Showing papers on "Quantum state published in 2011"
TL;DR: Sideband cooling of an approximately 10-MHz micromechanical oscillator to the quantum ground state is demonstrated and the device exhibits strong coupling, allowing coherent exchange of microwave photons and mechanical phonons.
Abstract: It has been a long-standing goal in the field of cavity optomechanics to cool down a mechanical resonator to its motional quantum ground state by using light. Teufel et al. have now achieved just that with a recently developed system in which a drum-like flexible aluminium membrane is incorporated in a superconducting circuit. Ground-state cooling of a mechanical resonator was demonstrated for the first time last year in a different type of device, but the quantum states in this new device should be much longer lived, allowing direct tests of fundamental principles of quantum mechanics. As a first step, the authors perform a quantum-limited position measurement that is only a factor of about five away from the Heisenberg limit. The advent of laser cooling techniques revolutionized the study of many atomic-scale systems, fuelling progress towards quantum computing with trapped ions1 and generating new states of matter with Bose–Einstein condensates2. Analogous cooling techniques3,4 can provide a general and flexible method of preparing macroscopic objects in their motional ground state. Cavity optomechanical or electromechanical systems achieve sideband cooling through the strong interaction between light and motion5,6,7,8,9,10,11,12,13,14,15. However, entering the quantum regime—in which a system has less than a single quantum of motion—has been difficult because sideband cooling has not sufficiently overwhelmed the coupling of low-frequency mechanical systems to their hot environments. Here we demonstrate sideband cooling of an approximately 10-MHz micromechanical oscillator to the quantum ground state. This achievement required a large electromechanical interaction, which was obtained by embedding a micromechanical membrane into a superconducting microwave resonant circuit. To verify the cooling of the membrane motion to a phonon occupation of 0.34 ± 0.05 phonons, we perform a near-Heisenberg-limited position measurement3 within (5.1 ± 0.4)h/2π, where h is Planck’s constant. Furthermore, our device exhibits strong coupling, allowing coherent exchange of microwave photons and mechanical phonons16. Simultaneously achieving strong coupling, ground state preparation and efficient measurement sets the stage for rapid advances in the control and detection of non-classical states of motion17,18, possibly even testing quantum theory itself in the unexplored region of larger size and mass19. Because mechanical oscillators can couple to light of any frequency, they could also serve as a unique intermediary for transferring quantum information between microwave and optical domains20.
1,702 citations
TL;DR: In this paper, it was shown that the quantum Hall effect without an external magnetic field can be achieved in a known ferromagnetic compound HgCr2Se4, with a single pair of Weyl fermions separated in momentum space.
Abstract: In 3D momentum space, a topological phase boundary separating the Chern insulating layers from normal insulating layers may exist, where the gap must be closed, resulting in a "Chern semimetal" state with topologically unavoidable band crossings at the Fermi level. This state is a condensed-matter realization of Weyl fermions in (3+1)D, and should exhibit remarkable features, such as magnetic monopoles and Fermi arcs. Here we predict, based on first principles calculations, that such a novel quantum state can be realized in a known ferromagnetic compound HgCr2Se4, with a single pair of Weyl fermions separated in momentum space. The quantum Hall effect without an external magnetic field can be achieved in its quantum-well structure.
1,162 citations
TL;DR: The basic circuit architecture presented here provides a feasible path to ground-state cooling and subsequent coherent control and measurement of long-lived quantum states of mechanical motion and is in excellent quantitative agreement with recent theoretical predictions.
