Pith. sign in

REVIEW 4 major objections 4 minor 1 cited by

Bound on efficiency of heat engine from uncertainty relation viewpoint

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read The paper derives upper and lower bounds on a quantum Stirling engine's efficiency directly from the position-momentum uncertainty relation.

desk verdict The efficiency bounds are not derived from uncertainty: Eq. (17) is a fitted reparameterization, and D and E just rescale the bath temperatures. read the letter →

arxiv 1908.06804 v2 pith:DXEG2VDS submitted 2019-08-19 quant-ph

Pritam Chattopadhyay , Ayan Mitra , Goutam Paul , Vasilios Zarikas This is my paper · ORCID
classification quant-ph PACS 03.65.-w05.70.-a
keywords quantumheatengineuncertaintyrelationStirlingcycleinfinitepotentialwellefficiencyboundpartitionfunctionthermodynamicvariablesentropy
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

The paper tries to show that the thermodynamic performance of a quantum heat engine can be expressed directly through the uncertainty relation for position and momentum, with no measurement performed on the working medium. Using a particle in a one-dimensional infinite well as the working substance of a Stirling cycle, it derives an approximate identity linking the partition function to the thermal sum uncertainty $\Delta X_T+\Delta P_T+C_T$, and from that identity writes the Helmholtz free energy, entropy, work, and efficiency in uncertainty terms. The resulting efficiency formula, Eq. (30), yields upper and lower bounds that the paper reports as consistent with the Carnot bound for the stated hot and cold bath temperatures. If the derivation holds, uncertainty relations become a predictive handle on engine performance, and a single known uncertainty relation could replace repeated state preparation and measurement in analysing quantum cycles.

What carries the argument

The load-bearing object is the approximate identity Eq. (17), $Z=\frac{L\sqrt{2}}{\hbar\sqrt{\pi}}(\Delta X_T+\Delta P_T+C_T)$, which converts the partition function of the trapped particle into a linear function of the thermal sum uncertainty, with $C_T$ a temperature-dependent constant obtained by expanding Eq. (10) and dropping higher-order terms in $\alpha\beta$. This identity lets the paper rewrite the partition functions at each stage of the Stirling cycle as uncertainty expressions, and the resulting $D$ and $E$ coefficients carry those expressions into the efficiency formula. Supporting it are the sum-uncertainty bounds: the lower bound on $\Delta X^2+\Delta P^2$ for incompatible observables and the reverse-uncertainty upper bound derived from an inequality relating the sum of uncertainties to the uncertainty of the difference. The Stirling cycle enters through the standard four-stage analysis of isothermal barrier insertion, isochoric cooling, isothermal removal, and isochoric heating, whose work and efficiency are re-expressed in terms of $D$ and $E$.

What would settle it

Compute the exact partition function $Z=\sum_n e^{-\beta E_n}$ and the exact sum uncertainty $\Delta X_T+\Delta P_T$ at $T_1=320$ K and $T_2=80$ K for $L$ from 1 to 5 nm, then test whether Eq. (17) holds; if the relative error is not small, the efficiency bounds in Eq. (30) do not hold as stated.

Watch

Extended reading notes

Core claim

The central claim is Eq. (30): for a quantum Stirling engine whose working substance is a particle of mass $m$ in a box of width $2L$ coupled to hot and cold baths at temperatures $T_1$ and $T_2$, the efficiency is $$\eta = \frac{D\ln(Z_B/Z_A)+E\ln(Z_D/Z_C)}{D(\ln(Z_B/Z_A)+1/2)-E/2},$$ where $D=\frac{8L^2}{\pi^3\hbar^2}(\Delta X_{T_1}+\Delta P_{T_1}+C_{T_1})^2$ and $E$ is the same combination evaluated at $T_2$. Because $D$ and $E$ are built solely from the thermal position-momentum sum uncertainty plus a temperature-dependent constant, the paper takes this to show that the efficiency bounds are governed by the uncertainty relation. The paper also reports that the upper bound so obtained stays consistent with the Carnot bound $1-T_2/T_1$, and that the lower bound decreases as uncertainty grows, saturating at large uncertainty.

Load-bearing premise

The argument leans on Eq. (17)'s truncated expansion: if the discarded higher-order terms are not negligible for the stated temperatures and box lengths, then D and E are not faithful measures of the bath temperatures and the efficiency bounds lose thermodynamic meaning.

Editorial extensions

If this is right

  • Efficiency bounds for the quantum Stirling engine can be stated using $\Delta X_T+\Delta P_T+C_T$, so the cycle can be analysed without projective measurements or multiple state copies.
  • The upper bound follows the Carnot limit $1-T_2/T_1$, so the uncertainty formulation reproduces the second-law ceiling for this model.
  • The lower efficiency bound decreases as sum uncertainty increases and saturates at large uncertainty, making the conversion ratio a function of the working medium's quantum spread.
  • Helmholtz free energy and entropy are written as functions of the uncertainty terms, giving an uncertainty-based route into the rest of the engine's thermodynamics.

