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On the Information Required for Feedback Control

T0 review · 2 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read This paper establishes that any stationary feedback controller must receive information from the controlled system at a rate no smaller than the entropy rate of the uncontrolled dynamics.

desk verdict Equation (12) is a real, clean bound; the explicit time-reversal protocol for state-independent passive dynamics is the genuinely new result, but the state-dependent 'Pareto frontiers' are proven lower bounds, not achieved trade-offs. read the letter →

arxiv 2607.16639 v1 pith:C3GKBLRY submitted 2026-07-18 cond-mat.stat-mech physics.bio-ph

Matthew P. Leighton , Jose M. Betancourt , Thierry Emonet , Benjamin B. Machta , Michael C. Abbott This is my paper · ORCID
classification cond-mat.stat-mechphysics.bio-ph MSC 60J7093E2094A1782C31 PACS 05.40.-a89.70.Cf
keywords information-limitedcontrolfeedbacktransferentropyratepassiveParetofrontierstime-reversalprotocolsstochasticdynamicsbiologicalnavigation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper asks how much information a controller must receive to hold a noisy stochastic system near a target steady state. It proves a lower bound: the steady-state transfer-entropy rate from system to controller is never less than the entropy growth rate of the uncontrolled passive dynamics evaluated at the target distribution. This bound depends only on the target distribution and the passive drift, diffusion, and jumps, so it converts a hard history-dependent information calculation into a local, computable quantity. For passive dynamics that do not depend on the state, the paper constructs an explicit information-optimal protocol that probabilistically time-reverses the passive dynamics, and shows the same time-reversal idea is near-optimal more generally. The result yields performance-information Pareto frontiers for examples ranging from bit control and particle localization to microbial navigation and feedback information engines.

What carries the argument

The argument revolves around two quantities: the passive entropy rate (the entropy growth of the uncontrolled dynamics, expressed as a sum of diffusion, drift, and jump terms evaluated at the target distribution) and the transfer-entropy rate, which measures the causal information flow from the system's past to the controller's action. Eq. (12) connects these two rates. Tightness is achieved by the probabilistic time-reversal protocol of Eq. (17), which draws the post-control state from the Bayesian posterior of the passive kernel; for state-independent passive dynamics this forces both gap terms in Eq. (14) to vanish. A Schrödinger-equation mapping then converts fixed-information performanc

What would settle it

Run the bit-control example with passive flip rate γ=1 and target ρ0=0.7 using a controller whose control error rate is set below the time-reversal value λ0=γ(1−ρ0)/ρ0, then compute the transfer-entropy rate numerically by path-weight sampling; if it falls below (2ρ0−1)γ ln(ρ0/(1−ρ0)), Eq. (12) would be refuted.

Watch

Extended reading notes

Core claim

The central discovery is Eq. (12): for any stationary feedback controller, the transfer-entropy rate from state to control action is bounded below by the passive entropy rate, the rate at which the Shannon entropy of the target distribution would grow if control were suddenly stopped. The inequality follows from stationarity, the data-processing inequality, and the Markovianity of the passive dynamics. When the passive dynamics are state-independent, the explicit protocol in Eq. (17), which probabilistically time-reverses the passive kernel, saturates the bound and is therefore information-optimal among controllers with full state observability. The paper further maps fixed-information perfo

Load-bearing premise

Tightness of the bound and the optimality of the time-reversal protocol rest on the passive dynamics being state-independent, the controller observing the full state without delay and being able to implement arbitrary stochastic actions including exact passive noise, and the state space having no reflecting boundaries—if any of these fails, the inequality still holds but the constructed protocol need not be optimal.

