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52 Los Alamos Science Number 27 2002

“When two systems, of which we know thestates by their respective representatives,enter into temporary physical interactiondue to known forces between them, andwhen after a time of mutual influence thesystems separate again, then they can nolonger be described in the same way asbefore, viz. by endowing each of them witha representative of its own. I would not call that one but rather the characteristictrait of quantum mechanics, the one thatenforces its entire departure from classicallines of thought. By the interaction, the two representatives (or ψ-functions) havebecome entangled.”—Erwin Schrödinger (1935)

Entanglement, a strong andinherently nonclassical correlation between two or

more distinct physical systems, wasdescribed by Erwin Schrödinger,a pioneer of quantum theory, as “the characteristic trait of quantummechanics.” For many years, entan-gled states were relegated to beingthe subject of philosophical argu-ments or were used only in experi-ments aimed at investigating thefundamental foundations of physics.In the past decade, however, entan-gled states have become a centralresource in the emerging field ofquantum information science, whichcan be roughly defined as the appli-cation of quantum physics phenom-ena to the storage, communication,and processing of information.

The direct application of entan-gled states to quantum-based tech-nologies, such as quantum state tele-portation or quantum cryptography,is being investigated at Los AlamosNational Laboratory, as well as otherinstitutions in the United States andabroad. These new technologiesoffer exciting prospects for commer-cial applications and may haveimportant national-security implica-tions. Furthermore, entanglement isa sine qua non for the more ambi-

tious technological goal of practicalquantum computation.

In this article, we will describewhat entanglement is, how we havecreated entangled quantum states ofphoton pairs, how entanglement canbe measured, and some of its appli-cations to quantum technologies.

Classical Correlation andQuantum State Entanglement

To describe the concept of quantum entanglement, we are firstgoing to describe what it is not! Let us imagine the simple experi-ment illustrated in Figure 1. In thatexperiment, a source S1 continuallyemits pairs of photons in two direc-tions. As seen in the figure, onephoton goes toward an observernamed Alice, while the other goestoward Bob.

First, imagine that the photonsemitted by S1 are always polarizedin the horizontal direction.Mathematically, we say that eachphoton is in the pure state denotedby the ket |H⟩, that is, the “represen-tative” of the state Schrödingerreferred to in the quotation on theopposite page. Because the photonsare paired, the combined state of the

two photons is denoted |HH⟩, wherethe first letter refers to Alice’s pho-ton and the second to Bob’s.

Alice and Bob want to measurethe polarization state of theirrespective photons. To do so, eachuses a rotatable, linear polarizer, adevice that has an intrinsic trans-mission axis for photons. For agiven angle φ between the photon’spolarization vector and the polariz-er’s transmission axis, the photonwill be transmitted with a probabili-ty equal to cos2φ. (See the box“Photons, Polarizers, andProjection” on page 76.) Formally,the polarizer acts like a quantum-mechanical projection operator Pφselecting out the component of thephoton wave function that lines upwith the transmission axis. We saythat the polarizer “collapses” thephoton wave function to a definitestate of polarization. If, for exam-ple, the polarizer is set to an angle θwith respect to the horizontal, thena horizontally polarized photon iseither projected into the state |θ⟩with probability cos2θ or absorbedwith probability 1 – cos2θ = sin2θ.The bizarre aspect of quantummechanics is that the projectionprocess is probabilistic.The fate ofany given photon is completely

Number 27 2002 Los Alamos Science 53

Quantum State EntanglementCreation, characterization, and application

Daniel F. V. James and Paul G. Kwiat

This article is dedicated to the memory of Professor Leonard Mandel,one of the pioneers of experimental quantum optics, whose profound scientific insights and gentlemanly bearing will be sorely missed.

unknown. Furthermore, any informa-tion about the photon’s previouspolarization state is lost.

Getting back to the experiment,we assume that Alice and Bob’s polar-izers are always aligned in the sameway: When Alice sets her polarizer toa certain angle, she communicates herchoice to Bob, who uses the same setting. Behind each polarizer is adetector. In our experiment, Alice andBob rotate their polarizers to a certainangle with respect to the horizontaland record whether they detect a photon. Then, they repeat the proce-dure for different polarizer settings. If Alice looks only at her own data (or Bob looks only at his), she candetermine the polarization state of thephotons emitted by the source—seeFigure 2(a). But Alice and Bob canalso make a photon-per-photon comparison of their data and deter-mine the probability that they have the

same result, that is, they can examinethe photon–photon correlations.

Suppose Alice has her polarizer oriented to transmit horizontally polarized photons. In that case, eachphoton coming to her from S1 will betransmitted, and her detector will“click,” indicating a photon hasarrived. Subsequent communicationwith Bob would reveal that he alsodetected each photon, so at this polarizer setting, there is a perfect correlation between Alice’s detectionof a photon and Bob’s. Similarly,by rotating the polarizer to the verticalposition, the two would again discovera perfect correlation, namely, neitherparty would detect his or her photons.

The correlation changes when Aliceand Bob have their polarizers oriented,say, at +45° to the horizontal. In thatcase, the photon sent to Alice has a50 percent chance of passing throughher polarizer, and independently, the

photon sent to Bob has a 50 percentchance of passing through his. The probability is therefore 25 percentthat both Alice and Bob detect a photon, 25 percent that neither detectsa photon, and thus 50 percent that theyobtain the same result.

The correlation function G is shown in Figure 2(a′). It is equal to the product of the independent probabilities for detecting a photon[(cos2θ)A × (cos2θ)B], plus the prod-uct of the probabilities for not detectingone [(sin2θ)A × (sin2θ)B], where sub-scripts A and B are for Alice and Bob,respectively. Thus, Alice and Bobdeduce that the two photons are independent of each other and the wave function is in fact separable:|HH⟩ = |H⟩|H⟩. In other words, the correlation is entirely consistent withclassical probability theory. The pho-tons are classically correlated.

Now, consider performing the


Quantum State Entanglement

Figure 1. A Simple Two-Photon Correlation Experiment In this experiment, a source emits pairs of photons: One photon is going to Alice and the other to Bob. Each photon passesthrough a linear polarizer on its way to its respective detector. Both Alice and Bob’s polarizers are rotatable and can bealigned to any angle with respect to the horizontal, but Bob’s is always kept parallel to Alice’s. For a given polarizer setting,Alice and Bob record those instances when they have the same results, that is, when both detect photons or when theydon’t. The figure shows the source emitting two horizontal photons in the state |Ψ⟩ = |HH⟩. The experiment can be performedwith other sources to examine differences between other two-photon states. (Picture of Bob is courtesy of Hope Enterprises, Inc.)

