Non-asymptotic entanglement distillation arXiv:1706.06221 Kun Fang - - PowerPoint PPT Presentation

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Non-asymptotic entanglement distillation arXiv:1706.06221 Kun Fang - - PowerPoint PPT Presentation

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Non-asymptotic entanglement distillation arXiv:1706.06221 Kun Fang Joint work with Xin Wang, Marco Tomamichel, Runyao Duan Centre for Quantum Software and Information U niversity of T echnology S ydney Entanglement distillation [Bennett,

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SLIDE 1

Non-asymptotic entanglement distillation

arXiv:1706.06221

Kun Fang

Joint work with Xin Wang, Marco Tomamichel, Runyao Duan Centre for Quantum Software and Information University of Technology Sydney

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SLIDE 2

Entanglement distillation

[Bennett, DiVincenzo, Smolin, Wootters, 1996]

Alice Bob Alice Bob Π ρ⊗n

AB

  • |00+|11

√ 2

⊗m ED

  • ρAB
  • : sup

m n : lim

n→∞ inf Π∈Ω

  • Π
  • ρ⊗n

AB

  • − φ⊗m
  • 1
  • error
  • .

|00+|11 √ 2

Asymptotically, the number of copies of Bell state we can get from per given state ρ.

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
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SLIDE 3

Entanglement distillation

[Bennett, DiVincenzo, Smolin, Wootters, 1996]

ED

  • ρAB
  • : sup

m n : lim

n→∞ inf Π∈Ω

  • Π
  • ρ⊗n

AB

  • − φ⊗m
  • 1
  • error
  • .

|00+|11 √ 2

⊚ Theoretically, fundamental and interesting. ⊚ But not easy to calculate in general. ⊚ From practical point of view, limn→∞ is not possible.

How to do estimation when we only have finite copies of state?

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
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SLIDE 4

Concrete example

ρAB 0.7 · |v1v1| + 0.3 · |v2v2|, |v1 1 √ 2 (|00 + |11) , |v2 1 √ 2 (|01 + |10) .

Question:

How many copies of Bell state we can get at most from 222 copies of the state ρ (within the error tolerance 0.01) ?

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
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SLIDE 5

One-shot entanglement distillation

Alice Bob Alice Bob Π ρAB

  • |00+|11

√ 2

⊗m ⊚ Fidelity of distillation [Rains, 2001]: FΩ

  • ρAB, m

: max

Π∈Ω F

Π ρAB

  • , φ⊗m

, where φ |00 + |11 √ 2 . ⊚ One-shot distillable entanglement: E(1)

Ω,ε

  • ρAB
  • : max

m : 1 − FΩ

  • ρAB, m

≤ ε . ⊚ Asymptotic rate: EΩ

  • ρAB
  • lim

ε→0 lim n→∞

1 n E(1)

Ω,ε

  • ρ⊗n

AB

  • .

Ω ∈ {1-LOCC, LOCC, SEP, PPT}

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
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SLIDE 6

A hierarchy of operation classes LOCC

Local operations and classical communication A ←→ B

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
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SLIDE 7

A hierarchy of operation classes LOCC 1-LOCC

Local operations and classical communication A −→ B

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
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SLIDE 8

A hierarchy of operation classes SEP JΠ ΠA1B1→A2B2

  • φA1B1:A′

1B′ 1

  • JΠ separable (A′

1A2 : B′ 1B2)

LOCC 1-LOCC

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
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SLIDE 9

A hierarchy of operation classes PPT JΠ ΠA1B1→A2B2

  • φA1B1:A′

1B′ 1

  • J

TB′

1B2

Π

≥ 0

SEP LOCC 1-LOCC

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
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SLIDE 10

One-shot SDP characterization

For any state ρAB and error tolerance ε ∈ (0, 1), E(1)

PPT,ε

  • ρAB
  • − log

min η s.t. 0 ≤ M ≤ 1, Tr ρM ≥ 1 − ε, − η1 ≤ MTB ≤ η1. Efficiently computable Main ingredient of this proof: Symmetry of maximally entangled state φ, i.e., φ is invariant under U ⊗ U.

