Spin-squeezing inequalities
for entanglement detection in cold gases
Phys. Rev. Lett. 107, 240502 (2011)
G. Tóth1,2,3 , G. Vitagliano1 , P. Hyllus1 , and I.L. Egusquiza1 ,
1 Theoretical
Physics, University of the Basque Country UPV/EHU, Bilbao, Spain
Basque Foundation for Science, Bilbao, Spain
3 Wigner Research Centre for Physics, Budapest, Hungary
2 IKERBASQUE,
ICSSUR 2013, Nürnberg,
24 June 2013
1 / 27
Outline
1
Motivation
Why spin squeezing inequalities are important?
2
Multipartite entanglement
Definition of entanglement
3
Spin squeezing entanglement criteria for j = 1/2
Collective measurements
The original criterion
Generalized criteria for j = 12
4
Spin squeezing inequality for an ensemble of spin-j atoms
Conditions with the angular momentum coordinates for j > 12
The usual spin squeezing inequality for j > 12
Conditions with the SU(d) generators
Detection of singlets
3 / 27
Why spin squeezing inequalities for j >
important?
1
2
is
Many experiments are aiming to create entangled states with
many atoms.
Only collective quantities can be measured.
Most experiments use atoms with j > 12 .
Outline
1
Motivation
Why spin squeezing inequalities are important?
2
Multipartite entanglement
Definition of entanglement
3
Spin squeezing entanglement criteria for j = 1/2
Collective measurements
The original criterion
Generalized criteria for j = 12
4
Spin squeezing inequality for an ensemble of spin-j atoms
Conditions with the angular momentum coordinates for j > 12
The usual spin squeezing inequality for j > 12
Conditions with the SU(d) generators
Detection of singlets
5 / 27
Entanglement
Definition
A multiparticle state is (fully) separable if it can be written as
X
(k )
(k )
(k )
pk %1 ⊗ %2 ⊗ ... ⊗ %N .
k
If a state is not fully separable, then it is called entangled.
Outline
1
Motivation
Why spin squeezing inequalities are important?
2
Multipartite entanglement
Definition of entanglement
3
Spin squeezing entanglement criteria for j = 1/2
Collective measurements
The original criterion
Generalized criteria for j = 12
4
Spin squeezing inequality for an ensemble of spin-j atoms
Conditions with the angular momentum coordinates for j > 12
The usual spin squeezing inequality for j > 12
Conditions with the SU(d) generators
Detection of singlets
7 / 27
Many-particle systems for j=1/2
For spin- 12 particles, we can measure the collective angular
momentum operators:
Jl :=
1
2
N
X
(k )
σl ,
k =1
(k )
where l = x, y , z and σl
a Pauli spin matrices.
We can also measure the variances
(∆Jl )2 := hJl2 i − hJl i2 .
Outline
1
Motivation
Why spin squeezing inequalities are important?
2
Multipartite entanglement
Definition of entanglement
3
Spin squeezing entanglement criteria for j = 1/2
Collective measurements
The original criterion
Generalized criteria for j = 12
4
Spin squeezing inequality for an ensemble of spin-j atoms
Conditions with the angular momentum coordinates for j > 12
The usual spin squeezing inequality for j > 12
Conditions with the SU(d) generators
Detection of singlets
9 / 27
The standard spin-squeezing criterion
The spin squeezing criteria for entanglement detection is
(∆Jx )2
1
≥ .
2
2
N
hJy i + hJz i
If it is violated then the state is entangled.
[A. Sørensen, L.M. Duan, J.I. Cirac, P. Zoller, Nature 409, 63 (2001).]
States violating it are like this:
Variance of Jx is small
Jz is large
z
x
y
Outline
1
Motivation
Why spin squeezing inequalities are important?
