Pith. sign in

REVIEW 3 major objections 4 minor 2 cited by

Erasure cost of a quantum process: A thermodynamic meaning of the dynamical min-entropy

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The thermodynamic cost to erase a quantum gate's output equals minus the gate's min-entropy, up to one-shot finite-size corrections.

desk verdict Genuinely useful channel-level thermodynamics result with a real quantifier gap in Theorem 2; the zero-error resource-theoretic part is solid and worth citing. read the letter →

arxiv 2506.05307 v5 pith:BAWHRHPU submitted 2025-06-05 quant-ph math-phmath.MP

Himanshu Badhani , Dhanuja GS , Swati Choudhary , Vishal Anand , Siddhartha Das This is my paper
classification quant-phmath-phmath.MP MSC 81P4580A10 PACS 03.67.-a05.70.-a
keywords quantumthermodynamicsdynamicalmin-entropychannelerasureone-shotcostdecouplingtheoremresourcetheoryofconditionalnonuniformityPPTchannelsLandauer
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to give the dynamical min-entropy of a quantum channel a concrete thermodynamic meaning: it claims the minimal work needed to erase the output of a single-use quantum gate, even when the input is worst-case, is (up to a smoothing error) minus the channel's min-entropy times $k_B T\ln 2$. The setting is a bipartite unitary gate whose ancillary input starts in the ground state; the eraser may keep the ancilla but cannot touch the reference that purifies the logical input. If the claim holds, the resetting and reuse of logical qubits inside a quantum circuit can be priced one-shot per gate in units of $k_B T\ln 2$, with no need to average over many uses. It also predicts that some channels, those with nonnegative min-entropy including all PPT channels, can be erased while producing work rather than consuming it.

What carries the argument

The load-bearing mechanism is the decoupling theorem for quantum processes, a channel-level analogue of the one-shot decoupling theorem: after applying a Haar-random unitary to the output $A$ of a channel $\mathcal{N}$ and then a trace-subpreserving postprocessing $\mathcal{T}$, the resulting channel is close, in diamond norm and on average over the unitary, to the completely depolarizing channel postprocessed by $\mathcal{T}$, with error bounded by $2^{-(S^\varepsilon_{\min}[\mathcal{N}] + S^\varepsilon_{\min}(A|B)_{\Phi_{\mathcal{T}}})/2} + 12\varepsilon$. This makes $S^\varepsilon_{\min}[\mathcal{N}]$ the quantity that controls how well a process can decouple its output from a purifying reference. A second supporting identity writes $S_{\min}[\mathcal{N}]$ dually as a singlet fidelity after local operations and as the degree to which an isometric extension decouples the output from the environment, which is why the same number appears in both the thermodynamic and resource-theoretic cost formulas.

What would settle it

Consider a qubit depolarizing channel $\mathcal{N}_p$ with a fixed $p$, compute $S_{\min}[\mathcal{N}_p]$ from its Choi state, implement its isometric dilation with a qubit ancilla, and run the erasure protocol repeatedly on worst-case inputs. If the measured work cost exceeds $-S_{\min}[\mathcal{N}_p]k_B T\ln 2$ at zero error, or exceeds $(-S^\varepsilon_{\min}[\mathcal{N}]+\Delta)k_B T\ln 2$ more often than the claimed failure probability $\delta$, the adversarial bound is false; observing work extraction for a PPT channel such as a measurement channel would support it.

Watch

Extended reading notes

Core claim

The central discovery is an identity: for a quantum channel $\mathcal{N}$ obtained from a bipartite unitary gate by tracing out the ancilla, the zero-error resource-theoretic adversarial erasure cost, and also the preparation cost under the dual access pattern, equals $-S_{\min}[\mathcal{N}] k_B T\ln 2$, where $S_{\min}[\mathcal{N}]$ is the channel's min-entropy. For the thermodynamic erasure protocol based on decoupling, the one-shot adversarial cost is bounded above by $(-S^\varepsilon_{\min}[\mathcal{N}] + \Delta) k_B T\ln 2$ with probability at least $1-\delta$ over the protocol's randomness. Both statements make the same entropy functional operational: $S_{\min}[\mathcal{N}]$ quantifies, in thermodynamic units, the irreducible cost, or the extractable work, of resetting the logical output of a quantum process. Its sign separates channels that consume work, such as isometric and unitary channels where $S_{\min}[\mathcal{N}]<0$, from channels that can yield work, such as PPT channels where $S_{\min}[\mathcal{N}]\ge 0$.

Load-bearing premise

The proof of the high-probability worst-case bound assumes that a decoupling statement that holds with high probability for each fixed input also holds uniformly over all inputs, and that the eraser can choose the random unitary after seeing the input; this uniformization step is not proved.

