Entanglement-assisted non-local optical interferometry in a quantum network

Abstract

The sensitivity of non-local optical measurements at low light intensities, such as those involved in long-baseline telescope arrays^1,2, is limited by fundamental quantum noise and photon losses³. Distributed quantum entanglement has been proposed as a route towards overcoming these limitations and accessing new regimes of non-local optical sensing^4–6. Here we demonstrate the use of entangled quantum memories in a quantum network of silicon–vacancy centres in diamond nanocavities^7–9 to experimentally perform such non-local phase measurements. Specifically, we combine the generation of event-ready remote quantum entanglement, photon mode erasure that hides the ‘which-path’ information of temporally and spatially separated incoming optical modes and non-local, non-destructive photon heralding enabled by remote entanglement to perform a proof-of-concept entanglement-assisted differential phase measurement of weak incident light between two spatially separate stations. Demonstrating successful operation of the remote phase sensing protocol with a fibre link baseline up to 1.55 km, our results provide an opportunity for a new class of quantum-enhanced optical imaging methods with potential applications ranging from long-baseline interferometry and astronomy to microscopy^10,11.

Entangled quantum memories are used in a quantum network of silicon–vacancy centres in diamond nanocavities to experimentally perform non-local phase measurements.

Main

Optical interferometry is a well-established method for high-resolution imaging with wide-ranging applications from physics and astronomy to biological and medical imaging^12–14. For instance, astronomical interferometry is routinely used for the observation of stellar objects in which the light signal from multiple physically separated telescopes is combined to increase the imaging resolution^1,2,15,16. In such a case, an array of optical receivers forms a synthetic aperture whose resolution scales with their separation (the baseline)¹³. However, increasing the baseline of receiver arrays in practice is challenging¹⁷. In the limit of weak signals typical of the optical domain, the optimal method for observation is direct interference of incident electromagnetic fields³, which is hindered by the exponential loss of signal light associated with optical-fibre-based connections¹⁸. Quantum networks^19–22 provide a way to perform non-local interference measurements. The key idea is to use quantum entanglement to effectively teleport the quantum state of the electromagnetic field modes between remote receiver stations (Fig. 1a, right), thereby enabling direct interference^4,23. Although a scheme involving entanglement for non-local interference has been recently demonstrated in an all-photonic setting²⁴, the use of quantum memories presents a practical path towards overcoming photon loss through event-ready (heralded) entanglement⁴ and efficient information processing together with local photon mode erasure, enabling an exponential reduction in the number of required entangled pairs^5,6.

Fig. 1. A quantum-memory-assisted non-local interferometer based on a quantum network.

a, Three remote phase sensing methods and their SNR scalings from left to right: direct interferometric detection, local measurements with an LO, and entanglement-assisted non-local phase sensing. b, The electron-spin-state-dependent cavity reflectance near the SiV optical transition is used for signal photon storage and quantum operations. The dashed line indicates the frequency used for electron spin state readout, photonic entanglement and signal light collection. c, Entangled qubits shared between the stations are used to improve the sensitivity of a non-local interferometer. d–f, Once entanglement has been generated, the steps of the quantum-memory-assisted remote phase sensing protocol are signal light collection through local operations (d), signal photon mode erasure to complete photon state storage (e), and non-local photon heralding through electron state measurement and phase probing through nuclear state measurement (f).

Here we demonstrate quantum-memory-assisted non-local interferometry with a two-station network separated by a line-of-sight distance of about 6 m (Fig. 1b,c). Our approach uses atom-like defects in solid state^21,25–31, particularly silicon–vacancy centre (SiV) integrated in diamond nanophotonic cavities. These systems recently emerged as a promising platform for quantum networking because of their access to long-lived spin quantum memories, high gate and readout fidelities and strong light–matter interaction, enabling efficient spin–photon operations^7,8. These properties have enabled experimental implementations of quantum-memory-enhanced communication³², entanglement generation over a metropolitan-scale deployed fibre⁹ and blind quantum computation³³. In our experiments, each SiV constitutes a two-qubit register with a communication qubit (electron spin) and a memory qubit (²⁹Si nuclear spin). Signal fields are reflected off the fibre-coupled SiV-cavity systems, and an optical fibre network is used for readout, entanglement generation and signal light collection^7,8. We use an improved parallel instead of serial⁹ entanglement scheme to reach higher entanglement rates for both electron–electron and nucleus–nucleus entanglement and demonstrate non-destructive photon heralding, first locally with a time-bin photonic qubit on a single station, then non-locally for a photon in superposition between two remote spatial modes by using remote entanglement. This photon heralding filters out vacuum fluctuations to achieve optimal interferometer sensitivity³. We combine this method with photon mode erasure by interfering the incident fields locally with a coherent state of light to hide the ‘which-path’ information for interferometric measurements. Finally, we integrate these elements to demonstrate the operation of a long-baseline quantum-memory-assisted interferometer with a fibre separation length of up to 1.55 km, five times larger than the current state-of-the-art optical telescope array baseline of 330 m (ref. ²).

Non-local phase sensing

The signal in a non-local interferometer, such as the angle of the incident light from a distant object at two detector stations, is typically proportional to the sine or cosine of the differential phase ϕ between the detector stations (Fig. 1c). The goal is, therefore, to determine this differential phase ϕ (from which the spatial information of this distant object can be inferred) with the highest possible efficiency and precision. In conventional systems, two approaches to measure ϕ can be distinguished for thermal light. The first involves direct interference of non-locally collected field modes (Fig. 1a, left), whereas the second involves local measurements (Fig. 1a, middle). In the first approach, direct interference of the signal light collected from each station is enabled by routing the light to a central beam splitter. This method achieves optimal interference visibility with signal-to-noise ratio (SNR) scaling as √μ_sig (ref. ³), where μ_sig ≪ 1 is the average photon number of the incident light field. However, for long-baseline measurements, this approach introduces signal attenuation that typically scales exponentially with the distance between the stations¹⁸. The second approach consists of interfering the collected field modes with a distributed local oscillator (LO) at each station. The phase difference between the two stations can then be determined by comparing the local measurement results. However, as the signal light is mixed with the LO, the local measurements cannot distinguish the vacuum component of the signal field modes from their single-photon component. The vacuum component of the field (which contains no photons) carries no useful phase information but introduces shot noise (vacuum fluctuation noise), reducing the interference visibility and resulting in an SNR scaling as μ_sig (ref. ³). Although it is also possible to independently measure the local phase of the incident light at each station using higher-order correlations^34,35, for thermal light, this requires the simultaneous arrival of a signal photon at each station, which maintains the unfavourable scaling³.

Entangled quantum memories provide a route to achieve optimal non-local measurements without the exponential field attenuation with the baseline size^4,5 (Fig. 1a, right). Specifically, pre-generated entanglement between the stations can be used as a resource to perform non-local photon heralding, in which the arrival of a signal photon can be detected without revealing at which station it arrives⁵. This allows us to distinguish vacuum from signal photons without destroying the phase information ϕ. By keeping only measurement results with successful non-local heralding, vacuum fluctuations can be effectively filtered out to increase the visibility and SNR (see Methods for details).

