论文：Entanglement Witness

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Horodecki, R., Horodecki, P., Horodecki, M., & Horodecki, K. (2009). Quantum entanglement. Reviews of Modern Physics, 81(2), 865–942. https://doi.org/10.1103/RevModPhys.81.865

Necessary condition for separability

LOO: Local orthogonal observables.
PPT: Positive partial transposition

PPT criterion: a necessary condition for separability.
For , if is still a density matrix (i.e. it is hermitian and has non-negative spectrum), then is separable.

（非常有意思的一点是，因为是只对做转置，所以取了转置符号的一半，来表示这样的操作，即将记作.）

PPT condition is equivalent to demanding the positivity of , where is the transposition map acting on B. is a positive (P) map, but not a completely positive (CP) map.

The map is completely positive if and only if is positive for identity map on any finite-dimensional system.

Any P but not CP map can form a nontrivial necessary condition for separation:

$$(I_{A} \otimes \Lambda_{B}) \rho_{A B} \geq 0\tag{1}$$

Then we can provide a necessary and sufficient condition: is separable if and only if is satisfied for all P but not CP maps. However to find all the P but not CP maps is very hard for systems with dimension bigger than 6.

Entanglement witness

An entanglement witness is an oberservable that has:

• (i) non-negative expectation values in all separable states.
• (ii) at least one negative eigenvalue.
  

Optimizing Witness