††thanks: Electronic address<EMAIL_ADDRESS>
# Quantum backaction effects in sequential measurements

Le Bin Ho  Frontier Research Institute for Interdisciplinary Sciences, Tohoku
University, Sendai 980-8578, Japan  Department of Applied Physics, Graduate
School of Engineering, Tohoku University, Sendai 980-8579, Japan

###### Abstract

Quantum backaction refers to the disturbance of a quantum system caused by
measuring it. In sequential measurements, this effect can accumulate and
become significant, leading to nontrivial modifications of the system state
and the measurement results. This paper explores the ways in which quantum
backaction can manifest in sequential measurements, including the role of
measurement strength and the nature of the measurement process. The paper
highlights the insight quantum foundation and the implications for quantum
measurement and information processing.

##  I Introduction and preliminary definitions

Quantum mechanics predicts that measuring a quantum state results in its
collapse and reveals hidden properties of the measuring system Wheeler and
Zurek (2014); Bohm (1989). This fundamental aspect is known as the
“measurement problem,” implying the state changes through measurements. The
von Neumann mechanism explains the changing state via quantum backaction (QBA)
caused by a quantum probe interacts with the measuring system von Neumann
(2018). This effect is evident in sequential measurements, where later
measurements are disturbed by the QBA of previous ones Busch  _et al._ (1990);
Gudder and Nagy (2001). This causal process was widely explored in recent
works, including sequential weak measurements Mitchison  _et al._ (2007);
Piacentini  _et al._ (2016); Avella  _et al._ (2017); Brodutch and Cohen
(2017); Ma  _et al._ (2018); Kim  _et al._ (2018); Chen  _et al._ (2019);
Pfender  _et al._ (2019), reduction of QBA effects Wiseman (1995); Wu  _et
al._ (2019, 2020); Shomroni  _et al._ (2019); Møller  _et al._ (2017); Tsang
and Caves (2012), and manipulation of qubits Blok  _et al._ (2014). Other
works also include violating noncontextual realism Larsson  _et al._ (2011)
and extracting maximal knowledge Nagali  _et al._ (2012) through sequential
measurements, which are noteworthy.

In some cases, later measurements can noncasually affect the system’s
behaviors in the past. For example, in a delayed-choice quantum eraser
experiment, a measurement made after photon detection can affect its past
behavior Kim  _et al._ (2000). Another vital example is postselection
measurements, where the measurement results at present are affected by future
postselections Aharonov  _et al._ (1964, 1988). Yet, studying the QBA caused
by subsequent measurements is still limited, e.g., the quantum trajectory in
continuous measurements Zhang and Mølmer (2017); Tan  _et al._ (2015); Ho
(2019); Campagne-Ibarcq  _et al._ (2014), quantum foundation Thompson  _et
al._ (2018); Guevara and Wiseman (2015); Chantasri  _et al._ (2013),
uncertainty relations Bao  _et al._ (2020a); Khanahmadi and Mølmer (2021), and
quantum measurements Bao  _et al._ (2020b); Chantasri  _et al._ (2021).
Remarkably, Tsang et al. observed sub-Heisenberg uncertainty via past
measurements Tsang  _et al._ (2009); Tsang (2009). Similarly, Bao et al.
experimentally observed the beyond Heisenberg uncertainty by using past
quantum states of preparation and postselection Bao  _et al._ (2020a). These
works push for scrutinizing the QBA in sequential measurements.

This work examines QBA effects in sequential measurements of two observables
$\bm{A}$ and $\bm{B}$ in a given system, where $\bm{B}$ is measured after
$\bm{A}$. The system is first coupled with a quantum probe $\mathcal{P}_{1}$
to measure $\bm{A}$, then with a probe $\mathcal{P}_{2}$ to measure $\bm{B}$.
The probes initialize as Gaussian states with zero mean and are measured on
continuous variables, e.g., $\bm{x}_{1,2}$. We discuss the QBA caused by the
first measurement affecting the second via the change in variance upon
measurement of $\bm{B}$, and the QBA caused by the second measurement
affecting the first via the change in variance of $\bm{A}$. We conduct
sequential measurements in two models: joint measurement and conditional
measurement (refer to Fig. 1). In the joint measurement model, the measuring
system couples to two quantum probes, and the joint state is measured
afterward. We find that only the first measurement causally affects the second
one in this model. In the conditional measurement model, measurements are
conditioned on each other, presenting QBA effects for both observables,
including causal and noncausal disturbances. We demonstrate these observations
using a spin system, and uncover a lower bound for the stronger uncertainty
relation proposed by Maccone and Pati Maccone and Pati (2014), enabling high-
precision measurements. We finally present a general framework for sequential
$N$ measurements.

QBA effects impede certain quantum technologies like quantum computing and
metrology. Understanding and controlling QBA in sequential measurements could
improve the quantum measurement foundation, increase precision, and offer
noise mitigation techniques.

The paper is structured as follows. Section II presents the sequential
measurements framework, including the joint and conditional measurement
models. Illustrative results are shown in Section III, and the general
framework for $N$-sequential measurements is provided in Section IV. The paper
is summarized in Section V.

##  II Sequential measurements

Figure 1:  Schematic of sequential measurements: (a) joint measurement model,
(b) conditional measurement model. System $\mathcal{S}$ is initially prepared
in $\rho_{0}$. The measurement of $\bm{A}$ in the system is done via a set of
measurement operators $\bm{M}_{m}$ in the first quantum probe
$\mathcal{P}_{1}$. The initial state of the probe is $\psi_{1}(x)$, and it
transforms to $\psi_{1}(x-a)$ after the interaction (figure c), which imprints
the information about the eigenvalue $a$ of the operator $\bm{A}$. As a QBA
effect, the interaction disturbs system $\mathcal{S}$ and transforms the
system state to $\rho_{1}$ in the joint measurement model (a) or conditional
$\bar{\rho}_{1}$ in the conditional measurement model (b). Likewise, the
measurement of $\bm{B}$ is taken in the same manner. It is given by a set of
measurement operators $\bm{N}_{n}$, where the initial state of the second
probe $\psi_{2}(x)$ transforms to $\psi_{2}(x-b)$, which corresponds to the
eigenvalue $b$ of operator $\bm{B}$. In the conditional measurement model, we
only keep the results when the outcomes $m=x_{1}$ and $n=x_{2}$, while the
others are discarded.

For a given system $\mathcal{S}$, we consider a measurement of $\bm{A}$
followed by an incompatible measurement of $\bm{B}$, where $\bm{A}$ and
$\bm{B}$ are selfadjoint operators. These measurements are performed by
coupling $\mathcal{S}$ with two quantum probes $\mathcal{P}_{1}$ and
$\mathcal{P}_{2}$, respectively. The measurement scheme is schematically shown
in Fig. 1. Concretely, let us consider an atom-light interaction scheme
Hammerer  _et al._ (2010), where $\mathcal{S}$ is an atomic ensemble that
interacts with light (quantum probes). The interactions are governed by

$\displaystyle\bm{U}_{1}=e^{-i\bm{A}\otimes\bm{p}_{1}},\text{ and
}\,\bm{U}_{2}=e^{-i\bm{B}\otimes\bm{p}_{2}},$ (1)

where $\bm{p}_{1}$ and $\bm{p}_{2}$ are canonical momentum operators of
$\mathcal{P}_{1}$ and $\mathcal{P}_{2}$, respectively. The initial state for
each probe is described by a Gaussian wave function with a mean of zero, such
as $|\psi_{i}(x)\rangle=\int\psi_{i}(x)|x\rangle\ {\rm d}x$, where

$\displaystyle\psi_{i}(x)=\Bigl{(}\dfrac{1}{2\pi\sigma_{i}^{2}}\Bigr{)}^{1/4}\exp\Bigl{(}-\frac{x^{2}}{4\sigma_{i}^{2}}\Bigr{)},$
(2)

is the quadrature distribution of the input probe state, $i=1,2$ correspond to
$\mathcal{P}_{1}$ and $\mathcal{P}_{2}$, respectively, $|x\rangle$ is a
continuous eigenstate of the canonical position operator $\bm{x}$, and
$\sigma_{i}$ is the width of the wave function, which stands for the
measurement strength: $\sigma_{i}\to 0$ is a strong measurement, while
$\sigma_{i}\to\infty$ is a weak measurement. Such interaction models were
wildely implemented in atom-light interaction concepts Hammerer  _et al._
(2010); Julsgaard  _et al._ (2003); Colangelo  _et al._ (2017); Jasperse  _et
al._ (2017); Atatüre  _et al._ (2007); Vasilyev  _et al._ (2012); Robb  _et
al._ (2007); Bao  _et al._ (2020a); Muñoz Arias  _et al._ (2020); Madsen and
Mølmer (2004); Liu  _et al._ (2009); Ho and Edamatsu (2022), and extensively
applied for simultaneous measurement of spin angle and amplitude Colangelo
_et al._ (2017), beyond Heisenberg uncertainty Bao  _et al._ (2020a), and
error-disturbance uncertainty in Faraday measurement Ho and Edamatsu (2022).

Below, we examine the QBA effects by analyzing the variances of measured
operators in a joint measurement model and a conditional measurement model.

