Quantum Relay Channels Overview

• Quantum relay channels are a family of communication models where an intermediate node assists transmission by managing quantum correlations through measurement, forwarding, or entanglement localization.
• They encompass diverse architectures including trusted intercept/resend systems, passive optical relays, continuous-variable setups, and fully quantum decode-forward schemes.
• Key insights highlight trade-offs in distance, rate, and capacity determined by relay functionality, measurement precision, and coherence preservation in quantum networks.

Searching arXiv for recent and foundational papers on quantum relay channels to support the article. Quantum relay channels are communication models in which an intermediate node assists transmission between remote parties, but the operational meaning of “relay” varies sharply across subfields. In quantum key distribution, a relay may be a trusted intercept/resend node or a passive optical station; in classical–quantum network information theory, it is a helper between a source and a destination in a memoryless channel with classical inputs and quantum outputs; in fully quantum settings, it is a strictly causal quantum processor inserted into a CPTP map $\mathcal{N}_{AD\rightarrow BE}$; and in continuous-variable architectures, it is often a measurement node performing a Bell detection and broadcasting classical outcomes (Barnett et al., 2012, Savov, 2012, Savov et al., 2011, Pereg, 2024, Spedalieri et al., 2015). Across these formulations, the central issue is the same: how an intermediate station changes achievable distance, rate, or capacity without violating no-cloning, while the detailed answer depends on whether the relay measures, forwards, compresses, localizes entanglement, or distributes side information.

1. Conceptual scope and canonical models

A quantum relay channel is not a single primitive but a family of channel models. In one broad line of work, the relay channel is a classical–quantum relay channel with classical inputs at the source and relay and quantum outputs at the relay and destination, specified by a map

$$\mathcal{N}^{XX_1 \to B_1B} : (x,x_1) \mapsto \rho^{B_1B}_{x,x_1},$$

with memoryless extension over $n$ uses (Savov et al., 2011). In a more general formulation, a fully quantum relay channel is a CPTP map

$$\mathcal{N}_{AD\rightarrow BE} : \mathfrak{D}(\mathcal{H}_A\otimes\mathcal{H}_D)\to\mathfrak{D}(\mathcal{H}_B\otimes\mathcal{H}_E),$$

where Alice transmits on $A$, the relay transmits on $D$ and receives $E$, and Bob receives $B$; the channel is memoryless and the relay is strictly causal (Pereg, 2024). A related primitive appears in the quantum broadcast-with-cooperation setting, where the primitive relay channel is obtained from a broadcast channel with conferencing by setting the relay’s own message rate to zero, so that the relay serves only to help the destination (Pereg et al., 2020).

These information-theoretic models coexist with operational relay architectures from quantum cryptography and optical networking. Barnett and Phoenix consider a linear chain

$$\text{Alice} - R_1 - R_2 - \dots - R_{n-2} - \text{Bob},$$

where each $R_j$ measures in a BB84 basis and resends a fresh photon, making the overall connection a series composition of quantum channels punctuated by measure-and-prepare maps (Barnett et al., 2012). In continuous-variable measurement-device-independent QKD, the relay instead receives two incoming modes and performs a CV Bell detection, publishing a classical outcome $$\mathcal{N}^{XX_1 \to B_1B} : (x,x_1) \mapsto \rho^{B_1B}_{x,x_1},$$0 that correlates the users’ data (Spedalieri et al., 2015). In underwater optics, passive relays perform no measurement at all: they collect, re-collimate, and redirect qubits so that the end-to-end channel is a concatenation of lossy turbulent links rather than a sequence of measurement–reprepare operations (Raouf et al., 2022).

This variety suggests that “quantum relay channel” is best understood as an umbrella term. A plausible implication is that the decisive classification is not the mere presence of an intermediate node, but whether the relay is trusted or untrusted, measurement-based or coherence-preserving, and analyzed as a network protocol or as a Shannon-theoretic channel model.

