Information gain versus interference in Bohr&#x02019;s principle of complementarity

Yan Liu; Jing Lu; Lan Zhou

doi:10.1364/OE.25.000202

1. Introduction

Quantum properties which are equally real but mutually exclusive are called complementary [1]. The wave-particle duality is one of them, and well described in Bohr’s complementarity principle. It is sometimes phrased as follows: waves and particles are two distinct types of complementary properties in nature, and the experimental situation determines the particle or wave nature of a quantum system; however, the simultaneous observation of wave and particle behavior is impossible. Mutual exclusiveness is regarded by Bohr as a “necessary” element in the complementarity principle to ensure its inner consistency [2]. The usual discussion about wave-particle duality starts from a physical system with two alternatives, typically, a two-way interferometer such as Young’s double-slit experiment or a Mach-Zehnder interferometer (MZI). If one performs quantum measurements to determine which way a quantum particle is taken (particle-like property), the interference pattern (wave-like property) is partially or completely destroyed by the partial or complete knowledge of the “which-way” information. The more one obtains the which-way information, the more the loss of interference [3–7]. Many experiments have demonstrated this complementarity with different quantum systems, like atoms [8], lasers [9], nuclear magnetic resonance [10, 11], and single photons [12–17]. Recently, the nonlocal property of particle wave duality is studied in Refs. [18, 19]. The particle wave duality property has been extended to living object [20], and exploited to perform computation [21]. Obviously, the concept of measurement plays an important role in a logically-consistent description of the wave-particle properties in a two-way interferometer.

Bohr’s interpretation of the interferometry experiments invokes the concept of wave-function collapse. Measurements perturb the wave function, so the collapse hypothesis is responsible for mutually exclusive quantities. Based on classical concepts and intuitions, Bohr thought that the stages of preparation and registering the quantum objects require classical apparatuses, which draw an obvious border between the quantum and the classical world. However, this concept is contrary to the belief that quantum theory is universally applicable and classical reality may be reconstructed or reconstituted from quantum dynamics. By treating the apparatus as a large number of particles or a large number of degrees of freedom obeying the Schödinger equation, the loss of interference is explained by the nonseparable correlation between the measuring apparatus and the system being observed, where the information of the measured system is stored in the pointer states [22, 23] of the apparatus. Such a correlation between two systems is called entanglement. Mathematically, the loss of interference is described by the elimination of the off-diagonal elements of the system’s reduced density matrix, which is called decoherence in the terminology of quantum measurement theory. By utilizing entanglement, the position-momentum uncertainty relation is found to play no role in the principle of complementarity [6, 24]. Note that a quantitative formulation of Bohr’s complementarity principle can be derived from the position-momentum uncertainty [3, 25]. To find out what makes the quantum apparatus have a number of states (which correspond to a number of possible distinct outcomes of the measurement), an environment is necessarily introduced to interact with the quantum apparatus [26].

In the duality relation [6], the wave nature is described in terms of the visibility V of the interference pattern. The particle nature is characterized by the path distinguishability D, which is a measure of the which-way information. The wave-particle duality is demonstrated by the visibility of fringes setting limits on the which-way information. In the quantum information theory, the total correlation in a bipartite system can be written as a sum of classical correlation [27, 29] and quantum correlation (which has been called quantum discord) [30, 31]. Both classical and quantum correlation are originally introduced as an information-theoretic approach to decoherence mechanisms in a quantum measurement process. Notice that the quantitative description of the wave-particle duality is close related to quantum measurement process. In this paper, we will study the accessible quantum and classical correlation in a MZI to find out the relation between the accessible information and quantities V and D in the wave-particle duality. Both the von Neumann measurement model and the Zurek’s “triple model” [22, 26] are considered. It is found that based on von Neumann measurement model, the total correlation between the photon and the detector is equally divided into the classical and quantum correlation. Both of them are monotone decreasing function of the visibility V. In Zurek’s quantum measurement theory, the classical correlation remains consistent with the path distinguishability D and can be used to describe the particle-like property of the photon. The relationship between quantum correlations and the visibility V is also obtained.