Editorial extensions

If this is right

  • A single use of a gate suffices to set a work budget: the one-shot, adversarial nature of the bounds means they apply without invoking many-copy or averaged limits.
  • Isometric and unitary channels have $S_{\min}[\mathcal{N}] = -\log d$, so resetting their outputs is the most expensive, costing $(\log d)k_B T\ln 2$, while the completely depolarizing channel has $S_{\min}[\mathcal{N}]=\log|A|$ and gives the largest work gain on erasure.
  • PPT channels, including measurement and entanglement-breaking channels, have nonnegative min-entropy, so the framework predicts work can be extracted when their outputs are erased.
  • The preparation cost of a channel obeys the same zero-error identity as erasure, forcing the sum of erasure plus preparation costs of a channel to be nonnegative.
  • The smoothed bound quantifies finite-size corrections: as the protocol approaches zero error and zero failure probability, the added term $\Delta$ in $(-S^\varepsilon_{\min}[\mathcal{N}] + \Delta)k_B T\ln 2$ grows, making the cost of certainty explicit.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the one-shot bound composes under tensor products and the asymptotic equipartition property for channel min-entropy holds at the stated rate, the average work per gate use as $n$ grows would settle on the channel's von Neumann entropy, making the one-shot identity the finite-size refinement of a per-bit erasure law for processes.
  • Because $S_{\min}[\mathcal{N}]<0$ implies the channel is NPT, the erasure cost could serve as a thermodynamic witness of non-PPT-ness, turning a work measurement into a channel-property test.
  • The uniform-over-inputs gap in the probabilistic bound could be probed numerically on qubit channels: if a single randomly drawn unitary fails on some input more often than the stated failure probability while the per-input bound holds, the adversarial statement would require the stronger uniformization guarantee.
  • A direct example such as the swap-based replacer channel shows the same $S_{\min}$ appearing as a cost and as an information quantity, suggesting the result might extend to conditional entropies of bipartite quantum channels.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies the one-shot thermodynamic cost of erasing the output system A of a bipartite unitary gate, modeled as a quantum channel N, when the eraser has access to the ancillary output E but not to the purifying reference R of the input. The main claims are that the adversarial erasure cost W_eras[A|E]_N is bounded by (−S^ε_min[N] + Δ) k_B T ln 2 with probability greater than 1−δ (Theorem 2), and that in the resource-theoretic framework the zero-error adversarial erasure and preparation costs both equal −S_min[N] k_B T ln 2 (Proposition 2). The paper also states a decoupling theorem for channels (Theorem 1), derives a dual expression for the dynamical min-entropy (Proposition 1), proves continuity and monotonicity properties, and gives numerical illustrations for qubit channels.

Significance. The potential significance is high if the main results hold: the paper would give the dynamical min-entropy a direct thermodynamic operational meaning and connect channel decoupling with one-shot thermodynamics. The zero-error resource-theoretic equality, Eq. (38), is well supported by the state-level results of [32] together with Proposition 1, and the paper is generally careful in defining entropic quantities and in providing proofs of supporting lemmas. The main obstacle is the unsupported uniform-convergence step in the proof of Theorem 2, which is load-bearing for the abstract's headline claim and for Eq. (40). The paper also makes a useful contribution by spelling out properties of the dynamical min-entropy and by presenting explicit qubit-channel examples.

major comments (3)
  1. [Appendix C.5, Eqs. (C35)-(C37)] The proof of Theorem 2 proceeds from a bound that is stated for each fixed pure input ψ with probability at least 1−δ over the random unitary U. Taking the supremum over ψ on both sides is not justified: the success event can depend on ψ, and the intersection of the individual success sets over all pure inputs can have arbitrarily small measure. Since W_eras[A|E]_N is defined in Eq. (23) as a supremum over inputs, the conclusion requires a single set of unitaries of measure at least 1−δ on which the bound holds simultaneously for every ψ. No ε-net, union bound, or uniform-convergence argument is supplied. As written, Theorem 2 is not established, and this gap propagates to Eq. (40).
  2. [Appendix C.3, Eq. (C19)] The proof of Theorem 1 contains the same quantifier problem. From the pointwise estimate ∫ ||T∘U∘N(ψ)−T∘R^π(ψ)||_1 dU ≤ ... for each pure ψ, one cannot conclude ∫ ||T∘U∘N−T∘R^π||_⋄ dU ≤ sup_ψ ..., because the diamond norm on the left-hand side requires taking a supremum over ψ inside the integral. The sentence 'Taking the supremum over all pure states ψ on both sides' is therefore not valid without an additional argument, such as a minimax theorem or a direct treatment of the Choi state of the difference map.
  3. [Section IV A, Theorem 2 and Eq. (23)] The statement that the bound holds 'with the probability greater than 1−δ' is ambiguous for the adversarial cost. Since W_eras[A|E]_N is a deterministic supremum over inputs, the probability in Theorem 2 must refer either to a random unitary chosen before seeing the input, in which case a uniform success event across all inputs must be proved, or to a random unitary chosen after seeing the input, in which case the probability is over a random choice that depends on the input and does not give a meaningful guarantee on the adversarial cost. The protocol and the interpretation of δ should be clarified, and the corresponding uniform bound must be proved.
minor comments (4)
  1. [Appendix C.4, Eq. (C33)] The displayed expression contains 'log|A| log F(...)', which appears to be a typo; the intended formula is log(|A| F(...)), matching Proposition 1.
  2. [Appendix D] The phrase 'form the basis of the 2 d Hilbert space' is unclear; it should presumably read '2^d-dimensional Hilbert space' or 'd-dimensional Hilbert space' as appropriate for the N = 2^d − 1 convention implied by the protocol.
  3. [Appendix A heading] The heading 'Review of Rèyni entropies' misspells Rényi.
  4. [Fig. 3 and Section V A] The figure is informative, but the text could state clearly that the cost values are reported in units of k_B T ln 2; this would remove a small ambiguity in the vertical axis.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: dynamical min-entropy and the erasure costs are independently defined, and the main results follow from external decoupling and state-level work-cost theorems.

