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

Alice Bob

A

A' B'

A communication task

Alice Bob

A

A' B'

Quantum teleportation

Alice Bob

A

A' B'

Quantum teleportation

Alice Bob

A

A' B'

Quantum teleportation

Alice Bob

A

A' B'

m

Quantum teleportation

Alice Bob

A

A' B'

m

Quantum teleportation

Alice Bob

A

A' B'

m

Port-based teleportation (PBT)

Alice Bob

A

A1 B1 Ai Bi AN BN

... ... ... ... Ishizaka & Hiroshima ‘08

Port-based teleportation (PBT)

Alice Bob

A

A1 B1 Ai Bi AN BN

... ... ... ... : number of “ports” : dimension of Alice’ input

N d

Ishizaka & Hiroshima ‘08

Alice Bob

A

A1 B1 Ai Bi AN BN

... ... ... ...

Port-based teleportation (PBT)

: number of “ports” : dimension of Alice’ input

N d

Ishizaka & Hiroshima ‘08

Alice Bob

A

A1 B1

port number i

Ai Bi AN BN

... ... ... ...

Port-based teleportation (PBT)

: number of “ports” : dimension of Alice’ input

N d

Ishizaka & Hiroshima ‘08

Alice Bob

A

A1 B1

port number i

Ai Bi AN BN

... ... ... ...

Port-based teleportation (PBT)

: number of “ports” : dimension of Alice’ input

N d

Ishizaka & Hiroshima ‘08

Why port-based teleportation?

Why port-based teleportation?

Alice Bob

A

A1 B1

port number i

Ai Bi AN BN

... ... ... ... Can break time order of quantum operations!

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?

How expensive is PBT? 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)

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 (

)

FEPR, pEPR

• Optimized resource state (

)

F*, p*

Trade-off between number of ports and /

N F p

Symmetries

Alice Bob

A

A1 B1

port number i

Ai Bi AN BN

... ... ... ...

Previous results

Previous results

Closed-form:

• (Ishizaka & Hiroshima ’08)

FEPR ≥ 1 − d2 − 1 N

• (Ishizaka ’15)

F* ≤ 1 − 1 4(d − 1)N2 + O(N−3)

• (Studzinski et al. ‘16)

p* = 1 − d2 − 1 d2 − 1 + N

Previous results

Previous results

Exact formulas (Studzinski et al. ’16, Mozrzymas et al. 17’):

FEPR

d

(N) = d−N−2 ∑

α⊢dN−1

∑

μ=α+□

dμmμ

2

with

pEPR

d

(N) = 1 dN ∑

α⊢dN−1

m2

α

dμ* mμ* μ* = maxargμ=α+□ mμdα mαdμ F*

d (N) = d−2 sup q

∑

μ, μ′ ⊢d N μ′ = μ + □ − □

q(μ)q(μ′ )

Previous results

Exact formulas (Studzinski et al. ’16, Mozrzymas et al. 17’):

FEPR

d

(N) = d−N−2 ∑

α⊢dN−1

∑

μ=α+□

dμmμ

2

with

pEPR

d

(N) = 1 dN ∑

α⊢dN−1

m2

α

dμ* mμ* μ* = maxargμ=α+□ mμdα mαdμ F*

d (N) = d−2 sup q

∑

μ, μ′ ⊢d N μ′ = μ + □ − □

q(μ)q(μ′ )

Evaluating these formulas scales exponential in ! Asymptotics for arbitrary and ?

d d N → ∞

Introduction II

Representation theory

Group representation of a group : group homomorphism from to

G G GL(n, ℂ)

Representation theory

Group representation of a group : group homomorphism from to

G G GL(n, ℂ)

Irreducible representations (irreps): no non-trivial invariant subspace

Representation theory

Group representation of a group : group homomorphism from to

G G GL(n, ℂ)

Irreducible representations (irreps): no non-trivial invariant subspace All representations of finite and compact groups are direct sums

• f irreps.

