Broadcasting quantum coherence via cloning

Quantum coherence has recently emerged as a key candidate for use as a resource in various quantum information processing tasks. Therefore, it is of utmost importance to explore the possibility of creating a greater number of coherent states from an existing coherent pair. In other words, we start with an initial incoherent pair and induce coherence via quantum cloning. More specifically, we start with a genuinely incoherent state that remains incoherent with the change of basis and make it a coherent state at the end. This process is known as broadcasting of coherence via cloning, which can either be optimal or nonoptimal. Interestingly, in this work, we are able to give a method by which we can introduce coherence in the genuinely incoherent state. We use the computational basis representation of the most general two-qubit mixed state, shared between Alice and Bob, as the input state for the universal symmetric optimal Buzek-Hillery cloner. First of all we show that it is impossible to ensure optimal broadcast of coherence. Second, in the case of nonoptimal broadcasting, we show that the coherence introduced in the output states of the cloner will always be lesser than the initial coherence of the input state. Finally, we take the examples of statistical mixture of the most coherent state and most incoherent state, and then the Bell-diagonal states, to obtain their respective ranges of nonoptimal broadcasting in terms of their input state parameters.

Figure 1

Application of local cloning operations $U_l^1$ and $U_l^2$ on qubits 1 and 2 of the input state ${\rho }_{12}$ and the blank states $|{\mathrm{\Sigma }}_3\rangle \langle {\mathrm{\Sigma }}_3|$ and $|{\mathrm{\Sigma }}_4\rangle \langle {\mathrm{\Sigma }}_4|$ on Alice's and Bob's labs, respectively, to get the output states ${\tilde{\rho }}_{12}$ and ${\tilde{\rho }}_{34}$.

Figure 2

Application of nonlocal cloning operation $U_{nl}$ on the input state ${\rho }_{12}$ and blank state ${\rho }_{34}$ to get the output states ${\tilde{\rho }}_{12}$ and ${\tilde{\rho }}_{34}$.

Figure 3

$\mathrm{\Delta }ABC$ in the $xy$ plane, with point D in its interior.

Figure 4

Broadcasting of coherence in Bell diagonal states via local state-independent cloning. The hue index represents the trend in hue with respect to the value of nonlocal coherence $[C\left({\tilde{\rho }}_{12}\right)$ or $C\left({\tilde{\rho }}_{34}\right)]$ after normalized in a scale of 0 to 1.

Figure 5

Broadcasting ranges for nonlocal state-independent cloning in the case of Bell diagonal states. The hue index represents the trend in hue with respect to the value of the nonlocal coherence $[C\left({\tilde{\rho }}_{12}\right)$ or $C\left({\tilde{\rho }}_{34}\right)]$ after normalized in a scale of 0–1.

Figure 6

$\mathrm{\Delta }ABC$ with point M in its interior.