Abstract
Taking advantage of quantum mechanics for executing computational tasks faster than classical computers^{1} or performing measurements with precision exceeding the classical limit^{2,3} requires the generation of specific large and complex quantum states. In this context, cluster states^{4} are particularly interesting because they can enable the realization of universal quantum computers by means of a ‘oneway’ scheme^{5}, where processing is performed through measurements^{6}. The generation of cluster states based on subsystems that have more than two dimensions, dlevel cluster states, provides increased quantum resources while keeping the number of parties constant^{7}, and also enables novel algorithms^{8}. Here, we experimentally realize, characterize and test the noise sensitivity of threelevel, fourpartite cluster states formed by two photons in the time^{9} and frequency^{10} domain, confirming genuine multipartite entanglement with higher noise robustness compared to conventional twolevel cluster states^{6,11,12,13}. We perform proofofconcept highdimensional oneway quantum operations, where the cluster states are transformed into orthogonal, maximally entangled dlevel twopartite states by means of projection measurements. Our scalable approach is based on integrated photonic chips^{9,10} and optical fibre communication components, thus achieving new and deterministic functionalities.
Main
Cluster states are a particularly important class of multipartite states (that is, those formed by more than two parties, such as multiple atoms, photons and so on) characterized by two unique properties: maximal connectedness^{4} (that is, any two parties of the state can be projected into a maximally entangled state through measurements on the remaining parties), as well as by the highest persistency of entanglement^{4} (that is, cluster states require a maximal number of projection measurements to fully destroy entanglement in the system). These properties make cluster states equivalent to universal oneway (also called measurementbased) quantum computers^{5}, where different algorithms can be implemented by performing measurements on the individual parties of the cluster states^{5,6}. This approach greatly simplifies quantum processing, since measurement settings can be usually implemented more easily than the gate operations required in other approaches^{1,14}. Furthermore, cluster states have structural properties that protect the quantum information; that is, they enable topological quantum error correction^{11} for minimizing computation errors. Due to the significant importance of cluster states, they have been studied in many different platforms. In particular, experimental realizations can be separated into two classes: discrete twolevel^{6,11,12,13} (that is, qubit) and continuousvariable^{15,16} cluster states. Continuousvariable systems are intrinsically highdimensional, and they can be achieved using squeezed states^{15,16}. However, the quantum resource of these states relies on the level of squeezing, which is very sensitive to noise^{17}. In contrast, discrete quantum states are less sensitive to noise than squeezed states^{17}, and even more importantly, their individual modes can be fully accessed and individually manipulated, making their use especially appealing.
Nevertheless, increasing the number of particles to boost the computational resource comes at the price of significantly reduced coherence time and detection rates, and increased sensitivity to noise, restricting the realization of discrete cluster states to a record of eight qubits^{11}. In a novel approach, the use of discrete yet dlevel (that is, qudit) entangled states has the potential to address several limitations of qubit cluster states. First, the quantum resource can be increased without changing the number of particles^{7}; second, dlevel quantum states enable the implementation of highly efficient computation protocols^{8}; and third, higher dimensions reduce the noise sensitivity of cluster states. The experimental realization of dlevel multipartite cluster states is the key missing piece required to exploit these important benefits for highdimensional quantum computation^{7}. Unfortunately, today’s established quantum systems are illsuited for increasing the dimensionality of discrete multipartite entangled states. For example, atomic^{18} and superconducting^{19} systems are mainly based on qubit schemes, and demonstrated highdimensional photonic systems^{20,21} become experimentally complex and inefficient and typically suffer from degraded state purity when the number of photons used increases. The only dlevel quantum states with more than two parties generated so far are threelevel, threephoton states in a bulk, freespace setup^{22}. Cluster states with two and three parties are locally equivalent to Bell and Greenberger–Horne–Zeilinger (GHZ) states, respectively^{4}, while genuine discrete dlevel cluster states require at least four parties^{4}. To date, these states have not been realized, nor have their entanglement properties and their tolerance to noise been investigated.
Here, we present a general approach to prepare and coherently manipulate discrete dlevel multipartite quantum systems based on the simultaneous entanglement—that is, hyperentanglement^{23}—of two photons in time and frequency, by exploiting integrated photonic chips combined with fibreoptics telecommunications components^{9,10}. Using this method, we present the first experimental realization and characterization of qudit cluster states as well as the first hyperentangled state employing only a single degree of freedom. Further, we use these states to perform proofofconcept highdimensional oneway quantum processing.
As a basis for the generation of dlevel cluster states, we create hyperentangled states that support higher dimensions. Hyperentangled photon states usually employ simultaneously multiple different and independent degrees of freedom, which can be described as distinct parties of the state. These states can therefore be treated as multipartite although the number of physical photons is not increased (see Methods). Until now, such states have been realized only using combinations of fully independent degrees of freedom, described by commuting operators, such as polarization, optical paths and temporal modes^{23,24,25,26}. Here we used intrinsically linked and noncommuting observables, two discrete forms of energytime entanglement, namely timebin^{9} and frequencybin^{10}. Timebin entanglement refers to states where the photons are in a superposition of welldefined and distinct temporal modes, while frequencybin entangled states are characterized by discrete and nonoverlapping frequency modes. Timebin entanglement can be generated by exciting a spontaneous parametric process in an optical nonlinear medium with multiple phaselocked pulses^{9} (see Fig. 1a). On the other hand, frequencybin entanglement can be realized when the nonlinear medium is placed within an optical resonator, where the emission bandwidth covers multiple resonances^{10} (see Fig. 1b). Remarkably, if the timefrequency product corresponding to the individual modes is well above the quantum limit (see Methods), frequencybin and timebin entanglement become independently controllable, allowing one to generate hyperentangled, multipartite states (see Methods). Such hyperentangled states can be produced by exciting the nonlinear element, placed inside the resonator, with a coherent set of multiple pulses (see Fig. 1c), as long as the pulse separation is much larger than the photon lifetime in the resonator. In this case, the timebin component can be fully controlled in the temporal domain, while the frequencybin component can be completely and independently controlled in the frequency domain.