Editorial extensions

If this is right

  • Any feedback controller that maintains a given steady-state distribution must receive information at a rate at least as large as the passive entropy rate, so performance improvements beyond that bound are impossible without more information.
  • For state-independent passive dynamics, probabilistic time reversal is exactly information-optimal; no controller using memory of past states or actions can do better under full observability.
  • Performance-information Pareto frontiers can be obtained from the ground state of a Schrödinger-like operator, yielding analytic frontiers in the bit, particle-localization, and navigation examples.
  • In microbial navigation with full state observability, memory cannot improve up-gradient velocity; with only heading measurements available, a memoryless Run-Reverse strategy is the unique optimal strategy at low information rates.
  • For an experimentally realized information engine, approaching the theoretical maximum rate of heat extraction from a thermal bath requires a diverging information rate, as confirmed by simulations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the bound depends only on the passive dynamics and the target distribution, an observer could estimate the minimal information rate of an unobserved biological controller by measuring how fast the uncontrolled system relaxes, without monitoring the controller's internal decisions.
  • If a Landauer-style relation between transfer-entropy rate and the controller's energy consumption holds, this bound would also imply energetic lower bounds on feedback control; the paper raises this as an open conjecture rather than a theorem.
  • The paper proves optimality of time reversal only for state-independent passive dynamics and full observability; extending the framework to time-dependent targets, finite-time control, or first-passage objectives is a natural next step the paper itself flags.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper studies information-limited feedback control of Markovian stochastic systems maintained at a target steady state. The main theoretical result is Eq. (12): for any feedback controller that maintains a stationary distribution ρ(x), the transfer-entropy rate from the state history to the control action, ˙T_{x→dx^c}, is bounded below by the passive entropy rate ˙S_p of the uncontrolled dynamics evaluated at ρ(x). The paper then shows that when the passive dynamics are state-independent (Eq. (15)), an explicit probabilistic time-reversal protocol (Eq. (17)) saturates the bound, establishing exact information-optimality within that class. Applications are given to bit control, Brownian localization in a Mexican-hat potential, microbial navigation, and a feedback information engine; the latter three use the bound to derive performance–information trade-off curves, with the navigation case also proving that a memoryless Run-Reverse strategy is asymptotically optimal under limited observability. The paper also derives a Schrödinger-equation mapping for constructing optimal steady-state distributions at fixed ˙S_p and provides numerical transfer-entropy computations via path weight sampling.

Significance. If the results hold, Eq. (12) is a valuable and broadly applicable lower bound: it depends only on the target distribution and the local passive dynamics, and it yields a simple information-theoretic obstruction to control performance. The explicit time-reversal saturating protocol for state-independent passive dynamics is an elegant constructive result, and the navigation application, including the proof of asymptotic optimality of Run-Reverse in a restricted-observability limit, is of independent interest. The paper is also commendable for providing Python code for reproducing the figures and for making the gap analysis explicit rather than hiding it. The main caveat, discussed below, is that the term 'Pareto frontier' is used for curves that, in the state-dependent applications, are rigorously established only as lower bounds, not as achievable trade-offs.

major comments (2)
  1. [III.B, III.D; Figs. 3, 5] The curves labeled 'Pareto frontier' in the Mexican-hat example and the information-engine example are rigorous lower bounds on the information rate required for a given performance, but they are not shown to be achievable. Exact saturation is proven only for state-independent passive dynamics (Eq. (15)); the Mexican-hat dynamics (Eq. (25)) and the information-engine dynamics (Eq. (29)) have state-dependent drift, so the time-reversal protocol's optimality is not established for them. The gap bound in Eq. (C17) is an upper bound on ˙T−˙S_p for the time-reversal protocol, and it does not vanish except in limiting regimes. Thus the black curves in Figs. 3 and 5 overstate the proven content. I recommend either proving achievability (or at least showing the gap vanishes for the plotted regimes) or relabeling these curves as 'lower-bound frontiers' and qualifying the abstract's claim that the
  2. [II.B, App. C (Eqs. C12–C17)] The proof that the time-reversal protocol sets Term 2 of Eq. (14) to zero uses, in an essential way, the translation-invariance of the passive kernel K^p(x'|y)=q(x'−y) for state-independent passive dynamics (Eq. (C12)–(C14)). For general state-dependent F(x) and λ(x,Δx), this argument fails, and the paper's 'near-optimality' conjecture for Eq. (C16) is not supported by a quantitative statement. The gap bound (C17) involves ⟨F·D^{-1}F⟩_ρ and a jump log-term; while these vanish when ρ becomes very narrow, the paper does not provide a precise condition or rate. Since the high-information near-optimality claim is used to justify the application curves, this is a load-bearing gap. I ask that the authors either prove the relevant gap vanishes for the specific examples (with explicit bounds), or clearly restrict the optimality claims to the state-independent class and present the state-dependen
minor comments (5)
  1. [Eq. (16)] The notation p(y_t) in Eq. (16) is undefined; it should be the post-control distribution, denoted ρ'(y) in App. C. Please define it explicitly.
  2. [Eq. (22)] The sum over λ_i Re(λ_i) is ambiguous when some eigenvalues of A=−H_* have negative real parts (as for a stable fixed point). Please specify whether the sum runs over all eigenvalues or only those with positive real part, and discuss the sign.
  3. [Fig. 5 caption] The caption says parameters are 'from the experimental setup in Ref. [10]', but the protocol is from Ref. [43]. Please correct the reference.
  4. [App. B] The Schrödinger mapping is presented for the case without jumps; the jump case is said to lead to a nonlinear/nonlocal equation. Since jumps appear in the bit-control example, it would be helpful to state explicitly how the linear mapping fails there and what is done instead.
  5. [App. H, Eq. (H20)] The measurement-noise restoration works only when inequality (H20) holds. The main text should mention this limitation in the discussion of the information-engine application, especially because experimental implementations inevitably have measurement noise.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Eq. (12) is derived self-contained; saturation protocols are constructed, not fitted; citations to prior work are re-derived, not load-bearing.

