Quantum Teleportation

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Seminar Report was presented by Anand Shekhar from SOE, CUSAT (2005-2009 batch).

Teleportation - the transmission and reconstruction of objects over
arbitrary distances - is a spectacular process, which actually has been
invented by science fiction authors some decades ago. Unbelievable as it
seems in 1993 a theoretical scheme has been found by Charles Bennett that
predicts the existence of teleportation in reality - at least for quantum systems

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Quantum Teleportation

  1. 1. QUANTUM TELEPORTATION A SEMINAR REPORT Submitted by ANAND SHEKHAR in partial fulfillment for award of the degree of BACHELOR OF TECHNOLOGY in COMPUTER SCIENCE & ENGINEERING SCHOOL OF ENGINEERINGCOCHIN UNIVERSITYUNIVERSITY OF SCIENCE & TECHNOLOGY,KOCHI-682022 AUGUST 2008
  2. 2. DIVISION OF COMPUTER ENGINEERING SCHOOL OF ENGINEERING COCHIN UNIVERSITY OF SCIENCE AND TECHNOLOGY KOCHI-682022 Certificate Certified that this is a bonafide record of the seminar entitled “QUANTUM TELEPORTATION” done by the following student ANAND SHEKHARof the VIIth semester, Computer Science and Engineering in the year 2008 inpartial fulfillment of the requirements to the award of Degree of Bachelor ofTechnology in Computer Science and Engineering of Cochin University ofScience and Technology.Mrs Sheikha Chenthara Dr. David Peter SSeminar Guide Head of Department
  3. 3. ACKNOWLEDGEMENT I thank my seminar guide Mrs. Sheikha Chenthara, Lecturer, CUSAT, for herproper guidance and valuable suggestions. I am greatly thankful to Mr. David Peter, theHOD, Division of Computer Engineering & other faculty members for giving me anopportunity to learn and do this seminar. If not for the above mentioned people, myseminar would never have been completed successfully. I once again extend my sincerethanks to all of them. Anand Shekhar i
  4. 4. ABSTRACT Teleportation - the transmission and reconstruction of objects overarbitrary distances - is a spectacular process, which actually has beeninvented by science fiction authors some decades ago. Unbelievable as itseems in 1993 a theoretical scheme has been found by Charles Bennett thatpredicts the existence of teleportation in reality - at least for quantumsystems. This scheme exploits some of the most essential and mostfascinating features of quantum theory, such as the existence of entangledquantum states. Only four years after its prediction, for the first timequantum teleportation has been experimentally realized by Anton Zeilinger ,who succeeded in teleporting the polarization state of photons. Apart fromthe fascination that arises from the possibility of teleporting particles,quantum teleportation is expected to play a crucial role in the construction ofquantum computers in future. Teleportation promises to be quite useful as an informationprocessing primitive, facilitating long range quantum communication andmaking it much easier to build a working quantum computer. ii
  5. 5. Table of contentsChapter Title Page No. No. Abstract ii List of figures iv 1 Introduction 1 2 History 6 3 How quantum teleportation works 8 3.1 Bell-state measurements 8 3.2 The teleporter 11 3.3 Working 12 3.4 Teleportation with squeezed light 14 3.5 Fidelity(quantum vs classic) 15 4 Concept 16 4.1 Description 16 4.2 Entanglement swapping 17 4.3 N-state particles 18 4.4 Result 19 4.5 Remarks 22 5 General teleportation scheme 23 5.1 General description 23 5.2 Further details 24 6 Applications 26 6.1 Quantum information 26 6.2 Quantum cryptography 27 7 References 30 iii
  6. 6. List of figuresSl. Images PageNo. No.1.1 Researchers 21.2 Quantum Teleportation 31.3 Conventional method of transmission 53.1.1 Photons just before colliding 93.1.2 Photons reflected and transmitted 93.1.3 Photons are either transmitted or reflected 93.2.1 Photon being Teleported 113.3.1 Flowchart showing Teleportation 123.3.2 River Danube Experiment 123.4.1 Teleportation Apparatus 14 iv
  7. 7. Quantum Teleportation 1. INTRODUCTION Teleportation - the transmission and reconstruction of objects over arbitrarydistances - is a spectacular process, which actually has been invented by sciencefiction authors some decades ago. Unbelievable as it seems in 1993 a theoreticalscheme has been found by Charles Bennett that predicts the existence of teleportationin reality - at least for quantum systems. This scheme exploits some of the mostessential and most fascinating features of quantum theory, such as the existence ofentangled quantum states. Only four years after its prediction, for the first timequantum teleportation has been experimentally realized by Anton Zeilinger, whosucceeded in teleporting the polarization state of photons. Apart from the fascinationthat arises from the possibility of teleporting particles, quantum teleportation isexpected to play a crucial role in the construction of quantum computers in future. Quantum teleportation, or entanglement-assisted teleportation, is atechnique used to transfer information on a quantum level, usually from one particle(or series of particles) to another particle (or series of particles) in another location viaquantum entanglement. It does not transport energy or matter, nor does it allowcommunication of information at superluminal (faster than light) speed, but is usefulfor quantum communication and computation. More precisely, quantum teleportation is a quantum protocol by which a qubita (the basic unit of quantum information) can be transmitted exactly (in principle)from one location to another. The prerequisites are a conventional communicationchannel capable of transmitting two classical bits (i.e. one of four states), and anentangled pair (b,c) of qubits, with b at the origin and c at the destination. (So whereasb and c are intimately related, a is entirely independent of them other than beinginitially colocated with b.) The protocol has three steps: measure a and b jointly toyield two classical bits; transmit the two bits to the other end of the channel (the onlypotentially time-consuming step, due to speed-of-light considerations); and use thetwo bits to select one of four ways of recovering c. The upshot of this protocol is topermute the original arrangement ((a,b),c) to ((b′,c′),a), that is, a moves to where cwas and the previously separated qubits of the Bell pair turn into a new Bell pair (b′,c′) at the origin. 1Division of Computer Engineering