In our experimental implementation, we produced photon pairs using the nonlinear process of spontaneous fourwave mixing within a microring resonator (see Methods). By exciting the resonator with three phaselocked pulses and considering three frequency mode pairs, we generated photon states simultaneously entangled in time and frequency, described by the following expression (equation (1)):
where numbers indicate time bins and letters indicate frequency bins, with the indices s and i referring to the signal and idler photons, respectively (the normalization is not shown for compactness). This hyperentangled state is biseparable, since any projection measurement performed in, for example, the timebin basis, does not affect the frequencybin entangled substate (and vice versa). In contrast, a cluster state is characterized by the fact that a projection measurement of one party affects the remaining portion of the state. An ideal compact threelevel cluster state (which is locally equivalent to a linear cluster in a onedimensional lattice, as well as a box cluster in a twodimensional lattice, see Methods) can be obtained by judiciously modifying the phase terms in equation (1), which then reads (equation (2)):
To experimentally transform the hyperentangled state (equation (1)) into this cluster state (equation (2)), access to the individual terms of the state is necessary, while maintaining coherence. For multiparticle states, this is technically very challenging, requiring twoparty quantum gates, which are usually probabilistic^{27}. Since we are employing two different types of discrete energy–time entanglement associated with different timescales (that is, timebin and frequencybin), it is possible to fully map the entangled state into the time domain to perform coherent state manipulations using synchronized electrooptic modulation. The frequencytotime mapping (see Fig. 2) was performed by a fibre Bragg grating array placed in a selfreferenced and phasestable loop configuration (see Methods). By choosing the appropriate phase pattern, the biseparable hyperentangled state was transformed into a threelevel fourpartite cluster state.
To confirm genuine multipartite entanglement, we determined an optimal entanglement witness for the cluster state (see Methods). This witness detects the presence of a cluster state when its expectation value is negative (a minimum of −1 is reached by a theoretically optimal cluster state; that is, in the absence of imperfection or noise contributions). This witness can be reduced to a measurable witness \({\cal W}_{2{\mathrm{S}}}^{\left( {{\cal C}_{4,3}} \right)}\), containing two orthogonal measurement settings represented by the generalized three dimensional Pauli matrices X and Z (see Methods):
$$\begin{array}{l}{\left\langle{\cal W}_{2{\mathrm{S}}}^{\left( {{\cal C}_{4,3}} \right)}\right\rangle} = \frac{5}{3}  \frac{1}{3}{\mathrm{Re}}\left(\left\langle {{{\Bbb{I}}}_1{{\Bbb{I}}}_2Z_3Z_4^\dagger } \right\rangle + \left\langle Z_1^\dagger Z_2{{{\Bbb{I}}}}_3{{{\Bbb{I}}}}_4 \right\rangle + \left\langle {{{{\Bbb{I}}}}_1Z_2X_3X_4} \right\rangle + \left\langle {{\mathrm{X}}_1X_2Z_3{{{\Bbb{I}}}}_4} \right\rangle\right.\\ \left. + \left\langle {Z_1{{{\Bbb{I}}}}_2X_3X_4} \right\rangle + \left\langle {Z_1^\dagger Z_2^\dagger X_3X_4} \right\rangle + \left\langle {X_1X_2{{{\Bbb{I}}}}_3Z_4} \right\rangle + \left\langle {X_1X_2Z_3^\dagger Z_4^\dagger } \right\rangle\right)\end{array}$$We measured a witness expectation value of \(\left\langle{\cal W}_{2{\mathrm{S}}}^{\left( {{\cal C}_{4,3}} \right)}\right\rangle =  0.28 \pm 0.04\) (see Fig. 3a,b and Methods), confirming (within the range of 7 standard deviations) the presence of a cluster state exhibiting genuine threelevel fourpartite entanglement.
We then tested the impact of incoherent, phase and amplitude noise on the measured state with respect to the expectation value of the witness via Monte Carlo simulations (see Fig. 3c–e). We also calculated the threshold for which the states lose their cluster properties due to the impact of noise. We found that our dlevel cluster states are highly robust towards incoherent noise (see Methods). The prepared cluster state can tolerate up to 66.6% of incoherent noise with respect to the optimal witness and 37.5% for the measured witness. In addition, they can also endure high amounts of amplitude and phase noise within the state itself; that is, as much as 83% (55% for the measurement witness) average amplitude fluctuations for the involved state components, and up to 37% (25%) error in their phase terms (see Fig. 3c–e). Most remarkably, our finding show that dlevel cluster states are significantly more robust towards noise compared to twolevel states. In comparison, a fourqubit cluster state can be mixed only with up to 50% of incoherent noise when employing the optimal witness and 33% when using the measurement witness to successfully detect the presence of the state. In comparison, a sixqubit cluster state (having slightly lower computational power compared to the fourqutrit state demonstrated here) can tolerate only 50% for the optimal and 28.5% for the measurement witness (see Methods).
To confirm the potential of dlevel cluster states for quantum computation, we demonstrated that different and orthogonal entangled states can be generated by simply performing projection measurements on the parties of the cluster, which is the working principle of oneway quantum computers^{5,7}. We here show this shaping of the cluster states and transform them into different orthogonal bipartite states. For this, we carried out twopartite projections in either the frequency or time domain (see Fig. 4), and verified, via quantum interference measurements, that the resultant states are mutually orthogonal and entangled (see Methods). The orthogonality of these target states was confirmed by the relative phase shift in the respective quantum interference patterns (see Methods). Furthermore, all measured raw visibilities—listed in Supplementary Table 1—violated their respective twopartite Bell inequalities (see Methods). Therefore, these projection measurements performed on a cluster state represent highdimensional oneway quantum computing operations. Universal quantum computation will require the application of dlevel Hadamard gates to turn the state generated here into a linear or box cluster, necessary for algorithm implementation (see Methods). Such gates have already been achieved for timebin qubits^{28}, as well as frequencybin qubits and qutrits^{29}.
