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Locality and a bound on entanglement assistance to classical - - PowerPoint PPT Presentation

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Locality and a bound on entanglement assistance to classical communication Mih aly Weiner (work in progress; joint with P.E. Frenkel ) Quantum Information and Operator Algebras Rome, 16 february 2018 Mih aly Weiner Locality and

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SLIDE 1

Locality and a bound on entanglement assistance to classical communication

Mih´ aly Weiner

(work in progress; joint with P.E. Frenkel)

Quantum Information and Operator Algebras Rome, 16 february 2018

Mih´ aly Weiner Locality and entanglement assistance 1 / 15

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SLIDE 2

2 headed oracles

X1 ⇒ Z1 ⇐ ⇐ X2 ⇒ Z2

Mih´ aly Weiner Locality and entanglement assistance 2 / 15

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SLIDE 3

2 headed oracles

X1 ⇒ Z1 ⇐ ⇐ X2 ⇒ Z2 aa

Mih´ aly Weiner Locality and entanglement assistance 2 / 15

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SLIDE 4

2 headed oracles

X1 ⇒ Z1 ⇐ ⇐ X2 ⇒ Z2 aa aa

Mih´ aly Weiner Locality and entanglement assistance 2 / 15

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SLIDE 5

2 headed oracles

X1 ⇒ Z1 ⇐ ⇐ X2 ⇒ Z2 aa aa user’s point of view:

Mih´ aly Weiner Locality and entanglement assistance 2 / 15

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SLIDE 6

2 headed oracles

X1 ⇒ Z1 ⇐ ⇐ X2 ⇒ Z2 aa aa user’s point of view: p(Z1 = z1, Z2 = z2 | X1 = x1, X2 = x2)

Mih´ aly Weiner Locality and entanglement assistance 2 / 15

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SLIDE 7

2 headed oracles

X1 ⇒ Z1 ⇐ ⇐ X2 ⇒ Z2 aa aa user’s point of view: p(Z1 = z1, Z2 = z2 | X1 = x1, X2 = x2) aa aaa. ⇒ a point in Rn1 × Rn2 × Rm1 × Rm2

Mih´ aly Weiner Locality and entanglement assistance 2 / 15

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SLIDE 8

2 headed oracles

X1 ⇒ Z1 ⇐ ⇐ X2 ⇒ Z2 aa aa user’s point of view: p(Z1 = z1, Z2 = z2 | X1 = x1, X2 = x2) aa aaa. ⇒ a point in Rn1 × Rn2 × Rm1 × Rm2 aa aaa. ⇒ all 2-headed oracles form a polytope

Mih´ aly Weiner Locality and entanglement assistance 2 / 15

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SLIDE 9

Realizations

p(Z1 = z1, Z2 = z2 | X1 = x1, X2 = x2) = ?

Mih´ aly Weiner Locality and entanglement assistance 3 / 15

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SLIDE 10

Realizations

p(Z1 = z1, Z2 = z2 | X1 = x1, X2 = x2) = ? Classical:

Mih´ aly Weiner Locality and entanglement assistance 3 / 15

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SLIDE 11

Realizations

p(Z1 = z1, Z2 = z2 | X1 = x1, X2 = x2) = ? Classical:

  • pI,λ(z1|x1) pII,λ(z2|x2) dµ(λ)

Mih´ aly Weiner Locality and entanglement assistance 3 / 15

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SLIDE 12

Realizations

p(Z1 = z1, Z2 = z2 | X1 = x1, X2 = x2) = ? Classical:

  • pI,λ(z1|x1) pII,λ(z2|x2) dµ(λ)

Mih´ aly Weiner Locality and entanglement assistance 3 / 15

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SLIDE 13

Realizations

p(Z1 = z1, Z2 = z2 | X1 = x1, X2 = x2) = ? Classical:

  • pI,λ(z1|x1) pII,λ(z2|x2) dµ(λ)

Mih´ aly Weiner Locality and entanglement assistance 3 / 15

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SLIDE 14

Realizations

p(Z1 = z1, Z2 = z2 | X1 = x1, X2 = x2) = ? Classical:

  • pI,λ(z1|x1) pII,λ(z2|x2) dµ(λ)

Quantum: ϕ(Ax1,z1Bx2,z2)

Mih´ aly Weiner Locality and entanglement assistance 3 / 15

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SLIDE 15

Realizations

p(Z1 = z1, Z2 = z2 | X1 = x1, X2 = x2) = ? Classical:

  • pI,λ(z1|x1) pII,λ(z2|x2) dµ(λ)

Quantum: ϕ(Ax1,z1Bx2,z2) where

ϕ is a positive normalized functional Ax1,z1 ≥ 0,

z1 Ax1,z1 = I & sim. cond. for B

[Ax1,z1, Bx2,z2] = 0

Mih´ aly Weiner Locality and entanglement assistance 3 / 15

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SLIDE 16

No-signaling

Input at 1 should have no effect on output at 2:

Mih´ aly Weiner Locality and entanglement assistance 4 / 15

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SLIDE 17

No-signaling

Input at 1 should have no effect on output at 2:

  • zk

p(Z2 =z2 | X1 =x1, X2 =x2) = p(Z2 =z2 | X1 = ˜ x1, X2 =x2)

Mih´ aly Weiner Locality and entanglement assistance 4 / 15

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SLIDE 18

No-signaling

Input at 1 should have no effect on output at 2:

  • zk

p(Z2 =z2 | X1 =x1, X2 =x2) = p(Z2 =z2 | X1 = ˜ x1, X2 =x2)

Mih´ aly Weiner Locality and entanglement assistance 4 / 15

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SLIDE 19

No-signaling

Input at 1 should have no effect on output at 2:

  • zk

p(Z2 =z2 | X1 =x1, X2 =x2) = p(Z2 =z2 | X1 = ˜ x1, X2 =x2)

Mih´ aly Weiner Locality and entanglement assistance 4 / 15

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SLIDE 20

No-signaling

Input at 1 should have no effect on output at 2:

  • zk

p(Z2 =z2 | X1 =x1, X2 =x2) = p(Z2 =z2 | X1 = ˜ x1, X2 =x2)

Mih´ aly Weiner Locality and entanglement assistance 4 / 15

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SLIDE 21

No-signaling

Input at 1 should have no effect on output at 2:

  • z1

p(Z1 =z1, Z2 =z2 | X1 =x1, X2 =x2) = func. of z2 & x2 only

Mih´ aly Weiner Locality and entanglement assistance 4 / 15

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SLIDE 22

No-signaling

Input at 1 should have no effect on output at 2:

  • z1

p(Z1 =z1, Z2 =z2 | X1 =x1, X2 =x2) = func. of z2 & x2 only

Mih´ aly Weiner Locality and entanglement assistance 4 / 15

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SLIDE 23

No-signaling

Input at 1 should have no effect on output at 2:

  • z1

p(Z1 =z1, Z2 =z2 | X1 =x1, X2 =x2) = func. of z2 & x2 only Input at 2 should have no effect on output at 1:

Mih´ aly Weiner Locality and entanglement assistance 4 / 15

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SLIDE 24

No-signaling

Input at 1 should have no effect on output at 2:

  • z1

p(Z1 =z1, Z2 =z2 | X1 =x1, X2 =x2) = func. of z2 & x2 only Input at 2 should have no effect on output at 1:

  • z2

p(Z1 =z1, Z2 =z2 | X1 =x1, X2 =x2) = func. of z1 & x1 only

Mih´ aly Weiner Locality and entanglement assistance 4 / 15

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SLIDE 25

No-signaling

Input at 1 should have no effect on output at 2:

  • z1

p(Z1 =z1, Z2 =z2 | X1 =x1, X2 =x2) = func. of z2 & x2 only Input at 2 should have no effect on output at 1:

  • z2

p(Z1 =z1, Z2 =z2 | X1 =x1, X2 =x2) = func. of z1 & x1 only XX “NS-oracle” ∈ “No-signaling polytope”

Mih´ aly Weiner Locality and entanglement assistance 4 / 15

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SLIDE 26

No-signaling quantum

No-signaling polytope: f.i

Mih´ aly Weiner Locality and entanglement assistance 5 / 15

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SLIDE 27

No-signaling quantum

No-signaling polytope: ,,

Mih´ aly Weiner Locality and entanglement assistance 5 / 15

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SLIDE 28

No-signaling quantum

No-signaling polytope: ,

Mih´ aly Weiner Locality and entanglement assistance 5 / 15

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SLIDE 29

No-signaling quantum

No-signaling polytope:

Mih´ aly Weiner Locality and entanglement assistance 5 / 15

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SLIDE 30

No-signaling quantum

No-signaling polytope:

Mih´ aly Weiner Locality and entanglement assistance 5 / 15

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SLIDE 31

No-signaling quantum

No-signaling polytope:

