TY - JOUR T1 - Correlations in excited states of local Hamiltonians JF - Physical Review A Y1 - 2012 A1 - Jianxin Chen A1 - Zhengfeng Ji A1 - Zhaohui Wei A1 - Bei Zeng AB - Physical properties of the ground and excited states of a $k$-local Hamiltonian are largely determined by the $k$-particle reduced density matrices ($k$-RDMs), or simply the $k$-matrix for fermionic systems---they are at least enough for the calculation of the ground state and excited state energies. Moreover, for a non-degenerate ground state of a $k$-local Hamiltonian, even the state itself is completely determined by its $k$-RDMs, and therefore contains no genuine ${>}k$-particle correlations, as they can be inferred from $k$-particle correlation functions. It is natural to ask whether a similar result holds for non-degenerate excited states. In fact, for fermionic systems, it has been conjectured that any non-degenerate excited state of a 2-local Hamiltonian is simultaneously a unique ground state of another 2-local Hamiltonian, hence is uniquely determined by its 2-matrix. And a weaker version of this conjecture states that any non-degenerate excited state of a 2-local Hamiltonian is uniquely determined by its 2-matrix among all the pure $n$-particle states. We construct explicit counterexamples to show that both conjectures are false. It means that correlations in excited states of local Hamiltonians could be dramatically different from those in ground states. We further show that any non-degenerate excited state of a $k$-local Hamiltonian is a unique ground state of another $2k$-local Hamiltonian, hence is uniquely determined by its $2k$-RDMs (or $2k$-matrix). VL - 85 UR - http://arxiv.org/abs/1106.1373v2 CP - 4 J1 - Phys. Rev. A U5 - 10.1103/PhysRevA.85.040303 ER - TY - JOUR T1 - Ground-State Spaces of Frustration-Free Hamiltonians JF - Journal of Mathematical Physics Y1 - 2012 A1 - Jianxin Chen A1 - Zhengfeng Ji A1 - David Kribs A1 - Zhaohui Wei A1 - Bei Zeng AB - We study the ground-state space properties for frustration-free Hamiltonians. We introduce a concept of `reduced spaces' to characterize local structures of ground-state spaces. For a many-body system, we characterize mathematical structures for the set $\Theta_k$ of all the $k$-particle reduced spaces, which with a binary operation called join forms a semilattice that can be interpreted as an abstract convex structure. The smallest nonzero elements in $\Theta_k$, called atoms, are analogs of extreme points. We study the properties of atoms in $\Theta_k$ and discuss its relationship with ground states of $k$-local frustration-free Hamiltonians. For spin-1/2 systems, we show that all the atoms in $\Theta_2$ are unique ground states of some 2-local frustration-free Hamiltonians. Moreover, we show that the elements in $\Theta_k$ may not be the join of atoms, indicating a richer structure for $\Theta_k$ beyond the convex structure. Our study of $\Theta_k$ deepens the understanding of ground-state space properties for frustration-free Hamiltonians, from a new angle of reduced spaces. VL - 53 U4 - 102201 UR - http://arxiv.org/abs/1112.0762v1 CP - 10 J1 - J. Math. Phys. U5 - 10.1063/1.4748527 ER - TY - JOUR T1 - The LU-LC conjecture is false Y1 - 2007 A1 - Zhengfeng Ji A1 - Jianxin Chen A1 - Zhaohui Wei A1 - Mingsheng Ying AB - The LU-LC conjecture is an important open problem concerning the structure of entanglement of states described in the stabilizer formalism. It states that two local unitary equivalent stabilizer states are also local Clifford equivalent. If this conjecture were true, the local equivalence of stabilizer states would be extremely easy to characterize. Unfortunately, however, based on the recent progress made by Gross and Van den Nest, we find that the conjecture is false. UR - http://arxiv.org/abs/0709.1266v2 J1 - Quantum Inf. Comput. ER -