Quantum Transduction:
Enabling Quantum Networking ^††thanks:

In the last decade, the field of quantum transduction has advanced significantly from a hardware standpoint [6, 17, 18]. The physics and hardware-engineering communities have been active in investigating schemes and technologies enabling such an interface, with multiple solutions.

In this context, the aim of this paper is to model quantum transduction from a complementary yet fundamental perspective, namely, from the communication engineering perspective. To this aim, we introduce and analyse different transducer strategies, by translating the quantum transduction process into a proper functional block within the quantum communication model. This analysis reveals that the transducer plays the functionality of adapting the quantum source output to the transmission channel at the source side, while at the receiver side it implements the opposite functionality. Thus, by resorting to a classical communication terminology, the transducer functionality is reminiscent of the modulator (transmission side) and de-modulator (receiver side) functional blocks within the classical communication system model [19].

Counter-intuitively, while in the classical world there exists only one scheme for implementing modulation/demodulation – namely, direct modulation – in a quantum network, direct modulation is one possibility, indeed, not even the most promising one due to the state-of-the-art limitations of transducer hardware.

Stemming from the above considerations, the aim of this paper is to drive the reader, with a tutorial approach, to grasp the fundamental research challenges underlying quantum transduction within the communication engineering domain. To the best of authors’ knowledge, a tutorial of this type is the first of its own.

The paper is structured as depicted in Figure 2.

Specifically, in Section II, we introduce and detail the core challenge for enabling the interaction between the different considered hardware-platforms, namely, the significant frequency gap between microwave and optical quantum carriers. In this same section, we describe how quantum transduction can operate either on quantum-information carrier or on entanglement carrier for entanglement generation and distribution¹¹1We suggest an unfamiliar reader to read the Boxes named Entanglement and Quantum Teleportation to grasp the importance of entanglement as a communication resource, before delving into the field of transduction acting on entanglement carrier.. Finally, we show that the same hardware device used for transduction of quantum-information or entanglement carriers – namely, for the so-called direct transduction – can be remarkably used as an entanglement source. This allows us to distinguish²²2It is worthwhile to note that the jargon direct quantum transduction is mainly limited by the literature to the process of transducing quantum information carrier [17, 20], whereas in the following we extend its use to the transduction of entanglement carrier, being entanglement the fundamental resource of quantum networks [21]. Furthermore, in the same works, EGT coupled with quantum teleportation is also referred to as Entanglement-Based Quantum Transduction (EQT). between Direct Quantum Transduction (DQT) and Entanglement Generation Transduction (EGT).

In Section III, we unpacks the specifics of DQT, by discussing individually the two cases: the conversion of a quantum-information carrier (i.e., of a qubit) in Section III-A and the conversion of an entanglement carrier (i.e., of an ebit) in Section III-B, respectively. This section is enriched with an in-depth analysis of the performance of DQT in terms of the key performance indicator from a communication engineering perspective, the quantum channel capacity, by taking into account the current state-of-the-art of transducer hardware. Through this analysis, we are able to highlight the reasons for which DQT acting on ebits is preferable with respect to DQT acting on informational qubits.

In Section IV we delve into the details of EGT, by introducing and discussing the ability of the transducer hardware to enable entanglement generation between the microwave and the optical domains. Specifically, we present two different physical interactions that can be exploited for entanglement generation: the two-mode squeezing interaction in Section IV-A and the beam splitter interaction in Section IV-B, respectively.

Then, stemming on the material presented so far, in Section V we provide some guidelines for elucidating and analysing how transduction is exploited for quantum information transmission. Specifically, we deepen different source-destination link archetypes, by exploiting different QT techniques. Our objective is to configure QT into network architecture considerations for a more comprehensive overview.

In Section VI, we introduce the transducer within a communication system model, designing it as a modulator/demodulator block.

Specifically, in Section  VII, we give a brief outlook of the noise source that can affect the transduction process, while in Section VII-A we focus on the transduction between photons in the same frequency domain, namely, the so-called intra-band transduction.

