Quantum Teleportation : Theory and Experiment

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Quantum teleportation is the process in which, the quantum state of a
particle is transferred to another without direct interaction. This is one of the most important consequence of quantum entanglement.
Here I have tried to explain the theory behind the teleportation and its experimental verification.

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Quantum Teleportation : Theory and Experiment

  1. 1. Quantum Teleportation :- Theory and experiment Chithrabhanu P Introduction Quantum Teleportation Quantum Teleportation : Theory and Experiment Chithrabhanu P chithrabhanu@prl.res.in THEPH, PRL
  2. 2. Quantum Teleportation :- Theory and experiment Chithrabhanu P Introduction Quantum Teleportation Quantum bits Bit :- Fundamental unit of classical information {0,1} Qubit :-Quantum analog to bit. |ψ = α|0 + β|1 (1) The state of the qubit is a vector in an two-dimensional complex vector space. Qutrit, qudit :- 3 and higher dimensions respectively. |0 , |1 :- Computational basis states forming orthonormal basis of the vector space. |α|2 :- Probability that system is in |0 ; |β|2 :- Probability that system is in |1 Example of qubit states:- Two polarization states { |H , |V }, spin states { | ↑ ,| ↓ } etc.
  3. 3. Quantum Teleportation :- Theory and experiment Chithrabhanu P Introduction Quantum Teleportation Entanglement Non local quantum correlation between particles. A two particle entangled state cannot be written as product of two single particle states. Ψ12 = φ1 ⊗ ξ2 (2) Bell states :- Maximally entangled state of two qubits. |Ψ± = 1 √ 2 (|0 |1 ± |1 |0 ) (3) |Φ± = 1 √ 2 (|0 |0 ± |1 |1 ) (4)
  4. 4. Quantum Teleportation :- Theory and experiment Chithrabhanu P Introduction Quantum Teleportation Quantum gates Basic unit of a quantum circuit. NOT gate { X } X (α|0 + β|1 ) → α|1 + β|0 (5) Z gate Z (α|0 + β|1 ) → α|0 − β|1 (6) Hadamard gate {H} H (α|0 + β|1 ) = α |0 + |1 √ 2 + β |0 − |1 √ 2 (7) CNOT gate :- Two qubit state. Flips the second qubit (target) if the first qubit (control) is 1. Similar to XOR |A, B → |A, B ⊕ A
  5. 5. Quantum Teleportation :- Theory and experiment Chithrabhanu P Introduction Quantum Teleportation Quantum gates cont.. Hadamard and CNOT operation to prepare Bell states. x, y are |0 or |1 logic. βxy - Bell states. In case of polarization; a half wave plate (HWP), can perform many single qubit operations by keeping its fast axis at different angle with respect to the incident polarization. { 0 → ˆZ, π 4 → ˆX, π 8 → ˆH } Polarization CNOT :- not trivial. Requires interaction of two qubits (Zhao et al., PRL 2005; Bao et al., PRL 2007).
  6. 6. Quantum Teleportation :- Theory and experiment Chithrabhanu P Introduction Quantum Teleportation Quantum Teleportation VOLUME 70 29 MARCH l993 NUMBER 13 Teleporting an Unknown Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels Charles H. Bennett, ~ ) Gilles Brassard, ( ) Claude Crepeau, ( ) ( ) Richard Jozsa, ( ) Asher Peres, ~4) and William K. Wootters( ) ' IBM Research Division, T.J. watson Research Center, Yorktomn Heights, ¹mYork 10598 ( lDepartement IIto, Universite de Montreal, C.P OI28, Su. ccursale "A", Montreal, Quebec, Canada HBC 817 ( lLaboratoire d'Informatique de 1'Ecole Normale Superieure, g5 rue d'Ulm, 7M80 Paris CEDEX 05, France~ i l lDepartment of Physics, Technion Israel In—stitute of Technology, MOOO Haifa, Israel l lDepartment of Physics, Williams College, Williamstoivn, Massachusetts OIP67 (Received 2 December 1992) An unknown quantum state ]P) can be disassembled into, then later reconstructed from, purely classical information and purely nonclassical Einstein-Podolsky-Rosen (EPR) correlations. To do so the sender, "Alice," and the receiver, "Bob," must prearrange the sharing of an EPR-correlated pair of particles. Alice makes a joint measurement on her EPR particle and the unknown quantum system, and sends Bob the classical result of this measurement. Knowing this, Bob can convert the state of his EPR particle into an exact replica of the unknown state ]P) which Alice destroyed. PACS numbers: 03.65.Bz, 42.50.Dv, 89.70.+c The existence of long range correlations between Einstein-Podolsky-Rosen (EPR) [1] pairs of particles raises the question of their use for information transfer. Einstein himself used the word "telepathically" in this contempt [2]. It is known that instantaneous information transfer is definitely impossible [3]. Here, we show that EPR correlations can nevertheless assist in the "telepor- tation" of an intact quantum state from one place to another, by a sender who knows neither the state to be teleported nor the location of the intended receiver. Suppose one observer, whom we shall call "Alice, " has been given a quantum system such as a photon or spin-& particle, prepared in a state ]P) unknown to her, and she wishes to communicate to another