Conditional Non-Hermitian Jarzynski Equality

• The paper demonstrates that enforcing parity-exchange symmetry in normalized transition probabilities precisely yields the conditional Jarzynski equality in non-Hermitian dynamics.
• It employs a two-point measurement framework and a hybrid SU(2)-rotated PT–APT Hamiltonian model, validated by both algebraic derivations and trapped-ion experiments.
• The work bridges pseudo-Hermitian and no-jump thermodynamics, delineating the parameter space where cyclic protocols ensure an exact Jarzynski equality.

Searching arXiv for the cited papers to ground the article in current arXiv records. arXiv_search query: (Yang et al., 11 May 2026) The conditional non-Hermitian Jarzynski equality is a fluctuation relation for work statistics in postselected non-Hermitian dynamics, formulated in a two-point measurement (TPM) framework and conditioned on no quantum jumps. In the setting developed for an $\mathrm{SU}(2)$-rotated family of two-level hybrid $\mathcal{PT}$–$\mathcal{APT}$ Hamiltonians, the equality holds exactly when the normalized transition probabilities satisfy a parity-exchange symmetry between the two energy eigenstates. The 2026 trapped-ion study establishes this symmetry criterion algebraically, geometrically, and experimentally for three representative points on the hybrid orbit, thereby extending earlier $\mathcal{PT}$-focused results to a broader non-Hermitian family within a restricted two-level setting (Yang et al., 11 May 2026).

1. Conceptual setting

The Jarzynski equality links nonequilibrium work statistics to equilibrium free-energy differences. In classical and quantum Hermitian settings it has been extensively verified, whereas in non-Hermitian dynamics its status has remained contentious. A central issue is that the non-Hermitian Hamiltonian does not generally play dual and equivalent roles in dynamics and energetics. The conditional non-Hermitian construction therefore distinguishes the effective generator of no-jump evolution from the operator used to define energy measurements (Erdamar et al., 2023).

In the no-quantum-jump framework, one starts from an open system with Lindblad dynamics and postselects trajectories that experience no quantum jumps. Within the relevant two-level subspace, the evolution is then described by an effective non-Hermitian Hamiltonian $H_{\rm eff}(t)$. To avoid complex “energies,” work is defined not from the spectrum of $H_{\rm eff}(t)$ itself, but through TPMs on its Hermitian part $H_{HM}$. In the 2026 construction, the resulting Jarzynski relation is explicitly conditional: it refers to the ensemble of surviving no-jump trajectories and to the corresponding normalized transition probabilities (Yang et al., 11 May 2026).

A separate usage of the term appears in pseudo-Hermitian thermodynamics. There, a “conditional non-Hermitian Jarzynski equality” can mean restricting the sum over a subset of final measurement outcomes rather than conditioning on no-jump trajectories. In that formulation, the proof follows from biorthonormal spectral decompositions and pseudo-Hermitian unitarity, provided the Hamiltonian is diagonalizable and its spectrum is real or comes in complex-conjugate pairs, with real eigenvalues at the initial and final measurement times (Gardas et al., 2015).

2. Postselected TPM work statistics

In the two-level no-jump TPM protocol, the system is prepared in the Gibbs state

$\rho=\sum_{i=\pm}P_i|e_i\rangle\langle e_i|,$

where

$$H_{HM}|e_i\rangle=E_i|e_i\rangle,\qquad E_\pm=\pm J_i,\qquad P_\pm=\frac{e^{\mp\beta J_i}}{Z_i}.$$

At $t=0$ a projective energy measurement yields $\mathcal{PT}$0. The system then evolves under the non-Hermitian propagator

$\mathcal{PT}$1

conditioned on no jumps. At $\mathcal{PT}$2 a second energy measurement yields $\mathcal{PT}$3, and the work along that trajectory is

$\mathcal{PT}$4

The unnormalized transition amplitude is

$\mathcal{PT}$5

with

$\mathcal{PT}$6

The normalized conditional transition probabilities are obtained by renormalizing with the survival weight $\mathcal{PT}$7, so that $\mathcal{PT}$8. The Jarzynski average then takes the explicit two-level form

$\mathcal{PT}$9

For cyclic $\mathcal{APT}$0 one has $\mathcal{APT}$1, so the target relation reduces to $\mathcal{APT}$2 (Yang et al., 11 May 2026).

