⇦Chinese Physics Letters, 2019, Vol. 36, No. 5, Article code 050302 Complementarity via Minimum Error Measurement in a Two-Path Interferometer * Junzhao Liu (刘君昭), Yanjun Liu (刘彦军), Jing Lu (卢竞)** Affiliations Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Department of Physics and Synergetic Innovation Center of Quantum Effects and Applications, Hunan Normal University, Changsha 410081 Received 25 December 2018, online 17 April 2019 *Supported by the National Natural Science Foundation of China under Grant Nos 11434011 and 11575058.
**Corresponding author. Email: lujing@hunnu.edu.cn Citation Text: Liu J Z, Liu Y J and Lu J 2019 Chin. Phys. Lett. 36 050302    Abstract We study the fringe visibility and the which-path information (WPI) of a general Mach–Zehnder interferometer with an asymmetric beam splitter (BS). A minimum error measurement in the detector is used to extract the WPI. Both the fringe visibility $V$ and the WPI $I_{\rm path}$ are affected by the initial state of the photon and the second asymmetric BS. The condition in which the WPI takes the maximum is obtained. The complementarity relationship $V^{2}+I_{\rm path}^{2}\leq 1$ is found, and the conditions for equality are also presented. DOI:10.1088/0256-307X/36/5/050302 PACS:03.67.-a, 03.65.Ta, 07.60.Ly © 2019 Chinese Physics Society Article Text Bohr's complementarity principle[1] indicates that a single quantum system has mutually exclusive properties, and the appearance of these two properties is determined by the experimental instrument. The wave-particle duality is often used to exhibit Bohr's complementarity principle. For a microscopic particle, the wave-like and particle-like nature exist simultaneously but cannot be observed simultaneously. Once the fringe visibility, which is used to quantify the wave property duality, is observed, the particle properties will disappear. In turn, the fringe visibility will disappear when the path of the particle is determined accurately. Based on the recoiling-slit gedanken experiment from Einstein and Bohr, Wootters and Zurek[2] first proposed their quantitative formulation of the wave-particle duality in the recoiling-slit gedanken experiment. In the standard Mach–Zehnder interferometer (MZI) setup, a priori fringe visibility $V_{0}$ of the interference pattern and the predictability $P_{0}$ were first introduced to quantify wave-particle duality,[3] which satisfy the complementary relationship $V_{0}^{2}+P_{0}^{2}\leq1$. Later, by introducing the concept of entanglement, the wave-like and particle-like properties were characterized by the visibility $V$ of the interferometer fringe and the path distinguishability $D$,[4] respectively, which satisfy the complementary relationship $V^{2}+D^{2}\leq1$. The quantification of wave-particle duality has been studied greatly in theory and experiment.[3–21] In recent years, the complementary relationship of asymmetric MZI interferometers has also attracted great attention,[22–25] and additional a priori which-path information (WPI) is introduced when the BS2 is asymmetrical. A two-path interferometer, such as an MZI, is usually used to study wave-particle duality. Here the mutual information, the WPI,[26] is used to characterize the particle nature of a quantum system. To obtain the WPI, a detector is placed on one of the paths of the MZI. The information cannot be obtained until a measurement is performed, and an error-minimum state-distinguishing measurement[27,28] is performed on the detector after the particle interacts with the detector to acquire the WPI. The minimum error measurement can obtain a conclusive result, but the error is unavoidable. In this work, we study the trade-off between the fringe visibility and WPI in a general MZI with an asymmetric beam splitter (BS). The minimum error measurement is used to obtain the WPI. It is found that the magnitude of the WPI is affected by the asymmetric BS and the input state of the particle. The upper bound of the WPI is obtained, and the condition of reaching the upper limit is found. A complementarity relationship between the fringe visibility and the WPI is also proposed. A general MZI has two BSs and phase shifters (PSs) as shown in Fig. 1. An incident photon is split into two paths by the 50:50 BS1. Two possible paths $a$ and $b$ are denoted by orthogonal normalized states $|a\rangle$, $|b\rangle$. States $|a\rangle$ and $|b\rangle$ support a two-dimensional $H_{\rm q}$. The phase of states $|a\rangle$ and $|b\rangle$ are adjusted by two phase shifts PS1 and PS2. When the photon propagates in these two paths, PS1 and PS2 perform a rotation on the path qubit $$ U_{\rm P}(\phi)=\exp (-i\phi\sigma_{z}),~~ \tag {1} $$ where the Pauli operator $\sigma_{z}=|b\rangle\langle b|-|a\rangle\langle a|$. Finally, these paths are recombined by the asymmetric BS2. The effect of the BS2 on the photon is described by the operator $$ U_{\rm B}(\beta)=\exp \Big(-i\frac{\beta}{2}\sigma_{y}\Big) =\left(\begin{array}{cc} \sqrt{t} & -\sqrt{r} \\ \sqrt{r} & \sqrt{t} \end{array}\right),~~ \tag {2} $$ which denotes a rotation around the $y$-axis by an angle $\beta$. Here $r$ and $t=1-r$ are the reflection coefficient and transmission coefficient of the BS, respectively.
