Manipulating, Drawing and Entangling Qubits
Marco B. Enrıquezjoint with Oscar Rosas-Ortiz
Chaos i Informacja Kwantowa
January 20, 2014
Department of PhysicsCenter for Research and Advanced Studies, Mexico
Outline
Preliminaries and notation
Mathematical interlude: Kronecker meets Hubbard
Atom + Field: The semiclassical approach
Atom + Field: The fully quantized version
Atom + Atom + Field: Sudden death of the Entanglement
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Qubits
A two-level atom: Upper level: |+〉 = |e1〉 =
(1
0
), Ground level:
|−〉 = |e2〉 =
(0
1
).
The Hamiltonian: H = ~ωa2σ3. Where ωa is the tansition frequency.
Hilbert space: Ha = span|e1〉, |e2〉.Any state is written as a linear combination:
|ψ〉 = c1|e1〉+ c2|e2〉, ci ∈ C.
The dynamics
σ+ =(
0 10 0
), σ− =
(0 01 0
), σ3 =
(1 00 −1
). (1)
su(2) algebra: [σ3, σ±] = 2σ± and [σ+, σ−] = σ3.
Pauli matrices σ1 = σ+ + σ−, σ− = i(σ− − σ+).
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The Bloch ball and the qubits
Density matrix of a qubit
ρa =( ρ11 ρ12
ρ21 ρ22
)é Normalized: trρa = ρ11 + ρ22 = 1.
é Hermitian: ρ11, ρ22 ∈ R and ρ12 = ρ21.
é Positive: |ρ12| = |ρ21| ≤√ρ11ρ22.
é In the Pauli matrices basis:
ρa =1
2(I + ~τ · ~σ), ~σ = (σ1, σ2, σ3). (2)
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The Bloch ball and the qubits
Where
τ1 = ρ12 + ρ21, τ2 = i(ρ12 − ρ21), τ3 = ρ11 − ρ22.
Since ρa describes a pure or a mixed state: trρ2a = τ21 + τ22 + τ23 ≤ 1.
Equality holds for pure states. Surface of the sphere: S2.
Mixed states within the ball.
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Composite systems
é A system made up two subsystems S = S1 + S2.
é Take H1 = span|e(1)k 〉
k∈I
and H2 = span|e(2)` 〉
`∈J
the Hilbert spaces
associated to S1 and S2, respectively. The Hilbert space of S is
H = span|e(1)k 〉 ⊗ |e
(2)` 〉, k, ` ∈ I,J
.
Example: Two qubits system
|e(1)1 〉 ⊗ |e(2)2 〉 =
(1
0
)⊗ |e(2)2 〉 =
1
(1
0
)0
(0
1
) =
0
1
0
0
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Composite systems
Observables. Let A1 an observable of S1. Then the action of A1 could beextended to H
A := A1 ⊗ I2 → A(|ψ(1)〉 ⊗ |ψ(2)〉
)= A1
(|ψ(1)〉 ⊗ |ψ(2)〉
).
Example.
I2 ⊗ σ3 =
(1 0
0 1
)⊗ σ3 =
(1σ3 0σ3
0σ3 1σ3
)=
1 0 0 0
0 −1 0 0
0 0 1 0
0 0 0 −1
(3)
In general, the operators acting on H will be written in terms of products
like C1 ⊗ C2.
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Hubbard’s playgroundHubbard operators properties. They act on a given Hilbert space ofdimension n. A set of n2 operators that satisfies
1. Multiplication rule. Xi,jXk,m = δjkXi,m
2. Completness.∑k
Xk,k = I.
3. Non-hermiticity (Xi,j)† = Xj,i.
4. Commutation and anti-commutation rules.
[Xi,j , Xk,m]± = δjkXi,m ± δmiXk,j .
5. Action on basis elements
Xi,j |ek〉 = δj,k|ei〉.
