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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.
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)