Entanglement concentration protocol using linear optics

1. Entanglement concentration protocol using linear optics IPQI 2014 Anindita Banerjee CAPSS, Bose Institute Kolkata Collaborators: Chitra Shukla, Anirban Pathak

2. Outline Introduction Motivation Linear optics Example:Purification using PBS ECP for CAT state ECP for GHZ-like state ECP for Four qubit state ECP for n+1 qubit state of a particular form Single qubit assisted ECP Transformation Efficiency

3. Entanglement Applications of entanglement Teleportation Dense coding Quantum key distribution Secure direct communication (require max entangled state between two parties) IPQI 2014

4. ! Distributed Qubits interact with the environment Problems Gets noisy due to storage processing and transmission Less entangled state Mixed state The IDEA is that the two distant parties Alice and Bob are supplied with finite ensemble of pure states from which they wish to extract the maximally entangled states (MESs). Entanglement concentration transforms a pure non maximally entangled state into MES Entanglement distillation transforms a mixed non maximally entangled state into MES IPQI 2014

5. Motivation ECP/EP Bell state: Bose et al., Zhao et al., Yamamoto et al., Sheng et al. , Sheng and Zhou, Gu et al., Deng GHZ state: Chaudhury and Dhara [GHZ] and Zhou et al.[GHZ] W state: Sheng et al.[W state], Ling-yan He Cluster state: Chaudhury and Dhara [Cluster], Ting-Ting Xu et al. , Zhau et al.[Cluster] IPQI 2014

6. Cross kerr nonlinearities Bell state W state Zhaop et al. Sheng et al[W] Schmidt Projective method Bell state Bennet et al. Bell state Zhao et al.,[Experiment] Yamamoto et al [Experiment] QED Bell state Romero et al. POVM Bell state Gu et al. Linear optics Bell state Zhao et al., Yamamoto et al., Sheng et al. S. Banyopadhyay Entanglement swapping Bell state Bose et al. IPQI 2014

7. I H1 > Polarized Beam Splitter I V3 > I V1 > Horizontally polarized photon is transmitted and Vertically polarized photon is reflected I H4 > IPQI 2014

8. Purification of Bell state using PBS a1 b1 Source pair Target pair a2b2 NATURE |VOL 423 | 22 MAY 2003 417--421 IPQI 2014

9. Three ingredients involved V. Vedral and M. B. Plenio Local operation Classical communication Post selection IPQI 2014

10. Entanglement Swapping Bose et al. Alice Bob 1 2 3 4 Bell measurement IPQI 2014

11. ECP for partially entangled cat state Bell and GHZ are special cases IPQI 2014

12. ECP for partially entangled GHZ-like state GHZ-like state example and are orthogonal to each other And belong to bell state IPQI 2014

14. General state Non-maximally entangled (n + 1)-qubit state where and are arbitrary n-qubit states that are mutually orthogonal. Bell state GHZ GHZ-like CAT states Why is it important? Bidirectional quantum teleportation Hierarchical quantum communication schemes (HQIS), Hierarchical quantum secret sharing (HQSS) Applications IPQI 2014

15. Four-qubits entangled states There exist nine failies of states corresponding to nine different ways of entangling four qubits. F. Verstraete, J. Dehaene, B. De Moor and H. Verschelde, “Four qubits can be entangled in nine different ways”, Phys. Rev. A 65 (2002) 052112. L. Borsten, D. Dahanayake, M. J. Duff, A. Marrani and W. Rubens, “Four-qubit entanglement classification from string theory”, Phys. Rev. Lett. 105 (2010) 100507. IPQI 2014

16. Four-qubit entangled states IPQI 2014

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19. Sigle qubit assisted ECP for general state IPQI 2014

20. Optical circuit using linear optics Bell measurement Bell states Further, the CNOT can be implemented using optical circuits implemented by J. L. O’Brien et al. Thus in general ECPs proposed here can be realized optically. These ECPs may be practically realized using NMR as Bell measurement is possible in NMR based technologies. IPQI 2014

21. Two alternative ECPs for quantum states IPQI 2014

22. Entanglement transformation efficiency is the amount of entanglement in the initial partially entangled state is the amount of entanglement of the state after concentration. is the amount of entanglement of the state to be concentrated OR is the amount of entanglement of the entire initial state. Ambiguity be the total initial entanglement higher efficiency of single photon assisted ECPs over Bell-type state assisted ECPs IPQI 2014