DQT acting on an informational qubit is not a trivial process, since it should be able to preserve the encoded quantum information [42]. To this aim, it is necessary that the quantum information is preserved through both the transmission on the quantum channel and during the up- and down-conversion processes. However, if the qubit is lost during the transmission due to the channel attenuation or is corrupted by noise, the associated quantum information cannot be recovered by a measurement process or by re-transmitting a copy of the original information, due to the quantum measurement postulate and the No-Cloning Theorem [24]. Accordingly, by taking into account both the impairments induced by the channel transmission and the failure probability of conversion processes, we conclude that DQT on informational qubits is not yet a viable strategy for the today technology.

By considering the limitations of DQT on informational qubits, an alternative approach is to apply up- and down-conversions on the entanglement resource itself, i.e. on the ebits. With this strategy the impact of noisy quantum transduction and noisy optical propagation shifts from quantum information to entanglement resource. Thus, its main advantage lies in the possibility of entanglement to be regenerated. Indeed, differently from informational qubits, entanglement, being a communication resource rather than information, is not constrained by the no-cloning theorem [22]. Thus, even if the ebit carrying quantum correlation is lost during the channel transmission or the transduction conversion failure, it can be regenerated without restrictions, until the conversions succeed and the entanglement is correctly distributed between the remote nodes. Once the entanglement distribution is successful, an informational qubit can then be “transmitted” via quantum teleportation. This allows to overcome the stringent requirements of DQT on quantum information. The main differences between DQT on quantum information and on ebits are summarized in Tab. I.

To implement both up- and down- conversions, an input laser pump is required to initiate the conversion of the photon associated to the qubit/ebit to be converted.

The conversion of the input photon into the output photon at the desired frequency can be performed through one or more intermediate steps, such as mechanics [28, 29, 33, 34, 38] or magnonics [43, 44], depending on the transducer hardware. A direct conversion between microwave and optical frequencies is instead realized through electro-optic transducers [45, 46, 35, 47, 36, 30, 31], which reduce device complexity and avoid intermediate noise sources. The trade-off though is the weaker nonlinearity compared with optomechanical schemes.

For the sake of clarity, in this paper we focus on electro-optic quantum transducers, but developed the theoretical analysis can be easily extended to different transduction hardware, by properly accounting for the particulars of the hardware parameters⁴⁴4See as instance the conversion efficiency (eq. 6) in [17] for electro-optomechanical transducers.. In a nutshell, electro-optic transducers implement the transduction process by exploiting an input pump laser that initialize the Pockels effect [45, 46] to create a beam splitter interaction between optical and microwave signals. Schematically, an input laser pump at frequency $\omega_{p}=\omega_{o}-\omega_{m}$ interacts with an input photon at microwave frequency frequency $\omega_{m}$ (at optical frequency $\omega_{o}$) to produce an optical photon at $\omega_{o}$ (a microwave photon at $\omega_{m}$).

The main parameter governing electro-optic transduction is the conversion efficiency $\eta$, which is the probability of successful conversion. In the resolved-sideband limit where undesired amplification is negligible, the system is reciprocal and the efficiency is the same for up- and down- conversions. Under resonant conditions, the efficiency can be expressed as [46, 17]:

In (3), $\zeta_{x}$ denotes the so-called extraction ratio of mode $x$⁵⁵5We denote optical mode with subscript $o$ whereas microwave mode with subscript $m$., given by the ration between the external coupling rates and the total dissipation rates: $\zeta_{x}=\frac{\kappa_{x,e}}{\kappa_{x}}$ [17], and $C$ denotes cooperativity, related to the interaction of microwave and optical field within the transducer, defined as [46, 17]:

In (3), $g$ denotes the single-photon electro-coupling rate and $n_{p}$ is the pump photon number. It’s worth pointing out that the added noise, in general, can be different for different conversion directions, depending on how the noise sources couple to the conversion process. Detailed analysis of added noise can be found in Ref. [17]. In the following texts, we use $\eta_{\uparrow}$ and $\eta_{\downarrow}$ to denote the efficiencies of up- and down- conversion, respectively. We choose this notation to enhance clarity of the paper, even though the formulation of the conversion efficiency, in terms of dependence on the transducer parameters, remains unchanged for both conversion directions. The main transducer parameters are summarized in Tab. II.