cpl-36-5-050302-fig1.png
Fig. 1. The schematic sketch of the general MZI with the second asymmetric BS.
To obtain the WPI, a detector is introduced. As long as the photon goes through the general MZI, the operator which acts on the detector is $$ M=\frac{1+\sigma_{Z}}{2}I+\frac{1-\sigma_{Z}}{2}U,~~ \tag {3} $$ where $I$ and $U$ are the identical and unitary operator, respectively. The initial state of the photon is described by the density operator $$ \rho_{\rm in}^{Q}=\frac{1}{2}(1+S_{x}\sigma_{x}+S_{y}\sigma _{y}+S_{z}\sigma_{z}),~~ \tag {4} $$ with the Bloch vector $\overrightarrow{S}=({S_{x}},{S_{y}},{S_{z}})$. After the photon passes through the BS2, the initial state of the photon and the detector is evolved into $$\begin{align} \rho_{\rm f}=\,&U_{\rm B}(\beta)MU_{\rm P}(\varphi)U_{\rm B}\Big(\frac{\pi}{2}\Big)\rho_{\rm in}^{Q}\rho_{\rm in}^{\rm D}\\ &\cdot U_{\rm B}^†\Big(\frac{\pi}{2}\Big)U_{\rm P}^†(\varphi)M^†U_{\rm B}^†(\beta) \\ =\,&\frac{1}{4}(1-S_{x}) (1+\sigma_{z}\cos \beta+\sigma_{x}\sin \beta) \otimes \rho_{\rm in}^{\rm D} \\ &-\frac{1}{4}e^{-i\phi}(S_{z}-iS_{y}) (\sigma_{z}\sin\beta -\sigma_{x}\cos \beta\\ &-i\sigma_{y}) \otimes \rho_{\rm in}^{\rm D}U^†-\frac{1}{4}e^{i\phi}(S_{z}+iS_{y}) (\sigma_{z}\sin \beta\\ &-\sigma_{x}\cos\beta +i\sigma_{y}) \otimes U\rho_{\rm in}^{\rm D}+\frac{1}{4}(1+S_{x}) (1\\ &-\sigma_{z}\cos\beta -\sigma_{x}\sin \beta) \otimes U\rho_{\rm in}^{\rm D}U^†,~~ \tag {5} \end{align} $$ where $\rho_{\rm in}^{\rm D}$ is the initial state of the detector. The probability that the photon is detected at output port $a$ reads $$\begin{align} p(\phi)=\,&{\rm tr}_{\rm QD}[|a\rangle \langle a|\rho_{\rm f}] \\ =\,& \frac{1}{2}(1+ S_{x}\cos\beta)+\frac{1}{2}\sqrt{S_{z}^{2}+S_{y}^{2}}\\ &\cdot\sin\beta|{\rm tr}_{\rm D}(U\rho_{\rm in}^{\rm D})| \cos (\alpha +\gamma +\phi),~~ \tag {6} \end{align} $$ where $\alpha$ and $\gamma$ are defined as $$ \alpha= \arctan\frac{S_{y}}{S_{z}},~~\gamma= -i\ln\frac{{\rm tr}_{\rm D}(U\rho_{\rm in}^{\rm D})}{|{\rm tr}_{\rm D}(U\rho_{\rm in}^{\rm D})|}.~~ \tag {7} $$ The maximum and minimum of $p(\phi)$ are adjusted by $\phi$. The fringe visibility, which characterizes the wave-like property of the particle, is defined via the probability in Eq. (6) as $$\begin{align} V =\,&\frac{\max P(\phi) -\min P(\phi)}{\max P(\phi) +\min P(\phi)} \\ =\,&\frac{\sin \beta}{1+S_{x}\cos \beta}\sqrt{S_{z}^{2}+S_{y}^{2}}|{\rm tr}_{\rm D}[ U\rho_{\rm in}^{\rm D}]|\\ \leq &|{\rm tr}_{\rm D}[ U\rho_{\rm in}^{\rm D}]|.~~ \tag {8} \end{align} $$ We notice that the fringe visibility measured in either output port $a$ or $b$ is different in a general MZI with an asymmetric BS2. Equation (8) shows that both the BS2 and the initial state of the photon have an influence on the fringe visibility. In Ref.  [24] we found that the fringe visibility obtains the maximum $C\equiv|{\rm tr}_{\rm D}(U\rho_{\rm in} ^{\rm D})|$ when the input photon is initially in a pure state and $\cos \beta=-S_{x}$. Li et al.[23] proposed that the additional a priori WPI is introduced when the BS2 is asymmetrical, thus we cannot obtain the WPI by simply removing the BS2. To study the WPI in the general MZI with the asymmetric BS2, the setup with four input and output ports is introduced.[24] Using the setup with four input and output ports, we cannot obtain the final state of the detector by tracing over the degree of the photon in Eq. (5). In Ref.  [24] we acquire the expression $$\begin{align} \rho^{\rm QD}_{\rm f}=\,&\omega_{\rm b}|b\rangle \langle b|\rho_{\rm in}^{\rm D} +\omega_{\rm a}|a\rangle\langle a|U\rho_{\rm in}^{\rm D}U^† \\ &+\frac{\sqrt{rt}}{1+S_{x}(t-r)}e^{i\phi}(S_{z}+iS_{y})|a\rangle \langle b|U\rho_{\rm in}^{\rm D} \\ &+\frac{\sqrt{rt}}{1+S_{x}(t-r)}e^{-i\phi}(S_{z}-iS_{y})|b\rangle \langle a|\rho_{\rm in}^{\rm D}U^†~~ \tag {9} \end{align} $$ of the final state of the total system when we only detect the photon at the output $a$, where $$ \omega_{\rm a}=\frac{\cos ^{2}\frac{\beta}{2}(1+S_{x})}{1+S_{x}\cos \beta}, ~\omega_{\rm b}=\frac{\sin ^{2}\frac{\beta}{2}(1-S_{x})}{1+S_{x}\cos \beta}.~~ \tag {10} $$ After tracing over the degree of the photon in Eq. (10), we obtain the final state of the detector $$\begin{align} \rho^{\rm D}_{\rm f}= \omega_{\rm b}\rho_{\rm in}^{\rm D}+\omega_{\rm a}\rho_{\rm out}^{\rm D}.~~ \tag {11} \end{align} $$ For simplicity, we take the initial state of the detector as a pure state, which is denoted by $\rho_{\rm in}^{\rm D}=|r\rangle\langle r|$. Since the unitary operator $U$ is arbitrary, states $|r\rangle$ and $|s\rangle \equiv U|r\rangle$ can be assumed to be linearly independent. To obtain the WPI, we discriminate the states $\rho_{\rm in}^{\rm D}$ and $\rho_{\rm out}^{\rm D}=|s\rangle\langle s|$ with minimum error. Mathematically, a discrimination among two nonorthogonal states $|s\rangle$ and $|r\rangle$ is introduced by the positive operator-valued measure $\{{{\it \Pi}_{k},k=a,b}\}$, which satisfies $\sum_{k}{\it \Pi}_{k}=I$. The probability of error is obtained as $$ P_{\rm err}=\omega_{\rm a}{\rm Tr}(\rho_{\rm out}^{\rm D}{\it \Pi}_{\rm b}) +\omega _{\rm b}{\rm Tr}(\rho_{\rm in}^{\rm D}{\it \Pi}_{\rm a}),~~ \tag {12} $$ where we have introduced the Hermitian operator ${\it \Lambda} =\omega_{\rm a}\rho_{\rm out}^{\rm D}-\omega_{\rm b}\rho_{\rm in}^{\rm D}$. The spectral