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The Minimal Work Cost of Information Processing
Philippe Faist, Frédéric Dupuis, Jonathan Oppenheim, Renato Renner
TL;DR
The paper asks how much work is fundamentally required to implement arbitrary logical processes, including computations and measurements. It models implementations with an information battery and joint unital operations, deriving a smooth-entropy bound. The minimal cost is determined by information discarded conditional on the output and becomes an initial-final entropy difference asymptotically.
Problem
The central problem is determining the minimal thermodynamic work required to implement arbitrary logical processes beyond standard average-case settings.
Method
The paper models work with an information battery and optimizes joint unital operations subject to exact logical implementation, using smooth-entropy characterizations.
Results
The minimal work is given by the entropy of information discarded conditional on the output and becomes an initial-final entropy difference in the asymptotic i.i.d. regime.
Takeaways & Limitations
The result interprets logical-process work as the cost of storing discarded information in an ancillary system and applies to general quantum logical processes.
Takeaways & Limitations
The framework assumes free pure ancillas and that transforming a state to a same-rank maximally mixed state yields no work.
Abstract
from arXiv · showhide
Irreversible information processing cannot be carried out without some inevitable thermodynamical work cost. This fundamental restriction, known as Landauer's principle, is increasingly relevant today, as the energy dissipation of computing devices impedes the development of their performance. Here we determine the minimal work required to carry out any logical process, for instance a computation. It is given by the entropy of the discarded information conditional to the output of the computation. Our formula takes precisely into account the statistically fluctuating work requirement of the logical process. It enables the explicit calculation of practical scenarios, such as computational circuits or quantum measurements. On the conceptual level, our result gives a precise and operationally justified connection between thermodynamic and information entropy, and explains the emergence of the entropy state function in macroscopic thermodynamics.
METHODS
The paper models logical processes using an information battery and optimizes over unital physical operations. This yields an explicit work minimization that can be expressed through conditional and smooth entropy quantities.
- The Framework: The framework represents work using an information battery whose mixed-qubit count changes from λ1 to λ2.The battery begins with λ1 maximally mixed qubits and ends with λ2; the work cost is proportional to kT ln 2 · (λ2 − λ1).
- The Framework: The framework assumes free pure ancillas and a final battery state consisting of maximally mixed qubits plus pure qubits.The maximally mixed final form relies on the assumption that transforming a state to a same-rank maximally mixed state cannot yield work.
- The Framework: A logical process is implemented by a physical map that reproduces the required input-output behavior, including correlations with a reference system.The reference-system condition ensures the implementation realizes the completely positive, trace-preserving map on arbitrary inputs, not only on reduced states.
- Mathematical formulation and proof: The optimization over unital maps reduces to minimizing λ2 − λ1 subject to ∥E(ΠX)∥∞ ≤ 2^(λ2−λ1).The reduction follows through subunital maps and their dilation to unital maps.
- Entropic form of the bound: The resulting bound is expressed using smooth entropy, with the projector on the state support entering the equivalent Rényi-zero formulation.The paper relates the Rényi-zero conditional entropy to the conditional max-entropy, up to logarithmic factors in the approximation parameter.
- Entropic form of the bound: In the asymptotic i.i.d. regime, the average work cost equals the entropy difference between the initial and final states.For repeated independent processes, the total work is obtained by applying the entropy expression to the joint state of all particles.
Appendix A: Motivation. Relation of Our Result to Previous Work.
The paper places its result within efforts to connect thermodynamics and information theory, extending prior average-case and quantum analyses to a general logical-process framework. It interprets minimal work as the cost of storing discarded information.
- Relation to Previous Work: Earlier work connected thermodynamic entropy with information-theoretic entropy through Maxwell’s demon, Szilard’s engine, and Jaynes’s statistical-mechanics interpretation.These developments established information as central to understanding thermodynamic behavior.
- Relation to Previous Work: Quantum extensions replaced Gibbs or Shannon entropy with quantum von Neumann entropy and were motivated partly by advances in microscopic thermodevices.The paper situates its contribution among broader quantum-thermodynamic generalizations of Landauer’s principle.
- Relation to Previous Work: Prior information-theoretic approaches largely studied averages over many independent repetitions, while single-instance tasks use smooth entropies.The paper connects its treatment of individual information-processing tasks to the smooth-entropy framework.
- The Present Result: The paper gives a general formula for the minimal work requirement of any quantum logical process.The result also characterizes the minimal ancillary-system size needed to store information discarded by the process.
