Entanglement distribution with minimal memory requirements using time-bin photonic qudits

Quantum networks bring new opportunities for secure communication [1, 2], distributed sensing [3, 4, 5], and distributed quantum computing [6, 7]. The ability to faithfully transmit quantum information over long distances is a key prerequisite for the construction of large-scale quantum networks. Quantum teleportation [8] provides a means to overcome transmission loss and noise given that high-quality entanglement can be created between the sender and the receiver. As such, there have been significant efforts for creating entanglement between remote qubit systems across a plethora of physical systems including trapped ions [9], diamond defect centers [10, 11], neutral atoms [12, 13] and quantum dots [14].

In canonical entanglement generation protocols, spin-photon entanglement is first created and then extended to spin-spin entanglement by either a photonic Bell measurement [9, 10, 11, 13, 14] or a direct spin-photon interaction [12]. Successful entanglement generation is heralded by the detection of transmitted photons, and the quantum state of the spin qubits need to stay coherent for at least the time of one entanglement generation attempt, which is generally set by the signaling time between the nodes.

Key functionalities in a quantum network such as entanglement purification [15, 16] and multi-qubit state teleportation [17, 18, 19] require the availability of more than a single entangled qubit pair to be executed. The latter, in particular, when quantum information encoded across multiple physical qubits in a quantum error correcting code has to be transmitted since all physical qubits of the code has to be transmitted simultaneously in order to perform error correction locally.

Multi-qubit memories are therefore necessary, and the general approach so far has been to consider the generation of multiple entangled pairs either sequentially [20, 21] or in parallel [22, 18] using the same photonic qubit based protocols as for the generation of a single entangled pair. High transmission loss, however, leads to much more demanding requirements for the coherence time of the quantum memories than for the single pair setup. For a feasible generation rate, successfully entangled pairs have to be stored in quantum memories while they wait for remaining pairs to entangle successfully. As a result, the required memory time increases as the inverse of the photon transmission probability, which decreases exponentially with the distance between the nodes for standard optical fiber propagation. To add insult to injury, the continued entanglement attempts of neighboring qubits may further decrease the memory time of a successful pair due to unwanted crosstalk [23].

In this article, we propose a fundamentally different scheme for the generation of multiple entangled pairs between two distant qubit registers, which exploits the use of high-dimensional photonic qudit states. Specifically, we show that a single photonic time-bin qudit in dimension $d=2^{m}$ allows for the heralded and simultaneous generation of $m$ entangled pairs between two distant multi-qubit registers. Photonic qudit encodings have previously been considered for quantum networks [24] and quantum repeaters [25] due to their higher information capacity  [26], which can lead to higher loss tolerance and more efficient quantum key generation [27]. In our qudit-based protocol, the required coherence time of the qubit memories is independent of the transmission probability and amounts only to the time of a single entanglement generation attempt. Furthermore, the rate of entanglement generation in our protocol is more robust to transmission loss between the distant registers than other qubit approaches. Our scheme opens up new opportunities for the generation of multiple high-fidelity entangled qubit pairs in an extended quantum network, where transmission loss between the nodes becomes a limiting factor both on generation rate and quantum memory requirements.

A high-level sketch of our protocol is shown in Fig. 1(a) to be compared with a similar qubit-based protocol in Fig. 1(b). We note that multiple other qubit-based protocols exist [9, 10, 11, 13, 14] but they all share the same fundamental scaling of success probability and necessary coherence time with the transmission probability. Without loss of generality, we can therefore compare our qudit protocol with the specific qubit-based protocol considered here.

We will first describe our qudit protocol at an abstract level to convey the general idea and then proceed with discussion of specific implementations in sec. III. In the first step, a photonic qudit is generated. We specifically consider a photonic time-bin qudit prepared in an equal superposition across $2^{m}$ modes as described by the state

where $|l\rangle_{ph}$ denotes the state where the photon is located in the $l$’th time-bin.