###  II.1 Joint measurement model

We first consider the joint measurement approach shown in Fig. 1 (a). Assume
the initial joint state of
$\mathcal{S}\otimes\mathcal{P}_{1}\otimes\mathcal{P}_{2}$ is

$\displaystyle\rho_{\rm
in}=\rho_{0}\otimes|\psi_{1}(x)\rangle\langle\psi_{1}(x)|\otimes|\psi_{2}(x)\rangle\langle\psi_{2}(x)|,$
(3)

where $\rho_{0}$ is the initial system state. The joint state transforms to
$\rho_{\rm fin}=\bm{U}_{2}\bm{U}_{1}\rho_{\rm
in}\bm{U}_{1}^{\dagger}\bm{U}_{2}^{\dagger}$. Inserting identity operators
$\bm{I}_{1}=\int|m\rangle\langle m|{\rm d}m$ and
$\bm{I}_{2}=\int|n\rangle\langle n|{\rm d}n$ into the probes, we obtain

$\displaystyle\rho_{\rm
fin}=\int\bm{N}_{n}\bm{M}_{m}\rho_{0}\bm{M}^{\dagger}_{m^{\prime}}\bm{N}_{n^{\prime}}^{\dagger}\otimes|m\rangle\langle
m^{\prime}|\otimes|n\rangle\langle n^{\prime}|\ {\rm d}^{4}\bm{t},$ (4)

where $\bm{t}$ represents a 4-tuple $(m,m^{\prime},n,n^{\prime})$ and ${\rm
d}^{4}\bm{t}$ is a 4-dimensional volume differential (see App. A). The
measurement operators $\bm{M}$ and $\bm{N}$ for the first and second
measurements are given by

$\displaystyle\bm{M}_{m}$ $\displaystyle=\int\psi_{1}(m-a)|a\rangle\langle
a|{\rm d}a,$ $\displaystyle\bm{N}_{n}$
$\displaystyle=\int\psi_{2}(n-b)|b\rangle\langle b|{\rm d}b,$ (5)

where $|a\rangle$ is an eigenstate of $\bm{A}$ per an eigenvalue $a$, and
similar for $\bm{B}$. These measurement operators satisfy the completeness
equations $\int\bm{M}_{m}^{\dagger}\bm{M}_{m}{\rm d}m=\bm{I}$ and
$\int\bm{N}_{n}^{\dagger}\bm{N}_{n}{\rm d}n=\bm{I}$. The final state of the
two probes is given by $\eta={\rm Tr}_{\mathcal{S}}[\rho_{\rm fin}]$, where
Tr${}_{\mathcal{S}}[*]$ is a partial trace w.r.t. system $\mathcal{S}$. It
explicitly yields (see App. B)

$\displaystyle\eta=\iiint\langle b|a\rangle\langle
a|\rho_{0}|a^{\prime}\rangle\langle
a^{\prime}|b\rangle\times|\psi_{1}(x-a)\rangle\langle\psi_{1}(x-a^{\prime})|\otimes|\psi_{2}(x-b)\rangle\langle\psi_{2}(x-b)|\
{\rm d}a{\rm d}a^{\prime}{\rm d}b.$ (6)

After the interaction, we measure the position operator $\bm{x}_{1}$ in
pointer $\mathcal{P}_{1}$ and get the expectation value as

$\displaystyle\langle\bm{x}_{1}\otimes\bm{I}_{2}\rangle={\rm
Tr}_{\mathcal{P}_{1}\mathcal{P}_{2}}\bigl{[}(\bm{x}_{1}\otimes\bm{I}_{2})\eta\bigr{]}={\rm
Tr}_{\mathcal{S}}[\bm{A}\rho_{0}].$ (7)

The variance of observed values upon measurement of $\bm{x}_{1}$ is given by

$\displaystyle{\rm Var}(\bm{x}_{1})_{\eta}$
$\displaystyle\equiv\langle(\bm{x}_{1}\otimes\bm{I}_{2})^{2}\rangle-\langle\bm{x}_{1}\otimes\bm{I}_{2}\rangle^{2}$
$\displaystyle=\sigma_{1}^{2}+{\rm Var}(\bm{A})_{\rho_{0}},$ (8)

which is the spread of the observed values upon measurement of $\bm{x}_{1}$
concerning $\eta$ (shorthanded as the variance of $\bm{x}_{1}$). Here, ${\rm
Var}(\bm{A})_{\rho_{0}}={\rm Tr}_{\mathcal{S}}[\bm{A}^{2}\rho_{0}]-({\rm
Tr}_{\mathcal{S}}[\bm{A}\rho_{0}])^{2}$ is the variance of $\bm{A}$ concerning
$\rho_{0}$ (see App. C.1). Experimentally, one can infer ${\rm
Var}(\bm{A})_{\rho_{0}}={\rm Var}(\bm{x}_{1})_{\eta}-\sigma_{1}^{2}$, where
$\sigma_{1}^{2}$ stands for shot noise fluctuations Bao  _et al._ (2020a).

The QBA effect caused by the first measurement can be informed by the change
in variance of the second measurement. To evaluate this QBA, we examine the
variance of $\bm{x}_{2}$ in the second measurement as follows

$\displaystyle{\rm Var}(\bm{x}_{2})_{\eta}$
$\displaystyle\equiv\bigl{\langle}(\bm{I}_{1}\otimes\bm{x}_{2})^{2}\bigr{\rangle}-\bigl{\langle}\bm{I}_{1}\otimes\bm{x}_{2}\bigr{\rangle}^{2},$
$\displaystyle=\sigma_{2}^{2}+{\rm Var}(\bm{B})_{\rho_{1}},$ (9)

where the expectation value is $\langle\bm{I}_{1}\otimes\bm{x}_{2}\rangle={\rm
Tr}_{\mathcal{P}_{1}\mathcal{P}_{2}}[(\bm{I}_{1}\otimes\bm{x}_{2})\eta]$, and

$\displaystyle{\rm Var}(\bm{B})_{\rho_{1}}={\rm
Tr}_{\mathcal{S}}[\bm{B}^{2}\rho_{1}]-({\rm
Tr}_{\mathcal{S}}[\bm{B}\rho_{1}])^{2},$ (10)

is the variance of $\bm{B}$ concerning $\rho_{1}$. Here, $\rho_{1}={\rm
Tr}_{\mathcal{P}_{1}}[\bm{U}_{1}(\rho_{0}\otimes|\psi_{1}(x)\rangle\langle\psi_{1}(x)|)\bm{U}_{1}^{\dagger}]$
is the system state after interacting with $\mathcal{P}_{1}$.
Straightforwardly, it yields (see App. C.2)

$\displaystyle\rho_{1}=\iint|a\rangle\langle
a|\rho_{0}|a^{\prime}\rangle\langle
a^{\prime}|e^{-\frac{(a-a^{\prime})^{2}}{8\sigma_{1}^{2}}}{\rm d}a{\rm
d}a^{\prime}.$ (11)

Obviously, if the initial state $\rho_{0}$ is an eigenstate of $\bm{A}$, i.e.,
$\rho_{0}=|a_{i}\rangle\langle a_{i}|$, then $\rho_{1}=\rho_{0}$, meaning that
system $\mathcal{S}$ is not disturbed after the first measurement Wu  _et al._
(2020). In general, however, the system is disturbed through the first
measurement, and thus, $\rho_{1}\neq\rho_{0}$. Hence, Eq. (10) explicitly
gives

$\displaystyle{\rm Var}(\bm{B})_{\rho_{1}}=\int b^{2}\langle
b|\rho_{1}|b\rangle{\rm d}b-\Bigg{(}\int b\langle b|\rho_{1}|b\rangle{\rm
d}b\Bigg{)}^{2},$ (12)

which significantly depends on the first measurement (via $\sigma_{1}$) for
all $a\neq a^{\prime}$.

In general, ${\rm Var}(\bm{B})_{\rho_{1}}\neq{\rm Var}(\bm{B})_{\rho_{0}}$ due
to the QBA caused by the first measurement, where ${\rm
Var}(\bm{B})_{\rho_{0}}={\rm Tr}_{\mathcal{S}}[\bm{B}^{2}\rho_{0}]-({\rm
Tr}_{\mathcal{S}}[\bm{B}\rho_{0}])^{2}$ is the variance of $\bm{B}$ concerning
$\rho_{0}$. The dependence of ${\rm Var}(\bm{B})_{\rho_{1}}$ on the first
measurement illustrates an in situ causal effect, where later measurement
results are affected by previous measurements.

When the first measurement is weak, i.e., $\sigma_{1}\to\infty$, Eq. (II.1)
simplifies to ${\rm Var}(\bm{x}_{2})_{\eta}=\sigma_{2}^{2}+{\rm
Var}(\bm{B})_{\rho_{0}}$, which is not affected by the first measurement. This
means the QBA is eliminated Šindelka and Moiseyev (2018); Abbott  _et al._
(2019). However, the cost is that the fluctuation of the first measurement
will increase, i.e., ${\rm Var}(\bm{x}_{1})_{\eta}$ is large Šindelka and
Moiseyev (2018); Aharonov and Vaidman (2009). Discussing this increasing
fluctuation is out of the scope, where we instead focus on the mutual
influence in sequential measurements.

###  II.2 Conditional measurement model

In this approach, assume that we first measure $\bm{x}_{1}$ in
$\mathcal{P}_{1}$ and get an outcome $x_{1}$. After the first measurement, the
initial system state $\rho_{0}$ transforms into an unnormalized state

$\displaystyle\bar{\rho}_{1}(x_{1})$
$\displaystyle=\bm{M}_{x_{1}}\rho_{0}\bm{M}^{\dagger}_{x_{1}}$
$\displaystyle=\iint|a\rangle\langle a|\rho_{0}|a^{\prime}\rangle\langle
a^{\prime}|\psi_{1}(x_{1}-a)\psi_{1}^{*}(x_{1}-a^{\prime}){\rm d}a{\rm
d}a^{\prime}.$ (13)

Here, we used $\bar{\rho}_{1}(x_{1})$ to specify that the system state is
given according to the outcome $x_{1}$ of the first measurement (see, for
example, Chap. II in Ref. Nielsen and Chuang (2010)). After the measurement of
$\bm{x}_{2}$ with an outcome $x_{2}$, the system state becomes
$\bar{\rho}_{2}(x_{1},x_{2})\propto\bm{N}_{x_{2}}\bar{\rho}_{1}(x_{1})\bm{N}^{\dagger}_{x_{2}}$.
For convenience, we introduce an effective matrix
$\bm{E}(x_{2})=\bm{N}^{\dagger}_{x_{2}}\bm{N}_{x_{2}}$, which is the past
quantum state Gammelmark  _et al._ (2013).

