Quantum Noise Fraction Diagnostics

• Quantum noise fraction is a dimensionless metric that compares the quantum-induced noise to the total noise, operationally defined per measurement task.
• It employs common mathematical forms such as additive decompositions and task-based normalization to relate quantum fluctuations to overall uncertainty.
• The concept bridges diverse fields—ranging from optical interferometry to quantum channels—by quantifying how quantum effects limit sensitivity and induce decoherence.

Quantum noise fraction denotes a family of dimensionless diagnostics that quantify how much of an observed uncertainty, fluctuation budget, or resource degradation is attributable to quantum effects rather than technical or thermal mechanisms. In the literature, the term appears as a variance fraction in optical interferometry, a ratio of noise to parameter response in quantum metrology, a fraction of total strain noise due to zero-point motion in gravitational-wave detection, a survival or destruction fraction for coherence under noisy channels, and a probability or fraction of logical failures in quantum annealing (Weyrauch et al., 2010, Escher, 2012, Gaudio, 29 Apr 2026, Char et al., 27 Apr 2026, Jeong et al., 6 Oct 2025). This suggests that the concept is operational rather than unique: its precise definition is fixed by the measurement task, the noise model, and the resource being monitored.

1. Conceptual scope and common mathematical forms

Across subfields, quantum noise fraction is typically realized as a ratio, probability, or threshold quantity comparing a quantum contribution with a total budget or a task-relevant benchmark. The common structure is dimensionless normalization: quantum variance divided by total variance, quantum-limited information divided by total information, or entanglement-destroying noise divided by an allowed noise budget.

Setting	Quantity	Interpretation
Optical interferometry	$$F_Q = (\Delta n_{\text{quantum}})^2 / (\Delta n_{\text{tot}})^2$$	Fraction of dark-port counting variance that is quantum
Quantum metrology	$\delta x_{\rm nsr}$, ${\cal F}^{\rm nsr}_Q$	Noise per unit sensibility, and its optimized inverse square
Gravitational-wave detection	$\beta$	Total quantum noise fraction of strain noise
Coherence theory	$\eta_{\text{surv}}, \eta_{\text{noise}}$	Fraction of coherence fraction preserved or destroyed by a channel
Quantum channels	$\mu_c(\Phi), n_c(\Phi)$	Fraction of added depolarizing noise, or number of uses before entanglement breaking
Quantum annealing	$\mathrm{CBP}, \mathrm{CBF}$	Probability or fraction of chains dominated by noise
Mesoscopic transport	$F_0, F_K$	Linear or nonlinear shot-noise fraction, often read as effective charge

Two broad mathematical patterns recur. The first is an additive decomposition, such as $$S_{\text{tot}} = S_{\text{quantum}} + S_{\text{classical}}$$, followed by normalization of the quantum term. The second is a task-based normalization, in which the relevant quantity is not variance itself but performance degradation, as in coherence survival, signal-to-noise limits, or entanglement-breaking thresholds. These patterns are explicit in interferometric counting noise, channel-noise quantification, and high-frequency gravitational-wave proposals (Weyrauch et al., 2010, Pasquale et al., 2012, Gaudio, 29 Apr 2026).

2. Metrological and detector-theoretic formulations

In single-parameter quantum estimation, the relevant object is Escher’s noise-to-sensibility ratio, defined for an observable $\hat{\cal M}$ and parameter $\delta x_{\rm nsr}$0 as

$\delta x_{\rm nsr}$1

It measures noise in the readout per unit response of the signal to the parameter. Optimizing over observables yields

$\delta x_{\rm nsr}$2

and the central result is

$\delta x_{\rm nsr}$3

with the symmetric logarithmic derivative as the optimal observable (Escher, 2012). In this formulation, quantum noise fraction is not a variance fraction but a calibrated metrological ratio: uncertainty in $\delta x_{\rm nsr}$4 per unit signal slope.

A complementary detector-theoretic formulation uses binary-state discrimination. For a detector observable $\delta x_{\rm nsr}$5, the signal is the difference of detector expectation values,

$\delta x_{\rm nsr}$6

and the noise is the sum of the two variances,

$\delta x_{\rm nsr}$7

A natural noise fraction is then

$\delta x_{\rm nsr}$8

with $\delta x_{\rm nsr}$9. The universal bound

${\cal F}^{\rm nsr}_Q$0

implies

${\cal F}^{\rm nsr}_Q$1

so fidelity fixes a lower bound on how small the detector noise fraction can be (Katsube et al., 2019). This suggests a general principle: whenever distinguishability is fundamentally limited, the irreducible noise fraction is bounded from below by state overlap.