Mih´ aly Weiner Locality and entanglement assistance 5 / 15

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SLIDE 32

Information Causality

aa d: n cbits data aaaaa aaaaaaa aaaaaaa aaaaa r ∈ {1, . . . n} aaaa ⇓ aaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇓

g: guess

⇒ aaaa ⇓ aaaaaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇑ aaaaaaa ⇒ ⇒ ⇒ ⇒

h: m cbits help

⇒ ⇒ ⇒ ⇒

n

  • k=1

I(dk : g |r = k) ≤ m

Mih´ aly Weiner Locality and entanglement assistance 6 / 15

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SLIDE 33

Information Causality

aa d: n cbits data aaaaa aaaaaaa aaaaaaa aaaaa r ∈ {1, . . . n} aaaa ⇓ aaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇓

g: guess

⇒ aaaa ⇓ aaaaaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇑ aaaaaaa ⇒ ⇒ ⇒ ⇒

h: m cbits help

⇒ ⇒ ⇒ ⇒

n

  • k=1

I(dk : g |r = k) ≤ m

Mih´ aly Weiner Locality and entanglement assistance 6 / 15

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SLIDE 34

Information Causality

aa d: n cbits data aaaaa aaaaaaa aaaaaaa aaaaa r ∈ {1, . . . n} aaaa ⇓ aaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇓

g: guess

⇒ aaaa ⇓ aaaaaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇑ aaaaaaa ⇒ ⇒ ⇒ ⇒

h: m cbits help

⇒ ⇒ ⇒ ⇒

n

  • k=1

I(dk : g |r = k) ≤ m

Mih´ aly Weiner Locality and entanglement assistance 6 / 15

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SLIDE 35

Information Causality

aa d: n cbits data aaaaa aaaaaaa aaaaaaa aaaaa r ∈ {1, . . . n} aaaa ⇓ aaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇓

g: guess

⇒ aaaa ⇓ aaaaaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇑ aaaaaaa ⇒ ⇒ ⇒ ⇒

h: m cbits help

⇒ ⇒ ⇒ ⇒

n

  • k=1

I(dk : g |r = k) ≤ m

Mih´ aly Weiner Locality and entanglement assistance 6 / 15

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SLIDE 36

Information Causality

aa d: n cbits data aaaaa aaaaaaa aaaaaaa aaaaa r ∈ {1, . . . n} aaaa ⇓ aaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇓

g: guess

⇒ aaaa ⇓ aaaaaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇑ aaaaaaa ⇒ ⇒ ⇒ ⇒

h: m cbits help

⇒ ⇒ ⇒ ⇒

n

  • k=1

I(dk : g |r = k) ≤ m

Mih´ aly Weiner Locality and entanglement assistance 6 / 15

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SLIDE 37

Information Causality

aa d: n cbits data aaaaa aaaaaaa aaaaaaa aaaaa r ∈ {1, . . . n} aaaa ⇓ aaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇓

g: guess

⇒ aaaa ⇓ aaaaaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇑ aaaaaaa ⇒ ⇒ ⇒ ⇒

h: m cbits help

⇒ ⇒ ⇒ ⇒

n

  • k=1

I(dk : g |r = k) ≤ m

Mih´ aly Weiner Locality and entanglement assistance 6 / 15

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SLIDE 38

Information Causality

aa d: n cbits data aaaaa aaaaaaa aaaaaaa aaaaa r ∈ {1, . . . n} aaaa ⇓ aaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇓

g: guess

⇒ aaaa ⇓ aaaaaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇑ aaaaaaa ⇒ ⇒ ⇒ ⇒

h: m cbits help

⇒ ⇒ ⇒ ⇒

n

  • k=1

I(dk : g |r = k) ≤ m

Mih´ aly Weiner Locality and entanglement assistance 6 / 15

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SLIDE 39

Information Causality

aa d: n cbits data aaaaa aaaaaaa aaaaaaa aaaaa r ∈ {1, . . . n} aaaa ⇓ aaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇓

g: guess

⇒ aaaa ⇓ aaaaaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇑ aaaaaaa ⇒ ⇒ ⇒ ⇒

h: m cbits help

⇒ ⇒ ⇒ ⇒

n

  • k=1

I(dk : g |r = k) ≤ m

Mih´ aly Weiner Locality and entanglement assistance 6 / 15

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SLIDE 40

Information Causality

aa d: n cbits data aaaaa aaaaaaa aaaaaaa aaaaa r ∈ {1, . . . n} aaaa ⇓ aaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇓

g: guess

⇒ aaaa ⇓ aaaaaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇑ aaaaaaa ⇒ ⇒ ⇒ ⇒

h: m cbits help

⇒ ⇒ ⇒ ⇒

n

  • k=1

I(dk : g |r = k) ≤ m

Mih´ aly Weiner Locality and entanglement assistance 6 / 15

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SLIDE 41

Information Causality

aa d: n cbits data aaaaa aaaaaaa aaaaaaa aaaaa r ∈ {1, . . . n} aaaa ⇓ aaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇓

g: guess

⇒ aaaa ⇓ aaaaaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇑ aaaaaaa ⇒ ⇒ ⇒ ⇒

h: m cbits help

⇒ ⇒ ⇒ ⇒

n

  • k=1

I(dk : g |r = k) ≤ m

Mih´ aly Weiner Locality and entanglement assistance 6 / 15

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SLIDE 42

Information Causality

aa d: n cbits data aaaaa aaaaaaa aaaaaaa aaaaa r ∈ {1, . . . n} aaaa ⇓ aaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇓

g: guess

⇒ aaaa ⇓ aaaaaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇑ aaaaaaa ⇒ ⇒ ⇒ ⇒

h: m cbits help

⇒ ⇒ ⇒ ⇒

n

  • k=1

I(dk : g |r = k) ≤ m

Mih´ aly Weiner Locality and entanglement assistance 6 / 15

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SLIDE 43

Information Causality

aa d: n cbits data aaaaa aaaaaaa aaaaaaa aaaaa r ∈ {1, . . . n} aaaa ⇓ aaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇓

g: guess

⇒ aaaa ⇓ aaaaaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇑ aaaaaaa ⇒ ⇒ ⇒ ⇒

h: m cbits help

⇒ ⇒ ⇒ ⇒

n

  • k=1

I(dk : g |r = k) ≤ m

Mih´ aly Weiner Locality and entanglement assistance 6 / 15

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SLIDE 44

Information Causality

aa d: n cbits data aaaaa aaaaaaa aaaaaaa aaaaa r ∈ {1, . . . n} aaaa ⇓ aaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇓

g: guess

⇒ aaaa ⇓ aaaaaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇑ aaaaaaa ⇒ ⇒ ⇒ ⇒

h: m cbits help

⇒ ⇒ ⇒ ⇒

n

  • k=1

I(dk : g |r = k) ≤ m

Mih´ aly Weiner Locality and entanglement assistance 6 / 15

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SLIDE 45

Information Causality

aa d: n cbits data aaaaa aaaaaaa aaaaaaa aaaaa r ∈ {1, . . . n} aaaa ⇓ aaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇓

g: guess

⇒ aaaa ⇓ aaaaaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇑ aaaaaaa ⇒ ⇒ ⇒ ⇒

h: m cbits help

⇒ ⇒ ⇒ ⇒

n

  • k=1

I(dk : g |r = k) ≤ m

Mih´ aly Weiner Locality and entanglement assistance 6 / 15

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SLIDE 46

Information Causality

Information Causality was proposed by Pawlowski, Paterek, Kaszlikowski, Scarani, Winter and Zukowski (Nature, 2009)

Mih´ aly Weiner Locality and entanglement assistance 7 / 15

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SLIDE 47

Information Causality

Information Causality was proposed by Pawlowski, Paterek, Kaszlikowski, Scarani, Winter and Zukowski (Nature, 2009) holds in the classical

Mih´ aly Weiner Locality and entanglement assistance 7 / 15

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SLIDE 48

Information Causality

Information Causality was proposed by Pawlowski, Paterek, Kaszlikowski, Scarani, Winter and Zukowski (Nature, 2009) holds in the classical / quantum case (nontrivial!)

