#  Dynamical topology of chiral and nonreciprocal state transfers in a non-
Hermitian quantum system

Pengfei Lu1,2,4, Yang Liu1,2,3,4, Qifeng Lao1, Teng Liu1, Xinxin Rao1, Ji
Bian1,
Hao Wu1, Feng Zhu1,2, Le Luo1,2,3,∗
1School of Physics and Astronomy, Sun Yat-sen University, Zhuhai 519082,
China.
2Center of Quantum Information Technology,
Shenzhen Research Institute of Sun Yat-Sen University, Shenzhen 518087, China.
3Quantum Science Center of Guangdong-HongKong-Macao Greater Bay Area,
Shenzhen 518048, China.
4These authors contributed equally to this work.
∗Corresponding author. E-mail<EMAIL_ADDRESS>
###### Abstract

The fundamental concept underlying topological phenomena posits the geometric
phase associated with eigenstates. In contrast to this prevailing notion,
theoretical studies on time-varying Hamiltonians allow for a new type of
topological phenomenon, known as topological dynamics, where the evolution
process allows a hidden topological invariant associated with continuous
flows. To validate this conjecture, we study topological chiral and
nonreciprocal dynamics by encircling the exceptional points (EPs) of non-
Hermitian Hamiltonians in a trapped ion system. These dynamics are
topologically robust against external perturbations even in the presence
dissipation-induced nonadiabatic processes. Our findings indicate that they
are protected by dynamical vorticity—an emerging topological invariant
associated with the energy dispersion of non-Hermitian band structures in a
parallel transported eigenbasis. The symmetry breaking and other key features
of topological dynamics are directly observed through quantum state
tomography. Our results mark a significant step towards exploring topological
properties of open quantum systems.

## Introduction

Geometric phases play a crucial role in classifying gapped quantum systems
that are protected by the symmetry of Hamiltonians [1, 2], often resulting in
robust physical properties that are resilient to perturbations. In time-
varying cases, if the evolution is sufficiently slow, the system remains in
the eigenstate of the instantaneous Hamiltonian, thereby preserving the
geometric phases during dynamic processes. This phenomenon, validated in
various Hermitian systems [3, 4, 5, 6], relies on the preservation of symmetry
and adherence to the adiabatic condition. One might expect that this similarly
applies to time-varying non-Hermitian systems; However, this expectation does
not hold. When encircling the exceptional points (EPs) in a non-Hermitian
system, the state deviates from the eigenstate of the instantaneous
Hamiltonian regardless of how slowly the system evolves [7, 8]. This deviation
is known as dissipation-induced nonadiabatic transitions (DNATs). The
occurrence of DNATs disrupts the continuous accumulation of geometric phases
on the Riemann surfaces of complex eigenvalues. Consequently, characterizing
the topological properties of dynamical non-Hermitian systems remains a
significant and unresolved question.

Previous research on encircling EPs in non-Hermitian systems reveals
fascinating topological behaviors, which have been experimentally explored in
both classical and quantum systems, including microwave/optical setups [9, 7,
10, 11, 12, 13, 14], optomechanical oscillators [8], acoustic cavities [15,
16], superconducting circuits [17], NV centers [18], and cold atoms [19]. Some
of these experiments have demonstrated that in the absence of DNATs, a
parameter variation encircling an EP causes two states to switch positions
after one cycle and acquires a geometric phase of $\pi$ after two cycles. With
DNATs, however, the geometric phase becomes elusive during dynamic processes.
Instead, the vorticity of energy eigenvalues may serve as a new type of
topological invariant in non-Hermitian systems [20, 21], but its validity
depends on the evolution trajectories remaining on the Riemann surfaces, which
is contradicted by the presence of DNATs. The most intriguing topological
behaviors in these systems [7, 8, 10], such as chiral state transfers and
nonreciprocal mode switching, depend on the occurrence of DNATs. Therefore, it
is essential to uncover a profound relationship between DNATs and hidden
topological invariants in dynamics. This understanding could help identify the
topological structure of non-Hermitian dynamics.

In this work, we experimentally study the topological chiral and nonreciprocal
state transfers by dynamically encircling the EPs of both $\mathcal{PT}$
(parity-time) ($[H,\mathcal{PT}]=0$) and $\mathcal{APT}$ (anti-parity-time)
symmetric ($\\{H,\mathcal{PT}\\}=0$) Hamiltonians in a trapped-ion system. The
outcomes unveil a universal rule for determining the behavior of state
transfers associated with chiral and time-reversal symmetries. Chiral state
transfers, where encircling an EP in a clockwise or counterclockwise direction
results in different final states, are exclusively dictated by the symmetry of
the initial effective Hamiltonian in the parallel transported eigenbasis.
Nonreciprocal state transfers, where the processes depend on the initial
states and the direction of encircling the EP, are dictated by the time
derivative of the initial states. We find that these state transfers are
robust against substantial external noise introduced during the encircling
processes. To reveal the underlying topological mechanism of this robustness,
we transform the original Hamiltonian into the case in the parallel
transported eigenbasis and define a new topological invariant, dynamic
vorticity. This dynamic vorticity remains invariant during the encircling
process, regardless of whether dissipation-induced nonadiabatic transitions
(DNATs) occur, and is solely determined by the number of EPs that the
trajectories encircle. This reveals that both chiral and nonreciprocal state
transfers are protected by dynamic vorticity, which can be considered a
universal topological invariant in time-varying non-Hermitian systems. These
discoveries validate that topological dynamics can arise from the interplay of
dissipation and coherence, opening new avenues to explore the topological
properties of open quantum systems.

## Results

### Dynamically encricling the EPs of $\mathcal{PT}$ and $\mathcal{APT}$
symmetric Hamiltonians

The experiments utilize a passive $\mathcal{PT}$-symmetric system involving a
single trapped 171Yb+ ion, building upon the methodologies detailed in our
previous works [22, 23, 24]. The ion is confined and laser-cooled in a linear
Paul trap. Then, it is initialized to the hyperfine state
$\mid\downarrow\rangle=|F=0,m_{F}=0,{}^{2}S_{1/2}\rangle$ of the ground state
by optical pumping. The qubit levels, hyperfine states $\mid\downarrow\rangle$
and $\mid\uparrow\rangle=|F=1,m_{F}=0,{}^{2}S_{1/2}\rangle$, are driven by a
microwave with a coupling rate $J$. The dissipation is introduced by
resonantly driving a transition from $\mid\uparrow\rangle$ to
$|F=1,m_{F}=0,{}^{2}P_{1/2}\rangle$ of the excited state, resulting in the
spontaneous decay to three ground states $|F=1,m_{F}=0,\pm 1\rangle$ in
${}^{2}S_{1/2}$ with equal probability. Taken the Zeeman sublevels
$|F=1,m_{F}=\pm 1\rangle$ in ${}^{2}S_{1/2}$ as an auxiliary state
$|a\rangle$, the system is equivalent to a spin-dependent loss from the qubit
state $\mid\uparrow\rangle$ to state $|a\rangle$ with a rate at $4\gamma$, as
the coupling of the qubit to the environment. When the coupling strength of
the dissipation beam is significantly lower than the linewidth of the
${}^{2}P_{1/2}$ excited state, the involved energy levels can be approximated
to a dissipative two-level system, as illustrated in Fig. 1A. Consequently, a
passive $\mathcal{PT}$-symmetric non-Hermitian Hamiltonian is derived [25, 22,
26], $H_{eff}=J\hat{\sigma}_{x}+i\gamma\hat{\sigma}_{z}-i\gamma\hat{I}$, where
$\hat{\sigma}_{x}=\mid\downarrow\rangle\langle\uparrow\mid+\mid\uparrow\rangle\langle\downarrow\mid$,
$\hat{\sigma}_{z}=\mid\downarrow\rangle\langle\downarrow\mid-\mid\uparrow\rangle\langle\uparrow\mid$,
and
$\hat{I}=\mid\downarrow\rangle\langle\downarrow\mid+\mid\uparrow\rangle\langle\uparrow\mid$,
which can be mapped to a $\mathcal{PT}$-symmetric Hamiltonian by adding an
extra term $i\gamma\hat{I}$.

To encircle the EP of the $\mathcal{PT}$ symmetric Hamiltonian, the driving
microwave features time-varying detuning and intensity, resulting in a time-
dependent Hamiltonian expressed as ($\hbar=1$)

$H(t)=\begin{pmatrix}\Delta(t)/2+i\gamma&J(t)\\\
J(t)&-\Delta(t)/2-i\gamma\end{pmatrix},$ (1)

where $\Delta(t)$ and $J(t)$ represent the detuning and the coupling rate of
the microwave, respectively, and $\gamma$ denotes the dissipation rate
dependent on the laser intensity. The eigenvalues of Eq. (1) are complex
numbers given by
$\lambda_{1,2}(t)=\pm\sqrt{J(t)^{2}-\gamma(t)^{2}+i\gamma(t)\Delta(t)+\Delta(t)^{2}/4}$,
with the EP occurring at $\Delta=0$ and $J=\gamma$.

The system evolves by varying $\Delta(t)$ and $J(t)$, while keeping $\gamma$
fixed. The real and imaginary components of $\lambda_{1,2}(t)$ form two
intersecting Riemann sheets wrapped around the EP, as shown in Fig. 1B. We
drive the qubit through a parameter loop defined by
$\Delta(t)=r\sin[\theta(t)+\theta_{0}]$ and
$J(t)=J_{0}+r\cos[\theta(t)+\theta_{0}]$, where $r$ represents the encircling
radius, $\theta(t)$ denotes the time-dependent encircling angle, and
$\theta_{0}=0$ ($\theta_{0}=\pi$) determines the starting point in the
$\mathcal{PT}$-symmetric (broken) regime, abbreviated as the $\mathcal{PTS}$
($\mathcal{PTB}$) regime. More details regarding the encircling procedure can
be found in Materials and Methods.

Figure 1: Generation of chirality and nonreciprocity in a dissipative trapped-
ion qubit. (A) Schematic diagram of the energy levels of a 171Yb+ ion. The
five involved levels include $|F=0,m_{F}=0\rangle$ and $|F=1,m_{F}=0,\pm
1\rangle$ in the electronic ground state ${}^{2}S_{1/2}$, and
$|F=0,m_{F}=0\rangle$ in the electronic excited state ${}^{2}P_{1/2}$. (B)
Encircling paths in the parameter space (coupling strength $J$ and detuning
$\Delta$) and the state evolution trajectories projected onto the eigenvalues’
Riemann sheets, starting in the $\mathcal{PT}$-symmetric ($\mathcal{PTS}$)
regime of the $\mathcal{PT}$ Hamiltonian. The solid (dashed) trajectory in (B)
denotes the clockwise (counterclockwise) evolution of $|\alpha_{A}(0)\rangle$.
(C) Chiral symmetry (depicted by red and green curves in the left panel) and
broken chiral symmetry (depicted by red and blue curves in the left panel)
serve as mathematical criteria for determining chiral dynamics. Time-reversal
symmetry (illustrated by red and green curves in the right panel) and broken
time-reversal symmetry (illustrated by red and blue curves in the right panel)
serve as mathematical criteria for determining nonreciprocal dynamics.

