. However, if such a system is coupled to its environment, it becomes possible to obtain conditional measures of visibility and predictability, i.e.~measures that are conditional on the state of the environment. We show that in this case, the predictability and visibility values can lead to an apparent violation of the duality principle. We experimentally realize this apparent violation in a controlled manner by enforcing a fair-sampling-like loophole via postselection. Given the ability to simultaneously obtain high predictability information and high visibility interference fringes for a wide range of coupling strengths and postselected states, this work highlights the role of fair-sampling in tests of the duality principle.

DOI: 10.48550/arxiv.1406.4300

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\mathcal V

and predictability

\mathcal P

within a two-alternative system, which are bound by the inequality

\mathcal V^2+ \mathcal P^2 \leq 1

The notion of wave-particle duality may be quantified by the inequality

{V}^{2}{+K}^{2}<~1,

relating interference fringe visibility V, and path knowledge K. With a single-photon interferometer in which polarization is used to label the paths, we have investigated the relation for various situations, including pure, mixed, and partially mixed input states. A quantum-eraser scheme has been realized that recovers interference fringes even when no which way information is available to erase.