Ex4 10

$\def\abs#1{|#1|}\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\def\tr{\mathord{\mbox{tr}}}\mathbf{Exercise\ 4.10}$

Design a measurement on a $3$-qubit system that distinguishes between states in which the number of $1$ bits is even, and those in which the number of $1$ bits is odd, and gives no other information. Write all operators in bra/ket notation.

page revision: 0, last edited: 18 Nov 2012 18:23

Assigning the eigenvalue of $1$ to the "even" outcome and $0$ to "odd":

(1)ReplyOptionsNote that Example 4.3.5 in Rieffel and Polak designs a similar measurement for a two-qubit system. Expanding this to three qubits, we can write the Hermitian operator

(1)which specifies measurement with respect to the decomposition of the vector space into subspaces $S_\textrm{even}$ and $S_\textrm{odd}$, which are generated by $\{ \ket{000},\ket{011},\ket{101},\ket{110} \}$ and $\{ \ket{001},\ket{010},\ket{100},\ket{111} \}$, respectively, where we have (arbitrarily) assigned the eigenvalue 2 to “even” and the eigenvalue 3 to “odd”.

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