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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).
slide9

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
  • 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
  • Our general method for constructing witnesses for states close to
  • Here Sk stabilize
stabilizing operators
Stabilizing operators
  • For an N-qubit GHZ stateFor an N-qubit cluster state
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
Stabilizer witnesses III
  • Optimal witness for N-qubit GHZ state
  • Optimal witness for N-qubit cluster state
stabilizer witnesses v i
Stabilizer witnesses VI

Stabilizer witnesses VI

  • The projector witness is also the sum of stabilizing operators
  • An alternative witness with the fewest terms:
slide18

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.
slide19

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%

slide20

Connection to Bell inequalities

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

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!!! *************
stabilizer witnesses v
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
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?
slide24

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.

...

stabilizer witnesses iii1
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%)

slide26

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)

entanglement witnesses i i1
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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