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

49 Symposium on Mathematical Physics Long time behavior of nonlocal quantum gates in collaboration with K. Ż yczkowski and T. Linowski Toru ń Antonio Mandarino 17 VI 2017 Center for Theoretical Physics - PAN mandarino@cft.edu.pl

Quantum gates Entanglement is a fundamental resource for quantum information and computation purposes Quantum gates are essential to entangle two separated systems M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 2000

Operator Schmidt decomposition Given a 2 quNits gate is it possible to find such a decomposition N 2 X p U = N 2 λ i A i ⊗ B i i =1 orthonormal basis in the Tr [ A † i A j ] = Tr [ B † i B j ] = δ ij space operators N 2 ~ X Schmidt vector � ≡ ( � 1 , ..., � N 2 ) λ i = 1 λ i ≥ 0 i =1 N 2 S L ( U ) := S L ( ~ X Linear Entropy � 2 � U ) = 1 − i i =1 N 2 S ( U ) := S ( ~ X von Neumann Entropy � U ) = − � i ln( � i ) i =1

Local equivalence of gates in SU(N 2 ) Given two gates acting on 2 quNits they are locally equivalent U ∼ loc V if there exist local operations such that U = ( W A ⊗ W B ) V ( W C ⊗ W D ) K.Hammerer, G.Vidal and J.I.Cirac, Phys.Rev.A 66, 062321 (2002)

Local equivalence of gates in SU(N 2 ) Given two gates acting on 2 qudits they are locally equivalent U ∼ loc V if there exist local operations such that U = ( W A ⊗ W B ) V ( W C ⊗ W D ) SU ( N 2 ) SU ( N ) ⊗ SU ( N ) W A & W C act on 1st subsystem Local gates W B & W D act on 2nd subsystem K.Hammerer, G.Vidal and J.I.Cirac, Phys.Rev.A 66, 062321 (2002)

Local equivalence of gates in SU(4)- Weyl Chamber 3 loc V ( α 1 , α 2 , α 3 ) = e ( i P 3 k =1 α k σ k ⊗ σ k ) = Y e ( i α k σ k ⊗ σ k ) U ∼ k =1 M. Musz, M. Kus,and K. Zyczkowski, Phys.Rev.A87, 022111 (2013)

Local equivalence of gates in SU(4)- Weyl Chamber 3 loc V ( α 1 , α 2 , α 3 ) = e ( i P 3 k =1 α k σ k ⊗ σ k ) = Y e ( i α k σ k ⊗ σ k ) U ∼ k =1 V ( α 1 , α 2 , α 3 ) .V ( α 0 3 ) = V ( α 1 + α 0 1 , α 0 2 , α 0 1 , α 2 + α 0 2 , α 3 + α 0 3 ) M. Musz, M. Kus,and K. Zyczkowski, Phys.Rev.A87, 022111 (2013)

Local equivalence of gates in SU(4)- Weyl Chamber 3 loc V ( α 1 , α 2 , α 3 ) = e ( i P 3 k =1 α k σ k ⊗ σ k ) = Y e ( i α k σ k ⊗ σ k ) U ∼ k =1 V ( α 1 , α 2 , α 3 ) .V ( α 0 3 ) = V ( α 1 + α 0 1 , α 0 2 , α 0 1 , α 2 + α 0 2 , α 3 + α 0 3 ) π 4 ≥ α 1 ≥ α 2 ≥ α 3 ≥ 0 M. Musz, M. Kus,and K. Zyczkowski, Phys.Rev.A87, 022111 (2013)

