The Edge Of Physics
harnessingquantaCOPYRIGHT 2003 SCIENTIFIC AMERICAN, INC.
ComplexQuantumWorldSimple Rules for aAn exciting newfundamental disciplineof research combinesinformation science andquantum mechanicsBy Michael A. NielsenUpdated from the November 2002 issueCOPYRIGHT 2003 SCIENTIFIC AMERICAN, INC.25
Over the past few decades, scientists have learned that simplegoal of quantum information science, afundamental field that is opening up in response to a new way of comprehendingthe world. Many articles about quantuminformation science focus on technological applications: research groups “teleport” quantum states from one locationto another. Other physicists use quantumstates to create cryptographic keys thatare absolutely secure from eavesdropping. Information scientists devise algorithms for hypothetical quantum-mechanical computers, much faster than thebest known algorithms for conventional,or classical, computers.These technologies are fascinating,but they obscure the fact that they are aby-product of investigations into deep newscientific questions. Applications such asquantum teleportation play a role similarto the steam engines and other machinesthat spurred the development of thermodynamics in the 18th and 19th centuries.Thermodynamics was motivated by profound, basic questions about how energy,heat and temperature are related, the transformations among these quantities in phys-Overview/Quantum InformationInformation is not purely mathematical. Instead it always has a physicalembodiment. In traditional information science the embodiment followsclassical, or nonquantum, physics. The burgeoning field of quantum informationscience puts information in a quantum context. The basic resource of classical information is the bit, which is always eithera 0 or a 1. Quantum information comes in quantum bits, or qubits (pronounced“cue-bits”). Qubits can exist in superpositions, which simultaneously involve0 and 1, and groups of qubits can be “entangled,” which gives themcounterintuitive correlations. Quantum computers processing qubits, particularly entangled qubits, canoutperform classical computers. Entanglement behaves like a resource, similarto energy, that can be used to do quantum information processing. The goal of quantum information science is to understand the general high-levelprinciples that govern complex quantum systems such as quantum computers.These principles relate to the laws of quantum mechanics in the way thatheuristics for skillful play at chess relate to the game’s basic rules. 26SCIENTIFIC AMERICANical processes, and the key role of entropy.Similarly, quantum information scientists are fathoming the relation betweenclassical and quantum units of information, the novel ways that quantum information can be processed, and the pivotalimportance of a quantum feature calledentanglement, which entails peculiar connections between different objects.Popular accounts often present entanglement as an all-or-nothing propertyin which quantum particles are eitherentangled or not. Quantum informationscience has revealed that entanglementis a quantifiable physical resource, likeenergy, that enables information-processing tasks: some systems have a littleentanglement; others have a lot. Themore entanglement available, the bettersuited a system is to quantum information processing.Furthermore, scientists have begun todevelop powerful quantitative laws of entanglement (analogous to the laws of thermodynamics governing energy), whichprovide a set of high-level principles for understanding the behavior of entanglementand describing how we can use it to do information processing.Quantum information science is newenough that researchers are still comingto grips with its very nature, and they disagree about which questions lie at itsheart. From my point of view, the centralgoal of quantum information science isto develop general principles, like thelaws of entanglement, that will enable usto understand complexity in quantumsystems.Complexity and Quantain complexityconcentrate on systems, such as the weather or piles of sand, that are described byclassical physics rather than quantumphysics. That focus is natural becausecomplex systems are usually macroscop-NUMEROUS STUDIESTHE EDGE OF PHYSICSCOPYRIGHT 2003 SCIENTIFIC AMERICAN, INC.SLIM FILMS (preceding pages)rules can give rise to very rich behavior. Agood example is chess. Imagine you’re anexperienced chess player introduced tosomeone claiming to know the game. Youplay a few times and realize that althoughthis person knows the rules of chess, hehas no idea how to play well. He makesabsurd moves, sacrificing his queen for apawn and losing a rook for no reason atall. He does not truly understand chess:he is ignorant of the high-level principlesand heuristics familiar to any knowledgeable player. These principles are collective or emergent properties of chess,features not immediately evident from therules but arising from interactions amongthe pieces on the chessboard.Scientists’ current understanding ofquantum mechanics is like that of a slowlearning student of chess. We’ve knownthe rules for more than 70 years, and wehave a few clever moves that work insome special situations, but we’re onlygradually learning the high-level principles that are needed to play a skillfuloverall game.The discovery of these principles is the