We examine the QBA effect caused by the first measurement, i.e., the outcome
of the second measurement is affected by the first measurement. We consider
the conditional probability for obtaining the outcome $\bm{x}_{2}=x_{2}$
conditioned on $\bm{x}_{1}=x_{1}$ as (see App. D)

$\displaystyle p(x_{2}|x_{1})$ $\displaystyle=$
$\displaystyle=\dfrac{\bigintssss\big{|}\psi_{2}(x_{2}-b)\big{|}^{2}\langle
b|\bar{\rho}_{1}(x_{1})|b\rangle\ {\rm
d}b}{\bigintssss\Bigl{[}\bigintssss\big{|}\psi_{2}(x^{\prime}_{2}-b)|^{2}\langle
b|\bar{\rho}_{1}(x_{1})|b\rangle\ {\rm d}b\Bigr{]}{\rm d}x^{\prime}_{2}},$
(14)

where $p(x_{2}|x_{1})$ is the shorthand of
$p(\bm{x}_{2}\\!\\!=\\!x_{2}|\bm{x}_{1}\\!\\!=\\!x_{1})$. Then, the
expectation value of $\bm{x}_{2}$ conditioned on the outcome
$\bm{x}_{1}=x_{1}$ reads

$\displaystyle E(\bm{x}_{2}|\bm{x}_{1}\\!\\!=\\!x_{1})=\int x_{2}\
p(x_{2}|x_{1}){\rm d}x_{2},$ (15)

where ‘$E$’ stands for $\langle\cdots\rangle$ as an expectation value.
Generally, this expectation value is affected by the QBA caused by the first
measurement. The conditional variance of $\bm{x}_{2}$ gives

$\displaystyle{\rm Var}(\bm{x}_{2}|\bm{x}_{1}\\!\\!=\\!x_{1})$
$\displaystyle=E(\bm{x}_{2}^{2}|\bm{x}_{1}\\!\\!=\\!x_{1})-[E(\bm{x}_{2}|\bm{x}_{1}\\!\\!=\\!x_{1})]^{2}$
$\displaystyle=\sigma_{2}^{2}+{\rm Var}(\bm{B}|\bm{A}),$ (16)

where ${\rm Var}(\bm{B}|\bm{A})$ is the variance of $\bm{B}$ conditioned on
the presence of $\bm{A}$. In indirect measurements, this variance is informed
by Eq. (II.2) as

$\displaystyle{\rm Var}(\bm{B}|\bm{A})={\rm
Var}(\bm{x}_{2}|\bm{x}_{1}\\!\\!=\\!x_{1})-\sigma_{2}^{2}.$ (17)

Generally, it depends on the strength of the first and second measurements.
The QBA effect appears when ${\rm Var}(\bm{B}|\bm{A})\neq{\rm
Var}(\bm{B})_{\rho_{0}}$.

Likewise, the second measurement can noncausally affect back the first
measurement. To evaluate the noncausal backaction, we consider the conditional
probability for obtaining $\bm{x}_{1}=x_{1}$ conditioned on $\bm{x}_{2}=x_{2}$

$\displaystyle p(x_{1}|x_{2})$ $\displaystyle=\dfrac{{\rm
Tr}\bigl{[}\bar{\rho}_{1}(x_{1})\bm{E}(x_{2})\bigr{]}}{\bigintssss{\rm
Tr}\bigl{[}\bar{\rho}_{1}(x^{\prime}_{1})\bm{E}(x_{2})\bigr{]}{\rm
d}x^{\prime}_{1}},$
$\displaystyle=\dfrac{\bigintssss\big{|}\psi_{2}(x_{2}-b)\big{|}^{2}\langle
b|\bar{\rho}_{1}(x_{1})|b\rangle\ {\rm
d}b}{\bigintssss\Bigl{[}\bigintssss\big{|}\psi_{2}(x_{2}-b)\big{|}^{2}\langle
b|\bar{\rho}_{1}(x^{\prime}_{1})|b\rangle\ {\rm d}b\Bigr{]}{\rm
d}x^{\prime}_{1}}.$ (18)

The conditional expectation value and conditional variance are given by
$E(\bm{x}_{1}|\bm{x}_{2}\\!\\!=\\!{x_{2}})=\int x_{1}p(x_{1}|x_{2}){\rm
d}x_{1}$ and ${\rm
Var}(\bm{x}_{1}|\bm{x}_{2}\\!\\!=\\!x_{2})=E(\bm{x}_{1}^{2}|\bm{x}_{2}\\!\\!=\\!{x_{2}})-\bigl{[}E(\bm{x}_{1}|\bm{x}_{2}\\!\\!=\\!{x_{2}})]^{2}$,
respectively. Similarly, the variance of $\bm{A}$ conditioned on $\bm{B}$ can
be informed from

$\displaystyle{\rm Var}(\bm{A}|\bm{B})={\rm
Var}(\bm{x}_{1}|\bm{x}_{2}\\!\\!=\\!x_{2})-\sigma_{1}^{2}.$ (19)

Here, ${\rm Var}(\bm{A}|\bm{B})\neq{\rm Var}(\bm{A})_{\rho_{0}}$ is the result
of the noncausal QBA effect caused by the second measurement. Equations (17,
19) explicitly show the impacts of the first measurement on the second
measurement and vice versa, respectively.

Intuitively, the conditional state $\bar{\rho}_{1}$ interferes with the
effective matrix $\bm{E}$ in the same way as the interference of pre- and
post-selection states, which was experimentally observed in the resonance
fluorescence Campagne-Ibarcq  _et al._ (2014). The interference here relies on
the conditional outcomes $x_{1},x_{2}$ (like postselection), allowing
measurements of $\bm{A}$ and $\bm{B}$ to affect each other. This is a purely
quantum feature of the reciprocal past-future information Chantasri  _et al._
(2021). The interference can be observed through continuous monitoring of the
system under different observables. Zhang and Mølmer (2017); Campagne-Ibarcq
_et al._ (2014); Ho (2019).

###  II.3 Conditional stronger uncertainty relations

It is unfeasible to get precise results in incompatible measurements, whether
they are simultaneous or sequential Robertson (1929); Arthurs and Goodman
(1988); Šindelka and Moiseyev (2018). The Heisenberg-Robertson uncertainty
relation for two incompatible observables $\bm{A}$ and $\bm{B}$ can be given
in terms of the commutator ${\rm Var}(\bm{A}){\rm
Var}(\bm{B})\geq\bigl{|}\frac{1}{2}\langle[\bm{A},\bm{B}]\rangle\bigr{|}^{2}$,
which is the very first investment on this issue Robertson (1929); Arthurs and
Goodman (1988). However, the lower bound in this relation is trivial, such as
it can reach zero when the system state is an eigenstate of either $\bm{A}$ or
$\bm{B}$. Maccone and Pati later proposed a stronger (nontrivial) uncertainty
relation in terms of the summation of variances Maccone and Pati (2014)

$\displaystyle{\rm Var}(\bm{A})+{\rm Var}(\bm{B})\geq\max(\mathcal{R}_{\rm
a},\mathcal{R}_{\rm b}),$ (20)

which we name as Maccone-Pati uncertainty relation (MPUR), where

$\displaystyle\mathcal{R}_{\rm a}$ $\displaystyle=\pm
i\langle[\bm{A},\bm{B}]\rangle+\bigl{|}\langle\psi|\bm{A}\pm
i\bm{B}|\psi^{\perp}\rangle\bigr{|}^{2},$ $\displaystyle\mathcal{R}_{\rm b}$
$\displaystyle=\dfrac{1}{2}\bigl{|}\langle\psi^{\perp}_{\bm{A}+\bm{B}}|\bm{A}+\bm{B}|\psi\rangle\bigr{|}^{2}.$
(21)

Here, $|\psi^{\perp}\rangle$ is an arbitrary orthogonal state and
$|\psi^{\perp}_{\bm{A}+\bm{B}}\rangle\propto(\bm{A}+\bm{B}-\langle\bm{A}+\bm{B}\rangle)|\psi\rangle$.
The sign should be chosen so that $\pm i\langle[\bm{A},\bm{B}]\rangle$ is
positive. The MPUR is nontrivial, i.e., nonzero lower bound. Such an MPUR has
been experimentally verified in Ref. Wang  _et al._ (2016).

In this framework, we examine the MPUR in terms of conditional variances,
where the summation ${\rm Var}(\bm{A})+{\rm Var}(\bm{B})$ is replaced by ${\rm
Var}(\bm{A}|\bm{B})+{\rm Var}(\bm{B}|\bm{A})$. In the following section, we
observe that ${\rm Var}(\bm{A}|\bm{B})+{\rm
Var}(\bm{B}|\bm{A})<\max(\mathcal{R}_{\rm a},\mathcal{R}_{\rm b})$. This
observation is a result of the conditional measurement, which analogizes to
the postselection measurement Abbott  _et al._ (2019); Bao  _et al._ (2020a).
We emphasize that this result does not contradict Eq. 20 since it only applies
to conditional cases. Significantly, it allows for getting higher measurements
precision and minimizing the trade-off between ${\rm Var}(\bm{A}|\bm{B})$ and
${\rm Var}(\bm{B}|\bm{A})$. The result is thus significant for minimizing QBA
effects in quantum measurements.

Figure 2:  The variance of $\bm{S}_{x}$ after the first measurement, ${\rm
Var}(\bm{S}_{x})_{\rho_{1}}$. The measurement of $\bm{S}_{x}$ is affected by
the first measurement via $\sigma_{1}$. When the first measurement is strong,
i.e., $\sigma_{1}\to 0$, the variance ${\rm Var}(\bm{S}_{x})_{\rho_{1}}$ is
large, indicating a large disturbance. When the first measurement is weak,
i.e., large $\sigma_{1}$, ${\rm Var}(\bm{S}_{x})_{\rho_{1}}$ reduces to zero
and equals to ${\rm Var}(\bm{S}_{x})_{\rho_{0}}$, which means that the system
is not disturbed. In the strong-to-weak transition regime, ${\rm
Var}(\bm{S}_{x})_{\rho_{1}}$ reduces from 1/4 to 0, resulting in a decreased
disturbance.  Figure 3:  (a) The conditional variance ${\rm
Var}(\bm{S}_{x}|\bm{S}_{z})$ in Eq. (30) as a function of $x_{1}$ and
$\sigma_{1}$. When ${\rm Var}(\bm{S}_{x}|\bm{S}_{z})={\rm
Var}(\bm{S}_{x})_{\rho_{0}}=0$, there is no disturbance for the system, such
as when $x_{1}=0$. For small $\sigma_{1}$ and $x_{1}\neq 0$, ${\rm
Var}(\bm{S}_{x}|\bm{S}_{z})$ reaches 1/4, the maximum disturbance resulting
from the QBA. For large $\sigma_{1}$, ${\rm Var}(\bm{S}_{x}|\bm{S}_{z})$
reduces to zero due to the weak measurement. (b) Plot of ${\rm
Var}(\bm{S}_{x}|\bm{S}_{z})$ as a function of $x_{1}$ for
$\sigma_{1}\in\\{0.01,0.2,0.5,1.0\\}$. (c) Plot of ${\rm
Var}(\bm{S}_{x}|\bm{S}_{z})$ as a function of $\sigma_{1}$ for
$x_{1}\in\\{0.0,0.2,0.5,1.0\\}$.

