Repeaterless Rate-Distance Bounds

How fast can two distant parties share secret key or entanglement through a lossy optical fibre, given unlimited two-way classical talk but no quantum hardware in between? Every QKD protocol ever built — BB84, E91, six-state, continuous-variable QKD — answers some version of this question, and they all hit the same wall. The wall has an exact height. In 2017 Pirandola, Laurenza, Ottaviani and Banchi proved that the two-way assisted secret-key capacity of a pure-loss bosonic channel of transmissivity $\eta$ is exactly $-\log_2(1-\eta)$ secret bits per channel use. This is the PLOB bound, the tightest repeaterless rate–distance limit, and it does not care what detector, source, or post-processing you use. What makes it remarkable is a closed-form, single-letter expression with matching achievability and a strong converse: higher rates are not just asymptotically forbidden — any attempt to exceed the line forces the secrecy to collapse. Because fibre loss makes $\eta$ fall exponentially with distance, the best key rate decays linearly in $\eta$ and so exponentially in kilometres — intercontinental QKD over a single direct fibre is not merely hard, it is capped below any useful rate by physics. The proof rests on two ideas that recur throughout quantum information. Teleportation simulation: a large class of channels, the lossy one included, is reproduced by teleporting through a single fixed resource — the channel's Choi state — collapsing an adaptive, many-round protocol into a one-shot quantity. And the relative entropy of entanglement, a monotone that cannot grow under local operations and classical communication, which upper-bounds the distillable key. Combine them and the two-way capacity is pinned to one number — and everything modern about long-distance quantum networking is a footnote to this curve.

Key takeaways

The PLOB bound fixes the two-way secret-key capacity of a pure-loss channel of transmissivity $\eta$ at exactly $K=-\log_2(1-\eta)$ bits per use, with no intermediate quantum node — a single-letter, strong-converse capacity.
Fibre loss makes $\eta=10^{-\alpha L/10}$ with $\alpha\approx0.2$ dB/km, so at long distance $K\approx1.44\,\eta$ : the rate falls linearly in $\eta$ and exponentially in length $L$ .
Every point-to-point protocol — BB84, E91, six-state, CV-QKD — lives strictly below this curve, whatever the hardware; it is the universal benchmark for a direct link.
It is proved by teleportation-simulating the channel into its Choi state, then capping the distillable key by the relative entropy of entanglement of that fixed state.
Twin-field QKD appears to beat it but does not: a relay splits one $\eta$ -channel into two $\sqrt{\eta}$ -arms, so the ceiling is the single-repeater bound $-\log_2(1-\sqrt{\eta})\approx1.44\sqrt{\eta}$ .
Key idea: teleportation-simulate the channel into a fixed Choi state, then upper-bound the two-way distillable key by the relative entropy of entanglement of that state; for the pure-loss channel this gives the exact secret-key capacity $K=-\log_2(1-\eta)$ .

The core insight

Teleportation-simulate the channel into a fixed Choi state, then upper-bound the two-way distillable key by the relative entropy of entanglement of that state; for the pure-loss channel this collapses to the exact secret-key capacity $K=-\log_2(1-\eta)$ — a linear-in- $\eta$ , hence exponential-in-distance, ceiling on every repeaterless protocol.

Where this lives today

The PLOB curve is now the universal yardstick for every long-distance QKD demonstration, and the headline experimental story of recent years has been beating it. Twin-field QKD, proposed in 2018, was engineered to operate against the single-repeater $-\log_2(1-\sqrt{\eta})$ bound: by interfering dim phase-locked pulses at an untrusted midpoint, laboratory groups have surpassed the repeaterless bound over hundreds of kilometres of real fibre, pushing past 500 km and, in the most extreme results, beyond 1000 km — where the direct-link ceiling is a handful of bits per second or less. Each experiment reports its rate against the PLOB line at the same loss, and clearing that line is the accepted signature of a genuinely relay-assisted channel. The bound is equally the justification for quantum repeaters: by proving exponential decay unavoidable without intermediate quantum nodes, it is the precise reason quantum memories and entanglement swapping are worth their cost. Beyond fibre, the same loss problem drives satellite QKD — free-space links from spacecraft like Micius trade fibre attenuation for diffraction to distribute entanglement over a thousand kilometres. The open frontier is closing the gap between these ceilings and what real, finite-key hardware achieves.