Simplest representation. Let H a n-dimensional Hilbert space over the fieldK. The basis elements of H will be denoted as |enk 〉, k ∈ 1, 2, . . . , n
We define the Hubbard operator of order n
Xi,jn = |ei〉〈ej |, i, j = 1, 2, . . . , n. (4)
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Hubbard’s playground
Since Hubbard operators are complete, any operator A can be written as follows
A =∑i,j
ai,jXi,jn , ai,j ∈ K, (5)
Example. a 0 0 0
0 b c 0
0 c d 00 0 0 e
0 0 0 −1
0 0 1 00 1 0 0−1 0 0 0
= ?
Deal it with Hubbard!(aX1,1 + bX2,2 + cX2,3 + cX3,2 + dX3,3 + eX4,4
) (−X1,4 +X2,3 +X3,2 −X4,1
)= −aX4,1 + bX2,3 + cX2,2 + cX3,3 + dX3,2 − eX4,1
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Kronecker product highlights
Proposition K1. Let Xi,jm and Xk,`
n be two Hubbard operators of order n
and m respectively. The Kronecker product Xi,jm ⊗Xk,`
n is the mn-Hubbard
operator Xn(i−1)+k,n(j−1)+`mn . That is,
Xi,jm ⊗Xk,`
n = Xn(i−1)+k,n(j−1)+`mn . (6)
Theorem M1. The Kronecker product of A = [ai,j ] and B = [bk,`],respectively n and m-square matrices, can be written as
A⊗B =
nm∑p,q=1
cp,qXp,qnm ≡ C, cp,q := ap′,q′bp′′,q′′
where x′ =⌈xm
⌉and x′′ = x+m−mx′
M. Enrıquez and O. Rosas-Ortiz, “The Kronecker product in terms of Hubbard
operators and the Clebsch-Gordan decomposition of SU(2)× SU(2)”, Annals
of Physics, 339 (2013) 218
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Some applications
Permutation matrices: Let π a bijection of the set of natural numbersS = 1, . . . , n onto itself
π =(
1 2 · · · nπ(1) π(2) · · · π(n)
).
The corresponding permutation matrix in terms of Hubbard operators reads
Pπ =
n∑j=1
Xj,π(j)n . (7)
SU(2) irreducible representation: Let j be a integer or semi-integer
J3 =
n∑k=1
mkXk,kn , n = 2j + 1.
J+ =
n−1∑k=1
√k(2j + 1− k)Xk,k+1
n , J− =
n−1∑k=1
√k(2j + 1− k)Xk+1,k
n .
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Some applications
Haddamard matrix:
H =1√
2
2∑i,j=1
(−1)(i−1)(j−1)Xi,j2 =
1√
2
(1 11 −1
), (8)
Note
H⊗k+1 =1
√2k+1
2k+1∑p,q=1
(−1)~p·~q Xp,q
2k+1 , k ≥ 1, (9)
with
~p · ~q :=
k∑s=0
(ps − 1)(qs − 1), with xs =⌈ x
2s
⌉. (10)
It can be shown that
~p · ~q =
k∑s=0
(p− 1)s(q − 1)s,
where (p− 1)s and (q − 1)s are the s-th binary coefficients of p− 1 and q − 1
respectively.
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Qubit + Classical Field
Qubit Hamiltonian: H0 = ωa2 σ3
Field: E(t) = E(e−iωf t + eiωf t)ef
Interaction Hamiltonian: HI = −p ·E(t), where p = p(σ+ + σ−)ep isthe dipole operator of the atom
Moving to the rotating frame we have
H =∆
2σ3 + g(σ+ + σ−), g = pEep · ef , ∆ = 1−
ωfωa. (11)
In terms of X-operators
H =∆
2
2∑p=1
Xp,p2 + g
2∑p=1
Xp,3−p2 (12)
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Qubit + Classical Field
The time evolution operator in terms of X-operators
U(t) =
2∑p,q=1
up,q(t)Xp,q2 +
i∆
2Ω
2∑p=1
(−1)pXp,3−p2 , Ω =
√Ω2/4 + g2
and the coefficients
up,q(t) =( g
Ω
)|p−q|ei π2|p−q|
cos(
Ωt−π
2|p− q|
).