The clockwise and counterclockwise encircling trajectories from the eigenstate
$\left|\alpha_{A}(0)\right\rangle$ ($\left|\beta_{A}(0)\right\rangle$) are
demonstrated in Fig. 2, A and C (E and G), where the black lines show the
state evolution projected onto the Riemann surfaces. Fig. 2, B and D (F and H)
show the overlap between the instantaneous eigenstate $|\alpha(t)\rangle$
($|\beta(t)\rangle$) with the evolutionary state
$|\psi(t)\rangle=C_{1}(t)|\alpha(t)\rangle+C_{2}(t)|\beta(t)\rangle$, i.e.
$\left\langle\alpha_{A}(t)|\psi(t)\right\rangle$
($\left\langle\beta_{A}(t)|\psi(t)\right\rangle$). When a DNAT occurs, the
qubit invariably jumps from the loss sheet (blue) to the gain sheet (red) on
the Riemann surface. Such jump is characterized by the crossing between
$\left\langle\alpha_{A}(t)|\psi(t)\right\rangle$ and
$\left\langle\beta_{A}(t)|\psi(t)\right\rangle$ in the cyan shaded region. Its
occurrence is measured through quantum state tomography of $|\psi(t)\rangle$
during these experiments. We also investigate the dynamical encircling with
$\mathcal{APT}$ symmetric Hamiltonians, constructed by sandwiching a passive
$\mathcal{PT}$-symmetric Hamiltonian $H_{eff}$ between two $\pi/2$ pulses
along the $\pm Y$ axis on the Bloch sphere [23]. The details are provided in
Supplementary Material S2.

### Chiral and nonreciprocal state transfer

The analyses of chiral behaviors in state transfer are depicted in the left
panel of Fig. 1C. The clockwise evolution of
$|\alpha_{A}(0)\rangle\xrightarrow{\text{CW}}|\beta_{A}(0)\rangle$ (trajectory
1) and the counterclockwise case of
$|\alpha_{A}(0)\rangle\xrightarrow{\text{CCW}}|\alpha_{A}(0)\rangle$
(trajectory 2) exhibit the chiral state transfer, whose evolution trajectories
are shown in Fig. 2, A to D. Another example of chiral state transfer involves
the pair of of
$|\beta_{A}(0)\rangle\xrightarrow{\text{CW}}|\beta_{A}(0)\rangle$ (trajectory
3) and $|\beta_{A}(0)\rangle\xrightarrow{\text{CCW}}|\alpha_{A}(0)\rangle$
(trajectory 4), as shown in Fig. 2, E to H. It is noted that the chiral state
transfers are associated with starting points located in the $\mathcal{PTS}$
regime. The paths in opposite directions experience unequal numbers of DNATs,
preserving the chiral symmetry. Conversely, for the starting point in the
$\mathcal{PTB}$ regime, whether starting from $|\alpha_{B}(0)\rangle$ (Fig. 2,
I to L) or $|\beta_{B}(0)\rangle$ (Fig. 2, M to P), both clockwise and
counterclockwise paths experience the same number of DNATs, breaking the
chiral symmetry.

Figure 2: The time-varying evolutionary state $|\psi(t)\rangle$ starts from
either the $\mathcal{PTS}$ or $\mathcal{PTB}$ regime with the $\mathcal{PT}$
Hamiltonian. The solid (dashed) box depict the state evolution trajectories in
the visual four-dimensional picture of the eigenspectrum as a function of the
$\Delta$ and $J$, with the colors on the Riemann sheet corresponding to the
respective imaginary (real) values. Clockwise and counterclockwise encircling
the EP starting from $|\alpha_{A}(0)\rangle$ ($|\alpha_{B}(0)\rangle$ ) and
$|\beta_{A}(0)\rangle$ ($|\beta_{B}(0)\rangle$) are represented by trajectory
1-4 in (A), (C), (E) and (G) (trajectory 5-8 in (I), (K), (M) and (O)). Figs.
(B), (D), (F) and (F) ((J), (L), (N) and (P)) represent the overlap between
the two instantaneous eigenstates and the evolutionary state
$\langle\alpha_{A}(t)|\psi(t)\rangle$, $\langle\beta_{A}(t)|\psi(t)\rangle$
($\langle\alpha_{B}(t)|\psi(t)\rangle$, $\langle\beta_{B}(t)|\psi(t)\rangle$ )
from $\mathcal{PTS}$ ($\mathcal{PTB}$) regime, where the nonadiabatic dynamics
emerge in the cyan shaded regions. Circles and lines represent experimental
measurements and numerical simulation, respectively.

The right panel of Fig. 1C explains how nonreciprocity is generated due to
broken time-reversal symmetry (TRS). Starting from the eigenstates in the
$\mathcal{PTS}$ regime, the forward-time evolution (trajectory 1) and the
backward-time evolution (trajectory 4) constitute a pair of reciprocal
processes, the same as trajectory 5 and trajectory 6 with the initial
eigenstate in the $\mathcal{PTB}$ regime. For both cases, the same number of
DNATs occurs in the forward-time and backward-time paths. On the contrary, if
the forward-time and backward-time encircling have different numbers of the
DNATs, as seen in trajectories 3 and 4, trajectories 2 and 1, trajectories 7
and 6, and, trajectories 8 and 5, TRS is broken, resulting in nonreciprocal
state transfers.

In addition, we systematically investigate the chirality and nonreciprocity
with the $\mathcal{APT}$ Hamiltonian. The emergence of chiral (nonchiral)
behavior in state transfer occurs when the starting point is in the
$\mathcal{PTB}$ ($\mathcal{PTS}$) regime, opposite to the
$\mathcal{PT}$-symmetric Hamiltonian. Regarding the chiral (nonreciprocal)
properties, we discover a dual relationship between the
$\mathcal{PT}$-Hamiltonian and the $\mathcal{APT}$ one, where the
$\mathcal{PTS}$ regime of the $\mathcal{APT}$ Hamiltonian is equivalent to the
$\mathcal{PTB}$ regime of the $\mathcal{PT}$ Hamiltonian, and vice versa (See
Supplementary Material S2).

Based on the aforementioned observations, we ascertain that the unequal
numbers of DNATs appearing in a pair of encircling paths result in chiral
(nonreciprocal) state transfers. The crucial role of DNATs stems from the
dynamical phase accumulated as the parameters of the Hamiltonian evolve
adiabatically [27, 28]. The evolutionary state is expressed as
$|\psi_{n}(t)\rangle$ =$e^{-\frac{i}{\hbar}\int
E_{n}(\mathbf{R}(t^{\prime}))dt^{\prime}}\cdot
e^{i\gamma(t)}|n(\textbf{R}(t))\rangle$, where $e^{-\frac{i}{\hbar}\int
E_{n}(\mathbf{R}(t^{\prime}))dt^{\prime}}$ is dynamical phase term,
$e^{i\gamma(t)}$ is geometric phase term, and $|n(\textbf{R}(t))\rangle$ is
instantaneous eigenstate state of Hamiltonian. When the state evolves on the
loss sheet, the dynamical phase contributes a decaying factor. Consequently,
the state evolving on the loss sheet will transition to the gain sheet,
whereas the state on the gain sheet will remain unaffected.

### Classification of nonadiabatic transition

We have developed a quantitative approach to examine the dependence of
chirality (nonreciprocity) on the adiabaticity of the encircling process. In
this method, the chiral (nonreciprocal) behavior is characterized by the the
fidelity of state transfer during one encircling of EP, defined as the squared
overlap between the final state $\psi(T)$ and the initial state. The
adiabaticity of the encircling is quantified by $\tau_{\textrm{crit}}$=
max$(\frac{1}{\lambda_{1}(t)-\lambda_{2}(t)})$. In the experiments, the
fidelities of state transfer
$|\alpha_{A}(0)\rangle\xrightarrow{\text{CW}}|\beta_{A}(0)\rangle$ (trajectory
1), i.e., $|\langle\alpha_{A}(0)|\psi(T)\rangle|^{2}$ and
$|\langle\beta_{A}(0)|\psi(T)\rangle|^{2}$, are studied with various
encircling periods and radii, as shown in Fig. 3. These results demonstrate
the dependence of chirality (nonreciprocity) on the adiabaticity of the
encircling process.

Figure 3: Investigation of the nonadiabatic effects with clockwise encircling
the EP starting from the initial eigenstate $|\alpha_{A}(0)\rangle$
(trajectory 1). (A-B) The dependence of the ratio $\tau_{\textrm{crit}}/T$ on
the encircling period and radius. (C-D) The change in the fidelity of state
transfer with varying periods and radii. The contour map and solid lines
denote the simulation results, while the circles represent the experimental
data. (E-F) The overlap $\langle\alpha_{A}(t)|\psi(t)\rangle$ (solid line) and
$\langle\beta_{A}(t)|\psi(t)\rangle$ (dash line) with varying period for the
initial state (E) $|\alpha_{A}(0)\rangle$ and (F) $|\beta_{A}(0)\rangle$,
respectively. The blue, red, and black circles correspond to $T=16.67~{}\mu
s$, $100~{}\mu s$, and $250~{}\mu s$, respectively, which match the vertical
lines in (A). (G-H) The overlap with varying radii for the initial state (E)
$|\alpha_{A}(0)\rangle$ and (F) $|\beta_{A}(0)\rangle$, respectively. The
blue, red, and black squares correspond to $r=$0.003 MHz, 0.008 MHz, and 0.03
MHz, respectively, which correspond to the vertical lines in (B).

The results for various periods are shown in Fig. 3A.
$\tau_{\textrm{crit}}=11.8~{}\mu s$ for all selected periods. For an
encircling period $T=250~{}\mu$s $\gg\tau_{\textrm{crit}}$ that satisfies the
adiabatic criteria, the fidelity of state transfer
$\langle\beta_{A}(0)|\psi(T)\rangle^{2}$ is nearly unity, representing almost
perfect state swaps of $|\alpha_{A}(0)\rangle$ and $|\beta_{A}(0)\rangle$.
When the encircling period reduces, the deviations from the ideal fidelities
get larger and larger, as shown in shown in the olive curve of the side plane
in Fig. 3, C-D. For example, for $T=100~{}\mu$s,
$\langle\beta_{A}(0)|\psi(T)\rangle$ becomes smaller than the unity,
indicating that the final state fails to fully reach $|\beta_{A}(0)\rangle$.
This decay of state overlap is more obvious for $T=16.67~{}\mu$s. Such
deviation from $|\beta_{A}(0)\rangle$ lead to the breakdown of the chiral
(nonreciprocal) state transfer.

The results for various radii are depicted in the dark yellow curve on the
side plane in Fig. 3, C-D, showing chiral (nonreciprocal) behavior primarily
observed at larger radii. For $r=$0.03 MHz, the fidelity of
$\langle\beta_{A}(0)|\psi(T)\rangle^{2}$ approaches unity. However, as the
radius decreases, $\tau_{crit}/T<0.1$, resulting in a significant deviation
from the predictions of the adiabatic theorem. The state transfer for the
$\mathcal{APT}$-symmetric Hamiltonian, with varying periods and radii, can be
found in Supplemental Material S2.