Reading between the lines

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

  • A natural next test, not given in the paper, is to check whether the same uncertainty-based bounds survive when the potential is finite or anharmonic; if they do, the method would generalize beyond particle-in-a-box.
  • The paper's entropy-uncertainty link suggests that entanglement estimates could be read off from measured sum uncertainty for any exactly solvable model, but the paper only raises this as an open direction.
  • Treating $C_T$ as a fitting parameter rather than a computed constant would turn Eq. (17) into an empirical relation, allowing experiment to decide how much of the efficiency bound is genuinely uncertainty-driven.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The manuscript attempts to connect the thermal sum uncertainty relation of position and momentum for a particle in a one-dimensional infinite potential well to thermodynamic quantities, and claims to derive upper and lower bounds on the efficiency of a quantum Stirling engine from that uncertainty relation. The central construction is Eq. (17), which expresses the partition function Z in terms of (ΔX_T + ΔP_T + C_T), and Eqs. (29)-(30), which express work and efficiency through constants D and E built from the same uncertainty sum. The paper asserts that the resulting efficiency bounds are consistent with the Carnot bound and that this establishes a direct link between quantum uncertainty and thermodynamic performance.

Significance. The question addressed is meaningful: if a genuine two-sided connection between uncertainty relations and heat-engine efficiency existed, it would provide a fundamental quantum constraint on thermodynamic cycles and could be of broad interest. The paper also contains explicit analytic expressions for thermal variances in a box, which are useful intermediate results. However, the central bridge is not a derived consequence of an uncertainty relation but an identity constructed by defining C_T; consequently, the claimed uncertainty-based bounds reduce to a relabeling of the bath temperatures under the exact form of Eq. (17), or lose thermodynamic meaning if the truncation is not exact. As presented, the central claim is therefore unsupported, and the manuscript does not establish a new uncertainty-relation constraint on efficiency.

major comments (4)
  1. [IV, Eq. (17)] The bridge Z = π n̄/2 = L√2/(ℏ√π)(ΔX_T + ΔP_T + C_T) is not a theorem but an identity by construction: C_T is defined by expanding Eq. (10) and discarding higher-order terms, with no error estimate. If the discarded terms are not negligible for the stated T1=320 K, T2=80 K and L in the nanometer range, then D and E in Eq. (30) are not proportional to the bath temperatures and the efficiency expression has no thermodynamic meaning. If the equality were exact, D and E would simply be rescaled values of Z^2 ∝ k_B T, and Eq. (30) would reproduce the standard Stirling efficiency with temperatures relabeled, so the uncertainty relation would impose no independent constraint. Either way, the central claim that the uncertainty relation determines the efficiency bounds is not supported.
  2. [II, Eq. (10) and IV, Eq. (17)] Equation (10) adds ΔX_T, which has dimensions of length, and ΔP_T, which has dimensions of momentum; no unit convention (for example ℏ=m=1) is stated that makes this addition meaningful. The constant C_T is also a length, so the right-hand side of Eq. (17) has dimensions of (length)^2/(ℏ), which is not the dimensionless partition function. The same dimensional problem propagates into D and E in Eq. (30), and into the work expression in Eq. (29), where with Eq. (17) the prefactor 8L^2α/(ℏ^2π^2)=1/m leaves W with dimensions of inverse mass unless unstated units are assumed.
  3. [III, Eqs. (13)-(15)] The upper bound is misapplied. For the symmetric box eigenstates, Cov(X,P)=(1/2)⟨XP+PX⟩−⟨X⟩⟨P⟩=0, so the denominator in Eq. (14) is 1 and the Dunkl–Williams inequality reduces to the trivial statement (ΔX−ΔP)^2 ≥ 0. The right-hand side of Eq. (15), L^2/3 − 2L^2/(nπ)^2 + π^2ℏ^2n^2/(4L^2), is exactly ΔX^2+ΔP^2, the quantity under bound, and it does not follow from Eq. (14). Thus the claimed upper bound on the sum of variances is not an independent uncertainty-relation constraint.
  4. [V, Eq. (30) and Fig. 6] The paper claims upper and lower bounds on efficiency from the uncertainty relation, but no procedure is given for converting Eq. (30) into bounds. D and E are determined by the temperatures and the box length, not by an uncertainty inequality, and the bounds on ΔX^2+ΔP^2 developed in Section III do not translate into two-sided bounds on (ΔX_T+ΔP_T+C_T)^2 that would bound the efficiency. Figure 6 plots two curves labeled 'upper bound' and 'lower bound' without specifying which parameter choices or which uncertainty bounds produce them, so the main claim of the paper is not demonstrated.
minor comments (4)
  1. [II, Eqs. (4), (7)-(10)] Notation is inconsistent: the paper uses ΔxΔp, (ΔX)_T, ΔX_T, and ΔP_T without clearly distinguishing the standard deviation from the variance; clarifying this would improve readability.
  2. [Figs. 1 and 2] The axis labels say 'Sum uncertainty (X2 + P2)', but the plotted quantity is ΔX_T + ΔP_T, not ΔX^2 + ΔP^2; the labels should be corrected.
  3. [V, Eq. (30)] The first displayed expression for efficiency uses terms such as n̄^2 T2 ln(ZD/ZC), while the final expression uses D ln(ZB/ZA) + E ln(ZD/ZC); the relation between these forms, especially the placement of T1 and T2 in numerator and denominator, is not explained and appears to contain a typographical error.
  4. [IV, Eq. (19)] The entropy expression contains the auxiliary quantities ν and γ whose definitions are dense and not derived; a short derivation or a reference would help the reader verify the result.