  8. 8. Quantum Teleportation Teleportation is the name given by science fiction writers to the feat of makingan object or person disintegrate in one place while a perfect replica appearssomewhere else. How this is accomplished is usually not explained in detail, but thegeneral idea seems to be that the original object is scanned in such a way as to extractall the information from it, then this information is transmitted to the receivinglocation and used to construct the replica, not necessarily from the actual material ofthe original, but perhaps from atoms of the same kinds, arranged in exactly the samepattern as the original. A teleportation machine would be like a fax machine, exceptthat it would work on 3-dimensional objects as well as documents, it would producean exact copy rather than an approximate facsimile, and it would destroy the originalin the process of scanning it. A few science fiction writers consider teleporters thatpreserve the original, and the plot gets complicated when the original and teleportedversions of the same person meet; but the more common kind of teleporter destroysthe original, functioning as a super transportation device, not as a perfect replicator ofsouls and bodies.In 1993 an international group of sixscientists, including IBM FellowCharles H. Bennett, confirmed theintuitions of the majority of sciencefiction writers by showing that perfectteleportation is indeed possible inprinciple, but only if the original isdestroyed. In subsequent years, otherscientists have demonstratedteleportation experimentally in a variety Fig 1.1 Researchersof systems, including single photons, coherent light fields, nuclear spins, and trappedions. Teleportation promises to be quite useful as an information processingprimitive, facilitating long range quantum communication (perhaps unltimatelyleading to a "quantum internet"), and making it much easier to build a workingquantum computer. But science fiction fans will be disappointed to learn that no oneexpects to be able to teleport people or other macroscopic objects in the foreseeablefuture, for a variety of engineering reasons, even though it would not violate anyfundamental law to do so. 2Division of Computer Engineering
  9. 9. Quantum Teleportation In the past, the idea of teleportation was not taken very seriously by scientists,because it was thought to violate the uncertainty principle of quantum mechanics,which forbids any measuring or scanning process from extracting all the informationin an atom or other object. According to the uncertainty principle, the more accuratelyan object is scanned, the more it is disturbed by the scanning process, until onereaches a point where the objects original state has been completely disrupted, stillwithout having extracted enough information to make a perfect replica. This soundslike a solid argument against teleportation: if one cannot extract enough informationfrom an object to make a perfect copy, it would seem that a perfect copy cannot bemade. But the six scientists found a way to make an end run around this logic, using acelebrated and paradoxical feature of quantum mechanics known as the Einstein-Podolsky-Rosen effect. In brief, they found a way to scan out part of the informationfrom an object A, which one wishes to teleport, while causing the remaining,unscanned, part of the information to pass, via the Einstein- Podolsky-Rosen effect. Inbrief, they found a way to scan out part of the information from an object A, whichone wishes to teleport, while causing the remaining, unscanned, part of theinformation to pass, via the Einstein-Podolsky-Rosen effect, into another object Cwhich has never been in Contact with A. Fig 1.2 Quantum Teleportation 3Division of Computer Engineering
  10. 10. Quantum Teleportation Later, by applying to C a treatment depending on the scanned-out information,it is possible to maneuver C into exactly the same state as A was in before it wasscanned. A itself is no longer in that state, having been thoroughly disrupted by thescanning, so what has been achieved is teleportation, not replication. As the figure above suggests, the unscanned part of the information isconveyed from A to C by an intermediary object B, which interacts first with C andthen with A. What? Can it really be correct to say "first with C and then with A"?Surely, in order to convey something from A to C, the delivery vehicle must visit Abefore C, not the other way around. But there is a subtle, unscannable kind ofinformation that, unlike any material cargo, and even unlike ordinary information, canindeed be delivered in such a backward fashion. This subtle kind of information, alsocalled "Einstein-Podolsky-Rosen (EPR) correlation" or "entanglement", has been atleast partly understood since the 1930s when it was discussed in a famous paper byAlbert Einstein, Boris Podolsky, and Nathan Rosen. In the 1960s John Bell showedthat a pair of entangled particles, which were once in contact but later move too farapart to interact directly, can exhibit individually random behavior that is too stronglycorrelated to be explained by classical statistics. Experiments on photons and otherparticles have repeatedly confirmed these correlations, thereby providing strongevidence for the validity of quantum mechanics, which neatly explains them. Anotherwell-known fact about EPR correlations is that they cannot by themselves deliver ameaningful and controllable message. It was thought that their only usefulness was inproving the validity of quantum mechanics. But now it is known that, through thephenomenon of quantum teleportation, they can deliver exactly that part of theinformation in an object which is too delicate to be scanned out and delivered byconventional methods. 4Division of Computer Engineering
