Loss & Errors

A quantum link is bounded by two impairments. Loss — how many photons survive the medium — is what heralded entanglement generation (HEG) fights with its retry-until-success pattern. Errors — how accurately the surviving qubits encode the intended state — is what heralded entanglement purification (HEP) trims by trading pairs for higher fidelity. Every other engineering choice (repeater spacing, swap depth, fibre vs satellite, purification cost) falls out of how these two impairments compose along a link ^{Cacciapuoti et al. 2020}.

Errors decompose further into fidelity (how close the delivered Bell pair is to the ideal target) and decoherence (how fast a stationary qubit loses its state in memory while it's waiting). We cover them in that order because HEP addresses fidelity directly; longer memory just extends the window in which fidelity is still high enough to purify.

The numbers below come from current standards and the most-cited experiments. Treat them as the back-of-envelope figures every later section in the series leans on, not as the last word — vendor- and platform-specific numbers move year over year.

Loss — the photon-survival metric

Optical channels attenuate exponentially: a length $L$ of medium with attenuation $α$ dB/km transmits a fraction $η = 1 0^{- α L /10}$ of the input photons. Loss is the single most punishing metric for long links; an entanglement-distribution rate that scales as $η$ per attempt drops by a factor of ten every time you add 10 dB to the link.

Medium	Attenuation	Reach @ 20 dB total loss	Source
Standard single-mode fibre (SMF, ITU-T G.652.D)	0.20 dB/km @ 1550 nm (C-band)	100 km	^{ITU-T G.652.D}
Hollow-core fibre — commercial	0.091 dB/km (Lumenisity / Microsoft, 2024)	~220 km	^{Microsoft / Lumenisity 2024}
Hollow-core fibre — academic best (DNANF)	0.124 dB/km (Jasion et al., OFC 2022)	~160 km	^{Jasion et al. 2022}
Free-space optical (FSO), clear sky horizontal link	~0.5 dB/km (Schmitt-Manderbach et al., 144 km QKD)	~40 km	^{Schmitt-Manderbach et al. 2007}
Satellite to ground (LEO downlink)	~50 dB single-pass to LEO, atmosphere-dominated (Micius)	1200 km demonstrated	^{Yin et al. 2017}

Two patterns to take from the table. First, hollow-core fibre roughly halves standard-fibre attenuation, which doubles the reach at any given loss budget; this is why hollow-core is treated as the leading medium-term roadmap item for long-haul quantum links. Second, satellite links route around the per-km attenuation problem entirely — the bulk of the loss is concentrated in the few tens of kilometres of atmosphere, after which vacuum is essentially lossless. Micius distributed Bell pairs over 1200 km of ground separation by paying a one-time ~50 dB atmospheric tax rather than a continuous 0.20 dB/km fibre tax ^{Yin et al. 2017}.

Fidelity — quality of the delivered Bell pair

For an output state $ρ$ that should be the ideal Bell state $∣ Φ^{+} ⟩$ , fidelity is $F = ⟨ Φ^{+} ∣ ρ ∣ Φ^{+} ⟩$ — the overlap of the actual state with the target. $F = 1$ is perfect; $F = 1/4$ is the maximally mixed state and corresponds to no entanglement. The closely-related singlet fraction is the maximum overlap with any of the four Bell states under local Cliffords ^{Horodecki et al. 1999}.

Application thresholds

What counts as "high enough" fidelity depends entirely on what the application is going to do with the pair:

Teleportation of an arbitrary qubit through a Bell pair has average output fidelity $(2 F + 1) /3$ . At $F = 1/2$ , the output drops to $2/3$ — what a classical measure-and-prepare strategy already achieves with no entanglement. Practical teleportation needs $F$ well above $1/2$ ^{Horodecki et al. 1999}.
Entanglement-based QKD (E91 / BBM92) needs the pair clean enough that the QBER (quantum bit-error rate) sits below the secret-key-rate threshold. The Shor–Preskill / CSS analysis keeps the asymptotic key rate positive while QBER < $\sim 11%$ in either basis; for a Werner-state Bell pair with fidelity $F$ the QBER in either basis is $2 (1 - F) /3$ , which puts the cut-off at $F \approx 0.835$ — hence the $F ≳ 0.85$ rule of thumb ^{Pirandola et al. 2020}.
Distributed quantum computing is the most demanding. A two-qubit gate between processors mediated by a teleported Bell pair inherits $1 - F$ directly into the gate's error budget, so the Bell-pair fidelity has to sit below the surface-code fault-tolerance threshold (around 1 % per gate in standard analyses). That puts the post-purification target at $F > 0.99$ ^{Fowler et al. 2012}.