##  III Illustration in a spin system

We consider the sequential measurements of a spin system using an atom-light
interaction scheme Hammerer  _et al._ (2010); Julsgaard  _et al._ (2003);
Colangelo  _et al._ (2017); Jasperse  _et al._ (2017); Atatüre  _et al._
(2007); Vasilyev  _et al._ (2012); Robb  _et al._ (2007); Bao  _et al._
(2020a); Muñoz Arias  _et al._ (2020); Madsen and Mølmer (2004); Liu  _et al._
(2009); Ho and Edamatsu (2022). To simplify, we consider the system to be a
single spin-1/2 and choose $\bm{A}=\bm{S}_{z}=\bm{\sigma}_{z}/2$ and
$\bm{B}=\bm{S}_{x}=\bm{\sigma}_{x}/2$, where $\bm{\sigma}_{i},(i=x,y,z)$ is a
Pauli matrix. The initial system state is $\rho_{0}=|+\rangle\langle+|$, where
$|\pm\rangle=(|\uparrow\rangle\pm|\downarrow\rangle)/\sqrt{2}$, with
$|\uparrow\rangle$ and $|\downarrow\rangle$ are the eigenstates of
$\bm{S}_{z}$. This choice gives ${\rm Var}(\bm{S}_{z})_{\rho_{0}}=1/4$ and
${\rm Var}(\bm{S}_{x})_{\rho_{0}}=0$. Additionally, we expand operators
$\bm{S}_{z}$ and $\bm{S}_{x}$ into their eigenvalues and eigenstates as
follows

$\displaystyle\bm{S}_{z}$
$\displaystyle\equiv\sum_{i}a_{i}|a_{i}\rangle\langle
a_{i}|=\frac{1}{2}|\uparrow\rangle\langle\uparrow|-\frac{1}{2}|\downarrow\rangle\langle\downarrow|),$
(22) $\displaystyle\bm{S}_{x}$
$\displaystyle\equiv\sum_{i}b_{i}|b_{i}\rangle\langle
b_{i}|=\frac{1}{2}|+\rangle\langle+|-\frac{1}{2}|-\rangle\langle-|).$ (23)

Since these spectrums are discrete, we replace the integrals (w.r.t d$a$ and
d$b$) with the summation over the discrete eigenstates.

###  III.1 Joint measurement model

First, we calculate $\rho_{1}$ in Eq. (11):

$\displaystyle\rho_{1}$
$\displaystyle=\sum_{i,j\in\\{\uparrow,\downarrow\\}}|a_{i}\rangle\langle
a_{i}|+\rangle\langle+|a_{j}\rangle\langle
a_{j}|e^{-\frac{(a_{i}-a_{j})^{2}}{8\sigma_{1}^{2}}}$
$\displaystyle=\dfrac{1}{2}\begin{pmatrix}1&e^{-\frac{1}{8\sigma_{1}^{2}}}\\\
e^{-\frac{1}{8\sigma_{1}^{2}}}&1\end{pmatrix}.$ (24)

Substituting Eq. (III.1) into Eq. (12), we obtain

$\displaystyle{\rm Var}(\bm{S}_{x})_{\rho_{1}}$
$\displaystyle=\sum_{i\in\\{+,-\\}}b_{i}^{2}\langle
b_{i}|\rho_{1}|b_{i}\rangle-\Bigl{[}\sum_{i\in\\{+,-\\}}b_{i}\langle
b_{i}|\rho_{1}|b_{i}\rangle\Bigr{]}^{2}$
$\displaystyle=\dfrac{1}{4}\Bigl{(}1-e^{-\frac{1}{4\sigma_{1}^{2}}}\Bigr{)}.$
(25)

In Fig. 2, we plot ${\rm Var}(\bm{S}_{x})_{\rho_{1}}$ versus the strength of
the first measurement, $\sigma_{1}$. For strong measurement ($\sigma_{1}\ll
1$), ${\rm Var}(\bm{S}_{x})_{\rho_{1}}$ reaches the maximum of 1/4, indicating
that the system is maximum disturbed by the first measurement. For weak
measurement ($\sigma_{1}\to\infty$), ${\rm Var}(\bm{S}_{x})_{\rho_{1}}$
decreases and approaches 0, which means that the first measurement has minimal
impact on the second one. During the strong-to-weak transition regime
Piacentini  _et al._ (2018); Pan  _et al._ (2020), ${\rm
Var}(\bm{S}_{x})_{\rho_{1}}$ reduces from 1/4 to 0, resulting in a decrease in
QBA from strong to weak interactions.

###  III.2 Conditional measurement model

First, let us calculate $\bar{\rho}_{1}(x_{1})$ in Eq. (II.2):

$\displaystyle\bar{\rho}_{1}(x_{1})$
$\displaystyle=\sum_{i,j\in\\{\uparrow,\downarrow\\}}|a_{i}\rangle\langle
a_{i}|\rho_{0}|a_{j}\rangle\langle
a_{j}|\psi_{1}(x_{1}-a_{i})\psi_{1}^{*}(x_{1}-a_{j})$
$\displaystyle=\dfrac{1}{2}\Bigl{(}\frac{1}{2\pi\sigma_{1}^{2}}\Bigr{)}^{1/2}e^{-\frac{x_{1}^{2}+1/4}{2\sigma_{1}^{2}}}\begin{pmatrix}e^{\frac{x_{1}}{2\sigma_{1}^{2}}}&1\\\
1&e^{-\frac{x_{1}}{2\sigma_{1}^{2}}}\end{pmatrix}.$ (26)

This state significantly depends on the outcome $x_{1}$ and the interaction
strength $\sigma_{1}$ of the first measurement. The conditional expectation
values in Eq. (15) yield

$\displaystyle E(\bm{x}_{2}|\bm{x}_{1}\\!\\!=\\!x_{1})$
$\displaystyle=\dfrac{{\rm exp}({\frac{x_{1}}{2\sigma_{1}^{2}}})}{1+{\rm
exp}({\frac{x_{1}}{\sigma_{1}^{2}}})},$ (27) $\displaystyle
E(\bm{x}_{2}^{2}|\bm{x}_{1}\\!\\!=\\!x_{1})$
$\displaystyle=\dfrac{1}{4}+\sigma_{2}^{2},$ (28)

and the conditional variance of $\bm{x}_{2}$ is given by

$\displaystyle{\rm
Var}(\bm{x}_{2}|\bm{x}_{1}\\!\\!=\\!x_{1})=\dfrac{1}{4}\tanh\Bigl{(}\dfrac{x_{1}}{2\sigma_{1}^{2}}\Bigr{)}^{2}+\sigma_{2}^{2}.$
(29)

Inversely, the variance of $\bm{S}_{x}$ conditioned on $\bm{S}_{z}$ gives

$\displaystyle{\rm
Var}(\bm{S}_{x}|\bm{S}_{z})=\dfrac{1}{4}\tanh\Bigl{(}\dfrac{x_{1}}{2\sigma_{1}^{2}}\Bigr{)}^{2}.$
(30)

We show the result in Fig. 3. When the first measurement is strong
($\sigma_{1}\to 0$), the conditional variance reaches the maximum of 1/4 for
$x_{1}\neq 0$, indicating that the system is maximally disturbed by the first
measurement due to the QBA. In other words, before the first measurement, the
variance ${\rm Var}(\bm{S}_{x})_{\rho_{0}}=0$; and after the measurement, it
reaches the maximum ${\rm Var}(\bm{S}_{x}|\bm{S}_{z})=1/4$. When the first
measurement is weak ($\sigma_{1}\to\infty$), it disturbs the system less, and
thus ${\rm Var}(\bm{S}_{x}|\bm{S}_{z})$ reduces to ${\rm
Var}(\bm{S}_{x})_{\rho_{0}}$. This observation is consistent with the joint
measurement approach above, where the first measurement affects the results of
the second.

Figure 4:  (a) The conditional variance ${\rm Var}(\bm{S}_{z}|\bm{S}_{x})$ in
Eq. (31) as a function of $x_{2}$ and $\sigma_{2}$ when $\sigma_{1}\to\infty$.
When ${\rm Var}(\bm{S}_{z}|\bm{S}_{x})={\rm Var}(\bm{S}_{z})_{\rho_{0}}=1/4$,
there is no disturbance for the system, such as when $x_{2}=0$. For small
$\sigma_{2}$ and $x_{2}<0$, then ${\rm Var}(\bm{S}_{z}|\bm{S}_{x})>1/4$, which
implies the QBA caused by the second measurement affecting the first. Here,
${\rm Var}(\bm{S}_{z}|\bm{S}_{x})\to\infty$ when $\sigma_{2}\to 0$, however,
we show the result up to ${\rm Var}(\bm{S}_{z}|\bm{S}_{x})=0.5$. For large
$\sigma_{2}$ ${\rm Var}(\bm{S}_{z}|\bm{S}_{x})$ towards to 1/4 as a result of
the weak interaction in the second measurement. Likewise, for $x_{2}>0$, ${\rm
Var}(\bm{S}_{z}|\bm{S}_{x})$ ranges from 1/8 to 1/4 when $\sigma_{2}$ goes
from strong to weak interaction, respectively. (b) Plot of ${\rm
Var}(\bm{S}_{z}|\bm{S}_{x})$ as a function of $x_{2}$ for
$\sigma_{2}\in\\{0.1,0.5,1.0,2.0\\}$. (c) Plot of ${\rm
Var}(\bm{S}_{z}|\bm{S}_{x})$ as a function of $\sigma_{2}$ for
$x_{2}\in\\{-1.0,-0.5,0.0,0.5,1.0\\}$.

