Violations of the Leggett–Garg inequality for coherent and cat states

Abstract

We show that in some cases the coherent state can have a larger violation of the Leggett–Garg inequality (LGI) than the cat state by numerical calculations. To achieve this result, we consider the LGI of the cavity mode weakly coupled to a zero-temperature environment as a practical instance of the physical system. We assume that the bosonic mode undergoes dissipation because of an interaction with the environment but is not affected by dephasing. Solving the master equation exactly, we derive an explicit form of the violation of the inequality for both systems prepared initially in the coherent state $|\alpha \rangle$ and the cat state $(|\alpha \rangle +|-\alpha \rangle )$. For the evaluation of the inequality, we choose the displaced parity operators characterized by a complex number $\beta$. We look for the optimum parameter $\beta$ that lets the upper bound of the inequality be maximum numerically. Contrary to our expectations, the coherent state occasionally exhibits quantum quality more strongly than the cat state for the upper bound of the violation of the LGI in a specific range of three equally spaced measurement times (spacing $\tau$). Moreover, as we let $\tau$ approach zero, the optimized parameter $\beta$ diverges and the LGI reveals intense singularity.

Graphic Abstract

Appendices

The mathematical forms used in Sect. 3

An explicit form of $w_{1\pm }(\tau )$, the time evolution of $w_{1\pm }(0)$ given by Eqs. (17) and (21), is written down as follows:

$$\begin{aligned} w_{1\pm }(\tau )= & {} \frac{1}{4} \Biggl \{ |\alpha e^{-i\varOmega \tau }\rangle \langle \alpha e^{-i\varOmega \tau }| \nonumber \\&\pm \exp \{ \beta ^{*}\alpha -\beta \alpha ^{*} -(1/2)[|2\beta -\alpha |^{2}+|\alpha |^{2}\nonumber \\&-2(2\beta -\alpha )\alpha ^{*}](1-e^{-2\varGamma \tau }) \} |(2\beta -\alpha )e^{-i\varOmega \tau }\rangle \langle \alpha e^{-i\varOmega \tau }| \nonumber \\&\pm (\text{ the } \text{ Hermitian } \text{ conjugate } \text{ of } \text{ the } \text{ above } \text{ term}) \nonumber \\&+ |(2\beta -\alpha )e^{-i\varOmega \tau }\rangle \langle (2\beta -\alpha )e^{-i\varOmega \tau }| \Biggr \}. \end{aligned}$$

(74)

Explicit forms of $p_{1\pm ,2+}$ and $p_{1\pm ,2-}$, the probabilities that $O_{2}=1$ and $O_{2}=-1$ are observed with the measurement on the above $w_{1\pm }(\tau )$ at time $t_{2}$, respectively, are given by

$$\begin{aligned}&p_{1\pm ,2+} = \text{ Tr } [\varPi ^{(+)}(\beta )w_{1\pm }(\tau )] \nonumber \\&\quad = \frac{1}{4} \Biggl \{ \exp (-|\alpha e^{-i\varOmega \tau }-\beta |^{2}) \cosh |\alpha e^{-i\varOmega \tau }-\beta |^{2} \nonumber \\&\qquad \pm 2\text{ Re } \Biggr [ \exp \{ \beta ^{*}\alpha -\beta \alpha ^{*} -(1/2)[|2\beta -\alpha |^{2}+|\alpha |^{2}\nonumber \\&\qquad -2(2\beta -\alpha )\alpha ^{*}](1-e^{-2\varGamma \tau }) \nonumber \\&\qquad -\beta (\beta ^{*}-\alpha ^{*})\exp (i\varOmega ^{*}\tau )+\beta ^{*}(\beta -\alpha )\exp (-i\varOmega \tau ) \nonumber \\&\qquad -(1/2)[|\alpha e^{-i\varOmega \tau }-\beta |^{2}+|(2\beta -\alpha )e^{-i\varOmega \tau }-\beta |^{2}] \}\nonumber \\&\qquad \times \cosh [(\alpha ^{*}\exp (i\varOmega ^{*}\tau )-\beta ^{*}) ((2\beta -\alpha )\exp (-i\varOmega \tau )-\beta )] \Biggl ] \nonumber \\&\qquad + \exp [-|(2\beta -\alpha )e^{-i\varOmega \tau }-\beta |^{2}] \cosh |(2\beta -\alpha )e^{-i\varOmega \tau }-\beta |^{2} \Biggr \}, \end{aligned}$$

