Characterizing Ground and Thermal States of Few-Body Hamiltonians

Abstract

The question whether a given quantum state is a ground or thermal state of a few-body Hamiltonian can be used to characterize the complexity of the state and is important for possible experimental implementations. We provide methods to characterize the states generated by two-and, more generally, k-body Hamiltonians as well as the convex hull of these sets. This leads to new insights into the question which states are uniquely determined by their marginals and to a generalization of the concept of entanglement. Finally, certification methods for quantum simulation can be derived.PACS numbers: 03.65. Ud, 03.67.Mn Introduction.-Interactions in quantum mechanics are described by Hamilton operators. The study of their properties, such as their symmetries, eigenvalues, and ground states, is central for several fields of physics. Physically relevant Hamiltonians, however, are often restricted to few-body interactions, as the relevant interaction mechanisms are local. But the characterization of generic few-body Hamiltonians is not well explored, since in most cases one starts with a given Hamiltonian and tries to find out its properties.In quantum information processing, ground and thermal states of local Hamiltonians are of interest for several reasons: First, if a desired state is the ground or thermal state of a sufficiently local Hamiltonian, it might be experimentally prepared by engineering the required interactions and cooling down or letting thermalise the physical system [1]. For example, one may try to prepare a cluster state, the resource for measurement-based quantum computation, as a ground state of a local Hamiltonian [2]. Second, on a more theoretical side, ground states of k-body Hamiltonians are completely characterized by their reduced k-body density matrices. The question which states are uniquely determined by their marginals has been repeatedly studied and is a variation of the representability problem, which asks whether given marginals can be represented by a global state [3]. It has turned out that many pure states have the property to be uniquely determined by a small set of their marginals [4,5], and for practical purposes it is relevant that often entanglement or non-locality can be inferred by considering the marginals only [6].In this paper we present a general approach to characterize ground and thermal states of few-body Hamiltonians. We use the formalism of exponential families, a concept first introduced for classical probability distributions by Amari [7] and extended to the quantum setting in Refs. [8][9][10][11]. This offers a systematic characterization of the complexity of quantum states in a conceptionally pleasing way. We derive two methods that can be used to compute various distances to thermal states of kbody Hamiltonians: The first method is general and uses semidefinite programming, while the second method isSchematic view of the state space, the exponential families Q1 and Q2, and their convex hulls. While the whole space of mixed states is convex, the exponential famil...

A complete hierarchy for the pure state marginal problem in quantum mechanics

“…Proof On the one hand, if there exists a pure state φ j i φ h j 2 C, one can easily verify that Φ AB ¼ φ j i φ h j φ j i φ h j satisfies the constraints in Eqs. (8) and 9as well as TrðV AB Φ AB Þ ¼ 1.…”

“…The first candidate is the positive partial transpose (PPT) criterion [24,25], which is an SDP relaxation of the optimization in Eq. (7). The PPT relaxation provides a pretty good approximation when the local dimension and the number of parties are small.…”

“…A variation of the marginal problem is the question whether or not the marginals determine the global state uniquely or not [5,6]. This is relevant in condensed matter physics, where one may ask whether a state is the unique ground state of a local Hamiltonian [7,8].…”

“…A variation of the marginal problem is the question of whether or not the marginals determine the global state uniquely or not [5][6][7] . This is relevant in condensed matter physics, where one may ask whether a state is the unique ground state of a local Hamiltonian 8,9 . Many other cases, such as marginal problems for Gaussian and symmetric states 10,11 and applications in quantum correlation 12 , quantum causality 13 , and interacting quantum many-body systems 14,15 have been studied.…”

Learning the dynamics of open quantum systems from their steady states

“…An isolated quantum system can be characterized by learning its underlying Hamiltonian. This can be achieved by monitoring the dynamics that the Hamiltonian generates [18][19][20][21][22][23][24][25][26][27][28][29][30][31], or by measuring local observables in one of its eigenstates or thermal states [32][33][34][35][36][37][38][39][40]. However, realistic quantum systems are never fully isolated.…”