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
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SLIDE 11

One-shot SDP characterization

For any state ρAB and error tolerance ε ∈ (0, 1), E(1)

PPT,ε

  • ρAB
  • − log

min η s.t. 0 ≤ M ≤ 1, Tr ρM ≥ 1 − ε, − η1 ≤ MTB ≤ η1. Efficiently computable

Are we done? How about large number of copies E(1)

PPT,ε

  • ρ⊗n

AB

  • ?

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
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SLIDE 12

Quantum hypothesis testing

? ρ ∈ {ρ1, ρ2}

Null: ρ ρ1 Alternative: ρ ρ2

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
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SLIDE 13

Quantum hypothesis testing

? ρ ∈ {ρ1, ρ2}

Null: ρ ρ1 Alternative: ρ ρ2

{M1, M2}

ρ i

i 1, accept ρ ρ1 i 2, accept ρ ρ2

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
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SLIDE 14

Quantum hypothesis testing

? ρ ∈ {ρ1, ρ2}

Null: ρ ρ1 Alternative: ρ ρ2

{M1, M2}

ρ i

i 1, accept ρ ρ1 i 2, accept ρ ρ2

{M1, M2}

ρ1 2

Type-I error: Tr M2ρ1

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
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SLIDE 15

Quantum hypothesis testing

? ρ ∈ {ρ1, ρ2}

Null: ρ ρ1 Alternative: ρ ρ2

{M1, M2}

ρ i

i 1, accept ρ ρ1 i 2, accept ρ ρ2

{M1, M2}

ρ1 2

Type-I error: Tr M2ρ1

{M1, M2}

ρ2 1

Type-II error: Tr M1ρ2

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
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SLIDE 16

Quantum hypothesis testing

{M1, M2}

ρ1 2

Type-I error: Tr M2ρ1

{M1, M2}

ρ2 1

Type-II error: Tr M1ρ2 Dε

H

  • ρ1||ρ2
  • : − log

min Tr M1ρ2 s.t. Tr M2ρ1 ≤ ε, M1, M2 ≥ 0, M1 + M2 1. Type-II error Type-I error

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
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SLIDE 17

One-shot Hypothesis testing characterization

Build a connection, E(1)

PPT,ε

  • ρAB
  • min

GTB1≤1

Dε

H

  • ρABG

. Distillation Hypothesis testing

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
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SLIDE 18

One-shot Hypothesis testing characterization

Build a connection, E(1)

PPT,ε

  • ρAB
  • min

GTB1≤1

Dε

H

  • ρABG

. Distillation Hypothesis testing Hermitian

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
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SLIDE 19

One-shot Hypothesis testing characterization

Build a connection, E(1)

PPT,ε

  • ρAB
  • min

GTB1≤1

Dε

H

  • ρABG

. Distillation Hypothesis testing Hermitian ρ G "Distance measure" → Dε

H

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
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SLIDE 20

One-shot Hypothesis testing characterization

Build a connection, E(1)

PPT,ε

  • ρAB
  • min

GTB1≤1

Dε

H

  • ρABG

. Distillation Hypothesis testing Hermitian ρ G "Distance measure" → Dε

H

Main ingredient of this proof: Norm duality between · 1 and · ∞.

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
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SLIDE 21

One-shot Hypothesis testing characterization

Build a connection, E(1)

PPT,ε

  • ρAB
  • min

GTB1≤1

Dε

H

  • ρABG

. Distillation Hypothesis testing Two Applications:

⊚ Recover the Rains bound. ⊚ Second-order estimation.

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
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SLIDE 22

Recover the Rains bound

E(1)

PPT,ε

  • ρ
  • min

GTB 1≤1 Dε

H

  • ρG

. ⊚ Rains bound [Rains, 2001; Audenaert, Moor, Vollbrecht, Werner, 2002] R ρ

  • min

σ≥0,σTB 1≤1

D ρσ , EPPT

  • ρ

≤ R ρ .

