ORGANIGRAMA FUNCIONAL

If the density operator of a state is known, then the Peres-Horodecki criterion is a very easy-to-use tool to test entanglement between two subsets of particles. To experimentally obtain the density operator means to gather the complete information about a quantum state. This requires a lot of experimental effort in the sense that data for a lot of different

8_{These states are called bound entangled as it was shown, that one cannot distill an negative partial}

transpose state (NPT) from many copies of such a state.

9_{The logarithmic negativity of a tensor product of states equals the sum of the logarithmic negativities}

Figure 2.3: An entanglement witness is a hyperplane in the convex space of mixed states (dashed line), separating all separable states from a subset of entangled states. An opti- mal witness (non-dashed line) is tangent to the separable states and provides the optimal separation for the state that one seeks test (ρideal).

measurement settings need to be collected, growing approximately exponentially with the number of qubits (a detailed discussion follows in section 4.3.2. Thus, tools for a more efficient detection of entanglement are required.

An elegant solution to this problem are entanglement witnesses. Using the convexity of the space of mixed states they allow to efficiently detect entanglement. In addition, the criteria and measures introduced so far were, apart from the tangle, tailored to study entanglement between two systems, including multipartite subsets of multiqubit states. Entanglement witnesses are not based on this kind of bisplitting and can be tailored such that genuine multipartite entanglement is studied directly. It makes entanglement witnesses a powerful tool for the analysis of multipartite states. Here, we will shortly present the idea of entanglement witnesses and their connection to the geometric measure of entanglement, a measure that applies to mixed multipartite entanglement.

The space of all density matrices (figure 2.3) is a convex space, as the decomposition into pure states is always a convex linear combination. In this space we can, using an operatorW, define a hyperplane of all states ρ that fulfill the equality:

T r(Wρ) = 0 (2.50)

The main idea is that the space of all separable states ρ_sep is a convex subspace, too. This holds due to the fact that they are given by convex combinations of pure separable states. Therefore it is always possible to define the hyperplane such that the whole space is cut into a part that contains only entangled states and another one that includes all separable statesρsep:

T r(Wρsep)>0 (2.51)

The operatorW is called an entanglement witness, because it is constructed such that: T r(Wρ)<0 =⇒ρentangled. (2.52) A negative expectation value of a state with this operator proves its entanglement. Fur- thermore, it was shown that a particular state ρ0 is entangled if and only if such an entanglement witness exists, i.e. if one can find an operatorW that fulfills the mentioned

2.3 Separability criteria and entanglement measures

conditions withT r(Wρ0)<0. A disadvantage is that, while the Peres-Horodecki criterion detects all distillable entangled states with only one test, entanglement witnesses do not detect the entanglement of all states, i.e. the states in the dark region on the left hand side of the witness hyperplane in figure 2.3. In practice we need to use a witness that is tailored to the state we want to detect. This is, however, not a major disadvantage for the experimentalist, as we try to achieve a certain known entangled stateρ_ideal and expect the actually prepared state to be in its vicinity, detectable by the same entanglement witness. Still, it is important to find entanglement witnesses such that as many entangled states as possible are detected. In the ideal case the corresponding hyperplane is a tangent to the set of separable states; then the witness is called optimal (figure 2.3). The generic way to define a witness W_ψ detecting entanglement in the vicinity of a certain pure state ρideal =|ψihψ|is the following:

Wψ =α11− |ψihψ|, (2.53)

where, for an optimal entanglement witness,α = sup_φsepkhφsep|ψikwith the supremum taken over all separable pure states |φsepi.

The generic form of the operator is merely one possibility of building an entanglement witness. In the experimental part of this thesis, several witnesses, that rely on properties of the quantum states we desire to analyze experimentally, are used. The goal is, in general, to obtain witnesses that give a strong separation from separable states with little experimental effort, i.e. few measurement settings. For example, the entanglement of both of the four-party entangled quantum states obtained in this thesis can be detected with 2 measurement settings. In comparison, 16 and 21 settings are needed for the generic witness, respectively and 81 for a complete state estimation.

Intuitively one might guess that there should be a relation between the witness expecta- tion value and the strength of entanglement in a certain quantum state. It took, however, quite long until, only recently, these relations where revealed [71, 72]. Conclusion from the expectation values of entanglement witnesses can be drawn for several entanglement measures. Out of those, the geometric measure of entanglement (GME, [73, 74]) is par- ticularily closely related to the generic form of entanglement witnesses [75] and will be introduced in the following. It is a multipartite entanglement measure and has a intuitive meaning: The GME is simply the distance of a given pure quantum state to the next separable state:

EG(ψ) = min

|φi_sep(k |ψi − |φik) (2.54) Goldbart and Wei [75] generalized this measure of entanglement to mixed states via a convex roof construction (similar to the method to obtain the entanglement of formation from the von Neumann entropy). In general it is not a simple task to calculate the GME for a given density matrix, as several optimizations have to be performed. Only recently G¨uhne and coworkers [72] presented a simple way to calculate lower bounds on the GME out of the results of witness measurements and applied it to some experiments. There is an particularily simple expression to obtain this bound on the GME based on generic entanglement witness:

EG(ρ)≥max

r (rhWψρi − ˆ

where ˆEG(rWψ) is defined as: ˆ E_G(rW_ψ) = 1−r 2 + 1 2 p (1−r)2_{+ 4rE} G(ψ) +rα−1. (2.56) Wψ,αand |ψiare defined as in equation 2.53 and the geometric measure of the theoretical stateEG(ψ) has to be known. If the optimal generic entanglement witness is known this measure can be obtained from the simple relationE_G(ψ) = 1−α_opt, presented in [75].