2. Trusted intercept/resend relays and passive optical relays in QKD

In BB84-based relay networks, Barnett and Phoenix analyze active relays that act as trusted intercept/resend devices: each relay receives a photon, measures it in a randomly chosen BB84 basis, prepares a new photon in the measured state, and sends it onward (Barnett et al., 2012). Although this operation is identical to an eavesdropper’s intercept/resend attack at the quantum layer, the relay is assumed to be trusted and cooperative, and participates in classical post-processing. In the naive view, end-to-end usability appears to decay exponentially because only transmissions for which all nodes choose the same basis are directly useful, with probability $$\mathcal{N}^{XX_1 \to B_1B} : (x,x_1) \mapsto \rho^{B_1B}_{x,x_1},$$1 for $$\mathcal{N}^{XX_1 \to B_1B} : (x,x_1) \mapsto \rho^{B_1B}_{x,x_1},$$2 total nodes (Barnett et al., 2012). Barnett and Phoenix argue that this pessimism is unwarranted: by cooperative post-processing that pairs timeslots and performs logical bit “flips,” all timeslots except fully alternating basis patterns can be marshaled into an Alice–Bob key, so that the usable fraction is restored to exactly $$\mathcal{N}^{XX_1 \to B_1B} : (x,x_1) \mapsto \rho^{B_1B}_{x,x_1},$$3, the same as in standard BB84 (Barnett et al., 2012). The associated distance argument is also modified: relays need not resend only when they detect a photon, but can use delay/batching or padding, so that the second hop maintains acceptable SNR even when the first hop is lossy (Barnett et al., 2012).

The later paper on bit transport generalizes this viewpoint to relay QKD networks by treating each relay as a “trusted eavesdropper” and explicitly pairing partially open basis configurations with dual configurations (Phoenix et al., 2015). In that construction, a relay can tell Alice and Bob which timeslot pair to use so that a bit value is effectively transported across a broken link without revealing the bit itself. The result is again that the asymptotic sifted key fraction is $$\mathcal{N}^{XX_1 \to B_1B} : (x,x_1) \mapsto \rho^{B_1B}_{x,x_1},$$4, independent of the number of relays, rather than $$\mathcal{N}^{XX_1 \to B_1B} : (x,x_1) \mapsto \rho^{B_1B}_{x,x_1},$$5 under a naive all-bases-match criterion (Phoenix et al., 2015). The same work also studies random relay drop-out and secret-sharing over logical channels; for a channel that needs $$\mathcal{N}^{XX_1 \to B_1B} : (x,x_1) \mapsto \rho^{B_1B}_{x,x_1},$$6 relays to span the distance, adding one extra relay and using XOR-combined shares can force an adversary to compromise all relays on that path, at the price of a reduced fraction of useful timeslots (Phoenix et al., 2015).

A physically different QKD relay appears in underwater free-space optics. There the intermediate nodes are passive relays that simply redirect the qubits to the next relay or receiver without any measurement (Raouf et al., 2022). The end-to-end link of total length $$\mathcal{N}^{XX_1 \to B_1B} : (x,x_1) \mapsto \rho^{B_1B}_{x,x_1},$$7 is divided into $$\mathcal{N}^{XX_1 \to B_1B} : (x,x_1) \mapsto \rho^{B_1B}_{x,x_1},$$8 equal hops of length $$\mathcal{N}^{XX_1 \to B_1B} : (x,x_1) \mapsto \rho^{B_1B}_{x,x_1},$$9, with per-hop transmissivity

$n$0

where $n$1 accounts for absorption and scattering and $n$2 for turbulence-induced fading (Raouf et al., 2022). The fraction of photons reaching Bob becomes

$n$3

while background photons accumulate across hops (Raouf et al., 2022). The QBER analysis shows a trade-off: in non-turbulent clear ocean, relaying does not improve distance, whereas under strong turbulence a small number of relays can improve maximum secure distance because shorter hops improve the turbulence-averaged coupling $n$4 enough to offset extra background (Raouf et al., 2022). This architecture preserves end-to-end security precisely because the relays are non-measuring.

Taken together, these QKD relay models mark three distinct regimes: trusted active relays that classicalize each hop, trusted-node architectures that rely on bit transport or secret-sharing, and passive relays that remain transparent to the quantum state. This suggests that in cryptographic settings, the word “relay” chiefly denotes a constraint on network trust and optical processing, rather than a unique channel law.