To realize this method experimentally, our approach consists of first ‘arming’ the interferometer by preparing the nuclear qubits in an entangled state between two stations. The entanglement is event-ready as it is heralded independently from the subsequent signal measurement, a key improvement over all-photonic approaches²⁴. We model the distributed signal light with a weak laser pulse with average photon number μ_sig ≪ 1. The local signal phase at each station is averaged over a uniform distribution (with fixed differential phase ϕ), such that the signal effectively behaves like a two-mode thermal state in the weak-signal regime: ${\rho }_{\mathrm{sig}}\approx |0_{\mathrm{L}}{0}_{\mathrm{R}}⟩⟨0_{\mathrm{L}}{0}_{\mathrm{R}}|+{\mu }_{\mathrm{sig}}/2(|0_{\mathrm{L}}{1}_{\mathrm{R}}⟩+{\mathrm{e}}^{\mathrm{i}\varphi }|1_{\mathrm{L}}{0}_{\mathrm{R}}⟩)(⟨0_{\mathrm{L}}{1}_{\mathrm{R}}|+$ ${\mathrm{e}}^{-\mathrm{i}\varphi }|1_{\mathrm{L}}{0}_{\mathrm{R}}⟩)$, where |0_L, 0_R⟩ corresponds to vacuum, and |1_L, 0_R⟩ corresponds to a single photon arriving at the left station and |0_L, 1_R⟩ corresponds to that arriving at the right station³ (Methods and Extended Data Fig. 7). The photonic signal is collected through local quantum operations (Fig. 1d) that entangle the photonic state with the qubits at each station. We then erase the photonic mode information⁶ (Fig. 1e) and subsequently implement non-local, non-destructive photon heralding by measuring the parity of electron qubit spins at the two stations. This correlated parity measurement heralds the arrival of a photon without revealing which station the photon arrived at (Fig. 1f). Finally, the differential phase ϕ of the photonic modes imprinted on the initial Bell state between the nuclear spins is obtained through a two-qubit nuclear parity measurement performed locally.

Extended Data Fig. 7. Signal state preparation.

(a) Two weak coherent states are sent to both stations and combined with local oscillators. The phase differences are probed for each experimental shot, and the data are grouped according to fixed differential phase ΔΦ_L−R, so that the terms depending on the local phases δϕ_L(R) average out to zero. (b) Scatter plot showing all local phases in purple, and the subset corresponding to the bin with differential phase ΔΦ_L−R = 0. c) and d) are histograms of the local phases for the left (δϕ_L) and right (δϕ_R) stations for the binning of differential phases ΔΦ_L−R = 0, respectively.

Parallel entanglement generation

In previous work, SiV remote entanglement generation has relied on serial entangling schemes^9,33. Here we implement a parallel entangling scheme^36,37 with a 7.5 times higher efficiency⁹. This is realized by connecting the two stations in a Mach–Zehnder interferometer configuration that must be phase-stable with each path reflecting off one SiV-cavity system (Fig. 2a). In this approach, we generate entanglement between electron spin qubits by splitting a weak laser pulse on a beam splitter to send two weak coherent states ${|\alpha /\sqrt{2}⟩}_{\mathrm{L}}{|\alpha /\sqrt{2}⟩}_{\mathrm{R}}\approx$$|0_{\mathrm{L}}{0}_{\mathrm{R}}⟩+\alpha /\sqrt{2}(|0_{\mathrm{L}}{1}_{\mathrm{R}}⟩+|1_{\mathrm{L}}{0}_{\mathrm{R}}⟩)+O\left({\alpha }^2\right)$ (with ∣α∣² = μ_ent, where μ_ent is the average photon number) to each station with electron spins initially in |+⟩ state (where $|\pm ⟩=(|\downarrow ⟩\pm |\uparrow ⟩)/\sqrt{2}$). We then perform a single-mode spin–photon gate (SMSPG; Fig. 2b), which relies on the spin-state-dependent conditional reflection amplitude between the photon and the electron spin qubit $|1_{\gamma }{+}_{\mathrm{e}}⟩\to |1_{\gamma }{\uparrow }_{\mathrm{e}}⟩/\sqrt{2}$ at each station (Fig. 1b). We note that this gate is non-unitary because of photon loss (Supplementary Information), corresponding to the transformation

$$\begin{array}{l}\left(\right|0_{\mathrm{L}}{0}_{\mathrm{R}}⟩+\alpha /\sqrt{2}\left(\right|0_{\mathrm{L}}{1}_{\mathrm{R}}⟩+|1_{\mathrm{L}}{0}_{\mathrm{R}}⟩))|{+}_{{\mathrm{e}}_{\mathrm{L}}}{+}_{{\mathrm{e}}_{\mathrm{R}}}⟩\\ \to |0_{\mathrm{L}}{0}_{\mathrm{R}}⟩|{+}_{{\mathrm{e}}_{\mathrm{L}}}{+}_{{\mathrm{e}}_{\mathrm{R}}}⟩+\alpha /2\left(\right|0_{\mathrm{L}}{1}_{\mathrm{R}}⟩|{+}_{{\mathrm{e}}_{\mathrm{L}}}{\mathrm{\uparrow }}_{{\mathrm{e}}_{\mathrm{R}}}⟩+|1_{\mathrm{L}}{0}_{\mathrm{R}}⟩|{\mathrm{\uparrow }}_{{\mathrm{e}}_{\mathrm{L}}}{+}_{{\mathrm{e}}_{\mathrm{R}}}⟩).\end{array}$$ 1

The two photonic paths are then recombined at a second beam splitter and on detecting a photon at its outputs, the photonic modes are projected onto the $|01⟩\pm {\mathrm{e}}^{\mathrm{i}\mathrm{\delta }{\varphi }_{\mathrm{e}}}|10⟩$ basis. This step heralds a successful entanglement attempt, ensuring that the fidelity does not degrade at low success probabilities. For measurement outcome $|01⟩+{\mathrm{e}}^{\mathrm{i}\mathrm{\delta }{\varphi }_{\mathrm{e}}}|10⟩$ the resulting electrons’ state (up to normalization) is

$$(1+{\mathrm{e}}^{\mathrm{i}\mathrm{\delta }{\varphi }_{\mathrm{e}}})(\mid \uparrow \uparrow ⟩+\mid {\Psi }^{+}⟩/2)+(1-{\mathrm{e}}^{\mathrm{i}\mathrm{\delta }{\varphi }_{\mathrm{e}}})\mid {\Psi }^{-}⟩/2,$$ 2

where $|{\Psi }^{\pm }⟩$ corresponds to the entangled Bell state |↑↓⟩ ± |↓↑⟩. By locking the interferometer phase δϕ_e to π, we prepare the Bell state |Ψ⁻⟩ across the two stations (Fig. 2d,f) with a fidelity of F = 0.83(3). By tuning the average photon number μ_ent of the incident weak coherent state, we can increase the entanglement success probability per trial at the cost of reduced entangled state fidelity because of multi-photon contributions. We reach an entanglement rate of 13 Hz for F ≥ 0.5 and 1.9 Hz for F = 0.79(3) (Fig. 2g), enabling practical entanglement-based sensing experiments.