Mih´ aly Weiner Locality and entanglement assistance 7 / 15

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SLIDE 49

Information Causality

Information Causality was proposed by Pawlowski, Paterek, Kaszlikowski, Scarani, Winter and Zukowski (Nature, 2009) holds in the classical / quantum case (nontrivial!) sometimes one needs to consider very high values of n, m to rule out a specific ns-oracle

Mih´ aly Weiner Locality and entanglement assistance 7 / 15

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SLIDE 50

Information Causality

Information Causality was proposed by Pawlowski, Paterek, Kaszlikowski, Scarani, Winter and Zukowski (Nature, 2009) holds in the classical / quantum case (nontrivial!) sometimes one needs to consider very high values of n, m to rule out a specific ns-oracle implies the Tsirelson-bound

Mih´ aly Weiner Locality and entanglement assistance 7 / 15

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SLIDE 51

Task: information sending

aaaa X aaaaa aaaaaaa aaaaaaa aaaaa r ∈ {1, . . . n} aaaa ⇓ aaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇓

Z

⇒ aaaa ⇓ aaaaaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇑ aaaaaaa ⇒ ⇒ ⇒ ⇒

m cbits

⇒ ⇒ ⇒ ⇒

Mih´ aly Weiner Locality and entanglement assistance 8 / 15

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SLIDE 52

Task: information sending

aaaa X aaaaa aaaaaaa aaaaaaa aaaaa r ∈ {1, . . . n} aaaa ⇓ aaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇓

Z

⇒ aaaa ⇓ aaaaaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇑ aaaaaaa ⇒ ⇒ ⇒ ⇒

m cbits

⇒ ⇒ ⇒ ⇒

Mih´ aly Weiner Locality and entanglement assistance 8 / 15

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SLIDE 53

Task: information sending

aaaa X aaaaa aaaaaaa aaaaaaa aaaaa r ∈ {1, . . . n} aaaa ⇓ aaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇓

Z

⇒ aaaa ⇓ aaaaaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇑ aaaaaaa ⇒ ⇒ ⇒ ⇒

m cbits

⇒ ⇒ ⇒ ⇒

Mih´ aly Weiner Locality and entanglement assistance 8 / 15

slide-54
SLIDE 54

Task: information sending

aaaa X aaaaa aaaaaaa aaaaaaa aaaaa r ∈ {1, . . . n} aaaa ⇓ aaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇓

Z

⇒ aaaa ⇓ aaaaaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇑ aaaaaaa ⇒ ⇒ ⇒ ⇒

m cbits

⇒ ⇒ ⇒ ⇒

Mih´ aly Weiner Locality and entanglement assistance 8 / 15

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SLIDE 55

Task: information sending

aaaa X aaaaa aaaaaaa aaaaaaa aaaaa r ∈ {1, . . . n} aaaa ⇓ aaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇓

Z

⇒ aaaa ⇓ aaaaaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇑ aaaaaaa ⇒ ⇒ ⇒ ⇒

m cbits

⇒ ⇒ ⇒ ⇒ X Z is a classical channel

Mih´ aly Weiner Locality and entanglement assistance 8 / 15

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SLIDE 56

Task: information sending

aaaa X aaaaa aaaaaaa aaaaaaa aaaaa r ∈ {1, . . . n} aaaa ⇓ aaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇓

Z

⇒ aaaa ⇓ aaaaaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇑ aaaaaaa ⇒ ⇒ ⇒ ⇒

m cbits

⇒ ⇒ ⇒ ⇒ X Z is a classical channel

  • x p(Z = X|X = x)

Mih´ aly Weiner Locality and entanglement assistance 8 / 15

slide-57
SLIDE 57

Task: information sending

aaaa X aaaaa aaaaaaa aaaaaaa aaaaa r ∈ {1, . . . n} aaaa ⇓ aaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇓

Z

⇒ aaaa ⇓ aaaaaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇑ aaaaaaa ⇒ ⇒ ⇒ ⇒

m cbits

⇒ ⇒ ⇒ ⇒ X Z is a classical channel

  • x p(Z = X|X = x)≤ 2m (i.e. no ns-oracle can help here)

Mih´ aly Weiner Locality and entanglement assistance 8 / 15

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SLIDE 58

Task: information sending

aaaa X aaaaa aaaaaaa aaaaaaa aaaaa r ∈ {1, . . . n} aaaa ⇓ aaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇓

Z

⇒ aaaa ⇓ aaaaaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇑ aaaaaaa ⇒ ⇒ ⇒ ⇒

m cbits

⇒ ⇒ ⇒ ⇒ X Z is a classical channel

  • x p(Z = X|X = x)≤ 2m (i.e. no ns-oracle can help here)

C ≤ m

Mih´ aly Weiner Locality and entanglement assistance 8 / 15

slide-59
SLIDE 59

Task: information sending

aaaa X aaaaa aaaaaaa aaaaaaa aaaaa r ∈ {1, . . . n} aaaa ⇓ aaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇓

Z

⇒ aaaa ⇓ aaaaaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇑ aaaaaaa ⇒ ⇒ ⇒ ⇒

m cbits

⇒ ⇒ ⇒ ⇒ X Z is a classical channel

  • x p(Z = X|X = x)≤ 2m (i.e. no ns-oracle can help here)

C ≤ m (i.e. again, no ns-oracle can help here)

Mih´ aly Weiner Locality and entanglement assistance 8 / 15

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SLIDE 60

An example

X: shown to Alice

Mih´ aly Weiner Locality and entanglement assistance 9 / 15

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SLIDE 61

An example

X: shown to Alice Z : choice of Bob (only one can be taken!)

