STEH Trade-Off Theorem: Hard Physical Limits for Space, Time, Energy, and Entropy in Computation

1. Introduction

Computation is a physical process. Three mature pillars constrain its cost: (i) Landauer’s principle lower-bounds dissipated heat for logically irreversible steps; (ii) quantum speed limits (QSL) lower-bound the time to traverse distinguishable quantum states given available energy; and (iii) the Bekenstein bound upper-bounds information capacity for finite-energy systems in bounded regions. We show how to combine these, with no additional heuristic approximations, into two explicit inequalities that jointly forbid the simultaneous minimization of space, time, exported entropy, and energy for nontrivial computations.

A central contribution is the introduction of three well-defined complexity parameters for a function $f:{\{0,1\}}^n\to {\{0,1\}}^m$ at target error $\epsilon$: (1) the reversible working-set $W^{*}(f,\epsilon )$, (2) the forced erasures under S space $G^{*}(f,S,\epsilon )$, and (3) the orthogonalization depth $D^{*}(f,\epsilon )$. We formalize these in Section 2, relate them to classical/quantum complexity notions, and prove the lemmas needed to derive the main result.

Physical regime and simultaneous applicability

Our derivations hold for devices satisfying:

(R1)

Thermal bath: Coupling to an environment at temperature T (near equilibrium) so that Landauer applies to logically irreversible operations [1,5,6,7].

(R2)

Quantum dynamics: The device evolution over runtime $\tau$ can be modeled by a (possibly open-system) CPTP map with well-defined average energy above ground $E_{\mathrm{tot}}$; we use QSLs valid for closed or Markovian/non-Markovian dynamics [8,9,10,11,12].

(R3)

Bounded geometry and weak gravity: Degrees of freedom are essentially confined to a ball of radius R, with total energy well below the gravitational-collapse threshold ($R\gg 2G{E}_{\mathrm{tot}}/c^4$), so Bekenstein’s bound controls information capacity [13,14,15].

These conditions are commonly met in laboratory and technological settings; therefore the three primitive bounds are consistent and simultaneously applicable.

2. Definitions of the Complexity Parameters

Let $f:{\{0,1\}}^n\to {\{0,1\}}^m$ and target worst-case error $\epsilon \in [0,1/2)$. Computations are implemented by families of physical processes indexed by input $x\in {\{0,1\}}^n$ acting on a working medium; mathematically, by CPTP maps ${\mathcal{E}}_t$ over $t\in [0,\tau ]$ whose induced input-output relation approximates f.

Definition 1

(Reversible working-set $W^{*}(f,\epsilon )$). Consider all logically reversible implementations of f with error $\le \epsilon$, i.e., injective encodings $x\mapsto (x,a_0)\mapsto (x,a_T)$ and a final clean-up that restores ancillae while outputting $f\left(x\right)$ (Bennett’s method). For each such implementation, let $S_{max}$ be the peak number of information-bearing bits (IBBs) that are simultaneously logically live (i.e., their value is needed by some future reversible step). Then

$$W^{*}(f,\epsilon )=\underset{\mathrm{reversible}\mathrm{implementations}\mathrm{of}f}{inf}{S}_{max}.$$

Remarks. 

This is the reversible pebbling number of a computation DAG for f [3,20,21]. It is finite for all computable f; constructive reversible simulations ensure existence.

Definition 2

(Forced erasures under space S: $G^{*}(f,S,\epsilon )$). Fix a space budget S (IBBs). Over all (reversible or irreversible) implementations of f with error $\le \epsilon$ that never exceed S live IBBs, let G be the total number of logically irreversible bit erasures performed. Then

$$G^{*}(f,S,\epsilon )=\underset{\mathrm{implementations}\mathrm{within}\mathrm{space}S}{inf}G.$$

Definition 3

(Orthogonalization depth $D^{*}(f,\epsilon )$). Let ${\rho }_t\left(x\right)$ be the device state (possibly mixed) at time t given input x. For a pair of inputs $x,y$ with $f\left(x\right)\ne f\left(y\right)$, define the Bures angle (quantum Fubini–Study distance for mixed states)

$$\Theta (\rho ,\sigma )=arccos\sqrt{F(\rho ,\sigma )}\in [0,\pi /2],$$

where F is the Uhlmann fidelity. Let $\mathcal{L}(x,y)$ be the minimal geodesic length (in Bures angle) a valid implementation must traverse between ${\rho }_0\left(x\right)$ and some ${\rho }_{\tau }\left(x\right)$ that produces an ε-accurate output and likewise for y. Then define

$$D^{*}(f,\epsilon )=\underset{x,y:f\left(x\right)\ne f\left(y\right)}{sup}⌈{\displaystyle \frac{\mathcal{L}(x,y)}{\pi /2}}⌉.$$

Remarks. 

