Detecting Genuine Multiqubit Entanglement with Two Local Measurement

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

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

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

Entanglement witnesses I l Entanglement witnesses are observables which have positive expectation values for separable states l l 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 Biseparable entangled states Genuine multi-qubit entangled states ted tec de s#1 tes nes Sta Wit by S states Separable

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

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

Decomposing the witness l For an experiment, the witness must be decomposed into locally measurable terms l 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 l l l 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.

Entanglement witnesses based on the stabilizer formalism

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

Stabilizer witnesses II l Our general method for constructing witnesses for states close to l Here Sk stabilize

Stabilizing operators l For an N-qubit GHZ state For an N-qubit cluster state

Cluster state l 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 l See Briegel, Raussendorf, PRL 86, 910 (2001). l l

Stabilizer witnesses III l Optimal witness for N-qubit GHZ state l Optimal witness for N-qubit cluster state

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

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

Noise l In an experiment the GHZ state is never prepared perfectly l 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 l Witness for N-qubit GHZ state for N=3 : 40% for large N : >33% l Witness for N-qubit cluster state for N=4 : 33% for large N : >25%

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

quant-ph/0405165 Summary l l 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 l Why do these witnesses detect genuine N-qubit entanglement? Because l Any state detected by our witness is also detected by the projector witness. Later detects genuine N-qubit entanglement.

Main topic of the talk: How can one decrease the number of local terms l l 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?

What is a measurement setting? l Measurement setting is the basic unit of experimental effort. At each qubit operator Ok is measured. O 1 O 2 O 3 O 4 O 5 l . . . After repeating the measurements several times, two-point correlations , three-point correlations , etc. , can be obtained.

Stabilizer witnesses III l 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%)

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

Entanglement witnesses II Witness#1 ted tec de s#1 tes nes Sta Wit by S states Separable Biseparable entangled states Genuine multi-qubit entangled states Witness#2