Asymmetric broadcasting of quantum correlations

In this work, we exhaustively investigate $1\to 2$ local and nonlocal broadcasting of entanglement as well as correlations beyond entanglement (geometric discord) using asymmetric Pauli cloners with the most general two-qubit state as the resource. We exemplify asymmetric broadcasting of entanglement using “maximally entangled mixed states.” We demonstrate the variation of broadcasting range with the amount of entanglement present in the resource state as well as with the asymmetry in the cloner. We show that it is impossible to optimally broadcast geometric discord with the help of these asymmetric Pauli cloning machines. We also study the problem of $1\to 3$ broadcasting of entanglement using nonmaximally entangled state as the resource. For this task, we introduce a method we call successive broadcasting which involves application of $1\to 2$ optimal cloning machines multiple times. We compare and contrast the performance of this method with the application of direct $1\to 3$ optimal cloning machines. We show that $1\to 3$ optimal cloner does a better job at broadcasting than the successive application of $1\to 2$ cloners and the successive method can be beneficial in the absence of $1\to 3$ cloners. We also bring out the fundamental difference between the tasks of cloning and broadcasting in the final part of the paper. We create examples to show that there exist local unitaries which can be employed to give a better range for broadcasting. Such unitary operations are not only economical, but also surpass the best possible range obtained using existing cloning machines enabling broadcasting of lesser entangled states. This result opens up a direction in exploration of methods to facilitate broadcasting which may outperform the standard strategies implemented through cloning transformations.

Figure 1

The figures depict broadcasting of entanglement. The boxes with dotted boundary highlight the local output pairs ${\tilde{\rho }}_{13}$ and ${\tilde{\rho }}_{24}$ [53]. (a) Local broadcasting. Only the diagonal nonlocal output pairs (${\tilde{\rho }}_{14}$ and ${\tilde{\rho }}_{23}$) have been highlighted for clarity. (b) Nonlocal broadcasting. Only the horizontal nonlocal output pairs (${\tilde{\rho }}_{12}$ and ${\tilde{\rho }}_{34}$) have been highlighted for clarity.

Figure 2

The figure shows broadcastable region for maximally entangled mixed states (MEMS). The brown (dark gray) region corresponds to the case for local broadcasting when horizontal nonlocal pairs are used, whereas the magenta (medium gray) region corresponds to the use of diagonal nonlocal pairs. The yellow (light gray) region represents nonlocal broadcasting using asymmetric cloners. The horizontal axis demonstrates the variation in asymmetry of the cloner with $p=\frac{1}{2}$ representing a completely symmetric cloner. The vertical axis represents the concurrence of the input state.

Figure 3

The figure shows the two-step procedure to carry out $1\to 3$ broadcasting by successive application of asymmetric cloners locally. In step 1, the cloner $U_a^l$ parametrized by $p_1$ is applied on both sides. In step 2, the cloner $U_a^l$ parametrized by $p_2$ is applied on both sides of ${\rho }_{34}^{\left(1\right)}$, which is a nonlocal pair obtained after first step. The outputs in the first step are superscripted with (1) while the final output states in second step are superscripted by (2). Here, only the horizontal nonlocal pairs have been highlighted for clarity. The dark blue dotted box comprises the local output pairs obtained after application of the first cloner while the dashed envelope contains those after application of the second cloner [53].

Figure 4

The figure depicts the two-step procedure to carry out $1\to 3$ broadcasting by successive application of asymmetric cloners nonlocally on the input resource. In step 1, the cloner $U_a^{nl}$ parametrized by $p_1$ is applied to the input state. In step 2, the cloner $U_a^{nl}$ parametrized by $p_2$ is applied to ${\rho }_{12}^{\left(1\right)}$, which is a nonlocal pair obtained after first step. The intermediate output pairs after the first cloning step are superscripted with (1), while the final output states are superscripted by (2), and only the horizontal nonlocal pairs (that are inseparable) have been highlighted for clarity. The dotted boxes comprise the local output pairs obtained after each step: dark blue box contains those from step 1 while dashed envelope contains those from step 2 [53].

Figure 5

The figure shows a contour plot of the allowed range of values of $k$ for a successful broadcasting, given the successive application of cloners parametrized by $p_1$ and $p_2$. The hue depicts the value of $\sigma$ such that the broadcasting range is $(0.5-\sigma )\le k\le (0.5+\sigma )$.

Figure 6

The figures depict $1\to 3$ asymmetric broadcasting of ${\rho }_{12}$ via direct 1 to 3 cloning. Only the horizontal nonlocal pairs (${\tilde{\rho }}_{12}$, ${\tilde{\rho }}_{34}$, and ${\tilde{\rho }}_{56}$) have been highlighted for clarity. (a) Local broadcasting. (b) Nonlocal broadcasting.

Figure 7

The figure shows a contour plot of the allowed range of values of $k$ for a successful broadcasting operation, given the application of a $1\to 3$ nonlocal cloner parametrized by $\beta$ and $\gamma$. The hue depicts the value of $\sigma$ such that the broadcasting range is $(0.5-\sigma )\le k\le (0.5+\sigma )$.

Figure 8

The figure shows the broadcastable region for Werner-type states when using arbitrary unitary and symmetric cloner. The yellow (light gray) region of input state space covered by the arbitrary unitary is strictly larger than and engulfs the cyan (dark gray) region covered by the symmetric Buzek-Hillary (BH) cloner.

Figure 9

The figure shows the state space for Bell-diagonal states. At the vertex tuples ($-1,-1,-1$), ($1,1,-1$), ($1,-1,1$), and ($-1,1,1$) of the translucent tetrahedron lie the Bell states $|{\gamma }_{uv}\rangle$. The cones emerging from these corners depict the broadcastable regions. The small (blue) cones which are associated with the use of symmetric Buzek-Hillary (BH) cloner are engulfed by the large (chrome-yellow) cones which correspond to the use of arbitrary unitary. The (black) octahedron in the middle of the tetrahedron depicts the separable region within the Bell-diagonal state space.