Физики создали термодинамический инструмент для доказательства квантовых корреляций через заряд батареи

Abstract We introduce a battery-explicit thermodynamic witness of post-quantum Bell correlations. In each round, a single supplied excitation is routed into an explicit two-level battery if and only if a Bell-game condition is satisfied. The routing operation is implemented by an energy-preserving controlled SWAP, with all logical control registers taken to be degenerate. Thus the correlation resource does not create energy; it only determines the probability that the supplied excitation reaches the battery. The construction is first formulated for finite two-player XOR games. For any such game, the mean battery charge is exactly the game success probability multiplied by the battery gap. Optimizing over local, quantum, or nonsignalling behaviours therefore turns the corresponding game values into local, quantum, or nonsignalling battery-charge ceilings. For the CHSH game, Tsirelson’s bound becomes a strict quantum ceiling on the mean battery charge, while a PR-box behaviour reaches the single-excitation cap. The witness is trusted-module rather than device-independent: it assumes calibrated Hamiltonians, correct classical wiring, and a trusted energy-preserving battery module. We also discuss a reversible-controller implementation, finite-statistics certification from work data, robustness to imperfect battery readout, and cyclic bookkeeping showing that no positive net work is obtained once fuel restoration and memory erasure are included. Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 license. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. 1. Introduction Bell inequalities show that different physical theories allow different performances in distributed information-processing tasks. The CHSH inequality is the standard example: local behaviours obey the classical bound, quantum behaviours are limited by Tsirelson’s bound, and nonsignalling post-quantum behaviours such as PR boxes can reach the algebraic value without allowing communication [1–4]. More generally, two-player XOR games provide a compact language for Bell-type tasks with binary outputs [5]. They include CHSH and the chained Bell games as important examples [6, 7]. A separate line of work in quantum thermodynamics emphasizes that work, information, and control must be accounted for explicitly. In resource-theoretic formulations, thermodynamic transformations are described by specifying allowed operations, nonequilibrium resources, [8–27], batteries, and memories [18, 28–35]. Likewise, Maxwell-demon and feedback-control settings show that information can modify operational work balances, provided that memories and erasure costs are included [36–42]. These considerations are especially important whenever a controller or a battery is used: if it is not modelled explicitly, an apparent thermodynamic advantage may come from an unaccounted work source. Thermodynamic observables have also been proposed as witnesses of quantum properties. In particular, heat exchange with a thermal environment, assisted by a quantum memory, can witness resources such as entanglement and coherence [43]. The present work follows a different route. We do not use heat generation or thermalization as the witnessing signal. Instead, a pre-existing excitation is routed into an explicit battery by an energy-preserving operation, and the mean battery charge is compared with Bell-game resource-class bounds. In this paper we connect these two viewpoints in a deliberately conservative way. We construct a trusted battery witness of post-quantum Bell correlations. The witness does not extract work from Bell correlations and does not derive Tsirelson’s bound from thermodynamics. Instead, it implements a calibrated energy-preserving transducer: a single supplied excitation is routed into an explicit two-level battery if and only if a Bell-game winning condition is satisfied. The construction is formulated first for finite two-player XOR games. Let be such a game, and let be the behaviour generated by the correlation resource. The referee samples and an independent uniform bit , and defines The players’ answers define Then Thus equality of the two logical bits is exactly the XOR-game winning event. The random pad makes the individual bits unbiased; the game performance is stored in their correlation. The thermodynamic module consists of two degenerate logical registers storing and , a fuel qubit , and a battery qubit . The fuel and battery have the same energy gap . At the beginning of a round the fuel is excited and the battery is empty. If the equality condition is satisfied, an equality-controlled SWAP transfers the excitation from to ; otherwise the fuel-battery state is left unchanged. Since the SWAP only exchanges equal-energy states, and the logical registers are degenerate, the operation is energy preserving. We denote by the increase of the battery energy in one round. In the ideal protocol this random variable takes only the two values It is equal to precisely on those rounds in which the game is won. Taking the average over many rounds gives the basic work–game identity where is the success probability of the behaviour in the game . Thus the mean battery charge is not an additional thermodynamic quantity independent of the Bell experiment. It is the game success probability written in energy units. Now fix a class of admissible behaviours, for instance the local, quantum, or nonsignalling class. Let denote the optimal success probability of the game within that class. Equation (1) immediately turns this game value into a battery ceiling, An observed mean battery charge above the quantum ceiling of the game is then a trusted-module witness that the effective behaviour is not quantum-realizable. For the CHSH game, let be the standard CHSH expression of the behaviour . Since the CHSH winning