  11. 11. Quantum Teleportation Fig 1.3 Conventional Method of Transmission This figure compares conventional facsimile transmission with quantumteleportation (see above). In conventional facsimile transmission the original isscanned, extracting partial information about it, but remains more or less intact afterthe scanning process. The scanned information is sent to the receiving station, whereit is imprinted on some raw material (eg paper) to produce an approximate copy of theoriginal. By contrast, in quantum teleportation, two objects B and C are first broughtinto contact and then separated. Object B is taken to the sending station, while objectC is taken to the receiving station. At the sending station object B is scanned togetherwith the original object A which one wishes to teleport, yielding some informationand totally disrupting the state of A and B. The scanned information is sent to thereceiving station, where it is used to select one of several treatments to be applied toobject C, thereby putting C into an exact replica of the former state of A. 5Division of Computer Engineering
  12. 12. Quantum Teleportation 2. HISTORY Teleportation is a term created by science fiction authors describing a process,which lets a person or object disappear while an exact replica appears in the bestcase immediately at some distant location. The first idea how the dream ofteleportation could be realized in practice might be the following: From a classicalpoint of view the object to be teleported can fully be characterized by its properties,which can be determined by measurement. To create a copy of the object one does notneed the original parts and pieces, but all that is needed is to send the scannedinformation to the place of destination, where the object can be reconstructed. Havinga closer look at that scheme, we realize that the weak point is the measuring process.If we want to get a perfect replica of the object, it would be inevitable to determinethe states of molecules, atoms and electrons - in a word: we would have to measurequantum properties. But according to Heisenberg’s uncertainty principle, these cannotbe determined with arbitrary precision not even in principle. We see that teleportationis not practicable in this way. And even more: it seems as if the laws of quantummechanics prohibit any teleportation scheme in general. It is the more surprising that in 1993 CharlesH. Bennett et al. have suggestedthat it is possible to transfer the quantum state of a particle onto another provided onedoes not get any information about the state in the course of this transformation. Thecentral point of Bennett’s idea is the use of an essential feature of quantummechanics: entanglement . Entanglement describes correlations between quantumsystems much stronger than any classical correlation could be. With the help of a so-called pair of entangled particles it is possible to circumvent the limitations caused byHeisen-berg’s uncertainty principle. Quite soon after its theoretical prediction in 1997 Anton Zeilinger et al.succeeded in the first experimental verification of quantum teleportation. Byproducing pairs of entangled photons with the process of parametric down-conversionand using two-photon interferometry for analyzing entanglement, they were able totransfer a quantum property (the polarization state) from one photon to another. Though the prediction and experimental realization of quantum teleportationare surely a great success of modern physics, we should be aware of the differencesbetween the physical quantum teleportation and its science fiction counterpart. We 6Division of Computer Engineering
  13. 13. Quantum Teleportationwill see that quantum teleportation transfers the quantum state from one particle toanother, but doesn’t transfer mass. Furthermore the original state is destroyed in thecourse of teleportation, which means that no copy of the original state is produced.This is due to the no-cloning theorem, which says that it is impossible within quantumtheory to produce a clone of a given quantum system . Finally we will learn thatteleporting a quantum state has a natural speed limit. In the best case it is possible toteleport at the speed of light - in accordance with Einstein’s theory of relativity. The two parties are Alice (A) and Bob (B), and a qubit is, in general, asuperposition of quantum state labeled and . Equivalently, a qubit is a unitvector in two-dimensional Hilbert space.Suppose Alice has a qubit in some arbitrary quantum state . Assume that thisquantum state is not known to Alice and she would like to send this state to Bob.Ostensibly, Alice has the following options:1. She can attempt to physically transport the qubit to Bob.2. She can broadcast this (quantum) information, and Bob can obtain the informationvia some suitable receiver.3. She can perhaps measure the unknown qubit in her