Circularity Check

2 steps flagged · score 8.0 of 10

Eq. (17) constructs an ad hoc bridge; D and E simply rename n̄², so the uncertainty relation does not determine the efficiency bounds.

  1. fitted input called prediction [Section IV, Eq. (17)]
    "Z = π¯n/2 = L√2/(ℏ√π)(∆XT + ∆PT +CT), where CT =−L/3 + 2L/π^{5/2}√(αβT) [αβT −√π(αβT)^{3/2} −1] is a constant for a specific temperature, which is derived by expanding Equation 10, and neglecting the higher order terms as the products of α and β are small."

    This is the only bridge between the thermodynamic object Z and the uncertainty combination. C_T is constructed by expansion and truncation so that the bracket equals Z up to discarded terms; it is not derived from the uncertainty principle. Inserting Eq. (17) into the later definitions gives D = n̄²_T1 and E = n̄²_T2 exactly. Thus the claimed uncertainty-based prediction inherits a fitted identity rather than an independent constraint. The paper itself defers the required error control: 'Computing the error margins would help us understand this behavior better, which we intend to do in a future paper.'

  2. renaming known result [Section V, Eq. (30) and the definitions of D and E]
    "η = [D ln(ZB/ZA)+E ln(ZD/ZC)]/[−E/2 +D(ln(ZB/ZA)+1/2)], where D = 8L2/π3ℏ2 (∆XT1 + ∆PT1 + CT1)2 and E = 8L2/π3ℏ2 (∆XT2 + ∆PT2 +CT2)2."

    By Eq. (17), (∆X_T+∆P_T+C_T) = ℏπ^{3/2} n̄/(2√2 L), so D = n̄²_T1 and E = n̄²_T2. The line immediately preceding Eq. (30) already gives the same efficiency in terms of n̄² at the two temperatures. Hence Eq. (30) is the standard partition-function Stirling efficiency with n̄² relabeled as an 'uncertainty combination'; the uncertainty relation adds no independent bound. The plotted upper and lower efficiency curves are therefore the familiar thermodynamic formula plotted against a rescaled variable.

full rationale

The central claim of the paper is that the uncertainty relation determines upper and lower bounds on the efficiency of the heat engine. The chain of derivation rests entirely on Eq. (17), where Z is equated to a combination of thermal position and momentum uncertainties plus an expressly constructed constant C_T. Because C_T is defined so that the equality holds after expanding Eq. (10) and dropping higher-order terms, Eq. (17) is a fitted bridge rather than a consequence of the uncertainty principle. Once that bridge is accepted, D and E in Eq. (30) reduce exactly to n̄² at T1 and T2, so the efficiency formula is identical to the conventional partition-function result that appears one line earlier in the paper. The uncertainty relation therefore does no independent work; the 'bounds from uncertainty' are a renaming of the known Stirling-engine efficiency. No load-bearing self-citation or uniqueness argument is involved, and the issue is not a question of scientific consensus but of a constructed identity whose error is unquantified and explicitly deferred. This is partial-to-strong circularity: the prediction reduces by construction to the input thermodynamic formula, yielding a score of 8.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central derivation rests on one ad hoc bookkeeping parameter, C_T, and on an unquantified high-temperature approximation. No new physical entities are introduced.

