Quantum dynamics under simultaneous and continuous measurement of noncommutative observables

The Arthurs-Kelly model is a single-shot measurement model which can be used to describe the simultaneous measurement of the position and the momentum of a particle in a one-dimensional quantum system E. Arthurs ; S. L. Braunstein ; A. J. Scott . It consists of two detectors and the system to be measured. The pointers of the two detectors are prepared in the Gaussian initial state, and, at an instance of time $t_{r}$, the position $\hat{x}$ and the momentum $\hat{p}$ of the particle are coupled with the pointers of the two detectors, respectively, when the measurement starts. After the coupling at $t_{r}$, the positions of the two pointers are influenced by the system; therefore, we can obtain measured values of $\hat{x}$ and $\hat{p}$ from the readouts of the positions of the two pointers by projective measurements of their positions. The coupling between the system and the detectors is described by the time-dependent Hamiltonian $\hat{H}_{I}(t)$, which is chosen in the following form:

$\displaystyle\hat{H}_{I}(t)=(s_{1}\hat{x}\hat{p}_{1}+s_{2}\hat{p}\hat{p}_{2})\,\delta(t-t_{r}),$

(1)

where $\hat{p}_{i}$ $(i=1,\,2)$ is the momentum of detector $i$’s pointer, $s_{i}$ is the coupling strength between the system and detector $i$, and all the quantities here (i.e., $\hat{H}_{I}$, $s_{i}$, $\hat{x}$, $\hat{p}$, and $\hat{p}_{i}$) are dimensionless.

We now consider simultaneous measurements of two arbitrary observables $\hat{A}$ and $\hat{B}$ of the system. This generalized measurement can be performed by the following analogical interaction Hamiltonian between the system and two detectors:

$\displaystyle\hat{H}_{I}(t)$

$\displaystyle=(s_{1}\hat{A}\hat{p}_{1}+s_{2}\hat{B}\hat{p}_{2})\,\delta(t-t_{r}).$

(2)

Similarly, all the physical quantities discussed here such as $\hat{H}_{I}$, $s_{i}$, and $\hat{A}$ are also dimensionless. Moreover, $\hbar$ is set to be unity throughout the whole paper for simplicity note:units . Again, detector $i$ is still prepared in the Gaussian initial state $\ket{d_{i}}$,

$\displaystyle\left\langle x_{i}|d_{i}\right\rangle=(\pi\Delta_{i})^{-1/4}\,\exp{\left(-\frac{x_{i}^{2}}{2\Delta_{i}}\right)},$

(3)

where $\ket{x_{i}}$ is the position eigenstate of detector $i$’s pointer, $\Delta_{i}\equiv s_{i}\sigma^{2}$ $(\textrm{with }s_{i}>0)$, and $\sigma^{2}$ is a parameter characterizing the measurement accuracy.

Before the coupling at $t_{r}$, the system and the two detectors are assumed to be uncorrelated with each other. Therefore, the density operator of the total system including the system and the detectors initially takes the following form:

$\displaystyle\hat{\rho}_{T}\equiv\hat{\rho}_{s}\otimes|d_{1}d_{2}\rangle\langle d_{1}d_{2}|,$

(4)

where $\hat{\rho}_{s}$ is the partial density operator of the system and $|d_{1}d_{2}\rangle\equiv|d_{1}\rangle\otimes|d_{2}\rangle$ is the uncorrelated Gaussian initial state of two detectors.

After the coupling between the system and the detectors, the total system is in the state

$\displaystyle\hat{\rho}_{T}^{\prime}=\hat{U}_{I}\hat{\rho}_{T}\hat{U}_{I}^{\dagger},$

(5)

where

$\displaystyle\hat{U}_{I}\equiv\exp\left[-i(s_{1}\hat{A}\hat{p}_{1}+s_{2}\hat{B}\hat{p}_{2})\right]$

(6)

is the evolution operator during the measurement. As a result, the average of observable $\hat{A}$ and the position of the pointer of detector 1 after the coupling are given as

	$\displaystyle\left\langle\hat{A}\right\rangle^{\prime}$	$\displaystyle=\mathrm{Tr}\,(\hat{A}\hat{\rho}_{T}^{\prime})=\mathrm{Tr}\,(\hat{U}_{I}^{\dagger}\hat{A}\hat{U}_{I}\hat{\rho}_{T})\equiv\left\langle\hat{U}_{I}^{\dagger}\hat{A}\hat{U}_{I}\right\rangle,$		(7)
	$\displaystyle\left\langle\hat{x}_{1}\right\rangle^{\prime}$	$\displaystyle=\mathrm{Tr}\,(\hat{x}_{1}\hat{\rho}_{T}^{\prime})=\mathrm{Tr}\,(\hat{U}_{I}^{\dagger}\hat{x}_{1}\hat{U}_{I}\hat{\rho}_{T})\equiv\left\langle\hat{U}_{I}^{\dagger}\hat{x}_{1}\hat{U}_{I}\right\rangle.$		(8)

