Detecting Genuine Multi-qubit Entanglement with Two Local

Measurement Settings

Géza Tóth (MPQ)

Otfried Gühne (Innsbruck)

Quantum Optics II,Cozumel, Dec 2004quant-ph/0405165

Outline

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

A mixed entangled state is biseparable if it is the mixture of biseparabe states (of possibly different partitions).

Biseparable entanglement

000 111

001 111

00 11 1

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.

SSeparable states

Genuine multi-qubit entangled states

Biseparable entangled states

Witness#1

States detected

by Witness#1

Entanglement witnesses II

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.

11

2PROJGHZN N NW GHZ GHZ

Entanglement witnesses IV

So if

then the system is genuinely multi-qubit entangled.

Question: how can we measure the witness operator?

0PROJGHZNW

Decomposing the witness

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

See O. Gühne, P. Hyllus, quant-ph/0301162; M. Bourennane et. al., PRL 92 087902 (2004).

1 2 1 3 2 3 1 2 33

1 1 2 2 3 3

1 1 2 2 3 3

1(3 1 2 )

81

41

4

PROJGHZ z z z z z z x x x

x y x y x y

x y x y x y

W

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.

Entanglement witnesses based on the stabilizer formalism

Stabilizer witnesses

1 2 3 1 2 2 3 1 33

3 11

2 2GHZ x x x z z z z z zW

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

1 kk

W c S

kS

Stabilizing operators

For an N-qubit GHZ state

For an N-qubit cluster state

kS

(1) (2) (3) ( )1 ... ,N

x x x xS ( 1) ( ) ; 2,3,..., .k k

k z zS k N

(1) (2)1

( 1) ( ) ( 1)

( 1) ( )

,

; 2,3,..., 1,

.

x z

k k kk z x z

N NN z x

S

S k N

S

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).

4 0000 1100 0011 1111C

Stabilizer witnesses III

Optimal witness for N-qubit GHZ state

Optimal witness for N-qubit cluster state

( )( )1

1

113 2

2 2

NN GHZGHZk

GHZNk

SSW

( ) ( )1 13 2

2 2

N N

N

C Ck k

Ck even k odd

S SW

Stabilizer witnesses VI

The projector witness is also the sum of stabilizing operators

An alternative witness with the fewest terms:

( )111

2 2

NGHZPROJ kGHZN

k

SW

Stabilizer witnesses VI

( )

1

' ( 1) 1 N

N

NGHZ

GHZ kk

W N S

( ) ( )1 13 2

2 2

N N

N

C Ck k

Ck even k odd

S SW

z z zx x z zx x x( )( )

1

1

113 2

2 2

NN GHZGHZk

GHZNk

SSW

x x xx x z zz z z

Only the minimal two measurement settings are needed

Noise

3 3 _(1 )noise noise totally mixedp GHZ GHZ p

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%

Noise tolerance: 40% (2 settings)

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

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

1 2 3 1 2 3 1 2 3 1 2 33 2GHZ x x x y y x x y y y x yW

1 2 2 3 1 33 3 1PROJ

GHZ GHZ z z z z z zW W

Connection to Bell inequalities

1 2 3 1 2 2 3 1 33

3 11

2 2GHZ x x x z z z z z zW

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

quant-ph/0405165

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.

2 0PROJGHZN GHZNW W

11

2PROJGHZN N NW GHZ GHZ

Stabilizer witnesses V

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?

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.

O1 O2 O3 O4 O5 ...

1k kO O

1 2k k kO O O

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%)

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 II

SSeparable states

Genuine multi-qubit entangled states

Biseparable entangled states

Witness#1

Witness#2

States detected

by Witness#1