Composition under chaining

Fidelity composes multiplicatively as a Bell pair is extended across hops by entanglement swapping. Two pairs at $F = 0.95$ joined by a swap give an end-to-end pair whose Bell-state overlap drops below either input's; the rough rule (depolarising noise model) is $F_{1 + 2} \approx (4 F_{1} F_{2} - F_{1} - F_{2} + 1) /3$ , which collapses to $F_{1} F_{2}$ in the high-fidelity limit. Ten hops at $F = 0.95$ reach the end-to-end pair at $F \sim 0.6$ — well below threshold for most applications. This is the multiplicative-decay mechanism that forces purification into long chains, covered in the purification subject.

Decoherence — $T_{1}$ and $T_{2}$

A qubit in memory does not stay in its prepared state indefinitely. Two characteristic times bound how long it lasts:

$T_{1}$ — energy relaxation. The time over which the excited-state population decays towards the ground state. Sets a hard ceiling on how long any state involving $∣1 ⟩$ survives.
$T_{2}$ — phase coherence. The time over which a superposition $α ∣0 ⟩ + β ∣1 ⟩$ retains the relative phase between its components. Always $T_{2} \leq 2 T_{1}$ , and typically much shorter — coupling to the environment scrambles phase faster than it depletes population. $T_{2}$ is the relevant number for entanglement-based protocols.

Storage times vary by ten orders of magnitude across platforms. Meddeb's 2025 survey tracks the per-platform best-published storage times year over year; the table below pulls from his Table 5 ^{Meddeb 2025}.

Memory platform	Best storage time (current)	A decade ago	TRL band
Atomic ensembles	~1 hour (atomic frequency comb)	0.6 ms	7–8
Trapped ions	~10 s (ion-photon entanglement transfer)	~50 s in long-lived isolated experiments	6–7
Quantum dots	~4 days (charge memory at room temperature)	~20 ms	5–6
Solid-state defects (NV / SiV)	~100 μs spin coherence; ms range with isotopic engineering	~50 ns	5–6
Superconducting (RAQM)	~34 ms in resonator-coupled designs	~35 μs	5–6
Photonic (telecom-band cavity)	~2 μs on-demand retrieval	~0.85 ms (atomic-vapour-coupled)	5–6

Storage time matters in proportion to two other latencies in the network. A memory needs to outlast the classical-correction round-trip (entanglement swapping and teleportation both wait on a 2-bit classical message at light-speed: about 1 ms per 200 km of fibre), and it needs to outlast the entanglement-generation attempt time the network is willing to spend per delivered pair. Sub-millisecond memories rule out anything beyond metro-scale repeater chains; second-scale and longer memories open the door to continental-scale chains where multiple swaps wait on each other across hundreds of kilometres ^{Kumar et al. 2025}.

A real link budget multiplies and adds more terms than the three above. The callouts below cover the line items that hide inside vendor spec sheets but show up in any honest measurement — expand the ones you need.

Further details on the loss + error budget — detector η, heralding, HOM, gates, PLOB

Detector efficiency, dark counts, and coupling — the silent loss budget

Channel attenuation is only part of the loss budget. Detector and interface efficiencies multiply with it:

Detector efficiency $η_{det}$ . Superconducting nanowire single-photon detectors (SNSPDs) reach 80–98 % at telecom wavelengths; silicon APDs sit around 30–60 %; the gap is 1–3 dB of additional link loss depending on the choice.
Dark counts. Detector clicks with no photon present — typically 1–1000 Hz depending on cryostat temperature and bias. Erode signal-to-noise; effectively a noise floor inside the loss budget, especially limiting at high attenuation.
Coupling and mode-matching losses. Fibre-to-chip interfaces typically cost 1–3 dB per junction; free-space-to-fibre coupling can be 1–6 dB depending on collimation. These usually do not appear in headline channel-loss numbers but compose into the end-to-end η that every repeaterless rate scales by.

Effective transmittance is the product: $η_{eff} = η_{chan} \cdot η_{couple} \cdot η_{det}$ . A 100 km fibre at 0.20 dB/km is 1 % channel transmittance; with a 50 % detector and two 2 dB coupling interfaces, end-to-end η drops to ~0.2 %.

