1. INTRODUCTION

Quantum coherence, which exhibits the fundamental signature of superposition in quantum mechanics, has been exploited in many fields of quantum physics, such as biological systems [1–3], transport theory [4,5], thermodynamics [6–12], nanoscale physics [13], and other scientific work associated with quantum theory [14–21]. Recently, a rigorous and unambiguous framework for quantifying quantum coherence has also been put forward, which has enhanced the exploitation of its operational significance in the context of quantum resource theory [22–25].

Quantifying quantum coherence starts from the definition of incoherent states (free states) and incoherent operations (free operations) [22,23,25,26]. A quantum state $\rho$ is incoherent if it is diagonal in a given reference basis $\{|i⟩\}$, i.e., $\rho =\sum _i{p}_i|i⟩⟨i|$, with some probability $\{p_i\}$ [23,27]. Incoherent operators are required to fulfill ${\widehat{K}}_n\mathcal{I}{\widehat{K}}_n^{†}\subset \mathcal{I}$ for all $n$ represented by the set of Kraus operators $\{{\widehat{K}}_n\}$, where $\mathcal{I}$ is the set of incoherent states. Moreover, in a $d$-dimensional Hilbert space $\mathcal{H}$, the maximally coherent state is $|{\mathrm{\varphi }}_d⟩=\sqrt{1/d}\sum _i|i⟩$, and $|\mathrm{\varphi }⟩≔|{\mathrm{\varphi }}_2⟩$ denotes the unit coherence resource state [22].

As a quantification and measure of quantum superposition in a fixed basis $\{|i⟩\}$, various coherence measures have been proposed [22,24,27,28]. In this paper, we choose relative entropy of coherence [22] as the measure of this property. Relative entropy of coherence of a quantum state $\rho$ is given by $C_r(\rho )=S(\mathrm{\Delta }(\rho ))-S(\rho )$, where $\mathrm{\Delta }(\rho )={\mathrm{\Sigma }}_i|i⟩⟨i|\rho |i⟩⟨i|$ is the dephasing in the reference basis. In many recent works, this kind of measure has been endowed with special significance as well as operational meaning [22–24]. Winter and Yang [23] showed that asymptotically the standard distillable coherence of a general quantum state is given by the relative entropy of quantum coherence. Likewise, Yuan et al. [24] has shown that the intrinsic randomness as a measure of coherence is just equal to the coherence of formation.

Recently, the manipulation and conversion of quantum coherence in bipartite or multipartite systems has been a hot topic [29–31]. Ma et al. [30] studied the interconversion of coherence and correlation between two parties under incoherent operations. The manipulation of coherence in a bipartite system was discussed by Chitambar et al. [31] and Streltsov et al. [32] when performing assisted distillation of quantum coherence. The task is considered with protocols where Alice can perform arbitrary operations and Bob is restricted to only incoherent operations while classical communication between Alice and Bob is allowed, which is referred as local quantum-incoherent operations and classical communication (LQICC). With this LQICC protocol, coherence can not be generated on Bob’s subsystem without classical communication from Alice, thus the theoretical framework forms a new kind of resource theory [26] in which the distillable coherence on one subsystem is demanded and manipulated by the set of free operations LQICC. Also, the key conclusion is that for pure bipartite states ${\rho }^{AB}$, the maximal distillable coherence after assisted distillation on Bob’s subsystem is equal to the von Neumann entropy of the reduced density matrix of Bob after it is dephased in the incoherent basis, i.e., $S(\mathrm{\Delta }({\rho }^B))$.

Quantum coherence is a highly demanded resource in many physical systems, especially in multipartite systems; however, to date there is no relevant experimental implementation of assisted distillation of quantum coherence. To fill this gap, we experimentally implement a class of such LQICC protocols based on a linear optic system in our laboratory, for extracting the maximal distillable coherence on one subsystem in our single-shot photonic scheme. Our results show that even in the presence of experimental imperfections and limitations, a maximal increase in distillable coherence can be observed even in the single-copy scenario for pure states. Furthermore, we also reported the experimental study with a specific class of mixed states, which showed that distillable coherence can be increased with less demand than entanglement distillation.

