Consider the Hamiltonian H=Z⊗3 acting on 3 qubits.
What is the ground energy of the system? (I.e. the lowest eigenvalue of H?)
What is the dimension of the ground energy subspace? (I.e. multiplicity of that eigenvalue?)
Which of the following correctly describes the ground energy states of this system?
|000⟩ is the only ground energy state.
|111⟩ is the only ground energy state.
|000⟩,|111⟩ and their linear combinations
|001⟩,|010⟩,|100⟩ and their linear combinations
|000⟩,|001⟩,|010⟩,|100⟩ and their linear combinations
|000⟩,|011⟩,|110⟩,|101⟩ and their linear combinations
|001⟩,|010⟩,|100⟩,|111⟩ and their linear combinations
last option
last option worked for me
000> is the only ground energy state, I think... don't know others
sure last option?justify
To find the ground energy of the system, you need to determine the lowest eigenvalue of the Hamiltonian H. In this case, the Hamiltonian H is given by H = Z ⊗ 3, where Z is the Pauli-Z matrix.
To find the eigenvalues of H, you can start by writing down the matrix representation of H. Since H acts on 3 qubits, the matrix representation will be a 8x8 matrix. Each element of the matrix can be obtained by taking the tensor product of the Z matrix with itself three times.
After obtaining the matrix representation of H, you can find its eigenvalues. The lowest eigenvalue will correspond to the ground energy of the system.
To find the dimension of the ground energy subspace, you can count the number of linearly independent eigenvectors corresponding to the lowest eigenvalue. Each eigenvector in the ground energy subspace will be a valid ground energy state.
Once you have determined the eigenvalues and corresponding eigenvectors, you can match them with the given options to find the correct description of the ground energy states of the system.
Note: Since the Hamiltonian is given by H = Z ⊗ 3, the ground energy states will be the eigenstates of the Z operator for each qubit.