Landauer's principle

In plain terms

The story starts with a demon. Maxwell imagined a tiny being that sorts fast molecules from slow ones, apparently creating a temperature difference for nothing and so breaking the second law of thermodynamics, the law that says the disorder of an isolated system never decreases. For a century the demon dodged every exorcism, as each proposed fix turned out to have a loophole.

Landauer found the real answer, and it's about memory. The demon must record what it learns, its memory is finite, and so eventually it must erase, and erasing is the step that can't be free. Resetting a memory to a blank state compresses many possible configurations into one, and that compression must be paid for in heat released to the surroundings.

The price is universal. It depends on the temperature and on nothing else, not the technology and not the ingenuity of the engineering. Forgetting one bit of information costs, at minimum, a tiny but fixed quantity of heat, and with that payment the demon's books balance exactly.

In CSD

Thermodynamics isn't bolted onto this programme, as it grows from the same soil as the Born rule. The motion of the substrate preserves volume, so at the finest grain nothing is ever lost and entropy, the measure of missing information, is conserved. Irreversibility appears only when a record is coarse-grained, which means many detailed configurations are lumped together under one description. This is Boltzmann's picture, with reversible microscopic dynamics and irreversibility living in the description rather than in the dynamics.

Given that setup, an erasure bound isn't an import from another field. It's what the record layer, the part of CSD that describes how outcomes are recorded, owes and pays. The corpus derives temperature and free energy first, from a variational principle, a rule that singles out a state by making some quantity as small or as large as it can be, and then proves the Landauer bound in those earned terms. It's the tightest link the development has between information and physical cost.

The bound is also stated on the constraint surface. Take a preparation of a system together with a bath, in product form, whose bath component has a Gibbs state as its density operator, and let a volume-preserving flow lifting a unitary act on the pair. The entropy the system loses is then bounded by the heat the bath receives, with the same constant as the matrix-level theorem. Nothing about the flow is posited beyond its lifting the joint unitary, so the thermodynamic cost of erasure is read off the substrate's own dynamics.

Mathematically

The entropy removed from a system is bounded by beta times the heat delivered to the bath. The proof runs in three steps, as unitary evolution on an initially product state conserves total entropy, subadditivity bounds the system's entropy drop by the bath's entropy rise, and the bath's Clausius inequality converts entropy rise into heat.

The statement is in the Reeb-Wolf form, which forces the assumptions into the open. The bath starts in a Gibbs state, everything is finite-dimensional, the initial system state is positive-definite, and both final marginals are full-rank. The last two aren't decoration, as they're what the subadditivity step consumes, and that step is proved only at positive-definite marginal scope. Naming them is the point, because several informal derivations of the principle are known to smuggle assumptions at exactly the step this version makes visible.

landauer_flow (Thermo/SigmaSecondLaw.lean) is landauer_bound applied to the barycentres of a product preparation flowed by a lift of the joint unitary, with the bath preparation's barycentre equal to the Gibbs state, and the full-rank hypotheses of the matrix theorem are inherited and not removed.

The name

Rolf Landauer (1927-1999) was born in Stuttgart, and his family fled to New York in 1938 after his father's death. He took his degrees at Harvard and joined IBM in 1952, spending his whole career at the Watson Research Center in Yorktown Heights, which was corporate research at its mid-century best, where a physicist could spend decades on the question of why computing costs energy.

The principle appeared in 1961 in the IBM Journal of Research and Development, a house journal, under the resolutely unglamorous title of a paper about heat generation. His slogan, that information is physical, was a deliberate provocation. His colleague Charles Bennett drew the astonishing corollary a decade later, that computation itself can be free and only forgetting costs. Experimental confirmation came in 2012, thirteen years after Landauer died.

Related entries

• von Neumann entropyA quantum whole can be perfectly known while its parts are as uncertain as they could possibly be, and entropy is where that stops being a metaphor and becomes arithmetic.
• Entropy subadditivityThe information missing from a whole never exceeds the sum of what's missing from its parts, and that's von Neumann's subadditivity inequality, proved at the matrix level with the operator-logarithm step informal proofs usually wave through.
• Gibbs stateThe state a system settles into at a given temperature is the unique solution to a trade-off between energy and disorder.
• Klein's inequalityOne quantum state is never negatively distinguishable from another, as the relative entropy of a state with respect to a second is at least zero, and that single inequality drives the second law, subadditivity, concavity and the Holevo bound in the corpus.
• De-isolationNothing collapses, as a measured system stops being isolated, and measurement is what de-isolation dynamics looks like from inside.
• Jarzynski equalityRush a system between two equilibrium conditions as violently as you like, and the average of a particular exponential of the work you did still returns the equilibrium free energy difference exactly. Irreversible experiments measure a reversible quantity.