3. Interferometric counting noise and squeezed-light reduction

In the dark-port analysis of optical interferometers, a strong coherent state ${\cal F}^{\rm nsr}_Q$2 enters one input port and a squeezed vacuum ${\cal F}^{\rm nsr}_Q$3 enters the unused port. The dark-port configuration is

${\cal F}^{\rm nsr}_Q$4

with photon counting at output port 1 and observable ${\cal F}^{\rm nsr}_Q$5. For optimal phase alignment ${\cal F}^{\rm nsr}_Q$6, the dark-port mean and variance are

${\cal F}^{\rm nsr}_Q$7

${\cal F}^{\rm nsr}_Q$8

For strong coherent input and realistic squeezing, the ${\cal F}^{\rm nsr}_Q$9 terms can often be neglected, so that

$\beta$0

Squeezing therefore reduces shot noise by $\beta$1 in the variance, or $\beta$2 in the standard deviation, while the signal term remains $\beta$3 and does not depend on $\beta$4 (Weyrauch et al., 2010).

Within this framework, a natural quantum noise fraction is obtained by introducing technical noise,

$\beta$5

and defining

$\beta$6

In the ideal limit treated in the paper, $\beta$7: the interferometer is fully quantum-limited. With squeezing,

$\beta$8

so the absolute quantum noise falls, but the mean signal is not amplified. This directly rules out “signal amplification without concurrent noise amplification” in the dark-port case; the gain is sensitivity improvement by noise reduction, not signal enhancement (Weyrauch et al., 2010).

4. Gravitational-wave detectors, $\beta$9, and the thermal frontier

A particularly explicit definition appears in high-frequency gravitational-wave detection, where the total strain-noise PSD is decomposed as

$\eta_{\text{surv}}, \eta_{\text{noise}}$0

and the total quantum noise fraction is

$\eta_{\text{surv}}, \eta_{\text{noise}}$1

If a quantum protocol reduces the amplitude of source $\eta_{\text{surv}}, \eta_{\text{noise}}$2 by $\eta_{\text{surv}}, \eta_{\text{noise}}$3, the strain sensitivity enhancement is

$\eta_{\text{surv}}, \eta_{\text{noise}}$4

so perfect removal of all quantum noise yields

$\eta_{\text{surv}}, \eta_{\text{noise}}$5

For a mechanical mode with thermal occupation

$\eta_{\text{surv}}, \eta_{\text{noise}}$6

the thermal and zero-point contributions satisfy

$\eta_{\text{surv}}, \eta_{\text{noise}}$7

hence

$\eta_{\text{surv}}, \eta_{\text{noise}}$8

This makes $\eta_{\text{surv}}, \eta_{\text{noise}}$9 a direct measure of whether the detector is thermally dominated or quantum dominated (Gaudio, 29 Apr 2026).

The thermal frontier is defined by $\mu_c(\Phi), n_c(\Phi)$0, equivalently $\mu_c(\Phi), n_c(\Phi)$1, which gives

$\mu_c(\Phi), n_c(\Phi)$2

Expressed as a frequency-dependent temperature,

$\mu_c(\Phi), n_c(\Phi)$3

Below this frontier, resonant mass detectors operating through tidal coupling are thermally dominated; at dilution temperatures they remain so below $\mu_c(\Phi), n_c(\Phi)$4 MHz. Above it, the quantum regime becomes accessible. The paper identifies a bulk acoustic wave resonator at $\mu_c(\Phi), n_c(\Phi)$5 GHz and $\mu_c(\Phi), n_c(\Phi)$6 mK with $\mu_c(\Phi), n_c(\Phi)$7, and an array of $\mu_c(\Phi), n_c(\Phi)$8 such resonators with $\mu_c(\Phi), n_c(\Phi)$9 dB mechanical squeezing reaches

$\mathrm{CBP}, \mathrm{CBF}$0

still a factor $\mathrm{CBP}, \mathrm{CBF}$1 above the BBN bound at $\mathrm{CBP}, \mathrm{CBF}$2 GHz. The stated implication is that the remaining sensitivity gap is predominantly classical in origin (Gaudio, 29 Apr 2026).

5. Resource-theoretic and channel-based definitions

In coherence theory, the relevant quantity is the coherence fraction

$\mathrm{CBP}, \mathrm{CBF}$3

with $\mathrm{CBP}, \mathrm{CBF}$4 a maximally coherent state in a fixed incoherent basis. For a noisy channel $\mathrm{CBP}, \mathrm{CBF}$5, the paper introduces the coherence survival fraction

$\mathrm{CBP}, \mathrm{CBF}$6

and the corresponding coherence noise fraction

$\mathrm{CBP}, \mathrm{CBF}$7

This makes quantum noise fraction a resource-destruction ratio rather than a variance ratio. Catalytic incoherent preprocessing can increase $\mathrm{CBP}, \mathrm{CBF}$8 relative to $\mathrm{CBP}, \mathrm{CBF}$9 for broad classes of channels, but the completely dephasing channel remains a hard limit: $F_0, F_K$0 for every input, so no preprocessing improves the output coherence fraction (Char et al., 27 Apr 2026).