Local equivalence of gates in SU(4)- Weyl Chamber 3 loc V ( α 1 , α 2 , α 3 ) = e ( i P 3 k =1 α k σ k ⊗ σ k ) = Y e ( i α k σ k ⊗ σ k ) U ∼ k =1 V ( α 1 , α 2 , α 3 ) .V ( α 0 3 ) = V ( α 1 + α 0 1 , α 0 2 , α 0 1 , α 2 + α 0 2 , α 3 + α 0 3 ) π 4 ≥ α 1 ≥ α 2 ≥ α 3 ≥ 0 SU (4) Γ III ≡ SU (2) ⊗ SU (2) ↵ = ( ↵ 1 , ↵ 2 , ↵ 3 ) ≡ ↵ III ~ Information or Interaction content M. Musz, M. Kus,and K. Zyczkowski, Phys.Rev.A87, 022111 (2013)

Weyl Chamber - 2qubit gates V α I = exp ( i α 1 σ 1 ⊗ σ 1 ) π 4 ≥ α 1 ≥ 0 ↵ = ( ↵ 1 , 0 , 0) ≡ ↵ I ~ � = 1 ~ 2(1 + cos ↵ 1 , 1 − cos ↵ 1 , 0 , 0) Γ I M. Musz, M. Kus,and K. Zyczkowski, Phys.Rev.A87, 022111 (2013)

Dynamics in 1D Weyl Chamber V t α I = exp ( if ( t α 1 ) σ 1 ⊗ σ 1 ) Particle bouncing V 5 against two walls V 4 V 3 V 2 V α 3 π π π π 0 16 16 8 4 α π 4 Irrational rotation map π Known Gates have 8 periodic orbits 0 t 0 2 4 6 8 10 Typical gate will explore all the space in t ∞

Weyl Chamber - 2qubit gates V α II = exp ( i α 1 σ 1 ⊗ σ 1 + α 2 σ 2 ⊗ σ 2 ) π 4 ≥ α 1 ≥ α 2 ≥ 0 ↵ = ( ↵ 1 , ↵ 2 , 0) ≡ ↵ II ~ rank Schmidt vector can be 1, 2, or 4. Not 3. Γ II M. Musz, M. Kus,and K. Zyczkowski, Phys.Rev.A87, 022111 (2013)

Dynamics in 2D Weyl Chamber Billiard dynamics in a isosceles right triangle α � π The corresponding dynamical � system is: 1. integrable 2. ergodic 3. non mixing π � α � � π π � � � N.E. Hurt, Quantum chaos and mesoscopic systems - 2013

Dynamics in 2D Weyl Chamber Billiard dynamics in a isosceles right triangle α � π The corresponding dynamical � system is: 1. integrable 2. ergodic 3. non mixing π � α � � π π � � � N.E. Hurt, Quantum chaos and mesoscopic systems - 2013

Dynamics in 2D Weyl Chamber Billiard dynamics in a isosceles right triangle α 2 π The corresponding dynamical 4 system is: V 8 1. integrable V 7 2. ergodic 3. non mixing π 8 V 4 V 3 V 6 V 2 V V 5 0 α 1 π π 0 8 4 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 ↵ ( V t ) = Sg( f ( t ↵ 1 ) , f ( t ↵ 2 ) , f ( t ↵ 3 )) ~ Sorting function: components in decreasing order Bouncing function of the 1D case

A naive formulation of ergodicity M x t = 1 X x ( U t ) M t =1 Average of a quantity over the trajectory of a single gate (or particle) M h x i = 1 X x ( U n ) M n =1 Average of a quantity over an ensemble of a huge number of point in the phase space

Dynamics of Linear entropy 2qubit gate α I ) = 1 S L ( V t 4(1 − c α 1 ) α II ) = 1 − 1 S L ( V t 16 (3 + c α 1 ) (3 + c α 2 ) α III ) = 9 16 − 1 S L ( V t 4 c α 1 + c − α 12 + 4 c α 2 + c + α 12 + 2(2 + c α 2 + c α 1 ) c α 3 � � 32 c α i = cos [4 f ( t α i )] c ± α ij = cos [4 f ( t α i ) ± 4 f ( t α j )] Z T 1 Time average S L ( V t S Lt = lim ↵ ) dt ~ T T →∞ 0 1 Z Space average ↵ ) p ( S L ) d m α h S L i m = S L ( V ~ N m Γ m

