1. Distillation and determination of unknown two-qubit entanglement: Construction of optimal witness operator Heung-Sun Sim Physics, KAIST ESF conference: Quantum Engineering of States and Devices June 9, 2010, Obergurgl (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 Iterative distillation and quantification 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

4. Entanglement detection and quantifcation Detecting or quantifying entanglement?  Separability criteria  Entanglement measures  Two-qubit entanglement: Concurrence Pure state CH Bennett et al PRA 54, 3824 (1996) Example Concurrence 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  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 Thank you for your attention! (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)