Multipartite secret key distillation and bound entanglement

Recently it has been shown that quantum cryptography beyond pure entanglement distillation is possible and a paradigm for the associated protocols has been established. Here we systematically generalize the whole paradigm to the multipartite scenario. We provide constructions of new classes of multipartite bound entangled states, i.e., those with underlying twisted Greenberger-Horne-Zeilinger (GHZ) structure and nonzero distillable cryptographic key. We quantitatively estimate the key from below with the help of the privacy squeezing technique.

Figure 1

(Color online) An exemplary plot of $K_{DW}\equiv I(A_1:A_2)\left({\Theta }_{A_1{A}_{2}E}^{\left(\mathrm{ccq}\right)}\right)-I(A_1:E)\left({\Theta }_{A_1{A}_{2}E}^{\left(\mathrm{ccq}\right)}\right)$ with $N=3$ (a) and for comparison in the case of $N=2$ (b), which was discussed in Ref. 27. For the sake of clarity, zero is put whenever the plotted function is less than zero. Notice also that even though $k$ and $D$ are discrete parameters, the graph is made as if $K_{DW}$ were a function of continuous parameters. It follows from both figures that the number of parties $N$ significantly influences the obtained lower bound. Namely, for $N=3$ one needs to spend more copies of a given state to get nonzero values of $K_{DW}$.

Figure 2

(Color online) An exemplary plot of ${\tilde{K}}_{DW}\equiv p_{D,3}^{\left(k\right)}$ $[I(A_1:A_2)\left({\Theta }_{A_1{A}_{2}E}^{\left(\mathrm{ccq}\right)}\right)-I(A_1:E)\left({\Theta }_{A_1{A}_{2}E}^{\left(\mathrm{ccq}\right)}\right)]$ with $N=3$. For comparison it is also presented for the case with $N=2$ (b). For the sake of clarity, zero is put whenever the plotted function is less or equal to zero. Also, though both the parameters $k$ and $D$ are integer, for convenience, the function ${\tilde{K}}_{DW}$ is plotted as if it were a function of continuous $k$ and $D$. It is clear that for $N=3$ the lower bound on distillable key is considerably smaller.

Figure 3

(Color online) The function appearing on the right-hand side of Eq. (118) (denoted here by $K_{DW}$) in the function of number of copies $k$ and the dimension $D$. Zero is put whenever the function is less than zero. Notice that both the parameters $k$ and $D$ are discrete; however, continuous plot is made to indicate better the behavior of $K_{DW}$. It is clear from the plot that for larger $k$ the distillable key of ${\tilde{\theta }}_{\mathrm{A}}^{(\mathcal{N},k)}$ approaches one bit (this is actually a maximal value obtainable from two-qubit states) and the convergence depends on $D$. Namely, for higher dimensions $D$ the convergence to the maximal value is faster.

Figure 4

(Color online) Lower bounds on $K_D$ of ${\tilde{\varrho }}_{{\mathrm{AA}}^{\prime }}^{(D,N)}$ in the function of $k$ and $D$. The upper plot (a) presents lower bound (denoted here by ${\tilde{K}}_{DW}^{\left(1\right)}$) on $K_D$ in the case of unitary Hermitian matrices $U_D$, while in the second plot (b) lower bound $\left({\tilde{K}}_{DW}^{\left(2\right)}\right)$ in the case of unitary but not Hermitian matrices is given. Both are just a product of probability ${\tilde{p}}_{D,N}^{\left(k1\right)}$ (left) or ${\tilde{p}}_{D,N}^{(k,2)}$ (right) and $K_{DW}$ plotted in Fig. 3. One infers that in the case of unitary but not Hermitian matrix $U_D$, the region of nonzero values of the plotted function is wider than in the case of unitary Hermitian matrices.