- The Present Result: For quantum measurements, including memory initialization makes the measurement cost work, whereas transferring information to an initially pure memory costs no work.The distinction depends on whether initialization is included in the measurement process.
Appendix B: Some Initial Remarks and Clarifications.
The paper distinguishes abstract logical mappings from their thermodynamic implementations and clarifies when minimal work depends on the process rather than only on endpoint states. In the thermodynamic limit, the work cost becomes an entropy-state-function difference.
- Appendix B: Some Initial Remarks and Clarifications: Logical processes map input states to output states independently of their physical implementation, while physical realizations can use different thermodynamic strategies and incur different work costs.The logical specification alone does not determine the work actually used by a device.
- Appendix B: Some Initial Remarks and Clarifications: The minimal work for a microscopic logical process generally cannot be represented by a state function of the initial and final states alone.The process itself matters because distinct logical mappings can connect the same endpoint macrostates.
- Appendix B: Some Initial Remarks and Clarifications: Two gas processes reaching half the original volume have the same optimal work cost despite implementing different logical mappings.One randomizes particle positions, while the other maps x to x/2 using separators and slice-wise compression.
- Appendix B: Some Initial Remarks and Clarifications: In the i.i.d. thermodynamic limit, the minimal work becomes the difference between final and initial entropy, regardless of the specific logical process.This recovers the state-function behavior associated with macroscopic thermodynamics.
- Appendix B: Some Initial Remarks and Clarifications: Irreversible thermodynamic implementations require additional work beyond the optimal implementation of the requested logical process.Fast compression followed by thermalization is an example of an avoidably irreversible strategy.
- Appendix B: Some Initial Remarks and Clarifications: For an and-gate-like mapping, work fluctuates with the input region: kT ln 2 · log2 3 occurs with probability 3/4, while no work occurs with probability 1/4.The worst-case work is kT ln 2 · log2 3, and many independent particles yield the average 3/4·kT ln 2·log2 3 ≈ 1.2 kT ln 2.
- Appendix B: Some Initial Remarks and Clarifications: With unequal input probabilities, reproducing only the correct output distribution can require less work than implementing the full logical mapping.In the stated example, the output distribution can be reproduced with kT ln 2 work.
Appendix D: Additional Comments. Applications of our Main Result.
The main result lower-bounds the work required to implement a logical process using a conditional entropy quantity, with a smoothed form when approximation is allowed.
- For ε-approximate implementation, the bound is replaced by its smoothed ε-dependent counterpart.
- The work bound is discussed alongside smoothing conventions that exclude very unlikely events from consideration.
1. On the Tightness of the Minimal Work Bound.
The paper establishes an achievable implementation whose work cost approaches the lower bound, but exact tightness remains unresolved because of a gap associated with general unital operations.
- A process constructed using the proposed method achieves the bound up to an error term of order log(1/ε).
- For resetting 1 MB with ε = 10^-10, the error term is about 30 bits, small relative to approximately 10^7 original bits.
- The same error term can become overwhelming for small systems consisting of several qubits.
- Whether the lower bound can be exactly achieved remains an open question, although the gap is sublinear in the number of systems.
2. Simple examples: the AND and XOR gates.
The AND and XOR examples show that minimal work depends on the particular computation and can differ from average entropy-based work requirements.
- AND gate: The best implementation of the AND gate costs kT ln 2 · log2 3 ≈ 1.6 kT ln 2.
- AND gate: The AND gate's exact value applies when the input distribution does not have eigenvalues comparably small to ε.
- The minimal work requirement depends on the specific computation, not only on the input and output states.
- AND gate: For uniformly random input, the AND gate's average work requirement is approximately 1.2 kT ln 2, differing from its minimal work value.
- XOR gate: For a uniformly random input, the XOR example's value coincides with its average work requirement, but the values differ for another input distribution.
3. Arbitrarily large dependence on the computation, with same input and output states.
Two processes can share identical input and output states while having radically different minimal work costs: identity costs nothing, whereas resetting and recreating the state can scale with system size.
- Figure 6 depicts a distribution combining one random qubit with n qubits that are either all pure or uniformly random.
- The identity process E1 leaves the input unchanged, while E2 resets the input and prepares a fresh copy of ρ.
- Both processes have exactly the same input and output states, but identity requires zero work because it can be implemented by doing nothing.
- For E2, the worst-case strategy must reset approximately n bits and therefore costs approximately n kT ln 2, despite extracting at most one bit when preparing the output.
- The exact cost is optimal and can become arbitrarily large as the number of qubits n increases.