(75)

$$\begin{aligned}&p_{1\pm ,2-} = \text{ Tr } [\varPi ^{(-)}(\beta )w_{1\pm }(\tau )] \nonumber \\&\quad = \frac{1}{4} \Biggl \{ \exp (-|\alpha e^{-i\varOmega \tau }-\beta |^{2}) \sinh |\alpha e^{-i\varOmega \tau }-\beta |^{2} \nonumber \\&\qquad \pm 2\text{ Re } \Biggr [ \exp \{ \beta ^{*}\alpha -\beta \alpha ^{*} -(1/2)[|2\beta -\alpha |^{2}+|\alpha |^{2}\nonumber \\&\qquad -2(2\beta -\alpha )\alpha ^{*}](1-e^{-2\varGamma \tau }) \nonumber \\&\qquad -\beta (\beta ^{*}-\alpha ^{*})\exp (i\varOmega ^{*}\tau )+\beta ^{*}(\beta -\alpha )\exp (-i\varOmega \tau ) \nonumber \\&\qquad -(1/2)[|\alpha e^{-i\varOmega \tau }-\beta |^{2}+|(2\beta -\alpha )e^{-i\varOmega \tau }-\beta |^{2}] \} \nonumber \\&\qquad \times \sinh [(\alpha ^{*}\exp (i\varOmega ^{*}\tau )-\beta ^{*}) ((2\beta -\alpha )\exp (-i\varOmega \tau )-\beta )] \Biggl ] \nonumber \\&\qquad + \exp [-|(2\beta -\alpha )e^{-i\varOmega \tau }-\beta |^{2}] \sinh |(2\beta -\alpha )e^{-i\varOmega \tau }-\beta |^{2} \Biggr \}. \end{aligned}$$

(76)

The mathematical forms used in Sect. 4

Explicit forms of $\{K^{(j)}(0):j=1,2,4\}$, $\{L^{(j)}(0):j=1,2,3,4\}$, and $\{M^{(j)}(0):j=1,2,4\}$ appearing in Eq. (35) are given by

$$\begin{aligned} K^{(1)}(0)= & {} |\alpha \rangle \langle \alpha |, \nonumber \\ K^{(2)}(0)= & {} \exp (\beta \alpha ^{*}-\beta ^{*}\alpha )|\alpha \rangle \langle -\alpha +2\beta |, \nonumber \\ K^{(4)}(0)= & {} |-\alpha +2\beta \rangle \langle -\alpha +2\beta |, \end{aligned}$$

(77)

$$\begin{aligned} L^{(1)}(0)= & {} |\alpha \rangle \langle -\alpha |, \nonumber \\ L^{(2)}(0)= & {} \exp (-\beta \alpha ^{*}+\beta ^{*}\alpha )|\alpha \rangle \langle \alpha +2\beta |, \nonumber \\ L^{(3)}(0)= & {} \exp (-\beta \alpha ^{*}+\beta ^{*}\alpha )|-\alpha +2\beta \rangle \langle -\alpha |, \nonumber \\ L^{(4)}(0)= & {} \exp [2(-\beta \alpha ^{*}+\beta ^{*}\alpha )]|-\alpha +2\beta \rangle \langle \alpha +2\beta |, \nonumber \\ \end{aligned}$$