Next, we examine the conditional variance of the first measurement
$\bm{S}_{z}$ conditioned on the presence of the second measurement
$\bm{S}_{x}$. It reads

$\displaystyle{\rm
Var}(\bm{S}_{z}|\bm{S}_{x})=\dfrac{1}{4}\dfrac{(s_{1}-1)s_{2}}{s_{1}s_{2}-2},$
(31)

where $s_{1}=1+\exp({\frac{1}{8\sigma_{1}^{2}}})$ and
$s_{2}=1+\exp({\frac{x_{2}}{\sigma_{2}^{2}}})$. There is an interplay between
$\sigma_{1}$ and $\sigma_{2}$. When $\sigma_{1}\to 0$ (the first measurement
is strong), then $s_{1}\to\infty$, it will dominate the result, and thus the
variance ${\rm Var}(\bm{S}_{z}|\bm{S}_{x})=1/4={\rm
Var}(\bm{S}_{z})_{\rho_{0}}$, regardless $s_{2}$. In this case, the second
measurement does not disturb the first because ${\rm
Var}(\bm{S}_{z}|\bm{S}_{x})={\rm Var}(\bm{S}_{z})_{\rho_{0}}$. Inversely, when
$\sigma_{1}\to\infty$ (the first measurement is weak), then $s_{1}\approx 2$,
and thus ${\rm
Var}(\bm{S}_{z}|\bm{S}_{x})\approx\frac{1}{8}\frac{s_{2}}{s_{2}-1}=\frac{1}{8}[1+{\rm
exp}({-\frac{x_{2}}{\sigma_{2}^{2}}})].$ In this case, the first measurement
is disturbed by the second via $x_{2}$ and $\sigma_{2}$. The results are shown
in Fig. 4. Unlike the previous case, the variance ${\rm
Var}(\bm{S}_{z}|\bm{S}_{x})$ is not symmetrical via the outcome $x_{2}$. When
$x_{2}=0$, ${\rm Var}(\bm{S}_{z}|\bm{S}_{x})$ reaches 1/4, indicating that the
first measurement is not affected by the second. When $x_{2}<0$, it will be
strongly disturbed, wherein $1/4<{\rm Var}(\bm{S}_{z}|\bm{S}_{x})<\infty$.
Here, ${\rm Var}(\bm{S}_{z}|\bm{S}_{x})\to\infty$ when $\sigma_{2}\to 0$ (we
show the result up to ${\rm Var}(\bm{S}_{z}|\bm{S}_{x})=0.5$). When $x_{2}>0$,
the variance is bound in the range $1/8\leq{\rm
Var}(\bm{S}_{z}|\bm{S}_{x})<1/4$ when $\sigma_{2}$ goes from strong to weak
interaction.

Finally, when both measurements are weak, such as sequential weak measurements
Mitchison  _et al._ (2007); Piacentini  _et al._ (2016); Avella  _et al._
(2017); Brodutch and Cohen (2017); Ma  _et al._ (2018); Kim  _et al._ (2018);
Chen  _et al._ (2019); Pfender  _et al._ (2019), the QBA effects are
eliminated, i.e., ${\rm Var}(\bm{S}_{x}|\bm{S}_{z})\to{\rm
Var}(\bm{S}_{x})_{\rho_{0}}$, and ${\rm Var}(\bm{S}_{z}|\bm{S}_{x})\to{\rm
Var}(\bm{S}_{z})_{\rho_{0}}$.

###  III.3 Conditonal stronger uncertainty relations

We study the MPUR in the spin model with conditional measurement. The initial
system state is $|\psi\rangle=|+\rangle$, an eigenstate of $\bm{S}_{x}$. We
thus choose
$|\psi^{\perp}\rangle=(\bm{S}_{z}-\langle\bm{S}_{z}\rangle)|+\rangle/\Delta(\bm{S}_{z})=|-\rangle$
Maccone and Pati (2014), and obtain
$|\psi^{\perp}_{\bm{S}_{z}+\bm{S}_{x}}\rangle=|-\rangle$. Then,
$\mathcal{R}_{\rm a}$ and $\mathcal{R}_{\rm b}$ in Eq. (II.3) yield

$\displaystyle\mathcal{R}_{\rm a}=\dfrac{1}{4},\text{and }\mathcal{R}_{\rm
b}=\dfrac{1}{8}.$ (32)

The MPUR for the system before the measurements reads

$\displaystyle{\rm Var}(\bm{S}_{z})_{\rho_{0}}+{\rm
Var}(\bm{S}_{x})_{\rho_{0}}=\dfrac{1}{4}=\max(\mathcal{R}_{\rm
a},\mathcal{R}_{\rm b}),$ (33)

in which reaches the lower bound in Eq. (20).

Next, we consider the MPUR for the conditional variances. Since ${\rm
Var}(\bm{S}_{z}|\bm{S}_{x})$ and ${\rm Var}(\bm{S}_{x}|\bm{S}_{z})$ can be
archived individually, we obtain the inequality ${\rm
Var}(\bm{S}_{z}|\bm{S}_{x})+{\rm Var}(\bm{S}_{x}|\bm{S}_{z})\geq\frac{1}{8}$.
The equality is attainable when ${\rm Var}(\bm{S}_{z}|\bm{S}_{x})=\frac{1}{8}$
and ${\rm Var}(\bm{S}_{x}|\bm{S}_{z})=0$, as previously discussed. This is the
lower bound for MPUR in the conditional sequential measurements, resulting in
enhanced precision measurements and reduced QBA effects.

##  IV $N$-sequential measurements

We extend the framework to $N$-sequential measurements. Let
$\bm{U}_{k}=e^{-i\bm{A}_{k}\otimes\bm{p}_{k}}$ be the evolution operator of
the $k^{\rm th}$ measurement, where $\bm{A}_{k}=\int a_{k}|a_{k}\rangle\langle
a_{k}|\ {\rm d}a_{k}$ is the system operator, and $\bm{p}_{k}$ is the $k^{\rm
th}$ probe operator. Let $\bm{\Omega}_{k}$ be the measurement operator of the
$k^{\rm th}$ measurement, we have

$\displaystyle\bm{\Omega}_{k}=\int\psi_{k}(x_{k}-a_{k})|a_{k}\rangle\langle
a_{k}|\ {\rm d}a_{k}.$ (34)

The system state after $k$ measurements is given by

$\displaystyle\bar{\rho}_{k}=\bm{\Omega}_{k}\bar{\rho}_{k-1}\bm{\Omega}^{\dagger}_{k},\
\text{ for }\ k=1,\cdots,N.$ (35)

The probability of obtaining outcome $x_{k}$ conditional on the rest of other
outcomes is given by

$\displaystyle p(x_{k}|\bar{x}_{k})$ $\displaystyle=\dfrac{{\rm
Tr}[\bar{\rho}_{k}\bm{E}]}{\int{\rm Tr}[\bar{\rho}_{k}\bm{E}]\ {\rm
d}x^{\prime}_{k}},$ (36)

where $\bar{x}_{k}=\\{x_{1},\cdots,x_{N}\\}\backslash x_{k}$ stands for the
rest of other outcomes, i.e., a set of all elements from $x_{1}$ to $x_{N}$
except $x_{k}$, and

$\displaystyle\bm{E}=\bm{\Omega}^{\dagger}_{k+1}\bm{\Omega}^{\dagger}_{k+2}\cdots\bm{\Omega}^{\dagger}_{N}\bm{\Omega}_{N}\bm{\Omega}_{N-1}\cdots\bm{\Omega}_{k+1}.$
(37)

See the proof in App. E. This result is similar to the one given in Ref. Bao
_et al._ (2020a). However, in Ref. Bao  _et al._ (2020a), the authors
approximate the Gaussian distribution of the conditional probability to deduce
the variance and use a hypothesis projective measurement at the $k^{\rm th}$
measurement. Here, we derive the exact variance from the expectation value of
the position operator $\bm{x}_{k}$. At first, the conditional expectation
value is given by

$\displaystyle E(\bm{x}_{k}|{\bar{x}_{k}})=\int x_{k}\
p(x_{k}|\bar{x}_{k}){\rm d}x_{k}.$ (38)

Then, the conditional variance is expressed as

$\displaystyle{\rm Var}(\bm{x}_{k}|\bar{x}_{k})$
$\displaystyle=E(\bm{x}_{k}^{2}|{\bar{x}_{k}})-[E(\bm{x}_{k}|{\bar{x}_{k}})]^{2}$
$\displaystyle=\sigma_{k}^{2}+{\rm Var}(\bm{A}_{k}\big{|}\bar{\bm{A}}_{k}),$
(39)

where
$\bar{\bm{A}}_{k}=\\{\bm{A}_{1},\cdots,\bm{A}_{N}\\}\backslash\bm{A}_{k}$. We
thus can extract the conditional variance ${\rm
Var}(\bm{A}_{k}|\bar{\bm{A}}_{k})$.

For example, in App. E, we provide a useful Mathematica code for calculating
the conditional variance ${\rm Var}(\bm{x}_{2}|x_{1},x_{3},x_{4})$ of
4-sequential measurements:
$\bm{S}_{z}\to\bm{S}_{x}\to\bm{S}_{x}\to\bm{S}_{z}$.

##  V Conclusion and outlook

We studied the impact of quantum backaction (QBA) on sequential measurements
of incompatible operators. We considered the indirect measurement of
observables $\bm{A}$ and $\bm{B}$ using two quantum probes $\mathcal{P}_{1}$
and $\mathcal{P}_{2}$ in the joint and conditional measurement models. In the
joint measurement, only the QBA caused by the first measurement is presented
due to the influence of former measurements on later ones. Otherwise, in the
conditional measurement approach, there is a mutual effect in sequential
measurements expressed through the QBAs of the first and second measurements.
We illustrated the results in a spin system, showing the dependence of
variance in subsequent measurements on prior measurements and vice versa. This
observation results from the interplay between the measurement strengths of
the first and second measurements. We further derived a new bound for the
stronger uncertainty relation with conditional measurements. The framework was
then extended to $N$-sequential measurements.