probability is Equation (1) gives The algebraic part of the construction is simple by design: a Boolean winning predicate is converted into a binary control signal. The point of the present work is not that this Boolean computation is difficult, but that it can be realized as an explicit energy-preserving battery operation with no hidden work source in the controller. The contribution is the calibrated thermodynamic transduction of Bell-game performance into a battery degree of freedom, together with the associated resource-class thresholds, finite-statistics witness, and cyclic bookkeeping showing that no positive net work is produced. The result should be read with its assumptions visible. The witness is not a loophole-free Bell test and is not device independent. It assumes calibrated Hamiltonians, correct classical wiring, a trusted equality-controlled SWAP, and a calibrated battery readout. It also does not produce positive cyclic work. The supplied excitation is the energetic resource. The correlation resource only determines whether that excitation reaches the battery. If the device is used cyclically, consumed fuel excitations must be restored, and any persistent memory record must be reset. Recent work has also studied work-extraction tasks in which quantum structure itself gives an advantage over classical commuting implementations. For example, [44] shows that incompatible Hamiltonian settings can exceed classical commuting limits in an average-work task. Our setting is different: the Hamiltonians of the fuel and battery are fixed and commuting, and the nonclassicality appears through the correlation resource controlling an energy-preserving routing operation. The paper is organized as follows. Section 2 states the physical setting and trusted-module assumptions. Section 3 discusses stored battery energy, passivity, ergotropy, coherent excitation transfer, and dissipative corrections. Section 4 introduces the XOR-game battery transducer and proves the main identity. Section 5 applies the construction to CHSH and chained Bell games. Section 6 discusses reversible controller implementations and cyclic thermodynamic bookkeeping. Section 7 treats finite-statistics certification and imperfect battery readout. Section 8 summarizes the scope and limitations. 2. Physical setting and trusted-module assumptions We now specify the physical model used throughout the paper. The aim is to isolate the thermodynamic part of the construction from the source of correlations. The correlation resource may be local, quantum, nonsignalling, or post-quantum. The energy module is trusted and has explicitly specified Hamiltonians. The overall architecture of the trusted battery-witness protocol is summarized in figure 1. 2.1. Correlation resource and game data The correlation resource is used only during the distributed game stage. The referee samples questions , sends to Alice and to Bob, and receives output bits . This defines a behaviour No thermodynamic assumption is made about the physical device that produces this behaviour. In particular, the battery module does not require a quantum model of the correlation resource. After the game outputs are produced, the relevant classical data are available to the trusted local controller that operates the battery module. This is a trusted-module setting: the thermodynamic stage is not required to be space like separated and is not itself a Bell test. 2.2. Logical registers The logical registers used to store the target bit, the guess bit, and any controller memory are taken to be degenerate. Thus their Hamiltonian is idealized as This assumption is used only for bookkeeping: it ensures that logical operations on these registers do not themselves raise or lower the explicit energy of the fuel-battery system. This is an idealization. Physically, it means that the logical energy splittings are negligible compared with the battery gap , or that they are compensated by a calibrated control system that is not being counted as a hidden work source. 2.3. Fuel and battery The energetic resource in each round is one excitation in a fuel qubit . The fuel Hamiltonian is The battery qubit has the same gap, At the beginning of a round, Thus the initial fuel-battery state contains one transferable excitation. The one-round battery-energy increase introduced in the Introduction is defined more explicitly as where and are the battery states before and after the routing operation. In the ideal protocol, 2.4. Energy-preserving routing The only nontrivial energetic operation is a controlled SWAP between and . When the control condition is satisfied, the operation maps When the condition is not satisfied, it acts as the identity. Since have the same total energy , the SWAP is energy preserving. The transducer therefore routes an already supplied excitation; it does not create one. The routing operation is compatible with the thermal operations paradigm in the trivial-bath sense: the core step is a global energy-preserving unitary. A thermal bath is needed only if one includes irreversible memory reset in a cyclic implementation. 2.5. Trusted-module interpretation The witness assumes that the following ingredients are calibrated and trusted: the Hamiltonians and , the battery gap , the logical wiring that computes the control condition, the controlled SWAP, and the battery readout. Under these assumptions, a battery charge above the quantum ceiling of a game implies that the effective behaviour is not quantum-realizable. Equivalently, in an actual experiment, observing such a violation means that at least one of two things is true: either the effective behaviour is outside the quantum set, or one of the trusted-module assumptions has failed. The witness therefore certifies post-quantumness only relative to independent validation of the energy module and readout. 