arXiv:quant-ph/9605035v1 23 May 1996 aaaHC37 mi:[email protected] email: 3J7. H3C Canada ml eino pc.Qieteopst,teinterest the opposite, the Quite space. a of within region speed—and primary small its computational quickly—indeed increased very is place takes purpose it if even the ful of complexity the with only concerned circuitry. required are comput- we quantum if than ing implement to is exclusive- teleportation quantum easier quantum two significantly Thus, as few gates. bit) (controlled-not) as (quantum or with qubit arbitrary teleported an be computa- of can state quantum simpler the nontrivial much task: any is tional by that required circuit that quantum than a with plemented teleportation. quantum of that with of clea feasibility computing less technological quantum is the comparing situation to The comes 10]. it 13, when [12, kilometres of distances tens over of information confidential reli transmitting of to ably capable prototypes respect experimental several with by demon- correct strated been is has feasibility ranking whose teleport- cryptography, This quantum the to [11]! is comes goulash it teleportation ing when quantum cryp- outrageous—especially quantum while most that easiest think is to tempting tography is it difficulty, cal introduced. overtone first were strong they a when had fiction trans- concepts science these be of of to Each of information impossibility [3]. the mitted the broadcasting despite infor- or location quantum measuring distant of a transmission to tele- the mation Quantum for [6]. allows to cryptography portation im- computers classical dramatic quantum for with [14], plications of efficiently ability very par- numbers the of factorize 7]; is [9, place hardware interest take of ticular piece to single computation allows a of in computing simultaneously amount Quantum exponential 4]. an 1, for technolog- [2, or sophistication power computing ical her eaves- of an of regardless nose dropper, the under trans- information confidential classical cryptogra- of the mission for quantum [5]. allows information teleportation of cryptography quantum Quantum and fond and computing computation particularly quantum phy, of am realm I quantum of theory, the applications in new exciting physics many the Among Introduction 1 1996 May 12 Draft, abstract Extended PhysComp96 nteohrhn,qatmcmuigi meaning- is computing quantum hand, other the On im- be can teleportation quantum hand, one the On technologi- of scale a on ideas these rank to asked If ∗ † upre npr by part in Supported ´ preetIO ..62,scusl etevle Mon centre–ville, succursale 6128, C.P. D´epartement IRO, Nserc and Fcar tr´eal, r - 1 rw nprso olcleet neett quantum described to circuit inherent teleportation in effects quantum The nonlocal on mechanics. parts in draws measurements these make should damaging. more which even circuit, these carried by the qubits affected other by qubits the the with entangled that This are fact measurements place. that despite takes true measurement is is not result the or final with whether same interaction obtained the of effects: ill types environment—without some by which disrupted measurement—or at completely circuit un- a be the can the in information with points quantum are circuit the there quantum that a feature interesting obtain usual As we computation. product, this quantum side of of very terms teleportation purpose in quantum achieve The simply to how show mysterious. to is very makesnote seem which measurements, process nonlocal Bell the Einstein–Podolsky–Rosen and ex- [8] usually of states is quantum terms This in entanglement. plained quantum prior share Bob. and Alice as to convention- transmission referred parties, faithful ally two the between information allow quantum to of circumvent as to so is impossibility [3] that this teleportation also quantum quantum Recall of transmit purpose channel. the to classical impossible a infor- through it incomplete information makes yields This and irreversibly mation. informa- it quantum measuring disturbs at tion attempt any that Recall teleportation Quantum 2 teleporta- quantum entangled! that and are computing see tion quantum we of Thus fates in- the computers. quantum quantum transporting inside in role formation a informa- play quantum short-distance will quantum that teleportation be of may it storage Nevertheless, long-term allows tion. that efficient technology a the await integer to for have significant small will across space very and teleportation time a quantum before but seen even achieved, be of is to factorization demonstra- likely quantum working is the teleportation a quantum Thus, of after tion immediately preparation. place required take the the to if had reduced greatly teleportation be actual would teleportation quantum of n esr olclsae.Nvrhls tsesnew sheds it create Nevertheless to computation states. quantum nonlocal uses