full rationale

The central inequality, Eq. (12), is derived entirely within the paper from stationarity, the Shannon-entropy chain rule, the data-processing inequality, and the definition of transfer entropy (Eqs. 6–12). No parameter is fitted to data, and no target result is used as an input: Ḍp is an independently defined functional of the passive dynamics and target distribution ρ(x). The saturation protocol, Eq. (17), is constructed in App. C by setting the active kernel to the Bayesian reverse of the passive kernel and explicitly proving that both gap terms vanish for state-independent passive dynamics; it is a derivation, not a restatement of the bound. The Schrödinger-mapping Pareto frontiers (App. B, Eq. B8) follow from extremizing P at fixed Ḍp and are therefore lower bounds when tightness is not proven; the paper explicitly limits the exact saturation claim to Eq. (15) and labels the general near-optimality of time reversal as a conjecture (“we conjecture that probabilistically time-reversing the passive dynamics constitutes a near-optimal control protocol more generally”). This is an honest scope limitation, not circularity. The navigation application cites the authors’ prior work [19], but the frontier and saturation are re-derived in App. G and extended to the full-history transfer entropy, so [19] is not load-bearing. Eq. (A7) is also cited to [19], but it is a parameter-free algebraic identity for the instantaneous information rate and is used only to bound the gap in Eq. (C17), not to establish Eq. (12). The paper checks its results against external benchmarks (Data Rate Theorem, second law of information thermodynamics, the experimental information engine [43], and TE-PWS [38]). The main non-circular concern is that the state-dependent “Pareto frontier” curves in Figs. 3–5 are proven lower bounds, not demonstrated achievable trade-offs, when the bound is not tight; this is a correctness/overclaim risk, not a reduction of the result to its inputs.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No data-fitted parameters and no invented entities. The central bound rests on standard information-theoretic inequalities, Markovian passive dynamics, stationarity of the controlled joint process, and, for the saturation result, full observability and state-independent passive dynamics. Model parameters in the examples are physical inputs from the problem setup or prior experiments.

assumptions (6)
  • domain assumption Passive dynamics are Markovian and follow Eq. (1) with drift, diffusion, and jumps.
    The Fokker-Planck equation (2) and the entire entropy-rate computation Eq. (5) assume Markovian dynamics with no hidden variables.
  • domain assumption The state space has no reflecting boundaries; boundary terms from integration by parts vanish.
    Explicitly stated in Section II before Eq. (1); needed for the divergence term and Fisher-information identity in Eq. (5).
  • domain assumption The controlled state-control-history joint process is stationary.
    The cancellation I[x_t; past] - I[x_{t+dt}; current] = 0 in Eqs. (10)-(11) uses stationarity; the paper's setup targets steady states.
  • domain assumption Full observability of x_t and unconstrained actuation for the saturating protocol.
    Eqs. (16)-(17) construct the optimal reverse kernel using exact x_t; App. H shows measurement noise is handled only under inequality (H20).
  • domain assumption For Pareto frontiers, performance is optimized at fixed dotS_p and the bound is assumed tight.
    Schrodinger mapping in App. B; exact tightness is proven for state-independent dynamics, while for state-dependent cases it is supported by examples and numerics and is the subject of a conjecture.
  • standard math Data-processing inequality and standard conditional-mutual-information identities.
    Used throughout and explicitly in Eqs. (C1)-(C2); standard information theory.