Here, we observe DNATs during the adiabatic evolution of system parameters,
which are distinctly different from the conventional speed-dependent
nonadiabatic transitions (SNATs) where the evolution does not satisfy the
adiabatic theorem. In Fig. 3E, as the period reduces from $T=250~{}\mu$s to
$T=16.67~{}\mu$s, we observe a crossing of
$\langle\alpha_{A}(t)|\psi(t)\rangle$ and $\langle\beta_{A}(t)|\psi(t)\rangle$
(for $T=16.67~{}\mu$s blue curves), indicating the occurrence of a SNAT.
Additionally, reducing the encircling period deviates the path from the
prediction dictated by the adiabatic theorem, as demonstrated by the different
crossings in Fig. 3F. The new crossing of
$\langle\alpha_{A}(t)|\psi(t)\rangle$ and $\langle\beta_{A}(t)|\psi(t)\rangle$
does not appear due to the dominant DNAT. Notably, decreasing the encircling
radius does not generate SNATs but only decreases the fidelity of the DNAT, as
shown in Fig. 3, G-H.

### Robustness of the state transfer

Figure 4: Investigation of the robustness of dynamically encircling an EP. (A,
C, E and G) The color map shows the overlap between the instantaneous
eigenstate and the evolutionary state under varying noise intensity. (B, D, F,
and H). The experimental results corresponding to the three dashed lines in
(A, C, E, and G) demonstrate robustness against noise, as all data match the
predicted values (solid lines). The insets in (B) and (F) display clockwise
encircling trajectories in the two-dimensional parameter space
$\\{\Delta,J\\}$ with random noise intensities $ir=$ 0 (black), 0.3 (red), and
0.5 (blue), respectively. In Fig. A-D, the encircling starts from point A in
the $\mathcal{PTS}$ regime, while in Fig. E-H, it starts from point B in the
$\mathcal{PTB}$ regime.

We experimentally verify the topological robustness of state transfers using
$\mathcal{PT}$ and $\mathcal{APT}$-symmetric Hamiltonians against the physical
noises in a dissipative trapped-ion qubit. For the $\mathcal{PT}$ Hamiltonian,
we introduce noises into the detuning and the coupling rate along the
encircling path

$\begin{cases}\Delta(t)=r(1+\kappa(t))Sin[\omega t+\theta_{0}]\\\
J(t)=\gamma+r(1+\kappa(t))Cos[\omega t+\theta_{0}],\end{cases}$ (2)

where the encircling radius $r=0.03$ MHz, the dissipation rate $\gamma=0.06$
MHz, and $\kappa(t)$ represents the noise in the encircling process.
$\kappa(t)\in[-ir,ir]$ is generated by pseudorandom real number, where $ir$ is
defined as the intensity of the noise.

Figure. 4 illustrates the final state outcomes of the encircling process,
showing that it remains unaffected by random noise, regardless of the noise
intensity. Figs. 4, A to D (E to H) display the experimental results for the
$\mathcal{PT}$-symmetric (broken) regime. Remarkably, we find that chiral
(nonreciprocal) quantum state transfers exhibit robustness against such noise,
provided the adiabatic encircling condition is satisfied. Even in the
adiabatic limit of evolution, this robustness persists near the EP because the
external noise fails to eliminate the degeneracy at the EP. We have also
experimentally verified the topological robustness of the quantum state
transfer with the $\mathcal{APT}$-symmetric Hamiltonian, as detailed in
Supplementary Material S2. In both cases, the experimental data match pretty
well with the theoretically predicted values, even under varying noise
intensities. This validation confirms the robustness of topological state
transfer with $\mathcal{APT}$-symmetric Hamiltonians.

## Discussion

In this study, we have observed chiral (nonreciprocal) quantum dynamics in a
trapped-ion qubit, wherein the EPs of both $\mathcal{PT}$-symmetric and
$\mathcal{APT}$-symmetric Hamiltonians are dynamically encircled. These
dynamics are rooted in the unique topological structure of intersecting
Riemann sheets around the EP. During adiabatic state transfers, this structure
leads to the swapping of two eigenstates after one cycle and the acquisition
of a geometric phase of $\pi$ after two cycles [9, 15]. The topological
invariant of the adiabatic process is characterized by this geometric phase,
providing the protection against perturbations. For nonadiabatic processes, it
is usually believed that the nonadiabaticity would cause the evolution to
deviate from the eigenstates of the instantaneous Hamiltonian, and
consequently undermine topological protection from the Riemann surface.
However, previous experiments have confirmed the robust state transfer not
only for adiabatic processes, but also existing in ones involving dissipation-
induced nonadiabatic transitions (DNATs) [29, 30, 16, 14]. This raises one
critical question: is there a topological invariant to elucidate the robust
state transfers in nonadiabatic processes , which could unveil the topological
structure of DNATs. Addressing this question is crucial for fully
understanding chiral and nonreciprocal state transfers in non-Hermitian
systems. To date, this question remains unexplored.

Based on the above results, we provide a scheme to address this question, in
which a new topological invariant that we dub as “dynamic vorticity” is used
to characterize topological protections in the processes involving DNATs.
Previously, the vorticity
$\mathcal{V}=-\frac{1}{2\pi}\oint_{C}\nabla_{\theta}Arg[E_{+}(\theta)-E_{-}(\theta)]\cdot
d\theta$, also known as the spectral or eigenvalue winding number[31, 32], has
been applied to explain the topological protection of the adiabatic state
transfers [20, 21]. $C$ is a closed loop in the complex energy plane, $\theta$
is the encircling angle, $E_{\pm}$ are the eigenvalue of the Hamiltonian.
Remarkably, we find that this concept can be extended to the process involving
DNATs, if we adopt the parallel transported eigenbasis [33]. This
transformation into the parallel transported eigenbasis gives an effective
time-dependent Hamiltonian

$\widetilde{H}_{eff}(t)=\begin{pmatrix}-i\gamma-\lambda(t)&f(t)\\\
-f(t)&-i\gamma+\lambda(t)\end{pmatrix},$ (3)

where $f(t)$ is the term corresponding to the nonadiabatic transition (Details
in Materials and methods). By extending the definition of the vorticity to the
parallel transported eigenbasis, the dynamic vorticity is given by

$\mathcal{V_{D}}=-\frac{1}{2\pi}\oint_{C}\nabla_{\theta}\textrm{Arg}[\widetilde{E}_{+}(\theta)-\widetilde{E}_{-}(\theta)]\cdot
d\theta,$ (4)

where $\widetilde{E}_{\pm}$ is the eigenvalue of $\widetilde{H}_{eff}(t)$. The
dynamic vorticity associates with the energy dispersion of
$\widetilde{H}_{eff}(t)$, and inherits the topology of Riemann surfaces. The
essence of its validity lies in the fact that nonadiabatic transitions will
not cause the deviations of the instantaneous state from the Riemann surfaces
in the parallel transported basis. Consequently, the system’s evolution
consistently remains in the eigenstate of $\widetilde{H}_{eff}(t)$. Therefore,
it can be treated as a universal topological invariant for dissipation-induced
nonadiabatic state transfers near the EP.

In our experiments shown in Fig. 4, we find that $\mathcal{V_{D}}=\pm 1/2$,
i.e., $-1/2$ for trajectory 1 and 1/2 for the trajectory 2, confirming the
chiral (nonreciprocal) state transfer is topologically protected. It is
further confirmed by the state transfers investigated in Refs [10, 13], where
$\mathcal{V_{D}}=0$ characterizes the topological property of the processes
that do not encircle an EP. From theoretical calculations, we conclude that
$\mathcal{V_{D}}=\pm n/2$, where $n$ represents the number of EPs being
encircled and the sign $\pm$ depends on the orientation of the encircling
curve.

Another advantage of using the parallel transported eigenbasis is that the
effective Hamiltonian in this basis can be used to determine the occurrence of
chiral behavior. If the initial effective Hamiltonian satisfies
$\\{\widetilde{H}_{eff}(0),\mathcal{CPT}\\}=0$, where we define the chiral
operator $\mathcal{S}=\mathcal{CPT}$, our experimental results indicate a pair
of encircling dynamics have the chiral symmetry (i.e., trajectories 1 and 2 in
Fig. 2 have different final states). Here, parity operator
$\mathcal{P}=\sigma_{x}$, $\mathcal{C}=\sigma_{z}$, and time-reversal operator
$\mathcal{T}$ represents the conjugate operator. Conversely, if
$\\{\widetilde{H}_{eff}(0),\mathcal{CPT}\\}\neq 0$, the dynamics breaks the
chiral symmetry (i.e., trajectories 5 and 6 in Fig. 2 have the same final
states). Note that $\mathcal{S}=\mathcal{CPT}$ instead of
$\mathcal{S}=\mathcal{CT}$ in the Hermitian case [2, 34] due to non-
Hermiticity [35]. We also find that the nonreciprocal behavior depends on the
initial density matrix $\rho(0)$. The positive (negative) value of
$\textrm{Tr}(\sigma_{z}\rho(0))$ corresponds to the preservation (breakdown)
of TRS for a pair of the forward-time and backforward-time encircling
processes. When $\textrm{Tr}(\sigma_{z}\rho(0))=0$, if
$d[\textrm{Tr}(\sigma_{z}\rho(t))]/dt|_{t=0}>0$, the processes preserve the
TRS (i.e., trajectories 1 and 4). Otherwise, they exhibits nonreciprocal
behavior. Therefore, we conclude that the chiral and nonreciprocal dynamics
are dictated by the symmetry of the effective Hamiltonian
$\widetilde{H}_{eff}(0)$ and the initial state.

It is worth mentioning that the dynamic vorticity is limited to the
dissipation-induced nonadiabatic processes and is not suitable for speed-
induced nonadiabatic processes, where the evolution period is shorter than the
critical time stipulated by the adiabatic theorem. Our results in Fig. 3 show
that the occurrence of SNATs disrupts the chiral and nonreciprocal dynamics
due to the nonadiabaticity of the Landau-Zener tunneling from an eigenstate to
another at the avoided-crossing.

## Materials and methods

### Encircling method

The dynamically encircling of the EP is realized by using the time-dependent
detuning $\Delta(t)=r\sin[\theta(t)+\theta_{0}]$ (in MHz) and the time-
dependent coupling rate $J(t)=J_{0}+r\cos[\theta(t)+\theta_{0}]$ (in MHz),
while keeping the dissipation rate fixed [18]. Here, $r$ is the encircling
radius, $\theta(t)=\omega t$ is the encircling angle, and $\omega$ is the
angular speed of encircling whose sign determines the encircling direction
(“+” for clockwise encircling and “-” for counterclockwise encircling). The
parameter $\theta_{0}=0$ ( $\theta_{0}=\pi$) indicates that the starting point
lies at the $\mathcal{PTS}$ ($\mathcal{PTB}$) regime.

The evolutionary state at each time step, i.e.
$|\psi(t)\rangle=C_{1}(t)|\alpha(t)\rangle+C_{2}(t)|\beta(t)\rangle$ is the
coherent superposition of the eigenstates $|\alpha(t)\rangle$ and
$|\beta(t)\rangle$. The coefficients $C_{1}(t)$ and $C_{2}(t)$ are probability
amplitudes of the $|\alpha(t)\rangle$ and $|\beta(t)\rangle$, respectively.
The trajectory of the encircling, i.e.
$(|C_{1}(t)|^{2}\lambda_{1}+|C_{2}(t)|^{2}\lambda_{2})/(|C_{1}(t)|^{2}+|C_{2}(t)|^{2})$
is derived, where the sudden transitions between $|\alpha(t)\rangle$ and
$|\beta(t)\rangle$ can be analyzed through the adiabatic multipliers [36, 37].
With the derived trajectory, the state evolution is projected onto the complex
Riemann sheets of the eigenvalues, shown as the black line in Fig. 1B of the
main text.