Heralding efficiency — how often a click is a real Bell pair

Heralding efficiency $η_{h}$ is the probability that the heralding pattern at a midpoint (or the matter qubit's photonic interface at an end node) correctly indicates a real Bell pair, as opposed to a dark-count coincidence or a single-photon ambiguous click. Distinct from raw detector efficiency: a high-η detector can still herald poorly if the optical setup admits noise paths.

$η_{h}$ sets the HEG retry rate. Pompili et al. (2021), the three-node NV-centre repeater demonstration, reported per-link heralding efficiencies in the few-percent range; pair-generation rates and link budgets are quoted against $η_{h}$ , not against the channel η alone ^{Kumar et al. 2025}.

Memory write/read efficiency — loss inside the memory

A quantum memory can have a long $T_{2}$ and still lose most photons. Write efficiency is the probability that an incident photon successfully maps onto the matter qubit; read efficiency is the probability that retrieval produces a photon back into the network mode. Both are orthogonal to storage time and to fidelity.

Numbers vary widely by platform. Atomic-frequency-comb memories in rare-earth doped crystals reach ~50–80 % read efficiency at short storage times but drop with delay. SiV-centre photonic-bus memories report ~20 % read efficiency. Trapped-ion ion-photon interfaces sit in the 1–10 % range. These multiply with channel and detector efficiencies on every hop a Bell pair takes through a memory-based repeater ^{Meddeb 2025}.

HOM visibility — the photonic-BSM fidelity knob

Two photons arriving at a 50:50 beam splitter from opposite ports interfere if and only if they are indistinguishable in every degree of freedom — spectral envelope, polarisation, arrival time, spatial mode. Perfect indistinguishability gives the Hong–Ou–Mandel (HOM) dip with visibility $V = 1$ ; spectral mismatch, timing jitter, or polarisation drift drops $V$ below 1 and degrades both the success rate and the output fidelity of every photonic Bell-state measurement that uses it ^{Hong et al. 1987}.

For a swap or entanglement-generation BSM, the output fidelity is roughly $F_{out} \approx \frac{1}{2} (1 + V)$ in the dominant noise model — a $V = 0.9$ dip ceilings the BSM output fidelity at $\sim 0.95$ , before any further channel or memory contributions. Indistinguishability between two independent sources (different vendors, different platforms, even different temperatures) is a much harder engineering problem than high $V$ from a single source.

Gate and SPAM errors — distinct from Bell-pair fidelity

The Bell-pair fidelity $F$ on the entanglement page measures the state of the delivered pair, but not the operations that consume it. A two-qubit gate mediated by a teleported Bell pair sees three error contributions stacked:

Bell-pair fidelity — the resource — directly hits the gate's accuracy.
Local-operation error — the CNOT + Hadamard + measurement that perform the BSM at the matter side. Best-in-class two-qubit gates are 0.999 fidelity; single-qubit gates $≳ 0.9999$ .
State-prep and measurement (SPAM) errors — the endpoints' qubit-initialisation and read-out steps. Typically 0.001–0.01 per shot.

These compose: the surface-code threshold sits at ~1 % per gate, so a Bell pair at $F = 0.99$ joined with a 0.999 BSM and 0.99 SPAM aggregates to a total error around 2 %, above threshold. Procurement-grade specifications quote per-operation fidelities precisely because the headline Bell-pair number does not capture this stack ^{Fowler et al. 2012}.

PLOB bound — the repeaterless capacity ceiling

For a pure-loss channel with transmittance $η$ , the secret-key (and entanglement-distribution) capacity of any point-to-point quantum protocol is bounded by $C (η) \leq - lo g_{2} (1 - η)$ bits per channel use — the PLOB bound ^{Pirandola et al. 2017}.

At 0.20 dB/km fibre, 200 km gives $η \approx 1 0^{- 4}$ , and the PLOB ceiling sits at ~1.4 × 10⁻⁴ bits per channel use; at 300 km it drops to ~10⁻⁶. No amount of clever encoding raises this without inserting a quantum node between the endpoints. This is the theorem that forces repeaters into the long-haul roadmap. A repeater chain breaks the bound by adding intermediate states that extend entanglement across each hop independently — every working quantum-network architecture in the rest of the series is a way of engineering past this ceiling.

