Squashed entanglement - PDF Free Download

Squashed Entanglement 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)

(Classical) intrinsic information The intrinsic information between two random variables X, Y, given a third r.v. Z, is given by I (X ; Y Z) := inf(i (X ; Y Z)) 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).

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 inf Λ:B(H C ) B(H E ) 2 I (A; B E) : ρabe = (id Λ) Ψ Ψ ABC Here, Ψ ABC is a purification of ρ AB, and ρ ABE is simply a tripartite extension; ρ AB = Tr E (ρ ABE ).

Equivalently, ( ) 1 E sq (ρ AB ) = inf ρ ABE 2 I (A; B E) : Tr E (ρ ABE ) = ρ AB (equivalence follows by the equivalence of purifications up to a unitary map)

Equivalently, ( ) 1 E sq (ρ AB ) = inf ρ ABE 2 I (A; B E) : Tr E (ρ ABE ) = ρ AB (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 ).

Some basic properties E sq has basic properties we want in an entanglement measure, e.g.:

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])

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.

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.)

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.

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 2I (A; B E) log k f (ɛ) log(k) for some function f with lim ɛ 0 f (ɛ) = 0.

Relations to other entanglement measures Recall that the entanglement of formation is given by E F (ρ AB ) = inf ρ= p i E( ψ i ) = inf 1 p i I ( ψ i A ; ψ i B ) i p i ψ i ψ i 2 i i

Relations to other entanglement measures Recall that the entanglement of formation is given by E F (ρ AB ) = inf ρ= p i E( ψ i ) = inf 1 p i I ( ψ i A ; ψ i B ) i p i ψ i ψ i 2 Consider the following extension i i ρ ABE = i p i ψ i ψ i AB i i E

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.

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 E C (ρ AB 1 ) = lim n n E F ((ρ AB ) n ) is also bounded below by E sq (ρ AB ).

Another interpretation of squashed entanglement (Oppenheim, arxiv:0801.0458) Suppose Alice and Bob share a state ρ AB, and Alice wants to send her part to Eve. How many qubits does she need? We allow Alice and Eve to have as much side information (e.g., ancillas) as they like. However, the amount of useful information is limited, since Alice s share of the state may be entangled with Bob s.

State redistribution and merging (Oppenheim, arxiv:0805.1065, and Lecture 16) Using state merging, the minimum amount of quantum communication required is Q A C = 1 I (A : B C) 2 E (entanglement) = 1 2 I (A : A ) 1 I (A : C) 2

Back to squashed entanglement Allowing arbitrary side information, the entanglement E is certainly attainable.

Back to squashed entanglement Allowing arbitrary side information, the entanglement E is certainly attainable. So, let ρ S be the side information shared between Alice and Eve (so that ρ ABS is pure), with Alice s share being ρ A and Eve s ρ C. Then inf Q A ρ C = inf A C ρ A C 1 I (A : B C) 2 = inf Λ:B(S) B(C) = E sq (ρ AB ) 1 I (A : B Λ(S)) 2

Sanity check If ρ AB = ψ ψ, E sq (ρ(ab)) = E( ψ ). Likewise, any pure state extension ρ ABS must have S completely uncorrelated with AB.

Sanity check If ρ AB = ψ ψ, E sq (ρ(ab)) = E( ψ ). Likewise, any pure state extension ρ ABS must have S completely uncorrelated with AB. If Alice and Bob share a separable state ρ AB, written as a convex combination of separable pure states, E sq (ρ AB ) = 0: The extension ρ ABE = i p i ψ i ψ i A φ i φ i B i i E has zero quantum conditional mutual information.

Why squashed entanglement?

Why squashed entanglement? Christandl and Winter: Our name for this functional comes from the idea that the right choice of a conditioning system reduces the quantum mutual information between A and B, thus squashing out the nonquantum correlations. Oppenheim defines the puffed entanglement as the supremum of I (A : B E) over all extensions E, with the operational interpretation of the rate required given the side information is as useless as possible.

Thank you! Questions?