1. Information Processing by Single Particle Hybrid Entangled States Archan S. Majumdar S. N. Bose National Centre for Basic Sciences Kolkata, India Collaborators: S. Adhikari D. Home A. K. Pan T. Pramanik International Program on Quantum Information, IOP Bhubaneswar, 2010

2. Perspective Hybrid entangled states: The theoretical framework of Q.M. allows for the existence of entangled states involving Hilbert spaces with distinctly different properties, e.g., spin & linear momentum; polarization & angular momentum; energy level & spin. etc… [c.f. Zukowski, Zeilinger, et. al. (2009), etc.] Single particle (intra-particle) entanglement: Entanglement between different degrees of freedom of the same particle. [ c.f. Blasone et. al. (2009) for neutrino oscillations; Dunningham & Vedral (2007) nonlocality for photons ; Home et al (2001) & Hasegawa et. al. (2003) for neutron path and spin, etc..] Motivations: Is it possible to perform information processing with such states ? These states could be more robust against noise and dissipative effects. Could have applications in complex systems. e.g., CM physics, BH physics ? Difficulty: Entanglement NOT shared between spatially separated entities

3. Plan Setting up path-spin entangled state of a single particle Swapping path-spin entanglement with spin-spin entanglement of two qubits Teleportation using path-spin entanglement of a single particle as resource A demonstration of contextuality using single-particle energy-spin entangled states Encoding information using the superposition of path and spin degrees of freedom Summary and Conclusions

4. Setting up a path-spin entangled state of a single particle Initial state: Beam-splitter Spin-flipper (or CNOT) Emergent path-spin entangled state

5. Path-spin hybrid entangled state of a single particle Salient features: Dichotomic path and spin variables – essentially two qubits carried by a single particle Intra-particle entanglement (as distinct from the entanglement of spins of two different particles, or even hybrid entanglement of say the polarization of one photon & spin of another spatially separated photon Entanglement NOT of similar modes, e.g., as in single particle neutrino flavour states This state is a true example of a single particle hybrid entangled state where the entanglement is between the path and spin of the same particle

6. Swapping path-spin entanglement onto spin-spin entanglement of two qubits Aim: To swap the entanglement of the created path-spin entangled state with that of two spatially separated qubits. The two qubits are disentangled initially, and have no interaction among them. Protocol: Alice has the path-spin entangled state “1”, and a Second particle “2” in a spin (up) state. Bob has particle “3” also in spin (up) state. Through some LOCC, the state of “2” And “3” get entangled, while the path-spin entangled state “1” is destroyed.

7. Protocol for entanglement swapping Alice performs CNOT with “2” (target) & “1” (source). “1” is sent to Bob, and “2” to Charlie. Bob performs CNOT with “1” (source) & “3” (target). Bob combines the two channels of “1” using BS2, and performs measurements with SG1 & SG2 Bob applies unitary operations on “3”. Result: “2” (with Charlie) and “3” (with Bob) get entangled.

8. Protocol for entanglement swapping Alice possesses path-spin entangled state, and another particle in spin up state After CNOT operation: Particle “1” is sent to Bob. His CNOT: Particle “2” is sent to Charlie. Bob uses a 50-50 beam splitter: SG1 & SG2 are placed using which the qubit “s” undergoes the transformation: Depending upon the following possibilities, Bob makes the corresponding unitary operations:

9. Measurement outcomes & Unitary operations Final outcome: Bob and Charlie share two-qubit entangled state with the same information content as the original path-spin entangled state. Intra-particle entanglement of “1” is swapped onto the spin-spin inter-particle entanglement of “2” & “3”. “2” & “3” never interact with each other, but entanglement is set-up by their separate interactions with “1”. S. Adhikari, A. S. Majumdar, D. Home, A. K. Pan, EPL 89, 10005 (2010)

10. Teleportation through single particle path-spin entangled state Perspective: Path-spin entangled state is NOT shared non-locally between two distant parties. How can such entanglement be used as a resource ? Aim: To transfer the state of an unknown qubit to a distant location using particle “1” (path-spin entangled) state as resource. Method: Since entanglement is not shared a-priori between the two distant parties, particle “1” has to be transported at some stage.

11. Protocol for information transfer using single particle path-spin entangled state Alice: “1” & “2” (unknown) & auxiliary “a”. Performs two CNOTs Sends particle “1” to Bob. Bob: Performs CNOT Alice: Measures “1” & “a”, communicates with Bob. Bob: Unitary transformation on “3” to recover teleported state.