In conclusion, our work shows that integration is not, as it is often regarded, simply limited to miniaturizing devices and reducing cost (typically at the expense of lower performance), but can in fact enable novel and powerful capabilities. Furthermore, our approach has an important scaling potential and is advantageous compared to current methods based on twolevel cluster states, since it can provide a better noise tolerance, as well as a significant improvement in terms of an effective quantum resource rate (EQRR). In particular, multiphoton states that are generated by multiple spontaneous parametric processes are hampered by a decrease in state purity with increasing photon number^{30}. Moreover, in general, quantum systems suffer significant reduction in their coherence time or detection rate when the number of entangled particles becomes larger, ultimately limiting the number of physical photons to well below the number of parties required for meaningful quantum computation tasks. It is therefore important to employ large cluster states with a low photon number. As an example, the largest twolevel cluster states realized to date were comprised of six^{13} and eight^{11} qubits. These states had Hilbert space sizes (H_{N,d} = d^{N}) of H_{6,2} = 64 and H_{8,2} = 256, and were featured by moderate (yet impressive for current technology) six and eightphoton detection rates (DR) of DR_{6,2} = 12 mHz and DR_{8,2} = 0.89 mHz, resulting in effective quantum resource rates (EQRR_{H} = H × DR) of EQRR_{64} = 0.768 Hz and EQRR_{256} = 0.228 Hz, respectively. In the multiphoton cluster state approach, the detection rate diminishes more than the gain obtained through the increase in Hilbert space size, thus reducing the EQRR. In contrast, we achieved a Hilbert space size of H_{4,3} = 81 (corresponding to 6.34 qubits), yet at a much higher detection rate of DR_{4,3} = 1 Hz, resulting in an EQRR_{81} = 81 Hz. This corresponds to a one (three) hundredfold increase with respect to the six (eight)photon cluster state. In addition, it has recently been shown that multiphoton states can also carry hyperentanglement^{31}. While this GHZtype state^{31} is not practically useful for quantum computation, its realization suggests that highdimensional cluster states based on hyperentangled multiphoton states can become a reality (see Methods). These would enable quantum computation based on many dlevel parties, yet with a manageable number of photons, and therefore high EQRRs. In this system, the computational flow could be achieved in such a way that the hyperentangled parties are measured simultaneously, and feedforward^{32} is implemented between the different photons of the state (see Methods and Supplementary Fig. 2). Our work provides an important step towards achieving powerful and noisetolerant quantum computation in a scalable and massproducible platform.
Methods
Experimental setup
The full experimental setup including all components is shown in Supplementary Fig. 1. To generate phaselocked triple pulses, we used a carrierenvelope phase stabilized modelocked laser (Menlo Systems Inc.) operating at 250 MHz repetition rate (that is, 4 ns pulse separation) and locked the carrierenvelope phase frequency to 250/6 = 41.667 MHz. We then employed an electrooptic intensity modulator, driven by an arbitrary waveform generator (Tektronix), to temporally gate triple pulses that were separated by 24 ns (that is, taking each 6th pulse from the initial pulse train), where the set of pulses was repeated at a rate of 10 MHz (that is, every 100 ns). The triple pulses were then spectrally filtered, amplified and coupled into the microring resonator at the resonance wavelength of 1,555.93 nm. The microring resonator was fabricated from a highrefractiveindex glass^{10}, with a free spectral range of 200 GHz and a Q factor of 235,000^{33,34,35}. The input and output waveguides were featured by mode converters and were connected to polarizationmaintaining fibres, resulting in coupling losses of <1.6 dB per facet. The average pump power coupled through the chip was 2.4 mW, measured at the drop port. After the microring resonator, the excitation field was removed using a highisolation (150 dB) notch filter, and the entangled photons were coupled into the controlled phase gate (see Fig. 2). This allowed us to generate a hyperentangled state^{23,36} simultaneously exploiting the timebin^{37} and frequencybin^{38} approaches. The phasegate achieved frequencytotime mapping via a custommade fibre Bragg grating (FBG) array, consisting of six independent FBGs matched to the photon wavelengths (1,551.08, 1,552.70, 1,554.31, 1,557.55, 1,559.17 and 1,560.80 nm) and spatially separated in the fibre to achieve a 3.96 ns temporal delay between adjacent frequency modes (that is, to introduce the frequencytime mapping). The FBGs were written into a photosensitive and deuteriumloaded polarizationmaintaining fibre (Nufern) using a 213nmwavelength laser (fifth harmonic of a Nd:YBO laser, Xiton Photonics Inc.) by means of a continuous writing scheme, realized in a tunable Talbot interferometer with a moving fibre^{39}. The FBGs were apodized using a squared cosine function, implemented by means of moving the Talbot phase mask with a piezo actuator^{39}. The FBGs were then heated to 80 °C for 48 h to remove the deuterium, thus decreasing losses. They were subsequently spliced to standard polarizationmaintaining Panda fibres using a tapered splice to decrease modematching losses. To implement the optical phase modulation within the gate, we used an electrooptic phase modulator (EOSpace), which was driven by an arbitrary waveform generator (Tektronix), synchronized to the 10 MHz reference clock of the modelocked laser. After the phasegate, the photons were sent to a computercontrolled amplitude and phase filter with two output ports (Finisar Waveshaper). This filter was used to route the photons to different characterization setups. One output was connected to a stabilized fibre interferometer with 24 ns unbalance to perform temporal projection measurements^{9,40}, and the other output was connected to an electrooptic phase modulator, which mixes different frequency components to perform frequency projection measurements^{10,41,42}. Finally, the photons were separated using spectral programmable filters and sent to superconducting singlephoton detectors (Quantum Opus).