Photon source

Polarized photons

Detector

Alice

Bob

Rotatable polarizer



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Figure 2. Quantum States, Polarization, and Correlation The three sets of graphs show the results of the three experiments discussed in the text. In each case, the leftmost graph showsthe probability that Alice alone detects a photon and reveals information about the net polarization state of her photon. The right-most graph shows the probability that Alice and Bob have the same result, which reveals information about the two-photon state.

(a) S1 emits photons in the purestate |HH⟩. Alice measures a cos2θfunction for her polarization dataand deduces that photons comingto her are horizontally polarized.(A different linear polarizationwould shift the curve to the left orright.) (a′) We define the correla-tion function G as the probabilitythat both Alice and Bob detect aphoton, plus the probability thatneither detects a photon. For thissource, G is completely consis-tent with classical probability theory for independent events;that is, the correlation function is the product of the detectionprobability of each photon in the pair.

(b) The source S2 emits photonsin the partially mixed state1/2(|HH⟩⟨HH| + |VV⟩⟨VV|). Photonsfrom this source do not have a net polarization. Alice receives atrandom either an |H⟩ or a |V⟩ pho-ton, so the sum of her measure-ments averages to a 50 percentdetection probability independentof angle. (b′) The correlation func-tion G, however, is the same as in (a), revealing that the photonsin each pair are independent ofeach other and have polarizationH or V. Therefore, the two photonsexhibit the same classical corre-lations seen in (a).

(c) The source S3 emits photonsin the maximally entangled state1/√2(|HH⟩ + |VV⟩). Unlike the photons in the mixed state,each photon is unpolarized.Nevertheless, if Alice and Bobalign their polarizers the sameway, they always get the sameresult independent of angle.(c′) Polarization measurements ofthe two photons are 100 percentcorrelated. The photons exhibit“quantum” correlations.

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Alice’s Polarization Measurements Correlation between Alice and Bob’sMeasurements

Polarizer setting (°) Polarizer setting (°)

G = cos4 + sin4

p+ = cos2 θ

θ θ

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θ

(a) (a′)

Probability that Alice (or Bob) detects a photon: p+ = cos2θ. Probability that Alice (or Bob) does not detect a photon: p– = sin2θ.

For independent photons: G = GHH = p+A × p+

B + p–A × p–

B = cos4θ + sin4θ.

For this mixed state, G = 1/2(GHH + GVV) = GHH .

experiment with a second source S2that has a 50 percent chance to emittwo horizontally polarized photons|HH⟩ and a 50 percent chance to emittwo vertically polarized photons |VV⟩.This type of source emits photons in amixed state, which cannot be writtenas a single “ket.” Instead, a mixedstate must be analyzed in terms ofseveral kets, each representing a par-ticular, distinct pure state that has aprobability associated with it. Makinga measurement on a mixed state is

equivalent to probing an ensemble of pure states. The likelihood of measuring a particular pure state isgiven by the appropriate probability.(More-detailed, mathematical descrip-tions of pure and mixed quantumstates are found in the box “Pure,Entangled, or Mixed?” above.)

The output of S2 is random (either|HH⟩ or |VV⟩), so Alice receives atrandom either an |H⟩ or a |V⟩ photon.Because the probability of detecting|H⟩ is 1/2 cos2θ, and the probability of

detecting |V⟩ is 1/2 sin2θ, Alice has a50 percent chance of detecting a pho-ton regardless of how she sets herpolarizer. The same is true for Bob.Each observer, therefore, deduces thatthe photons coming from S2 have nonet polarization. But as seen inFigure 2(b′), the correlation functiontells a different story. In fact, the cor-relation function for this source isidentical to the one obtained for S1because, in both cases, the individualphotons leave the source in definite



A pure state is a vector in a system’s Hilbert space. Forexample, the most general, pure two-photon polarizationstate can be written as

|ψpure⟩ = α|HH⟩ + β|HV⟩ + γ |VH⟩ + δ |VV⟩ . (1)

This state is specified by the four probability amplitudes α,β, γ, and δ (expressed by four complex numbers or eightreal numbers) although these parameters are subject to twoconstraints. The first is that the mean-square amplitudesmust equal unity, that is,

|α |2 + |β |2 + |γ |2 + |δ |2 = 1 . (2)

The second relates to the fact that the overall phase of awave function has no physical relevance. The net result ofthese constraints is that any pure two-photon state dependson only six independent real numbers.

In general, however, any physical system contains a greateror lesser degree of randomness and disorder, and one mustadapt the formalism of quantum mechanics to take this randomness into account. We do so by averaging over the fluctuations. It is convenient to represent states as density operators, or density matrices, formally defined as

ρ = |ψ⟩⟨ψ | , (3)

where the overbar denotes an ensemble average over therandomness. All the measurable properties of the state aredetermined by ρ.

The density matrix must be used when representing mixedstates, which can be thought of as probabilistic combina-

tions of pure states. Mathematically, the density matrixcan always be decomposed into an incoherent sum overpure states,

ρ = Σipi|ψi⟩⟨ψi| , (4)

where each |ψi⟩ is a pure state and pi are probabilities with values that lie between 0 and 1 and whose sum is 1.In general, this decomposition is not unique. To characterize mixed states, one uses mean values and classical coherences; that is, one must specify the fourmean-square amplitudes (subject to the constraint |α |2 + |β |2 + |γ |2 + |δ |2 = 1) and the six independent classical complex correlations α∗β—–

, α∗γ—–, and so on.

For example, the source S2 mentioned in the text emits apartially mixed state that is 50 percent |HH⟩ and 50 per-cent |VV⟩, so that

ρmix = 0.5 |HH⟩⟨HH| + 0.5 |VV⟩⟨VV| , (5)

or in matrix form

(6)

This state is neither pure nor completely random; it is partially mixed.

We next consider whether quantum states involving two or more systems (for example, two photons), are separable

ρmix

.5 0 0 0

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Pure, Entangled, or Mixed?

polarization states. For S1, the polar-ization information is “carried” indi-vidually by each photon. For S2,the polarization information is carriedby the photon pairs. By examining thecorrelations, Alice and Bob candeduce that information.

A different situation occurs for asource S3 that emits pairs of photonsin the state |Φ+⟩ = 1/√2 (|HH⟩ + |VV⟩).Like the mixed state from S2, thisstate is a combination of two horizon-tally polarized photons and two verti-

cally polarized photons. Unlike themixed state, |Φ+⟩ is a coherent, quan-tum mechanical superposition: A prob-ability amplitude is associated witheach component, |HH⟩ and |VV⟩, andthe two components have a fixed phaserelationship. An important property ofthis particular state is that we canrotate the axes of polarization (H andV) and not change the state’s essentialproperties.