The physical picture

Think of entanglement as a fragile, non-replenishable currency poured into one end of the fibre. Each photon reaches Bob with probability $\eta$ and leaks to the environment with probability $1-\eta$ — and whatever leaks is, in the worst case, collected by an eavesdropper. With no intermediate station to catch and re-launch the signal, the surviving correlation can only decay, so the best key per use is capped by how much entanglement remains in the channel's Choi state — for the lossy channel, exactly $-\log_2(1-\eta)$ . The first figure contrasts the only two architectures that matter: a bare direct link, and the same distance split by one relay.

Figure

Direct vs relay-split link — a bare channel of transmissivity η is capped by K = −log₂(1−η) ≈ 1.44 η, while inserting one interfering relay turns it into two arms of √η and lifts the ceiling to −log₂(1−√η) ≈ 1.44 √η; at η = 10⁻⁴ (about 200 km) that is a 100× gap

The trick that makes the ceiling computable is to never touch the adaptive back-and-forth directly: replace every channel use by teleporting through one copy of the Choi state, so a whole $n$ -round protocol becomes local operations and classical communication (LOCC) on $n$ identical resources. Because the relative entropy of entanglement cannot increase under LOCC and is subadditive, the per-copy value caps the whole key. The second figure is the payoff on the plane engineers stare at: secret key per use against fibre length, with the direct PLOB line and the single-repeater line diverging as distance grows.

Figure

The rate–distance plane on a log axis — the repeaterless PLOB rate −log₂(1−η) falls one decade every 50 km (0.2 dB/km), crashing through 10⁻⁶ bits per use near 300 km, while the single-repeater √η line falls half as fast and is still alive past 500 km; at 100 km the ceiling is 0.0145 and at 200 km it is 1.4×10⁻⁴ bits per use

Why linear in $\eta$ ? At long distance $-\log_2(1-\eta)\approx1.44\,\eta$ , so halving $\eta$ halves the rate; and $\eta$ itself falls tenfold every 50 km, making the key rate exponential in kilometres — the single fact that forced the field toward repeaters and twin-field schemes. As a strong converse the curve is a genuine no-go line: never literally zero at finite distance, but dropping below any useful rate so fast that "direct link" and "throughput-starved" become synonyms past a few hundred kilometres.

The core equations

The headline is a single closed-form capacity: the exact two-way assisted secret-key capacity of the pure-loss channel — the supremum over all protocols with unlimited two-way classical communication of secret bits distillable per channel use.

K(\mathcal{N}_\eta) = -\log_2(1-\eta) \qquad (1)

where: $K$ — two-way assisted secret-key capacity, bits per channel use (per bosonic mode); $\mathcal{N}_\eta$ — pure-loss bosonic channel of transmissivity $\eta$ ; $\eta$ — transmissivity, $0\le\eta\le1$ , the fraction of input energy reaching Bob. Reading: the whole rate–distance story is one function of one number, diverging only in the lossless limit $\eta\to1$ . To turn the per-use capacity into a curve in kilometres, convert decibel loss into transmissivity: the dB definition is a base-ten logarithm of the input/output power ratio, which exponentiates into $\eta(L)$ .

\eta = 10^{-\alpha L/10} \qquad (2)

where: $L$ — fibre length in km; $\alpha$ — attenuation coefficient, $\approx0.2$ dB/km at the 1550 nm telecom window. Reading: every 15 km halves $\eta$ , every 50 km divides it by ten — the exponential decay equation (1) passes straight to the key rate. The capacity is proved as a converse: the two-way key can never exceed the channel's relative entropy of entanglement, optimised over input probes.