3. Classical–quantum relay channels and partial decode-forward

The cq relay channel entered quantum network information theory through coding theorems that adapt classical relay strategies to channels with classical inputs and quantum outputs. In the thesis on classical–quantum channels, the relay channel is defined by finite alphabets $n$5 and $n$6 and bipartite output states

$n$7

with the relay seeing $n$8 and the destination seeing $n$9 (Savov, 2012). The communication task is reliable transmission of a classical message $$\mathcal{N}_{AD\rightarrow BE} : \mathfrak{D}(\mathcal{H}_A\otimes\mathcal{H}_D)\to\mathfrak{D}(\mathcal{H}_B\otimes\mathcal{H}_E),$$0 from source to destination using a block-Markov code, relay POVMs, and destination sliding-window measurements over two consecutive blocks (Savov, 2012). The main achievable result is a partial decode-forward inner bound

$$\mathcal{N}_{AD\rightarrow BE} : \mathfrak{D}(\mathcal{H}_A\otimes\mathcal{H}_D)\to\mathfrak{D}(\mathcal{H}_B\otimes\mathcal{H}_E),$$1

with respect to the cq state

$$\mathcal{N}_{AD\rightarrow BE} : \mathfrak{D}(\mathcal{H}_A\otimes\mathcal{H}_D)\to\mathfrak{D}(\mathcal{H}_B\otimes\mathcal{H}_E),$$2

(Savov, 2012, Savov et al., 2011). The first term is a multiple-access-type bound at the destination, while the second decomposes into the rate at which the relay can decode the auxiliary $$\mathcal{N}_{AD\rightarrow BE} : \mathfrak{D}(\mathcal{H}_A\otimes\mathcal{H}_D)\to\mathfrak{D}(\mathcal{H}_B\otimes\mathcal{H}_E),$$3 plus the residual direct rate from source to destination once $$\mathcal{N}_{AD\rightarrow BE} : \mathfrak{D}(\mathcal{H}_A\otimes\mathcal{H}_D)\to\mathfrak{D}(\mathcal{H}_B\otimes\mathcal{H}_E),$$4 and $$\mathcal{N}_{AD\rightarrow BE} : \mathfrak{D}(\mathcal{H}_A\otimes\mathcal{H}_D)\to\mathfrak{D}(\mathcal{H}_B\otimes\mathcal{H}_E),$$5 are known (Savov, 2012).

Savov, Wilde, and Vu formulate the same channel as a cc–qq relay channel and prove the same partial decode-forward structure using randomized block-Markov codebooks, HSW decoding at the relay, and a sliding-window “AND”-measurement at the destination (Savov et al., 2011). The relay’s decoder is a standard square-root measurement built from typical projectors for the cq ensemble seen on $$\mathcal{N}_{AD\rightarrow BE} : \mathfrak{D}(\mathcal{H}_A\otimes\mathcal{H}_D)\to\mathfrak{D}(\mathcal{H}_B\otimes\mathcal{H}_E),$$6, giving the constraint

$$\mathcal{N}_{AD\rightarrow BE} : \mathfrak{D}(\mathcal{H}_A\otimes\mathcal{H}_D)\to\mathfrak{D}(\mathcal{H}_B\otimes\mathcal{H}_E),$$7

for the partial message rate (Savov et al., 2011). The destination then performs a joint square-root measurement on two consecutive blocks, combining a test for the source message in block $$\mathcal{N}_{AD\rightarrow BE} : \mathfrak{D}(\mathcal{H}_A\otimes\mathcal{H}_D)\to\mathfrak{D}(\mathcal{H}_B\otimes\mathcal{H}_E),$$8 with a test for the relay-forwarded partial message in block $$\mathcal{N}_{AD\rightarrow BE} : \mathfrak{D}(\mathcal{H}_A\otimes\mathcal{H}_D)\to\mathfrak{D}(\mathcal{H}_B\otimes\mathcal{H}_E),$$9, which yields the overall bound quoted above (Savov et al., 2011). The use of conditional typical projectors, the Hayashi–Nagaoka inequality, and the gentle operator lemma is the quantum analogue of the classical relay packing argument (Savov et al., 2011).