The state |Φ+⟩ is a fully entangledquantum state. It cannot be factorized,

or separated, into a part describing oneof the photons and a part describingthe other. The two photons are inextri-cably linked to each other and theirproperties are always correlated. A measurement of one of the photonsmakes the two-photon state instantlydisappear, and the remaining photonassumes a definite state that is perfectlycorrelated with the measured photon.Neither photon carries definite infor-mation by itself—all the information iscarried in the joint two-photon state.



or entangled. If the state is separable and pure, it can bewritten (in some basis) as a product of the states of the indi-vidual systems, that is, as

|ψ⟩ = |ψA⟩ ⊗ |ψB⟩ , (7)

where ⊗ denotes the tensor product. The state |ψ1⟩ = |HH⟩ isone such product of pure states and can be written as

|ψ1⟩ = |HA⟩ ⊗ |HB⟩ . (8)

Another example is the state

|ψ⟩ = (|HH⟩ + |HV⟩ + |VH⟩ + |VV⟩)/2 , (9)

which can be written as the product state

|ψ⟩ = 1/√2(|H⟩ + |V⟩)A ⊗ 1/√2 (|H⟩ + |V⟩)B . (10)

A third example is the matrix ρmix on the opposite page,which represents a separable mixed state.

In contrast, if there is no way to write the two-photon stateas a direct product of states, the state is said to be entangled.This definition leads to a quantity called concurrence, whichis defined for the general pure state |ψpure⟩ by

C = 2|αδ – βγ | . (11)

If and only if C is zero is the state separable. If C is equal tounity (its maximum value), the state is maximally entangled.

For example, consider any one of the four Bell states

|Φ±⟩ = 1/√2(|ΗΗ⟩ ± |VV⟩) , and

|Ψ±⟩ = 1/√2(|HV⟩ ± |VH⟩) . (12)

These states are a basis for the two-photon Hilbert space,and linear combinations of the four states can be used torepresent any two-photon state. If we compare, say, |Φ+⟩with the general state |ψpure⟩, we have α = δ = 1/√2, andβ = γ = 0. Thus C = 1, and this Bell state is maximallyentangled (as are the other three).

The value of C provides a good metric for the amountof entanglement in a pure two-qubit system.Equivalently, some researchers use C2 (a quantity knownas the tangle) to characterize the degree of entanglement.

The concurrence can also be defined for mixed states,although the definition is much more complicated.Indeed, calculating the concurrence for mixed states ofmore than two qubits is currently a hot topic of research.

In the everyday world, it is common to ascribe two (ormore) variables to the same object (for example, a hot,sweet cup of coffee). Similarly, quantum states aredescribed by the two characteristics discussed above,so that it is possible to have a pure entangled state,a pure separable state, a mixed separable state, orsomething in between, such as a partially mixed,partially entangled state.

Thus, when Alice and Bob repeatthe experiment using the source S3,the correlation is 100 percent regard-less of polarizer orientation (assumingBob’s polarizer is always set the sameway as Alice’s). Figure 2(c) illustratesthe striking difference between theclassical correlations of the photonsgenerated by the sources S1 and S2and the nonclassical correlationsexhibited by entangled photons.

To better understand the correlationcurve shown for |Φ+⟩, consider thatquantum mechanics allows us toexpress that state in any basis; that is,|Φ+⟩ = 1/√2 (|XX⟩ + |YY⟩), where |X⟩ isan arbitrary linear-basis state and |Y⟩is the orthogonal-basis state. SupposeAlice has her polarizer set to +45°. Inthe diagonal (+45/–45) basis, theentangled state will be |Φ+⟩ =1/√2 (|+45,+45⟩ + |–45,–45⟩). If Alicedetects her photon (a 50–50 proposi-tion), then Bob’s photon will collapseto the |+45⟩ state, and he will detecthis photon as well. Likewise, if Alicedoesn’t detect her photon, Bob won’tdetect his. The same deductions canbe made for any polarizer setting.

According to quantum mechanics,the correlation occurs regardless ofthe distance separating the two pho-tons. For example, suppose one of twoentangled photons from the state |Φ+⟩is sent to Alice, who “stores” it in her laboratory at Los Alamos,New Mexico. The other photon is sentto Bob, who is in orbit about the starα-Centauri, nearly 4 light-years away.After some time, Alice performs a measurement on her photon anddetermines that it is |H⟩. Her measure-ment selects the |HH⟩ part of the state|Φ+⟩ and eliminates the |VV⟩ part sothat Bob’s photon is necessarily in the state |H⟩. If, instead, Alice hasdetermined that her photon was |+45⟩,the state of Bob’s photon will beinstantly collapsed to |+45⟩ as well. Inother words, the state of Bob’s photonhas been nonlocally influenced byAlice’s measurement. By nonlocal, we

mean that the correlation betweenAlice and Bob’s measurements occurseven if there is not enough time for alight signal (or any signal) to propa-gate between the two experimentalists.This is not to say that special relativityhas been violated: Because Alice cannot predetermine the outcome ofher measurement, she cannot use thenonlocal quantum correlations to sendany information to Bob. In fact, entan-glement can never be used to send signals faster than the speed of light.Nonetheless, Bob’s photon “knows”the outcome of Alice’s measurement.

Nonlocality was the central point ofa famous argument raised by AlbertEinstein, Boris Podolsky, and NathanRosen in 1935, now known as theEPR paradox. The three physicists dis-agreed with the Copenhagen interpreta-tion of quantum mechanics, accordingto which the state of a quantum systemis indeterminate until it is projectedinto a definite state as a result of ameasurement. Einstein, Podolsky, andRosen argued that even unmeasuredquantities corresponded to definite “elements of reality.” The quantumstate only appeared to be indeterminatebecause some of the parameters thatcharacterize the system were unknownand unmeasurable. These local parame-ters, or “hidden variables,” determinedthe outcome of the experiment.

In 1964, John Bell showed that thecorrelations between measured prop-erties of any classical two-particle system would obey a mathematicalinequality, but the same measured cor-relations would violate the inequalityif the two particles were an entangledquantum system. Experiments couldtherefore determine if nature exhibitednonlocal features. Following the

development of laboratory sources ofentangled photons, experimental testsof Bell’s inequality were pursued withvigor. The results to date suggest thatthe observed photon correlations can-not be explained by any local hidden-variable theory,1 and most physicistsagree that quantum mechanics is trulya nonlocal theory.

Entanglement and Quantum Information

Entanglement, a measurable prop-erty of quantum systems, can beexploited for specific goals. Here, wepresent three potential applications, allof which have been shown to work asproof-of-principle demonstrations inthe laboratory.

Quantum Cryptography. Considertwo bank managers, Alice and Bob,who want to have a secret conversa-tion. If they are together in the sameroom, they can simply whisper dis-cretely to each other, but when Aliceand Bob are in their respective cross-town offices, their best chance forsecret communication is to encrypttheir messages.

