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

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

2 headed oracles X 1 ⇒ ⇐ X 2 Z 1 ⇐ ⇒ Z 2 Mih´ aly Weiner Locality and entanglement assistance 2 / 15

2 headed oracles X 1 ⇒ ⇐ X 2 Z 1 ⇐ ⇒ Z 2 aa Mih´ aly Weiner Locality and entanglement assistance 2 / 15

2 headed oracles X 1 ⇒ ⇐ X 2 Z 1 ⇐ ⇒ Z 2 aa aa Mih´ aly Weiner Locality and entanglement assistance 2 / 15

2 headed oracles X 1 ⇒ ⇐ X 2 Z 1 ⇐ ⇒ Z 2 aa user’s point of view: aa Mih´ aly Weiner Locality and entanglement assistance 2 / 15

2 headed oracles X 1 ⇒ ⇐ X 2 Z 1 ⇐ ⇒ Z 2 aa user’s point of view: p ( Z 1 = z 1 , Z 2 = z 2 | X 1 = x 1 , X 2 = x 2 ) aa Mih´ aly Weiner Locality and entanglement assistance 2 / 15

2 headed oracles X 1 ⇒ ⇐ X 2 Z 1 ⇐ ⇒ Z 2 aa user’s point of view: p ( Z 1 = z 1 , Z 2 = z 2 | X 1 = x 1 , X 2 = x 2 ) aa aaa. ⇒ a point in R n 1 × R n 2 × R m 1 × R m 2 aa Mih´ aly Weiner Locality and entanglement assistance 2 / 15

2 headed oracles X 1 ⇒ ⇐ X 2 Z 1 ⇐ ⇒ Z 2 aa user’s point of view: p ( Z 1 = z 1 , Z 2 = z 2 | X 1 = x 1 , X 2 = x 2 ) aa aaa. ⇒ a point in R n 1 × R n 2 × R m 1 × R m 2 aa aaa. ⇒ all 2-headed oracles form a polytope aa Mih´ aly Weiner Locality and entanglement assistance 2 / 15

Realizations p ( Z 1 = z 1 , Z 2 = z 2 | X 1 = x 1 , X 2 = x 2 ) = ? Mih´ aly Weiner Locality and entanglement assistance 3 / 15

Realizations p ( Z 1 = z 1 , Z 2 = z 2 | X 1 = x 1 , X 2 = x 2 ) = ? Classical: Mih´ aly Weiner Locality and entanglement assistance 3 / 15

Realizations p ( Z 1 = z 1 , Z 2 = z 2 | X 1 = x 1 , X 2 = x 2 ) = ? � Classical: p I ,λ ( z 1 | x 1 ) p II ,λ ( z 2 | x 2 ) d µ ( λ ) Mih´ aly Weiner Locality and entanglement assistance 3 / 15

Realizations p ( Z 1 = z 1 , Z 2 = z 2 | X 1 = x 1 , X 2 = x 2 ) = ? � Classical: p I ,λ ( z 1 | x 1 ) p II ,λ ( z 2 | x 2 ) d µ ( λ ) Mih´ aly Weiner Locality and entanglement assistance 3 / 15

Realizations p ( Z 1 = z 1 , Z 2 = z 2 | X 1 = x 1 , X 2 = x 2 ) = ? � Classical: p I ,λ ( z 1 | x 1 ) p II ,λ ( z 2 | x 2 ) d µ ( λ ) Mih´ aly Weiner Locality and entanglement assistance 3 / 15

Realizations p ( Z 1 = z 1 , Z 2 = z 2 | X 1 = x 1 , X 2 = x 2 ) = ? � Classical: p I ,λ ( z 1 | x 1 ) p II ,λ ( z 2 | x 2 ) d µ ( λ ) Quantum: ϕ ( A x 1 , z 1 B x 2 , z 2 ) Mih´ aly Weiner Locality and entanglement assistance 3 / 15

Realizations p ( Z 1 = z 1 , Z 2 = z 2 | X 1 = x 1 , X 2 = x 2 ) = ? � Classical: p I ,λ ( z 1 | x 1 ) p II ,λ ( z 2 | x 2 ) d µ ( λ ) Quantum: ϕ ( A x 1 , z 1 B x 2 , z 2 ) where ϕ is a positive normalized functional A x 1 , z 1 ≥ 0 , � z 1 A x 1 , z 1 = I & sim. cond. for B [ A x 1 , z 1 , B x 2 , z 2 ] = 0 Mih´ aly Weiner Locality and entanglement assistance 3 / 15

No-signaling Input at 1 should have no effect on output at 2: Mih´ aly Weiner Locality and entanglement assistance 4 / 15

No-signaling Input at 1 should have no effect on output at 2: � p ( Z 2 = z 2 | X 1 = x 1 , X 2 = x 2 ) = p ( Z 2 = z 2 | X 1 = ˜ x 1 , X 2 = x 2 ) z k Mih´ aly Weiner Locality and entanglement assistance 4 / 15

