Entanglement measures Squashed Entanglement Nicholas LeCompte based on ”Squashed Entanglement” - An Additive Entanglement Measure (M. Christandl, A. Winter, quant-ph/0308088 ), and A paradigm for entanglement theory based on quantum communication (J. Oppenheim arXiv:0801.0458 ) Nicholas LeCompte Squashed entanglement
Entanglement measures (Classical) intrinsic information ◮ The intrinsic information between two random variables X , Y , given a third r.v. Z , is given by Z ( I ( X ; Y | ¯ I ( X ; Y ↓ Z ) := inf Z )) ¯ with ¯ Z governed by a conditional probability distribution P ¯ Z | Z . ◮ Idea: two parties have access to X and Y , and want to generate a secret key even if an adversary has access to some additional knowledge ( Z ). Nicholas LeCompte Squashed entanglement
Entanglement measures (Classical) intrinsic information ◮ The intrinsic information between two random variables X , Y , given a third r.v. Z , is given by Z ( I ( X ; Y | ¯ I ( X ; Y ↓ Z ) := inf Z )) ¯ with ¯ Z governed by a conditional probability distribution P ¯ Z | Z . ◮ Idea: two parties have access to X and Y , and want to generate a secret key even if an adversary has access to some additional knowledge ( Z ). ◮ Mauer, Wolf 1999: The intrinsic information upper-bounds the rate at which the key can be extracted. Nicholas LeCompte Squashed entanglement
Entanglement measures A quantum analogue (Christandl, Winter, quant-ph/0308088) ◮ Replace classical mutual conditional infos with quantum, P ¯ Z | Z with quantum channel: E sq ( ρ AB ) = � 1 � 2 I ( A ; B | E ) : ρ ABE = ( id ⊗ Λ) | Ψ �� Ψ | ABC inf Λ: B ( H C ) →B ( H E ) ◮ Here, | Ψ � ABC is a purification of ρ AB , and ρ ABE is simply a tripartite extension; ρ AB = Tr E ( ρ ABE ). Nicholas LeCompte Squashed entanglement
Entanglement measures A quantum analogue (Christandl, Winter, quant-ph/0308088) ◮ Replace classical mutual conditional infos with quantum, P ¯ Z | Z with quantum channel: E sq ( ρ AB ) = � 1 � 2 I ( A ; B | E ) : ρ ABE = ( id ⊗ Λ) | Ψ �� Ψ | ABC inf Λ: B ( H C ) →B ( H E ) ◮ Here, | Ψ � ABC is a purification of ρ AB , and ρ ABE is simply a tripartite extension; ρ AB = Tr E ( ρ ABE ). ◮ E sq ( ρ AB ) is the squashed entanglement . Nicholas LeCompte Squashed entanglement
Entanglement measures Squashed entanglement ◮ Equivalently, � 1 � E sq ( ρ AB ) = inf 2 I ( A ; B | E ) : Tr E ( ρ ABE ) = ρ AB ρ ABE ◮ (equivalence follows by the equivalence of purifications up to a unitary map) Nicholas LeCompte Squashed entanglement
Entanglement measures Squashed entanglement ◮ Equivalently, � 1 � E sq ( ρ AB ) = inf 2 I ( A ; B | E ) : Tr E ( ρ ABE ) = ρ AB ρ ABE ◮ (equivalence follows by the equivalence of purifications up to a unitary map) ◮ Sanity check : If ρ AB = | ψ �� ψ | AB , all tripartite extensions are given by ρ AB ⊗ ρ E , so E sq ( ρ AB ) = S ( ρ A ) = E ( | ψ � ). Nicholas LeCompte Squashed entanglement
Entanglement measures Some basic properties ◮ E sq has basic properties we want in an entanglement measure, e.g.: Nicholas LeCompte Squashed entanglement
Entanglement measures Some basic properties ◮ E sq has basic properties we want in an entanglement measure, e.g.: ◮ does not increase under LOCC ◮ is convex ◮ is continuous (by Alicki-Fannes theorem, quant-ph/0312081 [or Lecture 10]) Nicholas LeCompte Squashed entanglement
Entanglement measures Some basic properties ◮ E sq has basic properties we want in an entanglement measure, e.g.: ◮ does not increase under LOCC ◮ is convex ◮ is continuous (by Alicki-Fannes theorem, quant-ph/0312081 [or Lecture 10]) ◮ It’s also superadditive, and additive on tensor products. Nicholas LeCompte Squashed entanglement
Entanglement measures Relations to other entanglement measures ◮ Recall that the intrinsic information upper-bounds the rate at which the key can be extracted given an adversary has access to some information. Nicholas LeCompte Squashed entanglement
Entanglement measures Relations to other entanglement measures ◮ Recall that the intrinsic information upper-bounds the rate at which the key can be extracted given an adversary has access to some information. ◮ Prop. E D ( ρ AB ) ≤ E sq ( ρ AB ), where E D is the distillable entanglement. ◮ (Recall E D ( ρ ) is the limiting ratio n / m of n Bell pairs created from m copies of ρ with LOCC.) Nicholas LeCompte Squashed entanglement
