Frustration-free quantum Hamiltonian

A quantum Hamiltonian $H={\sum }_j{h}_j$ with positive-semidefinite local terms is frustration free when one state minimizes every $h_j$ simultaneously. After shifting each local minimum to zero, that state lies in every kernel $\ker h_j$ and has zero total energy.

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.

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 $⟨{\psi }_0|L{A}_x=⟨{\psi }_0|A_x$.

Combine a local projector cone in a frustration-free chain with the contraction bound from the detectability lemma. If the localized operator misses a second observable's support, their commutation and the exact ground-state bra identity convert the connected correlation function into an approximation error. The two-projection power estimate yields spatial decay rate $\alpha =\frac{1}{3}\ln (1+\Delta /2)$ for the stated detectability contraction.

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.