Fig. 2. Parallel entanglement generation between SiVs.

a, SiV stations connected in a Mach–Zehnder interferometer setup. b, Electron–electron entanglement circuit using the SMSPG. c, Nucleus–nucleus entanglement circuit using the SMPHONE. For both b and c, all rotations are around the y-axis. d,e, Bell state tomography for electron–electron (d) and nucleus–nucleus entanglement (e). f, Electron–electron entanglement fidelity as a function of interferometer phase lock point. g, Electron–electron entanglement fidelity as a function of entanglement rate, sweeping the average photon number in the weak laser pulse used for entangling electron spins from 0.1 to 1. Error bars in d–g are 1 s.d.

The non-local phase measurement protocol requires entanglement between the nuclear spins, which we generate by replacing the SMSPG with single-mode photon–nucleus entangling gates (SMPHONE, Fig. 2c), which generates the state $|{\mathrm{\uparrow }}_{{\mathrm{e}}_{\mathrm{L}}}{\mathrm{\uparrow }}_{{\mathrm{e}}_{\mathrm{R}}}⟩{|{\Psi }^{-}⟩}_{{\mathrm{n}}_{\mathrm{L}},{\mathrm{n}}_{\mathrm{R}}}$. Similar to the PHONE gate^8,9, the electron ends in the |↑⟩ state unless a gate error occurred (see the Methods for details). By reading out the electron state, we can detect microwave (MW) errors and discard electron |↓⟩ measurements, reaching a nuclear Bell state fidelity of F = 0.73(4) at 0.25 Hz (Fig. 2e).

Photon erasure and non-local heralding

Photon mode erasure^6,38 is a crucial step in our protocol to store the signal phase information onto the quantum memories while being compatible with efficient signal storage techniques⁵. After the interaction with the SiV at each station, we must ensure—by performing photon mode erasure—that the which-path information (that is, whether there is a photon at the left or right station) is not extracted to preserve the differential signal phase information (ϕ). We achieve this by interfering the signal with an LO coherent state on a beam splitter at each station and measuring the output ports with photon-number-resolving detectors (Fig. 3b). We apply feedback on the nuclear qubits depending on the photon number measurement outcomes at the first i and second i′ port of the left station mode and first j and second j′ port of the right station mode, and post-select on click outcomes⁶ (i ≠ i′) and (j ≠ j′) (Supplementary Information). Because the LO contains vacuum, single-photon and multi-photon components, detection cannot distinguish whether the photons originated from the LO or from the signal. To identify the photon arrival and thus eliminate the vacuum, we then use the electron spins to herald the presence of a signal photon—without revealing in which mode this photon is located—and imprint its phase on the nuclear spins.

Fig. 3. Erasing and heralding photons in two temporal modes on a single station.

a, Experimental sequence to measure phase difference between two temporal modes of a photon using erasure and heralding, with all rotations around the y-axis. For the C_nNOT_e gates, the black circle gate is conditional on the nucleus being in state |↓⟩ and the white circle gate is conditional on the nucleus being in state |↑⟩. b, Photon-erasure protocol for any two photonic modes (such as temporal and spatial). c, Nuclear state measurement result after the sequence shown in a. d,e, Success probability (d) and signal visibility (e) of sequence a as a function of coherent state strength of LO. Error bars in c–e are 1 s.d.

As a first step towards realizing this protocol, we demonstrate the local implementation of erasure and heralding of a photon in a superposition of two temporal modes (|1_e0_l⟩ + |0_e1_l⟩, with |i_e⟩ the photonic state of the early temporal mode and |i_l⟩ the photonic state of the late temporal mode, separated by 1 μs). As the photon heralding is local in this case, it does not require a Bell pair. Instead, we use an electron X-basis measurement to herald the arrival of a signal photon while hiding the temporal information. We note that this experimental sequence can be applied for the implementation of quantum receivers for classical communication³⁹.

In our experiment, starting with the photon state $|0_{\mathrm{e}}{0}_{\mathrm{l}}⟩+$ $\sqrt{{\mu }_{\mathrm{sig}}/2}(|1_{\mathrm{e}}{0}_{\mathrm{l}}⟩+{\mathrm{e}}^{\mathrm{i}\theta }|0_{\mathrm{e}}{1}_{\mathrm{l}}⟩)$, we implement the gate sequence shown in Fig. 3a, resulting in the state (right before the photon mode erasure step; see Supplementary Information for details)

$$|0_{\mathrm{e}}{0}_{\mathrm{l}}⟩{|+⟩}_{\mathrm{elec}}{|+⟩}_{\mathrm{n}}+\sqrt{{\mu }_{\mathrm{sig}}}/2\left[\right|1_{\mathrm{e}}{0}_{\mathrm{l}}⟩{|\mathrm{\downarrow }⟩}_{\mathrm{elec}}{|+⟩}_{\mathrm{n}}+{\mathrm{e}}^{\mathrm{i}\theta }|0_{\mathrm{e}}{1}_{\mathrm{l}}⟩({|\mathrm{\downarrow }⟩}_{\mathrm{elec}}{|\mathrm{\downarrow }⟩}_{\mathrm{n}}+{|\mathrm{\uparrow }⟩}_{\mathrm{elec}}{|\mathrm{\uparrow }⟩}_{\mathrm{n}})/\sqrt{2}].$$ 3

We then perform the photon erasure with feedback $X^{\left(1-\mathrm{sign}\left[\left(i-i^{\prime }\right)\left(j-j^{\prime }\right)\right]\right)/2}$ on the nucleus, which is equivalent to measuring each temporal mode in the X-basis. This results in the state

$$|{+⟩}_{\mathrm{elec}}{|+⟩}_{\mathrm{n}}+\sqrt{\mu }/2\left[\right|{\mathrm{\downarrow }⟩}_{\mathrm{elec}}|{+⟩}_{\mathrm{n}}+{\mathrm{e}}^{\mathrm{i}\theta }\left(\right|{\mathrm{\downarrow }⟩}_{\mathrm{elec}}{|\mathrm{\downarrow }⟩}_{\mathrm{n}}+\left|{\mathrm{\uparrow }⟩}_{\mathrm{elec}}\right|{\mathrm{\uparrow }⟩}_{\mathrm{n}})/\sqrt{2}],$$ 4

effectively realizing a one-bit teleportation⁴⁰ such that the signal photon differential phase remains imprinted on the nuclear spin state. By measuring the electron spin state in the X-basis and selecting the events with outcome |−⟩_elec only, we then non-destructively herald the presence of a signal photon, regardless of whether it was in the early or late temporal mode (Extended Data Fig. 2), yielding the nuclear state

$$(1+{\mathrm{e}}^{\mathrm{i}\theta })/2{|\mathrm{\downarrow }⟩}_{\mathrm{n}}+(1-{\mathrm{e}}^{\mathrm{i}\theta })/2{|\mathrm{\uparrow }⟩}_{\mathrm{n}}.$$ 5

Experimentally, we determine θ by measuring the nucleus in the Z-basis (Fig. 3c). Increasing the strength $\left|{\alpha }_{\mathrm{LO}}\right|=\sqrt{{\mu }_{\mathrm{LO}}}$ of the LO pulses in the erasure increases the efficiency (Fig. 3d) and reaches a maximum visibility of 0.36(3) at α_LO = 0.32 (Fig. 3e). The visibility is limited by photon loss (including the spin–photon gate efficiency) between the SiV and the final photon detectors at large |α_LO| and detector dark counts at small |α_LO|, as well as MW errors (Extended Data Table 1 and Supplementary Information).