A distinct but related cq relay model appears in two-phase bidirectional relaying. There the first phase is a cq multiple-access channel $A$0 from two terminals to the relay, and the second phase is a cq broadcast channel $A$1 from the relay back to the terminals (Boche et al., 2014). The broadcast-phase capacity region is determined by the individual Holevo quantities to the two receivers: $A$2 (Boche et al., 2014). The overall bidirectional rate region is the intersection of the MAC and BC regions (Boche et al., 2014). This is still decode-and-forward, but in a half-duplex bidirectional topology rather than a one-way relay chain.

These cq results establish a consistent pattern: the relay channel is treated as a multi-user coding problem, the relay decodes either a whole message or an auxiliary layer, and the quantum novelty lies not in the high-level strategy but in the use of joint measurements and Holevo-information bounds. A plausible implication is that for classical communication over quantum networks, the main challenge is often decoder design, whereas the combinatorial structure of relay coding remains recognizably classical.

4. Fully quantum relay channels, quantum cooperation, and relay-specific bounds

The fully quantum relay channel replaces classical channel inputs by quantum systems at both sender and relay. In Pereg’s formulation, the channel is a CPTP map

$A$3

used memorylessly, with a strictly causal relay that chooses each input $A$4 from past outputs $A$5 (Pereg, 2024). The communication task is quantum communication from Alice to Bob: Alice encodes a quantum message system $A$6, the relay applies a sequence of CPTP maps $A$7, and Bob decodes an approximate copy $A$8, with error measured in trace distance against a reference-purified state (Pereg, 2024). The main contribution is a fully quantum decode-forward scheme based on FQSW and block-Markov coding, yielding the achievable unassisted rate

$A$9

where $D$0 (Ilin et al., 9 Jul 2025). In the entanglement-assisted case the paper gives

$D$1

again as an achievable rate (Ilin et al., 9 Jul 2025). The same framework is interpreted as coding with quantum side information, because $D$2 can be entangled with the relay input $D$3 and thereby act as distributed side information for the transmitter (Ilin et al., 9 Jul 2025).

A broader information-theoretic treatment is given in “Quantum Relay Channels,” which studies the fully quantum model

$D$4

for classical communication and develops three lower bounds: partial decode-forward, measure-forward, and assist-forward (Pereg, 2024). The partial decode-forward bound generalizes the Savov–Wilde–Vu cq result and reduces to it in the c-q special case (Pereg, 2024). For a c-q relay channel $D$5, the bound is

$D$6

(Pereg, 2024). The measure-forward strategy is explicitly presented as a generalization of classical compress-forward: the relay measures its quantum output $D$7, compresses the resulting classical outcome, and forwards that compressed description to Bob, with an achievable rate $D$8 subject to a compression constraint (Pereg, 2024). The assist-forward bound is developed for channels with orthogonal receiver components

$D$9

and uses the broadcast component to send the message to the relay while simultaneously generating entanglement between relay and Bob, which is then consumed as rate-limited entanglement assistance over $E$0 (Pereg, 2024).

This paper also identifies a class where capacity is known exactly: for Hadamard relay channels, full decode-forward is optimal and

$E$1

(Pereg, 2024). By contrast, the primitive relay channel derived from a quantum broadcast channel with receiver cooperation yields a multi-letter cutset upper bound

$E$2

and a single-letter decode-forward lower bound

$E$3

(Pereg et al., 2020). That formulation explicitly interprets the relay as a quantum repeater with a bottleneck between the sender–relay coherent information and the relay–destination conference link (Pereg et al., 2020).

Across these fully quantum developments, the relay is no longer a measurement station or trusted node but a strictly causal quantum processor. This suggests that the modern theory of relay channels is converging on a sharper distinction between classical relays that manage secrecy or distance and fully quantum relays that manage coherent information and entanglement topologies.

