Long time behavior of nonlocal quantum gates in collaboration with - - PowerPoint PPT Presentation

Antonio Mandarino

Center for Theoretical Physics - PAN

Long time behavior of nonlocal quantum gates

Toruń 17 VI 2017

mandarino@cft.edu.pl

49 Symposium on Mathematical Physics

in collaboration with K. Życzkowski and T. Linowski

Quantum gates

• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 2000

Entanglement is a fundamental resource for quantum information and computation purposes Quantum gates are essential to entangle two separated systems

Operator Schmidt decomposition U = N 2

N 2

X

i=1

p λiAi ⊗ Bi Tr[A†

iAj] = Tr[B† i Bj] = δij

~ ≡ (1, ..., N 2) SL(U) := SL(~ U) = 1 −

N 2

X

i=1

2

i

S(U) := S(~ U) = −

N 2

X

i=1

i ln(i)

Given a 2 quNits gate is it possible to find such a decomposition Schmidt vector Linear Entropy von Neumann Entropy

• rthonormal basis in the

space operators

N 2

X

i=1

λi = 1 λi ≥ 0

Local equivalence of gates in SU(N2)

K.Hammerer, G.Vidal and J.I.Cirac, Phys.Rev.A 66, 062321 (2002)

U ∼

loc V

U = (WA ⊗ WB)V (WC ⊗ WD)

Given two gates acting on 2 quNits they are locally equivalent if there exist local operations such that

Local equivalence of gates in SU(N2)

K.Hammerer, G.Vidal and J.I.Cirac, Phys.Rev.A 66, 062321 (2002)

U ∼

loc V

U = (WA ⊗ WB)V (WC ⊗ WD) SU(N 2) SU(N) ⊗ SU(N)

Given two gates acting on 2 qudits they are locally equivalent if there exist local operations such that WA & WC act on 1st subsystem WB & WD act on 2nd subsystem Local gates

Local equivalence of gates in SU(4)- Weyl Chamber

U ∼

loc V(α1,α2,α3) = e(i P3

k=1 αkσk⊗σk) =

3

Y

k=1

e(iαkσk⊗σk)

• M. Musz, M. Kus,and K. Zyczkowski, Phys.Rev.A87, 022111 (2013)

Local equivalence of gates in SU(4)- Weyl Chamber

U ∼

loc V(α1,α2,α3) = e(i P3

k=1 αkσk⊗σk) =

3

Y

k=1

e(iαkσk⊗σk) V(α1,α2,α3).V(α0

1,α0 2,α0 3) = V(α1+α0 1,α2+α0 2,α3+α0 3)

• M. Musz, M. Kus,and K. Zyczkowski, Phys.Rev.A87, 022111 (2013)

Local equivalence of gates in SU(4)- Weyl Chamber

U ∼

loc V(α1,α2,α3) = e(i P3

k=1 αkσk⊗σk) =

3

Y

k=1

e(iαkσk⊗σk) V(α1,α2,α3).V(α0

1,α0 2,α0 3) = V(α1+α0 1,α2+α0 2,α3+α0 3)

• M. Musz, M. Kus,and K. Zyczkowski, Phys.Rev.A87, 022111 (2013)

π 4 ≥ α1 ≥ α2 ≥ α3 ≥ 0

Local equivalence of gates in SU(4)- Weyl Chamber SU(4) SU(2) ⊗ SU(2) π 4 ≥ α1 ≥ α2 ≥ α3 ≥ 0

U ∼

loc V(α1,α2,α3) = e(i P3

k=1 αkσk⊗σk) =

3

Y

k=1

e(iαkσk⊗σk)

~ ↵ = (↵1, ↵2, ↵3) ≡ ↵III

V(α1,α2,α3).V(α0

1,α0 2,α0 3) = V(α1+α0 1,α2+α0 2,α3+α0 3)

• M. Musz, M. Kus,and K. Zyczkowski, Phys.Rev.A87, 022111 (2013)

ΓIII ≡

Information or Interaction content

Weyl Chamber - 2qubit gates ~ ↵ = (↵1, 0, 0) ≡ ↵I VαI = exp (iα1σ1 ⊗ σ1)

ΓI

• M. Musz, M. Kus,and K. Zyczkowski, Phys.Rev.A87, 022111 (2013)

π 4 ≥ α1 ≥ 0

~ = 1 2(1 + cos ↵1, 1 − cos ↵1, 0, 0)

Dynamics in 1D Weyl Chamber

2 4 6 8 10 t

π 8 π 4

α

V t

αI = exp (if(tα1)σ1 ⊗ σ1)

V V 2 V 3 V 4 V 5

π 16 π 8 3π 16 π 4

α

Irrational rotation map Known Gates have periodic orbits Particle bouncing against two walls Typical gate will explore all the space in t ∞

Weyl Chamber - 2qubit gates

• M. Musz, M. Kus,and K. Zyczkowski, Phys.Rev.A87, 022111 (2013)

~ ↵ = (↵1, ↵2, 0) ≡ ↵II π 4 ≥ α1 ≥ α2 ≥ 0 ΓII

VαII = exp (iα1σ1 ⊗ σ1 + α2σ2 ⊗ σ2)

rank Schmidt vector can be 1, 2, or 4. Not 3.

