Which one of the following are stabilizer states of X⊗Z⊗Z𝑋⊗𝑍⊗𝑍 ?Group of answer choices|+00⟩|+00⟩ |+01⟩|+01⟩ |−10⟩|−10⟩ |−11⟩|−11⟩
Question
Which one of the following are stabilizer states of X⊗Z⊗Z𝑋⊗𝑍⊗𝑍 ?Group of answer choices|+00⟩|+00⟩ |+01⟩|+01⟩ |−10⟩|−10⟩ |−11⟩|−11⟩
Solution 1
The stabilizer states of X⊗Z⊗Z are |+00⟩ and |-00⟩.
Here's why:
The X operator flips the sign of the state it acts on, while the Z operator leaves the sign of the state unchanged. Therefore, the states that are stabilized by X⊗Z⊗Z are those that remain unchanged when the X operator acts on the first qubit and the Z operator acts on the second and third qubits.
The state |+00⟩ is stabilized because the X operator flips the sign of the first qubit, resulting in |-00⟩, and the Z operators leave the sign of the second and third qubits unchanged.
Similarly, the state |-00⟩ is stabilized because the X operator flips the sign of the first qubit, resulting in |+00⟩, and the Z operators leave the sign of the second and third qubits unchanged.
The states |+01⟩, |-01⟩, |+10⟩, |-10⟩, |+11⟩, and |-11⟩ are not stabilized by X⊗Z⊗Z because the Z operator would flip the sign of the second and/or third qubits in these states.
Solution 2
The stabilizer states of X⊗Z⊗Z are |+00⟩ and |-00⟩.
Here's why:
The X operator flips the sign of the state it acts on, while the Z operator leaves the sign of the state it acts on unchanged.
So, if we apply X⊗Z⊗Z to |+00⟩, we get |-00⟩, and if we apply X⊗Z⊗Z to |-00⟩, we get |+00⟩.
Therefore, |+00⟩ and |-00⟩ are stabilizer states of X⊗Z⊗Z.
The other states, |+01⟩, |-10⟩, and |-11⟩, are not stabilizer states of X⊗Z⊗Z because applying X⊗Z⊗Z to these states will not return the original state.
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