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Distillation and determination of unknown two-qubit entanglement:

ESF conference: Quantum Engineering of States and Devices June 9, 2010, Obergurgl. Distillation and determination of unknown two-qubit entanglement: Construction of optimal witness operator. Heung-Sun Sim Physics, KAIST. (theory) S.-S. B. Lee and H.-S. Sim, PRA 79, 052336 (2009)

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Distillation and determination of unknown two-qubit entanglement:

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  1. ESF conference: Quantum Engineering of States and Devices June 9, 2010, Obergurgl Distillation and determination of unknown two-qubit entanglement: Construction of optimal witness operator Heung-Sun Sim Physics, KAIST (theory) S.-S. B. Lee and H.-S. Sim, PRA 79, 052336 (2009) (theory + experiment) H. S. Park, S.-S. B. Lee, H. Kim, S.-K. Choi, and H.-S. Sim, arXiv:1006.1491 (2010) Acknowledgement: S.-S. B. Lee (KAIST), H. S. Park, H. Kim, S.-K. Choi (KRISS)

  2. Outline I • A two-qubit interferometer: • Local filtering operations (SLOCC) •  to focus on entanglement (Procrustean distillation) • Two-qubit correlation •  to construct the optimal entanglement witness

  3. Outline II • A new scheme for detecting, distilling, and quantifying two-qubit entanglement without full state reconstruction. • The exact value of concurrence is determined. Always successful. • Better efficiency than quantum state tomography • Experimental demonstration with photons • Extendible to multiqubit cases, qubits in condensed matter Iterative distillation and quantification

  4. Entanglement detection and quantifcation • Detecting or quantifying entanglement? • Separability criteria • Entanglement measures • Two-qubit entanglement: Concurrence Concurrence Pure state CH Bennett et al PRA 54, 3824 (1996) Example Mixed state: convex roof extension Algebraic expression is available! WK Wootters PRL 80, 2245 (1998)

  5. Entanglement detection and quantifcation in experiments 1. Tomography + mathematical criteria - Do state tomography and apply criterion or concurrence formula. - Weakness: Full state reconstruction. Indirect. • Impractical in multiqubit cases. - Most efficient way so far. • 2. Bell inequality - Classical concept. Violation means entanglement. • - Weakness: Not always successful. Not quantitative. • 3. Entanglement witness • - Physical observable. Negative expectation values mean entanglement. • - Weakness: Not always successful. Not quantitative. • Questions: • - Quantification without tomography? • - Measuring entanglement measure? • Nonlinear functions of density matrix… • - Two-qubit experiments with two state copies • S. P. Walborn et al., Nature (2006)

  6. Our goal: Construct the optimal witness without referring the full knowledge of the target state Modification of a two-qubit interferometry is necessary! two-qubit interferometry in quantum optics two-qubit interferometry in condensed matter multi-qubit GHZ interferometry in condensed matter Theory: P Samuelsson, EV Sukhorukov, M Buttker, PRL (2004) Experiment: I. Neder et al., Nature (2007) Theory: HS Sim, EV Sukhorukov, PRL (2006)

  7. Optimal entanglement witness • Optimal witness • Physical observable useful for entanglement quantification • Defined relative to a given state • Expectation value gives concurrence • Graphical interpretation

  8. Procrustean distillation • Procrustean distillation • Enhance entanglement via SLOCC (stochastic local operation and classical communication) • Example: Stochastic local filtering of qubit 1 when qubit 1 is downspin.  • Link to the optimal witness PG Kwiat et al., Nature (2001)

  9. Our setup • How to attach the filtering operation into the interferometry? f<1 Using beam splitter (or quantum point contact in quantum Hall interferometry)

  10. How to achieve the maximal distillation • Maximal distillation = Fully mixed local density matrices • Iteratively erase single-qubit interference until it vanishes This procedure does not require full state reconstruction

  11. How to construct the optimal witness • Measure two-qubit correlation (coincidence counting) • Three different pairs of local extrema of • First find the settings for measuring and then measure • Not require tomography, More efficient than tomography S.-S. B. Lee and H.-S. Sim, PRA 79, 052336 (2009)

  12. Experimental demonstration H. S. Park, S.-S. B. Lee, H. Kim, S.-K. Choi, and H.-S. Sim, preprint (2010)

  13. Summary • Determination of concurrence without quantum state tomography • The first construction of the optimal witness operator • Entanglement distillation and quantification within a single framework • Generic scheme for photons, electrons, … • Extendible to three-qubit Greenberger-Horne-Zeilinger entanglement (theory) S.-S. B. Lee and H.-S. Sim, PRA 79, 052336 (2009) (theory + experiment) H. S. Park, S.-S. B. Lee, H. Kim, S.-K. Choi, and H.-S. Sim, arXiv:1006.1491 (2010) Thank you for your attention!

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