full rationale

The paper's central objects are introduced independently. The dynamical min-entropy S_min[N] is defined in Eq. (6) via the Choi state and max-relative entropy, following the external reference [31]; it is not defined in terms of any erasure cost. The thermodynamic adversarial erasure cost W_eras[A|E]_N is defined in Eq. (23) as a supremum of state-level erasure costs built on the framework of Ref. [21], and the resource-theoretic costs in Eqs. (34)-(35) use the state-level costs of Ref. [32]. The main inequalities are then derived: Theorem 2 combines the pointwise bound (22) from [21] with Lemma 1, which follows from a max-min inequality; Proposition 2 uses the exact state-level formulas (30)-(31) from [32] together with duality of conditional entropies. No fitted parameter is renamed as a prediction, no target result is inserted into a definition, and no uniqueness claim is imported from the authors' prior work. The self-citations that appear (e.g., Refs. [52], [53], [54], [58], [62]) are used in side remarks, examples, or auxiliary monotoniciy properties and are not load-bearing for the headline result. The reader-flagged step in Appendix C.5—taking the supremum over input states inside a probabilistic bound—is a technical correctness concern about uniform convergence or quantifier order, not a circularity: even if the proof gap were real, the claimed bound would be unsupported rather than true by construction. The derivation is therefore self-contained against external benchmarks and no circular step was found.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central results rest on standard quantum information axioms, the stated domain assumptions (trivial Hamiltonian, ground-state ancilla, single use, no reference access), and the cited external results: the state decoupling theorem [44] and the state-level resource-theoretic cost formulas [32]. No new physical entities are introduced, and no numbers are fitted to data.

free parameters (1)
  • slack parameter Δ
    Introduced in Theorem 2 to trade off the bound's slack against the failure probability δ. It is an error-tradeoff parameter, not fitted to data, but it is a free choice in the statement.
assumptions (5)
  • domain assumption Hamiltonian of the logical output system A is trivial.
    Stated in Section I; all erasure and preparation cost formulas are derived under this assumption.
  • domain assumption Ancilla input E' is initialized in the pure ground state |0><0|.
    Stated in Section I; this makes the reduced dynamics a channel and defines the isometric extension.
  • standard math The state decoupling theorem of [44] (Eq. 12) is valid.
    Used as the starting point for the channel decoupling theorem (Theorem 1) and the erasure cost bound (Theorem 2).
  • domain assumption The resource-theoretic zero-error preparation and erasure cost formulas for states from [32] (Eqs. 30-31) are correct.
    Proposition 2 directly applies these formulas to channel-output states; if they are incorrect, the equality (38) fails.
  • domain assumption The work extraction protocol from a pure state (Appendix D, following [21]) extracts exactly log-dimension k_B T ln 2 work.
    This is the standard model of work extraction from a pure state with a thermal reservoir, used in the derivation of the erasure cost.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Erasure cost of a quantum process: A thermodynamic meaning of the dynamical min-entropy." pith.science (2026). https://pith.science/paper/BAWHRHPU

@misc{pith2026250605307,
  author       = {Pith},
  title        = {Pith review of: Erasure cost of a quantum process: A thermodynamic meaning of the dynamical min-entropy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BAWHRHPU}},
  note         = {Machine review of arXiv:2506.05307}
}
read the original abstract

The erasure of information is fundamentally an irreversible logical operation, carrying profound consequences for the energetics of computation and information processing. We investigate the thermodynamic costs associated with erasing (and preparing) quantum processes. Specifically, we analyze an arbitrary bipartite unitary gate acting on logical and ancillary input-output systems, where the ancillary input is always initialized in the ground state. We focus on the adversarial erasure cost of the reduced dynamics - that is, the minimal thermodynamic work cost to erase the logical output of the gate for any logical input, assuming full access to the ancilla but no access to any purifying reference of the logical input state. We determine that this adversarial erasure cost is directly proportional to the negative min-entropy of the reduced dynamics, thereby giving the dynamical min-entropy a clear operational meaning. The dynamical min-entropy can take positive and negative values, depending on the underlying quantum dynamics. The negative value of the erasure cost implies that the extraction of thermodynamic work is possible instead of its consumption during the process. A key foundation of this result is the quantum process decoupling theorem, which quantitatively relates the decoupling ability of a process with its min-entropy. This insight bridges thermodynamics, information theory, and the fundamental limits of quantum computation.

