Multipartite dense coding versus quantum correlation: Noise inverts relative capability of information transfer

A highly entangled bipartite quantum state is more advantageous for the quantum dense coding protocol than states with low entanglement. Applications of quantum channels are most likely to be commercially important only in the multiparty regime, where such a correspondence does not exist even for pure quantum states. We establish a connection between the multiparty capacity of classical information transmission in quantum dense coding and several multipartite quantum correlation measures of the shared state, used in the dense coding protocol. In particular, we show that for the noiseless channel, if multipartite quantum correlations of an arbitrary multipartite state of arbitrary number of qubits are the same as those of the corresponding generalized Greenberger-Horne-Zeilinger state, then the multipartite dense coding capability of the former is the same as or better than that of the generalized Greenberger-Horne-Zeilinger state. Interestingly, in a noisy-channel scenario, where we consider both uncorrelated and correlated noise models, the relative abilities of the quantum channels to transfer classical information can get inverted by administration of a sufficient amount of noise. When the shared state is an arbitrary multipartite mixed state, we also establish a link between the classical capacity for the noiseless case and multipartite quantum correlation measures.

Figure 1

Schematic of the change of status of the $g\mathrm{GHZ}$ state in comparison to other multipartite states with respect to the multipartite DC capacity in the presence of noise. Comparison was made with states which possess the same value of multipartite quantum correlations as the $g\mathrm{GHZ}$ state. Results obtained in this paper show that the $g\mathrm{GHZ}$ state is more robust against noise compared to arbitrary states for the dense coding protocol. This is independent of whether the noise in the system is from the source or in the channel after the encoding.

Figure 2

GGM vs multipartite DC capacity. The GGM is plotted as the ordinate, while the multipartite DC capacity is plotted as the abscissa for ${10}^5$ randomly chosen three-qubit pure states, according to the uniform Haar measure over the corresponding space (blue) triangles. The solid (red) line represents the generalized GHZ states. There is a set of states for which, if the capacity matches that of a $g\mathrm{GHZ}$ state, then their GGMs are also equal. For the remaining states, if the capacity is equal to that of a $g\mathrm{GHZ}$ state, its GGM is bounded above by that of the $g\mathrm{GHZ}$ state. Note that the range of the horizontal axis is considered only when the states are dense codable. Quantities represented on both axes are dimensionless. We consider the case where postencoded states are sent through noiseless channels.

Figure 3

Left: Tangle (vertical axis) vs multiparty DC capacity (horizontal axis) for randomly generated three-qubit pure states [(blue) triangles]. Right: Discord monogamy score (vertical axis) vs DC capacity (horizontal axis) for the same states. In both cases, the $g\mathrm{GHZ}$ states give the boundary [solid (red) line]. The capacity is dimensionless, while the tangle and discord monogamy score are measured in ebits and bits, respectively. All other considerations are the same as in Fig. 2.

Figure 4

Discord monogamy score vs multipartite DC capacity for Haar uniformly generated rank 2 three-qubit states. The solid (red) line represents the $g\mathrm{GHZ}$ states. About 1.85% of the randomly generated states lie below the solid line and are represented by (blue) triangles. The remainder, represented by (green) circles, lie above the solid line. The horizontal axis is dimensionless, while the vertical one is measured in bits.

Figure 5

Discord monogamy score and raw DC capacity are plotted against the mixing parameter $p$, for the rank 8 state, ${\rho }_8=(1-p)\rho +\frac{p}{8}I$. Here $\rho =q|\mathrm{GHZ}\rangle \langle \mathrm{GHZ}|+(1-q)|{\mathrm{GHZ}}^{\prime }\rangle \langle {\mathrm{GHZ}}^{\prime }|$. Each value of $q$ provides a curve, and we present several exemplary curves here. All quantities plotted are dimensionless, except ${\delta }_D$, which is measured in bits.

Figure 6

GGM (vertical axis) vs raw DC capacity (horizontal axis) under the fully correlated Pauli channel, when the shared state is an arbitrary three-qubit pure state [(orange) squares, (green) circles, and (blue) triangles] or the $g\mathrm{GHZ}$ state [solid (red) line]. For all states, the noise in the channel is fixed at 0.19. We choose $q_0=q_3=0.485$ and $q_1=q_2=0.015$ as noise parameters for the arbitrary as well as the $g\mathrm{GHZ}$ state. This corresponds to Case 1 in Sec. 5a. See the discussion there for further details. Both axes are dimensionless. The vertical line at $C_c^{\mathrm{noisy}}=2/3$ helps to readily read out the actual capacity from the raw capacity.

Figure 7

GGM (vertical axis) vs DC capacity (horizontal axis) under the fully correlated Pauli channel. In this case, we choose $\left\{q_i\right\}$ as $q_0=0.93,q_1=0.01,q_2=0.02$, and $q_3=0.04$. This corresponds to Case 2 in Sec. 5a. We randomly (Haar uniformly) generate $5\times {10}^4$ three-qubit pure states. See text for further details. Both axes represent dimensionless quantities. The vertical line at $C_c^{\mathrm{noisy}}=2/3$ has the same function as in Fig. 6.

Figure 8

DC capacity vs noise for various choices of $\alpha$ in the $g\mathrm{GHZ}$ state. Top: The DC capacity is plotted against the noise of the depolarizing channel for the $g\mathrm{GHZ}$ state. Bottom: The DC capacity is plotted with respect to the noise in the fully correlated Pauli channel, for the $g\mathrm{GHZ}$ state. Different curves correspond to different values of $\alpha$. The vertical axis starts from 2/3, below which the states are not dense codable. The states remain dense codable in the presence of moderate to high Pauli noise, while this is not the case for the uncorrelated depolarizing channel. Horizontal axes are measured in bits. All other quantities are dimensionless.

Figure 9

GGM vs raw DC capacity, $C_{\mathrm{uc}}^{\mathrm{noisy}}$, in the presence of uncorrelated noise. See text for further details. Both axes represent dimensionless quantities. The vertical line at $C_{\mathrm{uc}}^{\mathrm{noisy}}=2/3$ again helps to read the actual capacity from the raw capacity.