2. THEORY

A. Resource Theory of Assisted Distillation of Quantum Coherence

In the context of quantum resource theory, the optimal rate of generating resources (distillation rate) from a quantum state is always bounded as the manipulation and transformation are under certain restrictions [26]. For instance, distillable coherence of a general quantum state $\rho$ under incoherent operations is equal to the relative entropy of coherence [23].

The framework considered in the present task involves a bipartite system (denoted as Alice and Bob) in which coherence on Bob’s side is viewed as a resource, and the set of free operations is restricted to LQICC. Under this constraint, a resource can never be generated from any quantum-incoherent state [31,32] (free states), which has the form of ${\mathrm{\Sigma }}_i{p}_i{\sigma }_i^A\otimes |i⟩{⟨i|}^B$, where $\{{|i⟩}^B\}$ is the incoherent basis with respect to Bob. Our goal is to maximize distillable coherence on Bob’s side with the assistance of Alice under such constraint. To quantify the optimal rate of preparing $|\mathrm{\varphi }⟩$ that can be possibly obtained, distillable coherence of collaboration [31,32] is defined as

Similarly, the bound of distillable coherence of collaboration $C_d^{A|B}(\rho )$ is given by the quantum-incoherent relative entropy [31,32], i.e.,

However, for a general bipartite resource state ${\rho }^{AB}$, whether the inequality in Eq. (2) can be an equality is still unknown. As shown in the relevant theoretical work, at least for pure states, the equal sign is affirmative [31,32].

B. Linking One-Copy Scenario to Asymptotic Settings

The task elaborated in the context of resource theory challenges one greatly even for a pure resource state as the optimal rate may only be obtained by working in the asymptotic settings, which involves complicated collective measurements on many copies of the shared resource states and hinders its operational significance as well as its physical implementation. Thus, linking the one-copy scenario to asymptotic settings is essential for experimentally obtaining the maximal increase in coherence.

Suppose that Alice and Bob share a pure resource state ${|\mathrm{\Psi }⟩}^{AB}$, through local measurement $M_A$ on Alice together with broadcast of the measurement outcomes, any possible pure decomposition of ${\rho }^B$ can be realized, i.e., ${\rho }^B={\mathrm{\Sigma }}_i{p}_i|{\mathrm{\Psi }}_i⟩⟨{\mathrm{\Psi }}_i|$ for any set of $\{p_i\}$ and pure state $\{|{\mathrm{\Psi }}_i⟩\}$. It is obvious that Alice can help Bob get an average distillable coherence of $C_d^A({\rho }^B)={\mathrm{\Sigma }}_i{p}_i{C}_r(|{\mathrm{\Psi }}_i⟩)$ in a one-copy scenario (Note that $C_d(\rho )=C_r(\rho )$), which is beyond the original distillable coherence $C_d({\rho }^B)$ on Bob’s side, as the relative entropy of coherence never increases under the mixing of quantum states [22].

To quantify the best performance that Alice can achieve to help Bob get the maximal distillable coherence when she is restricted to just local measurement on her system and classical broadcast processing, we use the coherence of assistance (COA) [31] defined as