By accounting for eq. 3, it follows that high conversion efficiency $\eta$ requires both cooperativity $C$ and extraction ratios $\zeta_{x}$ close to 1. This is clearly depicted in Fig. 4, which shows $\eta$ as a function of $C$, and the product of the extraction ratios $\zeta_{o}\zeta_{m}$. However, reaching high values of both these parameters is still an open and crucial challenge. Indeed, while there is a wide-scientific consensus in considering unitary values for $\zeta_{x}$ feasible to achieve in the near-future⁶⁶6Typical values assumed in theoretical studies are around $\zeta_{x}=0.9$ [48], whereas experimental values in the order of $0.1-0.2$ have already been measured [47]., experimentally measured values for $C$ only recently reached $0.3$. Therefore, the cooperativity parameter constitutes the bottleneck of the transducer electro-optical efficiency.

Since the conversion efficiency is the probability of having a successful conversion, low values of $\eta$, in turn, affect also the achievable quantum channel capacity. In other words, for having a non-zero quantum capacity, stringent conditions on $\eta$ should hold [48, 20]. These conditions change whether DQT occurs on the informational qubit or on the ebit.

More into details, the cascade of up-conversion, quantum channel and down-conversion, in point-to-point communication link leveraging DQT on informational qubits, can be modeled as an overall equivalent quantum erasure channel. Accordingly and as highlighted in [49], the one-way quantum capacity is the right metric to adopt for capturing the communication performance. As proved in [2], by neglecting the length effects of the fiber connecting the source and the destination, assuring a unitary capacity requires unitary cooperativity $C=1$ for both the up- and the down- conversions. Yet, such a value, as aforementioned, exceeds current state-of-the-art technologies. By relaxing the hypothesis of unitary capacity, i.e., by requiring only a non-null one-way quantum capacity, the up- and down efficiencies should satisfy the condition $\eta_{\uparrow}\eta_{\downarrow}>\frac{1}{2}$. By accounting for eq. (3), this, in turn, implies that the cooperativity should approximately satisfy $C>0.3$ [2]. In Fig. 4, the efficiency values enabling non-null one-way quantum capacity are highlighted with the dotted black curve.

Similarly, when in a point-to-point communication link, DQT acts on ebits, the cascade of up-conversion, fiber channel and down-conversion can be still modelled as an equivalent quantum erasure channel [4]. However, in such a case [20, 42], the two-way quantum capacity [49] should be considered as performance metric rather than the one-way capacity. In fact, the ebits of the EPR pairs – distributed for eventually teleporting the informational qubit to the destination – can be regenerated and re-distributed in case of losses, without affecting the informational qubit. Thus, for assuring a no-null two-way quantum capacity, the up- and down efficiencies should satisfy the condition: $\eta_{\uparrow}\eta_{\downarrow}>0$. In a nutshell, the key advantage of DQT applied on ebits with respect to DQT on informational qubits is the less stringent requirement on the transducer hardware parameters for assuring a no-null quantum capacity. We will delve deeper on this in Sec. V-D.

In the case of EGT via two-mode squeezing interaction, the input pump, exciting the transducer, drives a spontaneous parametric down-conversion (SPDC), which generates entanglement between optical and microwave fields [48, 50, 51, 27]. Specifically, entanglement is generated whenever the quantum transducer is initialized with no input microwave photon, as depicted in Fig. 5, and the input pump frequency is set to the sum of the frequencies of the optical and microwave photons, i.e., $\omega_{p}=\omega_{m}+\omega_{o}$, (aka “blue detuning”). Ideally, the output state can be expressed with Fock state notation as[51, 27, 52]:

with the subscripts $(\cdot_{m})$ and $(\cdot_{o})$ denoting the photon domain, i.e., microwave or optical. Accordingly, in Eq. (5) the term $\left|1_{m}1_{o}\right\rangle$ denotes the generation of both microwave and optical photons, and the term $\left|0_{m}0_{o}\right\rangle$ denotes no photon generation [53]. The coefficients $\alpha$ and $\beta$ depend on the hardware parameters as the effective squeezing factor [51] and the cooperativity parameter $C$. It is important to note that Eq.(5) assumes that the generation of higher order photon pairs is negligible, which is a good approximation in the low power regime. In general, the SPDC produces a two-mode squeezed vacuum that gives continuous-variable entanglement [53].