Irreducible representations of SN, U(d), Young diagrams

Group Sn U(d) Irreducible representations [α] Vd,α Dimension dα md,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 ,

Sn U(d)

Irreducible representations of SN, U(d), Young diagrams

Group Sn U(d) Irreducible representations [α] Vd,α Dimension dα md,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 ,

Sn U(d)

For , formally

α ⊢d n α ∈ ℕd ⊂ ℝd

Irreducible representations of SN, U(d), Young diagrams

Group Sn U(d) Irreducible representations [α] Vd,α Dimension dα md,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 ,

Sn U(d)

For , formally

α ⊢d n α ∈ ℕd ⊂ ℝd

Example: representing the partition 13=6+4+3

α ⊢4 13 α = (6,4,3,0) ∈ ℕ4

The Schur-Weyl distribution

, have natural representations on :

Sn U(d) (ℂd)

⊗n

The Schur-Weyl distribution

, have natural representations on :

Sn U(d) (ℂd)

⊗n

• acts by

U ∈ U(d) U⊗n

The Schur-Weyl distribution

, have natural representations on :

Sn U(d) (ℂd)

⊗n

• acts by

U ∈ U(d) U⊗n

• permutes the tensor factors

Sn

The Schur-Weyl distribution

, have natural representations on :

Sn U(d) (ℂd)

⊗n

• acts by

U ∈ U(d) U⊗n

• permutes the tensor factors

Sn

The two actions commute is representation of

⟹ (ℂd)

⊗n

Sn × U(d)

The Schur-Weyl distribution

, have natural representations on :

Sn U(d) (ℂd)

⊗n

• acts by

U ∈ U(d) U⊗n

• permutes the tensor factors

Sn

The two actions commute is representation of

⟹ (ℂd)

⊗n

Sn × U(d)

Schur-Weyl duality theorem:

(ℂd)

⊗n ≅ ⨁ α⊢dn

[α] ⊗ Vd,α

The Schur-Weyl distribution

, have natural representations on :

Sn U(d) (ℂd)

⊗n

• acts by

U ∈ U(d) U⊗n

• permutes the tensor factors

Sn

The two actions commute is representation of

⟹ (ℂd)

⊗n

Sn × U(d)

Schur-Weyl duality theorem:

(ℂd)

⊗n ≅ ⨁ α⊢dn

[α] ⊗ Vd,α

Schur-Weyl distribution: pd,n(α) =

dαmd,α dn

Asymptotics

Schur-Weyl distribution: pd,n(α) =

dαmd,α dn

Change of variables:

̂ α = α n ̂ pd,n( ̂ α) := pd,n(α)

Asymptotics

Schur-Weyl distribution: pd,n(α) =

dαmd,α dn

Change of variables:

̂ α = α n ̂ pd,n( ̂ α) := pd,n(α)

Theorem (Alicki 88’, Keyl & Werner ’01): Let . Then in distribution.

X(n) ∼ ̂ pd,n X(n) ⟶ 1 d(1,...,1)

Asymptotics

Schur-Weyl distribution: pd,n(α) =

dαmd,α dn

Change of variables:

̂ α = α n ̂ pd,n( ̂ α) := pd,n(α)

Theorem (Alicki 88’, Keyl & Werner ’01): Let . Then in distribution.

X(n) ∼ ̂ pd,n X(n) ⟶ 1 d(1,...,1)

Theorem (Johansson ’01): Let and set . Then in distribution, where and .

X(n) ∼ ̂ pd,n A(n) = n d (X(n) − 1 d(1,...,1)) A(n) ⟶ A A = spec(G) G ∼ GUE0

d

Asymptotics

Schur-Weyl distribution: pd,n(α) =

dαmd,α dn

Change of variables:

̂ α = α n ̂ pd,n( ̂ α) := pd,n(α)

Theorem (Alicki 88’, Keyl & Werner ’01): Let . Then in distribution.

X(n) ∼ ̂ pd,n X(n) ⟶ 1 d(1,...,1)

Theorem (Johansson ’01): Let and set . Then in distribution, where and .

X(n) ∼ ̂ pd,n A(n) = n d (X(n) − 1 d(1,...,1)) A(n) ⟶ A A = spec(G) G ∼ GUE0

d

Strengthen convergence? (needed for PBT…)

Results

Strengthened convergence

X(n) ∼ ̂ pd,n A(n) = n d (X(n) − 1 d(1,...,1))

Strengthened convergence

X(n) ∼ ̂ pd,n A(n) = n d (X(n) − 1 d(1,...,1))

Theorem (CLMSSW ’19): converges in moments, i.e. for all where and .

A(n) 𝔽 [(A(n)

i ) s

] → 𝔽 [As

i]

s ∈ ℕ A = spec(G) G ∼ GUE0

d

Strengthened convergence

X(n) ∼ ̂ pd,n A(n) = n d (X(n) − 1 d(1,...,1))

Theorem (CLMSSW ’19): converges in moments, i.e. for all where and .

A(n) 𝔽 [(A(n)

i ) s

] → 𝔽 [As

i]

s ∈ ℕ A = spec(G) G ∼ GUE0

d

Expectations of rational functions?

