How much memory does it take to apply the same quantum channel many times, when each output must be released before the next input arrives and every retained qubit counts? One-use invariants do not decide it. Two explicit channels on dimension 1664 have identical normalized Choi spectra (32 eigenvalues 1/32), the same maximal complementary entropy 5, and zero asymptotic purity cost. At the same optimal exchange rate, one is implemented exactly with five memory qubits for any number of uses, while the other needs Omega(n^(1/(2 log_2(107)+2))) memory qubits for n uses, at every fixed error at most 1/16 and whatever purity is available; memory O(n^(1/3) log^(4/3) n) suffices for it. The hard channel is built from Houghton's group H_3. Behind the example is a general theory: the achievable exchange and purity rates, the purity cost kappa, which vanishes exactly on the closure of finite tracial factorizations, and memory bounds from how well a channel's environment is approximated by finite ones, and, for channels built from finite presentations, from the approximation profile of the group. Version 1.1 adds where the expensive channels lie: on the relative boundary of the closure; any fixed depolarizing admixture brings them down to polylogarithmic memory at the rank rate; on a face containing the Houghton channel a single moment of a trace on the group decides expense; and one free fermion per site already forces polynomial memory. Version 1.1.1 adds a map of what the main separation depends on (published inputs, companion results and new arguments), sets finite-budget and asymptotic-rate statements side by side, cites the companion papers at their published versions, pins the tested software versions, and makes every check refuse to run under optimized Python. The mathematical results are unchanged. Version 1.1.2 corrects attributions and makes two headline statements precise. The separation of Theorem A is stated at the exchange budget 5n/2, and the channels that need polynomial memory at every exchange rate are those with purity limited to O(log n). Credits added or corrected: Bisio, D'Ariano, Perinotti and Chiribella, and Bisio, D'Ariano, Perinotti and Sedlak; Boes et al. for the square-root dephasing construction; Lie, Son, Boes, Ng and Wilming for the exact case of the tracialization lemma; Faist, Berta and Brandao for the relation to the thermodynamic capacity; Haagerup, Musat and Rordam for the moment criteria; Thoma, Johnson, Watrous, Horodecki-Oppenheim-Winter and Lie-Jeong; Cornulier for the Baumslag-Solitar bound. Known results are moved out of numbered statements. The mathematical results and the numbering are unchanged. All operational bounds use complete-experiment error against adaptive observers with quantum references. The general rate-region achievability result uses a cited closed-device theorem; the explicit separation does not. This preprint archive includes the PDF, TeX source, finite verification scripts, and scoped Lean arithmetic proofs. The artifacts do not formally verify the entire manuscript; their scope and external dependencies are documented in verification/README.md.
How much memory does it take to apply the same quantum channel many times, when each output must be released before the next input arrives and every retained qubit counts? One-use invariants do not decide it. Two explicit channels on dimension 1664 have identical normalized Choi spectra (32 eigenvalues 1/32), the same maximal complementary entropy 5, and zero asymptotic purity cost. At the same optimal exchange rate, one is implemented exactly with five memory qubits for any number of uses, while the other needs Omega(n^(1/(2 log_2(107)+2))) memory qubits for n uses, at every fixed error at most 1/16 and whatever purity is available; memory O(n^(1/3) log^(4/3) n) suffices for it. The hard channel is built from Houghton's group H_3. Behind the example is a general theory: the achievable exchange and purity rates, the purity cost kappa, which vanishes exactly on the closure of finite tracial factorizations, and memory bounds from how well a channel's environment is approximated by finite ones, and, for channels built from finite presentations, from the approximation profile of the