12. Steps of the teleportation protocol Alice possesses “1”: “2”: and “a”: Performs two CNOTS: Alice sends “1” to Bob. After Bob confirms receipt, Alice Measures “2” & “a”.

13. Teleportation protocol…. {Case i} Bob performs suitable unitary transformations based on Alice’s measurement results communicated to him, e.g., if the outcome is & Bob combines the two channels of “1” using BS2, and performs a CNOT involving “1” & “3”: Bob measures spin of “1” by SG2/SG3. The possible outcomes are: occurring with equal probability ¼. Bob performs the corresponding unitary operations

14. Fidelity of teleportation Unitary operations performed by Bob: Average Fidelity: If the path-spin state “1” is maximally entangled, i.e., teleportation is perfect. Bob possesses the same state “3” as the one “2” originally held by Alice. T.Pramanik, S.Adhikari, A.S.Majumdar, D.Home, A.K.Pan, Phys. Lett. A 374, 1121 (2010)

15. Security of the Protocol What happens when particle “1” is lost in transit ? “1” is held by Eve (now), and “2” & “a” are possessed by Alice. Eve can perform a spin measurement on “1”. However, she fails to uncover information about the state “2”: When Bob does NOT receive particle “1”, Alice performs the following operations to recover the state to be teleported: Alice makes a spin measurement on her particle “a” in the basis Then she makes (i) unitary operation (ii) does nothing to retrieve the unknown state to be teleported. Information in “2” is never lost even if “1” is lost in transit. T. Pramanik, S. Adhikari, A. S. Majumdar, D. Home, A. K. Pan, Phys. Lett. A (2010)

16. Energy-spin entangled state of a single atom in cavity-QED Two-(energy) level atom in (energy & spin) state Apply in vacuum Cavity: Stern-Gerlach along x-axis: Place in state in the spin up path Place in state in the spin down path With suitable chosen interaction times, energy-spin entangled state of atom:

17. Proposal for testing contextuality with single atom energy-spin entangled state Energy-spin entangled state Emerging from & pulses in, Measurements through SG and four detectors

18. Violation of Bell-CHSH inequality Non-contextual model: Q.M. correlations:

19. Encoding information using the quantum superposition principle Perspective: Dense coding requires quantum entanglement; Is it possible to encode information using the superposition principle ? We propose a scheme using the path and spin degrees of freedom of a single particle. Operational difficulty of Bell State Analysis in standard quantum dense coding scheme leads to maximum channel capacity of about 1.6 bits in stead of the theoretically attainable 2 bits. Is it possible to do better using the spin and path variables of a spin-1/2 particle ?

20. Set-up for encoding information Alice prepares state by choosing particular values of the phase of SR & PS Bob obtains particle in channel corresponding to Alice’s unitary operation

21. Protocol for encoding information Initial state: SR: Phase-shifter: Final state on emerging from BS2:

22. The scheme Alice has two possible choices for the parameters and The four possible combination of choices (unitary operations): U1 ( ) U2 This leads to the following distinct possibilities for the channel where Bob finds the particle. Thus, two bits of information can be communicated by Alice to Bob without using quantum entanglement. This scheme is solely based on the superposition principle. D.Home, A.S.Majumdar, S. Adhikari, A.K.Pan, M.Whitaker, arXiv:0906.0270

23. Summary & Conclusions Information processing through single particle hybrid entangled states Generation of path-spin entangled state by BS and SF (or CNOT) Swapping of path-spin intra-particle entanglement onto spin-spin inter- particle entanglement Teleportation of a spin qubit using single particle path-spin entanglement as resource. Secure protocol with ideal fidelity. Proposal to test quantum contextuality using single atom energy-spin entangled state. Other forms of hybrid entanglement ? Encoding information through the superposition of path & spin of a single particle

24. Perspectives & future directions A.S.M., D.H., S.A., A.K.P., T.P., EPL (2010); PLA (2010); arXiv:0906.0270 Nature of quantum entanglement independent of the particular realization of Hilbert space – single particle hybrid entangled states could be useful information processing resource. Path (or linear momentum) degrees of freedom ubiquitous in experiments. Our protocols are based on exploiting these hitherto unexplored resources. Entanglement at the level of a single particle could be less susceptible to decoherence effects – this feature needs further development and testing in real experiments. Reliance of information processing protocols on entanglement vis-à- vis the superposition principle ?