Present address: ]Departamento de Física, Facultad de Ciencias Físicas y
Matemáticas, Universidad de Concepción, Concepción, Chile.

# Testing precision and accuracy of weak value measurements in an IBM quantum
system

David R. A. Ruelas Paredes<EMAIL_ADDRESS>Mariano Uria  [  Eduardo
Massoni  Francisco De Zela<EMAIL_ADDRESS>Departamento de Ciencias,
Sección Física
Pontificia Universidad Católica del Perú
Lima 15088, Peru

###### Abstract

Historically, weak values have been associated with weak measurements
performed on quantum systems. Over the past two decades, a series of works
have shown that weak values can be determined via measurements of arbitrary
strength. One such proposal by Denkmayr et al. [Phys. Rev. Lett. 118, 010402
(2017)], carried out in neutron interferometry experiments, yielded better
outcomes for strong than for weak measurements. We extend this scheme and
explain how to implement it in an optical setting as well as in a quantum
computational context. Our implementation in a quantum computing system
provided by IBM confirms that weak values can be measured, with varying
degrees of performance, over a range of measurement strengths. However, at
least for this model, strong measurements do not always perform better than
weak ones.

##  I Introduction

Quantum weak values are complex numbers associated to an operator, an initial
(preselection) state, and a final (postselection) state. More concretely, let
$S$ be the quantum system of interest, $A_{S}$ an observable, $|i_{S}\rangle$
the initial state, and $|f_{S}\rangle$ the final state. The weak value of
$A_{S}$ for these pre- and postselection states is defined as

$\left\langle A_{S}\right\rangle_{w}\equiv\frac{\langle
f_{S}|A_{S}|i_{S}\rangle}{\langle f_{S}|i_{S}\rangle}.$ (1)

Evidently, there is nothing inherently “weak” about the above quantity. A more
descriptive label, such as $\langle A_{S}\rangle_{fi}$, would have the benefit
of suggesting the natural interpretation as a generalized mean value. The
adjective “weak” dates back to the introduction of weak values in the context
of weak measurements [1]. To make sense of this qualifier, let us consider von
Neumann’s model of quantum measurements, wherein $P$ is the pointer (an
ancillary system that represents the measurement apparatus), and
$H_{\text{int}}=gA_{S}\otimes B_{P}=gA_{S}B_{P}$ describes the coupling
between $S$ and $P$ while $A_{S}$ is being measured. In this interaction
Hamiltonian, $B_{P}$ is a momentum operator, so it is conjugate to the
position of the pointer space: $\left[X_{P},B_{P}\right]=i$, with $\hbar=1$
and $X_{P}|x\rangle=x|x\rangle$; $g$ accounts for the coupling between both
systems, and may be considered a function of time. A measurement of $A_{S}$ is
characterized by a shift in the pointer’s position. Indeed, let
$|i_{S}\rangle=\sum_{n}\alpha_{n}|a_{n}\rangle$ be the state of the system,
with $A_{S}|a_{n}\rangle=a_{n}|a_{n}\rangle$, and
$|i_{P}\rangle=|\Phi(0)\rangle$ be the state of the pointer, a Gaussian state
centered at $x=0$ with spread $\sigma$:

$\langle
x|\Phi(0)\rangle=\phi(x)=(2\pi\sigma^{2})^{-1/4}\exp\left(-x^{2}/2\sigma^{2}\right).$

When $S$ and $P$ interact, their states evolve jointly under the action of

$U(\epsilon)\equiv\exp\\!\left(-i\int_{0}^{\Delta
t}\mathrm{d}t^{\prime}\,H_{\text{int}}\right)=\exp\left(-i\epsilon
A_{S}B_{P}\right).$ (2)

The parameter $\epsilon\equiv\int_{0}^{\Delta t}g\,\mathrm{d}t^{\prime}$,
which we will assume positive, quantifies the interaction strength, whence the
“weak” or “strong” character of a measurement can be ascertained. It can be
shown [[][(tobepublished).]ruelas2023] that the system-pointer state then
becomes

$U(\epsilon)|i_{S}\rangle|i_{P}\rangle=\sum_{n}\alpha_{n}|a_{n}\rangle|\Phi({\epsilon
a_{n}})\rangle,$ (3)

a superposition of eigenstates of $A_{S}$ coupled to Gaussian states centered
at $x=\epsilon a_{n}$, each with spread $\sigma$. If
$\epsilon(a_{n+1}-a_{n})\gg\sigma$, the pointer wave packets do not overlap,
so a reading of the pointer’s position $X_{P}$ reveals the measured eigenvalue
of $A_{S}$ with no ambiguity; such a measurement is called strong. For
$\epsilon(a_{n+1}-a_{n})\not\gg\sigma$, conversely, the Gaussians in the
evolved state do overlap, and the pointer’s position is not unambiguously
correlated with any single $a_{n}$; these are the so-called weak measurements.

Immediately after a weak measurement, the evolved state can be projected onto
a final system state. This procedure, called postselection, yields the
(unnormalized) final pointer state $|f_{P}\rangle=\langle
f_{S}|U(\epsilon)|i_{S}\rangle|i_{P}\rangle$. Since the evolution operator of
a weak measurement can be approximated as $U(\epsilon)=1_{SP}-i\epsilon
A_{S}B_{P}+\mathcal{O}\left(\epsilon^{2}\right)$, the final pointer state
reads

$|f_{P}\rangle=\langle f_{S}|i_{S}\rangle\left(1_{P}-i\epsilon\left\langle
A_{S}\right\rangle_{w}B_{P}+\mathcal{O}\\!\left(\epsilon^{2}\right)\right)\\!\int\\!\mathrm{d}p\,\tilde{\phi}(p)\,|p\rangle,$
(4)

where $1_{SP}$ and $1_{P}$ are the identity operators of the system-pointer
and pointer spaces, respectively, and we have introduced the momentum
representation of $|\Phi(0)\rangle$. Under suitable weakness conditions [3,
2], the truncated sum in Eq. (4) can be rewritten as an exponential. In this
approximation, after an inverse Fourier transform, $|f_{P}\rangle$ takes the
form of a Gaussian centered at the complex number $\epsilon\\!\left\langle
A_{S}\right\rangle_{w}$:

$\displaystyle|f_{P}\rangle$ $\displaystyle\approx\langle
f_{S}|i_{S}\rangle\int\\!\mathrm{d}p\,\exp\\!\left(-i\epsilon\langle
A_{S}\rangle_{w}\,p\right)\tilde{\phi}(p)\,|p\rangle$ $\displaystyle=\langle
f_{S}|i_{S}\rangle\int\\!\mathrm{d}x\,\phi\\!\left(x-\epsilon\langle
A_{S}\rangle_{w}\right)|x\rangle$ $\displaystyle=\langle
f_{S}|i_{S}\rangle|\Phi\\!\left(\epsilon\langle
A_{S}\rangle_{w}\right)\rangle.$ (5)

However, as it turned out, weak measurements are not a necessary condition for
so-called weak values to emerge. Without need for the weakness hypothesis,
Johansen [4] proposed a way to measure the real and imaginary parts of a weak
value; Zou et al. [5] presented an algorithm that uses experimentally obtained
weak values to determine the complex amplitudes of pure states; and Zhang et
al. developed a theoretical framework wherein, by probing pointer observables
that are functions of the interaction strength, weak values [6] as well as
mixed states [7] can be reconstructed. Along these lines, Denkmayr et al. [8,
9] put forward a model for measuring the weak values of a system qubit’s Pauli
spin operator $\sigma_{z}$ via measurements of arbitrary strength on a pointer
qubit. In neutron interferometry experiments, they measured
$\left\langle\sigma_{z}\right\rangle_{w}$ once in a weak setting and once in a
strong setting, and concluded that “experimental evidence is given that strong
interactions are superior to weak interactions in terms of accuracy and
precision, as well as required measurement time” [9]. Later experimental works
[10, 11] arrived at similar conclusions, which had been first surmised from
Vallone and Dequal’s numerical simulations [12]. In contrast, a series of
theoretical results prior to Denkmayr et al.’s papers had shown that precision
and accuracy are not always maximized at higher coupling strengths [13, 14,
7].