Figures

Figures reproduced from arXiv: 2506.05307 by the authors.

Figure 1
Figure 1. FIG. 1. Pictorial representation of the main problem: We consider a bipartite unitary gate (channel) [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The picture illustrates the mechanism of the decoupling theorem for quantum channels. Decoupling theorem provides a fundamental [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. We plot the numerical values for the negative of the dynamical min-entropy [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

Discussion (0). Sign in to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Thermodynamics of quantum processes: An operational framework for free energy and reversible athermality

    quant-ph 2025-10 conditional novelty 5.0 of 10

    For quantum channels, athermality distillation and formation under Gibbs-preserving superchannels both converge asymptotically to the channel's relative-entropy free energy, making the resource theory asymptotically r...

  2. Maximum entropy principle for quantum processes

    quant-ph 2025-06 reject novelty 4.0 of 10

    The paper's central theorem, that energy-constrained quantum channels maximize channel entropy if and only if they are absolutely thermalizing, is false: other channels can also reach the maximum.

Reference graph

Works this paper leans on

91 extracted references · 60 canonical work pages · cited by 2 Pith papers

  1. [32]

    Strasberg and A

    P. Strasberg and A. Winter, First and second law of quantum thermodynamics: A consistent derivation based on a microscopic definition of entropy, PRX Quantum 2, 030202 (2021)

    2021

  2. [1]

    Review of Rèyni entropies In this appendix, we recall the entropic quantities and their properties from [27, 38, 39, 63–66] necessary to understand the results. Sandwiched Rényi relative entropies .– The family of quantum sandwiched Rényi relative entropy between ρ∈ St(A) and σ∈ Pos(A) is given by [65, 67], Dα(ρ∥σ) = 1 α− 1 log tr n σ 1−α 2α ρσ 1−α 2α αo ...

  3. [2]

    Monotonically nondecreasing under the action of R-preserving superchannels Ω, i.e., S[N]≤ S[Ω(N)], for an arbitrary quantum channelN. 15

  4. [3]

    LetM,N be a pair of quantum channels, then S[N⊗M ] = S[N] + S[M]

  5. [4]

    For a replacer channelRω, S[Rω] = S(ω). The sandwiched Rènyi relative channel entropy [73] between two quantum channelsNA′→A andMA′→A is given by Dα[N∥M] = sup ψ∈St RA′ Dα(idR⊗N(ψRA′)∥ idR⊗M(ψRA′)), (A26) where it suffices to optimize over pure statesψRA′ and R≃ A′. The sandwiched Rényi entropy of a quantum channelNA′→A is defined as [31], forα∈ (1,∞) Sα[...

  6. [5]

    Given a quantum channelNA′→A and limα∈{ 1,∞}, we have Sε α[N]≤ inf |ψ⟩⟨ψ|∈St(RA′) Sε α(A|R)N(ψ)

    Proof of Lemma 1 Lemma. Given a quantum channelNA′→A and limα∈{ 1,∞}, we have Sε α[N]≤ inf |ψ⟩⟨ψ|∈St(RA′) Sε α(A|R)N(ψ). (C1) Proof. We note that ifM∈B ε[N] for a quantum channelNA′→A, thenM(φRA′)∈B ε(N(φRA′)). This follows from the definition of the purified distance. For an arbitrary stateφRA′ andα∈ (1,∞), we have Sε α(A|R)N(φ) = sup ρ∈Bε(N(φRA′ )) Sα(A...

  7. [6]

    Given two quantum channelsNA′→A andMA′→A such that 1 2∥N−M∥ ⋄≤ δ, the respective dynamical min-entropies satisfy |S min[N]− S min[M]|≤ 1 ln 2|A| min{|A|,|A′|}δ

    Proof of Lemma 2 Lemma. Given two quantum channelsNA′→A andMA′→A such that 1 2∥N−M∥ ⋄≤ δ, the respective dynamical min-entropies satisfy |S min[N]− S min[M]|≤ 1 ln 2|A| min{|A|,|A′|}δ. (C8) Proof. Note that 1 2∥N−M∥ ⋄≤ δ implies 1 2∥N(ρRA′)−M (ρRA′)∥1≤ δ for all states ρRA′, and it suffices to consider R≃ A′ (in general|R|≥| A′|). From the uniform continu...

  8. [7]

    LetNA′→A be a quantum channel, TA→B a completely positive map such that tr(ΓT AB)≤| A|, and ε∈ (0, 1)

    Proof of Theorem 1 Theorem. LetNA′→A be a quantum channel, TA→B a completely positive map such that tr(ΓT AB)≤| A|, and ε∈ (0, 1). The distance of the channelN post-processed byT◦U A when UA is chosen uniformly at random from the Haar measure over the full unitary group U on A, with the uniformly randomizing channelRπ post-processed byT is upper bounded a...