Applying the Baker-Campbell-Hausdorff relation to these two equations, we get

$\displaystyle\left\langle\hat{A}\right\rangle^{\prime}=\left\langle\hat{A}\right\rangle-\frac{s_{2}}{4\sigma^{2}}\left\langle[\hat{B},[\hat{B},\hat{A}]]\right\rangle+O\left(\frac{s_{i}^{2}}{\sigma^{4}}\right),$

(9)

and

$\displaystyle\left\langle\hat{x}_{1}\right\rangle^{\prime}=s_{1}\left[\left\langle\hat{A}\right\rangle-\frac{s_{2}}{12\sigma^{2}}\left\langle[\hat{B},[\hat{B},\hat{A}]]\right\rangle+O\left(\frac{s_{i}^{2}}{\sigma^{4}}\right)\right].$

(10)

Here, we only have evenfold commutators since $\left\langle\hat{p}_{i}^{2n+1}\right\rangle=0$ for the Gaussian state for non-negative integer $n$. Following the same procedure for $\hat{B}$ and $\hat{x}_{2}$, we can obtain $\left\langle B\right\rangle^{\prime}$ and $\left\langle x_{2}\right\rangle^{\prime}$,

$\displaystyle\left\langle\hat{B}\right\rangle^{\prime}=\left\langle\hat{B}\right\rangle-\frac{s_{1}}{4\sigma^{2}}\left\langle[\hat{A},[\hat{A},\hat{B}]]\right\rangle+O\left(\frac{s_{i}^{2}}{\sigma^{4}}\right),$

(11)

and

$\displaystyle\left\langle\hat{x}_{2}\right\rangle^{\prime}=s_{2}\left[\left\langle\hat{B}\right\rangle-\frac{s_{1}}{12\sigma^{2}}\left\langle[\hat{A},[\hat{A},\hat{B}]]\right\rangle+O\left(\frac{s_{i}^{2}}{\sigma^{4}}\right)\right].$

(12)

If $\left[\hat{B},\left[\hat{B},\hat{A}\right]\right]=\left[\hat{A},\left[\hat{A},\hat{B}\right]\right]=0$, for instance $\hat{A}=\hat{x}$ and $\hat{B}=\hat{p}$, then all the multifold commutators vanish so that we get $\left\langle\hat{A}\right\rangle^{\prime}=s_{1}^{-1}\left\langle\hat{x}_{1}\right\rangle^{\prime}$ and $\left\langle\hat{B}\right\rangle^{\prime}=s_{2}^{-1}\left\langle\hat{x}_{2}\right\rangle^{\prime}$. However, for a general case in which $[\hat{B},[\hat{B},\hat{A}]]$ and $\left[\hat{A},\left[\hat{A},\hat{B}\right]\right]$ are nonzero, all the higher-order terms remain and thus the measurement result $s_{i}^{-1}\left\langle\hat{x}_{i}\right\rangle^{\prime}$ deviates from $\left\langle\hat{A}\right\rangle^{\prime}$ and $\left\langle\hat{B}\right\rangle^{\prime}$. From Eqs. (9) – (12), we see that the leading order of the deviations is $s_{i}/\sigma^{2}$, and thus the deviations are negligible only when $s_{i}/\sigma^{2}\ll 1$. To discuss the deviations, it is convenient to introduce relative deviations $\epsilon_{1}$ and $\epsilon_{2}$ defined as

	$\displaystyle\epsilon_{1}$	$\displaystyle\equiv\frac{\left\langle\hat{x}_{1}\right\rangle^{\prime}-s_{1}\left\langle\hat{A}\right\rangle^{\prime}}{s_{1}\left\langle\hat{A}\right\rangle^{\prime}},$		(13)
	$\displaystyle\epsilon_{2}$	$\displaystyle\equiv\frac{\left\langle\hat{x}_{2}\right\rangle^{\prime}-s_{2}\left\langle\hat{B}\right\rangle^{\prime}}{s_{2}\left\langle\hat{B}\right\rangle^{\prime}}.$		(14)

