Quantum Repeaters

A quantum repeater distributes usable quantum correlations across a long, lossy, or noisy path by dividing it into shorter elementary links and joining their quantum resources at intermediate nodes. Depending on the architecture, the node stores heralded entanglement, performs entanglement swapping and distillation, applies quantum error correction, or prepares a photonic graph state whose measurements accomplish the same logical task.

The word repeater does not mean that an unknown qubit is measured, copied, and regenerated. No-Cloning and No-Signaling forbids that classical strategy. Nor is every middle station a repeater. A trusted key relay, optical amplifier, measurement-device-independent QKD relay, postselected swap, and satellite downlink have different resources and trust models.

The key idea is to keep a successful short link while waiting for its neighbor, then connect them. That changes a product of simultaneous success probabilities into a waiting-time problem. The benefit depends on how long the stored state remains useful, whether swaps succeed, and how many physical resources are available. Briegel et al. (1998) combined nested connection with purification to control accumulated errors.

Required background. Quantum communication introduces channels; entanglement swapping supplies the Bell measurement and Pauli frame. Distillation and memory hardware explain the two main resources used below.

Why Direct Transmission Fails at Long Distance

For a fiber with attenuation coefficient $\alpha$ in decibels per unit length, the channel transmissivity is

$$\eta(L) = 10^{-\alpha L/10}.$$

At $\alpha=0.2\ \mathrm{dB/km}$, every additional $50\ \mathrm{km}$ costs a factor of ten in transmission probability. Faster sources improve the number of attempts per second, but they do not change this exponential dependence on $L$.

For an ideal single-mode pure-loss bosonic channel with no input-energy constraint and unlimited two-way classical assistance, Pirandola et al. (2017) give the ultimate secret-key rate in bits, or entanglement-distribution rate in ebits, per channel use as the repeaterless value

$$K_{\mathrm{PLOB}}(\eta) = -\log_2(1-\eta) \simeq \frac{\eta}{\ln 2}$$

at high loss. A practical direct apparatus can operate far below this bound. Beating that apparatus is useful engineering progress; beating the bound under matched modes, channel uses, directions, and trust assumptions is a stronger claim.

Segmentation alone changes nothing

Suppose a path is divided into $m$ independent segments and all segments must succeed in the same clock cycle because no quantum state can wait. If segment $j$ succeeds with probability $p_j$, then

$$p_{\mathrm{sim}} = \prod_{j=1}^{m}p_j.$$

For ideal transmission factors with $\prod_j\eta_j=\eta(L)$, the product has the same exponential loss as the original path, with additional interface and detector penalties. Shorter spans become useful only when the protocol can retain or encode progress rather than demanding simultaneous end-to-end survival.

A useful target is a Bell pair stored at the endpoints with specified fidelity and a known Pauli frame. Its delivered rate counts completed usable pairs per wall-clock time, including retries, cutoffs, swaps and readout. A conditional fidelity characterizes the retained state; it does not include the probability of obtaining that state. Count optical modes separately when comparing against a capacity per mode. A detector click alone does not establish that a usable pair remains available at the endpoints.

Elementary-Link Generation

An elementary link joins neighboring nodes separated by $L_0$. Many protocols place an optical Bell analyzer near the midpoint, so photons travel about $L_0/2$ from each node. A schematic two-photon heralding probability is

$$p_0 \approx{} p_{\mathrm{emit}}^2 \eta_{\mathrm{int}}^2 \eta_{\mathrm f}(L_0/2)^2 \times \eta_{\mathrm{det}}^2 p_{\mathrm{BSM}}.$$

$p_{\mathrm{emit}}$ includes successful spin–photon or memory–photon state generation, $\eta_{\mathrm{int}}$ includes collection and conversion, $\eta_{\mathrm f}$ is fiber transmission, $\eta_{\mathrm{det}}$ is detector efficiency, and $p_{\mathrm{BSM}}$ is the accepted Bell-measurement fraction. The expression is not universal. Single-photon protocols, direct absorption, satellite links, cavity gates, and multiplexed schemes have different powers and correlated failure terms.