4. Erasure of a Quantum System Using a Quantum Memory.
Erasing a quantum system while preserving its correlations with a quantum memory has a work cost governed by conditional max-entropy. The reverse preparation process can instead extract work according to conditional min-entropy, producing a single-shot irreversibility gap.
- Erasure with quantum memory: The erasure task maps a correlated state σ_SM to |0⟩⟨0|_S ⊗ σ_M while preserving correlations with a reference system.The process is formally defined using a purification σ_SMR and requires preserving σ_MR.
- Erasure with quantum memory: The framework applies the general logical-process bound to this erasure mapping and its purified output.The joint input system is taken as S and M, with the environment carrying information transferred from S.
- Erasure with quantum memory: The minimal erasure cost is at least kT ln 2 · Hε_max(S|M)_σ.When M is trivial, this reduces to standard Landauer erasure with cost Hε_max(S).
- Reverse preparation: The reverse process prepares σ_SM from a pure S and memory state, but its exact cost depends on which correlations with M are required.Without specifying the completely positive map or preserved correlations, the preparation scenario is not uniquely defined.
- Reverse preparation: The reverse process can extract kT ln 2 · Hε_min(S|M)_σ, and the min–max entropy difference quantifies the single-shot irreversibility gap.The two values can differ arbitrarily because both processes are required to succeed with high probability.
6. The Minimal Work Cost of a Quantum Measurement.
Quantum measurements are treated as logical processes with a pure outcome register and a work cost determined by the paper’s entropy bound. Measurement itself can yield work in important cases, while resetting its outcome register generally incurs a positive cost that can be reduced using retained quantum information.
- Measurement process: A quantum measurement is modeled as a process that writes outcomes into a pure memory register and produces post-measurement states.The measurement is represented by a POVM, with outcome k occurring with probability tr(Q_kσ).
- Measurement process: The minimal measurement work cost is given by the paper’s entropy expression evaluated on the corresponding purified measurement state.The bound is generally only approximately achievable, as indicated by the paper’s ≈ notation.
- Measurement work cost: Measurements whose collapse operators are sub-unital have W_meas ≤ 0, so they require no work and may yield work.Projective measurements are included among the examples satisfying this condition.
- Resetting the outcome register: Resetting the outcome register directly costs kT ln 2 · Hε_max(C)_ρ, but access to the post-measurement state can lower this cost.The post-measurement system S′ can serve as a memory for resetting C.
- Resetting the outcome register: For single-Kraus collapse operators, the reset cost exceeds the measurement work yield by the difference between conditional max- and min-entropies.The total measurement-plus-reset work is nonnegative.
- Examples: In a maximally mixed single-qubit example, measurement extracts one bit of work, while resetting costs zero using S′ and one bit using a trivial reference system R.The different reset costs arise from the available correlations with the retained systems.
7. State Transformation while Decoupling from the Reference System.
State transformation with complete decoupling from the input reference requires erasing the input information before preparing the output independently. Correlations can therefore determine whether a transformation costs work, even when the input and output reduced states have the same spectrum.
- Decoupled state transformation: A replacement map erases the input and prepares the output independently, requiring the final output to be uncorrelated with the reference system.The condition is ρ_X′R = ρ_X′ ⊗ ρ_R.
- Decoupled state transformation: Under complete decoupling, the minimal work bound corresponds to erasing the input state to a pure state before preparing the required output.The paper relates this cost to the entropy of the input state.
- W-state example: For the W-state example, preserving correlations while erasing S conditioned on M requires at least H0(S|M)_σ = log 2/3 ≈ 0.59 work.Because the system is small, the paper does not assert achievability at this bound, but excludes zero-work implementation.
- W-state example: The same reduced-state transformation can be performed by a unitary at no work cost when correlations with the reference are not preserved.The unitary changes the correlations between the memory and reference systems.
Appendix E: Alternative Proof Using Lambda-Majorization.
The appendix gives an alternative proof of the main result using majorization and semidefinite programming, imposing the required logical process after analyzing possible state transitions.
- Alternative proof: The alternative proof studies state transitions with majorization and semidefinite-programming techniques before enforcing the specified logical process.This proof strategy considers possible state transitions independently of the logical process at first.
1. The Framework. Work cost or yield as generating or absorbing randomness.
The framework defines work through changes in ancilla randomness: creating mixedness yields work, while disposing of randomness costs work. It formalizes these transitions with lambda-majorization and permits reversible information processing subject to restoring ancillas.
- The framework allows erasure of n qubits at n kT ln 2 work cost and the reverse transformation of pure qubits into fully mixed qubits to extract the same amount.