23. Sheng et al. (von Neumann entropy ) as a measure of entanglement, But von Neumann entropy is a good measure of entanglement for bipartite systems only. How to find for an ECP that is designed for multipartite case? Interestingly, the problem is equivalent to provide a quantitative measure of multiparite entanglement. Let us choose tangle as a measure of entanglement. Yu and Song established that any good measure MA-Bof bi-partite entanglement can be generalized to multipartite systems, by considering bipartite partitions of the multipartite system. Yu and Song defined a simple measure of tripartite entanglement as where Mi-jkis a measure of entanglement between subsystem iand subsystem jk. IPQI 2014

24. Thus Sabın and Garca-Alcaine’s measure of tripartite entanglement For the Bell measurement protocol For the single qubit assisted protocol IPQI 2014

25. References • C. Shukla, A. Pathak and R. Srikanth, “Beyond the Goldenberg-Vaidman protocol: Secure and efficient quantum communication using arbitrary orthogonal, multi-particle quantum states”, IJQI 10 (2012) 1241009. • C. H. Bennett, H. J. Bernstein, S. Popescu and B. Schumacher, “Concentrating partial entanglement by local operations”, Phys. Rev. A 53 (1996) 2046. • C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels”, Phys. Rev. Lett. 76 (1996) 722. • S. Bose, V. Vedral and P. L. Knight, “Purification via entanglement swapping and conserved entanglement”, Phys. Rev. A 60 (1999) 194. • Z. Zhao, J. W. Pan and M. S. Zhan, “Practical scheme for entanglement concentration”, Phys. Rev. A 64 (2001) 014301. • T. Yamamoto et al., “Concentration and purification scheme for two partially entangled photon pairs”, Phys. Rev. A 64 (2001) 012304. • Y.-B. Sheng, L. Zhou, S.-M. Zhao and B.-Y. Zheng, “Efficient single-photon-assisted entanglement concentration for partially entangled photon pairs”, Phys. Rev. A 85 (2012) 012307. • Y.-B. Sheng and L. Zhou, “Quantum entanglement concentration based on nonlinear optics for quantum communications” Entropy 15 (2013) 1776. • Y.-J. Gu, W.-D. Li and G.-C. Guo, “Protocol and quantum circuits for realizing deterministic entanglement concentration”, Phys. Rev. A 73 (2006) 022321. • F.-G. Deng, “Optimal nonlocal multipartite entanglement concentration based on projection measurements”, ' Phys. Rev. A 85 (2012) 022311. • Y.-B. Sheng, L. Zhou and S.-M. Zhao, “Efficient two-step entanglement concentration for arbitrary W states”, Phys. Rev. A 85 (2012) 042302. • L.-y. He, C. Cao and C. Wang, “Entanglement concentration for multi-particle partially entangled W state using nitrogen vacancy center and microtoroidal resonator system”, Opt. Commun. 298–299 (2013) 260. • B. S. Choudhury and A. Dhara, “A three-qubit state entanglement concentration protocol assisted by two-qubit systems”, Int. J. Theor. Phys. 52 (2013) 3965.

