The elements of the set{X⊗Z⊗Z,X⊗I⊗I,I⊗X⊗X}{𝑋⊗𝑍⊗𝑍,𝑋⊗𝐼⊗𝐼,𝐼⊗𝑋⊗𝑋} are generators of a stabilizer group.Group of answer choicesTrueFalse

To determine whether the given elements of the set $\{X \otimes Z \otimes Z, X \otimes I \otimes I, I \otimes X \otimes X\}$ are generators of a stabilizer group, we should consider the properties of stabilizer groups in quantum mechanics.

Understanding Stabilizer Groups

1. Stabilizer Group Definition: Stabilizer groups are subgroups of the Pauli group where each element has eigenvalues of $+1$ for a certain set of quantum states. The stabilizer group is typically generated by elements that commute with each other.

2. Pauli Operators:

   • The Pauli operators $X$, $Y$, and $Z$ satisfy certain commutation relations:
     • $XZ = -ZX$
     • $YZ = -ZY$
     • $XY = -YX$
   • The identity operator $I$ acts as a neutral element.

3. Conditions for Generators:

   • Each generator should commute with each other.
   • The number of generators should match the number of qubits involved.

Let's analyze the elements step by step.

Step 1: Elements in the Set

• The elements are:
  1. $E_1 = X \otimes Z \otimes Z$
  2. $E_2 = X \otimes I \otimes I$
  3. $E_3 = I \otimes X \otimes X$

Step 2: Check Commutativity

• We need to check if the generators commute with each other.

Commuting Pairs:

1. Commutativity of $E_1$ and $E_2$:

   • $(X \otimes Z \otimes Z)(X \otimes I \otimes I) = X^2 \otimes Z \otimes Z = I \otimes Z \otimes Z$
   • $(X \otimes I \otimes I)(X \otimes Z \otimes Z) = X^2 \otimes I \otimes I = I \otimes I \otimes I$
   • They commute, since both equal to elements that can be factored.

2. Commutativity of $E_1$ and $E_3$:

   • Checking $(X \otimes Z \otimes Z)(I \otimes X \otimes X)$ and vice versa leads to non-commuting pairs because $Z$ and $X$ do not commute.

3. Commutativity of $E_2$ and $E_3$:

   • $(X \otimes I \otimes I)(I \otimes X \otimes X) = X \otimes X \otimes I$ and $(I \otimes X \otimes X)(X \otimes I \otimes I) = X^2 \otimes I \otimes I = I \otimes I \otimes I$, they do not commute either.

Step 3: Conclusion on Generators

• Since not all of the generators commute with one another, the set of elements does not form a stabilizer group.

Final Answer

The statement is False. The given elements are not generators of a stabilizer group.