free parameters (1)
  • C_T = temperature-dependent expression from expansion of Eq. (10)
    Introduced in Eq. (17) to force the partition function to equal the uncertainty sum; it is derived by expanding and dropping higher-order terms, so it is an ad hoc bookkeeping term rather than a measured or independently derived constant.
assumptions (4)
  • standard math One-dimensional infinite potential well eigenstates and energies, including the Delta x Delta p relation in Eq. (4).
    Standard quantum mechanics; used as the working substance for the engine.
  • domain assumption Partition function can be replaced by the Gaussian integral Z approximately (1/2) sqrt(pi/(beta alpha)) because beta alpha is small.
    Invoked in Eq. (5) and used throughout; the error is said to be negligible but never quantified.
  • domain assumption The Stirling cycle with barrier insertion and removal is modeled by partition-function ratios Z_B = 2 sum e^{-beta E_{2n}} and Z_C similarly, following Ref. [67].
    This is the physical model of the engine; the efficiency formula inherits its validity.
  • standard math The Dunkl-Williams inequality (Eq. (13)) can be squared and applied to give an upper bound on DeltaX^2 + DeltaP^2.
    The inequality is a cited theorem, but for Cov(X,P)=0 in the box states the resulting bound collapses to an equality, so it provides no actual upper bound. The paper does not address this.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Bound on efficiency of heat engine from uncertainty relation viewpoint." pith.science (2026). https://pith.science/paper/DXEG2VDS

@misc{pith2026190806804,
  author       = {Pith},
  title        = {Pith review of: Bound on efficiency of heat engine from uncertainty relation viewpoint},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DXEG2VDS}},
  note         = {Machine review of arXiv:1908.06804}
}
read the original abstract

Quantum cycles in established heat engines can be modeled with various quantum systems as working substances. For example, a heat engine can be modeled with an infinite potential well as the working substance to determine the efficiency and work done. However, in this method, the relationship between the quantum observables and the physically measurable parameters i.e., the efficiency and work done is not well understood from the quantum mechanics approach. A detailed analysis is needed to link the thermodynamic variables (on which the efficiency and work done depends) with the uncertainty principle for better understanding. Here, we present the connection of the sum uncertainty relation of position and momentum operators with thermodynamic variables in the quantum heat engine model. We are able to determine the upper and lower bounds on the efficiency of the heat engine through the uncertainty relation.

Figures

Figures reproduced from arXiv: 1908.06804 by the authors.

Figure 2
Figure 2. FIG. 2. Similarly, this shows variation for different values of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 1
Figure 1. FIG. 1. The variation of sum uncertainty relation (Equa [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The bound of uncertainty relation (Equation (12) [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The variation of entropy (Equation (19)) for differ [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: Similar to [67], we calculate the work done and the efficiency but in terms of uncertainty relation. To de￾termine the work done of the engine, a one-dimensional well of length 2L is considered with a particle of mass m at a temperature T1. The energy of this system is…
Figure 5
Figure 5. Figure 5: FIG. 5. The figure shows the four stages (two isothermal and [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. The bounds on the efficiency by heat engine in terms [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

  1. Relativistic quantum heat engine from uncertainty relation standpoint

    quant-ph 2019-08 reject novelty 4.0 of 10

    Relativistic Stirling engine efficiency bounds claimed via uncertainty relation are dimensionally inconsistent and vanish under the paper's own partition-function approximations.

Reference graph

Works this paper leans on

76 extracted references · 76 canonical work pages · cited by 1 Pith paper

  1. [67]

    Realization of a micrometre- sized stochastic heat engine

    Blickle, V.; Bechinger, C. Realization of a micrometre- sized stochastic heat engine. Nat. Phys. 2011, 8, 143

    work page 2011

  2. [1]

    Black holes and thermodynamics

    Hawking, S.W. Black holes and thermodynamics. Phys. Rev. D 1976, 13, 191

    work page 1976

  3. [2]

    The sum of uncertainty [59] was introduced to capture uncertainty in the observables when the system happens to be an eigenstate of the ob- servables

    (9) The product uncertainty relation loses its importance when the system under consideration is an eigenstate of the observable under study. The sum of uncertainty [59] was introduced to capture uncertainty in the observables when the system happens to be an eigenstate of the ob- servables. The sum of uncertainty for this system at a particular temperatu...

    work page

  4. [3]

    Gravity and the thermodynamics of 8 horizons

    Padmanabhan, T. Gravity and the thermodynamics of 8 horizons. Phys. Rep. 2005, 406, 49–125

    work page 2005

  5. [4]

    Hologra- phy and thermodynamics of 5D dilaton-gravity

    G¨ ursoy,U.; Kiritsis, E.; Mazzanti, L.; Nitti, F. Hologra- phy and thermodynamics of 5D dilaton-gravity. J. High Energy Phys. 2009, 2009, 033

    work page 2009

  6. [5]

    The Quantum Open System as a Model of the Heat Engine

    Alicki, R. The Quantum Open System as a Model of the Heat Engine. J. Phys. A 1979, 12, L103

    work page 1979

  7. [6]