For the present work, we extended Denkmayr et al.’s scheme in order to measure
weak values of Pauli spin operators ${\sigma}_{\mathbf{n}}$, with $\mathbf{n}$
a real unit vector. Prompted by their conclusions, we set out to test our
proposal across a wider range of system-pointer interaction strengths, for
which we recurred to cloud-based quantum computation systems provided by the
IBM Corporation. Since these systems can be operated remotely via Python
scripts and allow for the experimental setup to be varied effortlessly, they
constitute a more versatile experimental context than neutron interferometry
and optical settings. Their basic building blocks are so-called
superconducting transmon qubits, which are acted upon by quantum gates
(operators). Made of materials such as niobium and aluminum, these qubits are
kept in a dilution refrigerator at 15 mK with the goal of minimizing
environmental effects 111IBM Quantum, The qubit, currently available at
https://web.archive.org/web/20230605160728/https://quantum-
computing.ibm.com/composer/docs/iqx/guide/the-qubit. In the Bloch-Redfield
model, times $T_{1}$ and $T_{2}$ characterize the time scale in which a qubit,
on average, undergoes decoherence [16]. To successfully realize quantum
circuits (sequences of operators), a qubit state must be preserved for
sufficient time for the required gates to be implemented and measurement
results to be read. From 2012 on, IBM’s quantum systems achieved decoherence
times that enabled useful computations [17]. Nowadays, the IBM Quantum
platform gives free access to quantum systems. Since its release in 2016,
researchers have employed the platform as a test tool for multiple
experimental proposals, such as tests of Mermin inequalities [18]; protocols
for “quantum error correction, quantum arithmetic, quantum graph theory, and
fault-tolerant quantum computation” [19]; dynamical decoupling protocols that
mitigate decoherence [20]; simulations of a one-dimensional Ising spin chain
[21]; detector tomography [22]; and protocols for entanglement purification
and swapping [23], to name a few.

Experiments in present-day quantum computers can be thought of as analogous to
experiments in more traditional contexts such as classical optics. Consider a
generic interferometric setup, where laser light’s polarization and
propagation path are the two qubits of interest. To prepare a target state,
wave plates, polarizers, and other instruments transform the beam’s
polarization. Their action is far from ideal, for they produce unwanted
reflections, reduce the beam’s power, and increase the state’s uncertainty
degree, which grows with the number of instruments and their precision. Upon
encountering a beam splitter, the beam separates into two paths, whose lengths
experience irregular changes due to mechanical perturbations. Both beams
undergo more transformations, are reflected on mirrors, which introduce
additional phase shifts, and are finally recombined in a beam splitter. The
extent to which the resulting state resembles the target state depends on the
magnitudes of the alluded factors. With regard to IBM’s quantum devices, the
state of transmon qubits also suffers from environmental effects, albeit on a
much shorter time scale, and from imperfect gate execution and measurement
readout. At the beginning of a run of a circuit, each qubit is initialized in
the ground state $|0\rangle$. One-qubit (two-qubit) gates are then applied,
with error rates ranging from 0.02% to 0.06% (0.6% to 1%) and a time duration
of a few tens (hundreds) of nanoseconds. Measurement results are read in
slightly less than one thousand nanoseconds, with a small but non-negligible
error probability between 1% and 14%. The decoherence times $T_{1}$ and
$T_{2}$ of each qubit range from tens to hundreds of microseconds. As in the
optical case, the final state differs from the target state, but is found on
average to resemble it depending, largely, on the total time required to run
the circuit. Because this analogy suggests that the two contexts allow for the
experimental realization of similar processes with comparable degrees of
uncertainty, we carried out our proposal in the latter.

This paper is organized as follows. We introduce our proposal in Sec. II, and
describe its implementation in an optical as well as in a quantum
computational setting. Our results, presented in Sec. III, provide
experimental evidence that strong measurements are not universally better than
their weaker counterparts, and attest to the usefulness of cloud-based quantum
computation devices for experimental tests. We conclude in Sec. IV by
contrasting our results to Denkmayr et al.’s and placing our work in the
broader context.

##  II Weak value measurements

Consider two interacting qubits, $a$ and $b$. Let
$\left\\{|0_{a}\rangle,|1_{a}\rangle\right\\}$ be the basis of qubit $a$, the
system, and $\left\\{|0_{b}\rangle,|1_{b}\rangle\right\\}$ the basis of qubit
$b$, the pointer. Measurements of the system operator
$\sigma^{a}_{\mathbf{n}}$ are described by
$U(\epsilon)=\exp\left(-i\epsilon\sigma^{a}_{\mathbf{n}}\sigma^{b}_{z}\right)$,
where $\epsilon$ is a positive number that quantifies the strength of the
measurement, $\mathbf{n}$ is a real unit vector, and

$\begin{gathered}\sigma^{a}_{\mathbf{n}}=\mathbf{n}\cdot\bm{\sigma}^{a},\quad\bm{\sigma}^{a}=\left(\sigma^{a}_{x},\sigma^{a}_{y},\sigma^{a}_{z}\right),\\\
\sigma^{a}_{x}=|0_{a}\rangle\langle 1_{a}|+|1_{a}\rangle\langle 0_{a}|,\\\
\sigma^{a}_{y}=-i|0_{a}\rangle\langle 1_{a}|+i|1_{a}\rangle\langle 0_{a}|,\\\
\sigma^{a}_{z}=|0_{a}\rangle\langle 0_{a}|-|1_{a}\rangle\langle
1_{a}|,\end{gathered}$ (6)

with $\sigma^{b}_{x,y,z}$ defined analogously. Weak measurements allow us to
approximate $U(\epsilon)$ as
$\sigma^{a}_{0}\sigma^{b}_{0}-i\epsilon\sigma^{a}_{\mathbf{n}}\sigma^{b}_{z}+\mathcal{O}(\epsilon^{2})$,
where $\sigma^{a,b}_{0}$ is the identity operator in each qubit space. It is
possible, however, to go beyond the weak regime. As described in [2, 24],
$U(\epsilon)$ can be expressed in closed form,

$U(\epsilon)=\cos\epsilon\,\sigma^{a}_{0}\sigma^{b}_{0}-i\sin\epsilon\,\sigma^{a}_{\mathbf{n}}\sigma^{b}_{z}.$
(7)

With an eye on the specific proposals to be discussed later, we rewrite the
evolution operator as

$\displaystyle\left(\cos\epsilon\,\sigma^{a}_{0}\\!-\\!i\sin\epsilon\,\sigma^{a}_{\mathbf{n}}\right)|0_{b}\rangle\langle
0_{b}|\\!+\\!\left(\cos\epsilon\,\sigma^{a}_{0}\\!+\\!i\sin\epsilon\,\sigma^{a}_{\mathbf{n}}\right)|1_{b}\rangle\langle
1_{b}|$
$\displaystyle=\exp\left(-i\epsilon\sigma^{a}_{\mathbf{n}}\right)|0_{b}\rangle\langle
0_{b}|+\exp\left(i\epsilon\sigma^{a}_{\mathbf{n}}\right)|1_{b}\rangle\langle
1_{b}|$ $\displaystyle\equiv
U^{\dagger}_{a}(\mathbf{n},\epsilon)\Pi_{0_{b}}+U_{a}(\mathbf{n},\epsilon)\Pi_{1_{b}}.$
(8)

The prescription for measuring the weak value
$\left\langle\sigma^{a}_{\mathbf{n}}\right\rangle_{w}$ goes as follows: first,
prepare the preselection (system) state $|A_{i}\rangle$ and couple it to the
(pointer) state $|B_{i}\rangle=|+_{b}\rangle$, with $|\pm_{b}\rangle\equiv
2^{-1/2}(|0_{b}\rangle\pm|1_{b}\rangle)$. Then, apply the evolution operator
in Eq. (7) to obtain

