Local Invariant Spin Squeezing
Criteria for multiqubit states
A. R. Usha Devi, Sudha and B. G. Divyamani
Department of Physics, Bangalore University, Bangalore
Department of Physics, Kuvempu University, Shankaraghatta
Inspire Institute, Alexandria, VA, USA
Tunga Mahavidyalaya, Thirthahalli
4th December 2013, HRI,
Allahabad
1
Outline of the talk
 Introduction: Spin Squeezing
 Exchange Symmetry, Local Invariance and Spin Squeezing:
The Need for Local Invariant Spin Squeezing Criteria (LISS):
LISS for symmetric and non-symmetric multiqubit states
 Relationship between LISS and Quantum Entanglement
 Connection between local invariants and LISS in symmetric
multiqubit states
 Conclusion and Future directions
2
Introduction
Recent
years
have
witnessed
revolutionary
improvement in the production, manipulation,
characterization and quantification of multiqubit
states because of their promising applications in high
precision atomic clocks, atomic interferrometry,
quantum metrology and quantum information
protocols.
Spin
Squeezing
and
Quantum
Entanglement are two important concepts in the
characterization and quantification of non-classical
atomic correlations.
3
Well before the focus on quantifying entanglement
begun, the notion of Spin Squeezing had caught
much attention both in the context of low-noise
spectroscopy
as
well
as
in
high
precision
interferrometry. While entanglement has its root
in superposition principle, squeezing originates
from uncertainty principle.
D.J. Wineland et.al., Phys. Rev. A 50, 67 (1994)
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Spin Squeezing Criteria
Several definitions of spin squeezing have been proposed in
the literature. In particular, Kitagawa and Ueda developed a
spin squeezing criteria based on the uncertainty relation
between collective angular momentum components of a
multiqubit state
K. Wodkiewicz and J. H. Eberly, J. Opt. Soc. Am. B. 2, 458 (1985)
M. Kitagawa and M. Ueda, Phys. Rev. A 47, 5138 (1993)
D. J. Wineland et. al, Phys. Rev. A 50, 67 (1994)
A. Sorenson et.al., Nature (London), 409, 63 (2001)
5
Spin Squeezing--Basic definition
6
Schematic representation of coherent and
spin squeezed states
Coherent state
Spin Squeezed
state
7
Kitagawa Ueda spin squeezing criteria
Kitagawa and Ueda also made an important observation
that identifying a mean spin direction of the multiqubit
state is essential for an unambiguous determination of
spin squeezing in multiqubit states.
M. Kitagawa and M. Ueda, Phys. Rev. A 47, 5138 (1993)
8
Kitegawa-Ueda Spin Squeezing Criteria
9
Kitegawa-Ueda Spin Squeezing Criteria
10
Wineland Squeezing Criteria
All Wineland Squeezed states are Kitegawa-Ueda squeezed
There exist Kitagawa-Ueda squeezed states that are not Wineland
squeezed
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Exchange Symmetry, Local Invariance and
Spin Squeezing
Spin squeezing, in the original sense, is defined for
multiqubit states that are invariant under the exchange
of particles. Such states, the so-called symmetric states
.
belong to the maximal multiplicity subspace of the
collective angular momentum operator. The possibility of
extending the concept of spin squeezing to multi-qubit
systems that are not necessarily symmetric under
interchange of particles and that are accessible not just to
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collective operations but also to local operations was
explored by Usha Devi et. al. This requires a criterion for
spin squeezing that exhibits invariance under local
unitary operations on the qubits.
.
It is important to notice that both the spin squeezing
parameters are not invariant under arbitrary local unitary
transformations on the qubits.
A.R. Usha Devi, X. Wang and B.C. Sanders, Quant. Infn. Proc. 2, 207 (2003)
13
Exchange Symmetry, Local Invariance and Spin Squeezing
.
14
Exchange Symmetry, Local Invariance and Spin Squeezing
.
15
Exchange Symmetry, Local Invariance and Spin Squeezing
From these two examples, we can conclude the following:
1) Both the criteria for spin squeezing do not exhibit local unitary
invariance. This aspect makes them inconvenient to relate with
the local unitary invariant entanglement criterion.