No-signaling Input at 1 should have no effect on output at 2: � p ( Z 2 = z 2 | X 1 = x 1 , X 2 = x 2 ) = p ( Z 2 = z 2 | X 1 = ˜ x 1 , X 2 = x 2 ) z k Mih´ aly Weiner Locality and entanglement assistance 4 / 15

No-signaling Input at 1 should have no effect on output at 2: � p ( Z 2 = z 2 | X 1 = x 1 , X 2 = x 2 ) = p ( Z 2 = z 2 | X 1 = ˜ x 1 , X 2 = x 2 ) z k Mih´ aly Weiner Locality and entanglement assistance 4 / 15

No-signaling Input at 1 should have no effect on output at 2: � p ( Z 2 = z 2 | X 1 = x 1 , X 2 = x 2 ) = p ( Z 2 = z 2 | X 1 = ˜ x 1 , X 2 = x 2 ) z k Mih´ aly Weiner Locality and entanglement assistance 4 / 15

No-signaling Input at 1 should have no effect on output at 2: � p ( Z 1 = z 1 , Z 2 = z 2 | X 1 = x 1 , X 2 = x 2 ) = func. of z 2 & x 2 only z 1 Mih´ aly Weiner Locality and entanglement assistance 4 / 15

No-signaling Input at 1 should have no effect on output at 2: � p ( Z 1 = z 1 , Z 2 = z 2 | X 1 = x 1 , X 2 = x 2 ) = func. of z 2 & x 2 only z 1 Mih´ aly Weiner Locality and entanglement assistance 4 / 15

No-signaling Input at 1 should have no effect on output at 2: � p ( Z 1 = z 1 , Z 2 = z 2 | X 1 = x 1 , X 2 = x 2 ) = func. of z 2 & x 2 only z 1 Input at 2 should have no effect on output at 1: Mih´ aly Weiner Locality and entanglement assistance 4 / 15

No-signaling Input at 1 should have no effect on output at 2: � p ( Z 1 = z 1 , Z 2 = z 2 | X 1 = x 1 , X 2 = x 2 ) = func. of z 2 & x 2 only z 1 Input at 2 should have no effect on output at 1: � p ( Z 1 = z 1 , Z 2 = z 2 | X 1 = x 1 , X 2 = x 2 ) = func. of z 1 & x 1 only z 2 Mih´ aly Weiner Locality and entanglement assistance 4 / 15

No-signaling Input at 1 should have no effect on output at 2: � p ( Z 1 = z 1 , Z 2 = z 2 | X 1 = x 1 , X 2 = x 2 ) = func. of z 2 & x 2 only z 1 Input at 2 should have no effect on output at 1: � p ( Z 1 = z 1 , Z 2 = z 2 | X 1 = x 1 , X 2 = x 2 ) = func. of z 1 & x 1 only z 2 XX � “NS-oracle” ∈ “No-signaling polytope” Mih´ aly Weiner Locality and entanglement assistance 4 / 15

No-signaling � quantum No-signaling polytope: f.i Mih´ aly Weiner Locality and entanglement assistance 5 / 15

No-signaling � quantum No-signaling polytope: ,, Mih´ aly Weiner Locality and entanglement assistance 5 / 15

No-signaling � quantum No-signaling polytope: , Mih´ aly Weiner Locality and entanglement assistance 5 / 15

No-signaling � quantum No-signaling polytope: Mih´ aly Weiner Locality and entanglement assistance 5 / 15

No-signaling � quantum No-signaling polytope: Mih´ aly Weiner Locality and entanglement assistance 5 / 15

No-signaling � quantum No-signaling polytope: Mih´ aly Weiner Locality and entanglement assistance 5 / 15

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 ⇑ h : m cbits help aaaaaaa ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ n � I ( d k : g | r = k ) ≤ m k =1 Mih´ aly Weiner Locality and entanglement assistance 6 / 15

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 ⇑ h : m cbits help aaaaaaa ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ n � I ( d k : g | r = k ) ≤ m k =1 Mih´ aly Weiner Locality and entanglement assistance 6 / 15

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 ⇑ h : m cbits help aaaaaaa ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ n � I ( d k : g | r = k ) ≤ m k =1 Mih´ aly Weiner Locality and entanglement assistance 6 / 15

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 ⇑ h : m cbits help aaaaaaa ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ n � I ( d k : g | r = k ) ≤ m k =1 Mih´ aly Weiner Locality and entanglement assistance 6 / 15

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 ⇑ h : m cbits help aaaaaaa ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ n � I ( d k : g | r = k ) ≤ m k =1 Mih´ aly Weiner Locality and entanglement assistance 6 / 15

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 ⇑ h : m cbits help aaaaaaa ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ n � I ( d k : g | r = k ) ≤ m k =1 Mih´ aly Weiner Locality and entanglement assistance 6 / 15

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 ⇑ h : m cbits help aaaaaaa ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ n � I ( d k : g | r = k ) ≤ m k =1 Mih´ aly Weiner Locality and entanglement assistance 6 / 15

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 ⇑ h : m cbits help aaaaaaa ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ n � I ( d k : g | r = k ) ≤ m k =1 Mih´ aly Weiner Locality and entanglement assistance 6 / 15