$\displaystyle U(\epsilon)|A_{i}\rangle|B_{i}\rangle$
$\displaystyle=\left(\cos\epsilon\,\sigma^{a}_{0}\sigma^{b}_{0}-i\sin\epsilon\,\sigma^{a}_{\mathbf{n}}\sigma^{b}_{z}\right)|A_{i}\rangle|+_{b}\rangle$
$\displaystyle=\cos\epsilon|A_{i}\rangle|+_{b}\rangle-i\sin\epsilon\,\sigma^{a}_{\mathbf{n}}|A_{i}\rangle|-_{b}\rangle.$
(9)

Afterwards, project the evolved state onto $|A_{f}\rangle|B_{f}\rangle$, the
postselection state coupled to an auxiliary pointer state, and rewrite the
resulting expression as

$\displaystyle\langle A_{f}|\langle
B_{f}|U(\epsilon)|A_{i}\rangle|B_{i}\rangle$
$\displaystyle=\cos\epsilon\langle A_{f}|A_{i}\rangle\langle
B_{f}|+_{b}\rangle-i\sin\epsilon\big{\langle}A_{f}\big{|}\sigma^{a}_{\mathbf{n}}\big{|}A_{i}\big{\rangle}\langle
B_{f}|-_{b}\rangle$ $\displaystyle=\langle
A_{f}|A_{i}\rangle\left(\cos\epsilon\langle
B_{f}|+_{b}\rangle-i\sin\epsilon\left\langle\sigma^{a}_{\mathbf{n}}\right\rangle_{w}\langle
B_{f}|-_{b}\rangle\right).$ (10)

The quantity in Eq. (10) is a complex number, as is the weak value of
interest, which we can write in terms of its real and imaginary parts:

$\left\langle\sigma^{a}_{\mathbf{n}}\right\rangle_{w}=R+iI.$ (11)

The square of the norm of Eq. (10) gives a measurable quantity, from which we
can extract all required information by selecting $|B_{f}\rangle$ to be

$\displaystyle|+_{b}\rangle\\!\longrightarrow i_{0}=|\langle
A_{f}|A_{i}\rangle|^{2}\cos^{2}\\!\epsilon,$ (12a)
$\displaystyle|-_{b}\rangle\\!\longrightarrow i_{1}=|\langle
A_{f}|A_{i}\rangle|^{2}\sin^{2}\\!\epsilon\,\left|\left\langle\sigma^{a}_{\mathbf{n}}\right\rangle_{w}\right|^{2},$
(12b) $\displaystyle|0_{b}\rangle\longrightarrow i_{2}=\frac{1}{2}|\langle
A_{f}|A_{i}\rangle|^{2}\left[\left(\cos\epsilon+I\sin\epsilon\right)^{2}+R^{2}\sin^{2}\\!\epsilon\right],$
(12c) $\displaystyle|1_{b}\rangle\longrightarrow i_{3}=\frac{1}{2}|\langle
A_{f}|A_{i}\rangle|^{2}\left[\left(\cos\epsilon-I\sin\epsilon\right)^{2}+R^{2}\sin^{2}\\!\epsilon\right],$
(12d) $\displaystyle|r_{b}\rangle\longrightarrow i_{4}=\frac{1}{2}|\langle
A_{f}|A_{i}\rangle|^{2}\left[\left(\cos\epsilon+R\sin\epsilon\right)^{2}+I^{2}\sin^{2}\\!\epsilon\right],$
(12e) $\displaystyle|l_{b}\rangle\longrightarrow i_{5}=\frac{1}{2}|\langle
A_{f}|A_{i}\rangle|^{2}\left[\left(\cos\epsilon-R\sin\epsilon\right)^{2}+I^{2}\sin^{2}\\!\epsilon\right],$
(12f)

with

$|r_{b}\rangle\equiv
2^{-1/2}(|0_{b}\rangle+i|1_{b}\rangle)\,\,\,\text{and}\,\,\,|l_{b}\rangle\equiv
2^{-1/2}(|0_{b}\rangle-i|1_{b}\rangle).$

We call $i_{0,\ldots,5}$ intensities, regardless of whether they represent
laser power, photon coincidence counts, or other physical quantities.
Subtracting Eqs. (12d) and (12f) from Eqs. (12c) and (12e), respectively,
yields

$\displaystyle i_{2}-i_{3}=I|\langle A_{f}|A_{i}\rangle|^{2}\sin 2\epsilon,$
(13a) $\displaystyle i_{4}-i_{5}=R|\langle A_{f}|A_{i}\rangle|^{2}\sin
2\epsilon.$ (13b)

When the overlap $\langle A_{f}|A_{i}\rangle$ is non-zero and $\sin
2\epsilon\neq 0$, it is possible to solve Eqs. (12a), (12b), (13a), and (13b)
for the squared magnitude, real part, and imaginary part of
$\left\langle\sigma^{a}_{\mathbf{n}}\right\rangle_{w}$:

$\left|\left\langle\sigma^{a}_{\mathbf{n}}\right\rangle_{w}\right|^{2}=\frac{i_{1}}{i_{0}}\cot^{2}\\!\epsilon,\,R=\frac{i_{4}-i_{5}}{2\,i_{0}}\cot\epsilon,\,I=\frac{i_{2}-i_{3}}{2\,i_{0}}\cot\epsilon.$
(14)

The above equations are periodic in $\epsilon$ with period $\pi$. We emphasize
that, while these results hold for both weak and strong couplings, the values
$\epsilon=0,\pi/2,$ and $\pi$ are exceptional, as will be seen in Sec. III.

###  II.1 All-optical realization

To implement the weak value measurement proposal in an optical setting, with
either classical or quantum light, consider the Mach-Zehnder type
interferometer shown in Fig 1. Let qubit $a$ represent the polarization of
light—so that $|0_{a}\rangle$ and $|1_{a}\rangle$ represent horizontally and
vertically polarized light, respectively—and qubit $b$ its propagation
direction, with $|0_{b}\rangle$ representing the horizontal and
$|1_{b}\rangle$ the vertical direction. Incoming light on the horizontal path
undergoes a transformation given by the operator $V_{i}$, whereas incoming
light on the vertical path experiences no transformation. Both beams then
enter the interferometer through a 50:50 beam splitter (BS), are reflected on
mirrors (M), and are acted upon by $V_{0}$ in the horizontal path and by
$V_{1}$ in the vertical path. Subsequently, they are recombined in another BS,
and the beam in the horizontal path undergoes one last transformation,
$V_{f}$. The evolution operator given by Eq. (II) can be realized in such an
interferometer with

$V_{0}=V_{1}=U_{a}(\mathbf{n},\epsilon)\,\,\,\text{and}\,\,\,V_{i}=V_{f}=U_{a}^{\dagger}(\mathbf{n},\epsilon),$

where $U_{a}(\mathbf{n},\epsilon)$ is a sequence of quarter-wave ($Q$) and
half-wave ($H$) plates,

$\displaystyle U_{a}(\mathbf{n},\epsilon)=$ $\displaystyle
Q\\!\left(\frac{\pi+\varphi}{2}\right)Q\\!\left(\frac{\vartheta+\varphi}{2}\right)H\\!\left(\frac{-\pi+\vartheta+\varphi}{2}+\frac{\epsilon}{2}\right)$
$\displaystyle\times
Q\\!\left(\frac{\vartheta+\varphi}{2}\right)Q\\!\left(\frac{\varphi}{2}\right),$
(15)

the unit vector is
$\mathbf{n}=(\sin\vartheta\cos\varphi,\sin\vartheta\sin\varphi,\cos\vartheta)$,
and the argument of each plate represents the angle made by its major axis
with respect to the vertical direction. The adjoint operator,
$U_{a}^{\dagger}(\mathbf{n},\epsilon)$, is given by the same plates in reverse
order and with an additional $+\pi/2$ in each argument. The initial and final
system-pointer states, $|A_{i}\rangle|B_{i}\rangle$ and
$|A_{f}\rangle|B_{f}\rangle$, respectively, can be implemented by using
similar interferometric arrangements at both the input and output ports of the
interferometer shown in Fig 1. (See [2], [25], and
[[][(tobepublished).]deZela2022proceedings] for more details on this setup.)
Finally, intensities and photon counts can be measured at the output ports,
whereupon weak values can be ascertained through Eqs. (14).

