A single quantum system has mutually exclusive properties, and these characteristics can be converted to each other depending on the method of observation, which is known as Bohr’s complementarity principle.^[1,2] The well-known example of the complementarity principle is wave-particle duality. A two-path interferometer, such as Young’s double slit or Mach–Zehnder interferometer (MZI), is used to quantify the wave-particle duality. The wave-like property and the particle-like property are shown by the fringe visibility and the which-path information (WPI) of the interferometer, respectively.^[3–6] If the path of the particle is known accurately, the fringe visibility will disappear. The more WPI is obtained, the less the fringe visibility is shown. Wootters and Zurek^[3] proposed the quantitative formulation of the wave-particle duality in the recoiling-slit gedanken experiment. A priori fringe visibility V₀ of the interference pattern and the predictability P₀ were first introduced to quantify the wave-particle duality,^[4] which satisfied the complementary relationship . The inequality V² + D² ≤ 1 is derived by Englert in 1996,^[6] where V is the fringe visibility and D is the path distinguishability. The inequality V² + D² ≤ 1 is also confirmed experimentally.^[7] Since the which-path knowledge can be defined in many ways, the complementarity between the fringe visibility and the which-path knowledge has been studied extensively in theory and experiment.^[5–29]

The complementary relationship between the fringe visibility and the distinguishability has been obtained in the standard MZI.^[6] Later, the complementary relationship in a general MZI with an asymmetric beam splitter (BS) has also attracted great attention.^[7,26,29] The asymmetry of the BS1 is equivalent to changing the initial state of the input particles. Since the complementary relationship is established for all the initial state of the input particles, the asymmetry of the BS1 does not affect the complementary relationship. Reference [26] has proposed that additional a priori WPI is introduced when the BS2 is asymmetrical. Unlike the symmetric MZI, the fringe visibility obtained at two output ports are different in the general MZI, and the a priori WPI is also different when the particle is detected at one or the other output port in the general MZI. In this paper, we study the fringe visibility and the distinguishability in the general MZI with an asymmetric BS. It is found that the magnitudes of fringe visibility and the distinguishability are affected by the asymmetric BS and the input state of the particle. The conditions of the upper bound of the fringe visibility and the lower bound of the distinguishability are also obtained, respectively.

The paper is organized as follows. In Section 2, we introduce the general MZI with an asymmetric BS, and study the state evolution of the particle and the the detector in the apparatus. In Section 3, we obtain the upper bound of the fringe visibility, and present the condition of obtaining the upper bound. In Section 4, the lower bound of the distinguishability is found, and the condition for obtaining the lower bound is also proposed. In Section 5, we make our conclusion.

A general MZI consists of two BSs and two phase shifters (PSs) as shown in Fig. 1. The incident particles are split into two paths by the symmetrical beam splitter BS1. Orthogonal normalized states |a〉 and |b〉 are used to denote two possible paths, which support a two-dimensional H_q. When the particles propagate in these two paths, PS1 and PS2 perform an rotation 1 on the path qubit, where pauli matrix σ_z = |b〉〈b| − |a〉〈a|. Finally, these two paths are recombined by the asymmetric beam splitter BS2. The effect of the BS2 on the particles is denoted by 2 which is equivalent to performing a rotation around the y axis by angle β. The BS2 is symmetrical when β = π/2.

Fig. 1.

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	Fig. 1. The schematic sketch of the general Mach–Zehnder interferometer with the second BS asymmetric, and a WPD is placed in path a.

To obtain the WPI, a which-path detector (WPD) is placed on the path a. When a particle is initially in state 3 go through the general MZI, the operator M = |b〉〈b| ⊗ I + |a〉〈a| ⊗ U is performed on the initial state of the WPD, where I and U are the identical and unitary operator, respectively. In Eq. (3), the Bloch vector S = (S_x,S_y,S_z). After the particle passes through the general MZI, the state of the particle and the detector becomes 4

The probability that the particle is detected at output port a reads 5 where , , α and γ are the phases of S_x + iS_y and , respectively. When λ = 1, the particle is in a pure state, when λ < 1, the particle is in a mixed state. The fringe visibility, which characterizes the wave-like property of the particle, is defined via the probability in Eq. (5) as 6 where the maximum and the minimum is achieved by adjusting ϕ. We note that the expression of the fringe visibility measured in either output port a or b is different in the general MZI.