Show all 91 references
  1. [8]

    Proof of Proposition 1 Proposition. The min-entropy S min[N] of a quantum channelNA′→A is S min[N] =− sup |ψ⟩⟨ψ|RA′ sup M∈Ch(R, ¯A) log(|A|F(M⊗N (ψRA′), ΦA ¯A)) (C24) =− sup ρ∈St(A′) sup σ∈St(E) log |A|F(VN A′→AE(ρA′),π A⊗σE) , (C25) whereVN A′→AE is an isometric extension cha...

  2. [9]

    compression of correlations

    Proof of Theorem 2 Theorem. The adversarial erasure cost Weras[A|E]N of a quantum channelNA′→A is bounded as Weras[A|E]N≤ −Sε min[N] + ∆ kBT ln 2, (C34) with the probability greater than 1−δ, whereδ := √ 2−∆/2 + 12ε, for allδ,ε> 0. Proof. Employing the results of [21], we obta...

  3. [10]

    1 4 1− µ 1−µ2 !# . (C44) Considering thatρ∈ St(AB) andσAB = 1 A⊗ρB, we get log

    Proof of Lemma 3 Lemma. Given a state ρAB, the sum of the work cost of erasing and preparing the system A conditioned on the system B, for errorµ∈ [0, 1], is bounded from below as eWµ prep(A|B)ρ + eWµ eras(A|B)ρ≥ " log 1− µ 1−µ2 ! − 2 # kBT ln 2, (C41) and forµ = 0, i.e., zero...

  4. [11]

    − inf σ∈St(E) Dµ H(VN(ρA′)∥1 A⊗σE) # , (C54) then sup ρA′ S↑,µ H (A|E)VN (ρ)≤ sup ρ∈St(A′)

    Proof of Proposition 2 Proposition. Given a quantum channelNA′→A with an isometric extensionVN A′→AE and a reference R, the work costs of prepar- ing and erasing a channel when a reservoir (bath) is at a fixed temperature T, are bounded from the above as, forµ∈ (0, 1) eWµ prep...

  5. [12]

    We can do this by manipulating the parameters of the Hamiltonian, e.g

    Increase the energy levels of the states{|φi⟩}N−1 1 to a very high value E→∞ , where the ground state energy is taken to be E0 = 0. We can do this by manipulating the parameters of the Hamiltonian, e.g. the strength of the magnetic field. Since these energy levels are empty, t...

  6. [13]

    In this configuration, the higher energy levels are occupied with probability p(E) = N/(N + e E kBT ), which vanishes in the limit E→∞ for any finite temperature

    Attach the system to a thermal bath at temperature T. In this configuration, the higher energy levels are occupied with probability p(E) = N/(N + e E kBT ), which vanishes in the limit E→∞ for any finite temperature

  7. [14]

    Doing this isothermally yields the energy of dkBT ln 2 joules

    Lowering the energy of the states {|φi⟩}N−1 1 , their occupancy increases. Doing this isothermally yields the energy of dkBT ln 2 joules. The erasure process is exactly the opposite of the above protocol, where one converts the maximally mixed state into a pure state, resultin...

  8. [15]

    Peres, Reversible logic and quantum computers, Physical Review A 32, 3266 (1985)

    A. Peres, Reversible logic and quantum computers, Physical Review A 32, 3266 (1985)

    1985

  9. [16]

    G. M. Palma, K.-A. Suominen, and A. K. Ekert, Quantum computers and dissipation, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 452, 567 (1996)

    1996

  10. [17]

    Verstraete, M

    F. Verstraete, M. M. Wolf, and J. Ignacio Cirac, Quantum computation and quantum-state engineering driven by dissipation, Nature Physics 5, 633 (2009)

    2009

  11. [18]

    Sagawa, Thermodynamics of information processing in small systems, Progress of theoretical physics 127, 1 (2012)

    T. Sagawa, Thermodynamics of information processing in small systems, Progress of theoretical physics 127, 1 (2012)

    2012

  12. [19]

    Klatzow, J

    J. Klatzow, J. N. Becker, P. M. Ledingham, C. Weinzetl, K. T. Kaczmarek, D. J. Saunders, J. Nunn, I. A. Walmsley, R. Uzdin, and E. Poem, Experimental demonstration of quantum e ffects in the operation of microscopic heat engines, Physical Review Letters 122, 110601 (2019)

    2019

  13. [20]

    Au ffèves, Quantum technologies need a quantum energy initiative, PRX Quantum 3, 020101 (2022)

    A. Au ffèves, Quantum technologies need a quantum energy initiative, PRX Quantum 3, 020101 (2022)

    2022

  14. [21]

    Das and U

    S. Das and U. Sen, Maximum entropy principle for quantum processes (2025), arXiv:2506.24079

    2025 arXiv

  15. [22]

    A. Y . Kitaev, Quantum computations: algorithms and error correction, Russian Mathematical Surveys52, 1191 (1997)

    1997

  16. [23]

    L. J. Schulman and U. V . Vazirani, Molecular scale heat engines and scalable quantum computation, in Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing , STOC ’99 (Association for Computing Machinery, New York, NY , USA, 1999) p. 322–329

    1999

  17. [24]

    Shizume, Heat generation required by information erasure, Physical Review E 52, 3495 (1995)