Figure 1: Mach-Zehnder type interferometer with mirrors (M), beam splitters
(BS), and unitary operators $V_{0,1,i,f}$ for implementing the evolution
operator $U(\epsilon)$. The horizontal (vertical) path corresponds to the
state $|0_{b}\rangle$ $\left(|1_{b}\rangle\right)$.

###  II.2 Quantum computational realization

Figure 2: Quantum circuit for measuring
$\left\langle\sigma^{a}_{\mathbf{n}}\right\rangle_{w}$ with interaction
strength given by $\epsilon$. It prepares the initial state
$|A_{i}\rangle|B_{i}\rangle$ and acts $U(\epsilon)$ on it. Then,
$i_{0,\ldots,5}$ are measured by choosing the appropriate gates G, followed by
projections onto $\langle 0_{a}|\langle 0_{b}|$ and $\langle 0_{a}|\langle
1_{b}|$ (see text).

To implement the proposal in a quantum computing system, consider, for
instance, a Pauli spin operator given by $\mathbf{n}=(-\sin\phi,\cos\phi,0)$.
The operators $\exp\left(\pm i\epsilon\sigma^{a}_{\mathbf{n}}\right)$ can be
decomposed into a product of rotations:

$\displaystyle\exp\left(\pm i\epsilon\sigma^{a}_{\mathbf{n}}\right)$
$\displaystyle=\cos\epsilon\pm
i\sin\epsilon\left(-\sin\phi\,\sigma_{x}^{a}+\cos\phi\,\sigma_{y}^{a}\right)$
$\displaystyle=\exp(-i\phi\sigma_{z}^{a}/2)\exp(\pm
i\epsilon\sigma_{y}^{a})\exp(i\phi\sigma_{z}^{a}/2)$ $\displaystyle\equiv
R_{z}(\phi)R_{y}(\mp 2\epsilon)R_{z}(-\phi).$ (16)

With $\exp\left(\pm i\epsilon\sigma^{a}_{\mathbf{n}}\right)$ given by Eq.
(II.2), the evolution operator $U(\epsilon)$ of Eq. (II) can be realized in
the quantum circuit shown in Fig. 2. The system state is prepared as the
desired preselection state by applying a rotation $R_{y}$ to the state
$|0_{a}\rangle$:
$|A_{i}\rangle=R_{y}(\alpha_{i})|0_{a}\rangle=\cos(\alpha_{i}/2)|0_{a}\rangle+\sin(\alpha_{i}/2)|1_{a}\rangle$.
The pointer state, in turn, is set to the auxiliary state
$|B_{i}\rangle=|+_{b}\rangle=H|0_{b}\rangle$, where $H$ is the Hadamard gate,

$\displaystyle H$ $\displaystyle=\frac{1}{\sqrt{2}}\left(|0_{b}\rangle\langle
0_{b}|+|0_{b}\rangle\langle 1_{b}|+|1_{b}\rangle\langle
0_{b}|-|1_{b}\rangle\langle 1_{b}|\right)$ $\displaystyle=|+_{b}\rangle\langle
0_{b}|+|-_{b}\rangle\langle 1_{b}|.$ (17)

To optimize the number of gates employed in the circuit, we substitute the
exponentials of Eq. (II.2) into Eq. (II) and regroup them as follows:

$\displaystyle U(\epsilon)$
$\displaystyle=R_{z}(\phi)R_{y}(2\epsilon)R_{z}(-\phi)\Pi_{0_{b}}$
$\displaystyle\quad+R_{z}(\phi)R_{y}(-2\epsilon)R_{z}(-\phi)\Pi_{1_{b}}$
$\displaystyle=R_{z}(\phi)R_{y}(2\epsilon)\mathrm{C}\\!R_{y}(-4\epsilon)R_{z}(-\phi),$
(18)

where we have defined the controlled $R_{y}$ gate,

$\mathrm{C}\\!R_{y}(\vartheta)\equiv\Pi_{0_{b}}+R_{y}(\vartheta)\Pi_{1_{b}},$

which applies a rotation on qubit $a$ depending on the state of $b$. The
evolution operator first subjects qubit $a$ to the rotation $R_{z}(-\phi)$.
Then, if the state of qubit $b$ is $|0_{b}\rangle$, $a$ is acted on by
$R_{y}(2\epsilon)$; otherwise, $R_{y}(-4\epsilon)$ followed by
$R_{y}(2\epsilon)$ are applied on qubit $a$. Lastly, $R_{z}(\phi)$ acts on
$a$. To project the evolved state onto $\langle A_{f}|\langle B_{f}|$, we
choose the postselection state $|A_{f}\rangle=|0_{a}\rangle$, let different
gates G act on qubit $b$ [in order to select the different $|B_{f}\rangle$ of
Eqs. (12)], and measure both qubits, as indicated in Fig. 2. The observables
$i_{0,1}$ result from applying a Hadamard gate to qubit $b$, i.e.
$\mathrm{G}=H$, and projecting onto $\langle 0_{a}|\langle 0_{b}|$ and
$\langle 0_{a}|\langle 1_{b}|$, respectively. To measure $i_{2,3}$,
$\mathrm{G}$ is set equal to the identity and is followed by the same
respective projections. For $i_{4,5}$ we set $\mathrm{G}=H\,S^{\dagger}$, with

$S^{\dagger}=|0_{b}\rangle\langle 0_{b}|+e^{-i\pi/2}|1_{b}\rangle\langle
1_{b}|=|0_{b}\rangle\langle 0_{b}|-i|1_{b}\rangle\langle 1_{b}|.$ (19)

Thus, $S^{\dagger}$ introduces a relative phase shift of $-\pi/2$ between
$|0_{b}\rangle$ and $|1_{b}\rangle$. The intensities, again, are obtained by
projecting onto $\langle 0_{a}|\langle 0_{b}|$ and $\langle 0_{a}|\langle
1_{b}|$.

Moreover, knowledge of $\left\langle\sigma^{a}_{\mathbf{n}}\right\rangle_{w}$
allows us to determine the preselection state [27, 28, 29, 30]. Any Pauli spin
operator has two eigenvectors,

$\sigma^{a}_{\mathbf{n}}|\mathbf{n}_{\pm}\rangle=\pm|\mathbf{n}_{\pm}\rangle,$
(20)

whose eigenprojectors can be expressed as

$\Pi_{\pm}^{a}=|\mathbf{n}_{\pm}\rangle\langle\mathbf{n}_{\pm}|=\frac{1}{2}\left(\sigma^{a}_{0}\pm\sigma^{a}_{\mathbf{n}}\right).$
(21)

For our choice of $\mathbf{n}$, preselection, and postselection states, the
theoretical real and imaginary parts of the weak value are given by

$\left\langle\sigma^{a}_{\mathbf{n}}\right\rangle_{w}=R_{t}+iI_{t}\equiv-\tan\\!\left(\alpha_{i}/2\right)(\sin\phi+i\cos\phi),$
(22)

the eigenvectors of $\sigma^{a}_{\mathbf{n}}$ are found to be

$|\mathbf{n}_{\pm}\rangle=\frac{1}{\sqrt{2}}\left(\pm|0_{a}\rangle+ie^{i\phi}|1_{a}\rangle\right),$
(23)

and the weak values of the eigenprojectors in Eq. (21) take the form

$\left\langle\Pi_{\pm}^{a}\right\rangle_{w}=\frac{1}{2}\left(1\pm\left\langle\sigma^{a}_{\mathbf{n}}\right\rangle_{w}\right)=\frac{1}{2}\left(1\mp
ie^{-i\phi}\tan(\alpha_{i}/2)\right).$ (24)

With the goal of comparing our results to Denkmayr et al.’s [8], we now
describe how to characterize the preselection state’s parameters. The
components of $|A_{i}\rangle$ in the basis of $\sigma^{a}_{\mathbf{n}}$ are