(78)

$$\begin{aligned} M^{(1)}(0)= & {} |-\alpha \rangle \langle -\alpha |, \nonumber \\ M^{(2)}(0)= & {} \exp (-\beta \alpha ^{*}+\beta ^{*}\alpha )|-\alpha \rangle \langle \alpha +2\beta |, \nonumber \\ M^{(4)}(0)= & {} |\alpha +2\beta \rangle \langle \alpha +2\beta |. \end{aligned}$$

(79)

Because of Eq. (9), time evolution from time $t_{1}=0$ to time $t_{2}=\tau$ of the above operators, $\{K^{(j)}(\tau ):j=1,2,4\}$, $\{L^{(j)}(\tau ):j=1,2,3,4\}$, and $\{M^{(j)}(\tau ):j=1,2,4\}$, are described in the forms,

$$\begin{aligned} K^{(1)}(\tau )= & {} |\alpha \exp (-i\varOmega \tau )\rangle \langle \alpha \exp (-i\varOmega \tau )|, \nonumber \\ K^{(2)}(\tau )= & {} \exp \{\beta \alpha ^{*}-\beta ^{*}\alpha -(1/2)[|\alpha |^{2}\nonumber \\&\quad +|-\alpha +2\beta |^{2}\nonumber \\&\quad -2\alpha (-\alpha ^{*}+2\beta ^{*})] [1-\exp (-2\varGamma \tau )]\} \nonumber \\&\quad \times |\alpha \exp (-i\varOmega \tau )\rangle \langle (-\alpha +2\beta ) \exp (-i\varOmega \tau )|, \nonumber \\ K^{(4)}(\tau )= & {} |(-\alpha +2\beta ) \exp (-i\varOmega \tau )\rangle \langle (-\alpha +2\beta ) \exp (-i\varOmega \tau )|, \nonumber \\ \end{aligned}$$

(80)

$$\begin{aligned} L^{(1)}(\tau )= & {} \exp \{-2|\alpha |^{2}[1-\exp (-2\varGamma \tau )]\} \nonumber \\&\quad \times |\alpha \exp (-i\varOmega \tau )\rangle \langle -\alpha \exp (-i\varOmega \tau )|, \nonumber \\ L^{(2)}(\tau )= & {} \exp \{-\beta \alpha ^{*}+\beta ^{*}\alpha -(1/2)[|\alpha |^{2}+|\alpha +2\beta |^{2}\nonumber \\&\quad -2\alpha (\alpha ^{*}+2\beta ^{*})] [1-\exp (-2\varGamma \tau )]\} \nonumber \\&\quad \times |\alpha \exp (-i\varOmega \tau )\rangle \langle (\alpha +2\beta )\exp (-i\varOmega \tau )|, \nonumber \\ L^{(3)}(\tau )= & {} \exp \{-\beta \alpha ^{*}+\beta ^{*}\alpha \nonumber \\&\quad -(1/2)[|-\alpha +2\beta |^{2}\nonumber \\&\quad +|\alpha |^{2}+2(-\alpha +2\beta )\alpha ^{*}] [1-\exp (-2\varGamma \tau )]\} \nonumber \\&\quad \times |(-\alpha +2\beta )\exp (-i\varOmega \tau )\rangle \langle -\alpha \exp (-i\varOmega \tau )|, \nonumber \\ L^{(4)}(\tau )= & {} \exp \{2(-\beta \alpha ^{*}+\beta ^{*}\alpha ) -(1/2)[|-\alpha +2\beta |^{2}\nonumber \\&\quad +|\alpha +2\beta |^{2} -2(-\alpha +2\beta )(\alpha ^{*}+2\beta ^{*})]\nonumber \\&\quad \times [1-\exp (-2\varGamma \tau )]\} \nonumber \\&\quad \times |(-\alpha +2\beta )\exp (-i\varOmega \tau )\rangle \langle (\alpha +2\beta )\exp (-i\varOmega \tau )|, \nonumber \\ \end{aligned}$$