Quantum vs classical metrics

A quantum network is measured against a different vocabulary from a classical one, and the differences are forced by physics rather than by engineering choice ^{Cacciapuoti et al. 2020}^{Wehner et al. 2018}:

Concern	Classical network	Quantum network
Unit of service	Bit (or packet) delivered from sender to receiver	Bell pair delivered between two named endpoints; the application then teleports, distils, or measures it
Throughput	Bits per second on the link	Entanglement-generation rate — Bell pairs per second
Quality	Bit-error rate / signal-to-noise ratio	Fidelity — closeness of the delivered pair to the ideal Bell state; below a protocol-specific threshold, downstream uses stop working
Information source	Can be read without altering it; can be copied and re-transmitted on corruption	Cannot be read without altering it (measurement collapse) and cannot be copied (no-cloning) — so it cannot be re-transmitted
Loss recovery	Amplify-and-forward repeaters; retransmit on packet loss	No optical amplification possible (no-cloning forbids it); lost photons cannot be recovered, only re-attempted from the source. Quantum repeaters use entanglement swapping + purification, not amplification
Reach without intermediaries	Thousands of km via amplify-and-forward chains	~100–200 km in standard fibre, set by 0.20 dB/km attenuation ^{ITU-T G.652.D}; longer reach requires repeaters or satellite links
Latency budget	Soft preference; long latency degrades user experience	Hard budget — Bell pairs decohere in memory, so generation-to-use time competes with memory coherence ( $T_{2}$ ) and the classical-correction round-trip
Auxiliary classical channel	None required	Always required — teleportation and entanglement swapping both consume 2 bits of classical communication per Bell pair to complete the protocol

The framing comes from Cacciapuoti et al.'s communication-system view of the quantum internet, which models classical and quantum networks against the same block diagram and lets the differences fall out where the physics demands them (no-cloning forces a classical side-channel; measurement collapse forces a fidelity metric instead of an SNR; loss cannot be recovered the classical way and forces repeaters of a different kind) ^{Cacciapuoti et al. 2020}.

Worked example — a 200 km hop, three media

Single-photon survival on one shot, no repeaters, no memories — just the channel loss. Probabilities below are for a heralded photon delivered intact end-to-end; photon-pair generation at the source and detector quantum efficiency are set aside for clarity.

Medium	$α$ (dB/km)	Total loss @ 200 km	Photon-survival fraction $η$	Per-Hz attempt rate to deliver 1 pair/s
SMF (G.652.D)	0.20	40 dB	10⁻⁴ (1 in 10 000)	10 kHz at the source
Hollow-core (Lumenisity 0.091)	0.091	18.2 dB	1.5 × 10⁻² (~1 in 65)	~65 Hz
LEO satellite, ~200 km slant range	— (atmosphere-dominated, not per-km)	~30–50 dB total link budget	10⁻³ to 10⁻⁵ depending on aperture	10³–10⁵ Hz

The numbers explain why nobody runs naked single-photon links beyond ~150 km of standard fibre: the source clock would have to push tens of kilohertz of attempts to the link to surface a single Bell pair per second at the far end, and any added pair-generation or detector inefficiency (each typically a factor of 10) compounds that further. Hollow-core lifts the budget by roughly two orders of magnitude per 200 km hop. Satellite links beat both for intercontinental reach but pay a one-time pointing-and-acquisition cost the per-km media don't.

For longer reach, the engineering response is to break the link into shorter hops and chain entanglement swaps — each hop sees a manageable $η$ , but the end-to-end success rate now multiplies hop rates and consumes memory at every node. The composition of loss, decoherence, and fidelity along that chain is the calculation the repeaters and distribution subjects work through in detail.

Where these numbers go next

Three subjects pick up the threads:

Memories grounds the per-platform decoherence numbers in the actual storage technologies — atomic ensembles, trapped ions, NV centres, quantum dots, photonic, superconducting — with TRL bands and integration trade-offs ^{Meddeb 2025}.
Links drills further into each medium: ITU fibre bands, hollow-core architectures, free-space turbulence, and Micius-style satellite links, with the operational caveats this page abstracts over.
Repeaters and purification show how the three metrics compose along a chain, why exponential fidelity decay forces purification, and where the 1G / 2G / 3G architectural families fall on the loss / decoherence / fidelity plane.

The link-layer service — what the network promises to deliver in terms of (rate, fidelity, latency) — is formalised in RFC 9340 and re-stated in Kumar 2025; both treat these three numbers as the parameter vector against which any quantum-internet service contract is written ^{RFC 9340}^{Kumar et al. 2025}.