A generic, classical encryption protocol would begin when Alice andBob convert their messages to sepa-rate binary streams of 0s and 1s.Encryption (locking up the messages)and decryption (unlocking the mes-sages) are then performed with a setof secret “keys” known only to thetwo bankers. Each key is a randomstring of 0s and 1s that is as long asthe binary string comprising eachmessage. To encrypt, Alice (thesender) sequentially adds each bit of



1 There were two loopholes to the EPR tests. The first stemmed from the fact that the detectors were not efficient enough. Consequently, the observed correlations couldhave been the result of some new physics that did not require nonlocal interactions. The second loophole stemmed from the researchers’ inability to choose rapidly and randomly a basis for photon measurement. This inability allowed for a potential communication conspiracy between Alice and Bob’s systems. Both of these loopholeshave recently been closed but, so far, not in the same experiment.

the key to each bit of her message,using modulo 2 addition (0 + 0 = 0,0 + 1 = 1, 1 + 0 = 1, and 1 + 1 = 0).She then sends the encrypted message toBob, who decrypts it simply by repeat-ing the operation, that is, by performinga sequential, bit-by-bit modulo 2 addi-tion of the key to the message.

This type of encryption protocol,known as a one-time pad, is currentlythe only provably secure protocol. Butthe one-time pad is effective only ifAlice and Bob never reuse the key,and more obviously, if the key remainssecret. A potential eavesdropper, Eve,cannot be allowed to glean any part of

the bit stream that makes up the key.Therein lies a central problem of cryp-tography: How can secret keys be cre-ated and then securely distributed?The nonlocal correlations of entangledphotons can play a role in this regard.(One can also exploit the properties ofnonentangled photons in cryptographicschemes. See the article “QuantumCryptography” on page 68.)

In the entangled-state quantumcryptography scheme, Alice and Bobperform an experiment similar to theone described in the first section ofthe paper. They use a source S3 thatemits entangled photons in the general

state |Φ+⟩ = 1/√2 (|XX⟩ +|YY⟩), where|X⟩ is an arbitrary linear-basis stateand |Y⟩ is the orthogonal-basis state.One photon goes to Alice and the otherto Bob. In this protocol, however,either banker can choose—at randomand independent of each other—to usea half-wave plate (HWP) to rotatephoton polarization by a set amount.The bankers then detect the photon inthe H/V basis using a polarizing beamsplitter, which transmits horizontallypolarized photons and reflects verti-cally polarized photons (see Figure 3).Detection of a horizontally polarizedphoton is recorded as a 0; of a verti-

Entangled-photon source

PBSH-detector V-detector

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(1) +⟩ = 1/√// 2 (√√ X,X ⟩ + Y,Y ⟩)

(2) '+⟩ √√2 (√√ (X + 45),X ⟩ + (Y + 45),Y ⟩)

(4) B⟩ – 5 ⟩(3) A⟩ H ⟩

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Figure 3. Quantum Cryptography Using Entangled Photons

Alice and Bob can use the properties ofentangled photons to create a pair ofidentical cryptographic keys. (1) Thesource emits entangled photons in amaximally entangled state |Φ⟩ = 1/√2(|XX⟩+ |YY⟩), where |X⟩ is an arbitrary linear-basis state and |Y⟩ = |X + 90⟩ is theorthogonal-basis state. One photon goesto Alice and the other to Bob. (2) Alicechooses at random either to let her pho-ton pass or to insert a half-wave plate(HWP), which will rotate her photon by+45°. The latter choice changes the rela-tive orientation between the two photonsby +45°. In the case shown, she choosesto rotate her photon. The new entangledstate is |Φ′⟩. (3) Alice uses a polarizing

beam splitter (PBS) to measure her pho-ton in the H/V basis. This optical elementtransmits horizontally polarized photonsand reflects vertically polarized photons,and her unpolarized photon can collapseto either a horizontal or vertical polariza-tion with equal probability. In this case, itcollapses to a horizontal polarization.Alice records a bit value of 0. (5) Bob’sphoton was entangled with Alice’s, so asa result of her measurement, his photonassumed the definite polarization state |H – 45⟩ = |–45⟩. If Bob makes the samechoice as Alice and inserts his HWP, hewill rotate his photon’s polarization by+45° and into a horizontal polarization.His photon will register in the H-detector,

and he will record a bit value of 0. If hemakes the opposite choice and doesn’trotate his photon, the photon polarizedat –45° has an equal probability of goingto either detector (bit value either 0 or 1).As seen in Table I on the next page,whenever Bob and Alice make the samechoice, they keep the bit because theirbit values coincide. If they make oppo-site choices, they discard the bit sincethe values are not correlated. Alice andBob can construct an identical sequenceof random bits—a cryptographic key—simply by declaring their sequence ofchoices. The discussion can be publicbecause the bit values are neverrevealed.

cally polarized photon, as a 1. After a sufficient number of meas-

urements (that number is dictated bythe length of the key), Alice and Bobhave a public discussion, duringwhich they reveal whether they insert-ed the HWP before each measure-ment. At no time do they reveal theactual measurement results. WheneverAlice and Bob make the same choice(50 percent of the time), they knowfrom the properties of entangled pho-tons that they will have completelycorrelated results. By contrast, if oneof them uses the HWP and the otherdoesn’t, they will discard the resultsbecause their measurements would becompletely uncorrelated (see Table I).Following this public discussion, eachbanker will be able to privately con-struct the same random string of 0sand 1s—an ideal key for cryptography.

What about the eavesdropper Eve?She is completely foiled in herattempts to know the secret key.Certainly, she cannot tap the photonline, as she might with conventional,classical communications. A single,indivisible quantum object—namely,a photon—is the conveyor of infor-mation in this cryptographic proto-col. If Eve steals Bob’s photon (a“denial-of-service” attack), the pho-ton’s information never becomes partof the key. Thus, although a wiretap

would reduce the rate of the trans-mission, it would not jeopardize thesecurity of the key.

Eve can try to intercept the photon,measure it, and send another one toBob. But any measurement Eve wouldmake to determine the photon’s polarization state would necessarilyperturb the photon and collapse theentangled state. The photon she sendsto Bob would therefore be classicallycorrelated with Alice’s photon.Consequently, Eve’s interventionwould necessarily induce additionalerrors into Bob’s key.

This last point is significant.Unlike their theoretical counterparts,the encryption keys created by anactual quantum cryptography systeminitially have a small fraction oferrors, because real equipment isalways less than perfect. To makesure their key is secure, Alice andBob ascribe all errors to Eve andthen use this “bit error rate” to esti-mate the maximum amount of infor-mation available to the eavesdropper.They then use a privacy amplificationscheme (discussed in the cryptogra-phy article on page 68) to reduceEve’s knowledge of the secret key toless than one bit.