2) Both the criteria are defined for symmetric states alone and
using them to assess the spin squeezing in non-symmetric
states may lead to unphysical results.
Thus there arises a definite need for defining spin squeezing
criteria that remains local unitary invariant and applicable for
non-symmetric states also. We accomplish this through the
Local Invariant Spin Squeezing criteria (LISS).
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Local Invariant Spin Squeezing criteria
A.R. Usha Devi, X. Wang and B.C. Sanders, Quant. Infn. Proc. 2, 207 (2003)
A.R. Usha Devi and Sudha, Asian Journal of Physics,19, 1 (2010)
17
Local Invariant Spin Squeezing criteria
18
Local Invariant Spin Squeezing criteria
19
Local Invariant nature of spin squeezing parameters
Having defined the generalized spin squeezing parameters as
establishing their local invariant nature is of utmost importance.
To do this we only have to establish the local invariant nature of
the quantities
and
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Local Invariant nature of spin squeezing parameters
21
Local Invariant nature of spin squeezing parameters
22
Local Invariant nature of spin squeezing parameters
23
Operational approach towards the evaluation of
the local invariant spin-squeezing parameters
.
24
Operational approach towards the evaluation of the local invariant
spin-squeezing parameters
25
Operational approach towards the evaluation of the local invariant
spin-squeezing parameters
26
Local Invariant Spin Squeezing parameters for
non-summetric states
27
Local Invariant Spin Squeezing parameters for
non-summetric states
28
The generalized collective spin operators of a multi-qubit
spin squeezed state can be represented geometrically by
an elliptical cone of local invariant height
centered
about the z-axis (common qubit orientation direction in
the case of symmetric multiqubit states). The semi-minor
and semi-major axes of the ellipse being
corresponding value
and the
respectively. In contrast, the
coherent spin state is depicted by a circular cone.
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Spin Squeezed State
Coherent State
This geometric picture of the spin squeezed state versus that of a spin
coherent state illustrates collective spin-squeezing feature in symmetric as
well as non-symmetric states.
Relationship between spin squeezing and
quantum entanglement
A deeper understanding between spin squeezing and quantum
entanglement has been explored in the literature and it has been
established
that the presence of spin squeezing
essentially
reflects pairwise entanglement in symmetric multiqubit states.
Here we obtain a relationship
between local invariant spin
squeezing parameters and quantum entanglement quantified
through concurrence in two-qubit pure states.
A. Sorenson et.al., Nature (London), 409, 63 (2001)
D. Ulam-Orgikh and M. Kitagawa, Phys. Rev. A 64, 052106 (2001)
X. Wang and B.C Sanders, Phys. Rev. A 68, 03382 (2003)
X. Wang and K. Molmer, Eur. Phys. J. D 68, 03382 (2003)
31
1. Spin squeezing and quantum entanglement in
pure 2-qubit states
32
1. Spin squeezing and quantum entanglement in
pure 2-qubit states
33
1. Spin squeezing and quantum entanglement in
pure 2-qubit states
34
Graphs depicting the variation of the LISS squeezing parameters and concurrence
as a function of time for the N-qubit spin-down state evolving under one-axis
twisting Hamiltonian
1
N=2
1
2
C
0.5
0
0
2
3
2
2
t
1
N>2
N 3
N 5
N 8
1
0.96
0.92
0.88
0
2
3
t
2
2
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Graphs depicting the variation of the LISS squeezing parameters and concurrence as
a function of time for the pure state
evolving under Ising
interaction
1
0.8
0.6
0.4
0.2
0
1
2
C
0
2
3 2
2
t
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Spin squeezing in essentially non-symmetric
qubit states
 We now wish to examine essentially non-symmetric
multiqubit states and the nature of spin squeezing in
them.
 The loss of symmetry in initially symmetric
multiqubit pure states with
under Ising chain
evolution has been established earlier and we
consider the 4-qubit spin down state subjected to
Ising interaction.