Strengthened convergence

• X(n) ∼ ̂

pd,n A(n) = n d (X(n) − 1 d(1,...,1))

Sum-free ordered cone Cd = {x ∈ ℝd xi ≥ xi+1, ∑

i

xi = 0}

A(n), spec(G) ∈ Cd

Strengthened convergence

• X(n) ∼ ̂

pd,n A(n) = n d (X(n) − 1 d(1,...,1))

Sum-free ordered cone Cd = {x ∈ ℝd xi ≥ xi+1, ∑

i

xi = 0}

A(n), spec(G) ∈ Cd

:

˜ A(n) ˜ A(n)

i

= A(n)

i

+ d n (d − i) ⇒ ˜ A(n) ∈ int(Cd)

Strengthened convergence

• X(n) ∼ ̂

pd,n A(n) = n d (X(n) − 1 d(1,...,1))

Sum-free ordered cone Cd = {x ∈ ℝd xi ≥ xi+1, ∑

i

xi = 0}

A(n), spec(G) ∈ Cd

:

˜ A(n) ˜ A(n)

i

= A(n)

i

+ d n (d − i) ⇒ ˜ A(n) ∈ int(Cd)

for same convergence

→ 0 n → ∞ ⇒

Strengthened convergence

Theorem (CLMSSW ’19, informal): Let be a continuous function that diverges at most as where

• is the distance from a facet of

, and

• .

Then where and .

g : int(Cd) → ℝ δη

δ Cd η > − 2 − 1 d − 1

𝔽 [g ( ˜ A(n))] → 𝔽 [g(A)] A = spec(G) G ∼ GUE0

d

Previous results

Exact formulas (Studzinski et al. ’16, Mozrzymas et al. 17’): with Evaluating these formulas scales exponential in ! Asymptotics for arbitrary and ?

FEPR

d

(N) = d−N−2 ∑

α⊢dN−1

∑

μ=α+□

dμmμ

2

pEPR

d

(N) = 1 dN ∑

α⊢dN−1

m2

α

dμ* mμ* μ* = maxargμ=α+□ mμdα mαdμ F*

d (N) = d−2 sup q

∑

μ, μ′ ⊢d N μ′ = μ + □ − □

q(μ)q(μ′ ) d d N → ∞

Previous results

Exact formulas (Studzinski et al. ’16, Mozrzymas et al. 17’): with

FEPR

d

(N) = d−N−2 ∑

α⊢dN−1

∑

μ=α+□

dμmμ

2

pEPR

d

(N) = 1 dN ∑

α⊢dN−1

m2

α

dμ* mμ* μ* = maxargμ=α+□ mμdα mαdμ F*

d (N) = d−2 sup q

∑

μ, μ′ ⊢d N μ′ = μ + □ − □

q(μ)q(μ′ )

Expectation w.r.t. the Schur-Weyl distribution! Asymptotics in terms of GUE0

d

Port-based teleportation: EPR

Theorem (CLMSSW ’19): The fidelity

• f plain port-based teleportation with maximally

entangled states is given by . For , the success probability of heralded port-based teleportation with maximally entangled states is given by .

FEPR FEPR = 1 − d2 − 1 4N + o(N−1) G ∼ GUE0

d

pEPR = 1 − 𝔽[λmax(G)] d N + o(N−1/2)

: largest eigenvalue

λmax

Port-based teleportation: optimized

Theorem (CLMSSW ’19): There exist constants , such that the fidelity

• f plain port-

based teleportation with optimized entangled resource is given by for large enough.

cd c′

d > 0

F* 1 − cdN−2 ≤ F* ≤ 1 − c′

dN−2

N

Ishizaka ’15 Completely different techniques. Formula by Mozrzymas et al. principal eigenvalue of graph Laplacian on simplex discretized with root lattice of scaled by .

∝ 𝔱𝔳(d) N−1

Conclusion

• Strengthened convergence theorem for Schur-Weyl distribution
• Characterized asymptotic performance of the standard port-

based teleportation protocols

• Heralding comes at a cost, optimized resources help.

Summary Open problems

• Close gap for plain PBT with optimized resource
• Directly characterize cost of instantaneous non-local quantum

computation

• General asymptotics for N, d → ∞

Thanks!

FEPR = 1 − d2 − 1 4N + o(N−1) pEPR = 1 − 𝔽[λmax(G)] d N + o(N−1/2) ≤ 1 − c′

dN−2

1 − cdN−2 ≤ F* p* = 1 − d2 − 1 d2 − 1 + N