3. EXPERIMENTAL SETUP

The whole experimental setup is shown in Fig. 1 (details are provided in the Methods section below). It involves two modules–a state preparation module and an LQICC module. In the state preparation module, the pure entangled state ${|\mathrm{\Psi }⟩}^{AB}=\cos 2\theta |HH⟩+\sin 2\theta |VV⟩$ can be generated with an arbitrary $\theta$ ranging from 0° to 45°. The optical arrangement in Bob’s path (see the state preparation module in Fig. 1) is used to generate Werner states as ${\rho }^{AB}=p|{\mathrm{\Psi }}^{-}⟩⟨{\mathrm{\Psi }}^{-}|+(1-p)\frac{I}{4}$ with an arbitrary $p$. The LQICC module includes an assisted distillation part and a tomography [33] part. In the assisted distillation part, we implement the optimal local measurement on every copy of Alice’s state. On Bob’s side, however, the operation is restricted to incoherent operations. Finally, tomography is used to evaluate the final distillable coherence of Bob’s state. For pure states, the distillable coherence of collaboration $C_d^{A|B}({|\mathrm{\Psi }⟩}^{AB})$ can be evaluated directly by final distillable coherence of Bob. For Werner states, we replaced $C_d^{A|B}({\rho }^{AB})$ with $C_d({\rho }_1^B)$ because we didn’t know whether the distillable coherence of collaboration could be reached in single-shot experiments.

 

Fig. 1. Experimental setup has two modules: state preparation and LQICC. In the state preparation module, two classes of pure states ${|\mathrm{\Psi }⟩}^{AB}=\cos 2\theta |HH⟩+\sin 2\theta |VV⟩$ and ${|\mathrm{\Psi }⟩}^{AB}=\frac{1}{\sqrt{2}}(\cos 2\theta |HH⟩+\cos 2\theta |HV⟩+\sin 2\theta |VH⟩-\sin 2\theta |VV⟩)$, and arbitrary Werner states in the form ${\rho }^{AB}=p|{\mathrm{\Psi }}^{-}⟩⟨{\mathrm{\Psi }}^{-}|+(1-p)\frac{I}{4}$ can be generated. After preparation of the desired two-photon resource states, the two photons are distributed to Alice and Bob. In the LQICC module, optimal assisted operations are then performed by Alice on her photon and the measurement results are sent to Bob via a classical communication channel. According to the classical message from Alice, desired corresponding incoherent operations are performed on Bob’s photon. After the assisted distillation protocol, quantum state tomography is used to characterize the final state of Bob’s photon to identify the final distillable coherence. Key to components: HWP, half-wave plate; QWP, quarter-wave plate; BS, beam splitter; IF, interference filter; SPD, single photon detector; and PBS, polarizing beam splitter.

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4. EXPERIMENTAL RESULTS

In the experimental test, we divided the procedure into two parts, respectively, testing pure and mixed states in an assisted distillation process.

In the first part of the experiments, we generated two classes of pure states.

The first class of pure states was prepared as ${|\mathrm{\Psi }⟩}^{AB}=\cos 2\theta |HH⟩+\sin 2\theta |VV⟩$, the rotation angle $\theta$ of HWP1 was set from 0° to 45°. Without assisted distillation, the state of Bob’s photon was ${\rho }^B={\cos }^{2}2\theta |H⟩⟨H|+{\sin }^{2}2\theta |V⟩⟨V|$, whose distillable coherence was theoretically evaluated as $C_d({\rho }^B)=0$. In the experiment, tomography was used for further calculation of distillable coherence on the state of Bob’s photon [see red upward-pointing triangles in Fig. 2(a)]. In the assisted distillation protocol, Alice’s optimal operation was to perform von Neumann measurements in the basis ${|{\eta }_{\pm }⟩}^A$, which only need to be mutually unbiased with $|H⟩$ and $|V⟩$ for the first class of pure states (details in the Methods section below). Here our experiment chose ${|{\eta }_{\pm }⟩}^A={|y_{\pm }⟩}^A=\frac{1}{\sqrt{2}}(|H⟩\pm i|V⟩)$. When Alice measured $|y_{+}⟩$, the state of Bob’s photon collapsed to $\cos 2\theta |H⟩-i\sin 2\theta |V⟩$ with a distillable coherence of $S(\mathrm{\Delta }({\rho }^B))=-{\cos }^{2}2\theta {\log }_2{\cos }^{2}2\theta -{\sin }^{2}2\theta {\log }_2{\sin }^{2}2\theta$. When $|y_{-}⟩$ was measured by Alice, the state of Bob’s photon collapsed to $\cos 2\theta |H⟩+i\sin 2\theta |V⟩$ with the same amount of distillable coherence $S(\mathrm{\Delta }({\rho }^B))$. After receiving Alice’s measurement result (classical information) via classical communication channel, we just need to divide Bob’s photons into two ensembles for future applications according to Alice’s two measurement results. Since the photon in either ensemble had a distillable coherence of $S(\mathrm{\Delta }({\rho }^B))$, the final distillable coherence on Bob’s side after assisted distillation is $S(\mathrm{\Delta }({\rho }^B))$ per photon. The distillable coherence of collaboration was evaluated using tomography as the final distillable coherence on Bob’s side and denoted as $C_d^{A|B}({|\mathrm{\Psi }⟩}^{AB})$ according to the two measurement results [see pink downward-pointing triangles in Fig. 2(a)]. The increase in distillable coherence can be obtained from $\delta C_d({\rho }^B)=C_d^{A|B}({|\mathrm{\Psi }⟩}^{AB})-C_d({\rho }^B)$ and equals $S({\rho }^B)$ theoretically.

