Broadcasting of quantum correlations in qubit-qudit systems

Quantum mechanical properties like entanglement, discord, and coherence act as fundamental resources in various quantum information processing tasks. Consequently, the technique of generating more resources from a few, typically termed as broadcasting, serves as a promising candidate for the design of quantum networks. One way to broadcast resources could be using a cloning operation. In this article, broadcasting of quantum resources beyond $2\otimes 2$ systems is investigated. In particular, in $2\otimes 3$ dimensions, a class of states not useful for broadcasting of entanglement is characterized considering an optimal universal Heisenberg cloning machine. The broadcasting ranges for maximally entangled mixed states and two-parameter class of states are obtained to exemplify our protocol. A significant derivative of our protocol is that the cloning operation generates a qutrit ($3\otimes 3$) entangled pair with positive partial transpose on one of the local sides, and an absolutely separable qubit ($2\otimes 2$) pair on the other side of the input bipartite $2\otimes 3$-dimensional resource state. Moving beyond entanglement, in $2\otimes d$ dimensions, the impossibility to optimally broadcast quantum discord and quantum coherence ($l_1$ norm) is established. However, some significant illustrations are provided to highlight that nonoptimal broadcasting of discord and coherence is still possible.

Figure 1

A schematic depicting the application of local cloning unitaries $U_a$ and $U_b$ on a qubit-qutrit state shared between Alice and Bob. The qutrit system ($d=3$) on Bob's side is illustrated with a three-sphere or glome [65] structure in the four-dimensional Euclidean space. Stereographic projection of the hypersphere's parallels (magenta), meridians (cyan), and hypermeridians (black) are illustrated separately, as subfigures on the right column, beside the glome qutrit for clarity. The projection being conformal, the circular curves intersect each other orthogonally at the yellow points. The rectangular box with a solid olive green boundary represents the input state ${\rho }_{12}$; the dotted rectangular envelopes in blue and orange highlight the cloned local output pairs ${\tilde{\rho }}_{13}$ and ${\tilde{\rho }}_{24}$, respectively. Further, the dotted oval-shaped envelopes in red and green depict the cloned nonlocal output pairs ${\tilde{\rho }}_{14}$ and ${\tilde{\rho }}_{23}$, respectively.

Figure 2

Blue colored circles denote the states that could be successfully characterized to be not useful for broadcasting of entanglement in $2\otimes 3$ dimensions out of $5\times {10}^4$ states generated uniformly and randomly using the Haar measure. Red colored circles mark the ones out of the $5\times {10}^4$ states which didn't pass the separability test in Eq. (13) and so couldn't be characterized either way.

Figure 3

Plot depicting the suboptimal (in dark brown) as well as nonoptimal (in light yellow) broadcastable region for the input TPCS in terms of two input state parameters: $\alpha$ and $\gamma$.

Figure 4

The 3D plot shows the variation of geometric discord ($D_G$) of the nonlocal output state ${\tilde{\rho }}_{14}$ as a function of input state parameters $\alpha$ and $\gamma$ for the two-parameter class of states.

Figure 5

The 3D plot shows the variation of coherence ($l_1$ norm) of nonlocal output state ${\tilde{\rho }}_{14}$ as a function of input state parameters $\alpha$ and $\gamma$ for the two-parameter class of states.