But the bit error rate alone can leadto a false sense of security. If nonentan-gled photons with a definite polariza-

tion are sent to Bob, it is conceivablethat some other degree of freedom mayalso be coupled to the polarizationstate. For example, if separate lasersare used to produce the two polariza-tion states, the photons from each lasermay have slightly different timingcharacteristics or frequency spectra.Such a difference would in principleallow an eavesdropper to distinguishbetween photons without disturbingthe polarization state and, hence,without affecting the bit error rate.

When the photons are entangled,however, any leakage of informationto other degrees of freedom can beshown to automatically manifest itselfin the error rate detected by Alice andBob. In other words, any degree offreedom with which the polarizationmight be coupled will cause notice-able effects on the polarization corre-lations. Therefore, using only thedetected error rates, one can set anupper limit on the information avail-able to an eavesdropper, even onewho is not directly measuring thepolarization of the photons, and thenuse privacy amplification to eliminatethat information.

As a last resort, Eve may think of“cloning” Alice’s photon. She couldmeasure the clone while allowing theoriginal to continue on to Bob, thuscompletely covering her tracks. But she



Table I. Constructing a Cryptographic Key with Entangled Photons

First Receiver(Alice)

Second Receiver(Bob)

Angle ofRotation

(°)

Detector BitValue

Polarization toSecond Receiver

Angle ofRotation

(°)

Detector BitValue

CommunicationResults

0 H 0 H 0 H 0 Keep bit

0 V 1 V 0 V 1 Keep bit

0 H 0 H +45 H or V 0 or 1 Discard bit

0 V 1 V +45 H or V 0 or 1 Discard bit

+45 H 0 –45° 0 H or V 0 or 1 Discard bit

+45 V 1 +45° 0 H or V 0 or 1 Discard bit

+45 H 0 –45° +45 H 0 Keep bit +45 V 1 +45° +45 V 1 Keep bit

is again foiled by quantum mechanics.According to the no-cloning theorem,it is impossible to make a copy of aphoton in an unknown state whilesimultaneously preserving the original.(See the box “The No-CloningTheorem” on page 79.) Eve is clearlyout of business.

Teleportation. In 1993, CharlesBennett of IBM, Yorktown Heights,and his colleagues proposed aremarkable experiment with entangled particles, namely, the “teleportation” of a pure quantumstate from one location to another.

Charlie wants to send his friendBob a photon in an arbitrary, pure

quantum state |ψ⟩ = α |H⟩ + β |V⟩. Heenlists the aid of Alice, who happensto run the Teleportation Laboratoryshown in Figure 4. Inside the lab, asource S3 is emitting a pair of entan-gled photons, one of which goes offto Bob. The other photon is input intoAlice’s “teleporter.” Charlie isinstructed to send his photon into theteleporter as well.

Alice then performs a special jointmeasurement of the polarization stateof the two photons in the teleporter.She relays the result to Bob, who subsequently performs a simple trans-formation of the polarization state ofhis photon. As if by magic, the stateof Bob’s photon is transformed into

the state of Charlie’s original photon. Mathematically, this magic is

described as follows. The three-photon initial state (that is, Charlie’sphoton plus the two entangled pho-tons) can be represented as

|ψ0⟩ = (α |H⟩ + β |V⟩)C× 1/√2(|HH⟩ + |VV⟩)A,B , (1)

where the subscripts C, A, and B referto Charlie’s, Alice’s, and Bob’s pho-tons, respectively. But |ψ0⟩ can alsobe represented as a superposition ofstates, each constructed in the follow-ing way: Charlie and Alice’s photonsare represented by one of the Bellstates |Φ±⟩ = 1/√2 (|ΗΗ⟩ ± |VV⟩) and




Entangled photons To Bob

To Bob

Alice’s “Teleporter”(Bell state analyzer)

Charlie’sunknownphoton

(a) Before Bell-State Measurement

(b) After Bell State Measurement

(c) After Classical Communication

Delaycavity

Opticalelements

Bob’s photon is projected into a pure state

Bob’s photon assumes the same polarization

state as Charlie’s

Alice relays results to Bob

Classical information

Classical information Bob selects

optical elements

Figure 4. Quantum StateTeleportation (a) Alice’s teleportation lab consists ofan entangled photon source and a Bellstate analyzer (the teleporter). Oneentangled photon goes to Bob and theother to the teleporter. Charlie sends aphoton of unknown polarization stateinto the teleporter. (b) Alice performs ajoint polarization measurement of thetwo photons in the teleporter andrelays the result to Bob using two clas-sical bits of information. The photongoing to Bob is projected into a purestate as a result of Alice’s measure-ment. (c) Upon receiving Alice’s classi-cal information, Bob performs a simpletransformation on his photon, such asa rotation of the polarization vector. Heduplicates the polarization state ofCharlie’s photon without knowing any-thing about its original state.

|ψ±⟩ = 1/√2(|ΗV⟩ ± |VH⟩),

and Bob’s photon is represented as aphoton in a pure state. Thus,

|ψ0⟩ = 1/2{|Φ–⟩C,A (α|H⟩ – β|V⟩)B

+ |Φ+⟩C,A (α|H⟩ + β|V⟩)B

+ |Ψ–⟩C,A (–β|H⟩ + α|V⟩)B

+ |Ψ+⟩C,A (β|H⟩ + α|V⟩)B} . (2)

Technically speaking, this repre-sentation is possible because the Bellstates are a basis for the two-photonHilbert space and any state of twophotons can be represented as a linearsuperposition of these states. It isimportant to point out that Alice’sphoton remains entangled with Bob’s.Teleportation relies on Alice’s abilityto perform a joint polarization meas-urement that explicitly projects the two photons in the teleporter into oneof the four Bell states. Once Alicecompletes her measurement, Bob’sphoton (which is totally correlated toAlice’s) will assume the correspondingpure state. For example, if the Bellstate measurement produces the result|Ψ–⟩C,A, then Bob’s photon would beprojected into the pure state |ψ⟩ = (–β |H⟩ + α |V⟩)B. By using asimple optical element, Bob canrotate the polarization state of hisphoton by 90° and transform it intothe state |ψ′⟩ = (α |Η⟩ + β |V⟩)B, thatis, the original input state. ProvidedAlice can specify which Bell statewas measured (a specification thatrequires two bits of classical infor-mation), Bob can always choose anappropriate optical element to effectthe proper rotation.

