# Uniquely determined pure quantum states need not be unique ground states of quasi-local Hamiltonians

## TL;DR

This paper investigates the relationship between uniquely determined pure quantum states and their realization as non-degenerate ground states of quasi-local Hamiltonians, revealing that uniqueness in state determination does not guarantee a corresponding Hamiltonian realization.

## Contribution

It introduces a duality framework linking state determination by reduced density matrices to the existence of quasi-local Hamiltonians with that state as a unique ground state, and provides a counterexample disproving a common implication.

## Key findings

- Uniquely determined pure states are not necessarily ground states of quasi-local Hamiltonians.
- A dual semidefinite program characterizes the relationship between state uniqueness and Hamiltonian ground states.
- Counterexample shows the implication from unique determination to being a non-degenerate ground state is false.

## Abstract

We consider the problem of characterizing states of a multipartite quantum system from restricted, quasi-local information, with emphasis on uniquely determined pure states. By leveraging tools from dissipative quantum control theory, we show how the search for states consistent with an assigned list of reduced density matrices may be restricted to a proper subspace, which is determined solely by their supports. The existence of a quasi-local observable which attains its unique minimum over such a subspace further provides a sufficient criterion for a pure state to be uniquely determined by its reduced states. While the condition that a pure state is uniquely determined is necessary for it to arise as a non-degenerate ground state of a quasi-local Hamiltonian, we prove the opposite implication to be false in general, by exhibiting an explicit analytic counterexample. We show how the problem of determining whether a quasi-local parent Hamiltonian admitting a given pure state as its unique ground state is dual, in the sense of semidefinite programming, to the one of determining whether such a state is uniquely determined by the quasi-local information. Failure of this dual program to attain its optimal value is what prevents these two classes of states to coincide.