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 FieldThe quantum analogy of Rabi oscillations

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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

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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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The Bar Quantum

QuantumFest 2013You are invited to QuantumFest 2014!!!

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Mexico city and the Cinvestav

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