[Paper Review] Energy conservation and fluctuation theorem are incompatible for quantum work

Karen V. Hovhannisyan, Alberto Imparato|arXiv (Cornell University)|Apr 19, 2021

Advanced Thermodynamics and Statistical Mechanics74 references4 citations

TL;DR

This paper proves that energy conservation and the Jarzynski fluctuation theorem (JE) are fundamentally incompatible for quantum work measurements. It shows that no physically reasonable work measurement scheme—especially state-dependent ones—can simultaneously satisfy both conditions, except for a narrow class of exotic schemes, and that only the two-point measurement scheme is compatible with JE under state-independent conditions.

ABSTRACT

Characterizing fluctuations of work in coherent quantum systems is notoriously problematic. Here we reveal the ultimate source of the problem by proving that ($\mathfrak{A}$) energy conservation and ($\mathfrak{B}$) the Jarzynski fluctuation theorem cannot be observed at the same time. Condition $\mathfrak{A}$ stipulates that, for any initial state of the system, the measured average work must be equal to the difference of initial and final average energies, and that untouched systems must exchange deterministically zero work. Condition $\mathfrak{B}$ is only for thermal initial states and encapsulates the second law of thermodynamics. We prove that $\mathfrak{A}$ and $\mathfrak{B}$ are incompatible for work measurement schemes that are differentiable functions of the state and satisfy two mild structural constraints. This covers all existing schemes and leaves the theoretical possibility of jointly observing $\mathfrak{A}$ and $\mathfrak{B}$ open only for a narrow class of exotic schemes. For the special but important case of state-independent schemes, the situation is much more rigid: we prove that, essentially, only the two-point measurement scheme is compatible with $\mathfrak{B}$.

Motivation & Objective

• To resolve the long-standing problem of defining work fluctuations in coherent quantum systems.
• To investigate whether energy conservation and the Jarzynski fluctuation theorem (JE) can coexist in quantum work measurements.
• To determine the theoretical limits on work measurement schemes that satisfy both physical principles.
• To clarify the foundational incompatibility between quantum work and classical thermodynamic principles like the second law and energy conservation.

Proposed method

• Formalizing energy conservation as two conditions: (1) average work equals energy difference (A₁), and (2) zero work for untouched systems (A₂), collectively called detailed energy conservation (A).
• Defining the Jarzynski fluctuation theorem (B) as a requirement that JE holds for all thermal initial states, encoding the second law in quantum work statistics.
• Analyzing work measurement schemes as differentiable functions of the system's state, under mild structural constraints to exclude unphysical or pathological cases.
• Proving that for state-independent schemes, JE (B) implies compatibility only with the two-point measurement (TPM) scheme, establishing JE ⇔ TPM.
• Using spectral analysis and majorization theory to compare work operator statistics, particularly through eigenvalue ordering and trace inequalities.
• Applying operator concavity of the logarithm and trace inequalities to derive bounds on work expectation values and to prove majorization relations between work statistics.

Experimental results

Research questions

• RQ1 Can a quantum work measurement scheme simultaneously satisfy energy conservation and the Jarzynski fluctuation theorem?
• RQ2 What are the necessary conditions for a work measurement scheme to be compatible with the Jarzynski equality for thermal states?
• RQ3 Is the two-point measurement scheme the only viable option for satisfying the Jarzynski equality under state-independent measurement rules?
• RQ4 How does the inclusion of state-dependent schemes affect the compatibility between energy conservation and the fluctuation theorem?
• RQ5 What is the role of differentiability and structural constraints in ruling out exotic schemes that might otherwise satisfy both A and B?

Key findings

• Energy conservation (A) and the Jarzynski fluctuation theorem (B) are fundamentally incompatible for all physically reasonable quantum work measurement schemes.
• For state-independent schemes, only the two-point measurement (TPM) scheme satisfies the Jarzynski equality (B), establishing JE ⇔ TPM in this class.
• When requiring both A₁ and A₂ (detailed energy conservation), the compatibility with B breaks down for all differentiable state-dependent schemes, leaving only a narrow class of exotic schemes as theoretically possible.
• The paper proves that any scheme satisfying B and producing correct average work for thermal states must reproduce the same statistics as the TPM scheme.
• For non-commuting Hamiltonians (i.e., [h,H] ≠ 0), the work expectation value of any scheme satisfying B violates A₁ for thermal states, demonstrating a direct incompatibility.
• The work operator Υρ is shown to majorize the TPM work operator Ω in eigenvalue ordering (Spec(Υτ)↓ ≻ Spec(Ω)↓), indicating a stronger statistical spread, and satisfies ⟨Ω⟩τ − ⟨Υτ⟩τ ≤ β⁻¹S(τ∥U†τ′U), linking it to relative entropy and free energy differences.