Converting Coherence to Quantum Correlations

Recent results in quantum information theory characterize quantum coherence in the context of resource theories. Here, we study the relation between quantum coherence and quantum discord, a kind of quantum correlation which appears even in nonentangled states. We prove that the creation of quantum discord with multipartite incoherent operations is bounded by the amount of quantum coherence consumed in its subsystems during the process. We show how the interplay between quantum coherence consumption and creation of quantum discord works in the preparation of multipartite quantum correlated states and in the model of deterministic quantum computation with one qubit.

Figure 1

Preparation of a quantum correlated state. A sequence of controlled-z gates is applied to generate quantum correlations in a quantum system that is initialized in a product state. The sequence represents a global incoherent operation for the register; i.e., the coherence of the global state is nonincreasing (it is constant in such a case). Each controlled gate consumes the coherence of the control and target qubits and increases the global discord of the state.

Figure 2

Coherence consumption bounds discord creation. We plot the values of the total coherence consumption of the subsystems $l\times \mathrm{\Delta }C({\rho }_{A_k})+\mathrm{\Delta }{C}^l({\rho }_{A_1})$ (the solid linked diamonds), the created global discord $\overline{D}({\rho }_{A_1,\dots ,A_n}^l)$ (the dotted linked squares) obtained by numerical optimization, and the coherence consumption (the dashed linked disks) on the control qubit $\mathrm{\Delta }{C}^l({\rho }_{A_1})$ after $l$ controlled-z gates for $p=0.2$ and $\theta =0.45$.

Figure 3

DQC1 model. The algorithm consists of (1) preparing coherence in an ancillary pure qubit; (2) applying a controlled operation, which, in general, creates discord between the ancilla and a maximally mixed register by consuming the coherence of the ancilla; (3) obtaining information about $\mathrm{Tr}[U]$ by a polarization measurement in the ancilla: $⟨{\sigma }_x+i{\sigma }_y{⟩}_{{\tilde{\rho }}_A}=\mathrm{Tr}[U]/2^n$.