Quantum Memories for Repeaters

A single photon launched into a telecom fiber is attenuated exponentially, and no-cloning forbids the amplifier that rescues a fading classical signal. The transmissivity is $\eta_{\mathrm{fib}}=e^{-L/L_{\mathrm{att}}}$ with an attenuation length $L_{\mathrm{att}}\approx 22\,\mathrm{km}$ at $1550\,\mathrm{nm}$ ; over $1000\,\mathrm{km}$ this is about $10^{-20}$ , so even a $10\,\mathrm{GHz}$ source would deliver roughly one photon every three centuries. Amplification is out because copying an unknown state is impossible, so the quantum repeater takes the only remaining route: break the span into short elementary links, generate entanglement on each one, and fuse the links with entanglement swapping — the entangled-state relay that teleportation makes possible. The scheme works only because each link can be heralded — it announces success with a classical bit (a detector click in a Bell-state measurement, a DLCZ Stokes photon) so that failures are simply retried rather than silently corrupting the data. But heralding is probabilistic and asynchronous: link 1 may fire long before link 5. This is exactly where a quantum memory earns its keep. It holds the herald-confirmed half of an entangled pair, idle and coherent, until the neighbouring link also succeeds, at which point a swap fuses the two. The memory is the temporal buffer that turns many independent low-probability events into one synchronized chain. A memory's usefulness is captured by a handful of numbers: the storage-and-retrieval efficiency $\eta$ , the coherence time $T_2$ that bounds how long the state stays usable, the time-bandwidth product $T_2 B$ that counts how many modes fit inside that window, and the fidelity $F$ that measures distortion. This article develops the loss argument, the rate scaling that makes memories mandatory, the DLCZ heralding mechanism, the standard figures of merit, and how imperfections compound across a swapped chain — the physics behind long-distance QKD and the quantum internet.

Key takeaways

Optical fiber attenuates a photon exponentially, $\eta_{\mathrm{fib}}=e^{-L/L_{\mathrm{att}}}$ with $L_{\mathrm{att}}\approx 22\,\mathrm{km}$ , and no-cloning forbids the amplifier that saves a classical signal — so a quantum repeater splits the span into short heralded links joined by entanglement swapping.
A quantum memory stores a flying photonic qubit as a stationary excitation (an atomic spin wave, a rare-earth-ion ensemble, a trapped ion, an NV or SiV centre) and re-emits it on demand; it is characterised by efficiency $\eta$ , coherence time $T_2$ , and fidelity $F$ .
Without memories an $n$ -link chain needs simultaneous success and its rate collapses as $p^{\,n}$ ; memories that hold a heralded link while neighbours catch up convert this to polynomial-in-distance scaling — the reason memories are "required".
Key figures of merit are the time-bandwidth product $T_2 B$ (how many modes fit within coherence) and the efficiency- and dark-count-limited generation rate; a useful node needs $\eta$ high and $T_2$ long enough that swapping outpaces decoherence.
Memory protocols include electromagnetically induced transparency (EIT), the atomic frequency comb (AFC), controlled reversible inhomogeneous broadening (CRIB), and the Duan–Lukin–Cirac–Zoller (DLCZ) heralded spin-wave scheme.
Heralded entanglement plus swapping plus memory is the architecture for QKD beyond the repeaterless secret-key bound and for a future quantum internet.
Key idea: A quantum memory is the temporal buffer that turns probabilistic, asynchronous heralded entanglement on many short links into a deterministic, synchronized long-distance entangled pair — replacing the impossible amplifier with storage plus entanglement swapping.

Where this lives today

Memory-based repeaters have moved from proposal to laboratory reality. The decisive proof of principle came in 2020, when a Harvard–MIT team (Bhaskar and co-workers, Lukin group) used a single silicon-vacancy centre in a nanophotonic diamond cavity as an integrated memory-and-swap node and demonstrated memory-enhanced quantum communication — a key rate exceeding what the same channel allows by direct transmission, the first experiment to beat the loss-equivalent bound using a genuine memory. In parallel, rare-earth-ion crystals push the storage frontier: europium-doped Y₂SiO₅ has shown nuclear-spin coherence of six hours, and atomic-frequency-comb memories store hundreds of temporal, spectral and spatial modes, directly cashing in the time-bandwidth product for rate. Solid-state spin platforms have built small networks: the Delft group entangled nitrogen-vacancy centres across $1.3\,\mathrm{km}$ in the 2015 loophole-free Bell test and, in 2021–2022, operated a three-node quantum network with memory qubits buffering entanglement between non-neighbouring nodes. Atomic-ensemble memories have been entangled over tens of kilometres of metropolitan fiber. Meanwhile the target line itself has been attacked from another direction: twin-field QKD surpasses the PLOB bound without a memory by using single-photon interference, sharpening the question of exactly when a stored quantum state is the right tool. The open problems now are engineering ones — raising $\eta$ toward unity, extending $T_2$ into the useful-distance regime, and integrating memories with telecom-wavelength photons — each a direct assault on the figures of merit above.

