Experimental Cyclic Interconversion between Coherence and Quantum Correlations

Quantum resource theories seek to quantify sources of nonclassicality that bestow quantum technologies their operational advantage. Chief among these are studies of quantum correlations and quantum coherence. The former isolates nonclassicality in the correlations between systems, and the latter captures nonclassicality of quantum superpositions within a single physical system. Here, we present a scheme that cyclically interconverts between these resources without loss. The first stage converts coherence present in an input system into correlations with an ancilla. The second stage harnesses these correlations to restore coherence on the input system by measurement of the ancilla. We experimentally demonstrate this interconversion process using linear optics. Our experiment highlights the connection between nonclassicality of correlations and nonclassicality within local quantum systems and provides potential flexibilities in exploiting one resource to perform tasks normally associated with the other.

Figure 1

(i) Quantum circuit and (ii) conceptual diagram for the cyclic interconversion process. This scheme has two stages. Stage (a) takes a coherent state ${\rho }_A$ as input (system $A$), along with an incoherent ancilla ${\tau }_B$ (system $B$). Coherence in $A$ is converted into discord between $A$ and $B$ using incoherent operations, yielding a state ${\rho }_{AB}^{I_{\max }}$ when the conversion is maximal (such that all coherence is consumed, and the resulting marginal ${\rho }_A^{I_{\max }}$ is incoherent). In (b), the coherence in $A$ is restored by application of LQICC operations on the output from stage (a). This involves first measuring on $B$ ($M_B$) and then applying an appropriate incoherence operation (${\mathrm{\Delta }}_{A^i}$) on $A$ conditioned on the resulting outcome. The ancilla can then be reset to its initial state via incoherent operations (${\mathrm{\Delta }}_{B^i}$).

Figure 2

Experimental setup. The setup divides into three modules: (a) state preparation, (b) coherence to discord conversion, and (c) coherence restoration. Module (a) initializes one photon in some arbitrary state ${\rho }_A$ (named control) and a second photon in state ${|0⟩⟨0|}_B$ (named target), in particular, sandwichlike type II beamlike phase-matching $\beta$-barium borate (BBO) crystals are used to generate two photon entangled sources (ES), while the YVO4 and LiNO3 crystals in each arm are used for temporal and spatial compensation, respectively. Module (b) implements a cnot gate between control and target, which converts the coherence in ${\rho }_A$ into discordant correlations between control and target photons. This results in a discorded state ${\rho }_{AB}^{\mathrm{I}\max }$. Module (c) implements a von Neumann measurement on the target photon using QWP2 and HWP8 together with a PBS and two SPDs. The final coherence on $A$ is then detected by quantum state tomography using a combination of QWP1, HWP9, PBS, and two SPDs. Note that at the end of module (b), we can also measure the discord in ${\rho }_{AB}^{\mathrm{I}\max }$ via state tomography by using the aforementioned measurement apparatus. Key elements include half-wave plate (HWP), quarter-wave plate (QWP), partially polarizing beam splitter (PPBS), polarizing beam splitter (PBS), single photon detector (SPD), entangled source (ES), fiber coupler (FC), interference filter (IF), quartz plate (QP).

Figure 3

Discord generated and coherence restored for various pure state inputs. We conducted the interconversion protocol for various pure state inputs of the form ${|\varphi ⟩}_{AB}=(\cos 2\theta |0⟩+\sin 2\theta |1⟩)|0⟩$, for various values of $\theta$ between 0° and 45°. The coherence of this input state is given by $h({\cos }^{2}2\theta )$, where $h(p)=-p\log p-(1-p)\log (1-p)$ is the binary entropy (brown line). The first stage of the experiment converts this coherent into discord (blue squares). After the second stage of the experiment, the amount of coherence restored in the input photon is given by the red triangles. The discrepancy from theoretical ideals (brown line) is due to the imperfect fidelity of the cnot gate (of approximately 0.885) employed in the conversion process. These imperfections are fully characterized by quantum process tomography (see the Supplemental Material [25]). The resulting predictions—using models that take this tomographic data into account—agree well with observed results (blue and red lines).

Figure 4

Discord generated and coherence restored for various mixed state inputs. Input states of various purity are generated by subjecting a pure maximally coherent input $|+⟩$ to quartz crystals of different thicknesses, resulting in mixed states ${\rho }_A$ with varying levels of initial coherence. The exact values are determined by quantum state tomography. For each crystal thickness, we plot the amount of discord successfully generate through the conversion process (blue squares) and the amount of coherence restored after one cycle (red triangle). Theoretical curves taking into account the imperfect fidelity of the cnot gate (blue and red line) are plotted for comparison. Meanwhile, the brown line represents the amount of coherence we would have achieved in the ideal case of perfect unitary gates (this, of course, aligns with the initial coherence).