Port-based teleportation and asymptotic representation theory - PowerPoint PPT Presentation

Port-based teleportation and asymptotic representation theory Christian Majenz Based on joint work with Matthias Christandl, Felix Leditzky, Graeme Smith, Florian Speelman and Michael Walter 20.01.2020 Korteweg-de Vries Institute for Mathematics

Outline Introduction I ‣ Port-based teleportation Introduction II ‣ The Schur-Weyl distribution ‣ Asymptotics: spectrum estimation, fluctuations Results ‣ Convergence result ‣ Asymptotics of port-based teleportation Summary, open problems

Introduction I

A communication task A A' B' Bob Alice

A communication task A A' B' Bob Alice

Quantum teleportation A A' B' Bob Alice

Quantum teleportation A A' B' Bob Alice

Quantum teleportation A A' B' Bob Alice m

Quantum teleportation A A' B' Bob Alice m

Quantum teleportation A A' B' Bob Alice m

Port-based teleportation (PBT) A A 1 B 1 ... ... Bob Alice A i B i ... ... A N B N Ishizaka & Hiroshima ‘08

Port-based teleportation (PBT) A A 1 B 1 ... ... Bob Alice A i B i ... ... A N B N : number of “ports” N : dimension of Alice’ input d Ishizaka & Hiroshima ‘08

Port-based teleportation (PBT) A A 1 B 1 ... ... Bob Alice A i B i ... ... A N B N : number of “ports” N : dimension of Alice’ input d Ishizaka & Hiroshima ‘08

Port-based teleportation (PBT) A A 1 B 1 ... ... Bob Alice A i B i ... ... A N B N port number i : number of “ports” N : dimension of Alice’ input d Ishizaka & Hiroshima ‘08

Port-based teleportation (PBT) A A 1 B 1 ... ... Bob Alice A i B i ... ... A N B N port number i : number of “ports” N : dimension of Alice’ input d Ishizaka & Hiroshima ‘08

Why port-based teleportation?

Why port-based teleportation? Can break time order of quantum operations! A A 1 B 1 ... ... Bob Alice A i B i ... ... A N B N port number i

Why port-based teleportation? Can break time order of quantum operations! Applications:

Why port-based teleportation? Can break time order of quantum operations! Applications: ‣ Universal programmable quantum processors

Why port-based teleportation? Can break time order of quantum operations! Applications: ‣ Universal programmable quantum processors ‣ Instantaneous non-local quantum computation (Beigi & König ’11)

Why port-based teleportation? Can break time order of quantum operations! Applications: ‣ Universal programmable quantum processors ‣ Instantaneous non-local quantum computation (Beigi & König ’11) ‣ Generic attack on any position-based cryptographic scheme (Buhrman et al. ’14)

Why port-based teleportation? Can break time order of quantum operations! Applications: ‣ Universal programmable quantum processors ‣ Instantaneous non-local quantum computation (Beigi & König ’11) ‣ Generic attack on any position-based cryptographic scheme (Buhrman et al. ’14) How expensive is PBT?

PBT variants

PBT variants Port-based teleportation is necessarily imperfect.

PBT variants Port-based teleportation is necessarily imperfect. Two variants: ‣ plain, figure of merit: fidelity F ‣ heralded, figure of merit: success probability p

PBT variants Port-based teleportation is necessarily imperfect. Two variants: ‣ plain, figure of merit: fidelity F ‣ heralded, figure of merit: success probability p Two types of protocols: ‣ Maximally entangled resource state ( ) F EPR , p EPR ‣ Optimized resource state ( ) F *, p * Trade-off between number of ports and / N F p

Symmetries A A 1 B 1 ... ... Bob Alice A i B i ... ... A N B N port number i

Previous results

Previous results Closed-form: F EPR ≥ 1 − d 2 − 1 (Ishizaka & Hiroshima ’08) ‣ N 1 (Ishizaka ’15) 4( d − 1) N 2 + O ( N − 3 ) ‣ F * ≤ 1 − d 2 − 1 (Studzinski et al. ‘16) ‣ p * = 1 − d 2 − 1 + N

Previous results

Previous results Exact formulas (Studzinski et al. ’16, Mozrzymas et al. 17’): 2 ∑ ∑ F EPR ( N ) = d − N − 2 d μ m μ d α ⊢ d N − 1 μ = α + □ d μ * m μ d α ( N ) = 1 ∑ with p EPR m 2 μ * = maxarg μ = α + □ α d d N m μ * m α d μ α ⊢ d N − 1 d ( N ) = d − 2 sup ∑ F * q ( μ ) q ( μ ′ ) q μ , μ ′ ⊢ d N μ ′ = μ + □ − □

Previous results Exact formulas (Studzinski et al. ’16, Mozrzymas et al. 17’): 2 ∑ ∑ F EPR ( N ) = d − N − 2 d μ m μ d α ⊢ d N − 1 μ = α + □ d μ * m μ d α ( N ) = 1 ∑ with p EPR m 2 μ * = maxarg μ = α + □ α d d N m μ * m α d μ α ⊢ d N − 1 d ( N ) = d − 2 sup ∑ F * q ( μ ) q ( μ ′ ) q μ , μ ′ ⊢ d N μ ′ = μ + □ − □ Evaluating these formulas scales exponential in ! d Asymptotics for arbitrary and ? N → ∞ d

Introduction II

Representation theory Group representation of a group : group homomorphism from G to GL( n , ℂ ) G

Representation theory Group representation of a group : group homomorphism from G to GL( n , ℂ ) G Irreducible representations (irreps): no non-trivial invariant subspace

Representation theory Group representation of a group : group homomorphism from G to GL( n , ℂ ) G Irreducible representations (irreps): no non-trivial invariant subspace All representations of finite and compact groups are direct sums of irreps.