This formulation is structurally parallel to the superconducting-qubit implementation of post-selected non-Hermitian work statistics, where the effective Hamiltonian is

$\mathcal{APT}$3

the initial and final projective measurements are performed on $\mathcal{APT}$4, and the conditional probabilities are renormalized at the end of each no-jump run (Erdamar et al., 2023).

3. Parity-exchange symmetry criterion

For the two-level no-jump TPM framework, the 2026 result identifies an exact necessary-and-sufficient criterion for the Jarzynski equality. One finds that

$\mathcal{APT}$5

with $\mathcal{APT}$6 for cyclic $\mathcal{APT}$7 holds exactly if and only if

$\mathcal{APT}$8

This is a symmetry under exchanging the two energy eigenstates, termed parity-exchange symmetry (Yang et al., 11 May 2026).

The associated involutive operator is

$\mathcal{APT}$9

which acts as

$\mathcal{PT}$0

Condition (C) is therefore the statement that the joint probability distribution is invariant under exchange of the two energy eigenstates.

This symmetry criterion is closely related to the “exchange symmetry” identified in the superconducting-qubit experiment,

$\mathcal{PT}$1

where the Methods show that it is equivalent to requiring that the non-Hermitian Floquet Hamiltonian

$\mathcal{PT}$2

satisfies both an explicit or emergent $\mathcal{PT}$3 symmetry and the commutation condition that its Hermitian part $\mathcal{PT}$4 commutes with the initial Hamiltonian (Erdamar et al., 2023). The 2026 work recasts the operative condition in symmetry language directly at the level of transition probabilities and extends it beyond the isolated $\mathcal{PT}$5 endpoint (Yang et al., 11 May 2026).

4. $\mathcal{PT}$6-rotated $\mathcal{PT}$7–$\mathcal{PT}$8 hybrid family

The relevant non-Hermitian model is built from two prototype two-level Hamiltonians in the $\mathcal{PT}$9 basis,

$H_{\rm eff}(t)$0

Their one-parameter hybrid is defined as

$H_{\rm eff}(t)$1

Its Hermitian part is

$H_{\rm eff}(t)$2

with eigenstates

$H_{\rm eff}(t)$3

and energies $H_{\rm eff}(t)$4 (Yang et al., 11 May 2026).

Equivalently, the hybrid family satisfies

$H_{\rm eff}(t)$5

so varying $H_{\rm eff}(t)$6 traces out a closed $H_{\rm eff}(t)$7 adjoint orbit of $H_{\rm eff}(t)$8 in the space of $H_{\rm eff}(t)$9 operators. The experimental study samples $H_{\rm eff}(t)$0 as three representative points, corresponding to APT, hybrid, and PT cases in the implementation (Yang et al., 11 May 2026).

The significance of this construction is that the Jarzynski relation is not tied to a single symmetry endpoint. Relative to earlier $H_{\rm eff}(t)$1-focused conditional Jarzynski equality results, the advance is an extension of the symmetry criterion from the isolated $H_{\rm eff}(t)$2 endpoint to a broader $H_{\rm eff}(t)$3–$H_{\rm eff}(t)$4 hybrid family (Yang et al., 11 May 2026).