possession. The results of thismeasurement would be communicated to Bob, who then prepares a qubit in hispossession accordingly, to obtain the desired state. (This hypothetical process is calledclassical teleportation.) Option 1 is highly undesirable because quantum states are fragile and anyperturbation en route would corrupt the state. The unavailability of option 2 is the statement of the no-broadcast theorem. Similarly, it has also been shown formally that classical teleportation, aka.option 3, is impossible; this is called the no teleportation theorem. This is another wayto say that quantum information cannot be measured reliably. Thus, Alice seems to face an impossible problem. A solution was discoveredby Bennet et al. The parts of a maximally entangled two-qubit state are distributed toAlice and Bob. The protocol then involves Alice and Bob interacting locally with thequbit(s) in their possession and Alice sending two classical bits to Bob. In the end, thequbit in Bobs possession will be in the desired state. 7Division of Computer Engineering
  14. 14. Quantum Teleportation 3. HOW QUANTUM TELEPORTATION WORKS3.1 BELL-STATE MEASUREMENTS In previous discussions we almost always talked about the spin state ofelectrons, although we regularly pointed out that the same situations exist for thepolarization of light, albeit with a difference of a factor of 2 in the angles being used.Here we will reverse the situation, and mostly talk about polarization states forphotons, although the arguments also apply to spin states of electrons.The fact that we may talk about light polarization in almost the same way that wediscuss electron spin is not a coincidence. It turns out that photons have spins whichcan exist in only two different states. And those different spins states are related to thepolarization of the light when we think of it as a wave. Here we shall prepare pairs of entangled photons with opposite polarizations;we shall call them E1 and E2. The entanglement means that if we measure a beam of,say, E1 photons with a polarizer, one-half of the incident photons will pass the filter,regardless of the orientation of the polarizer. Whether a particular photon will pass thefilter is random. However, if we measure its companion E2 photon with a polarizeroriented at 90 degrees relative to the first, then if E1 passes its filter E2 will also passits filter. Similarly if E1 does not pass its filter its companion E2 will not.Earlier we discussed the Michelson-Morley experiment, and later the Mach-Zehnderinterferometer. You will recall that for both of these we had half-silvered mirrors,which reflect one-half of the light incident on them and transmit the other half withoutreflection. These mirrors are sometimes called beam splitters because they split a lightbeam into two equal parts. We shall use a half-silvered mirror to perform Bell State Measurements. Thename is after the originator of Bells Theorem. We direct one of the entangled photons, say E1, to the beam splitter. Meanwhile, we prepare another photon with a polarization of 450, and direct it to the same beam splitter from the other side, as shown. This is the 8Division of Computer Engineering
  15. 15. Quantum Teleportationphoton whose properties will be transported; we label it K (for Kirk). We time it sothat both E1 and K reach the beam splitter at the same time.Fig 3.1.1 Photons just before colliding The E1 photon incident from above will be reflected by the beam splitter some of the time and will be transmitted some of the time. Similarly for the K photon that is incident from below. So sometimes both photons will end up going up and to the right as shown above.Fig 3.1.2 Photons reflected and transmittedSimilarly, sometimes both photons will end up going down and to the right.But sometimes one photon will end up goingupwards and the other will be going downwards, asshown. This will occur when either both photonshave been reflected or both photons have beentransmitted.Thus there are three possible arrangements for thephotons from the beam splitter: both upwards, bothdownwards, or one upwards and one downwards. Fig 3.1.3 Photons are either transmitted or reflected Which of these three possibilities has occurred can be determined if we putdetectors in the paths of the photons after they have left the beam splitter.However, in the case of one photon going upwards and the other going downwards,we can not tell which is which. Perhaps both photons were reflected by the beamsplitter, but perhaps both were transmitted. This means that the two photons have become entangled. 9Division of Computer Engineering
  16. 16. Quantum Teleportation If we have a large beam of identically prepared photon pairs incident on thebeam splitter, the case of one photon ending up going upwards and the otherdownwards occurs, perhaps surprisingly, 25% of the time. Also somewhat surprisingly, for a single pair of photons incident on the beamsplitter, the photon E1 has now collapsed into a state where its polarization is -450, theopposite polarization of the prepared 450 one. This is because the photons havebecome entangled. So although we dont know which photon is which, we know thepolarizations of both of them. The explanation of these two somewhat surprising results is beyond the levelof this discussion, but can be explained by the phase shifts the light experiences whenreflected, the mixture of polarization states of E1, and the consequent interferencebetween the two photons.3.2 THE TELEPORTERNow we shall think about the E2 companion to E1.25 percent of the time, the Bell-state measurementresulted in the circumstance shown, and in thesecases we have collapsed the state of the E1 photoninto a state where its polarization is -450. 10Division of Computer Engineering