Irreps of , S n U ( d ) Irreducible representations of S N , U ( d ), Young diagrams Group U( d ) S n Irreducible representations [ α ] V d , α Dimension d α m d , α Partition range α ` n α ` d k , k 2 N Examples: α = (5 , 3 , 2 , 1) ` 11 α = (5 , 3 , 3 , 3 , 1) ` 15 α = (2 , 1) ` 3

Irreps of , S n U ( d ) Irreducible representations of S N , U ( d ), Young diagrams Group U( d ) S n Irreducible representations [ α ] V d , α Dimension d α m d , α Partition range α ` n α ` d k , k 2 N Examples: α ∈ ℕ d ⊂ ℝ d For , formally α ⊢ d n α = (5 , 3 , 2 , 1) ` 11 α = (5 , 3 , 3 , 3 , 1) ` 15 α = (2 , 1) ` 3

Irreps of , S n U ( d ) Irreducible representations of S N , U ( d ), Young diagrams Group U( d ) S n Irreducible representations [ α ] V d , α Dimension d α m d , α Partition range α ` n α ` d k , k 2 N Examples: α ∈ ℕ d ⊂ ℝ d For , formally α ⊢ d n Example: representing the partition 13=6+4+3 α ⊢ 4 13 α = (6,4,3,0) ∈ ℕ 4 α = (5 , 3 , 2 , 1) ` 11 α = (5 , 3 , 3 , 3 , 1) ` 15 α = (2 , 1) ` 3

The Schur-Weyl distribution ⊗ n , have natural representations on : ( ℂ d ) S n U ( d )

The Schur-Weyl distribution ⊗ n , have natural representations on : ( ℂ d ) S n U ( d ) ‣ acts by U ⊗ n U ∈ U ( d )

The Schur-Weyl distribution ⊗ n , have natural representations on : ( ℂ d ) S n U ( d ) ‣ acts by U ⊗ n U ∈ U ( d ) ‣ permutes the tensor factors S n

The Schur-Weyl distribution ⊗ n , have natural representations on : ( ℂ d ) S n U ( d ) ‣ acts by U ⊗ n U ∈ U ( d ) ‣ permutes the tensor factors S n ⊗ n ⟹ ( ℂ d ) The two actions commute is representation of S n × U ( d )

The Schur-Weyl distribution ⊗ n , have natural representations on : ( ℂ d ) S n U ( d ) ‣ acts by U ⊗ n U ∈ U ( d ) ‣ permutes the tensor factors S n ⊗ n ⟹ ( ℂ d ) The two actions commute is representation of S n × U ( d ) Schur-Weyl duality theorem: ⊗ n ≅ ⨁ ( ℂ d ) [ α ] ⊗ V d , α α ⊢ d n

The Schur-Weyl distribution ⊗ n , have natural representations on : ( ℂ d ) S n U ( d ) ‣ acts by U ⊗ n U ∈ U ( d ) ‣ permutes the tensor factors S n ⊗ n ⟹ ( ℂ d ) The two actions commute is representation of S n × U ( d ) Schur-Weyl duality theorem: ⊗ n ≅ ⨁ ( ℂ d ) [ α ] ⊗ V d , α α ⊢ d n d α m d , α Schur-Weyl distribution: p d , n ( α ) = d n

̂ ̂ Asymptotics d α m d , α Schur-Weyl distribution: p d , n ( α ) = d n α = α Change of variables: p d , n ( ̂ α ) := p d , n ( α ) n

̂ ̂ Asymptotics d α m d , α Schur-Weyl distribution: p d , n ( α ) = d n α = α Change of variables: p d , n ( ̂ α ) := p d , n ( α ) n Theorem (Alicki 88’, Keyl & Werner ’01): X ( n ) ⟶ 1 X ( n ) ∼ ̂ Let . Then in distribution. p d , n d (1,...,1)

̂ ̂ Asymptotics d α m d , α Schur-Weyl distribution: p d , n ( α ) = d n α = α Change of variables: p d , n ( ̂ α ) := p d , n ( α ) n Theorem (Alicki 88’, Keyl & Werner ’01): X ( n ) ⟶ 1 X ( n ) ∼ ̂ Let . Then in distribution. p d , n d (1,...,1) Theorem (Johansson ’01): d ( X ( n ) − 1 d (1,...,1) ) n X ( n ) ∼ ̂ A ( n ) = Let and set . Then p d , n A ( n ) ⟶ A in distribution, where and . G ∼ GUE 0 A = spec( G ) d

̂ ̂ Asymptotics d α m d , α Schur-Weyl distribution: p d , n ( α ) = d n α = α Change of variables: p d , n ( ̂ α ) := p d , n ( α ) n Theorem (Alicki 88’, Keyl & Werner ’01): X ( n ) ⟶ 1 X ( n ) ∼ ̂ Let . Then in distribution. p d , n d (1,...,1) Strengthen convergence? Theorem (Johansson ’01): (needed for PBT…) d ( X ( n ) − 1 d (1,...,1) ) n X ( n ) ∼ ̂ A ( n ) = Let and set . Then p d , n A ( n ) ⟶ A in distribution, where and . G ∼ GUE 0 A = spec( G ) d

Results