2.6. Thermodynamic boundary of the cycle The single-round transducer is not a heat engine. It does not convert heat into work, and it does not convert Bell correlations into net work. It only transfers a supplied excitation to the battery on winning rounds. If the device is used cyclically, then successful rounds leave the fuel in , and the fuel excitation must be restored before reuse. This restoration costs at least on those rounds. Failed rounds leave the fuel excitation unused and do not require re-excitation of the same fuel qubit. If the controller stores a persistent success/failure record, that memory must also be reset before the next cycle. For blind erasure, Landauer’s principle gives the corresponding erasure cost. If the full game transcript is stored inside the thermodynamic cycle, then its reset cost must also be included. In the main cyclic bookkeeping below, the transcript registers are treated as external game data, while the local fuel, battery, and possible success/failure memory are included explicitly. 3. Battery energy, passivity, and coherent charging 3.1. Stored energy, passivity, and ergotropy The battery variable used in the witness is the increase of the energy of an explicit two-level work-storage system. Since in the quantum-battery literature one often distinguishes stored energy from ergotropy, we now spell out this distinction for the present model. The battery Hamiltonian is In one round of the ideal protocol, the battery is either left in or charged to . Thus the run-resolved battery-energy increase is This is the quantity entering the witness. If the charge record is kept, then the interpretation is simple. Conditional on a charged round, the battery state is . This state has ergotropy , since the unitary flip lowers the battery energy by . Conditional on an uncharged round, the battery state is , whose ergotropy is zero. Therefore the average event-resolved extractable energy is If, instead, the charge record is ignored and only the average battery state is considered, then after many rounds the battery state is Its mean stored energy is However, its unconditional ergotropy is not generally equal to . For a two-level Hamiltonian, the passive state is obtained by putting the larger eigenvalue of the density matrix on the ground state. Hence If , the averaged battery state is passive. If , it is population-inverted and a unitary swap of the two levels extracts . Thus the witness should be read as a statement about stored battery energy, or about event-resolved extractable energy when the charge record is retained. It is not a statement that the ergotropy of the averaged battery state is always . This is the standard distinction between passivity and maximal unitary work extraction [45, 46]. The same terminology also clarifies the role of temperature. For a reservoir at inverse temperature , the thermal state of the isolated battery is For positive temperature this state is passive. The ideal battery-routing step does not thermalize the battery; it is a closed energy-preserving excitation transfer. Temperature enters only when one models relaxation, thermal preparation, or erasure/reset operations. 3.2. Coherent charger-battery coupling The excitation transfer used in the battery module can be realized by a coherent resonant coupling between the fuel and the battery. On the system consider where . This Hamiltonian acts only in the one-excitation subspace Since both basis states in this subspace have total energy , one has Thus the coupling is energy preserving. Starting from an excited fuel and an empty battery, At the interaction time this gives Hence the excitation is transferred from the fuel to the battery, up to a phase. For the classical work statistics considered here this phase has no effect. If one wants a coherent implementation on superpositions of different control histories, an additional energy-preserving phase correction may be included. The equality-controlled version is obtained by switching on the exchange only on the branch : The logical registers are degenerate, and commutes with . Therefore Evolution under for time implements the desired equality-controlled excitation transfer, again up to a phase irrelevant for the classical battery population. For completeness, the ideal controlled-SWAP unitary can also be written as a finite-time Hamiltonian evolution. Let denote the SWAP on . Since is Hermitian and unitary, define where is a chosen gate time. On the eigenspace of , has eigenvalue , while on the eigenspace it has eigenvalue . Thus The corresponding controlled generator is and satisfies This formal expression answers how the ideal unitary may be generated. The resonant exchange Hamiltonian in equation (11) is the more direct charger-battery implementation in the one-excitation sector used by the protocol. The same expression also shows how timing errors enter. If the exchange pulse lasts then the transfer probability is For small , Thus an imperfect pulse reduces the true-positive efficiency of the energy-routing module. In the readout-imperfection model used later, this contributes to the calibrated parameter . 3.3. Bath, temperature, and dissipative corrections The ideal witness identity is derived for the closed routing step. No dissipative environment is needed to obtain The role of a thermal bath is different: it enters when one models relaxation of the fuel or battery, thermal preparation, or erasure of a persistent memory record. In particular, the Landauer term in the cyclic bookkeeping section is the place where the bath temperature enters the ideal thermodynamic balance. If the battery is weakly coupled to a reservoir during or after the charging step, the ideal unitary dynamics should be replaced by an open-system evolution. A simple phenomenological description is given by the standard GKSL form [47, 48] where Here describes relaxation of the battery excitation, describes thermal excitation, and describes dephasing. For a thermal reservoir at inverse temperature , the rates satisfy the detailed-balance relation The corresponding stationary state of the isolated battery is the Gibbs state which is passive for positive temperature. Such dissipative effects are not part of the ideal witness theorem. They change the experimentally observed charging probability by producing losses, false charges, or relaxation before readout. In the notation of section 7, they contribute to the calibrated parameters and . Thus dissipation does not change the logical form of the witness; it changes the calibration needed to infer the underlying game success probability from the observed battery signal. 