    K. Shizume, Heat generation required by information erasure, Physical Review E 52, 3495 (1995)

    1995

  18. [25]

    C. H. Bennett, Notes on Landauer’s principle, reversible computation, and Maxwell’s demon, Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 34, 501 (2003), Quantum Information and Computation

    2003

  19. [26]

    Landauer, Irreversibility and heat generation in the computing process, IBM Journal of Research and Development 5, 183 (1961)

    R. Landauer, Irreversibility and heat generation in the computing process, IBM Journal of Research and Development 5, 183 (1961)

    1961

  20. [27]

    C. H. Bennett, Logical reversibility of computation, IBM Journal of Research and Development 17, 525 (1973)

    1973

  21. [28]

    Bérut, A

    A. Bérut, A. Arakelyan, A. Petrosyan, S. Ciliberto, R. Dillenschneider, and E. Lutz, Experimental verification of Landauer’s principle linking information and thermodynamics, Nature 483, 187 (2012)

    2012

  22. [29]

    Cabello, M

    A. Cabello, M. Gu, O. Gühne, J.-A. Larsson, and K. Wiesner, Thermodynamical cost of some interpretations of quantum theory, Physical Review A 94, 052127 (2016)

    2016

  23. [30]

    Misra, U

    A. Misra, U. Singh, S. Bhattacharya, and A. K. Pati, Energy cost of creating quantum coherence, Physical Review A 93, 052335 (2016)

    2016

  24. [31]

    Alicki and M

    R. Alicki and M. Horodecki, Information-thermodynamics link revisited, Journal of Physics A: Mathematical and Theoretical52, 204001 (2019)

    2019

  25. [33]

    J. M. R. Parrondo, Thermodynamics of information (2023), arXiv:2306.12447

    2023 arXiv

  26. [34]

    Reeb and M

    D. Reeb and M. M. Wolf, An improved Landauer principle with finite-size corrections, New Journal of Physics 16, 103011 (2014)

    2014

  27. [35]

    L. d. Rio, J. Åberg, R. Renner, O. Dahlsten, and V . Vedral, The thermodynamic meaning of negative entropy, Nature474, 61–63 (2011), see also arXiv:1009.1630v2

    2011 arXiv

  28. [36]

    Faist, F

    P. Faist, F. Dupuis, J. Oppenheim, and R. Renner, The minimal work cost of information processing, Nature Communications 6, 10.1038/ncomms8669 (2015)

    2015 doi

  29. [37]

    M. N. Bera, A. Riera, M. Lewenstein, and A. Winter, Generalized laws of thermodynamics in the presence of correlations, Nature communications 8, 2180 (2017)

    2017

  30. [38]

    Galindo and M

    A. Galindo and M. A. Martín-Delgado, Information and computation: Classical and quantum aspects, Reviews of Modern Physics 74, 347 (2002)

    2002

  31. [39]

    S. Das, S. Bäuml, M. Winczewski, and K. Horodecki, Universal limitations on quantum key distribution over a network, Physical Review X 11, 041016 (2021)

    2021

  32. [40]

    Bernstein and U

    E. Bernstein and U. Vazirani, Quantum complexity theory, in Proceedings of the twenty-fifth annual ACM symposium on Theory of computing (1993) pp. 11–20

    1993

  33. [41]

    Tomamichel, Quantum information processing with finite resources – mathematical foundations (2021), arXiv:1504.00233v5

    M. Tomamichel, Quantum information processing with finite resources – mathematical foundations (2021), arXiv:1504.00233v5

    2021 arXiv

  34. [42]

    Meier and H

    F. Meier and H. Yamasaki, Energy-consumption advantage of quantum computation, PRX Energy 4, 023008 (2025)

    2025

  35. [43]

    W. F. Stinespring, Positive functions on C*-algebras, Proceedings of the American Mathematical Society 6, 211 (1955)

    1955

  36. [44]

    Lindblad, Completely positive maps and entropy inequalities, Communications in Mathematical Physics 40, 147 (1975)

    G. Lindblad, Completely positive maps and entropy inequalities, Communications in Mathematical Physics 40, 147 (1975). 22

    1975

  37. [45]

    Gour and M

    G. Gour and M. M. Wilde, Entropy of a quantum channel, Physical Review Research 3, 10.1103/physrevresearch.3.023096 (2021)

    2021 doi

  38. [46]

    K. Ji, G. Gour, and M. M. Wilde, Fundamental work costs of preparation and erasure in the presence of quantum side information (2025), arXiv:2503.09012 [quant-ph]

    2025 arXiv

  39. [47]

    Horodecki, P

    M. Horodecki, P. W. Shor, and M. B. Ruskai, Entanglement breaking channels, Reviews in Mathematical Physics 15, 629–641 (2003)

    2003

  40. [48]

    E. P. Hanson, C. Rouzé, and D. S. França, Eventually entanglement breaking Markovian dynamics: Structure and characteristic times, Annales Henri Poincaré 21, 1517 (2020)

    2020

  41. [49]