 

Fig. 2. Experimental results for the pure states. As shown in Fig. 1, $C_d({\rho }^B)$ (red upward-pointing triangles) represent the distillable coherence of Bob without distillation, $C_d^{A|B}({|\mathrm{\Psi }⟩}^{AB})$ (pink downward-pointing triangles) represent the distillable coherence of Bob after assisted distillation, $\delta C_d({\rho }^B)$ (blue squares) represent an increase in coherence, and black dashed-dotted lines represent the theoretical curve. In Fig. 1, the resource state has the form of ${|\mathrm{\Psi }⟩}^{AB}=\cos 2\theta |HH⟩+\sin 2\theta |VV⟩$ and in Fig. 3 below, ${|\mathrm{\Psi }⟩}^{AB}=\frac{1}{\sqrt{2}}(\cos 2\theta |HH⟩+\cos 2\theta |HV⟩+\sin 2\theta |VH⟩-\sin 2\theta |VV⟩)$.

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The second class of pure states for assisted distillation was prepared as ${|\mathrm{\Psi }⟩}^{AB}=\frac{1}{\sqrt{2}}(\cos 2\theta |HH⟩+\cos 2\theta |HV⟩+\sin 2\theta |VH⟩-\sin 2\theta |VV⟩)$. Similarly, without assisted distillation, the state of Bob’s photon is ${\rho }^B={\cos }^{2}2\theta |x_{+}⟩⟨x_{+}|+{\sin }^{2}2\theta |x_{-}⟩⟨x_{-}|$ with $|x_{\pm }⟩=\frac{1}{\sqrt{2}}(|H⟩\pm |V⟩)$, which had a distillable coherence of $C_d({\rho }^B)=\frac{1}{2}[(1+\cos 4\theta ){\log }_2(1+\cos 4\theta )+(1-\cos 4\theta ){\log }_2(1-\cos 4\theta )]$ [black dash-dotted line in Fig. 2(b)]. The experimental results [red upward-pointing triangles in Fig. 2(b)] were also obtained from quantum state tomography and agreed well with the theoretical predictions. In the assisted protocol, Alice’s optimal operation was to perform von Neumann measurements mutually unbiased with $\cos 2\theta |H⟩+\sin 2\theta |V⟩$ and $\cos 2\theta |H⟩-\sin 2\theta |V⟩$ for the second class of pure states, and we also chose ${|{\eta }_{\pm }⟩}^A={|y_{\pm }⟩}^A$. The state of Bob’s photon collapsed to $\frac{1}{\sqrt{2}}(\cos 2\theta \mp i\sin 2\theta )|H⟩+\frac{1}{\sqrt{2}}(\cos 2\theta \pm i\sin 2\theta )|V⟩$ according to Alice’s two measurement results $|y_{\pm }⟩$ and the distillable coherence of collaboration $C_d^{A|B}({|\mathrm{\Psi }⟩}^{AB})=1$ theoretically. We used quantum state tomography to obtain experimental results of $C_d^{A|B}({|\mathrm{\Psi }⟩}^{AB})$ (pink downward-pointing triangles) and $\delta C_d({\rho }^B)$ (blue squares).