$\Theta =\pi /2$ corresponds to orthogonality; thus $D^{*}$ counts, in units of “orthogonalization-length,” how many decisively distinguishable transitions are unavoidably required for worst-case inputs. In circuit models, $D^{*}$ lower-bounds depth; in decision trees, $D^{*}$ lower-bounds the number of decisive queries; in quantum models, $D^{*}$ relates to adversary/Hybrid-argument lower bounds [22,23]. Well-definedness follows since $\mathcal{L}$ is finite for CPTP paths and the supremum is over a finite domain.

Relationships and computability (high level)

• $W^{*}$ equals (up to constant factors) the reversible pebbling number of a suitable computation DAG for f; lower bounds are known for FFT, GEMM, sorting, Krylov, etc. [16,17,18,19].
• $G^{*}(f,S,\epsilon )\ge {[W^{*}(f,\epsilon )-S]}_{+}$ (Lemma 1).
• $D^{*}(f,\epsilon )\ge$ classical decision-tree depth for f; for quantum, $D^{*}$ is lower-bounded by adversary methods [22].

3. Key Lemmas

Lemma 1

(Space-deficit forces erasure). For any f and ε, and space budget S, one has

$$G^{*}(f,S,\epsilon )\ge {[W^{*}(f,\epsilon )-S]}_{+}.$$

Proof. 

Consider a reversible implementation achieving $W^{*}$. If $S\ge W^{*}$, history can be retained and uncomputed with $G=0$. If $S<W^{*}$, at some peak step at least $W^{*}-S$ logically live bits must be discarded to remain within budget S. Any such discard is a logically irreversible many-to-one map on IBBs, contributing at least one bit erasure. Taking the infimum over all implementations yields the bound. This is the standard reversible pebbling trade-off [3,20,21]. □

Lemma 2

(Landauer dissipation). At temperature T, any logically irreversible bit erasure exports at least $k_{B}Tln2$ heat to the environment. Therefore,

$$Q\ge k_{B}Tln2G^{*}(f,S,\epsilon ).$$

Lemma 3

(Quantum speed limit as step-count bound). Let $E_{\mathrm{tot}}$ be the average available energy above ground over runtime τ. For any implementation of f with worst-case orthogonalization-depth $D^{*}(f,\epsilon )$,

$$\tau \ge {\displaystyle \frac{\pi \hslash }{2}}{\displaystyle \frac{D^{*}(f,\epsilon )}{E_{\mathrm{tot}}}}.$$

Proof. 

For closed systems, concatenate $D^{*}$ minimal orthogonalizations; each requires time at least $\pi \hslash /\left(2E\right)$ by Margolus–Levitin; additivity yields the bound. For open systems, use mixed-state QSLs (e.g., Deffner–Lutz; Taddei et al.; del Campo et al.) on the Bures angle, noting that orthogonalization corresponds to angle $\pi /2$; summing $D^{*}$ segments gives the same scaling (constants may vary but only by universal factors) [10,11,12]. □

Lemma 4

(Bekenstein information-capacity). For degrees of freedom essentially confined in radius R with total energy $E_{\mathrm{tot}}$,

$$S\le {\displaystyle \frac{2\pi E_{\mathrm{tot}}R}{\hslash cln2}}.$$

Equivalently,

$$E_{\mathrm{tot}}\ge {\displaystyle \frac{\hslash cln2}{2\pi R}}S.$$

4. Deriving the Combined STEH Inequalities

We now show the algebraic steps explicitly.