K(\mathcal{N}) \;\le\; E_R(\mathcal{N}) \;\equiv\; \sup_{\rho_{AB}} E_R\big((\mathcal{I}\otimes\mathcal{N})\rho_{AB}\big) \qquad (3)

where: $E_R$ — relative entropy of entanglement, an LOCC-monotone entanglement measure; $\rho_{AB}$ — an input bipartite state feeding system $B$ into the channel; $\mathcal{I}$ — identity channel on the untouched reference $A$ . Reading: any key you can distil is entanglement you must already have had, so an entanglement monotone caps it. The monotone measures how far a state sits from the nearest unentangled state in quantum relative entropy — the natural information-theoretic "distance", contractive under quantum channels.

E_R(\rho)=\min_{\sigma\in\mathrm{SEP}} S(\rho\,\|\,\sigma),\qquad S(\rho\,\|\,\sigma)=\mathrm{Tr}[\rho\log_2\rho]-\mathrm{Tr}[\rho\log_2\sigma] \qquad (4)

where: $\mathrm{SEP}$ — separable (unentangled) states; $S(\rho\,\|\,\sigma)$ — quantum relative entropy of $\rho$ with respect to $\sigma$ ; $\sigma$ — the closest separable state. Reading: $E_R$ vanishes exactly on separable states and grows with entanglement — why it is a legitimate ceiling on distillable key. The bridge from "adaptive protocol" to "one fixed state" is teleportation simulation: for a teleportation-covariant channel, a single use is reproduced by teleporting the input through one copy of the Choi state.

\mathcal{N}(\rho) = \mathrm{Tr}_M\big[\,\mathcal{T}\big(\rho\otimes\rho_{\mathcal{N}}\big)\big] \qquad (5)

where: $\rho_{\mathcal{N}}=(\mathcal{I}\otimes\mathcal{N})\Phi$ — the Choi (resource) state, half of a maximally entangled $\Phi$ sent through the channel; $\mathcal{T}$ — the generalised teleportation LOCC (Bell-type measurement plus correction); $\mathrm{Tr}_M$ — trace over the measured ancillary modes. Reading: with a fixed resource and an LOCC wrapper, stacking $n$ uses is just LOCC on $\rho_{\mathcal{N}}^{\otimes n}$ , and the single-letter cap $E_R(\rho_{\mathcal{N}})$ follows. In the long-distance regime the capacity is essentially linear in $\eta$ ; the series makes the $1.44\,\eta$ asymptote explicit.

-\log_2(1-\eta) = \frac{\eta}{\ln 2} + \frac{\eta^2}{2\ln 2} + O(\eta^3) \qquad (6)

where: $\ln2\approx0.6931$ converts nats to bits, so $1/\ln2\approx1.4427$ . Reading: to leading order $K\approx1.44\,\eta$ , the quadratic correction negligible once $\eta\lesssim0.1$ (beyond ~50 km). Finally, the hierarchy that saves twin-field QKD: a relay that measures and splits the channel is bounded not by (1) but by the single-repeater capacity, in which each half-channel carries $\sqrt{\eta}$ .

K_{\mathrm{1\text{-}rep}} = -\log_2\!\big(1-\sqrt{\eta}\big)\approx\frac{\sqrt{\eta}}{\ln2} \qquad (7)

where: $\sqrt{\eta}$ — transmissivity of each of the two equal arms created by a midpoint relay. Reading: since $\sqrt{\eta}\gg\eta$ for small $\eta$ , the single-repeater ceiling towers over the repeaterless one at long distance — the square-root advantage twin-field hardware chases.

Derivation AFrom adaptive protocols to a single Choi state

The goal is to justify the converse $K(\mathcal{N})\le E_R(\rho_{\mathcal{N}})$ — that the two-way secret key of a teleportation-covariant channel cannot exceed the relative entropy of entanglement of one copy of its Choi state — by showing how teleportation simulation ("Choi-stretching") collapses an arbitrary adaptive protocol into LOCC on a fixed stack of identical resource states, where an entanglement monotone finishes the job.