$\langle\mathbf{n}_{\pm}|A_{i}\rangle=\pm\frac{1}{\sqrt{2}}\left(\cos(\alpha_{i}/2)\mp
ie^{-i\phi}\sin(\alpha_{i}/2)\right),$ (25)

so the preselection state can be written as

$|A_{i}\rangle=\frac{\big{\langle}\Pi_{+}^{a}\big{\rangle}_{w}|\mathbf{n}_{+}\rangle-\big{\langle}\Pi_{-}^{a}\big{\rangle}_{w}|\mathbf{n}_{-}\rangle}{\sqrt{\big{|}\big{\langle}\Pi_{+}^{a}\big{\rangle}_{w}\big{|}^{2}+\big{|}\big{\langle}\Pi_{-}^{a}\big{\rangle}_{w}\big{|}^{2}}}.$
(26)

The relative minus sign in Eq. (26) is a consequence of having

$|A_{f}\rangle=|0_{a}\rangle=\frac{1}{\sqrt{2}}\left(|\mathbf{n}_{+}\rangle-|\mathbf{n}_{-}\rangle\right)$
(27)

instead of

$|1_{a}\rangle=\frac{1}{\sqrt{2}}e^{-i(\phi+\pi/2)}\left(|\mathbf{n}_{+}\rangle+|\mathbf{n}_{-}\rangle\right)$

as postselection state 222The state $|1_{a}\rangle$ is, up to a global phase,
the “diagonal” state of $\sigma_{\mathbf{n}}^{a}$, just as the postselection
state in [8],
$|P_{x};+\rangle=2^{-1/2}\left(|P_{z};+\rangle+|P_{z};-\rangle\right)$, was
the diagonal state of $\sigma_{z}$ (cf. Eqs. (5) and (8) of [8]). “Diagonal”
and “antidiagonal” states (e.g., $|1_{a}\rangle$ and $|0_{a}\rangle$) and the
basis states ($|\mathbf{n}_{+}\rangle$ and $|\mathbf{n}_{-}\rangle$) are said
to form mutually unbiased bases, of which the “diagonal” states are special
members (see, for instance, [27, 29]).. Then, we define the normalization
factor

$\nu\\!\equiv\\!\left(\big{|}\big{\langle}\Pi_{+}^{a}\big{\rangle}_{w}\big{|}^{2}\\!+\\!\big{|}\big{\langle}\Pi_{-}^{a}\big{\rangle}_{w}\big{|}^{2}\right)^{-1/2}\\!=\\!\left(\frac{1}{2}\\!+\\!\frac{1}{2}\big{|}\big{\langle}\sigma^{a}_{\mathbf{n}}\big{\rangle}_{w}\big{|}^{2}\right)^{-1/2}\\!\\!.$
(28)

For our choice of states and unit vector, Eq. (28) yields the theoretical
normalization factor $\nu_{t}=\sqrt{2}\cos(\alpha_{i}/2)$. Therefore, the
characterization of Eq. (26) [or, equivalently, of Eqs. (24) and (28)] is
achieved by compounding Eqs. (12) as

$\displaystyle\nu$
$\displaystyle=\left(\frac{1}{2}+\frac{1}{2}\frac{i_{1}}{i_{0}}\cot^{2}\\!\epsilon\right)^{-1/2}\\!,$
(29a) $\displaystyle\alpha$
$\displaystyle=2\arctan\sqrt{\frac{i_{1}}{i_{0}}\cot^{2}\\!\epsilon},$ (29b)
$\displaystyle\phi$
$\displaystyle=\arctan\\!\left(\frac{i_{4}-i_{5}}{i_{2}-i_{3}}\right).$ (29c)

Equations (14) and (29) express all the parameters of interest in terms of the
measurable quantities $i_{0,\ldots,5}$. We employed them to obtain the results
presented in the following section.

##  III Results

Figure 3: Weak values $\left\langle\sigma^{a}_{\mathbf{n}}\right\rangle_{w}$
obtained in ibm_oslo as functions of the rotation angle $\phi$, with
$\alpha_{i}=\pi/4$, for twelve measurement strengths $\epsilon$. The solid
blue lines (circles) stand for the theoretical (experimental) real part of the
weak value, $R_{t}$ ($R_{m}$). The dashed red lines (squares) represent the
theoretical (experimental) imaginary part of the same weak value, $I_{t}$
($I_{m}$). Error bars represent $\sigma_{R}$ or $\sigma_{I}$ [see Eqs. (30)].
Figure 4: Logarithmic plots of $\bar{\sigma}$, a measure of statistical
uncertainty for assessing precision, and $\bar{\Delta}$, a metric for
systematic errors that reflects accuracy, as defined in Eq. (32), of the real
(blue circles) and imaginary (red triangles) parts of
$\big{\langle}\sigma^{a}_{\mathbf{n}}\big{\rangle}_{w}$, as functions of the
measurement strength $\epsilon$. The vertical dotted lines represent
$\epsilon=\pi/24$ and $\pi/4$, the values analyzed by Denkmayr et al. [8].
Figure 5: Statistical uncertainty (top) and systematic error (bottom), as
quantified by $\ln\\!\left(\bar{\sigma}\right)$ and
$\ln\\!\left(\bar{\Delta}\right)$, respectively, of the normalization factor
$\nu$ (left), amplitude angle $\alpha$ (center), and phase $\phi$ (right), as
functions of the measurement strength $\epsilon$. The vertical dotted lines
represent $\epsilon=\pi/24$ and $\pi/4$, the values analyzed by Denkmayr et
al. [8].

Our work was conducted in “ibm_oslo”, an open-access quantum system with seven
qubits, labeled 0 to 6, from which we chose qubits 0 and 1 to carry out our
experiments. During data recollection, qubit 0 (1) had an average $T_{1}$ of
142 $\mu$s (135 $\mu$s) and an average $T_{2}$ of 101 $\mu$s (26 $\mu$s)
333Preliminary tests revealed that ibm_oslo, as well as other open-access
quantum systems, produced data similar to that reported in Sec. III whenever
their relaxation times differed by a few tens of microseconds from our average
values. These parameters sometimes dropped significantly, however. As a
consequence, the multiple error sources inherent to the hardware rendered all
measurement outcomes meaningless.. As explained in Sec. II, three different
circuits are required to compute
$\left\langle\sigma^{a}_{\mathbf{n}}\right\rangle_{w}$; their outcomes are a
measurement of either $i_{0}$ and $i_{1}$, $i_{2}$ and $i_{3}$, or $i_{4}$ and
$i_{5}$. For each value of $\epsilon$ and $\phi$ sampled, we ran each circuit
2000 times, repeated this procedure 20 times, averaged the intensities, and
computed the parameters of interest in accordance with Eqs. (14) and (29).

To distinguish between theoretical and experimental parameters, let us denote
the theoretical intensities as ${i_{0t},\ldots,i_{5t}}$, which are given by
Eqs. (12) with $|\langle
A_{f}|A_{i}\rangle|^{2}\to\cos^{2}\\!\left(\alpha_{i}/2\right)$, $I\to I_{t}$,
and $R\to R_{t}$. The experimental mean values of these parameters will be
denoted as $R_{m}$ and $I_{m}$. Each mean intensity has an associated standard
deviation $\sigma_{i_{0}},\ldots,\sigma_{i_{5}}$. A measure of uncertainty for
the experimental weak values is given by the propagation formulae

