an experimentally accessible geometric measure for ...iscqi/talks/pjoag-talk.pdf · an...
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An Experimentally AccessibleGeometric Measure for Entanglement
in 3-qubit Pure states
Pramod S. JoagDepartment of Physics, University of Pune,
Pune, India-411007.
In collaboration with
Ali Saif M. Hassan
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Right from its inception, quantuminformation fraternity is confronted
with two basic questions :
(i) Given a multipartite quantum state(possibly mixed), how to find out
whether it is entangled or separable?
(ii) Given an entangled state, how todecide how much entangled it is?
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Answers to both these questions areknown for bipartite pure states.
(i) If ρ2A = ρA then ρ is separable.
(ii) Typical measure is theentanglement entropyE (|ψ〉) = S(ρA) = −∑
i λi lnλi
Zero for separable states, lnN formaximally entangled states.
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Multipartite statesGeneral answers to both these questions are
not knowns. Different types of entanglement.
Many separability criteria are proposed
Example: Generalizations of Peres-Horodecki
criterion. The genuine entanglement of pure
multipartite quantum state is established by
checking whether it is entangled in all
bipartite cuts, which can be tested using
Peres-Horodecki criterion.
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For mixed states this strategy does not work
because there are mixed states which are
separable in all bipartite cuts but are
genuinely entangled [PRL 1999, 82, 5385].
A direct and independent detection of
genuine multipartite entanglement is lacking.
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In this talk I present a new measure ofentanglement for 3-qubit pure states.I present all the results for N-qubitpure states except one which we couldprove only for two and three qubitpure states.
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Let ρ act on H ; dim(H) = d . ρ ∈ L(H);
scalar product (A, B) = Tr(A†B).
dim(L(H)) = d2.
ρ can be expanded in any orthonormal basis
of L(H).
The basis comprising d2 − 1 generators of
SU(d) is particularly useful:
{Id , λi ; i = 1, 2, · · · , d2 − 1}.
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{λi} are traceless Hermition operators
satisfying
Tr(λiλj) = 2δij
and λiλj = 2d δij Id + ifijkλk + gijkλk
fijk , gijk are completely antisymmetics
(symm.) tensors.
d = 2 :
λi ↔ σi ; fijk = εijk (Levi-civita) gijk = 0.
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ρ expanded in this basis:
ρ = 1d (Id +
∑i siλi) (A)
where si = 〈λi〉 = Tr(ρλi) is the averagevalue of the ith generator λi in thestate ρ.
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Bloch Vectors
The vector s = (s1, s2, . . . , sd2−1); si = 〈λi〉is called the Bloch vector of state.
The correspondence s ↔ ρ
via the expansion of ρ in (A) is one-to-one .
Thus we can use s to specify a quantum
state.
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Note that s is very easily accessible
experimentally because all the averages can
be directly computed using the outputs of
measurements of {λi}; i = 1, 2, · · · , d2 − 1.
In fact the Bloch vector s can be obtaind
experimenttally even if the form of ρ is not
known.
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Bloch vector space
Bloch vectors for a given system live in
Rd2−1.
If we put an arbitrary vector ∈ Rd2−1 in
equation (A) we may not get a valid density
operator.
A density operator has to satisfy
(i) Trρ = 1 (ii) ρ = ρ†
(iii) x†ρx ≥ 0 ∀x ∈ C
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So the problem is to find the set of Bloch
vectors in Rd2−1, called Bloch vector space
B(Rd2−1).
This problem is solved only for d = 2:
The Bloch vector space is a ball of unit
radius in R3, known as the Bloch ball.
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For d > 2, the problem is still open.
However for pure states (ρ2 = ρ) the
following relations hold,
||s||2 =√
d(d−1)2 ; sisjgijk = (d−2)sk (A′)
It is known that
Dr(Rd2−1) ⊆ B(Rd2−1) ⊆ DR(Rd2−1)
r =√
d2(d−1) R =
√d(d−1)
2
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Bloch Representation of a Multipartitestate
We construct the basis of L(H) which is the
product of individual bases comprising
generators of SU(dk); k = 1, 2, · · · , N .
k ,ki : a subsystem chosen from N subsys.
{Idki, λαki
}; αki= 1, 2, · · · , d2
ki− 1 is the
basis of Cd2k−i , comprising the generetors of
SU(ddki).
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Define, for subsystems k1 and k2
λ(k1)αk1
= (Id1⊗Id2⊗· · ·⊗λαk1⊗Idk1+1
⊗· · ·⊗IdN)
λ(k2)αk2
= (Id1⊗Id2⊗· · ·⊗λαk2⊗Idk2+1
⊗· · ·⊗IdN)
λ(k1)αk1
λ(k2)αk2
= (Id1 ⊗ Id2 ⊗ · · · ⊗ λαk1⊗ Idk1+1
⊗· · · ⊗ λαk2
⊗ Idk2+1⊗ IdN
λαk1and λαk2
occur at the k1th and k2th
places and are the λαk1th and λαk2
th
generators of SU(dk1), SU(dk2) respectively.
