# Axiomatic approach

The aim of the axiomatic approach to entanglement measures is to find, classify and study all functions that capture our intuitive notion of what it means to measure entanglement. The approach sets out axioms, i.e. properties, that an entanglement measure should or should not satisfy. This intuitive notion may be based on more practical grounds such as operational definitions. The most striking applications of the axiomatic approach are upper and lower bounds on operational measures such as distillable entanglement, entanglement cost and most recently distillable key.

### Properties

The following is a list of properties that have been studied in the context of entanglement measures. E(ρ) denotes an entanglement measure.

Norm: normalised on maximally entangled states

For all $\psi=\frac{1}{\sqrt{\dim A}}\sum_i |i\rangle^A|i\rangle^B$, E(∣ψ⟩⟨ψ∣) = logdimA with {∣iA} an orthonormal basis.

Van Sep: vanishing on separable states

For all separable states ρ, E(ρ) = 0 .

PPT Mon: PPT monotone

For all PPT operations ρ → {pi, ρi}, E(ρ) ≥ ∑ipiE(ρi).

SEP Mon: SEP monotone

For all separable operations ρ → {pi, ρi}, E(ρ) ≥ ∑ipiE(ρi).

LOCC Mon: LOCC monotone

For all LOCC operations ρ → {pi, ρi}, E(ρ) ≥ ∑ipiE(ρi).

Loc Mon: local monotone

For all (strictly) local instruments (i.e. an instrument that acts either on A or B) ρ → {pi, ρi}, E(ρ) ≥ ∑ipiE(ρi).

LOq Mon: LOq monotone

For all LOq operations ρ → {pi, ρi}, E(ρ) ≥ ∑ipiE(ρi).

As Cont: asymptotic continuity

There are c, cʹ ≥ 0 s.th. for all ρ, σ with δ(ρ, σ) ≤ ε, ∣E(ρ) − E(σ)∣ ≤ cεlogd + cʹ.

As Cont Pure: asympt. cont. near pure states

There are c, cʹ ≥ 0 s.th. for all ρ, σ = ∣ψ⟩⟨ψ with δ(ρ, σ) ≤ ε, ∣E(ρ) − E(σ)∣ ≤ cεlogd + cʹ.

Conv: convex

For all ρ, σ and p ∈ [0, 1] , pE(ρ) + (1 − p)E(σ) ≥ E(pρ + (1 − p)σ).

Conv Pure: convex on pure states

For all {pi, ∣ψi⟩} with pi ≥ 0 and ipi = 1, ∑ipiE(ρi) ≥ E(ρ).

For all ρAAʹBBʹ, E(ρAAʹ, BBʹ) ≥ E(ρAB) + E(ρAʹBʹ).

For all ρ, σ, E(ρ ⊗ σ) = E(ρ) + E(σ).

Ext (Add i.i.d.): extensive

For all ρ and N, NE(ρ) = E(ρ ⊗ N).

For all ρ, σ, E(ρ ⊗ σ) ≤ E(ρ) + E(σ).

For all ρ and m, n, E(ρ ⊗ (m + n)) ≤ E(ρ ⊗ m) + E(ρ ⊗ n).

Regu: regularisable

For all ρ, the limit $E^\infty(\rho)=\lim_{n \rightarrow \infty} \frac{E(\rho^{\otimes n})}{n}$ exists.

Non Lock: not lockable

There is c ≥ 0 s.th. for all ρAAʹB, E(ρAAʹB) ≤ E(ρAB) + clogrankρAʹ.