Equation (6) shows that the fringe visibility is affected by the initial state of the particle and the BS2. It is found that for a given β, more of the wave’s nature appear when the input particle is in a pure state, i.e., λ = 1. To demonstrate the effect of the initial state of the particle and the BS2 on the fringe visibility, the fringe visibility as a function of the parameter S_x(β) for a given β(S_x) with λ = 9/25 or 1 is shown in Fig. 2. In Fig. 2, we have chosen the parameter A = 1/3. In Figs. 2(a) (2(c)), we plot the fringe visibility as a function of the parameter S_x for λ = 9/25 (1) and β = π/4, π/2, 3π/4 in solid line, dash line, and dotted line. From Figs. 2(a) (2(c)), we can obtain that the fringe visibility first increases and then decreases as S_x increases for a given β. The S_x varies from to . The value of the fringe visibility is zero when , since the particle is determined to propagate only in the path a or the path b, which corresponds to or . The position of the peak changes as β changes, and the peak of the fringe visibility appears when S_x = −λ cos β. For a given β, the change of λ affects the position where the upper bound of the fringe visibility reaches. In Fig. 2(b) (2(d)), we plot the fringe visibility as a function of the parameter β for λ = 9/25 (1) and a given S_x. The S_x = −0.5, 0, 0.5 are shown by solid line, dash line, and dotted line, respectively. The fringe visibility first increases and then decreases as β increases for a given S_x, as shown in Figs. 2(b) and 2(d). The β varies from 0 to π. The value of the fringe visibility is zero when β = 0 or π, since the effect of the BS2 for the particle is full transmission or full reflection, which is corresponding to β = 0 or π. The position of the peak changes as S_x changes, and the peak of the fringe visibility appears when S_x = −cos β. Figures 2(b) and 2(d) show that for a given S_x, the change of λ does not affect the position where the upper bound of the fringe visibility reaches. Comparing Figs. 2(a) and 2(c) or Figs. 2(b) and 2(d), we find that when the input particle is initially in a mixed state, the maximum of the fringe visibility is reached only if S_x = 0 and β = π/2, i.e., and the BS2 is symmetrical. The maximum A of the fringe visibility is reached when the input particle is initially in a pure state with S_x = −cos β. It is also found that the upper bound of the fringe visibility changes as β(S_x) changes for a given S_x(β) when the input particle is initially in a mixed state, and the upper bound of the fringe visibility does not change as β(S_x) changes for a given S_x(β) when the input particle is initially in a pure state. To reflect more clearly the conclusions obtained in Fig. 2, we have listed these parameters that make the fringe visibility reach the upper bound in Table 1. Mathematically, by solving the second-order partial derivative of S_x of Eq. (6), it is found that the fringe visibility obtains the upper bound with S_x = −λ cos β. The fringe visibility reaches the upper bound with S_x = −cos β by solving the second-order partial derivative of β of Eq. (6). By simple calculations, we can obtain the maximum A of the fringe visibility when the input particle is initially in a pure state with cos β = −S_x. It is also found that the fringe visibility obtain the maximum when the input particle is initially in a mixed state, and the maximum is reached only if S_x = 0 and β = π/2.

Table 1.

Table 1.

Table 1. The maximum of the fringe visibility shown in Fig. 2. . A λ β Sx Vmax Fig. 2(a) π/4 π/2 0 1/5 3π/4 Fig. 2(b) π/3 −1/2 π/2 0 1/5 2π/3 1/2 Fig. 2(c) π/4 π/2 0 1 3π/4 Fig. 2(d) π/3 −1/2 π/2 0 2π/3 1/2

Table 1. The maximum of the fringe visibility shown in Fig. 2. .

Fig. 2.

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	Fig. 2. The fringe visibility as a function of Sx for a given β = π/4 (solid line), β = π/2 (dash line), β = 3π/4 (dotted line) with λ = 9/25 (a) and λ = 1 (c). The fringe visibility as a function of β for a given Sx = −0.5 (solid line), Sx = 0 (dash line), Sx = 0.5 (dotted line) with λ = 9/25 (b) and λ = 1 (d). Throughout, we have set A = 1/3.

To study the particle-like property in the general MZI, the apparatus with four input and output ports has been introduced in Ref. [29]. The state of the WPD was obtained as 7 where 8 We note that the probabilities ω_a and ω_b are only dependent on the parameters S_x and β.

