Show that measuring the observable | 1 ⟩ ⟨ 1 | is equivalent to measuring the observable Z up to a relabelling of the measurement outcomes.What does it mean by relabelling of the measurement outcomes and how do I prove it? Could anyone help me?

Question

Show that measuring the observable       |    1  ⟩  ⟨  1      |   is equivalent to measuring the observable Z up to a relabelling of the measurement outcomes.What does it mean by relabelling of the measurement outcomes and how do I prove it? Could anyone help me?

arhaitategr · Accepted Answer

Measuring the observable Z means that you collapse to either   |  0  ⟩  ⟨  0  | or to   |  1  ⟩  ⟨  1  |, this is exactly the same as measuring the observable   |  1  ⟩  ⟨  1  |, in which case you get that outcome or the opposite outcome   1  −  |  1  ⟩  ⟨  1  |  =  |  0  ⟩  ⟨  0  |.

Show that measuring the observable <mrow class="MJX-TeXAtom-ORD"> | </mrow> 1 <mo

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