The physical picture

Picture relaying a fragile glass sculpture across a continent by a chain of couriers, under two cruel rules: each handoff succeeds only occasionally, and you may neither destroy nor copy the piece. The only workable scheme is for each courier to wait — holding the sculpture — until the next courier is ready, then hand it over. That waiting is a memory; the handoff is an entanglement swap. And if waiting degrades the piece faster than the handoffs complete, nothing arrives intact. That single tension — how long you can store versus how long you must wait — is the $\eta$ -versus- $T_2$ tradeoff that governs the whole field. Heralding is the trick that turns a one-shot fragile transmission into a repeatable one. Because each link announces "I hold a good pair" with a classical bit, you never advance the protocol on a corrupted link; you simply retry until it clicks. The memory's job is then narrow and precise: freeze the herald-confirmed state in place while the rest of the chain catches up, so that successes occurring at random, uncorrelated times can still be combined. The first figure contrasts the hopeless direct route with the repeatered one and puts the loss numbers on the page.

Figure

Direct transmission versus a quantum repeater: over 1000 km of fiber the transmissivity η = e^(−L/L_att) with L_att ≈ 22 km falls to about 10⁻²⁰, so a 10 GHz source delivers roughly one photon every three centuries — while splitting the span into four heralded elementary links of 250 km each raises the per-link transmissivity to about 10⁻⁵, and memories hold each success until its neighbour is ready to swap

A useful mental scaling law separates the hopeless from the feasible. With $n$ independent links each succeeding with probability $p$ , requiring all of them to fire in the same time bin has probability $p^{\,n}$ — exponentially small, and no better than direct transmission. But if a memory lets each link be retried independently until it succeeds and then holds it, the chain assembles in a time that grows only polynomially with the number of links. The memory is precisely what breaks the $p^{\,n}$ curse. The catch is that storage is never free: the moment a link is heralded, its stored qubit begins to decohere, and the second figure quantifies how fast that clock runs down.

Figure

How a stored qubit forgets: the depolarizing fidelity F(t) = ¼(1 + 3 e^(−t/T₂)) of one half of a Bell pair waiting in memory decays from 1 toward the maximally mixed floor ¼. At t = T₂ the fidelity is only 0.526, and it drops below the entanglement threshold F = ½ once t exceeds T₂ ln3 ≈ 1.10 T₂ — the hard deadline the swap must beat

The core equations

The starting point is the loss that makes the whole enterprise necessary. Fiber attenuation is quoted in decibels per kilometre, but the physics is a bare exponential; converting one to the other fixes the attenuation length that sets every distance scale in the field.

\eta_{\mathrm{fib}}(L)=e^{-L/L_{\mathrm{att}}},\qquad L_{\mathrm{att}}=\frac{10}{\alpha\ln 10}\approx 22\,\mathrm{km} \qquad (1)

where: $L$ — channel length; $\alpha$ — attenuation coefficient, $\approx 0.2\,\mathrm{dB/km}$ at $1550\,\mathrm{nm}$ ; $L_{\mathrm{att}}$ — attenuation length, the distance for $1/e$ transmission. Reading: transmissivity falls by a factor $e$ every $\approx 22\,\mathrm{km}$ , so at $1000\,\mathrm{km}$ it is $\approx 10^{-20}$ — the exponential wall a repeater must climb over. The resource a repeater manufactures on each elementary link is a maximally entangled pair shared between two memory nodes. It is the currency that swapping and teleportation spend.

|\Phi^+\rangle=\tfrac{1}{\sqrt2}\big(|0\rangle_A|0\rangle_B+|1\rangle_A|1\rangle_B\big) \qquad (2)

where: $A,B$ — the two memory nodes terminating one elementary link; $|0\rangle,|1\rangle$ — the logical basis of each stored memory mode. Reading: a perfect link puts $A$ and $B$ into this state; every imperfection below is measured as a departure from it. How far a stored or retrieved state has drifted from the ideal is quantified by fidelity. For a pure target the expression is a simple overlap; for two general mixed states one needs the Uhlmann form.

F=\langle\psi|\rho|\psi\rangle,\qquad \mathcal{F}(\rho,\sigma)=\Big(\mathrm{Tr}\sqrt{\sqrt{\rho}\,\sigma\sqrt{\rho}}\Big)^2 \qquad (3)

where: $|\psi\rangle$ — the ideal stored/retrieved state; $\rho$ — the actual (mixed) state; $\mathcal{F}$ — Uhlmann fidelity for two general states. Reading: $F=1$ is perfect recall, and a two-qubit Werner state is entangled only while $F>\tfrac12$ — the threshold marked on the second figure. While a heralded qubit waits for its neighbour, the dominant effect of memory imperfection is depolarization: the stored state is gradually replaced by the featureless maximally mixed state, with a weight that grows on the coherence timescale $T_2$ .