The attempt period cannot be inferred from a source clock alone. If the node must wait for a remote success or failure message before reusing a memory,

$$\tau_0 \gtrsim \tau_{\mathrm{prepare}} + \tau_{\mathrm{optical}} + \tau_{\mathrm{herald}} + \tau_{\mathrm{reset}}.$$

Depending on where detection and control decisions occur, $\tau_{\mathrm{herald}}$ contains one-way or round-trip propagation over part of $L_0$. A quoted megahertz emitter does not imply a megahertz independent link-attempt rate if only one memory mode is available.

Heralding must be trustworthy

A useful elementary event is heralded online and leaves a quantum state available at both nodes. Dark counts, multiphoton emission, double excitation, mode mismatch, and readout errors create false heralds. The link is characterized by both

$$p_0 \quad\text{and}\quad \rho_0\mid\mathrm{herald},$$

not one without the other. Background subtraction may diagnose hardware, but raw false-herald probability determines how the repeater behaves.

Synchronizing Probabilistic Links

Let each of two links be attempted once per interval and succeed independently with probability $p$. The waiting times $N_1,N_2\in\{1,2,\ldots\}$ are geometric:

$$\Pr(N_i=n) = (1-p)^{n-1}p.$$

Both links are ready after $N_{\max}=\max(N_1,N_2)$ intervals. Its exact mean is obtained from $\max(N_1,N_2)=N_1+N_2-\min(N_1,N_2)$. The minimum is geometric with success probability $1-(1-p)^2=p(2-p)$, giving

$$\mathbb E[N_{\max}] = \frac{3-2p}{p(2-p)} \simeq \frac{3}{2p}$$

for $p\ll1$. The earlier pair waits, on average,

$$\mathbb E[|N_1-N_2|] = \frac{2(1-p)}{p(2-p)} \simeq \frac1p$$

additional intervals. Thus a memory must usually survive much longer than one attempt. For $m$ equal links prepared in parallel, the small-$p$ behavior is

$$\mathbb E[N_{\max}] \simeq \frac{H_m}{p}, \qquad H_m = \sum_{j=1}^{m}\frac1j.$$

The logarithmic growth of $H_m$ is far better than demanding simultaneous success with probability $p^m$, but only if enough memories, modes, and local switching exist to keep the successes.

Memory age and cutoff policy

Memory quality is a function of age. One phenomenological description is

$$v(t) = \exp\left[-\left(\frac{t}{T_2}\right)^\beta\right], \qquad \eta_{\mathrm m}(t),$$

where $v(t)$ tracks coherence visibility and $\eta_{\mathrm m}(t)$ tracks retrieval. Conditioning only on successful retrieval can hide a low unconditional rate.

A scheduler may discard an old link after $k_c$ additional attempts. Conditioned on exactly one link being ready and its independent partner still missing, the probability that the partner arrives within the next $k_c$ trials is

$$P_{\mathrm{match}}(k_c) = 1-(1-p)^{k_c}.$$

Larger $k_c$ wastes fewer early successes but admits older, noisier states. The optimum depends on the fidelity threshold, latency objective, swap policy, and memory channel. There is no universally optimal timeout.

Averaging the stored coherence

Let $D=|N_1-N_2|$ and let the early link’s Bell coherence decay by $e^{-\lambda D}$, where $\lambda=\tau_0/T_{\mathrm{coh}}$ is dimensionless. Here $T_{\mathrm{coh}}$ describes the relevant stored-pair coherence; it need not equal one constituent qubit’s measured $T_2$. Ignore retrieval loss, swapping error and any additional final communication wait. Summing the two geometric arrival distributions gives

$$\Pr(D=0)=\frac{p}{2-p},\qquad \Pr(D=d)=\frac{2p(1-p)^d}{2-p}\quad(d\ge1).$$

Hence, with $z=e^{-\lambda}$,

$$\overline v=\mathbb E[z^D]=\frac{p}{2-p}\frac{1+(1-p)z}{1-(1-p)z}.$$

It is generally wrong to substitute the mean age into the exponential. Since $e^{-\lambda D}$ is convex, $\overline v\ge e^{-\lambda\mathbb E[D]}$. For a pure phase-noise model the averaged endpoint state is a mixture of $\Phi^+$ and $\Phi^-$, with target fidelity $(1+\overline v)/2$ after ideal swapping. If retrieval probability also depends on age, the accepted state is weighted by that probability and its delivered rate changes too.