measure it and since [3] idea nal fcus,teucnypwro unu computation quantum of power uncanny the course, Of must Bob and Alice teleportation, achieve to order In § sntral ieeti rnil rmteorigi- the from principle in different really not is 4 unu computation quantum e eotto sa as Teleportation ilsBrassard, Gilles nvri´ eMontr´ealUniversit´e de frsc † ∗ α light on teleportation, at least from a pedagogical point if α 0 + β 1 is represented by vector (β ). Similarly the of view, since it makes the process completely straight- quantum| i exclusive-or| i operation is given by matrix forward to anyone who believes that quantum computa- tion is a reasonable proposition. Moreover, this circuit 1 00 0  0 10 0  could genuinely be used for teleportation purposes inside XOR = 0 00 1 a quantum computer. Finally, the surprising resilience of    0 01 0  this circuit to measurements performed while it is pro-   cessing information may turn out to have relevance to if α 00 + β 01 + γ 10 + δ 11 is represented by the quantum error correction. transpose| i of| vectori (|α,β,γ,δi |). i

3 The basic ingredients 4 The teleportation circuit As is often the case with quantum computation, we shall Consider the following quantum circuit. Please disregard need two basic ingredients: the exclusive-or gate (also the dashed line for the moment. known as controlled-not), which acts on two qubits at once, and arbitrary unitary operations on single qubits. a t R S i S i x Let 0 and 1 denote basis states for single qubits and recall| i that pure| i states are given by linear combination of basis states such as ψ = α 0 + β 1 where α and β are b L ti t y complex numbers such| i that | αi 2 + |βi2 = 1. The quantum exclusive-or| (XOR),| | | denoted as follows, c i it T t z a t x Alice Bob b i y Let ψ be an arbitrary one-qubit state. Consider what sends 00 to 00 , 01 to 01 , 10 to 11 and 11 to 10 . | i In other| words,i | providedi | i the| i input| i states| i at a and| ib are| ini happens if you feed ψ00 in this circuit, that is if you set upper input a| to iψ and both other inputs b basis states, the output state at x is the same as the input | i state at a, and the output state at y is the exclusive-or and c to 0 . It is a straightforward exercise to verify that state| ψi will be transferred to the lower output z, of the two input states at a and b. This is also known | i as the controlled-not gate because the state carried by whereas both other outputs x and y will come out in the control wire “ax” is not disturbed whereas the state state φ = ( 0 + 1 )/√2. In other words the output will be φφψ| i . If| thei two| i upper outputs are measured in the carried by the controlled wire “by” is flipped if and only | i if the state on the control wire was 1 . Note that the standard basis ( 0 versus 1 ), two random classical bits | i will be obtained| ini addition| i to quantum state ψ on the classical interpretation given above no longer holds if the | i input qubits are not in basis states: it is possible for the lower output. output state on the control wire (at x) to be different from Now, let us consider the state of the system at the its input state (at a). Moreover, the joint state of the dashed line. A simple calculation shows that all three output qubits can be entangled even if the input qubits qubits are entangled. We should therefore be especially were not, and vice versa. careful not to disturb the system at that point. Never- In addition to the quantum exclusive-or, we shall need theless, let us measure the two upper qubits, leaving the two single-qubit rotations L and R, and two single-qubit lower qubit undisturbed. This measurement results in two conditional phase-shifts S and T. Rotation L sends 0 purely random classical bits u and v, bearing no correla- | i tion whatsoever with the original state ψ . Let us now to ( 0 + 1 )/√2 and 1 to ( 0 + 1 )/√2, whereas | i R sends| i 0| ito ( 0 1|)/i√2 and−| i1 to| i ( 0 + 1 )/√2. turn u and v back into quantum bits and reinject u and | i | i − | i | i | i | i v in the circuit immediately after the dashed line.| i Note that LR ψ = RL ψ = ψ for any qubit ψ . Condi- | i tional phase-shift| i S sends| i 0| toi i 0 and leaves| i1 undis- Needless to say that the quantum state carried at turbed, whereas T sends | 0i to | i 0 and 1 |toi i 1 . the dashed line has been completely disrupted by this In terms of unitary matrices,| i the operations−| i | arei − | i measurement-and-resend process. We would therefore ex- pect this disturbance to play havoc with the final output 1 1 1 1 1 1 of the circuit. Not at all! In the end, the state carried at L = − R = √2  1 1  √2  1 1  xyz is uvψ . In other words, ψ is still obtained at z and − the other| twoi qubits, if measured,| i are purely random pro- i 0 1 0 vided we forget the measurement outcomes at the dashed S = and T = −  0 1   0 i  line. Another way of seeing this phenomenon is that the − 2 outcome of the circuit will not be altered if the state of References the upper two qubits leaks to the environment (in the Bennett Bessette Bras- standard basis) at the dashed line. [1] , Charles H., Fran¸cois , Gilles sard, Louis Salvail and John Smolin, “Experimen- To turn this circuit into a quantum teleportation tal quantum cryptography”, Journal of Cryptology 5:1 device, we need the ability to store qubits. Assume Alice (1992), 3 – 28. prepares two qubits in state 0 and pushes them through Bennett Brassard the first two gates of the circuit.