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 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
v1 when amplitudes from two paths are combined. It is shown that [maxD(scrP)
]2+
v12=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
v12 of two-particle fringes:
v122+
v12\ensuremath{\le}1. With a suitable extension of operations on the pair of particles, this inequality is strengthened to an equality.
Bohr's complementarity principle is quantitatively formulated in terms of the distinguishability of various paths a quanton can take and the measure of the interference it produces. This phenomenon results from the interference of single-quanton amplitudes for various paths. The distinguishability of paths puts a bound on the sharpness of the interference the quanton can produce. However, there exist other kinds of quantum phenomena where interference of two-particle amplitudes results in a two-particle interference, if the particles are indistinguishable. The Hong-Ou-Mandel (HOM) effect and the Hanbury-Brown-Twiss (HBT) effect are two well-known examples. However, two-particle interference is not as easy to define as its single-particle counterpart, and the realization that it involves interference of two-particle amplitudes came much later. In this work, a duality relation, between the particle distinguishability and the visibility of two-particle interference, is derived. The distinguishability of the two particles, arising from some internal degree of freedom, puts a bound on the sharpness of the two-particle interference they can produce, in a HOM or HBT kind of experiment. It is argued that the existence of this kind of complementarity can be used to characterize two-particle interference, which in turn leads one to the conclusion that the HOM and the HBT effects are equivalent in essence and may be treated as a single two-particle-interference phenonmenon.
Photons in a two-path interferometer best embody wave-particle duality (WPD), which is a core concept of quantum theory. So far, the WPD relation is commonly written as
V2+D2≤1, where
V is the interference fringe visibility and
D is path distinguishability, i.e., the distinguishability of which path a photon passed. This inequality is saturated only when the which-way marker (WWM), which embodies which-path information (WPI) via an internal degree of freedom of photons, such as polarization, is in a pure state. For mixed-state WWM, conventionally defined distinguishability underestimates the amount of WPI and thus does not saturate the WPD relation. Here, we introduce a generalized measure of distinguishability
D that properly quantifies the WPI and saturates the WPD relation for all pure- and mixed-state WWM within a purification-based framework. To this end, mixed-state WWM is treated as a result of entanglement formation between the WWM and an external degree of freedom, e.g., environment, and
D is defined so that it incorporates the total WPI shared between the WWM and the environment. We show that
D thus defined is experimentally quantifiable, independently of
V, without access to the environment. We experimentally evaluate
V and
D using true single photons generated in the completely mixed (unpolarized) state, and thus verify the saturated WPD relation.
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.