In the second part of the experiments, the task of assisted distillation of coherence for mixed states is much more complicated because it is still an open problem that needs answers to two questions: Is it possible to achieve the upper bound of the inequality in Eq. (2) and what are the optimal LQICC operations. For theoretical and experimental simplicity, the optimal LQICC operation was considered for assisted distillation in one-copy instead of $n$-copy scenarios. In this part, Werner states were prepared as ${\rho }^{AB}=p|{\mathrm{\Psi }}^{-}⟩⟨{\mathrm{\Psi }}^{-}|+(1-p)\frac{I}{4}$ with $p$ ranging from 0.05 to 0.95 by the adjustment of apertures. We denoted ${\rho }_0^B$ and ${\rho }_1^B$ as the reduced density operator for Bob before and after the assistance of Alice. Without Alice’s assistance, Bob’s state was a maximally mixed state, i.e., ${\rho }_0^B=\frac{I}{2}$, whose distillable coherence was $C_d({\rho }_0^B)=0$ (see the upward-pointing triangles).

First we found out optimal von Neumann measurements in the single-shot experiment of assisted coherence distillation, which turned out to be a maximally coherent basis $|{\eta }_{\pm }⟩=\frac{1}{\sqrt{2}}(|H⟩\pm e^{i\phi }|V⟩)$ with arbitrary $\phi$ for Werner states. For simplicity we chose $|y\pm ⟩$. After measurements along $|y\pm ⟩$ were performed on Alice’s photons, Bob’s state collapsed to ${\rho }_1^B=p|y_{\mp }⟩⟨y_{\mp }|+(1-p)\frac{I}{2}$ according to Alice’s two measurement results, respectively. Then Bob’s distillable coherence after Alice’s assistance theoretically was $C_d({\rho }_1^B)=\frac{1}{2}[(1+p){\log }_2(1+p)+(1-p){\log }_2(1-p)]$ (black dash-dotted line in Fig. 3). The experimental results for $C_d({\rho }_1^B)$, denoted as pink downward-pointing triangles, are shown in Fig. 3, which agreed well with the theoretical curve. The increase in distillable coherence $\delta C_d({\rho }^B)$ can be easily obtained from $\delta C_d({\rho }^B)=C_d({\rho }_1^B)-C_d({\rho }_0^B)$ (relevant experimental results denoted as blue squares in Fig. 3). We also calculated the upper bound QI relative entropy in Eq. (2) as $C_r^{A|B}({\rho }^{AB})=\frac{1}{4}[(1-p){\log }_2(1-p)+(1+3p){\log }_2(1+3p)-2(1+p){\log }_2(1+p)]$ and the theoretical curve is shown in Fig. 3 as a brown dash-dotted line.

 

Fig. 3. Experimental results for Werner states [the colored symbols have the same meaning as shown in Fig. 2, except the brown dash-dotted line represents the theoretical calculated upper bound in inequality in Eq. (2)]. The resource state has the form ${\rho }^{AB}=p|{\mathrm{\Psi }}^{-}⟩⟨{\mathrm{\Psi }}^{-}|+(1-p)\frac{I}{4}$.