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In this basis we can expand ρ as
ρ = 1ΠN
k dk{⊗N
k Idk+
∑k∈N
∑αk
sαkλ
(k)αk +
∑{k1,k2}
∑αk1
αk2tαk1
αk2λ
(k1)αk1
λ(k2)αk2
+ · · · +∑
{k1,k2,··· ,kM}∑
αk1αk2
···αkMtαk1
αk2···αkM
λ(k1)αk1
λ(k2)αk2· · ·
λ(kM)αkM
+ · · · +∑α1α2···αN
tα1α2···αNλ
(1)α1 λ
(2)α2 · · ·λ(N)
αN }. (B)
(B) is called the Bloch representation of ρ.
s(k) = [sαk]d2k−1
αk=1 : Bloch vector for kth
subsystem.
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(NM
)terms in the sum
∑{k1,k2,··· ,kM}
Each contains a tensor (M-way array) of
order M
tαk1αk2
...αkM=
dk1dk2
...dkM2M Tr [ρλ
(k1)αk1
λ(k2)αk2· · ·λ(kM)
αkM]
=dk1
dk2...dkM
2M Tr [ρk1k2...kM(λαk1
⊗ λαk2⊗ · · · ⊗
λαkM)]
T {k1,k2,··· ,kM} = [tαk1αk2
···αkM]
The tensor in the last term is T (N).
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Outer product of vectors
Let u(1),u(2), . . . ,u(M) be vectors in
Rd21−1,Rd2
2−1, · · · ,Rd2M−1.
The outer product u(1) ◦ u(2) ◦ · · · ◦ u(M) is a
tensor of order M , (M-way array), defined by
ti1i2···iM = u(1)i1
u(2)i2
. . .u(M)iM
; 1 ≤ ik ≤d2
k − 1, k = 1, 2, · · · , M .
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We need the following result
A pure N-partite state with Blochrepresentation (B) is fully separable(product state) If and only ifT (N) = s(1) ◦ s(2) ◦ · · · ◦ s(N) where s(k) isthe Bloch vector of kth subsystemreduced density matrix.
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We propose the following measure for
N-qubit pure state entanglement.
E (ρ) = (||T (N)||−1)R
where the normalization constant R is given
by
R = (1 + 14(1 + (−1)N)2 +
∑bN2 c
k=1
(N2k
))1/2− 1
where R = ||T (N)|| − 1 calculated for
N-qubit GHZ state as shown below.
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The general GHZ state is
|ψ〉 =√
p|0 · · · 0〉 +√
1− p|1 · · · 1〉For this state the elements of T (N) are given
by ti1i2···iN = 〈ψ|σi1 ⊗ · · · ⊗ σiN |ψ〉Using this, the norm of T (N) for the state
|ψ〉〈ψ| is given by
||T (N)||2 = 4p(1− p) + (p + (−1)N(1−p))2 + 4p(1− p)
∑bN2 c
k=1
(N2k
)
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For maximally entangled state p = 12
R = ||T (N)|| − 1
= (1 + 14(1 + (−1)N)2 +
∑bN2 c
k=1
(N2k
))1/2 − 1
E (ρGHZ ) = 1R [(4p(1− p) + (p + (−1)N(1−
p))2 + 4p(1− p)∑bN
2 ck=1
(N2k
))1/2 − 1]
E as a function of p is plotted in the next
slide. Note that E (ρ) ≥ 0 for general
N-qubit GHZ state.
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0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
p
E
figure 1
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|W 〉 state
|W 〉 = 1√N
∑j |0 · · · 01j0 · · · 0〉
|W̃ 〉 = 1√N
∑j |1 · · · 10j1 · · · 1〉
where jth summand has a single 1 for |W 〉and sigle 0 for |W̃ 〉 at the jth bit.
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For both the states we get
||T (N)||2 = 1 + 4N−1N
so that,
E (|W 〉) = E (|W̃ 〉) = 1R (
√1 + 4N−1
N − 1).
E (|W 〉) = E (|W̃ 〉) is to be expected as
these are LU equivalent.
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Figure 2 shows the variation of E with
weight s in the state
|ψs〉 =√
s|W 〉 +√
1− s e iφ|W̃ 〉Note that the entanglement is independent
of φ.
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
s
E
figure 2
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Figure 3 shows the variation of E with
weight s in the state
|χs〉 =√
s|GHZ 〉 +√
1− s e iφ|W 〉Note agian that the entanglement is
independent of φ.
