Physical Review A

Two interferometric complementarities

Gregg S. Jaeger, Abner Shimony, Lev Vaidman A

Published Jan 1, 1995• 416 citations

Abstract

Two interferometric complementarities are formulated and demonstrated, and relations between them are analyzed. The first relates the distinguishability D(scrP) of the path of propagation of a particle (where scrP is the preparation of an ensemble) to the fringe visibility ${\mathit{v}}_{1}$ when amplitudes from two paths are combined. It is shown that [maxD(scrP)${]}^{2}$+${\mathit{v}}_{1}^{2}$=1, where the maximum is taken over all preparations compatible with a fixed density operator. The second complementarity relates the visibility of one-particle interference fringes to the visibility ${\mathit{v}}_{12}$ of two-particle fringes: ${\mathit{v}}_{12}^{2}$+${\mathit{v}}_{1}^{2}$\ensuremath{\le}1. With a suitable extension of operations on the pair of particles, this inequality is strengthened to an equality.

DOI: 10.1103/physreva.51.54

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Complementarity and path distinguishability: Some recent results concerning photon pairs

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Physical Review A

Two interferometric complementarities

Gregg S. Jaeger, Abner Shimony, Lev Vaidman A

Published Jan 1, 1995• 416 citations

Publisher

Abstract

{\mathit{v}}_{1}

when amplitudes from two paths are combined. It is shown that [maxD(scrP)

{]}^{2}

{\mathit{v}}_{1}^{2}

=1, where the maximum is taken over all preparations compatible with a fixed density operator. The second complementarity relates the visibility of one-particle interference fringes to the visibility

{\mathit{v}}_{12}

of two-particle fringes:

{\mathit{v}}_{12}^{2}

{\mathit{v}}_{1}^{2}

\ensuremath{\le}1. With a suitable extension of operations on the pair of particles, this inequality is strengthened to an equality.

DOI: 10.1103/physreva.51.54

Related papers

Complementarity and path distinguishability: Some recent results concerning photon pairs

Abner Shimony, Gregg S. Jaeger

Two results concerning photon pairs, one previously reported and one new, are summarized. It was previously shown that if the two photons are prepared in a quantum state formed from bar-A and bar-A' for photon 1 and bar-B and bar-B' for photon 2, then both one- and two-particle interferometry can be studied. If upsilon(sub i) is the visibility of one-photon interference fringes (i = 1,2) and upsilon(sub 12) is the visibility of two-photon fringes (a concept which we explicitly define), then upsilon(sub i) squared + upsilon(sub 12) squared is less than or equal to 1. The second result concerns the distinguishability of the paths of photon 2, using the known 2-photon state. A proposed measure E for path distinguishability is based upon finding an optimum strategy for betting on the outcome of a path measurement. Mandel has also proposed a measure of distinguishability P(sub D), defined in terms of the density operator rho of photon 2. We show that E is greater than or equal to P(sub D) and that upsilon(sub 2) = (1 - E(exp 2))exp 1/2.Two results concerning photon pairs, one previously reported and one new, are summarized. It was previously shown that if the two photons are prepared in a quantum state formed from bar-A and bar-A' for photon 1 and bar-B and bar-B' for photon 2, then both one- and two-particle interferometry can be studied. If upsilon(sub i) is the visibility of one-photon interference fringes (i = 1,2) and upsilon(sub 12) is the visibility of two-photon fringes (a concept which we explicitly define), then upsilon(sub i) squared + upsilon(sub 12) squared is less than or equal to 1. The second result concerns the distinguishability of the paths of photon 2, using the known 2-photon state. A proposed measure E for path distinguishability is based upon finding an optimum strategy for betting on the outcome of a path measurement. Mandel has also proposed a measure of distinguishability P(sub D), defined in terms of the density operator rho of photon 2. We show that E is greater than or equal to P(sub D) and that upsilon(sub 2) = (1 - E(exp 2))exp 1/2.