In a series of groundbreaking exper-iments conducted at the University ofInnsbruck, Austria, Anton Zeilinger

and coworkers were the first to demon-strate quantum teleportation. The groupis now able to determine two of thefour Bell states unambiguously (theother two states give the same experi-mental signature2) and prove for thosecases that the state of Charlie’s photoncould indeed be transferred to Bob’s.

Several points should be madeabout quantum teleportation. First,during the entire procedure, neitherAlice nor Bob has any idea what thevalues are for the parameters α and βthat specify Charlie’s photon. The ini-tial, arbitrary pure state remainsunknown. Second, teleportation is not cloning. The original state ofCharlie’s photon is necessarilydestroyed by Alice’s measurement,so the photon that Bob ends up with is still one of a kind.

Finally, hopeful sci-fi fans may bea little disappointed by this realizationof teleportation. Unlike the TV show“Star Trek,” in which Captain Kirkcould be transported body and soulfrom the starship Enterprise to thesurface of an alien planet,3 here onlycertain information about the photonis transferred to a photon in some faraway location. Because photonshave numerous degrees of freedom inaddition to their polarization, the orig-inal and the teleported photons aretwo different entities. And it goeswithout saying that an even simplerway for Charlie to send his quantumstate to Bob would be to dispatch theoriginal photon directly to him.

Nevertheless, teleportation remainsan interesting application of quantumstate entanglement. Furthermore,researchers have discussed how itmight form the basis of a distributednetwork of quantum communicationchannels and how this basic informa-

tion protocol might be useful forquantum computing.

Quantum Microscopy andLithography. The general topic ofquantum metrology involves capitaliz-ing on the ultrastrong correlations ofentangled systems to make measure-ments more precisely than would bepossible with classical tools. The twomain photon-based applications underinvestigation are quantum microscopyand quantum lithography.

At present, two-photon microscopyis widely used to produce high-resolution images, often of biologicalsystems. However, the classical lightsources (lasers) used for the imaginghave random spreads in the temporaland spatial distributions of the pho-tons, and the light intensity must bevery high if two photons are to inter-sect within a small enough volume andcause a detectable excitation. The highintensity can damage the system underinvestigation. Because the temporaland spatial correlations may be muchstronger between members of anentangled photon pair, one could con-ceivably get away with much weakerlight sources, which would be muchless damaging to the systems beingobserved. Moreover, entangled-photonsources may also enable obtainingenhanced spatial resolution.

Lithography, in which a pattern isoptically imaged onto some photoresistmaterial, is the primary method of manufacturing microscale or nanoscaleelectronic devices. An inherent limita-tion of this process is that details smallerthan a wavelength of light cannot bewritten reliably. However, quantum state entanglement might circumventthis limitation. Under the right circum-stances, the interference pattern formedby beams of entangled photon pairs canhave half the classical fringe spacing.

Quantum lithography requires two beams of photons, which we will call A and B, but in this case,the type of entanglement is different



2 Distinguishing between the four Bell states is still an unsolved technical problem. Itrequires a strong nonlinear interaction between two photons, which is extremely difficultto achieve in practice. 3 “Teleportation” (though it was not explicitly called that) was supposedly introduced inthis TV show because the producer, Gene Roddenberry, wished to save the expense ofsimulating the landing of a starship on a planet.

from the one discussed in the previoussections. What is needed is a coherentsuperposition consisting of the state inwhich two photons are in beam Awhile none are in B and the state inwhich no photon is in beam A whiletwo photons are in B. Such number-entangled states can be made in the

laboratory, and the predictions aboutfringe spacings have been verified.However, other obstacles must be over-come in order to surpass current classi-cal-lithography techniques. Researcherscontinue to explore the potential of thisideawith the hope of achieving a viablecommercial technology.

Creating and MeasuringEntangled States

If quantum state entanglement is such a remarkable propertybecause it allows one to performsecret communications, teleportstates, or test the nonlocality of quantum mechanics, one naturallywonders how to make entangledstates. Currently, scientists can create entangled states of particles in a controlled manner by using severaltechnologies such as ion traps, cavityquantum electrodynamics (QED),and optical down-conversion. Here,we will concentrate on the opticalrealization.

Crystals of a certain chemicalstructure, such as beta-barium borate(BBO), have the property of opticalnonlinearity, which means that thepolarizability of these crystalsdepends on the square (or higherpowers) of an applied electric field.The practical upshot of this propertyis that, when passing through such acrystal, a single-parent photon cansplit (or down-convert) into a pair ofdaughter photons. The probabilitythat this event occurs is extremelysmall; on average, it happens to onlyone out of every 10 billion photons!

When down-conversion doesoccur, energy and momentum areconserved (as they must be for anisolated system). The daughter pho-tons have lower frequencies (longerwavelengths) than the parent photonand emerge from the crystal on oppo-site sides of a cone that is centeredabout the direction traveled by theparent. For what is known as Type Iphase matching, the daughtersemerge from a specifically orientedBBO crystal with identical polariza-tions that are aligned perpendicularto the parent polarization—seeFigure 5(a). Because each photon isin a definite state of polarization, thetwo photons are not in an entangledstate but are classically correlated.



(a)

Daughterphotons

Parentphoton

BBO Crystal

(b)

Entangleddaughterphotons

Parentphoton

Crossed BBO Crystals

Figure 5. Entangled-Photon Source(a) For a given orientation of the beta-barium borate (BBO) crystal, a horizontally

polarized parent photon produces a pair of vertically polarized daughters.

The daughters emerge on opposite sides of an imaginary cone. The cone’s axis is

parallel to the original direction taken by the parent photon. The two daughter

photons are not in an entangled state. Reorienting the BBO crystal by 90° will

produce a pair of horizontally polarized daughters if a vertically polarized pump

beam is used. (b) Passing a photon polarized at +45° through two crossed BBO

crystals can produce two photons in an entangled state. Because of the

Heisenberg uncertainty principle, there is no way to tell in which crystal the parent

photon “gave birth,” and so a coherent superposition of two possible outcomes

results: a pair of vertically polarized photons or a pair of horizontally polarized

photons. The photons are in the maximally entangled state |Φ+⟩ = 1/√2(|HH ⟩ + |VV ⟩).

(The crystal acts like the source S1described earlier.)