    Equivalence of quantum heat machines, and quantum-thermodynamic signatures

    Uzdin, R.; Levy, A.; Kosloff, R. Equivalence of quantum heat machines, and quantum-thermodynamic signatures. Phys. Rev. X 2015, 5, 031044

    work page 2015

  8. [7]

    A Quantum Mechanical Open System as a Model of a Heat Engine

    Kosloff, R. A Quantum Mechanical Open System as a Model of a Heat Engine. J. Chem. Phys. 1984, 80, 1625

    work page 1984

Show all 76 references
  1. [8]

    Quantum Heat Engines and Refrig- erators: Continuous Devices

    Kosloff, R.; Levy, A. Quantum Heat Engines and Refrig- erators: Continuous Devices. Annu. Rev. Phys. Chem. 2014, 65, 365

    2014

  2. [9]

    Work Extraction and Thermodynamics for Individual Quantum Systems

    Skrzypczyk, P.; Short, A.J.; Popescu, S. Work Extraction and Thermodynamics for Individual Quantum Systems. Nat. Commun. 2014, 5, 4185

    2014

  3. [10]

    Quantum Bath Refrigeration Towards Absolute Zero: Challenging the Unattainability Principle

    Kol´ aˇ r, M.; Gelbwaser-Klimovsky, D.; Alicki, R.; Kur- izki, G. Quantum Bath Refrigeration Towards Absolute Zero: Challenging the Unattainability Principle. Phys. Rev. Lett. 2012, 109, 090601

    2012

  4. [11]

    Quantum Thermodynamic Cycles and Quantum Heat Engines

    Quan, H.T.; Liu, Y.X.; Sun, C.P.; Nori, F. Quantum Thermodynamic Cycles and Quantum Heat Engines. Phys. Rev. E 2007, 76, 031105

    2007

  5. [12]

    Nanoscale Heat Engine Beyond the Carnot Limit

    Roßnagel, J.; Abah, O.; Schmidt-Kaler, F.; Singer, K.; Lutz, E. Nanoscale Heat Engine Beyond the Carnot Limit. Phys. Rev. Lett. 2014, 112, 030602

    2014

  6. [13]

    Quantum-Enhanced Absorption Refrigerators

    Correa, L.A.; Palao, J.P.; Alonso, D.; Adesso, G. Quantum-Enhanced Absorption Refrigerators. Sci. Rep. 2014, 4, 3949

    2014

  7. [14]

    Extracting Quantum Work Statistics and Fluctuation Theorems by Single-Qubit Interferometry

    Dorner, R.; Clark, S.R.; Heaney, L.; Fazio, R.; Goold, J.; Vedral, V. Extracting Quantum Work Statistics and Fluctuation Theorems by Single-Qubit Interferometry. Phys. Rev. Lett. 2013, 110, 230601

    2013

  8. [15]

    Thermalization in Na- ture and on a Quantum Computer

    Riera, A.; Gogolin, C.; Eisert, J. Thermalization in Na- ture and on a Quantum Computer. Phys. Rev. Lett. 2012, 108, 080402

    2012

  9. [16]

    Single-Ion Heat Engine at Maximum Power

    Abah, O.; Roßnagel, J.; Jacob, G.; Deffner, S.; Schmidt Kaler, F.; Singer, K.; Lutz, E. Single-Ion Heat Engine at Maximum Power. Phys. Rev. Lett. 2012, 109, 203006

    2012

  10. [17]

    All-Optical Nano- mechanical Heat Engine

    Dechant, A.; Kiesel, N.; Lutz, E. All-Optical Nano- mechanical Heat Engine. Phys. Rev. Lett. 2015, 114, 183602

    2015

  11. [18]

    Quantum Opto- mechanical Heat Engine

    Zhang, K.; Bariani, F.; Meystre, P. Quantum Opto- mechanical Heat Engine. Phys. Rev. Lett. 2014, 112, 150602

    2014

  12. [19]

    Cooling by Heating: Very Hot Ther- mal Light Can Significantly Cool Quantum Systems

    Mari, A.; Eisert, J. Cooling by Heating: Very Hot Ther- mal Light Can Significantly Cool Quantum Systems. Phys. Rev. Lett. 2012, 108, 120602

    2012

  13. [20]

    Maxwell’s demon, Szilard’s engine and quantum measurements

    Zurek, W.H. Maxwell’s demon, Szilard’s engine and quantum measurements. arXiv 2003, arXiv:quant- ph/0301076

    2003

  14. [21]

    ¨Uber die Entropieverminderung in einem ther- modynamischen System bei Eingriffen intelligenter We- sen