Extended Data Fig. 2. Impact of photon heralding on single stationtemporal mode phase sensing.

Nuclear state measurement result after the sequence shown in Fig. 3a with (red) and without (blue) signal photon heralding.

Extended Data Table 1.

Visibility reduction from different sources

Time-bin sensing corresponds to the experiment in Fig. 3, and Spatial (non-local) sensing corresponds to the experiment in Fig. 4. V_Z is the Z-basis visibility of the initial nucleus $|+⟩$ state, and V_XX is the XX-basis visibility of the initial nuclei $\mid {\mathit{\Psi }}^{-}⟩$ state.

Quantum-memory-assisted interferometry

We now demonstrate the complete quantum-memory-assisted remote phase sensing protocol. Figure 4a,b shows the experimental setup and circuit. We first generate entanglement between the nuclear qubits at each station using SMPHONE gates, reading out the electron qubit states mid-circuit after the SMPHONE gate application to detect MW errors. We then reset the electron spin qubits on the Bloch sphere equator: $|{+}_{{\mathrm{e}}_{\mathrm{L}}}{+}_{{\mathrm{e}}_{\mathrm{R}}}⟩{|{\Psi }^{-}⟩}_{{\mathrm{n}}_{\mathrm{L}},{\mathrm{n}}_{\mathrm{R}}}$. After this, the signal light is collected by performing an SMSPG at each station (Fig. 4b), resulting in

$$\begin{array}{l}[|0_{\mathrm{L}}{0}_{\mathrm{R}}⟩|{+}_{{\mathrm{e}}_{\mathrm{L}}}{+}_{{\mathrm{e}}_{\mathrm{R}}}⟩+\sqrt{\mu /2}(|0_{\mathrm{L}}{1}_{\mathrm{R}}⟩|{+}_{{\mathrm{e}}_{\mathrm{L}}}{\mathrm{\uparrow }}_{{\mathrm{e}}_{\mathrm{R}}}⟩\\ +{\mathrm{e}}^{\mathrm{i}\varphi }|1_{\mathrm{L}}{0}_{\mathrm{R}}⟩|{\mathrm{\uparrow }}_{{\mathrm{e}}_{\mathrm{L}}}{+}_{{\mathrm{e}}_{\mathrm{R}}}⟩)]{|{\Psi }^{-}⟩}_{{\mathrm{n}}_{\mathrm{L}},{\mathrm{n}}_{\mathrm{R}}}.\end{array}$$ 6

We then erase the reflected photonic mode information with feedback $Z_{\mathrm{L}}^{(1-\mathrm{sign}[i-i^{\prime }])/2}{Z}_{\mathrm{R}}^{(1-\mathrm{sign}[j-j^{\prime }])/2}$ on the nuclei. To implement non-local signal photon heralding, we apply local two-qubit gates to entangle the electron and nuclear qubits, and then measure the electron qubits and keep only the outcomes in which the two-qubit parity is even (↑↑) or (↓↓). The vacuum state yields parity outcomes |↑↓⟩ or |↓↑⟩, and thus these measurements can be discarded (Methods). We note that the non-local heralding is specifically enabled by the nuclear-spin entanglement between the stations, without which the signal photon which-path information would be revealed by the heralding step. We measure the non-local heralding probability as a function of μ_sig, which scales with the probability of at least one photon arriving at the stations (Fig. 4c). We note that μ_sig is defined as the average photon number that reaches the SiVs, so it does not include losses due to in-coupling, such as the circulator (η = 70%) and the nanophotonic cavity fibre coupling (η = 70%).

Fig. 4. Implementation of quantum-memory-assisted interferometry.

a, Signal light collection and routing to photon erasure. The shaded components are those used in the entanglement generation. b, Circuit diagram of the non-local phase sensing protocol implementation. All rotations are around the y-axis. c, Left, non-local signal photon heralding probability on the electron |↑_e↑_e⟩ state as a function of the average signal photon number. Right, non-local signal photon heralding probability on the electron |↓_e↓_e⟩ state as a function of the average signal photon number. d, Nuclear two-qubit XX-parity expectation value as a function of station differential phase ΔΦ_L–R (including LO local phases; Methods) with (red) and without (blue) non-local signal photon heralding, averaged over all μ_sig = 0.25, … ,2. e, Nuclear parity measurement visibility as a function of average signal photon number μ_sig arriving at the SiVs with (red) and without (blue) non-local signal photon heralding. The solid blue (dashed red) curve corresponds to the visibility scaling without (with ideal) heralding, and the solid red curve corresponds to imperfect heralding due to mis-heralding (see Methods for more details). Error bars in c–e are 1 s.d.

The resulting nuclear state (|↓↑⟩ − e^±iϕ|↑↓⟩)/√2 contains the phase information ϕ that can be extracted by evaluating the two-qubit nuclear XX parity through local measurements (Fig. 4b,d). Figure 4d shows that the nuclear parity amplitude improves with the non-local signal photon heralding, demonstrating the benefit of filtering out the vacuum fluctuation noise. By changing the signal photon number μ_sig and measuring the nuclear XX parity without non-local heralding, we find the visibility decreases with smaller μ_sig in which vacuum contributions dominate. With non-local heralding, the visibility improves to 0.090(26) averaged over all μ_sig in Fig. 4e (from 0.031(18) without heralding). With perfect single-photon heralding, we would expect the signal visibility to remain roughly constant as a function of μ_sig (with an expected decrease due to multi-photon contamination for large μ_sig ≳ 1), but due to errors in the pre-generated Bell state, there is a probability (about 30%) to mis-herald photons (equivalent to ‘dark-count detection’ of a photon when there was none, red curve in Fig. 4e), which reintroduces sensitivity to vacuum fluctuations at lower μ_sig. We note that the average data collection rate over all μ_sig = 0.25, … , 2 is about 12 mHz, set by a 0.41 Hz entanglement rate and further reduced by the photon mode erasure and non-local heralding success probabilities.

The visibility enhancement due to vacuum filtering translates to an improved interferometric SNR scaling³ (Methods and Extended Data Fig. 5). We note, however, that mis-heralding events reduce the SNR scaling when the effective mis-heralding probability ${\tilde{\epsilon }}_{\mathrm{mh}}$ is larger than the signal μ_sig. Improving the Bell state fidelity is thus central to increasing the range for which optimal SNR can be maintained in the weak-signal regime (Methods and Extended Data Table 1).