Finally, we remark that these findings can be demonstrated using current
technologies, such as trapped ions, macroscopic vapor cells, and nuclear
magnetic resonance. As we showed theoretically, the measurement operators
$\bm{M}_{m}$ and $\bm{N}_{n}$ can be implemented in the phase-space
representation of the driven field (e.g., light, magnetic fields, etc), and
the measurement results can be inferred from the position-shift of the final
state. Potential future research could include controlling and eliminating the
QBA effects, experimental verifying the stronger uncertainty relation.

##  Appendix A Final joint state in the joint measurement model

In this section, we derive the final state of sequential measurements in the
joint measurement model:

$\displaystyle\rho_{\rm fin}=\bm{U}_{2}\bm{U}_{1}\rho_{\rm
int}\bm{U}_{1}^{\dagger}\bm{U}_{2}^{\dagger}.$ (A.1)

Inserting $\bm{I}_{1}=\int|m\rangle\langle m|{\rm d}m$ and
$\bm{I}_{2}=\int|n\rangle\langle n|{\rm d}n$, we have

$\displaystyle\rho_{\rm fin}=\int|n\rangle\langle n|$
$\displaystyle\bm{U}_{2}\int|m\rangle\langle
m|\bm{U}_{1}\Bigl{(}\rho_{0}\otimes|\psi_{1}(x)\rangle\langle\psi_{1}(x)|\otimes|\psi_{2}(x)\rangle\langle\psi_{2}(x)|\Bigr{)}\bm{U}_{1}^{\dagger}$
$\displaystyle\times\int|m^{\prime}\rangle\langle
m^{\prime}|\bm{U}_{2}^{\dagger}\int|n^{\prime}\rangle\langle n^{\prime}|{\rm
d}m{\rm d}m^{\prime}{\rm d}n{\rm d}n^{\prime}.$ (A.2)

Set $\bm{M}_{m}=\bigl{(}\bm{I}_{\mathcal{S}}\otimes\langle
m|\bigr{)}\bm{U}_{1}\bigl{(}\bm{I}_{\mathcal{S}}\otimes|\psi_{1}(x)\rangle\bigr{)}$
is the measurement operator for the first measurement, where
$\bm{I}_{\mathcal{S}}$ is an identity operator in the system space and can be
expressed in terms of eigenstates of $\bm{A}$ (or $\bm{B}$), i.e.,
$\bm{I}_{\mathcal{S}}=\int|a\rangle\langle a|{\rm d}a$. Particularly, we have

$\displaystyle\bm{M}_{m}$
$\displaystyle=\bigl{(}\bm{I}_{\mathcal{S}}\otimes\langle
m|\bigr{)}e^{-i\bm{A}\otimes\bm{p}_{1}}\int|a\rangle\langle
a|\otimes|\psi_{1}(x)\rangle{\rm d}a$
$\displaystyle=\bigl{(}\bm{I}_{\mathcal{S}}\otimes\langle
m|\bigr{)}\int|a\rangle\langle a|\otimes
e^{-ia\bm{p}_{1}}|\psi_{1}(x)\rangle{\rm d}a$
$\displaystyle=\int|a\rangle\langle a|\otimes\langle
m|e^{-ia\bm{p}_{1}}|\psi_{1}(x)\rangle{\rm d}a.$ (A.3)

Submitting $|\psi_{1}(x)\rangle=\int\psi_{1}(x)|x\rangle{\rm d}x$ into Eq.
(A), we obtain

$\displaystyle\bm{M}_{m}$ $\displaystyle=\int\Bigl{[}|a\rangle\langle
a|\otimes\langle m|e^{-ia\bm{p}_{1}}\int\psi_{1}(x)|x\rangle{\rm
d}x\Bigr{]}{\rm d}a$ $\displaystyle=\int\psi_{1}(m-a)|a\rangle\langle a|{\rm
d}a.$ (A.4)

The measurement operator $\bm{M}_{m}$ satisfies the completeness relation as
$\int\bm{M}_{m}^{\dagger}\bm{M}_{m}{\rm d}m=\bm{I}$:

$\displaystyle\int\bm{M}_{m}^{\dagger}\bm{M}_{m}{\rm d}m$
$\displaystyle=\int\Big{[}\int\psi_{1}^{*}(m-a)|a\rangle\langle a|{\rm
d}a\times\int\psi_{1}(m-a^{\prime})|a^{\prime}\rangle\langle a^{\prime}|{\rm
d}a^{\prime}\Big{]}{\rm d}m$
$\displaystyle=\int\Big{[}\int\psi_{1}^{*}(m-a)\psi_{1}(m-a){\rm
d}m\Big{]}|a\rangle\langle a|{\rm d}a$ $\displaystyle=\bm{I}.$ (A.5)

Similarly, for the second measurement, we define a measurement operator
$\bm{N}_{n}=\bigl{(}\bm{I}_{\mathcal{S}}\otimes\langle
n|\bigr{)}\bm{U}_{2}|\bigl{(}\bm{I}_{\mathcal{S}}\otimes\psi_{2}(x)\rangle\bigr{)}$,
which particularly gives

$\displaystyle\bm{N}_{n}=\int\psi_{2}(n-b)|b\rangle\langle b|{\rm d}b,$ (A.6)

and obeys the relation $\int\bm{N}_{n}^{\dagger}\bm{N}_{n}{\rm d}n=\bm{I}.$
Inserting $\bm{M}_{m}$ and $\bm{N}_{n}$ into Eq. (A), we obtain

$\displaystyle\rho_{\rm fin}$
$\displaystyle=\iiiint\bm{N}_{n}\bm{M}_{m}\rho_{0}\bm{M}^{\dagger}_{m^{\prime}}\bm{N}^{\dagger}_{n^{\prime}}\
|m\rangle\langle m^{\prime}|\otimes|n\rangle\langle n^{\prime}|{\rm d}m{\rm
d}m^{\prime}{\rm d}n{\rm d}n^{\prime},$ (A.7)

which is given in Eq. (4) in the main text.

##  Appendix B Final joint probes state in the joint measurement model

Here, we derive the joint state of $\mathcal{P}_{1}\otimes\mathcal{P}_{2}$ by
partially tracing out the system space

$\displaystyle\eta={\rm Tr}_{\mathcal{S}}[\rho_{\rm fin}].$ (B.1)

We first consider the cyclic property of the trace as ${\rm
Tr}_{\mathcal{S}}\bigl{[}\bm{N}_{n}\bm{M}_{m}\rho_{0}\bm{M}^{\dagger}_{m^{\prime}}\bm{N}^{\dagger}_{n^{\prime}}\bigr{]}={\rm
Tr}_{\mathcal{S}}\bigl{[}\bm{M}_{m}\rho_{0}\bm{M}^{\dagger}_{m^{\prime}}\bm{N}^{\dagger}_{n^{\prime}}\bm{N}_{n}\bigr{]},$
where

$\displaystyle\bm{N}^{\dagger}_{n^{\prime}}\bm{N}_{n}$
$\displaystyle=\int\psi_{2}^{*}(n^{\prime}-b^{\prime})|b^{\prime}\rangle\langle
b^{\prime}|\int\psi_{2}(n-b)|b\rangle\langle b|{\rm d}b{\rm d}b^{\prime}$
$\displaystyle=\int\psi_{2}(n-b)\psi_{2}^{*}(n^{\prime}-b)|b\rangle\langle
b|{\rm d}b,$ (B.2)

and

$\displaystyle\bm{M}_{m}\rho_{0}\bm{M}^{\dagger}_{m^{\prime}}$
$\displaystyle=\Bigl{[}\int\psi_{1}(m-a)|a\rangle\langle a|{\rm
d}a\Bigr{]}\times\rho_{0}\times\Bigl{[}\int\psi_{1}^{*}(m^{\prime}-a^{\prime})|a^{\prime}\rangle\langle
a^{\prime}|{\rm d}a^{\prime}\Bigr{]}$ $\displaystyle=\iint|a\rangle\langle
a|\rho_{0}|a^{\prime}\rangle\langle
a^{\prime}|\times\psi_{1}(m-a)\psi_{1}^{*}(m^{\prime}-a^{\prime}){\rm d}a{\rm
d}a^{\prime}.$ (B.3)

From Eqs. (B, B), we have

$\displaystyle{\rm
Tr}_{\mathcal{S}}\bigl{[}\bm{M}_{m}\rho_{0}\bm{M}^{\dagger}_{m^{\prime}}\bm{N}^{\dagger}_{n^{\prime}}\bm{N}_{n}\bigr{]}=\iiint\langle
b|a\rangle\langle a|\rho_{0}|a^{\prime}\rangle\langle
a^{\prime}|b\rangle\times\psi_{1}(m-a)\psi_{1}^{*}(m^{\prime}-a^{\prime})\
\times\psi_{2}(n-b)\psi_{2}^{*}(n^{\prime}-b){\rm d}a{\rm d}a^{\prime}{\rm
d}b.$ (B.4)

Submitting it into Eq. (A.7) and taking trace over system $\mathcal{S}$, we
get Eq. (B.1), which explicitly gives

$\displaystyle\eta=\iiint$ $\displaystyle\langle b|a\rangle\langle
a|\rho_{0}|a^{\prime}\rangle\langle
a^{\prime}|b\rangle\times|\psi_{1}(m-a)\rangle\langle\psi_{1}(m^{\prime}-a^{\prime})|\otimes|\psi_{2}(n-b)\rangle\langle\psi_{2}(n^{\prime}-b)|{\rm
d}a{\rm d}a^{\prime}{\rm d}b,$ (B.5)

where we used $\int\psi_{1}(m-a)|m\rangle{\rm d}m=|\psi(m-a)\rangle$ and
similarly for these others. Noting that this formula does not explicitly
depend on $m$, we thus can replace $m$ by $x$, and recast Eq. (B.5) by

$\displaystyle\eta=\iiint\langle b|a\rangle\langle
a|\rho_{0}|a^{\prime}\rangle\langle
a^{\prime}|b\rangle\times|\psi_{1}(x-a)\rangle\langle\psi_{1}(x-a^{\prime})|\otimes|\psi_{2}(x-b)\rangle\langle\psi_{2}(x-b)|{\rm
d}a{\rm d}a^{\prime}{\rm d}b,$ (B.6)

which is shown in Eq. (6) in the main text.

