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(b) Prove that performing a projective measurement with respect to the 'parity' measurement (discussed in class and in the text- book) Po, P on an n-qubit state is equivalent to measuring the observable Z®n.

(b) Prove that performing a projective measurement with respect to the 'parity' measurement (discussed in class and in the text- book) Po, P on an n-qubit state is equivalent to measuring the observable Z®n.

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
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Practice Problem 24.) Recall the discussion in Section 3.4 of the textbook on viewing the Pauli operator Z = |0X0| – |1X1| as an observable. (a) Show that measuring the observable |1X1| is equivalent to mea- suring the observable Z, up to a relabelling of the measurement outcomes. (b) Prove that performing a projective measurement with respect to the 'parity' measurement (discussed in class and in the text- book) Po, P on an n-qubit state is equivalent to measuring the observable Z®n
Transcribed Image Text:Practice Problem 24.) Recall the discussion in Section 3.4 of the textbook on viewing the Pauli operator Z = |0X0| – |1X1| as an observable. (a) Show that measuring the observable |1X1| is equivalent to mea- suring the observable Z, up to a relabelling of the measurement outcomes. (b) Prove that performing a projective measurement with respect to the 'parity' measurement (discussed in class and in the text- book) Po, P on an n-qubit state is equivalent to measuring the observable Z®n
Note that the Von Neumann measurement as described in the Measurement Postulate (which can be described as a 'complete' or 'maximal' measurement) is the special case of a projective measurement where all the projectors P; have rank one (in other words, are of the form |;)(½¿| for a normalized state The simplest example of a complete Von Neumann measurement is a complete measurement in the computational basis. This can be viewed as a projective measurement with respect to the following decomposition of the identity I = > Pi ie{0,1}" where P = |i)(i]- A simple example of an incomplete projective measurement is a 'parity' mea- surement, where Po = Eparity(2)=0 |2){x| and P1 = Eparity(æ)=1 |2)(x|, where Po sums over all strings with an even number of 1s and P, with an odd number of 1s (Section 4.5 shows how to implement this projective measurement). Projective measurements are often described in terms of an observable. An ob- servable is a Hermitean operator M acting on the state space of the system. Since M is Hermitean, it has a spectral decomposition M = > m;P; (3.4.6) where P; is the orthogonal projector on the eigenspace of M with real eigenvalue m;. Measuring the observable corresponds to performing a projective measure- ment with respect to the decomposition I = D; P: where the measurement outcome i corresponds to the eigenvalue m;.
Transcribed Image Text:Note that the Von Neumann measurement as described in the Measurement Postulate (which can be described as a 'complete' or 'maximal' measurement) is the special case of a projective measurement where all the projectors P; have rank one (in other words, are of the form |;)(½¿| for a normalized state The simplest example of a complete Von Neumann measurement is a complete measurement in the computational basis. This can be viewed as a projective measurement with respect to the following decomposition of the identity I = > Pi ie{0,1}" where P = |i)(i]- A simple example of an incomplete projective measurement is a 'parity' mea- surement, where Po = Eparity(2)=0 |2){x| and P1 = Eparity(æ)=1 |2)(x|, where Po sums over all strings with an even number of 1s and P, with an odd number of 1s (Section 4.5 shows how to implement this projective measurement). Projective measurements are often described in terms of an observable. An ob- servable is a Hermitean operator M acting on the state space of the system. Since M is Hermitean, it has a spectral decomposition M = > m;P; (3.4.6) where P; is the orthogonal projector on the eigenspace of M with real eigenvalue m;. Measuring the observable corresponds to performing a projective measure- ment with respect to the decomposition I = D; P: where the measurement outcome i corresponds to the eigenvalue m;.
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