The presence of spin-dependent loss causes the qubit population to
exponentially decay in a sinusoidal manner for small $\gamma/J$ ratios [22,
23, 38], leading to an extremely low state population that is challenging to
detect experimentally. In response to this challenge, we employ a piecewise
strategy. Encircling starts at $t=0$ and ends at $T=\frac{15}{\gamma}$.
Throughout the encircling process, the dissipation rate remains constant at
$\gamma=0.06$ MHz, the angular speed is maintained at
$\omega=\pm\frac{2\pi}{T}$, and the radius is fixed at $r=0.03$ MHz (unless
specified otherwise in subsequent experiments).

Figure 5: Experimental verification of the piecewise strategy with the initial
eigenstate $|\alpha_{A}(0)\rangle$ in the $\mathcal{PT}$-symmetric regime. (A)
The overlap between the evolutionary state $|\psi(t)\rangle$ and instantaneous
eigenstate $|\alpha_{A}(t)\rangle$. (B) The overlap between the evolutionary
state $|\psi(t)\rangle$ and instantaneous eigenstate $|\beta_{A}(t)\rangle$.
The experimental results with $N=$10, 20, 30, 50, 80 and 100 are represented
by black, red, green, blue, cyan, and magenta squares, respectively. The
yellow shaded region represents the numerical calculation with $5\%$
uncertainty in the density matrix of the measured state $|\psi(t)\rangle$,
accounting for fluctuations in the dissipation strength and the coupling
strength.

The entire encircling path, with a period of $T=2\pi/\omega$, is divided into
$N$ segments. In each of the $N$ segments ($1\leq n\leq N$), the qubit state
is prepared to evolve for $t_{n}=(n-1)T/N$ time according to theoretical
prediction. Then, it continues to evolve for $t=T/N$ under the Hamiltonian
$H(t_{n})$. With this scheme, we map out the whole encircling process by
quantum state tomography of $|\psi(t)\rangle$, which is carried out at the end
point of each segment. The overlap (i.e., the inner product) of the measured
evolutionary state $|\psi(t)\rangle$ and the instantaneous eigenstates
$|\alpha_{A}(t)\rangle$ (or $|\beta_{A}(t)\rangle$) of the time-varying
Hamiltonian $H_{eff}(t)$ is used to evaluate whether the measured state
$|\psi(t)\rangle$ matches with the theoretical calculation.

We adjust the size of $N$ to verify the effectiveness of the piecewise
strategy as shown in Fig. 5, where $J=0.06$ MHz and $\omega=2\pi\times$ 4 rad
ms-1, respectively. The encircling starts from $|\alpha_{A}(0)\rangle$, and
$N$ varies with 10 (black squares), 20 (red squares), 30 (green squares), 50
(blue squares), 80 (cyan squares) and 100 (magenta squares). The numerical
calculation has a $5\%$ uncertainty (orange error band) in the density matrix
of the measured evolutionary state $|\psi(t)\rangle$, accounting for the
fluctuations of the dissipation strength and the coupling strength. The
nonzero overlap between $|\beta_{A}(0)\rangle$ and $|\psi(0)\rangle$ is due to
the nonorthogonality of the two eigenstates of $H_{eff}(0)$. The experimental
results for $N=100$ agree well with the numerical simulation, while the others
do not, proving the validness of the proposed scheme for $N\geq 100$.
Therefore, we choose $N=100$ in the following experiments to investigate the
dynamics of the encircling.

### Mapping nonadiabatic transition amplitudes

Figure 6: The time-dependent nonadiabatic transition amplitude when the
$\mathcal{PT}$ Hamiltonian encircles the EP. (A-D) Encircling starting from
the $\mathcal{PT}$-symmetric regime: (A) Clockwise with the initial state
$|\alpha_{A}(0)\rangle$, (B) Counterclockwise with the initial state
$|\alpha_{A}(0)\rangle$, (C) Clockwise with the initial state
$|\beta_{A}(0)\rangle$, (D) Counterclockwise with the initial state
$|\beta_{A}(0)\rangle$. (E-H) Encircling starting from the
$\mathcal{PT}$-broken regime: (E) Clockwise with the initial state
$|\alpha_{B}(0)\rangle$, (F) Counterclockwise with the initial state
$|\alpha_{B}(0)\rangle$, (G) Clockwise with the initial state
$|\beta_{B}(0)\rangle$, (H) Counterclockwise with the initial state
$|\beta_{B}(0)\rangle$. The color maps for the time-dependent nonadiabatic
transition amplitude when changing the encirlcing period. The cyan circles
with error bars, solid and dashed lines on the encircling angle-$|R_{i}|$
plane correspond to the experimental data, theoretical calculation (Eq. 17)
and numerical simulation from $C_{1}(t)$ and $C_{2}(t)$, where the encircling
period $T=250~{}\mu s$.

The nonadiabatic transitions that occur during the encircling of the EP can be
attributed to the Stokes phenomenon of asymptotics [39, 40] or stability loss
delay [28, 41]. Here, we analyze DNATs observed in our experiment using the
method of parallel transported basis [28]. The state evolution for a passive
$\mathcal{PT}$ Hamiltonian is determined by ($\hbar$=1)

$i\frac{\partial}{\partial
t}\left|\psi\left(t\right)\right\rangle=H_{eff}\left|\psi\left(t\right)\right\rangle,$
(5)

where
$\left|\psi(t)\right\rangle=C_{1}(t)\left|\alpha(t)\right\rangle+C_{2}(t)\left|\beta(t)\right\rangle$
and $H_{eff}=\begin{pmatrix}\Delta/2&J\\\ J&-\Delta/2-2i\gamma\end{pmatrix}$.

The eigenvalues of $H_{eff}$ are
$\widetilde{\lambda}_{\pm}=-i\gamma\pm\sqrt{(\Delta/2+i\gamma)^{2}+J^{2}}=-i\gamma\pm\lambda$,
where $\lambda$ is the eigenvalue of the $\mathcal{PT}$-symmetric Hamiltonian
$H_{\mathcal{PT}}$ with detuning. The normalized eigenstates of $H_{eff}$ are

$\begin{split}\chi_{1}=\frac{1}{\sqrt{(\lambda-(\Delta/2+i\gamma))^{2}+J^{2}}}\begin{pmatrix}-\lambda+(\Delta/2+i\gamma)\\\
J\end{pmatrix}\\\
\chi_{2}=\frac{-1}{\sqrt{(\lambda+(\Delta/2+i\gamma))^{2}+J^{2}}}\begin{pmatrix}\lambda+(\Delta/2+i\gamma)\\\
J\end{pmatrix},\end{split}$ (6)

which can be expressed as the parallel transported eigenbasis

$\left|\alpha\right\rangle=\chi_{1}=\begin{pmatrix}cos(\theta/2)\\\
sin(\theta/2)\end{pmatrix},\\\
\left|\beta\right\rangle=\chi_{2}=\begin{pmatrix}-sin(\theta/2)\\\
cos(\theta/2)\end{pmatrix},$ (7)

where

$\begin{split}\sin(\theta/2)=\sqrt{\frac{\lambda+(\Delta/2+i\gamma)}{2\lambda}}\\\
\cos(\theta/2)=-\sqrt{\frac{\lambda-(\Delta/2+i\gamma)}{2\lambda}}\\\
\tan\theta=J/(\Delta/2+i\gamma).\end{split}$ (8)

Then, we use $\left|\alpha\right\rangle=T\begin{pmatrix}1\\\ 0\end{pmatrix}$,
$\left|\beta\right\rangle=T\begin{pmatrix}0\\\ 1\end{pmatrix}$ to represent
the parallel transported eigenbasis of Eq. 7. The rotation matrix
$T=\begin{pmatrix}\cos(\theta/2)&-\sin(\theta/2)\\\
\sin(\theta/2)&\cos(\theta/2)\end{pmatrix}$, where $TT^{\top}=T^{\top}T=1$.

Now $|\psi(t)\rangle=T|\psi^{\prime}(t)\rangle$, where
$|\psi^{\prime}(t)\rangle=C_{1}(t)\begin{pmatrix}1\\\
0\end{pmatrix}+C_{2}(t)\begin{pmatrix}0\\\
1\end{pmatrix}=\begin{pmatrix}C_{1}(t)\\\ C_{2}(t)\end{pmatrix}$. Substituting
the $|\psi(t)\rangle=T|\psi^{\prime}(t)\rangle$ into Eq. 5, we obtain

$i\frac{\partial}{\partial
t}(T|\psi^{\prime}(t)\rangle)=H_{eff}(T|\psi^{\prime}(t)\rangle).$ (9)

We consider the evolution $U^{\prime}(t)$ defined by
$|\psi^{\prime}(t)\rangle=U^{\prime}(t)|\psi^{\prime}(0)\rangle$. Eq. 9 can be
rewritten as $\frac{\partial}{\partial
t}(TU^{\prime}(t))=-iH_{eff}(TU^{\prime}(t))$, which is simplified as

$\dot{T}U^{\prime}(t)+T\dot{U^{\prime}}(t)=-iH_{eff}(TU^{\prime}(t)).$ (10)

Multiplying $T^{\top}$ to the left of both sides in Eq. 10, we obtain

$T^{\top}\dot{T}U^{\prime}(t)+\dot{U^{\prime}}(t)=-i(T^{\top}H_{eff}T)U^{\prime}(t),$
(11)

where

$\dot{T}=\frac{\lambda(\dot{\Delta}/2+i\dot{\gamma})-\dot{\lambda}(\Delta/2+i\gamma)}{(2\lambda)^{2}}\begin{pmatrix}-\frac{1}{\cos(\theta/2)}&-\frac{1}{\sin(\theta/2)}\\\
\frac{1}{\sin(\theta/2)}&-\frac{1}{\cos(\theta/2)}\end{pmatrix}.$ (12)

Then,

$\begin{split}T^{\top}\dot{T}=\frac{\lambda(\dot{\Delta}/2+i\dot{\gamma})-\dot{\lambda}(\Delta/2+i\gamma)}{2\lambda
J}\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}\\\
T^{\top}H_{eff}T=\begin{pmatrix}-i\gamma-\lambda&0\\\
0&-i\gamma+\lambda\end{pmatrix}.\end{split}$ (13)

Substituting Eq. 13 into Eq. 11, we obtain

$\dot{U^{\prime}}(t)=-i(-iT^{\top}\dot{T}+T^{\top}H_{eff}T)U^{\prime}(t)=-i\widetilde{H}_{eff}U^{\prime}(t),$
(14)

where

$\displaystyle\widetilde{H}_{eff}$
$\displaystyle=-iT^{\top}\dot{T}+T^{\top}H_{eff}T=\begin{pmatrix}-i\gamma-\lambda&f\\\
-f&-i\gamma+\lambda\end{pmatrix},$ (15) $\displaystyle f$
$\displaystyle=\frac{J(\dot{\Delta}/2+i\dot{\gamma})-\dot{J}(\Delta/2+i\gamma)}{2i\lambda^{2}}.$