26. • L. Zhou, “Efficient entanglement concentration for arbitrary less-entangled N-atom state”, Int. J. Theor. Phys. (2014) DOI 10.1007/s10773-013-1974-8. • B. S. Choudhury and A. Dhara, “An entanglement concentration protocol for cluster states”, Quant. Inf. Process. 12 (2013) 2577. • T.-T. Xu, W. Xiong and L. Ye, “An efficient scheme for entanglement concentration of arbitrary four-photon states”, Int. J. Theor. Phys. 52 (2013) 2981. • S.-Y. Zhao, J. Liu, L. Zhou and Y.-B. Sheng, “Two-step entanglement concentration for arbitrary electronic cluster state”, Quant. Inf. Process. 12 (2013) 3633. • L. Zhou, Y.-B. Sheng, W.-W. Cheng, L.-Y. Gong and S.-M. Zhao, “Efficient entanglement concentration for arbitrary less-entangled NOON states”, Quant. Inf. Process. 12 (2013) 1307. • Z. Zhao, T. Yang, Y. A. Chen, A. N. Zhang and J. W. Pan, “Experimental realization of entanglement concentration and a quantum repeater”, Phys. Rev. Lett. 90 (2003) 207901. • T. Yamamoto, M. Koashi, S. K. Ozdemir and N. Imoto, “Experimental extraction of an entangled photon pair from two identically decohered pairs”, Nature 421 (2003) 343. • W. {\rm D\ddot{u}r} , G. Vidal and J. I. Cirac, “Three qubits can be entangled in two ways”, Phys. Rev. A 62 (2000) 062314. • A. Banerjee, K. Patel and A. Pathak, “Comment on quantum teleportation via GHZ-like state”, Int. J. Theor. Phys. 50 (2011) 507. • C. Shukla, V. Kothari, A. Banerjee and A. Pathak, “On the group-theoretic structure of a class of quantum dialogue protocols”, PLA 377 (2013) 518. • K. Yang, L. Huang, W. Yang and F. Song, “Quantum teleportation via GHZ-like state”, Int. J. Theor. Phys. 48 (2009) 516. • F. Verstraete, J. Dehaene, B. De Moor and H. Verschelde, “Four qubits can be entangled in nine different ways”, Phys. Rev. A 65 (2002) 052112. • O. Chterental and D. Z. Djokovic, “Normal forms and tensor ranks of pure states of four qubits”, arxiv:quant-ph/0612184. • L. Borsten, D. Dahanayake, M. J. Duff, A. Marrani and W. Rubens, “Four-qubit entanglement classification from string theory” , Phys. Rev. Lett. 105 (2010) 100507. • Z. Zhao, J. W. Pan and M. S. Zhan, “Practical scheme for enatnglement concentration”, Phys. Rev. A 64 (2001) 014301.

27. • J. L. Romero, L. Roa, J. C. Retamal and C. Saavedra, “Entanglement purification in cavity QED using local operations”, Phys. Rev. A 65 (2002) 052319. • J. R. Samal, M. Gupta, P. K. Panigrahi and A. Kumar, “Non-destructive discrimination of Bell states by NMR using a single ancilla qubit”, J. Phys. B 43 (2010) 095508. • C. Shukla, A. Banerjee and A. Pathak, “Bidirectional controlled teleportation by using 5-Qubit states: A generalized view”, Int. J. Theor. Phys. 52 (2013) 3790. • C. Shukla and A. Pathak, “Hierarchical quantum communication”, Phys. Lett. A 377 (2013) 1337. • B. Pradhan, P. Agrawal and A. K. Pati, “Teleportation and superdense coding with genuine quadripartite entangled states”, arxiv:0705.1917v1 [quant-ph]. • S.-W. Lee and H. Jeong, “Bell-state measurement and quantum teleportation using linear optics: two-photon pairs, entangled coherent states, and hybrid entanglement”, arxiv:1304.1214 [quant-ph]. • S.-H. Xiang, et al. “Concentration scheme for partially entangled photon states via entanglement reflector and no Bell-state analysis” , Opt. Commun. 284 (2011) 2402. • J. L. O'Brien, G. J. Pryde, A. G. White, T. C. Ralph and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate”, Nature 426 (2003) 264.C. • Sabın and G. Garca-Alcaine, “A classification of entanglement in three-qubit systems”, Eur. Phys. J. D 48 (2008) 435. • C. Eltschka, A. Osterloh, J. Siewert and A. Uhlmann, “Three-tangle for mixtures of generalized GHZ and generalized W states” , New Journal of Physics 10 (2008) 043014 • .W. K. Wootters, “The rebit three-tangle and its relation to two-qubit entanglement”, arxiv:1402.2219v1 [quant-ph]. • C-s. Yu, H.-s. Song, “Free entanglement measure of multiparticle quantum states”, Phys. Lett. A 330 (2004) 377. • V. Coffman, J. Kundu and W. K. Wootters, “Distributed Entanglement”, Phys. Rev. A 61 (2000) 052306. • P. Rungta, V. Buzek, C. M. Caves, M. Hillery and G. J. Milburn, “Universal state inversion and concurrence in arbitrary dimensions”, Phys. Rev. A 64 (2001) 042315. • V. Vedral and M. B. Plenio, “Entanglement measures and purifcation procedures”, Phys. Rev. A 57 (1998) 1619.

28. Thank you