Mih´ aly Weiner Locality and entanglement assistance 9 / 15

slide-62
SLIDE 62

An example

X: shown to Alice Z : choice of Bob (only one can be taken!) m = 1 cbit is allowed to be transmitted

Mih´ aly Weiner Locality and entanglement assistance 9 / 15

slide-63
SLIDE 63

An example

X: shown to Alice Z : choice of Bob (only one can be taken!) m = 1 cbit is allowed to be transmitted without oracle: max(p(win)) = 5

6

Mih´ aly Weiner Locality and entanglement assistance 9 / 15

slide-64
SLIDE 64

An example

X: shown to Alice Z : choice of Bob (only one can be taken!) m = 1 cbit is allowed to be transmitted without oracle: max(p(win)) = 5

6

with ↑↓ shared previously, p(win) = 4+

√ 2 6

is achievable

Mih´ aly Weiner Locality and entanglement assistance 9 / 15

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SLIDE 65

An example

X: shown to Alice Z : choice of Bob (only one can be taken!) m = 1 cbit is allowed to be transmitted without oracle: max(p(win)) = 5

6

with ↑↓ shared previously, p(win) = 4+

√ 2 6

is achievable with a PR-box, p(win) = 1 is achievable!

Mih´ aly Weiner Locality and entanglement assistance 9 / 15

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SLIDE 66

Exact classical simulation

Definition Cj,k(t): set of all j × k channel matrices realizable by transmitting a classical t-level system (that is, m = log2(t) cbits) and using a common source of randomness

Mih´ aly Weiner Locality and entanglement assistance 10 / 15

slide-67
SLIDE 67

Exact classical simulation

Definition Cj,k(t): set of all j × k channel matrices realizable by transmitting a classical t-level system (that is, m = log2(t) cbits) and using a common source of randomness In our example, transmitting 1 cbit + using ↑↓, Alice and Bob actually realized a 6 × 4 channel matrix / ∈ C6,4(2)

Mih´ aly Weiner Locality and entanglement assistance 10 / 15

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SLIDE 68

Exact classical simulation

Definition Cj,k(t): set of all j × k channel matrices realizable by transmitting a classical t-level system (that is, m = log2(t) cbits) and using a common source of randomness In our example, transmitting 1 cbit + using ↑↓, Alice and Bob actually realized a 6 × 4 channel matrix / ∈ C6,4(2) But e.g. C3,3(2) is characterized by the “trivial inequalities”

3

  • k=1

p(σ(k)|k) ≤ 2 ∀σ ∈ Perm{1, 2, 3}

Mih´ aly Weiner Locality and entanglement assistance 10 / 15

slide-69
SLIDE 69

Exact classical simulation

Definition Cj,k(t): set of all j × k channel matrices realizable by transmitting a classical t-level system (that is, m = log2(t) cbits) and using a common source of randomness In our example, transmitting 1 cbit + using ↑↓, Alice and Bob actually realized a 6 × 4 channel matrix / ∈ C6,4(2) But e.g. C3,3(2) is characterized by the “trivial inequalities”

3

  • k=1

p(σ(k)|k) ≤ 2 ∀σ ∈ Perm{1, 2, 3} In a game with 3 inputs / outputs, ns-oracles can be of any help

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SLIDE 70

Classical capacities of convex bodies

state space a convex body K; e.g. classical simplices

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SLIDE 71

Classical capacities of convex bodies

state space a convex body K; e.g. classical simplices

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SLIDE 72

Classical capacities of convex bodies

state space a convex body K; e.g. classical simplices

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SLIDE 73

Classical capacities of convex bodies

state space a convex body K; e.g. classical simplices . . .