To create photons in the entangledstate, one can use two crystals thatare aligned with their axes of sym-metry oriented at 90° to each other,as shown in Figure 5(b). Withcrossed crystals, two competingprocesses are possible: The parentphoton can down-convert in the firstcrystal to yield two vertically polar-ized photons (|VV⟩), or it can down-convert in the second to yield twohorizontally polarized photons(|HH⟩). It is impossible to distinguishwhich of these processes hasoccurred. Thus, the state of the daughter photons is a coherent quantum-mechanical superposition ofthe states that would arise from eachcrystal alone; the crossed crystals

produce photons in the state |Ψout⟩ = 1/√2(|ΗΗ⟩ + |VV⟩), which ismaximally entangled.4

Figure 6 shows how this basicsource can be adapted to produce anypure quantum state of two photons byplacing rotatable half- and quarter-wave plates (which can be used totransform the polarization state of asingle photon) before the crystal and inthe paths of the two daughter photons.To create more general quantum

states—mixed states—a long birefrin-gent crystal can be used to delay onepolarization component with respect tothe other. If the relative delay is longerthan the coherence time of the photons,the horizontal and vertical componentshave been effectively decohered; thatis, the phase relationship between thedifferent states is destroyed.Researchers are still discovering howto combine sources and polarization-transforming elements to create allpossible two-photon quantum states.

Characterizing Entanglement:The Map of Hilbert Space

As discussed in the box on page 56,a mixed state of two photons (or ingeneral, a mixed state of two qubits)

PBS

HWPBeam stop

Prism

Ar+

laser


QWP

Decoherers

State Selector Tomographic Analyzer Detectors

Filter Lens

ApertureSingle-photondetector



Figure 6. Creating and Measuring Two-Photon Entangled States (a) The “parent” photons are created in an argon ion laser and are linearly polarizedwith a polarizing beam splitter (PBS). The half-wave plate (HWP) rotates the polar-ization state before the photon enters the entangled-photon source. The entangledphotons produced diverge as they exit. Each photon’s polarization state can bealtered at will by the subsequent HWP and quarter-wave plate (QWP). The decoher-ers following the state selection allow us to produce (partially) mixed photon states.The optical elements (QWP, HWP, and PBS) in the tomographic analyzer allow us tomeasure each photon in an arbitrary basis, for example in H/V or +45/–45.Combining the measurements on both photons allows us to determine the quantumstate. (b) In the photo, Paul Kwiat is shown with the two-photon entangled source at Los Alamos.

4 In an alternative approach known as“Type II phase matching,” only one crys-tal is needed to create the entangled state.The crystal has a different orientation,and each of the daughter photons emergesfrom the crystal on one of two possibleexit cones. Entangled photons created bythis approach were used in the firstdemonstration of quantum teleportation.

(a)

(b)

is represented by a 4 × 4 densitymatrix, which is described by 15 inde-pendent parameters (15 real numbers).To determine the independent parame-ters, we make 15 coincidence measurements on the ensemble ofphoton pairs emitted from the source.Each measurement is similar to theone used in the simple experimentdescribed at the start of this article.The measurement may be made withthe tomographic analyzer shown inFigure 6. Using such a system, wewere able to determine the densitymatrices of many types of states. An example is shown in Figure 7.

Whereas 15 numbers fully describea two-photon mixed state, the densitymatrix for N photons needs 4N – 1 real numbers. Thus, the density matrix of a 4-photon state contains

255 parameters and requires 255 sepa-rate measurements just to characterizethe state. Note that, if each parameteris allowed to assume one of, say,10 possible values, those 4 photons canbe in any of 10255 distinct quantumstates! This number of states is manyorders of magnitude greater than thetotal number of particles in our universe. The mathematical space in which the quantum states rest (the Hilbert space) is unfathomablylarge, and in order to have any hope of navigating it, one needs to introduce a simpler representation for quantumstates.

Two characteristics of centralimportance for quantum informationprocessing are the extent of entanglement and the degree of purity of an arbitrary state. A quantity

called the von Neumann entropy has been introduced to characterize the degree of puritya. (See the box“Characterizing Mixed States” on the next page.) However, for the analysis of two-photon states,we found it easier to use a relatedquantity, known as the linear entropy.When the linear entropy equals zero,the state is pure. When it reaches its maximum value of 1, the state iscompletely random.

Measuring the entanglement of amixed state is more complicated and,in general, is an unsolved researchproblem when more than two qubitsare involved. Any mixed quantum statecan be thought of as an incoherentcombination of pure states: The systemis in a number of possible pure states,each of which has some probability



(a) Real—Theoretical (b) Imaginary—Theoretical

(c) Real—Measured (d) Imaginary—Measured

0.4

0.2

0

0.4

0.2

0

0.4

0.2

0

0.4

0.2

0

HHHV

VHVV

VV

VH

HV

HH

VV

VH

HV

HH

HHHV

VHVV

VV

VH

HV

HH

HHHV

VHVV

HHHV

VHVV

VV

VH

HV

HH

Figure 7. Density MatricesTheoretical and experimental densitymatrices for the entangled state|Φ+⟩ = 1/√2 (|HH⟩ + |VV⟩) are illustratedhere. Both real and imaginary parts ofthe matrix are shown. The value of eachmatrix element is derived from theresults of thousands of two-photon correlation experiments (simulatedexperiments for the theoretical matrix.)The experimental matrix indicates thatour source can output a state close to amaximally entangled one. Written out“longhand,” the density matrix describ-ing the state |Φ+⟩ is

ρ = |Φ+⟩⟨Φ+|= 1/2( |HH⟩⟨HH| + |VV⟩⟨VV|

+ |HH⟩⟨VV| + |VV⟩⟨HH| ) .

The first two terms, which lie on thediagonal of the matrix (dashed line),give the probability of the result (forexample, 50% HH and 50% VV). Theother two terms describe the quantumcoherence between the states |HH⟩ and|VV⟩. For a classical mixed state (suchas the source S2 described in the text),these off-diagonal terms in the densitymatrix would equal zero. Notice that allcoefficients in this density matrix arereal, so that all terms in the imaginary part of the matrix should be zero.

between 0 and 1 associated with it(rather than the complex numbersdefining the probability amplitudesthat specify a particular superpositionof pure states). A reasonable measureof the entanglement of such a mixedstate is to take the average value ofthe entanglement (for example, asmeasured by the concurrence dis-cussed in the box on this page) for allthose pure states.

One must, however, use this proce-dure carefully because the decomposi-tion of the mixed state into an incoherent sum of pure states is notunique. For this “average entangle-ment” to make any sense as a measureof entanglement of the mixed state,one must use the decomposition forwhich the average is a minimum. The square of this minimized quantityis called the “tangle.” It has a value of zero for entirely unentangled,separable states and of unity for com-pletely entangled states.

