When looking at multipartite entanglement, a very natural class of states to consider are the Dicke states. The Dicke states areN-partite states in a perfect superposition of k excitations, i.e.
|N, ki= n
k −12
sym
|0i⊗n−k|1i⊗k
, (5.3)
where sym denotes symmetrization over all parties, e.g. sym[|0i|1i] = |01i+|10i. This state naturally occurs when considering the absorption of k photons by an atomic ensemble. There are of course several assumptions that come along with this; the atoms were all previously in their energy ground state |0i, the wavelength of the photons is such that only a transition from |0i to a specific excited state |1i can absorb a photon, each atom absorbs at most one photon and all atoms have the same probability of absorbing one.
Dicke states are particularly interesting when considering specific quantum mem-ories. When photons are stored within a quantum memory given by a crystal, the state of the memory can in first approximation be thought of as a Dicke state. This makes the study of Dicke states very interesting; given the huge number of atoms involved, such a quantum memory could be said to be macroscopically entangled1. One way to show such entanglement would be by testing these Dicke states for nonlocality.
In the following, we will denote by ρN,k = |N, kihN, k| the density matrix cor-responding to the Dicke state |N, ki. It should be noted that the Dicke state with one single excitation, |N,1i, is known as the W state.
Together with collaborators [6], we studied how resistant the nonlocality that Dicke states can exhibit is to two specific types of noise: noise due to loss of exci-tations and noise due to loss of particles. The first type corresponds to each party having a probabilitypto lose its excitation if it has one. In other words, we consider the non-unitary transformation T which maps |1ito |0i with probabilityp for each party and does nothing otherwise. We denote the post-loss state by
ρf =T(ρ).
In the case of an atomic ensemble, this corresponds to the atoms individually spon-taneously decaying into the ground state. We want to determine how large pcan be such that the individual Dicke states retain their nonlocal properties.
The noise due to loss of particles corresponds to the case of there being a chance to lose a certain number m of the parties, e.g. some of the atoms in our atomic ensemble. In such a case we are left with performing our experiment on the remaining particles. For simplicity, we assume that we know which parties have been lost. The transformation describing this type of loss is simply the partial trace over the m parties that have been lost. We denote the final state by
τf = trm(ρ).
1Note that the question as to what is and is not ’macroscopic’ is a subject of much debate that we will not get into here.
τf is an Nf =N −m partite state. We wish to determine how large the fraction of lost particles mN can be such that the nonlocality of a given Dicke state is preserved.
In order to study this multipartite nonlocality, we used three families of inequal-ities. The first is an extension of Hardy’s paradox [34] given by
SN =P(~0|~0)− XN
j=1
P(~0|0. . .0 1|{z}
j
0. . .0)−P(~1|~1)≤0
first discussed in [39]. The second is the MABK inequality [3, 12, 42]
MN =
X
~ x
cos
"
π
4 1 +N −2X
i
xi
!#
h~xi
≤2N−21 with
h~xi=X
~a
(−1)PiaiP(~a|~x).
Finally, we considered the WWZB inequality [37, 54]
BN = X
~ r∈{0,1}N
X
~x
(−1)~r·~xh~xi ≤2N.
We conducted numerical optimizations in order to find good measurements for lossy Dicke states of different number of excitations for different numbers of parties N and for both types of losses. In order for the problem to remain tractable, we made some assumptions on the optimal measurements, motivated by more complete numerics for low numbers of parties. For the technical details, we refer to [6]. This allows us to give lower bounds on the threshold probabilitypthr(N, k) and the thresh-old fraction fthr(N, k) = mthrN(N,k), i.e. the values such that for all p > pthr(N, k) or f > fthr(N, k), the corresponding lossy state cannot produce any nonlocal correla-tions.
It should be noted that all three inequality families turn out to have their uses.
The WWZB inequalities generally provide the best bounds. Unfortunately, due to them being nonlinear, optimizations are difficult and we were only able to determine good measurements for relatively low numbers of parties, N ≤46. The MABK in-equalities can be optimized for more parties and outperform the generalised Hardy inequalities when we consider noise due to loss of excitations. The generalised Hardy inequalities, by virtue of only having a small number of terms2 can be dealt with even for very large numbers of parties N ≈ 104 −105. Additionally, they seem to outperform the MABK inequalities when considering noise due to loss of particles.
In the case of the W state, it turns out that the numerical optimization for both types of losses is the same. This is due to the fact that the post-loss states are given by
ρf = (1−p)ρN,1+pρN,0 τf = N−m
N ρNf,1+ m NρNf,0.
2Specifically, the number of terms scales linearly with the number of parties.
Figure 5.1: Lower bounds on the threshold loss probability pthr(N,1) for the W state. It can be seen that for the cases for which we could optimize the measurements for the WWZB inequality, it outperforms both the MABK and Hardy inequality, leading to a lower bound on pthr(46,1)≥ 31.4%. For larger numbers of parties N, we managed to show pthr(N,1)≥27.4% using the MABK inequality.
The only difference is that while pthr can take any value in [0,1], fthr = mNthr with N fixed andmthr a positive integer smaller thanN. We thus have that fthr(N,1) =
dN pthre
N . Our numerical optimizations show that the pthr(N,1) appears to be non-decreasing as a function of the total number of parties N, see Fig. 5.1. ForN = 46, the WWZB inequality witnesses the nonlocality of the lossy W state for p≥31.4%, giving us a lower bound on the actual pthr.
For general Dicke states, we have to distinguish between the two noisetypes. We are interested in how the thresholds behave with changing numbers of parties and excitations. Our numerics suggest that the best resistance to noise for both types of noise is achieved for a low number of excitations, see Fig. 5.2. This may be surprising since it seems intuitive that a larger number of total excitations would suffer less from losing some of them. Additionally, for noise due to loss of excitation, we used the fact that the generalized Hardy inequalities allow us to perform our optimizations for large numbers of parties in order to see how the resistance to noise depends on the total number of parties involved. We find that the resistance to noise increases with the total number of parties N. Whether or not these observations are an artifact of our chosen inequalities or potentially even just due to the assumptions we had to make to keep the problem numerically tractable remains an open question.
(a) (b)
(c)
Figure 5.2: For Dicke states ofN qubits with k excitations, we show that the lower bounds on the threshold loss probabilitypthr(N, k) for noise due to loss of excitations and the threshold loss fraction fthr(N, k) for noise due to loss of particles that we derived in this work take their optimal values for low numbers of excitations k.
In Fig. 5.2a, we plot the bounds given by the generalized Hardy inequalities for different number of parties for noise due to loss of excitations. It can be seen that the bound increases with the number of parties. For the comparison of the WWZB, MABK and generalized Hardy inequality in the case of loss of excitation, we show in Fig. 5.2b the case of N = 30. The WWZB performs best, with the MABK inequality being almost as good. For the case of noise due to loss of particles, we show in Fig. 5.2c the bounds due to the generalized Hardy inequality forN = 200.