letters to nature 7. Wootters, W. K. & Zurek, W. H. A single quantum cannot be cloned. Nature 299, 802–803 (1982). surprising as precise specification of a general qubit state requires an 8. Nielsen, M. A. & Chuang, I. J. Quantum Computation and Quantum Information (Cambridge Univ. infinite amount of classical information. Aside from the obvious Press, Cambridge, 2000). 9. Gottesman, D. & Chuang, I. L. Demonstrating the viability of universal quantum computation using differences in the various experimental demonstrations, the basic teleportation and single-qubit operations. Nature 402, 390–393 (1999). teleportation protocol is the same1. Alice is in possession of a qubit 10. Furusawa, A. et al. Unconditional quantum teleportation. Science 282, 706–709 (1998). (here labelled 2) that is in an unknown state jwl2 ; aj"l2 þ bj#l2; 11. Nielsen, M. A., Knill, E. & Laflamme, R. Complete quantum teleportation using nuclear magnetic j#l j"l resonance. Nature 396, 52–55 (1998). where and denote eigenstates of the qubit in the measure- 12. Barrett, M. D. et al. Quantum teleportation with atomic qubits. Nature (this issue). ment basis. In addition, Alice and Bob each possess one qubit of a 13. Schmidt-Kaler, F. et al. How to realize a universal quantum gate with trapped ions. Appl. Phys. B 77, two-qubit entangled pair that we take to be a singlet jSl1;3 ; j"l1j#l3 789–796 (2003). 2j#l j"l (where, for simplicity, we omit normalization factors). 14. Roos, C. F. et al. Bell states of atoms with ultra long lifetimes and their tomographic state analysis. 1 3 Phys. Rev. Lett. (in the press) Preprint at khttp://arXiv.org/abs/ physics/0307210l (2003). Therefore, Alice possesses qubits 1 and 2, while Bob holds qubit 3. 15. Massar, S. & Popescu, S. Optimal extraction of information from finite quantum ensembles. Phys. Rev. Alice wishes to transmit the state of qubit 2 to Bob’s qubit using only Lett. 74, 1259–1263 (1995). classical communication. The initial joint state of all three qubits is 16. Gisin, N. Nonlocality criteria for quantum teleportation. Phys. Lett. A 210, 157–159 (1996). 17. Kielpinski, D., Monroe, C. & Wineland, D. J. Architecture for a large-scale ion-trap quantum ^ computer. Nature 417, 709–711 (2002). jFl ¼jSl1;3 jwl2: ð1Þ 18. Gulde, S. et al. Implementation of the Deutsch-Jozsa algorithm on an ion-trap quantum computer. Nature 412, 48–50 (2003). 19. Hahn, E. L. Spin Echoes. Phys. Rev. 80, 580–594 (1950). This state can be rewritten using an orthonormal basis of Bell states14 jWkl1;2ðk ¼ 1–4Þ for the first two qubits and unitary transformations Acknowledgements We thank H. Briegel and P.Zoller for a critical reading of the manuscript. We U k acting on jwl3ð¼ aj"l3 þ bj#l3Þ so that jFl ¼ gratefully acknowledge support by the Austrian Science Fund (FWF), by the European 4 Commission (QUEST, QUBITS and QGATES networks), by the Institut fu¨r Sk¼1jWkl1;2ðUkjwl3Þ: A measurement in the Bell-state basis {jWkl} Quanteninformation, and by the Los Alamos LDRD Program. This material is based upon work by Alice then leaves Bob with one of the four possibilities Ukjwl3: supported in part by the US Army Research Office. H.H is funded by the Marie-Curie program of Once Bob learns of Alice’s measurement outcome (through classical the European Union. communication), he can recover the original unknown state by 21 applying the appropriate unitary operator, Uk ; to his state Competing interests statement The authors declare that they have no competing financial j l interests. Uk w 3: We note that Alice’s Bell-state measurement can be accom- plished by transforming from the basis {jWkl1;2} into the measure- Correspondence and requests for materials should be addressed to R.B. ([email protected]) ment basis {j""l1;2; j"#l1;2; j#"l1;2; j##l1;2}beforethemeasurement. Our implementation uses atomic qubits (9Beþ ions) that are confined in a linear radiofrequency Paul trap similar to that used in ref. 15. The control electrodes are segmented into eight sections as shown schematically in Fig. 1, providing a total of six trapping zones .............................................................. (centred on electrode segments 2 to 7). Potentials applied to these electrodes can be varied in time to separate ions and move them to Deterministic quantum teleportation different locations. The qubits are composed of the ground-state of atomic qubits hyperfine levels j"l;jF ¼ 1;m ¼ 21l and j#l;jF ¼ 2;m ¼ 22l; whichareseparatedbyq0 ø 2p £ 1:25GHz: These states are coupled through stimulated Raman transitions16–18 from two laser M. D. Barrett1*, J. Chiaverini1, T. Schaetz1, J. Britton1, W. M. Itano1, J. D. Jost1, E. Knill2, C. Langer1, D. Leibfried1, R. Ozeri1 & D. J. Wineland1 1Time and Frequency Division, NIST, Boulder, Colorado 