I'm attempting to understand some of the characteristics of Posiitive Operator Value Measurement (POVM). For instance in Nielsen and Chuang, they obtain a set of measurement operators { E m } for states | ψ 1 ⟩ = | 0 ⟩ , | ψ 2 ⟩ = ( | 0 ⟩ + | 1 ⟩ ) / 2 . The end up obtaining the following set of operators: E 1 ≡ 2 1 + 2 | 1 ⟩ ⟨ 1 | , E 2 ≡ 2 1 + 2 ( | 0 ⟩ − | 1 ⟩ ) ( ⟨ 0 | − ⟨ 1 | ) 2 , E 3 ≡ I − E 1 − E 2 Basically, I'm oblivious to how they were able to obtain these. I thought that perhaps they found E 1 by utilizing the formula: E 1 = I − | ψ 2 ⟩ ⟨ ψ 2 | 1 + | ⟨ ψ 1 | ψ 2 ⟩ | However, when working it out, I do not obtain the same result. I'm sure it's something dumb and obvious I'm missing here. Any help on this would be very much appreciated.

Question

I&#039;m attempting to understand some of the characteristics of Posiitive Operator Value Measurement (POVM). For instance in Nielsen and Chuang, they obtain a set of measurement operators   {      E    m    } for states       |        ψ    1    ⟩  =      |    0  ⟩  ,      |        ψ    2    ⟩  =  (      |    0  ⟩  +      |    1  ⟩  )      /        2  . The end up obtaining the following set of operators:                              E          1                                    ≡                              2                                1            +                          2                                                |                1        ⟩        ⟨        1                  |                ,                                      E          2                                    ≡                              2                                1            +                          2                                                            (                          |                        0            ⟩            −                          |                        1            ⟩            )            (            ⟨            0                          |                        −            ⟨            1                          |                        )                    2                ,                                      E          3                                    ≡        I        −                  E          1                −                  E          2                    Basically, I&#039;m oblivious to how they were able to obtain these. I thought that perhaps they found       E    1   by utilizing the formula:                              E          1                =                              I            −                          |                                      ψ              2                        ⟩            ⟨                          ψ              2                                      |                                            1            +                          |                        ⟨                          ψ              1                                      |                                      ψ              2                        ⟩                          |                                          However, when working it out, I do not obtain the same result. I&#039;m sure it&#039;s something dumb and obvious I&#039;m missing here. Any help on this would be very much appreciated.

asafand2c · Accepted Answer

Yes, those are the results but you have the subindexes swaped.

I'm attempting to understand some of the characteristics of Posiitive Operator Value Measurement (PO

Answered question

Answer & Explanation

New Questions in Pre-Algebra