Small Phase Condition in Signal, Control, and Quantum

• Small Phase Condition is a set of criteria that enforce phase uniqueness in analytic signals, feedback loops, and quantum systems.
• It provides practical tests—such as Nyquist-style winding numbers, phase-sector inequalities, and SRG metrics—to guarantee stability and reliable phase retrieval.
• This concept underpins methodologies from minimum-phase signal reconstruction to robust stability analysis in MIMO and cyclic feedback systems.

Searching arXiv for recent and foundational papers on the Small Phase Condition across control, signal processing, and optics. The expression Small Phase Condition denotes a family of phase-constrained criteria that arise in several technically distinct literatures. In band-limited analytic signal theory, it is tantamount to the minimum-phase property and characterizes when the phase of a complex field is uniquely determined by its intensity (Mecozzi, 2016). In linear feedback theory for SISO and MIMO LTI systems, it refers to frequency-wise phase-sector inequalities that guarantee closed-loop stability, originally under sectoriality assumptions and subsequently in formulations based on Scaled Relative Graphs (SRGs), segmental phase, and symmetry-restricted necessity results (Zhao et al., 2022, Baron-Prada et al., 17 Mar 2025, Chen et al., 2023, Yang et al., 9 Jul 2025). In quantum-optical amplifier analysis, a distinct usage concerns a small-noise or small phase fluctuation regime in which phase diffusion can be treated by replacing random photon number by its mean (Chia et al., 2018).

1. Terminological scope and domain-specific meanings

The term is not monolithic. Its technical content depends on the analytic object whose phase is being constrained: a scalar field, a transfer matrix, a cyclic interconnection, or a stochastic optical amplitude.

Domain	Object	Small phase content
Phase retrieval	Band-limited analytic signal $E(t)$	Minimum phase; zero winding about the origin (Mecozzi, 2016)
Two-block feedback	$H_1(s),H_2(s)\in\mathcal{RH}_\infty$	Frequency-wise phase sums remain within $(-\pi,\pi)$ (Zhao et al., 2022)
MIMO feedback without sectoriality	SRG/singular-angle description	$$\hat{\alpha}_{\max}(H_1(j\omega))+\hat{\alpha}_{\max}(H_2(j\omega))<\pi$$ (Baron-Prada et al., 17 Mar 2025)
Cyclic feedback	$P_1(s),\dots,P_m(s)$	Sum of segmental phases lies in $(-\pi,\pi)$ (Chen et al., 2023)
Quantum amplifier noise	$N(t),\Phi(t)$	Narrow number distribution enabling small-noise phase analysis (Chia et al., 2018)

Two distinctions are especially important. First, minimum phase in signal theory and small phase in feedback theory are not the same concept, even though both prevent phase ambiguity and both admit Nyquist-like interpretations. Second, in control, multiple non-equivalent MIMO phase notions coexist: sectorial phases from numerical ranges, singular-angle phases from SRGs, and segmental phases from normalized numerical ranges.

A recurrent structural theme is exclusion of a critical angular event. In signal theory, the excluded event is encirclement of the origin by the field trajectory $E(t)$. In feedback theory, it is crossing or encirclement of the critical point $-1$ by loop-induced frequency-domain geometry. This suggests a common geometric principle, although the underlying objects and proofs differ substantially.

2. Minimum phase as the Small Phase Condition in band-limited analytic signals

For band-limited analytic signals, the Small Phase Condition is precisely the minimum-phase condition (Mecozzi, 2016). The signal model is

$E(t)=E_s(t)+\bar E,$

with complex bias $H_1(s),H_2(s)\in\mathcal{RH}_\infty$0, $H_1(s),H_2(s)\in\mathcal{RH}_\infty$1, and single-sideband spectrum

$H_1(s),H_2(s)\in\mathcal{RH}_\infty$2

supported in $H_1(s),H_2(s)\in\mathcal{RH}_\infty$3. The paper considers signal classes constructed from square-root raised-cosine pulses and their sinc limit, including

$H_1(s),H_2(s)\in\mathcal{RH}_\infty$4

A signal $H_1(s),H_2(s)\in\mathcal{RH}_\infty$5 is minimum phase when its analytic continuation $H_1(s),H_2(s)\in\mathcal{RH}_\infty$6 has no zeros in the lower half-plane $H_1(s),H_2(s)\in\mathcal{RH}_\infty$7. Equivalently, the signal is outer and its inner all-pass factor is trivial. Under this condition, and assuming $H_1(s),H_2(s)\in\mathcal{RH}_\infty$8 for all $H_1(s),H_2(s)\in\mathcal{RH}_\infty$9, the logarithm