decomposition form is $$\begin{align} {\it \Lambda} =\lambda_{\rm a}|M_{\rm a}\rangle \langle M_{\rm a}|+\lambda_{\rm b}|M_{\rm b}\rangle \langle M_{\rm b}|,~~ \tag {13} \end{align} $$ where $\lambda_{\rm a}(\lambda_{\rm b})$ represents an eigenvalue with a positive (negative) value and the corresponding eigenstate is $|M_{\rm a}\rangle (|M_{\rm b}\rangle)$. Using the spectral decomposition of ${\it \Lambda}$, we can obtain the relationship $$\begin{alignat}{1} \!\!\!\!\!\!\!\!\!\!\!P_{\rm err}=\omega_{\rm a}\!-\!\lambda_{\rm a}\langle M_{\rm a}|{\it \Pi} _{\rm a}|M_{\rm a}\rangle =\omega_{\rm b}\!-\!\lambda_{\rm b}\langle M_{\rm b}|{\it \Pi}_{\rm b}|M_{\rm b}\rangle.~~ \tag {14} \end{alignat} $$
cpl-36-5-050302-fig2.png
Fig. 2. (a) The WPI as functions of $S_{x}$ and $\beta$ with $C=1/2$. (b) The cross-section of 3D surface for $S_{x}=-0.5$, 0, and 0.5. (c) The cross-section of 3D surface for $\beta=\pi/4$, $\pi/2$, and $3\pi/4$.
Apparently, when the projection operators ${\it \Pi}_{\rm a}=|M_{\rm a}\rangle \langle M_{\rm a}|$, ${\it \Pi}_{\rm b}=|M_{\rm b}\rangle \langle M_{\rm b}|$, the error probability $P_{\rm err}$ takes the minimum value. Here $$\begin{alignat}{1} |M_{\rm a}\rangle =\,&-\frac{1-2\omega_{\rm a}+\sqrt{1-4\omega_{\rm a}\omega_{\rm b}C^{2}}}{2\omega_{\rm a}S^{2}A_{\rm a}}|r\rangle \\ &+\frac{1-2\omega_{\rm a}C^{2}+\sqrt{1-4\omega_{\rm a}\omega_{\rm b}C^{2}}}{2\omega _{\rm a}CS^{2}A_{\rm a}}|s\rangle,~~ \tag {15} \end{alignat} $$ $$\begin{alignat}{1} |M_{\rm b}\rangle =\,&-\frac{1-2\omega_{\rm a}-\sqrt{1-4\omega_{\rm a}\omega_{\rm b}C^{2}}}{2\omega_{\rm a}S^{2}A_{\rm b}}|r\rangle \\ &+\frac{1-2\omega_{\rm a}C^{2}-\sqrt{1-4\omega_{\rm a}\omega_{\rm b}C^{2}}}{2\omega _{\rm a}CS^{2}A_{\rm b}}|s\rangle,~~ \tag {16} \end{alignat} $$ where $A_{\rm a}$ ($A_{\rm b}$) is a normalization coefficient of the eigenstate $|M_{\rm a}\rangle$ ($|M_{\rm b}\rangle$), which is denoted by $$\begin{alignat}{1} \!\!\!\!\!\!\!\!A_{\rm a}^{2}=\,&\frac{1\!-\!4\omega_{\rm a}\omega_{\rm b}C^{2} \!+\! (1-2\omega _{\rm a}C^{2}) \sqrt{1\!-\!4\omega_{\rm a}\omega_{\rm b}C^{2}}}{2\omega _{\rm a}^{2}C^{2}(1\!-\!C^{2})},~~ \tag {17} \end{alignat} $$ $$\begin{alignat}{1} \!\!\!\!\!\!\!\!A_{\rm b}^{2}=\,&\frac{1\!-\!4\omega_{\rm a}\omega_{\rm b}C^{2}\!-\!(1\!-\!2\omega _{\rm a}C^{2}) \sqrt{1\!-\!4\omega_{\rm a}\omega_{\rm b}C^{2}}}{2\omega _{\rm a}^{2}C^{2}(1\!-\!C^{2})}.~~ \tag {18} \end{alignat} $$
cpl-36-5-050302-fig3.png
Fig. 3. (a) The WPI as functions of $\beta$ and $C$ with $S_{x}=1/2$. (b) The cross-section of 3D surface for $\beta=2\pi/3$, $3\pi/4$, and $\pi/3$. (c) The cross-section of 3D surface for $C=0.7$, 0.5, and 0.3.