2
Multipartite entanglement
Definition of entanglement
3
Spin squeezing entanglement criteria for j = 1/2
Collective measurements
The original criterion
Generalized criteria for j = 12
4
Spin squeezing inequality for an ensemble of spin-j atoms
Conditions with the angular momentum coordinates for j > 12
The usual spin squeezing inequality for j > 12
Conditions with the SU(d) generators
Detection of singlets
11 / 27
Generalized spin squeezing criteria for j =
1
2
Let us assume that for a system we know only
~J := (hJx i, hJy i, hJz i),
~ := (hJ 2 i, hJ 2 i, hJ 2 i).
K
x
y
z
Then any state violating the following inequalities is entangled.
hJx2 i + hJy2 i + hJz2 i ≤
(∆Jx )2 + (∆Jy )2 + (∆Jz )2 ≥
N(N+2)
,
4
N
2,
hJk2 i + hJl2 i ≤ (N − 1)(∆Jm )2 +
h
i
N(N−2)
2
(N − 1) (∆Jk )2 + (∆Jl )2 ≥ hJm
i+ 4 ,
where k , l, m take all the possible permutations of x, y , z.
[GT, C. Knapp, O. Gühne, and H.J. Briegel, PRL 99, 250405 (2007)]
N
2,
Generalized spin squeezing criteria for j =
1
2
〈 J2z 〉
The previous inequalities, for fixed hJx/y /z i, describe a polytope in
2
the hJx/y
/z i space.
D E
For ~J = 0 and N = 6 the polytope is the following:
10
Az
5
By
〈 J2 〉
x
Ay
Bx
0
0S
Bz
Ax
5
10 0
5
10
〈 J2 〉
y
Completeness
4
3.5
<J2z >
2
<Jz >
Random separable states:
3
3
4
2.5
2
2
1.5
1
1
0
0
0.5
2
0
0
1
2
3
4
<J2x >
<J2x > 4 0
The completeness can be proved for large N.
1
2
3
4
<J2y >
J = (0, 0, 2.5),
25
25
〈 J2z 〉
〈 J2z 〉
The polytope for N = 10 and
J = (0, 0, 0),
20
20
15
15
10
10
5
5
0
0
0
0
y
10
20
0
5
and J = (0, 0, 4.5).
20
15
10
〈 J2z 〉
10
〈 J2 〉
〈 J2y 〉
25
〈 J2x 〉
20
0
25
20
15
10
5
0
0
10
〈 J2y 〉
20
0
5
10
15
20 2 25
〈 Jx 〉
5
10
15
25
20
〈 J2x 〉
Outline
1
Motivation
Why spin squeezing inequalities are important?
2
Multipartite entanglement
Definition of entanglement
3
Spin squeezing entanglement criteria for j = 1/2
Collective measurements
The original criterion
Generalized criteria for j = 12
4
Spin squeezing inequality for an ensemble of spin-j atoms
Conditions with the angular momentum coordinates for j > 12
The usual spin squeezing inequality for j > 12
Conditions with the SU(d) generators
Detection of singlets
16 / 27
“Modified” quantities for j >
1
2
For the j = 12 case, the SSIs were developed based on the first
and second moments and variances of the such collective
operators.
For the j >
1
2
case, we define the modified second moment
hJ̃k2 i := hJk2 i − h
X
(n)
(jk )2 i =
n
X
(n) (m)
hjk jk i
m,n
and the modified variance
˜ k )2 := (∆Jk )2 − h
(∆J
X
(n)
(jk )2 i.
n
These are essential to get tight equations for j > 12 .
The inequalities for j> 12 with the angular
momentum coordinates
For fully separable states of spin-j particles, all the following
inequalities are fulfilled
hJx2 i + hJy2 i + hJz2 i ≤ Nj(Nj + 1),
(∆Jx )2 + (∆Jy )2 + (∆Jz )2 ≥ Nj,
˜ m )2 ,
hJ̃k2 i + hJ̃l2 i − N(N − 1)j 2 ≤ (N − 1)(∆J
h
i
2
˜ k )2 + (∆J
˜ l )2 ≥ hJ̃m
(N − 1) (∆J
i − N(N − 1)j 2 ,
where k , l, m take all possible permutations of x, y , z.
