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With the development of quantum many-body simulator, Hamiltonian tomography has become an increasingly important technique for verification of quantum devices. Here we investigate recovering the Hamiltonians of two spin chains with 2-local interactions and 3-local interactions by measuring local observables. For these two models, we show that when the chain length reaches a certain critical number, we can recover the local Hamiltonian from its one steady state by solving the homogeneous operator equation (HOE) developed in Ref. [1]. To explain the existence of such a critical chain length, we develop an alternative method to recover Hamiltonian by solving the energy eigenvalue equations (EEE). By using the EEE method, we completely recovered the numerical results from the HOE method. Then we theoretically prove the equivalence between the HOE method and the EEE method. In particular, we obtain the analytical expression of the rank of the constraint matrix in the HOE method by using the EEE method, which can be used to determine the correct critical chain length in all the cases.
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A broad range of quantum optimisation problems can be phrased as the question whether a specific system has a ground state at zero energy, i.e. whether its Hamiltonian is frustration free. Frustration-free Hamiltonians, in turn, play a central role for constructing and understanding new phases of matter in quantum many-body physics. Unfortunately, determining whether this is the case is known to be a complexity-theoretically intractable problem. This makes it highly desirable to search for efficient heuristics and algorithms in order to, at least, partially answer this question. Here we prove a general criterion - a sufficient condition - under which a local Hamiltonian is guaranteed to be frustration free by lifting Shearers theorem from classical probability theory to the quantum world. Remarkably, evaluating this condition proceeds via a fully classical analysis of a hard-core lattice gas at negative fugacity on the Hamiltonians interaction graph which, as a statistical mechanics problem, is of interest in its own right. We concretely apply this criterion to local Hamiltonians on various regular lattices, while bringing to bear the tools of spin glass physics which permit us to obtain new bounds on the SAT/UNSAT transition in random quantum satisfiability. These also lead us to natural conjectures for when such bounds will be tight, as well as to a novel notion of universality for these computer science problems. Besides providing concrete algorithms leading to detailed and quantitative insights, this underscores the power of marrying classical statistical mechanics with quantum computation and complexity theory.
Remote state preparation (RSP) is a quantum information protocol which allows preparing a quantum state at a distant location with the help of a preshared nonclassical resource state and a classical channel. The efficiency of successfully doing this task can be represented by the RSP-fidelity of the resource state. In this paper, we study the influence on the RSP-fidelity by applying certain local operations on the resource state. We prove that RSP-fidelity does not increase for any unital local operation. However, for nonunital local operation, such as local amplitude damping channel, we find that some resource states can be enhanced to increase the RSP-fidelity. We give the optimal parameter of symmetric local amplitude damping channel for enhancing Bell-diagonal resource states. In addition, we show RSP-fidelity can suddenly change or even vanish at instant under local decoherence.
We study whether one can write a Matrix Product Density Operator (MPDO) as the Gibbs state of a quasi-local parent Hamiltonian. We conjecture this is the case for generic MPDO and give supporting evidences. To investigate the locality of the parent Hamiltonian, we take the approach of checking whether the quantum conditional mutual information decays exponentially. The MPDO we consider are constructed from a chain of 1-input/2-output (`Y-shaped) completely-positive maps, i.e. the MPDO have a local purification. We derive an upper bound on the conditional mutual information for bistochastic channels and strictly positive channels, and show that it decays exponentially if the correctable algebra of the channel is trivial. We also introduce a conjecture on a quantum data processing inequality that implies the exponential decay of the conditional mutual information for every Y-shaped channel with trivial correctable algebra. We additionally investigate a close but nonequivalent cousin: MPDO measured in a local basis. We provide sufficient conditions for the exponential decay of the conditional mutual information of the measured states, and numerically confirmed they are generically true for certain random MPDO.
The steady state of a driven dense ensemble of two-level atoms is determined from the competition of coherent laser excitation and decay that acts in a correlated way on several atoms simultaneously. We show that the presence of this non-local dissipation lifts the direct link between the density of excited atoms and the photon emission rate which is typically present when atoms decay independently. The non-locality disconnects these static and dynamic observables so that a dynamical transition in one does not necessarily imply a transition in the other. Furthermore, the collective nature of the quantum jump operators governing the non-local decay results in the formation of spatial coherence in the steady state which can be measured by analyzing solely global quantities - the photon emission rate and the density of excited atoms. The experimental realization of the system with strontium atoms in a lattice is discussed.
Generalizing Amaris work titled Information geometry on hierarchy of probability distributions, we define the degrees of irreducible multiparty correlations in a multiparty quantum state based on quantum relative entropy. We prove that these definitions are equivalent to those derived from the maximal von Neaumann entropy principle. Based on these definitions, we find a counterintuitive result on irreducible multiparty correlations: although the degree of the total correlation in a three-party quantum state does not increase under local operations, the irreducible three-party correlation can be created by local operations from a three-party state with only irreducible two-party correlations. In other words, even if a three-party state is initially completely determined by measuring two-party Hermitian operators, the determination of the state after local operations have to resort to the measurements of three-party Hermitian operators.
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