---
title: Conditional Non-Hermitian Jarzynski Equality
url: https://www.emergentmind.com/topics/conditional-non-hermitian-jarzynski-equality
type: topic
---

# Conditional Non-Hermitian Jarzynski Equality

Searching arXiv for the cited papers to ground the article in current arXiv records.
arXiv_search query: 2605.10099
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 [2605.10099].

## 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 [2309.12393].

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 [2605.10099].

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 [1511.06256].

## 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 $|e_i\rangle$. The system then evolves under the non-Hermitian propagator
$$
K(T)=\mathcal{T}\exp\bigl[-i\!\int_0^T\!H_{\rm eff}(t)\,dt\bigr]
$$
conditioned on no jumps. At $t=T$ a second energy measurement yields $|e_f\rangle$, and the work along that trajectory is
$$
W=E_f-E_i.
$$

The unnormalized transition amplitude is
$$
A_{fi}=\langle e_f|K(T)|e_i\rangle,
$$
with
$$
p_{fi}=|A_{fi}|^2,\qquad S_i(T)=\sum_f p_{fi}=\langle e_i|K^\dagger K|e_i\rangle.
$$
The normalized conditional transition probabilities are obtained by renormalizing with the survival weight $S_i(T)$, so that $\sum_f P_{fi}=1$. The Jarzynski average then takes the explicit two-level form
$$
\langle e^{-\beta W}\rangle
=\sum_{i,f}e^{-\beta(E_f-E_i)}\,P_{fi}\,P_i
=\frac{1}{Z_i}\Big[e^{-\beta J_f}(1+P_{++}-P_{--})
+e^{\beta J_f}(1+P_{-+}-P_{+-})\Big].
$$
For cyclic $H_{HM}$ one has $\Delta F=0$, so the target relation reduces to $\langle e^{-\beta W}\rangle=1$ [2605.10099].

This formulation is structurally parallel to the superconducting-qubit implementation of post-selected non-Hermitian work statistics, where the effective Hamiltonian is
$$
H_{\rm eff}(t)=J(t)\,\sigma_x+\Delta(t)\,\tfrac{\sigma_z}{2}+i\frac{\gamma}{4}\,\sigma_z,
$$
the initial and final projective measurements are performed on $H(0)=H(\tau)=J_{\max}\sigma_x$, and the conditional probabilities are renormalized at the end of each no-jump run [2309.12393].

## 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
$$
\langle e^{-\beta W}\rangle=e^{-\beta\Delta F}
$$
with $\Delta F=0$ for cyclic $H_{HM}$ holds exactly **if and only if**
$$
P_{++}=P_{--},\qquad P_{+-}=P_{-+}. \tag{C}
$$
This is a symmetry under exchanging the two energy eigenstates, termed parity-exchange symmetry [2605.10099].

The associated involutive operator is
$$
\mathcal{P}_{\rm ex}
=|e_+\rangle\langle e_-|+|e_-\rangle\langle e_+|,\qquad
\mathcal{P}_{\rm ex}^2=I,
$$
which acts as
$$
\mathcal{P}_{\rm ex}|e_\pm\rangle=|e_\mp\rangle.
$$
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,
$$
P_{++}=P_{--},\quad P_{+-}=P_{-+},
$$
where the Methods show that it is equivalent to requiring that the non-Hermitian Floquet Hamiltonian
$$
H_{\rm eff}^F=H^F+i\,\Gamma^F=\sum_{k=x,y,z}(h_k+i\gamma_k)\sigma_k
$$
satisfies both an explicit or emergent $\mathcal{PT}$ symmetry and the commutation condition that its Hermitian part $H^F=h_x\sigma_x$ commutes with the initial Hamiltonian [2309.12393]. The 2026 work recasts the operative condition in symmetry language directly at the level of transition probabilities and extends it beyond the isolated $\mathcal{PT}$ endpoint [2605.10099].

