Entropy Power Conjecture Hierarchy

• The Entropy Power Conjecture is a claim on the complete monotonicity of entropy power under Gaussian smoothing, extending Costa’s second-derivative result into an alternating-sign hierarchy.
• It connects de Bruijn’s identity with Fisher information and employs Bell polynomial techniques to derive rigorous derivative sign conditions, influencing related conjectures like McKean’s.
• The research informs diverse areas including semidefinite programming approaches, Rényi and Tsallis entropy extensions, and discrete as well as quantum analogues in information theory.

Entropy Power Conjecture commonly denotes an all-orders strengthening of Costa’s entropy-power concavity along Gaussian heat flow. Let

$$\mu_t=\mathrm{Law}(X_0+\sqrt{t}\,G), \qquad G\sim \mathcal N(0,I_D),$$

and define

$$y(t)=-\frac{2}{D}H(\mu_t), \qquad N(t)=e^{y(t)}=e^{\frac{2}{D}h(\mu_t)},$$

where $H$ is negative differential entropy and $h=-H$ is the usual differential entropy. The conjecture asserts that for every $m\ge 1$,

$(-1)^{m-1}\frac{d^m}{dt^m}N(t)\ge 0.$

Thus $N'(t)\ge 0$, $N''(t)\le 0$, $N^{(3)}(t)\ge 0$, and so on. In this form, the conjecture extends the classical $m=2$ Costa concavity statement into a complete alternating-sign hierarchy and serves as a reference point for broader programs involving Fisher information, Rényi entropies, discrete thinning, bosonic beamsplitting, and Brunn–Minkowski analogies (Wang, 2024).

1. Classical heat-flow formulation

The conjecture is posed along Gaussian smoothing. Starting from a probability measure $$y(t)=-\frac{2}{D}H(\mu_t), \qquad N(t)=e^{y(t)}=e^{\frac{2}{D}h(\mu_t)},$$0 on $$y(t)=-\frac{2}{D}H(\mu_t), \qquad N(t)=e^{y(t)}=e^{\frac{2}{D}h(\mu_t)},$$1, one considers

$$y(t)=-\frac{2}{D}H(\mu_t), \qquad N(t)=e^{y(t)}=e^{\frac{2}{D}h(\mu_t)},$$2

so that $$y(t)=-\frac{2}{D}H(\mu_t), \qquad N(t)=e^{y(t)}=e^{\frac{2}{D}h(\mu_t)},$$3 is the Gaussian noise variance. In the normalization used in "The entropy power conjecture implies the McKean conjecture" (Wang, 2024), one writes

$$y(t)=-\frac{2}{D}H(\mu_t), \qquad N(t)=e^{y(t)}=e^{\frac{2}{D}h(\mu_t)},$$4

with Fisher information

$$y(t)=-\frac{2}{D}H(\mu_t), \qquad N(t)=e^{y(t)}=e^{\frac{2}{D}h(\mu_t)},$$5

De Bruijn’s identity takes the form

$$y(t)=-\frac{2}{D}H(\mu_t), \qquad N(t)=e^{y(t)}=e^{\frac{2}{D}h(\mu_t)},$$6

or equivalently $$y(t)=-\frac{2}{D}H(\mu_t), \qquad N(t)=e^{y(t)}=e^{\frac{2}{D}h(\mu_t)},$$7.

Under this convention, the Entropy Power Conjecture is the assertion that

$$y(t)=-\frac{2}{D}H(\mu_t), \qquad N(t)=e^{y(t)}=e^{\frac{2}{D}h(\mu_t)},$$8

Its first two instances are familiar: $$y(t)=-\frac{2}{D}H(\mu_t), \qquad N(t)=e^{y(t)}=e^{\frac{2}{D}h(\mu_t)},$$9 is entropy-power monotonicity under Gaussian smoothing, and $H$0 is Costa’s entropy-power concavity. The conjecture therefore strengthens the usual second-derivative statement into a completely alternating hierarchy.

This formulation is distinct from the convolutional Shannon EPI $H$1, but it is closely related. Gaussian smoothing may be viewed as a one-parameter convolutional interpolation, and the conjecture asks for a much finer regularity statement than ordinary concavity: not only should entropy power be concave, its successive derivatives should continue alternating in sign.

2. Relation to McKean and Gaussian complete monotonicity

A central structural result is that the entropy-power hierarchy implies a corresponding hierarchy for Fisher information derivatives. Assume

$H$2

The McKean conjecture, in the same notation, states that for all $H$3,

$H$4

The weaker Gaussian completely monotone conjecture keeps only the sign condition

$H$5

The implication

$H$6

is proved order-by-order in (Wang, 2024).