Expressing $U^{\prime}(t)$ in matrix form
$U^{\prime}(t)=\begin{pmatrix}U^{\prime}_{\alpha\alpha}(t)&U^{\prime}_{\alpha\beta}(t)\\\
U^{\prime}_{\beta\alpha}(t)&U^{\prime}_{\beta\beta}(t)\\\ \end{pmatrix}$. Its
differential form can be described as

$\begin{split}\dot{U^{\prime}}_{\alpha\alpha}(t)=-i((-i\gamma-\lambda)U^{\prime}_{\alpha\alpha}(t)+fU^{\prime}_{\beta\alpha}(t))\\\
\dot{U^{\prime}}_{\alpha\beta}(t)=-i((-i\gamma-\lambda)U^{\prime}_{\alpha\beta}(t)+fU^{\prime}_{\beta\beta}(t))\\\
\dot{U^{\prime}}_{\beta\alpha}(t)=-i((-i\gamma+\lambda)U^{\prime}_{\beta\alpha}(t)-fU^{\prime}_{\alpha\alpha}(t))\\\
\dot{U^{\prime}}_{\beta\beta}(t)=-i((-i\gamma+\lambda)U^{\prime}_{\beta\beta}(t)-fU^{\prime}_{\alpha\beta}(t))\end{split}$
(16)

If the initial state is $|\alpha(0)\rangle$, meaning $C_{1}(0)=1,C_{2}(0)=0$,
then the final state
$|\psi^{\prime}(t)\rangle=U^{\prime}(t)\begin{pmatrix}1\\\
0\end{pmatrix}=\begin{pmatrix}U^{\prime}_{\alpha\alpha}(t)\\\
U^{\prime}_{\beta\alpha}(t)\end{pmatrix}=\begin{pmatrix}C_{1}(t)\\\
C_{2}(t)\end{pmatrix}$. This implies that $U^{\prime}_{\alpha\alpha}(t)$ and
$U^{\prime}_{\beta\alpha}(t)$ serve as the amplitudes of the corresponding
eigenstates. Therefore, the relative nonadiabatic transition amplitudes can be
defined as
$R_{1}(t)=\frac{U^{\prime}_{\beta\alpha}(t)}{U^{\prime}_{\alpha\alpha}(t)}$.
Initially set at $R_{1}(0)=0$, if the evolution remains adiabatic,
$R_{1}(t)<1$. However, in the case of a nonadiabatic transition, where two
eigenstates exchange position, $R_{1}(t)>1$. The same applies to the initial
state $|\beta\rangle$, where $C_{1}(0)=0$ and $C_{2}(0)=1$. In this case,
$R_{2}(t)=\frac{U^{\prime}_{\alpha\beta}(t)}{U^{\prime}_{\beta\beta}(t)}$ is
employed to indicate the occurrence of nonadiabatic transitions. Utilizing Eq.
16, we derive the differential equations governing the relative nonadiabatic
transition amplitudes.

$\begin{split}\dot{R_{1}}(t)=2i\lambda R_{1}(t)-if(1+R^{2}_{1}(t))\\\
\dot{R_{2}}(t)=-2i\lambda R_{2}(t)+if(1+R^{2}_{2}(t)).\end{split}$ (17)

We can also calculate $R_{1}(t)$ ($R_{2}(t)$) by measuring $C_{1}(t)$ and
$C_{2}(t)$, giving $R_{1}(t)=\frac{C_{2}(t)}{C_{1}(t)}$
($R_{2}(t)=\frac{C_{1}(t)}{C_{2}(t)})$. For clockwise encircling from
$|\alpha_{A}(0)\rangle$ in Fig. 6(A), $R_{1}(t)=\frac{C_{2}(t)}{C_{1}(t)}<1$
in the whole period, indicating $|C_{1}(t)|$ and $|C_{2}(t)|$ do not cross, so
no DNAT occurs. Similarly, in Fig. 6(B), $R_{1}(t)=\frac{C_{2}(t)}{C_{1}(t)}$
evolves from $R_{1}(t)<1$ to $R_{1}(t)>1$, indicating $C_{1}(t)$ and
$C_{2}(t)$ cross, hence a DNAT occurs. Both the theoretical and experimental
results for $\mathcal{PTS}$ and $\mathcal{PTB}$ regimes are shown in Fig. 6.

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## Acknowledgments

Acknowledgments: We thank Profs. Shidong Liang and Wei Yi for valuable
discussions. Funding: This work is supported by the Key-Area Research and
Development Program of Guang Dong Province under Grant No.2019B030330001, the
National Natural Science Foundation of China under Grant No.11774436,
No.11974434 and No.12074439, Natural Science Foundation of Guangdong Province
under Grant 2020A1515011159, Science and Technology Program of Guangzhou,
China 202102080380, the Shenzhen Science and Technology Program under Grant
No.2021Szvup172 and JCYJ20220818102003006. Le Luo acknowledges the support
from Guangdong Province Youth Talent Program under Grant No.2017GC010656. Ji
Bian acknowledges the support from the China Postdoctoral Science Foundation
under Grant No.2021M703768. Author contributions: L.L. conceived the original
idea and supervised the overall research. P.L, Y.L, Q.L, T.L, and L.L.
designed the experiments. P.L, Y.L, H.W and F.Z contributed the spin-dependent
dissipation technique. P.L, J.B, and Y.L conducted the experimental work. P.L
and Y.L analyzed data. Q.L, P.L, T.L, Y.L, and J.B carried out theoretical
modeling of encircling. Y.L, P.L, Q.L, T.L and L.L prepared the manuscript.
All authors provided helpful discussion. Competing interests: The authors
declare no competing interests. Data and materials availability:All data are
available in the main text or the supplementary materials.

Dynamical topology of chiral and nonreciprocal state transfers in a non-
Hermitian quantum system

## S1 Dynamically encircling of $\mathcal{PT}$ Hamiltonian

### S1.1 Time-varying nonadiabatic transition amplitude

The nonadiabatic transition amplitudes with a $\mathcal{PT}$-symmetric
Hamiltonian have been measured in our experiments, as depicted in Fig. S1. The
time-dependent amplitude coefficients of $C_{1}(t)$ and $C_{2}(t)$ are
determined by
$|\psi(t)\rangle=C_{1}(t)|\alpha(t)\rangle+C_{2}(t)|\beta(t)\rangle$. The
crossing of $C_{1}(t)$ and $C_{2}(t)$ serves as a characterization of the
nonadiabatic transition. According to stability loss delay, the time-varying
relative nonadiabatic transition amplitude $R_{1}(t)$ and $R_{2}(t)$ can also
be obtained by measuring $C_{1}(t)$ and $C_{2}(t)$, Specifically, these are
given by $R_{1}(t)=\frac{C_{2}(t)}{C_{1}(t)}$ and
$R_{2}(t)=\frac{C_{1}(t)}{C_{2}(t)}$, as discussed in the main text.

Figure S1: The amplitude coefficients of the instantaneous eigenstates when
the $\mathcal{PT}$ Hamiltonian encircles the EP. (a-d) Encircling starting
from the $\mathcal{PT}$-symmetric regime, with (a) Clockwise with the initial
state $|\alpha_{A}(0)\rangle$, (b) Counterclockwise with the initial state
$|\alpha_{A}(0)\rangle$, (c) Clockwise with the initial state
$|\beta_{A}(0)\rangle$, (d) Counterclockwise with the initial state
$|\beta_{A}(0)\rangle$. (e-h) Encircling starting from the
$\mathcal{PT}$-broken regime, with (e) Clockwise with the initial state
$|\alpha_{B}(0)\rangle$, (f) Counterclockwise with the initial state
$|\alpha_{B}(0)\rangle$, (g) Clockwise with the initial state
$|\beta_{B}(0)\rangle$, (h) Counterclockwise with the initial state
$|\beta_{B}(0)\rangle$.

### S1.2 The winding number of the complex-energy bands

The singularity of complex-energy Riemann surface allows for the half-integer
windings of the paired bands.Figure 2A in the main text illustrates such a
scenario with a winding number $\mathcal{V}=-1/2$. It is important to note
that the loop in Fig. 2A does not involve DNAT. To calculate the winding
number of a loop that involves DNATs, the evolution processes must be
transformed into the parallel transported eigenbasis. In this basis, loops
involving DNATs also can be parametrized by the encircling angle
$\theta\in[0,2\pi]$ around the EP. The theoretical results for $\theta$ in the
complex-energy plane are shown in Fig. S2. The solid and dashed curves
represent the theoretical loop and the projection of energy trajectories,
respectively. Each band’s winding number is 1/2, defining a topological
invariant associated with the non-Hermitian band structures [20]. The positive
or negative value of the winding number depends on the orientation of
encircling curve; it is negative if the curve travels around the EP in a
clockwise direction.

Figure S2: Windings of the complex-energy band structure in parallel
transported basis. (a) The encirclement starting from the
$\mathcal{PT}$-symmetric regime; (b) The encirclement starting from the
$\mathcal{PT}$-broken regime. The encircling angle $\theta\in[0,2\pi]$
parametrizes the closed loop, and dashed curves are the projection of the
energy trajectory. The red and blue colors represent the different bands, and
the olive circles represent the EPs.

## S2 Dynamically encircling of $\mathcal{APT}$ Hamiltonian

### S2.1 $\mathcal{APT}$ symmetric Hamiltonian

The construction of the $\mathcal{APT}$-symmetric Hamiltonian has been
demonstrated in our previous work [23]. The evolution operator of the
$\mathcal{APT}$ Hamiltonian $U(\tau)=e^{-iH^{\prime}_{eff}t}$, is constructed
by sandwiching a passive $\mathcal{PT}$-symmetric Hamiltonian $H_{eff}$
between two $\pi/2$ pulses along the $\pm Y$ axis on the Bloch sphere. First,
a $-\pi/2$ microwave pulse is applied along the $Y$ axis in the rotating
frame. Then, identical to the construction of the $\mathcal{PT}$-symmetric
Hamiltonian, a microwave field along the $X$ axis with strength $J$ and a
dissipation laser beam are applied simultaneously for a duration $t$. Finally,
a $\pi/2$ pulse along the $Y$ is implemented. The whole evolution is described
as

$R_{y}(\frac{\pi}{2})e^{-iH_{eff}t}R_{y}(-\frac{\pi}{2})=e^{-iR_{y}(\frac{\pi}{2})H_{eff}R_{y}(-\frac{\pi}{2})t}=e^{-i(2i\gamma
I_{x}-2JI_{z}-i\gamma\mathbf{I})t}=e^{-iH^{\prime}_{eff}t},$ (S1)

where the passive $\mathcal{APT}$-symmetric Hamiltonian is

$H^{\prime}_{eff}=\begin{pmatrix}-J-i\gamma&i\gamma\\\
i\gamma&J-i\gamma\end{pmatrix}.$ (S2)

It differs from the $\mathcal{APT}$-symmetric Hamiltonian

$H_{APT}=\begin{pmatrix}-J&i\gamma\\\ i\gamma&J\end{pmatrix}.$ (S3)

by a term $i\gamma$I. $H_{APT}$ satisfies the anti-commutation relation
$\\{H_{\mathcal{APT}},\mathcal{PT}\\}=0$ and is a pseudo-Hermitian
[zhang2020synthetic], satisfying
$\sigma_{z}^{-1}H_{\mathcal{APT}}\sigma_{z}=H_{\mathcal{APT}}^{\dagger}$.

