arxiv: 2605.10099 · v1 · submitted 2026-05-11 · 🪐 quant-ph · cond-mat.stat-mech

Recognition: no theorem link

Symmetry-Enforced Non-Hermitian Jarzynski Equality in an SU(2)-Rotated Family of Hybrid mathcal{PT}--mathcal{APT} Systems

Zongru Yang , Teng Liu , Xiaodong Tan , Feng Zhu , Le Luo

Authors on Pith no claims yet

Pith reviewed 2026-05-12 03:15 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mech

keywords non-Hermitian Jarzynski equalityparity-exchange symmetryPT symmetryAPT symmetrySU(2) rotationtrapped ionconditional thermodynamicsnonequilibrium thermodynamics

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The pith

A parity-exchange symmetry extends the conditional Jarzynski equality from isolated PT systems to an entire SU(2)-rotated family of hybrid PT-APT Hamiltonians.

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

The paper shows that in postselected non-Hermitian evolution without quantum jumps, the Jarzynski equality relating work statistics to free-energy differences holds conditionally provided the transition probabilities satisfy a parity-exchange symmetry. This symmetry criterion, previously known only for pure PT-symmetric cases, is proven to survive throughout a continuous family of two-level hybrid Hamiltonians that mix PT and APT terms via SU(2) rotations. The authors supply both geometric and algebraic arguments for the persistence and verify the resulting equality experimentally by implementing three representative rotation angles in a single trapped ion under cyclic driving with zero free-energy change.

Core claim

In the postselected no-quantum-jump framework, a conditional non-Hermitian Jarzynski equality holds when transition probabilities obey a parity-exchange symmetry. For the constructed family of two-level hybrid PT-APT Hamiltonians, this symmetry persists throughout the corresponding SU(2)-rotated orbit, as shown by complementary geometric and algebraic arguments. The result therefore extends the symmetry criterion from the isolated PT endpoint to the full hybrid family.

What carries the argument

The parity-exchange symmetry of transition probabilities, which is shown to be invariant under SU(2) rotations of the hybrid PT-APT Hamiltonian and thereby enforces the conditional Jarzynski equality.

If this is right

The conditional Jarzynski equality applies to every point in the continuous SU(2) orbit of hybrid Hamiltonians rather than only isolated PT endpoints.
Geometric and algebraic proofs establish the symmetry persistence independently of particular parameter values.
Cyclic protocols with vanishing free-energy difference can be used to test the equality at any chosen rotation angle.
The symmetry-based criterion provides a concrete template for identifying similar relations in other non-Hermitian two-level systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Analogous parity-exchange or related discrete symmetries could be sought in higher-dimensional or multi-level non-Hermitian platforms to enlarge the domain of conditional fluctuation theorems.
The orbit-wide invariance suggests that other nonequilibrium relations might likewise survive SU(2) mixing of PT and APT terms.
Direct experimental scans at rotation angles other than the three tested points would supply an independent check of the algebraic persistence argument.

Load-bearing premise

The transition probabilities must obey the parity-exchange symmetry in the postselected framework, and the specific two-level hybrid Hamiltonian construction must be representative of the broader non-Hermitian case.

What would settle it

A measurement of the work distribution for a cyclic protocol at an intermediate SU(2) rotation angle in the trapped-ion system that deviates from the predicted equality while the parity-exchange symmetry is intact would falsify the claim.

Figures

Figures reproduced from arXiv: 2605.10099 by Feng Zhu, Le Luo, Teng Liu, Xiaodong Tan, Zongru Yang.

**Figure 1.** Figure 1: Evolution trajectories for quantum states |e−(θk)⟩ and |e+(θk)⟩ under different values of θk. (a) Trajectories initialized at |e−(θk)⟩. (b) Trajectories initialized at |e+(θk)⟩. Different colors correspond to different θk values: blue (θk = 0), purple (θk = π/4) and black (θk = π/2). Initial and final states are marked in red and green, respectively. The evolution trajectories for various θk (representing… view at source ↗

**Figure 2.** Figure 2: Experimental setup for verifying the Jarzynski equality. (a) Schematic of the trapped ion system, where the coherent driving terms J(t) and ∆(t) are realized via applied microwave fields. (b) Implementation of the non-Hermitian Hamiltonian on a 171Yb+ ion. The dissipation γ is engineered by a 369.5 nm laser coupling the |1⟩ state to the 2P1/2 excited state, followed by spontaneous decay to the uncoupled … view at source ↗

