A fundamental problem in quantizing general relativity

We point out a fundamental problem that hinders the quantization of general relativity: quantum mechanics is formulated in terms of systems, typically limited in space but infinitely extended in time, while general relativity is formulated in terms of events, limited both in space and in time. Many of the problems faced while connecting the two theories stem from the difficulty in shoe-horning one formulation into the other. A solution is not presented, but a list of desiderata for a quantum theory based on events is laid out.Comment: 3.4 pages, one figur

Entropic information-disturbance tradeoff

We show the flaws found in the customary fidelity-based definitions of disturbance in quantum measurements and evolutions. We introduce the "entropic disturbance" D and show that it adequately measures the degree of disturbance, intended essentially as an irreversible change in the state of the system. We also find that it complies with an information-disturbance tradeoff, namely the mutual information between the eigenvalues of the initial state and the measurement results is less than or equal to D.Comment: 4 pages, 1 figure. Revised versio

A simple proof of Bell's inequality

Bell's theorem is a fundamental result in quantum mechanics: it discriminates between quantum mechanics and all theories where probabilities in measurement results arise from the ignorance of pre-existing local properties. We give an extremely simple proof of Bell's inequality: a single figure suffices. This simplicity may be useful in the unending debate of what exactly the Bell inequality means, since the hypothesis at the basis of the proof become extremely transparent. It is also a useful didactic tool, as the Bell inequality can be explained in a single intuitive lecture.Comment: Version accepted for publication on American Journal of Physics. Appendix B added, along with various clarifications on the nomenclatur

The Pauli objection

Schroedinger's equation says that the Hamiltonian is the generator of time translations. This seems to imply that any reasonable definition of time operator must be conjugate to the Hamiltonian. Then both time and energy must have the same spectrum since conjugate operators are unitarily equivalent. Clearly this is not always true: normal Hamiltonians have lower bounded spectrum and often only have discrete eigenvalues, whereas we typically desire that time can take any real value. Pauli concluded that constructing a general a time operator is impossible (although clearly it can be done in specific cases). Here we show how the Pauli argument fails when one uses an external system (a "clock") to track time, so that time arises as correlations between the system and the clock (conditional probability amplitudes framework). In this case, the time operator is not conjugate to the system Hamiltonian, but its eigenvalues still satisfy the Schroedinger equation for arbitrary Hamiltonians.Comment: 6 page

Stronger uncertainty relations for the sum of variances

Heisenberg-Robertson's uncertainty relation expresses a limitation in the possible preparations of the system by giving a lower bound to the product of the variances of two observables in terms of their commutator. Notably, it does not capture the concept of incompatible observables because it can be trivial, i.e., the lower bound can be null even for two non-compatible observables. Here we give two stronger uncertainty relations, relating to the sum of variances, whose lower bound is guaranteed to be nontrivial whenever the two observables are incompatible on the state of the system.Comment: Version accepted for publication on Phys. Rev. Let

State estimation: direct state measurement vs. tomography

We compare direct state measurement (DST or weak state tomography) to conventional state reconstruction (tomography) through accurate Monte-Carlo simulations. We show that DST is surprisingly robust to its inherent bias. We propose a method to estimate such bias (which introduces an unavoidable error in the reconstruction) from the experimental data. As expected we find that DST is much less precise than tomography. We consider both finite and infinite-dimensional states of the DST pointer, showing that they provide comparable reconstructions.Comment: 4 pages, 4 figure

Using entanglement against noise in quantum metrology

We analyze the role of entanglement among probes and with external ancillas in quantum metrology. In the absence of noise, it is known that unentangled sequential strategies can achieve the same Heisenberg scaling of entangled strategies and that external ancillas are useless. This changes in the presence of noise: here we prove that entangled strategies can have higher precision than unentangled ones and that the addition of passive external ancillas can also increase the precision. We analyze some specific noise models and use the results to conjecture a general hierarchy for quantum metrology strategies in the presence of noise.Comment: 7 pages, 4 figures, published versio

Quantum metrology: why entanglement?

We show why and when entanglement is needed for quantum-enhanced precision measurements, and which type of entanglement is useful. We give a simple, intuitive construction that shows how entanglement transforms parallel estimation strategies into sequential ones of same precision. We employ this argument to generalize conventional quantum metrology, to identify a class of noise whose effects can be easily managed, and to treat the case of indistinguishable probes (such as interferometry with light).Comment: 5 pages, 3 figure