  17. 17. Quantum TeleportationBut since the two photon system E1 and E2 wasprepared with opposite polarizations, this meansthat the companion to E1, E2, now has apolarization of +450. Thus the state of the K photonhas now been transferred to the E2 photon. Wehave teleported the information about the K photonto E2.Although this collapse of E2 into a 450 polarizationstate occurs instantaneously, we havent achieved Fig 3.2.1 Photon being Teleportedteleportation until we communicate that the Bell-state measurement has yielded theresult shown. Thus the teleportation does not occur instantaneously. Note that the teleportation has destroyed the state of the original K photon. Quantum entanglements such as exist between E1 and E2 in principle areindependent of how far apart the two photons become. This has been experimentallyverified for distances as large as 10km. Thus, the Quantum Teleportation is similarlyindependent of the distance. The Original State of the Teleported Photon Must Be Destroyed Above we saw that the K photons state was destroyed when the E2 photonacquired it. Consider for a moment that this was not the case, so we end up with twophotons with identical polarization states. Then we could measure the polarization ofone of the photons at, say, 450 and the other photon at 22.50. Then we would know thepolarization state of both photons for both of those angles. As we saw in our discussion of Bells Theorem, the Heisenberg UncertaintyPrinciple says that this is impossible: we can never know the polarization of a photonfor these two angles. Thus any teleporter must destroy the state of the object beingteleported.3.3 WORKINGBefore going further, here is how quantum teleportation works. 11Division of Computer Engineering
  18. 18. Quantum Teleportation First, an entangled state of ions A and B is generated, then the state to be teleported -- a coherent superposition of internal states -- is created in a third ion, P. Fig 3.3.1 Flowchart showing TeleportationThe third step is a joint measurement of P and A, with the result sent to the location ofion B, where it is used to transform the state .Now, lets look at the BBC News article. Long distance teleportation is crucial if dreams of superfast quantumcomputing are to be realised. When physicists say "teleportation", they are describingthe transfer of key properties from one particle to another without a physical link. Researchers from the University of Vienna and the Austrian Academy ofScience used an 800m-long optical fibre fed through a public sewer system tunnel toconnect labs on opposite sides of the River Danube. The link establishes a channel between the labs, dubbed Alice and Bob. Thisenables the properties, or "quantum states", of light particles to be transferred betweenthe sender (Alice) and the receiver (Bob). This illustration shows how the experiment was conducted. Fig 3.3.2 River Danube Experiment In "Teleportation Takes Quantum Leap," National Geographic explainswhy this experiment is a worlds premiere. 12Division of Computer Engineering
  19. 19. Quantum Teleportation "We were able to perform a quantum teleportation experiment for the firsttime ever outside a university laboratory," said Rupert Ursin, a researcher at theInstitute for Experimental Physics at the University of Vienna in Austria. The science is not new, said Mark Kuzyk, a physics professor at WashingtonState University in Pullman. But this is the first time "researchers have demonstratedthat teleportation works in the kinds of real-life conditions that are found in telecomapplications." Efficient long-distance quantum teleportation is crucial for quantumcommunication and quantum networking schemes. Here we describe the high-fidelityteleportation of photons over a distance of 600 metres across the River Danube inVienna, with the optimal efficiency that can be achieved using linear optics. Ourresult is a step towards the implementation of a quantum repeater, which will enablepure entanglement to be shared between distant parties in a public environment andeventually on a worldwide scale.3.4 TELEPORTATION WITH SQUEEZED LIGHT 13Division of Computer Engineering
  20. 20. Quantum Teleportation We have implemented quantum teleportation with light beams serving as boththe entangled pair and the input (and output) state. Squeezed light is used to generatethe entangled (EPR) beams which are sent to Alice and Bob. A third beam, the input,is a coherent state of unknown complex amplitude. This state is teleported to Bobwith a high fidelity only achievable via the use of quantum entanglement. TeleportationApparatusEntangled EPR beamsare generated bycombining two beamsof squeezed light at a50/50 beamsplitter.EPR beam 1 propagatesto Alices sendingstation, Fig3.4.1 TeleportationApparatus where it is combined at a 50/50 beamsplitter with the unknown input state, in thiscase a coherent state of unknown complex amplitude. Alice uses two sets of balancedhomodyne detectors to make a Bell-state measurement on the amplitudes of thecombined state. Because of the entanglement between the EPR beams, Alicesdetection collapses Bobs field (EPR beam 2) into a state conditioned on Alicesmeasurement outcome. After receiving the classical result from Alice, Bob is able toconstruct the teleported state via a simple phase-space displacement of the EPR field2.3.5 FIDELITY(QUANTUM VS CLASSIC) 14Division of Computer Engineering