$(-\pi,\pi)$0

belongs to $(-\pi,\pi)$1 and has nonnegative-frequency support, so its real and imaginary parts are Hilbert transforms of each other. The phase is then uniquely determined, up to a constant, by the intensity $(-\pi,\pi)$2 through the logarithmic Kramers–Kronig relation

$(-\pi,\pi)$3

The key result is a necessary and sufficient criterion formulated entirely on the real axis: $(-\pi,\pi)$4 The number of zeros of $(-\pi,\pi)$5 in the lower half-plane equals the winding number of the trajectory $(-\pi,\pi)$6 about the origin. This avoids analytic continuation and root finding in the complex plane, replacing them by a Nyquist-style winding test on measured or synthesized real-axis data (Mecozzi, 2016).

Several practical sufficient conditions follow. If there exists $(-\pi,\pi)$7 such that

$(-\pi,\pi)$8

then $(-\pi,\pi)$9 is minimum phase. Likewise,

$$\hat{\alpha}_{\max}(H_1(j\omega))+\hat{\alpha}_{\max}(H_2(j\omega))<\pi$$0

implies no encirclement of the origin, hence minimum phase. The paper emphasizes that this magnitude bound is sufficient but not necessary: a shifted 8-level amplitude modulation example with small bias violates $$\hat{\alpha}_{\max}(H_1(j\omega))+\hat{\alpha}_{\max}(H_2(j\omega))<\pi$$1 at some times while still remaining minimum phase because the trajectory does not wind around the origin (Mecozzi, 2016).

When minimum phase fails, ambiguity is exactly characterized by an inner factor. If $$\hat{\alpha}_{\max}(H_1(j\omega))+\hat{\alpha}_{\max}(H_2(j\omega))<\pi$$2 is the minimum-phase representative with the same intensity, then

$$\hat{\alpha}_{\max}(H_1(j\omega))+\hat{\alpha}_{\max}(H_2(j\omega))<\pi$$3

where $$\hat{\alpha}_{\max}(H_1(j\omega))+\hat{\alpha}_{\max}(H_2(j\omega))<\pi$$4 are lower-half-plane zeros. These are localized Lorentzian all-pass distortions. Consequently, intensity-only phase reconstruction returns the minimum-phase representative; non-minimum-phase departures are confined to neighborhoods associated with the lower-half-plane zeros.

The same framework yields a classification of intensity-equivalence classes within the band-limited space $$\hat{\alpha}_{\max}(H_1(j\omega))+\hat{\alpha}_{\max}(H_2(j\omega))<\pi$$5. Each class

$$\hat{\alpha}_{\max}(H_1(j\omega))+\hat{\alpha}_{\max}(H_2(j\omega))<\pi$$6

contains exactly one minimum-phase representative up to global phase. This establishes a one-to-one correspondence between distinguishable receiver states under intensity-only detection and minimum-phase representatives of those classes (Mecozzi, 2016).

3. Classical feedback Small Phase Condition for SISO and MIMO LTI systems

In control theory, the Small Phase Condition is a feedback-stability condition stated in terms of operator phases of $$\hat{\alpha}_{\max}(H_1(j\omega))+\hat{\alpha}_{\max}(H_2(j\omega))<\pi$$7 and $$\hat{\alpha}_{\max}(H_1(j\omega))+\hat{\alpha}_{\max}(H_2(j\omega))<\pi$$8. The classical setting assumes $$\hat{\alpha}_{\max}(H_1(j\omega))+\hat{\alpha}_{\max}(H_2(j\omega))<\pi$$9, i.e. rational, proper, stable transfer matrices (Baron-Prada et al., 17 Mar 2025).

Its traditional MIMO formulation is built on the numerical range

$P_1(s),\dots,P_m(s)$0

and on sectoriality: $P_1(s),\dots,P_m(s)$1 is sectorial if $P_1(s),\dots,P_m(s)$2 lies in an angular sector and $P_1(s),\dots,P_m(s)$3. For a sectorial operator, there exists a decomposition

$P_1(s),\dots,P_m(s)$4

where $P_1(s),\dots,P_m(s)$5 is invertible and $P_1(s),\dots,P_m(s)$6 is diagonal unitary with diagonal entries lying on an arc of the unit circle of length smaller than $P_1(s),\dots,P_m(s)$7. The associated phases are ordered as

$P_1(s),\dots,P_m(s)$8

with

$P_1(s),\dots,P_m(s)$9

The classical small-phase theorem states that the negative-feedback interconnection is stable if, for each $(-\pi,\pi)$0,

