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Interpolating Between Quantum Teleportation Protocols

Explore Quantum Teleportation Protocols, Convex-Split Lemma, unified schemes, and multipartite teleportation. Understand resource states, convex-split lemma, and generalizing teleportation protocols. Discuss new developments and goals for efficient teleportation.

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Interpolating Between Quantum Teleportation Protocols

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  1. InterpolatingBetween Quantum TeleportationProtocols MICHAŁ STUDZIŃSKI, UNIVERSITY OF GDAŃSK 17.06.2019, 51th Symposium on Mathematical Physics, Toruń Joint work with: M. Horodecki (University of Gdańsk) M. M. Wilde (LouisianaState University)

  2. Outline of the talk: • VariousTypes of Quantum TeleportationProtocols • Convex Split Lemma • UnifiedTeleportationScheme • InterpolatingBetweenTeleportationSchemes • Multipartite Quantum TeleportationSchemes • EntanglementFidelity of the Protocols

  3. QUANTUM TELEPORTATION PROTOCOLS

  4. Standard QuatumTeleportation (Bennet et al. PRL 70(13):1895-1899, 1993) It worksalso for ‚A completebasis of generalized Bell states’, NJP 11(1):013006 (2009)

  5. Port-basedTeleportationFor d=2; PRL 101(24):240501, 2008For d>2; Sci. Rep. (2017); 7:10871 & NJP 20.5 (2018): 053006 • Anyentangledstateiscalled port • No unitarycorrection !! • Idealonly in asymptotic limit! • Complicatedmeasurements (square-rootmeasurements)

  6. TwoTypes of PBT Scheme for M. Christandl, F. Leditzky, Ch. Majenz, G. Smith, F. Speelman, M. Walter, arXiv:1809.10751v1

  7. Why do we care? Identification of cause-effect relations Someaspect of commuicationcomplexity Port-basedTeleportation Instantenous non-local quantum computations Universal programmable quantum processor Cryptographicattacks Quantum channelsdiscrimination

  8. QUANTUM TELEPORTATION PROTOCOLS • Isthereanyunifiedscheme? • Can we producenew quantum teleportation • protocols? • Our list of wishes: • To reduceamount of entaglement • To teleportmoreparticles with higherfidelity • To usesimplermeasurements • Possiblynewapplications

  9. General Idea Behind the Quantum Teleportation ALICE BOB Resource State

  10. CONVEX-SPLIT LEMMA A.Anshu, V. K. Devabathini, R. Jain, PRL 119, 120506 (2017)

  11. Convex-split Lemma Definition (Quantum RelativeEntropy) For anytwo quantum states quantum relativeentropyisgiven as Definition (max-relativeentropy) For anytwo quantum states max-relativeentropyisgiven as Definition (Fidelity) Fidelitybetween quantum statesisrepresented as

  12. Convex-split Lemma Lemma Letand be quantum statessuchthat. Let. Define the followingstate: Onregisterswhereand. Then,

  13. THE GENERALISED TELEPORTATION PROTOCOL

  14. The generalisedteleportationprotocol • Let ustake, everyacts on the Hilbert space. • The goal: Teleport system of a state to Bob. • Preparing of the purestate • Alice prepares the state and then the controlledunitary

  15. The generalisedteleportationprotocol • The totalstateisnow of the form • Alice performsanisometric channel (purification) • Measures the systems and getting and • Classicalcommunication with Bob • The state of the system • Bob has to apply to recover the state

  16. Whatabout the efficiency? • Let usfind non-trivialbound on the fidelity • We define the followingstate and

  17. Whatabout the efficiency? • Applying Convex-Split Lemma we arrive with where

  18. INTERPOLATING BETWEEN TELEPORTATION SCHMES

  19. EXAMPLES • Standard quantum teleportation: - max. entangledstate, - Heisenberg-Weylgroup of qudit • Original PBT scheme: -max. entangledstate, - max. entangledstate, - only the identity element

  20. EXAMPLES • Other example: - max. entangledstates, - only the phase-shiftoperators where

  21. GENERALISATIONS TO NEW PROTOCOLS GOAL: We wouldlike to teleport quantum states in one go usingports. • For the simplicityletusfocus on the casewhen. • Alice wishes to teleportsystems from where

  22. FURTHER DEVELOPMENTS DIRECTIONS OF POSSIBLE RESEARCH • Findexactexpression for fidelity for variouschoices of • Try to formulateprobabilistic version for generalised PBT • (Dis)provesimulability by projectivemeasurements • Find the tradeoffbetweenamount of entanglement and correction POSSIBLE APPLICATIONS (?) • Information compression • Multipartiteinstantenous non-local quantum computations • Multipartiteposition-basedcryptography

  23. REFERENCES Port-basedTeleportation • S. Ishizaka, T. Hiroshima, PRL 101, 240501 (2008) - solution for qubits • S. Ishizaka, T. Hiroshima, PRA 79, 042306 (2009) – solution for qubita + optimality • M. Studziński et al. Sci. Rep. 7, 10871 (2017)- solution for higherdimensions + optimality • M. Mozrzymas et al. NJP 20.5, 053006 (2018)- solutions for higherdimensions + optimality • M. Christandl et al. arXiv.1809.10751-full description of asymptotic performance Applications of Port-basedTeleportation • S.Beigi, R. Koenig, NJP 13, 093036 (2011) - positionbased quantum cryptography • H. Buhrman et al. PNAS 113, 3191-3196 (2016) – Bell inequalityviolations vs complexitytheory • G. Chiribella, D. Ebler, Nature Communication 10, 1472 (2018) – identification of cause-effect relations • S. Pirandola et al. NPJ Quantum Information 5,3 (2019) – limitations on quantum channelsdiscrimination

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