A channel-theoretic formulation uses entanglement breaking as the threshold of maximal noise. Given a CPT map $F_0, F_K$1, one mixes it with a completely depolarizing channel

$F_0, F_K$2

and defines

$F_0, F_K$3

This is literally the minimum fraction of extra noise required to force $F_0, F_K$4 into the entanglement-breaking set. A second quantifier is

$F_0, F_K$5

the smallest integer such that $F_0, F_K$6 becomes entanglement-breaking. For unital qubit channels with Bloch matrix $F_0, F_K$7,

$F_0, F_K$8

The same work also introduces entanglement-breaking channels of order $F_0, F_K$9 and amendable channels, i.e. channels that can be prevented from becoming entanglement-breaking after iterations by interposing proper quantum transformations (Pasquale et al., 2012).

6. Probabilistic, spectral, and transport realizations

In embedded quantum annealing, the paper does not define a literal quantum noise fraction, but it introduces closely related metrics. For a chain $$S_{\text{tot}} = S_{\text{quantum}} + S_{\text{classical}}$$0 of length $$S_{\text{tot}} = S_{\text{quantum}} + S_{\text{classical}}$$1, the chain-level control error is

$$S_{\text{tot}} = S_{\text{quantum}} + S_{\text{classical}}$$2

with

$$S_{\text{tot}} = S_{\text{quantum}} + S_{\text{classical}}$$3

The chain break probability is

$$S_{\text{tot}} = S_{\text{quantum}} + S_{\text{classical}}$$4

and the chain break fraction is

$$S_{\text{tot}} = S_{\text{quantum}} + S_{\text{classical}}$$5

CBF is explicitly described as the fraction of chains that break, and hence a global noise fraction at the logical level. The core dimensionless ratio

$$S_{\text{tot}} = S_{\text{quantum}} + S_{\text{classical}}$$6

acts as an effective signal-to-noise parameter controlling CBP through an $$S_{\text{tot}} = S_{\text{quantum}} + S_{\text{classical}}$$7 law (Jeong et al., 6 Oct 2025).

In open-system and spectral formulations, the same idea appears as a fractional contribution of one spectral component to total decoherence. For quantum $$S_{\text{tot}} = S_{\text{quantum}} + S_{\text{classical}}$$8 noise in the spin-boson model, a natural quantity is

$$S_{\text{tot}} = S_{\text{quantum}} + S_{\text{classical}}$$9

with weak-coupling zero-temperature dephasing rate

$\hat{\cal M}$0

for $\hat{\cal M}$1, $\hat{\cal M}$2, and $\hat{\cal M}$3. Because the infrared cutoff satisfies

$\hat{\cal M}$4

the low-frequency quantum contribution grows with measurement time, so the fraction of decoherence attributable to quantum $\hat{\cal M}$5 noise increases as the observation window lengthens (Otterpohl et al., 18 Jul 2025). A different spectral viewpoint derives a lower bound on voltage-noise power from operator noncommutativity: $\hat{\cal M}$6 For InGaAs quantum wells, the measured noise was reported to be only a few times larger than this bound, implying that the fundamental quantum contribution can account for a substantial fraction of observed $\hat{\cal M}$7 noise (Kazakov, 2020).

In mesoscopic transport, shot-noise fractions are encoded in Fano factors. For an SU(N) Kondo system, the linear factor is

$\hat{\cal M}$8

while the nonlinear Kondo contribution is

$\hat{\cal M}$9

At half filling, the cited universal values are $\delta x_{\rm nsr}$00 for SU(2), $\delta x_{\rm nsr}$01 for SU(4), and $\delta x_{\rm nsr}$02 for SU(6). The same study finds that nonlinear shot noise is dominated by two-quasiparticle scattering near electron-hole symmetry, whereas three-body correlations must be revealed in occupation regions distant from the electron-hole symmetry point (Krychowski et al., 9 Jun 2026).

These realizations make clear that quantum noise fraction is not tied to a single ontology. It can denote a variance share, a sensitivity-limiting ratio, a coherence-loss fraction, a minimum added-noise threshold, a logical failure probability, a spectral-band contribution to decoherence, or an effective charge extracted from shot noise. The common content is the same: a normalized measure of how strongly quantum fluctuations govern the observable consequences of noise.