Tighter quantum uncertainty relations follow from a general probabilistic bound

Uncertainty relations (URs) like the Heisenberg-Robertson or the time-energy UR are often considered to be hallmarks of quantum theory. Here, a simple derivation of these URs is presented based on a single classical inequality from estimation theory, a Cram\'er-Rao-like bound. The Heisenberg-Robertson UR is then obtained by using the Born rule and the Schr\"odinger equation. This allows a clear separtion of the probabilistic nature of quantum mechanics from the Hilbert space structure and the dynamical law. It also simplifies the interpretation of the bound. In addition, the Heisenberg-Robertson UR is tightened for mixed states by replacing one variance by the so-called quantum Fisher information. Thermal states of Hamiltonians with evenly-gapped energy levels are shown to saturate the tighter bound for natural choices of the operators. This example is further extended to Gaussian states of a harmonic oscillator. For many-qubit systems, we illustrate the interplay between entanglement and the structure of the operators that saturate the UR with spin-squeezed states and Dicke states.Comment: 8 pages, 1 figure. v2: improved presentation, references added, results on the connection between saturated inequality and entanglement structure for multi-qubit states adde

Certifiability criterion for large-scale quantum systems

Can one certify the preparation of a coherent, many-body quantum state by measurements with bounded accuracy in the presence of noise and decoherence? Here, we introduce a criterion to assess the fragility of large-scale quantum states which is based on the distinguishability of orthogonal states after the action of very small amounts of noise. States which do not pass this criterion are called asymptotically incertifiable. We show that, if a coherent quantum state is asymptotically incertifiable, there exists an incoherent mixture (with entropy at least log 2) which is experimentally indistinguishable from the initial state. The Greenberger-Horne-Zeilinger states are examples of such asymptotically incertifiable states. More generally, we prove that any so-called macroscopic superposition state is asymptotically incertifiable. We also provide examples of quantum states that are experimentally indistinguishable from highly incoherent mixtures, i.e., with an almost-linear entropy in the number of qubits. Finally we show that all unique ground states of local gapped Hamiltonians (in any dimension) are certifiable.Comment: 21 pages, 1 figure; V2: title changed plus minor changes in the text and some additional reference

Linking Measures for Macroscopic Quantum States via Photon-Spin Mapping

We review and compare several measures that identify quantum states that are "macroscopically quantum". These measures were initially formulated either for photonic systems or spin ensembles. Here, we compare them through a simple model which maps photonic states to spin ensembles. On the one hand, we reveal problems for some spin measures to handle correctly photonic states that typically are considered to be macroscopically quantum. On the other hand, we find significant similarities between other measures even though they were differently motivated.Comment: 12 pages, 1 figure; published in a special issue of Optics Communications: "Macroscopic quantumness: theory and applications in optical sciences"; v2: minor change

Stable macroscopic quantum superpositions

We study the stability of superpositions of macroscopically distinct quantum states under decoherence. We introduce a class of quantum states with entanglement features similar to Greenberger-Horne-Zeilinger (GHZ) states, but with an inherent stability against noise and decoherence. We show that in contrast to GHZ states, these so-called concatenated GHZ states remain multipartite entangled even for macroscopic numbers of particles and can be used for quantum metrology in noisy environments. We also propose a scalable experimental realization of these states using existing ion-trap set-ups.Comment: 4 pages, 1 figure; v2: minor changes due to referee report