group. Version 1.1 adds where the expensive channels lie: on the relative boundary of the closure; any fixed depolarizing admixture brings them down to polylogarithmic memory at the rank rate; on a face containing the Houghton channel a single moment of a trace on the group decides expense; and one free fermion per site already forces polynomial memory. Version 1.1.1 adds a map of what the main separation depends on (published inputs, companion results and new arguments), sets finite-budget and asymptotic-rate statements side by side, cites the companion papers at their published versions, pins the tested software versions, and makes every check refuse to run under optimized Python. The mathematical results are unchanged. Version 1.1.2 corrects attributions and makes two headline statements precise. The separation of Theorem A is stated at the exchange budget 5n/2, and the channels that need polynomial memory at every exchange rate are those with purity limited to O(log n). Credits added or corrected: Bisio, D'Ariano, Perinotti and Chiribella, and Bisio, D'Ariano, Perinotti and Sedlak; Boes et al. for the square-root dephasing construction; Lie, Son, Boes, Ng and Wilming for the exact case of the tracialization lemma; Faist, Berta and Brandao for the relation to the thermodynamic capacity; Haagerup, Musat and Rordam for the moment criteria; Thoma, Johnson, Watrous, Horodecki-Oppenheim-Winter and Lie-Jeong; Cornulier for the Baumslag-Solitar bound. Known results are moved out of numbered statements. The mathematical results and the numbering are unchanged. All operational bounds use complete-experiment error against adaptive observers with quantum references. The general rate-region achievability result uses a cited closed-device theorem; the explicit separation does not. This preprint archive includes the PDF, TeX source, finite verification scripts, and scoped Lean arithmetic proofs. The artifacts do not formally verify the entire manuscript; their scope and external dependencies are documented in verification/README.md.
How much memory does a device need to apply one quantum channel n times, if each output must leave before the next input arrives? The paper counts the qubits the device keeps, the qubits it exchanges for fresh ones, and the purity it is given, with error measured on the whole experiment against an adaptive observer. Immediate release has a price. For the qutrit Werner-Holevo channel, vanishing error requires log2 3 bits of purity per use, a known rate; the paper adds a bound at finite error. If all outputs may leave together, O(log n) bits suffice in total at polynomially small error, refining a known zero rate. At error n^-2 the change comes when outputs wait in batches of about log n, and the zero supplied-purity-rate side holds for every unital channel. Purity separates cheap from expensive repeated use along one set of channels, the closure of those that a finite, maximally mixed bath implements exactly: outside it, linear purity is known to be needed at small error, hence linear memory when fresh qubits are maximally mixed. The paper proves a converse for memory, keeping maximally mixed fresh qubits and the optimal exchange rate: a channel in the closure with a flat probe (an input that leaves the minimal environment maximally mixed) can be served with sublinear memory at any error that is not exponentially small. For channels built from finitely many group relations, memory at fixed error is at most logarithmic, divergent but sublinear, or linear, and some are linear exactly when some finitely presented group is not hyperlinear. Along the way, Rybar and Ziman's question on repeating a channel without resets is answered. At high precision, the memory of n uses of such a channel equals, up to squared logarithms, the cost of the best single use that never leaves the channel's support. That cost is the bath entropy plus n times the entropy one use adds to the bath. For Houghton's group, devices acting through the group need and attain memory n^(1/3) up to logarithmic factors. Whether the single-use law holds at fixed error is open. This archive includes the paper, its LaTeX source and the finite numerical checks; they support but do not replace the proofs.