The joint probability[16] $$ Q(\mu,k)=Tr_{\rm D}\langle \mu|{\it \Pi}_{k}\rho^{\rm QD}_{\rm f}|\mu \rangle~~ \tag {19} $$ indicates that the photon travels on the path $\mu \in \{a,b\}$, and a measurement on the detector yields the which-path result $k$. The joint probability is obtained from Eq. (19) as $$\begin{alignat}{1} \!\!\!\!\!\!\!\!\!\!\!\!Q(b,b) =\,&\frac{\omega_{\rm b}}{A_{\rm b}^{2}}, ~Q(a,a)=\frac{(1+\sqrt{1-4\omega_{\rm a}\omega_{\rm b}C^{2}})^{2}}{4\omega_{\rm a}C^{2}A_{\rm a}^{2}},~~ \tag {20} \end{alignat} $$ $$\begin{alignat}{1} \!\!\!\!\!\!\!\!\!\!\!\!Q(b,a) =\,&\frac{\omega_{\rm b}}{A_{\rm a}^{2}}, ~Q(a,b)=\frac{(1-\sqrt{1-4\omega_{\rm a}\omega_{\rm b}C^{2}})^{2}}{4\omega_{\rm a}C^{2}A_{\rm b}^{2}}.~~ \tag {21} \end{alignat} $$ The summation of the which-path results gives the probability of the photon traveling on path $\mu$, $$ Q(\mu) =\sum_{k}Q(\mu,k),~~ \tag {22} $$ and the summation of the photon traveling on path $\mu$ gives the probability of the which-path result being obtained after measuring the detector $$ Q(k) =\sum_{\mu}Q(\mu,k).~~ \tag {23} $$
cpl-36-5-050302-fig4.png
Fig. 4. (a) The WPI as functions of $S_{x}$ and $C$ with $\beta=\pi/3$. (b) The cross-section of 3D surface for $S_{x}=-0.5$, 0, and 0.5. (c) The cross section of 3D surface for $C=0.7$, 0.5, and 0.3.
