1. Introduction

Quantum networks consisting of multipartite entangled states shared between many nodes provide a setting for a wide variety of quantum computing and quantum communication tasks [1–6]. Recent works have experimentally realized some of the basic features of distributed quantum computation [7–9] and quantum communication schemes, including quantum secret sharing [10,11], open-destination teleportation [12] and multiparty quantum key distribution [13, 14]. These experiments employed networks of small-sized entangled resources and showed the potential of distributed quantum information processing in realistic scenarios [15]. Individual photons serve as a viable platform for implementation of quantum networks, since they can be easily transmitted over free-space or fiber links in order to distribute the necessary resources [4,16].

A common problem in quantum networks is that once the entangled resource is shared among the nodes it is fixed and can only be used for a given set of quantum tasks [5, 6, 15]. A different task then requires conversion of the available multipartite entangled state into another state. When the separation between the nodes is large, such conversion can employ only local operations and classical communication, which severely limits the class of potentially available states [17–20]. Fortunately, in some cases two nodes of the network may be close enough for application of a non-local operation between them. This relaxes the constraint and opens an interesting question: what types of entangled states are convertible in this scenario?

Recently, a non-local conversion gate was proposed for exactly this setting of two nodes in close proximity [21]. It was shown that one can employ a single probabilistic two-qubit gate to convert a four-qubit linear cluster state [22] into many other forms of four-qubit entangled states that are inequivalent to each other under local operations and classical communication. The gate therefore enables one to convert between different states so that different tasks can be performed. For instance, the four-qubit linear cluster state can be used for a variety of quantum protocols, such as blind quantum computation [7] and quantum algorithms [23–28]. On the other hand, a four qubit GHZ state [29] can be used for open-destination teleportation [12] and multiparty quantum key distribution [14], and a four-qubit Dicke state can be used for telecloning [30] and quantum secret sharing [31, 32]. In this work, we experimentally realize the non-local conversion gate suggested by Tashima et al. in [21] with single photons using a linear optical setup and characterize its performance using quantum process tomography. We find that the conversion gate operates with high quality under realistic conditions and show its potential for converting a four-qubit linear cluster state into a GHZ state, a Dicke state [30], and a product of two Bell states [1]. The conversion gate can also be used to generate quantum correlations that are not associated with entanglement, but whose presence is captured by the notion of discord [33]. The generated states with discord may also be used as resources in distributed quantum tasks [34–37]. Furthermore, the conversion gate can be used for ‘re-wiring’ the entanglement connections in a larger graph state network [22,38]. The experimental results match the theory expectations well and highlight the suitability of the conversion gate as a flexible component in photonic-based quantum networks.

2. Theoretical background

The non-local conversion gate for polarization encoded photonic qubits is depicted in Fig. 1. The gate operation is based on postselection where one photon is detected at each of the output ports. The gate itself is based on a Mach-Zehnder interferometer and it is created from two polarizing beam splitters (PBSs) and four half-wave plates (HWPs). Two of the HWPs labeled as HWP(45°) are rotated to a fixed angle of 45°, the other two HWPs labeled as HWP₁(θ₁) and HWP₂(θ₂) are used to adjust the gate to a particular setting. The total operator characterizing the action of the gate in the computational basis of horizontally (|H〉) and vertically (|V〉) polarized photons is given by

 

Fig. 1 The non-local conversion gate using linear optics. The gate takes two photons as inputs, one in each input mode, and performs a non-local operation when the photons exit from different output ports. PBS - polarizing beam splitter, and HWP - half-wave plate.

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The main feature of the non-local conversion gate is the ability to convert quantum states from one type to another, even though such a conversion is impossible with local operations and classical communication. This specific goal differentiates the gate from other established two-qubit gates, such as the control-NOT or control-phase gate. These more universal gates cannot realize the non-local conversion without additional ancillary modes and measurements, due to the non-unitary nature of the non-local conversion gate’s operation. To demonstrate the conversion gate’s capabilities consider a four-qubit linear cluster state |C₄〉 given by

Table 1. Non-local conversion gate angle settings, its success probabilities p_s, and the maximal theoretical success probabilities p_s,max for converting a linear cluster state to GHZ, Dicke and two Bell states. Each kind of conversion can be realized by several settings. The angle θ_± is found from the relation $\text{sin}(2{\theta }_{\pm })={\left[(5\pm \sqrt{5})/10\right]}^{1/2}$.

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The non-local nature of the gate can also be effectively utilized to generate classical or nonclassical correlations in a pair of initially separable states [21]. For example, the gate G(3π/8, π/8) transforms a pair of factorized pure states into a maximally entangled state, while the gate G(π/3, 0) transforms the state $\frac{1}{2}𝕀\otimes |+〉〈+|$, where $|+〉=\left(|H〉+|V〉\right)/\sqrt{2}$, into a state with quantum correlations but no entanglement.

