Detecting genuine m ulti qubit entanglement with two local measurement settings
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Detecting Genuine M ulti -qubit Entanglement with Two Local Measurement Settings. G é za T ó th (MPQ) Otfried Gühne (Innsbruck ). Quantum Optics II, Cozumel, Dec 2004. quant-ph/0 405165. Outline. Genuine multi-qubit entanglement Entanglement detection with entanglement witnesses

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Detecting genuine m ulti qubit entanglement with two local measurement settings

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
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
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 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 i i
Entanglement witnesses II

Witness#1

S

States detected

by Witness#1

Separable states

Biseparable entangled states

Genuine multi-qubit entangled states


Entanglement witnesses iii
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
Entanglement witnesses IV

  • So if then the system is genuinely multi-qubit entangled.

  • Question: how can we measure the witness operator?


Decomposing the witness
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).


Detecting genuine m ulti qubit entanglement with two local measurement settings

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
Stabilizer witnesses local terms

  • 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
Stabilizer witnesses II local terms

  • Our general method for constructing witnesses for states close to

  • Here Sk stabilize


Stabilizing operators
Stabilizing operators local terms

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


Cluster state
Cluster state local terms

  • 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
Stabilizer witnesses III local terms

  • Optimal witness for N-qubit GHZ state

  • Optimal witness for N-qubit cluster state


Stabilizer witnesses v i
Stabilizer witnesses V local termsI

Stabilizer witnesses VI

  • The projector witness is also the sum of stabilizing operators

  • An alternative witness with the fewest terms:


Only the minimal two measurement settings are needed
Only the minimal two measurement settings are needed local terms

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Detecting genuine m ulti qubit entanglement with two local measurement settings

Noise local 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.


Detecting genuine m ulti qubit entanglement with two local measurement settings

Noise tolerance: our witnesses are optimal local terms

  • 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%


Detecting genuine m ulti qubit entanglement with two local measurement settings

Connection to Bell inequalities local terms

  • Noise tolerance: 40% (2 settings)

  • Noise tolerance: 50% (4 settings) Bell ineq.!

  • Noise tolerance: 57% (4 settings)Projector!!


Summary

quant-ph/0 local terms405165

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 local terms

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
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?


Detecting genuine m ulti qubit entanglement with two local measurement settings

O local terms3

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 iii1
Stabilizer witnesses III local terms

  • 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%)


Detecting genuine m ulti qubit entanglement with two local measurement settings

Entanglement witnesses I local terms

  • 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 i i1
Entanglement witnesses I local termsI

Witness#1

S

States detected

by Witness#1

Separable states

Biseparable entangled states

Genuine multi-qubit entangled states

Witness#2