Hyperentanglement within energy–time entangled states
Hyperentangled states are quantum states that are entangled in two or more independent degrees of freedom^{23,26,36,43}, such as polarization, optical paths or orbital angular momentum. Since these degrees of freedom can be controlled autonomously, such systems can be best described as multipartite states, where each degree of freedom represents an independent party^{23,36}. Such a system could also be mathematically mapped into a highdimensional state representation; however, this requires a rather complex, nonintuitive description of the experimentally accessible operations. In particular, the underlying physical quantum system and the accessible operations determine which state representation should be used. For hyperentangled systems, a multipartite representation treats individual degrees of freedom as separate parties, where a photon is seen as a carrier of multiple parties. Past realizations of hyperentanglement made use of independent degrees of freedom, which require very different control mechanisms for each degree of freedom^{31}. In this work, we generated hyperentanglement between two energytime realizations (that is, a single degree of freedom). The temporal and frequency observables in energy–time entangled states are noncommuting and form Einstein–Podolsky–Rosen^{44} (EPR)type quantum correlations^{45}. In the chosen discrete energy–time entanglement realization, the timefrequency product is much larger than the EPR limit, thus allowing independent control described by commuting operators and ultimately hyperentanglement. In particular, the time and frequencybin mode separation were chosen to be 24 ns and 200 GHz, respectively, resulting in a time–frequency product of 4,800 ≫ 1, which is three orders of magnitude above the EPR limit. Remarkably, even though time and frequencybin entanglement are independent, they both belong to the same degree of freedom, which enabled the implementation of the controlled phase gate, where the complete state was mapped into the time domain for coherent control.
State structure and oneway quantum computation with multipartite hyperentangled cluster states
The structure or topology of cluster states is important for useful applications. The experimentally realized cluster state was measured in its compact form, which is locally equivalent to a fourpartite linear cluster in a onedimensional lattice (if Hadamard gates are applied to qudits 1 and 4), as well as a fourpartite box cluster state in a twodimensional lattice (if Hadamard gates are applied to all four qudits and qudits 2 and 3 are swapped). Both the linear and box cluster can be used for universal quantum computation^{6}.
To perform meaningful algorithms, the number of parties has to be increased. This is still a significant challenge for all known quantum platforms. Highdimensional hyperentangled multiphoton cluster states can provide a path to achieve such scalability. In particular, with current and foreseeable technology, it is not realistic to achieve, for example, a 100qubit cluster state comprised of 100 photons, since the combination of multiphoton states is still probabilistic, which would lead to generation rates that are too low for functional processing speeds. It is therefore of paramount importance to reduce the number of physical photons, while maintaining the Hilbert space size. With the method we propose, this becomes possible through the realization of parties via hyperentanglement and highdimensional superposition. An important advantage of using hyperentangled qudits in terms of scalability is that operations can be performed in a deterministic way, while multiphoton gates are commonly probabilistic. A judicious compromise between multiphotons and hyperentangled parties is thus advantageous in terms of performing efficient operations and scalability. For efficient quantum computation, a hyperentangled multiphoton cluster state with highdimensional superposition can be envisioned. For example, a cluster state formed by 6 photons and hyperentangled in 4 degrees of freedom, each in a superposition of 16 levels, would result in an equivalent 96qubit system.
The suggested computational flow would involve a twodimensional lattice of parties in which the hyperentangled degrees of freedom are measured simultaneously (as they belong to the same photon), and feedforward is performed between the involved photons (see Supplementary Fig. 2). To potentially realize such a state, the dlevel multipartite cluster demonstrated here has to be extended to include more hyperentangled parties, as well as more photons. Additional hyperentangled parties could be obtained by making use of mutually independent energy–time entangled timescales, which could be controlled through concatenated interferometers, fast switches and spectral filters. The advantage of using multiple energytime hyperentangled states is that the controlled quantum gates can be used also for additional parties. Increasing the number of photons could be achieved either directly in the state generation^{6,11,13}, or through state fusion gates^{46}, which have already been proposed for timebin entangled states^{47}.
Entanglement witness for cluster states of qutrits
Multipartite entangled quantum systems provide powerful resources for the implementation and advancement of many applications^{48}. The presence of a genuine pure multipartite quantum state \(\left \psi \right\rangle\), and those states close to them (for example, states degraded by noise), can be identified by measuring the expectation value of an entanglement witness operator^{49}:
$$\widehat {\cal W} = \delta \times {{\Bbb{I}}}  \left \psi \right\rangle \left\langle \psi \right$$with \({\Bbb I}\) being the identity. The factor \(\delta\) has to be chosen in such a way to exclude all biseparable quantum states^{49}. This witness detects the presence of the target state if the expectation value of the witness is negative. For convention, we normalize this witness such that the optimal state results in an expectation value of minus one:
