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

DevicesJune 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)

Outline IA two-qubit interferometer:

• Local filtering operations (SLOCC) to focus on entanglement (Procrustean

distillation)

• Two-qubit correlation to construct the optimal entanglement witness

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

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 extensionAlgebraic 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 copiesS. P. Walborn et al., Nature

(2006)

Our goal:

Construct the optimal witnesswithout 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)

Optimal entanglement witness

Optimal witness Physical observable useful for entanglement

quantification Defined relative to a given state

Expectation value gives concurrence

Graphical interpretation

Procrustean distillationProcrustean 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)

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 tomographyS.-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

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)