$(-\pi,\pi)$1

$(-\pi,\pi)$2

and both $(-\pi,\pi)$3 and $(-\pi,\pi)$4 satisfy the sectorial property (Baron-Prada et al., 17 Mar 2025). In SISO, this reduces to the familiar loop-phase bound

$(-\pi,\pi)$5

A broader formulation allows a semi-stable plant $(-\pi,\pi)$6 with imaginary-axis poles and a stable controller $(-\pi,\pi)$7, provided $(-\pi,\pi)$8 is frequency-wise semi-sectorial and the phase constraints hold along an indented contour around the $(-\pi,\pi)$9-axis singularities (Zhao et al., 2022). In that setting, the small-phase theorem is

$N(t),\Phi(t)$0

for all $N(t),\Phi(t)$1, where $N(t),\Phi(t)$2 denotes the imaginary-axis poles of $N(t),\Phi(t)$3.

This sectorial formulation places the Small Phase Condition alongside small-gain and passivity-type criteria. It is a phase-domain analogue of the gain constraint

$N(t),\Phi(t)$4

but it is neither reducible to singular-value bounds nor equivalent to positive realness. The phase description becomes especially useful when low-frequency dynamics are phase-favorable but gain-large, a regime in which pure small-gain certification can fail (Zhao et al., 2022).

A common misconception is that the magnitude of the loop is central to the theorem. In the classical small-phase result it is not: the key hypothesis is sectorial phase geometry, not gain attenuation. Magnitude enters only in mixed gain–phase extensions.

4. Mixed gain–phase formulations and SRG-based removal of sectoriality

A major limitation of the classical theorem is the sectoriality assumption. Recent work removes that assumption by replacing numerical-range phases with SRG-based singular angles (Baron-Prada et al., 17 Mar 2025).

For a square matrix $N(t),\Phi(t)$5, the Scaled Relative Graph is

$N(t),\Phi(t)$6

From SRG geometry, the maximum gain and maximum phase are defined by

$N(t),\Phi(t)$7

and

$N(t),\Phi(t)$8

For SISO systems, $N(t),\Phi(t)$9 coincides with $E(t)$0; for MIMO systems, $E(t)$1 and $E(t)$2 generally occur at different SRG boundary points (Baron-Prada et al., 17 Mar 2025).

The SRG-based small-phase theorem states that if

$E(t)$3

then the closed-loop system is $E(t)$4-stable. This entirely removes sectoriality. An equivalent set-separation condition is

$E(t)$5

The geometric meaning is direct: scaled SRG images must exclude the critical point $E(t)$6 at every frequency (Baron-Prada et al., 17 Mar 2025).

The same paper gives a mixed criterion: $E(t)$7 or

$E(t)$8

pointwise in frequency. If this disjunctive condition covers the entire spectrum, the feedback interconnection is $E(t)$9-stable. This interpolates between pure small phase and pure small gain (Baron-Prada et al., 17 Mar 2025).

An earlier mixed framework formulated the same philosophy via sectorial phases, “fan” sets $-1$0, and gain balls $-1$1 (Zhao et al., 2022). In particular, for a cutoff $-1$2, the loop is stable if phase inequalities hold on $-1$3 and the gain product is less than one on $-1$4. The paper also develops a “small vase theorem,” a necessary-and-sufficient robustness characterization for vase-shaped uncertainty sets, and a bounded & sectored real lemma expressed as a triple of generalized KYP LMIs (Zhao et al., 2022).

These developments reduce conservatism relative to single-metric criteria, but they do not eliminate it entirely. The SRG separation test is evaluated frequency by frequency and is stated to be more conservative than the Generalized Nyquist Criterion, because the latter also uses the global trajectory of eigenvalues over the full spectrum (Baron-Prada et al., 17 Mar 2025).

5. Cyclic interconnections, segmental phase, and necessity for symmetric systems

For cyclic feedback systems with $-1$5 subsystems, existing MIMO phase notions do not provide the product eigen-phase bound needed for a direct extension of small-phase reasoning. To address this, the cyclic theory introduces the segmental phase, built from the normalized numerical range

$-1$6

which lies in the unit disk (Chen et al., 2023).