For $\hat{A}=\hat{L}_{x}$ and $\hat{B}=\hat{L}_{y}$ as an example, where $\hat{L}_{x}$ and $\hat{L}_{y}$ are $x$ and $y$ components of the angular momentum, respectively note:angularmom , $\epsilon_{1}$ and $\epsilon_{2}$, with $s_{i}^{-1}\left\langle\hat{x}_{i}\right\rangle^{\prime}$, $\left\langle\hat{A}\right\rangle^{\prime}$, and $\left\langle\hat{B}\right\rangle^{\prime}$ up to the second order of $s_{i}/\sigma^{2}$, read

	$\displaystyle\epsilon_{1}$	$\displaystyle=\frac{s_{2}}{6\sigma^{2}}\frac{1-(s_{1}+3s_{2})/20\sigma^{2}}{1-s_{2}/4\sigma^{2}+s_{2}(s_{1}+3s_{2})/96\sigma^{4}},$		(15)
	$\displaystyle\epsilon_{2}$	$\displaystyle=\frac{s_{1}}{6\sigma^{2}}\frac{1-(s_{2}+3s_{1})/20\sigma^{2}}{1-s_{1}/4\sigma^{2}+s_{1}(s_{2}+3s_{1})/96\sigma^{4}}.$		(16)

In Eqs. (15) and (16), $\epsilon_{1}$ and $\epsilon_{2}$ are independent of the state of the system due to the closed algebra of the angular momentum and the symmetry of the Gaussian state. Take $\epsilon_{1}$ for example: because of the closed algebra of the angular momentum, the numerator and the denominator of $\epsilon_{1}$ can be written as a linear combination of $\left\langle\hat{L}_{x}\right\rangle$, $\left\langle\hat{L}_{y}\right\rangle$, and $\left\langle\hat{L}_{z}\right\rangle$. However, due to the symmetry of the Gaussian state, the coefficients of $\left\langle\hat{L}_{y}\right\rangle$ and $\left\langle\hat{L}_{z}\right\rangle$, which are averages of odd powers of $\hat{x}_{i}$ and $\hat{p}_{i}$, are zero. Therefore, the numerator and the denominator of $\epsilon_{1}$ are proportional to $\left\langle\hat{L}_{x}\right\rangle$, which are canceled with each other finally. The relative deviations $\epsilon_{1}$ and $\epsilon_{2}$ are monotonically increasing with parameters $s_{1}/\sigma^{2}$ and $s_{2}/\sigma^{2}$. When $s_{i}/\sigma^{2}\simeq 0.5$, the relative deviations $\epsilon_{1}$ and $\epsilon_{2}$ reach around $10\%$, which are non-negligible. Consequently, $s_{i}/\sigma^{2}$ must be much smaller than 1 in order to obtain an accurate measurement outcome. Details are shown in Fig. 1.

We now go back to the arbitrary observables $\hat{A}$ and $\hat{B}$, and consider the second moment of the positions of the two pointers $\hat{x}_{1}$ and $\hat{x}_{2}$ after the coupling,

	$\displaystyle\left\langle\hat{x}_{1}^{2}\right\rangle^{\prime}=$	$\displaystyle\frac{s_{1}\sigma^{2}}{2}\left[1+\frac{2s_{1}}{\sigma^{2}}\left\langle\hat{A}^{2}\right\rangle-\frac{s_{1}s_{2}}{12\sigma^{4}}\Big{(}\left\langle[\hat{B},[\hat{B},\hat{A}^{2}]]\right\rangle\right.$
		$\displaystyle\left.+\left\langle\hat{A}\,[\hat{B},[\hat{B},\hat{A}]]\right\rangle+\left\langle[\hat{B},\hat{A}\,[\hat{B},\hat{A}]]\right\rangle\Big{)}+O\left(\frac{s_{i}^{3}}{\sigma^{6}}\right)\right],$		(17)

and

	$\displaystyle\left\langle\hat{x}_{2}^{2}\right\rangle^{\prime}=$	$\displaystyle\frac{s_{2}\sigma^{2}}{2}\left[1+\frac{2s_{2}}{\sigma^{2}}\left\langle\hat{B}^{2}\right\rangle-\frac{s_{1}s_{2}}{12\sigma^{4}}\Big{(}\left\langle[\hat{A},[\hat{A},\hat{B}^{2}]]\right\rangle\right.$
		$\displaystyle\left.+\left\langle\hat{B}\,[\hat{A},[\hat{A},\hat{B}]]\right\rangle+\left\langle[\hat{A},\hat{B}\,[\hat{A},\hat{B}]]\right\rangle\Big{)}+O\left(\frac{s_{i}^{3}}{\sigma^{6}}\right)\right].$		(18)