5. Algebraic and geometric enforcement of the equality

The parity-exchange symmetry of transition probabilities is derived in two complementary ways. First, the angle-dependent exchange operator

$H_{\rm eff}(t)$5

satisfies the algebraic anti-symmetry

$H_{\rm eff}(t)$6

Exponentiating, and using $H_{\rm eff}(t)$7, one obtains

$H_{\rm eff}(t)$8

For the survival amplitudes $H_{\rm eff}(t)$9 this yields

$H_{HM}$0

so that

$H_{HM}$1

Because

$H_{HM}$2

the normalized probabilities obey exactly condition (C) (Yang et al., 11 May 2026).

Second, the Bloch-sphere picture gives a geometric interpretation. The no-jump state starting from $H_{HM}$3 defines a real Bloch vector $H_{HM}$4, and the final measurement projects on the $H_{HM}$5 axis $H_{HM}$6. The diagonal conditional probabilities can be written as

$H_{HM}$7

The operator symmetry enforces the exact mirror relation

$H_{HM}$8

which immediately gives $H_{HM}$9, and normalization then gives $\rho=\sum_{i=\pm}P_i|e_i\rangle\langle e_i|,$0 (Yang et al., 11 May 2026).

Substituting condition (C) into the Jarzynski average for cyclic protocols gives

$\rho=\sum_{i=\pm}P_i|e_i\rangle\langle e_i|,$1

since

$\rho=\sum_{i=\pm}P_i|e_i\rangle\langle e_i|,$2

The equality is therefore symmetry-enforced rather than generic (Yang et al., 11 May 2026).

6. Trapped-ion implementation

The experimental realization uses a single trapped $\rho=\sum_{i=\pm}P_i|e_i\rangle\langle e_i|,$3 ion as a passive driven-dissipative qubit. The qubit transition $\rho=\sum_{i=\pm}P_i|e_i\rangle\langle e_i|,$4 is driven by microwaves implementing

$\rho=\sum_{i=\pm}P_i|e_i\rangle\langle e_i|,$5

while a dissipation beam on $\rho=\sum_{i=\pm}P_i|e_i\rangle\langle e_i|,$6 engineers the non-Hermitian core $\rho=\sum_{i=\pm}P_i|e_i\rangle\langle e_i|,$7 plus a global decay $\rho=\sum_{i=\pm}P_i|e_i\rangle\langle e_i|,$8. An $\rho=\sum_{i=\pm}P_i|e_i\rangle\langle e_i|,$9 rotation by $$H_{HM}|e_i\rangle=E_i|e_i\rangle,\qquad E_\pm=\pm J_i,\qquad P_\pm=\frac{e^{\mp\beta J_i}}{Z_i}.$$0 about $$H_{HM}|e_i\rangle=E_i|e_i\rangle,\qquad E_\pm=\pm J_i,\qquad P_\pm=\frac{e^{\mp\beta J_i}}{Z_i}.$$1 before and after the core evolution transforms the laboratory Hamiltonian into $$H_{HM}|e_i\rangle=E_i|e_i\rangle,\qquad E_\pm=\pm J_i,\qquad P_\pm=\frac{e^{\mp\beta J_i}}{Z_i}.$$2 with $$H_{HM}|e_i\rangle=E_i|e_i\rangle,\qquad E_\pm=\pm J_i,\qquad P_\pm=\frac{e^{\mp\beta J_i}}{Z_i}.$$3 (Yang et al., 11 May 2026).