• π
• π
• α
• π
• π
• α

Dynamics in 2D Weyl Chamber

Billiard dynamics in a isosceles right triangle The corresponding dynamical system is:

• 1. integrable
• 2. ergodic
• 3. non mixing

N.E. Hurt, Quantum chaos and mesoscopic systems - 2013

• π
• π
• α
• π
• π
• α

Dynamics in 2D Weyl Chamber

Billiard dynamics in a isosceles right triangle The corresponding dynamical system is:

• 1. integrable
• 2. ergodic
• 3. non mixing

N.E. Hurt, Quantum chaos and mesoscopic systems - 2013

Dynamics in 2D Weyl Chamber

V V 3 V 2 V 8 V 4 V 5 V 6 V 7

π 8 π 4

α1

π 8 π 4

α2

Billiard dynamics in a isosceles right triangle The corresponding dynamical system is:

• 1. integrable
• 2. ergodic
• 3. non mixing

N.E. Hurt, Quantum chaos and mesoscopic systems - 2013

Dynamics in 3D Weyl Chamber

Billiard dynamics in a tetrahedron

~ ↵(V t) = Sg(f(t↵1), f(t↵2), f(t↵3))

Dynamics in 3D Weyl Chamber

Billiard dynamics in a tetrahedron Sorting function: components in decreasing order Bouncing function

• f the 1D case

~ ↵(V t) = Sg(f(t↵1), f(t↵2), f(t↵3))

A naive formulation of ergodicity

xt = 1 M

M

X

t=1

x(U t)

hxi = 1 M

M

X

n=1

x(Un)

Average of a quantity over the trajectory of a single gate (or particle) Average of a quantity over an ensemble

• f a huge number of point in the phase space

Dynamics of Linear entropy 2qubit gate

cαi = cos [4f(tαi)] c±αij = cos [4f(tαi) ± 4f(tαj)]

SLt = lim

T →∞

1 T Z T SL(V t

~ ↵)dt

hSLim = 1 Nm Z

Γm

SL(V~

↵)p(SL)dmα

SL(V t

αI) = 1

4(1 − cα1) SL(V t

αII) = 1 − 1

16 (3 + cα1) (3 + cα2) SL(V t

αIII) = 9

16 − 1 32

• 4cα1 + c−α12 + 4cα2 + c+α12 + 2(2 + cα2 + cα1)cα3
• Time average

Space average

Entropies mean values

For any α ⃗ ∈ Γm with m = {I, II, III} the following relations, for the two first moment j = 1, 2 are satisfied

hSjim = Sjt hSj

Lim = Sj Lt

Prob Distribution linear entropy

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• ()

α α α

Prob Distribution linear entropy

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• ()

α α α

PαI(SL) = 1 π q SL( 1

2 − SL)

Invariant Measure of SL

empty markers filled markers

SLt

hSLiΓm

the probability distribution in time is equal to the space probability

Random Matrix Theory in Pills

• M. L. Mehta, Random Matrices, 2nd ed. (Academic Press, New York, 1991).

COE is defined on the set of all symmetric unitary matrices by the property of being invariant under all transformations by an arbitrary unitary matrix O Ujk = δjkeiφj φj ∈ [0, 2π) CPE diagonal unitary matrices with independent unimodular eigenvalues, CUE consists of all unitary matrices with the (normalized) Haar measure on the unitary group SU(N).

W = W T = (W †)−1 W → OT WO

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▲ ▲ ▲ ▲ △ △ △

• <>

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• N: subsystem dimension

empty markers filled markers

SLt

hSLiens

Mean linear Entropy 2 quNits gate

hSLiCUE = N 2 1 N 2 + 1 hSLiCP E = (N 1)2 N 2

Mean linear Entropy 2 quNits gate

N: subsystem dimension empty markers filled markers

SLt

hSLiens

b = hSiCOE hSiCUE

• P. Zanardi, PRA 63, 040304(R), (2001)

A.Lakshminarayan,Z.Puchala andK.Zyczkowski, PRA 90, 032303 (2014)

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• <>

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• hSLiCUE = N 2 1

N 2 + 1 hSLiCP E = (N 1)2 N 2

Summary and future perspective

Mean values for a typical CPE4 are equal to mean values over1D Weyl Chamber How does the local dynamics effects the nonlocal properties? Analytical expressions for the gate dynamics in the Weyl chamber Ergodicity of the entropy for gate in the Weyl Chamber Signature of ergodicity for 2-quNits gate

Thanks for your attention

Thanks for your attention