The leading term of the variance $\left\langle\hat{x}_{i}^{2}\right\rangle^{\prime}-\left(\left\langle\hat{x}_{i}\right\rangle^{\prime}\right)^{2}$ of the measurement result depends on the parameter $s_{i}\sigma^{2}$. In order to get a stable readout, i.e., the variances are small so that the measurement results are less scattered, $s_{i}\sigma^{2}$ need to be set as small as possible. There is a tradeoff between the stability characterized by $s_{i}\sigma^{2}$ and the accuracy characterized by $s_{i}/\sigma^{2}$ [see Eqs. (15) and (16)]. Therefore, $\sigma^{2}$ cannot be too large for a given $s_{i}$, although $\sigma^{2}$ is intrinsically a large quantity in the weak measurements, which is assumed to hold $s_{i}/\sigma^{2}\ll 1$. On the other hand, for a given $\sigma^{2}$, $s_{i}$ should be sufficiently small so that both $s_{i}\sigma^{2}$ and $s_{i}/\sigma^{2}$ are smaller than unity. From this point of view, the generalized Arthurs-Kelly measurement model is valid only for a weak measurement.

A continuous and simultaneous measurement of observables $\hat{A}$ and $\hat{B}$ can be realized by the following interaction Hamiltonian A. J. Scott :

$\displaystyle\hat{H}_{I}(t)=\sum_{n=1}^{\infty}(s_{1}\hat{A}\hat{p}_{1}+s_{2}\hat{B}\hat{p}_{2})\,\delta(t-n\delta t),$

(19)

where $\delta t$ is the time interval between two consecutive measurements and will finally be set to be infinitesimal. After each single-shot measurement, the readouts of $x_{1}$ and $x_{2}$ are recorded, and then the total composite system is reset to the decoupled initial state given by Eq. (4) based on the Born-Markov assumption. The state of the system immediately before and after the $n$th measurement is denoted by $\hat{\rho}_{s}(n\delta t)$ and $\hat{\rho}_{s}^{\prime}(n\delta t)$ respectively, which satisfy the following relationship:

$\displaystyle\hat{\rho}^{\prime}_{s}(n\delta t)=\mathrm{Tr}\,_{d}\left[\hat{U}_{I}\hat{\rho}_{s}(n\delta t)\otimes|d_{1}d_{2}\rangle\langle d_{1}d_{2}|\hat{U}_{I}^{\dagger}\right],$

(20)

where $\mathrm{Tr}\,_{d}$ means the trace over the degrees of freedom of the two detectors.

We introduce the following measurement operator:

	$\displaystyle\hat{M}(x_{1},x_{2})\equiv$	$\displaystyle\left\langle x_{1}x_{2}|\hat{U}_{I}|d_{1}d_{2}\right\rangle$
	$\displaystyle=$	$\displaystyle\int dp_{1}dp_{2}\,\left\langle x_{1}x_{2}|\hat{U}_{I}|p_{1}p_{2}\right\rangle\left\langle p_{1}p_{2}|d_{1}d_{2}\right\rangle$
	$\displaystyle=$	$\displaystyle(2\pi)^{-1}\int dp_{1}dp_{2}\ \exp\left\{i[(x_{1}-s_{1}\hat{A})p_{1}\right.$
		$\displaystyle\left.+(x_{2}-s_{2}\hat{B})p_{2}]\right\}\left\langle p_{1}p_{2}|d_{1}d_{2}\right\rangle,$		(21)

where $\ket{x_{1}x_{2}}\equiv\ket{x_{1}}\otimes\ket{x_{2}}$, $\ket{p_{1}p_{2}}\equiv\ket{p_{1}}\otimes\ket{p_{2}}$, and $\ket{p_{i}}$ is the eigenstate of the momentum of detector $i$’s pointer. Equation (20) can be rewritten explicitly as

	$\displaystyle\hat{\rho}^{\prime}_{s}(n\delta t)=$	$\displaystyle\int dx_{1}dx_{2}\ \hat{M}(x_{1},x_{2})\,\hat{\rho}_{s}(n\delta t)\,\hat{M}^{\dagger}(x_{1},x_{2})$
	$\displaystyle=$	$\displaystyle\int dp_{1}dp_{2}\ \exp\left[-i(s_{1}p_{1}\hat{A}+s_{2}p_{2}\hat{B})\right]\,\hat{\rho}_{s}(n\delta t)$
		$\displaystyle\exp\left[i(s_{1}p_{1}\hat{A}+s_{2}p_{2}\hat{B})\right]\,\left|\left\langle p_{1}p_{2}|d_{1}d_{2}\right\rangle\right|^{2}$
	$\displaystyle=$	$\displaystyle\hat{\rho}_{s}(n\delta t)-\frac{s_{1}}{4\sigma^{2}}[\hat{A},[\hat{A},\hat{\rho}_{s}(n\delta t)]]$
		$\displaystyle-\frac{s_{2}}{4\sigma^{2}}[\hat{B},[\hat{B},\hat{\rho}_{s}(n\delta t)]]+O\left(\frac{s_{i}^{2}}{\sigma^{4}}\right).$		(22)