4. XOR-game battery transducer We now give the main construction. The starting point is a finite two-player XOR game where . The referee samples , Alice and Bob output bits , and the game is won when For a behaviour , the success probability is 4.1. Equality construction The trusted referee/controller also samples an independent uniform bit . This bit is not supplied to the devices producing and , and the behaviour is assumed not to depend on . We define The bit is the target bit and is the guess bit. Lemma 1(Winning is equality). For every realization , Consequently, because . Therefore if and only if , which is equivalent to . Averaging over the distribution of all variables gives □ The random pad also has a useful thermodynamic role. It removes local bias from the logical bits. Lemma 2(No local bias in the equality registers). The target bit and the guess bit are both uniform: Since is uniform, for both . Hence is uniform. Now define the error bit From the same cancellation as above, Thus depends on the game transcript but not on . The one-time pad makes independent of the transcript and hence independent of . Since we get, for , □ Remark 1(Correlation rather than local free energy). Because and are individually uniform and stored in degenerate logical registers, they carry no local nonequilibrium free energy in this idealized model. The relevant information is their correlation: The battery module below converts this correlation, not a local bias of either bit, into a charging probability. 4.2. Battery module The target and guess bits are stored in degenerate logical registers, also called and . The fuel and battery have Hamiltonians At the beginning of the round, Let be the two-qubit SWAP on fuel and battery: Since and have the same gap, Indeed, the only nontrivial action in the one-excitation sector is and both states have total energy . Define the equality-controlled battery unitary where Since the branches are controlled on orthogonal projectors and each branch is unitary, is unitary. Since the logical registers are degenerate and each branch commutes with , Theorem 1(XOR-game battery transduction). For any finite two-player XOR game and any behaviour , the battery module satisfies in every run. Therefore Proof. If , the SWAP branch is applied, and The battery gains energy . If , the identity branch is applied and the battery remains in , so it gains no energy. Hence Taking expectations and using lemma 1 gives □ Remark 2(Why we focus on XOR games). The battery mechanism itself is more general than XOR games. Any finite classical binary predicate , computed from a transcript , can be used to control the same energy-preserving SWAP, giving We focus on XOR games because their local, quantum, and nonsignalling values are standard and give familiar post-quantum thresholds, including CHSH and chained Bell inequalities. The fully general binary-predicate statement is recorded in appendix Corollary 1(Resource-class ceilings). Let be any class of behaviours for the game , and define Then the maximal mean battery charge achievable within is Proof. By theorem 1, for every behaviour . Taking the supremum over gives □ Thus, under the trusted-module assumptions, if then either the effective behaviour is not quantum-realizable, or one of the trusted assumptions about the battery module has failed. With independently validated module assumptions, this becomes a witness of post-quantumness. 5. CHSH and chained-game witnesses We now specialize the general theorem to two standard families of XOR games. The CHSH game gives the clearest post-quantum separation. The chained games show that the construction is not tied to the four-setting CHSH algebra. 5.1. CHSH For CHSH, The winning condition is Define correlators and the CHSH expression With this convention, local behaviours satisfy , quantum behaviours satisfy Tsirelson’s bound , and nonsignalling behaviours satisfy , with the algebraic value attained by a PR box [1–4]. For CHSH, the success probability is Indeed, the XOR-game bias is and . Therefore theorem 1 gives The standard CHSH bounds become the following battery ceilings: and Thus a trusted observation of is a post-quantum witness for the effective CHSH behaviour, subject to the trusted-module assumptions. A noisy PR-box interpolation gives a simple benchmark. Let where and . Then and hence This remains above the quantum CHSH ceiling exactly when or Figure 2 shows the affine relation between the CHSH value and the normalized mean battery charge. Download figure: Standard image High-resolution image 5.2. Chained Bell games The chained Bell games form a standard family of XOR games generalizing CHSH [6, 7]. Fix . The question sets are The referee samples uniformly from the input pairs where and addition is modulo . All edges require equal outputs except one wrap-around edge, which requires unequal outputs: and For this game, the standard values are and The local value follows because deterministic assignments cannot satisfy all parity constraints around the cycle, while they can satisfy of them. The quantum value is the chained Tsirelson value [6, 7]. The nonsignalling value is one, because a nonsignalling box can satisfy each allowed XOR constraint with uniformly random local marginals. By corollary 1, the corresponding battery ceilings are and Thus, under the trusted-module assumptions, witnesses that the effective chained-game behaviour is not quantum-realizable. For large , the quantum-to-nonsignalling battery gap is Thus the gap becomes smaller as grows. Within this family, CHSH gives the largest and cleanest quantum-to-PR separation. 