##  Appendix C Variances in the joint measurement model

###  C.1 Derivation of the variance ${\rm Var}(\bm{x}_{1})_{\eta}$

We calculate the expectation value of $\bm{x}_{1}$ in Eq. (7) in the main text
as

$\displaystyle\langle\bm{x}_{1}\otimes\bm{I}_{2}\rangle$ $\displaystyle={\rm
Tr}_{\mathcal{P}_{1}\mathcal{P}_{2}}\bigl{[}(\bm{x}_{1}\otimes\bm{I}_{2})\eta\bigr{]}$
$\displaystyle=\iiint\langle b|a\rangle\langle
a|\rho_{0}|a^{\prime}\rangle\langle
a^{\prime}|b\rangle\times\langle\psi_{1}(x-a^{\prime})|\bm{x}_{1}|\psi_{1}(x-a)\rangle{\rm
d}a{\rm d}a^{\prime}{\rm d}b.$ (C.1)

Wherein, we have

$\displaystyle\int\langle b|a\rangle\langle
a|\rho_{0}|a^{\prime}\rangle\langle a^{\prime}|b\rangle{\rm d}b$
$\displaystyle={\rm Tr}_{\mathcal{S}}\Bigl{[}|a\rangle\langle
a|\rho_{0}|a^{\prime}\rangle\langle a^{\prime}|\Bigr{]}$ $\displaystyle={\rm
Tr}_{\mathcal{S}}\bigl{[}|a\rangle\langle a|\rho_{0}\bigr{]},$ (C.2)

where we used the cyclic property of the trace, then, only $a=a^{\prime}$ is
non-vanish, and

$\displaystyle\langle\psi_{1}(x-a^{\prime})|\bm{x}_{1}|\psi_{1}(x-a)\rangle$
$\displaystyle=\Bigl{[}\int\Bigl{(}\dfrac{1}{2\pi\sigma_{1}^{2}}\Bigr{)}^{1/4}e^{-\frac{(x^{\prime}-a^{\prime})^{2}}{4\sigma_{1}^{2}}}\langle
x^{\prime}|{\rm d}x^{\prime}\Bigr{]}\times\Bigl{[}\int
x_{1}|x_{1}\rangle\langle x_{1}|{\rm
d}x_{1}\Bigr{]}\times\Bigl{[}\int\Bigl{(}\dfrac{1}{2\pi\sigma_{1}^{2}}\Bigr{)}^{1/4}e^{-\frac{(x-a)^{2}}{4\sigma_{1}^{2}}}|x\rangle{\rm
d}x\Bigr{]}$
$\displaystyle=\dfrac{a+a^{\prime}}{2}e^{-\frac{(a-a^{\prime})^{2}}{8\sigma_{1}^{2}}}$
$\displaystyle=a\ \text{ for all }a=a^{\prime}.$ (C.3)

Then, the expectation value (C.1) is recast as

$\displaystyle\langle\bm{x}_{1}\otimes\bm{I}_{2}\rangle$ $\displaystyle=\int
a\ {\rm Tr}_{\mathcal{S}}\bigl{[}|a\rangle\langle a|\rho_{0}\bigr{]}{\rm d}a$
$\displaystyle={\rm Tr}_{\mathcal{S}}[\bm{A}\rho_{0}].$ (C.4)

Obviously, the expectation value obtained from the first measurement does not
depend on the second measurement.

We next calculate $\langle(\bm{x}_{1}\otimes\bm{I}_{2})^{2}\rangle$ for the
non-vanish terms, i.e., $a=a^{\prime}$. We first evaluate

$\displaystyle\langle\psi_{1}(x-a)|\bm{x}_{1}^{2}|\psi_{1}(x-a)\rangle$
$\displaystyle=\Bigl{[}\int\Bigl{(}\dfrac{1}{2\pi\sigma_{1}^{2}}\Bigr{)}^{1/2}e^{-\frac{(x^{\prime}-a)^{2}}{4\sigma_{1}^{2}}}\langle
x^{\prime}|{\rm d}x^{\prime}\Bigr{]}\times\Bigl{[}\int
x_{1}^{2}|x_{1}\rangle\langle x_{1}|{\rm
d}x_{1}\Bigr{]}\times\Bigl{[}\int\Bigl{(}\dfrac{1}{2\pi\sigma_{1}^{2}}\Bigr{)}^{1/2}e^{-\frac{(x-a)^{2}}{4\sigma_{1}^{2}}}|x\rangle{\rm
d}x\Bigr{]}$ $\displaystyle=\sigma_{1}^{2}+a^{2}.$ (C.5)

Combining with Eq. (C.1), we obtain

$\displaystyle\langle(\bm{x}_{1}\otimes\bm{I}_{2})^{2}\rangle$
$\displaystyle=\int(\sigma_{1}^{2}+a^{2})\ {\rm
Tr}_{\mathcal{S}}\bigl{[}|a\rangle\langle a|\rho_{0}\bigr{]}{\rm d}a$
$\displaystyle=\sigma_{1}^{2}+{\rm
Tr}_{\mathcal{S}}\bigl{[}\bm{A}^{2}\rho_{0}\bigr{]}.$ (C.6)

Then, the variance is

$\displaystyle{\rm Var}(\bm{x}_{1})_{\eta}$
$\displaystyle=\langle(\bm{x}_{1}\otimes\bm{I}_{2})^{2}\rangle-\langle\bm{x}_{1}\otimes\bm{I}_{2}\rangle^{2}$
$\displaystyle=\sigma_{1}^{2}+{\rm Var}(\bm{A})_{\rho_{0}},$ (C.7)

where ${\rm Var}(\bm{A})_{\rho_{0}}={\rm
Tr}_{\mathcal{S}}[\bm{A}^{2}\rho_{0}]-({\rm
Tr}_{\mathcal{S}}[\bm{A}\rho_{0}])^{2}$. This result is casually trivial,
where the second measurement does not affect the first one.

###  C.2 Derivation of the variance ${\rm Var}(\bm{x}_{2})_{\eta}$

We next examine the backaction effect of the first measurement affects the
second measurement. We start from the expectation value:

$\displaystyle\langle\bm{I}_{1}\otimes\bm{x}_{2}\rangle$ $\displaystyle={\rm
Tr}_{\mathcal{P}_{1}\mathcal{P}_{2}}\bigl{[}(\bm{I}_{1}\otimes\bm{x}_{2})\eta\bigr{]}$
$\displaystyle=\iiint\langle b|a\rangle\langle
a|\rho_{0}|a^{\prime}\rangle\langle
a^{\prime}|b\rangle\times\langle\psi_{1}(x-a^{\prime})|\psi_{1}(x-a)\rangle\
\times\langle\psi_{2}(x-b)|\bm{x}_{2}|\psi_{2}(x-b)\rangle{\rm d}a{\rm
d}a^{\prime}{\rm d}b.$ (C.8)

The middle term is

$\displaystyle\langle\psi_{1}(x-a^{\prime})|\psi_{1}(x-a)\rangle=e^{-\frac{(a-a^{\prime})^{2}}{8\sigma_{1}^{2}}}.$
(C.9)

The last term is (similar to (C.1))

$\displaystyle\langle\psi_{2}(x-b)|\bm{x}_{2}|\psi_{2}(x-b)\rangle=b.$ (C.10)

Then, Eq. (C.2) explicitly gives

$\displaystyle\langle\bm{I}_{1}\otimes\bm{x}_{2}\rangle$ $\displaystyle=\iiint
b\langle b|a\rangle\langle a|\rho_{0}|a^{\prime}\rangle\langle
a^{\prime}|b\rangle e^{-\frac{(a-a^{\prime})^{2}}{8\sigma_{1}^{2}}}{\rm
d}a{\rm d}a^{\prime}{\rm d}b$ $\displaystyle={\rm
Tr}_{\mathcal{S}}\Bigl{[}\bm{B}\rho_{1}\Bigr{]},$ (C.11)

where

$\displaystyle\rho_{1}$ $\displaystyle={\rm
Tr}_{\mathcal{P}_{1}}\Bigl{[}e^{-i{\bm{A}}\otimes\bm{p}_{1}}\Bigl{(}\rho_{0}\otimes|\psi_{1}(x)\rangle\langle\psi_{1}(x)|\Bigr{)}e^{i{\bm{A}}\otimes\bm{p}_{1}}\Bigr{]}$
$\displaystyle=\iint|a\rangle\langle a|\rho_{0}|a^{\prime}\rangle\langle
a^{\prime}|e^{-\frac{(a-a^{\prime})^{2}}{8\sigma_{1}^{2}}}{\rm d}a{\rm
d}a^{\prime},$ (C.12)

is the system state after the first interaction. For weak measurement,
$\sigma_{1}\gg|a-a^{\prime}|$, i.e.,
$e^{-\frac{(a-a^{\prime})^{2}}{8\sigma_{1}^{2}}}\approx 1$, we have
$\langle\bm{I}_{1}\otimes\bm{x}_{2}\rangle\approx{\rm
Tr}_{\mathcal{S}}[\bm{B}\rho_{0}]$. We next calculate the term
$\langle(\bm{I}_{1}\otimes\bm{x}_{2})^{2}\rangle$ and obtain

$\displaystyle\langle(\bm{I}_{1}\otimes\bm{x}_{2})^{2}\rangle=\sigma_{2}^{2}+{\rm
Tr}_{\mathcal{S}}[\bm{B}^{2}\rho_{1}]$ (C.13)

Finally, the variance is given by

$\displaystyle{\rm Var}({\bm{x}_{2}})_{\eta}=\sigma_{2}^{2}+{\rm
Var}(\bm{B})_{\rho_{1}},$ (C.14)

where ${\rm Var}(\bm{B})_{\rho_{1}}={\rm
Tr}_{\mathcal{S}}[\bm{B}^{2}\rho_{1}]-({\rm
Tr}_{\mathcal{S}}[\bm{B}\rho_{1}])^{2}$ and particularly gives

$\displaystyle{\rm Var}(\bm{B})_{\rho_{1}}=\int b^{2}\langle
b|\rho_{1}|b\rangle{\rm d}b-\Bigl{[}\int b\langle b|\rho_{1}|b\rangle{\rm
d}b\Bigr{]}^{2}.$ (C.15)

For the weak measurement, ${\rm Var}(\bm{B})_{\rho_{1}}={\rm
Var}(\bm{B})_{\rho_{0}}$, which implies that the weak measurement will not
affect the later measurement results.