The segmental phase of $-1$7 is the smallest circular-segment interval covering $-1$8: $-1$9 Its crucial property is the product eigen-phase bound: $E(t)=E_s(t)+\bar E,$0 for all nonzero eigenvalues of the product. Hence

$E(t)=E_s(t)+\bar E,$1

is invertible if

$E(t)=E_s(t)+\bar E,$2

For a cyclic interconnection of semi-stable MIMO LTI subsystems $E(t)=E_s(t)+\bar E,$3, the cyclic small phase theorem states that the loop is stable if

$E(t)=E_s(t)+\bar E,$4

under the paper’s continuity and full-rank assumptions around $E(t)=E_s(t)+\bar E,$5-axis poles (Chen et al., 2023). The corresponding $E(t)=E_s(t)+\bar E,$6-style sufficient condition is

$E(t)=E_s(t)+\bar E,$7

Angular scaling generalizes this construction. For a chosen center $E(t)=E_s(t)+\bar E,$8,

$E(t)=E_s(t)+\bar E,$9

with

$H_1(s),H_2(s)\in\mathcal{RH}_\infty$00

By coordinating $H_1(s),H_2(s)\in\mathcal{RH}_\infty$01 across known subsystems, the summed phase intervals can be shortened, reducing conservatism in robust cyclic stability tests (Chen et al., 2023).

A different line of development concerns necessity. For symmetric systems $H_1(s),H_2(s)\in\mathcal{RH}_\infty$02, the Small Phase Condition is not only sufficient but also necessary against symmetric phase-bounded uncertainties (Yang et al., 9 Jul 2025). The system theorem states that if $H_1(s),H_2(s)\in\mathcal{RH}_\infty$03 is symmetric and frequency-wise quasi-sectorial, then

$H_1(s),H_2(s)\in\mathcal{RH}_\infty$04

if and only if $H_1(s),H_2(s)\in\mathcal{RH}_\infty$05 is stable and

$H_1(s),H_2(s)\in\mathcal{RH}_\infty$06

The proof relies on a structural result specific to complex symmetric semi-sectorial matrices: their generalized sectorial decomposition can be written with a real nonsingular congruence matrix $H_1(s),H_2(s)\in\mathcal{RH}_\infty$07. This allows the destabilizing phase-bounded symmetric perturbation to be constructed explicitly when SPC fails (Yang et al., 9 Jul 2025).

For general asymmetric systems, necessity remains incomplete. The obstruction is an unresolved phase interpolation problem involving complex positive semidefinite target matrices under arbitrarily small phase sectors. The paper gives partial necessity results for inner systems but presents the general case as open (Yang et al., 9 Jul 2025).

6. Distinct optical and quantum usages

In quantum linear amplifiers, the relevant phrase is not a feedback-stability theorem but a small-noise approximation, also described as a small phase fluctuation regime (Chia et al., 2018). The underlying optical field amplitude is written as

$H_1(s),H_2(s)\in\mathcal{RH}_\infty$08

and the phase SDE contains the nonlinear factor $H_1(s),H_2(s)\in\mathcal{RH}_\infty$09: $H_1(s),H_2(s)\in\mathcal{RH}_\infty$10 The exact phase variance satisfies

$H_1(s),H_2(s)\in\mathcal{RH}_\infty$11

The small-noise approximation replaces $H_1(s),H_2(s)\in\mathcal{RH}_\infty$12 by $H_1(s),H_2(s)\in\mathcal{RH}_\infty$13, which is valid only when the photon-number distribution is narrow. For coherent input, the required condition is

$H_1(s),H_2(s)\in\mathcal{RH}_\infty$14

equivalently, a high signal-to-noise ratio at the input that remains high during amplification. In the ideal amplifier case $H_1(s),H_2(s)\in\mathcal{RH}_\infty$15, this gives a closed-form phase-variance formula, but the paper shows that the approximation breaks down for coherent inputs with only a few photons on average and becomes much better only in the tens-of-photons regime (Chia et al., 2018).

A Jensen inequality argument yields

$H_1(s),H_2(s)\in\mathcal{RH}_\infty$16

so the small-noise prediction is a lower bound on the true phase variance. This use of “small phase” is therefore probabilistic and approximation-theoretic, not topological or Nyquist-like.

A related but distinct condition appears in quantum metrology, where a phase matching condition maximizes the quantum Fisher information of a Mach–Zehnder interferometer with an arbitrary state in one port and an odd/even state in the other (Liu et al., 2013). There the optimal relation is

$H_1(s),H_2(s)\in\mathcal{RH}_\infty$17

which is not termed a Small Phase Condition in the paper but is often adjacent in discussions of phase-sensitive quantum performance.

Taken together, these usages show that the phrase “Small Phase Condition” does not identify a single theorem. In signal theory it characterizes outer/minimum-phase structure and phase retrievability from intensity; in feedback theory it defines angular exclusion conditions for stability, with sectorial, SRG-based, segmental, and symmetry-based variants; and in quantum optics it can denote a regime of sufficiently small phase diffusion to justify mean-field treatment of photon-number fluctuations (Mecozzi, 2016, Baron-Prada et al., 17 Mar 2025, Chen et al., 2023, Yang et al., 9 Jul 2025, Chia et al., 2018).