$${\cal W} = \frac{1}{{1  \delta }}\left( {\delta \times {{\Bbb{I}}}  \left \psi \right\rangle \left\langle \psi \right} \right)$$The factor δ is given by the square of the largest Schmidt coefficient of all singlet states between any combination of qutrits^{49,50}. Determining this for cluster states is straightforward, since a cluster state can be projected under local operations on maximally entangled singlet states between any combination of qutrits. This immediately leads to the fact that the maximal Schmidt coefficient of a threelevel cluster state when performing Schmidtmode decomposition with respect to an arbitrary bipartite split does not exceed the maximal Schmidt coefficient of the singlet^{51}, which is given by \(1{\mathrm{/}}\sqrt 3\) for threelevel cluster states. This means that \(\delta = \frac{1}{3}\) for a fourqutrit cluster state. The optimal witness operator for a fourpartite, threelevel cluster state \(\left {C_{4,3}} \right\rangle\) is therefore given by:
$${\cal W}_{\rm{opt}}^{\left( {{\cal C}_{4,3}} \right)} = \frac{1}{2}{{\Bbb{I}}}  \frac{3}{2}\left {C_{4,3}} \right\rangle \left\langle {C_{4,3}} \right$$Measuring the optimal witness would require full state tomography, which is experimentally very demanding. For this reason, the optimal witness is usually reduced to a witness that includes only two orthogonal measurement settings^{50}, which can be achieved for cluster states using the stabilizer formalism^{50}. Stabilizers are operators that are expressed as products of (generalized) Pauli matrices and are thus locally measurable by means of singlequdit projection measurements. Following the stabilizer formalism for dlevel cluster states, developed in ref. ^{7}, we determined the stabilizers of the generated threelevel, fourpartite cluster state. In general, a cluster state \(\left {C_{N,d}} \right\rangle\) can be uniquely defined by a set of main eigenvalue equations, where N is the number of parties and d is the number of levels. These equations are:
$$\begin{array}{ccc} X_a & \otimes & Z_b\left {{\cal C}_{N,d}} \right\rangle = \left {{\cal C}_{N,d}} \right\rangle \\&{{b \in {\cal N}(a)}}&\end{array}$$where a denotes the qudit and
$${\cal N}\left( a \right) = \left\{ {\begin{array}{*{20}{l}} {\left\{ 2 \right\},} \hfill & {a = 1} \hfill \\ {\left\{ {N  1} \right\},} \hfill & {a = N} \hfill \\ {\left\{ {a  1,\,a + 1} \right\},} \hfill & {a \notin \left\{ {1,N} \right\}} \hfill \end{array}} \right.$$denotes the neighbours of the qudit a. In the case of four qutrits, there is a set of four main eigenvalue equations that uniquely describe the cluster state \(\left {C_{4,3}} \right\rangle\) generated here; that is,
$$S_i^{(C_{4,3})}\left {C_{4,3}} \right\rangle = \left {C_{4,3}} \right\rangle$$with
$$\begin{array}{l}S_1^{(C_{4,3})} = Z_1^\dagger Z_2{{\Bbb{I}}}_3{{\Bbb{I}}}_4,\\ S_2^{(C_{4,3})} = X_1X_2Z_3{{\Bbb{I}}}_4,\\ S_3^{(C_{4,3})} = {{\Bbb{I}}}_1Z_2X_3X_4,\\ S_4^{(C_{4,3})} = {\Bbb{I}}_1{{\Bbb{I}}}_2Z_3Z_4^\dagger \end{array}$$where X and Z are the generalized Pauli matrices, \({{\Bbb{I}}}\) is the identity and † denotes the transpose conjugate. In particular,
$$Z = \left( {\begin{array}{*{20}{c}} 1 & 0 & 0 \\ 0 & q & 0 \\ 0 & 0 & {q^2} \end{array}} \right),\,X = \left( {\begin{array}{*{20}{c}} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}} \right),\,{{\Bbb{I}}} = \left( {\begin{array}{*{20}{c}} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}} \right)$$with \(q = {\rm{e}}^{i2{\rm{\pi}} {\mathrm{/}}3}\). The matrices composing the stabilizers belong to the orthonormal Pauli group^{52} \(P = \{{{\Bbb{I}}},X,X^\dagger ,Z,Z^\dagger ,Y,Y^\dagger ,V,V^\dagger \}\), with Y = XZ, V = XZ^{†}.
Using the stabilizers, the density matrix of the cluster state can be written as
$$\left {C_{4,3}} \right\rangle \left\langle {C_{4,3}} \right = \mathop {\prod}\limits_{k = 1}^4 {\frac{{S_k^{(C_{4,3})} + S_k^{\dagger (C_{4,3})} + {\Bbb I}}}{3}}$$We can separate the stabilizers into two orthogonal sets that include only X and Z^{50}, which leads to a witness operator that contains only two measurement settings:
$$\begin{array}{ll}{\cal W}_{ 2S}^{\left( {{\cal C}_{4,3}} \right)}& = 2{\Bbb I}  \frac{3}{2}\mathop {\prod}\limits_{\rm even}^4 {\frac{{S_k^{\left( {C_{4,3}} \right)} + S_k^{\dagger \left( {C_{4,3}} \right)} + {\Bbb I}}}{3}} \\&  \frac{3}{2}\mathop {\prod }\limits_{\rm odd}^4 \frac{{S_k^{(C_{4,3})} + S_k^{\dagger (C_{4,3})} + {\Bbb I}}}{3}\end{array}$$Considering the stabilizers listed above, this witness has an expectation value of:
$$\begin{array}{*{20}{l}}\left\langle{\cal W}_{2{{S}}}^{\left( {{\cal C}_{4,3}} \right)}\right\rangle & = & \frac{5}{3}  \frac{1}{3}{\mathrm{Re}}\left(\left\langle {{\mathbb{I}}_1{\mathbb{I}}_2Z_3Z_4^\dagger } \right\rangle + \left\langle {Z_1^\dagger Z_2{\mathbb{I}}_3{\mathbb{I}}_4} \right\rangle + \left\langle {{\mathbb{I}}_1Z_2X_3X_4} \right\rangle\right.\\ &&\left. + \left\langle {X_1X_2Z_3{\mathbb{I}}_4} \right\rangle \right. \left. + \left\langle {Z_1{\mathbb{I}}_2X_3X_4} \right\rangle + \left\langle {Z_1^\dagger Z_2^\dagger X_3X_4} \right\rangle\right.\\ &&\left. + \left\langle {X_1X_2{\mathbb{I}}_3Z_4} \right\rangle + \left\langle {X_1X_2Z_3^\dagger Z_4^\dagger } \right\rangle \right)\end{array}$$where Re() refers to the real part of the operators, also considering that the real part of an imaginary number C is given by \({\mathrm{Re}}(C) = \frac{{C + C^\dagger }}{2}\). The witness operator therefore always has a real expectation value, as required for a measurable value with physical meaning. Note that the generalized Pauli matrices (that is, for dlevel systems) are nonHermitian and have complex eigenvalues, and therefore the expectation values of the individual stabilizers can have imaginary components.
In a similar way as described above, witnesses were previously derived for cluster states of qubits^{50}. The optimal witness for all cluster states of qubits is given by
$${\cal W}_{{\mathrm{opt}}}^{\left( {{\cal C}_{N,2}} \right)} = {{\Bbb{I}}}  2\left {C_{N,2}} \right\rangle \left\langle {C_{N,2}} \right$$which results in a noise tolerance for a cluster state of qubits mixed with linear incoherent noise of 50%. The reduced witness for cluster states of qubits was derived in ref. ^{50}, and reads:
$${\cal W}_{\rm 2S}^{\left( {{\cal C}_{N,2}} \right)} = 3{{\Bbb{I}}}  2\mathop {\prod }\limits_{\rm even}^N \frac{{S_k^{\left( {C_{N,2}} \right)} + {{\Bbb{I}}}}}{2}  2\mathop {\prod }\limits_{\rm odd}^N \frac{{S_k^{(C_{N,2})} + {{\Bbb{I}}}}}{2}$$This witness for qubits has a tolerance with respect to incoherent linear noise of 33.33% for four qubits, and 28.57% for six qubits^{50}.