Suppose the initial state is given by |ψ(0)〉 = |ek〉 with k = 1, 2. Thus
|ψ(t)〉 = U(t)|ek〉 =
2∑p=1
up,k(t)|ep〉+i∆
2Ω(−1)k+1|e3−k〉.
Any operator can be written in terms of Hubbards
σ3 =
2∑`=1
(−1)`+1X`,`2 ,
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Qubit + Classical Field
Then the atomic population inversion reads
〈σ(t)〉 = (−1)k+1
[( gΩ
)2cos(2Ωt) +
(∆
2Ω
)2]. (13)
Figure: Atomic population inversion for: k = 1 the initial state is |+〉 and ∆ = 0
(black), ∆ = g (red) and ∆ = 4g (blue).
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Qubit + Classical FieldThe correspondent density matrix reads
ρ(t) =∑p,q
up,kuq,kXp,q2 +
(∆
2Ω
)2
|e3−k〉〈e3−k|+
(i∆
Ω(−1)
k∑p
up,kXp,3−p
)sim
.
Figure: Bloch vectors trajectories on S2. The initial state is |+〉 and ∆ = 0 (black),
∆ = g (red) and ∆ = 4g (blue).
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Qubit + Quantized Radiation Field
Electric field: E(t) = E(e−iωta+ eiωta†)
In the RWA
H =
H0︷ ︸︸ ︷ωfa†a+
ωf
2+ωa
2σ3
+ g(σ+a+ σ−a†)︸ ︷︷ ︸
HI
We will suppose ∆ = 0, (ωf = ωa)
Such a Hamiltonian acts on states like
|e1〉n := |+, n〉, |e2〉n = |−, n+ 1〉,
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Qubit + Quantized Radiation Field
The following set of 4 operators is a representation of the Hubbard operatorsof order 2
X1,1 =
( I 0
0 0
), X1,2 =
(0
∑n
|n〉〈n+ 1|
0 0
),
X2,1 =
(0 0∑
n
|n+ 1〉〈n| 0
), X2,2 =
(0 0
0 I
) (14)
The Hamiltonian is written as
H0 = (N + 1)X1,1 +NX2,2, HI = γ
2∑p=1
Np Xp,3−p, Np =
√N + 2− p
The time evolution operator reads
U(t) =
2∑p,q=1
up,q(Np) Xp,q , up,q(Np) = eiπ2|p−q| cos
(γtNp −
π
2|p− q|
).
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Qubit + Quantized Radiation Field
In general, a initial state could be written as a linear combination of basiselements |e1〉n, |e2〉n for any n
|ψ(0)〉 =
2∑i=1
∞∑m=0
Qim|ei〉m,∑i
∑m
|Qim|2 = 1
Hence|ψ(t)〉 =
∑p,q
∑n
Qqn up,q (gn) |ep〉n. (15)
For instance
Taking Qin = δi,kδm,n then |ψ(0)〉 = |ek〉n. The atomic populationinvertion
σ3(t) = (−1)k+1 cos [2gnt] , gn = γ√n+ 1.
The state |ψ(0)〉 =∑n αn|ek〉n is obtained by Qin = δi,2 αn. The atomic
population invertion
σ3(t) = (−1)k+12∑p=1
∑n=0
|αn|2(−1)p+1|up,2(gn)|2 = (−1)k+1∑n=0
|αn|2 cos(2gnt).
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Qubit + Quantized Radiation Field
The reduced density matrix of the atom
ρat(t) =∑m=0
〈m|ρ(t)|m〉 =∑k,p
ak,pXk,p2 , ak,p =
2∑q,`=1
∑n=0
QqnQ`nup,q(gn)uk,`(gn)
Figure: Left: Atomic population inversion. Right: Trajectories within the Bloch ball.