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
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SLIDE 23

Recover the Rains bound

E(1)

PPT,ε

  • ρ
  • min

GTB 1≤1 Dε

H

  • ρG

. ⊚ Rains bound [Rains, 2001; Audenaert, Moor, Vollbrecht, Werner, 2002] R ρ

  • min

σ≥0,σTB 1≤1

D ρσ , EPPT

  • ρ

≤ R ρ . 1 n E(1)

PPT,ε

  • ρ⊗n

1 n min

  • GTBn

1≤1

Dε

H

  • ρ⊗nG

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
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SLIDE 24

Recover the Rains bound

E(1)

PPT,ε

  • ρ
  • min

GTB 1≤1 Dε

H

  • ρG

. ⊚ Rains bound [Rains, 2001; Audenaert, Moor, Vollbrecht, Werner, 2002] R ρ

  • min

σ≥0,σTB 1≤1

D ρσ , EPPT

  • ρ

≤ R ρ . 1 n E(1)

PPT,ε

  • ρ⊗n

1 n min

  • GTBn

1≤1

Dε

H

  • ρ⊗nG

≤ 1 n Dε

H

  • ρ⊗nσ⊗n

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
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SLIDE 25

Recover the Rains bound

E(1)

PPT,ε

  • ρ
  • min

GTB 1≤1 Dε

H

  • ρG

. ⊚ Rains bound [Rains, 2001; Audenaert, Moor, Vollbrecht, Werner, 2002] R ρ

  • min

σ≥0,σTB 1≤1

D ρσ , EPPT

  • ρ

≤ R ρ . 1 n E(1)

PPT,ε

  • ρ⊗n

1 n min

  • GTBn

1≤1

Dε

H

  • ρ⊗nG

≤ 1 n Dε

H

  • ρ⊗nσ⊗n

Quantum Stein′s lemma

− − − − − − − − − − − − − − − − − − − − →

[Hiai & Petz,1991]

D ρσ

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
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SLIDE 26

Recover the Rains bound

E(1)

PPT,ε

  • ρ
  • min

GTB 1≤1 Dε

H

  • ρG

. ⊚ Rains bound [Rains, 2001; Audenaert, Moor, Vollbrecht, Werner, 2002] R ρ

  • min

σ≥0,σTB 1≤1

D ρσ , EPPT

  • ρ

≤ R ρ . 1 n E(1)

PPT,ε

  • ρ⊗n

1 n min

  • GTBn

1≤1

Dε

H

  • ρ⊗nG

≤ 1 n Dε

H

  • ρ⊗nσ⊗n

Quantum Stein′s lemma

− − − − − − − − − − − − − − − − − − − − →

[Hiai & Petz,1991]

D ρσ R ρ .

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
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SLIDE 27

Recover the Rains bound

E(1)

PPT,ε

  • ρ
  • min

GTB 1≤1 Dε

H

  • ρG

. ⊚ Rains bound [Rains, 2001; Audenaert, Moor, Vollbrecht, Werner, 2002] R ρ

  • min

σ≥0,σTB 1≤1

D ρσ , EPPT

  • ρ

≤ R ρ . EPPT

  • ρ

← 1 n E(1)

PPT,ε

  • ρ⊗n

1 n min

  • GTBn

1≤1

Dε

H

  • ρ⊗nG

≤ 1 n Dε

H

  • ρ⊗nσ⊗n

Quantum Stein′s lemma

− − − − − − − − − − − − − − − − − − − − →

[Hiai & Petz,1991]

D ρσ R ρ .