\rho(t)=e^{-t/T_2}\,|\psi\rangle\langle\psi| + \big(1-e^{-t/T_2}\big)\,\tfrac{I}{2},\qquad F(t)=\tfrac12\big(1+e^{-t/T_2}\big) \qquad (4)

where: $t$ — storage (waiting) time; $T_2$ — memory coherence time; $I/2$ — the single-qubit maximally mixed state reached at long times. Reading: a lone stored qubit's fidelity falls from $1$ toward $\tfrac12$ ; keeping it useful demands $t\ll T_2$ , i.e. fast swapping or long-lived memories. Not all of a memory's coherence is spent on one mode. The product of coherence time and acceptance bandwidth counts how many independent modes can share the memory at once — the multiplexing resource that lifts the effective link rate.

\mathrm{TBP}=T_2\,B \qquad (5)

where: $B$ — memory acceptance bandwidth (the inverse of the shortest storable pulse); $T_2$ — coherence time. Reading: $T_2 B$ is roughly the number of distinct time-bin (or spectral, or spatial) modes stored within one coherence window; rare-earth AFC crystals reach $10^3$ – $10^6$ modes, which is why they are pursued for multiplexed nodes. The reason memories are not a mere convenience but a necessity is a scaling law. Without storage, every link must fire in the same shot; with storage, each link is retried independently and held, and the swap schedule pays only a polynomial overhead.

R_{\mathrm{no\,mem}}\propto\frac{1}{T_0}\big(p_0\,\eta_{\mathrm{fib}}(L/n)\big)^{n},\qquad R_{\mathrm{mem}}\propto\frac{1}{T_0}\,\frac{p_0\,\eta_{\mathrm{fib}}(L/n)}{(3/2)^{\log_2 n}} \qquad (6)

where: $n$ — number of elementary links; $p_0$ — per-attempt prefactor from sources and detectors; $T_0$ — attempt repetition period; $(3/2)^{\log_2 n}=n^{0.585}$ — the schematic overhead from waiting for the slower of two child links at each of $\log_2 n$ swap levels. Reading: the memoryless rate is exponentially small in $n$ ; the memory-assisted rate carries only a single link's loss times a polynomial factor. Heralding on an atomic ensemble has a canonical microscopic model, DLCZ, in which a weak write pulse simultaneously scatters a Stokes photon and creates one collective spin excitation. Detecting the photon projects the ensemble into a stored single excitation.

|\psi_{\mathrm{DLCZ}}\rangle \propto |0_a 0_s\rangle + \sqrt{p_c}\,|1_a 1_s\rangle + O(p_c) \qquad (7)

where: $|1_s\rangle$ — one collective spin excitation in the ensemble; $|1_a\rangle$ — the emitted Stokes photon that heralds it; $p_c$ — the small write-excitation probability. Reading: interfering the Stokes photons from two ensembles $L,R$ at a beam splitter erases which-ensemble information and heralds the entangled spin-wave state $|1\rangle_L|0\rangle_R+e^{i\phi}|0\rangle_L|1\rangle_R$ — one elementary link. Finally, the bar a repeater must clear is a proven bound: without a repeater, the two-way secret-key capacity of a pure-loss channel is fixed by its transmissivity alone.

R_{\mathrm{PLOB}} = -\log_2(1-\eta)\approx\frac{\eta}{\ln 2}\ (\eta\ll1),\qquad \eta=\eta_{\mathrm{fib}}(L) \qquad (8)

where: $\eta$ — end-to-end transmissivity; $R_{\mathrm{PLOB}}$ — the ultimate repeaterless secret-key capacity (Pirandola, Laurenza, Ottaviani, Banchi). Reading: the key rate falls linearly in $\eta$ and hence exponentially in distance; a genuine repeater is any scheme whose rate-versus-distance curve beats this line.

Derivation AFrom decibels to an exponential attenuation length

The goal is to turn the engineer's unit — decibels of loss per kilometre — into the physicist's exponential, extracting the single length $L_{\mathrm{att}}$ that sets every distance scale in the repeater problem, and then to feed a real span through it to expose just how brutal the loss is. We do this because the entire justification for memories rests on one number, the $\approx 10^{-20}$ transmissivity at continental distance, and a reader should be able to reproduce it from the tabulated $0.2\,\mathrm{dB/km}$ alone.

Step 1. Loss in decibels over a length $L$ is $\mathrm{dB}=\alpha L$ , and by the definition of the decibel the transmissivity is $\eta=10^{-\alpha L/10}$ . Why: decibels are a base-ten logarithmic measure of a power ratio, so undoing the logarithm returns a power of ten.