The mechanism is elementary but highly structured. Since $H$7, Faà di Bruno’s formula gives

$H$8

where $H$9 is the complete exponential Bell polynomial. Introducing

$h=-H$0

the entropy-power sign condition becomes

$h=-H$1

A Bell-polynomial lemma then yields

$h=-H$2

Applying the Cramér–Rao bound

$h=-H$3

upgrades this to the McKean inequality.

This implication is conceptually significant because it places entropy-power complete monotonicity above Fisher-information complete monotonicity in the logical hierarchy. It also clarifies what “for each order of the time-derivative” means: the $h=-H$4-th derivative sign of $h=-H$5 controls the $h=-H$6-st derivative of $h=-H$7, with an explicit Gaussian lower bound.

3. Proven cases and proof technology

The full hierarchy remains conjectural, but several low-order and low-dimensional regimes are known. As summarized in (Wang, 2024), the entropy power conjecture and the McKean conjecture hold for any dimension $h=-H$8 when $h=-H$9 and $m\ge 1$0 is log-concave, citing Toscani. The McKean conjecture holds in dimension $m\ge 1$1 for $m\ge 1$2 and log-concave $m\ge 1$3, citing Zhang. The Gaussian completely monotone conjecture holds when $m\ge 1$4 and $m\ge 1$5, and also when $m\ge 1$6 and $m\ge 1$7, citing Cheng, Geng, and Guo.

A computational-algebraic line of work replaces ad hoc square completions by a systematic semidefinite-programming procedure. "Prove Costa's Entropy Power Inequality and High Order Inequality for Differential Entropy with Semidefinite Programming" (Guo et al., 2020) studies

$m\ge 1$8

with

$m\ge 1$9

The paper gives a new proof of Costa’s entropy power inequality, proves $(-1)^{m-1}\frac{d^m}{dt^m}N(t)\ge 0.$0 for $(-1)^{m-1}\frac{d^m}{dt^m}N(t)\ge 0.$1, and recovers $(-1)^{m-1}\frac{d^m}{dt^m}N(t)\ge 0.$2 and $(-1)^{m-1}\frac{d^m}{dt^m}N(t)\ge 0.$3.

Its method begins from the heat equation

$(-1)^{m-1}\frac{d^m}{dt^m}N(t)\ge 0.$4

and represents higher entropy derivatives as integrals of differential forms: $(-1)^{m-1}\frac{d^m}{dt^m}N(t)\ge 0.$5 One then generates integral-zero constraints from repeated integration by parts, divergence identities, and time differentiation of lower-order constraints. After introducing monomials of degree $(-1)^{m-1}\frac{d^m}{dt^m}N(t)\ge 0.$6 and total differential order $(-1)^{m-1}\frac{d^m}{dt^m}N(t)\ge 0.$7 as auxiliary variables, the target form is rewritten as a quadratic form. The remaining step is to show positivity modulo the constraints by a sum-of-squares certificate, equivalently by a positive-semidefinite matrix found through SDP.

The same paper also identifies explicit barriers. With the currently known constraints, its procedure does not prove $(-1)^{m-1}\frac{d^m}{dt^m}N(t)\ge 0.$8 or $(-1)^{m-1}\frac{d^m}{dt^m}N(t)\ge 0.$9. This does not disprove those inequalities; it shows that the present SOS/SDP architecture is incomplete for those cases. In the entropy-power landscape, that negative evidence is informative: low-order success does not automatically scale to a uniform all-orders proof.

4. Rényi and Tsallis extensions

A major generalization asks whether Costa-type concavity and complete monotonicity persist for Rényi entropy. In one dimension, "On the Complete Monotonicity of Rényi Entropy" (Wu et al., 2023) studies the heat flow

$N'(t)\ge 0$0

with Rényi entropy

$N'(t)\ge 0$1

and one-dimensional Rényi entropy power

$N'(t)\ge 0$2

The paper proves

$N'(t)\ge 0$3

$N'(t)\ge 0$4

and

$N'(t)\ge 0$5

where $N'(t)\ge 0$6 is the unique real root of $N'(t)\ge 0$7. It also proves Costa-type concavity for suitable powers: $N'(t)\ge 0$8 and

$N'(t)\ge 0$9

These are not full concavity results for $N''(t)\le 0$0 itself except at $N''(t)\le 0$1. The same paper proves exact complete monotonicity for Tsallis entropy of order $N''(t)\le 0$2,

$N''(t)\le 0$3

and shows that complete monotonicity fails uniformly over all Rényi orders by an explicit $N''(t)\le 0$4 counterexample.

