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Dicke states are the eigenstates of the total spin operator ˆJ2_{and the spin operator in the z-} direction ˆJz in a coupled system of N spin-1₂ particles. The eigenvalues arej(j+1) andm,

3.2 Phenomenology of quantum states

Figure 3.3: Scheme of all Dicke states of four qubits sorted by the eigenvalues of the ˆJ2 (lines) and ˆJz (columns). The two states forj= 0 span a two dimensional subspace, while

the three subspaces forj= 1 are three-dimensional spanned by the three states obtained by qubit permutation of depicted state. Our major interest concerns the states with maximal total spin, i.e. the symmetric Dicke states in the top row.

respectively. These Dicke states form a basis of the N-spin Hilbert space. Figure 3.3 shows all Dicke states of four qubits (up to permutations of qubits). Here, we are interested in the Dicke states that are completely symmetric under particle exchange. They are depicted in the upper row, as all of them have a maximal total spin. Thus, these states may either be described by the spin-z component m, or by the number of V’s, which corresponds in other physical systems, like e.g. atoms in a cavity, to the number of excitations. Here, the latter notation is used and an N-qubit symmetric Dicke state with E excitations is denoted by: |D(_NE)i= µ N E ¶_−1/2X k P_k(|V₁, V₂, ..., V_E, H_E+1, ..., H_Ni), (3.16)

where Pk denote all possible permutations of qubits. Note, that a ˆσx-transformation (spin-flip in the computational basis) on each qubit transforms |D(_NE)i to |D_N(N−E)i, i.e. |D_N(E)i ≡LU |D(NN−E)i. Therefore, they have exactly the same entanglement prop- erties. W states can, in this notation, be written as |WNi = |D1Ni, their spin flipped counterpart as |W¯Ni = |D_N(N−1)i. Note that GHZ4 states are not symmetric Dicke states, but can be expressed as superposition thereof and are equivalent under LU to the state with j=1 and m=0: |GHZ0

4i= 1/ √

2(|HHV V i − |V V HHi).

In the following, a short overview of the ideas how to observe Dicke states in experi- mental setups will be presented. Concerning the creation of arbitrary superpositions of symmetric Dicke states in quantum dots, there was a scheme proposed in [110]. Xiao et al. [111] presented a scheme for the generation of Dicke state entanglement between the excitations of atoms in a cavity. A way how to dynamically create Dicke states with atoms in a cavity or trapped ions was presented in [112, 113]. Another way to obtain these states, via an adiabatic process, in trapped ions was proposed by Unanyan et al. [114]. Further, Stockton et al. [80] proposed the preparation of those states via collective spin measure- ments of an atomic ensemble. Experimentally, the focus was so far put on the observation

of W states. We observed the first three-photon W state via an interferometric linear op- tics setup [88, 90, 115] based on non-collinear parametric down conversion (section 4.1.1). A more efficient scheme based on collinear down conversion and a weak coherent beam was presented by Mikami et al. [116] recently. Photonic experiments, despite many proposals, did to our knowledge not demonstrate four-qubit W state entanglement up to now. With trapped ions, however, W states of up to eight particles have been prepared by H¨affner et al. [21]. Concerning the applications of symmetric Dicke states, so far most of the research focused on W states. For those, quantum telecloning [117, 118], generation of the uni- versal entangled state [119], quantum teleportation [120–125], quantum key distribution [126], and dense coding [124] have been proposed.

In chapter 7, the first experiment on the analysis of a symmetric Dicke state with more than one excitation [127], the stateD(2)₄ is presented. There have been a few theoretical investigations on this type of quantum states. Entanglement properties of symmetric Dicke states are studied extensively in [40], in particular the behavior for a growing number of qubits. I would like to note that some theoretical studies on symmetric entangled states in general are presented in [51], which include also all superpositions of symmetric Dicke states, in particular also GHZ states. Apart from this, there have been some general studies on entanglement detection for these states, which I will discuss shortly in the following. Thereby a focus is put on results that are useful for our investigation of D₄(2).

Just like for graph states one can find entanglement witnesses for the analysis of symmet- ric Dicke states in general [128]. One particularly peculiar example are the entanglement witnesses of the form:

Ws=hJˆx2i+hJˆy2i ≤B, (3.17) where ˆJ_x/y = 1/2P_kσk

x/y with e.g. σ3x =11⊗11⊗σx⊗11. The boundB depends on the number of qubits and which type of separability should be excluded (complete separability, biseparability, ...). The witness can easily be rewritten to the standard form, where a negative expectation value proves entanglement: W0

s =B−Ws. We preferred the notation in equation 3.17, as it can be interpreted physically. To do so, we rewritehJˆ_x2i+hJˆ_y2i = hJˆ2_{i − hJ}ˆ2

zi where ˆJ = ( ˆJx,Jˆy,Jˆz). As for symmetric states hJˆ2i = N/2(N/2 + 1) our criterion requires hJˆ2

zi ≥ B, i.e. the collective spin squared of biseparable symmetric states in any direction cannot be arbitrarily small [129–131]. For Dicke states, however, the expectation value hJˆ2

zi can even vanish when the number of excitations approaches N/2. Thus, it is the difference between maximal total spin and zero spin squared in one direction that cannot be achieved by separable, but only by entangled quantum states. One example where this contrast is maximal is the state |D₄(2)i, which is subject of this thesis. It should also be noted that the application of this criterion does not require the ability to access individual qubits, but is intrinsically collective. For other physical implementations of qubits, this might be a big advantage.

The behavior of entangled Dicke states under photon loss is completely different in comparison to Graph states. The residual photons’ state after tracing out one qubit is a mixed state of the following kind:

T rA(|D(_NE)ihD_N(E)|) =E/N |D(_NE₋−₁1)ihD(_NE₋−₁1)|+(N −E)/N |D(_NE₋)₁ihD(_NE₋)₁|. (3.18) At least one of the two terms is genuinely (N-1)-partite entangled, and so is the resulting

3.2 Phenomenology of quantum states

mixed state. Thus, the entanglement persistency in Dicke states is very high, i.e. the entanglement cannot be destroyed by qubit loss.

The behavior under projective measurements is also different from graph states. If both terms in the previous equation are entangled, then any projective measurement will result in a superposition of Dicke states. Otherwise, i.e. in case we start from a W state, there is the possibility to disentangle the state with a certain probability. This is in strong contrast to the behavior of graph states, where one can choose whether the entanglement should be kept by choice of the measurement basis. Here, this is additionally dependent on the measurement outcome.

In the following, we will restrict our investigation to four particles. This is the smallest number of qubits where in addition to theW4states also a Dicke state of the form |D(_NN/2)i exists. In other words, there exists a symmetric state with a z-spin component m = 0. The two kinds of states show very different behavior when studied for a large number of particles [40]. On the one hand, when the entanglement between the pairs after tracing out all other qubits was studied, the W states showed the maximal entanglement compared to all other Dicke states (it was conjectured even in comparison to all quantum states [23]). On the other hand the maximal entanglement between bisplittings4 _{is maximized for the} states D_N(N/2). This is, however, only true with respect to other Dicke states. There are other states [51] that show a higher degree of entanglement in this respect.