Step 1. Start from the most permissive protocol: Alice and Bob alternate channel uses with arbitrary rounds of two-way classical communication and local quantum processing, adapting each move to everything heard so far. Why: a bound proved at this generality binds every concrete scheme — BB84, E91, CV-QKD — as a special case.

Step 2. Replace each use $\mathcal{N}$ by its teleportation simulation, equation (5): teleport the signal through one copy of the Choi state $\rho_{\mathcal{N}}$ via a Bell-type measurement and correction, which for a teleportation-covariant channel reproduces $\mathcal{N}$ exactly. Why: it converts a dynamical object — a channel used mid-protocol — into a static resource prepared in advance, and states are far easier to bound than adaptive strategies.

Step 3. "Stretch" the protocol: since every use is teleportation through a copy of $\rho_{\mathcal{N}}$ , the whole $n$ -round interaction rearranges into a single LOCC map on $\rho_{\mathcal{N}}^{\otimes n}$ plus local registers — all adaptivity absorbed into the wrapper. Why: the decisive simplification, turning an $n$ -use protocol of any complexity into LOCC on $n$ identical states.

Step 4. Use the two properties of $E_R$ : LOCC-monotone and subadditive, $E_R(\rho^{\otimes n})\le n\,E_R(\rho)$ . The output key cannot exceed $E_R$ of the input, so the per-use rate is capped by $E_R(\rho_{\mathcal{N}})$ ; a supremum over probes gives $K(\mathcal{N})\le E_R(\mathcal{N})$ , equation (3). Why: monotonicity forbids LOCC from manufacturing entanglement and subadditivity turns the many-copy bound into a single-letter one — the converse the next derivation evaluates.

Go deeper · Derivation B · 1 minThe exact secret-key capacity of the lossy channel

Derivation BThe exact secret-key capacity of the lossy channel

The goal is to sharpen the general converse into an exact number for the pure-loss channel by evaluating $E_R$ of its Choi state and proving a matching achievability, squeezing the capacity from above and below to the single value $-\log_2(1-\eta)$ . This "sandwich" is what upgrades an inequality into a capacity theorem with a strong converse — the technical heart of the Pirandola–Laurenza–Ottaviani–Banchi result.

Step 1. The lossy channel is teleportation-covariant — it commutes appropriately with the phase-space displacements of continuous-variable teleportation — so Choi-stretching applies and $K(\mathcal{N}_\eta)\le E_R(\rho_{\mathcal{N}_\eta})$ holds. Why: covariance is the licence to reduce the adaptive protocol to LOCC on the lossy Choi state; without it equation (3) would be unavailable here.

Step 2. Evaluate the upper bound: construct an explicit separable $\sigma$ close to the lossy Choi state and compute $S(\rho_{\mathcal{N}_\eta}\,\|\,\sigma)$ ; in the infinite-energy, maximally entangled probe limit this gives exactly $E_R(\rho_{\mathcal{N}_\eta})=-\log_2(1-\eta)$ . Why: a concrete near-optimal $\sigma$ turns an abstract minimisation into a closed number — where the form $-\log_2(1-\eta)$ appears.

Step 3. Prove a matching lower bound: reverse-reconciliation continuous-variable QKD attains the reverse coherent information of the pure-loss channel, which also equals $-\log_2(1-\eta)$ . Why: a converse says only "no higher"; a concrete achievability says "this high is reachable", and together they pin the capacity rather than bracket it.