**Figure 4.** Figure 4: Experimental and theoretical results of the exponential work ⟨e−βW ⟩ and transition probabilities under PT and APT evolution, where J varies linearly from 0.03 µs−1 to 0.06 µs−1 and then returns to 0.03 µs−1 . The data points with error bars represent experimental results, while the solid or dashed lines denote theoretical results. The vertical axes in (b) and (d) are broken to make the experimental error … view at source ↗

**Figure 3.** Figure 3: Experimental and theoretical results of the exponential work ⟨e−βW ⟩ and transition probabilities under PT and APT evolution with constant J. Panels (b) and (e) show the transition probabilities at J = 0.03 µs−1 , while panels (c) and (f) show the corresponding results at J = 0.06 µs−1 . The data points with error bars represent experimental results, while the solid or dashed lines denote theoretical resu… view at source ↗

**Figure 5.** Figure 5: The theoretical and experimental results of the exponential work ⟨e−βW ⟩ and transition probabilities with detuning added to the PT and APT Hamiltonians. (a) and (c) show the cases where a sinusoidal modulation of detuning ∆(t) = 0.5 sin 2πt T µs−1 is introduced into the PT and APT Hamiltonians, respectively. The points with error bars represent the experimental results, while the solid lines correspond… view at source ↗

read the original abstract

The Jarzynski equality is a cornerstone of nonequilibrium thermodynamics, linking work statistics to equilibrium free-energy differences. Although it has been extensively verified in classical and quantum Hermitian settings, its status in non-Hermitian dynamics remains under debate. Here we show that, in a postselected no-quantum-jump framework, a conditional non-Hermitian Jarzynski equality holds when the transition probabilities obey a parity-exchange symmetry. We study a constructed family of two-level hybrid Hamiltonians formed as linear combinations of parity-time ($\mathcal{PT}$) and anti-parity-time ($\mathcal{APT}$) symmetric terms, and demonstrate using complementary geometric and algebraic arguments that the parity-exchange symmetry persists throughout the corresponding $\mathrm{SU}(2)$-rotated orbit. Relative to previous $\mathcal{PT}$-focused conditional Jarzynski equality results, the advance here is an extension of the symmetry criterion from the isolated $\mathcal{PT}$ endpoint to a broader $\mathcal{PT}$--$\mathcal{APT}$ hybrid family. Experimentally, we implement three representative points, $\theta_k = 0, \pi/4, \pi/2$, in a single trapped $^{171}\mathrm{Yb}^+$ ion and measure the resulting work distributions under cyclic protocols with $\Delta F = 0$, confirming the predicted symmetry criterion at those points. Our results establish a symmetry-based extension of the conditional non-Hermitian Jarzynski relation within this restricted two-level setting.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper extends the conditional Jarzynski equality to an SU(2) family of hybrid PT-APT systems with geometric arguments and three-point ion-trap checks, but the postselection normalization may break the required symmetry.

read the letter

The core advance is taking the parity-exchange symmetry that enforces a conditional Jarzynski equality in pure PT-symmetric non-Hermitian systems and showing it survives across the full SU(2) orbit of hybrid PT-APT two-level Hamiltonians. They give geometric and algebraic reasons why the symmetry holds for any mixing angle, then measure work distributions at theta = 0, pi/4, and pi/2 in a trapped Yb+ ion under cyclic protocols with zero free-energy change, confirming the equality at those points. That continuous-family construction is new relative to the PT-only results they cite, and the experiment is a straightforward, relevant test in a controllable platform. The setup lets them dial the hybrid parameter and extract the conditional probabilities cleanly enough to check the claim at discrete locations. The soft spot sits in the postselected no-jump framework. The normalization factor for the effective transition probabilities comes from the no-jump evolution operator and depends on the imaginary parts of the eigenvalues, which vary with theta. Nothing in the abstract shows that this factor commutes with the parity-exchange operation for arbitrary theta; if it does not, the conditional probabilities lose the symmetry even when the bare Hamiltonian orbit preserves it. The paper asserts the symmetry persists by construction, but the derivation needs to be checked line by line on how the norm is handled. This is a narrow but real gap for a result that rests on the conditional probabilities obeying the symmetry. The work is for people already inside non-Hermitian quantum thermodynamics who care about fluctuation theorems beyond Hermitian or pure PT cases. A reader looking for incremental extensions with some experimental backing will get value from the family construction and the data points. It is not broad enough to interest people outside that subfield. I would send it to peer review. The idea is well-motivated, the experiment is doable to verify, and the normalization question is concrete enough that referees can settle it with the full equations.

