The standard 1+1d Z₂×Z₂ cluster model has a non-invertible global symmetry described by the fusion category Rep(D₈). This means the cluster state is not only a Z₂×Z₂ symmetry protected topological (SPT) phase but also a non-invertible SPT phase. Two new commuting Pauli Hamiltonians are found for the other two Rep(D₈) SPT phases on a tensor product Hilbert space of qubits, matching the classification in field theory and mathematics. Edge modes and local projective algebras at the interfaces between these non-invertible SPT phases are identified. It is shown that there does not exist a symmetric entangler that maps between these distinct SPT states. The cluster state is in a distinct Z₂×Z₂ SPT phase compared to the product state. The cluster model has a non-invertible symmetry, which is described by the fusion category Rep(D₈). This symmetry leads to new selection rules and constraints on the phase diagram. The cluster state is a topological phase protected by a non-invertible symmetry. The non-invertible symmetry is shown to be described by the fusion category Rep(D₈), whose fusion algebra is given by the tensor product of the irreducible representations of the group D₈. The cluster model also has another non-invertible symmetry, which mixes with the lattice translation and does not form a fusion category on the lattice. Three Rep(D₈) SPT phases are identified, distinguished by their symmetry breaking patterns after the KT transformation. These phases are related to the cluster state by finite-depth circuits. The three SPT phases are distinguished by the projective algebra involving the local factors of D and η^e, η^o. The projective algebra implies that the defect of the non-invertible symmetry D carries a non-trivial charge under η^e, η^o, or η^d. There is no symmetric entangler between the three different SPT states |cluster>, |odd>, and |even>. This is because the existence of an entangler would imply the existence of a unitary operator in the TST gauged/KT transformed system that is invariant under the dual symmetry and maps the two symmetry breaking phases to each other. However, this is impossible since the two SPT phases correspond to two different patterns of symmetry breaking after gauging and hence cannot be related by a symmetric unitary operator. The cluster state is in the same SPT phase as the product state with respect to the Z₂×Z₂ symmetry, but is distinguished by the non-invertible symmetry. This is reflected in the edge modes from the local projective algebra of the non-invertible symmetry. The product state is not invariant under D, and there is no canonical notion of a trivial non-invertible SPT phase. The lattice construction provides some of theThe standard 1+1d Z₂×Z₂ cluster model has a non-invertible global symmetry described by the fusion category Rep(D₈). This means the cluster state is not only a Z₂×Z₂ symmetry protected topological (SPT) phase but also a non-invertible SPT phase. Two new commuting Pauli Hamiltonians are found for the other two Rep(D₈) SPT phases on a tensor product Hilbert space of qubits, matching the classification in field theory and mathematics. Edge modes and local projective algebras at the interfaces between these non-invertible SPT phases are identified. It is shown that there does not exist a symmetric entangler that maps between these distinct SPT states. The cluster state is in a distinct Z₂×Z₂ SPT phase compared to the product state. The cluster model has a non-invertible symmetry, which is described by the fusion category Rep(D₈). This symmetry leads to new selection rules and constraints on the phase diagram. The cluster state is a topological phase protected by a non-invertible symmetry. The non-invertible symmetry is shown to be described by the fusion category Rep(D₈), whose fusion algebra is given by the tensor product of the irreducible representations of the group D₈. The cluster model also has another non-invertible symmetry, which mixes with the lattice translation and does not form a fusion category on the lattice. Three Rep(D₈) SPT phases are identified, distinguished by their symmetry breaking patterns after the KT transformation. These phases are related to the cluster state by finite-depth circuits. The three SPT phases are distinguished by the projective algebra involving the local factors of D and η^e, η^o. The projective algebra implies that the defect of the non-invertible symmetry D carries a non-trivial charge under η^e, η^o, or η^d. There is no symmetric entangler between the three different SPT states |cluster>, |odd>, and |even>. This is because the existence of an entangler would imply the existence of a unitary operator in the TST gauged/KT transformed system that is invariant under the dual symmetry and maps the two symmetry breaking phases to each other. However, this is impossible since the two SPT phases correspond to two different patterns of symmetry breaking after gauging and hence cannot be related by a symmetric unitary operator. The cluster state is in the same SPT phase as the product state with respect to the Z₂×Z₂ symmetry, but is distinguished by the non-invertible symmetry. This is reflected in the edge modes from the local projective algebra of the non-invertible symmetry. The product state is not invariant under D, and there is no canonical notion of a trivial non-invertible SPT phase. The lattice construction provides some of the