The photon state $|\psi\rangle_{ph}$ will then interact with the first $m$-qubit register (Alice) which is initialized in state $|0\rangle^{\otimes m}$ in the following way: If the photon is in state $|l\rangle$ where the binary representation of the decimal value $l$ is $[l]_{10}=[l_{m-1}l_{m-2}...l_{0}]_{2}$ ($l_{i}\in\{0,1\}$), the photon will interact with the $i$’th qubit of the register and flip it from $|0\rangle$ to $|1\rangle$ if $l_{i}=1$. If $l_{i}=0$, the qubit remains in state $|0\rangle$. In section III, we will outline how such an interaction can be achived with optical switches and cavity-based scattering gates [28, 29]. This interaction scheme results in the following transformation

where $|1_{l}\rangle_{A}=|l_{m-1}\rangle|l_{m-2}\rangle...|l_{1}\rangle|l_{0}\rangle_{A}$ as shown in Fig. 2(a). For example, a state $|5\rangle_{ph}\otimes|000\rangle$ would transform to

as illustrated in Fig. 2(b). For the input state in Eq. (1), the collective state after the photon interacting with Alice’s register is

which is a maximally entangled state between the $2^{m}$ dimensional time-bin photon and the $m$-qubit register.

In the next step, the photonic qudit is sent to Bob’s register by means of direct transmission. In practice, the photon most likely will get lost during the transmission, but we will continue our description assuming that the photon is successfully transmitted since we will herald the protocol on a final detection of the photon on Bob’s side.

On Bob’s side, the photonic qudit will interact with his $m$-qubit register in the same way as described for Alice’s register. Thus, the state of both qubit registers (Alice and Bob) and the photonic qudit after the interaction will be

where $|1_{l}\rangle_{B}=|l_{m-1}\rangle|l_{m-2}\rangle...|l_{1}\rangle|l_{0}\rangle_{B}$ denotes the state of Bob’s register.

We now wish to perform a measurement that confirms that the photonic qudit was transmitted and heralds the simultaneous generation of entanglement between the two qubit registers. The measurement should confirm the arrival of the photon without extracting its time-bin information in order to avoid collapsing the state of the two $m$-qubit registers to a product state. To this end, Bob can perform a generalized X-basis measurement on the photonic qudit, which amounts to projecting the photon state on to the time-bin Fourier basis states defined as

where $l=0,1,...,2^{m}-1$. Such a measurement can be implemented using optical switches and linear optics as we detail in Appendix D). Successfully detecting the photonic qudit in any of the Fourier basis states will herald the entangling operation and prepare the two qubit registers in a state

up to single-qubit phase corrections dependent on the measurement outcome (see details in Appendix D)). Thus, Alice and Bob can create $m$ entangled qubit pairs by only transmitting a single photonic qudit between them.

Importantly, the multiple entangled pairs are created simultaneously, and the minimal required coherence time of the qubits in the registers only depends on the duration of a single entanglement attempt. For distant registers in an extended quantum network, this will be determined by the signaling time between the registers. Furthermore, assuming an overall transmission probability of $\eta$ between the two qubit registers, the entanglement generation rate of the protocol scales as $\eta$ since only one single photon is transmitted.

The above protocol should be compared to the standard qubit approach where entanglement between two $m$-qubit registers would be attempted in parallel (see Fig. 1(b)). In the qubit approach, entanglement between a photonic qubit and a memory qubit would be generated separately for each of the memory qubits in the register. For applications that require the presence of all entangled pairs before execution such as the teleportation of a logical qubit composed of multiple physical qubits [17, 18, 19] or entanglement purification protocols [15, 16], successful pairs need to be stored while waiting for the remaining pairs to succeed. Otherwise, simultaneous success in all qubit links would be necessary, leading to a success probability scaling as $\eta^{m}$ and thus causing an impractical rate for entanglement generation. Storing all successful pairs leads to a necessary coherence time of the qubits that scales as $1/\eta$ while the rate would scale as $\sim(2/3)^{\log(m)}\eta$ for $\eta\ll 1$ [30]. Compared with the qubit protocol, our qudit protocol both removes the unfavorable scaling of the memory time with $\eta$ ($t_{0}\ \ {\rm v.s.}\ \ t_{0}/\eta$), and offers a rate that is more robust to transmission loss in terms of scaling with $m$ ($\eta\ \ {\rm v.s.}\ \ (2/3)^{\log(m)}\eta$). However, we do note that the local photon loss at the nodes can be different for the qubit and qudit approach depending on the specific implementation, which we will discuss in the following section.