Entanglement measures Relations to other entanglement measures ◮ Recall that the intrinsic information upper-bounds the rate at which the key can be extracted given an adversary has access to some information. ◮ Prop. E D ( ρ AB ) ≤ E sq ( ρ AB ), where E D is the distillable entanglement. ◮ (Recall E D ( ρ ) is the limiting ratio n / m of n Bell pairs created from m copies of ρ with LOCC.) ◮ The squashed entanglement is an analogous upper-bound (viewing maximally entangled states as secret quantum correlations). Nicholas LeCompte Squashed entanglement
Entanglement measures Proof sketch: E D ( ρ AB ) ≤ E sq ( ρ AB ) ◮ Start with an arbitrary distillation protocol, taking ( ρ AB ) ⊗ n → LOCC Ψ AB such that || Ψ AB − | k �� k | AB || 1 ≤ ε , where | k � is a rank- k maximally entangled state. Nicholas LeCompte Squashed entanglement
Entanglement measures Proof sketch: E D ( ρ AB ) ≤ E sq ( ρ AB ) ◮ Start with an arbitrary distillation protocol, taking ( ρ AB ) ⊗ n → LOCC Ψ AB such that || Ψ AB − | k �� k | AB || 1 ≤ ε , where | k � is a rank- k maximally entangled state. ◮ By superadditivity and monotonicity under LOCC, nE sq ( ρ AB ) ≥ E sq (Ψ AB ). Nicholas LeCompte Squashed entanglement
Entanglement measures Proof sketch: E D ( ρ AB ) ≤ E sq ( ρ AB ) ◮ Start with an arbitrary distillation protocol, taking ( ρ AB ) ⊗ n → LOCC Ψ AB such that || Ψ AB − | k �� k | AB || 1 ≤ ε , where | k � is a rank- k maximally entangled state. ◮ By superadditivity and monotonicity under LOCC, nE sq ( ρ AB ) ≥ E sq (Ψ AB ). ◮ Note E sq ( | k �� k | ) = log k (since | k � is maximally entangled), and using Fannes’ inequality, one can show that for any extension Ψ ABE , 1 2 I ( A ; B | E ) ≥ log k − f ( ǫ ) log( k ) for some function f with lim ǫ → 0 f ( ǫ ) = 0. Nicholas LeCompte Squashed entanglement
Entanglement measures Proof sketch: E D ( ρ AB ) ≤ E sq ( ρ AB ) ◮ Start with an arbitrary distillation protocol, taking ( ρ AB ) ⊗ n → LOCC Ψ AB such that || Ψ AB − | k �� k | AB || 1 ≤ ε , where | k � is a rank- k maximally entangled state. ◮ By superadditivity and monotonicity under LOCC, nE sq ( ρ AB ) ≥ E sq (Ψ AB ). ◮ Note E sq ( | k �� k | ) = log k (since | k � is maximally entangled), and using Fannes’ inequality, one can show that for any extension Ψ ABE , 1 2 I ( A ; B | E ) ≥ log k − f ( ǫ ) log( k ) for some function f with lim ǫ → 0 f ( ǫ ) = 0. ◮ Hence E sq ( ρ AB ) ≥ 1 n (1 − f ( ǫ )) log k . Nicholas LeCompte Squashed entanglement
Entanglement measures Relations to other entanglement measures ◮ Recall that the entanglement of formation is given by p i E ( | ψ i � ) = inf 1 � � E F ( ρ AB ) = p i I ( | ψ i � A ; | ψ i � B ) inf 2 ρ = � i p i | ψ i �� ψ i | i i Nicholas LeCompte Squashed entanglement
Entanglement measures Relations to other entanglement measures ◮ Recall that the entanglement of formation is given by p i E ( | ψ i � ) = inf 1 � � E F ( ρ AB ) = p i I ( | ψ i � A ; | ψ i � B ) inf 2 ρ = � i p i | ψ i �� ψ i | i i ◮ Consider the following extension ρ ABE = p i | ψ i �� ψ i | AB ⊗ | i �� i | E � i Nicholas LeCompte Squashed entanglement
Entanglement measures Relations to other entanglement measures ◮ Recall that the entanglement of formation is given by p i E ( | ψ i � ) = inf 1 � � E F ( ρ AB ) = p i I ( | ψ i � A ; | ψ i � B ) inf 2 ρ = � i p i | ψ i �� ψ i | i i ◮ Consider the following extension ρ ABE = p i | ψ i �� ψ i | AB ⊗ | i �� i | E � i ◮ Then 1 p i I ( | ψ i � A ; | ψ i � B ) = 1 � 2 I ( A ; B | E ) . 2 i Nicholas LeCompte Squashed entanglement
Entanglement measures Relations to other entanglement measures ◮ Hence the entanglement of formation can be thought of as the infinum of I ( A ; B | E ) over a certain type of tripartite extensions. Nicholas LeCompte Squashed entanglement
Entanglement measures Relations to other entanglement measures ◮ Hence the entanglement of formation can be thought of as the infinum of I ( A ; B | E ) over a certain type of tripartite extensions. ◮ It follows that E sq ( ρ AB ) ≤ E F ( ρ AB ). Nicholas LeCompte Squashed entanglement
Entanglement measures Relations to other entanglement measures ◮ Hence the entanglement of formation can be thought of as the infinum of I ( A ; B | E ) over a certain type of tripartite extensions. ◮ It follows that E sq ( ρ AB ) ≤ E F ( ρ AB ). ◮ Invoking additivity, the entanglement cost 1 E C ( ρ AB ) = lim nE F (( ρ AB ) ⊗ n ) n →∞ is also bounded below by E sq ( ρ AB ). Nicholas LeCompte Squashed entanglement
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