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No Universal Process Currying: The Intercept Principle, Environmental Currying, and Finite-Dimensional Obstructions to Exact Process Storage, Evaluation, and Recovery

Amin Abaee

Published Aug 4, 2026• 0 citations
Publisher

Abstract

We ask whether coupling a fixed finite-dimensional environment to a quantum process can be undone functorially: does tensoring with an environment E admit an adjoint in the category of quantum channels, and in related categories of instruments and stochastic maps? We prove that it does not, unless the environment is trivial. Our central new tool is an intercept principle: an enriched adjunction must preserve the affine dimension of hom-sets coefficient by coefficient, and probability normalization contributes a dimension-counting intercept that no environment-independent representing object can absorb. Specialized to the environment endofunctor, this gives a single coefficient-rigidity obstruction that governs channels, ordered instruments, finite stochastic maps, and higher-order superchannels uniformly, and yields an exact classification in each case: a left or right adjoint exists precisely when the environment is isomorphic to the tensor unit. We introduce an operational counterpart of this categorical obstruction. No finite-dimensional quantum memory with a fixed processor can store and exactly evaluate all channels on a nontrivial system, and any natural finite-memory encoding of processes would induce the adjoint just shown to be impossible. For approximate compression into insufficient memory we prove a new quantitative floor on the achievable diamond-norm error, sharp for the argument used and extended from linear to arbitrary continuous encoders by a Borsuk–Ulam argument. Finally, we place these obstructions against the standard theory of exact recovery: a channel is Petz-recoverable exactly when its complementary channel is constant, a morphism-level splitting that we show is strictly weaker than the functorial currying structure ruled out above. We delimit what these results do and do not establish, and accompany the mathematical claims with a machine-checkable verification suite.

DOI
10.5281/zenodo.21789865

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A finite-dimensional physical processor cannot exactly program a continuous family of distinct unitary channels. We show that this obstruction becomes quantitative when the target channel is stored in a normalized Choi state and its output observables are reconstructed by sampling physical channels and classically post-processing their measurement outcomes. For arbitrary ddd-dimensional channels, we construct a target-independent exact reconstruction protocol and prove the optimal one-copy sampling overhead, which grows quadratically with system dimension. We further prove the sharp fixed-ddd law that the excess overhead vanishes inversely with the number of identical Choi programs. The upper bound combines deterministic port-based teleportation with a quasi-decomposition that corrects its depolarizing distortion. The converse maps any low-overhead reconstruction protocol to a physical learner of unknown unitaries and uses local quantum estimation to recover the same leading coefficient. These results recast the universal no-programming obstruction as a quantitative trade-off between quantum program memory and classical sampling, with a leading cost that reflects the locally learnable unitary degrees of freedom.
We study the task of lifting arbitrary quantum states and channels to purifications and Stinespring dilations, respectively, in both the probabilistic exact and deterministic approximate settings. We formalize this task through a general framework of quantum purification machines that, given a finite number of copies or uses of a black-box input, aim to output a corresponding purification or Stinespring dilation. In the probabilistic exact setting, we show that universality is not necessary to rule out such transformations: the simple requirement that a machine purifies two inputs of different rank with non-zero probability already implies that it cannot be described by a linear positive map. This simple argument captures a fundamental obstruction of quantum theory and recovers the impossibility of universal probabilistic purification from finitely many copies. In the approximate setting, we allow for general machines that are not required, in general, to produce a pure output. Using the minimum average error as our figure of merit, we derive analytical expressions for the performance of several physically motivated strategies as well as a general upper bound on the achievable error, which is tight in a specific regime. Our analysis reveals a trade-off: strategies that produce a pure output - among which we prove the optimal to be a strategy that produces as a fixed output a maximally entangled purification of the fully depolarizing channel - perform optimally between those considered for large environment dimension, while append-environment strategies that generally produce non-pure outputs perform better at small environment dimension.
Can an external environment be absorbed universally into a quantum process? This article identifies trace normalization as the obstruction and tracks it through higher orders. Exact ranks are derived for sequential-comb normalization at every order. Two concrete families of higher-order maps are defined, and their transformation-space dimensions are computed. In the family stable under arbitrary auxiliary combs, adding an environment to an input has neither a left nor a right adjoint from order three onward. Together with the channel and superchannel cases, this yields an all-order family of no-go theorems. The claim is restricted to specified process types: broader normalized type systems may still admit internal homs. Without normalization, a two-sided adjunction returns for normal completely positive maps. Exactly the finite-dimensional Hilbert spaces are dualizable, but trace scaling and rank-one evaluation prevent restriction to deterministic channels. The continuous scalar laws underlying composition- and tensor-additive dynamics are proved to be logarithmic boundary terms. Nonadditive quadratic actions are analyzed through exact stationarity equations and several exact qubit phase classifications. Finally, continuous channel encoding below the full channel-space dimension has worst-case diamond-norm error at least 1/(dAdB ), while normalized weighted cups have sharp defect 1 − 1/δ2. The results identify three ways around the obstruction: enlarge the type universe, remove normalization, or accept controlled approximation
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