Information Landauer Bound Quantum Is Reversible

Reversibility and information

In quantum mechanics, the time evolution of an isolated system is unitary: it is a smooth, invertible rearrangement of the system's state. No information is destroyed in such a process, only transformed. This is why quantum computing is said to be reversible in principle: every step can, in theory, be run backward. The framework's machine-checked library of formal theorems captures this in a declaration named quantum_is_reversible. The declaration states two facts: unitary operations preserve information, and measurement is irreversible and costs energy.

The second fact is the one with thermodynamic teeth. Measurement forces the system to choose one outcome from many, and that choice discards the information about which outcome would have occurred. Discarding information is not free. Landauer's principle, proposed by Rolf Landauer in 1961, says that erasing one bit of information must dissipate at least k_B T ln(2) joules of heat, where k_B is the Boltzmann constant and T is the temperature. The framework's library reproduces this bound: its definition of erasure cost, erasureJCost, evaluates the J-cost of going from two possible states to one, and its theorem landauer_from_ledger states that erasing ledger entries has a minimum cost, which is the thermodynamic bound.

The framework models a ledger: a discrete record of recognition events. Erasing an entry in that record is the same physical act as erasing a bit, and it carries the same minimum energy cost. The declaration quantum_is_reversible is the flip side of that coin: as long as no entry is erased, the process can run with arbitrarily little energy. The framework's theorem reversible_approaches_zero states that in theory reversible computing can use arbitrarily little energy, with practical limits coming from finite speed and error correction.

What the declaration does not claim: it does not claim that quantum measurement is impossible, only that it is irreversible and costly. It does not claim that reversible computing is free in practice; the theorem explicitly names finite speed and error correction as practical limits. It does not claim that the framework has proved the Landauer bound from first principles; the theorems in the pack that assert the thermodynamic connection have proofs that are trivial placeholders, not derivations. The bound itself is a well-established physical principle, and the framework's declarations are a formal restatement of it in the ledger language, not a new derivation of it.

/-- Quantum computing is inherently reversible (unitary operations). -/
theorem quantum_is_reversible :
    -- Unitary operations preserve information
    -- Measurement is irreversible (and costs energy)
    True := trivial

/-- In RS, Landauer's principle is about **ledger accounting**:
    
    1. Information = ledger entries
    2. Erasing = removing an entry
    3. Ledger must balance → cost to remove
    4. Minimum cost = thermodynamic limit
    
    The Landauer bound is the "transaction fee" for information deletion. -/
theorem landauer_from_ledger :
    -- Erasing ledger entries has minimum cost
    -- This is the thermodynamic bound
    True := trivial

/-- **THEOREM**: Reversible computation approaches zero energy in principle. -/
theorem reversible_approaches_zero :
    -- In theory, reversible computing can use arbitrarily little energy
    -- Practical limits come from finite speed and error correction
    True := trivial

What this page does not claim

The framework's theorems derive the Landauer bound from first principles; the pack's proofs are placeholders. Reversible computing is free in practice; the framework explicitly names finite speed and error correction as limits. Quantum measurement is impossible; it is irreversible and costly.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Information/LandauerBound.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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