The assumption of neglecting the generation of higher order photon pairs in EGT with two-mode squeezing interaction is satisfied without any restriction in case of exploiting beam splitter interaction for entanglement generation. In this case the transducer requires a specific initialization of a microwave photon inside the cavity [27], as schematically depicted in Fig. 5, and the input pump field is set to operate on a frequency that is the difference of the frequencies of the optical and microwave photons, i.e. $\omega_{p}=\omega_{o}-\omega_{m}$. This leads to an entangled state in the form[2]:

where the term $\left|0_{M}1_{O}\right\rangle$ denotes that the microwave photon of the initizalization was converted into an optical one and the term $\left|1_{M}0_{O}\right\rangle$ denotes that the microwave photon was not converted. Specifically, if the transducer conversion efficiency is $50\%$ the state in eq. (6) becomes [2]:

that constitutes a Bell State between different frequency domains.

It is important to notice that the beam splitter interaction exploited in EGT is the same interaction exploited in DQT for frequency conversions (up- and down-) presented in Sec. III. The main differences between the beam splitter interaction exploited for DQT and EGT lie in two key points. First, the photon to be converted in the DQT is the informational qubit, we aim to transmit, or the ebit of the EPR pair, we aim to distribute. Therefore, these quantum states have to be preserved in the conversion. In contrast, in the EGT, the entanglement is generated in the so-called path-entanglement [54, 55, 56]. Thus, the preservation of the quantum states is not of concern.

The second key point is that, while low values of the efficiency $\eta$ imply higher conversion-failure probability in the DQT (either for informational qubit or ebit), in EGT low values of the efficiency $\eta$ do not imply a failure of the process, as evident from (6). Indeed, $\eta$ determines how much the generated entangled state deviates from being a maximally entangled one. According to this last consideration, the great advantage of the EGT over DQT is therefore related to the hardware limitations itself. In other words, while the quality of DQT is strictly related to high values of conversion efficiency, the EGT process can generate entanglement with values of $C$ that are reachable with current state-of-the-art technology. The main differences between DQT and EGT are summarized in Tab. I.

It is worthwhile to note that in EGT with beam splitter interaction the role played by the microwave initialization at hardware level is reminiscent of a basis change. Indeed, (5) and (6) are equivalent quantum states up-to a basis change. This consideration allows us to use interchangeably the two EGT interactions, since they produce LOCC-equivalent states, from a communication perspective.

The first archetype we consider is a source-destination link, where entanglement distribution exploits two DQTs, as depicted in Fig. 6(a). For the sake of clarity, we assume that the entangled state, to be distributed, is an EPR pair. The state is locally generated at the source at microwave frequencies, which can be expressed in the Fock-state notation as [57]:

with the superscript $(\cdot^{s})$ denotes the “location” of the photons, i.e., at the source. The EPR is then distributed with a sequence of up- and down-conversions. Specifically, one microwave photon is up-converted at the source, sent over a fiber channel, and down-converted at the destination. If both the conversions are successful, the resulting EPR state shared between the source and the destination is:

where the superscript $(\cdot^{d})$ denoting the “location” of the photons at the destination. Once the EPR is distributed the teleportation protocol can be performed.

As mentioned in Sec. III, if one or both conversions fail, the entanglement generation and distribution process can be re-executed again until the distribution is successful. Moreover, it is important to empathize that the informational qubit is not involved in the transduction process, but only in the local operations and classical communications (LOCC) [1] required by the quantum teleportation protocol. This, in turn, implies that the noise introduced by DQT impacts on the quantum state to be transmitted.

In Sec. V-E, we generalize the described e-DQT archetype, by removing the hypothesis of entanglement generated at the source.

In this section, we present a second archetype for a source-destination link, exploiting EGT, as depicted in Fig. 6(b). This archetype allows to reduce the number of direct conversions for distributing entanglement between source and destination, with respect to the e-DQT archetype.