This paper is organized as follows. In Sec. 2, we briefly review the well-known method of quantifying wave–particle duality. In Sec. 3, we study the amount of information gain from the quantum measurement process by using von Neumann measurement model and Zurek’s “triple model” in a which-way experiment. Finally, we conclude this work in Sec. 4.

2. Wave–particle duality relation

A symmetric MZI has two 50:50 beam splitters (BSs) and a phase shifter (PS) as shown in Fig. 1. Between the BSs, two possible routes a and b are macroscopically well separated, which are represented by orthogonal unit vectors |a〉, |b〉. Hereafter, the states |a〉 and |b〉 are called the path states. The wave-function of a photon incident on path a (b) is changed to an equally-weighted superposition

\begin{array}{l} | a 〉 \to \frac{1}{\sqrt{2}} (| a 〉 + | b 〉), \\ | b 〉 \to \frac{1}{\sqrt{2}} (| a 〉 - | b 〉) \end{array}

by the first BS. As a single photon propagates along these paths, a relative phase ϕ is accumulated between states |a〉 and |b〉. To obtain the knowledge of the actual path a photon has taken, a which-way detector (WWD) is introduced.

Fig. 1 Schematic of a symmetric Mach-Zehnder interferometer. Here, PS refers to phase shifter, BS to beam splitters, and HR to high reflector.

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If a photon propagates along the path a, the initial state $ρ_{in}^{D}$ of the WWD remains unchanged; however, $ρ_{in}^{D}$ is changed to $U ρ_{in}^{D} U^{†}$ if the photon propagates on path b. States $ρ_{in}^{D}$ and $U ρ_{in}^{D} U^{†}$ are not always orthogonal. By defining the operators σ_α (α = x, y, x) in terms of the states |a〉 and |b〉, e.g., σ_z = |a〉〈a| − |b〉〈b|, the degree of freedom described by the path states is analogous to a spin. Therefore, the initial state of a photon is generally characterized by the density matrix

ρ_{in}^{Q} = \frac{1}{2} (1 + S_{x} σ_{x} + S_{y} σ_{y} + S_{z} σ_{z})

with an initial Bloch vector

\vec{S} = (S_{x}, S_{y}, S_{z})

. The initial product state

ρ_{in}^{Q} \otimes ρ_{in}^{D}

of the two subsystems is evolved into

\begin{array}{l} ρ_{f} = \frac{1}{4} (1 - S_{x}) (1 + σ_{x}) \otimes ρ_{in}^{D} \\ + \frac{1}{4} (1 + S_{x}) (1 - σ_{x}) \otimes U ρ_{in}^{D} U^{†} \\ - \frac{1}{4} e^{- i ϕ} (S_{z} - i S_{y}) (σ_{z} - i σ_{y}) \otimes ρ_{in}^{D} U^{†} \\ - \frac{1}{4} e^{i ϕ} (S_{z} + i S_{y}) (σ_{z} + i σ_{y}) \otimes U ρ_{in}^{D}, \end{array}

after the photon has gone through the MZI, where the correlation between the photon and WWD is established, and ϕ is the phase shift caused by the PS. Then, the which-way information is stored in the WWD. The state of the single photon reads

\begin{array}{l} ρ_{f}^{Q} = \frac{1}{4} (1 - S_{x}) (1 + σ_{x}) + \frac{1}{4} (1 + S_{x}) (1 - σ_{x}) \\ - \frac{1}{4} e^{- i ϕ} (S_{z} - i S_{y}) (σ_{z} - i σ_{y}) {Tr}_{D} (ρ_{in}^{D} U^{†}) \\ - \frac{1}{4} e^{i ϕ} (S_{z} + i S_{y}) (σ_{z} + i σ_{y}) {Tr}_{D} (U ρ_{in}^{D}) \end{array}

by tracing over the degrees of freedom of the WWD. The probability for the photon emerging from the output a

\begin{array}{l} P^{a} = {Tr}_{Q} [\frac{1}{2} (1 + σ_{z}) ρ_{f}^{Q}] \\ = \frac{1}{2} - \frac{1}{2} \sqrt{S_{y}^{2} + S_{z}^{2}} | {Tr}_{D} (U ρ_{in}^{D}) | \cos (α + β + ϕ) \end{array}