Step 2. Rewrite the base-ten power as a natural exponential using $10^{x}=e^{x\ln 10}$ , giving $\eta=e^{-(\alpha\ln 10/10)L}$ and identifying $1/L_{\mathrm{att}}=\alpha\ln 10/10$ . Why: casting the loss as $e^{-L/L_{\mathrm{att}}}$ isolates a single length scale, which is far more portable than a per-kilometre figure.

Step 3. Insert $\alpha=0.2\,\mathrm{dB/km}$ and $\ln 10=2.3026$ : $L_{\mathrm{att}}=10/(0.2\times 2.3026)=21.7\,\mathrm{km}$ , commonly rounded to $22\,\mathrm{km}$ . Why: this is the $1/e$ distance, the natural yardstick against which any link length $L/n$ should be compared.

Step 4. Evaluate a $1000\,\mathrm{km}$ span: $\eta=e^{-1000/21.7}=e^{-46.1}\approx 10^{-20}$ ; at a $10\,\mathrm{GHz}$ rate this is $10^{10}\times 10^{-20}=10^{-10}$ photons per second, or about one every three centuries. Why: this single number is the whole motivation — direct transmission is not slow, it is physically hopeless, so no engineering of sources or detectors can rescue it. *Verified: $L_{\mathrm{att}}=10/(0.2\ln 10)=21.72\,\mathrm{km}$ ; $10^{-0.2\times 1000/10}=10^{-20}$ exactly; $1/(10^{10}\cdot 10^{-20}\cdot 3.16\times 10^{7})\approx 317$ years per photon.*

Go deeper · Derivation B · 1 minHow a stored qubit fades while it waits for a swap

Derivation BHow a stored qubit fades while it waits for a swap

The aim is to compute the fidelity of a heralded Bell pair when one of its two qubits sits in memory for a time $t$ before its partner link is ready, modelling the memory's imperfection as single-qubit depolarization. Only one qubit waits and decoheres; the other is freshly heralded and undisturbed. We want the explicit $F(t)$ because it converts the abstract coherence time $T_2$ into an operational deadline — the moment past which the swapped pair is no longer even entangled — and it supplies the per-link fidelity that the chain formula will later compound.

Step 1. Apply a depolarizing channel to the stored qubit $A$ alone, $\mathcal{D}_p(\rho_A)=(1-p)\rho_A+p\,\tfrac{I}{2}$ with $p=1-e^{-t/T_2}$ , leaving partner $B$ untouched. Why: the maximally mixed state is the fixed point of decoherence, and the exponential weight $p$ encodes that memory errors accumulate on the timescale $T_2$ .

Step 2. Evaluate the Bell-state overlap under this one-sided channel, $F(t)=\langle\Phi^+|\mathcal{D}_p(|\Phi^+\rangle\langle\Phi^+|)|\Phi^+\rangle=(1-p)+p\cdot\tfrac14=\tfrac14(1+3(1-p))$ . Why: a one-sided depolarizing channel on a Bell state keeps the state with weight $1-p$ and replaces it with a state having overlap $\tfrac14$ with weight $p$ — the $\tfrac14$ is the overlap of $|\Phi^+\rangle$ with the two-qubit maximally mixed state.

Step 3. Substitute $1-p=e^{-t/T_2}$ to get the boxed result $F(t)=\tfrac14\big(1+3\,e^{-t/T_2}\big)$ . At $t=0$ , $F=1$ ; at $t\gg T_2$ , $F\to\tfrac14$ . Why: this is the two-qubit analogue of the single-qubit $\tfrac12(1+e^{-t/T_2})$ — the extra qubit drops the floor from $\tfrac12$ to $\tfrac14$ .

Step 4. Locate the entanglement deadline by setting $F=\tfrac12$ : $e^{-t/T_2}=\tfrac13$ , so $t=T_2\ln 3\approx 1.10\,T_2$ ; beyond this the Werner state is separable and useless. Why: a repeater does not merely want high fidelity, it needs $F>\tfrac12$ to preserve any entanglement at all, which turns $T_2\ln 3$ into a hard scheduling constraint on how long a link may wait. *Verified: $F(0)=1$ , $F(\infty)=\tfrac14$ ; $F(T_2)=\tfrac14(1+3/e)=0.526$ ; the $F=\tfrac12$ crossing at $t/T_2=\ln 3=1.099$ ; consistent with the Werner picture $w=e^{-t/T_2}$ .*

Go deeper · Derivation C · 1 minEnd-to-end fidelity of a chain of swapped Werner links

Derivation CEnd-to-end fidelity of a chain of swapped Werner links

The purpose is to track how depolarizing memory errors compound across $n$ identical elementary links joined by $n-1$ ideal entanglement swaps, producing the fidelity of the final long-distance Bell pair. We model each link, after storage, as a Werner state of fidelity $F_0$ to $|\Phi^+\rangle$ and use the fact that swapping multiplies the depolarization weights. This matters because it reveals a geometric — not merely additive — decay in the number of links, which is the quantitative case for inserting entanglement purification between swap levels rather than trusting raw fidelity to survive the chain.