Extended Data Fig. 5. Signal to noise ratio (SNR) of the remote phase measurement in function of μ_sig with (in red) and without (in blue) non-local heralding (NLH).

The SNR is calculated by taking the square root of the Fisher information $\sqrt{{\mathcal{F}}_I}$. The solid blue (dashed red) curve corresponds to the SNR scaling without (with ideal) heralding, and the solid red curve corresponds to the scaling with imperfect heralding due to mis-heralding.

Finally, we extend the effective baseline by placing spools of fibre between the stations within the entanglement interferometer (Fig. 5a). The additional length of fibre increases the phase noise of the entanglement interferometer, worsening the locking performance. We maintain a nuclear qubit Bell pair fidelity well above the verifiable entanglement limit with F = 0.63(3) for an inter-station fibre length of 1.55 km (3.1 km fibre length inside the interferometer) (Fig. 5b). Repeating the non-local phase sensing protocol (Fig. 4b) with a fibre length of 1.55 km, we measure a ϕ-dependent two-qubit nuclear parity oscillation visibility of 0.11(4) (Fig. 5c). As after the entanglement generation all operations are local, these steps are not affected by the increased distance between the stations, although there is an increased overhead resulting from the entanglement generation itself.

Fig. 5. Extending the baseline of the interferometer to 1.55 km.

a, Additional spools of length L added in the entanglement interferometer. A baseline length of L requires a fibre length of 2L within the entanglement interferometer. b, Nucleus–nucleus entanglement Bell state tomography for different baseline lengths. The dashed line shows the classical limit of 1/3. c, Nuclear two-qubit XX-parity expectation value as a function of station differential phase ΔΦ_L−R for a baseline of 1.55 km. Error bars in b and c are 1 s.d.

Outlook

Our experiments demonstrate entanglement-assisted non-local interferometry by combining key ingredients and techniques, including photon erasure and non-local, non-destructive photon heralding, an enabling step for removing vacuum fluctuations to achieve optimal phase measurement sensitivity³. Although in our proof-of-concept setting we can extend the measurement baseline to 1.55 km, demonstrating the ability to realize long-baseline quantum memory-assisted interferometry, several system improvements will be necessary to achieve practical gains over a large baseline. Specifically, entanglement rates should be increased using quantum repeaters¹⁸ and entanglement multiplexing^30,41. The number of qubits at each station can be increased with additional devices per station and by incorporating additional quantum memories with ¹³C nuclear spin control^7,42. This would allow for efficient storage of incoming photons, enabling the logarithmic compression method described in ref. ⁵, and the extraction of both temporal⁴³ and polarization⁴⁴ information. Furthermore, using spin–photon phase gates instead of amplitude gates used here opens pathways to higher efficiency deterministic operations⁴⁵ (Supplementary Information). The Purcell-enhanced linewidth of the SiV of approximately 1 GHz constrains the spectral window, but this can be substantially extended by increasing the number of devices per station with efficient and scalable fibre packaging⁴⁶, combined with wavelength division multiplexing⁴⁷, and strain-induced SiV optical frequency tuning⁴⁸.

Our experiments establish a new approach for advancing quantum-enhanced optical imaging by demonstrating coherent storage and manipulation of weak optical signals using quantum devices. Encoding these signals into qubit memories and coupling them to modest-scale quantum-processing units can further allow for the application of advanced quantum algorithms to extract information beyond the reach of direct detection and classical post-processing. For example, by extending the present approach to multiple detectors, quantum algorithms can be used to overcome tomographic constraints and shot-noise accumulation in classical techniques, leading to fundamental improvements of SNR scaling with system dimensionality^10,49. For instance, these systems can be used to improve the performance in demanding imaging tasks such as exoplanet detection¹⁰. Therefore, our experiments provide opportunities for realizing quantum-enhanced imaging in the weak-signal regime, with potentially transformative applications ranging from curved spacetime proper-time interferometry⁵⁰ and deep-space optical communication^39,51 to more general weak-signal imaging tasks^11,52,53. After submission of our paper, a new experimental realization of non-local interferometry based on entangled atomic ensemble quantum memories has been reported⁵⁴.

Methods

Experimental setup

The experiment encompasses two labs, each containing one station with an SiV inside a dilution refrigerator (BlueFors BF-LD250) at about 100 mK and connected as shown in Extended Data Fig. 1. Light signals are prepared in the laser setup, including for entanglement and erasure interferometer locking (NewFocus TLB-6700 Velocity), right station SiV readout, entanglement qubit generation, signal light generation, and erasure LO generation (MSquared SolsTiS Ti:Sapphire), left station SiV readout and filter cavity locking (Toptica DLPro) and SiV de-ionization (Thorlabs Green diode LP520-SF15). All free space and in-fibre acousto-optic modulators (AOM) are driven with 215 MHz. The entanglement qubit, signal light and erasure LO pulses are shaped by modulating the radiofrequency signal sent to in-fibre AOMs. We bridge the frequency difference between the SiVs at each station of Δf_L−R ≈ 10 GHz (Extended Data Fig. 8) by generating sidebands with electro-optic modulators (EOM) driven with a radiofrequency signal at Δf_L−R inside the entanglement interferometer and filtering the light with a Fabry–Perot cavity⁹. Free space AOMs at the left and right stations act both as a switch between the entanglement path and signal-erasing path, as well as frequency shifters for entanglement interferometer phase locking. Photon counts for erasure are detected with pairs of superconducting nanowire single-photon detectors (SNSPD) (Photon Spot) at each station, and entanglement photon heralding clicks are detected with a single-photon avalanche photodiode (APD). We note that the erasure SNSPDs are not instantaneously photon-number-resolving but can effectively resolve photon number when the detector deadtime is much lower than the photon length. All counts are logged with a time tagger (Swabian Instruments Time Tagger Ultra), and two Zurich Instrument HDAWG 2.4 GSa/s arbitrary waveform generators are used for sequence logic, control of the AOMs and EOMs, as well as MW and radiofrequency pulse generation for SiV control.

Extended Data Fig. 1. Experimental setup.

The Toptica laser is set to the left station SiV optical frequency f_L, the Ti:Sa laser to the right station SiV optical frequency f_R and the Velocity laser to f_R + 5 ⋅ FSR, where FSR is the free spectral range of the entanglement filter cavity of  ~46 GHz. APD and SNSPD stand for avalanche photodiode and superconducting nanowire single-photon detector, and FC is filter cavity.

Extended Data Fig. 8. Cavity-QED parameters and SiV properties.

a) Energy level structure of the SiV b) Left station and c) Right station. Left: Measured normalized reflectivity as a function of optical frequency. Right: Extracted cavity QED parameters and transitions: total cavity decay constant κ_tot, cavity input decay constant κ_in, SiV-cavity coupling strength g, cavity resonance frequency ω_c, SiV resonance frequency ω_SiV (for $\mid \downarrow ⟩$), cooperativity C, microwave transitions MW₁, MW₂, and radio-frequency transitions RF₁, and RF₂.