So far, we have only considered the high-level outline of our proposed protocol. Arguably, we have assumed a number of operations, which might be experimentally more demanding than standard spin-photon entanglement operations. Furthermore, the entangling operation between the photonic qudit and multiple memory-qubits can lead to correlated errors across the entangled qubit pairs that would not be present for parallel qubit approaches.

To investigate the impact of these effects, we consider a specific implementation of our scheme based on single quantum emitters such as neutral atoms [31] or diamond defect centers [29] coupled to optical resonators, which has already demonstrated some of the key functionality required for our scheme. In particular, we will exploit the photon-induced atomic phase gates based on single-sided cavities with a strongly coupled emitter [28]. We will refer to such systems as a spin-cavity system. The same system can also be used to generate high-dimensional photonic time-bin qudits by means of a pulsed, cavity-assisted Raman scheme [32].

A sketch of the proposed implementation is shown in Fig. 3(a). In the first step, a photonic qudit is generated by a pulsed driving of a cavity-assisted Raman transition. Control of the driving power allows to tailor the amplitudes in the qudit state. This is crucial for the specific implementation considered here since the photon will experience different loss depending on which time-bin it is emitted in due to e.g. non-perfect interaction with a different number of spin-cavity systems in the spin-photon entangling step. For an initially even amplitude state (see Eq. (1)), this would decrease the fidelity of the entangled pairs. However, we can compensate the uneven loss by generating a qudit state with uneven amplitudes in such a way that the time-bin experiencing the most loss initially has the highest amplitude. This allows to move the effect from decreasing the fidelity to a modest decrease in rate.

Dominant imperfections in the qudit generation step amount to finite spin coherence time of the emitter, imperfect pulse shaping of the driving laser, spontaneous emission from the excited state, and general photon loss (e.g. from absorption/material scattering). The latter will simply decrease the rate of the protocol given that no photon will be detected in the heralding step. The other imperfections will in general lead to an effective dephasing of the photonic qudit state due to leak of information to the environment about the emission time. Furthermore, imperfect driving may also lead to errors in the amplitude shaping of the qudit state. We refer to Appendix A for a detailed optical model of the photon generation step.

In the second step of the protocol, optical switches are used to route the photon to the spin-cavity systems as dictated the binary encoding of the time-bins (see Fig. 3(b)-(c)). We model the imperfections of the switches as consisting of both general loss and wrong switching (see Appendix C for details). The spin qubits are initialized in the state $|+\rangle=(|0\rangle+|1\rangle)/\sqrt{2}$ and ideally only the state $|1\rangle$ is coupled to an excited state, $|e_{1}\rangle$, by the cavity field. However, for systems such as SiV defect centers and quantum dots, the state $|0\rangle$ would also be coupled off-resonantly by the cavity field to another excited state, $|e_{0}\rangle$ (not shown in Fig. 3(b)). However, by tuning the frequency of the incoming photon and the cavity wrt. to the optical spin transitions, it is possible to realize a high-fidelity controlled phase gate where an incoming photon will flip the atomic state from $|+\rangle\to|-\rangle$. The combination of the switches and the spin-photon gates, followed by Hadamard gates on the spins, leads to the generation of the photon-spin entangled state in Eq. (4) in the ideal case.

The gate will ultimately be limited by the optical splitting between the $|0\rangle\leftrightarrow|e_{0}\rangle$ and $|1\rangle\leftrightarrow|e_{1}\rangle$ transitions, finite cooperativity of the emitter-cavity system, spectral width of the photon, and cavity loss assuming that the emitters and cavities can be tuned into the specific resonance conditions [33]. In Appendix B we give a detailed quantum optical model of the spin-photon interaction encompassing these imperfections.

