Directly Measuring the Degree of Quantum Coherence using Interference Fringes

Quantum coherence is the most distinguished feature of quantum mechanics. It lies at the heart of the quantum-information technologies as the fundamental resource and is also related to other quantum resources, including entanglement. It plays a critical role in various fields, even in biology. Nevertheless, the rigorous and systematic resource-theoretic framework of coherence has just been developed recently, and several coherence measures are proposed. Experimentally, the usual method to measure coherence is to perform state tomography and use mathematical expressions. Here, we alternatively develop a method to measure coherence directly using its most essential behavior—the interference fringes. The ancilla states are mixed into the target state with various ratios, and the minimal ratio that makes the interference fringes of the “mixed state” vanish is taken as the quantity of coherence. We also use the witness observable to witness coherence, and the optimal witness constitutes another direct method to measure coherence. For comparison, we perform tomography and calculate $l_1$ norm of coherence, which coincides with the results of the other two methods in our situation. Our methods are explicit and robust, providing a nice alternative to the tomographic technique.

Figure 1

Experimental setup. (a) The preparation of the target state. (b) The direct coherence-measurement method using the interference fringes. The same setups as those in (a) prepare the various ancilla states and the half-wave plate (HWP) and polarizing beam splitter (PBS) before it adjusts the mixing ratio $s$ by tuning the number of the ancilla photons. Then, the beam splitters (BSs) mix the target and ancilla states. The following rotated phase plates (RPPs), HWP, PBS, and the electronic module detect and monitor the interference fringes. The inset figure on the top of the box are actually two sample fringe data that we obtain. The blue and red dots, respectively, correspond to the “mixed state” with and without coherence. (c) The method using coherence-witness observable. The quarter-wave plates (QWPs) and HWP can map the bases of the PBS to the eigenstates of any witness observable. Each of the eigenstates corresponds to the detection events with the value of the corresponding eigenvalue. Then, the expectation value of the events can be calculated. (d) Tomography method. Details can be referred to in Ref. [31] and Supplemental Material [29]. PP, phase plate; SPAD, single-photon avalanche diode.

Figure 2

Searching process for ${\tau }^{*}$ (part of the interference-fringe method). (a) The typical results of sweeping ${\phi }_{\tau }$ (${\phi }_{\tau }=n\pi /4$, with $n=0$ to 7). $V$ is the visibility of the interference fringes of the “mixed state,” and the mixing ratio $s$ is determined by the finally detected photon numbers of, respectively, the target and ancilla states. The dots are the experimental data, and the lines are the corresponding theoretical simulations. The error bars originate from the fitting errors of the interference fringes. (b) The enlargement of the red dashed rectangle in (a). The minimum visibility is marked using red color, and it vanishes within the error bar. The corresponding $s$ is denoted as $s_{V=0}$, which only appears when ${\phi }_{\tau }={\phi }_{\rho }+\pi$. (c) The typical results of sweeping $u_{\tau }$ [along the blue dashed arrow in (e)]. The minimum $s_{V=0}$ appears when $u_{\tau }=\sin {\theta }_{\tau }$ (or $r_{\tau }=1$), i.e., the pure state situation. (d) The typical results of sweeping ${\theta }_{\tau }$ [along the green dotted arrow in (e)]. The minimum $s_{V=0}$ appears when ${\theta }_{\tau }=\pi /2$. (e) The cross section of the Bloch sphere in the $r_x-r_z$ plane.

Figure 3

Searching process for ${\mathcal{W}}^{*}$ (part of the witness-observable method). (a) The typical results of sweeping ${\phi }_{\mathcal{W}}$. The dots are experimental data, and the line is the theoretical simulation. The error bars originate from the standard deviation. The maximum $-\mathrm{Tr}[\rho \mathcal{W}]$ appears when ${\phi }_{\mathcal{W}}={\phi }_{\rho }+\pi$. (b) The typical results of sweeping ${\theta }_{\mathcal{W}}$ in the $0\le r_{\mathcal{W}}\le 1$ case [along the purple dashed arrow in (f)]. The maximum $-\mathrm{Tr}[\rho \mathcal{W}]$ appears when ${\theta }_{\mathcal{W}}=\pi -{\theta }_{\rho }$. (c) The typical results of sweeping ${\theta }_{\mathcal{W}}$ in the $r_{\mathcal{W}}>1$ case [along the green dotted arrow in (f)]. The maximum $-\mathrm{Tr}[\rho \mathcal{W}]$ appears between $\pi -{\theta }_{\rho }$ and $\pi /2$. All the maximums of (b),(c) are marked in (f) using the blue dot-dashed arrow. (d) The results of sweeping $r_{\mathcal{W}}=2/a-1$ ($a$) along the blue dot-dashed arrow. (e) The enlargement of the red dashed rectangle in (d). The maximum $-\mathrm{Tr}[\rho \mathcal{W}]$ appears when $a=0$, i.e., ${\mathcal{W}}^{*}=-(\cos {\phi }_{\rho }{\sigma }_x+\sin {\phi }_{\rho }{\sigma }_y)$. (f) The cross section of the $\mathcal{W}$’s space in the $r_x-r_z$ plane. All possible $\mathcal{W}$’s are sandwiched between the planes of $r_z=\pm 1$.

Figure 4

Results of the three coherence-measurement methods ($\mathcal{C}$, ${\mathcal{C}}_{\mathcal{W}}$, and ${\mathcal{C}}_{l_1}$ that are shown in (a), (b), and (c), respectively). The target state is varied by going through a pure-dephasing evolution. The lines are the corresponding theoretical fits for these data.