Quantum entanglement of mechanical systems emerges when distinct objects move with such a high degree of correlation that they can no longer be described separately. Although quantum mechanics presumably applies to objects of all sizes, directly observing entanglement becomes challenging as masses increase, requiring measurement and control with a vanishingly small error. Here, using pulsed electromechanics, we deterministically entangle two mechanical drumheads with masses of 70 picograms. Through nearly quantum-limited measurements of the position and momentum quadratures of both drums, we perform quantum state tomography and thereby directly observe entanglement. Such entangled macroscopic systems are poised to serve in fundamental tests of quantum mechanics, enable sensing beyond the standard quantum limit, and function as long-lived nodes of future quantum networks.

Elastic and thermal properties of the TiO(2) lattice in anatase and rutile phases were studied in the framework of density functional perturbation theory within the local density approximation (LDA) and generalized-gradient approximation (GGA). The full elastic constant tensors of the polymorphs were calculated by linear fits to the acoustic branches of the phonon band structure near the center of the first Brillouin zone in symmetry directions of the crystals. It was observed that the elastic constants within the GGA are in better agreement with experiment. In addition, the Born effective charges, dielectric tensor, heat capacity, mean sound velocity and Debye temperature were calculated. The heat capacity at room temperature and the Debye temperature within the LDA are in better agreement with the experimental results. Therefore, using the phonon band structure and the density of states, one can obtain the important mechanical and thermal properties of materials.

We have studied the high-pressure cubic fluorite polymorph of TiO2 (c-TiO2) using the diffusion Monte Carlo (DMC) method. The estimated bulk modulus is within the range reported previously in density functional studies, high, but does not rival that of diamond. The calculated excitation energies within DMC are consistent with the results of GW approximation. The infrared frequency of c-TiO2, obtained via the frozen phonon method within DMC, shows non-negligible anharmonicity. This suggests that c-TiO2 might be stabilized if this anharmonicity is considered. Our DMC results could help to establish more accurate results for c-TiO2 compared with the widely-scattered mean-field results.

The spin-coherent-state positive-operator-valued-measure (POVM) is a fundamental measurement in quantum science, with applications including tomography, metrology, teleportation, benchmarking, and measurement of Husimi phase space probabilities. We prove that this POVM is achieved by collectively measuring the spin projection of an ensemble of qubits weakly and isotropically. We apply this in the context of optimal tomography of pure qubits. We show numerically that through a sequence of weak measurements of random directions of the collective spin component, sampled discretely or in a continuous measurement with random controls, one can approach the optimal bound. PACS numbers:In the standard paradigm of quantum tomography, one is given N copies of a quantum state that one seeks to estimate. When limited only by these finite quantum statistics and no other systematic experimental errors, what is the measurement that achieves the optimal average estimation fidelity? For the case of qubits, given a priori knowledge that the state is pure, this problem was solved long ago in a seminal paper by Massar and Popescu (MP) [1]. The optimal average fidelity is F opt = (N + 1)/(N + 2), and one can only reach this bound with a measurement that acts collectively on all N copies. "Local" measurements acting nonadaptively on one copy at a time can only achieve at best a scaling ofThe MP bound is achieved by a measurement whose positive-operator-valued-measure (POVM) is an overcomplete basis whose elements are proportional to projectors onto spin-coherent states (SCS) of the collective spin J in the symmetric subspace of N = 2J qubits. The SCS-POVM is a fundamental measurement in quantum information science, with applications including metrology [5,6], teleportation [7], benchmarking [8], and measurement of Husimi phase space probabilities [9]. While the Glauber-coherent-state-POVM in infinite dimensions has a well-known implementation via heterodyne measurement [10], despite various attempts [11][12][13], there is no known implementation of POVMs over generalizedcoherent-states for other Lie groups [14,15], such as the SU(2)-coherent-states considered here (except for one qubit, N = 1, J = 1/2) [13].The SCS-POVM has been considered physically unattainable and previous works have constructed alternative POVMs that also attain the optimal bound for tomography of qubits and qudits [16][17][18][19][20]. While in principle one can use the Neumark extension to realize these POVMs consisting of a finite number of measurement outcomes, such constructions have limited applicability, particularly as N grows beyond a few qubits.In this Letter we show that the SCS-POVM is in fact physically realizable in a direct manner for the application of optimal tomography and other quantum information protocols. In particular, we show that we can realize the SCS-POVM by measuring the collective spin, J = N i=1 σ (i) /2, weakly and isotropically over a sufficiently long time. This sequence of weak measurements is in a similar spirit to continuous col...