A separate Rényi direction concerns convolution inequalities rather than heat-flow derivatives. "Rényi entropy power inequality and a reverse" (Li, 2017) emphasizes that for $N''(t)\le 0$5 the naive Shannon-form inequality

$N''(t)\le 0$6

fails in general, where in $N''(t)\le 0$7 dimensions

$N''(t)\le 0$8

For $N''(t)\le 0$9, the paper proves the powered inequality

$N^{(3)}(t)\ge 0$0

with

$N^{(3)}(t)\ge 0$1

improving the previously known exponent $N^{(3)}(t)\ge 0$2. It also studies a reverse Rényi EPI,

$N^{(3)}(t)\ge 0$3

and confirms conjectured cases at $N^{(3)}(t)\ge 0$4 and at $N^{(3)}(t)\ge 0$5 under log-concavity.

Taken together, these results suggest that Rényi analogues of the entropy power conjecture are intrinsically parameter-restricted. The Shannon case sits at a structurally privileged point: for $N^{(3)}(t)\ge 0$6, Costa concavity is exact, whereas outside that point the natural statements frequently involve modified exponents, partial derivative orders, or extra structural assumptions.

5. Discrete and quantum analogues

The discrete setting requires a reformulation of both scaling and entropy power. "Thinning, photonic beamsplitting, and a general discrete entropy power inequality" (Guha et al., 2016) argues that a natural unrestricted discrete analogue should be organized around the geometric distribution, which maximizes Shannon entropy on $N^{(3)}(t)\ge 0$7 at fixed mean. For a geometric law with mean $N^{(3)}(t)\ge 0$8,

$N^{(3)}(t)\ge 0$9

its entropy is

$m=2$0

The proposed discrete entropy power is

$m=2$1

Discrete scaling is modeled by thinning: $m=2$2 with $m=2$3 i.i.d. Bernoulli$m=2$4, so that $m=2$5. The paper then defines a beamsplitter-inspired scaled addition

$m=2$6

where the right-hand side transports continuous scaled addition through a transform $m=2$7 between discrete laws and circularly symmetric densities on $m=2$8. This operation reduces to thinning when one input is deterministic zero.

The conjectured discrete EPI is

$m=2$9

The full statement remains open, but the paper proves several supporting results. It proves the linear entropy inequality

$$y(t)=-\frac{2}{D}H(\mu_t), \qquad N(t)=e^{y(t)}=e^{\frac{2}{D}h(\mu_t)},$$00

for arbitrary independent discrete $$y(t)=-\frac{2}{D}H(\mu_t), \qquad N(t)=e^{y(t)}=e^{\frac{2}{D}h(\mu_t)},$$01, establishes a discrete CLT in which normalized $$y(t)=-\frac{2}{D}H(\mu_t), \qquad N(t)=e^{y(t)}=e^{\frac{2}{D}h(\mu_t)},$$02-sums converge to a geometric distribution, and obtains entropy monotonicity along powers-of-two subsequences. It also proves an alternative EPI under the normalization

$$y(t)=-\frac{2}{D}H(\mu_t), \qquad N(t)=e^{y(t)}=e^{\frac{2}{D}h(\mu_t)},$$03

while treating this normalization as secondary to $$y(t)=-\frac{2}{D}H(\mu_t), \qquad N(t)=e^{y(t)}=e^{\frac{2}{D}h(\mu_t)},$$04.

The same construction is explicitly linked to the Entropy Photon-number Inequality (EPnI). When the inputs are number-diagonal quantum states, the paper’s $$y(t)=-\frac{2}{D}H(\mu_t), \qquad N(t)=e^{y(t)}=e^{\frac{2}{D}h(\mu_t)},$$05 is the output photon-number law of a bosonic beamsplitter, and the discrete conjecture becomes a special case of EPnI. The discrete entropy-power program is therefore not merely combinatorial; it sits at the interface of classical discrete information theory and bosonic quantum channel theory.