3. Experimental setup

We experimentally demonstrated and characterized the two-qubit photonic non-local conversion gate using the linear optical setup shown in Fig. 2. Here, orthogonally polarized time-correlated photon pairs with central wavelength of 810 nm were generated in the process of degenerate spontaneous parametric down-conversion in a BBO crystal pumped by a continuous-wave laser diode [39, 40] and fed into single mode optical fibers guiding photons to the signal and idler input ports of the linear optical setup. The linearly polarized signal and idler photons were decoupled into free space and directed into polarization qubit state preparation blocks (dotted boxes), each consisting of a quarter-wave plate (QWP) and a half-wave plate (HWP). In contrast to the theoretical proposal by Tashima et al. in [21] shown in Fig. 1, the experimental conversion gate was implemented using a displaced Sagnac interferometer and a single polarizing beam splitter (PBS), where the interferometric phase was controlled by tilting one of the glass plates (GP). This construction provides passive stabilization of the Mach-Zehnder interferometer [41]. HWP₁(θ₁) and HWP₂(θ₂) were used to configure the conversion gate for its different settings. Outputs from the conversion gate were analyzed using the detection blocks (DB), which consist of a HWP, a QWP, and a PBS followed by an avalanche photodiode (APD). The scheme operated in the coincidence basis and the operation succeeded upon detecting a two-photon coincidence at the output ports.

 

Fig. 2 Experimental setup for the charaterization of the non-local conversion gate. QWP - quarter-wave plate, HWP - half-wave plate, PBS - polarizing beam splitter, GP - glass plate, DB - detection block, and APD - avalanche photodiode. The dotted boxes represent preparation stages for encoding different inputs. The dashed box represents the analysis stages (DB) for characterizing the output states of the gate.

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By measuring the coincidences we were able to carry out complete quantum process tomography [42] of the non-local conversion gate for all the settings in Table 1. Each input qubit was prepared in six states {|H〉, |V〉, |+〉, |−〉, |R〉, |L〉}, and each output qubit was measured in three bases {|H〉, |V〉}, {|+〉, |−〉} and {|R〉, |L〉}, where $|\pm 〉=\left(|H〉\pm |V〉\right)/\sqrt{2}$, $|R〉=\left(|H〉+i|V〉\right)/\sqrt{2}$ and $|L〉=\left(|H〉-i|V〉\right)/\sqrt{2}$. Two-photon coincidences corresponding to the measurement in any chosen product of two-qubit bases were recorded sequentially and the measurement time of each basis was set to 10 s. Using the measured coincidence counts as a mean value of the Poisson distribution from the down-conversion we numerically generated 1000 samples in order to estimate the uncertainty of the experimental results. The quantum process matrices χ were reconstructed from this data using a Maximum Likelihood estimation algorithm [42,43].

The process matrix χ can be used to fully characterize any operation. Based on the Jamiolkowski-Choi isomorphism [44, 45], the matrix χ corresponds to a density matrix of a bipartite maximally entangled quantum state with one of the states subjected to the operation in question. When the operation maps input Hilbert space ℋ_in into output Hilbert space ℋ_out, the matrix χ is defined on their tensor product. In the case of our two-qubit gate, both of these Hilbert spaces, which describe polarization states of two photons, are four-dimensional and the process matrix is a 16 × 16 matrix. An arbitrary quantum operation represented by a set of two-qubit Kraus operators {A_k} can be represented by the process matrix $\chi ={\sum }_k\left[\left({𝕀}_{\mathit{in}}\otimes A_{k,\mathit{out}}\right)|{\psi }^{+}〉〈{\psi }^{+}|\left({𝕀}_{\mathit{in}}\otimes A_{k,\mathit{out}}^{†}\right)\right]$, where $|{\psi }^{+}〉={\sum }_{a,b=H}^V{|ab〉}_{\mathit{in}}{|ab〉}_{\mathit{out}}$ is the unnormalized maximally entangled state. Physically, the χ matrix describes the maximally entangled quantum state with the quantum operation applied to the modes of the output Hilbert space only. The state can be then used for quantum teleportation, which transcribes the desired input state to the output modes, but also subjects the state to the quantum operation that was pre-coded onto the entangled state. The transformation of any input quantum state ρ_in into an output state ρ_out can be then obtained as

In our scenario, a possible source of reduction in the process fidelity is the introduction of phase shifts experienced by one or more modes in the setup, which are caused by the imperfect nature of the realistic experimental components. These can be compensated for by suitable phase corrections. To reflect this we calculated two kinds of process fidelity for each scenario.

The first is the raw fidelity, which was calculated directly from the reconstructed process matrix. The second is the optimized fidelity, which was calculated from the process matrix subject to four phase shifts, one in each of the two input and two output modes. The four phases were optimized over and ultimately chosen in such a way that the resulting fidelity is maximal. The relevant process purity and process fidelity for all the considered scenarios are given in Table 2, while the process matrices are shown in Fig. 3.

Table 2. Purity and process fidelity of the non-local conversion gate, including one standard deviation related to the last significant digit (represented by the number in brackets). The values were obtained by numerical simulation based on the obtained experimental results.