SNR and Fisher information

The SNR can be evaluated through the Fisher information ${\mathcal{F}}_I$ of the measurement, which is equivalent to (SNR)² and bounds the ϕ estimation variance as $\mathrm{var}\left({\varphi }_{\mathrm{est}}\right)\ge 1/{\mathcal{F}}_I$ (refs. ^3,55):

$${\mathcal{F}}_I=\sum _y\frac{1}{P(y\mid \varphi )}{\left(\frac{\mathrm{\delta }P(y\mid \varphi )}{\mathrm{\delta }\varphi }\right)}^2,$$ 7

where for our experiment P(y|ϕ) is the probability of obtaining a nuclear two-qubit parity measurement outcome y for a given ϕ. The probabilities P(y|ϕ) are

$$\begin{array}{l}P\left(\mathrm{discarded}\right|\varphi )=1-P_{\mathrm{succ}}\\ P(\pm \mathrm{parity}|\varphi )=P_{\mathrm{succ}}(1\pm V\cos \left(\varphi \right))/2,\end{array}$$ 8

where V is the visibility of the measurement and P_succ is the probability to herald a photon (including the photon presence probability itself). Using equations (7) and (8) for small visibility V² ≪ 1, we get ${\mathcal{F}}_I\propto P_{\mathrm{succ}}{V}^2$.

For our implemented protocol, $P_{\mathrm{succ}}={\eta }_{\mathrm{erasure}}{\eta }_{\mathrm{herald}}P(n_{\mathrm{photon}}\ge 1)$, where η_{erasure(herald)} are constant factors given by the erasure (signal photon heralding) efficiency. The signal photon heralding efficiency is limited to 50% by the use of amplitude-based SMSPG, but can be increased to 100% by using phase-based gates instead (Supplementary Information). The sequence heralds whether there was at least one signal photon but does not distinguish between single and multi-photon events. As the protocol fails when more than one photon is collected, the visibility is given by $V=\overline{V}P(n_{\mathrm{photon}}=1|n_{\mathrm{photon}}\ge 1)$ (Fig. 4e, dashed red curve), where $\overline{V}$ is the constant overhead factor due to fidelity reduction from imperfect photon erasure, gate errors and initial qubit state fidelities (Extended Data Table 1). For a light signal obeying Poissonian photon-number statistics (as our signal results from an attenuated laser with scrambled local phase but constant intensity) with average photon number μ_sig arriving at the stations, the Fisher information is

$${\mathcal{F}}_I={\eta }_{\mathrm{erasure}}{\eta }_{\mathrm{herald}}{\overline{V}}^2\frac{{\mu }_{\mathrm{sig}}^2{\mathrm{e}}^{-2{\mu }_{\mathrm{sig}}}}{1-{\mathrm{e}}^{-{\mu }_{\mathrm{sig}}}},$$ 9

which reduces to ${\mathcal{F}}_I\propto {\mu }_{\mathrm{sig}}$ for small signal μ_sig ≪ 1.

Without signal photon heralding, $P_{\mathrm{succ}}={\eta }_{\mathrm{erasure}}$ and $V={\eta }_{\mathrm{herald}}$ $\overline{V}P\left(n_{\mathrm{photon}}=1\right)$ (Fig. 4e, solid blue curve), so that the resulting Fisher information is

$${\mathcal{F}}_I={\eta }_{\mathrm{erasure}}{\left({\eta }_{\mathrm{herald}}\overline{V}\right)}^2{\mu }_{\mathrm{sig}}^2{\mathrm{e}}^{-2{\mu }_{\mathrm{sig}}},$$ 10

which reduces to ${\mathcal{F}}_I\propto {\mu }_{\mathrm{sig}}^2$ for μ_sig ≪ 1. This precisely shows that the key feature that enables SNR scaling enhancement is the non-destructive non-local signal photon heralding. This step, enabled by pre-generated entanglement, is what gives the remote phase sensing protocol its non-local character.

By contrast, when using non-local signal photon heralding, mis-heralding events (with probability ε_mh) corrupt the signal, modifying P_succ to ${\eta }_{\mathrm{erasure}}P(\mathrm{herald}\cup \text{mis-herald})$ and V to $\overline{V}P(n_{\mathrm{photon}}=$ $1|(\mathrm{herald}\cup \text{mis-herald}))$ (Fig. 4e, solid red curve). This results in

$${\mathcal{F}}_I={\eta }_{\mathrm{erasure}}{\eta }_{\mathrm{herald}}{\overline{V}}^2\frac{{\mu }_{\mathrm{sig}}^2{\mathrm{e}}^{-2{\mu }_{\mathrm{sig}}}}{1-{\mathrm{e}}^{-{\mu }_{\mathrm{sig}}}(1-{\epsilon }_{\mathrm{mh}}/{\eta }_{\mathrm{herald}})},$$ 11

which scales as ${\mathcal{F}}_I\propto {\mu }_{\mathrm{sig}}$ for ${\mu }_{\mathrm{sig}}\gtrsim {\tilde{\epsilon }}_{\mathrm{mh}}$ (where ${\tilde{\epsilon }}_{\mathrm{mh}}={\epsilon }_{\mathrm{mh}}/{\eta }_{\mathrm{herald}}$ is the effective mis-heralding probability) but curves down to ${\mathcal{F}}_I\propto {\mu }_{\mathrm{sig}}^2$ for ${\mu }_{\mathrm{sig}}\lesssim {\tilde{\epsilon }}_{\mathrm{mh}}$ (Extended Data Fig. 5 and Supplementary Information). The visibility improvement from signal photon heralding can be seen both in Fig. 4d and Extended Data Fig. 2.

SMPHONE gate error detection

Similarly to the PHONE gate^8,9, the SMPHONE gate entangles a photonic qubit with the nuclear spin—but in the Fock basis instead of the time-bin basis—mediated by the electron spin. Starting the nucleus and the photon in superposition states (|↓⟩ + |↑⟩)_n/√2 and (|0⟩ + |1⟩)_phot/√2 and the electron in the |↑⟩_e state, we implement the SMPHONE gate (Extended Data Fig. 3a):

$${\left(\right|0⟩+|1⟩)}_{\mathrm{phot}}{|\mathrm{\uparrow }⟩}_{\mathrm{e}}{\left(\right|\mathrm{\downarrow }⟩+|\mathrm{\uparrow }⟩)}_{\mathrm{n}}\to {|\mathrm{\uparrow }⟩}_{\mathrm{e}}({|0⟩}_{\mathrm{phot}}{|+⟩}_{\mathrm{n}}+{|1⟩}_{\mathrm{phot}}{|\mathrm{\downarrow }⟩}_{\mathrm{n}}/\sqrt{2}).$$ 12

Here the nucleus is entangled with the photon and the electron is always in the |↑⟩ state, unless a MW error occurred during the SMPHONE gate operation. We note, however, that the nucleus does not directly interface with light, and the nucleus–photon entanglement generation is mediated by the electron (which does interface with light), so that MW errors on the electron translate to errors on the nucleus–photon entangled state. Therefore, by measuring the electron state we can detect these MW errors and post-select on |↑⟩ results to boost the nucleus entanglement fidelity (Extended Data Fig. 3b). As measuring the electron in the |↑⟩ state (as opposed to the |↑⟩ state) does not cause decoherence of the ²⁹Si nucleus state⁸, we can perform error detection mid-circuit, as in Fig. 4b.