Step 1. Parametrize each link as $\rho_0=w_0|\Phi^+\rangle\langle\Phi^+|+(1-w_0)\tfrac{I}{4}$ with depolarization parameter $w_0=(4F_0-1)/3\in[0,1]$ , so that $F_0=(1+3w_0)/4$ . Why: the Werner form is the unique two-qubit state invariant under the depolarizing decoherence a memory imposes, and $w_0$ is the clean variable in which swapping acts simply.

Step 2. An ideal Bell-state-measurement swap of two Werner states with parameters $w_1,w_2$ yields a Werner state with parameter $w_{\mathrm{out}}=w_1 w_2$ . Why: the maximally mixed component of either input survives the swap, so the surviving "signal" weights multiply — decoherence compounds multiplicatively, not additively.

Step 3. Iterate over the chain: joining $n$ identical links takes $n-1$ swaps, each multiplying in another factor of $w_0$ , giving $w_n=w_0^{\,n}$ . Why: every swap discards a little more of the coherent weight, and because the factors multiply the loss is geometric in the number of links.

Step 4. Convert back to fidelity with $F_n=(1+3w_n)/4$ , obtaining the boxed result $F_n=\tfrac14\big[1+3\big((4F_0-1)/3\big)^{n}\big]$ . Why: expressed in fidelity, the result shows $F_n\to\tfrac14$ for any $F_0<1$ as $n$ grows, so only near-unit link fidelity — or purification between swaps — keeps a long chain usable. *Verified: $F_0=1\Rightarrow w_0=1\Rightarrow F_n=1$ for all $n$ (lossless); $F_0=\tfrac14\Rightarrow w_0=0\Rightarrow F_n=\tfrac14$ (already mixed); numerically $F_0=0.95,\ n=8\Rightarrow F_8=0.682$ and $F_0=0.99,\ n=8\Rightarrow F_8=0.924$ .*

Limiting cases

Reading the machinery in its extreme regimes exposes what each figure of merit actually buys.

Case	Result	Meaning
Perfect memory	$\eta\to1,\ T_2\to\infty$	Heralded links never decay while waiting; the chain rate is limited only by the slowest link's generation probability, recovering the ideal repeater speedup.
No memory	$R\propto p^{\,n}\,e^{-L/L_{\mathrm{att}}}$	All $n$ links must succeed in one time bin; the probability is exponentially small, equivalent to direct transmission — memories are exactly what break this scaling.
Decoherence-limited	$\tau_{\mathrm{wait}}\gtrsim T_2$	When the typical wait for a neighbour exceeds the coherence time, the stored qubit decoheres before swapping; fidelity collapses toward $\tfrac14$ and the link must be regenerated.
Single elementary link	$R_0=\tfrac{1}{T_0}\,p_{\mathrm{gen}}\,\eta^{2}$	With $T_0$ the attempt period, $\eta$ enters twice (store then retrieve), so even one link's rate is quadratically sensitive to memory efficiency.
Fidelity threshold	$F>\tfrac12$ per pair	A two-qubit Werner state is entangled only above $F=\tfrac12$ ; below it, no swap or measurement can recover a usable pair, setting the $t<T_2\ln 3$ deadline.

Common trapA quantum memory is just a delay line or a buffer that stores a photon

It is neither. A fixed optical delay line can only hold light for a predetermined time, but a repeater must wait an unknown, success-dependent interval until a neighbouring link happens to herald — a duration no length of fiber can match. And no classical buffer will do, because the object being stored is an unknown quantum state whose amplitude and, crucially, phase must be preserved; copying or amplifying it is forbidden outright by no-cloning. "Storing a photon" is necessary but wildly insufficient: a memory that returns a photon in the right mode but scrambles its relative phase has fidelity $\tfrac12$ and is worthless for a swap. The figure of merit is coherent, on-demand recall of the full qubit within $T_2$ — not mere optical storage.

Worked examples

1 · Why the span must be broken up (fully worked)

A $1000\,\mathrm{km}$ telecom fiber carries single photons at a $10\,\mathrm{GHz}$ source rate. Find the end-to-end photon rate for direct transmission, then compare with splitting the span into $n=4$ elementary links.

Step 1. Direct transmissivity from equation (1): $\eta=e^{-1000/21.7}=e^{-46.1}$ . Converting, $\eta=10^{-0.2\times 1000/10}=\mathbf{10^{-20}}$ . The whole $200\,\mathrm{dB}$ of loss lands on a single photon.