Figure 8 shows how those twoparameters—tangle and linearentropy—can be used to create asimplified map of Hilbert space fortwo-photon states. The crosses (with error bars) are the states we



Characterizing Mixed States

It is convenient to characterize the extent of entanglement and the degreeof purity of a mixed state using two derived parameters: the tangle and thelinear entropy. The linear entropy, which gives a measure of the purity ofthe state, derives from the von Neumann entropy. The latter is given by theformula S = –Tr{ρlog2(ρ)}, where ρ is the density matrix. Here Tr{M} isthe trace of a matrix (that is, the sum of terms on the diagonal) and log2 isa logarithm base 2, which can be defined for matrices via a power series.The von Neumann entropy is zero for a pure state. When the von Neumannentropy has its maximum value (equal to the number of qubits), the state iscompletely random, with no information or entanglement being present.The linear entropy, defined for two qubits as SL = 4/3(1 – Tr{ρ2}), is simi-lar to the von Neumann entropy, but it is easier to calculate. Specifically, itequals 0 for a pure state and has a maximum value of 1 for completely random states.

Characterizing the degree of entanglement is more difficult. Mathematicallyspeaking, if one decomposes the density matrix into an incoherent sum of pure states, that is, ρ = Σi pi |ψi⟩ ⟨ψi|, where 0 ≤ pi ≤ 1 and ∑i pi = 1,then the average entanglement is E

–= ∑i piC(ψi) = 1, where C(ψi) is the

concurrence of the pure state |ψi⟩ (defined in the box on page 56). It is veryimportant to find the decomposition for which E

–takes its minimum possible

value; otherwise, one can infer a nonzero entanglement for states such as thecompletely mixed state, which is certainly not entangled! Fortunately, the wayto do that decomposition has been worked out for two qubits. Characterizingthe degree of entanglement for three or more qubits remains an unsolvedresearch problem.

1.0

0.8

0.6

0.4

0.2

0.20 0.4 0.5 0.8 1.0

Tang

le

Linear entropy

Classical statesalong T = 0 axis

Maximallymixed states

Forbidden region

Maximally entangled states(Bell states)

Figure 8. The Map of Hilbert Space The amount of entanglement (or thetangle) is plotted against the degree of purity (represented by the linearentropy) for a multitude of two-photonstates created and measured at Los Alamos. Each state is representedby a black spot with error bars.The boundary line, which representsthe class of states that have the maxi-mum possible entanglement for a givenvalue of the linear entropy, was firstdetermined theoretically but then confirmed by a numerical simulation of two million random density matrices.Important states, such as those that aremaximally entangled or completelymixed, are indicated. Efforts are underway to create states that lie along the boundary line.

have created and measured experi-mentally. Most display a high degreeof entanglement. States created byother technologies can be plotted onsuch a diagram as well.

Conclusions

Entangled states arise naturallywhenever two or more quantum systems interact. In fact, one of theprevalent theories of nature is that the universe is really one big, vastlycomplicated entangled state, describedby the “wave function of the uni-verse.” Despite their seeming ubiquity,however, entangled states are not gen-erally observed in the world at large.Only relatively recently have scientistsdeveloped the means to controllablyproduce, manipulate, and detect thismost bizarre quantum phenomenon.Initially, the fascination was limited toexperimental studies of the foundationsof quantum mechanics, especially thenotion of nonlocal “spooklike” influ-ences (to quote Einstein). However,even more recently, has come the real-ization that entanglement could lead toenhanced—sometimes vastlyenhanced—capabilities in the realm ofinformation processing.

This paper has discussed howentangled states could be a keyresource in applications as diverse as cryptography, lithography, andmetrology because they enable featsbeyond those possible with classicalphysics. In addition, the quest to create a quantum computer haspushed entangled systems to the fore-front of quantum research. Part of thepower of a quantum computer is thatit creates entangled states of N qubitsso that information can be stored andprocessed in the 2N-dimensional qubitspace. Quantum algorithms have been developed that would manipulate the complex entangled state and makeuse of the nonclassical correlations tosolve problems more efficiently than

could be done classically. Scientistswho work on developing quantumcomputers are envisioning systems ofthousands of entangled qubits.

We don’t know whether we will beable to create or maintain such a com-plex entangled state. At this point,we won’t even claim to know whether we will fully understand that state if it is created. More research is neededbefore those questions can beanswered. All that we can say now is that the once-hidden domain ofquantum entanglement has brokeninto our classical world. �

Further Reading

Bouwmeester, D., A. K. Ekert, and A. Zeilinger, eds. 2000. The Physics of Quantum Information. Berlin:Springer-Verlag.

Haroche, S. 1998. Entanglement, Decoherence and the Quantum/Classical Boundary. Phys. Today 51 (7): 36.

James, D. F. V., P. G. Kwiat, W. J. Munro,and A. G. White. 2001. Measurement of Qubits. Phys. Rev. A 64: 052312.

Mandel, L. and E. Wolf. 1995. Optical Coherence and Quantum Optics. Cambridge: Cambridge University Press.

Naik, D. S., C. G. Peterson, A. G. White,A. J. Berglund, and P. G. Kwiat. EntangledState Quantum Cryptography: Eavesdroppingon the Ekert Protocol. 2000. Phys. Rev. Lett.84: 4733.

Nielson, M. A., and I. L. Chuang. 2000. Quantum Computation and Quantum Information. Cambridge: Cambridge University Press.

Schrödinger, E. 1935. Discussion of Probability Relations Separated Systems. Proc. Cambridge Philos. Soc. 31: 555.

White, A. G., D. F. V. James, P. H. Eberhard,and P. G. Kwiat. 1999. Nonmaximally Entangled States: Production,Characterization, and Utilization.Phys. Rev. Lett. 83 (16): 3103.

Zeilinger, A. 2000. Quantum Teleportation. Sci. Am. 282 (4): 50.



Daniel F. V. James was born in rainyManchester, in northwest England, within sightof the Old Trafford (home of ManchesterUnited, the most suc-cessful sports team inrecorded human history). He was educated at NewCollege, Universityof Oxford, England,and at the Institute ofOptics, University ofRochester, NewYork, where heearned his Ph.D. inoptics in 1992. He came to Los AlamosNational Laboratory as a postdoctoral fellowin 1994. Three years later, he became a staffmember in the Atomic and Optical TheoryGroup at Los Alamos. Daniel specializes intheoretical optical physics, including coher-ence theory, diffraction, scattering, statisticaloptics, and quantum technologies.

Paul G. Kwiat comes from Ohio (not a placenoted for good sports teams). He was educatedat the Massachusetts Institute of Technologyand the University of California at Berkeley,

where he receivedhis Ph.D. inphysics. In 1995,after having completed a Lise Meitner post-doctoral fellowshipat the University of Innsbruck, inAustria, Paul cameto Los AlamosNational

Laboratory as an Oppenheimer Fellow andbecame a staff member in 1998. In January2001, he became the John Bardeen Professorof Physics and Electrical Engineering at theUniversity of Illinois at Urbana-Champaign.He specializes in experimental quantum optics,with an eye toward shedding light on quantuminformation protocols.

cover#27/for cd · toward bob. first,imagine that the photons 1 are always polarized in the...

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