Step 4. Sandwich: upper bound and achievable rate coincide, so every inequality is an equality and $K(\mathcal{N}_\eta)=-\log_2(1-\eta)$ exactly. The relative-entropy method makes this a strong converse — rates above the line force secrecy to vanish — and the asymptotics check out, $K\to\eta/\ln2\approx1.44\,\eta$ as $\eta\to0$ and diverging as $\eta\to1$ . Why: coincidence of the two bounds is what "determines a capacity" means, and the strong-converse property is what makes the curve a hard wall rather than a soft trend.

Go deeper · Derivation C · 1 minWhy twin-field QKD does not break the bound

Derivation CWhy twin-field QKD does not break the bound

The goal is to resolve the apparent paradox that twin-field QKD reports rates above the PLOB curve, by showing that a twin-field link is not a repeaterless channel at all: a central interfering station measures the field and splits one channel of transmissivity $\eta$ into two arms of $\sqrt{\eta}$ , placing the architecture under the single-repeater bound rather than the no-repeater one, with no contradiction anywhere.

Step 1. Geometry: Alice and Bob each send a dim phase-locked pulse to an untrusted midpoint node, so the total transmissivity factorises as $\eta=\eta_A\eta_B$ with each arm carrying $\eta_A=\eta_B=\sqrt{\eta}$ . Why: the photon that produces a detection travels only half the distance, so it experiences $\sqrt{\eta}$ , not $\eta$ — the origin of the square-root scaling.

Step 2. The node measures and effectively swaps single-photon coherence between the arms, so the applicable limit is the single-repeater capacity (7), $-\log_2(1-\sqrt{\eta})$ , not the repeaterless (1). Why: once an intermediate node measures and post-processes, the "no quantum equipment in between" premise is violated and a higher ceiling governs.

Step 3. The advantage is $\sqrt{\eta}/\eta=\eta^{-1/2}$ : a factor 100 at $\eta=10^{-4}$ (about 200 km), $10^4$ at $\eta=10^{-8}$ (about 400 km). Why: this is the diverging gap of the second figure — why twin-field beats a direct link by orders of magnitude at long range while respecting every bound.

Step 4. No violation: twin-field lives below its own ceiling $-\log_2(1-\sqrt{\eta})$ , which lies above the repeaterless $-\log_2(1-\eta)$ . Surpassing the repeaterless curve is the intended signature of a relay-assisted architecture, not evidence against the theorem. Why: the capacity hierarchy is the organising principle — knowing which bound applies to which architecture turns a "paradox" into a design specification.

Limiting cases

How the PLOB secret-key capacity behaves across the regimes that matter:

Case	Result	Meaning
Long distance / high loss	$K=-\log_2(1-\eta)\xrightarrow{\eta\to0}\dfrac{\eta}{\ln2}\approx1.44\,\eta$	Linear in transmissivity, hence exponential in length since $\eta=10^{-\alpha L/10}$ ; every 50 km divides both $\eta$ and the rate by ten.
Lossless channel	$\eta\to1\;\Rightarrow\;K\to+\infty$	The bound diverges: arbitrarily much key per use in principle, limited only by the source's mode budget — the mark of a perfect link.
Single-repeater (twin-field) bound	$K_{\mathrm{1\text{-}rep}}=-\log_2(1-\sqrt{\eta})\approx1.44\sqrt{\eta}$	One midpoint relay improves the scaling to $\sqrt{\eta}$ ; the ceiling twin-field QKD approaches, sitting above the repeaterless curve at long range.
Thermal-loss channel	$K\le -\log_2\!\big[(1-\eta)\eta^{\bar n}\big]-g(\bar n)$	Thermal noise of mean photon number $\bar n$ lowers the ceiling; $g(x)=(x+1)\log_2(x+1)-x\log_2 x$ is the bosonic entropy, and $\bar n=0$ recovers PLOB.

