Restrictions on shareability of classical correlations for random multipartite quantum states

Unlike quantum correlations, the shareability of classical correlations (CCs) between two parties of a multipartite state is assumed to be free since there exist states for which CCs for each of the reduced states can simultaneously reach their algebraic maximum value. However, when one randomly picks out states from the state space, we find that the probability of obtaining those states possessing the algebraic maximum value is vanishingly small. Therefore, the possibility of a nontrivial upper bound on the distribution of CCs that is less than the algebraic maxima emerges. We explore this possibility by Haar uniformly generating random multipartite states and computing the frequency distribution for various CC measures, conventional classical correlators, and two axiomatic measures of classical correlations, namely, the classical part of quantum discord and local work of work-deficit. We find that the distributions are typically Gaussian-like and their standard deviations decrease with the increase in number of parties. It also reveals that, among the multiqubit random states, most of the reduced density matrices possess a low amount of CCs which can also be confirmed by the mean of the distributions, thereby showing a kind of restrictions on the shareability of classical correlations for random states. Furthermore, we also notice that the maximal value for random states is much lower than the algebraic maxima obtained for a set of states, and the gap between the two increases further for states with a higher number of parties. We report that, for a higher number of parties, the classical part of quantum discord and local work can follow a monogamy-based upper bound on shareability while classical correlators have a different upper bound. The trends of shareability for classical correlation measures in random states clearly demarcate between the axiomatic definition of classical correlations and the conventional ones.

Figure 1

Frequency distribution of ${\sum }_i{C}_{1i}^{xx}$. The fraction $f$ of states (vertical axis) is plotted against ${\sum }_i{C}_{1i}^{xx}$ (horizontal axis) with a bin size of 0.01. Total number of random states generated for the analysis for each $N$ is ${10}^6$. Note here that the covariance of ${\sum }_i{C}_{1i}^{xx}$ also gives the similar frequency distribution, having almost the same mean and standard deviation. All the axes are dimensionless.

Figure 2

Maximum of $C_{12}^{kx}+C_{13}^{xx}$ ($y$ axis) vs $\theta$ ($x$ axis) where $k$ is the unit vector, $(cos\theta ,sin\theta ,0)$. Red points represent maximal values of $C_{12}^{kx}+C_{13}^{xx}$ for different $\theta$ and the blue line is the best fit, depicting the decreasing nature with the increase of noncommutativity as measured by $\theta$. Dotted line is the upper bound obtained in Eq. (14). All the axes are dimensionless.

Figure 3

(a) Frequency distributions of ${\sum }_i{C}_{1i}^D$, and (b) ${\sum }_i{C}_{1i}^{\mathrm{LW}}$ for $N=3$ to 6 parties. The other specifications are the same as in Fig. 1.

Figure 4

Frequency distribution of ${\sum }_{i=2}^N{\left(C_{1i}^{xx}\right)}^2$. The fraction $f$ of states (vertical axis) is plotted against ${\sum }_{i=2}^N{\left(C_{1i}^{xx}\right)}^2$ (horizontal axis) with a bin size of 0.01. Total number of random states generated for the analysis for each $N$ is ${10}^5$. All the axes are dimensionless.

Figure 5

Frequency distribution (f) of ${\sum }_{i=2}^N{C}_{1i}^I$ (vertical axis) vs ${\sum }_{i=2}^N{C}_{1i}^I$ (horizontal axis) as defined in Eq. (15). All other specifications are same as in Fig. 4.

Figure 6

Average (avg) and maximum values (max) of ${\sum }_{i=2}^N{C}_{1i}^{yy}$ (ordinate) for random $N$-party states possessing a definite amount of ${\sum }_{i=2}^N{C}_{1i}^{xx}$ (abscissa). Squares represent the average values while circles are for the maximum value. $N=3$ and 4 are in the upper panel while in the lower panel from left, $N=5$ and 6 are displayed. Here the bin size is set to 0.1. All axes are dimensionless.

Figure 7

Plot of an average and maximum of ${\sum }_{i=2}^N{C}_{1i}^{yy}$ ($y$ axis) for random states having a fixed genuine multiparty entanglement content as measured by GGM ($x$ axis). Other specifications are same as in Fig. 6, with the exception that the bin size in this case is 0.05. All axes are dimensionless.

Figure 8

For a given amount of ${\sum }_i^N{C}_{1i}^{\mathrm{LW}}$ (horizontal axis), average and maximal values of ${\sum }_{i=2}^N{C}_{1i}^D$ (vertical axis) for random states is plotted for different values of $N\le 6$. For other specifications, see Fig. 6.

Figure 9

Upper panel shows the maximum and average values of ${\sum }_{i=2}^N{C}_{1i}^D$ (ordinate) vs GGM (abscissa). From left to right, $N$ increases from 3 to 6. Lower panel shows ${\sum }_{i=2}^N{C}_{1i}^{\mathrm{LW}}$ against GGM. Here, the bin size for computation is 0.05. All axes are dimensionless.

Figure 10

Monogamy-motivated upper bounds from CC measures. From left to right: the frequency distribution $f$ on the vertical axis is plotted for ${\delta }_{C^{zz}}, {\delta }_{C^D}$, and ${\delta }_{C^{\mathrm{LW}}}$. $N$ also varies from 3 to 6. All axes are dimensionless.

Figure 11

Scatter plot for comparing ${\delta }_{C^D}$ and ${\delta }_{C^{\mathrm{LW}}}$ with GGM. We find that, with the increase of $N$, higher GGM values (ordinate) reveals the monogamous nature of ${\delta }_{C^D}$ (abscissa in left panel) and ${\delta }_{C^{\mathrm{LW}}}$ (abscissa in right panel). Circles, squares, triangles, and diamonds correspond to $N=3$, 4, 5, and 6. All axes are dimensionless.