## 4. $\mathrm{SU}(2)$-rotated $\mathcal{PT}$–$\mathcal{APT}$ hybrid family

The relevant non-Hermitian model is built from two prototype two-level Hamiltonians in the $\sigma$ basis,
$$
H_{PT}=i\gamma\,\sigma_z+J\,\sigma_x,\qquad
H_{APT}=i\gamma\,\sigma_x-J\,\sigma_z.
$$
Their one-parameter hybrid is defined as
$$
H_{hb}(\theta)
=\cos\theta\,H_{PT}+\sin\theta\,H_{APT}
=J(\sin\theta\,\sigma_x-\cos\theta\,\sigma_z)
+i\gamma(\cos\theta\,\sigma_x+\sin\theta\,\sigma_z).
$$
Its Hermitian part is
$$
H_{HM}(\theta)=J(\sin\theta\,\sigma_x-\cos\theta\,\sigma_z),
$$
with eigenstates
$$
|e_-(\theta)\rangle
=\begin{pmatrix}-\cos\frac\theta2\\\sin\frac\theta2\end{pmatrix},
\qquad
|e_+(\theta)\rangle
=\begin{pmatrix}\sin\frac\theta2\\\cos\frac\theta2\end{pmatrix},
$$
and energies $\pm J$ [2605.10099].

Equivalently, the hybrid family satisfies
$$
H_{hb}(\theta)=U(\theta)\,H_{APT}\,U^\dagger(\theta),
\qquad
U(\theta)=e^{-i(-\theta)\sigma_y/2}\in SU(2),
$$
so varying $\theta$ traces out a closed $S^1$ adjoint orbit of $H_{APT}$ in the space of $2\times2$ operators. The experimental study samples $\theta_k=\{0,\pi/4,\pi/2\}$ as three representative points, corresponding to APT, hybrid, and PT cases in the implementation [2605.10099].

The significance of this construction is that the Jarzynski relation is not tied to a single symmetry endpoint. Relative to earlier $\mathcal{PT}$-focused conditional Jarzynski equality results, the advance is an extension of the symmetry criterion from the isolated $\mathcal{PT}$ endpoint to a broader $\mathcal{PT}$–$\mathcal{APT}$ hybrid family [2605.10099].

## 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
$$
\mathcal{P}_{\rm ex}(\theta)
=-\sin\theta\,\sigma_z-\cos\theta\,\sigma_x
$$
satisfies the algebraic anti-symmetry
$$
\mathcal{P}_{\rm ex}\,H_{hb}(\theta)^*\,\mathcal{P}_{\rm ex}
=-\,H_{hb}(\theta).
$$
Exponentiating, and using $\mathcal{P}_{\rm ex}^2=I$, one obtains
$$
\mathcal{P}_{\rm ex}\,K(T)^*\,\mathcal{P}_{\rm ex}=K(T),
\qquad K(T)=e^{-iH_{hb}(\theta)T}.
$$
For the survival amplitudes $K_{fi}=\langle e_f|K|e_i\rangle$ this yields
$$
K_{++}=K_{--}^*,\qquad K_{+-}=K_{-+}^*,
$$
so that
$$
p_{++}=p_{--},\qquad p_{+-}=p_{-+}.
$$
Because
$$
S_+(T)=p_{++}+p_{-+}=S_-(T),
$$
the normalized probabilities obey exactly condition (C) [2605.10099].

Second, the Bloch-sphere picture gives a geometric interpretation. The no-jump state starting from $|e_\pm(\theta)\rangle$ defines a real Bloch vector $n_\pm(T)$, and the final measurement projects on the $H_{HM}(\theta)$ axis $\hat n_{HM}$. The diagonal conditional probabilities can be written as
$$
P_{++}=\tfrac12\bigl[1+n_+(T)\!\cdot\!\hat n_{HM}\bigr],\qquad
P_{--}=\tfrac12\bigl[1-n_-(T)\!\cdot\!\hat n_{HM}\bigr].
$$
The operator symmetry enforces the exact mirror relation
$$
n_+(T)\!\cdot\!\hat n_{HM}=-\,n_-(T)\!\cdot\!\hat n_{HM},
$$
which immediately gives $P_{++}=P_{--}$, and normalization then gives $P_{+-}=P_{-+}$ [2605.10099].