Common trapTwin-field QKD's √η scaling "violates" the PLOB bound

It does not, and the reason is a definition, not a loophole. The PLOB capacity $-\log_2(1-\eta)$ is the ceiling for a repeaterless link — a direct channel of transmissivity $\eta$ with no quantum equipment in between. Twin-field QKD places an interfering node at the midpoint that measures the fields and splits one $\eta$ -channel into two arms of $\sqrt{\eta}$ , so the architecture is governed by the single-repeater bound $-\log_2(1-\sqrt{\eta})$ , far above the repeaterless one at long distance. Experiments beating the PLOB curve behave exactly as theory predicts a relay-assisted system should. A quieter trap: $K$ is a capacity per channel use (per mode), not per second — multiply by the source's mode rate for bits per second, and real links fall well below once detector efficiency, dark counts, and finite-key effects enter.

Worked examples

1 · The rate ceiling of a real fibre link, in bits per second

A QKD system runs a 1 GHz source ( $10^9$ channel uses per second) over standard fibre at $\alpha=0.2$ dB/km. What is the maximum secret-key rate at 100, 200, 300 and 400 km, ignoring every device imperfection?

Step 1. Convert distance to transmissivity with equation (2): loss $=\alpha L$ dB, so $\eta=10^{-\alpha L/10}$ . At 100 km, 20 dB gives $\eta=10^{-2}$ ; at 200 km, 40 dB gives $10^{-4}$ ; at 300 km, $10^{-6}$ ; at 400 km, $10^{-8}$ .

Step 2. Apply capacity (1) in the linear form $K\approx1.4427\,\eta$ bits per use: $K=\mathbf{1.44\times10^{-2}}$ at 100 km (the exact $-\log_2 0.99=0.01449$ confirms it), $\mathbf{1.44\times10^{-4}}$ at 200 km, $\mathbf{1.44\times10^{-6}}$ at 300 km, $\mathbf{1.44\times10^{-8}}$ at 400 km — one decade lost per 50 km.

Step 3. Multiply by the mode rate $10^9$ /s for the ceiling in bits per second: $\mathbf{1.44\times10^{7}}$ bps ( $\approx14$ Mbit/s) at 100 km, $\mathbf{1.44\times10^{5}}$ bps ( $\approx144$ kbit/s) at 200 km, $\mathbf{1.44\times10^{3}}$ bps at 300 km, and just $\mathbf{14.4}$ bps at 400 km.

Step 4. Even with perfect source and detectors, a direct link is throughput-starved past a few hundred kilometres — at 400 km the ceiling is under fifteen bits per second, and real hardware sits well below. This is the quantitative case for repeaters: the bound says the direct route cannot be rescued by better engineering.

2 · The twin-field advantage and where it takes over

Compare the repeaterless ceiling with the single-repeater (twin-field) ceiling at 200 km and 400 km, and state the advantage.

Step 1. At 200 km, $\eta=10^{-4}$ and $\sqrt{\eta}=10^{-2}$ : the repeaterless ceiling is $K\approx1.44\times10^{-4}$ , the single-repeater ceiling $\approx1.44\times10^{-2}$ bits per use — a ratio $\sqrt{\eta}/\eta=\eta^{-1/2}=\mathbf{100}$ .

Step 2. At 400 km, $\eta=10^{-8}$ and $\sqrt{\eta}=10^{-4}$ : the repeaterless ceiling collapses to $\approx1.44\times10^{-8}$ while the single-repeater ceiling holds at $\approx1.44\times10^{-4}$ — a ratio $\eta^{-1/2}=\mathbf{10^{4}}$ .

Step 3. The advantage grows without bound with distance: the two log-plot lines have slopes in ratio 2:1 (one decade per 50 km vs per 100 km). Note the single-repeater ceiling at 400 km, $1.44\times10^{-4}$ , equals the repeaterless ceiling at 200 km — a relay doubles the reach at fixed rate. A midpoint relay buys a factor $\eta^{-1/2}$ in rate, or a doubling of distance at fixed rate — why twin-field QKD is the technology of choice past ~200 km. By the same algebra, halving the fibre loss coefficient also turns $\eta$ -scaling into $\sqrt{\eta}$ -scaling.