6. Reversible controller and cyclic bookkeeping The battery transducer can be described in feed-forward form: after the game outputs are produced, the equality condition is computed and used to control the fuel-battery SWAP. We now explain how this feed-forward step can be implemented reversibly on degenerate logical registers, and then discuss the cyclic thermodynamic bookkeeping. 6.1. Reversible-controller implementation Let denote the classical data registers storing the transcript We take these registers to be degenerate: Let be a degenerate memory bit initialized in , and let denote finitely many degenerate ancilla bits initialized in . Thus all logical controller registers have zero Hamiltonian. For a fixed game transcript, define the success bit Equivalently, Since the game is finite, is a finite Boolean function. Hence it can be computed reversibly using standard reversible logic, with enough temporary ancillas. We may therefore choose a reversible circuit such that All temporary workspace is uncomputed at the end of . The feedback SWAP is Since is degenerate and commutes with , Define Then is energy preserving with respect to the explicit fuel-battery Hamiltonian and implements the same work statistics as the feed-forward transducer. Indeed, writes the success bit into , moves the excitation from to if and only if , and restores and the temporary ancillas to their initial blank states. Thus and therefore Remark 3(Reversible, not fully autonomous). The construction above is a reversible-controller implementation. We do not claim here a fully autonomous clock-Hamiltonian model of the controller. The logical gates are trusted control operations on degenerate registers. Their role is to show that the feed-forward computation need not change the explicit fuel-battery energy bookkeeping. Remark 4(Hamiltonian implementation). A Hamiltonian implementation of the controlled excitation transfer, including the phase appearing in the resonant exchange pulse, is discussed in section 3. In the reversible-controller description used here the control registers are classical computational-basis registers, so this phase does not affect the battery statistics. 6.2. Boundary of the thermodynamic cycle The reversible-controller construction preserves the transcript registers . In the cyclic bookkeeping below, these registers are treated as external game data supplied by the referee/game interface. The thermodynamic cycle explicitly includes only the local fuel, battery, controller memory when it is persistently stored, and their reset or restoration operations. If a concrete implementation instead reuses the same physical registers for the full transcript, then their erasure or overwrite cost must also be included. That additional cost is at least the Landauer cost of the stored transcript and is no smaller than the cost of erasing the compressed success/failure bit. 6.3. Fuel restoration Let The mean battery charge is In a successful round, the fuel-battery state changes as The fuel excitation has been consumed. To reuse the same fuel system in the initial state , one must restore the excitation, which costs at least in the ideal energy-eigenstate model. In a failed round, the state remains The fuel excitation is still present and no fuel re-excitation is needed. Hence the minimal average fuel restoration cost is success-weighted: This is the relevant cyclic accounting when unused fuel excitations are retained and reused. 6.4. Reversible-controller cycle In the reversible-controller implementation, the success bit is computed, used, and uncomputed. No persistent success/failure memory remains. Thus there is no Landauer erasure cost for that bit. The fuel-battery contribution to the cycle satisfies In the ideal reversible limit this balance is saturated. Thus the reversible implementation gives no positive net work once the restoration of consumed fuel excitations is included. 6.5. Measured-memory implementation A different implementation may measure or persistently store the success/failure bit Then and If this memory is blindly reset before the next run, Landauer’s principle gives [36, 37, 41, 42] Therefore the measured-memory cycle obeys and hence For a perfect nonsignalling strategy, , the success/failure memory is deterministic and . Even then, the battery charge is exactly balanced by the fuel restoration cost. Thus neither quantum nor post-quantum correlations generate positive cyclic work in this model. 6.6. Summary of the thermodynamic role The battery transducer is an energy-routing witness. The supplied excitation is the energetic resource. The correlation resource determines the probability with which that excitation reaches the battery. The cyclic bookkeeping confirms that the protocol does not convert Bell nonlocality or post-quantumness into a thermodynamic fuel. 7. Finite statistics and readout imperfections In the ideal transducer, each round produces a binary work value Thus the game success probability can be estimated directly from battery data. This section gives simple finite-statistics and readout-robustness statements. 7.1. Finite statistics from work data Define the observed work bit and let In the ideal i.i.d. setting, Hoeffding’s inequality [49] gives Thus, for error probability , define Then, except with probability at most , A finite-data nonlocality certificate is obtained if and a finite-data post-quantumness certificate is obtained if Equivalently, in work units, where For CHSH, this becomes Equivalently, using a lower confidence bound on is and post-quantumness is certified if 7.2. Martingale version The i.i.d. assumption can be weakened. Let be the history before round , and define Let The variables are martingale differences with Azuma–Hoeffding gives [49, 50] Thus the same lower confidence bound applies to the time-averaged success probability: except with probability at most . This version is useful when the effective behaviour may drift between rounds. The conclusion then concerns the average success probability over the tested rounds. 