Note that the right-hand side of the last equality of Eq. (22) contains neither the terms $[\hat{A},[\hat{B},\hat{\rho}_{s}]]$ nor $[\hat{B},[\hat{A},\hat{\rho}_{s}]]$, which describe the interplay effect between the measurements of the two observables, because their coefficients are zero for Gaussian states.

Taking the unitary evolution by the system Hamiltonian $\hat{H}_{s}$ after each measurement into consideration, the state of the system becomes

$\displaystyle\hat{\rho}_{s}(n\delta t+\delta t)=\hat{U}_{s}\,\hat{\rho}^{\prime}_{s}(n\delta t)\,\hat{U}_{s}^{\dagger},$

(23)

where

$\displaystyle\hat{U}_{s}\equiv\exp\left(-i\hat{H}_{s}\delta t\right)$

(24)

is the unitary evolution operator between two consecutive measurements. After taking the continuous limit $\delta t\to 0$ and $\sigma\to\infty$ with $\zeta\equiv 1/(\delta t\,\sigma^{2})=\mathrm{const.}$, the unconditioned master equation of the system under the simultaneous and continuous measurement reduces to the Lindblad form H.-P. Breuer ; H. M. W. ; K. Jacobs2 ; Lindblad ; Gorini76 :

$\displaystyle\frac{d\hat{\rho}_{s}}{dt}=-i[\hat{H}_{s},\hat{\rho}_{s}]-\frac{\gamma_{1}}{8}[\hat{A},[\hat{A},\hat{\rho}_{s}]]-\frac{\gamma_{2}}{8}[\hat{B},[\hat{B},\hat{\rho}_{s}]].$

(25)

Here, $\gamma_{i}\equiv 2s_{i}\zeta$ is the measurement strength of $\hat{A}$ $(i=1)$ or $\hat{B}$ $(i=2)$. In addition, we can obtain the master equation for the measurement of a single observable $\hat{A}$ or $\hat{B}$ by setting $s_{2}$ or $s_{1}$ to be zero, respectively. Note that even for noninfinitesimal $s_{i}$, the master equation (25) for simultaneous and continuous measurement is still valid in the continuous limit $s_{i}/\sigma^{2}\to 0$, unlike the results for the single-shot measurement in the previous section.

The final master equation (25) of simultaneous measurement does not contain terms of the interplay effect such as $[\hat{A},[\hat{B},\hat{\rho}_{s}]]$ and $[\hat{B},[\hat{A},\hat{\rho}_{s}]]$, but just consists of a linear combination of the two independent measurement effects. This is because of the uncorrelated Gaussian initial state and the Born-Markovian approximation: Before each single-shot measurement, the total composite system is reset to the decoupled initial state given by Eq. (4). In addition, according to Eq. (22), the single-shot measurement discussed in the previous section does not introduce the terms of the interplay effect in the master equation if the initial state of the system and the two detectors are uncorrelated with each other and the detectors are initially prepared in the Gaussian state. Therefore, the terms of the interplay effects $[\hat{A},[\hat{B},\hat{\rho}_{s}]]$ and $[\hat{B},[\hat{A},\hat{\rho}_{s}]]$ are absent in the final master equation. Note that for $\hat{A}=\hat{x}$ and $\hat{B}=\hat{p}$ considered in Ref. A. J. Scott , such terms of the interplay effect have vanished, $[\hat{x},[\hat{p},\hat{\rho}_{s}]]-[\hat{p},[\hat{x},\hat{\rho}_{s}]]=0$, because of $[\hat{x},\hat{p}]=i$. Thus, even though the master equation of the simultaneous and continuous measurement of $\hat{x}$ and $\hat{p}$ does not contain the terms of the interplay effect of the two measurements A. J. Scott , it is still open whether this equation still holds for arbitrary observables $\hat{A}$ and $\hat{B}$. Here, we have shown that the master equation indeed does not contain the terms of the interplay effect, irrespective of the observables measured.

