Detecting Genuine M ulti -qubit Entanglement with Two Local Measurement Settings

1. Detecting Genuine Multi-qubit Entanglement with Two Local Measurement Settings Géza Tóth (MPQ) OtfriedGühne (Innsbruck) Quantum Optics II, Cozumel, Dec 2004 quant-ph/0405165

2. Outline • Genuine multi-qubit entanglement • Entanglement detection with entanglement witnesses • Witness based on projectors • Our proposal: witness with few localmeasurements (for GHZ & cluster states) • Connection to Bell inequalities

3. Genuine multi-qubit entanglement • A mixed entangled state is biseparable if it is the mixture of biseparabe states (of possibly different partitions). • Biseparable entanglement • Genuine three-qubit entanglement

4. Entanglement witnesses I • Entanglement witnesses are observables which have positive expectation values for separable states negative expectation values for some entangled states. • Witnesses can be constructed which detect entangled states close to a state chosen by us. • Witnesses can be constructed which detect only genuine multi-party entanglement.

5. Entanglement witnesses II Witness#1 S States detected by Witness#1 Separable states Biseparable entangled states Genuine multi-qubit entangled states

6. Entanglement witnesses III • It is possible to construct witnessesfor detecting entangled states close to a particular state with a projector. E.g.,detects N-qubit entangled states close to an N-qubit GHZ state.

7. Entanglement witnesses IV • So if then the system is genuinely multi-qubit entangled. • Question: how can we measure the witness operator?

8. Decomposing the witness • For an experiment, the witnessmust be decomposed into locally measurable terms • See O. Gühne, P. Hyllus, quant-ph/0301162; M. Bourennane et. al., PRL 92 087902 (2004).

9. Main topic of the talk: How can one decrease the number of local terms • The number of local terms increases rapidly (exponentially?) with the number of qubits. • More importantly, we need more and more measurement settings to measure. (This is also true for Bell inequalities.) • Entanglement detection becomes harder and harder for increasing number of qubits.

10. Entanglement witnesses based on the stabilizer formalism

11. Stabilizer witnesses • We propose new type of witnesses. E.g., for three-qubit GHZ states • All correlation terms are +1 for the GHZ state.

12. Stabilizer witnesses II • Our general method for constructing witnesses for states close to • Here Sk stabilize

13. Stabilizing operators • For an N-qubit GHZ stateFor an N-qubit cluster state

14. Cluster state • Can easily be obtained from Ising spin chain dynamics. Often encountered in error correction. • For N=3 qubits it is equivalent to a GHZ state • For N=4 qubits it is equivalent to • See Briegel, Raussendorf, PRL 86, 910 (2001).

15. Stabilizer witnesses III • Optimal witness for N-qubit GHZ state • Optimal witness for N-qubit cluster state

16. Stabilizer witnesses VI Stabilizer witnesses VI • The projector witness is also the sum of stabilizing operators • An alternative witness with the fewest terms:

17. Only the minimal two measurement settings are needed x z x x z x x x z x x z z z x z z z x z

18. Noise • In an experiment the GHZ state is neverprepared perfectly • For each witness there is a noise limit.For a noise larger than this limit the GHZ state is not detected as entangled.

19. Noise tolerance: our witnesses are optimal • Witness for N-qubit GHZ state  for N=3 : 40%  for large N : >33% • Witness for N-qubit cluster state  for N=4 :33%  for large N : >25%

20. Connection to Bell inequalities • Noise tolerance: 40% (2 settings) • Noise tolerance: 50% (4 settings) Bell ineq.! • Noise tolerance: 57% (4 settings)Projector!!

21. quant-ph/0405165 Summary • Detection of genuine N-qubit entanglement was considered with few local measurements. • The methods detect entangled states close to N-qubit GHZ and cluster states. • Home page: http://www.mpq.mpg.de/Theorygroup/CIRAC/people/toth • *************** THANK YOU!!! *************

22. Stabilizer witnesses V Stabilizer witnesses V • Why do these witnesses detect genuineN-qubit entanglement? Because • Any state detected by our witness is also detected by the projector witness. Later detectsgenuine N-qubit entanglement.

23. Main topic of the talk: How can one decrease the number of local terms • As the number of qubits increases, the number of local terms increases exponentially. Similar thing happens to Bell inequalities for the GHZ state. • Q: How can we construct entanglement witnesses with few locally measurable terms?

24. O3 O5 O1 O2 O4 What is a measurement setting? • Measurement setting is the basic unit of experimental effort. At each qubit operator Ok is measured. • After repeating the measurements several times, two-point correlations , three-point correlations , etc., can be obtained. ...

25. Stabilizer witnesses III • Characteristics for our N-qubit entanglement witnesses Usually the minimal 2 measurement settings For large N, tolerates noise pnoise<33% (GHZ) / 25% (cluster) For small N, noise tolerance is better (N=3; 40% / N=4; 33%)

26. Entanglement witnesses I • Bell inequalities Classical: no knowledge of quantum mechanics is used to construct them. Need many measurements. • Entanglement witnesses QM is used for constructing them. Can one detect entanglement with fewer measurements? (Yes)

27. Entanglement witnesses II Witness#1 S States detected by Witness#1 Separable states Biseparable entangled states Genuine multi-qubit entangled states Witness#2