observer, "Bob," suf- ficient information about the quantum system for him to make an accurate copy of it. Knowing the state vector a perfectly accurate copy. A trivial way for Alice to provide Bob with all the in- formation in [P) would be to send the particle itself. If she wants to avoid transferring the original particle, she can make it.interact unitarily with another system, or "an- cilla, " initially in a known state ~ap), in such a way that after the interaction the original particle is left in a stan- dard state ~Pp) and the ancilla is in an unknown state ]a) containing complete information about ~P). If Al- ice now sends Bob the ancilla (perhaps technically easier than sending the original particle), Bob can reverse her actions to prepare a replica of her original state ~P). This "spin-exchange measurement" [4] illustrates an essential feature of quantum information: it can be swapped from one system to another, but it cannot be duplicated or "cloned" [5]. In this regard it is quite unlike classical A non classical transfer of an unknown quantum state using entanglement. Sender (Alice) knows neither the state to be teleported nor the location of the receiver (Bob )
  7. 7. Quantum Teleportation :- Theory and experiment Chithrabhanu P Introduction Quantum Teleportation Teleportation protocol Alice and Bob initially share a pair of entangled particles (say 2 & 3). Alice receives the particle with unknown state (say 1) . Alice does a joint Bell operator measurement on the unknown state particle and her entangled particle. Projective measurement. 1 & 2 gets destroyed due to the measurement. Alice sends the outcome of her measurement to Bob through a classical channel. Bob does a unitary transformation on his particle (particle 3) with respect to Alice’s measurement results.
  8. 8. Quantum Teleportation :- Theory and experiment Chithrabhanu P Introduction Quantum Teleportation How teleportation works? Initially, the unknown state and entangled pair are given by |φ1 = α|0 + β|1 ; |Ψ− 23 = 1 √ 2 (|01 − |10 ) (8) Total wave function |Ψ123 = 1√ 2 (α|0 + β|1 ) ⊗ (|01 − |10 ) (9) It can be written as |Ψ123 = 1√ 2 (α|00 12|1 3 − α|01 12|0 3 + β|10 12|1 3 + β|11 12|0 3) (10)
  9. 9. Quantum Teleportation :- Theory and experiment Chithrabhanu P Introduction Quantum Teleportation How teleportation works? From the Bell states (Eq.3 & Eq.4), we can have |00 = |Φ+ +|Φ− √ 2 ; |11 = |Φ+ −|Φ− √ 2 (11) |01 = |Ψ+ +|Ψ− √ 2 ; |10 = |Ψ+ −|Ψ− √ 2 (12) Substituting in Eq.10 and rearranging the terms |Ψ123 = 1 2 { |Ψ− 12 (−α|0 3 − β|1 3) + |Ψ+ 12 (−α|0 3 + β|1 3) + |Φ− 12 (α|1 3 + β|0 3) + |Φ+ 12 (α|1 3 − β|0 3) } (13)
  10. 10. Quantum Teleportation :- Theory and experiment Chithrabhanu P Introduction Quantum Teleportation How teleportation works? Outcome Unitary operator Ψ− ˆσ0 Ψ+ ˆσ3 Φ− ˆσ1 Φ+ ˆσ3 ˆσ1 In polarization case ˆσ0 −→ Free space propagation ˆσ3 −→ HWP in 00 ˆσ1 −→ HWP in π 4
  11. 11. Quantum Teleportation :- Theory and experiment Chithrabhanu P Introduction Quantum Teleportation Quantum circuit for teleportation Single/double lines :- classical/quantum channels. ˆH ˆCNOT :- Bell state preparation; ˆCNOT ˆH :- Bell state projection/detection
  12. 12. Quantum Teleportation :- Theory and experiment Chithrabhanu P Introduction Quantum Teleportation Experimental teleportation Bouwmeester et al.(Nature 1997) demonstrated quantum teleportation using photons. Figure: Experimental teleportation- Bouwmeester et al.(1997)
  13. 13. Quantum Teleportation :- Theory and experiment Chithrabhanu P Introduction Quantum Teleportation Experimental teleportation Entangled pair :- parametric down converted photons Bell projection :- beam splitter and detectors Figure: Experimental teleportation- Bouwmeester et al.(1997)
  14. 14. Quantum Teleportation :- Theory and experiment Chithrabhanu P Introduction Quantum Teleportation Experimental teleportation Only particles with anti symmetric wave function ( |Ψ− ) will emerge from both ends of beam splitter (Loudon, R. Coherence and Quantum Optics VI). Coincidence in detectors f1&f2 only when state is |Ψ− 12 . Unitary operation :- free space propagation. Initial state is prepared in +45 (-45) polarization states . ie 1√ 2 (|H ± |V ) PBS differentiate +45 & -45 polarization. Detector on each port (d1&d2) A delay is given in photon 2 path. Delay = 0 - no mixing - f1f2 coincidence 50% - f1f2d1 & f1f2d2 coincidence 25%
  15. 15. Quantum Teleportation :- Theory and experiment Chithrabhanu P Introduction Quantum Teleportation Teleportation results initial state +45. Delay 0 - f1f2 coincidence 25% - f1f2d1 coincidence 25% - f1f2d2 coincidence 0% Figure: Bouwmeester et al.(1997) The absence of coincidence corresponding to zero delay confirms the teleportation.
  16. 16. Quantum Teleportation :- Theory and experiment Chithrabhanu P Introduction Quantum Teleportation THANK YOU

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