Moreover, we shall also derive the conditioned master equation of simultaneous measurement. The $n$th measurement outcome of detector $i$ is denoted by $x_{i}(n)$ and the final master equation is conditioned by a sequence of the outcome $\{x_{i}(n)\}$. The average and the variance of $x_{i}(n)$ can be obtained by keeping the leading terms of Eqs. (10), (12), (17), and (18),

	$\displaystyle E\left[x_{1}(n)\right]$	$\displaystyle\approx s_{1}\left\langle\hat{A}\right\rangle,$		(26)
	$\displaystyle E\left[x_{2}(n)\right]$	$\displaystyle\approx s_{2}\left\langle\hat{B}\right\rangle,$		(27)
	$\displaystyle V\left[x_{i}(n)\right]$	$\displaystyle=\left\langle\hat{x}_{i}^{2}\right\rangle^{\prime}-\left(\left\langle\hat{x}_{i}\right\rangle^{\prime}\right)^{2}\approx\frac{s_{i}}{2\zeta\delta t},$		(28)

where $E[\cdots]$ represents the ensemble average over all the experimental realizations and $V[\cdots]$ represents the variance of the measurement outcome. The correlation function of $\hat{x}_{1}$ and $\hat{x}_{2}$ after the coupling can also be obtained:

	$\displaystyle\left\langle\hat{x}_{1}\hat{x}_{2}\right\rangle^{\prime}=$	$\displaystyle\mathrm{Tr}\,\left(\hat{x}_{1}\hat{x}_{2}\hat{\rho}_{T}^{\prime}\right)$
	$\displaystyle=$	$\displaystyle\frac{s_{1}s_{2}}{2}\left\langle\hat{A}\hat{B}+\hat{B}\hat{A}\right\rangle.$		(29)

Comparing Eqs. (28) and (29), we find the covariance $C[x_{1}(n),x_{2}(n)]\equiv\langle\hat{x}_{1}\hat{x}_{2}\rangle^{\prime}-\langle\hat{x}_{1}\rangle^{\prime}\langle\hat{x}_{2}\rangle^{\prime}$ of $\hat{x}_{1}$ and $\hat{x}_{2}$ [which is of the order of $(\delta t)^{0}$] is much smaller than the variance of $\hat{x}_{1}$ and $\hat{x}_{2}$ (which is of the order of $1/\delta t$) in the continuous limit $\delta t\to 0$. Therefore, the correlation between $\hat{x}_{1}$ and $\hat{x}_{2}$ can be neglected within the current approximation where the terms of order $1/\delta t$ are kept for the second moment of $x_{i}$, and $x_{1}(n)$ and $x_{2}(n)$ can be treated as two uncorrelated random variables note:covar . Then, approximating $x_{i}(n)$ as a Gaussian random variable, $x_{i}(n)$ can be written as a summation of its average and fluctuation,

	$\displaystyle x_{1}(n)$	$\displaystyle=s_{1}\left\langle\hat{A}\right\rangle+\sqrt{\frac{s_{1}}{2\zeta}}\,d\xi_{1}(n)\cdot(\delta t)^{-1}$
		$\displaystyle\equiv s_{1}\left\langle\hat{A}\right\rangle+\lambda_{1}\,d\xi_{1}(n)\cdot(\delta t)^{-1},$		(30)

and

	$\displaystyle x_{2}(n)$	$\displaystyle=s_{2}\left\langle\hat{B}\right\rangle+\sqrt{\frac{s_{2}}{2\zeta}}\,d\xi_{2}(n)\cdot(\delta t)^{-1}$
		$\displaystyle\equiv s_{2}\left\langle\hat{B}\right\rangle+\lambda_{2}\,d\xi_{2}(n)\cdot(\delta t)^{-1},$		(31)

where $\lambda_{i}\equiv\sqrt{s_{i}/2\zeta}$ is the fluctuation of the measurement outcome of observables $\hat{A}$ ($i=1$) and $\hat{B}$ ($i=2$), and $d\xi_{i}$ is the Itô increment C. W. ; N. G. V. Kampen which satisfies

	$\displaystyle E[d\xi_{i}]=0,$		(32)
	$\displaystyle E[d\xi_{i}(m)\cdot d\xi_{i}(n)]=\delta_{mn}\cdot\delta t,$		(33)
	$\displaystyle E[d\xi_{1}(n)\cdot d\xi_{2}(n)]=0.$		(34)

For notational simplicity, we will treat $d\xi_{i}$ to be equal to $\delta t^{1/2}$ and omit the symbol $E[\cdots]$ in the following derivation. Then, the measurement operator can be expanded as

