Coherent State Assisted Entanglement Generation Between Quantum Memories

The gist: Weak-coherent-state-assisted protocols can generate entanglement near-deterministically between reflective-cavity-based quantum memories at a success rate that exceeds the 50% limit associated with single-photon-mediated schemes, offering a pronounced benefit in the low-channel loss regime and enabling deterministic multi-partite entanglement generation.

Introduction and Motivation

The vision for the quantum internet is to enable quantum communications between a diverse group of users supporting various applications [1–5]. Central to this vision, is the need for quantum networks that can reliably and faithfully establish distributed entanglement among quantum memories [6]. Most currently studied quantum network architectures are based on pairwise entanglement generation between two quantum memory banks separated by an optical channel, paired with entanglement distillation [5, 17, 18]. Experimental demonstrations rely on the transmission and interference of single photons [19–22]. Single-photon protocols often suffer from a low pairwise heralding success probability which results in an exponential drop in the rate at which multi-qubit entanglement is generated costly [40]. High-rate multi-qubit entanglement generation is crucial for many applications, such as preparing local ancillas for measurement-based quantum error correction for communications [41–43], quantum repeaters for multi-site entanglement [44–46], and memory-assisted quantum switches or routers [47–49].

Coherent Two-Way (CTW) Protocol

The CTW protocol is a coherent-two-way (CTW) protocol driven by the remote nondestructive parity measurement [51, 54, 66]. In this scheme, both sites first prepare the qubit state of their own quantum memory to +⟩ state and interact their memory qubit with a locally generated coherent state α⟩ in an optical mode [Fig. 1(c)]. The interaction imparts an optical πphase shift to the reflected optical mode, conditioned on the qubit state of the quantum memory, which has been theoretically investigated [34, 57–60] and experimentally realized in multiple platforms [35, 61–65]. The success outcomes are heralded by a click pattern from a 50:50 beamsplitter followed by one PNR detector at each output port (labeled D1 and D2) [Fig. 1(c)]. The first four outcomes are the success outcomes as they herald the generation of entanglement between the quantum memories – the corresponding Bell states are shown at the bottom of Fig. 1(c). The lower bound to the achievable distillable entanglement generation rate, RCTW, can be derived by evaluating the product of the total heralding probability and Hashing bound per heralded success [2]. For a given η, there is an optimal initial coherent state amplitude α⟩ that maximizes RCTW [3]. A larger amplitude α increases the success probability but also increases the information leakage to the environment (which degrades the Hashing bound) [3]. For a channel with very low loss, i.e., η → 1, the Hashing rate approaches 1 ebit/ch for the optimal α, which means the entanglement generation becomes deterministic, 3 dB higher than the highest-reachable rate of single-photon protocols [3].

Coherent One-Way (COW) Protocols

The COW protocol involves subsequent interactions between the traveling coherent pulse and the quantum memories [Fig. S2]. In this scheme, Alice generates a coherent state at amplitude α, which interacts with her quantum memory first [Fig. 1(d)]. The photonic state is transmitted over a lossy optical channel of transmissivity η to Bob [S17]. After the interaction with the second memory at Bob’s site, he carries out state discrimination (to distinguish the phase of the coherent state) [S20]. The unambiguous state discrimination (USD) measurement-assisted approach involves interfering the undetected photonic mode with a coherent pulse of amplitude α√η⟩ LO, generated by a local oscillator [S19]. The probability of success for the COW–USD protocol is given by PCOW,USD(α, η) = 1 − e −2ηα 2 [S23]. This protocol yields an achievable distillable entanglement generation rate of RUSD COW(α, η) = (1 − exp(−2ηα 2)) / (1 + h2(1 + exp(−4(1−η)α 2)) [S23].

Impact of Non-Idealities

The primary sources of non-ideality that are likely to limit performance are excess noise in the channel (e.g., from the dark-click probability of the detectors Pd), the subunity mode matching visibility (of the temporal-spectral mode), and imbalanced power at the beamsplitter caused by imperfect calibration [Fig. 2]. In excess noise, for example, for the CTW protocol, excess noise leads to three possible errors: (1) D1 and D2 click at the same time, causing a measurement ambiguity and decreasing the success rate; (2) An extra photon is detected at D1 or D2 measurement, switching the photon number parity, which causes a bit-flip error on the final state and (3) A dark count triggers either D1 or D2 when all photons are lost in transmission, equivalent to the occurrence of the depolarizing error [S30a]. For both protocols, pulse power interference with sub-unity visibility decreases the success rate by reducing the chance of detecting a photon at the bright port (i.e., the port that is supposed to collect an output coherent state after a constructive interference) or triggering an ambiguous click at the dark port (i.e., the port that is supposed to obtain a vacuum output after a destructive interference) [S30b].