Measurement of the witness expectation value
To measure the expectation value of the entanglement witness, the individual expectation values for eight stabilizers have to be measured separately. Each stabilizer expectation value can be extracted by means of 81 separate measurements, which can be performed by projecting the state on the respective combinations of stabilizer eigenvectors. In particular, the witness terms \(Z_1^\dagger Z_2{{\Bbb{I}}}_3{{\Bbb{I}}}_4\) and \({\Bbb{I}}_1{\Bbb{I}}_2Z_3Z_4^\dagger\) have eigenvectors \(\left 1 \right\rangle\), \(\left 2 \right\rangle\), \(\left 3 \right\rangle\), \(\left a \right\rangle\), \(\left b \right\rangle\) and \(\left c \right\rangle\); while \({\Bbb{I}}_1Z_2X_3X_4\), \(Z_1{\Bbb{I}}_2X_3X_4\) and \(Z_1^\dagger Z_2^\dagger X_3X_4\) have eigenvectors \(\left 1 \right\rangle\), \(\left 2 \right\rangle\), \(\left 3 \right\rangle\), \(\left {f1} \right\rangle\), \(\left {f2} \right\rangle\) and \(\left {f3} \right\rangle\), with \(\left {f1} \right\rangle = \left a \right\rangle + \left b \right\rangle + \left c \right\rangle ,\,\left {f2} \right\rangle = \left a \right\rangle + {\rm e}^{i2{\rm \pi} /3}\left b \right\rangle + {\rm e}^{  i2{\rm \pi} /3}\left c \right\rangle\) and \(\left {f3} \right\rangle = \left a \right\rangle + {\rm e}^{  i2{\rm \pi} {\mathrm{/}}3}\left b \right\rangle + {\rm e}^{i2{\rm \pi} /3}\left c \right\rangle\); finally, X_{1}X_{2}Z_{3}I_{4}, X_{1}X_{2}I_{3}Z_{4} and \(X_1X_2Z_3^\dagger Z_4^\dagger\) have eigenvectors \(\left {t1} \right\rangle ,\,\left {t2} \right\rangle ,\,\left {t3} \right\rangle ,\,\left a \right\rangle ,\,\left b \right\rangle \,{\mathrm{and}}\,\left c \right\rangle\), with \(\left {t1} \right\rangle = \left 1 \right\rangle + \left 2 \right\rangle + \left 3 \right\rangle ,\,\left {t2} \right\rangle = \left 1 \right\rangle + {\rm e}^{i2{\rm \pi} {\mathrm{/3}}}\left 2 \right\rangle + {\rm e}^{  i2{\rm \pi} {\mathrm{/3}}}\left 3 \right\rangle\) and \(\left {t3} \right\rangle = \left 1 \right\rangle + {\rm e}^{  i2{\rm \pi} /3}\left 2 \right\rangle + {\rm e}^{i2{\rm \pi} /3}\left 3 \right\rangle\). To extract all projection values, 3 × 81 = 243 parameters, which can take real values between 0 and 1, have to be experimentally determined. From these 243 measurement outcomes, the expectation values of the individual witness terms (stabilizers) can be calculated, which are complex numbers with an absolute value smaller than one. The witness is then calculated from the real parts of the eight stabilizer terms. Projections on time and frequency modes (\(\left 1 \right\rangle ,\,\left a \right\rangle\) and so on), as well as frequencybin superpositions (\(\left {f1} \right\rangle\), and so on), can be immediately obtained with the experimental setup. Projections on timebin superpositions were achieved as follows: we assessed the state phases through simultaneous projection measurements on the superposition of two time bins each, implemented by unbalanced twoarm interferometers. The timebin projections \(\left {t1} \right\rangle ,\,\left {t2} \right\rangle \,{\mathrm{and}}\,\left {t3} \right\rangle\) were then reconstructed considering the measured interference patterns.
Extraction of the wavefunction's amplitude and phase terms and orthogonality of the bipartite states after projection
The quantum interference measurements shown in Fig. 4 can be used to extract the amplitude and phase terms of the fourpartite cluster state, which can then be used to confirm that the bipartite states that remain after projection measurements are orthogonal. The most generic wavefunction to express the fourqutrit hyperentangled states is
$$\begin{array}{ll}\left {\mathit{\Psi }} \right\rangle &= m_{1,{{a}}}\left {1,1,{{a}},{{a}}} \right\rangle + m_{1,{{b}}}\left {1,1,{{b}},{{b}}} \right\rangle + m_{1,{{c}}}\left {1,1,{{c}},{{c}}} \right\rangle \\ &+\, m_{2,{{a}}}\left {2,2,{{a}},{\mathrm{a}}} \right\rangle + m_{2,{{b}}}\left {2,2,{{b}},{{b}}} \right\rangle + m_{2,{{c}}}\left {2,2,{{c}},{{c}}} \right\rangle \\ &+\, m_{3,{{a}}}\left {3,3,{{a}},{\mathrm{a}}} \right\rangle + m_{3,{{b}}}\left {3,3,{{b}},{{b}}} \right\rangle + m_{3,{{c}}}\left {3,3,{{c}},{{c}}} \right\rangle \end{array}$$where \(m_{t,f} = \left {m_{t,f}} \right{\rm e}^{i\phi _{t,f}}\) are complex numbers with amplitude \(m_{t,f}\) and phase ϕ_{t,f}. Here, t = 1, 2, 3 and f = a, b, c. We determined the amplitudes by performing 81 coincidence measurements between all combinations of temporal and frequency modes, and confirmed that, as expected, the wavefunction has only the abovestated nine nonzero elements (see Fig. 3a). To extract the phases ϕ_{t,f}, we used the nine quantum interference measurements shown in Fig. 4. In particular, we projected onto time or frequencybin bases by means of temporal gating (in the detection) or optical filtering. The remaining twopartite states were then measured in superpositions. For timebin measurements, we employed a twoarm interferometer, which simultaneously projects on timebin superpositions; that is \(\left 1 \right\rangle + {\rm e}^{  i\theta }\left 2 \right\rangle\), as well as \(\left 2 \right\rangle + {\rm e}^{  i\theta }\left 3 \right\rangle\) for each photon. The relative phase offset between the quantum interference measurements can be used to extract the relative phase between the coefficients of the wavefunction. For the frequencybin projections, we first added spectral phases, and then mixed the three frequency components using electrooptic modulation. The spectral phase was adjusted to perform projection measurements of the form \(\left a \right\rangle + {\rm e}^{  i\theta }\left b \right\rangle + {\rm e}^{i2\theta }\left c \right\rangle\). We then fitted the predicted functions to the quantum interference patterns to extract the individual phases of the wavefunction (listed in Supplementary Table 1), including the estimated error for these values. Note that all measured visibilities exceed the threshold (also shown in Supplementary Table 1) required to violate twopartite Bell inequalities^{53}.
