Fig. 1. Experimental setup. (a) Optical structure for the experiments. The entangled photon pairs are produced via the type-I spontaneous parametric down-conversion (SPDC) process by pumping two adjacent nonlinear crystals of BBO with a 405-nm laser diode. Two α-BBO crystals are inserted after the BBOs to compensate the walk-off effect. Mixed states emerge from the single-qubit noisy channel, such as the amplitude damping channel, the Pauli channel, the dephasing channel, and the depolarizing channel, acting on one of the qubits. Local measurements in M are carried out via a sequence of quarter-wave plate (QWP)-HWP-PBS and single-photon detection. Coincidence measurements are then performed via avalanche photodiodes (APDs). Total coincidence counts are about 120,000 over a collection time of 12 s within a 3-ns time window. (b) Schematic of the Pauli channel realized with two liquid crystals (LCs). Voltage sequences applied on the spatial path of photons. ti indicates the time interval of voltage to realize the gate σi. T is the collection time. VH is the corresponding applied voltage when an LC acts as an HWP, and VI is for an identity operator.

Fig. 2. Experimental results for two-qubit systems. (a) Entanglement witness value as a function of state parameter α for the pure states and of parameter p of the quantum noisy channels for the mixed states. (b) Negativities versus state parameter α or noisy parameter p. The solid curves indicate the theoretical predictions, and the symbols are for the experimental results. Error bar indicates the statistical uncertainty which is obtained via the Monte Carlo simulation method.

Fig. 3. Experimental results for higher-dimensional bipartite systems. (a) Values of the entanglement witness Tr(Wijkρ). (b) Tr(W[abc]ρ) as functions of the parameter θ1 of the states in Eq. (13). The solid curve indicates the theoretical predictions, and symbols are for the experimental results. (c) Logarithmic negativities of the states EN(ρ) versus the state parameter θ1. Error bar indicates the statistical uncertainty which is obtained via the Monte Carlo simulation.

Fig. 4. Concurrence and negativity of the states versus the noisy parameter p of the Pauli channel. The blue line indicates the theoretical predictions obtained by the assumption of the perfect pure initial-state |Φ+⟩, and the red line indicates the theoretical predictions obtained by the assumption of the mixed initial-state ρw. Symbols are for the experimental results. Error bar indicates the statistical uncertainty, which is obtained via the Monte Carlo simulation method.

Fig. 5. (a) Entanglement witness value as a function of the state parameter α for the pure states and of parameter p of the quantum noisy channels for the mixed states. (b) Negativities and (c) concurrence versus state parameter α or noisy parameter p. The solid curves indicate the theoretical predictions, and the symbols are for the experimental results. Error bar indicates the statistical uncertainty, which is obtained via the Monte Carlo simulation method.

Fig. 6. Experimental setup for two-qutrit systems. The two-qutrit states are generated in the state preparation module, consisting of a set of SPDC entangled photon sources, two BDs, and two HWPs by employing the spatial and polarization modes of the photons. The local measurements and the state tomographic measurements are executed by the measurement modules, consisting of a sequence of HWPs, QWPs, a BD, and a PBS.