Conclusion

The three discussed coherent-state-assisted entanglement distribution protocols between two reflective-cavity-based quantum memories are near-term solutions that could boost both the ebit rate per channel use and success probability from 0.5 to 1.0, which surpasses the previous bound of using single-photon protocols in the low-loss regime [Conclusion]. Based on this, they further enable the near-deterministic entanglement generation among more than two quantum memories, which can serve as a local entanglement resource hub for quantum network applications [5] and measurement-based quantum computing [78, 79]. Moreover, since the cavity-coupled quantum memory supports both the single-photon protocols and the coherent-state-assisted protocols, such a monolithic platform may be enough to demonstrate the errorcorrection-empowered quantum repeater with constant overhead inside a quantum network [75].

Supplemental Information

The state φ5⟩ is not an entangled state of the memories. Consequently the total probability of success is given by PCTW(α, η) = exp(−2√ηα 2) [S11e]. Tracing out the environment mode gives us (for the state φ1⟩) TrE(φ1⟩⟨φ1) = 1/2 0, 0⟩⟨0, 0M + 1, 1⟩⟨1, 1M + (0, 0⟩⟨1, 1 + 1, 1⟩⟨0, 0) × Θ(−α q 2√η, αq 2√η) /2 = (1 + T) squared Φ+⟩⟨Φ+M + (1 − T) squared Φ−⟩⟨Φ−M ≡ ρ(1) [S13]. The hashing rate of the CTW protocol is given by RCTW(α, η) = (1 − exp(−2√ηα 2)) / (1 − h2(1 + exp(−4(1−√η)α 2)) 2 [S15]. The analysis of the Dolinar receiver is a bit more complicated due to the requirement of feedback-assisted displacement - we shall omit it for the purposes of the current analysis [S26]. The effect of input power mismatch can be used to model an imperfect 50:50 beamsplitter of angle π/4 ± ε where ε → 0 [S36a]. The key difference is that the imperfect beamplitter leads to states with a lowered probability of success without any additional dephasing [S37]. The ‘dephasing strength’ (encapsulated in the T term or the Θ(·, ·) functions in the density operator definition) is only dependent on the input powers and the channel loss; imperfect beamsplitting ratio does not affect this [S37]. For both protocols, pulse power interference with sub-unity visibility decreases the success rate by reducing the chance of detecting a photon at the bright port (i.e., the port that is supposed to collect an output coherent state after a constructive interference) or triggering an ambiguous click at the dark port (i.e., the port that is supposed to obtain a vacuum output after a destructive interference) [S37]. The analysis of excess noise is straightforward - since our protocols rely on distinct click patterns where a specific detector (at the output port of the interferometer) clicks or doesn’t click, in the presence of noise, the output state will be a weighted mixture of the potential outcomes [S30a].

1. The system can achieve near-deterministic entanglement generation between quantum memories at a success rate that exceeds the 50% limit associated with single-photon-mediated schemes, specifically in the low-channel-loss regime where it outperforms single-photon protocols, as shown in Fig. 1(a).

2. The improved system can prepare a GHZ state among an array of memories, inferring that it yields an exponential speed-up compared to previous single-photon-based protocols, which is crucial for preparing local ancillas for measurement-based quantum error correction for communications.

3. The AI can utilize the Coherent Two-Way (CTW) protocol to generate entanglement with a rate bounded by RCTW = (1 − exp(−2√ηα2)) / [1 − h2(1 + exp(−4(1−√η)α2))2] in the low-loss regime, which is 3 dB higher than the highest-reachable rate of single-photon protocols.

4. The system can implement a Coherent One-Way (COW) protocol with Unambiguous State Discrimination (USD) to achieve a hashing rate of RUSD COW = (1 − exp(−2ηα2)) / [1 − h2(1 + exp(−2(1−η)α2))2], which can approach 1 ebit/ch at η → 1.

5. The system can be designed to maintain high-fidelity entanglement generation with sub-unity success probability even in the presence of channel loss, as CTW and COW-USD protocols are described as unambiguous in this regard.

6. The AI can model and mitigate imperfections like input power mismatch, where the resulting state is modified by terms such as β′1 = α q√η(1 + √V) when analyzing the CTW protocol's performance with non-ideal components.

7. The system can be optimized for robustness against excess noise by utilizing the derived final states, such as ρ˜(1) = (1 − Pd)2Pe,CTWρ(1) + Pd(1 − Pd)Po,CTWρ(2)/Pe,CTW, to determine the resulting entanglement quality in noisy channels.

8. The system can incorporate mode mismatch modeling by accounting for how imperfect mode matching modifies the post-interference state in Eq. (S7) of the CTW protocol, leading to modified outcome probabilities for heralded entangled states.

Generating entanglement deterministically at a capacity-approaching rate is critical for next-generation quantum networks. We revisit previous protocols and model a new coherent-state-assisted protocol variant that can generate entanglement near-deterministically between reflective-cavity-based quantum memories at a success rate that exceeds the 50% limit of single-photon-mediated schemes. The most pronounced benefit is shown in the low-channel-loss regime and persists even with moderate noise. We generalize our protocols from entangling two memories to entangling an array of memories in a GHZ state, and show an exponential increase in all-success probability compared to previous single-photon-based protocols under a common restart-on-failure rule.