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0 0.2 0.4 0.6 0.8 10.84
0.86
0.88
0.9
0.92
0.94
0.96
0.98
1
1.02
s
E
figure 3
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An entanglement measure must have the
following basic properties
(a) (i) E (ρ)) ≥ 0 (ii) E (ρ) = 0 if and only if
ρ is separable
(b) Monotonicity under probabilistic LOCC .
(c) Convexity,
E (pρ + (1− p)σ) ≤ pE (ρ) + (1− p)E (σ)
with p ∈ [0, 1].
We prove these properties for our measure
one by one.
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Proposition 1 : Let ρ be a N-qubit pure
state with Bloch representation (B). Then,
||T (N)|| = 1 if and only if ρ is a product
state.
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By the result we have quoted
T (N) = s(1) ◦ s(2) ◦ · · · ◦ s(N)
Taking norm on both sides
||T (N)||2 = 〈T (N), T (N)〉 = Πi〈si, si〉 =
Πi ||si||2 = 1
Immediatly it follows that N-qubit pure state
ρ has E (ρ) = 0 if and only if ρ is a product
state.
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Proposition 2 : For two and three qubit
states ||T (N)|| ≥ 1
We prove this by directly computing ||T (N)||for the general two and three qubit states.
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Consider, the general two qubit state
|ψ〉 = a1|00〉 + a2|01〉 + a3|01〉 + a4|11〉,∑
i |ai |2 = 1.
||T (2)||2 = 1 + 8(a2a3 − a1a4)2 ≥ 1
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Consider, the general Schmidt form of three
qubit state
|ψ〉 = λ0|000〉 + λ1eiφ|100〉 + λ2|101〉 +
λ3|110〉 + λ4|111〉, λi ≥ 0,∑
i |λi |2 = 1.
By direct calculation of ||T (3)|| we get
||T (3)||2 ≥ 1 + 12λ20λ
24 + 8λ2
0λ22 + 8λ2
0λ23 +
8(λ20λ3 − λ1λ4)
2 ≥ 1
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For any two and three qubit pure states ρ
E (ρ) ≥ 0.
We conjecture that ||T (N)|| ≥ 1 for any
N-qubit pure state.
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Proposition 3 : Let Ui be a local unitray
operator acting on the Hilbert space of ith
subsystem.
If ρ′ = (⊗Ni=1Ui)ρ(⊗N
i=1U†i )
then ||T ′(N)|| = ||T (N)||.
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Proposition 4 : E (ρ) is LOCC invariant.
This follows from proposition 3 and the
result due to Bennett et al. that N-partite
pure state is LOCC invariant if and only if it
is LU invariant [PRA 2000,63 012307].
![Page 40: An Experimentally Accessible Geometric Measure for ...iscqi/talks/pjoag-talk.pdf · An Experimentally Accessible Geometric Measure for Entanglement in 3-qubit Pure states Pramod S](https://reader033.vdocument.in/reader033/viewer/2022041613/5e38f6e4c82bed43cc325597/html5/thumbnails/40.jpg)
Convexity
E (p|ψ〉〈ψ| + (1− p)|φ〉〈φ|)
= 1R (||pT (N)
|ψ〉 + (1− p)T (N)|φ〉 || − 1)
≤ 1R (p||T (N)
|ψ〉 || + (1− p)||T (N)|φ〉 || − 1)
= pE (|ψ〉) + (1− p)E (|φ〉)
![Page 41: An Experimentally Accessible Geometric Measure for ...iscqi/talks/pjoag-talk.pdf · An Experimentally Accessible Geometric Measure for Entanglement in 3-qubit Pure states Pramod S](https://reader033.vdocument.in/reader033/viewer/2022041613/5e38f6e4c82bed43cc325597/html5/thumbnails/41.jpg)
Continuity
||(|ψ〉〈ψ| − |φ〉〈φ|)|| → 0
⇒∣∣∣E (|ψ〉)− E (|φ〉)
∣∣∣ → 0
||(|ψ〉〈ψ| − |φ〉〈φ|)|| → 0
⇒ ||T (N)|ψ〉 − T
(N)|φ〉 || → 0
But ||T (N)|ψ〉 − T
(N)|φ〉 || ≥
∣∣∣||T (N)|ψ〉 || − ||T
(N)|φ〉 ||
∣∣∣
![Page 42: An Experimentally Accessible Geometric Measure for ...iscqi/talks/pjoag-talk.pdf · An Experimentally Accessible Geometric Measure for Entanglement in 3-qubit Pure states Pramod S](https://reader033.vdocument.in/reader033/viewer/2022041613/5e38f6e4c82bed43cc325597/html5/thumbnails/42.jpg)
Therefore ||T (N)|ψ〉 − T
(N)|φ〉 || → 0
⇒ ||T (N)|ψ〉 || − ||T
(N)|φ〉 || → 0
⇒∣∣∣E (|ψ〉)− E (|φ〉)
∣∣∣ → 0