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SLIDE 74

Classical capacities of convex bodies

state space a convex body K; e.g. classical simplices . . . quantum case: S+

1 (Cd)

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SLIDE 75

Classical capacities of convex bodies

state space a convex body K; e.g. classical simplices . . . quantum case: S+

1 (Cd), or more gen.: states of a v.N. algebra

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SLIDE 76

Classical capacities of convex bodies

state space a convex body K; e.g. classical simplices . . . quantum case: S+

1 (Cd), or more gen.: states of a v.N. algebra

measurement with r poss. outcomes K affine → r-simplex e.g. in quantum case S+

1 (Cd) ∋ ρ → (Tr(ρE1), . . . Tr(ρEr))

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SLIDE 77

Classical capacities of convex bodies

state space a convex body K; e.g. classical simplices . . . quantum case: S+

1 (Cd), or more gen.: states of a v.N. algebra

measurement with r poss. outcomes K affine → r-simplex e.g. in quantum case S+

1 (Cd) ∋ ρ → (Tr(ρE1), . . . Tr(ρEr))

Question: what is the minimum t value for which Cj,k(t) contains all j × k channel matrices realizable by transmitting a single isolated system whose state space is K?

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SLIDE 78

Classical capacity of convex bodies

In general, Cj,k(t) is not characterized by the “trivial” bounds and the fact that all of its elements have channel capacity ≤ log2(t) Thus, even for K = S+

1 (Cd), the question is nontrivial!

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SLIDE 79

Classical capacity of convex bodies

In general, Cj,k(t) is not characterized by the “trivial” bounds and the fact that all of its elements have channel capacity ≤ log2(t) Thus, even for K = S+

1 (Cd), the question is nontrivial!

F.-W. (2015): for K = S+

1 (Cd) this minimal t value is d

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SLIDE 80

Classical capacity of convex bodies

In general, Cj,k(t) is not characterized by the “trivial” bounds and the fact that all of its elements have channel capacity ≤ log2(t) Thus, even for K = S+

1 (Cd), the question is nontrivial!

F.-W. (2015): for K = S+

1 (Cd) this minimal t value is d

Question For a qbit, K = 3-dim ball. What else it could be? For what K it is true, that the resulting set of j × k channel matrices always coincides with Cj,k(2)?

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SLIDE 81

Classical capacity of convex bodies

In general, Cj,k(t) is not characterized by the “trivial” bounds and the fact that all of its elements have channel capacity ≤ log2(t) Thus, even for K = S+

1 (Cd), the question is nontrivial!

F.-W. (2015): for K = S+

1 (Cd) this minimal t value is d

Question For a qbit, K = 3-dim ball. What else it could be? For what K it is true, that the resulting set of j × k channel matrices always coincides with Cj,k(2)? Answer A lot of other bodies would be still ok; nothing specific about the 3-dim ball. E.g. is also a “1-bit space”.

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SLIDE 82

From one part to bipartite

Suppose we have a bipartite physical system whose part II (in itself) has state space K. After preparation but previous to measurements, the (partial) state of part II is q ∈ K.

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SLIDE 83

From one part to bipartite

Suppose we have a bipartite physical system whose part II (in itself) has state space K. After preparation but previous to measurements, the (partial) state of part II is q ∈ K. x1 → z1 measurement on I

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SLIDE 84

From one part to bipartite

Suppose we have a bipartite physical system whose part II (in itself) has state space K. After preparation but previous to measurements, the (partial) state of part II is q ∈ K. x1 → z1 measurement on I q “changes” to qx1

z1 ∈ K

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SLIDE 85

From one part to bipartite

Suppose we have a bipartite physical system whose part II (in itself) has state space K. After preparation but previous to measurements, the (partial) state of part II is q ∈ K. x1 → z1 measurement on I q “changes” to qx1

z1 ∈ K

ns-condition:

z1 p(z1|x1)qx1 z1 = q

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SLIDE 86

From one part to bipartite

Suppose we have a bipartite physical system whose part II (in itself) has state space K. After preparation but previous to measurements, the (partial) state of part II is q ∈ K. x1 → z1 measurement on I q “changes” to qx1

z1 ∈ K

ns-condition:

z1 p(z1|x1)qx1 z1 = q

x2 measurement on II: Φx2 : K affine → simplex

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SLIDE 87

From one part to bipartite

Suppose we have a bipartite physical system whose part II (in itself) has state space K. After preparation but previous to measurements, the (partial) state of part II is q ∈ K. x1 → z1 measurement on I q “changes” to qx1

z1 ∈ K

ns-condition:

z1 p(z1|x1)qx1 z1 = q

x2 measurement on II: Φx2 : K affine → simplex p(z1, z2|x1, x2) = Φx2

z2(qx1 z1 )