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As shown in Figs. 2(a) and 2(b), for the two classes of pure states, both experimental results of $C_d^{A|B}({|\mathrm{\Psi }⟩}^{AB})$ and $\delta C_d({\rho }^B)$ are slightly smaller than the theoretical prediction around $\theta =22.5°$ due to difficulty in preparing highly entangled states.

For Werner state, experimental results in Fig. 3 show that the increased coherence on Bob’s side in single-shot experiments agreed well with the numerical simulation and was close to the upper bound $C_r^{A|B}({\rho }^{AB})$. However, as it is obvious that $C_d^{A|B}({\rho }^{AB})\ge C_d({\rho }^B)$, whether the upper bound can be reached in the asymptotic limit is still unknown. What’s interesting is that the distillable coherence on Bob’s side after assisted distillation is more than zero as long as $p>0$, i.e., $C_d^{A|B}({\rho }^{AB})\ge C_d({\rho }^B)>0$ if and only if $p>0$, which is much less demanding than the entanglement requirement with $p>1/3$ [34]. Thus the distillable coherence of collaboration $C_d^{A|B}({\rho }^{AB})$ is nonzero if and only if the state ${\rho }^{AB}$ is not quantum-incoherent in the reference basis. Hence, the coherence on Bob’s side can still be increased with Alice’s assistance even if the shared resource state ${\rho }^{AB}$ is not entangled.

5. CONCLUSION

We have reported what we believe is the first experimental study of assisted distillation of quantum coherence for both pure and mixed two-qubit states. In experiments for pure states, optimal assisted coherence distillation is achieved and the increased distillable coherence on Bob’s subsystem is equal to the von Neumann entropy of the reduced density matrix of Bob after it is dephased in the incoherent basis, i.e., $S(\mathrm{\Delta }({\rho }^B))$. For mixed states, optimal assisted coherence distillation is considered in a one-copy scenario and the maximal distillable coherence on Bob’s side after assistance is close to the upper bound $C_r^{A|B}({\rho }^{AB})$.

The task of assisted distillation of quantum coherence can be applied in many scenarios. Quantum coherence is a significant property of superposition that is highly demanded in quantum information processing and quantum computation, and one needs to distill as many copies of a pure maximally coherent state as possible for better performance in these tasks. However, when considering a system where coherence is demanded, and is remote or inaccessible, the theoretical framework gives an optimal solution to this problem. As the results of our first experiment showed, one can extract maximal coherence from a bipartite resource state in an assisted distillation process based on linear optical systems. Thus, the procedure can be physically implemented in most related tasks of the quantum information process with sufficiently high fidelity. Furthermore, the test for Werner states linked the quantum coherence to quantum correlation, which is beyond entanglement, i.e., for some special form of mixed states, one can always extract distillable coherence on one subsystem as long as correlation exists. This result sheds light on the strong relation between quantum coherence and nonclassical correlation.

6. METHODS

A. Selection of Measurement Basis

In this section, we will show how to select the optimal measurement basis when the shared state is a pure two-qubit state ${\rho }^{AB}$.

According to theoretical analysis [31], if we expand the purification ${|\mathrm{\Psi }⟩}^{AB}$ in the incoherent basis,

Hence we can use the theoretical analysis to maximize the distillable coherence on Bob’s system. In the case where the shared state is a pure two-qubit state, when we perform von Neumann measurement on Alice in the mutually unbiased basis, we can obtain the maximal increase in distillable coherence in Bob’s system. In our experiments, when we expand the two pure states in incoherent basis, we obtain the sets of states ${|{\mathrm{\Psi }}_k⟩}^A$ are $|H⟩$, $|V⟩$ for state ${|\mathrm{\Psi }⟩}^{AB}=\cos 2\theta |HH⟩+\sin 2\theta |VV⟩$ and $\cos 2\theta |H⟩\pm \sin 2\theta |V⟩$ for state ${|\mathrm{\Psi }⟩}^{AB}=\frac{1}{\sqrt{2}}(\cos 2\theta |HH⟩+\cos 2\theta |HV⟩+\sin 2\theta |VH⟩-\sin 2\theta |VV⟩)$. Because there obviously exists a shared mutually unbiased basis $|y\pm ⟩$ with respect to both cases, we can apply von Neumann measurement on Alice along the $|y\pm ⟩$ basis in the first part of our experiments.