Extended Data Fig. 3. SMPHONE gate operation.

a) SMPHONE gate sequence, after which the electron state can be measured to detect errors. b) Nuclear Bell state fidelity for different baseline lengths with and without error detection.

Entanglement interferometer phase

The entanglement interferometer phase δφ_e stability (Extended Data Fig. 4c) is limited by noise from the fibre link between the two labs in which the stations are located and vibrations from the pulse tube motor-head of the dilution refrigerators. We reduce phase noise introduced in the fibre link by packaging the fibre in a rubber tube filled with sand for vibration damping (Extended Data Fig. 4a). We limit the phase noise introduced by the pulse tube motor-head by clamping the motor-head between aluminium plates padded with foam. To reduce vibrations guided to the dilution refrigerator through the flexline connecting to the pulse tube head, we clamp the flexline in bags of sand that further damp vibrations (Extended Data Fig. 4b). With this passive stabilization, the interferometer phase auto-correlation time increases from about 4 ms to 500 ms (Extended Data Fig. 4c). When we add the two spools of 1.5 km for the long-baseline operation (Fig. 5), the auto-correlation time of the entanglement interferometer decreases again (Extended Data Fig. 4c, inset).

Extended Data Fig. 4. Entanglement interferometer phase passive stabilization and active locking.

a) Fiber packaging used for the inter-station labs fiber link. b) Pulse tube motor-head noise isolation. c) Entanglement interferometer accumulated phase auto-correlation function with (green) and without (purple) passive phase stabilization. Inset: zoomed-in auto-correlation curves, including for the interferometer with 2 spools of 1.5 km each. d) Entanglement APD counts as a function of locked interferometer phase lock setpoint.

We then lock the interferometer phase by alternating phase probing with SiV readout every 50 μs. A field-programmable gate array integrates the phase probing light for 1 ms and locks the interferometer phase by adjusting the drive frequency of acousto-optic modulators in each arm, resulting in a locked optical interference visibility of around 0.93 (Extended Data Fig. 4d).

Quantum-memory-assisted interferometry details

After generating entanglement and collecting the signal in the non-local phase sensing protocol, with the resulting state in equation (6), we erase the photonic spatial mode

$$\left[\right|{+}_{{\mathrm{e}}_{\mathrm{L}}}{+}_{{\mathrm{e}}_{\mathrm{R}}}⟩+\sqrt{{\mu }_{\mathrm{sig}}/2}\left(\right|{+}_{{\mathrm{e}}_{\mathrm{L}}}{\mathrm{\uparrow }}_{{\mathrm{e}}_{\mathrm{R}}}⟩+{\mathrm{e}}^{\mathrm{i}\varphi }|{\mathrm{\uparrow }}_{{\mathrm{e}}_{\mathrm{L}}}{+}_{{\mathrm{e}}_{\mathrm{R}}}⟩)]{\mid {\Psi }^{-}⟩}_{{\mathrm{n}}_{\mathrm{L}},{\mathrm{n}}_{\mathrm{R}}}.$$ 13

Then, with local C_nNOT_e and π/2 pulses at each station, we transform the state to

$$\begin{array}{l}\left(\right|{\mathrm{\downarrow }}_{{\mathrm{e}}_{\mathrm{L}}}{\mathrm{\uparrow }}_{{\mathrm{e}}_{\mathrm{R}}}⟩|{\mathrm{\downarrow }}_{{\mathrm{n}}_{\mathrm{L}}}{\mathrm{\uparrow }}_{{\mathrm{n}}_{\mathrm{R}}}⟩-|{\mathrm{\uparrow }}_{{\mathrm{e}}_{\mathrm{L}}}{\mathrm{\downarrow }}_{{\mathrm{e}}_{\mathrm{R}}}⟩|{\mathrm{\uparrow }}_{{\mathrm{n}}_{\mathrm{L}}}{\mathrm{\downarrow }}_{{\mathrm{n}}_{\mathrm{R}}}⟩)/\sqrt{2}\\ +\sqrt{\mu }/2[|{\mathrm{\downarrow }}_{{\mathrm{e}}_{\mathrm{L}}}{\mathrm{\downarrow }}_{{\mathrm{e}}_{\mathrm{R}}}⟩\left(\right|{\mathrm{\downarrow }}_{{\mathrm{n}}_{\mathrm{L}}}{\mathrm{\uparrow }}_{{\mathrm{n}}_{\mathrm{R}}}⟩-{\mathrm{e}}^{\mathrm{i}\varphi }|{\mathrm{\uparrow }}_{{\mathrm{n}}_{\mathrm{L}}}{\mathrm{\downarrow }}_{{\mathrm{n}}_{\mathrm{R}}}⟩)\\ +|{\mathrm{\uparrow }}_{{\mathrm{e}}_{\mathrm{L}}}{\mathrm{\uparrow }}_{{\mathrm{e}}_{\mathrm{R}}}⟩\left({\mathrm{e}}^{\mathrm{i}\varphi }\right|{\mathrm{\downarrow }}_{{\mathrm{n}}_{\mathrm{L}}}{\mathrm{\uparrow }}_{{\mathrm{n}}_{\mathrm{R}}}⟩-|{\mathrm{\uparrow }}_{{\mathrm{n}}_{\mathrm{L}}}{\mathrm{\downarrow }}_{{\mathrm{n}}_{\mathrm{R}}}⟩)\\ +|{\mathrm{\downarrow }}_{{\mathrm{e}}_{\mathrm{L}}}{\mathrm{\uparrow }}_{{\mathrm{e}}_{\mathrm{R}}}⟩(1+{\mathrm{e}}^{\mathrm{i}\varphi })|{\mathrm{\downarrow }}_{{\mathrm{n}}_{\mathrm{L}}}{\mathrm{\uparrow }}_{{\mathrm{n}}_{\mathrm{R}}}⟩+|{\mathrm{\uparrow }}_{{\mathrm{e}}_{\mathrm{L}}}{\mathrm{\downarrow }}_{{\mathrm{e}}_{\mathrm{R}}}⟩(1-{\mathrm{e}}^{\mathrm{i}\varphi })|{\mathrm{\uparrow }}_{{\mathrm{n}}_{\mathrm{L}}}{\mathrm{\downarrow }}_{{\mathrm{n}}_{\mathrm{R}}}⟩],\end{array}$$ 14

so the electron two-qubit parity is even (|↑↑⟩ or |↓↓⟩) only if a signal photon was present, and the probability of measuring these states scales with the probability of at least one signal photon arriving (Fig. 4c). We note that the mis-heralding probability is higher for heralding on the |↓_e↓_e⟩ than the |↑_e↑_e⟩ electron state due to experimental errors accumulating coherently in the |↓_e↓_e⟩ state. The nuclear two-qubit parity oscillation curves with and without non-local signal photon heralding in function of signal phase separated by signal strength are shown in Extended Data Fig. 6. These curves are combined to plot the curve in Fig. 4e. Figure 4c has 9,898 successful experimental trials for a 16 h 24 min run time. Figure 4d has 6,645 successful experimental trials with and 16,270 without non-local heralding for a 155 h 43 min run time. Figure 4e has 3,167 successful experimental trials with and 7,400 without non-local heralding for a 74 h 58 min run time.