A quantum channel says what happens to each input, not when inputs arrive, when outputs leave for good, or what must be handed back. We show that for every finite schedule in which each carrier returns with its Hamiltonian and the controller comes back, the optimal work per use at vanishing error is the least entropy one implementation must add to its own record, plus the energy it hands over, against the most demanding observer. Landauer's bound, the thermodynamic capacity of Faist, Berta and Brandão in the time-covariant case, and the real-time costs of channels and processes are this law on different schedules. The cost survives arbitrarily slow operation: order, not speed, sets it. Releasing each output of a repeated memoryless channel before the next input arrives costs nothing extra for classical channels, a positive amount for partial reset at every temperature, and kBT ln 3 per use for the Werner–Holevo qutrit at zero Hamiltonian, which is essentially free in batch. For a classical task with zero Hamiltonians that releases a past and later answers a question about it, the real-time and held prices are logarithms of two fiber counts; for reversible questions they differ by the logarithm of the number of classes of pasts that later questions tell apart. With a returned clock, the batch cost of every channel is a single-letter quantity, equal to the capacity when the environment can carry the channel's energy change; otherwise the clock's spread must grow at least linearly with the number of uses. Freeness is decidable in batch for exact algebraic data and undecidable in real time, already in a fixed dimension. A two-qubit process that is free when held costs every early-release device at least 0.87 kBT ln 2 per episode at one percent error and budgets of 0.025 kBT ln 2; for a classical version we propose a three-arm feedback-trap protocol. This record archives the manuscript, its TeX source and its machine checks (Python, and Lean with Mathlib). The manuscript is the authority for exact statements and hypotheses; the verification notes list what is checked by machine and what is proved only in the text. How the work was done, including the use of AI models, is described in the manuscript. Most of the results first appeared in four papers, which this paper brings together: The price of time order (doi:10.5281/zenodo.23310622); Only quantum channels save work by waiting (doi:10.5281/zenodo.23310624; its title concerns repeated uses of one memoryless channel, and classical tasks with memory can pay for early release); The work cost of answering in real time (doi:10.5281/zenodo.23310628); The price of keeping time (doi:10.5281/zenodo.23310632). A short letter summarizes them (doi:10.5281/zenodo.23323808). Scoped licenses: the manuscript, PDF and TeX source are licensed under Creative Commons Attribution 4.0 International (CC-BY-4.0). The software and machine-readable verification artifacts are additionally available under the MIT License. See RIGHTS.md in the archive.
A device that applies a quantum channel over and over must release each output before the next input arrives. How much memory does it need? Suppose every fresh qubit the device receives is maximally mixed. In companion work we showed that the answer then turns on one set: the channels that a unitary with a finite, maximally mixed bath implements, exactly or in the limit. A channel in this set with a flat probe, an input that leaves its environment maximally mixed, can be served with sublinear memory. A channel outside the set needs linear memory at small error, however many qubits the device exchanges. No algorithm can decide on which side of this line a channel lies. A computable map sends each program to an explicit Schur channel with Gaussian-rational entries. If the program halts, logarithmic memory suffices; if it runs forever, linear memory is needed at an error computed from the program. The same line recasts Connes' embedding problem. It has a positive answer exactly when every factorizable channel with a flat probe can be served with sublinear memory; Schur channels already suffice. Just past the line, the cost switches on at the statistical scale. Take the qutrit Werner-Holevo family at distance h/sqrt(n) beyond its boundary point. Independent uses of the boundary channel imitate it with error tending to 2 Phi(h/(2 sigma)) - 1, where Phi is the standard normal distribution function and sigma = 5 sqrt(2)/27. We prove that no device drawing on purity that grows more slowly than sqrt(n) does better, whatever its memory. This archive includes the paper, its LaTeX source, an exact certificate and the finite numerical checks; they support but do not replace the proofs.
We characterize the bath resources needed to supply repeated uses of a fixed finite-dimensional quantum channel in a closed device. For each horizon T, one bath, one initial state and one repeated unitary are fixed before the user. Each output is returned before the next input arrives; no reset, discard, fresh ancilla or uncounted controller is available. Approximation error must vanish against arbitrary adaptive users with quantum memory and references. Writing r=limlog2(RT)/T for the bath dimension rate and s=limS(ωT)/T for the actual initial entropy rate, we prove that the achievable region is exactly s≥0, r+s≥h and r−s≥κ. Here h is maximum entropy exchange and κ is a smoothed independent-reference extension cost, with the zero-error limit taken before the supremum over full-rank inputs. Its exact fixed-input form is an affine transform of the zero-leakage quantum privacy funnel. The minimum dimension rate is (h+κ)/2. The proof combines entropy converses, a bath-dimension-independent support repair, and a closed adaptive implementation of encoder-only fully quantum Slepian--Wolf recycling. All seeds, clocks, workspace and retained residues are counted. Worked examples include dephasing, pure replacement and a qubit channel with 0<κ<h. No computability of κ or efficient circuit synthesis is claimed.