Step 2. Photon rate: $R=10^{10}\times 10^{-20}=10^{-10}\,\mathrm{s^{-1}}$ . Inverting and dividing by $3.16\times 10^{7}\,\mathrm{s/yr}$ gives one photon every $\mathbf{\approx 317\ years}$ — direct transmission is not slow, it is hopeless.

Step 3. Now split into $n=4$ links of $L/n=250\,\mathrm{km}$ . Each link's transmissivity is $\eta_{\mathrm{link}}=e^{-250/21.7}=e^{-11.5}=\mathbf{1.0\times 10^{-5}}$ , fifteen orders of magnitude better than the end-to-end figure. Because heralding retries each link independently and memories hold each success, the chain rate tracks this single-link loss times a polynomial swap overhead — not the product of four such factors.

Step 4. The improvement factor is $10^{-5}/10^{-20}=\mathbf{10^{15}}$ , and midpoint heralding (each photon travelling only $125\,\mathrm{km}$ ) improves it further. This gap — feasible versus impossible — is the entire reason repeaters and their memories exist.

2 · How long a link may wait (fully worked)

A memory has coherence time $T_2=1\,\mathrm{ms}$ . One half of a heralded Bell pair waits in it while the neighbouring link is generated. (a) What is the pair fidelity after a mean wait of $t=0.5\,\mathrm{ms}$ ? (b) After exactly one $T_2$ ? (c) What is the longest wait that still leaves the pair entangled?

Step 1. Use the wait-time law $F(t)=\tfrac14(1+3e^{-t/T_2})$ from Derivation 2. For (a), $t/T_2=0.5$ , so $e^{-0.5}=0.6065$ and $F=\tfrac14(1+3\times 0.6065)=\tfrac14(2.820)=\mathbf{0.705}$ .

Step 2. For (b), $t=T_2$ : $e^{-1}=0.3679$ , so $F=\tfrac14(1+1.104)=\mathbf{0.526}$ — barely above the maximally mixed value $\tfrac14$ , and only just entangled.

Step 3. For (c), the pair is entangled while $F>\tfrac12$ , i.e. $e^{-t/T_2}>\tfrac13$ , giving $t<T_2\ln 3=\mathbf{1.10\ ms}$ . Past this the swap produces a separable, useless state no matter how good the hardware downstream.

Step 4. The lesson: a memory with $T_2=1\,\mathrm{ms}$ imposes a $\approx 1\,\mathrm{ms}$ scheduling budget on the whole neighbouring-link generation process. Doubling $T_2$ doubles the budget linearly — which is why memory coherence, not raw efficiency, often gates the achievable distance.

3 · Fidelity down a chain, and the case for purification

Elementary links are prepared at fidelity $F_0$ . Using $F_n=\tfrac14[1+3((4F_0-1)/3)^n]$ , find the end-to-end fidelity of an $n=8$ -link chain for $F_0=0.99$ and for $F_0=0.95$ .

Step 1. Convert to the depolarization parameter. For $F_0=0.99$ , $w_0=(4\times 0.99-1)/3=2.96/3=0.9867$ ; for $F_0=0.95$ , $w_0=(3.8-1)/3=0.9333$ .

Step 2. Raise to the eighth power: $w_8=0.9867^{8}=0.898$ and $0.9333^{8}=0.576$ respectively.

Step 3. Convert back: $F_8=\tfrac14(1+3w_8)$ gives $\mathbf{0.924}$ for $F_0=0.99$ and $\mathbf{0.682}$ for $F_0=0.95$ . A one-percentage-point drop in link fidelity costs a quarter of the end-to-end fidelity over eight links.

Step 4. Because the decay is geometric in $n$ , no chain of moderately imperfect links reaches long distance intact. The remedy is entanglement purification (distillation) inserted between swap levels, trading several lower-fidelity pairs for one higher-fidelity pair and resetting $F_0$ upward before it compounds away.

Physics of the portrait

The quantum repeater is not a device anyone has ever held; it is an architecture — a way of arranging entanglement, storage, and measurement so that quantum information survives a channel that would otherwise annihilate it. Its history is the story of assembling three primitives (heralding, memory, swapping) into one machine.