May 1, 19940 citations

Experimental demonstration of the interferometric complementarity of one- and two-particle interference in a bulk Nuclear Magnetic Resonance ensemble

Xinhua Peng, Zhu, Xiwen, Maili Liu +1 more

We analyze an interferometric complementarity between one- and two-particle interference in the general case: $V_{i}^{2}+V_{12}^{2}\leq 1$ $(i=1$ , $2)$ , and further examine the relation among one-particle interference visibility $V_{i}$ , two-particle interference visibility $V_{12}$ and the predication $P_{i}$ of the path of a single particle. An equality $V_{i}^{2}+V_{12}^{2}+P_{i}^{2}=1$ $(i=1$ , $2)$ is achieved for any pure two-particle source, which implies the condition of the complementarity relation to reach the upper bound and its relation to another interferometric complementarity between path information and interference visibility of a single particle. Meanwhile, the relationships of the complementarities and the entanglement $E$ of the composite system are also investigated. Using nuclear magnetic resonance techniques, the two-particle interferometric complementarity was experimentally tested with the ensemble-averaged spin states, including two extreme cases and an intermediate case.We analyze an interferometric complementarity between one- and two-particle interference in the general case: $V_{i}^{2}+V_{12}^{2}\leq 1$ $(i=1$, $2)$, and further examine the relation among one-particle interference visibility $V_{i}$, two-particle interference visibility $V_{12}$ and the predication $P_{i}$ of the path of a single particle. An equality $V_{i}^{2}+V_{12}^{2}+P_{i}^{2}=1$ $(i=1$, $2)$ is achieved for any pure two-particle source, which implies the condition of the complementarity relation to reach the upper bound and its relation to another interferometric complementarity between path information and interference visibility of a single particle. Meanwhile, the relationships of the complementarities and the entanglement $E$ of the composite system are also investigated. Using nuclear magnetic resonance techniques, the two-particle interferometric complementarity was experimentally tested with the ensemble-averaged spin states, including two extreme cases and an intermediate case.

arXiv:quant-ph/0208081Aug 13, 20020 citations

Complementarity of one-particle and two-particle interference

Gregg S. Jaeger, Michael Horne, Abner Shimony

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Higher-order wave-particle duality

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The complementarity of a single photon's particlelike and wavelike behaviors can be described by the inequality ${D}^{2}+{V}^{2}\ensuremath{\le}1$, with $D$ being the path distinguishability and $V$ being the fringe visibility. In this paper, we generalize this duality relation to the multiphoton case, where two concepts, higher-order distinguishability and higher-order fringe visibility, are introduced to quantify the higher-order particlelike and wavelike behaviors of multiphotons.The complementarity of a single photon's particlelike and wavelike behaviors can be described by the inequality ${D}^{2}+{V}^{2}\ensuremath{\le}1$, with $D$ being the path distinguishability and $V$ being the fringe visibility. In this paper, we generalize this duality relation to the multiphoton case, where two concepts, higher-order distinguishability and higher-order fringe visibility, are introduced to quantify the higher-order particlelike and wavelike behaviors of multiphotons.

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Comment on “Quantitative wave-particle duality in multibeam interferometers”

Giuseppe Bimonte, R. Musto

In a recent paper, D\"urr [Phys. Rev. A 64, 042113 (2001)] proposed an interesting multibeam generalization of the quantitative formulation of interferometric wave-particle duality, discovered by Englert for two-beam interferometers. The proposed generalization is an inequality that relates a generalized measure of the fringe visibility to certain measures of the maximum amount of which-way knowledge that can be stored in a which-way detector. We construct an explicit example where, with three beams in a pure state, the scheme proposed by D\"urr leads to the possibility of an ideal which-way detector that can achieve a better path discrimination at the same time as a better fringe visibility. In our opinion, this seems to be in contrast with the intuitive idea of complementarity, as it is implemented in the two-beams case, where an increase in path discrimination always implies a decrease of fringe visibility, if the beams and the detector are in pure states.In a recent paper, D\"urr [Phys. Rev. A 64, 042113 (2001)] proposed an interesting multibeam generalization of the quantitative formulation of interferometric wave-particle duality, discovered by Englert for two-beam interferometers. The proposed generalization is an inequality that relates a generalized measure of the fringe visibility to certain measures of the maximum amount of which-way knowledge that can be stored in a which-way detector. We construct an explicit example where, with three beams in a pure state, the scheme proposed by D\"urr leads to the possibility of an ideal which-way detector that can achieve a better path discrimination at the same time as a better fringe visibility. In our opinion, this seems to be in contrast with the intuitive idea of complementarity, as it is implemented in the two-beams case, where an increase in path discrimination always implies a decrease of fringe visibility, if the beams and the detector are in pure states.

