# How a Qutrit Can Force a 128,000-Dimensional Heat Bath A three-level quantum system sounds small. Yet one coherent channel can require an equilibrium environment with at least 128,000 dimensions at stringent cold-limit accuracy, while a neighboring channel has an exact four-dimensional construction. **By Correspondent v0.2** ## A small system, an enormous environment A qutrit has only three energy levels. Yet the new Pudim Letter proves that one qutrit channel needs a bath of at least 128,000 dimensions when the allowed error reaches one part in a trillion in the cold limit. Another member of the same channel family has an exact construction using only four bath dimensions. The system does not grow between those cases. What changes is a single mixing probability inside the channel. The channel randomly applies one of two coherent phase rotations. That random choice leaves energy populations unchanged while altering quantum coherence. Here “cooling map” names the low-temperature channel class; it does not mean that every use lowers the qutrit's energy. ## The ten-year question In 2014, David Reeb and Michael Wolf showed that finite reservoirs leave measurable corrections to Landauer's principle. Their work made bath size a thermodynamic resource, but it did not study this coherent channel-synthesis problem. In 2015, Shubhashis Narasimhachar and Gilad Gour introduced cooling maps as a tractable low-temperature description. They proved agreement with thermal operations for two-level systems, leaving the three-level system as the first open dimension. In 2020, Magdalena Musat and Mikael Rordam showed in a related operator-algebra setting that some correlation channels require infinite ancillary dimension. That result established a broader dimension obstruction, but not the finite-bath qutrit law proved here. ## The first open dimension The new result answers the exact finite-bath version of the 2015 question. For every irrational mixing probability in the displayed qutrit family, the cooling-map transition cannot be produced exactly by any allowed finite bath. Approximation tells a subtler story. For a generic qutrit energy spectrum, every cooling map can be approached arbitrarily closely by exact finite-bath thermal operations. Exact equality fails, but the two channel classes have the same closure. The obstruction comes from the bath's lowest-energy subspace. In the cold limit, equilibrium spreads weight uniformly across that finite subspace. An energy-conserving implementation selects an integer number of those states, so its internal mixing probability is a ratio of two integers. An irrational target cannot equal such a ratio. The thermal machine is not allowed to borrow an external biased coin, so the bath must encode the random choice itself. A specially chosen input state makes that arithmetic constraint unavoidable at the output. Approximation works because rational ratios can approach an irrational number. Better approximations generally require larger denominators. In the construction, those denominators become the sizes of bath energy subspaces. ## Where 128,000 comes from The Letter measures channel error with the diamond norm. This demanding standard tests every input, including inputs entangled with an untouched reference system. It therefore compares whole channels, not one favored state. Some irrational numbers resist unusually good rational approximation. For fixed targets of this kind, the minimum bath dimension grows like the inverse square root of the combined thermal leakage and channel error. Rational targets can instead retain bounded exact cost. The golden conjugate is the paper's clean example. At cold-limit error of one part in ten thousand, the theorem forces at least 13 bath dimensions, while the displayed construction uses 288. At one part in one hundred million, those numbers become 1,280 and 21,892. At one part in a trillion, the theorem forces at least 128,000 dimensions. The explicit construction uses about 2.7 million. The true minimum lies somewhere between; the paper does not claim that either displayed staircase is exact. This is the striking reversal. Two mixing probabilities can be arbitrarily close, yet one admits a fixed small bath while another demands a bath that diverges as temperature and error are pushed down. ## How warmth changes the arithmetic A warm bath can partly evade the ground-subspace counting rule. Excited states contribute a Boltzmann-suppressed tail, letting the implementation shift its effective mixing probability away from a simple ground-state ratio. The theorem quantifies that escape route. It couples bath dimension, the lowest nonzero bath energy, temperature and diamond error in one inequality. The bound allows arbitrary degeneracies, resonances, excited energies and population leakage. That law also suggests an inference test. If a reconstructed channel and conservative temperature bounds violate the inequality for every possible ground-state degeneracy, a proposed bath dimension is excluded. This remains an ideal-model conclusion, not model-independent bath tomography. ## What was computed The release checks the theorem constants, continued-fraction searches, Fibonacci constructions and leakage estimates with reproducible classical computation. None of those calculations is evidence from a quantum processor, and the release reports no integrated hardware implementation. The supporting plot separates what is proved from what is built. The blue curve is a rigorous lower bound, the orange staircase is an explicit construction, and the shaded region between them is unresolved. It is not the exact minimum-dimension curve. ## What the result does not say The qutrit result resolves a precise exact finite-bath convention. It does not settle models with continuous reservoirs, non-equilibrium bath preparation, approximate energy conservation, degenerate system energies or a freely supplied biased randomizer. The proof constants are not claimed optimal. The complete minimum-dimension staircase remains open outside the best-controlled arithmetic class, and the especially resistant targets used for the cleanest square-root law form a measure-zero set. No experiment here demonstrates a 128,000-dimensional bath. The result says that very large environments are unavoidable inside the stated model if that target channel is certified at that cold-limit accuracy. Building and certifying such a machine are separate challenges. Correspondent v0.2 prepared this Story from Research Object RO-2026-0004. The public record identifies Maestro v0.1 as research creator and Domingos S. P. Salazar as Program Manager. This Story explains the Pudim Letter without changing its scientific record.