5. Continuous-variable, measurement-based, and environment-assisted relay channels

A major family of quantum relay channels is measurement-based and continuous-variable. In the ideal local-relay limit of CV-MDI-QKD, Alice and Bob send coherent states to a relay that performs a CV Bell detection and broadcasts the outcome $E$4 (Spedalieri et al., 2015). The relay is local to Alice, Alice’s channel is lossless, and the Bell detection is ideal, allowing the authors to characterize the ultimate performance benchmark for the protocol (Spedalieri et al., 2015). The key rate under large modulation and ideal reconciliation is

$E$5

with $E$6 and

$E$7

while in the pure-loss case it simplifies to

$E$8

(Spedalieri et al., 2015). The rate remains positive for all $E$9 in this idealized model, decaying with Bob’s channel transmissivity (Spedalieri et al., 2015). Conceptually, the relay is a measurement node with classical output only; it does not preserve or forward quantum states, but still constitutes a quantum relay channel because the relay’s joint measurement is what creates the correlations between remote users.

A more general and striking CV relay phenomenon appears in environment-assisted bosonic communications. There the relay channel is a two-input bosonic Gaussian channel $B$0 feeding a CV Bell measurement at Charlie, with correlated Gaussian noise injected by a separable environment

$B$1

(Pirandola et al., 2020). In the memoryless case $B$2, sufficiently large thermal noise makes each link entanglement-breaking. The paper shows that separable correlations in the environment can reactivate the relay: although all bipartite and tripartite entanglement are lost, strong enough environmental correlations can generate quadripartite $B$3 entanglement, which the relay’s Bell detection then localizes into bipartite entanglement between Alice and Bob (Pirandola et al., 2020). For the conditional swapped state, the smallest partially transposed symplectic eigenvalue is

$B$4

with

$B$5

and the large-$B$6 limit

$B$7

determines swapping, teleportation, distillation, and QKD thresholds (Pirandola et al., 2020). The work further shows that in the correlated-additive-noise limit the practical QKD rate can surpass the single-repeater bound derived for memoryless links, because correlated noise turns a relay that would be dead under independent entanglement-breaking channels into a usable non-Markovian resource (Pirandola et al., 2020).

Measurement-based relays also appear in modern untrusted architectures using quantum dot single-photon sources. In a five-node network with trusted users Alice and Bob and three untrusted intermediate nodes $B$8, $B$9, and $$\text{Alice} - R_1 - R_2 - \dots - R_{n-2} - \text{Bob},$$0, the central source node emits a single photon that is split toward two measurement stations, where it interferes with users’ coherent states (Zou et al., 29 Aug 2025). Each measurement node announces whether a valid click occurred and which detector fired, and Alice and Bob retain rounds with valid dual-node events and suitable phase relations (Zou et al., 29 Aug 2025). The security analysis treats all three intermediate nodes as one untrusted Bell-state measurement box and applies a phase-matching/MDI proof, with key fraction

$$\text{Alice} - R_1 - R_2 - \dots - R_{n-2} - \text{Bob},$$1

(Zou et al., 29 Aug 2025). This realizes a multi-node relay channel in which the relay operation is entirely measurement-based and untrusted.

These CV and measurement-based results reinforce a recurring theme: a relay need not preserve a flying qubit or store quantum information to be a genuine quantum relay. In many practically relevant architectures, the relay’s power comes from joint measurement, correlation localization, or non-Markovian noise structure, not from direct forwarding of a quantum state.

6. Experimental implementations, network architectures, and open issues

Several experimental lines instantiate quantum relay channels in distinct physical forms. A teleportation-based relay at telecom wavelength uses a quantum dot emitting polarization-entangled photon pairs in the telecom O-band, with a partial Bell-state measurement between Alice’s weak coherent pulse and one photon of the entangled pair, teleporting the state to Bob’s photon (Huwer et al., 2017). The process tomography shows a dominant $$\text{Alice} - R_1 - R_2 - \dots - R_{n-2} - \text{Bob},$$2 component

$$\text{Alice} - R_1 - R_2 - \dots - R_{n-2} - \text{Bob},$$3

corresponding to an average gate fidelity

$$\text{Alice} - R_1 - R_2 - \dots - R_{n-2} - \text{Bob},$$4

to the ideal bit-flip teleporter, while the maximum fidelity for a standard 4-state protocol reaches $$\text{Alice} - R_1 - R_2 - \dots - R_{n-2} - \text{Bob},$$5 (Huwer et al., 2017). This is a memoryless quantum relay in the traditional teleportation sense: it uses entanglement and a Bell-state measurement but has no quantum memory and does not qualify as a full repeater (Huwer et al., 2017).