THE FUNDAMENTAL QUESTIONMUCH OF INFORMATION SCIENCE, both classical and quantum,can be summed up by analyzing variants of a basic question:“What quantity of an information resource is needed toperform a specific information-processing task?”For example: “How many computational steps are needed to findthe prime factors of a 300-digit number?” The best classicalalgorithm known would take about 5 1024 steps, or about 150,000years at terahertz speed. By taking advantage of innumerablequantum states, a quantum factoring algorithm would take only5 1010 steps, or less than a second at terahertz speed.Classical computer300-digit number2:30:00 P.M.Year: 20122:30:01 P.M.Year: 20122:30:00 P.M.Year: 154,267BRYAN CHRISTIE DESIGNQuantum computeric, containing many constituent parts, andmost systems lose their quantum natureas their size is increased. This quantumto-classical transition occurs becauselarge quantum systems generally interactstrongly with their environment, causinga process of decoherence, which destroysthe system’s quantum properties [see “100Years of Quantum Mysteries,” by MaxTegmark and John A. Wheeler; Scientific American, February 2001].As an example of decoherence, thinkof Erwin Schrödinger’s famous cat insidea box. In principle, the cat ends up in aweird quantum state, somewhere between dead and alive; it makes no sense todescribe it as either one or the other. In areal experiment, however, the cat interacts with the box by exchange of light,heat and sound, and the box similarly interacts with the rest of the world. In nanoseconds, these processes destroy the delicate quantum states inside the box and replace them with states describable, to agood approximation, by the laws of classical physics. The cat inside really is eitheralive or dead, not in some mysteriousnonclassical state that combines the two.The key to seeing truly quantum behavior in a complex system is to isolatethe system extremely well from the rest ofthe world, preventing decoherence andpreserving fragile quantum states. Thisisolation is relatively easy to achieve withsmall systems, such as atoms suspendedin a magnetic trap in a vacuum, but ismuch more difficult with the larger onesin which complex behavior may befound. Accidental laboratory discoveriesof remarkable phenomena such as superconductivity and the quantum Hall effectare examples in which physicists haveachieved large, well-isolated quantumsystems. These phenomena demonstratethat the simple rules of quantum mechanics can give rise to emergent principles governing complex behaviors.Resources and Tasksunderstand the highlevel principles that govern in those rareinstances when the quantum and theWE ATTEMPT TOwww.sciam.comcomplex meet by abstracting, adaptingand extending tools from classical information theory. In 2001 Benjamin W.Schumacher of Kenyon College proposedthat the essential elements of informationscience, both classical and quantum, canbe summarized as a three-step procedure:1. Identify a physical resource. A familiar classical example is a string of bits.Although bits are often thought of as abstract entities— 0’s and 1’s— all information is inevitably encoded in real physicalobjects, and thus a string of bits shouldbe regarded as a physical resource.2. Identify an information-processingtask that can be performed using thephysical resource of step 1. A classical example is the two-part task of compressingthe output from an information source(for example, the text in a book) into a bitstring and then decompressing it— that is,recovering the original information fromthe compressed bit string.3. Identify a criterion for successfulcompletion of the task of step 2. In ourexample, the criterion could be that theTHE EDGE OF PHYSICSCOPYRIGHT 2003 SCIENTIFIC AMERICAN, INC.27
QUBITS EXPLAINEDA BIT can have one oftwo states: 0 or 1. A bitcan be represented bya transistor switch setto “off” or “on” orabstractly by an arrowpointing up or down.version of a bit, hasmany more possiblestates. The states canbe represented by anarrow pointing to alocation on a sphere.The north pole isequivalent to 1, thesouth pole to 0. Theother locations arequantum superpositions of 0 and 1.N S output from the decompression stageperfectly matches the input to the compression stage.The fundamental question of information science is then “What is the minimal quantity of the physical resource (1)we need to perform the information-processing task (2) in compliance with thesuccess criterion (3)?” Although this question does not quite capture all of information science, it provides a powerful lensthrough which to view much research inthe field [see box on preceding page].The data-compression example corresponds to a basic question of classicalinformation science— namely, what is theminimum number of bits needed to storethe information produced by somesource? This problem was solved byClaude E. Shannon in his famous 1948papers founding information theory. Inso doing, Shannon quantified the information content produced by an information source, defining it to be the minimumnumber of bits needed to reliably storethe output of the source. His mathematical expression for the information content28E 32º 48′ 10.3476.