##  Appendix D Conditional measurement model

We calculate the conditional probability

$\displaystyle p(x_{2}\big{|}x_{1})=\dfrac{{\rm
Tr}\Bigl{[}\bar{\rho}_{1}(x_{1})\bm{E}(x_{2})\Bigr{]}}{\bigintssss{\rm
Tr}\Bigl{[}\bar{\rho}_{1}(x_{1})\bm{E}(x^{\prime}_{2})\Bigr{]}{\rm
d}x^{\prime}_{2}}.$ (D.1)

Using $\bm{E}(x_{2})=\bm{N}_{x_{2}}^{\dagger}\bm{N}_{x_{2}}$, we have

$\displaystyle{\rm Tr}\Bigl{[}\bar{\rho}_{1}(x_{1})\bm{E}(x_{2})\Bigr{]}$
$\displaystyle={\rm
Tr}\Bigl{[}\bm{N}_{x_{2}}\bar{\rho}_{1}(x_{1})\bm{N}^{\dagger}_{x_{2}}\Bigr{]}$
$\displaystyle={\rm Tr}\Bigl{[}\int\psi_{2}(x_{2}-b)|b\rangle\langle b|{\rm
d}b\times\bar{\rho}_{1}(x_{1})\times\int\psi_{2}^{*}(x_{2}-b^{\prime})|b^{\prime}\rangle\langle
b^{\prime}|{\rm d}b^{\prime}\Bigr{]}$
$\displaystyle=\int\big{|}\psi_{2}(x_{2}-b)\big{|}^{2}\langle
b|\bar{\rho}_{1}(x_{1})|b\rangle{\rm d}b.$ (D.2)

Submitting to Eq. (D.1), we obtain

$\displaystyle
p(x_{2}\big{|}x_{1})=\dfrac{\bigintssss\big{|}\psi_{2}(x_{2}-b)\big{|}^{2}\langle
b|\bar{\rho}_{1}|b\rangle\ {\rm
d}b}{\bigintssss\Bigl{[}\bigintssss\big{|}\psi_{2}(x^{\prime}_{2}-b)\big{|}^{2}\langle
b|\bar{\rho}_{1}|b\rangle\ {\rm d}b\Bigr{]}{\rm d}x^{\prime}_{2}},$ (D.3)

as shown in Eq. (14) in the main text.

##  Appendix E $N$-sequential measurements

Here, we prove Eq. (35) in the main text. Under $N$-sequential measurements,
the joint probability is given by

$\displaystyle p(x_{1},\cdots,x_{N})$ $\displaystyle={\rm
Tr}_{\mathcal{S}}\bigl{[}\bm{\Omega}_{N}\cdots\bm{\Omega}_{k+1}\bar{\rho}_{k}\bm{\Omega}_{k+1}^{\dagger}\cdots\bm{\Omega}_{N}^{\dagger}\bigr{]}$
$\displaystyle={\rm
Tr}_{\mathcal{S}}\bigl{[}\bar{\rho}_{k}\bm{\Omega}_{k+1}^{\dagger}\cdots\bm{\Omega}_{N}^{\dagger}\bm{\Omega}_{N}\cdots\bm{\Omega}_{k+1}\bigr{]}$
$\displaystyle={\rm Tr}_{\mathcal{S}}\bigl{[}\bar{\rho}_{k}E\bigr{]},$ (E.1)

where we used
$\bar{\rho}_{k}=\bm{\Omega}_{k}\bar{\rho}_{k-1}\bm{\Omega}_{k}^{\dagger}$, and
$E=\bm{\Omega}_{k+1}^{\dagger}\cdots\bm{\Omega}_{N}^{\dagger}\bm{\Omega}_{N}\cdots\bm{\Omega}_{k+1}$.
Then, the conditional probability is defined as

$\displaystyle p(x_{k}\big{|}\bar{x}_{k})$
$\displaystyle\equiv\dfrac{p(x_{1},\cdots,x_{N})}{\int
p(x_{1},\cdots,x_{N}){\rm d}x^{\prime}_{k}}$ $\displaystyle=\dfrac{{\rm
Tr}_{\mathcal{S}}\bigl{[}\bar{\rho}_{k}\bm{E}\bigr{]}}{\int{\rm
Tr}_{\mathcal{S}}\bigl{[}\bar{\rho}_{k}\bm{E}\bigr{]}{\rm d}x^{\prime}_{k}},$
(E.2)

where $\bar{x}_{k}=\\{x_{1},\cdots,x_{N}\\}\backslash x_{k}$.

Next, substituting
$\bm{\Omega}_{k}=\int\psi_{k}(x_{k}-a_{k})|a_{k}\rangle\langle a_{k}|\ {\rm
d}a_{k}$ into $\bar{\rho}_{k}$, we obtain

$\displaystyle\bar{\rho}_{k}$
$\displaystyle=\Bigl{[}\int\psi_{k}(x_{k}-a_{k})|a_{k}\rangle\langle
a_{k}|{\rm d}a_{k}\Bigr{]}\times\bar{\rho}_{k-1}\
\times\Bigl{[}\int\psi_{k}^{*}(x_{k}-a^{\prime}_{k})|a^{\prime}_{k}\rangle\langle
a^{\prime}_{k}|{\rm d}a^{\prime}_{k}\Bigr{]},$
$\displaystyle=\iint\Bigl{[}\psi_{k}(x_{k}-a_{k})\psi_{k}^{*}(x_{k}-a^{\prime}_{k})|a_{k}\rangle\langle
a_{k}|\bar{\rho}_{k-1}|a^{\prime}_{k}\rangle\langle
a^{\prime}_{k}|\Bigr{]}{\rm d}a_{k}{\rm d}a^{\prime}_{k}.$ (E.3)

Submitting Eq. (E) into Eq. (E), we have

$\displaystyle
p(x_{k}\big{|}\bar{x}_{k})=\dfrac{\iint\psi_{k}(x_{k}-a_{k})\psi_{k}^{*}(x_{k}-a_{k}^{\prime})\langle
a_{k}|\bar{\rho}_{k-1}|a_{k}\rangle\langle a_{k}^{\prime}|\bm{E}|a_{k}\rangle\
{\rm d}a_{k}{\rm
d}a^{\prime}_{k}}{\int\bigl{[}\iint\psi_{k}(x_{k}-a_{k})\psi_{k}^{*}(x_{k}-a_{k}^{\prime})\langle
a_{k}|\bar{\rho}_{k-1}|a_{k}\rangle\langle a_{k}^{\prime}|\bm{E}|a_{k}\rangle\
{\rm d}a_{k}{\rm d}a^{\prime}_{k}\bigr{]}{\rm d}x_{k}^{\prime}}.$ (E.4)

A useful Mathematica code for calculating the conditional variance is given is
Listing 1.

Listing 1:  Mathematica code for calculating the conditional variance ${\rm
Var}(\bm{x}_{2}|x_{1},x_{3},x_{4})$ of 4-sequential measurements:
$\bm{S}_{z}\to\bm{S}_{x}\to\bm{S}_{x}\to\bm{S}_{z}$.

⬇

1(* Eigenstates and eigenvectors *)

2ez1 = 0.5;

3ex2 = -0.5;

4MatrixForm[vecz1 = {{1},{0}}];

5MatrixForm[vecz2 = {{0},{1}}];

6

7ex1 = 0.5;

8ex2 = -0.5;

9MatrixForm[vecx1 = 1/Sqrt[2]*{{1},{1}}];

10MatrixForm[vecx2 = 1/Sqrt[2]*{{1},{-1}}];

11

12(* Measurement operators *)

13$\psi$[$\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}{\sigma\\_}$,${\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}\text{x}\\_}$,${\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}\text{a}\\_}$]:=$\Bigl{(}\dfrac{1}{2\pi*\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}{\sigma}^{2}}\Bigr{)}^{1/4}$*Exp[-$\dfrac{({\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}\text{x}}-{\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}\text{a}})^{2}}{4\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}{\sigma}^{2}}$];

14$\Omega$[$\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}{\sigma\\_}$,${\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}\text{x}\\_}$,${\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}\text{e}\\_}$,${\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}\text{vec}\\_}$]:=
${\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}\text{vec}}^{\dagger}.{\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}\text{vec}}*\psi$[$\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}{\sigma}$,x,a];

15$\Omega$Z[$\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}{\sigma\\_}$,${\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}\text{x}\\_}$]:=
$\Omega$[$\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}{\sigma}$,x,ez1,vecz1]+$\Omega$[$\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}{\sigma}$,x,ez2,vecz2];

16$\Omega$X[$\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}{\sigma\\_}$,${\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}\text{x}\\_}$]:=
$\Omega$[$\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}{\sigma}$,x,ex1,vecx1]+$\Omega$[$\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}{\sigma}$,x,ex2,vecx2];

17

18(* Evolution system state *)

19MatrixForm[$\rho 0$ = {{0.5, 0.5},{0.5, 0.5}}];

20$\rho 1$:=
FullSimplify[$\Omega$Z[$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma
1}$,x1].$\rho
0$.$\Omega$Z[$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma
1}$,x1]];

21$\rho 2$:=
FullSimplify[$\Omega$X[$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma
2}$,x2].$\rho
1$.$\Omega$X[$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma
2}$,x2]];

22fE:=$\Omega$X[$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma
3}$,x3].$\Omega$Z[$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma
4}$,x4].$\Omega$Z[$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma
4}$,x4].$\Omega$X[$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma
3}$,x3];

23

24(* Conditional probability *)

25Num = Tr[$\rho 2$.fE];

26Den= Integrate[Num, {x2,-$\infty,\infty$}];

27Pr = FullSimplify[Num / Den];

28

29(* Conditional variance *)

30Average = Simplify[Integrate[x2*Pr, {x2,-$\infty,\infty$}]];

31Average2 =
Simplify[Integrate[${\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}\text{x2}}^{2}$*Pr,
{x2,-$\infty,\infty$}]];

32Variances = FullSimplify[Average2 - Average2];

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