Exact two-link examples with independent success probability $p=0.1$ per cycle and instantaneous perfect swapping. Top: retaining a success changes completion from a geometric trial with probability $p^2$ to the maximum of two geometric waits. Bottom: the solid curve averages the coherence over all age differences; the dashed curve substitutes the mean age. At $\lambda=0.1$ they give about $0.514$ and $0.388$, respectively. This model omits readout loss and extra control delays.

Multiplexing

For $M$ independent usable modes in one cycle,

$$p_M=1-(1-p)^M\simeq Mp\qquad(Mp\ll1).$$

Temporal, spectral, spatial and multi-emitter modes can reduce waiting only if successes can be identified, stored, routed and read without exhausting shared detectors or control hardware. Mode count is a physical resource: it must also appear in a comparison per channel use.

Swapping and Nested Connection

Once adjacent links are ready, an intermediate node performs a Bell measurement or equivalent local operation. This consumes the two short pairs and creates a longer endpoint pair, conditioned on the swap result. The result also determines a Pauli frame that must be communicated or tracked. The state identity and noisy Bell-diagonal examples are derived in Entanglement Swapping.

For $2^k$ equal elementary links, a binary nested schedule has levels $\ell=0,1,\ldots,k$. Level 0 prepares length-$L_0$ pairs; level 1 joins pairs across $2L_0$; level 2 joins those across $4L_0$; and so on.

The often-used estimate $T_{\ell+1}\approx3T_\ell/(2q_\ell)$ treats each lower-level completion time as approximately exponential, with mean $T_\ell$, and assumes a swap succeeds with probability $q_\ell$ after both inputs arrive. It also assumes complete restart after failure and negligible local time. Small elementary success probability alone does not make every higher-level completion time exponential.

For independent fresh attempts, a more informative renewal expression is

$$T_{\ell+1}=\frac{\mathbb E[\max(W_\ell^{(1)},W_\ell^{(2)})]+t_{\mathrm{swap}}}{q_\ell},$$

where $W_\ell^{(i)}$ are the random preparation times and each attempted swap costs $t_{\mathrm{swap}}$. Extra signaling after the final success must be added separately. Repeated signaling during failed attempts belongs inside the numerator. The distribution, not just $T_\ell=\mathbb E[W_\ell]$, controls the maximum.

There is a useful exact limit. With $n$ elementary links attempted in parallel, perfect unlimited memories, deterministic instantaneous swaps and negligible additional communication, completion occurs at $\max_jN_j$. Thus

$$\Pr(N_{\max}\le k)=[1-(1-p)^k]^n, \qquad \mathbb E[N_{\max}]=\sum_{j=1}^{n}\frac{(-1)^{j+1}\binom nj}{1-(1-p)^j}.$$

For fixed $n$ and small $p$, this becomes $H_n/p$. A nesting tree does not change the identity $\max(\max(N_1,N_2),\max(N_3,N_4))=\max(N_1,N_2,N_3,N_4)$. Repeating a factor $3/2$ at every level would incorrectly predict a power of $n$ in this deterministic limit. Shchukin, Schmidt and van Loock (2019) analyze the full waiting-time problem, including probabilistic swaps and finite memories.

These ideal formulas are useful checks, not predictions for a complete device. Failed swaps, age-dependent quality, purification, limited buffers, classical propagation and competing requests require the corresponding stochastic process.