(81)

$$\begin{aligned} M^{(1)}(\tau )= & {} |-\alpha \exp (-i\varOmega \tau )\rangle \langle -\alpha \exp (-i\varOmega \tau )|, \nonumber \\ M^{(2)}(\tau )= & {} \exp \{-\beta \alpha ^{*}+\beta ^{*}\alpha -(1/2)[|\alpha |^{2}+|\alpha +2\beta |^{2}\nonumber \\&\quad +2\alpha (\alpha ^{*}+2\beta ^{*})] [1-\exp (-2\varGamma \tau )]\} \nonumber \\&\quad \times |-\alpha \exp (-i\varOmega \tau )\rangle \langle (\alpha +2\beta )\exp (-i\varOmega \tau )|, \nonumber \\ M^{(4)}(\tau )= & {} |(\alpha +2\beta )\exp (-i\varOmega \tau )\rangle \langle (\alpha +2\beta )\exp (-i\varOmega \tau )|.\nonumber \\ \end{aligned}$$

(82)

A mathematically rigorous form of $p_{1\pm ,2+}$, the probability that $O_{2}=1$ is obtained with the measurement on $w_{1\pm }(\tau )$ at time $t_{2}$, is given by

$$\begin{aligned} p_{1\pm ,2+}= & {} (1/4)q(\alpha )^{-1} \nonumber \\&\times \Bigl ( \text{ Tr }[K^{(1)}(\tau )\varPi ^{(+)}(\beta )] \pm 2\text{ Re } \{ \text{ Tr }[K^{(2)}(\tau )\varPi ^{(+)}(\beta )] \}\nonumber \\&+\text{ Tr }[K^{(4)}(\tau )\varPi ^{(+)}(\beta )] \nonumber \\&+ 2\text{ Re } \{ \text{ Tr }[L^{(1)}(\tau )\varPi ^{(+)}(\beta )] \pm \text{ Tr }[L^{(2)}(\tau )\varPi ^{(+)}(\beta )] \nonumber \\&\pm \text{ Tr }[L^{(3)}(\tau )\varPi ^{(+)}(\beta )] + \text{ Tr }[L^{(4)}(\tau )\varPi ^{(+)}(\beta )] \} \nonumber \\&+ \text{ Tr }[M^{(1)}(\tau )\varPi ^{(+)}(\beta )] \pm 2\text{ Re } \{ \text{ Tr }[M^{(2)}(\tau )\varPi ^{(+)}(\beta )] \}\nonumber \\&+\text{ Tr }[M^{(4)}(\tau )\varPi ^{(+)}(\beta )] \Bigr ), \end{aligned}$$

(83)

where explicit forms of $\{\text{ Tr }[K^{(j)}(\tau )\varPi ^{(+)}(\beta )]:j=1,2,4\}$, $\{\text{ Tr }[L^{(j)}(\tau )\varPi ^{(+)}(\beta )]:j=1,2,3,4\}$, and $\{\text{ Tr }[M^{(j)}(\tau )\varPi ^{(+)}(\beta )]:j=1,2,4\}$ are written as