Figure S3: The timing sequence of the coupling microwave and dissipation laser
for realizing the dynamical encircling of the EP with the
$\mathcal{APT}$-symmetric Hamiltonian. $Y(\pm\pi/2)$ denotes the rotation
along the $\pm Y$ axis by $\pi/2$ pulse.

### S2.2 Dynamically encircling the EP of $\mathcal{APT}$ Hamiltonian

The experimental timing sequence of dynamically encircling the EP of the
$\mathcal{APT}$-symmetric Hamiltonian is shown in Fig. S3. Here, the Riemann
surfaces defined by $\\{\frac{J}{\gamma},\frac{\Delta}{\gamma},Re[\lambda]\\}$
and $\\{\frac{J}{\gamma},\frac{\Delta}{\gamma},Im[\lambda]\\}$ are identical
to those of the $\mathcal{PT}$-symmetric Hamiltonian due to the same complex
eigenvalues.

Fig. S4 shows the state evolution when the initial states of encircling are in
the $\mathcal{PTS}$ regime of $\mathcal{APT}$ symmetric Hamiltonian, where the
nonreciprocal state transfer can be observed. For example, the forward-time
evolution
$|\beta^{\prime}_{A}(0)\rangle\xrightarrow{\text{CW}}|\alpha^{\prime}_{A}(0)\rangle$
(trajectory 7′) and its backward-time evolution
$|\alpha^{\prime}_{A}(0)\rangle\xrightarrow{\text{CCW}}|\alpha^{\prime}_{A}(0)\rangle$
(trajectory 6′) illustrate this nonreciprocal behavior. Similarly, the
forward-time evolution
$|\beta^{\prime}_{A}(0)\rangle\xrightarrow{\text{CCW}}|\alpha^{\prime}_{A}(0)\rangle$
(trajectory 8′) and its backward-time evolution
$|\alpha^{\prime}_{A}(0)\rangle\xrightarrow{\text{CW}}|\alpha^{\prime}_{A}(0)\rangle$
(trajectory 5′) further exemplify this nonreciprocal behavior.

Figure S4: Time evolution of $|\psi(t)\rangle$ starting from the
$\mathcal{PTS}$ regime of the $\mathcal{APT}$ symmetric Hamiltonian. The first
two columns (a,d,g,j) depict the trajectories of the state evolution on the
eigenvalue Riemann surface. The third column (b,e,h,k) depicts the overlap
between the evolutionary state and two time-dependent eigenstates
$\langle\alpha^{\prime}_{A}(t)|\psi(t)\rangle$ and
$\langle\beta^{\prime}_{A}(t)|\psi(t)\rangle$, where the cyan shaded regions
indicate the occurrence of nonadiabatic dynamics. The fourth column (c,f,i,l)
represents the overlap between the evolutionary state and two initial
eigenstates. (a-c): Clockwise encircling with the initial state
$|\alpha^{\prime}_{A}(0)\rangle$. (d-f): Counterclockwise encircling with the
initial state $|\alpha^{\prime}_{A}(0)\rangle$. (g-i): Clockwise encircling
with the initial state $|\beta^{\prime}_{A}(0)\rangle$. (j-l):
Counterclockwise encircling with the initial state
$|\beta^{\prime}_{A}(0)\rangle$. Circles and lines represent experimental
measurements and numerical simulations, respectively.

Fig. S5 shows the state evolution when the initial state of encircling is in
the $\mathcal{PTB}$ regime of the $\mathcal{APT}$ symmetric Hamiltonian, where
both the chiral and nonreciprocal state transfer can be observed. The
clockwise evolution
$|\alpha^{\prime}_{B}(0)\rangle\xrightarrow{\text{CW}}|\beta^{\prime}_{B}(0)\rangle$
(trajectory 1′) and counterclockwise evolution
$|\alpha^{\prime}_{B}(0)\rangle\xrightarrow{\text{CCW}}|\alpha^{\prime}_{B}(0)\rangle$
(trajectory 2′) exhibit the chiral behavior, similar to trajectory 3′
$|\beta^{\prime}_{B}(0)\rangle\xrightarrow{\text{CW}}|\beta^{\prime}_{B}(0)\rangle$
and trajectory 4′
$|\beta^{\prime}_{B}(0)\rangle\xrightarrow{\text{CCW}}|\alpha^{\prime}_{B}(0)\rangle$.
Nonreciprocal state transfer is also illustruated in the pair of the forward-
time evolution
$|\alpha^{\prime}_{B}(0)\rangle\xrightarrow{\text{CCW}}|\alpha^{\prime}_{B}(0)\rangle$
(trajectory 2′) and the backward-time evolution
$|\alpha^{\prime}_{B}(0)\rangle\xrightarrow{\text{CW}}|\beta^{\prime}_{B}(0)\rangle$
(trajectory 1′), as well as in the pair of trajectory 3′
$|\beta^{\prime}_{B}(0)\rangle\xrightarrow{\text{CW}}|\beta^{\prime}_{B}(0)\rangle$
and trajectory 4′
$|\beta^{\prime}_{B}(0)\rangle\xrightarrow{\text{CCW}}|\alpha^{\prime}_{B}(0)\rangle$.

Figure S5: Time evolution of $|\psi(t)\rangle$ starting from the
$\mathcal{PTB}$ regime with the $\mathcal{APT}$ symmetric Hamiltonian. The
first two columns (a,d,g,j) depict the trajectories of the state evolution on
the eigenvalue Riemann surface. The third column (b,e,h,k) depicts the overlap
between the evolutionary state and two time-dependent eigenstates
$\langle\alpha^{\prime}_{B}(t)|\psi(t)\rangle$ and
$\langle\beta^{\prime}_{B}(t)|\psi(t)\rangle$, where the cyan shaded regions
indicate the occurrence of non-adiabatic dynamics. The fourth column (c,f,i,l)
represents the overlap between the evolutionary state and two initial
eigenstates. (a-c): Clockwise encircling with the initial state
$|\alpha^{\prime}_{B}(0)\rangle$. (d-f): Counterclockwise encircling with the
initial state $|\alpha^{\prime}_{B}(0)\rangle$. (g-i): Clockwise encircling
with the initial state $|\beta^{\prime}_{B}(0)\rangle$. (j-l):
Counterclockwise encircling with the initial state
$|\beta^{\prime}_{B}(0)\rangle$. Circles and lines represent experimental
measurements and numerical simulations, respectively.

It is noticed that the chiral and nonreciprocal dynamics of the
$\mathcal{PTB}$ regime of the $\mathcal{APT}$ symmetric Hamiltonian are
analogous to those in the $\mathcal{PTS}$ regime of the $\mathcal{PT}$
symmetric Hamiltonian, and vice versa. We summarize the chiral (nonchiral) and
reciprocal (nonreciprocal) behaviors of both $\mathcal{PT}$ and
$\mathcal{APT}$ symmetric Hamiltonians in Table. LABEL:table1.

The experimental results of DNATs for the $\mathcal{APT}$-symmetric
Hamiltonian are shown in Fig. S6 and S7. The notation and procedure are
similar to those used in the experiments for measuring DNATs for the
$\mathcal{PT}$-symmetric Hamiltonian as described in the main text.

The DNATs with the $\mathcal{APT}$-symmetric Hamiltonian have also been
measured in our experiments. In this context, the nonadiabatic transition
amplitudes for the $\mathcal{APT}$-symmetric Hamiltonian can be defined using
the same method as that employed for the $\mathcal{PT}$-symmetric Hamiltonian.

$\begin{split}\dot{R^{\prime}_{1}}(t)=-2i\lambda
R^{\prime}_{1}(t)-if^{\prime}(1+(R_{1}^{\prime}(t))^{2})\\\
\dot{R^{\prime}_{2}}(t)=2i\lambda
R^{\prime}_{2}(t)+if^{\prime}(1+(R_{2}^{\prime}(t))^{2}),\end{split}$ (S4)

where

$f^{\prime}=\frac{\dot{J}(i\gamma+\Delta/2)-J(i\dot{\gamma}+\dot{\Delta}/2)}{2i\lambda^{2}}.$
(S5)

The time-varying relative nonadiabatic transition amplitude
$R^{\prime}_{1}(t)=\frac{C^{\prime}_{2}(t)}{C^{\prime}_{1}(t)}$ and
$R^{\prime}_{2}(t)=\frac{C^{\prime}_{1}(t)}{C^{\prime}_{2}(t)}$ can also be
obtained by numerically solving Eq. S4.

Table S1: Chiral (nonchiral) and reciprocal (nonreciprocal) dynamic behavior in a dissipative quantum system. |  Chirality |  Nonchirality |  Reciprocity |  Nonreciprocity
---|---|---|---|---
$[\mathcal{PT},H]=0$ $\mathcal{PT}|\psi\rangle=|\psi\rangle$ |  Trajectory 1 and 2 Trajectory 3 and 4 |  |  1 and 4 4 and 1 |  2 and 1 3 and 4
$\\{\mathcal{PT},H\\}=0$ $\mathcal{PT}|\psi\rangle\neq|\psi\rangle$ |  1′ and 2′ 3′ and 4′ |  |  1′ and 4′ 4′ and 1′ |  2′ and 1′ 3′ and 4′
$[\mathcal{PT},H]=0$ $\mathcal{PT}|\psi\rangle\neq|\psi\rangle$ |  |  5 and 6 7 and 8 |  5 and 6 6 and 5 |  7 and 6 8 and 5
$\\{\mathcal{PT},H\\}=0$ $\mathcal{PT}|\psi\rangle=|\psi\rangle$ |  |  5′ and 6′ 7′ and 8′ |  5′ and 6′ 6′ and 5′ |  7′ and 6′ 8′ and 5′
Figure S6: The amplitude coefficients of the instantaneous eigenstates when
the $\mathcal{APT}$ Hamiltonian encircles the EP. (a-d) Encircling starting
from the $\mathcal{PT}$ symmetric regime, with (a) Clockwise with the initial
state $|\alpha^{\prime}_{A}(0)\rangle$, (b) Counterclockwise with the initial
state $|\alpha^{\prime}_{A}(0)\rangle$, (c) Clockwise with the initial state
$|\beta^{\prime}_{A}(0)\rangle$, (d) Counterclockwise with the initial state
$|\beta^{\prime}_{A}(0)\rangle$. (e-h) Encircling starting from the
$\mathcal{PT}$ broken regime, with (e) Clockwise with the initial state
$|\alpha^{\prime}_{B}(0)\rangle$, (f) Counterclockwise with the initial state
$|\alpha^{\prime}_{B}(0)\rangle$, (g) Clockwise with the initial state
$|\beta^{\prime}_{B}(0)\rangle$, (h) Counterclockwise with the initial state
$|\beta^{\prime}_{B}(0)\rangle$. The amplitude coefficients of
$C^{\prime}_{1}(t)$ and $C^{\prime}_{2}(t)$ are determined by
$|\psi(t)\rangle=C^{\prime}_{1}(t)|\alpha^{\prime}(t)\rangle+C^{\prime}_{2}(t)|\beta^{\prime}(t)\rangle$.
Figure S7: The time-dependent nonadiabatic transition amplitude when the
$\mathcal{APT}$ Hamiltonian encircles the EP. (a-d) Encircling starting from
the $\mathcal{PT}$ symmetric regime, with (a) Clockwise with the initial state
$|\alpha^{\prime}_{A}(0)\rangle$, (b) Counterclockwise with the initial state
$|\alpha^{\prime}_{A}(0)\rangle$, (c) Clockwise with the initial state
$|\beta^{\prime}_{A}(0)\rangle$, (d) Counterclockwise with the initial state
$|\beta^{\prime}_{A}(0)\rangle$. (e-h) Encircling starting from the
$\mathcal{PT}$ broken regime, with (e) Clockwise with the initial state
$|\alpha^{\prime}_{B}(0)\rangle$, (f) Counterclockwise with the initial state
$|\alpha^{\prime}_{B}(0)\rangle$, (g) Clockwise with the initial state
$|\beta^{\prime}_{B}(0)\rangle$, (h) Counterclockwise with the initial state
$|\beta^{\prime}_{B}(0)\rangle$. The black squares with error bars, red and
blue lines correspond to the experimental data, theoretical calculation (Eq.
S4) and numerical simulations from $C^{\prime}_{1}(t)$ and
$C^{\prime}_{2}(t)$.