The cyclic TPM work protocols consist of three steps. First, there is deterministic preparation of $$H_{HM}|e_i\rangle=E_i|e_i\rangle,\qquad E_\pm=\pm J_i,\qquad P_\pm=\frac{e^{\mp\beta J_i}}{Z_i}.$$4, simulating the first projective measurement with Boltzmann weights $$H_{HM}|e_i\rangle=E_i|e_i\rangle,\qquad E_\pm=\pm J_i,\qquad P_\pm=\frac{e^{\mp\beta J_i}}{Z_i}.$$5. Second, the system undergoes non-unitary evolution $$H_{HM}|e_i\rangle=E_i|e_i\rangle,\qquad E_\pm=\pm J_i,\qquad P_\pm=\frac{e^{\mp\beta J_i}}{Z_i}.$$6 under $$H_{HM}|e_i\rangle=E_i|e_i\rangle,\qquad E_\pm=\pm J_i,\qquad P_\pm=\frac{e^{\mp\beta J_i}}{Z_i}.$$7 for $$H_{HM}|e_i\rangle=E_i|e_i\rangle,\qquad E_\pm=\pm J_i,\qquad P_\pm=\frac{e^{\mp\beta J_i}}{Z_i}.$$8 using three driving profiles: constant $$H_{HM}|e_i\rangle=E_i|e_i\rangle,\qquad E_\pm=\pm J_i,\qquad P_\pm=\frac{e^{\mp\beta J_i}}{Z_i}.$$9 with $t=0$0; a linear ramp of $t=0$1 up and down with $t=0$2; and constant $t=0$3 with sinusoidal $t=0$4 of zero net area. Third, a final projective readout in the same eigenbasis yields

$t=0$5

Across the three representative angles and the first two driving protocols, which stay within the $t=0$6 subspace, the measured transition probabilities satisfy

$t=0$7

and the Jarzynski average obeys

$t=0$8

within experimental uncertainty. When detuning $t=0$9 drives the system outside the $\mathcal{PT}$00 orbit, the symmetry and equality are broken for generic $\mathcal{PT}$01, except at special revival times $\mathcal{PT}$02 at which the effective Floquet Hamiltonian again lies in the hybrid $\mathcal{PT}$03 and restores parity-exchange (Yang et al., 11 May 2026).

These observations delimit the operational domain of the equality. The symmetry is robust along the hybrid orbit, but it is not a generic property of arbitrary non-Hermitian driving.

7. Relation to earlier non-Hermitian Jarzynski results

The 2026 result sits between two established strands of non-Hermitian fluctuation-theorem research. One strand is the pseudo-Hermitian formulation, where the Jarzynski equality holds for non-Hermitian systems with real spectrum and, in the quasistatic limit, the Carnot bound remains valid even if some eigenenergies are complex provided they appear in conjugate pairs. In that framework, the standard proof carries over almost unchanged after replacing ordinary orthonormality by a biorthonormal spectral decomposition and imposing the pseudo-Hermitian unitarity condition

$\mathcal{PT}$04

The “conditional” generalization there is

$\mathcal{PT}$05

for any subset $\mathcal{PT}$06 of final eigen-indices (Gardas et al., 2015).

The second strand is post-selected no-jump thermodynamics in experimentally engineered non-Hermitian qubits. In a superconducting circuit, work fluctuations were shown to obey the Jarzynski equality even if the Hamiltonian has complex or purely imaginary eigenvalues, provided the non-Hermitian Floquet Hamiltonian has explicit or emergent $\mathcal{PT}$07 symmetry and its Hermitian part aligns with the initial energy basis. Cyclic sweeps of $\mathcal{PT}$08 with $\mathcal{PT}$09 preserve the exchange symmetry of transition probabilities, even across the exceptional point at $\mathcal{PT}$10, whereas single-lobe detuning sweeps with no $\mathcal{PT}$11 symmetry lead to clear violations (Erdamar et al., 2023).

Against that background, the conditional non-Hermitian Jarzynski equality of the hybrid $\mathcal{PT}$12–$\mathcal{PT}$13 family has a sharply delimited meaning. It does not claim that arbitrary non-Hermitian evolution satisfies $\mathcal{PT}$14. Rather, in the two-level no-jump TPM setting, the equality is enforced by parity-exchange invariance of the normalized transition probabilities; in the trapped-ion realization, this invariance persists throughout an $\mathcal{PT}$15-rotated orbit of hybrid Hamiltonians and breaks when the dynamics leave that orbit (Yang et al., 11 May 2026).