangled. If the time required to complete this disentangling com-
putation is independent of the size of the system, then we say the input state is
short-range entangled; otherwise, it is long-range entangled. More generally
(loosely speaking), two states are in different quantum phases if no constant-time
quantum computation can convert one state to the other. This fundamental con-
nection between quantum computation and quantum order has many ramifications
which are explored in this book.
When is the right time for a book that summarizes the status of an ongoing
research area? It’s a subtle question. The subjectshould be sufficientlymature that
enduringconceptsandresultscanbeidentifiedandclearlyexplained.Ifthepaceof
progressissufficientlyrapid,andthetopicsemphasizedarenotwellchosen,thenan
ill-timedbookmightbecomeobsoletequickly.Ontheotherhand,thesubjectought
not to be too mature; only if there are many exciting open questions to attack will
the book be likely to attract a sizable audience eager to master the material.
IfeelconfidentthatQuantumInformationMeetsQuantumMatterisappearingat
an opportune time, and that the authors have made wise choices about what to
include. They are world-class experts and are themselves responsible for many
ofthescientificadvancesexplainedhere.Thestudentorseniorscientistwhostudies
this book closely will be well grounded in the tools and ideas at the forefront of
current research at the confluence of quantum information science and quantum
condensed matter physics.
Indeed, I expect that in the years ahead a steadily expanding community of
scientists, including computer scientists, chemists, andhigh-energy physicists, will
want to be well acquainted with the ideas at the heart of Quantum Information
Meets Quantum Matter. In particular, growingevidence suggests that thequantum
physics of spacetime itself is an emergent manifestation of long-range quantum
entanglementinanunderlyingmorefundamentalquantumtheory.Morebroadly,as
quantum technology grows ever more sophisticated, I believe that the theoretical
and experimental study of highly complex many-particle systems will be an
increasingly central theme of twenty-first-century physical science. If that’s true,
QuantumInformationMeetsQuantumMatterisboundtoholdanhonoredplaceon
the bookshelves of many scientists for years to come.
Pasadena, CA, USA John Preskill
September 2018
Preface
After decades of development, quantum information science and technology has
nowcometoitsgoldenage.Itisnotonlywidelybelievedthatquantuminformation
processing offers the secure and high rate information transmission, fast compu-
tationalsolutionofcertainimportantproblems,whichareattheheartofthemodern
informationtechnology.Butalso,itprovidesnewangles,tools,andmethodswhich
helpinunderstandingotherfieldsofscience,amongwhichoneimportantareaisthe
link to modern condensed matter physics.
For a long time, people believe that all phases of matter are described by
Landau’s symmetry-breaking theory, and the transitions between those phases are
described by the change of those symmetry-breaking orders. However, after the
discovery of fractional quantum Hall effect, it was realized in 1989 that the frac-
tional quantum Hall states contain a new type of order (named topological order)
whichisbeyondLandausymmetry-breakingtheory.Traditionalmany-bodytheory
for condensed matter systems is mostly based on various correlation functions,
which suite Landau symmetry-breaking theory very well. But this kind of
approaches is totally inadequate for topological orders, since all different topo-
logical orders have the similar short-range correlations.
The traditional condensed matter theory mostly only considers two kinds of
many-bodystates:productstates(suchasinvariousmean-fieldtheories)andstates
obtained by filling orbitals (such as in Fermi liquid theory). Those two types of
states fail to include the more general topologically ordered states. So the big
questionis,canweunderstandwhatismissingintheabovetwotypesofstates,so
that they fail to capture the topological order?
What quantum information science brings is the information-theoretic under-
standing of correlation, and a new concept called “entanglement”, which is a pure
quantum correlation that has no classical counterpart. Such input from quantum
information science led to a recent realization that the new topological order in
some strongly correlated systems is nothing but the pattern of many-body entan-
glement. The study of topological order and the related new quantum phases is
ix
x Preface
actually a study of patterns of entanglement. The nontrivial patterns of entangle-
mentaretherootofmanyhighlynovelphenomenaintopologicallyorderedphases
(such as fractional quantum Hall states and spin liquid states), which include
fractional charge, fractional statistics, protected gapless boundary excitations,
emergence of gauge theory, Fermi statistics from purely bosonic systems, etc.
The connection between quantum information science and condensed matter
physics is not accidental, but has a very deep root. Quantum theory has explained
and unified many microscopic phenomena, ranging from discrete spectrum of
Hydrogen atom, black-body radiation, to interference of electron beam, etc.
However, what quantum theory really unifies is information and matter. We know
that a change or frequency is a property of information. But according to quantum
theory, frequency corresponds to energy. According to the theory of relativity,
energycorrespondstomass.Energyandmassarepropertiesofmatter.Inthissense,
frequency leads to mass and information becomes matter.
Butdowebelievethatmatter(andtheelementaryparticlesthatformthematter)
all come from qubits? Is it possible that qubits are the building blocks of all the
elementary particles? If matter were formed by simple spin-0 bosonic elementary
particles,thenitwasquitepossiblethatthespin-0bosonicelementaryparticles,and
thematterthattheyform,allcamefromqubits.Wecansimplyviewthespaceasa
collection of qubits and the 0-state of qubits as the vacuum. Then, the 1-state of
qubit will correspond to a spin-0 bosonic elementary particle in space. But our
world is much more complicated. The matter in our world is formed by particles
that have two really strange properties: Fermi statistics and fractional angular
momentum (spin-1/2). Our world also have light, which correspond to spin-1
particles thatstrangelyonlyhavetwocomponents.Suchspin-1particles arecalled
gauge bosons.
Canspaceformedbysimplequbitsproducespin-1/2fermionsandspin-1gauge
bosons?Inthelast20years(andasexplainedinthisbook),westarttorealizethat
althoughqubitsareverysimple,theirorganization—theirquantumentanglement—
can be extremely rich and complex. The long-range quantum entanglement of
qubits makes it possible to use simple qubits to produce spin-1/2 fermions and
spin-1 gauge bosons, as well as the matter formed by those elementary particles.
Thousands of research papers studying the properties of quantum entanglement
have been published in the past two decades. Notable progress includes, but not
limited to, extensive study of correlation and entanglement properties in various
strongly correlated systems, development of concepts of entanglement area law
whichresultsinanewtoolcalledtensornetworkmethod,theroleofentanglement
play in quantumphase transitions, theconcept oflong-range entanglement, and its
useinthestudyoftopologicalphaseofmatter.Also,extensiveattentionshavebeen
attracted on the new states of quantum matter and the emergence of fractional
quantum numbers and fractional/Fermi statistics, with many published papers
during the last decades along these directions.
Itisnotpossibletoincludealltheseexcitingdevelopmentsinasinglebook.The
scopeofthisbookisrathertointroducesomegeneralconceptsandbasicideasand
methods that the viewpoints of quantum information scientists have on condensed