$$\begin{aligned}&\text{ Tr }[K^{(1)}(\tau )\varPi ^{(+)}(\beta )] \nonumber \\&\quad = \exp [-|\alpha \exp (-i\varOmega \tau )-\beta |^{2}]\cosh |\alpha \exp (-i\varOmega \tau )-\beta |^{2}, \nonumber \\&\text{ Tr }[K^{(2)}(\tau )\varPi ^{(+)}(\beta )] \nonumber \\&\quad = \exp \{ \beta \alpha ^{*}-\beta ^{*}\alpha -(1/2)[|\alpha |^{2}+|-\alpha +2\beta |^{2}\nonumber \\&\qquad -2\alpha (-\alpha ^{*}+2\beta ^{*})] [1-\exp (-2\varGamma \tau )] \nonumber \\&\qquad -[\beta \alpha ^{*}\exp (i\varOmega ^{*}\tau )-\beta ^{*}\alpha \exp (-i\varOmega \tau )] \nonumber \\&\qquad +2i|\beta |^{2}\sin \omega \tau \exp (-\varGamma \tau ) \nonumber \\&\qquad -(1/2)[|\alpha \exp (-i\varOmega \tau )-\beta |^{2}\nonumber \\&\qquad +|(-\alpha +2\beta )\exp (-i\varOmega \tau )-\beta |^{2}] \} \nonumber \\&\qquad \times \cosh \{[(-\alpha ^{*}+2\beta ^{*})\exp (i\varOmega ^{*}\tau ) -\beta ^{*}]\nonumber \\&\qquad \times [\alpha \exp (-i\varOmega \tau )-\beta ]\}, \nonumber \\&\text{ Tr }[K^{(4)}(\tau )\varPi ^{(+)}(\beta )] \nonumber \\&\quad = \exp [-|(-\alpha +2\beta )\exp (-i\varOmega \tau )-\beta |^{2}]\nonumber \\&\qquad \quad \times \cosh |(-\alpha +2\beta )\exp (-i\varOmega \tau )-\beta |^{2}, \end{aligned}$$

(84)

$$\begin{aligned}&\text{ Tr }[L^{(1)}(\tau )\varPi ^{(+)}(\beta )]\nonumber \\&\quad = \exp \{ -2|\alpha |^{2}[1-\exp (-2\varGamma \tau )]\nonumber \\&\qquad -[\beta \alpha ^{*}\exp (i\varOmega ^{*}\tau )-\beta ^{*}\alpha \exp (-i\varOmega \tau )] \nonumber \\&\qquad -(1/2)[|\alpha \exp (-i\varOmega \tau )-\beta |^{2}+|\alpha \exp (-i\varOmega \tau )+\beta |^{2}] \} \nonumber \\&\qquad \times \cosh \{-[\alpha ^{*}\exp (i\varOmega ^{*}\tau )+\beta ^{*}]\nonumber \\&\qquad \times [\alpha \exp (-i\varOmega \tau )-\beta ]\}, \nonumber \\&\text{ Tr }[L^{(2)}(\tau )\varPi ^{(+)}(\beta )] \nonumber \\&\quad = \exp \{ -\beta \alpha ^{*}+\beta ^{*}\alpha -(1/2)[|\alpha |^{2}+|\alpha +2\beta |^{2} \nonumber \\&\qquad -2\alpha (\alpha ^{*}+2\beta ^{*})] [1-\exp (-2\varGamma \tau )] \nonumber \\&\qquad +2i|\beta |^{2}\sin \omega \tau \exp (-\varGamma \tau ) -(1/2)[|\alpha \exp (-i\varOmega \tau )-\beta |^{2}\nonumber \\&\qquad +|(\alpha +2\beta )\exp (-i\varOmega \tau )-\beta |^{2}] \} \nonumber \\&\qquad \times \cosh \{[(\alpha ^{*}+2\beta ^{*})\exp (i\varOmega ^{*}\tau )-\beta ^{*}]\nonumber \\&\qquad \times [\alpha \exp (-i\varOmega \tau )-\beta ]\}, \nonumber \\&\text{ Tr }[L^{(3)}(\tau )\varPi ^{(+)}(\beta )] \nonumber \\&\quad = \exp \{ -\beta \alpha ^{*}+\beta ^{*}\alpha -(1/2)[|-\alpha +2\beta |^{2}\nonumber \\&\qquad +|\alpha |^{2}+2(-\alpha +2\beta )\alpha ^{*}] [1-\exp (-2\varGamma \tau )] \nonumber \\&\qquad -2i|\beta |^{2}\sin \omega \tau \exp (-\varGamma \tau )\nonumber \\&\qquad -(1/2)[|(-\alpha +2\beta )\exp (-i\varOmega \tau )-\beta |^{2}\nonumber \\&\qquad +|\alpha \exp (-i\varOmega \tau )+\beta |^{2}] \} \nonumber \\&\qquad \times \cosh \{-[\alpha ^{*}\exp (i\varOmega ^{*}\tau )+\beta ^{*}]\nonumber \\&\qquad \times [(-\alpha +2\beta )\exp (-i\varOmega \tau )-\beta ]\}, \nonumber \\&\text{ Tr }[L^{(4)}(\tau )\varPi ^{(+)}(\beta )] \nonumber \\&\quad = \exp \{ 2(-\beta \alpha ^{*}+\beta ^{*}\alpha ) -(1/2)[|-\alpha +2 \nonumber \\&\qquad \beta |^{2}+|\alpha +2\beta |^{2}\nonumber \\&\qquad -2(-\alpha +2\beta )(\alpha ^{*}+2\beta ^{*})] [1-\exp (-2\varGamma \tau )] \nonumber \\&\qquad -[-\beta \alpha ^{*}\exp (i\varOmega ^{*}\tau ) +\beta ^{*}\alpha \exp (-i\varOmega \tau )] \nonumber \\&\qquad -(1/2)[|(-\alpha +2\beta )\exp (-i\varOmega \tau )-\beta |^{2}\nonumber \\&\qquad +|(\alpha +2\beta )\exp (-i\varOmega \tau )-\beta |^{2}] \} \nonumber \\&\qquad \times \cosh \{[(\alpha ^{*}+2\beta ^{*})\exp (i\varOmega ^{*}\tau )-\beta ^{*}]\nonumber \\&\qquad \times [(-\alpha +2\beta )\exp (-i\varOmega \tau )-\beta ]\}, \end{aligned}$$