Specifically, a transducer located at the source generates hybrid entanglement, by exploiting one of the two physical interactions described in Sec. IV-A and Sec. IV-B. For the sake of clairty, by assuming a beam spitter interaction and a conversion efficiency of $\eta=50\%$, a Fock state in the form of eq. (7) is generated. The optical photon of the generated entangled pair is then transmitted to the destination through an optical fiber and down-converted to the microwave domain therein. The resulting shared between the source and the destination can be expressed as follows:

The teleportation protocol can now be applied to transfer the quantum information.

In this archetype, the entanglement distribution process requires that the destination is equipped with a quantum transducer capable of down-converting from optical to microwave one ebit of the generated hybrid entangled state. Thus, the strategy still suffers from the inefficiency of direct quantum transduction – although limited to a single conversion (optical to microwave) rather than both up- and down-conversions.

A third source-destination link archetype exploits EGT coupled with entanglement swapping⁷⁷7Entanglement swapping[65] is a strategy which extends the entanglement distribution distance. The reader may refer to the vast literature on the subject. , as depicted in Fig. 6(c).

Specifically, with two EGT at the source and destination side, two hybrid EPR states, in the form of eq. (7), are generated. Accordingly, the entanglement generation occurs “at both points” rather than at “source only” [24, 21].

The optical ebits of both the generated entangled states are then transmitted through optical fibers, by reaching a beam splitter followed by two detectors. The overall setup is unable to distinguishing the which-path information [27, 66, 67]. A click of one of the two detectors denotes the presence of an optical photon. However, due to the path-erasure – i.e., the impossibility of knowing whether the optical photon responsible for the detector-click has been generated at the source or at the destination – it is impossible to distinguish where the entanglement generation process has taken place (namely, whether at the source or at the destination), and thus it is impossible to distinguish whether a microwave photon is present at source or at destination. This results into the generation of another form of path-entanglement [68] between the microwave photons at the source and at the destination, in the form of (10). Basically, the famous Duan–Lukin–Cirac–Zoller protocol is performed [69].Thus, the overall effect of beam splitter and detectors is reminiscent of entanglement swapping, projecting the received optical photons into a Bell state.

The distributed microwave-microwave entanglement can now be exploited to transmit quantum information, through quantum teleportation.

It is worthwhile to highlight that a detector click does not always correspond to an EPR shared between source and destination. Indeed, while the purity of the hybrid generated EPR depends exclusively on the transducer hardware, as discussed in Sec. IV-B, the purity of heralded microwave EPR expressed in (10) depends also on the characteristics of the repeater node, i.e. on the beam splitter and optical detectors characteristics. Specifically, there exists the possibility that more than one photon reach the repeater node. In fact, when both the transducers generate optical photons, only one detector click is triggered due to path erasure. In such a case, a detector click corresponds to the presence of two microwave photons, one at the source and one at the destination. Hence, the state shared between the remote nodes is $\left|0^{s}_{m}\right\rangle\left|0^{d}_{m}\right\rangle$, which is definitely not an entangled state as in (10). However, if we reasonably assume the availability of photon-number-resolved detectors (PNRD), then it is possible to distinguish the event of receiving two optical photons – one for each transducer in each link – from the event where only one optical photon is received. And the double-photon event can be discarded in favour of a new distribution attempt.

It must be acknowledged that the key advantage of the archetype “EGT coupled with swapping” lies in the possibility of heralding entanglement via off-the-shelf hardware – i.e., via PNRD. Specifically, a detector click for each transduction attempt constitutes an indicator for identifying the generation of entanglement between the source and the destination, without destroying it. In the previous two archetypes, i.e. e-DQT (Sec. V-A) and EGT coupled with e-DQT (Sec. V-B), there exists the possibility of heralding entanglement, by exploiting another degree of freedom, different from the one for entanglement encoding.

For instance, time-bin entanglement of microwave photons have been experimentally demonstrated [70]. Also time bin of hybrid entanglement generated within a traducer (EGT) have been measured [53, 40]. These examples not only exploit two different degree of freedom (one for entanglement generation and the other for entanglement heralding), but also imply the introduction of additional hardware setups for the heralding. On the contrary, in EGT coupled with swapping, the entanglement heralding is embedded in the setup itself and, for this reason, it is not necessary to exploit other degree of freedom and/or to introduce any additional heralding equipment.

