When a negative weak value −1 plays the counterpart of a probability 1

Abstract

When the weak value of a projector is 1, a quantum system behaves as in that eigenstate with probability 1. By definition, however, the weak value may take an anomalous value lying outside the range of probability like −1. From the viewpoint of a physical effect, we show that such a negative weak value of −1 can be regarded as the counterpart of the ordinary value of 1. Using photons, we experimentally verify it as the symmetrical shift in polarization depending on the weak value given by pre-postselection of the path state. Unlike observation of a weak value as an ensemble average via weak measurements, the effect of a weak value is definitely confirmed in Hong-Ou-Mandel effect: the symmetrical shift corresponding to the weak value can be directly observed as the rotation angle of a half wave plate.

Negation of photon loss provided by negative weak value

“…Actually we experimentally demonstrated the negation provided by a negative weak value. Before proceeding to the discussion, we would like to point out the difference from our previous result [36] here. First, a weak value appears without 'weak' condition this time.…”

“…Figure 3 shows our experimental implementation composed of three Mach-Zehnder interferometers ( ñ |1 and ñ |2 , ñ |2 and ñ |3 , ñ |3 and ñ |1 ), the visibilities of which were at least 98.3±1.8% for horizontally polarized photons. The path lengths of the Mach-Zehnder interferometer composed of ñ |1 and ñ |3 ( ñ |2 and ñ |3 ) were adjusted so that the port of D1 became the bright (dart) port, which was controlled by using the extra laser diode (LD) and the photodiode (PD) [23,36]: in our experiment, we stopped counting photon at D1 every 5 seconds, and observed the fringe pattern of lights coming from LD at PD by moving the 50/50 BSs (beam splitters with reflectivity/ transmissivity of 50%) on the piezo stages (P). Then the location of the BSs were reset to achieve the appropriate path lengths of interferometers as we have mentioned, and restarted the photon counting.…”

“…According to the manner of weak measurement in [36,40], we experimentally observed the weak value of each path by HWP3, HWP4, and HWP5: which path a photon had passed was written into it's polarization. For example, we consider how to verify whether a photon has passed ñ |1 .…”

“…Such a measurement completely disturbs the path state due to the strong correlation with the polarization state. To achieve weak measurement, we should set the angle of HWP3 so that the polarization of a photon is almost H, namely, corresponds to the pointer of measurement, and b b = -G cos sin 2 2 is the measurement strength, where β=π/4−α [36]. Actually the normalized readout shows á ñá ñ | | Re 1 1 w as  G 0.…”

A possible solution to the which-way problem of quantum interference

“…The effects of local polarization rotations merely confirm that there is a correlation between the presence of each particle in the paths and the measurement outcome p of an interference measurement. Within the framework of the theory, this peculiar dependence of path presence on the outcomes p is possible because the quantum interferences between the two local polarization rotations result in a total change of the polarization that is fundamentally different from the two local rotations [23,24]. As explained above, this total change of the polarization cannot be explained in terms of a statistical distribution of the local rotation angles.…”