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From the normalized ground state annihilated by every $A_i$, the normalized Cartesian number states areThey have energy $\hslash \omega (n_1+n_2+n_3+3/2)$.

The condition $[g_Z,P_0]=0$ makes a unique global ground state an eigenvector of every filtered term, but its termwise eigenvalues can exceed the corresponding minima. For example $h_{12}=|1⟩⟨1{|}_1+|0⟩⟨0{|}_2$ and $h_{23}=|1⟩⟨1{|}_3+2|1⟩⟨1{|}_2$ commute with their sum. Their sum is uniquely minimized by $|000⟩$, yet $h_{12}$ is lower on $|010⟩$. Filtering leaves these commuting terms unchanged, so it cannot force frustration freeness.

A computational history state coherently records a circuit's intermediate states with orthogonal clock labels: $(T+1{)}^{-1/2}{\sum }_{t=0}^T|{\psi }_t⟩|c_t⟩$. A propagation term penalizes a mismatch between adjacent time labels and the corresponding gate. With correct propagation, uniform coefficients form a zero-energy ground state within the history subspace.

In a nearest-neighbour frustration-free Hamiltonian chain, multiply the bond orthogonal projections in two disjoint alternating layers. The resulting product $K$ fixes the common ground state space but contracts its orthogonal complement by a factor bounded away from one in terms of the spectral gap. A bound $\Vert K{|}_{\mathrm{supp}H}\Vert \le (1+\Delta /2{)}^{-1/3}$ implies exponentially convergent powers. The product is generally neither Hermitian nor normal.

An excited state is an energy eigenstate whose energy exceeds the ground state energy.

A connected isotropic spin-$S$ ferromagnet with positive exchanges has a fully polarized ground state and a maximal-total-spin multiplet. For $N$ sites the total spin is $NS$, with $2NS+1$ magnetic states generated by total-spin lowering. Global rotations span the same degenerate space. Slowly varying reorientations produce gapless ferromagnetic magnons; finite-volume degeneracy must be distinguished from a thermodynamic statement about spontaneous order.

If each local term is already minimized by the global ground state, subtract its minimum $a_Z$ to obtain a positive operator. Nonnegative normalized spectral filtering of Hamiltonian terms preserves positivity and leaves the ground eigenvalue $a_Z$ unchanged. Thus the same state minimizes every filtered term. This preserves frustration freeness but does not create it for an initially frustrated decomposition.

A decomposition $H={\sum }_Z{h}_Z$ has frustration freeness when a global ground state minimizes every term simultaneously. For finite-dimensional Hermitian terms this implies $E_0(H)={\sum }_Z{E}_0(h_Z)$. Shifting each term by its minimum produces positive operators with a common null vector. The property depends on the decomposition, not only the total operator.

For $M(s)=(1-s)D+sE$ in the unary history subspace, the first Gershgorin disc lies in $[0,s]$ and every other disc has real part at least $1-s$. For $s<1/2$, the first disc is isolated and contains the unique ground state energy. Thus the spectral gap is at least $1-2s$, in particular at least $1/3$ for $s\le 1/3$.

In the sudden approximation, an initial oscillator ground state overlaps the displaced ground state by $\exp [-m\omega d^2/(4\hslash )]$. Completing the square in the product of the two normalized Gaussian wavefunctions and using the Gaussian integral proves this amplitude. Squaring it gives the displayed probability. Only an irrelevant overall phase evolves before the sudden switch.

To evaluate a layered product on $A_x|{\psi }_0⟩$, scan the bond orthogonal projections from right to left. A projection disjoint from the current support commutes through the accumulated operator and disappears against the common ground state. Retained projections expand the support by at most one site per layer. Their product $L$ gives a local operator $L{A}_x$ with both $K^r{A}_x|{\psi }_0⟩=L{A}_x|{\psi }_0⟩$ and $⟨{\psi }_0|L{A}_x=⟨{\psi }_0|A_x$.