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
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SLIDE 28

Recover the Rains bound

E(1)

PPT,ε

  • ρ
  • min

GTB 1≤1 Dε

H

  • ρG

. ⊚ Rains bound [Rains, 2001; Audenaert, Moor, Vollbrecht, Werner, 2002] R ρ

  • min

σ≥0,σTB 1≤1

D ρσ , EPPT

  • ρ

≤ R ρ . EPPT

  • ρ

← 1 n E(1)

PPT,ε

  • ρ⊗n

1 n min

  • GTBn

1≤1

Dε

H

  • ρ⊗nG

≤ 1 n Dε

H

  • ρ⊗nσ⊗n

Quantum Stein′s lemma

− − − − − − − − − − − − − − − − − − − − →

[Hiai & Petz,1991]

D ρσ R ρ . ⊚ Can we improve it by taking other forms of feasible solution?

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
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SLIDE 29

Second-order estimation: upper bound

E(1)

PPT,ε

  • ρ
  • min

GTB 1≤1 Dε

H

  • ρG

.

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
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SLIDE 30

Second-order estimation: upper bound

E(1)

PPT,ε

  • ρ
  • min

GTB 1≤1 Dε

H

  • ρG

. [Tomamichel & Hayashi, 2013; Li 2014] Dε

H

  • ρ⊗n||σ⊗n

nD ρ||σ +

  • nV

ρ||σ Φ−1 (ε) + O log n .

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
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SLIDE 31

Second-order estimation: upper bound

E(1)

PPT,ε

  • ρ
  • min

GTB 1≤1 Dε

H

  • ρG

. [Tomamichel & Hayashi, 2013; Li 2014] Dε

H

  • ρ⊗n||σ⊗n

nD ρ||σ +

  • nV

ρ||σ Φ−1 (ε) + O log n . E(1)

PPT,ε

  • ρ⊗n

≤ nR ρ +

  • nVR
  • ρ

Φ−1 (ε) + O log n .

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
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SLIDE 32

Second-order estimation: upper bound

E(1)

PPT,ε

  • ρ
  • min

GTB 1≤1 Dε

H

  • ρG

. [Tomamichel & Hayashi, 2013; Li 2014] Dε

H

  • ρ⊗n||σ⊗n

nD ρ||σ +

  • nV

ρ||σ Φ−1 (ε) + O log n . E(1)

PPT,ε

  • ρ⊗n

≤ nR ρ +

  • nVR
  • ρ

Φ−1 (ε) + O log n .

where VR

  • ρAB

    maxσ∈Sρ V ρAB

  • σAB
  • if

0 < ε ≤ 1/2 minσ∈Sρ V ρAB

  • σAB
  • if

1/2 < ε < 1 , and Sρ is the set of operators that achieve the minimum of R ρ D ρσ : Tr ρ log ρ − log σ , V ρσ : Tr ρ log ρ − log σ2 − D ρσ2 , Φ−1 inverse of the cumulative distribution function of standard normal distribution.

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
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SLIDE 33

Second-order estimation: lower bound

[Wilde, Tomamichel, Berta, 2016] E(1)

→,ε

  • ρAB
  • ≥ −H

√ε−η max (A|B)ρ + 4 log η, where 0 ≤ η <

√ ε.

1-LOCC Smooth conditional max-entropy

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
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SLIDE 34

Second-order estimation: lower bound

[Wilde, Tomamichel, Berta, 2016] E(1)

→,ε

  • ρAB
  • ≥ −H

√ε−η max (A|B)ρ + 4 log η, where 0 ≤ η <

√ ε.

1-LOCC Smooth conditional max-entropy

[Tomamichel & Hayashi, 2013] Hε

max (An|Bn)ρ⊗n nH (A|B)ρ −

  • nV (A|B)ρΦ−1

ε2 + O log n .

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
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SLIDE 35

Second-order estimation: lower bound

[Wilde, Tomamichel, Berta, 2016] E(1)

→,ε

  • ρAB
  • ≥ −H

√ε−η max (A|B)ρ + 4 log η, where 0 ≤ η <

√ ε.

1-LOCC Smooth conditional max-entropy

[Tomamichel & Hayashi, 2013] Hε

max (An|Bn)ρ⊗n nH (A|B)ρ −

  • nV (A|B)ρΦ−1

ε2 + O log n . E(1)

→,ε

  • ρ⊗n

AB

  • ≥ nI (AB)ρ +
  • nV (A|B)ρ Φ−1 (ε) + O

log n . where I (AB)ρ : D ρAB1A ⊗ ρB

  • , V (A|B)ρ : V

ρAB1A ⊗ ρB

  • .