(85)

$$\begin{aligned}&\text{ Tr }[M^{(1)}(\tau )\varPi ^{(+)}(\beta )] \nonumber \\&\quad = \exp [-|\alpha \exp (-i\varOmega \tau )+\beta |^{2}]\nonumber \\&\qquad \quad \times \cosh |\alpha \exp (-i\varOmega \tau )+\beta |^{2}, \nonumber \\&\text{ Tr }[M^{(2)}(\tau )\varPi ^{(+)}(\beta )] \nonumber \\&\quad = \exp \{ -\beta \alpha ^{*}+\beta ^{*}\alpha \nonumber \\&\qquad -(1/2)[|\alpha |^{2}+|\alpha +2\beta |^{2}+2\alpha (\alpha ^{*}+2\beta ^{*})]\nonumber \\&[1-\exp (-2\varGamma \tau )] +[\beta \alpha ^{*}\exp (i\varOmega ^{*}\tau )-\beta ^{*}\alpha \exp (-i\varOmega \tau )]\nonumber \\&\qquad +2i|\beta |^{2}\sin \omega \tau \exp (-\varGamma \tau ) \nonumber \\&\qquad -(1/2)[|\alpha \exp (-i\varOmega \tau )+\beta |^{2}\nonumber \\&\qquad +|(\alpha +2\beta )\exp (-i\varOmega \tau )-\beta |^{2}] \} \nonumber \\&\qquad \times \cosh \{-[(\alpha ^{*}+2\beta ^{*})\exp (i\varOmega ^{*}\tau )-\beta ^{*}]\nonumber \\&\qquad \times [\alpha \exp (-i\varOmega \tau )+\beta ]\}, \nonumber \\&\text{ Tr }[M^{(4)}(\tau )\varPi ^{(+)}(\beta )] \nonumber \\&\quad = \exp [-|(\alpha +2\beta )\exp (-i\varOmega \tau )-\beta |^{2}]\nonumber \\&\qquad \quad \times \cosh |(\alpha +2\beta )\exp (-i\varOmega \tau )-\beta |^{2}. \end{aligned}$$