	$\displaystyle\hat{M}(x_{1}(n),x_{2}(n))=$	$\displaystyle(2\pi)^{-1}\int dp_{1}dp_{2}\ \exp\left\{i[x_{1}(n)p_{1}+x_{2}(n)p_{2}]\right\}\exp\left[-i(s_{1}p_{1}\hat{A}+s_{2}p_{2}\hat{B})\right]$
		$\displaystyle\cdot\frac{(\Delta_{1}\Delta_{2})^{-1/4}}{\pi^{1/2}}\exp\left(-\frac{\Delta_{1}p_{1}^{2}+\Delta_{2}p_{2}^{2}}{2}\right)$
	$\displaystyle\propto$	$\displaystyle\int dp_{1}dp_{2}\ \exp\left\{-i(s_{1}p_{1}\hat{A}+s_{2}p_{2}\hat{B})\right\}\exp\left[-\frac{\Delta_{1}\left(p_{1}-\frac{ix_{1}(n)}{\Delta_{1}}\right)^{2}+\Delta_{2}\left(p_{2}-\frac{ix_{2}(n)}{\Delta_{2}}\right)^{2}}{2}\right]$
	$\displaystyle=$	$\displaystyle\int dp_{1}dp_{2}\ \left\{\sum_{n=0}^{\infty}\frac{1}{n!}\left[-i(s_{1}p_{1}\hat{A}+s_{2}p_{2}\hat{B})\right]^{n}\right\}\cdot\exp\left[-\frac{\Delta_{1}\left(p_{1}-\frac{ix_{1}(n)}{\Delta_{1}}\right)^{2}+\Delta_{2}\left(p_{2}-\frac{ix_{2}(n)}{\Delta_{2}}\right)^{2}}{2}\right]$
	$\displaystyle=$	$\displaystyle\ \frac{2\pi}{(\Delta_{1}\Delta_{2})^{1/2}}\bigg{[}1+\frac{s_{1}\hat{A}}{\Delta_{1}}x_{1}(n)+\frac{s_{2}\hat{B}}{\Delta_{2}}x_{2}(n)-\frac{s_{1}^{2}\hat{A}^{2}}{2\Delta_{1}}+\frac{s_{1}^{2}\hat{A}^{2}}{2\Delta_{1}^{2}}x_{1}(n)^{2}-\frac{s_{2}^{2}\hat{B}^{2}}{2\Delta_{2}}+\frac{s_{2}^{2}\hat{B}^{2}}{2\Delta_{2}^{2}}x_{2}(n)^{2}\bigg{]}+O\left(\delta t^{3/2}\right)$
	$\displaystyle\propto$	$\displaystyle\ 1+\zeta\delta t\left(s_{1}\left\langle\hat{A}\right\rangle\hat{A}+s_{2}\left\langle\hat{B}\right\rangle\hat{B}-\frac{s_{1}\hat{A}^{2}}{4}-\frac{s_{2}\hat{B}^{2}}{4}\right)+\lambda_{1}\zeta\,\hat{A}\,d\xi_{1}+\lambda_{2}\zeta\,\hat{B}\,d\xi_{2}+O\left(\delta t^{3/2}\right).$		(35)

The unnormalized state of the system after monitoring then becomes

	$\displaystyle\hat{\rho}^{\prime}_{s}(n\delta t)=$	$\displaystyle\hat{M}\,\hat{\rho}_{s}(n\delta t)\,\hat{M}^{\dagger}$
	$\displaystyle=$	$\displaystyle\hat{\rho}_{s}+\zeta\,\delta t\bigg{[}s_{1}\left\langle\hat{A}\right\rangle(\hat{A}\hat{\rho}_{s}+\hat{\rho}_{s}\hat{A})+s_{2}\left\langle\hat{B}\right\rangle(\hat{B}\hat{\rho}_{s}+\hat{\rho}_{s}\hat{B})-\frac{s_{1}}{4}(\hat{A}^{2}\hat{\rho}_{s}+\hat{\rho}_{s}\hat{A}^{2})-\frac{s_{2}}{4}(\hat{B}^{2}\hat{\rho}_{s}+\hat{\rho}_{s}\hat{B}^{2})\bigg{]}$
		$\displaystyle+\lambda_{1}\,\zeta\,(\hat{A}\hat{\rho}_{s}+\hat{\rho}_{s}\hat{A})\,d\xi_{1}+\lambda_{2}\,\zeta\,(\hat{B}\hat{\rho}_{s}+\hat{\rho}_{s}\hat{B})\,d\xi_{2}+\zeta^{2}\,\delta t\,(\lambda_{1}^{2}\,\hat{A}\,\hat{\rho}_{s}\,\hat{A}+\lambda_{2}^{2}\,\hat{B}\,\hat{\rho}_{s}\,\hat{B})+O\left(\delta t^{3/2}\right).$		(36)