is used to define the visibility

V \equiv \frac{P_{\max}^{a} - P_{\min}^{a}}{P_{\max}^{a} + P_{\min}^{a}} \sqrt{S_{y}^{2} + S_{z}^{2}} | {Tr}_{D} (U ρ_{in}^{D}) |

of the interference pattern, where α and β are the phases of S_z+iS_y and

{Tr}_{D} (U ρ_{i n}^{D})

respectively. To extract the information from the final state of the WWD

ρ_{f}^{D} = \frac{1 - S_{x}}{2} ρ_{in}^{D} + \frac{1 + S_{x}}{2} U ρ_{in}^{D} U^{†},

an observable must be chosen for the readout. Englert [6] introduced the distinguishability

D = {Tr}_{D} | \frac{1 - S_{x}}{2} ρ_{in}^{D} - \frac{1 + S_{x}}{2} U ρ_{in}^{D} U^{†} |

to be the maximum amount of which-way information. Then the fringe visibility and distinguishable are bound in a trade-off relation

D^{2} + \frac{1 - P^{2}}{V_{0}^{2}} V^{2} \leq 1,

where P = |S_x | is the predictability and

V_{0} = \sqrt{S_{y}^{2} + S_{z}^{2}}

is a priori fringe visibility. Actually, the parameters P and V₀ construct a trade-off relation

P^{2} + V_{0}^{2} \leq 1

, which is known as the duality relationship for preparation [4].

Special attention is paid in Ref. [6] on the initial state with S_x = 0 and S_z + iS_y = e⁻^iθ (in this case, P = 0, V₀ = 1) to emphasize the quantum properties of the WWD, which enforce duality and make sure that the principle of complementarity is not circumvented. In this sense, we will set S_x = 0 and S_z + iS_y = e⁻^iθ in the rest of our paper.

3. Information gain versus interference

For simplicity, we take the initial state of the WWD as a pure state, i.e., $ρ_{in}^{D} = | d 〉〈 d |$ . In Ref. [6], the “likelihood for guessing the way right” is introduced to discriminate the states |d〉 and U |d〉 with minimum error [32, 33]. Mathematically, a discrimination among two nonorthogonal states can be performed by the von Neumann projection, which are the orthogonal completed operator ∏_k = |M_k〉〈M_k|, k ∈ {a, bs}. Here, vectors

\begin{array}{l} | M_{a} 〉 = \frac{\sin γ}{\sqrt{1 - V^{2}}} | d 〉 + e^{i φ} (\cos γ - \sin γ \frac{V}{\sqrt{1 - V^{2}}}) U | d 〉, \\ | M_{b} 〉 = e^{- i φ} \frac{\cos γ}{\sqrt{1 - V^{2}}} | d 〉 - (\sin γ + \cos γ \frac{V}{\sqrt{1 - V^{2}}}) U | d 〉 \end{array}

with Ve⁻^iφ = 〈d| U |d〉, 0 ≤ φ ≤ 2π and 0 ≤ γ ≤ 2π. The minium error is achieved when the vectors become

\begin{array}{l} | M_{a} 〉 = \frac{1}{m} \sqrt{\frac{1 + m}{2}} | d 〉 - e^{i φ} \frac{1}{m} \sqrt{\frac{1 - m}{2}} U | d 〉, \\ | M_{b} 〉 = e^{- i φ} \frac{1}{m} \sqrt{\frac{1 - m}{2}} | d 〉 - \frac{1}{m} \sqrt{\frac{1 + m}{2}} U | d 〉 \end{array}

with

m = \sqrt{1 - V^{2}}

. Actually, the vectors |M_k〉 in Eq. (11) are the eigenstates of the operator |d〉〈d| − U |d〉〈d| U^†. In this section, we will investigate the classical and quantum information in the MZI from the point view of the following quantum measurement theory: one is the von Neumann’s quantum theory of measurement; the other is Zurek’s “triple model” of quantum measurement.