Origins · 1 minquantum repeater and how this was discovered

quantum repeater

the memory-and-swapping architecture, proposed 1998 · long-distance quantum communication

The quantum repeater is one of the youngest central concepts in quantum information, and unlike a classical repeater it can never be a simple amplify-and-forward box — the no-cloning theorem sees to that. The idea crystallised in 1998, when Hans Briegel, Wolfgang Dür, Ignacio Cirac and Peter Zoller showed that a long channel could be divided into segments, each holding a modestly entangled pair, and that nested entanglement purification interleaved with entanglement swapping could stitch the segments together while keeping the error and resource cost polynomial in distance. Three years later Lu-Ming Duan, Mikhail Lukin, Cirac and Zoller (the DLCZ proposal) turned the abstraction into a hardware blueprint: atomic ensembles serving as memories, heralded by scattered Stokes photons, with linear optics performing the swaps — the first design robust to realistic loss and memory error. The following decade added the memory protocols that make it practical: electromagnetically induced transparency and "stopped light" (Lukin, Fleischhauer, Hau), controlled reversible inhomogeneous broadening, and the multimode atomic frequency comb of Afzelius and co-workers in rare-earth crystals. Nicolas Gisin's group and the 2011 Sangouard–Simon–de Riedmatten–Gisin review consolidated the figures of merit — $\eta$ , $T_2$ , $T_2 B$ , $F$ — that a memory must hit. The modern statement is exact: a repeater is any scheme whose secret-key rate beats the repeaterless PLOB bound $-\log_2(1-\eta)$ , and the memory is the component that makes beating it possible.

History

1998

— Briegel, Dür, Cirac and Zoller propose the quantum repeater with nested entanglement purification

2001

— Duan, Lukin, Cirac and Zoller introduce the DLCZ atomic-ensemble memory heralded by Stokes photons

2009

— Afzelius and co-workers propose the atomic frequency comb, a multimode rare-earth memory

2015

— a europium-doped crystal reaches six hours of nuclear-spin coherence

2020

— a silicon-vacancy memory node demonstrates communication beating the direct-transmission rate

Going further

Texts. Nielsen & Chuang, Quantum Computation and Quantum Information, 10th anniv. ed., Ch. 1 (no-cloning) and Ch. 12 (quantum channels, fidelity) · Preskill, Quantum Information lecture notes (Caltech Ph219), chapters on channels and error correction · Wilde, Quantum Information Theory, 2nd ed. (channel fidelity, no-cloning, capacity bounds) · Watrous, The Theory of Quantum Information (completely positive maps, fidelity, diamond norm).

Primary sources. Briegel, Dür, Cirac, Zoller, Phys. Rev. Lett. 81, 5932 (1998) — the quantum repeater · Duan, Lukin, Cirac, Zoller, Nature 414, 413 (2001) — the DLCZ atomic-ensemble scheme · Afzelius, Simon, de Riedmatten, Gisin, Phys. Rev. A 79, 052329 (2009) — atomic frequency comb · Sangouard, Simon, de Riedmatten, Gisin, Rev. Mod. Phys. 83, 33 (2011) — the memory-repeater review · Pirandola, Laurenza, Ottaviani, Banchi, Nat. Commun. 8, 15043 (2017) — the repeaterless secret-key bound · Bhaskar et al., Nature 580, 60 (2020) — memory-enhanced quantum communication.

Provenance. A verifier should recompute: (1) $L_{\mathrm{att}}=10/(\alpha\ln 10)\approx 21.7\,\mathrm{km}$ from $\alpha=0.2\,\mathrm{dB/km}$ , and $\eta(1000\,\mathrm{km})=10^{-20}$ ; (2) the wait-time fidelity $F(t)=\tfrac14(1+3e^{-t/T_2})$ , its value $0.526$ at $t=T_2$ , and the $F=\tfrac12$ crossing at $t=T_2\ln 3\approx 1.10\,T_2$ ; (3) the Werner-swap rule $w_{\mathrm{out}}=w_1 w_2$ and the chain formula $F_n=\tfrac14[1+3((4F_0-1)/3)^n]$ , with limit checks $F_0=1\Rightarrow F_n=1$ and $F_0=\tfrac14\Rightarrow F_n=\tfrac14$ , and the sample values $F_8=0.924$ ( $F_0=0.99$ ), $F_8=0.682$ ( $F_0=0.95$ ); (4) the PLOB bound $-\log_2(1-\eta)\approx\eta/\ln 2$ for small $\eta$ . The $(3/2)^{\log_2 n}=n^{0.585}$ swap overhead is schematic and model-dependent, and is presented as such rather than as an exact constant.

Prerequisites. Entanglement Swapping · Bell-State Measurement · Quantum Teleportation · No-Cloning Theorem in Cryptography. Leads to. E91 Protocol · BB84 Protocol · Quantum Bit Error Rate and Sifting.

Ask the tutor: Ask me to walk through why a memory with a given $\eta$ and $T_2$ can or cannot beat the repeaterless bound over a distance you choose, and we will run the numbers together.

Why can't I just amplify the signal like a classical fiber repeater does?