Stemming from the above described archetypes, here we conduct a performance comparison among them, by analysing some key communication metrics as functions of the main transducer-hardware parameters. Specifically, our analysis bridges the gap between different hardware platforms for quantum transducers, such as electro-optic and opto-mechanical systems. Indeed, by focusing on common characteristics shared by all transducer implementations, such as conversion efficiency, we establish communication metrics that are agnostic to specific hardware configuration. The key metric we define is the probability $p_{e}$ of successfully distributing an EPR from the source to the destination⁸⁸8For the theoretical details about the closed-form expressions of the success ebit distribution and for the achievable quantum capacity for all the three archetypes, we refer the reader to the appendices in [4]. [4].

For the e-DQT archetype, $p_{e}^{eDQT}$, as a function of transducer parameters, can be expressed as follows [4]:

where the superscripts $(\cdot^{s})$ and destination $(\cdot^{d})$ denote the “location” of the transducer process, i.e., at source and at destination side. In Eq. (11) the term $e^{-\frac{l_{s,d}}{L_{0}}}$ takes into account the fiber losses, with $l_{s,d}$ denoting the length of the fiber link between source and destination and $L_{0}$ denoting the attenuation length of the fiber⁹⁹9As for today, commercial fibers feature an attenuation lower than $1$db/km. As instance, optical photons with wavelength equal to $1550$nm – i.e., DWDM ITU $100$GHz channel number $35$ in the C band – experience an attenuation of $0.2$dB/km, which corresponds to $L_{0}=22$ km. [71]..

It is evident that in case of noiseless quantum teleportation, i.e, under the hypothesis of noise-free LOCC, $p_{e}$ can be seen as the probability of successful transmission of quantum information as well. Therefore, the probabilities of successfully transmitting quantum information and successfully distributing entanglement become equivalent. In other words, the two-way quantum capacity coincides with $p_{e}$.

For the “EGT coupled with e-DQT” archetype, the probability $p_{e}^{\text{\rm EGT}}$ of successfully distributing an EPR pair is:

where $S(\cdot)$¹⁰¹⁰10With a small abuse of notation, we have indicated in the argument of the Von Neuman entropy the eigenvalue determining its value rather than – as usually done – the density matrix on which the entropy is evaluated. denotes the Von Neuman entropy given by:

Finally, for the “EGT coupled with Swapping” archetype, the probability $p_{e}^{\text{\rm EGT-S}}$ of successfully distributing an EPR is given by [4]:

where $\tilde{\eta}_{\uparrow}$ denotes the efficiency between $\eta_{\uparrow}^{s}$ and $\eta_{\uparrow}^{d}$ that minimizes $S(\cdot)$.

Fig. 7 shows the probability $p_{e}$ of successful EPR distribution for the three considered archetypes, as function of the cooperativity $C$, namely, as discussed in Sec. III-C, the main hardware parameter limiting the transducer performances. Similarly to what was done with $\eta$, also for $C$ we indicate with the subscripts $(\cdot_{\uparrow})$ and $(\cdot_{\downarrow})$ the direction of the conversion (up or down) and with the superscripts $(\cdot^{s})$ and $(\cdot^{d})$ the location of the transduction (source and destination).

We note that for the e-DQT archetype, the presence of two e-DQT processes requires unitary values for both $C^{s}_{\uparrow}$ and $C^{d}_{\downarrow}$ in order to obtain $p_{e}=1$ (equivalently unitary two-way quantum capacity), as shown in Fig. LABEL:fig:07.1. This restriction can be relaxed in the case of EGT coupled with e-DQT. In fact, in this type of archetype, to obtain a unitary $p_{e}$, the value of $C$ strictly equal to 1 is required only for the transducer at the destination ($C^{d}_{\downarrow}$=1), responsible for the direct conversion, while $C^{s}_{\uparrow}$ has to satisfy $C^{s}_{\uparrow}\approx 3-2\sqrt{2}$, as shown in Fig. LABEL:fig:07.2.