$\displaystyle\sigma_{R}$ $\displaystyle=\sqrt{\left({\frac{\partial
R}{\partial i_{0}}}\right)^{2}\sigma_{i_{0}}^{2}+\left(\frac{\partial
R}{\partial i_{4}}\right)^{2}\sigma_{i_{4}}^{2}+\left(\frac{\partial
R}{\partial i_{5}}\right)^{2}\sigma_{i_{5}}^{2}}$
$\displaystyle=\frac{1}{\cos^{2}\\!\left(\alpha_{i}/2\right)}\sqrt{\frac{\tan^{2}\\!\left(\alpha_{i}/2\right)\sin^{2}\\!\phi}{\cos^{4}\\!{\epsilon}}\sigma_{i_{0}}^{2}+\frac{\sigma_{i_{4}}^{2}+\sigma_{i_{5}}^{2}}{\sin^{2}\\!{2\epsilon}}},$
(30a) $\displaystyle\sigma_{I}$ $\displaystyle=\sqrt{\left(\frac{\partial
I}{\partial i_{0}}\right)^{2}\sigma_{i_{0}}^{2}+\left(\frac{\partial
I}{\partial i_{2}}\right)^{2}\sigma_{i_{2}}^{2}+\left(\frac{\partial
I}{\partial i_{3}}\right)^{2}\sigma_{i_{3}}^{2}}$
$\displaystyle=\frac{1}{\cos^{2}\\!\left(\alpha_{i}/2\right)}\sqrt{\frac{\tan^{2}\\!\left(\alpha_{i}/2\right)\cos^{2}\\!\phi}{\cos^{4}\\!{\epsilon}}\sigma_{i_{0}}^{2}+\frac{\sigma_{i_{2}}^{2}+\sigma_{i_{3}}^{2}}{\sin^{2}\\!{2\epsilon}}},$
(30b)

where the derivatives are evaluated at the theoretical intensities. Similar
relations hold for the uncertainties of the preselection state parameters,
$\sigma_{\nu},\sigma_{\alpha},\sigma_{\phi}$.

For our experiments, we swept the measurement strength $\epsilon$ across a
full period. At each value of $\epsilon$, we took $N=9$ samples of $\phi$ from
$-\pi$ to $\pi$, and set $\alpha_{i}=\pi/4$. Following Denkmayr et al. [8], we
introduce figures of merit for our results at fixed $\epsilon$. Let $\mu$
refer to $R$, $I$, $\nu$, $\alpha$, or $\phi$, so that $\mu_{t,j}$ is given by
Eq. (22) evaluated at $\phi_{j}$, or by

$\displaystyle\nu_{t,j}$ $\displaystyle=\sqrt{2}\cos(\alpha_{i}/2),$ (31a)
$\displaystyle\alpha_{t,j}$ $\displaystyle=\pi/4,$ (31b)
$\displaystyle\phi_{t,j}$
$\displaystyle\in\\{-\pi,-3\pi/4,\ldots,3\pi/4,\pi\\}.$ (31c)

Similarly, $\sigma_{\mu,j}$ refers to Eqs. (30) or
$\sigma_{\nu},\sigma_{\alpha},\sigma_{\phi}$, and $\mu_{m,j}$ is the mean
value of $\mu$ obtained for sample $j$. Using this notation, the achieved
degrees of precision and accuracy, respectively, can be assessed by means of
the following parameters:

$\bar{\sigma}_{\mu}\equiv\sqrt{\frac{1}{N}\sum_{j}^{N}\sigma_{\mu,j}^{2}},\quad\bar{\Delta}_{\mu}\equiv\sqrt{\frac{1}{N}\sum_{j}^{N}\left(\mu_{t,j}-\mu_{m,j}\right)^{2}}.$
(32)

For a given strength $\epsilon$, $\bar{\sigma}_{\mu}$ is the root mean square
(RMS) value of the sample uncertainties. It serves as a proxy for precision,
in that it quantifies the statistical uncertainty of the sample. Likewise,
$\bar{\Delta}_{\mu}$ is the RMS value of the difference between theoretical
and experimental parameters in a sample. So defined, it appraises the accuracy
of our results by representing systematic errors in the data.

Figure 3 shows the real and imaginary parts of
$\left\langle\sigma^{a}_{\mathbf{n}}\right\rangle_{w}$ as functions of the
rotation angle $\phi$, for twelve different measurement strengths. For
$\epsilon$ near zero, $R_{m}$ and $I_{m}$ result from subtracting two very
close numbers [see Eqs. (12c)-(12f)] and dividing by $\sin 2\epsilon$, which
also tends to zero. The experimental results behave accordingly: both
parameters fluctuate about 0, with greater dispersion and error for smaller
$\epsilon$. As measurements become stronger, the results are in better
accordance with theoretical predictions, and fluctuations subside. When
$\epsilon$ approaches $\pi/2$, this trend reverses, with the weak values
behaving similarly to how they did close to $\epsilon=0$. For larger values of
$\epsilon$, the results improve until $\epsilon$ nears $\pi$, at which point
the previous erratic behavior re-emerges. Hence, critical values in the regime
of strong coupling exist, for which precision and accuracy behave the same as
in the weak coupling regime.

Figures 4 and 5 summarize the information contained in our results that
pertains to statistical uncertainty and systematic errors. In Fig. 4,
$\bar{\sigma}_{R,I}$ take relatively high values for very weak measurements,
but decrease as $\epsilon$ grows up to $\pi/4$, where they begin to increase.
Since $\bar{\sigma}_{R,I}$ are the RMS values of the uncertainties in Eqs.
(30), which diverge for $\epsilon=0,\pi/2$, and $\pi$, we can expect
$\bar{\sigma}_{R,I}$ to reach relative maxima at these strengths. This
prediction is confirmed by Fig. 4. The height of each maximum of
$\bar{\sigma}_{R,I}$ depends on both the standard deviations and their
respective coefficients in Eqs. (30). In the rest of the domain of $\epsilon$,
$\bar{\sigma}$ first drops and then rises until the last divergence is
reached. On the other hand, $\bar{\Delta}$ reflects how much the measured
values differ from the theoretical predictions. As we saw in Fig. 3 and argued
above, the resemblance is worse for $\epsilon$ in the vicinity of $0,\pi/2$,
and $\pi$ because Eqs. (14) diverge at these points. The results in Fig. 4
illustrate this trend, with $\bar{\Delta}$ first decreasing and then rising in
between divergences, albeit less smoothly than $\bar{\sigma}$. Figure 5
displays the same tendencies for the precision and accuracy of the
preselection state parameters $\nu$, $\alpha$, and $\phi$ 444In samples where
$\tan\phi_{j}=0$ (for example, $\phi_{j}=-\pi$) the experimental values of
$i_{4}$ and $i_{5}$ are two close numbers. Due to random fluctuations,
intensities such that $i_{4}-i_{5}>0$ or $i_{4}-i_{5}<0$ are both found. As a
consequence, the inverse tangent function introduces a spurious $\pm\pi$
factor whenever the difference has the wrong sign (positive, in our example).
To avoid this issue, we computed the angle $\phi$ as
$\arctan\left|\dfrac{i_{4}-i_{5}}{i_{2}-i_{3}}\right|$, thereby constraining
the results to the first quadrant and the theoretical phases $\phi_{t,j}$ to
$\left\\{0,\pi/4,\pi/2\right\\}$..

##  IV Summary and conclusions

In this work, we have extended Denkmayr et al.’s scheme for analysing the
behavior of the weak value $\left\langle\sigma_{z}\right\rangle_{w}$ with
measurements of arbitrary strength [8, 9]. We tested this extension with the
quantum computational tools hosted by IBM. Our results show that the model
works as expected: $\left\langle\sigma^{a}_{\mathbf{n}}\right\rangle_{w}$ and
the preselection state $|A_{i}\rangle$ can be computed for all allowed
strengths, but the statistical uncertainty and systematic errors present in
each reconstruction do not decrease monotonically with increasing strength.

In the paper that inspired the present work, Denkmayr et al. [8] compared
precision and accuracy for two measurement strengths, given by $\alpha=\pi/12$
and $\pi/2$ in their notation, which correspond to $\epsilon=\pi/24$ and
$\pi/4$ in ours. They concluded that strong measurements perform better than
their weaker counterparts. From both Fig. 4 and Fig. 5 we can conclude the
same for those cases—the difference in the quality of the results is
compelling. If statistical uncertainties and systematic errors were
monotonically decreasing functions of $\epsilon$, we could further assert that
stronger measurements are universally better than weaker ones. It must be
stressed, however, that this is not the case. (In the experimental setting of
[8], $\alpha/2=\pi/4$ was the actual maximum measurement strength possible.
This may have constrained the analysis.) Figures 4 and 5 also suggest that it
could be possible to minimize statistical uncertainties and systematic errors
(equivalently, maximize precision and accuracy) by choosing $\epsilon$
appropriately.