The extracted phase parameters can then be used to confirm that the individual twopartite states are orthogonal. In particular, the quantum interference measurements witness the generation of the following bipartite states:
$$\begin{array}{l}\left {{\mathrm{\psi }}_{{\mathrm{a}},{\mathrm{a}}}} \right\rangle = 0.581\left {1,1} \right\rangle + 0.577{\mathrm{e}}^{{\mathrm{i}}0.06{\mathrm{\pi }}}\left {2,2} \right\rangle + 0.574{\mathrm{e}}^{  {\mathrm{i}}0.06{\mathrm{\pi }}}\left {3,3} \right\rangle \\ \left {{\mathrm{\psi }}_{{\mathrm{b}},{\mathrm{b}}}} \right\rangle = 0.588\left {1,1} \right\rangle + 0.563{\mathrm{e}}^{{\mathrm{i}}0.65{\mathrm{\pi }}}\left {2,2} \right\rangle + 0.581{\mathrm{e}}^{  {\mathrm{i}}0.72{\mathrm{\pi }}}\left {3,3} \right\rangle \\ \left {{\mathrm{\psi }}_{{\mathrm{c}},{\mathrm{c}}}} \right\rangle = 0.596\left {1,1} \right\rangle + 0.574{\mathrm{e}}^{  {\mathrm{i}}0.67{\mathrm{\pi }}}\left {2,2} \right\rangle + 0.562{\mathrm{e}}^{{\mathrm{i}}0.69{\mathrm{\pi }}}\left {3,3} \right\rangle \\ \left {{\mathrm{\psi }}_{1,1}} \right\rangle = 0.574\left {{\mathrm{a,a}}} \right\rangle + 0.577{\mathrm{e}}^{{\mathrm{i}}0.087{\mathrm{\pi }}}\left {{\mathrm{b,b}}} \right\rangle + 0.581{\mathrm{e}}^{{\mathrm{i}}0.001{\mathrm{\pi }}}\left {{\mathrm{c,c}}} \right\rangle \\ \left {{\mathrm{\psi }}_{2,2}} \right\rangle = 0.586\left {{\mathrm{a,a}}} \right\rangle + 0.569{\mathrm{e}}^{{\mathrm{i}}0.594{\mathrm{\pi }}}\left {{\mathrm{b,b}}} \right\rangle + 0.576{\mathrm{e}}^{  {\mathrm{i}}0.667{\mathrm{\pi }}}\left {{\mathrm{c,c}}} \right\rangle \\ \left {{\mathrm{\psi }}_{3,3}} \right\rangle = 0.582\left {{\mathrm{a,a}}} \right\rangle + 0.586{\mathrm{e}}^{  {\mathrm{i}}0.638{\mathrm{\pi }}}\left {{\mathrm{b,b}}} \right\rangle + 0.564{\mathrm{e}}^{{\mathrm{i}}0.628{\mathrm{\pi }}}\left {{\mathrm{c,c}}} \right\rangle \end{array}$$Note that these measurements do not present full quantum state tomography, but are based on the realistic assumption that only the considered elements of the quantum state contribute to the wavefunction, and that the final actual state is an incoherent mixture of a pure state with linear noise. These assumptions are justified since we have an accurate experimental control over the generation process as well as the projection measurements, and the measured interference agrees well with the fitted curves (see Fig. 4). From the estimated wavefunctions, it is possible to confirm that the generated states are orthogonal within the measurement uncertainties:
$$\begin{array}{l}\left {\left\langle {{\mathrm{\psi }}_{{\mathrm{a}},{\mathrm{a}}}{\mathrm{\psi }}_{{\mathrm{b}},{\mathrm{b}}}} \right\rangle } \right^2 = 0.009 \pm 0.02 \approx 0\\ \left {\left\langle {{\mathrm{\psi }}_{{\mathrm{a}},{\mathrm{a}}}{\mathrm{\psi }}_{{\mathrm{c}},{\mathrm{c}}}} \right\rangle } \right^2 = 0.011 \pm 0.02 \approx 0\\ \left {\left\langle {{\mathrm{\psi }}_{{\mathrm{b}},{\mathrm{b}}}{\mathrm{\psi }}_{{\mathrm{c}},{\mathrm{c}}}} \right\rangle } \right^2 = 0.009 \pm 0.02 \approx 0\\ \left {\left\langle {{\mathrm{\psi }}_{1,1}{\mathrm{\psi }}_{2,2}} \right\rangle } \right^2 = 0.027 \pm 0.02 \approx 0\\ \left {\left\langle {{\mathrm{\psi }}_{1,1}{\mathrm{\psi }}_{3,3}} \right\rangle } \right^2 = 0.002 \pm 0.02 \approx 0\\ \left {\left\langle {{\mathrm{\psi }}_{2,2}{\mathrm{\psi }}_{3,3}} \right\rangle } \right^2 = 0.012 \pm 0.02 \approx 0\end{array}$$Witness distribution and noise characterization via Monte Carlo simulations
We performed Monte Carlo simulations to infer the distribution of the witness expectation value. For all noise calculations, we used a linear noise model^{50,53}, where the pure quantum state is incoherently mixed with linear, uncorrelated classical noise:
$$\rho _{{\mathrm{noise}}} = \varepsilon \frac{{{\Bbb{I}}}}{{d^N}} + \left( {1  \varepsilon } \right)\left \psi \right\rangle \left\langle \psi \right$$Here, ρ_{noise} is the density matrix of the state after the incoherent mixture with noise, ε measures the quantity of mixed noise and \(\left \psi \right\rangle\) is the wavefunction of the pure quantum state. The linear noise model is very useful for this analysis, as it allows one to separate different noise sources. The incoherent noise term is commonly used to describe the impact of state impurities such as losses and detection noise. Other sources such as amplitude and phase noise do not affect the purity of the state, and can be incorporated into the wavefunction of the pure state by adjusting the amplitude and phase terms within the wavefunction. Starting from the measured input values and their determined experimental errors (Supplementary Table 1), we assumed a Gaussian error distribution for each individual input parameter and calculated the witness expectation value one million times. These calculations were then summarized in normalized histograms (see Fig. 3). To determine the witness bound for different sources of input noise (that is, incoherent, amplitude and phase noise), we calculated the associated operator for input parameters with different noise sources and average strengths. In each calculation, only two noise sources were considered at a time, while the remaining noise source was kept at zero. For diagrams involving incoherent noise, a fixed value for this term was first set and ten million random input values for either the phases or amplitudes were generated (then the incoherent noise value was changed for different rounds of simulations). For the diagrams where both amplitude and phase noise were open parameters, we generated one billion random input settings. For each simulation input, we calculated the witness expectation value. The outcomes were then sorted according to positive and negative witness values as well as their average noise. We defined the amplitude noise as the average over the absolute deviations from the ideal value, normalized by the latter: \(\sigma _{\rm{ampl.