The normalization constant for the state after the measurement reads

	$\displaystyle\mathrm{Tr}\,(\hat{M}\,\hat{\rho}_{s}\,\hat{M}^{\dagger})=$	$\displaystyle 1+2\zeta\,\delta t\,\left(s_{1}\left\langle\hat{A}\right\rangle^{2}+s_{2}\left\langle\hat{B}\right\rangle^{2}\right)$
		$\displaystyle+2\lambda_{1}\,\zeta\,\left\langle\hat{A}\right\rangle\,d\xi_{1}+2\lambda_{2}\,\zeta\,\left\langle\hat{B}\right\rangle\,d\xi_{2}+O\left(\delta t^{3/2}\right).$		(37)

Applying $(1+x)^{-1}\approx 1-x+x^{2}$ and keeping terms up to the first order of $\delta t$, we obtain the normalized state after the measurement,

	$\displaystyle\hat{\rho}^{\prime}_{s}(n\delta t)=$	$\displaystyle\frac{\hat{M}\hat{\rho}_{s}\hat{M}^{\dagger}}{\mathrm{Tr}\,(\hat{M}\hat{\rho}_{s}\hat{M}^{\dagger})}$
	$\displaystyle\approx$	$\displaystyle\hat{\rho}_{s}-\frac{s_{1}\zeta}{4}(\hat{A}^{2}\hat{\rho}_{s}+\hat{\rho}_{s}\hat{A}^{2}-2\hat{A}\hat{\rho}_{s}\hat{A})\,\delta t-\frac{s_{2}\zeta}{4}(\hat{B}^{2}\hat{\rho}_{s}+\hat{\rho}_{s}\hat{B}^{2}-2\hat{B}\hat{\rho}_{s}\hat{B})\,\delta t+\sqrt{2s_{1}\zeta}\,\bigg{(}\frac{\hat{A}\hat{\rho}_{s}+\hat{\rho}_{s}\hat{A}}{2}-\left\langle\hat{A}\right\rangle\hat{\rho}_{s}\bigg{)}\,d\xi_{1}$
		$\displaystyle+\sqrt{2s_{2}\zeta}\,\bigg{(}\frac{\hat{B}\hat{\rho}_{s}+\hat{\rho}_{s}\hat{B}}{2}-\left\langle\hat{B}\right\rangle\hat{\rho}_{s}\bigg{)}\,d\xi_{2}$
	$\displaystyle=$	$\displaystyle\hat{\rho}_{s}-\frac{\gamma_{1}}{8}[\hat{A},[\hat{A},\hat{\rho}_{s}]]\,\delta t-\frac{\gamma_{2}}{8}[\hat{B},[\hat{B},\hat{\rho}_{s}]]\,\delta t+\sqrt{\gamma_{1}}\,\mathcal{H}\left[\left(\hat{A}-\left\langle\hat{A}\right\rangle\right)\,\hat{\rho}_{s}\right]\,d\xi_{1}+\sqrt{\gamma_{2}}\,\mathcal{H}\left[\left(\hat{B}-\left\langle\hat{B}\right\rangle\right)\,\hat{\rho}_{s}\right]\,d\xi_{2}.$		(38)

Here, the symbol $\mathcal{H}[\hat{O}]$ is defined as the Hermitian part of operator $\hat{O}$,

$\displaystyle\mathcal{H}[\hat{O}]\equiv\frac{1}{2}(\hat{O}+\hat{O}^{\dagger}).$

(39)

Including the unitary evolution between the two consecutive measurements and taking the continuous limit, the conditioned master equation reads

	$\displaystyle d\hat{\rho}_{s}=$	$\displaystyle-i[\hat{H}_{s},\hat{\rho}_{s}]\,dt-\frac{\gamma_{1}}{8}[\hat{A},[\hat{A},\hat{\rho}_{s}]]\,dt-\frac{\gamma_{2}}{8}[\hat{B},[\hat{B},\hat{\rho}_{s}]]\,dt$
		$\displaystyle+\sqrt{\gamma_{1}}\,\mathcal{H}\left[\left(\hat{A}-\left\langle\hat{A}\right\rangle\right)\hat{\rho}_{s}\right]\,d\xi_{1}$
		$\displaystyle+\sqrt{\gamma_{2}}\,\mathcal{H}\left[\left(\hat{B}-\left\langle\hat{B}\right\rangle\right)\hat{\rho}_{s}\right]\,d\xi_{2}.$		(40)