Physics of the portrait

The repeaterless bound is where Shannon's idea of a channel capacity met the peculiarities of quantum entanglement — and it took two decades of tools before the answer could be written in one line.

Origins · 1 minPLOB bound and how this was discovered

PLOB bound

the exact repeaterless secret-key capacity K = −log₂(1−η) · two-way quantum channel capacities and quantum key distribution

The repeaterless rate–distance limit is a young result built from old parts. Quantum teleportation, introduced by Bennett and five co-workers in 1993, supplied the LOCC primitive that would let a channel be "simulated" by a fixed resource state; the relative entropy of entanglement, defined by Vedral, Plenio and collaborators in 1997, supplied a monotone that cannot increase under local operations and classical communication — the property that makes it a legitimate ceiling on distillable key. The question these tools were aimed at resisted an exact answer for years. In 2014 Takeoka, Guha and Wilde produced the first general repeaterless bound via squashed entanglement, proving the rate must decay with loss — the "you need repeaters" verdict — though it was not tight. The exact capacity arrived in 2017, when Stefano Pirandola, Riccardo Laurenza, Carlo Ottaviani and Leonardo Banchi combined teleportation simulation ("Choi-stretching") with the relative entropy of entanglement and showed the two bounds meet at $-\log_2(1-\eta)$ , matching the reverse-coherent-information rate García-Patrón and colleagues had shown achievable in 2009. The result is a strong-converse, single-letter capacity — as clean as Shannon's noisy-channel theorem, but for entanglement and secrecy — and it drew the sharp line between what a direct link can do and what demands a quantum repeater. Within a year, twin-field QKD showed how to live on the higher, single-repeater side of that line.

History

1993

— Bennett and co-workers introduce quantum teleportation, the LOCC primitive behind channel simulation

1997

— the relative entropy of entanglement is defined as an LOCC-monotone entanglement measure

2009

— the reverse coherent information is shown achievable for the pure-loss channel

2014

— Takeoka, Guha and Wilde prove the first general repeaterless rate–loss bound via squashed entanglement

2017

— Pirandola, Laurenza, Ottaviani and Banchi fix the exact secret-key capacity at −log₂(1−η)

Going further

References. Pirandola, Laurenza, Ottaviani, Banchi, "Fundamental limits of repeaterless quantum communications", Nature Communications 8, 15043 (2017) · Takeoka, Guha, Wilde, "Fundamental rate-loss tradeoff for optical quantum key distribution", Nature Communications 5, 5235 (2014) · Lucamarini, Yuan, Dynes, Shields, "Overcoming the rate–distance limit of quantum key distribution without quantum repeaters", Nature 557, 400 (2018) · Wilde, "Quantum Information Theory", 2nd ed., Cambridge (2017), chapters on private capacity and squashed entanglement · Preskill, Lecture Notes for Physics 219, Ch. 10 (quantum channels and capacities) · Nielsen & Chuang, "Quantum Computation and Quantum Information", 10th anniv. ed., Ch. 12 · Watrous, "The Theory of Quantum Information", Cambridge (2018), entanglement measures.

Provenance. The capacity $K=-\log_2(1-\eta)$ and the single-repeater bound $-\log_2(1-\sqrt{\eta})$ are from Pirandola et al., Nat. Commun. 8, 15043 (2017); a verifier should reproduce the converse via teleportation simulation plus the relative entropy of entanglement and the achievability via the reverse coherent information (García-Patrón et al., 2009). The first general repeaterless bound and the $\sim1.44\,\eta$ order of magnitude are consistent with Takeoka–Guha–Wilde, Nat. Commun. 5, 5235 (2014). The fibre loss $\alpha\approx0.2$ dB/km at 1550 nm is the standard single-mode value, and $\eta=10^{-\alpha L/10}$ is the decibel definition; all numerical anchors (20 dB at 100 km, 40 dB at 200 km, $\eta=10^{-2}$ and $10^{-4}$ , $1/\ln2=1.4427$ ) are directly recomputable. The relative entropy of entanglement definition follows Vedral–Plenio and Watrous.