Sudha, B,G. Divyamani and A.R. Usha Devi , Chin. Phys. Lett. 2, 020305 (2011)
37
Variation of pairwise entanglement and KitegawaUeda generalized spin squeezing in an essentially
nonsymmetric 4-qubit state
1
1
C
0.5
0
0
2
t
38
Entanglement and spin-squeezing in
multiqubit states
39
Entanglement and spin-squeezing in
multiqubit states
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Entanglement and spin-squeezing in
multiqubit states
 That is, fully separable states are not spin squeezed in
the Wineland sense.
 But this does not mean that all states with no
Wineland squeezing are of fully separable form
 In particular, there can be entangled states with
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Connection between local invariants and Kitagawa-Ueda
spin squeezing in symmetric multiqubit states
It is well known that entanglement of a composite quantum system remain
invariant, when the subsystems are subjected to local unitary operations. Any
two quantum states are entanglementwise equivalent iff they are related to
each other through local unitary transformations. In fact, the non-local
properties associated with a quantum state can be represented in terms of a
complete set of local invariants. While a set of 18 local invariants are required
for the complete description of an arbitrary two-qubit mixed state, the
number of invariants reduce to 6, when the two qubit state obeys exchange
symmetry. Symmetric states indeed offer elegant mathematical analysis as
the dimension of the Hilbert space reduces drastically from
to N+1,
when N two-level systems respect exchange symmetry.
Y. Makhlin, Quantum.Inf.Proc 1, 243 (2003)
A. R. Usha Devi, M.S. Uma, R. Prabhu and Sudha, J. Opt. B. 7, S740 (2005) 42
Connection between local invariants and Kitagawa-Ueda
spin squeezing in symmetric multiqubit states
The parameters of a collective phenomena like spin-squeezing,
reflecting pairwise entanglement of symmetric qubits, should
be expressible in terms of two qubit local invariants. In fact,
the local invariant version of Kitagawa-Ueda spin squeezing
parameter can be expressed in terms of one of the local
invariant quantity
associated with the symmetric two-qubit
reduced density matrix of the N qubit symmetric state.
We proceed to describe an elegant connection between spin
squeezing and pairwise entanglement.
A. R. Usha Devi, M.S. Uma, R. Prabhu and Sudha, J. Opt. B. 7, S740 (2005)
A. R. Usha Devi, M.S. Uma, R. Prabhu and Sudha, Int. J. Mod. Phys. B 20, 1917 (2006)
43
Connection between local invariants and Kitagawa-Ueda
spin squeezing in symmetric multiqubit states
44
Connection between local invariants and Kitagawa-Ueda
spin squeezing in symmetric multiqubit states
45
Connection between local invariants and Kitagawa-Ueda
spin squeezing in symmetric multiqubit states
46
Connection between local invariants and Kitagawa-Ueda
spin squeezing in symmetric multiqubit states
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Variation of the local invariant parameter
as a function of time for the
N-qubit spin down state subjected to one-axis twisting Hamilotonian.
0
N
N
N
N
0.02
0.04
2
3
5
8
0.06
0.08
0
2
3
2
2
t
The reduction in the negativity of
as N increases is clearly seen.
This supports our observation that squeezing reduces as N increases for this state48
Variation of the local invariant parameter
state
I
ti
as a function of time for the
subjected to Ising chain evolution.
0
0.2
0.4
0
2
3
2
2
t
The state under consideration exhibits spin squeezing at all times as is
indicated by the negative value of
. Maximum squeezing is seen at
time t=0 and shows a periodic decrease and increase.
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Conclusion
 We have demonstrated the need for Local Invariant Spin
Squeezing criteria that exhibit local invariance in
addition to being applicable to non-symmetric states.
 We have explicitly obtained the expressions for local
invariant spin squeezing parameters for symmetric as
well as non-symmetric states.
 The connection between quantum entanglement and
local invariant spin squeezing has been established
(especially for 2-qubit pure states).
 Examples from both symmetric and nonsymmetric class
of states are worked out.
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Future Directions. . .
 Extending the local invariant spin squeezing criteria
to higher dimensional states ( Spin squeezing criteria
for qudits has recently been discussed in quant-ph.
arxiv: 1310.2269)
 Establishing the connection between local invariants
of a non-symmetric multiqubit state with the
corresponding local invariant spin squeezing
parameters and thereby with pairwise entanglement.
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