Swap-as-soon-as-possible is not always optimal

Immediately swapping any ready neighboring links reduces memory age, but it can create mismatched link lengths, consume scarce memories, or produce a long pair that waits even longer for a partner. Other policies reserve pairs for balanced nesting levels, prioritize youngest or highest-fidelity states, or optimize a deadline-aware application utility. The best policy depends on the complete network state, not only on which edges are occupied.

Controlling Accumulated Errors

Range extension normally reduces quality. Memory dephasing, imperfect local gates, readout faults, leakage, phase-reference errors, and false heralds accumulate with every level. A repeater therefore needs an explicit quality strategy.

Entanglement distillation

First-generation designs consume several noisy pairs to probabilistically produce fewer better pairs. Purification may occur at elementary and nested levels. It requires two-way classical communication and additional waiting, and a failed round may destroy both inputs. Entanglement Distillation derives recurrence protocols, acceptance probabilities and hashing yields, including noise and distillability limits.

Quantum error correction

Encoded repeaters protect a logical qubit or logical Bell pair against erasure and operational faults. Located photon loss is structurally different from an unflagged Pauli error, so code selection and decoder assumptions matter. A third-generation design must correct loss quickly enough for one-way operation and suppress local operation errors below a logical target.

The Threshold Theorem explains fault-tolerance conditions. A statement that a repeater uses error correction is incomplete without the code, erasure and gate error model, syndrome circuit, decoder latency, logical failure target, and physical overhead.

Repeater Generations and Architectures

The common three-generation classification asks how loss and operation errors are handled. It is a resource taxonomy, not a chronology or quality ranking.

Generation	Loss	Operation errors	Signaling
First	heralded links	purification	two-way through nested levels
Second	heralded links	quantum error correction	two-way elementary heralds
Third	quantum error correction	quantum error correction	one-way transmission

Muralidharan et al. (2016) compare these choices under explicit component and resource assumptions.

Which generation minimizes resources depends on distance, attenuation, coupling and detection efficiency, gate error and duration, memory coherence, node spacing, multiplexing, and the required output. A slower first-generation system can be preferable when excellent memories exist but local gate counts must stay small. A third-generation system can dominate only after its much larger node satisfies the relevant thresholds.

All-photonic repeaters

The all-photonic construction of Azuma, Tamaki and Lo (2015) replaces long-lived matter memories with large photonic cluster states, loss-tolerant tree encodings, and adaptive measurements. Photonic branching provides multiple chances to complete an effective Bell measurement or indirect measurement before the flying resource is lost.

This does not make the repeater resource-free. The burden moves to near-deterministic or highly multiplexed graph-state generation, low-loss delay, fast feed-forward, switching, number-resolving or high-efficiency detection, and management of a large number of optical modes. Calling the design “memoryless” should not hide short optical delays and synchronization needed within a station.

Hybrid and encoded-photonic designs

Many proposals combine matter communication qubits, long-lived local memories, multimode ensemble storage, telecom photons, bosonic loss codes, and photonic cluster states. “Hybrid” identifies an interface strategy, not a security or performance level. Every conversion and transfer must appear in the end-to-end efficiency and noise model.

Tracking age, resources and latency

The useful storage-to-attempt ratio $\Lambda=T_{\mathrm{useful}}/\tau_0$ counts how many trials fit before a stored state becomes unsuitable. It must be interpreted alongside write/read efficiency, mode count, reset time and local gate performance. A long isolated coherence measurement does not specify the complete memory interface.

An event-driven model should track each pair’s state or error summary, age, memory slot, Pauli frame and pending messages. Link attempts, herald arrivals, cutoffs, swaps, purification, retrieval and resets change those quantities together. A cutoff may improve fidelity while lowering rate; a delayed decoder may occupy a buffer needed by another link. Report uncertainty in rate and latency quantiles, and specify correlations between links instead of silently treating all failures as independent.

For independent repeated cycles that reset the same physical state, the long-run delivered rate is $\mathbb E[B]/\mathbb E[T]$, where $B$ is usable output per cycle and $T$ includes failures. Averaging individual ratios $B/T$ gives a different quantity. More complex network scheduling requires a steady-state analysis rather than an assumed renewal cycle. Network architectures discuss multiple users and routing.