7.3. Imperfect battery readout We now include a simple readout-error model. Let be the true-positive probability, and let be the false-positive probability. Assume If then the observed charging probability is Therefore If the calibration parameters are known, a lower confidence bound gives Thus the corrected post-quantumness condition is 7.4. Imperfect random pad The ideal construction assumes that the pad bit is uniform, private, and independent of the devices producing . Uniformity is used to remove local bias from the equality registers. It is not needed for the equality identity itself. Suppose instead that is still private and independent, but slightly biased: Then the algebraic cancellation still holds. Hence and the battery identity is unchanged. What changes is the local bias of the logical registers. Let Then Therefore Similarly, if then and hence Thus imperfect randomness of does not spoil the work–success-probability identity, provided remains private and independent of the devices. It does, however, weaken the statement that the individual logical registers have no local bias. If the logical registers are exactly degenerate, this bias does not change their internal energy. If one assigns an additional information bookkeeping cost to biased logical records, that cost must be accounted for separately. If is available to the devices producing , the situation is different: the behaviour may effectively depend on , and the usual resource-class values , , and for the original game no longer apply without modification. For this reason the pad is assumed to be generated inside the trusted referee/controller module and not supplied to the correlation resource. 7.5. Conservative calibration bounds If the calibration parameters are not known exactly, the conservative lower bound must be chosen carefully. Since the inferred value of decreases when increases, and it also decreases when increases, provided the numerator is positive. Therefore, if calibration gives with then a conservative lower bound is whenever the numerator is positive. If no useful upper bound on is available, one may use the trivial upper bound , giving The right-hand side should be truncated to the interval . For CHSH, the corrected post-quantumness condition is 7.6. Symmetric work-bit flips As a simple benchmark, suppose the observed work bit is obtained from the ideal work bit by a symmetric flip with probability . Then For an ideal PR-box behaviour, , so To remain above the CHSH quantum ceiling without statistical uncertainty, one needs Thus This is the maximum symmetric work-bit error rate for which an ideal PR-box signal remains above the quantum CHSH battery ceiling. 8. Discussion We have introduced a trusted battery witness of post-quantum Bell correlations. The construction takes the success event of a finite XOR game and writes it into an explicit work-storage degree of freedom. In each round, a supplied excitation is transferred from a fuel qubit to a battery qubit exactly when the game is won. The resulting mean battery charge is Thus the local, quantum, and nonsignalling values of the game become local, quantum, and nonsignalling battery-charge ceilings. For CHSH, this gives The classical, quantum, and nonsignalling thresholds are respectively A mean battery charge above the Tsirelson-calibrated threshold therefore witnesses post-quantumness of the effective behaviour, provided the thermodynamic module is independently trusted. The chained Bell games give a second family of examples, where the same transduction theorem converts the known game values into corresponding battery thresholds. The construction is intentionally modest in its thermodynamic claims. It is not a heat engine and it does not convert Bell correlations into net work. The energetic resource is the supplied fuel excitation. The correlation resource only determines the probability that this excitation is routed into the battery. When the protocol is made cyclic, successful rounds require restoration of the fuel excitation. If a persistent success/failure memory is used, Landauer’s principle adds an erasure cost. With these costs included, the full-cycle net work is non-positive. This distinction is important for interpreting the witness. The battery charge is not the thermodynamic value of the Bell correlation as a fuel. Rather, it is a calibrated energetic representation of the Bell-game success probability. The role of thermodynamics is to ensure that the readout is implemented without hiding energy in the controller: the battery is charged only by an energy-preserving transfer of a supplied excitation. As discussed in section 3, this stored energy is equal to event-resolved extractable energy when the charge record is retained. If the charge record is ignored, the ergotropy of the averaged battery state is instead , with . The one-time-pad bit also has a useful thermodynamic interpretation. It makes the target and guess bits individually unbiased. In the degenerate logical-register model, neither bit carries local nonequilibrium free energy. The game performance is instead encoded in the correlation between them, namely in the probability that . The battery transducer converts precisely this correlation into a charging probability. The witness is not device-independent. It assumes trusted Hamiltonians, trusted classical wiring, a calibrated battery gap, a correctly implemented energy-preserving controlled SWAP, and a calibrated battery