    Szilard, L. ¨Uber die Entropieverminderung in einem ther- modynamischen System bei Eingriffen intelligenter We- sen. Zeitschrift f¨ ur Physik1929, 53, 840–856

  15. [22]

    Quan- tum Szilard engine

    Kim, S.W.; Sagawa, T.; De Liberato, S.; Ueda, M. Quan- tum Szilard engine. Phys. Rev. Lett. 2011, 106, 070401

    2011

  16. [23]

    Revisiting the quantum Szilard engine with fully quantum consid- erations

    Li, H.; Zou, J.; Li, J.-G.; Shao, B.; Wu, L.-A. Revisiting the quantum Szilard engine with fully quantum consid- erations. Ann. Phys. 2012, 327, 2955

    2012

  17. [24]

    Szilard’s information heat engines in the deep quantum regime

    Kim, K.H.; Kim, S.W. Szilard’s information heat engines in the deep quantum regime. J. Korean Phys. Soc. 2012, 61, 1187

    2012

  18. [25]

    Multiparticle quantum Szilard engine with optimal cycles assisted by a Maxwell’s demon

    Cai, C.Y.; Dong, H.; Sun, C.P. Multiparticle quantum Szilard engine with optimal cycles assisted by a Maxwell’s demon. Phys. Rev. E 2012, 85, 031114

    2012

  19. [26]

    Quantum Szilard engines with arbitrary spin

    Zhuang, Z.; Liang, S.D. Quantum Szilard engines with arbitrary spin. Phys. Rev. E 2014, 90, 052117

    2014

  20. [27]

    Supremacy of the quantum many-body Szilard engine with attractive bosons

    Bengtsson, J.; Nilsson Tengstrand, M.; Wacker, A.; Samuelsson, P.; Ueda, M.; Linke, H.; Reimann, S.M. Supremacy of the quantum many-body Szilard engine with attractive bosons. arXiv 2017, arXiv:1701.08138

    2017 arXiv

  21. [28]

    The quantum harmonic Otto cycle

    Kosloff, R.; Rezek, Y. The quantum harmonic Otto cycle. Entropy 2017, 19, 136

    2017

  22. [29]

    Efficiency of harmonic quantum Otto engines at maximal power

    Deffner, S. Efficiency of harmonic quantum Otto engines at maximal power. Entropy 2018, 20, 875

    2018

  23. [30]

    Thermodynamics and steady state of quantum motors and pumps far from equi- librium

    Bustos-Mar´ un, R.A.; Calvo, H.L. Thermodynamics and steady state of quantum motors and pumps far from equi- librium. Entropy 2019, 21, 824

    2019

  24. [31]

    Optimal power and efficiency of multi-stage endoreversible quantum Carnot heat en- gine with harmonic oscillators at the classical limit

    Meng, Z.; Chen, L.; Wu, F. Optimal power and efficiency of multi-stage endoreversible quantum Carnot heat en- gine with harmonic oscillators at the classical limit. En- tropy 2020, 22, 457

    2020

  25. [32]

    The quantum friction and optimal finite- time performance of the quantum Otto cycle

    Insinga, A.R. The quantum friction and optimal finite- time performance of the quantum Otto cycle. Entropy 2020, 22, 1060

    2020

  26. [33]

    Quantum finite-time thermodynamics: Insight from a single qubit engine

    Dann, R.; Kosloff, R.; Salamon, P. Quantum finite-time thermodynamics: Insight from a single qubit engine. En- tropy 2020, 22, 1255

    2020

  27. [34]

    Quantum thermodynamics and open-systems modeling

    Kosloff, R. Quantum thermodynamics and open-systems modeling. J. Chem. Phys. 2019, 150, 204105

    2019

  28. [35]

    Exergy- based ecological optimization of an irreversible quantum Carnot heat pump with harmonic oscillators

    Chen, L.; Liu, X.; Wu, F.; Xia, S.; Feng, H. Exergy- based ecological optimization of an irreversible quantum Carnot heat pump with harmonic oscillators. Phys. A Stat. Mech. Appl. 2020, 537, 122597

    2020

  29. [36]

    Maximum power from fluid flow by applying the first and second laws of thermodynamics

    Diaz, G.A.; Forero, J.D.; Garcia, J.; Rincon, A.; Fontalvo, A.; Bula, A.; Padilla, R.V. Maximum power from fluid flow by applying the first and second laws of thermodynamics. J. Energy Resour. Technol. 2017, 139, 3

    2017

  30. [37]

    Shortcuts to adiabaticity in a time-dependent box

    Del Campo, A.; Boshier, M.G. Shortcuts to adiabaticity in a time-dependent box. Sci. Rep. 2012, 2, 1–6

    2012

  31. [38]

