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)
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
Outline II
A new scheme for detecting, distilling, and quantifying twoqubit 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
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)
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)
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
Theory: P Samuelsson, EV Sukhorukov, M Buttker, PRL (2004)
Experiment: I. Neder et al., Nature (2007)
multi-qubit GHZ interferometry in condensed matter
Theory: HS Sim, EV Sukhorukov, PRL (2006)
Optimal entanglement witness
Optimal witness
 Physical observable useful for entanglement quantification
 Defined relative to a given state
 Expectation value gives concurrence
 Graphical interpretation
Procrustean distillation
PG Kwiat et al., Nature (2001)
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
Our setup
 How to attach the filtering operation into the interferometry?
f<1
Using beam splitter
(or quantum point contact in
quantum Hall interferometry)
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
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)
Experimental demonstration
H. S. Park, S.-S. B. Lee, H. Kim, S.-K. Choi, and H.-S. Sim, preprint (2010)
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!