Quantum teleportation using three-particle entanglement Anders Karlsson* and Mohamed Bourennane Department of Electronics, Laboratory of Photonics and Microwave Engineering, Royal Institute of Technology (KTH), Electrum 229, 164 40 Kista, Sweden Received 13 February 1998 We investigate the ‘‘teleportation’’ of a quantum state using three-particle entanglement to either one of two receivers in such a way that, generally, either one of the two, but only one, can fully reconstruct the quantum state conditioned on the measurement outcome of the other. We furthermore delineate the similarities between this process and a quantum nondemolition measurement. S1050-29479808812-X PACS numbers: 03.67.Hk, 03.65.Bz, 42.50.Dv I. INTRODUCTION By quantum teleportation a process is denoted by which the complete information about a quantum state can be sent using a classical transmission of information with the aid of long-range Einstein-Podolsky-Rosen EPR correlations 1 in an entangled quantum state 2. The truly interesting as- pect of quantum teleportation is the light it sheds on the nature of classical and quantum information. Experimentally, quantum teleportation was recently demonstrated 3,4 using parametric down-conversion 5, in 3 interferometric Bell- state analyzers 6, and in 4 k -vector entanglement. Given that teleportation has been demonstrated using two- particle entanglement, and the general interest and quest to demonstrate a three-particle entangled Greenberger-Horne- Zeilinger GHZ state 8,9, we may ask the following: What new scheme can be developed using a three-particle en- tangled state? It is clear that it does not permit by any means the faithful transmission of an unknown quantum state to two locations. This would be forbidden by the no-cloning/ broadcast theorems 12,13. In view of this, we instead find that one may teleport to either of the two locations consid- ered, but not to both. However, there is an interesting mid- way case where both parties have some information about the original state. This, of course, is very similar to a quan- tum copier cloning device14–17. Recently, it was also brought to our attention that a scheme similar to ours had been studied in a more general context by Bruß et al. 18. We will comment on the similarity and difference between their proposal and ours. The paper is outlined as follows. In Sec. II, we briefly review quantum teleportation using two-particle entangle- ment. In Sec. III, we present the three-particle entanglement teleportation scheme. In both Secs. II and III, we only con- sider the case of polarization entanglement. In Sec. IV we discuss the similarities to a quantum nondemolition measure- ment, and in Sec. V we analyze how much information both receivers have on the state. Finally, in Sec. VI we discuss the results and present some conclusions. II. A BRIEF REVIEW OF QUANTUM TELEPORTATION USING TWO-PARTICLE ENTANGLEMENT Let us begin with a brief review of quantum teleportation using a two-particle polarization entanglement. Quantum teleportation can be accomplished using a two-particle en- tangled state, such as from a type II parametric down- conversion 5. The state generated from a type II down- conversion crystal can be written as 5 |  i , j  = 1 &  |   i | ↔ j +e i  | ↔ i |   j ), 2.1 where  is a birefringent phase shift of the crystal, and the subscripts denote particles i and j , respectively. Using ap- propriate birefringent phase shifts and polarization conver- sion, one may easily convert the above state into any of the four Bell states 5: |  i , j   = 1 &  |   i | ↔ j | ↔ i |   j ), 2.2 and |  i , j   = 1 &  |   i |   j | ↔ i | ↔ j ). 2.3 Experimentally, shifting between these states actually be- tween all four Bell states has been demonstrated in Bell state analysis 6 and quantum dense coding experiments 7. By making a shift of basis from a 0° and 90° base  | ↔ , |    to a 45° and a 135° polarization base  | ր , | տ  , the states become |  +  =( | տ | տ +| ր | ր )/ & or |  -  =( | տ | ր +| ր | տ )/ &. For the reader who is more versed in spin measurement, we may rewrite the state in terms of spin 1/2 particles putting | ↔ =| z + , |   =| z - . In the same terminology, a 45° and a 135° polarized photon become | ր =| x + and | տ = | x - , where | z + denotes a spin eigenstate in the positive z direction, etc. See any quantum mechanics textbook for de- tails. Suppose now that a person ‘‘Alice’’ wants to send a quan- tum state |  A  1 =a |   1 +b | ↔ 1 2.4 *Electronic address: andkar@ele.kth.se PHYSICAL REVIEW A DECEMBER 1998 VOLUME 58, NUMBER 6 PRA 58 1050-2947/98/586/43947/$15.00 4394 © 1998 The American Physical Society