After transmission to the second register, a similar system of optical switches and spin-cavity systems is used to create the state in Eq. (5) in the ideal case. As detailed above, the protocol is heralded by the generalized X-basis measurement of the qudit state. This can be implemented using optical switches and linear optics as detailed in Appendix D. In particular, the switches are used to convert the time-bin encoding into a spatial encoding, where delay lines are used to ensure temporal coincidence. The quantum Fourier transform can then be implemented by means of a beam splitter circuit [34], and the photon is finally detected with single photon detectors.

Notably, the physical resources of the detection step such as number of switches, beam splitters and detectors scale linearly with the dimension of the photonic qudit. This can, in principle, be circumvented with probabilistic implementations using linear optics where beam splitters can be used to interfere multiple time-bins using a single fiber loop or interfere the qudit photon with local single photons to erase the arrival time as outlined in Ref. [35]. Alternatively, deterministic approaches where a Raman-based absorption scheme in a three level emitter is attempted for each time-bin followed by atomic detection as described in Ref. [4] can be used to make the physical resources independent of the qudit dimension.

In the detection step, we model imperfect switches as in the entangling steps of the protocol (see Appendix C for details). Notably, both loss and wrong switching lead to detectable errors. The former is due to the absence of a detection and the latter is due to a wrongly timed detection. Furthermore, we model loss in the delay lines which leads to a decrease in rate and not fidelity when balanced correctly in the qudit generation step. Finally, we model phase fluctuations in the beam splitters and optical paths as a general dephasing channel on the qudit state, where the dephasing parameter scales with the dimension of the system $m$. We note that the general dephasing channel can also incorporate other phase errors from e.g. general phase instabilities in optical paths

We numerically simulate the simultaneous generation of multiple entangled pairs using our protocol and benchmark it against the parallel qubit approach using similar systems for increasing distance between the two nodes. As shown in Fig. 4, the qudit protocol quickly outperforms the qubit protocol (all-keep) for high-fidelity entanglement generation as the distance and hence transmission loss and signaling time increases. In general, we see that the qudit protocol has the same scaling in fidelity as the one-shot qubit protocol due to similar coherence time requirements while being as efficient as the all-keep qubit protocol.

In our simulations, we have assumed the spin qubits to be subject to pure dephasing with a dephasing time of $T_{\phi}=5{\rm ms}$ and a finite-temperature amplitude damping channel with decoherence time of $T_{1}=10{\rm ms}$ [36, 37]. In the qudit generation, we have assumed to be limited by amplitude and phase fluctuations in the laser modelled as Gaussian noise with variances $\sigma_{a}^{2}=\sigma_{p}^{2}=0.01$. We have assumed 10% loss for the photonic switches and 1% switching error, 5% loss in the input/output coupling from the cavity, a cooperativity of C=100 for the spin-cavity system and negligible coupling of the $|0\rangle$ state. We note that while our qudit protocol is more robust to transmission loss than the qubit approach, it is less robust to local loss. Letting $\eta_{0}$ be the efficiency of transmission of the photon when interacting with a single switch and cavity, the total transmission of the qudit protocol will scale $\eta_{0}^{m}$. Nonetheless, we see from Fig. 4 that even for $\eta_{0}\sim 86\%$ and $m=5$, the qudit protocol significantly outperforms the qubit protocol for high-fidelity entanglement generation for distances $\gtrsim 20$ km due to the much relaxed requirements on the qubit coherence times.

We also include a general qudit dephasing channel modelled as random phase shifts of the time-bin states following a Gaussian distribution with zero mean and variance $\sigma=0.1m$. We assume the scaling with $m$ to account for the additional complexity of the interferometer and phase instability in optical paths (See Appendix D for details). This affects the fidelity of the qudit protocol resulting in a worse performance as $m$ increases as visible from Fig. 4. We note that this increased dephasing, however, is somewhat arbitrary given that it models a technical noise and not a fundamental one.