The distinguishability, which is used to characterize the particle-like behavior, is introduced by Englert^[6] 9 The sign holds in Eq. (9) when the state of the WPD is in a pure state. To obtain the WPI as much as possible, we only study the state of the WPD in a pure state in this section. Here, the magnitude of the distinguishability is determined by the parameters β, S_x, and A. The fringe visibility is determined by all the components of Bloch vector, but only S_x appears in the expression of the distinguishability, indicating that the distinguishability is not dependent on the S_y and S_z . To demonstrate the effect of the initial state of the particle and the BS2 on the distinguishability, the distinguishability as a function of the parameter S_x(β) for a given β(S_x) with A = 1/3 or 4/5 is shown in Fig. 3. In Fig. 3(a) (3(c)), we plot the distinguishability as a function of the parameter S_x for A = 1/3 (4/5) and β = π/4, π/2, 3π/4 with solid line, dash line, and dotted line. From Figs. 3(a) and 3(c) we can obtain that the distinguishability first decreases and then increases as S_x increases for a given β. The S_x varies from −1 to 1. The value of the distinguishability is 1 when S_x = ±1, since the particle is determined to propagate only in the path a or the path b, which corresponds to S_x = 1 or −1. The position of the valley changes with β, and the valley appears at when β = π/4, S_x = 0 when β = π/2, and when β = 3π/4, as shown in Figs. 3(a) and 3(c). Figures 3(a) and 3(c) show that for a given β, the change of A does not affect the position where the lower bound of the distinguishability reaches. In Fig. 3(b) (3(d)), we plot the distinguishability as a function of the parameter β for A = 1/3 (4/5) and a given S_x. The S_x = −0.5, 0, 0.5 are shown by solid line, dash line, and dotted line, respectively. The distinguishability first decreases and then increases as β increases for a given S_x, as shown in Figs. 3(b) and 3(d). The β varies from 0 to π. The value of the distinguishability is 1 when β = 0 or π, since the effect of the BS2 for the particle is full transmission or full reflection, which corresponds to β = 0 or π. The valley appears at β = π/3 when S_x = −0.5, β = π/2 when S_x = 0, and β = 2π/3 when S_x = 0.5, as shown in Figs. 3(b) and 3(d). Figures 3(b) and 3(d) show that for a given S_x, the change of A does not affect the position where the lower bound of the distinguishability reaches. By comparing Figs. 3(a) and 3(c) or Figs. 3(b) and 3(d), we find that the lower bound of the distinguishability is determined by A, and the position of the lower bound is determined by S_x and β. It is also found that the distinguishability obtains the lower bound A when cos β = −S_x (see Fig. 3). To reflect more clearly the conclusions obtained in Fig. 3, we list these parameters that make the distinguishability reach the lower bound in Table 2. Mathematically, we can obtain the minimum of the distinguishability when cos β = −S_x by solving the second-order partial derivative of S_x or β of Eq. (9).

Fig. 3.

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	Fig. 3. The distinguishability as a function of Sx for a given β = π/4 (solid line), β = π/2 (dash line), β = 3π/4 (dotted line) with A = 1/3 (a) and A = 4/5 (c). The distinguishability as a function of β for a given Sx = −0.5 (solid line), Sx = 0 (dash line), Sx = 0.5 (dotted line) with A = 1/3 (b) and A = 4/5 (d).

Table 2.

Table 2.

Table 2. The minimum of the distinguishability shown in Fig. 3. . A β Sx Dmax Fig. 3(a) π/4 π/2 0 3π/4 Fig. 3(b) π/3 −1/2 π/2 0 2π/3 1/2 Fig. 3(c) π/4 3/5 π/2 0 3π/4 Fig. 3(d) π/3 −1/2 π/2 0 2π/3 1/2

Table 2. The minimum of the distinguishability shown in Fig. 3. .

We have investigated the effect of the initial state of the particle and the BS2 on both the fringe visibility and the distinguishability. The fringe visibility obtains the upper bound when the total system is initially in a pure state and cos β = −S_x, and the maximum A is determined by the initial state of the detector and the unitary operator performed on it. The upper bound of the fringe visibility is related to A, λ, and S_x when the input particle is initially in a mixed state, and the condition for obtaining the upper bound of the fringe visibility for a given β is different from that obtained for a given S_x. The fringe visibility obtains the maximum when the input particle is initially in a mixed state, and the maximum is reached only if S_x = 0 and β = π/2. The lower bound of the distinguishability can be achieved when cos β = −S_x. We find that the lower bound of the distinguishability is determined by A, and the position of the lower bound are determined by S_x and β. From Eqs. (6) and (9), we obtain the complementary relationship V² + D² = 1 − [A² sin²β(1 − λ)]/(1 + S_x cos β)² when the initially detector is in a pure state and the state of the initially particle is arbitrary. [A² sin²β(1 − λ)]/(1 + S_x cosβ)² = 0 in the following situations. (i) The effect of the BS2 for the particle is full transmission or full reflection, which is corresponding to β = 0 or π. (ii) The state of the input particle is in a pure state, which corresponds to λ = 1. (iii) The states of the detector |r〉 and |s〉 are orthogonal state, which corresponds to A = 0.