3.1. Classical correlation versus interference

In quantum information theory, the classical correlation (CC) [27] between the photon and the WWD is captured by

J (ρ) = \max [S (ρ^{Q}) - S (ρ^{Q} | {\prod_{k}})],

where ρ^Q is the reduced density operator for the photon, S(ρ^Q) is the von Neumann entropy, S(ρ^Q|{Π_k }) is the quantum conditional entropy, and {Π_k} is a set of projectors performed locally on the WWD, which have been given in Eq. (10). Here, we note that in Ref. [28], Hamith et. al have showed that for two-qubit system, the optimal measurement is a projective measurement.

(1) Based on von Neumann’s model of quantum measurement

In this case, the density matrix of the photon-detector system can be written as ρ′ = |Ψ〉〈Ψ| with

| Ψ 〉 = \frac{1}{\sqrt{2}} (| a 〉 \otimes | d 〉 + | b 〉 \otimes U | d 〉) .

The entanglement between the photon and the detector yields that S(ρ^Q|{Π_k}) = 0, which means

J (ρ^{'}) = - \frac{1 + V}{2} \log (\frac{1 + V}{2}) - \frac{1 - V}{2} \log (\frac{1 - V}{2})

does not dependent on the direction of measure. However, the distinguishability D is achieved by choosing a specific observable. Thus, we think the CC and D have essential distinction. Furthermore, the CC can be rewritten as

J (ρ^{'}) = ℋ (V) = ℋ (\sqrt{1 - D^{2}})

, where

ℋ (x)

denotes the entropy of the probability distribution (1 ± x)/2. In Fig. 2, we plot the CC as a function of the visibility V. It can be found that the CC decrease monotonously as the visibility V increases. It means the more the CC, the more the loss of visibility.

Fig. 2 The relationship between the classical correlations and the fringe visibility V. ρ′ refers to the entangles state in Eq. (13), and ρ″ to the correlated state in Eq. (15).

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(2) Based on Zurek’s “triple model” of quantum measurement

To realize the wave-function collapse of the measured system by establishing an entanglement between the system and the apparatus in quantum measurement theory, an observer must first select the state of the detector and then read it out. To avoid this subjective selection, Zurek introduced a “triple model” [22, 26] of quantum measurement process, which consists of a measured system (photon), an apparatus (WWD), and an environment. According to the environment-induced superselection, the photon and the detector are correlated to each other and in a state described by the density matrix

ρ^{″} = \frac{1}{2} (| a 〉 〈 a | \otimes | d 〉 〈 d | + | b 〉 〈 b | \otimes U | d 〉 〈 d | U^{†}),

where the path state |a〉 (|b〉) is correlated with the pointer state |d〉 (U|d〉), which is chosen by the environment. The observer just reads out the pointer states of WWD. If the WWD is in the state |d〉 (U |d〉), it infers that the photon passes through the path a (b). If |d〉 = U |d〉, the path the photon taking is not known, so D = 0. If |d〉 and U |d〉 are mutually orthogonal, a perfect knowledge of which-path the photon propagating is achieved, i.e., D = 1. However, when |d〉 and U |d〉 are not mutually orthogonal, it is impossible to discriminate them perfectly.

Starting from Eq. (15), after some algebra, we find

\begin{array}{l} S (ρ^{Q}) - S (ρ^{Q} | {\prod_{k}}) \\ = 1 + \frac{1}{2} [\cos^{2} γ] \log [\cos^{2} γ] + \frac{1}{2} [\sin^{2} γ] \log [\sin^{2} γ] \\ + \frac{1}{2} [{(\sqrt{1 - V^{2}} \cos γ - V \sin γ)}^{2}] \log [{(\sqrt{1 - V^{2}} \cos γ - V \sin γ)}^{2}] \\ + \frac{1}{2} [{(\sqrt{1 - V^{2}} \sin γ + V \cos γ)}^{2}] \log [{(\sqrt{1 - V^{2}} \sin γ + V \cos γ)}^{2}] \\ - \frac{1}{2} [{(\sqrt{1 - V^{2}} \sin γ + V \cos γ)}^{2} + \cos^{2} γ] \log [{(\sqrt{1 - V^{2}} \sin γ + V \cos γ)}^{2} + \cos^{2} γ] \\ - \frac{1}{2} [{(\sqrt{1 - V^{2}} \cos γ - V \sin γ)}^{2} + \sin^{2} γ] \log [{(\sqrt{1 - V^{2}} \cos γ - V \sin γ)}^{2} + \sin^{2} γ] \end{array}