}} = 3 \times \frac{1}{9}{\sum} {\left {a_i  1{\mathrm{/}}3} \right}\), where a_{i} are the nine different amplitude values in the wavefunction and 1/3 is the ideal value. We define the phase noise as the average over the absolute phase deviation from the ideal value, normalized by the optimal phase terms: \(\sigma_{\rm{ampl.}} = \frac{3}{{2{\uppi }}} \times \frac{1}{9}{\sum} {\left {\theta _i  \varphi _{\rm{ideal}}} \right}\), where \(\varphi _{\rm{ideal}} = \left[ {0,\frac{{2{\uppi} }}{3},\frac{{  2{\uppi} }}{3}} \right]\) are the ideal phase settings, and θ_{i} are the nine different determined phase values in the wavefunction. The stated bound for the witness was determined as the points where over 95% of all calculated witness values were negative.
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Acknowledgements
This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) through the Steacie, Strategic, Discovery and Acceleration Grants Schemes, by the MESI PSRSIIRI Initiative in Quebec, by the Canada Research Chair Program and by the Australian Research Council Discovery Projects scheme (DP150104327). C.R., P.R. and S.L. acknowledge the support of NSERC Vanier Canada Graduate Scholarships. M.K. acknowledges funding from the European Union’s Horizon 2020 Research and Innovation programme under the Marie SklodowskaCurie grant agreement number 656607. S.T.C. acknowledges support from the CityU APRC programme number 9610356. B.E.L. acknowledges support from the Strategic Priority Research Program of the Chinese Academy of Sciences (grant number XDB24030300). W.J.M. acknowledges support from the John Templeton Foundation (JTF) number 60478. R.M. acknowledges additional support by the Government of the Russian Federation through the ITMO Fellowship and Professorship Program (grant 074U 01) and from the 1000 Talents Sichuan Program. We thank R. Helsten for technical insights; A. Tavares and K. Nemoto for discussions; P. Kung from QPS Photronics for help and the use of processing equipment; and Quantum Opus and N. Bertone of OptoElectronics Components for their support and for providing us with stateoftheart photon detection equipment.
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These authors contributed equally: Christian Reimer, Michael Kues.
Affiliations
Institut National de la Recherche Scientifique (INRSEMT), Varennes, Quebec, Canada
 Christian Reimer
 , Stefania Sciara
 , Piotr Roztocki
 , Mehedi Islam
 , Luis Romero Cortés
 , Yanbing Zhang
 , Bennet Fischer
 , José Azaña
 , Michael Kues
 & Roberto Morandotti
John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA
 Christian Reimer
Department of Energy, Information Engineering and Mathematical Models, University of Palermo, Palermo, Italy
 Stefania Sciara
 & Alfonso Cino
Engineering Physics Department, Polytechnique Montreal, Montreal, Quebec, Canada
 Sébastien Loranger
 & Raman Kashyap
Electrical Engineering Department, Polytechnique Montreal, Montreal, Quebec, Canada
 Raman Kashyap
Department of Physics and Material Science, City University of Hong Kong, Hong Kong, China
 Sai T. Chu
State Key Laboratory of Transient Optics and Photonics, Xi’an Institute of Optics and Precision Mechanics, Chinese Academy of Science, Xi’an, China
 Brent E. Little
Centre for Micro Photonics, Swinburne University of Technology, Hawthorn, Victoria, Australia
 David J. Moss
Institute of Photonics, Department of Physics, University of Strathclyde, Glasgow, UK
 Lucia Caspani
NTT Basic Research Laboratories and NTT Research Center for Theoretical Quantum Physics, NTT Corporation, Kanagawa, Japan
 William J. Munro
National Institute of Informatics, Tokyo, Japan
 William J. Munro
School of Engineering, University of Glasgow , Glasgow, UK
 Michael Kues
Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu, China
 Roberto Morandotti
ITMO University, St Petersburg, Russia
 Roberto Morandotti
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Contributions
C.R. and M.K. contributed equally. M.K., C.R., P.R., and S.S. developed the idea. C.R., M.K. P.R., M.I., Y.Z., L.R.C., and B.F. performed the measurements and analyzed the data. S.S., C.R., M.K., L.C., and W.J.M. performed the theoretical analysis. S.T.C. and B.E.L. designed and fabricated the microring resonator. S.L. and R.K. designed and fabricated the fibre Bragg gratings. D.J.M. and A.C. contributed to discussions. R.M. and J.A. managed the project. All authors contributed to the writing of the manuscript.
Competing interests
The authors declare no competing interests.
Corresponding authors
Correspondence to Michael Kues or Roberto Morandotti.
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Quantum optical microcombs
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