Denkmayr et al. remarked that their model “can be used for any coupling
between two two-level quantum systems” [8, 9]. As we outlined in Sec. II.1,
our measurement scheme can be implemented in an all-optical interferometric
setup that admits classical light beams as well as single photons.
Furthermore, their results [8, 9] and those of other authors [10, 11, 12]
seemed to suggest that, in general, stronger measurements are superior to
weaker ones in terms of precision and accuracy. Previous theoretical analyses
[13, 14, 7] had shown that this need not always be the case. To the best of
our knowledge, here we have presented the first experimental observation of
such a behavior.

A recent survey of several IBM quantum systems has shed light on how their
performances compare to each other and how their individual qubits compare to
one another [34]. In particular, it revealed the degree to which their
relaxation times, gate error rates, and readout error rates vary across the
sampled machines and over time. As argued throughout this article, we can
expect our model to yield similar results when run on different devices—as
long as the relaxation times do not dwindle considerably. How the present
scheme fares in different experimental contexts and in contrast to, e.g.,
standard tomography is an open question whose undertaking we would welcome.

###### Acknowledgements.

The authors acknowledge the use of IBM Quantum services for this work. The
views expressed are ours, and do not reflect the official policy or position
of IBM or the IBM Quantum team. D. R. A. R. P. acknowledges funding by
FONDECYT through Grant 236-2015.

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KB190298 $(u_{0}>0)$ | $0.686_{-0.356}^{+0.395}$ | $1.787_{-0.926}^{+1.029}$ | $5.763_{-2.603}^{+3.165}$ | $6.555_{-1.854}^{+1.218}$ | 1.00 | 1.00
KB190298 $(u_{0}<0)$ | $0.710_{-0.343}^{+0.370}$ | $1.850_{-0.895}^{+0.966}$ | $5.525_{-2.129}^{+2.741}$ | $6.921_{-1.344}^{+1.032}$ | 0.72 | 0.95
Adopted | $0.70\pm 0.37$ | $1.81\pm 0.96$ | $5.67\pm 2.70$ | $6.71\pm 1.39$ |  |
KB191216 (Inner, standard) | $0.525_{-0.247}^{+0.271}$ | $0.134_{-0.063}^{+0.069}$ | $2.987_{-1.295}^{+1.838}$ | $3.573_{-1.338}^{+1.836}$ | 1.00 | 1.00
KB191216 (Outer, standard) | $0.521_{-0.239}^{+0.265}$ | $0.127_{-0.059}^{+0.065}$ | $2.859_{-1.233}^{+1.751}$ | $3.441_{-1.272}^{+1.782}$ | 0.93 | 0.95
KB191216 (Inner, $u_{0}>0$) | $0.413_{-0.172}^{+0.229}$ | $0.101_{-0.042}^{+0.056}$ | $2.560_{-0.944}^{+1.253}$ | $2.895_{-0.852}^{+1.134}$ | 0.54 | 1.00
KB191216 (Inner, $u_{0}<0$) | $0.361_{-0.162}^{+0.252}$ | $0.088_{-0.039}^{+0.061}$ | $2.323_{-0.896}^{+1.318}$ | $2.607_{-0.810}^{+1.230}$ | 0.93 | 0.95
KB191216 (Outer, $u_{0}>0$) | $0.422_{-0.176}^{+0.226}$ | $0.103_{-0.043}^{+0.055}$ | $2.591_{-0.957}^{+1.262}$ | $2.931_{-0.864}^{+1.140}$ | 0.64 | 1.00
KB191216 (Outer, $u_{0}<0$) | $0.370_{-0.169}^{+0.253}$ | $0.090_{-0.041}^{+0.062}$ | $2.359_{-0.927}^{+1.355}$ | $2.649_{-0.846}^{+1.272}$ | 1.00 | 0.90
Adopted | $0.39\pm 0.21$ | $0.094\pm 0.050$ | $2.44\pm 1.12$ | $2.74\pm 1.02$ |  |
KB192783 | $0.339_{-0.197}^{+0.254}$ | $1.160_{-0.672}^{+0.871}$ | $1.850_{-0.930}^{+0.937}$ | $5.913_{-2.118}^{+1.224}$ | 1.00 | 1.00
Adopted | $0.34\pm 2.25$ | $1.16\pm 0.77$ | $1.85\pm 0.93$ | $5.91\pm 1.67$ |  |
OB190249 (Close, $u_{0}>0$) | $0.912_{-0.204}^{+0.170}$ | $7.129_{-1.598}^{+1.326}$ | $1.837_{-0.438}^{+0.457}$ | $6.333_{-1.026}^{+0.672}$ | 0.89 | 1.00
OB190249 (Close, $u_{0}<0$) | $0.912_{-0.210}^{+0.165}$ | $7.124_{-1.645}^{+1.288}$ | $1.840_{-0.429}^{+0.419}$ | $6.357_{-1.002}^{+0.648}$ | 1.00 | 0.95
OB190249 (Wide, $u_{0}>0$) | $0.865_{-0.206}^{+0.179}$ | $7.090_{-1.684}^{+1.467}$ | $5.201_{-0.782}^{+0.938}$ | $6.633_{-0.636}^{+0.684}$ | 0.46 | 0.19
OB190249 (Wide, $u_{0}<0$) | $0.850_{-0.216}^{+0.163}$ | $6.954_{-1.767}^{+1.330}$ | $5.079_{-0.848}^{+0.989}$ | $6.687_{-0.654}^{+0.672}$ | 0.46 | 0.19
Adopted | $0.91\pm 0.19$ | $7.12\pm 1.47$ | $1.84\pm 0.44$ | $6.37\pm 0.82$ |  |
OB190679 | $0.665_{-0.350}^{+0.407}$ | $3.337_{-1.758}^{+2.039}$ | $6.988_{-3.034}^{+3.385}$ | $5.631_{-1.980}^{+1.452}$ | 1.00 | 1.00
Adopted | $0.66\pm 0.38$ | $3.34\pm 1.90$ | $6.99\pm 3.21$ | $5.63\pm 1.71$ |  |
Table 16: AnomalyFinder Planets in KMT Sub-prime Fields for 2019

Event Name | KMT Name | $\log q$ | $s$ | Reference
---|---|---|---|---
OB190960a | KB191591 | $-4.87$ | 1.00 | Yee et al. (2021)
KB191806a | KB191806 | $-4.72$ | 0.94 | Zang et al. (2023)
KB191367a | KB191367 | $-4.30$ | 0.94 | Zang et al. (2023)
KB191216a | KB191216 | $-3.62$ | 1.11 | This Work
KB190298 | KB190298 | $-2.53$ | 1.85 | This Work
KB192783 | KB192783 | $-2.48$ | 0.81 | This Work
OB190468cb | KB192696 | $-2.46$ | 0.85 | Han et al. (2022a)
OB190679 | KB192688 | $-2.36$ | 2.18 | This Work
OB190362c | KB190075 | $-2.13$ | 0.90 | Chung et al. (2022)
OB190249c | KB190109 | $-2.12$ | 0.54 | This Work
OB190299 | KB192735 | $-2.00$ | 0.99 | Han et al. (2021)
OB190468bb,c | KB192696 | $-1.98$ | 0.72 | Han et al. (2022a)
OB191470c,d,e | KB192814 | $-2.32$ | 1.17 | Kuang et al. (2022)
KB190414c,f | KB190414 | $-2.25$ | 0.35 | Han et al. (2022b)
KB190304c,f | KB190304 | $-1.84$ | 1.57 | This Work
OB190344c,f,g | KB190149 | $-1.52$ | 1.70 | This Work

Note. — Event names are abbreviations for, e.g., OGLE-2019-BLG-0960 and
KMT-2019-BLG-1216. a: Inconsequential $s$ degeneracy. b: 2-planet system. c:
$s$ degeneracy. d: large $q$ degeneracy. e: planet in binary system. f:
1L2S/2L1S degeneracy. g: planet/binary degeneracy.