80305, USA 2Mathematical and Computational Sciences Division, NIST, Boulder, Colorado 80305, USA * Present address: Department of Physics, University of Otago, PO Box 56, Dunedin, New Zealand ............................................................................................................................................................................. Quantum teleportation1 provides a means to transport quantum information efficiently from one location to another, without the physical transfer of the associated quantum-information carrier. This is achieved by using the non-local correlations of previously distributed, entangled quantum bits (qubits). Teleportation is expected to play an integral role in quantum communication2 and quantum computation3. Previous experimental demon- strations have been implemented with optical systems that used both discrete and continuous variables4–9, and with liquid- state nuclear magnetic resonance10. Here we report uncondi- tional teleportation5 of massive particle qubits using atomic Figure 1 Schematic representation of the teleportation protocol. The ions are numbered (9Be1) ions confined in a segmented ion trap, which aids left to right, as indicated at the top, and retain their order throughout. Positions, relative individual qubit addressing. We achieve an average fidelity of to the electrodes, are shown at each step in the protocol. The widths of the electrodes 78 per cent, which exceeds the fidelity of any protocol that vary, with the width of the separation electrode (6) being the smallest at 100 mm. The does not use entanglement11. This demonstration is also import- spacing between ions in the same trap is about 3 mm, and laser-beam spot sizes (in ant because it incorporates most of the techniques necessary traps 5 and 6) at the position of the ions are approximately 30 mm. In step 1 we prepare for scalable quantum information processing in an ion-trap the outer ions in an entangled (singlet) state and the middle ion in an arbitrary state system12,13. (equation (1)). Steps 2–4 constitute a measurement in a Bell-basis for ions 1 and 2 (Alice’s Quantum teleportation1 provides a means for transporting a qubits), teleporting the state of ion 2 onto ion 3 (Bob’s qubit), up to unitary operations quantum state between two separated parties, Alice and Bob, that depend on the measurement outcomes. In step 5 we invoke these conditional through the transmission of a relatively small amount of classical operations, recovering the initial state. Interspersed are spin-echo pulses applied in trap 6 information. For the case of a two-state quantum system or ‘qubit’, that protect the state from de-phasing due to fluctuating magnetic fields but do not affect only two bits of classical information are needed, which seems the teleportation protocol. 737 NATURE | VOL 429 | 17 JUNE 2004 | www.nature.com/nature © 2004 Nature Publishing Group letters to nature beams, which are used to implement the single-qubit rotations state of ion 1 to j"l1; a subsequent simultaneous detection of ions 1 Rðv;fÞ¼cosðv=2ÞI þ isinðv=2ÞcosðfÞj þ isinðv=2ÞsinðfÞj ; ð2Þ and 2 effectively measures the state of ion 2 with error less than 1% x y due to the presence of ion 1. where I is the identity operator, j x, jy and j z denote the Pauli spin To complete the teleportation, we apply unitary operations to ion matrices in the {j"l; j#l} basis (k "j; ð1;0Þ; k #j; ð0;1Þ), v is 3 that depend on the measurement outcomes for ions 1 and 2. We proportional to the duration of the Raman pulse, and f is the first move ions 1 and 2 to trap 2 and ion 3 to trap 5, where unitary relative phase between the Raman beams at the position of the ion. operations consisting of a p/2-pulse, R(p/2,p/2), followed by The Raman beams are also used to generate entanglement between the operators j x, j y, I, j z for the measurement outcomes j""l1;2; 18 two qubits by implementing the phase gate j"#l1;2; j#"l1;2; j##l1;2 respectively, are applied. As noted above, the aj""l þ bj"#l þ cj#"l þ dj##l ! aj""l inclusion of the spin-echo pulses does not fundamentally change ð3Þ the teleportation protocol; however for fSE ¼ p=2; we must reorder 2 ibj"#l 2 icj#"l þ dj##l: the operations following the p/2-pulse, R(p/2,p/2), to I, j z, j x, j y respectively. A complete experiment is about 4 ms in duration, Our teleportation scheme is shown schematically in Fig. 1. We predominantly due to three elements: the cooling of all three axial highlight the key elements of the protocol in bold lettering and also modes to the ground state (1 ms), implementing the ion separations 15,18 include the auxiliary ‘spin-echo’ pulses (R(p,f SE)) applied to and movements (2 ms), and the three detection durations (0.6 ms). ions in trap 6.