  21. 21. Quantum Teleportation Quantum teleportation is theoretically perfect, yielding an output state whichequals the input with a fidelity F=1. In practice, fidelities less than one are realizeddue to imperfections in the EPR pair, Alices Bell measurement, and Bobs unitarytransformation. By contrast, a sender and receiver who share only a classicalcommunication channel cannot hope to transfer an arbitrary quantum state with afidelity of one. For coherent states, the classical teleportation limit is F=0.5, while forlight polarization states it is F=0.67. The quantum nature of the teleportation achievedin this case is demonstrated by the experimentally determined fidelity of F=0.58,greater than the classical limit of 0.5 for coherent states. Note that the fidelity is anaverage over all input states and so measures the ability to transfer an arbitrary,unknown superposition from Alice to Bob. 4. CONCEPT 15Division of Computer Engineering
  22. 22. Quantum Teleportation Assume that Alice and Bob share an entangled qubit AB. That is, Alice hasone half, A, and Bob has the other half, B. Let C denote the qubit Alice wishes totransmit to Bob. Alice applies a unitary operation on the qubits AC and measures the result toobtain two classical bits. In this process, the two qubits are destroyed. Bobs qubit, B,now contains information about C; however, the information is somewhatrandomized. More specifically, Bobs qubit B is in one of four states uniformly chosenat random and Bob cannot obtain any information about C from his qubit.Alice provides her two measured qubits, which indicate which of the four states Bobpossesses. Bob applies a unitary transformation which depends on the qubits heobtains from Alice, transforming his qubit into an identical copy of the qubit C.4.1 DESCRIPTION In the literature, one might find alternative, but completely equivalent,descriptions of the teleportation protocol given above. Namely, the unitarytransformation that is the change of basis (from the standard product basis into theBell basis) can also be implemented by quantum gates. Direct calculation shows thatthis gate is given bywhere H is the one qubit Walsh-Hadamard gate and CN is the Controlled NOT gate.4.2 ENTANGLEMENT SWAPPING 16Division of Computer Engineering
  23. 23. Quantum Teleportation Entanglement can be applied not just to pure states, but also mixed states, oreven the undefined state of an entangled particle. The so-called entanglementswapping is a simple and illustrative example. If Alice has a particle which is entangled with a particle owned by Bob, andBob teleports it to Carol, then afterwards, Alices particle is entangled with Carols.A more symmetric way to describe the situation is the following: Alice has oneparticle, Bob two, and Carol one. Alices particle and Bobs first particle areentangled, and so are Bobs second and Carols particle: / Alice-:-:-:-:-:-Bob1 -:- Bob2-:-:-:-:-:-Carol ___/ Now, if Bob performs a projective measurement on his two particles in theBell state basis and communicates the results to Carol, as per the teleportation schemedescribed above, the state of Bobs first particle can be teleported to Carols. AlthoughAlice and Carol never interacted with each other, their particles are now entangled.4.3 N-STATE PARTICLES 17Division of Computer Engineering
  24. 24. Quantum Teleportation One can imagine how the teleportation scheme given above might be extendedto N-state particles, i.e. particles whose states lie in the N dimensional Hilbert space.The combined system of the three particles now has a N3 dimensional state space. Toteleport, Alice makes a partial measurement on the two particles in her possession insome entangled basis on the N2 dimensional subsystem. This measurement has N2equally probable outcomes, which are then communicated to Bob classically. Bobrecovers the desired state by sending his particle through an appropriate unitary gate.4.4 RESULT 18Division of Computer Engineering
  25. 25. Quantum Teleportation Suppose Alice has a qubit that she wants to teleport to Bob. This qubit can bewritten generally as:Our quantum teleportation scheme requires Alice and Bob to share a maximallyentangled state beforehand, for instance one of the four Bell states , , , . Alice takes one of the particles in the pair, and Bob keeps the other one. Thesubscripts A and B in the entangled state refer to Alices or Bobs particle. We willassume that Alice and Bob share the entangled state . So, Alice has two particles (C, the one she wants to teleport, and A, one of theentangled pair), and Bob has one particle, B. In the total system, the state of thesethree particles is given by Alice will then make a partial measurement in the Bell basis on the two qubitsin her possession. To make the result of her measurement clear, we will rewrite thetwo qubits of Alice in the Bell basis via the following general identities (these can beeasily verified): 19Division of Computer Engineering
  26. 26. Quantum Teleportation The three particle state shown above thus becomes the following four-termsuperposition: Notice all we have done so far is a change of basis on Alices part of thesystem. No operation has been performed and the three particles are still in the samestate. The actual teleportation starts when Alice measures her two qubits in the Bellbasis. Given the above expression, evidently the results of her (local) measurement isthat the three-particle state would collapse to one of the following four states (withequal probability of obtaining each):•••• Alices two particles are now entangled to each other, in one of the four Bellstates. The entanglement originally shared between Alices and Bobs is now broken.Bobs particle takes on one of the four superposition states shown above. Note howBobs qubit is now in a state that resembles the state to be teleported. The fourpossible states for Bobs qubit are unitary images of the state to be teleported.The crucial step, the local measurement done by Alice on the Bell basis, is done. It isclear how to proceed further. Alice now has complete knowledge of the state of thethree particles; the result of her Bell measurement tells her which of the four states thesystem is in. She simply has to send her results to Bob through a classical channel.Two classical bits can communicate which of the four results she obtained. After Bob receives the message from Alice, he will know which of the fourstates his particle is in. Using this information, he performs a unitary operation on hisparticle to transform it to the desired state :• If Alice indicates her result is , Bob knows his qubit is already in the desired state and does nothing. This amounts to the trivial unitary operation, the identity operator. 20Division of Computer Engineering
