Quantum entanglement, which is often counterintuitive and was originally questioned by Einstein, Podolsky, and Rosen（EPR）,¹ is now recognized as a versatile resource for quantum information protocols.^2,3 In order to meet various requirements for various applications, an important technological development is to increase the number of available entangled qubits. In particular, measurement-based quantum computation（MBQC）requires the cluster type of large-scale entangled state,^4,5 where individual components of the cluster state must be accessible by measuring devices. The number of entangled qubits with individual accessibility seems to be currently limited to a moderate scale.^6–8

Shifting to the continuous variable（CV）counterpart,^9,10 recently the number of fully inseparable qumodes of light fields drastically increased by introducing multiplexing either in the time domain^11–13 or in the frequency domain.^14–17 The successes of the CV approach are based on the deterministic nature of field-quadrature squeezing. This is a strong advantage over the photonic qubit-based approach, where photons or photons pairs are generated with low probabilities and rare success events are postselected.^8,18 On the flip side, the disadvantage of the CV MBQC has been considered as finite squeezing in resource cluster states, which results in accumulation of noises during computation.¹⁹ However, there has still been a possibility of a special encoding which can circumvent this problem, and indeed, a fault-tolerant threshold corresponding to about −20.5 dB of squeezing is recently proven by Menicucci²⁰ for the Gottesman-Kitaev-Preskill（GKP）type of encoding.²¹ Although the threshold of the squeezing levels is still unreached, it is good news that CV cluster states with finite squeezing can be a sufficient resource for quantum computation.

The multiplexing allows many qumodes to be assigned to a single beam, which drastically reduces complexity and physical size of the optical system from the conventional implementation assigning a single qumode to a single optical beam.^22–26 Although both of the time-domain multiplexing and the frequency-domain multiplexing would be important developmental directions, the time-domain multiplexing is advantageous because of time translation symmetry. With the time-domain multiplexing, a cluster state of qumodes can be an arbitrarily long chain in the longitudinal direction by extending the operating time of the cluster-state generator.

In spite of the in-principle unlimitation of the time-domain-multiplexing, the previous experimental demonstration of a dual-rail CV cluster state was limited in the number of qumodes,¹³ due to a technical reason as follows. The optical setup is a large Mach-Zehnder interferometer as depicted in Fig. 1, including a long optical delay line for a shift of qumodes on a rail. The mathematical derivation of the cluster state with this setup is contained in Sec. S1 of the supplementary material. In order to operate this cluster-state generator, the relative phases at all interference points must be properly locked. For this purpose, modulated bright beams are injected into the optical paths of the cluster state as phase probes, and their classical interference signals are exploited as error signals for the feedback control. However, the modulated bright beams are noisy, which obscures objective quantum-level correlation signals of the cluster state. The noises were circumvented in the previous demonstration by chopping the bright beams and by detecting the cluster state when the bright beams are absent. Due to the above previous situation, the optical system was uncontrollable during the cluster-state generation. Phase drifts gradually accumulated and quantum correlations gradually degraded at the rear part of the cluster state. Finally, the full-inseparability criterion²⁷ became unsatisfied after the creation of around 16 000 qumodes within 1.3 ms.¹³

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FIG. 1.

Schematic of the optical setup to generate a dual-rail CV cluster state multiplexed in the time domain. Accompanied is a graph image of the quantum state at each stage, where qumodes are represented by nodes and their correlations are represented by edges. Edge colors in（iv）distinguish positive and negative correlations, as explained in Sec. S1 of the supplementary material.¹²

Here we demonstrate a new time-domain-multiplexing experiment, where a dual-rail CV cluster state with full inseparability is deterministically generated without limitation in the number of qumodes. The previous limitation explained above is removed by continuing the feedback control during the generation of a cluster state. Instead, the noises of the modulated bright beams are eliminated electrically after homodyne detections. The resulting effective squeezing levels are −4.3 dB, which are not as high as −5.0 dB for the first thousands of qumodes in the previous demonstration,¹³ but in contrast the initial squeezing levels continue without degradation. The squeezing levels are sufficiently higher than −3.0 dB of the full-inseparability criterion.^13,27 In principle, there is no limitation in the number of fully inseparable qumodes. We test the full inseparability of about 1.2 × 10⁶ qumodes, which are generated within 100 ms. The reason why we stop at about 1.2 × 10⁶ qumodes is simply the data size for verification. In order to estimate the effective squeezing levels, we repeatedly acquire homodyne signals of the 1.2 × 10⁶ qumodes, by which the data size reaches 100 GB.

In Sec. II, we summarize theories regarding quantum correlations in the dual-rail CV cluster state, full inseparability criterion, and longitudinal mode functions. In Sec. III, we describe experimental methods. In Sec. IV, we show experimental results and make discussions. In Sec. V, we summarize our results. For self-containedness, in Sec. S1 of the supplementary material, we describe derivation of the quantum correlations in the dual-rail CV cluster state for an ideal case, and in Sec. S2 of the supplementary material, we describe derivation of inequalities for full inseparability.