Then, the amount of the WPI[26] obtained from the minimum error measurement is given by $$\begin{align} &I_{\rm path} =\sum_{\mu =a,b}\sum_{k=a,b}Q(\mu,k)\log \Big(\frac{Q(\mu,k)}{Q(\mu)Q(k)}\Big) \\ &=\frac{\cos \beta \!+\!S_{x}\!-\!\sqrt{(1+S_{x}\cos \beta)^{2}-\sin ^{2}\beta (1-S_{x}^{2}) C^{2}}}{2(1+S_{x}\cos\beta)}\\ &\cdot\log \!\Big\{\!\frac{\!1\!+\!S_{x}\!\cos \beta \!-\!\!\sqrt{(1\!+\!S_{x}\!\cos\beta) ^{2}\!-\!\sin ^{2}\beta (1\!-\!S_{x}^{2}) C^{2}}}{2\cos^{2}\frac{\beta}{2}(1\!+\!S_{x})}\Big\}\\ &+\frac{\cos \beta \!+\!S_{x}\!+\!\sqrt{(1\!+\!S_{x}\cos \beta) ^{2}-\sin^{2}\beta (1-S_{x}^{2}) C^{2}}}{2(1+S_{x}\cos \beta)}\\ &\cdot\log \!\Big\{\frac{\!1\!+\!S_{x}\!\cos \beta\!+\!\!\sqrt{\!(1\!+\!S_{x}\!\cos \beta) ^{2}\!-\!\sin ^{2}\beta (1\!-\!S_{x}^{2}) C^{2}}}{2\cos ^{2}\frac{\beta}{2}(1+S_{x})}\Big\}\\ &+\frac{\sin ^{2}\frac{\beta}{2}(1-S_{x})}{1+S_{x}\cos \beta}\log [C^{2}],~~ \tag {24} \end{align} $$ where $I_{\rm path}$ is determined by parameters $S_{x}$, $\beta$, and $C$. From Eq. (6), we note that the fringe visibility $V$ is determined by all the components of the Bloch vector. Only $S_{x}$ occurs in the expression of the WPI, indicating that $I_{\rm path}$ is independent of $S_{y}$ and $S_{z}$. In Fig. 2(a), we plot the WPI as functions of $S_{x}$ and $\beta$ with the overlap $C=1/2$. In Fig. 2(b) (Fig. 2(c)), we plot $I_{\rm path}$ as a function of the parameter $\beta$ ($S_{x}$) for $C=1/2$ and $S_{x}=-0.5$, 0, and 0.5 ($\beta=\pi/4$, $\pi/2$, and $3\pi/4$) with the solid line, dashed line, and dotted line. From Fig. 2(b) (Fig. 2(c)) we can obtain that $I_{\rm path}$ first increases and then decreases as $\beta$ ($S_{x}$) increases for a given $S_{x}$ ($\beta$). The position along the $S_{x}$ ($\beta$) axis shows that the peak of $I_{\rm path}$ varies with different given $\beta$ ($S_{x}$), and the peak of $I_{\rm path}$ appears at $\beta=\pi/4$ when $S_{x}=-1/\sqrt2$, $\beta=\pi/2$ when $S_{x}=0$, and $\beta=3\pi/4$ when $S_{x}=1/\sqrt2$ in Fig. 2(c); $S_{x}=-0.5$ when $\beta=\pi/3$, $S_{x}=0$ when $\beta=\pi/2$, and $S_{x}=0.5$ when $\beta=2\pi/3$ in Fig. 2(b). It can be observed from Figs. 2(b) and 2(c) that $I_{\rm path}$ has an upper bound when $\cos \beta =-S_{x}$. In Fig. 3(a) (Fig. 4(a)), we plot $I_{\rm path}$ as functions of parameters $\beta (S_{x})$ and $C$ for a given $S_{x}(\beta)$. In Fig. 3(b) (Fig. 3(c)), we plot $I_{\rm path}$ as a function of the parameter $C$ ($\beta$) for $S_{x}=0.5$ and a given $\beta=2\pi/3$, $3\pi/4$, and $\pi/3$ ($C=0.7$, 0.5, and 0.3). Similarly, we plot Figs. 4(b) and 4(c). From Figs. 3(b) and 4(b), we can obtain that $I_{\rm path}$ decreases as $C$ increases. The position at which the peak occurs is fixed for different overlaps $C$, as shown in Figs. 3(c) and 4(c). The peak appears at $\beta=2\pi /3$ when $S_{x}=1/2$, $S_{x}=-1/2$ when $\beta =\pi /3$, which satisfies the relationship $\cos \beta=-S_{x}$. To more clearly reflect the conclusions obtained in Figs. 2–4, we have listed these parameters that make the WPI reach the upper limit in Table 1. By analyzing Figs. 2–4, $I_{\rm path}$ is zero in the following situations: (1) The effect of the BS2 for the photon is full transmission or full reflection, when $\beta=0$ or $\pi$. (2) The photon travels only in the $a$ path or $b$ path, corresponding to $S_{x}=1$ or $S_{x}=-1$. (3) The value of the fringe visibility is 1, corresponding to $C=1$. We find that the peak of $I_{\rm path}$ is determined by $C$, and the position of the upper bound is determined by $S_{x}$ and $\beta$, and when $\cos \beta=-S_{x}$, we obtain the upper bound $$\begin{alignat}{1} I^{\rm max}_{\rm path} =\,&\frac{1+\sqrt{1-C^{2}}}{2}\log (1+\sqrt{1-C^{2}}) \\ &+\frac{1-\sqrt{1-C^{2}}}{2}\log (1-\sqrt{1-C^{2}}).~~ \tag {25} \end{alignat} $$
Table 1. The maximum of the WPI shown in Figs. 2, 3, and 4.