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Pith. "Pith review of On the Information Required for Feedback Control." pith.science (2026). https://pith.science/paper/C3GKBLRY

@misc{pith2026260716639,
  author       = {Pith},
  title        = {Pith review of: On the Information Required for Feedback Control},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C3GKBLRY}},
  note         = {Machine review of arXiv:2607.16639}
}
read the original abstract

Biological systems across scales, along with many engineering problems, must control noisy systems with limited information. Here we study information-limited feedback control of stochastic systems to achieve target steady states, and derive a lower bound on the information rate from controlled system to controller. This framework allows us to obtain performance-information Pareto frontiers for wide-ranging control problems with limited information. For systems with state-independent passive dynamics, the bound is saturated by an explicit optimal control protocol which probabilistically time-reverses the passive dynamics. We showcase these results through applications to nonlinear particle localization, microbial navigation, and experimentally realized information engines.

Figures

Figures reproduced from arXiv: 2607.16639 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic illustrating the setup and our bound. A [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Controlling a noisy bit. Left: schematic, with passive [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Controlling a Brownian particle in a Mexican–hat [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Up-gradient velocity vs information rate for navi [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Information rate vs heat extracted from the bath [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Control of a Brownian particle in a harmonic trap. (a) and (b) Information rate (blue) as a function of controller [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]

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    The Schr¨ odinger equation is −4∇2 + 2−4r 2 −λ|x−x ∗ 2|2 ψ(x) =E 0(λ)ψ(x).(F7) Usingx= (x, y) and completing the square, we can rewrite this as E0(λ)ψ(x) = −4∇2 + 2−4(x 2 +y 2)−λ ((x−1) 2 +y 2 ψ(x) = " −4∇2 + 2−(4 +λ) " x− λ 4 +λ 2 +y 2 # −λ+ λ2 4 +λ # ψ(x). (F8) The ground-st...

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    Optimal Control Protocol with F ull Observability For the navigation problem, the corresponding Schr¨ odinger equation is −4Dr∂2 θ −λcosθ ψ(θ) =E 0(λ)ψ(θ).(G1) This is the Hamiltonian for a quantum rotor which has been studied previously [51]. The ground-state distribution is ...

  49. [57]

    We consider the Markovian version of this strategy whereλ=λ(θ)

    Run-and-T umble Navigation Now consider a Run-and-Tumble strategy, where the only action available is to fully reorient the heading with rate λ({θ}t −∞,{dθ c}t −∞) such that the angle change is uniformly distributed on [0,2π]. We consider the Markovian version of this strategy...

  50. [58]

    Here we prove that Run-Reverse is the unique optimal control protocol in the low-information limit, even allowing the use of arbitrary memory, given that only cosθcan be measured

    Optimal Protocol with Limited Observability Run-and-Tumble is not the optimal control protocol using only measurements of cosθin the low-information limit. Here we prove that Run-Reverse is the unique optimal control protocol in the low-information limit, even allowing the use...

  51. [59]

    Deriving the Pareto F rontier Here we derive the Pareto frontier for the information engine, Eq. (30). Applying the bound (12) to the information engine, and evaluating the passive entropy rate, gives (usingu=z−x, and the steady-state probabilityρ(u)): DI[ρ]− κ γ ≤ ˙Tu→dzc .(H...

  52. [60]

    Letxandzbe bipartite degrees of freedom, as described in Ref

    Second Law for the Information Engine Here we show that the second law impliesβ ˙Q≤ ˙Tu→dzc . Letxandzbe bipartite degrees of freedom, as described in Ref. [29]. Then the second law of bipartite thermodynamics applied to thexsubsystem gives [27, 29] β ˙Q≤ ˙IZ ≡lim dt→0 I[x t;z...

  53. [61]

    The protocol implemented in Ref

    Experimentally Implemented Control Protocol As described in the main text, the controller makes noisy measurements of the particle position ˆxt =x t +N(0, σ2 m) (with measurement noiseσ m) at periodic time intervals of lengtht s. The protocol implemented in Ref. [43] updates t...

  54. [62]

    Probabilistic Time-Reversal Protocol Here we construct the time-reversal control protocol for the information engine shown in Fig. 5. We will initially neglect the measurement noise, then restore it at the end of this subsection. For notational simplicity, we use the dimension...

  55. [63]

    Information Rate for the Time-Reversal Protocol For the time-reversal protocol, we can compute the transfer entropy rate semi-analytically. For a single controller update, the transfer entropy is ˙Tξ→dξc = 1 ∆ I dξc τ ;{ξ} τ −∞|{dξc}τ−∆ −∞ = 1 ∆ I dξc τ ;ξ − τ |{dξc}τ−∆ −∞ = 1...

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.