Furthermore, we investigate the adiabatic theorem for dynamical encircling of
the EP with the $\mathcal{APT}$-symmetric Hamiltonian. The details are
presented in the Fig. S8 and Fig. S9. These observations are quite similar to
those with the $\mathcal{PT}$-symmetric Hamiltonian, as expected since they
share the same Riemann surface of the eigenvalues.

Figure S8: The encircling of the EP of the $\mathcal{APT}$ Hamiltonian with
various periods. (a-c) Clockwise encircling the EP with the initial eigenstate
$|\alpha^{\prime}_{B}(0)\rangle$. (a) and (b) are
$\langle\alpha^{\prime}_{B}(0)|\psi(t)\rangle$ and
$\langle\beta^{\prime}_{B}(0)|\psi(t)\rangle$, respectively. (c)
$\langle\alpha^{\prime}_{B}(t)|\psi(t)\rangle$ (solid line) and
$\langle\beta^{\prime}_{B}(t)|\psi(t)\rangle$ (dash line). (d-f) Clockwise
encircling the EP with the initial eigenstate $|\beta^{\prime}_{B}(0)\rangle$,
where the notation in (d-f) are the same as (a-c). The blue, red, and black
circles correspond to $T=16.67~{}\mu s$, $100~{}\mu s$, and $250~{}\mu s$,
respectively. The lines represent the numerical simulations. Figure S9: The
encircling of the EP of the $\mathcal{APT}$ Hamiltonian with various radii.
The notation for (a-f) is similar to Fig. S8 (a-f) as defined previously. The
blue, red, and black squares correspond to $r=$0.004 MHz, 0.009 MHz, and 0.03
MHz, respectively. The lines represent the numerical simulations.

### S2.3 Topological robustness of quantum state transfer with $\mathcal{APT}$
Hamiltonian

The results with the initial states of $|\beta^{\prime}_{A}(0)\rangle$ (in the
$\mathcal{PTS}$ regime) and $|\beta^{\prime}_{B}(0)\rangle$ (in the
$\mathcal{PTB}$ regime) are shown in Fig. S10(a-b) and Fig. S10(c-d),
respectively. In both cases, the experimental data match well with the
theoretically predicted values despite varying noise intensities, thereby
validating the robustness of topological state transfers with the
$\mathcal{APT}$-symmetric Hamiltonian.

Figure S10: Quantum state transfers while encircling the EP of the
$\mathcal{APT}$ Hamiltonian with various noise intensities. (a-b) The
clockwise encircling from the initial state $|\beta^{\prime}_{A}(0)\rangle$.
(c-d) The clockwise encircling from the initial state
$|\beta^{\prime}_{B}(0)\rangle$. The insets in (a) and (c) display the
encircling trajectories in the parameter space $\\{\Delta,J\\}$ with random
noise intensities $ir=$ 0 (black), 0.3 (red), and 0.7 (blue), respectively.
The dots are experimental results, while the solid lines represent the
numerical simulations.


# On the Formalism of The Screening Paradox

Dr. Jacques Balayla MD, MPH, CIP, FRCSC111To whom correspondence should be
addressed: Dr. Jacques Balayla MD, MPH, CIP, FRCSC. Quilligan Scholar. e-mail:
<EMAIL_ADDRESS>Osler Fellow. Department of Obstetrics and
Gynecology. Faculty of Medicine. McGill University, Montreal, Quebec, Canada

###### Abstract

Bayes’ Theorem imposes inevitable limitations on the accuracy of screening
tests by tying the test’s predictive value to the disease prevalence. The
aforementioned limitation is independent of the adequacy and make-up of the
test and thus implies inherent Bayesian limitations to the screening process
itself. As per the WHO’s $Wilson-Jungner$ criteria, one of the prerequisite
steps before undertaking screening is to ensure that a treatment for the
condition screened exists. However, in so doing, a paradox, henceforth termed
the “screening paradox”, ensues. If a disease process is screened for and
subsequently treated, its prevalence would drop in the population, which as
per Bayes’ theorem, would make the tests’ predictive value drop in return. Put
another way, a very powerful screening test would, by performing and
succeeding at the very task it was developed to do, paradoxically reduce its
ability to correctly identify individuals with the disease it screens for in
the future. Where $J$ is Youden’s statistic (sensitivity $a$ \+ specificity
$b$ \- 1), and $\phi$ is the prevalence, the ratio of positive predictive
values at subsequent time $k$, $\rho(\phi_{k})$, over the original
$\rho(\phi_{0})$ at $t_{0}$ is given by:

$\zeta(\phi_{0},k)=\frac{\rho(\phi_{k})}{\rho(\phi_{0})}=\frac{\phi_{k}(1-b)+J\phi_{0}\phi_{k}}{\phi_{0}(1-b)+J\phi_{0}\phi_{k}}$
(1)

In this manuscript, we explore the mathematical model which formalizes said
screening paradox and explore its implications for population level screening
programs. In particular, we define the number of positive test iterations
(PTI) needed to reverse the effects of the paradox as follows:

$n_{i\phi_{e}}=\left\lceil\frac{ln\left[\frac{\omega\phi_{e}\phi_{k}-\omega\phi_{e}}{\omega\phi_{e}\phi_{k}-\phi_{k}}\right]}{2ln\omega}\right\rceil$
(2)

where $\omega$ is the square root of the positive likelihood ratio (LR+).

##  1 Bayes’ Theorem and Predictive Values

Bayes’ Theorem describes the probability of an event occurring based on prior
knowledge of conditions related to that specific event [1]. The essence of the
Bayesian approach is to provide a mathematical model explaining how existing
beliefs change in light of new evidence. Remarkably, Bayes theorem’ has
applications in innumerable fields. Indeed, it has significant implications in
epidemiological modelling as well. From Bayes’ theorem, we can derive the
positive predictive value $\rho(\phi)$ (PPV) of a screening test, defined as
the percentage of patients with a positive screening test that do in fact have
the disease, as follows[1]:

$\rho(\phi)=\frac{a\phi}{a\phi+(1-b)(1-\phi)}$ (3)

where $\rho(\phi)$ = PPV, a = sensitivity, b = specificity and $\phi$ =
prevalence.

The PPV $\rho(\phi)$ is therefore a function of the disease prevalence,
$\phi$. As the prevalence increases, $\rho(\phi)$ also increases and vice-
versa [2].

##  2 Bayesian Dynamics of Predictive Values

Let us define a hypothetical disease present in a population. Said condition
has a preclinical phase and is amenable to screening through a given test
designed to detect it. The test therefore has all of the pertinent screening
parameters - sensitivity, specificity, and negative and positive predictive
values. Finally, as required by the WHO’s $Wilson-Jungner$ criteria, a
treatment for the condition screened exists. Let us denote $\phi_{0}$ as the
original or initial prevalence before screening is undertaken. As per equation
(3), we thus obtain a positive predictive value $\rho(\phi_{0})$ at $t_{0}$:

$\rho(\phi_{0})=\frac{a\phi_{0}}{a\phi_{0}+(1-b)(1-\phi_{0})}$ (4)

It follows that as the individual with a positive screening test is treated,
the disease prevalence $\phi_{0}$ drops by some magnitude $k$, which
represents the percentage reduction in prevalence. Consequently, the
predictive value $\rho(\phi)$ will drop by some factor as well, so that
individuals who test positive at some time $t_{x>0}$ experience a positive
predictive value of:

$\rho(\phi_{0}-k)=\frac{a(\phi_{0}-k)}{a(\phi_{0}-k)+(1-b)[1-(\phi_{0}-k)]}$
(5)

From the above equations, we define the ratio $\zeta$ as follows:

$\rho(\phi_{0})>\rho(\phi_{0}-k)\Rightarrow\frac{\rho(\phi_{0}-k)}{\rho(\phi_{0})}=\zeta(\phi_{0},k)$
(6)

where $0<\zeta<1$ and $k>0$. Given the shape of the screening curve, and the
principle of the prevalence threshold, even small changes in the prevalence
$\phi$ can have significant changes in the positive predictive value
$\rho(\phi)$. To determine the degree of reduction in the predictive value of
the screening test at time $t_{k}$, we take the ratio of $\rho(\phi)$ at two
different times, be it $t_{0}$, and some later time $t_{k}$ with a prevalence
reduction of $\phi_{0}-k$, where $k<\phi$ is the percentage reduction in
prevalence:

$\frac{\rho(\phi_{0}-k)}{\rho(\phi_{0})}=\frac{\left(\phi-k\right)\left[a\phi+\left(1-b\right)\left(1-\phi\right)\right]}{\phi\left[a\left(\phi-k\right)+\left(1-b\right)\left(1+k-\phi\right)\right]}$
(7)

Since $\phi_{0}-k$ yields a new, lower prevalence $\phi_{k}$, we can re-write
the above equation as:

$\frac{\rho(\phi_{k})}{\rho(\phi_{0})}=\frac{\phi_{k}\left[a\phi_{0}+\left(1-b\right)\left(1-\phi_{0}\right)\right]}{\phi_{0}\left[a\phi_{k}+\left(1-b\right)(1-\phi_{k})\right]}$
(8)

Expanding the parentheses and simplifying the expression where the sum of the
sensitivity $a$ and specificity $b$ is defined by $\varepsilon=a+b$, we
obtain:

$\zeta(\phi_{0},k)=\frac{\rho(\phi_{k})}{\rho(\phi_{0})}=\frac{\phi_{k}(1-b)+\phi_{0}\phi_{k}(\varepsilon-1)}{\phi_{0}(1-b)+\phi_{0}\phi_{k}(\varepsilon-1)}$
(9)

The term $\varepsilon-1=a+b-1$ has been previously defined in the context of
receiver-operating characteristics (ROC) curves, and is termed the Youden’s
$J$ statistic [3]. As such, we can re-write the above equation as:

$\zeta(\phi_{0},k)=\frac{\rho(\phi_{k})}{\rho(\phi_{0})}=\frac{\phi_{k}(1-b)+J\phi_{0}\phi_{k}}{\phi_{0}(1-b)+J\phi_{0}\phi_{k}}$
(10)

From the above relationship, we infer:

$\lim_{k\to 0}\frac{\rho(\phi_{k})}{\rho(\phi_{0})}=\lim_{k\to
0}\frac{\rho(\phi_{0}-k)}{\rho(\phi_{0})}=\lim_{k\to 0}\zeta=1$ (11)

$\zeta(\phi_{0},k)$ may be considered as the predictive value percentage loss
as the prevalence decreases from $\phi_{0}$ to $\phi_{k}$. If we consider both
$\phi_{0}$ and $k$ as independent variables, both of which affect
$\rho(\phi)$, we can establish the individual contributions of each variable
towards $\rho(\phi)$ relative to each other through the partial differential
equation $\zeta(\phi_{0},k)$.