    Eggeling, K

    T. Eggeling, K. G. H. V ollbrecht, R. F. Werner, and M. M. Wolf, Distillability via protocols respecting the positivity of partial transpose, Physical Review Letters 87, 257902 (2001)

    2001

  42. [50]

    Smith and J

    G. Smith and J. Yard, Quantum communication with zero-capacity channels, Science 321, 1812 (2008)

    2008

  43. [51]

    Datta, Min- and max-relative entropies and a new entanglement monotone, IEEE Transactions on Information Theory55, 2816 (2009)

    N. Datta, Min- and max-relative entropies and a new entanglement monotone, IEEE Transactions on Information Theory55, 2816 (2009)

    2009

  44. [52]

    Renner, Security of quantum key distribution (2006), arXiv:quant-ph/0512258 [quant-ph]

    R. Renner, Security of quantum key distribution (2006), arXiv:quant-ph/0512258 [quant-ph]

    2006 arXiv

  45. [53]

    Buscemi and N

    F. Buscemi and N. Datta, The quantum capacity of channels with arbitrarily correlated noise, IEEE Transactions on Information Theory 56, 1447 (2010)

    2010

  46. [54]

    Wang and R

    L. Wang and R. Renner, One-shot classical-quantum capacity and hypothesis testing, Physical Review Letters 108, 200501 (2012)

    2012

  47. [55]

    Piechocinska, Information erasure, Physical Review A 61, 062314 (2000)

    B. Piechocinska, Information erasure, Physical Review A 61, 062314 (2000)

    2000

  48. [56]

    Mohan, S

    B. Mohan, S. Das, and A. K. Pati, Quantum speed limits for information and coherence, New Journal of Physics 24, 065003 (2022)

    2022

  49. [57]

    Here, work cost is estimated per copy when asymptotically many copies (ρ⊗n for n→∞ ) of the stateρ is available

  50. [58]

    Dupuis, M

    F. Dupuis, M. Berta, J. Wullschleger, and R. Renner, One-shot decoupling, Communications in Mathematical Physics 328, 251 (2014)

    2014

  51. [59]

    Majenz, M

    C. Majenz, M. Berta, F. Dupuis, R. Renner, and M. Christandl, Catalytic decoupling of quantum information, Physical Review Letters 118, 080503 (2017)

    2017

  52. [60]

    Konig, R

    R. Konig, R. Renner, and C. Schaffner, The operational meaning of min- and max-entropy, IEEE Transactions on Information Theory55, 4337–4347 (2009)

    2009

  53. [61]

    Gour, Comparison of quantum channels by superchannels, IEEE Transactions on Information Theory 65, 5880 (2019)

    G. Gour, Comparison of quantum channels by superchannels, IEEE Transactions on Information Theory 65, 5880 (2019)

    2019

  54. [62]

    Tomamichel, R

    M. Tomamichel, R. Colbeck, and R. Renner, Duality between smooth min- and max-entropies, IEEE Transactions on Information Theory 56, 4674–4681 (2010), arXiv:0907.5238

    2010 arXiv

  55. [63]

    E. M. Rains, A semidefinite program for distillable entanglement (2001), quant-ph /0008047

    2001

  56. [64]

    E. Kaur, S. Das, M. M. Wilde, and A. Winter, Extendibility limits the performance of quantum processors, Physical Review Letters 123, 070502 (2019)

    2019

  57. [65]

    S. Das, S. Khatri, G. Siopsis, and M. M. Wilde, Fundamental limits on quantum dynamics based on entropy change, Journal of Mathe- matical Physics 59, 10.1063/1.4997044 (2018)

    2018 doi

  58. [66]

    Pandey, U

    Sohail, V . Pandey, U. Singh, and S. Das, Fundamental limitations on the recoverability of quantum processes, Annales Henri Poincaré 10.1007/s00023-025-01590-y (2025)

    2025 doi

  59. [67]

    Badhani, D

    H. Badhani, D. G. S, and S. Das, Thermodynamics of quantum processes: An operational framework for free energy and reversible athermality (2025), arXiv:2510.12790 [quant-ph]

    2025 arXiv

  60. [68]

    Badhani, D

    H. Badhani, D. G. S, and S. Das, Thermodynamic work capacity of quantum information processing (2025), arXiv:2510.23731 [quant- ph]

    2025

  61. [69]

    Devetak and P

    I. Devetak and P. W. Shor, The capacity of a quantum channel for simultaneous transmission of classical and quantum information, Communications in Mathematical Physics 256, 287 (2005)

    2005

  62. [70]

    Yu and J

    T. Yu and J. H. Eberly, Qubit disentanglement and decoherence via dephasing, Phys. Rev. B68, 165322 (2003)

    2003

  63. [71]

    D. P. DiVincenzo, Quantum computation and spin physics (invited), Journal of Applied Physics 81, 4602 (1997), https://pubs.aip.org/aip/jap/article-pdf/81/8/4602/18691873/4602_1_online.pdf

    1997

  64. [72]

    S. Das, K. Goswami, and V . Pandey, Conditional entropy and information of quantum processes (2024), arXiv:2410.01740

    2024 arXiv

  65. [73]