Referee Report

1 major / 1 minor

Summary. The paper claims that in a postselected no-quantum-jump framework, a conditional non-Hermitian Jarzynski equality holds when transition probabilities obey a parity-exchange symmetry. It constructs a family of two-level hybrid PT-APT Hamiltonians as SU(2) rotations of PT and APT terms, provides complementary geometric and algebraic arguments that the symmetry persists throughout the orbit, and experimentally confirms the equality (with ΔF=0) at three discrete points θ=0, π/4, π/2 in a trapped 171Yb+ ion.

Significance. If the symmetry preservation is rigorously established, the work extends prior PT-specific conditional Jarzynski results to a continuous hybrid PT-APT family within the two-level postselected setting. The experimental implementation on a quantum-optical platform supplies concrete verification at representative points and strengthens the symmetry criterion as a practical tool for nonequilibrium thermodynamics in non-Hermitian systems.

major comments (1)

[Algebraic and geometric arguments for symmetry persistence] The algebraic argument for symmetry persistence (invoked to enforce the conditional Jarzynski equality) must explicitly verify that the θ-dependent normalization of the no-jump projector commutes with the parity-exchange operation. Because the norm is set by the imaginary parts of the eigenvalues, which vary continuously with the mixing angle θ, it is not immediate that the conditional probabilities P(i→f | no jump) remain symmetric for arbitrary θ; the geometric argument alone does not resolve this normalization issue.

minor comments (1)

[Abstract] The abstract is dense; separating the theoretical symmetry criterion from the limited experimental sampling at three discrete θ values would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comment on the normalization issue within our algebraic argument. We address the point below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Algebraic and geometric arguments for symmetry persistence] The algebraic argument for symmetry persistence (invoked to enforce the conditional Jarzynski equality) must explicitly verify that the θ-dependent normalization of the no-jump projector commutes with the parity-exchange operation. Because the norm is set by the imaginary parts of the eigenvalues, which vary continuously with the mixing angle θ, it is not immediate that the conditional probabilities P(i→f | no jump) remain symmetric for arbitrary θ; the geometric argument alone does not resolve this normalization issue.

Authors: We appreciate the referee drawing attention to this subtlety. Our geometric argument shows that the parity-exchange symmetry is preserved under the full SU(2) orbit at the level of the Hamiltonian family, while the algebraic argument demonstrates that the unnormalized transition amplitudes obey the required parity-exchange relation for any θ. The θ-dependent normalization factor (arising from the imaginary parts of the eigenvalues) is itself invariant under the parity-exchange operation because the eigenvalue spectrum transforms symmetrically under the same SU(2) rotation that maps the PT and APT endpoints into each other. Consequently the conditional probabilities inherit the symmetry. Nevertheless, we agree that an explicit verification of the commutation between the normalization and the parity-exchange operator would strengthen the presentation. We will add a short dedicated subsection (or paragraph) in the revised manuscript that computes the no-jump projector norm explicitly as a function of θ and verifies its invariance under the parity-exchange map, thereby confirming that the conditional Jarzynski equality holds throughout the family. revision: yes

Circularity Check

0 steps flagged

No significant circularity; symmetry verified algebraically within constructed family

full rationale

The derivation begins from the standard conditional Jarzynski relation under an assumed parity-exchange symmetry on transition probabilities, then constructs an explicit two-level hybrid PT-APT Hamiltonian family parametrized by SU(2) rotation angle θ. Geometric and algebraic arguments are supplied to establish that the symmetry persists across the orbit. This verification step is independent of the Jarzynski statement itself and does not reduce to a tautology or self-citation; the family is defined first, after which the symmetry property is demonstrated rather than presupposed. Experimental measurements at three discrete θ points supply an external consistency check. No fitted parameters are relabeled as predictions, no load-bearing uniqueness theorems are imported from prior self-work, and no ansatz is smuggled via citation. The central claim therefore retains independent content beyond its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard mathematical structures (SU(2) group action on two-level systems) and domain assumptions about PT/APT symmetry definitions; no free parameters are introduced and no new physical entities are postulated.

axioms (2)

standard math SU(2) group structure for continuous rotations of two-level Hamiltonians
Invoked to define the family of hybrid Hamiltonians and to prove symmetry persistence.
domain assumption Standard definitions of PT and APT symmetry for non-Hermitian operators
Used to construct the hybrid family and to identify the parity-exchange symmetry.

pith-pipeline@v0.9.0 · 5590 in / 1525 out tokens · 69192 ms · 2026-05-12T03:15:35.718180+00:00 · methodology

discussion (0)

Reference graph

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