Prerequisites. Quantum Teleportation · Entanglement Swapping · Bell-State Measurement · E91 Protocol · Information Reconciliation and Privacy Amplification. Leads to. Quantum Repeaters · Twin-Field QKD · Bell-State Measurement.

Ask the tutor: Ask me to walk through the Choi-stretching argument line by line, or to plot exactly where twin-field QKD overtakes a direct link on your fibre's loss budget.

What exactly does "repeaterless" mean in the PLOB bound?

It means a single direct quantum channel from Alice to Bob with no quantum equipment in between — no quantum memory, no entanglement-swapping node, no measuring relay. The parties may exchange unlimited classical messages both ways and do arbitrary local processing at their ends, but the channel is a bare pipe. The instant an intermediate device acts on the quantum signal — even an untrusted one that only measures, as in twin-field QKD — the link is no longer repeaterless and the higher single-repeater bound applies. "Repeaterless" is a statement about the middle of the link, not the endpoints.

Why is the relative entropy of entanglement the right tool here?

Because secret key is, at bottom, distilled entanglement, and any good entanglement measure must not increase under local operations and classical communication — you cannot make entanglement for free by talking and acting locally. The relative entropy of entanglement has exactly that monotonicity and is subadditive on tensor copies, so a stack of $n$ identical resources is capped by $n$ times the single-copy value. Combined with teleportation simulation, which turns any adaptive protocol into LOCC on such a stack, it converts the messy two-way key rate into one computable number: $E_R$ of the channel's Choi state.

Is the PLOB bound really tight, or just an upper bound?

It is tight — an exact capacity, not merely a ceiling. The upper bound is the relative-entropy-of-entanglement converse; the matching lower bound is an explicit protocol, reverse-reconciliation continuous-variable QKD, which achieves the reverse coherent information, equal to $-\log_2(1-\eta)$ for the pure-loss channel. Because the two coincide, the inequality becomes an equality and the capacity is fixed. This is what distinguishes PLOB from the earlier Takeoka–Guha–Wilde bound, which correctly showed the rate must decay with loss but was not tight; PLOB closed the gap and proved the strong-converse property.

What does "strong converse" add over an ordinary capacity statement?

An ordinary (weak) converse says that above the capacity your error cannot be driven to zero — it stays bounded away from perfect. A strong converse is harsher: attempt any rate above the capacity and the secrecy does not merely fail to reach one, it collapses all the way to zero. There is no gentle degradation and no rate–reliability trade-off to exploit above the line. That is what makes the PLOB curve a genuine wall rather than a soft trend — you cannot buy a little extra rate by tolerating a little insecurity, because past the line there is essentially no security left.

Does the bound apply to entanglement distribution as well as key?

Yes, in the same breath. The quantity PLOB compute is the two-way assisted capacity for both secret key and distillable entanglement, and for the pure-loss channel these coincide at $-\log_2(1-\eta)$ . So the bound simultaneously limits QKD (secret bits) and entanglement-based networking (shared ebits for teleportation or distributed computing). Any resource you could spread over a direct lossy link — key, entanglement, or the ability to teleport — is capped by the same curve, which is why the result is treated as the fundamental limit of point-to-point quantum communication, not a QKD-specific technicality.

Why do we still build quantum repeaters if twin-field already beats the bound?

Because twin-field only reaches the single-repeater scaling $\sqrt{\eta}$ , still exponential in distance — just with half the exponent. It doubles your reach at a given rate, a huge gain, but it does not make the rate decay polynomially or restore continental-scale throughput. True quantum repeaters, using quantum memories and multi-node entanglement swapping, aim to break the exponential entirely and scale far more gently with distance, which is what a genuine intercontinental network needs. Twin-field squeezes the most out of a single relay; multi-node repeaters are the long-term architecture the bound proves we cannot ultimately do without.