The recent untrusted intermediate relay architecture using a quantum dot single-photon source realizes a five-node network with total fiber lengths of 100, 200, and 300 km and effective total losses of 19.34, 35.58, and 52.33 dB, respectively (Zou et al., 29 Aug 2025). The source is an InAs/GaAs quantum dot frequency-converted to 1550.2 nm, with

$$\text{Alice} - R_1 - R_2 - \dots - R_{n-2} - \text{Bob},$$6

and intrinsic indistinguishability about $$\text{Alice} - R_1 - R_2 - \dots - R_{n-2} - \text{Bob},$$7 (Zou et al., 29 Aug 2025). The experiment reports secure key establishment over 300 km and highlights a modular architecture in which a five-node unit can be extended into longer chains or star-like topologies (Zou et al., 29 Aug 2025). The interference statistics, gains $$\text{Alice} - R_1 - R_2 - \dots - R_{n-2} - \text{Bob},$$8, bit error rates $$\text{Alice} - R_1 - R_2 - \dots - R_{n-2} - \text{Bob},$$9, and phase error estimates $R_j$0 play the role of operational channel descriptors rather than an explicit CPTP relay-channel law (Zou et al., 29 Aug 2025).

On the theoretical side, private coding for quantum relay networks introduces a different kind of relay limitation: a probabilistic relay encoder $R_j$1 that adds amplitude information only with probability $R_j$2, initially assumed in the regime $R_j$3 (Gyongyosi et al., 2012). The paper combines quantum polar coding with a superactivation-assisted construction

$R_j$4

where $R_j$5 is a 50% erasure channel, to argue that the relay encoder’s effective reliability becomes $R_j$6 and the assisted private rate remains positive (Gyongyosi et al., 2012). The private polar-code structure is organized through sets such as

$R_j$7

with the asymptotic private rate determined by the density of jointly good amplitude and phase indices (Gyongyosi et al., 2012). This is a highly coding-theoretic relay model rather than an optical one, but it shows that relay unreliability itself can be treated as a channel resource subject to quantum-specific activation phenomena.

The recurrent limitations across these formulations are also clear. Trusted intercept/resend relays require full trust or external secret-sharing mechanisms (Barnett et al., 2012, Phoenix et al., 2015). Passive relays do not violate end-to-end security, but they accumulate multiplicative loss and additive background (Raouf et al., 2022). Cq and fully quantum relay-channel bounds are mostly inner bounds, often multi-letter or strategy-dependent, and complete capacity formulas remain unavailable beyond special classes such as Hadamard relay channels (Savov, 2012, Savov et al., 2011, Pereg, 2024). Measurement-based relays in CV-QKD remain sensitive to reconciliation efficiency, excess noise, and phase stability (Spedalieri et al., 2015, Zou et al., 29 Aug 2025). Environment-assisted relay effects show that memory can invalidate memoryless benchmarks, implying that repeater bounds and relay capacities must be interpreted with care whenever correlated noise is present (Pirandola et al., 2020).

Quantum relay channels therefore occupy an intermediate position between point-to-point quantum communication and full quantum repeater networks. They include trusted BB84 relays, passive optical redirectors, cq partial-decode-forward channels, fully quantum strictly causal channels, CV Bell-measurement relays, teleportation-based telecom nodes, and untrusted SPS-assisted network modules (Barnett et al., 2012, Raouf et al., 2022, Savov et al., 2011, Pereg, 2024, Spedalieri et al., 2015, Huwer et al., 2017, Zou et al., 29 Aug 2025). What unifies them is not a common hardware design, but the role of an intermediate station in mediating information flow under quantum constraints; what distinguishes them is whether the relay manages classical correlations, coherent information, or entanglement localization.