  27. 27. Quantum Teleportation• If the message indicates , Bob would send his qubit through the unitary gate given by the Pauli matrixto recover the state.• If Alices message corresponds to , Bob applies the gateto his qubit.• Finally, for the remaining case, the appropriate gate is given byTeleportation is therefore achieved. Experimentally, the projective measurement done by Alice may be achievedvia a series of laser pulses directed at the two particles. 21Division of Computer Engineering
  28. 28. Quantum Teleportation4.5 REMARKS After this operation, Bobs qubit will take on the state, and Alices qubitbecomes (undefined) part of an entangled state. Teleportation does not result in thecopying of qubits, and hence is consistent with the no cloning theorem. There is no transfer of matter or energy involved. Alices particle has not beenphysically moved to Bob; only its state has been transferred. The term "teleportation",coined by Bennett, Brassard, Crépeau, Jozsa, Peres and Wootters., reflects theindistinguishability of quantum mechanical particles. The teleportation scheme combines the resources of two separately impossibleprocedures. If we remove the shared entangled state from Alice and Bob, the schemebecomes classical teleportation, which is impossible as mentioned before. On theother hand, if the classical channel is removed, then it becomes an attempt to achievesuperluminal communication, again impossible. For every qubit teleported, Alice needs to send Bob two classical bits ofinformation. These two classical bits do not carry complete information about thequbit being teleported. If an eavesdropper intercepts the two bits, she may knowexactly what Bob needs to do in order to recover the desired state. However, thisinformation is useless if she cannot interact with the entangled particle in Bobspossession. 22Division of Computer Engineering
  29. 29. Quantum Teleportation5. GENERAL TELEPORTATION SCHEME5.1 GENERAL DESCRIPTION A general teleportation scheme can be described as follows. Three quantumsystems are involved. System 1 is the (unknown) state ρ to be teleported by Alice.Systems 2 and 3 are in a maximally entangled state ω that are distributed to Alice andBob, respectively. The total system is then in the stateA successful teleportation process is a LOCC quantum channel Φ that satisfieswhere Tr12 is the partial trace operation with respect systems 1 and 2, and denotes thecomposition of maps. This describes the channel in the Schrodinger picture. Taking adjoint maps in the Heisenberg picture, the success condition becomesfor all observable O on Bobs system. The tensor factor in is whilethat of is . 23Division of Computer Engineering
  30. 30. Quantum Teleportation5.2 FURTHER DETAILS The proposed channel Φ can be described more explicitly. To beginteleportation, Alice performs a local measurement on the two subsystems (1 and 2) inher possession. Assume the local measurement have effectsIf the measurement registers the i-th outcome, the overall state collapses to The tensor factor in is while that of is .Bob then applies a corresponding local operation Ψi on system 3. On the combinedsystem, this is described bywhere Id is the identity map on the composite system .Therefore the channel Φ is defined byNotice Φ satisfies the definition of LOCC. As stated above, the teleportation is said tobe successful if, for all observable O on Bobs system, the equalityholds. The left hand side of the equation is: 24Division of Computer Engineering
  31. 31. Quantum Teleportationwhere Ψi* is the adjoint of Ψi in the Heisenberg picture. Assuming all objects arefinite dimensional, this becomesThe success criterion for teleportation has the expression 25Division of Computer Engineering
  32. 32. Quantum Teleportation 6. APPLICATIONS Teleporting the polarization state of a single photon a quarter of the time is along long way from reliably teleporting Captain Kirk. However, there are otherapplications of the above sort of apparatus that may be closer to being useful.6.1 QUANTUM INFORMATION As you probably know, computers store information as sequences of 0s and1s. For example, in the ASCII encoding the letter A is represented by the number 65.As a binary number this is: 1,000,001 Inside the computer, there are transistors that are either on or off, and weassign the on-state be 1 and the off state 0. However, the same information can bestored in exactly the same way in any system that has two mutually exclusive binarystates. For example, if we have a collection photons we could represent the 1s asphotons whose polarization is +450 and the 0s as polarizations of -450. We couldsimilarly use electrons with spin-up and spin-down states to encode the information.These quantum bits of information are called qubits. Above we were thinking about an apparatus to do Quantum Teleportation.Now we see that we can think of the same apparatus as transferring QuantumInformation. Note that, as opposed to, say, a fax, when transferring QuantumInformation the original, the polarization of the K photon, is destroyed. 26Division of Computer Engineering