6. Geometric analogies and structural limits

A longstanding geometric analogue of entropy-power concavity is the Costa–Cover conjecture. For a bounded measurable set $$y(t)=-\frac{2}{D}H(\mu_t), \qquad N(t)=e^{y(t)}=e^{\frac{2}{D}h(\mu_t)},$$06, define the outer parallel volume

$$y(t)=-\frac{2}{D}H(\mu_t), \qquad N(t)=e^{y(t)}=e^{\frac{2}{D}h(\mu_t)},$$07

The conjecture asks whether

$$y(t)=-\frac{2}{D}H(\mu_t), \qquad N(t)=e^{y(t)}=e^{\frac{2}{D}h(\mu_t)},$$08

is concave on $$y(t)=-\frac{2}{D}H(\mu_t), \qquad N(t)=e^{y(t)}=e^{\frac{2}{D}h(\mu_t)},$$09, mirroring the concavity of

$$y(t)=-\frac{2}{D}H(\mu_t), \qquad N(t)=e^{y(t)}=e^{\frac{2}{D}h(\mu_t)},$$10

in the Shannon setting. "On the analogue of the concavity of entropy power in the Brunn-Minkowski theory" (Fradelizi et al., 2013) shows that this analogy is only partially valid.

The paper proves that in dimension $$y(t)=-\frac{2}{D}H(\mu_t), \qquad N(t)=e^{y(t)}=e^{\frac{2}{D}h(\mu_t)},$$11, $$y(t)=-\frac{2}{D}H(\mu_t), \qquad N(t)=e^{y(t)}=e^{\frac{2}{D}h(\mu_t)},$$12 is concave on $$y(t)=-\frac{2}{D}H(\mu_t), \qquad N(t)=e^{y(t)}=e^{\frac{2}{D}h(\mu_t)},$$13. In dimension $$y(t)=-\frac{2}{D}H(\mu_t), \qquad N(t)=e^{y(t)}=e^{\frac{2}{D}h(\mu_t)},$$14, if $$y(t)=-\frac{2}{D}H(\mu_t), \qquad N(t)=e^{y(t)}=e^{\frac{2}{D}h(\mu_t)},$$15 is connected then $$y(t)=-\frac{2}{D}H(\mu_t), \qquad N(t)=e^{y(t)}=e^{\frac{2}{D}h(\mu_t)},$$16 is concave on $$y(t)=-\frac{2}{D}H(\mu_t), \qquad N(t)=e^{y(t)}=e^{\frac{2}{D}h(\mu_t)},$$17, but there exist non-connected planar sets for which concavity fails. In dimensions $$y(t)=-\frac{2}{D}H(\mu_t), \qquad N(t)=e^{y(t)}=e^{\frac{2}{D}h(\mu_t)},$$18, the conjecture fails even for connected sets: the paper exhibits connected examples with nonconcave $$y(t)=-\frac{2}{D}H(\mu_t), \qquad N(t)=e^{y(t)}=e^{\frac{2}{D}h(\mu_t)},$$19. At the same time, it proves eventual concavity for several classes. In particular, if $$y(t)=-\frac{2}{D}H(\mu_t), \qquad N(t)=e^{y(t)}=e^{\frac{2}{D}h(\mu_t)},$$20 is $$y(t)=-\frac{2}{D}H(\mu_t), \qquad N(t)=e^{y(t)}=e^{\frac{2}{D}h(\mu_t)},$$21 near $$y(t)=-\frac{2}{D}H(\mu_t), \qquad N(t)=e^{y(t)}=e^{\frac{2}{D}h(\mu_t)},$$22, then there exists $$y(t)=-\frac{2}{D}H(\mu_t), \qquad N(t)=e^{y(t)}=e^{\frac{2}{D}h(\mu_t)},$$23 such that

$$y(t)=-\frac{2}{D}H(\mu_t), \qquad N(t)=e^{y(t)}=e^{\frac{2}{D}h(\mu_t)},$$24

is concave on $$y(t)=-\frac{2}{D}H(\mu_t), \qquad N(t)=e^{y(t)}=e^{\frac{2}{D}h(\mu_t)},$$25.

The significance of these results is negative as well as positive. They show that the Brunn–Minkowski analogy reproduces the entropy-power picture exactly only in restricted regimes, such as convexity or low-dimensional connectedness. The entropy-power conjecture, by contrast, lives in the analytic regularity of Gaussian smoothing, where de Bruijn identities and Fisher-information inequalities impose a structure stronger than the geometry of arbitrary parallel sets. The geometric program therefore illuminates the conjecture by contrast: it reveals which parts of the entropy-power paradigm are universal and which depend specifically on heat-flow analyticity rather than on coarse convexity principles.