is a periodic function of the angle γ with period π/2. For a given V, when

γ = \arcsin {\frac{1}{2} [1 + {(1 - V^{2})}^{\frac{1}{2}}]}^{\frac{1}{2}},

we can obtain the CC

\begin{array}{r} J (ρ^{″}) = \frac{1 + \sqrt{1 - V^{2}}}{2} \log (1 + \sqrt{1 - V^{2}}) \\ + \frac{1 - \sqrt{1 - V^{2}}}{2} \log (1 - \sqrt{1 - V^{2}}), \end{array}

which can also be written simply as

J (ρ^{″}) = 1 - ℋ (\sqrt{1 - V^{2}}) = 1 - ℋ (D)

.

The CC plotted as a function of the visibility V in Fig. 2. It can be observed that when the pointer states |d〉 = U |d〉, both classical correlation and distinguishability are equal to zero, i.e., CC = D = 0, while the fringe visibility V = 1, which means that the wave-like behavior of the photon can be perfectly observed. When the pointer states |d〉 and U |d〉 are mutually orthogonal, both classical correlation and distinguishability reach the maximum value, i.e., CC = D = 1, while the wave-like behavior of the photon disappears, i.e., V = 0. As the fringe visibility V increases, the CC decreases monotonously, so does the distinguishability $D = \sqrt{1 - V^{2}}$ . The more the which-way information (both CC and D) we obtain, the less the fringe visibility V. The CC’s monotone similar to the distinguishability indicates that the CC characterizes the which-way information of the photon in MZI. We note that the orthogonal projection vectors for the CC $J (ρ^{″})$ is the same to those for the distinguishability D. Comparing to the von Neumann’s model, the environment not only induces the emergence of the optimal measuring basis, but also reduces the CC between the photon and the WWD, i.e., $J (ρ^{″}) < J (ρ^{'})$ as shown in Fig. 2.

3.2. Quantum correlation versus interference

Quantum discord (QD), denoted by $D$ , is usually used to measure quantum correlation in a bipartite system, and it is defined as the difference between the total correlation, $ℐ$ , and the classical correlation, $J$ ,

D (ρ) = ℐ (ρ) - J (ρ) .

Here, the total correlation is equal to quantum mutual information

ℐ (ρ) = S (ρ^{Q}) + S (ρ^{D}) - S (ρ),

where S is the von Neumann entropy, and ρ^Q (ρ^D) is the reduced density matrix of the photon (WWD).

(1) Based on von Neumann’s model of quantum measurement

In von Neumann measurement theory, the photon-detector system is in the entangled state in Eq. (13). The quantum correlation between the photon and the WWD reads

D (ρ^{'}) = - \frac{1 + V}{2} \log (\frac{1 + V}{2}) - \frac{1 - V}{2} \log (\frac{1 - V}{2}) .

We can find that the quantum correlation is equal to the classical correlation, i.e., $D (ρ^{'}) = J (ρ^{'}) = ℋ (V) = ℋ (\sqrt{1 - D^{2}})$ . In fact, for any pure state, the total correlation is equally divided into the classical and quantum correlation [34]. In this case, the quantum correlation has the same behaviors as the classical correlation, i.e., the QD decreases monotonously with an increasing of the visibility V. It can be observed from the solid curve in Fig. 3 that the fringe visibility is responsible for the quantum correlation.

Fig. 3 The relationship between the quantum correlations and the fringe visibility V. ρ′ refers to the entangles state in Eq. (13), and ρ″ to the correlated state in Eq. (15).