Entropies mean values ⃗ ∈ Γ m with m = {I, II, III} the following relations, for the two For any α first moment j = 1, 2 are satisfied h S j i m = S jt h S j L i m = S j Lt

Prob Distribution linear entropy � ( � � ) α � α ��� � ▲ ○ □ ■ ● △ � ▲ △ ○ □ ▲ ● △ ▲ □ △ ○ ● ■ ▲ △ ▲ � △ ○ ■ ● □ ○ ● ▲ ■ α �� □ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ● ● ● ● ● ● ● ● ● ● ● ● ● ● □ □ □ □ □ ■ ■ ■ □ □ □ □ □ □ □ □ □ □ □ △ ■ ■ ■ ▲ ■ △ ■ � ▲ ■ △ ■ ■ ▲ ■ ■ △ ■ ▲ ■ ■ △ ▲ △ ▲ △ ▲ △ ▲ △ ▲ △ ▲ △ ▲ △ ▲ △ ▲ � � ��� ��� ��� ��� ��� ��� ��� ���

Prob Distribution linear entropy � ( � � ) α � α ��� � ▲ empty markers S Lt ○ □ ■ ● △ h S L i Γ m � ▲ filled markers △ ○ □ ▲ ● △ ▲ □ △ ○ ● ■ ▲ △ ▲ � △ ○ ■ ● □ ○ ● ▲ ■ α �� □ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ● ● ● ● ● ● ● ● ● ● ● ● ● ● □ □ □ □ □ ■ ■ ■ □ □ □ □ □ □ □ □ □ □ □ △ ■ ■ ■ ▲ ■ △ ■ � ▲ ■ △ ■ ■ ▲ ■ ■ △ ■ ▲ ■ ■ △ ▲ △ ▲ △ ▲ △ ▲ △ ▲ △ ▲ △ ▲ △ ▲ △ ▲ � � ��� ��� ��� ��� ��� ��� ��� ��� 1 Invariant Measure of S L P α I ( S L ) = q S L ( 1 2 − S L ) π the probability distribution in time is equal to the space probability

Random Matrix Theory in Pills CUE consists of all unitary matrices with the (normalized) Haar measure on the unitary group SU(N). COE is defined on the set of all symmetric unitary matrices W = W T = ( W † ) − 1 by the property of being invariant under all transformations by an arbitrary unitary matrix O W → O T WO CPE diagonal unitary matrices with independent unimodular eigenvalues, U jk = δ jk e i φ j φ j ∈ [0 , 2 π ) M. L. Mehta, Random Matrices, 2nd ed. (Academic Press, New York, 1991).

Mean linear Entropy 2 quNits gate < � � > ��� ��� ● ○ ● ○ ● empty markers S Lt ��� ○ ��� ▲ filled markers h S L i ens △ ▲ △ ��� ▲ ● △ � ��� � ◆ ◆ ◆ ◆ ◆ ◆ ���� ▲ ��� ���� ◆ � � � � � � � � � � � N : subsystem dimension h S L i CUE = N 2 � 1 N 2 + 1 h S L i CP E = ( N � 1) 2 N 2

Mean linear Entropy 2 quNits gate < � � > ��� ��� ● ○ ● ○ ● empty markers S Lt ��� ○ ��� ▲ filled markers h S L i ens △ ▲ △ ��� ▲ b = h S i COE ● △ h S i CUE � ��� � ◆ ◆ ◆ ◆ ◆ ◆ ���� ▲ ��� ���� ◆ � � � � � � � � � � � N : subsystem dimension h S L i CUE = N 2 � 1 N 2 + 1 P. Zanardi, PRA 63, 040304(R), (2001) h S L i CP E = ( N � 1) 2 A.Lakshminarayan,Z.Puchala andK.Zyczkowski, PRA 90, 032303 (2014) N 2

Summary and future perspective 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 Mean values for a typical CPE 4 are equal to mean values over1D Weyl Chamber How does the local dynamics effects the nonlocal properties?