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

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### Detecting Genuine Multi-qubit Entanglement with Two Local Measurement Settings

Géza Tóth (MPQ)

OtfriedGühne (Innsbruck)

Quantum Optics II,

Cozumel, Dec 2004

quant-ph/0405165

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

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

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.

Entanglement witnesses II

Witness#1

S

States detected

by Witness#1

Separable states

Biseparable entangled states

Genuine multi-qubit entangled states

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.

Entanglement witnesses IV

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

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).

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.

Stabilizer witnesses

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

Stabilizer witnesses II

- Our general method for constructing witnesses for states close to
- Here Sk stabilize

Stabilizing operators

- For an N-qubit GHZ stateFor an N-qubit cluster state

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).

Stabilizer witnesses III

- Optimal witness for N-qubit GHZ state
- Optimal witness for N-qubit cluster state

Stabilizer witnesses VI

Stabilizer witnesses VI

- The projector witness is also the sum of stabilizing operators
- An alternative witness with the fewest terms:

- 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.

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%

Connection to Bell inequalities

- Noise tolerance: 40% (2 settings)
- Noise tolerance: 50% (4 settings) Bell ineq.!
- Noise tolerance: 57% (4 settings)Projector!!

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!!! *************

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.

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?

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.

...

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%)

- 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)

Entanglement witnesses II

Witness#1

S

States detected

by Witness#1

Separable states

Biseparable entangled states

Genuine multi-qubit entangled states

Witness#2

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