In our simulations, we have assumed that the time of the local operations is negligible compared to the signaling time between the two stations, which is a valid assumption for extended networks. Node distances for realistic quantum networks are usually considered to be tens of kilometres corresponding to signaling times around 0.1-1 ms [38, 39, 40]. This is, in general, much longer than the time of local operations, which sets the duration of a photonic time-bin. Purcell enhanced emission and fast switching can be envisioned leading to time-bins in the regime of 10 nanoseconds [41] and the time of local operations thus only become relevant for qudit dimensions of $\gtrsim 10^{3}$ corresponding to $m\gtrsim 10$. We note that in practice $m$ will, however, most likely be limited to $<10$ due to the growing complexity of the implementation such as the generation of the many time-bins as well as calibration and phase stabilization of the optical path. Thus, the time of one entanglement generation attempt is $T_{\text{gen}}\sim L/c$, where $L$ is the distance between the stations and $c$ is the speed of light in an optical fiber. For fiber propagation, the transmission efficiency of a photon between the stations will be $\eta_{\text{f}}=e^{-L/L_{\text{att}}}$, where $L_{\text{att}}$ is the attentuation length of the fiber. In our simulations, we have assumed $L_{\text{att}}=20$ km corresponding to fiber loss for telecom light. We note that for systems such as diamond defect centers and neutral atoms, frequency conversion from optical to telecom would be required.

It is clear that if $T_{\text{coh}}\gg T_{\text{gen}}e^{L/L_{\text{att}}}$, the qudit protocol does not offer much advantage compared to the standard qubit approaches. We note, however, that this requires the coherence time to increase exponentially with the distance. This exponential dependence can be circumvented either with the use of quantum repeaters [18] or by employing multiplexing [42] where the number of qubit memories far exceeds the number of desired Bell pairs. In the latter case, the required number of memories will also increase with the transmission loss. Both approaches can, in principle, be combined with the proposed qudit protocol for entanglement generation to further alleviate memory requirements and decrease resources. In particular, both for entanglement purification in 1st generation repeaters and in the operation of 2nd generation repeaters, multiple and high-fidelity Bell pairs are required between neighboring stations [18, 19]. Employing the qudit protocol to alleviate the memory requirements can lead to longer distances between links in the repeater thereby also reducing the overall resources for long-distance entanglement distribution. This, in particular, may also be beneficial for highly-lossy links in satellite-based entanglement distribution [43].

We have proposed a novel entanglement distribution protocol using photonic qudits, which significantly alleviates the requirement for coherence time compared with the canonical protocol and provides benefits for near-term quantum networks.

We note that while we discussed a specific implementation based on scattering of single-sided cavities and optical switches, a number of alternative implementations can be imagined that align with current experimental efforts on quantum repeaters with solid state spins. The optical switches can be replaced by periodic spin rotations in between the time-bins as demonstrated in Ref. [29] with SiV centers and the multiple cavity systems can be replaced with a single emitter coupled to a nearby qubit register as demonstrated in Ref. [36]. If controlled rotations of the emitter spin system dependent on the state of the qubit register can be performed in between the time-bins, the same qudit-qubit entanglement can be generated. Our work thus integrates with current experimental state-of-the-art for quantum communication.

An alternative implementation as an emission-based scheme where the qubit register controls whether the emitter is in an optically bright or dark state similar to entangling schemes demonstrated with NV centers [10, 11] can also be envisioned. Performing a generalized Bell measurement at a middle station could swap the qubit-qudit entanglement to qubit-qubit entanglement.

For single emitter systems such as defect centers in diamond, coupled to nearby nuclear qubit registers, the coherence time of the nuclear qubits is often severely reduced when operations on the emitter are performed. Generation of multiple Bell pairs with such a system requires storage of already generated Bell pairs in the qubit register while attempting another Bell pair with the emitter. As a result, the stored Bell pairs will quickly decohere, limiting the performance of the qubit protocol [23]. The qudit approach circumvents this since all Bell pairs are generated simultaneously. We thus believe that our protocol is particularly relevant for enhancing the performance of current quantum network systems based on diamond defect centers such as NV and SiV centers.