Direct Benchmarks and Trust Models

A repeater claim needs a comparator with the same requested service.

• Direct transmission: count every optical mode, source attempt, detector, and temporal or spectral channel.
• Repeaterless capacity: state the channel model and whether the PLOB or another bound applies.
• Trusted relays: deployed QKD keys may be decrypted or combined at trusted nodes; this is not end-to-end untrusted entanglement distribution.
• Twin-field and related QKD: an untrusted middle measurement can improve rate–loss scaling without providing stored endpoint entanglement or a general quantum channel.
• Satellite links: long free-space spans can reduce attenuation relative to fiber, but a trusted satellite or source is not automatically a repeater.

An entanglement-based repeater can, in principle, leave intermediate stations untrusted for an end-to-end cryptographic task because the endpoints test the delivered correlations. A compromised node can still deny service, bias loss, or exploit implementation side channels. Classical control messages must be authenticated, and the application proof must cover postselection, finite statistics, node collusion, and device assumptions.

Comparing a component with a repeater

A remote memory pair is an elementary link. A memory-based connection additionally needs another usable link and a swap at an intermediate station. A full application then needs delivered endpoint quality and timing after all retries and corrections. These are different achievements. Network case studies discuss experimental comparisons; component results should not be combined into a hypothetical device unless their interfaces, clocks, bandwidths and operating conditions are compatible.

Exercises

1: Fiber loss and the direct bound

A fiber has $\alpha=0.2\ \mathrm{dB/km}$ and length $L=200\ \mathrm{km}$. Compute its ideal transmissivity and the high-loss approximation to the PLOB bound in bits per optical mode.

Solution

The total attenuation is $40\ \mathrm{dB}$, so

$$\eta = 10^{-40/10} = 10^{-4}.$$

At high loss,

$$K_{\mathrm{PLOB}} \simeq \frac{10^{-4}}{\ln2} \approx 1.44\times10^{-4}$$

bits per mode. A real apparatus normally lies below this ideal pure-loss capacity because of source, coupling, detector, and protocol inefficiencies.

2: Why memoryless segments do not help

The same path is split into four ideal equal segments, each with transmissivity $0.1$. If all four must succeed in one trial, what is the end-to-end success probability? Compare it with the unsplit path.

Solution

Without memory or encoding,

$$p_{\mathrm{sim}} = (0.1)^4 = 10^{-4}.$$

This equals the unsplit ideal transmissivity. Real intermediate interfaces would make it smaller. Segmentation becomes useful only when the system can retain separate successes or correct losses.

3: Two-link synchronization

Two links are attempted in parallel with $p=0.01$ per $1\ \mathrm{ms}$ cycle. Find the mean number of cycles until both are ready and the mean additional age of the earlier success.

Solution

Using the exact formulas,

$$\mathbb E[N_{\max}] = \frac{3-0.02}{0.01(1.99)} \approx 149.75,$$

so both links are ready after about $150\ \mathrm{ms}$ on average. The earlier pair waits

$$\mathbb E[|N_1-N_2|] = \frac{2(0.99)}{0.01(1.99)} \approx 99.50$$

cycles, or about $99.5\ \mathrm{ms}$. A memory specified only for the $1\ \mathrm{ms}$ attempt interval would be inadequate.

4: Multiplexing

Each optical mode succeeds with probability $p=10^{-3}$. Compute the chance of at least one success among $M=100$ independent modes in a cycle and compare it with the approximation $Mp$.

Solution

The exact probability is

$$p_M = 1-(1-10^{-3})^{100} \approx 0.0952.$$

The linear approximation gives $Mp=0.1$, about five percent high relative to the exact value. It is already beginning to leave the $Mp\ll1$ regime. This gain is realizable only if all 100 modes can be independently identified and routed.

5: Cutoff versus decoherence

After one link succeeds, its partner has $p=0.02$ per $1\ \mathrm{ms}$ trial. Alice permits $k_c=50$ more trials. Find the matching probability. If the stored visibility is $v(t)=e^{-t/T_2}$ with $T_2=100\ \mathrm{ms}$, what is the visibility at the cutoff?