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
slide-36
SLIDE 36

Examples: pure state

For any pure state ψ, with reduced state ρA TrB ψ,

E(1)

→,ε

  • ψ⊗n

E(1)

PPT,ε

  • ψ⊗n

nS ρA

  • +
  • n
  • Tr ρA
  • log ρA

2 − S ρA 2 Φ−1 (ε) + O log n .

Remark: Recover [Datta, Leditzky, 2015] ’s result about distillable entanglement via

LOCC operations for pure states, since 1-LOCC LOCC PPT.

102 104 106

Number of state copies, n

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Average distillation rate (qubit)

ψ |00+2|11

√ 5

, ε 0.01.

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
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SLIDE 37

Examples: mixed state

For the state ρAB p|v1v1| + 1 − p |v2v2|, where p ∈ (0, 1), |v1 1 √ 2 (|00 + |11) , |v2 1 √ 2 (|01 + |10) , its distillable entanglement is

E(1)

→,ε

  • ρ⊗n

AB

  • E(1)

PPT,ε

  • ρ⊗n

AB

  • n

1 − h2

  • p

+

  • np

1 − p log 1 − p p 2 Φ−1 (ε) + O log n .

where h2

  • p

−p log p − 1 − p log 1 − p .

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
slide-38
SLIDE 38

Examples: Isotropic state ρF (F 0.9, ε 0.001)

ρF (1 − F) 1 − φ (d) d2 − 1 + F · φ (d), F ∈ [0, 1], φ (d) 1 d

d−1

  • i,j0

|iijj|. Small number of copies:

10 20 30 40 50 60 70 80 90 100

Number of state copies, n

0.2 0.4 0.6 0.8 1

Average distillation rate (qubit)

Large number of copies:

102 104 106

Number of state copies, n

0.2 0.4 0.6 0.8 1

Average distillation rate (qubit) Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
slide-39
SLIDE 39

Example: Isotropic state ρF (F 0.9, ε 0.001)

102 104 106

Number of state copies, n

0.2 0.4 0.6 0.8 1

Average distillation rate (qubit)

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
slide-40
SLIDE 40

Example: Isotropic state ρF (F 0.9, ε 0.001)

102 104 106

Number of state copies, n

0.2 0.4 0.6 0.8 1

Average distillation rate (qubit) c1 + c2 1

√n + c3 log n n

+ c4 1

n

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
slide-41
SLIDE 41

Example: Isotropic state ρF (F 0.9, ε 0.001)

102 104 106

Number of state copies, n

0.2 0.4 0.6 0.8 1

Average distillation rate (qubit) c1 + c2 1

√n + c3 log n n

+ c4 1

n

1 n E(1)

PPT,ε

  • ρ⊗n

≤ R ρ + 1 √n

  • VR
  • ρ

Φ−1 (ε) + O log n n

  • .

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
slide-42
SLIDE 42

Example: Isotropic state ρF (F 0.9, ε 0.001)

102 104 106

Number of state copies, n

0.2 0.4 0.6 0.8 1

Average distillation rate (qubit) c1 + c2 1

√n + c3 log n n

+ c4 1

n

Conjecture: EPPT

  • ρF
  • R

ρF

  • .

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
slide-43
SLIDE 43

Summary

E(1)

PPT,ε

  • ρ

small scale estimation

SDP

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
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SLIDE 44

Summary

Hypothesis testing

E(1)

PPT,ε

  • ρ
  • min
  • GTB
  • 1≤1

Dε

H

  • ρG

E(1)

PPT,ε

  • ρ

small scale estimation

SDP

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
slide-45
SLIDE 45

Summary

Hypothesis testing

E(1)

PPT,ε

  • ρ
  • min
  • GTB
  • 1≤1

Dε

H

  • ρG

E(1)

PPT,ε

  • ρ

small scale estimation

SDP

Rains bound

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
slide-46
SLIDE 46

Summary

Hypothesis testing

E(1)

PPT,ε

  • ρ
  • min
  • GTB
  • 1≤1

Dε

H

  • ρG

E(1)

PPT,ε

  • ρ

small scale estimation

SDP

Rains bound improve?