Extended Data Fig. 6. Additional data for implementation of quantum memory-assisted interferometry.

For each average photon number μ_sig, the plot shows the nuclear 2-qubit parity ⟨XX⟩ in function of signal phase ΔΦ_L−R with (in red) and without (in blue) non-local heralding.

The phases of the LO pulses used in the photon erasure step at the left and right stations are also imprinted onto the nuclear state, so that the relevant phase is ΔΦ_L−R = δϕ_L − δϕ_R with δϕ_L the differential phase between the signal and the LO pulses at the left station and δϕ_R at the right station. ΔΦ_L−R reduces to simply ϕ when the phases of the LO pulses are locked to one another. However, it is enough to simply know the phases through calibration measurements, which we perform every four experimental trials (Supplementary Information).

Signal state preparation

A weak coherent state, used to emulate the signal light, is split on a beam splitter and sent to both stations. The optical phases δϕ_L and δϕ_R at the left and right stations, respectively, are not phase-locked and therefore fluctuate freely. After interacting with the SiVs, the signal undergoes a photon-erasure step, during which it is combined with an LO. The LOs at the two stations are also not phase-locked.

Rather than stabilizing the phases, we probe them using bright reference lasers for each experimental shot. This allows us to determine the phase at each station for every shot. The phase of the LO at each station can be taken as the zero reference, as we measure all quantities relative to it.

For local phases δϕ_L and δϕ_R at the left and right stations, respectively, and differential phase ΔΦ_L−R = δϕ_L − δϕ_R, the photonic state density matrix in the {|00⟩, |01⟩, |10⟩} basis for weak signals μ ≪ 1 is

$$\begin{array}{l}{\rho }_{\mathrm{sig}}=|{\alpha }_L⟩\otimes |{\alpha }_{\mathrm{R}}⟩=|\sqrt{\mu /2}{\mathrm{e}}^{\mathrm{\delta }{\varphi }_{\mathrm{L}}}⟩\otimes |\sqrt{\mu /2}{\mathrm{e}}^{\mathrm{\delta }{\varphi }_{\mathrm{R}}}⟩\\ \approx \left(\begin{array}{ccc}1& {\mathrm{e}}^{-\mathrm{i}\mathrm{\delta }{\varphi }_{\mathrm{R}}}\sqrt{\mu /2}& {\mathrm{e}}^{-\mathrm{i}\mathrm{\delta }{\varphi }_{\mathrm{L}}}\sqrt{\mu /2}\\ {\mathrm{e}}^{\mathrm{i}\mathrm{\delta }{\varphi }_{\mathrm{R}}}\sqrt{\mu /2}& \mu /2& {\mathrm{e}}^{-\mathrm{i}(\mathrm{\delta }{\varphi }_{\mathrm{L}}-\mathrm{\delta }{\varphi }_{\mathrm{R}})}\mu /2\\ {\mathrm{e}}^{\mathrm{i}\mathrm{\delta }{\varphi }_{\mathrm{L}}}\sqrt{\mu /2}& {\mathrm{e}}^{\mathrm{i}(\mathrm{\delta }{\varphi }_{\mathrm{L}}-\mathrm{\delta }{\varphi }_{\mathrm{R}})}\mu /2& \mu /2\end{array}\right).\end{array}$$

During data analysis, we group the measurements according to the same differential phase ΔΦ_L−R = δϕ_L − δϕ_R between the stations. On averaging, terms depending on the individual local phases δϕ_L and δϕ_R cancel out, whereas terms depending on the fixed differential phase remain (Extended Data Fig. 7), so that the density matrix becomes

$$\begin{array}{l}{\tilde{\rho }}_{\mathrm{sig}}\approx \left(\begin{array}{ccc}1& 0& 0\\ 0& \mu /2& {\mathrm{e}}^{-\mathrm{i}\mathrm{\Delta }{\Phi }_{\mathrm{L}-\mathrm{R}}}\mu /2\\ 0& {\mathrm{e}}^{\mathrm{i}\mathrm{\Delta }{\Phi }_{\mathrm{L}-\mathrm{R}}}\mu /2& \mu /2\end{array}\right)\approx {\rho }_{\mathrm{th}},\end{array}$$

which is approximately the density matrix of a weak thermal state (with mean photon number μ and complex visibility $g={\mathrm{e}}^{\mathrm{i\Delta }{\Phi }_{\mathrm{L}-\mathrm{R}}}$) (ref. ³). Although the photon number statistics of the light remain Poissonian rather than thermal, in the low-mean-photon-number regime and to first photon order, the corresponding density matrices are effectively the same.

We note that in our work, the generated signal visibility is unity, but this is often not the case in practical astronomical imaging³. The value of the visibility can, in principle, also be estimated with our protocol, although care must be taken to distinguish the intrinsic signal visibility from the measurement protocol infidelity, as well as from environmental noise sources. With additional quantum resources, more advanced quantum-processing techniques¹⁰ can be used to extract the intrinsic visibility from unknown and fluctuating noise contributions without needing to reconstruct the noise itself.

Online content

Any methods, additional references, Nature Portfolio reporting summaries, source data, extended data, supplementary information, acknowledgements, peer review information; details of author contributions and competing interests; and statements of data and code availability are available at 10.1038/s41586-026-10171-w.

Author contributions

P.-J.S., Y.-C.W., M.S. and A.S. planned the experiment. U.Y., B.M. and D.R. fabricated the devices. P.-J.S., Y.-C.W., M.S., Y.Q.H., F.A.A., E.K. and A.S. built the setup, and P.-J.S., Y.-C.W. and M.S. performed the experiment. P.-J.S., Y.-C.W., M.S., Y.Q.H., F.A.A., E.K., S.G. and G.B. analysed the data and interpreted the results. All work was supervised by H.P., M.L., J.B., A.S. and M.D.L. All authors discussed the results and contributed to the paper.

Peer review

Peer review information

Nature thanks Saikat Guha, Yunkai Wang and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

Data availability

All data related to the current study are available from the corresponding author upon reasonable request.

Code availability

All analysis codes related to the current study are available from the corresponding author upon reasonable request.

Competing interests

M.D.L. is a co-founder, shareholder and chief scientist of QuEra Computing. B.M. is the director of photonics technologies at IonQ. D.R. is a senior research science manager of IonQ. J.B. is a senior staff research scientist of IonQ. M.D.L. has no current financial or personal relationships with IonQ. M.D.L. previously provided consulting services to LightSynq, which was acquired by IonQ in May 2025. This past relationship did not influence the research reported in this paper.

Extended data

is available for this paper at 10.1038/s41586-026-10171-w.

Supplementary information

The online version contains supplementary material available at 10.1038/s41586-026-10171-w.