readout. If these assumptions are not independently validated, a work value above the quantum ceiling could indicate a failure of the trusted module rather than genuine post-quantumness. Thus the operational statement is conditional: under the trusted-module assumptions, a battery charge above the quantum game value certifies that the effective behaviour is not quantum-realizable. The model is idealized, but the idealizations are explicit. Imperfect exchange pulses, battery relaxation, thermal excitation, dephasing, biased pad randomness, and readout errors do not change the algebraic structure of the witness; they change the calibration relating the observed battery signal to the underlying game success probability. For this reason the witness should be used as a calibrated thermodynamic module. Its conclusion is conditional: a work value above the quantum threshold indicates either post-quantum effective correlations or failure of one of the trusted calibrations. The present construction is complementary to information-engine approaches. In a Szilard-engine setting, correlations or side information can change the work available to a feedback controller through mutual information. A complementary CHSH-based side-information construction was considered in [51]. The difference is that the present paper does not assign work value to side information. Instead, it routes an explicit fuel excitation into a battery according to the Bell-game winning event. It is also distinct from heat-based witnesses of quantum properties [43]: here the signal is not heat generated in contact with a reservoir, but the charge of an explicit battery receiving an excitation supplied at the beginning of the round. Several extensions are natural. One can study other XOR games for which the gap between the quantum and nonsignalling values is more robust to noise or finite statistics. One can replace the ideal two-level battery by more realistic work-storage systems with finite resolution, finite-time control, or small Hamiltonian mismatches. One can also refine the statistical analysis to include loss, input-dependent detection efficiency, or composable confidence bounds in the presence of memory effects. Finally, the same basic idea can be applied to more general finite games with binary winning predicates: an energy-preserving controlled SWAP can route a supplied excitation whenever the predicate is satisfied. XOR games were chosen here because they give a clean connection to CHSH, chained Bell inequalities, and standard resource-class game values. The general binary-predicate formulation also clarifies the scope of the result: the thermodynamic module realizes a calibrated energy readout of a classical predicate, while the physics enters through which predicate and which resource class determine the relevant threshold. The main message is therefore simple. Bell-game performance can be written directly into an explicit battery degree of freedom without making the controller an unaccounted work source. The result is a trusted thermodynamic witness: not a new principle limiting correlations, and not a work-extraction engine, but a battery-explicit way of representing local, quantum, and post-quantum correlation strengths. Acknowledgments We thank Marcin Pawłowski, Sumit Rout and Anubhav Chaturvedi for discussions. This work is supported by the IRAP/MAB programme, Project No. FENG.02.01IP.05-0006/23, financed by the MAB FENG program 2021-2027, Priority FENG.02, Measure FENG.02.01., with the support of the FNP (Foundation for Polish Science). Data availability statement All data that support the findings of this study are included within the article (and any supplementary files). Appendix A: Detailed proof of the battery transduction theorem This appendix gives the details behind the energy-preserving transduction statement used in section 4. The main text keeps only the short proof. A.1. Independence induced by the one-time pad Recall that the referee samples an independent uniform bit , and defines Define also the error bit Then Thus depends on , but not on . Lemma 3(Independence of and ). The target bit is independent of the error bit . Proof. Let . By the law of total probability, For fixed , there is exactly one value of satisfying namely Since is uniform, Therefore the sum over gives a factor , and Since is uniform, Thus Hence and are independent. □ Corollary 2(Binary symmetric channel). The relation between and is a binary symmetric channel: where is independent of . Its crossover probability is By lemma 3, is independent of . Moreover, Therefore and hence □ A.2. Energy preservation of the equal-gap SWAP The fuel and battery Hamiltonians are The computational basis vectors of have energies The SWAP unitary satisfies and It leaves the zero- and two-excitation sectors fixed, and it only exchanges the two states in the one-excitation sector. Since the one-excitation states have the same total energy, the SWAP preserves every eigenspace of . Therefore A.3. Unitarity of the equality-controlled battery operation The equality-controlled battery unitary is where Let The projectors are mutually orthogonal and resolve the identity: Each is unitary. Therefore The same calculation gives Thus is unitary. A.4. Energy preservation of the equality-controlled operation The logical registers are degenerate: Hence For each branch, is either the identity or . Both commute with . Therefore every block commutes with , and so does their sum: Appendix B: General binary-predicate battery routing The main text focuses on XOR games because they connect directly to CHSH, chained Bell inequalities, and standard local, quantum, and nonsignalling game values. However, the energy-routing step is more general. Let be any finite classical transcript, distributed according to some probability distribution, and let be a binary predicate. The event may represent winning a game, satisfying a Bell