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arXiv:1902.09481 · Physical Review A

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

Salini Karuvade, Peter D. Johnson, Francesco Ticozzi, Lorenza Viola

Published Jun 10, 2019• 10 citations
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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 nondegenerate 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 from coinciding.

DOI
10.1103/physreva.99.062104

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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 nondegenerate 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 from coinciding.
Abstract We investigate whether a generic pure state on a multipartite quantum system can be the unique asymptotic steady state of locality-constrained purely dissipative Markovian dynamics. In the tripartite setting, we show that the problem is equivalent to characterizing the solution space of a set of linear equations and establish that the set of pure states obeying the above property has either measure zero or measure one, solely depending on the subsystems’ dimension. A complete analytical characterization is given when the central subsystem is a qubit. In the N -partite case, we provide conditions on the subsystems’ size and the nature of the locality constraint, under which random pure states cannot be quasi-locally stabilized generically. Also, allowing for the possibility to approximately stabilize entangled pure states that cannot be exact steady states in settings where stabilizability is generic, our results offer insights into the extent to which random pure states may arise as unique ground states of frustration-free parent Hamiltonians. We further argue that, to a high probability, pure quantum states sampled from a t -design enjoy the same stabilizability properties of Haar-random ones as long as suitable dimension constraints are obeyed and t is sufficiently large. Lastly, we demonstrate a connection between the tasks of quasi-local state stabilization and unique state reconstruction from local tomographic information, and provide a constructive procedure for determining a generic N -partite pure state based only on knowledge of the support of any two of the reduced density matrices of about half the parties, improving over existing results.
We focus on two classes of pure states for a finite-dimensional multipartite quantum system: those that can be identified as unique ground states (UGS) of a locally-constrained Hamiltonian, and those that can be obtained as unique, asymptotically stable equilibria of a Markov quantum dynamics under the same quasi-locality constraints (QLS). While there is an equivalence between (strict) subsets of two classes, namely, states that are UGS of frustration-free Hamiltonians and unique equilibria of purely dissipative dynamics, two paradigmatic entangled states, GHZ and W, offer counterexamples to either implication in the general setting. The intuition that every UGS state can be obtained as the exact steady state of a Markovian cooling process is salvaged only if suitable non-local resources are added. We show that stochastic stabilization of the target UGS state can be achieved if we perform continuous measurement of the global Hamiltonian, while controlling the dynamics with a switching, filtering-based quantum feedback strategy.
We characterize time-independent Markovian dynamics that drive a finite-dimensional multipartite quantum system into a target (pure) entangled steady state, subject to physical locality constraints. New control schemes are introduced in situations where the desired stabilization task {\em cannot} be attained solely based on quasi-local dissipative means, as considered in previous analysis. The new schemes either allow for Hamiltonian control or, if the latter is not an option, suitably restrict the set of admissible initial states. In both cases, we provide explicit algorithms for constructing a Markovian master equation that achieves the intended objective and show how this genuinely extends the manifold of stabilizable states. In particular, we present dissipative quasi-local control protocols for deterministically engineering multipartite GHZ ``cat'' states and W states on nnn qubits. For GHZ states, we show that no scalable procedure exists for achieving stabilization from arbitrary initial states, whereas this is possible for a target W state by a suitable combination of a two-body Hamiltonian and dissipators. Interestingly, for both entanglement classes, we show that quasi-local stabilization may be {\em scalably} achieved conditional to initialization of the system in a large, appropriately chosen subspace.
Traditional quantum physics solves ground states for a given Hamiltonian, while quantum information science asks for the existence and construction of certain Hamiltonians for given ground states. In practical situations, one would be mainly interested in local Hamiltonians with certain interaction patterns, such as nearest-neighbor interactions on some types of lattices. A necessary condition for a space VVV to be the ground-state space of some local Hamiltonian with a given interaction pattern is that the maximally mixed state supported on VVV is uniquely determined by its reduced density matrices associated with the given pattern, based on the principle of maximum entropy. However, it is unclear whether this condition is in general also sufficient. We examine the situations for the existence of such a local Hamiltonian to have VVV satisfying the necessary condition mentioned above as its ground-state space by linking to faces of the convex body of the local reduced states. We further discuss some methods for constructing the corresponding local Hamiltonians with given interaction patterns, mainly from physical points of view, including constructions related to perturbation methods, local frustration-free Hamiltonians, as well as thermodynamical ensembles.
We provide a solution to the problem of determining whether a target pure state can be asymptotically prepared using dissipative Markovian dynamics under fixed locality constraints. Besides recovering existing results for a large class of physically relevant entangled states, our approach has the advantage of providing an explicit stabilization test solely based on the input state and constraints of the problem. Connections with the formalism of frustration-free parent Hamiltonians are discussed, as well as control implementations in terms of a switching output-feedback law.
We investigate under which conditions a mixed state on a finite-dimensional multipartite quantum system may be the unique, globally stable fixed point of frustration-free semigroup dynamics subject to specified quasi-locality constraints. Our central result is a linear-algebraic necessary and sufficient condition for a generic (full-rank) target state to be frustration-free quasi-locally stabilizable, along with an explicit procedure for constructing Markovian dynamics that achieve stabilization. If the target state is not full-rank, we establish sufficiency under an additional condition, which is naturally motivated by consistency with pure-state stabilization results yet provably not necessary in general. Several applications are discussed, of relevance to both dissipative quantum engineering and information processing, and non-equilibrium quantum statistical mechanics. In particular, we show that a large class of graph product states (including arbitrary thermal graph states) as well as Gibbs states of commuting Hamiltonians are frustration-free stabilizable relative to natural quasi-locality constraints. Likewise, we provide explicit examples of non-commuting Gibbs states and non-trivially entangled mixed states that are stabilizable despite the lack of an underlying commuting structure, albeit scalability to arbitrary system size remains in this case an open question.