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Entanglement of two JC atoms
The Hamiltonian of the whole system reads
H = H1 +H2,
H1 = H0;1 +HI;1, H0;1 = N1 + 12
+ 12σ3, HI;1 = γ1(σ+a+ σ−a†),
H2 = H0;2 +HI;2, H0;2 = N2 + 12
+ 12s3, HI;2 = γ2(s+b+ s−b†).
(16)
The basis state of the corresponding Hilbert space read|e1〉n,m := |e1〉n ⊗ |e1〉m = |+,+;n,m〉,
|e2〉n,m := |e1〉n ⊗ |e2〉m = |+,−;n,m+ 1〉,
|e3〉n,m := |e2〉n ⊗ |e1〉m = |−,+;n+ 1,m〉,
|e4〉n,m := |e2〉n ⊗ |e2〉m = |−,−;n+ 1,m+ 1〉.
The time evolution operator U(t) = U1(t)⊗ U2(t) with
U1(t) =
2∑p,q=1
up,q(Np;1) Xp,q , U2(t) =
2∑p,q=1
up,q(Np;2) Xp,q .
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Entanglement of two JC atoms
Then
U(t) =
4∑p,q=1
up′,q′(Np′;1
)up′′,q′′
(Np′′;2
)Xp,q
4 .
The time evolution of any state
|ψ(t)〉 =∑p,q
∑n,m
Qqn,m wp,k(n,m) |ek〉n,m, wp,k(n,m) = up′,k′ (gn)up′′,k′′ (gm),
The reduced density matrix of the two qubits
ρat(t) =
4∑p,k=1
ap,k Xp,k4 , ap,k =
4∑q,`=1
∑n,m=0
Qqn,mQ`i,j wp,q(n,m)wk,`(i, j).
For instance, suppose the initial state is given by|ψ(0)〉 = (cosα|+,−〉+ sinα|−,+〉)⊗ |r + 1, s+ 1〉. (17)
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Wootters and the ConcurrenceHow much entangled are two systems?
The concurrence C of a bipartite system whose density matrix is ρ, is givenby
C(ρ) = max 0,√λ1 −
√λ2 −
√λ3 −
√λ4,
where λ1, λ2, λ3 and λ4 are the eigenvalues of the matrix ρ(σ2 ⊗ σ2)ρ?(σ2 ⊗ σ2).Here ? means the complex conjugate. Besides, the eigenvalues λi satisfyλ1 ≥ λ2 ≥ λ3 ≥ λ4.
Example. For pure state given as|ϕ〉 = a1|+,+〉+ a2|+,−〉+ a3|−,+〉+ a4|−,−〉, (18)
the concurrence reads C (|ϕ〉〈ϕ|) = 2|a1a4 − a2a3|.
Bell state a1 = a4 = 1√2
. Then C (|ϕ〉〈ϕ|) = 1.
Separable statea1a4 − a2a3 = 0.
Thus C (|ϕ〉〈ϕ|) = 0.
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Entanglement of two JC atoms
The initial state of the two atoms is described by|ψa〉 = cosα|+,−〉+ sinα|−,+〉.
Figure: Concurrence of two JC atoms. With 0 photons in cavity A and 0photons in cavity B. Besides α = π
4 (blue), α = π12 (red) and α = π
6 (brown)
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Entanglement of two JC atoms
Figure: Concurrence of two JC atoms.
With 5 photons in cavity A and 4
photons in cavity B. Besides α = π4
Figure: Concurrence of two JC atoms.
With 1 photon in cavity A and 3
photons in cavity B in a Kerr medium.
Besides α = π4
(Red) α = π16
(Blue)
and α = 3π8
(Brown)
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Conclusions
We studied the dynamics of qubits using the Hubbard operators.Interacting qubits?
A geometrical representation of the qubits was given in terms of convexsets. Two qubits?
“Collapses and revivals” of Concurrence of two JC were analized.
Too much work to continue doing!
Thanks!
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