″A QUBIT MIGHT SEEM TO CONTAIN an infinite amount of informationbecause its coordinates can encode an infinite sequence of digits. Butthe information in a qubit must be extracted by a measurement. Whenthe qubit is measured, quantum mechanics requires that the result isalways an ordinary bit— a 0 or a 1. The probability of each outcomedepends on the qubit’s “latitude.”is now known as the Shannon entropy.Shannon’s entropy arises as the answer to a simple, fundamental questionabout classical information processing. Itis perhaps not surprising, then, thatstudying the properties of the Shannonentropy has proved fruitful in analyzingprocesses far more complex than datacompression. For example, it plays a central role in calculating how much information can be transmitted reliably througha noisy communications channel andeven in understanding phenomena suchas gambling and the behavior of the stockmarket. A general theme in informationscience is that questions about elementary processes lead to unifying conceptsthat stimulate insight into more complexprocesses.In quantum information science, allthree elements of Schumacher’s list takeon new richness. What novel physical resources are available in quantum mechanics? What information-processingtasks can we hope to perform? What areappropriate criteria for success? The resources now include superposition states,SCIENTIFIC AMERICANlike the idealized alive and dead cat ofSchrödinger. The processes can involvemanipulations of entanglement (mysterious quantum correlations) between widely separated objects. The criteria of success become more subtle than in the classical case, because to extract the result ofa quantum information-processing taskwe must observe, or measure, the system— which almost inevitably changes it,destroying the special superposition statesthat are unique to quantum physics.Qubitsscience begins by generalizing the fundamental resource of classical information— bits— toquantum bits, or qubits. Just as bits areideal objects abstracted from the principles of classical physics, qubits are idealquantum objects abstracted from theprinciples of quantum mechanics. Bitscan be represented by magnetic regionson disks, voltages in circuitry, or graphitemarks made by a pencil on paper. Thefunctioning of these classical physicalstates as bits does not depend on the de-QUANTUM INFORMATIONTHE EDGE OF PHYSICSCOPYRIGHT 2003 SCIENTIFIC AMERICAN, INC.BRYAN CHRISTIE DESIGNA QUBIT, the quantumN 23º 34′ 41.4422.″
BRYAN CHRISTIE DESIGNsingle bit that the measurement uncovers.The principles of quantum mechanicsprevent us from ever extracting morethan a single bit of information, no matter how cleverly we encode the qubit orhow ingeniously we measure it afterward.This surprising result was proved in 1973by Alexander S. Holevo of the SteklovMathematical Institute in Moscow, following a 1964 conjecture by J. P. Gordonof AT&T Bell Laboratories. It is asthough the qubit contains hidden information that we can manipulate but notaccess directly. A better viewpoint, however, is to regard this hidden informationas being a unit of quantum informationrather than an infinite number of inaccessible classical bits.Notice how this example followsSchumacher’s paradigm for informationscience. Gordon and Holevo asked howmany qubits (the physical resource) arerequired to store a given amount of classical information (the task) in such a waythat the information can be reliably recovered (the criterion for success). Furthermore, to answer this question, theyintroduced a mathematical concept, nowknown as the Holevo chi (represented bythe Greek letter χ), that has since beenused to simplify the analysis of more complex phenomena, similar to the simplifications enabled by Shannon’s entropy.For example, Michal Horodecki of theUniversity of Gdansk in Poland hasshown that the Holevo chi can be used toanalyze the problem of compressingquantum states produced by a quantuminformation source, which is analogousto the classical data compression considered by Shannon.Entangled Statesare interesting, butmore fascinating behavior arises whenseveral qubits are brought together. A keyfeature of quantum information science isthe understanding that groups of two orSINGLE QUBITSHERE THERE BE QUANTUM TYGERSQUANTUM INFORMATION SCIENTISTS are still mapping out the broad topography oftheir nascent field. Some simpler processes, such as teleportation and quantumcryptography, are well understood. In contrast, complex phenomena such asquantum error correction and Peter W. Shor’s factorization algorithm are