(86)

An explicit form of $p_{1\pm ,2-}$, the probability that $O_{2}=-1$ is obtained with the measurement on $w_{1\pm }(\tau )$ at time $t_{2}$, is described in the form,

$$\begin{aligned} p_{1\pm ,2-}= & {} (1/4)q(\alpha )^{-1} \nonumber \\&\times \Bigl ( \text{ Tr }[K^{(1)}(\tau )\varPi ^{(-)}(\beta )] \pm 2\text{ Re } \{ \text{ Tr }[K^{(2)}(\tau )\varPi ^{(-)}(\beta )] \}\nonumber \\&+\text{ Tr }[K^{(4)}(\tau )\varPi ^{(-)}(\beta )] \nonumber \\&+ 2\text{ Re } \{ \text{ Tr }[L^{(1)}(\tau )\varPi ^{(-)}(\beta )] \pm \text{ Tr }[L^{(2)}(\tau )\varPi ^{(-)}(\beta )] \nonumber \\&\pm \text{ Tr }[L^{(3)}(\tau )\varPi ^{(-)}(\beta )] + \text{ Tr }[L^{(4)}(\tau )\varPi ^{(-)}(\beta )] \} \nonumber \\&+ \text{ Tr }[M^{(1)}(\tau )\varPi ^{(-)}(\beta )] \pm 2\text{ Re } \{ \text{ Tr }[M^{(2)}(\tau )\varPi ^{(-)}(\beta )] \}\nonumber \\&+\text{ Tr }[M^{(4)}(\tau )\varPi ^{(-)}(\beta )] \Bigr ), \end{aligned}$$

(87)

where $\{\text{ Tr }[K^{(j)}(\tau )\varPi ^{(-)}(\beta )]:j=1,2,4\}$, $\{\text{ Tr }[L^{(j)}(\tau )\varPi ^{(-)}(\beta )]:j=1,2,3,4\}$, and $\{\text{ Tr }[M^{(j)}(\tau )\varPi ^{(-)}(\beta )]:j=1,2,4\}$ are given by $\{\text{ Tr }[K^{(j)}(\tau )\varPi ^{(+)}(\beta )]:j=1,2,4\}$, $\{\text{ Tr }[L^{(j)}(\tau )\varPi ^{(+)}(\beta )]:j=1,2,3,4\}$, and $\{\text{ Tr }[M^{(j)}(\tau )\varPi ^{(+)}(\beta )]:j=1,2,4\}$ in Eqs. (84), (85), and (86) with replacing $\cosh$ with $\sinh$, that is, substitution of hyperbolic sines for hyperbolic cosines.

Explicit forms of $p_{2\pm ,3+}$ and $p_{2\pm ,3-}$, the probabilities that $O_{3}=1$ and $O_{3}=-1$ are obtained with the measurement on $w_{2\pm }(2\tau )$ at $t_{3}$, respectively, are given by

