Quality of a Which-Way Detector

We introduce a measure Q of the "quality" of a quantum which-way detector, which characterizes its intrinsic ability to extract which-way information in an asymmetric two-way interferometer. The "quality" Q allows one to separate the contribution to the distinguishability of the ways arising from the quantum properties of the detector from the contribution stemming from a-priori which-way knowledge available to the experimenter, which can be quantified by a predictability parameter P. We provide an inequality relating these two sources of which-way information to the value of the fringe visibility displayed by the interferometer. We show that this inequality is an expression of duality, allowing one to trace the loss of coherence to the two reservoirs of which-way information represented by Q and P. Finally, we illustrate the formalism with the use of a quantum logic gate: the Symmetric Quanton-Detecton System (SQDS). The SQDS can be regarded as two qubits trying to acquire which way information about each other. The SQDS will provide an illustrating example of the reciprocal effects induced by duality between system and which-way detector.