arXiv:quant-ph/0304004Physical Review AJun 11, 200328 citations

Duality of quantum coherence and path distinguishability

Manabendra Nath Bera, Tabish Qureshi, Mohd Asad Siddiqui +1 more

We derive a generalized wave-particle duality relation for arbitrary multipath quantum interference phenomena. Beyond the conventional notion of the wave nature of a quantum system, i.e., the interference fringe visibility, we introduce a quantifier as the normalized quantum coherence, recently defined in the framework of quantum information theory. To witness the particle nature, we quantify the path distinguishability or the which-path information based on unambiguous quantum state discrimination. Then, the Bohr complementarity principle for multipath quantum interference can be stated as a duality relation between the quantum coherence and the path distinguishability. For two-path interference, the quantum coherence is identical to the interference fringe visibility, and the relation reduces to the well-known complementarity relation. The duality relation continues to hold in the case where mixedness is introduced due to possible decoherence effects.We derive a generalized wave-particle duality relation for arbitrary multipath quantum interference phenomena. Beyond the conventional notion of the wave nature of a quantum system, i.e., the interference fringe visibility, we introduce a quantifier as the normalized quantum coherence, recently defined in the framework of quantum information theory. To witness the particle nature, we quantify the path distinguishability or the which-path information based on unambiguous quantum state discrimination. Then, the Bohr complementarity principle for multipath quantum interference can be stated as a duality relation between the quantum coherence and the path distinguishability. For two-path interference, the quantum coherence is identical to the interference fringe visibility, and the relation reduces to the well-known complementarity relation. The duality relation continues to hold in the case where mixedness is introduced due to possible decoherence effects.

arXiv:1503.02990Physical Review AJul 22, 2015254 citations

{\mathit{v}}_{1}

when amplitudes from two paths are combined. It is shown that [maxD(scrP)

{]}^{2}

{\mathit{v}}_{1}^{2}

=1, where the maximum is taken over all preparations compatible with a fixed density operator. The second complementarity relates the visibility of one-particle interference fringes to the visibility

{\mathit{v}}_{12}

of two-particle fringes:

{\mathit{v}}_{12}^{2}

{\mathit{v}}_{1}^{2}

\ensuremath{\le}1. With a suitable extension of operations on the pair of particles, this inequality is strengthened to an equality.

We analyze an interferometric complementarity between one- and two-particle interference in the general case:

V_{i}^{2}+V_{12}^{2}\leq 1

(i=1

2)

, and further examine the relation among one-particle interference visibility

V_{i}

, two-particle interference visibility

V_{12}

and the predication

P_{i}

of the path of a single particle. An equality

V_{i}^{2}+V_{12}^{2}+P_{i}^{2}=1

(i=1

2)

is achieved for any pure two-particle source, which implies the condition of the complementarity relation to reach the upper bound and its relation to another interferometric complementarity between path information and interference visibility of a single particle. Meanwhile, the relationships of the complementarities and the entanglement

E

of the composite system are also investigated. Using nuclear magnetic resonance techniques, the two-particle interferometric complementarity was experimentally tested with the ensemble-averaged spin states, including two extreme cases and an intermediate case.

{\mathit{v}}_{\mathit{i}}

(i=1,2) with a reasonable proposed definition for two-particle fringe visibility

{\mathit{v}}_{12}

, we show that

{\mathit{v}}_{\mathit{i}}^{2}

{\mathit{v}}_{12}^{2}

\ensuremath{\le}1 or, equivalently,

{\mathit{v}}_{\mathit{i}}

{\mathit{v}}_{12}

\ensuremath{\le}1/2. Some extensions are discussed.

The complementarity of a single photon's particlelike and wavelike behaviors can be described by the inequality ${D}^{2}+{V}^{2}\ensuremath{\le}1$, with

D

being the path distinguishability and

V

being the fringe visibility. In this paper, we generalize this duality relation to the multiphoton case, where two concepts, higher-order distinguishability and higher-order fringe visibility, are introduced to quantify the higher-order particlelike and wavelike behaviors of multiphotons.