predicate, passing a test, or any other classical condition. Proposition 1(Battery routing for a binary predicate). Let be reversibly computable into a degenerate memory bit. Then there is an energy-preserving fuel-battery unitary such that in every run. Consequently, Proof. Let be a degenerate memory bit initialized in . Since is finite, the predicate can be computed reversibly using finitely many degenerate ancillas: Then apply This unitary is energy preserving because is degenerate and commutes with . If , the fuel excitation is moved to the battery and . If , the identity branch acts and . Therefore Taking expectations gives □ Remark 5. This proposition shows that the battery mechanism itself is not special to XOR games. The reason to use XOR games in the main text is that their local, quantum, and nonsignalling values are standard and yield transparent post-quantum thresholds. Appendix C: Details for the chained Bell games This appendix gives the elementary parts of the chained-game analysis. The quantum value is quoted from the standard chained Tsirelson bound [6, 7]. C.1. Classical value For the chained game , the tested constraints are and Here and are deterministic local outputs. A deterministic strategy cannot satisfy all constraints. Indeed, from we get for all . From for , we get Therefore Since we obtain contradicting the final condition Thus at least one of the constraints must fail. This upper bound is tight: choosing all deterministic outputs to be zero wins all equality constraints and loses only the final inequality constraint. Therefore C.2. Nonsignalling value For every allowed input pair , define Then the winning condition is satisfied with probability one. Alice’s marginal is uniform: for both , independently of . Bob’s marginal is also uniform: for both , independently of . Hence the behaviour is nonsignalling on the support of the game. It can be extended to all input pairs with uniform marginals. Therefore C.3. Quantum value The quantum value is the standard chained Tsirelson value: For , this gives which is the usual CHSH quantum winning probability. Appendix D: Convex-content bounds from battery data The battery value can also be used to lower-bound the fraction of a behaviour that must lie outside a chosen resource class. Let be two behaviour classes, and assume Suppose that where Proposition 2(Lower bound on non- content). If then whenever the numerator is positive. In battery units, Proof. The success probability is affine in the behaviour: Using the class values, and Therefore Rearranging gives the stated bound. The battery form follows from □ D.1. CHSH nonlocal and post-quantum content For CHSH, A lower bound on nonsignalling nonlocal content is obtained by taking Then Using this becomes For post-quantum content, take Then In terms of the CHSH value, Appendix E: CHSH monogamy in battery form For a tripartite nonsignalling behaviour , let be the CHSH value of the marginal shared by Alice and Bob, and let be the CHSH value of the marginal shared by Alice and Charlie. The standard CHSH monogamy relation gives [52] for consistently oriented CHSH expressions. For the battery transducer applied to the two marginals, and Adding these identities gives Using we obtain Thus the usual nonsignalling CHSH monogamy relation becomes a battery-monogamy bound. Appendix F: Exact binomial confidence intervals The main text uses Hoeffding and Azuma–Hoeffding bounds because they are simple and analytic. In the i.i.d. setting, the work bits are Bernoulli random variables, so one can also use exact or sharper binomial intervals. Let and let In the ideal i.i.d. transducer model, where F.1. Clopper–Pearson interval A two-sided Clopper–Pearson interval [53] with error probability is where, for , and Here is the -quantile of the beta distribution with parameters . The endpoint conventions are and A one-sided lower confidence bound is obtained by replacing by : F.2. Wilson interval The Wilson interval [54] is often shorter while maintaining good coverage. Let where is the standard normal cumulative distribution function. The Wilson interval is F.3. Mapping to CHSH Any confidence interval gives a battery interval For CHSH, Thus A finite-data post-quantumness certificate is obtained whenever or equivalently Appendix G: Memory reset variants The Landauer term in the main text refers to a compressed success/failure memory. Here we spell out how the cost depends on what is stored. G.1. Compressed success memory If the only persistent memory is then where The memory entropy is Blind erasure costs at least G.2. Full transcript memory If the implementation stores the full transcript then the erasure cost is governed by , not merely by . Since is a deterministic function of , Thus erasing the full transcript is at least as costly as erasing the compressed success/failure bit. G.3. Side-information-assisted reset If the erasing agent has side information correlated with the memory, then the relevant classical entropy can be reduced to a conditional entropy . The present work deliberately uses blind reset of the persistent local memory, so such side-information-assisted reductions are not used. G.4. Reversible uncomputation In the reversible-controller implementation, the success bit is computed, used, and uncomputed. No persistent remains. Therefore no Landauer erasure cost is assigned to the success bit in that implementation. Appendix H: Detailed fuel-battery balance The initial fuel-battery state is The initial fuel energy is and the initial battery energy is If the game is won, the SWAP branch gives Thus and If the game is lost, the identity branch gives Thus and Combining the two cases, and Therefore in every run. Averaging gives and Thus restoring the fuel to costs at least on average in the ideal energy-eigenstate model. This is why the minimal cyclic fuel-restoration cost is success-weighted rather than automatically equal to per attempt.