##  3 The Prevalence Threshold

We have previously defined the prevalence threshold as the prevalence level in
the screening curve below which screening tests start to fail [1]. In
technical terms, this is equivalent to the inflection point in the screening
curve below which the the rate of change of a test’s positive predictive value
drops at a differential pace relative to the prevalence[1]. This value, termed
$\phi_{e}$, is defined at the following point on the prevalence axis:

$\phi_{e}=\frac{\sqrt{a\left(-b+1\right)}+b-1}{(a+b-1)}=\frac{\sqrt{a\left(-b+1\right)}+b-1}{(\varepsilon-1)}$
(12)

The corresponding positive predictive value is given by plotting the above
equation into the positive predictive value equation. In so doing we obtain:

$\rho(\phi_{e})=\sqrt{\frac{a}{1-b}}\left[\frac{\sqrt{a\left(-b+1\right)}+b-1}{(\varepsilon-1)}\right]$
(13)

Interestingly, the above expression leads to the well known formulation for
the positive predictive value as a function of prevalence and the positive
likelihood ratio (LR+), defined as the sensitivity $a$ over the compliment of
the specificity $b$ [4].

$\rho(\phi_{e})=\phi_{e}\sqrt{\frac{a}{1-b}}$ (14)

From the above relationship we can identify three different points in the
screening curve, notably $\phi_{0}$, $\phi_{k}$, and $\phi_{e}$.

$0$$0.2$$0.4$$0.6$$0.8$$1$$0$$0.2$$0.4$$0.6$$0.8$$1$$\phi_{0}$ =
0.38$\phi_{e}$ = 0.25$\phi_{k}$ = 0.18$\phi$$\rho(\phi)$$\varepsilon=1.75$

Figure 1. Example illustration of $\phi_{0}$, $\phi_{k}$, and $\phi_{e}$ where
a = 0.85, and b = 0.90.

We can deduce important relationships between these values that contextualize
the screening paradox. First, we observe that though by definition
$\phi_{0}>\phi_{k}$, $\phi_{e}$ can either be outside or in between $\phi_{0}$
and $\phi_{k}$. As such three different scenarios may arise. Herein we explore
each.

###  3.1 First scenario: $\phi_{e}>\phi_{0}>\phi_{k}$

By design, $\phi_{0}>\phi_{k}$. Let $\phi_{e}$ define the prevalence threshold
such that $\phi_{e}>\phi_{0}>\phi_{k}$. It then follows that
$\phi_{e}-\phi_{k}>\phi_{0}-\phi_{k}$ and thus $\phi_{e}-\phi_{0}>0$. Since
$\phi_{0}$ = $\phi_{k}+k$, we obtain $\phi_{e}-\phi_{k}-k>0$ or otherwise
stated, $\phi_{e}>\phi_{k}+k$ and thus $\phi_{e}-\phi_{k}>k$. We thus infer
that:

$\lim_{k\to\phi_{0}}\zeta(\phi)\sim 0$ (15)

and

$\lim_{k\to 0}\zeta(\phi)\sim 1$ (16)

###  3.2 Second scenario: $\phi_{0}>\phi_{k}>\phi_{e}$

This scenario is akin to the first scenario in that the prevalence threshold
lies outside the range between $\phi_{0}$ and $\phi_{k}$. However, an
important difference arises. While $\lim_{k\to 0}\zeta(\phi)\sim 1$, by design
$\phi_{k}>\phi_{e}$, and thus the maximum value that $k$ can take cannot be
greater than $\phi_{0}-\phi_{e}$, as $\phi_{k}\rightarrow\phi_{e}$. We thus
infer that:

$\lim_{k\to\phi_{0}-\phi_{e}}\zeta(\phi)\sim 1$ (17)

and

$\lim_{k\to 0}\zeta(\phi)\sim 1$ (18)

The above relationships follow since for $\phi>\phi_{e}\rightarrow
d\zeta/d\phi\sim 0$, as per equation (11).

###  3.3 Third scenario: $\phi_{0}>\phi_{e}>\phi_{k}$

Perhaps the most interesting scenario is one where
$\phi_{0}>\phi_{e}>\phi_{k}$. By design $\phi_{0}>\phi_{k}$. Let $\phi_{e}$
define the prevalence threshold such that $\phi_{0}>\phi_{e}>\phi_{k}$. It
then follows that $\phi_{0}-\phi_{k}>\phi_{0}-\phi_{e}$ and thus
$0>\phi_{k}-\phi_{e}$. Since $\phi_{k}$ = $\phi_{0}-k$, we obtain
$0>\phi_{0}-k-\phi_{e}$ and thus:

$k>\phi_{0}-\phi_{e}$ (19)

We thus infer that:

$\lim_{k\to\phi_{0}-\phi_{e}}\zeta(\phi)\sim 1$ (20)

In other words, when the initial prevalence lies beyond the prevalence
threshold, changes in prevalence such that $k$ approaches the difference
between $\phi_{0}-\phi_{e}$, the ratio of positive predictive values as
determined by the $\zeta(\phi)$ function approaches 1. However, we can
theorize a case where $k$ is sufficiently large so that $\phi_{k}$ goes well
below $\phi_{e}$ and therefore:

$\lim_{k\to\phi_{0}}\zeta(\phi)\sim 0$ (21)

##  4 The Screening Paradox at the Population-Level

The mechanism by which the screening paradox arises is depicted through the
following arrow flow diagram:

Given the presence of a disease amenable to screening: As per the $Wilson-
Jungner$ criteria [5]: 1) $\uparrow$ Screening $\rightarrow$ $\uparrow$
Treatment As per the axiom of prevalence [6]: 2) $\uparrow$ Treatment
$\rightarrow$ $\downarrow$ Prevalence As per Bayes’ Theorem [1]: 3)
$\downarrow$ Prevalence $\rightarrow$ $\downarrow$ Positive Predictive Value
As per the principles of consumer value and utility [7]: 4) $\downarrow$
Positive Predictive Value $\rightarrow$ $\downarrow$ Screening

Given the screening paradox, an increase in screening eventually leads to
less, or more accurately lower quality, screening as the prevalence drops.
This paradox is inherently insurmountable unless acted upon by a subsequent
test - either the same test repeated serially or an altogether different,
better test [8].

##  5 Overcoming the Screening Paradox

As the prevalence in a population drops with successful population-level
screening and treatment, the positive predictive value of the screening test
drops, and the false discovery rate, which is equivalent to the complement of
the positive predictive value, increases. The aforementioned paradox occurs
any time that disease is successfully treated because while $d\phi/d\rho$
drops throughout the function’s domain, it never reaches 0. In other words,
the positive predictive value function always increases throughout its domain,
so even minute changes in prevalence will bring about changes in the positive
predictive value. That said, as we described above, the critical factor is
where lie the initial prevalence level $\phi_{0}$, the subsequent prevalence
level $\phi_{k}$, their difference $k$, and how they relate to the prevalence
threshold, $\phi_{e}$, below which the screening paradox becomes more
pronounced. In the presence of a screening paradox it is worth considering
potential solutions to overcoming the losses in predictive value as the
prevalence drops. Though many options exist, the most logical step would be to
undertake serial testing, be it with the same test undertaken serially or an
alternative test altogether. Herein we explore both scenarios.

###  5.1 Repeated testing with a single test

We have shown in previous work that a screening test carried out serially
improves the overall positive predictive value when each individual test
iteration is positive [8]. The number of serial iterations $n_{i}$ required to
achieve a desired $\rho(\phi)$ is given by the following ceiling function:

$n_{i}=\lim_{\rho\to
k}\left\lceil\frac{ln\left[\frac{\rho(\phi-1)}{\phi(\rho-1)}\right]}{ln\left[\frac{a}{1-b}\right]}\right\rceil$
(22)

The key question then becomes, how many serial positive tests are needed to
mitigate or reverse the effect of the screening paradox when
$\phi_{k}<\phi_{e}$? In other words, to achieve a positive predictive value
comparable to that under $\phi_{0}$? Given the geometry of the screening
curve, the answer should be that the PPV ought to at least attain the level at
the prevalence threshold as described in the third scenario in section 3.3 of
this manuscript so that $\zeta(\phi)\sim 1$. We can calculate the number of
iterations $n_{i\phi_{e}}$ needed using the formula above, by plugging
$\rho(\phi_{e})$ into $\rho$ as defined in equation (13), where
$\rho(\phi_{e})=\phi_{e}\sqrt{\frac{a}{1-b}}$.

$n_{i\phi_{e}}=\left\lceil\frac{ln\left[\frac{\phi_{e}\sqrt{\frac{a}{1-b}}(\phi-1)}{\phi(\phi_{e}\sqrt{\frac{a}{1-b}}-1)}\right]}{ln\left[\frac{a}{1-b}\right]}\right\rceil$
(23)

The above expression can be simplified by considering the square root of the
positive likelihood ratio $\sqrt{\frac{a}{1-b}}$ as $\omega$, and $\phi_{k}$
is the prevalence at subsequent time $k$ such that:

$n_{i\phi_{e}}=\left\lceil\frac{ln\left[\frac{\omega\phi_{e}(\phi-1)}{\phi(\omega\phi_{e}-1)}\right]}{2ln\omega}\right\rceil=\left\lceil\frac{ln\left[\frac{\omega\phi_{e}\phi_{k}-\omega\phi_{e}}{\omega\phi_{e}\phi_{k}-\phi_{k}}\right]}{2ln\omega}\right\rceil$
(24)

We take the ceiling function of the above equation to ensure that we obtain an
integer number of of positive test iterations (PTI) needed to surpass the
prevalence threshold [8].

###  5.2 Using a different screening test

We can likewise revert the effects of the screening paradox by using a
different screening test all together, which is the most common scenario in
clinical practice today. That said, it would be impractical to determine the
number of iterations of a different test for numerous reasons. First,
different tests would have different sensitivity/specificity parameters, so a
third test may be then needed in the rare scenario where two different
positive ones are insufficient - rendering the notion of iteration inadequate.
Likewise, and perhaps more importantly, there may not be an alternative
screening test for a particular condition altogether, so the above exercise
may be moot.

##  6 Conclusion

In this manuscript, we explore the mathematical model which formalizes the
screening paradox and explore its implications for population level screening
programs in the three possible scenarios - each as a function of the position
of the initial prevalence of a condition relative to the prevalence threshold
level of its screening test. Likewise, we provide a mathematical model to
determine the predictive value percentage loss as the prevalence decreases and
define the number of positive test iterations (PTI) needed to reverse the
effects of the paradox when a single test is undertaken serially. Given their
theoretical nature, clinical application of the concepts herein reported need
validation prior to implementation.

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