Substituting condition (C) into the Jarzynski average for cyclic protocols gives
$$
\langle e^{-\beta W}\rangle
=\frac{1}{Z_i}\Big[e^{-\beta J}(1+P_{++}-P_{--})
+e^{\beta J}(1+P_{-+}-P_{+-})\Big]
=\frac{1}{Z_i}\big[e^{-\beta J}+e^{\beta J}\big]
=1,
$$
since
$$
Z_i=e^{-\beta J}+e^{+\beta J}.
$$
The equality is therefore symmetry-enforced rather than generic [2605.10099].

## 6. Trapped-ion implementation

The experimental realization uses a single trapped $^{171}\mathrm{Yb}^+$ ion as a passive driven-dissipative qubit. The qubit transition $|0\rangle\leftrightarrow|1\rangle$ is driven by microwaves implementing
$$
J(t)\sigma_x+\Delta(t)\sigma_z/2,
$$
while a dissipation beam on $|1\rangle\to|a\rangle$ engineers the non-Hermitian core $i\gamma\sigma_z$ plus a global decay $-i\gamma I$. An $\mathrm{SU}(2)$ rotation by $\pm(\theta_k-\pi/2)$ about $y$ before and after the core evolution transforms the laboratory Hamiltonian into $H_{hb}(\theta_k)$ with $\Delta=0$ [2605.10099].

The cyclic TPM work protocols consist of three steps. First, there is deterministic preparation of $|e_\pm(\theta_k)\rangle$, simulating the first projective measurement with Boltzmann weights $P_\pm$. Second, the system undergoes non-unitary evolution $U'(\theta_k,T)$ under $H'_{\rm total}(\theta_k,t)$ for $T=10\ldots 50\,\mu{\rm s}$ using three driving profiles: constant $J$ with $\Delta=0$; a linear ramp of $J$ up and down with $\Delta=0$; and constant $J$ with sinusoidal $\Delta(t)$ of zero net area. Third, a final projective readout in the same eigenbasis yields
$$
P_{fi}(\theta_k,T)=\frac{|\langle f|U'|i\rangle|^2}{\langle i|U'^\dagger U'|i\rangle}.
$$

Across the three representative angles and the first two driving protocols, which stay within the $\mathrm{SU}(2)$ subspace, the measured transition probabilities satisfy
$$
P_{++}=P_{--},\qquad P_{+-}=P_{-+},
$$
and the Jarzynski average obeys
$$
\langle e^{-\beta W}\rangle\approx 1
$$
within experimental uncertainty. When detuning $\Delta(t)\neq 0$ drives the system outside the $\mathrm{SU}(2)$ orbit, the symmetry and equality are broken for generic $T$, except at special revival times $T_1,T_2$ at which the effective Floquet Hamiltonian again lies in the hybrid $S^1$ and restores parity-exchange [2605.10099].

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
$$
U_\tau^\dagger\,g_\tau\,U_\tau=g_0.
$$
The “conditional” generalization there is
$$
\sum_{n,m\in M}P(n,m)\,e^{-\beta(E_m^\tau-E_n^0)}
=\frac{1}{Z_0}\sum_{m\in M}e^{-\beta E_m^\tau}
\equiv \frac{Z_\tau(M)}{Z_0},
$$
for any subset $M$ of final eigen-indices [1511.06256].

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 $\mathcal{PT}$ symmetry and its Hermitian part aligns with the initial energy basis. Cyclic sweeps of $J(t)$ with $\Delta(t)=0$ preserve the exchange symmetry of transition probabilities, even across the exceptional point at $J_{\rm EP}=\gamma/4$, whereas single-lobe detuning sweeps with no $\mathcal{PT}$ symmetry lead to clear violations [2309.12393].

Against that background, the conditional non-Hermitian Jarzynski equality of the hybrid $\mathcal{PT}$–$\mathcal{APT}$ family has a sharply delimited meaning. It does **not** claim that arbitrary non-Hermitian evolution satisfies $\langle e^{-\beta W}\rangle=e^{-\beta\Delta F}$. 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 $\mathrm{SU}(2)$-rotated orbit of hybrid Hamiltonians and breaks when the dynamics leave that orbit [2605.10099].

Source: https://www.emergentmind.com/topics/conditional-non-hermitian-jarzynski-equality