In the experiments for Werner states, we considered the task in a one-copy scenario because the complex collective measurement on many copies of a state is hard to implement because of experimental limitations. Because the two-qubit Werner states have a form of ${\rho }^{AB}=p|{\mathrm{\Psi }}^{-}⟩⟨{\mathrm{\Psi }}^{-}|+(1-p)\frac{I}{4}$ with high symmetry, when Alice measures her system in the basis $|{\eta }_{\pm }⟩=\frac{1}{\sqrt{2}}(|H⟩\pm e^{i\phi }|V⟩)$ with arbitrary $\phi$, Bob’s state will collapse to ${\rho }_1^B=p|{\eta }_{\mp }⟩⟨{\eta }_{\mp }|+(1-p)\frac{I}{2}$, and we chose the same $|y_{\pm }⟩$ as that for pure states.

B. Detail Experimental Apparatus

We now move to the detailed experimental arrangement in our laboratory based on the polarization-entangled photon pairs.

In state preparation module in Fig. 1, two type I phase-matched $\beta$-barium borate (BBO) crystals, whose optic axes are normal to each other [35], are pumped by the continuous ${\mathrm{Ar}}^{+}$ laser at 351.1 nm with power up to 60 mW for the generation of photon pairs with a central wavelength at $\lambda =702.2\mathrm{nm}$ via spontaneous parametric down conversion process (SPDC). HWP1 working at $\lambda =351.1\mathrm{nm}$ is set in front of the BBO crystals to control the quantum state of the generated photon pairs encoded in polarization. Half-wave plates at both ends of the two single-mode fibers (SMF) are used to control polarization. In one branch, QWP3 is tilted to compensate the phase of the two-photon state for the generation of required states. This setup is capable of preparing two classes of pure states ${|\mathrm{\Psi }⟩}^{AB}=\cos 2\theta |HH⟩+\sin 2\theta |VV⟩$ and ${|\mathrm{\Psi }⟩}^{AB}=\frac{1}{\sqrt{2}}(\cos 2\theta |HH⟩+\cos 2\theta |HV⟩+\sin 2\theta |VH⟩-\sin 2\theta |VV⟩)$, where $\theta$ is the rotation angle of HWP1.

As for the generation of Werner states, two 50/50 beam splitters (BS) are inserted into one branch. In the transmission path, the two-photon state is prepared as the singlet state $|{\mathrm{\Psi }}^{-}⟩=\frac{1}{\sqrt{2}}(|HV⟩-|VH⟩)$ when the rotation angle of HWP1 is set as $\theta =22.5°$. In the reflected path, three $446\lambda$ quartz crystals and a half-wave plate with 22.5° are used to dephase the two-photon state into a completely mixed state $\frac{I}{4}$. The ratio of the two states mixed at the output port of the second BS can be changed by the two adjustable apertures for the generation of arbitrary Werner states with ${\rho }^{AB}=p|{\mathrm{\Psi }}^{-}⟩⟨{\mathrm{\Psi }}^{-}|+(1-p)\frac{I}{4}$. Out of the state preparation module, the two photons are distributed to Alice and Bob, as shown in Fig. 1.

In the LQICC module, on Alice’s side, QWP1, HWP3, PBS, and two single photon detectors are used to perform arbitrary assisted projective measurements and the measurement results are sent to Bob via a classical communication channel. In the next stage, QWP2, HWP4, PBS, and two single photon detectors are used to perform tomography on Bob’s photon. Actually, we used a beam displacer (BD) together with a 45° holophote, which can act as a PBS and has a rather high extinction ratio.