Violation of any of the inequalities implies entanglement.
Completeness
In the large N limit, the inequalities with the angular momentum
are complete.
It is not possible to find new entanglement conditions based on
hJk i and hJ̃k2 i that detect more states.
Outline
1
Motivation
Why spin squeezing inequalities are important?
2
Multipartite entanglement
Definition of entanglement
3
Spin squeezing entanglement criteria for j = 1/2
Collective measurements
The original criterion
Generalized criteria for j = 12
4
Spin squeezing inequality for an ensemble of spin-j atoms
Conditions with the angular momentum coordinates for j > 12
The usual spin squeezing inequality for j > 12
Conditions with the SU(d) generators
Detection of singlets
20 / 27
The usual spin squeezing inequality for j >
1
2
The standard spin-squeezing inequality becomes
(∆Jx )2
+
hJy i2 + hJz i2
P
2
(n)
− h(jx )2 i)
1
≥ .
2
2
N
hJy i + hJz i
n (j
Violated only if there is entanglement between the spin-j particles.
The second term on the LHS is nonnegative.
Outline
1
Motivation
Why spin squeezing inequalities are important?
2
Multipartite entanglement
Definition of entanglement
3
Spin squeezing entanglement criteria for j = 1/2
Collective measurements
The original criterion
Generalized criteria for j = 12
4
Spin squeezing inequality for an ensemble of spin-j atoms
Conditions with the angular momentum coordinates for j > 12
The usual spin squeezing inequality for j > 12
Conditions with the SU(d) generators
Detection of singlets
22 / 27
The inequalities for j >
1
2
with the Gk ’s
Collective operators:
Gl :=
N
X
(k )
gl ,
k =1
(k )
where l = 1, 2, ..., d 2 − 1 and gl
are the SU(d) generators.
We can also measure the
(∆Gl )2 := hGl2 i − hGl i2
variances.
Outline
1
Motivation
Why spin squeezing inequalities are important?
2
Multipartite entanglement
Definition of entanglement
3
Spin squeezing entanglement criteria for j = 1/2
Collective measurements
The original criterion
Generalized criteria for j = 12
4
Spin squeezing inequality for an ensemble of spin-j atoms
Conditions with the angular momentum coordinates for j > 12
The usual spin squeezing inequality for j > 12
Conditions with the SU(d) generators
Detection of singlets
24 / 27
An example: The criterion for SU(d) singlets
A condition for two-producibility (i.e., higher form of
entanglement) for N qudits of dimension d is
X
(∆Gk )2 ≥ 2N(d − 2).
k
A condition for separability is
X
(∆Gk )2 ≥ 2N(d − 1).
k
[ G. Vitagliano, P. Hyllus, I.L. Egusquiza, and G. Tóth,
Spin squeezing inequalities for arbitrary spin, PRL 2011. ]
Group
Philipp Hyllus
Zoltán Zimborás
Iñigo Urizar-Lanz
Giuseppe Vitagliano
Iagoba Apellaniz
Research Fellow (2011-2012)
Research Fellow (2012-)
Ph.D. Student
Ph.D. Student
Ph.D. Student
Topics
Multipartite entanglement and its detection
Metrology, cold gases
Collaborating on experiments:
Weinfurter group, Munich, (photons)
Mitchell group, Barcelona, (cold gases)
Funding:
European Research Council starting grant 2011-2016,
1.3 million euros
CHIST-ERA QUASAR collaborative EU project (H. Weinfurter)
Grants of the Spanish Government and the Basque Government
Summary
Full set of generalized spin squeezing inequalities with Jl with
l = x, y , z for j > 21 .
Large set of inequalities with the other collective operators.
These might make possible new experiments and make existing
experiments simpler.
See: G. Vitagliano, P. Hyllus, I.L. Egusquiza, and G. Tóth,
Phys. Rev. Lett. 107, 240502 (2011) + manuscript in preparation.
See www.gtoth.eu
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