| i [2] , Charles H. and Gilles , “Quantum cryptography: Public key distribution and coin tossing”, Proceedings of IEEE International Conference on Com- 0 L t σ | i puters, Systems and Signal Processing, Bangalore, India (1984), 175 – 179. 0 i ρ | i [3] Bennett, Charles H., Gilles Brassard, Claude Crepeau´ , Richard Jozsa, Asher Peres and William She keeps the upper qubit σ in quantum memory and Wootters, “Teleporting an unknown quantum state via gives the other, ρ, to Bob. [We do not denote these dual classical and Einstein–Podolsky–Rosen channels”, qubits by kets because they are not individual pure Physical Review Letters 70:13 (1993), 1895 – 1899. states: together they are in state Φ+ = ( 00 + 11 )/√2.] | i | i [4] Bennett, Charles H., Gilles Brassard and Artur At some later time, Alice receives a mystery qubit in Ekert, “Quantum cryptography”, Scientific American unknown state ψ . In order to teleport this qubit to Bob, (October 1992), 50 – 57. she releases σ from| i her quantum memory and pushes it [5] Brassard, Gilles, “A quantum jump in computer sci- together with the mystery qubit through the next two ence”, in Computer Science Today (Jan van Leeuwen gates of the circuit. She measures both output wires to ed.), Lecture Notes in Computer Science 1000, Springer- turn them into classical bits u and v. Verlag (1995), 1 – 14. Brassard ψ t R u [6] , Gilles, “The impending demise of RSA?”, | i RSA Laboratories CryptoBytes 1:1 (1995), 1 – 4. Deutsch σ i v [7] , David, “Quantum theory, the Church–Turing principle and the universal quantum computer”, Proceed- ings of the Royal Society, London A400 (1985), 97 – 117. To complete teleportation, Alice has to communicate u Einstein Podolsky Rosen and v to Bob by way of a classical communication chan- [8] , Albert, Boris and Nathan , nel. Upon reception of the signal, Bob creates quantum “Can quantum-mechanical description of physical real- Physical Review 47 states u and v from the classical information received ity be considered complete?”, (1935), 777 – 780. from Alice,| i he| releasesi the qubit ρ he had kept in quan- tum memory, and he pushes all three qubits into his part [9] Feynman, Richard P., “Quantum mechanical comput- of the circuit (on the right of the dashed line). Finally ers”, Optics News (1985); Reprinted in Foundations of Physics 16:6 (1986), 507 – 531. Bob may wish to measure the two upper qubit at x and y to make sure that he gets u and v; otherwise something [10] Hughes, Richard J., G. G. Luther, G. L. Morgan, Peterson Simmons went wrong in the teleportation apparatus. At this point, C. G. and C. , “Quantum cryptogra- teleportation is complete as Bob’s output z is in state ψ . phy over underground optical fibers”, Advances in Cryp- tology: Crypto ’96 Proceedings Note that this process works equally well if Alice’s mys-| i , Springer-Verlag (1996). tery qubit is not in a pure state. In particular, Alice can [11] International Business Machines, “Stand by: I’ll teleport to Bob entanglement with an arbitrary auxiliary teleport you some goulash”, Scientific American (Febru- system, possibly outside both Alice’s and Bob’s labora- ary 1996), 0 – 1 (sic). tories. [12] C. Marand and Paul D. Townsend, “Quantum key dis- In practice, Bob need not use the quantum circuit tribution over distances as long as 30 km”, Optics Letters 20 shown right of the dashed line at all. Instead, he may (1995). choose classically one of 4 possible rotations to apply to [13] Muller, Antoine, Hugo Zbinden and Nicolas Gisin, the qubit he had kept in quantum memory, depending on “Underwater quantum coding”, Nature 378 (1995), 449. the 2 classical bits he receives from Alice. (This would be [14] Shor, Peter W., “Algorithms for quantum computation: more in tune with the original teleportation proposal [3].) Discrete logarithms and factoring”, Proceedings of the This explains the earlier claim that quantum teleporta- 35th Annual IEEE Symposium on Foundations of Com- tion can be achieved at the cost of only two quantum puter Science (1994), 124 – 134. exclusive-ors: those of Alice. Nevertheless, the unitary version of Bob’s process given here may be more appeal- ing than choosing classically among 4 courses of action if teleportation is used inside a quantum computer.

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