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SLIDE 88

From one part to bipartite

Suppose we have a bipartite physical system whose part II (in itself) has state space K. After preparation but previous to measurements, the (partial) state of part II is q ∈ K. x1 → z1 measurement on I q “changes” to qx1

z1 ∈ K

ns-condition:

z1 p(z1|x1)qx1 z1 = q

x2 measurement on II: Φx2 : K affine → simplex p(z1, z2|x1, x2) = Φx2

z2(qx1 z1 )

Conclusion Considering all “initial states” q ∈ K together with every convex decomposition of it + every possible K affine → simplex map we can construct the set of all possible ns-oracles arising from bipartite physical systems where one part has state space K.

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SLIDE 89

From one part to bipartite

Further considerations:

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SLIDE 90

From one part to bipartite

Further considerations: Other conditions on realizability? (E.g. regarding the state space of the other part.)

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SLIDE 91

From one part to bipartite

Further considerations: Other conditions on realizability? (E.g. regarding the state space of the other part.) When K = simplex / K = S+

1 (Cd), this construction gives

precisely the set of classical / quantum ns-oracles.

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SLIDE 92

From one part to bipartite

Further considerations: Other conditions on realizability? (E.g. regarding the state space of the other part.) When K = simplex / K = S+

1 (Cd), this construction gives

precisely the set of classical / quantum ns-oracles. ֒ → Taken as a “principle” / “law of nature”, we get that everything is decided by specifying the state space of just one part!

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SLIDE 93

From one part to bipartite

Further considerations: Other conditions on realizability? (E.g. regarding the state space of the other part.) When K = simplex / K = S+

1 (Cd), this construction gives

precisely the set of classical / quantum ns-oracles. ֒ → Taken as a “principle” / “law of nature”, we get that everything is decided by specifying the state space of just one part! ֒ → Further restrictions on what the state space of a 1-bit system can be. E.g. K = would allow realization of the PR-box.

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SLIDE 94

Nontrivial bounds: results

aaaa X aaaaa aaaaaaa aaaaaaa aaaaa r ∈ {1, . . . n} aaaa ⇓ aaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇓

Z

⇒ aaaa ⇓ aaaaaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇑ aaaaaaa ⇒ ⇒ ⇒ ⇒

1 cbit

⇒ ⇒ ⇒ ⇒

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SLIDE 95

Nontrivial bounds: results

aaaa X aaaaa aaaaaaa aaaaaaa aaaaa r ∈ {1, . . . n} aaaa ⇓ aaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇓

Z

⇒ aaaa ⇓ aaaaaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇑ aaaaaaa ⇒ ⇒ ⇒ ⇒

1 cbit

⇒ ⇒ ⇒ ⇒ if = bipartite quant. sys. in state ρ and ρII =TrI(ρ) is a multiple of a projection, then exact classical simulation is always possible with 2 cbits to be sent instead of 1

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SLIDE 96

Nontrivial bounds: results

aaaa X aaaaa aaaaaaa aaaaaaa aaaaa r ∈ {1, . . . n} aaaa ⇓ aaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇓

Z

⇒ aaaa ⇓ aaaaaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇑ aaaaaaa ⇒ ⇒ ⇒ ⇒

1 cbit

⇒ ⇒ ⇒ ⇒ if = bipartite quant. sys. in state ρ and ρII =TrI(ρ) is a multiple of a projection, then exact classical simulation is always possible with 2 cbits to be sent instead of 1 ∀n ∃ example with some ns-oracle that cannot be simulated classically even if we allow n cbits to be sent instead of 1

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SLIDE 97

Nontrivial bounds: results

aaaa X aaaaa aaaaaaa aaaaaaa aaaaa r ∈ {1, . . . n} aaaa ⇓ aaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇓

Z

⇒ aaaa ⇓ aaaaaaa aaaaaaa aaaaaaa aaaaaa aaaaaa aaaaaa ⇑ aaaaaaa ⇒ ⇒ ⇒ ⇒

1 cbit

⇒ ⇒ ⇒ ⇒ if = bipartite quant. sys. in state ρ and ρII =TrI(ρ) is a multiple of a projection, then exact classical simulation is always possible with 2 cbits to be sent instead of 1 ∀n ∃ example with some ns-oracle that cannot be simulated classically even if we allow n cbits to be sent instead of 1 Possible principle? “God helps those who help themselves.”

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