Finally, in EGT coupled with swapping, no direct conversion is required. Therefore, the maximum amount of $p_{e}$ can be achieved with $C^{s}_{\uparrow}=C^{s}_{\uparrow}\approx 3-2\sqrt{2}$, as shown Fig. LABEL:fig:07.3. Thus, this archetype determines an improvement in terms of minimum cooperativity that allows a non-zero entanglement distribution probability. But, this comes at the cost of a probability $p_{e}$ that never reaches $1$. This, in turn, implies that the two-way quantum capacity does not reach one as well.

To further highlight the comparison between the performances of the different proposed archetypes, Fig. 8 presents the three $p_{e}$ as function of $C$ within the same plot¹¹¹¹11Here, for EGT coupled with e-DQT, $C$ is $min(C,3-2\sqrt{2})$.. It is evident that in archetypes exploiting hybrid entanglement generation transducers, it is possible to relax the constraints on the required hardware parameters, for reaching selected probability values (aka, both unitary cooperativity at source and destination). Therefore, the presence of a DQT acts as a bottleneck, limiting the overall performances of the system.

The analysis developed in the previous subsections is not exhaustive. Indeed, additional source-destination link archetypes can be considered, accordingly to i) the entanglement generation “location” and ii) the entanglement type.

More in detail, with reference to the “location”, the entanglement generator can be at source, at the middle or at both points [24, 21]. Instead, regarding the type, it is possible to distinguish archetypes whether the entanglement resource is generated within (aka EGT deeply described in Sec. IV) or outside the transducer, through an external entanglement source. In literature, it is common to refer to these two types of entanglement generation as intrinsic and extrinsic process [42], respectively.

Tab. III summarizes the possible source-destination link archetypes, accordingly to the “location”, frequencies and type of entanglement, by including also the archetypes discussed in the previous subsections. Teleportation process for quantum information transmission is not explicitly depicted to make the figures clearer. As done before, blue and red symbols refer to ebit at microwave and optical frequencies, respectively.

By observing the various possibilities, we can draw some considerations. First, it is worth to highlight that as the number of DQT processes increases, the risk of losing the entanglement increases accordingly. Indeed, any processing can only worsen entanglement. This directly implies that extrinsic entanglement generation at middle point requires at least two DQT processes that can scale up to four in the case of entanglement generated at microwave frequencies.

Additionally, we observe that it is meaningless to treat the generation of extrinsic optical entanglement at the “source”, since we are considering the network scenario where source and destination are superconducting nodes. Furthermore, the generation of extrinsic optical entanglement at “both point” introduces a communication scheme that differs from those introduced in [24, 21]. Indeed, the “both point” referred to are not source and destination nodes, but two external optical nodes, whose function is just to generate entanglement.

Clearly starting from the described source-destination link archetypes, it is possible to consider more complex network architectures, characterized by higher number of direct conversions or swapping nodes, accordingly to the specific network application and distance among the nodes.

In this work, we analysed microwave-optical transduction with the aim of interconnecting distant superconducting quantum nodes via optical quantum channels. However, as mentioned in Sec.I, there exist several qubit platforms. Among them, trapped ions[72, 73], quantum dots [74, 75, 76] and spin-qubits [77] directly interact with optical photons.

These qubit platforms emit entangled photons at visible/NIR wavelengths. Therefore, for long-distance entanglement distribution, it is necessary a frequency conversion to teleco frequencies. Indeed, O-band and C-band are commonly used for long-distance entanglement distribution [78, 79] with the ultimate goal of entanglement distribution on lit-fiber networks. [80, 81, 82]. This frequency conversion from optical to optical frequencies is referred to as intra-band transduction [17]. Therefore, also if some qubit platforms exploited for computation can spontaneously interact with optical photons, quantum transduction is inevitable for long-distance communications. And moreover, due to the intrinsically weak interactions between photons, a frequency converter for O- or C-band requires high power pump and coupling [17].

In a nutshell, quantum transduction is mandatory for quantum networks. We choose to focus on the transduction between microwave and optical domains, since at the time of the manuscript writing, superconducting technology constitutes the most promising platform for quantum computing. Indeed, superconducting quantum gates are fast [83] with high-fidelity levels [84], and their scalability allows to build quantum processors with hundreds of qubits [85].