    Optimal shortcuts to adiabaticity for a quantum piston

    Stefanatos, D. Optimal shortcuts to adiabaticity for a quantum piston. Automatica 2013, 49, 3079–3083

    2013

  32. [39]

    Exponential bound in the quest for abso- lute zero

    Stefanatos, D. Exponential bound in the quest for abso- lute zero. Phys. Rev. E 2017, 96, 042103

    2017

  33. [40]

    Relativistic quantum heat engine from uncertainty relation standpoint

    Chattopadhyay, P.; Paul, G. Relativistic quantum heat engine from uncertainty relation standpoint. Sci. Rep. 2019, 9, 16967

    2019

  34. [41]

    Quantum-state disturbance ver- sus information gain: Uncertainty relations for quantum information

    Fuchs, C.A.; Peres, A. Quantum-state disturbance ver- sus information gain: Uncertainty relations for quantum information. Phys. Rev. A 1996, 53, 2038

    1996

  35. [42]

    Unconditional security of quantum key dis- tribution and the uncertainty principle

    Koashi, M. Unconditional security of quantum key dis- tribution and the uncertainty principle. J. Phys. Conf. Ser. 2006, 36, 16

    2006

  36. [43]

    Simple security proof of quantum key distribution via uncertainty principle

    Koashi, M. Simple security proof of quantum key distribution via uncertainty principle. arXiv 2005, arXiv:0505108

    2005

  37. [44]

    Violation of local uncer- tainty relations as a signature of entanglement

    Hofmann, H.F.; Takeuchi, S. Violation of local uncer- tainty relations as a signature of entanglement. Phys. Rev. A 2003, 68, 032103

    2003

  38. [45]

    Entanglement and its facets in condensed matter systems

    Osterloh, A. Entanglement and its facets in condensed matter systems. arXiv 2008, arXiv:0810.1240

    2008 arXiv

  39. [46]

    Quantifying entanglement with scattering experiments

    Marty, O.; Epping, M.; Kampermann, H.; Bruß, D.; Ple- nio, M.B.; Cramer, M. Quantifying entanglement with scattering experiments. Phys. Rev. B 2014, 89, 125117

    2014

  40. [47]

    Characterizing entanglement via uncertainty 9 relations

    G¨ uhne, O. Characterizing entanglement via uncertainty 9 relations. Phys. Rev. Lett. 2004, 9, 117903

    2004

  41. [48]

    Advances in quantum metrology

    Giovannetti, V.; Lloyd, S.; Maccone, L. Advances in quantum metrology. Nat. Photonics 2011, 5, 222

    2011

  42. [49]

    Quantum coherence sets the quantum speed limit for mixed states.Phys

    Mondal, D.; Datta, C.; Sazim, S.K. Quantum coherence sets the quantum speed limit for mixed states.Phys. Lett. A 2016, 380, 689–695

    2016

  43. [50]

    Quantum speed limits, coherence, and asymmetry

    Marvian, I.; Spekkens, R.W.; Zanardi, P. Quantum speed limits, coherence, and asymmetry. Phys. Rev. A 2016, 93, 052331

    2016

  44. [51]

    Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control

    Deffner, S.; Campbell, S. Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control. J. Phys. A 2017, 50, 453001

    2017

  45. [52]

    Generalized geometric quantum speed limits

    Pires, D.P.; Cianciaruso, M.; C´ eleri, L.C.; Adesso, G.; Soares-Pinto, D.O. Generalized geometric quantum speed limits. Phys. Rev. X 2016, 6, 021031,

    2016

  46. [53]

    Experimental test of uncertainty relations for general unitary operators

    Xiao, L. Experimental test of uncertainty relations for general unitary operators. Opt. Express 2017, 25, 17904– 17910

    2017

  47. [54]

    Experimental test of Heisenberg’s measurement uncertainty relation based on statistical distances

    Ma, W. Experimental test of Heisenberg’s measurement uncertainty relation based on statistical distances. Phys. Rev. Lett. 2016, 116, 160405

    2016

  48. [55]

    Experimental violation and reformulation of the Heisen- berg’s error disturbance uncertainty relation

    Baek, S.Y.; ; Kaneda, F.; Ozawa, ; M.; Edamatsu, K. Experimental violation and reformulation of the Heisen- berg’s error disturbance uncertainty relation. Sci. Rep. 2013, 3, 2221

    2013

  49. [56]

    Tighter uncertainty and reverse uncertainty relations

    Mondal, D.; Bagchi, S.; Pati, A.K. Tighter uncertainty and reverse uncertainty relations. Phys. Rev. A 2017, 95, 052117

    2017

  50. [57]

    Quantum Mechanics, International series in pure and applied physics ; McGraw-Hill: New York, NY, USA, 1955