Table 17: Breakdown of Detections by 4 Questions Q&A | Year |  |  |
---|---|---|---|---
Field Type? | 2018 | 19 | 14 |
Prime, Subprime | 2019 | 13 | 12 |
| Total | 32 | 26 |
Identified By? | 2018 | 11 | 22 |
AnomalyFinder, Eye | 2019 | 12 | 13 |
| Total | 23 | 35 |
Caustic Crossing? | 2018 | 16 | 17 |
Yes, No | 2019 | 13 | 12 |
| Total | 29 | 29 |
Image Perturbed? | 2018 | 17 | 14 | 2
Major, Minor, Central | 2019 | 9 | 12 | 4
| Total | 26 | 26 | 6

Figure 1: Data (color-coded by observatory) together with the predictions and
residuals for the two planetary models of KMT-2019-BLG-0298 specified in Table
2, plus the 1L2S model. The caustic topologies are shown in the insets for
both the close and wide geometries, but the wide geometry is decisively
favored by $\Delta\chi^{2}=19$. The 1L2S model is excluded at
$\Delta\chi^{2}=125$.  Figure 2: Scatter plot on the
${\mbox{\boldmath$\pi$}}_{\rm E}$ plane derived from the MCMC of the two
parallax models of KMT-2019-BLG-0298 presented in Table 2, color coded (red,
yellow, green, cyan, blue) for $\Delta\chi^{2}<(1,4,9,16,25)$. The contours
are effectively 1-D, with widths $\sigma(\pi_{{\rm
E},\parallel})\equiv\sigma_{\parallel}\sim 0.039$ and 0.046. The black
contours show the mean and covariances ($\Delta\chi^{2}=1$) that are used in
Section 5.1.  Figure 3: Data (color-coded by observatory) together with the
predictions and residuals for the three planetary models of KMT-2019-BLG-1216
specified in Table 3, plus the 1L2S model. The caustic topologies are shown in
the insets for the inner, outer, and off-axis models. The first two are
perfectly degenerate but the last is decisively disfavored at
$\Delta\chi^{2}=24$. The 1L2S model is excluded at $\Delta\chi^{2}=57$.
Figure 4: Scatter plot on the ${\mbox{\boldmath$\pi$}}_{\rm E}$ plane derived
from the MCMC of the four parallax models of KMT-2019-BLG-1216 presented in
Table 4, color coded (red, yellow, green, cyan, blue) for
$\Delta\chi^{2}<(1,4,9,16,25)$. The black contours show the mean and
covariances ($\Delta\chi^{2}=1$) that are used in Section 5.2.  Figure 5: Data
(color-coded by observatory) together with the prediction and residuals for
the model of KMT-2019-BLG-2783 specified in Table 5. The caustic topology is
shown in the insets.  Figure 6: Data (color-coded by observatory) together
with the predictions and residuals for the two planetary models of
OGLE-2019-BLG-0249 specified in Table 7 with the caustic topologies shown as
insets. Also shown are the 1L1S and 1L2S models.  Figure 7: Scatter plot on
the ${\mbox{\boldmath$\pi$}}_{\rm E}$ plane derived from the MCMC of the four
parallax models of OGLE-2019-BLG-0249 presented in Table 7, color coded (red,
yellow, green, cyan, blue) for $\Delta\chi^{2}<(1,4,9,16,25)$. The contours
are effectively 1-D, with typical widths $\sigma_{\parallel}\sim 0.0080$ and
0.0065, for the close and wide solutions respectively. The black contours show
the mean and covariances that are used in Section 5.4.  Figure 8: Data (color-
coded by observatory) together with the prediction and residuals for the model
of OGLE-2019-BLG-0679 specified in Table 8. The caustic topology is shown in
the insets.  Figure 9: Scatter plot on the ${\mbox{\boldmath$\pi$}}_{\rm E}$
plane derived from the MCMC of the two parallax models of OGLE-2019-BLG-0679
presented in Table 8, color coded (red, yellow, green, cyan, blue) for
$\Delta\chi^{2}<(1,4,9,16,25)$. The black contours show the mean and
covariances that are used in Section 5.5.  Figure 10: Data (color-coded by
observatory) together with the predictions and residuals for the two planetary
models of OGLE-2019-BLG-0344 specified in Table 9 as well as the 1L2S model
specified in Table 11. All three models can explain the data equally well. In
addition, there are 6 other non-planetary 2L1S models, most of which also can
explain the data equally well. See Figure 11. Hence, this event cannot be
cataloged as “planetary”.  Figure 11: Data (color-coded by observatory)
together with the predictions and residuals for the three close-binary models
of OGLE-2019-BLG-0344 specified in Table 10, as well as the 1L2S model. Two of
the three close-binary models can explain the data about as well as the
planetary models. In addition, there are three wide-binary models that are not
shown. These viable binary-lens models provide an additional reason that this
event cannot be cataloged as “planetary”.  Figure 12: Data (color-coded by
observatory) together with the predictions and residuals for the two planetary
models of KMT-2019-BLG-0304 specified in Table 12, as well as the 1L2S model
specified in Table 13. All three models can explain the data equally well.
Hence, this event cannot be cataloged as “planetary”.  Figure 13: Color-
magnitude diagrams for 4 of the 7 events analyzed in this paper, each
identified by an abbreviation, e.g., KB190298 for KMT-2019-BLG-0298. The
centroid of the red clump and the lens position are always shown in red and
blue, respectively. Where relevant, the blended light is shown in green. When
there are multiple solutions, we show only the source and blend for the
lowest-$\chi^{2}$ solution.  Figure 14: Color-magnitude diagrams for the
remaining 3 events analyzed in this paper. See caption of Figure 13.  Figure
15: Envelope functions for $\chi^{2}$ versus $\rho$ for all 5 planetary events
in this paper.  Figure 16: Histograms of the host mass (left) and lens
distance (right) for 3 of the 5 unambiguously planetary events, as derived
from the Bayesian analysis. Disk (blue) and bulge (red) distributions are
shown separately, with their total shown in black.  Figure 17: Histograms of
the host mass (left) and lens distance (right) for the remaining 2
unambiguously planetary events, as derived from the Bayesian analysis. Disk
(blue) and bulge (red) distributions are shown separately, with their total
shown in black.  Figure 18: Comparison of the posterior distributions of
${\mbox{\boldmath$\mu$}}_{S}$ for OGLE-2019-BLG-0679, for four classes of
model (as indicated in the legend) to the Gaia DR2, Gaia DR3, and OGLE-IV
measurements of the same quantity. In particular, Gaia DR3 and OGLE-IV are
both in strong tension with all the other error ellipses, including Gaia DR2.
Figure 19: Scatter plot of $\log s^{\dagger}$ versus $\log q$ for 58
AnomalyFinder planets from 2018 and 2019. Here,
$s^{\dagger}_{\pm}=(\sqrt{u_{\rm anom}+4}\pm u_{\rm anom})/2$, $u_{\rm anom}$
is the lens-source separation normalized to $\theta_{\rm E}$ at the time of
the anomaly, and the “$\pm$” refers to major and minor image perturbations,
respectively. A majority of detections have $|\log s^{\dagger}|<0.05$ (inner
dashed lines), for which the same light-curve morphology can almost equally be
generated by positive or negative $\log s$, including with large absolute
values. Hence, there is no correspondence between the signs of $\log
s^{\dagger}$ and $\log s$ in this regime. By contrast, for $|s^{\dagger}|>0.2$
(outer dashed lines), the anomalies are generally associated with the
planetary caustics, so that $\log s^{\dagger}$ and $\log s$ have the same
sign. In this regime there is suggestive evidence (6 versus 2 detections) that
there is more overall senstivity to wide separation planets.  Figure 20:
6-dimensional scatter plot of 58 planets from 2018–2019. (1) Abscissa: log
mass ratio. (2) Ordinate: source magnitude of unperturbed event at time of
anomaly. The remaining four dimensions are shown in the legend and are the
same as in Figure 14 from Jung et al. (2022).