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I have a question about the phase modulation scheme described in Bennett et al. (1992). In figure 2 of this paper, it says

Source at left supplies the coherent pulse of M>1 expected photons intensity to Alice's half-interferometer, where unsymmetric beam spiitters (UBS), mirrors, and phase shifter (PSA 0 or 180 deg) produce a dim signal pulse, followed by a bright reference pulse. Sent to Bob through a single mode optical Sent to Bob through a single mode optical fiber, the pulses enter Bob's half-interferometer, where, depending on whether the sum of Alice's and Bob's phase shifts (PSA+PSB) is 0 or 180 deg, the signal pulse undergoes constructive or destructive. Arriving before this interference pulse is a very dim pulse. Arriving after the interference pulse is a bright twice-delayed reference pulse which Arriving after the interference pulse is a bright twice -delayed reference pulse which Bob monitors to be sure the reference pulses are not being suppressed. Also not shown are two unused beams leaving the rightmost beam splitter of each half interferometer in the downward direction.

It is understood that in the absence of an eavesdropper, Eve, random numbers can be shared by this protocol. In this and subsequent papers, the following arguments are made against Eve:

  • If Eve makes the same measurement as Bob and the measurement succeeds (i.e., constructive interference occurs), she can use the result to send a reference pulse and a signal pulse, and the eavesdropper will not notice.
  • If the measurement fails, neither the reference pulse nor the signal pulse is sent. In this case, however, the reference pulse does not reach Bob, so Bob is aware of the eavesdropping. Or, Eve sends only a reference pulse. In this case, the measurement result changes because of the interference with the vacuum, and Bob is aware of the eavesdropping.

My question is about the case where Eve fails to make a measurement. If she makes a measurement without phase shifting and the measurement fails, destructive interference has occurred so that she can send a reference pulse and a signal pulse with phase $\pi$ shifted to Bob, and Alice and Bob will not notice the eavesdropping. If Eve makes a measurement with a $\pi$ phase shift and the measurement fails, destructive interference occurs due to the phase shift, so if she sends the reference pulse and the signal pulse in the same phase to Bob, the eavesdropper will not notice the eavesdropping.

In this way, Eve would be able to eavesdrop without being aware of her presence. What is wrong with this interpretation?

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