Because amplification is copying in disguise. A classical repeater reads a fading pulse and re-emits a fresh strong one, which is fine for bits but fatal for qubits: producing a second, independent copy of an unknown quantum state is exactly what the no-cloning theorem forbids. Any device with gain that worked on arbitrary quantum inputs would let you clone, so channel loss cannot be undone by adding energy. The quantum repeater sidesteps this entirely — it never tries to restore the attenuated photon. Instead it distributes entanglement over short links where loss is survivable, stores each success in a memory, and uses entanglement swapping and teleportation to move the quantum information without ever copying it.

What exactly does a quantum memory store, and how is that different from a photon buffer?

It stores a qubit — a full quantum state including its relative phase — as a stationary excitation, typically a collective atomic spin wave, a rare-earth-ion coherence, or the spin of a single defect centre. The distinction from a photon buffer or delay line is twofold. First, timing: a repeater must hold the state for an unknown, success-dependent interval until a neighbouring link happens to herald, which no fixed-length delay can do. Second, faithfulness: returning "a photon" is not enough if its phase is scrambled — that gives fidelity one-half and is useless for a swap. A memory must offer coherent, on-demand recall of the entire qubit within its coherence time, releasing it precisely when the herald arrives.

Why do memories change the distance scaling from exponential to polynomial?

Without memories every elementary link must succeed in the same instant, so for $n$ links with per-link probability $p$ the joint success is $p^{\,n}$ — vanishingly small and no better than direct transmission. A memory removes the "same instant" requirement: each link is attempted repeatedly and independently until it heralds, and the confirmed state is frozen in place while its neighbours catch up. The chain then assembles level by nested level, and the only penalty is that each swap must wait for the slower of its two child links — a modest overhead that grows as a polynomial in $n$ , roughly $n^{0.585}$ , rather than an exponential. That swap from exponential to polynomial cost is the entire reason the word "required" attaches to memories.

What are the key numbers I should quote for a memory?

Four. The storage-and-retrieval efficiency $\eta$ is the probability a stored qubit is re-emitted into the wanted mode; because storing and retrieving each cost a factor of $\eta$ , single-link rate scales as $\eta^2$ , so efficiency is punishing. The coherence time $T_2$ sets how long the state stays usable and, via $F(t)=\tfrac14(1+3e^{-t/T_2})$ , imposes a hard wait deadline near $T_2\ln 3$ . The time-bandwidth product $T_2 B$ counts how many modes the memory can hold at once, which multiplexing turns directly into rate. And the fidelity $F$ measures distortion, with $F>\tfrac12$ the bare requirement for the stored pair to remain entangled at all. A good node needs all four simultaneously — no single number suffices.

How does DLCZ herald entanglement between two distant memories?

A weak "write" laser pulse hits an atomic ensemble and, with small probability, scatters one Stokes photon while simultaneously depositing exactly one collective spin excitation in the ensemble — photon number and excitation number are entangled, $|0_a0_s\rangle+\sqrt{p_c}|1_a1_s\rangle$ . Detecting that Stokes photon heralds, unambiguously, that the ensemble now holds one stored excitation. To entangle two ensembles $L$ and $R$ , their Stokes photons are sent to a central beam splitter that erases which-ensemble information; a single click there cannot say which side emitted, projecting the pair into $|1\rangle_L|0\rangle_R+e^{i\phi}|0\rangle_L|1\rangle_R$ — a heralded entangled spin-wave state, the elementary link. Because the excitation stays stored, this link can then wait for its neighbour.

What does it mean to "beat the repeaterless bound", and has anyone done it?

The PLOB bound $-\log_2(1-\eta)$ is a proven ceiling on the secret-key rate of a pure-loss channel with no repeater; it falls off linearly in transmissivity, hence exponentially in distance. A scheme earns the name "repeater" only if its rate-versus-distance curve crosses above this line. In 2020 a silicon-vacancy memory node in a diamond nanocavity did exactly that with a genuine stored qubit — memory-enhanced quantum communication that exceeded the direct-transmission rate for its channel. Separately, twin-field QKD beats the same bound using single-photon interference and no memory at all, which is a useful reminder that the bound is about the channel, not about any one technology: memories are one powerful way to surpass it, not the only way.

Why does end-to-end fidelity fall so fast down a long chain?

Because errors compound multiplicatively, not additively. Model each link as a Werner state with depolarization weight $w_0=(4F_0-1)/3$ ; an ideal swap of two such states produces weight $w_1w_2$ , so after $n$ links the surviving weight is $w_0^{\,n}$ and the fidelity is $F_n=\tfrac14(1+3w_0^{\,n})$ . That is geometric decay toward the useless value $\tfrac14$ : at $F_0=0.99$ an eight-link chain already sags to $0.92$ , and at $F_0=0.95$ to $0.68$ . There is no length of chain over which mediocre links stay good. The standard fix is entanglement purification inserted between swap levels — consuming several lower-fidelity pairs to distil one higher-fidelity pair — which resets $F_0$ upward before the exponent grinds it away.