    Schiff, L. Quantum Mechanics, International series in pure and applied physics ; McGraw-Hill: New York, NY, USA, 1955

    1955

  51. [58]

    Griffiths, D. J. Introduction to Quantum Mechanics ; Pearson Prentice Hall: Upper Saddle River, NJ, USA, 2005

    2005

  52. [59]

    Fundamentals of Statistical and Thermal Physics; Waveland Press: Long Grove, IL, USA, 2009

    Reif, F. Fundamentals of Statistical and Thermal Physics; Waveland Press: Long Grove, IL, USA, 2009

    2009

  53. [60]

    Stronger uncertainty relations for all incompatible observables

    Maccone, L.; Pati, A.K. Stronger uncertainty relations for all incompatible observables. Phys. Rev. Lett. 2014, 113, 260401

    2014

  54. [61]

    Probing Un- certainty Relations in Non-Commutative Space

    Chattopadhyay, P.; Mitra, A.; Paul, G. Probing Un- certainty Relations in Non-Commutative Space. Int. J. Theor. Phys. 2019, 58, 2619–2631

    2019

  55. [62]

    Mathematical Inequalities ; Chapman and Hall/CRC: London, UK, 2011; pp

    Cerone, P.; Dragomir, S.S. Mathematical Inequalities ; Chapman and Hall/CRC: London, UK, 2011; pp. 241– 313

    2011

  56. [63]

    Saha, M.; and Srivastava, B. N. Treatise on Heat ; Long- man, Rees, Orme, Brown, Green & Longman: New York, NY, USA, 1935

    1935

  57. [64]

    Quantum degeneracy effect on the work output from a Stirling cycle

    Saygin, H.; S ¸i¸ sman, A. Quantum degeneracy effect on the work output from a Stirling cycle. J. Appl. Phys. 2001, 90, 3086–3089

    2001

  58. [65]

    Quantum dynamical framework for Brownian heat engines

    Agarwal, G.S.; Chaturvedi, S. Quantum dynamical framework for Brownian heat engines. Phys. Rev. E 2013, 88, 012130

    2013

  59. [66]

    Quantum Stirling heat engine and refrigerator with single and cou- pled spin systems

    Huang, X.L.; Niu, X.Y.; Xiu, X.M.; Yi, X.X. Quantum Stirling heat engine and refrigerator with single and cou- pled spin systems. Eur. Phys. J. D 2014, 68, 32

    2014

  60. [68]

    Quantum heat en- gine based on level degeneracy

    Thomas, G.; Das, D.; Ghosh, S. Quantum heat en- gine based on level degeneracy. Phys. Rev. E 2019, 100, 012123

    2019

  61. [69]

    Quantum engine efficiency bound beyond the second law of thermodynamics

    Niedenzu, W.; Mukherjee, V.; Ghosh, A.; Kofman, A.G.; Kurizki, G. Quantum engine efficiency bound beyond the second law of thermodynamics. Nat. Commun. 2018, 9, 165

    2018

  62. [70]

    An introduction to entangle- ment measures

    Plenio, M.B.; Virmani, S. An introduction to entangle- ment measures. In Quantum Information and Coherence ; Springer: Cham, Switzerland, 2014; pp. 173–209

    2014

  63. [71]

    Relativistic particle in a box

    Alberto, P.; Fiolhais, C.; Gil, V.M.S. Relativistic particle in a box. Eur. J. Phys. 1996, 17, 19

    1996

  64. [72]

    Holographic derivation of entan- glement entropy from the anti–de sitter space/conformal field theory correspondence

    Ryu, S.; Takayanagi, T. Holographic derivation of entan- glement entropy from the anti–de sitter space/conformal field theory correspondence. Phys. Rev. Lett. 2006, 96, 181602

    2006

  65. [73]

    Hilbert space rep- resentation of the minimal length uncertainty relation

    Kempf, A.; Mangano, G.; Mann, R.B. Hilbert space rep- resentation of the minimal length uncertainty relation. Phys. Rev. D 1995, 52, 1108

    1995

  66. [74]

    Composite system in de- formed space with minimal length

    Quesne, C.; Tkachuk, V.M. Composite system in de- formed space with minimal length. Phys. Rev. A 2010, 81, 012106

    2010

  67. [75]

    Noncommutative differen- tial forms on the kappa-deformed space.J

    Meljanac, S.; Kreˇ si´ c-Juri´ c, S. Noncommutative differen- tial forms on the kappa-deformed space.J. Phys. A 2009, 42, 365204

    2009

  68. [76]

    Irreversible performance of a quan- tum harmonic heat engine

    Rezek, Y.; Kosloff, R. Irreversible performance of a quan- tum harmonic heat engine. New J. Phys. 2006, 8, 83

    2006

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.