- Global unitaries and pure ancillas support quantum information processing at no work cost, but ancillas must be restored to their initial pure states.
- Noisy operations can be implemented with zero net work and correspond mathematically to majorization, where σ ≻ ρ means that ρ is more mixed than σ.
- Lambda-majorization generalizes majorization by quantifying how much ancilla randomness must be absorbed or supplied when the desired state transition is not achievable by ordinary noisy operations.
- The framework counts work through the change in fully mixed ancilla qubits, with λ = λ1−λ2 positive for extracted work and negative for work cost.
2. The Main Result.
The main result characterizes the least work required to implement a logical process while preserving specified correlations with a reference system. The bound is determined by the conditional entropy of information discarded into the environment.
- The task is to minimize work for a process E that transforms σX, together with its purification, into the specified output state ρX′R.
- λ−→ρX′ holds exactly when λ ≤ −H0(E|X′)ρ, linking the optimal randomness balance to the Rényi-zero conditional entropy.
- Any implementation has work cost at least the entropy of information discarded by the process, conditioned on its output.
- The semidefinite-program formulation provides a broader optimization toolbox when the mapping or output correlations are not completely specified, although an entropy expression is then not established.
b. Proof of the Main Result. Formulation as a Semidefinite Program.
The proof formulates the lambda-majorization task as a semidefinite program and derives its optimum from a Stinespring representation of the logical process. The construction connects subunital maps, majorization, and ancilla randomness.
- b. Proof of the Main Result. Formulation as a Semidefinite Program.: The proof task is to find the maximal λ for a completely positive, 2−λ-subunital, trace-nonincreasing map sending σXR to ρX′R.
- b. Proof of the Main Result. Formulation as a Semidefinite Program.: The optimization is expressed as a semidefinite program over α = 2−λ and the Choi-Jamiołkowski representation of the map.
- b. Proof of the Main Result. Formulation as a Semidefinite Program.: The input and output purifications are related by a partial isometry VX→X′E, which supplies the Stinespring representation used to construct the feasible map.
- b. Proof of the Main Result. Formulation as a Semidefinite Program.: The dual construction attains the same value as the primal program, establishing optimality of the derived λ.
- 1. Preliminaries and Main Definition: Ordinary majorization and weak submajorization are characterized by doubly stochastic and doubly substochastic matrices, respectively, and majorization orders states by mixedness.
- 1. Preliminaries and Main Definition: Lambda-majorization is defined through weak submajorization of states augmented with differently sized fully mixed ancillas, with λ equal to the difference in their logarithmic sizes.
- 1. Preliminaries and Main Definition: The lambda-majorization definition is independent of the particular ancilla sizes chosen, because a fully mixed state cannot act as a catalyst.
2. Formulation of Lambda-Majorization in Terms of Maps
The paper reformulates lambda-majorization using completely positive, trace-nonincreasing maps with a tunable subunital normalization. This map formulation supports dilation to ordinary unital, trace-preserving dynamics on enlarged spaces.
- Weak submajorization is equivalent to the existence of a completely positive map that maps σ to ρ while being subunital and trace-nonincreasing.
- Projecting such an enlarged unital map onto input and output subspaces yields a subunital, trace-nonincreasing map between the original spaces.
- A subunital, trace-nonincreasing completely positive map can be embedded into a unital, trace-preserving completely positive map on a larger Hilbert space.
- An α-subunital map satisfies T(1X) ≤ α1Y, generalizing subunitality to arbitrary normalizations.
- The composition of an α-subunital map with a β-subunital map is α · β-subunital.
- Lambda-majorization is equivalent to a completely positive, trace-nonincreasing map that is 2−λ-subunital and maps σ to ρ.
3. Properties for quantum states
This section develops lambda-majorization properties for normalized quantum states, including criteria involving rank, maximum eigenvalue, and absorbed randomness. It relates absorbed randomness to single-shot entropy measures and gives bounds and special cases.
- For normalized states, weak lambda-majorization automatically implies regular majorization because both states have unit trace.
- A state lambda-majorizes a pure state exactly when its rank satisfies the corresponding bound rank σ ⩽ 2^-λ.
- A state is lambda-majorized by a state ρ exactly when λmax(ρ) ⩽ 2^-λ.
- Absorbed randomness is defined as the maximal randomness removable, or the minimal randomness generable, in a noisy transition.
- Absorbed randomness has tight relations to single-shot entropy measures, with explicit bounds and special-case equalities available.
- The bounds use the min-entropy of the target state and the zero-Rényi entropy of the source state as maximization candidates.