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(2) Based on Zurek’s triple model of quantum measurement

For the state ρ″ in Eq. (15), the QD between the photon and the WWD is calculated as

\begin{array}{l} D (ρ^{″}) = - \frac{1 + V}{2} \log (\frac{1 + V}{2}) - \frac{1 - V}{2} \log (\frac{1 - V}{2}) \\ - \frac{1 + \sqrt{1 - V^{2}}}{2} \log (1 + \sqrt{1 - V^{2}}) \\ - \frac{1 - \sqrt{1 - V^{2}}}{2} \log (1 - \sqrt{1 - V^{2}}) . \end{array}

It can be rewritten as $D (ρ^{″}) = ℋ (V) + ℋ (D) - 1$ , indicating that the QD is not a monotonic function of the fringe visibility V. In Fig. 3, we plot the QD as a function of the fringe visibility V (see the dashed curve). When V = 0 corresponding to pointer state |d〉 ⊥ U |d〉, the environment introduces a completely quantum-to-classical transition. Since the bipartite system is in a classical state [35, 36], there is no quantum correlation $(D (ρ^{″}) = 0)$ and only the classical correlation $(J (ρ^{″}) = 1)$ . Hence, “no fringe visibility” is responsible for “no quantum correlations”. As V increases, the QD first increases and then decreases. In this case, the bipartite system is in a separable state [37] due to the overlap between two pointer states. A separable state indicates no entanglement, but no vanishing CC and QD. When V = 1 corresponding to the pointer states |d〉 = U |d〉, the bipartite system is in a product state or an uncorrelated state, both the CC or the QD vanish. Here, the QD is different from the visibility.

By comparing the results in Zurek’s model with those in von Neumann’s model, one can find that the environment reduces both the classical and quantum correlation of the photon-WWD system. However, the relation $J (ρ^{″}) + D (ρ^{″}) = J (ρ^{'}) = D (ρ^{'})$ is obtained between these two models.

4. Conclusion

In the quantitative relation formulation of wave-particle duality, the wave nature is described by the visibility of the interference pattern, while the particle nature is characterized by the path distinguishability. In modern measurement theory, the Zurek’s “triple model” (which consists of a measured system, a detector and environment) is introduced, where the interaction between the quantum system and the detector produces a quantum entanglement between them, and later the coupling of the environment and the detector generates a triple entanglement among the system, the detector, and environment. By tracing the environment, the state of system and detector is no longer in an entangled state, but a correlated state. With the help of quantum information theory, the correlation between the measured system and detector [which includes classical correlation (CC) and quantum correlation (QD)] is related to the information gain about the measured system. In this paper, we use both the von Neumann’s measurement theory and the Zurek’s “triple model” to study the relation between the accessible information and the wave-particle duality in a MZI. It can be found that based on von Neumann measurement model, the total correlation between the photon and the detector is equally divided into the classical and quantum correlation. Both of them are monotone decreasing function of the visibility V. In Zurek’s quantum measurement theory, the CC is the information gain about the particle-like property of the measured system which is consistent with the path distinguishability, and the relationship between the QD and the visibility V can be obtained in the Eq. (22). By comparing the results in these two models, one can find the environment not only induces the emergence of the optimal measuring basis, but also reduces the CC and the QD between the measured system and the detector. We also find a way to calculate the QD for one type of two-qubit separable state other than the X-type.

Funding

National Natural Science Foundation of China (NSFC)(11374095,11422540,11434011,11575058); National Fundamental Research Program of China (Program 973) (2012CB922103).

Acknowledgments

We are grateful to F. Nori for useful discussions and many comments which improved this manuscript. Stimulating discussions with Y. J. Song are gratefully acknowledged.

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Information gain versus interference in Bohr’s principle of complementarity

Abstract

1. Introduction

2. Wave–particle duality relation

3. Information gain versus interference

3.1. Classical correlation versus interference

(1) Based on von Neumann’s model of quantum measurement

(2) Based on Zurek’s “triple model” of quantum measurement

3.2. Quantum correlation versus interference

(1) Based on von Neumann’s model of quantum measurement

(2) Based on Zurek’s triple model of quantum measurement

4. Conclusion

Funding

Acknowledgments

References and links

Cited By

Figures (3)

Equations (22)

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