$$\begin{aligned} p_{2\pm ,3+}= & {} (1/4)q(\alpha )^{-1} \nonumber \\&\times \Bigl ( \text{ Tr }[{\tilde{K}}^{(1)}(\tau )\varPi ^{(+)}(\beta )]\nonumber \\&\pm 2\text{ Re } \{ \text{ Tr }[{\tilde{K}}^{(2)}(\tau )\varPi ^{(+)}(\beta )] \}\nonumber \\&+\text{ Tr }[{\tilde{K}}^{(4)}(\tau )\varPi ^{(+)}(\beta )] \nonumber \\&+ 2\exp [-2|\alpha |^{2}(1-e^{-2\varGamma \tau })] \nonumber \\&\times \text{ Re } \{ \text{ Tr }[{\tilde{L}}^{(1)}(\tau )\varPi ^{(+)}(\beta )] \pm \text{ Tr }[{\tilde{L}}^{(2)}(\tau )\varPi ^{(+)}(\beta )] \nonumber \\&\pm \text{ Tr }[{\tilde{L}}^{(3)}(\tau )\varPi ^{(+)}(\beta )] \nonumber \\&+ \text{ Tr }[{\tilde{L}}^{(4)}(\tau )\varPi ^{(+)}(\beta )] \} \nonumber \\&+ \text{ Tr }[{\tilde{M}}^{(1)}(\tau )\varPi ^{(+)}(\beta )] \nonumber \\&\pm 2\text{ Re } \{ \text{ Tr }[{\tilde{M}}^{(2)}(\tau )\varPi ^{(+)}(\beta )] \}\nonumber \\&+\text{ Tr }[{\tilde{M}}^{(4)}(\tau )\varPi ^{(+)}(\beta )] \Bigr ), \end{aligned}$$

(88)

$$\begin{aligned} p_{2\pm ,3-}= & {} (1/4)q(\alpha )^{-1} \nonumber \\&\times \Bigl ( \text{ Tr }[{\tilde{K}}^{(1)}(\tau )\varPi ^{(-)}(\beta )]\nonumber \\&\pm 2\text{ Re } \{ \text{ Tr }[{\tilde{K}}^{(2)}(\tau )\varPi ^{(-)}(\beta )] \}\nonumber \\&+\text{ Tr }[{\tilde{K}}^{(4)}(\tau )\varPi ^{(-)}(\beta )] \nonumber \\&+ 2\exp [-2|\alpha |^{2}(1-e^{-2\varGamma \tau })] \nonumber \\&\times \text{ Re } \{ \text{ Tr }[{\tilde{L}}^{(1)}(\tau )\varPi ^{(-)}(\beta )] \nonumber \\&\pm \text{ Tr }[{\tilde{L}}^{(2)}(\tau )\varPi ^{(-)}(\beta )] \pm \text{ Tr }[{\tilde{L}}^{(3)}(\tau )\varPi ^{(-)}(\beta )] \nonumber \\&+ \text{ Tr }[{\tilde{L}}^{(4)}(\tau )\varPi ^{(-)}(\beta )] \} \nonumber \\&+ \text{ Tr }[{\tilde{M}}^{(1)}(\tau )\varPi ^{(-)}(\beta )] \nonumber \\&\pm 2\text{ Re } \{ \text{ Tr }[{\tilde{M}}^{(2)}(\tau )\varPi ^{(-)}(\beta )] \}\nonumber \\&+\text{ Tr }[{\tilde{M}}^{(4)}(\tau )\varPi ^{(-)}(\beta )] \Bigr ), \end{aligned}$$

(89)

where

$$\begin{aligned} \text{ Tr }[{\tilde{K}}^{(j)}(\tau )\varPi ^{(\pm )}(\beta )]= & {} \left. \text{ Tr }[K^{(j)}(\tau )\varPi ^{(\pm )}(\beta )] \right| _{\alpha \rightarrow \alpha \exp (-i\varOmega \tau )}, \nonumber \\ \text{ Tr }[{\tilde{L}}^{(j)}(\tau )\varPi ^{(\pm )}(\beta )]= & {} \left. \text{ Tr }[L^{(j)}(\tau )\varPi ^{(\pm )}(\beta )] \right| _{\alpha \rightarrow \alpha \exp (-i\varOmega \tau )}, \nonumber \\ \text{ Tr }[{\tilde{M}}^{(j)}(\tau )\varPi ^{(\pm )}(\beta )]= & {} \left. \text{ Tr }[M^{(j)}(\tau )\varPi ^{(\pm )}(\beta )] \right| _{\alpha \rightarrow \alpha \exp (-i\varOmega \tau )} \quad \nonumber \\&\quad \hbox { for}\ j=1,2,3,4. \end{aligned}$$

(90)