The question of which two-qubit states are steerable (i.e. permit a demonstration of EPR-steering) remains open. Here, a strong necessary condition is obtained for the steerability of two-qubit states having maximally-mixed reduced states, via the construction of local hidden state models. It is conjectured that this condition is in fact sufficient. Two provably sufficient conditions are also obtained, via asymmetric EPR-steering inequalities. Our work uses ideas from the quantum steering ellipsoid formalism, and explicitly evaluates the integral of n/(n An) 2 over arbitrary unit hemispheres for any positive matrix A.

Quantum metrology promises improved sensitivity in parameter estimation over classical procedures. However, there is an extensive debate over the question how the sensitivity scales with the resources (such as the average photon number) and number of queries that are used in estimation procedures. Here, we reconcile the physical definition of the relevant resources used in parameter estimation with the information-theoretical scaling in terms of the query complexity of a quantum network. This leads to a completely general optimality proof of the Heisenberg limit for quantum metrology. We give an example how our proof resolves paradoxes that suggest sensitivities beyond the Heisenberg limit, and we show that the Heisenberg limit is an information-theoretic interpretation of the Margolus-Levitin bound, rather than Heisenberg's uncertainty relation. 03.65.Ta, 42.50.Lc Parameter estimation is a fundamental pillar of science and technology, and improved measurement techniques for parameter estimation have often led to scientific breakthroughs and technological advancement. Caves [1] showed that quantum mechanical systems can in principle produce greater sensitivity over classical methods, and many quantum parameter estimation protocols have been proposed since [2]. The field of quantum metrology started with the work of Helstrom [3,4], who derived the minimum value for the mean square error in a parameter in terms of the density matrix of the quantum system and a measurement procedure. This was a generalisation of a known result in classical parameter estimation, called the Cramér-Rao bound. Braunstein and Caves [5] showed how this bound can be formulated for the most general state preparation and measurement procedures. While it is generally a hard problem to show that the Cramér-Rao bound can be attained in a given setup, at least it gives an upper limit to the precision of quantum parameter estimation.The quantum Cramér-Rao bound is typically formulated in terms of the Fisher information, an abstract quantity that measures the maximum information about a parameter ϕ that can be extracted from a given measurement procedure. One of the central questions in quantum metrology is how the Fisher information scales with the physical resources used in the measurement procedure. We usually consider two scaling regimes: First, in the standard quantum limit (SQL) [6] or shot-noise limit the Fisher information is constant, and the error scales with the inverse square root of the number of times T we make a measurement. Second, in the Heisenberg limit [7] the error is bounded by the inverse of the physical resources. Typically, these are expressed in terms of the size N of the probe system, e.g., (average) photon number. However, it has been clearly demonstrated that this form of the limit is not universally valid. For example, Beltrán and Luis [8] showed that the use of classical optical nonlinearities can lead to an error with average photon number scaling N −3/2 . Boixo et al. [9] devised a parameter estimation procedur...

It was pointed out by Dorje C. Brody that our Eq. (6), which was taken from Ref. [16], is false. One of our main results, given in Eq. (8), depends on this, and its validity is therefore put into question. A weaker form of Eq. (8) still holds, and this is sufficient to uphold the conclusion that the Heisenberg limit as defined in our Letter is optimal. However, we can no longer conclude that the Heisenberg limit is a form of the Margolus-Levitin bound [19].We now give the correct derivation of our result. From Ref.[17] we derived our Eq. (9):

A single quantum particle can be described by a wavefunction that spreads over arbitrarily large distances; however, it is never detected in two (or more) places. This strange phenomenon is explained in the quantum theory by what Einstein repudiated as 'spooky action at a distance': the instantaneous nonlocal collapse of the wavefunction to wherever the particle is detected. Here we demonstrate this single-particle spooky action, with no efficiency loophole, by splitting a single photon between two laboratories and experimentally testing whether the choice of measurement in one laboratory really causes a change in the local quantum state in the other laboratory. To this end, we use homodyne measurements with six different measurement settings and quantitatively verify Einstein's spooky action by violating an Einstein-Podolsky-Rosen-steering inequality by 0.042±0.006. Our experiment also verifies the entanglement of the split single photon even when one side is untrusted.