$C$ $\beta$ $S_{x}$ $I_{\rm path}^{\rm max}$
0.3 $\pi$/3 $-$0.5 0.8419
2$\pi$/3 0.5
0.5 $\pi$/4 $-\sqrt{2}/2$ 0.6454
$\pi$/3 $-$0.5
$\pi$/2 0
2$\pi$/3 0.5
3$\pi$/4 $\sqrt{2}/2$
0.7 $\pi$/3 $-$0.5 0.4081
$2\pi$/3 0.5
The relationship of $V_{\rm max}$ and the $I^{\rm max}_{\rm path}$ can be obtained as follows: $$ V_{\rm max}^{2}+I_{\rm path}^{\rm max2}\leq1,~~ \tag {26} $$ by solving the second-order partial derivative of $C$. The equals sign holds in Eq. (26) when $C=0$ or $C=1$. We can obtain the complementary relation of the fringe visibility and the WPI $$\begin{align} V^{2}+I_{\rm path}^{2}\leq 1.~~ \tag {27} \end{align} $$ The equals sign holds in Eq. (27) in the following situations: (1) The value of the fringe visibility is 1, corresponding to $C=1$ and $\cos \beta=-S_{x}$. (2) The value of the WPI is 1, corresponding to $C=0$ and $\cos \beta=-S_{x}$. In summary, we have investigated the complementarity of the fringe visibility and the WPI in a general MZI with the BS2 being asymmetric. The fringe visibility $V$ exhibits an upper bound $C$ when the total system is initially in a pure state and $\cos \beta=-S_{x}$. The WPI $I_{\rm path}$ is obtained via minimum error measurement on the state of the detector. The WPI is dependent on the asymmetric BS2 and the initial state of the photon. The WPI is bounded by Eq. (25), and the maximum of the WPI is achieved when $\cos \beta=-S_{x}$. We obtain the complementary relationship $V^{2}+I_{\rm path}^{2}\leq 1$ between the fringe visibility and the WPI. References The Quantum Postulate and the Recent Development of Atomic Theory1Complementarity in the double-slit experiment: Quantum nonseparability and a quantitative statement of Bohr's principleSimultaneous wave and particle knowledge in a neutron interferometerFringe Visibility and Which-Way Information: An InequalityCoherence and indistinguishabilityComplementarity of one-particle and two-particle interferenceTwo interferometric complementaritiesFringe Visibility and Which-Way Information in an Atom InterferometerQuantitative wave-particle duality in multibeam interferometersThree-beam interference and which-way information in neutron interferometryInformation-Theoretic Approach to Single-Particle and Two-Particle Interference in Multipath InterferometersComment on “Quantitative wave-particle duality in multibeam interferometers”Quantitative complementarity in two-path interferometryComplementarity and entanglement in bipartite qudit systemsDynamical approach to complementarity of interferometersComplementarity of interferometers interacting with the dynamical model of the quantum measurement processAn Inequality Concerning Quantitative Duality in an Asymmetry Two-Way InterferometerInterference and which-state information for nonorthogonal statesWave–Particle Duality: An Information-Based ApproachThe duality of a single particle with an n -dimensional internal degree of freedomDelayed-Choice Test of Quantum Complementarity with Interfering Single PhotonsDuality relations in a two-path interferometer with an asymmetric beam splitterComplementarity via error-free measurement in a two-path interferometerFringe visibility and distinguishability in two-path interferometer with an asymmetric beam splitterQuantum state discrimination
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