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 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 physical device acts in time: inputs arrive, outputs are released and can no longer be touched, some systems are kept, and some must come back. This paper shows that the optimal work rate of such a device, over many uses at vanishing error, is fixed by one quantity, the least entropy that one implementation must add to its own record, plus the energy it hands over to the carriers, against the most demanding observer its schedule allows. One law for every schedule in which each borrowed system comes back with the Hamiltonian it arrived with: Landauer's bound, the thermodynamic capacity of time-covariant channels, the real-time cost of channels, interactive processes and the batch cost of a channel whose clock must be returned are special cases. Order, not speed. The cost survives arbitrarily slow operation; an operation can cost more merely because some of its outputs must leave sooner. Which carriers to hold. For linear tasks the price of a schedule is a rank, and deciding whether holding a given number of carriers lowers it at all is NP-complete. A witness. A two-qubit process that is free when held costs every early-release device at least 0.87 kT ln 2 per episode at 1% error, when the energy and entropy budgets of its controller and battery total at most 0.025 (energies in kT ln 2, entropies in bits); the error that enters is fixed by three Pauli measurements on one probe. A classical version needs no coherence. Beyond thermodynamics. Order has a price in entanglement and coherence theory too, with no single exchange rate between theories. Holding is decidable, releasing is not. In thermodynamics, whether a channel is free is decidable when outputs may be held and undecidable when they must be released; in the other theories examined, both are decidable. 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: Only quantum channels save work by waiting (https://github.com/Apsiape/save-work-by-waiting); 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). 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.
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 device that runs a quantum operation which does not respect time translation needs its own clock. This paper asks what owning that clock costs. Gates. The clock is paid for in capital, in the energy range of the heat baths, and in an energy that comes back only as a lottery, each growing faster than the number of uses as the error vanishes. The batch law. With a returned clock, the batch work per use of every channel is a single-letter quantity B. It equals the thermodynamic capacity when the channel's environment can carry its energy change, and can exceed it otherwise, even for channels of capacity zero. How large the clock must be. At fixed error, the same condition sets how the clock's capital spread must grow: not at all for covariant channels, like the square root of the number of uses when the environment can carry the energy change (by a construction that does not attain the optimal work), and at least linearly when it cannot. A controller that comes back only in state. Counting the controller's free energy, no time reference does better in batch than a returned clock. A controller that comes back close in state but does not refund its free energy, as an embezzling state does, can lower the battery's work to the capacity or below it, but only by paying the difference from its own free energy, like a second battery; to pay a fixed amount per use within return error η, its energy range plus kT times the logarithm of its dimension must be at least of order the number of uses divided by η, and a classical ladder of that size suffices. Time reversal. In batch with a returned clock, a Gibbs-preserving qutrit channel costs 0.1547 kT per use, while its Petz reverse costs nothing. Noise horizon. Any fixed amount of full-rank thermal noise makes both directions free, but for a finite run what matters is the expected number of noisy uses. If it tends to zero, the original work rate remains; if it diverges, sublinear work is possible; if it tends to a positive finite value, the work rate stays positive, between two envelopes whose equality is open. 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, except in the free-energy ledger of a controller that comes back only in state, which counts its unrefunded free energy instead. 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: Only quantum channels save work by waiting (doi:10.5281/zenodo.23310624; https://github.com/Apsiape/save-work-by-waiting); The work cost of answering in real time (doi:10.5281/zenodo.23310628; https://github.com/Apsiape/answering-in-real-time); The price of time order (doi:10.5281/zenodo.23310622; 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.
Landauer's principle prices forgetting, and finite-time thermodynamics prices speed, a cost that patience removes. Neither prices order: a machine that computes in time must let some outputs go before later inputs arrive, and can then no longer use them to clean up. This letter shows that order has its own work price, which no patience removes, and gives it exactly. One entropy-plus-energy functional gives the optimal rate of deterministic work, over many uses at vanishing error, for a broad class of schedules of arrivals, releases and loans, and for devices that give back what they borrow, with its energy and entropy. At a fixed state, the price of releasing is the price of correlations whose partners have left. Together with results of the companion papers, the law settles three questions. Does answering promptly cost more? For memoryless channels, only if they are quantum. Can one decide whether a channel is free? Yes when outputs may be held, and no when they must be released. What does keeping time cost? A clock that must come back supplies phase but no net energy, so in batch, thermal operations with a returned clock do not reach the capacity of every channel, and a process can cost more than its time reverse. The price is testable: a two-qubit process that is free when held costs at least 0.87 kT ln 2 per episode at 1% error when its first output must leave early and the controller and battery budgets total at most 0.025 (energies in kT ln 2, entropies in bits); certifying the budgets is the remaining step. The letter summarizes four papers, which give the full proofs and carry the machine checks: The price of time order (doi:10.5281/zenodo.23310622); Only quantum channels save work by waiting (doi:10.5281/zenodo.23310624); The work cost of answering in real time (doi:10.5281/zenodo.23310628); The price of keeping time (doi:10.5281/zenodo.23310632). This record archives the letter, its TeX source and its build script. How the work was done, including the use of AI models, is described in the letter.
Many devices must answer each request before the next one arrives, while remembering what came before. This paper determines how much work such a device must spend per episode, one complete run of the process, for every finite interactive quantum process whose outputs carry the same Hamiltonians as its inputs, at every positive finite temperature. Work law for processes. The least work per episode is kT ln 2 times one quantity: an entropy change plus an energy change, both measured against the same adaptive observer, the most demanding one, and minimized over all ways to build the process in the order its outputs are due. For processes that commute with time translation, a strong converse holds for devices with thermal hardware and a sharp battery. What is free. A process costs nothing exactly when it is a limit of devices that run energy-conserving steps, in the required order, on one shared heat bath. Order costs work. A two-qubit process that is free when its outputs may wait costs kT ln 2 [1 - h(p)] per episode when they may not, where p is the bias of a coin the process draws, even when everything is classical. Without time covariance. The same law holds for every finite process when the device's whole controller need only come back approximately, with its actual mean energy and entropy refunded and its startup capital declared. 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: Only quantum channels save work by waiting (https://github.com/Apsiape/save-work-by-waiting); 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.
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.