Faist, Berta and Brandão showed that n uses of a time-covariant quantum channel cost, to leading order, n times its thermodynamic capacity, when the device may collect all inputs before it releases any output. Many devices cannot wait: they must release each output before the next input arrives. This paper shows that, for devices built from energy-conserving dynamics and heat baths, this can cost extra work, but only for quantum channels, and determines exactly how much. Real-time work law. For a time-covariant channel the real-time work per use is a maximum over inputs of an entropy change plus an energy change, with one extra term: the least correlation an implementation must leave between each output and what stays in the device. Below this rate, with thermal hardware and deterministic work, the worst-case error of every real-time device tends to one. Only quantum channels pay. Classical channels, including full reset, pay no surcharge. Partial reset of a qubit pays one at every temperature, and the Werner–Holevo qutrit channel pays kT ln 3 per use, against essentially nothing in batch. What is free, and undecidability. In real time a channel costs nothing exactly when it is a limit of thermal operations, and no algorithm decides which channels these are, even in a fixed dimension. Clocks. If the battery holds no phase reference of its own, a clock that must come back exactly cannot run a channel that is not time-covariant; one that may come back slightly changed, with its actual mean energy and entropy refunded, runs every channel at the same rate, and for a gate the heat baths must reach energies that grow faster than the number of uses. All rates here are optimal asymptotic work rates at vanishing error over the complete interaction, with hardware allowed to grow; where a controller comes back only approximately, its actual mean energy and entropy are also refunded up to sublinear changes. This record archives the manuscript, its TeX source and its machine checks (Python, and Lean with Mathlib). The manuscript is the authority for exact statements and hypotheses; the verification notes list what is checked by machine and what is proved only in the text. How the work was done, including the use of AI models, is described in the manuscript. This paper is one of four on the thermodynamics of time order: The work cost of answering in real time (https://github.com/Apsiape/answering-in-real-time); The price of keeping time (https://github.com/Apsiape/price-of-keeping-time); The price of time order (https://github.com/Apsiape/price-of-time-order). Scoped licenses: the manuscript, PDF and TeX source are licensed under Creative Commons Attribution 4.0 International (CC-BY-4.0). The software and machine-readable verification artifacts are additionally available under the MIT License. See RIGHTS.md in the archive. The aggregate Other (Open) category represents this mixed-license package.
A quantum channel can always be implemented by a unitary interaction with an environment that starts in a pure state. With a finite, maximally mixed environment, the implementable channels are the noisy operations. A tracial von Neumann algebra as environment gives the factorizable channels, and by a theorem of Haagerup and Musat combined with MIP* = RE, some of these lie outside the closure of the noisy operations. Version 1.0 of this record wrote down explicit rational channels of this kind, with no value for their distance from the closure. This version gives explicit lower bounds. Using Wang and Zhi's dimension-uniform reflection estimate, it constructs a factorizable unital channel on twelve qubits with Kraus entries in {0, 1, ±3/5, 4/5}, whose normalized Choi and diamond distances from the closure are at least 2^-(2J+50), with J = 2^(2^1122000). The bound comes from a compiler theorem: group relations that are rigid uniformly in the matrix dimension give a rational channel with an explicit distance bound. For the channels of version 1.0, a new spectral certificate for the root groups of their presentation gives explicit gaps: 2^-(2J+41) for the fifteen-qubit channel and its two variants and, by the bound of version 1.0, 2^-(2J+36) in Choi distance and 2^-(2J+21) in diamond distance for the fully referenced channel. The numbers are tiny because Wang and Zhi's tolerance is 2^-J. The results depend on recent preprints of Wang and Zhi, Thom, and Alekseev, Liu and Thom, which are not yet refereed. This archive includes the paper, the channel data, the certificates, and programs that rebuild and check them; python verify_all.py ends with OK ALL.