    Singh and N

    S. Singh and N. Datta, Detecting positive quantum capacities of quantum channels, npj Quantum Information 8, 10.1038/s41534-022- 00550-2 (2022)

    2022 doi

  66. [74]

    D. Yang, K. Horodecki, and A. Winter, Distributed private randomness distillation, Physical Review Letters 123, 170501 (2019)

    2019

  67. [75]

    Konig and R

    R. Konig and R. Renner, Sampling of min-entropy relative to quantum knowledge, IEEE Transactions on Information Theory 57, 4760 (2011)

    2011

  68. [76]

    Das, Bipartite quantum interactions: Entangling and information processing abilities (2019), arXiv:1901.05895

    S. Das, Bipartite quantum interactions: Entangling and information processing abilities (2019), arXiv:1901.05895

    2019 arXiv

  69. [77]

    Petz, Quasi-entropies for states of a von Neumann algebra, Publications of the Research Institute for Mathematical Sciences 21, 787 (1985)

    D. Petz, Quasi-entropies for states of a von Neumann algebra, Publications of the Research Institute for Mathematical Sciences 21, 787 (1985)

    1985

  70. [78]

    Petz, Quasi-entropies for finite quantum systems, Reports on Mathematical Physics 23, 57 (1986)

    D. Petz, Quasi-entropies for finite quantum systems, Reports on Mathematical Physics 23, 57 (1986)

    1986

  71. [79]

    M. M. Wilde, A. Winter, and D. Yang, Strong converse for the classical capacity of entanglement-breaking and Hadamard channels via a sandwiched Rényi relative entropy, Communications in Mathematical Physics331, 593 (2014), arXiv:1306.1586

    2014 arXiv

  72. [80]

    Tomamichel, M

    M. Tomamichel, M. Berta, and M. Hayashi, Relating di fferent quantum generalizations of the conditional Rényi entropy, Journal of Mathematical Physics 55 (2014)

    2014

  73. [81]

    MüllerLennert, F

    M. MüllerLennert, F. Dupuis, O. Szehr, S. Fehr, and M. Tomamichel, On quantum Rényi entropies: a new definition and some properties, Journal of Mathematical Physics 54, 122203 (2013), arXiv:1306.3142

    2013 arXiv

  74. [82]

    R. L. Frank and E. H. Lieb, Monotonicity of a relative Rényi entropy, Journal of Mathematical Physics 54, 122201 (2013)

    2013

  75. [83]

    Regula, L

    B. Regula, L. Lami, and N. Datta, Tight relations and equivalences between smooth relative entropies (2025), arXiv:2501.12447 [quant- ph]

    2025 arXiv

  76. [84]

    Anshu, M

    A. Anshu, M. Berta, R. Jain, and M. Tomamichel, A minimax approach to one-shot entropy inequalities, Journal of Mathematical Physics 60, https://doi.org/10.1063/1.5126723 (2019)

    2019 doi

  77. [85]

    Datta, M

    N. Datta, M. Mosonyi, M.-H. Hsieh, and F. G. S. L. Brandão, A smooth entropy approach to quantum hypothesis testing and the classical capacity of quantum channels, IEEE Transactions on Information Theory 59, 8014 (2013). 23

    2013

  78. [86]

    Dupuis, L

    F. Dupuis, L. Krämer, P. Faist, J. M. Renes, and R. Renner, Generalized entropies, in XVIIth International Congress on Mathematical Physics (WORLD SCIENTIFIC, 2013) p. 134–153, arXiv:1211.3141

    2013 arXiv

  79. [87]

    Cooney, M

    T. Cooney, M. Mosonyi, and M. M. Wilde, Strong converse exponents for a quantum channel discrimination problem and quantum- feedback-assisted communication, Communications in Mathematical Physics 344, 797 (2016), arXiv:1408.3373

    2016 arXiv

  80. [88]

    Yuan, Hypothesis testing and entropies of quantum channels, Physical Review A 99, 032317 (2019)

    X. Yuan, Hypothesis testing and entropies of quantum channels, Physical Review A 99, 032317 (2019)

    2019

  81. [89]

    Fan, Minimax theorems*, Proceedings of the National Academy of Sciences 39, 42 (1953), https://www.pnas.org/doi/pdf/10.1073/pnas.39.1.42

    K. Fan, Minimax theorems*, Proceedings of the National Academy of Sciences 39, 42 (1953), https://www.pnas.org/doi/pdf/10.1073/pnas.39.1.42

    1953 doi

  82. [90]

    J. M. Danskin, The theory of max-min, with applications, SIAM Journal on Applied Mathematics 14, 641 (1966), https://doi.org/10.1137/0114053

    1966 doi

  83. [91]

    Marwah and F

    A. Marwah and F. Dupuis, Uniform continuity bound for sandwiched Rényi conditional entropy, Journal of Mathematical Physics 63, 052201 (2022), https://pubs.aip.org/aip/jmp/article-pdf/doi/10.1063/5.0088507/16561116/052201_1_online.pdf

    2022 doi

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.