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
slide-47
SLIDE 47

Summary

Hypothesis testing

E(1)

PPT,ε

  • ρ
  • min
  • GTB
  • 1≤1

Dε

H

  • ρG

E(1)

PPT,ε

  • ρ

small scale estimation

SDP

Rains bound improve? Second-order bound large scale estimation

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
slide-48
SLIDE 48

Summary

Hypothesis testing

E(1)

PPT,ε

  • ρ
  • min
  • GTB
  • 1≤1

Dε

H

  • ρG

E(1)

PPT,ε

  • ρ

small scale estimation

SDP

Rains bound improve? Second-order bound large scale estimation EPPT

  • ρF

? R ρF

  • Non-asymptotic entanglement distillation (1706.06221)

|

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
slide-49
SLIDE 49

Summary

Hypothesis testing

E(1)

PPT,ε

  • ρ
  • min
  • GTB
  • 1≤1

Dε

H

  • ρG

E(1)

PPT,ε

  • ρ

small scale estimation

SDP

Rains bound improve? Second-order bound large scale estimation EPPT

  • ρF

? R ρF

  • Entanglement dilution?

Key distillation? Multi-partite cases?

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
slide-50
SLIDE 50

THE END THANK YOU!

slide-51
SLIDE 51

References

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  • Inf. Theory, vol. 47, no. 7, pp. 2921–2933, 2001.

4 K. Audenaert, B. De Moor, K. G. H. Vollbrecht, and R. F. Werner, “Asymptotic relative entropy of entanglement for orthogonally invariant states,” Phys. Rev. A,

  • vol. 66, no. 3, p. 32310, 2002.

5 F. Hiai and D. Petz, “The proper formula for relative entropy and its asymptotics in quantum probability,” Commun. Math. Phys., vol. 143, no. 1, pp. 99–114, 1991. 6 M. Tomamichel and M. Hayashi, “A hierarchy of information quantities for finite block length analysis of quantum tasks,” IEEE Trans. Inf. Theory, vol. 59, no. 11,

  • pp. 7693–7710, 2013.

7 K. Li, “Second-order asymptotics for quantum hypothesis testing,” Ann. Stat., vol. 42, no. 1, pp. 171–189, Feb. 2014.

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan
slide-52
SLIDE 52

References

8 M. M. Wilde, M. Tomamichel, and M. Berta, “Converse bounds for private communication over quantum channels,” IEEE Trans. Inf. Theory, vol. 63, no. 3,

  • pp. 1792–1817, Feb. 2016.

9 Y. Zinchenko, S. Friedland, and G. Gour, “Numerical estimation of the relative entropy of entanglement,” Phys. Rev. A, vol. 82, no. 5, p. 52336, 2010. 10 H. Fawzi and O. Fawzi, “Relative entropy optimization in quantum information theory via semidefinite programming approximations,” arXiv: 1705.06671 11 H. Fawzi, J. Saunderson, and P. A. Parrilo, “Semidefinite approximations of the matrix logarithm,” arXiv: 1705.00812 12 N. Datta and F. Leditzky, “Second-Order Asymptotics for Source Coding, Dense Coding, and Pure-State Entanglement Conversions,” IEEE Trans. Inf. Theory, vol. 61, no. 1, pp. 582–608, 2015. 13 X. Wang and R. Duan, “Nonadditivity of Rains’ bound for distillable entanglement,” Phys. Rev. A, vol. 95, no. 6, p. 62322, Jun. 2017.

Non-asymptotic entanglement distillation (1706.06221) |

  • K. Fang, X. Wang, M. Tomamichel, R. Duan