Solution

The partner arrives before cutoff with probability

$$P_{\mathrm{match}} = 1-(0.98)^{50} \approx 0.636.$$

At $t=50\ \mathrm{ms}$,

$$v(t_c) = e^{-50/100} = e^{-1/2} \approx 0.607.$$

Increasing the cutoff raises the match probability but admits lower-visibility pairs. The service threshold determines whether that trade is useful.

6: Four deterministic links

Four independent elementary links succeed with $p=0.01$ per cycle. Assume unlimited perfect memory, deterministic instantaneous swaps and no extra signaling. Find the small-$p$ approximation and exact mean completion time. Compare with repeating the factor $3/2$ twice from $T_0=1/p$.

Solution

The approximation is $H_4/p=(1+1/2+1/3+1/4)/0.01\approx208.33$ cycles. The exact result is

$$\mathbb E[N_{\max}]=\frac4p-\frac6{1-(1-p)^2}+\frac4{1-(1-p)^3}-\frac1{1-(1-p)^4}\approx207.79.$$

Repeating the factor $3/2$ gives $225$ cycles. After the first connection, the waiting time is a maximum of geometric variables; it is not a fresh geometric variable with that same mean. Its distribution must be used for the next maximum. For larger chains the discrepancy is structural: $H_n$ grows logarithmically, whereas $(3/2)^{\log_2n}=n^{\log_2(3/2)}$ grows as a power.

7: Classify the architecture

A proposal uses heralded photon transmission for elementary links, encodes each stored qubit to correct local gate errors, and waits for a classical herald only across each elementary span. Which generation best describes it?

Solution

It is a second-generation architecture: transmission loss is handled by heralded generation, while operation errors are handled by quantum error correction. Two-way signaling remains for elementary-link creation, but purification messages need not propagate through every nesting level. The label alone does not determine whether the proposed component parameters make it advantageous.

8: Average coherence or coherence at average age?

Use $p=0.1$ and $\lambda=\tau_0/T_{\mathrm{coh}}=0.1$ in the two-link phase-noise model. Compute $\mathbb E[D]$, $\overline v$ and the phase-only Bell fidelity. Compare with substituting the mean age into the exponential.

Solution

The mean age difference is $2(0.9)/(0.1\times1.9)\approx9.474$ cycles. With $z=e^{-0.1}$,

$$\overline v=\frac{0.1}{1.9}\frac{1+0.9e^{-0.1}}{1-0.9e^{-0.1}}\approx0.5144, \qquad F=\frac{1+\overline v}{2}\approx0.7572.$$

Substitution of the mean instead gives $e^{-0.1\times9.474}\approx0.3878$, or fidelity $0.6939$. The average output includes relatively young pairs with high coherence; replacing the whole distribution by its mean loses that contribution. These numbers describe the stated phase-only model, not additional retrieval, gate or heralding noise.

References

• K. Azuma, K. Tamaki, and H.-K. Lo, “All-photonic quantum repeaters,” Nature Communications 6, 6787 (2015), doi:10.1038/ncomms7787.
• H.-J. Briegel, W. Dür, J. I. Cirac, and P. Zoller, “Quantum repeaters: The role of imperfect local operations in quantum communication,” Physical Review Letters 81, 5932–5935 (1998), doi:10.1103/PhysRevLett.81.5932.
• S. Muralidharan et al., “Optimal architectures for long distance quantum communication,” Scientific Reports 6, 20463 (2016), doi:10.1038/srep20463.
• S. Pirandola, R. Laurenza, C. Ottaviani, and L. Banchi, “Fundamental limits of repeaterless quantum communications,” Nature Communications 8, 15043 (2017), doi:10.1038/ncomms15043.
• E. Shchukin, F. Schmidt, and P. van Loock, “On the waiting time in quantum repeaters with probabilistic entanglement swapping,” Physical Review A 100, 032322 (2019), arXiv:1710.06214.