Information complementarity in multipartite quantum states and security in cryptography

We derive complementarity relations for arbitrary quantum states of multiparty systems of any number of parties and dimensions between the purity of a part of the system and several correlation quantities, including entanglement and other quantum correlations as well as classical and total correlations, of that part with the remainder of the system. We subsequently use such a complementarity relation between purity and quantum mutual information in the tripartite scenario to provide a bound on the secret key rate for individual attacks on a quantum key distribution protocol.

Figure 1

The complementarity for three-qubit systems. The different panels exhibit histograms for the sum of two quantities, viz., the normalized purity $\mathcal{P}$ and a normalized correlation. The correlation is (a) the negativity $\mathcal{N}$, (b) logarithmic negativity $E_{\mathcal{N}}$, and (c) quantum mutual information $\left(\mathcal{I}\right)$ for rank-1 states. We Haar uniformly generate ${10}^4$ three-qubit states for generating the histograms. The histograms for normalized measured quantum mutual information, quantum discord, and quantum work deficit are almost identical to that of the normalized quantum mutual information. The vertical axis represents the relative frequency (R.F.) of the occurrence of a randomly generated three-qubit state in the corresponding range of the sum of the two quantities on the horizontal axis. All quantities are dimensionless.

Figure 2

The complementarity relation, again for three-qubit systems, but for rank-2 states. The different panels exhibit histograms for the sum of two quantities, viz., the normalized purity $\mathcal{P}$ and a normalized correlation. The correlation is (a) the negativity $\mathcal{N}$, (b) logarithmic negativity $E_{\mathcal{N}}$, (c) quantum mutual information $\mathcal{I}$, (d) measured quantum mutual information $\mathcal{J}$, (e) quantum discord $\mathcal{D}$, and (f) quantum work deficit $▵$. The vertical axis represents the relative frequency (R.F.) of the occurrence of a randomly generated three-qubit state in the corresponding range of the sum of the two quantities on the horizontal axis. All quantities are dimensionless.

Figure 3

The horizontal axis represents the quantity $\mathcal{P}+\mathcal{Q}$, where $\mathcal{Q}=min\{{\mathcal{Q}}_{A:C},{\mathcal{Q}}_{B:C}\}$. The different panels exhibit overlapping histograms of three-qubit states of different ranks for this quantity. The gray (white) bars correspond to the cases when $Q$ represents the normalized negativity (normalized quantum mutual information). The panels show ranks (a) 1, (b) 2, and (c) 3. For each rank, we Haar uniformly generate ${10}^4$ three-qubit states of that rank. The vertical axis represents the relative frequency (R.F.) of the occurrence of a randomly generated three-qubit state of the considered rank in the corresponding range of the sum of the two quantities on the horizontal axis. All quantities are dimensionless.