  33. 33. Quantum Teleportation6.2 Quantum Cryptography Cryptography depends on both the sender and receiver of the encryptedinformation both knowing a key. The sender uses the key to encrypt the informationand the receiver uses the same key to decrypt it. The key can be something very simple, such as both parties knowing that eachletter has been shifted up by 13 places, with letters above the thirteenth in the alphabetrotated to the beginning. Or they can be very complex, such as a very very long stringof binary digits. Here is an example of using binary numbers to encrypt and decrypt a message,in this case the letter A, which we have seen is 1,000,001 in a binary ASCII encoding.We shall use as the key the number 23, which in binary is 0,010,111. We will use thekey to encode the letter using a rule that if the corresponding bits of the letter and keyare the same, the result is a 1, and otherwise a 0. A 1000001 Key 0010111 Encrypted 0 1 0 1 0 0 1The encrypted value is 41, which in ASCII is the right parenthesis: )To decrypt the message we use the key and the same procedure: Encrypted 0 1 0 1 0 0 1 Key 0010111 A 1000001Any classical encryption scheme is vulnerable on two counts: • If the "bad guys" get hold of the key they too can decrypt the message. So- called public key encryptation schemes reveals on an open channel a long string of binary digits which must be converted to the key by means of a secret procedure; here security is based on the computational complexity of "cracking" the secret procedure. • Because there are patterns in all messages, such as the fact that the letter e predominates, then if multiple messages are intercepted using the same key the bad guys can begin to decipher them. To be really secure, then, there must be a unique secret key for each message. Sothe question becomes how can we generate a unique key and be sure that the bad guysdont know what it is. 27Division of Computer Engineering
  34. 34. Quantum Teleportation To send a key in Quantum Cryptography, simply send photons in one of fourpolarizations: -45, 0, 45, or 90 degrees. As you know, the receiver can measure, say,whether or not a photon is polarized at 90 degrees and if it is not then be sure than itwas polarized at 0 degrees. Similarly the receiver can measure whether a photon waspolarized at 45 degrees, and if it is not then it is surely polarized at -45 degrees.However the receiver can not measure both the 0 degree state and 45 degree state,since the first measurement destroys the information of the second one, regardless ofwhich one is performed first. The receiver measures the incoming photons, randomly choosing whether tomeasure at 90 degrees or 45 degrees, and records the results but keeps them secret.The receiver contacts the sender and tells her on an open channel which type ofmeasurement was done for each, without revealing the result. The sender tells thereceiver which of the measurements were of the correct type. Both the sender andreceiver keep only the qubits that were measured correctly, and they have now formedthe key. If the bad guys intercept the transmission of photons, measure their polarizations,and then send them on to the receiver, they will inevitably introduce errors becausethey dont know which polarization measurement to perform. The two legitimate usersof the quantum channel test for eavesdropping by revealing a random subset of thekey bits and checking the error rate on an open channel. Although they cannot preventeavesdropping, they will never be fooled by an eavesdropper because any, howeversubtle and sophisticated, effort to tap the channel will be detected. Whenever they arenot happy with the security of the channel they can try to set up the key distributionagain. By February 2000 a working Quantum Cryptography system using the abovescheme achieved the admittedly modest rates of 10 bits per second over a 30 cmlength. There is another method of Quantum Cryptography which uses entangled photons.A sequence of correlated particle pairs is generated, with one member of each pairbeing detected by each party (for example, a pair of photons whose polarisations aremeasured by the parties). An eavesdropper on this communication would have todetect a particle to read the signal, and retransmit it in order for his presence to remainunknown. However, the act of detection of one particle of a pair destroys its quantumcorrelation with the other, and the two parties can easily verify whether this has been 28Division of Computer Engineering
  35. 35. Quantum Teleportationdone, without revealing the results of their own measurements, by communicationover an open channel 7. REFERENCES 29Division of Computer Engineering
  36. 36. Quantum Teleportation • http://www.primidi.com/2004/08/24.html • http://www.sciam.com/article.cfm?id=why-teleporting-is-nothing-like- star-trek • http://www.upscale.utoronto.ca/GeneralInterest/Harrison/QuantTeleport/ QuantTeleport.html • http://www.its.caltech.edu/~qoptics/teleport.html • http://www.research.ibm.com/quantuminfo/teleportation/ • http://www.iop.org/EJ/article/1367630/9/7/211/njp7_7_211.html#nj248372 s4 • http://heart-c704.uibk.ac.at/publications/papers/nature04_riebe.pdf • http://quantum.at/research/photonentangle/teleport/index.html • http://www.quantum.physik.uni- mainz.de/lectures/2004/ss04_quantenoptikseminar/quantumteleportation.pdf • http://en.wikipedia.org/wiki/Quantum_teleportation 30Division of Computer Engineering

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