How can one prove that a given state is entangled? In this paper we review different methods that have been proposed for entanglement detection. We first explain the basic elements of entanglement theory for two or more particles and then entanglement verification procedures such as Bell inequalities, entanglement witnesses, the determination of nonlinear properties of a quantum state via measurements on several copies, and spin squeezing inequalities. An emphasis is given on the theory and application of entanglement witnesses. We also discuss several experiments, where some of the presented methods have been implemented.

Entanglement detection

Multiphoton interference reveals strictly nonclassical phenomena. Its applications range from fundamental tests of quantum mechanics to photonic quantum information processing, where a significant fraction of key experiments achieved so far comes from multiphoton state manipulation. The progress, both theoretical and experimental, of this rapidly advancing research is reviewed. The emphasis is given to the creation of photonic entanglement of various forms, tests of the completeness of quantum mechanics (in particular, violations of local realism), quantum information protocols for quantum communication (e.g., quantum teleportation, entanglement purification, and quantum repeater), and quantum computation with linear optics. The scope of the review is limited to ``few-photon'' phenomena involving measurements of discrete observables.

/pdf/multiphoton-entanglement-and-interferometry-36qb24ejfd.pdf

Multiphoton entanglement and interferometry

a ) p. 131 The discussion between eqs. (5.14) and (5.15) is incorrect (dA should be made as large as possible!). b ) p. 256 In the figure, the numbers 6) and 7) occur twice. c ) p. 292 At the end of section 12.5, it should be the space of isospectral 0Hermitian matrices. d ) p. 306 A ”Tr” is missing in eq. (13.43). e ) p. 327, Eq. (14.64b) is 〈Trρ〉B = N(14N+10) (5N+1)(N+3) should be 〈Trρ〉B = 8N+7 (N+2)(N+4)

https://chaos.if.uj.edu.pl/~karol/erratum10.pdf

Geometry of Quantum States

Geometry of quantum states: an introduction to quantum entanglement by Ingemar Bengtsson and Karol Zyczkowski

We define a polynomial measure of multiparticle entanglement which is scalable, i.e., which applies to any number of spin-12 particles. By evaluating it for three particle states, for eigenstates of the one dimensional Heisenberg antiferromagnet and on quantum error correcting code subspaces, we illustrate the extent to which it quantifies global entanglement. We also apply it to track the evolution of entanglement during a quantum computation.

/pdf/global-entanglement-in-multiparticle-systems-4ayjv6ssba.pdf

Global entanglement in multiparticle systems

We find a canonical form for pure states of a general multipartite system, in which the constraints on the coordinates (with respect to a factorizable orthonormal basis) are simply that certain ones vanish and certain others are real. For identical particles they are invariant under permutations of the particles. As an application, we find the dimension of the generic local equivalence class.

Multipartite generalization of the Schmidt decomposition

Entanglement types of pure states of three spin-½ particles are classified by means of their stabilizers in the group of local unitary transformations. It is shown that the stabilizer is generically discrete, and that a larger stabilizer indicates a stationary value for some local invariant. We describe all the exceptional states with enlarged stabilizers.

https://iopscience.iop.org/article/10.1088/0305-4470/33/28/303/pdf

Local symmetry properties of pure three-qubit states

We find a canonical form for pure states of a general multipartite system, in which the constraints on the coordinates (with respect to a factorisable orthonormal basis) are simply that certain ones vanish and certain others are real. For identical particles they are invariant under permutations of the particles. As an application, we find the dimension of the generic local equivalence class.

https://arxiv.org/pdf/quant-ph/0006125.pdf

H. A. Carteret

Papers

Multipartite generalization of the Schmidt decomposition

Local symmetry properties of pure three-qubit states

Multipartite generalisation of the Schmidt decomposition

Multi-particle entanglement