Step 1: Eliminate $E_{\mathrm{tot}}$ from the QSL

Start from Lemma 3:

$$\tau \ge {\displaystyle \frac{\pi \hslash }{2}}{\displaystyle \frac{D^{*}(f,\epsilon )}{E_{\mathrm{tot}}}}.$$

(1)

Apply Lemma 4 (lower bound on $E_{\mathrm{tot}}$ given S and R):

$$E_{\mathrm{tot}}\ge {\displaystyle \frac{\hslash cln2}{2\pi R}}S.$$

(2)

Substitute (2) into (1):

$$\begin{array}{cc}\hfill \tau & \ge {\displaystyle \frac{\pi \hslash }{2}}{D}^{*}(f,\epsilon )/\left({\displaystyle \frac{\hslash cln2}{2\pi R}}S\right)\hfill \\ \hfill & ={\displaystyle \frac{\pi \hslash }{2}}{D}^{*}(f,\epsilon )·{\displaystyle \frac{2\pi R}{\hslash cln2}}·{\displaystyle \frac{1}{S}}\hfill \\ \hfill & =\begin{array}{|c|}\hline {\displaystyle \frac{{\pi }^2}{cln2}}{\displaystyle \frac{R{D}^{*}(f,\epsilon )}{S}}\\ \hline\end{array}.\hfill \end{array}$$

(3)

Step 2: Product bound for $\tau Q$

From Lemma 2,

$$Q\ge k_{B}Tln2G^{*}(f,S,\epsilon ).$$

(4)

Multiply (3) and (4):

$$\begin{array}{cc}\hfill \tau Q& \ge \left({\displaystyle \frac{{\pi }^2}{cln2}}{\displaystyle \frac{R{D}^{*}}{S}}\right)·\left(k_{B}Tln2G^{*}\right)\hfill \\ \hfill & =\begin{array}{|c|}\hline {\displaystyle \frac{{\pi }^2{k}_{B}T}{c}}R{D}^{*}(f,\epsilon ){\displaystyle \frac{G^{*}(f,S,\epsilon )}{S}}\\ \hline\end{array}.\hfill \end{array}$$

(5)

This completes the explicit derivation of the two combined STEH inequalities. □

5. Examples and Sanity Checks

AND of n bits.

Reversible working set $W^{*}\approx n$; orthogonalization depth $D^{*}\ge n$. For $S<n$, $G^{*}\ge n-S$ (Lemma 1) hence $Q\ge k_{B}Tln2(n-S)$ and $\tau \ge {\displaystyle \frac{{\pi }^2}{cln2}}{\displaystyle \frac{Rn}{S}}$.

GEMM ($n\times n$).

Known I/O lower bounds imply $W^{*}\gtrsim \Theta \left(n^2\right)$ live tiles; the arithmetic workload gives $D^{*}\gtrsim \Theta \left(n^3\right)$ decisive updates [16,17,18,19]. Therefore $\tau \gtrsim {\displaystyle \frac{{\pi }^2}{cln2}}{\displaystyle \frac{R{n}^3}{S}}$, matching communication-optimal scaling (up to constants and logs).

FFT (N).

$W^{*}\gtrsim \Theta \left(N\right)$ and $D^{*}\gtrsim \Theta (NlogN)$ by circuit-depth arguments; thus $\tau \gtrsim {\displaystyle \frac{{\pi }^2}{cln2}}{\displaystyle \frac{RNlogN}{S}}$.

6. Assumptions, Scope, and Limitations

Simultaneous applicability. Conditions (R1)–(R3) delineate when Landauer, QSL, and Bekenstein can be applied together: (i) near-equilibrium thermalization for irreversible steps; (ii) CPTP dynamics with well-defined average energy; (iii) weak gravity and effective confinement within radius R. These cover standard digital/quantum platforms (CMOS, superconducting/ion, nanomechanical).

Model idealizations. The constants in QSLs vary among formulations (Margolus–Levitin vs. Mandelstam–Tamm vs. mixed-state extensions); our constants are chosen conservatively, and any version yields the same scaling and the same algebraic combination. Bekenstein’s bound has modern reformulations via relative entropy; any such form provides an $E_{\mathrm{tot}}$–S trade-off suitable for our derivation [15]. Landauer has repeated experimental support [6,7].

What is not claimed. We do not claim tightness of constants for specific hardware, nor do we assume a specific gate library. The theorem is agnostic to architecture and applies to any implementation achieving the input-output relation.

7. Conclusions

We provided precise definitions for the complexity parameters $W^{*},G^{*},D^{*}$, proved the lemmas linking them to Landauer, QSL, and Bekenstein, and derived combined STEH inequalities (3)–(5) step-by-step. The result formalizes the impossibility of driving all of (space, time, exported entropy, energy) to zero in nontrivial computation, and it recovers communication lower bounds for fundamental tasks. The STEH framework offers a quantitative checklist for evaluating algorithms against absolute physical limits.