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Fig. 3 Reconstructed process matrices χ of the conversion gate using the Jamiolkowski-Choi representation. The 16 × 16 matrices are written in the polarization basis of the input and output Hilbert spaces ({|0〉, |1〉} ↔ {|H〉, |V〉}) and correspond to the gate operations given in Table 1. They are arranged in columns: (a) Cluster state, (b) GHZ state, (c) Dicke state, and (d) two Bell states. The theoretical process matrix χ_th is depicted in the top row of each column followed by the real and imaginary part of the reconstructed process matrix in the middle and bottom row, respectively. Note that the theoretical process matrix has only real values.

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We also analyzed the conversion gate and its performance from a different angle. The gate is non-local and as such it should be able to transform a two-qubit factorized state into a state with non-zero entanglement. To see this entanglement generation, the input state was set to | − −〉_in and fed into the conversion gate with parameters θ₁ = 3π/8 and θ₂ = π/8, which in the ideal case transforms it into the entangled Bell state $|{\mathrm{\Phi }}^{+}〉=\left(|HH〉+|VV〉\right)/\sqrt{2}$. Using the non-local conversion gate we generated the maximally entangled Bell state with purity P = 0.946(7) and fidelity F = 0.966(3). The number in the brackets represents one standard deviation at the final decimal place.

We can also use the conversion gate to prepare a separable state, i.e. a state with no entanglement, but with non-zero quantum correlations that can be measured by the discord [33]. For this we started with a mixed factorized state ${\rho }_{\mathit{in}}=\frac{1}{2}𝕀\otimes |+〉〈+|$ and fed it into the conversion gate with parameters θ₁ = π/3 and θ₂ = 0. The experimental realization was similar to the previous case, only the totally mixed state was prepared by using an electronically driven fiber polarization controller. The polarization controller applied mechanical stress on the input single mode optical fiber in three orthogonal axes using three co-prime frequencies. This randomized the polarization state on a time scale of tens of ms, which is two orders of magnitude shorter than the projection-acquisition time of 1 s, thus effectively resulting in a partially mixed state. This preparation method lead to an output state with zero entanglement and non-zero discord. The output state was again determined by using full two-mode quantum state tomography, followed by a maximum likelihood estimation algorithm. Separability of a realistic reconstructed state is difficult to prove, but both entanglement measures we employed - logarithmic negativity LN = 0.019(20) and concurrence C = 0.015(15) [1] - show values separated from zero by less than one standard deviation. This points to a high probability that the state is indeed separable. On the other hand, the discord of the state is D = 0.066(7) [33], which is significantly positive. The confidence intervals were obtained by using a Monte Carlo method based on the measured data.

As the final step of our analysis we looked at how the conversion gate might perform in a realistic scenario. For this, we employed the reconstructed process matrices from Fig. 3 and numerically simulated the effect of the two-qubit conversion gate on a realistic version of a four-qubit linear cluster state generated in a four-qubit linear-optical quantum logic circuit [46]. The cluster state |C₄〈 〉C₄|, whose density matrix was reconstructed with the help of a maximum likelihood algorithm, is shown in Fig. 4. The state fidelity, given by $F=\text{Tr}{\left[{\left[\sqrt{\rho \mathit{th}}\rho \sqrt{\rho \mathit{th}}\right]}^{1/2}\right]}^2$, where ρ_th is the ideal theoretical state and ρ is the experimental state, gives a value of F = 0.915.

 

Fig. 4 Density matrix of the linear cluster state. (a) Ideal theoretical matrix ρ_th = |C₄〉 〈C₄|, (b) the real part of the experimental matrix ρ, (c) the imaginary part of the experimental matrix ρ. Note that the ideal theoretical density matrix has only real values.

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The simulated output state fidelities for the different conversions are given in Table 3. In the second column, the operation fidelity measures the overlap between the realistic state transformed by the experimental conversion gate and the ideal theoretical conversion gate. In the third column, the total fidelity measures the overall fidelity between the realistic state transformed by the experimental conversion gate and the ideal state transformed by the ideal theoretical conversion gate. The values in Table 3 show the various limits of the gate derived from current experimental technology.

Table 3. Numerically simulated fidelity of output states converted from a realistic |C₄〉 〈C₄| state.

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4. Summary and discussion

We have experimentally realized the non-local photonic conversion gate proposed by Tashima et al. in [21] and tested its performance for all four of its basic conversion settings. We performed quantum process tomography and characterized the individual conversion gate operations by their process matrices. We also directly tested the ability of the conversion gate to generate quantum correlations without entanglement between a pair of separable qubits, finding no entanglement, but non-zero discord present. Finally, we tested the limits of the setup by simulating its action on a realistic experimental four-qubit linear cluster state. In all the tests the conversion gate performed close to the theoretical predictions, with fidelities generally surpassing 90%. These experimental results are very promising with regards to the potential future applications of the conversion gate, which includes converting between different small-sized multipartite entangled states. This is an important problem in quantum networks, where different quantum protocols require different resource states and the conversion gate can be used to prepare them. The conversion gate can also be used to rewire the entanglement connections in larger multipartite entangled states in the form of extended graph states [21] and so it would provide a useful way to reconfigure a network for a given distributed quantum protocol.