surroundedby large tracts of terra incognita. One effort to bridge the gaps between the simpleand the complex is work on a comprehensive theory of entanglement, analogous tothe theory of energy embodied in GRAPHYQUANTUMERROR-CORRECTINGCODESQUTHEORY OFENTANGLEMENTGROVER’SSEARCHINGALGORITHMFOURIER TRANSFTUMOANSHOR’SFACTORINGALGORITHMRMtails of how they are realized. Similarly,the properties of a qubit are independentof its specific physical representation asthe spin of an atomic nucleus, say, or thepolarization of a photon of light.A bit is described by its state, 0 or 1.Likewise, a qubit is described by its quantum state. Two possible quantum statesfor a qubit correspond to the 0 and 1 of aclassical bit. In quantum mechanics, however, any object that has two differentstates necessarily has a range of other possible states, called superpositions, whichentail both states to varying degrees. Theallowed states of a qubit are precisely allthose states that must be available, in principle, to a classical bit that is transplantedinto a quantum world. Qubit states correspond to points on the surface of asphere, with the 0 and 1 being the southand north poles [see box on oppositepage]. The continuum of states between 0and 1 fosters many of the extraordinaryproperties of quantum information.How much classical information canwe store in a qubit? One line of reasoningsuggests the amount is infinite: To specify a quantum state we need to specify thelatitude and longitude of the corresponding point on the sphere, and in principleeach may be given to arbitrary precision.These numbers can encode a long stringof bits. For example, 011101101. couldbe encoded as a state with latitude 01 degrees, 11 minutes and 01.101. seconds.This reasoning, though plausible, isincorrect. One can encode an infiniteamount of classical information in a single qubit, but one can never retrieve thatinformation from the qubit. The simplestattempt to read the qubit’s state, a standard direct measurement of it, will give aresult of either 0 or 1, south pole or northpole, with the probability of each outcome determined by the latitude of theoriginal state. You could have chosen adifferent measurement, perhaps using the“Melbourne–Azores Islands” axis instead of north-south, but again only onebit of information would have been extracted, albeit one governed by probabilities with a different dependence on thestate’s latitude and longitude. Whichever measurement you choose erases all theinformation in the qubit except for SING COMPLEXITYwww.sciam.comTHE EDGE OF PHYSICSCOPYRIGHT 2003 SCIENTIFIC AMERICAN, INC.29
DISENTANGLING ENTANGLEMENTIF DICE COULD BE “entangled” in the manner of quantum particles,each entangled pair would give the same outcome, even if rolledlight-years apart or at very different times.BobAliceThe Standard E-BitWeighing EntanglementWHEN TWO QUBITS are entangled, they noINCOMPLETELY ENTANGLED PAIRS carry less than one e-bit. If Alice and Bob share two partiallyentangled pairs, they can try to “distill” the entanglement onto a single pair. If distillationproduces a maximally entangled pair, then Alice and Bob know their pairs originally carrieda total of at least one e-bit of entanglement.longer have individual quantum states.Instead a relation between the qubits isdefined. For example, in one type ofmaximally entangled pair, the qubits giveopposite results when measured. If onegives 0, the other returns 1, and vice versa.A maximally entangled pair carries one“e-bit” of etely entangled pairBy using distillation (and theinverse process, entanglementdilution), one constructs a virtualset of scales for weighing theentanglement of various statesagainst the standard e-bit.2 3 e-bitAliceBEFOREBobQuantum TeleportationIF ALICE AND BOB share one e-bit,they can teleport one qubit. Theshared e-bit is “used up,” in that theyno longer share it after teleporting.If Bob teleports a member (b) of anentangled pair to Alice, that particle’sentanglement with its originalpartner (c) is transferred to Alice’sparticle (a). Alice and Bob cannotuse teleportation, however, toincrease their stock of shared e-bits.AliceBobAFTERAliceQubit to beteleportedbQubit to beteleportedaBRYAN CHRISTIE DESIGNMaximallyentangled pairc30SCIENTIFIC AMERICANTHE EDGE OF PHYSICSCOPYRIGHT 2003 SCIENTIFIC AMERICAN, INC.
Entangled quantum systems behave in waysimpossible in any classical world.more quantum objects can have statesthat are entangled. These entangled stateshave properties fundamentally unlikeanything in classical physics and are coming to be thought of as an essentially newtype of physical resource that can be usedto perform interesting tasks.
classical physics rather than quantum physics. That focus is natural because complex systems are usually macroscop- Information is not purely mathematical. Instead it always has a physical embodiment. In traditional information science the embodiment follows classical, or nonquantum, physics. The burgeoning field of quantum information
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