Abstract: Demonstrating and exploiting the quantum nature of macroscopic mechanical objects would help us to investigate directly the limitations of quantum-based measurements and quantum information protocols, as well as to test long-standing questions about macroscopic quantum coherence. Central to this effort is the necessity of long-lived mechanical states. Previous efforts have witnessed quantum behaviour, but for a low-quality-factor mechanical system. The field of cavity optomechanics and electromechanics, in which a high-quality-factor mechanical oscillator is parametrically coupled to an electromagnetic cavity resonance, provides a practical architecture for cooling, manipulation and detection of motion at the quantum level. One requirement is strong coupling, in which the interaction between the two systems is faster than the dissipation of energy from either system. Here, by incorporating a free-standing, flexible aluminium membrane into a lumped-element superconducting resonant cavity, we have increased the single-photon coupling strength between these two systems by more than two orders of magnitude, compared to previously obtained coupling strengths. A parametric drive tone at the difference frequency between the mechanical oscillator and the cavity resonance dramatically increases the overall coupling strength, allowing us to completely enter the quantum-enabled, strong-coupling regime. This is evidenced by a maximum normal-mode splitting of nearly six bare cavity linewidths. Spectroscopic measurements of these 'dressed states' are in excellent quantitative agreement with recent theoretical predictions. The basic circuit architecture presented here provides a feasible path to ground-state cooling and subsequent coherent control and measurement of long-lived quantum states of mechanical motion.
705 citations
TL;DR: The experiment demonstrates that active control can generate non-classical states of this oscillator and combat their decoherence, and is a significant step towards the implementation of complex quantum information operations.
Abstract: Feedback loops are central to most classical control procedures A controller compares the signal measured by a sensor (system output) with the target value or set-point It then adjusts an actuator (system input) to stabilize the signal around the target value Generalizing this scheme to stabilize a micro-system's quantum state relies on quantum feedback, which must overcome a fundamental difficulty: the sensor measurements cause a random back-action on the system An optimal compromise uses weak measurements, providing partial information with minimal perturbation The controller should include the effect of this perturbation in the computation of the actuator's operation, which brings the incrementally perturbed state closer to the target Although some aspects of this scenario have been experimentally demonstrated for the control of quantum or classical micro-system variables, continuous feedback loop operations that permanently stabilize quantum systems around a target state have not yet been realized Here we have implemented such a real-time stabilizing quantum feedback scheme following a method inspired by ref 13 It prepares on demand photon number states (Fock states) of a microwave field in a superconducting cavity, and subsequently reverses the effects of decoherence-induced field quantum jumps The sensor is a beam of atoms crossing the cavity, which repeatedly performs weak quantum non-demolition measurements of the photon number The controller is implemented in a real-time computer commanding the actuator, which injects adjusted small classical fields into the cavity between measurements The microwave field is a quantum oscillator usable as a quantum memory or as a quantum bus swapping information between atoms Our experiment demonstrates that active control can generate non-classical states of this oscillator and combat their decoherence, and is a significant step towards the implementation of complex quantum information operations
556 citations
TL;DR: This work has created a few-body quantum system with complete control over its quantum state using ultracold fermionic atoms in an optical dipole trap and has observed the interaction-induced energy shift for a pair of repulsively interacting atoms.
Abstract: Systems consisting of few interacting fermions are the building blocks of matter, with atoms and nuclei being the most prominent examples. We have created a few-body quantum system with complete control over its quantum state using ultracold fermionic atoms in an optical dipole trap. Ground-state systems consisting of 1 to 10 particles are prepared with fidelities of ∼90%. We can tune the interparticle interactions to arbitrary values using a Feshbach resonance and have observed the interaction-induced energy shift for a pair of repulsively interacting atoms. This work is expected to enable quantum simulation of strongly correlated few-body systems.
484 citations
TL;DR: A simple method for certifying that an experimental device prepares a desired quantum state ρ, and it provides an estimate of the fidelity between ρ and the actual (arbitrary) state in the lab, up to a constant additive error.
Abstract: We describe a simple method for certifying that an experimental device prepares a desired quantum state ρ. Our method is applicable to any pure state ρ, and it provides an estimate of the fidelity between ρ and the actual (arbitrary) state in the lab, up to a constant additive error. The method requires measuring only a constant number of Pauli expectation values, selected at random according to an importance-weighting rule. Our method is faster than full tomography by a factor of d, the dimension of the state space, and extends easily and naturally to quantum channels.
413 citations
TL;DR: It is shown that even complex systems, with more than 1,000 internal degrees of freedom, can be prepared in quantum states that are sufficiently well isolated from their environment to avoid decoherence and to show almost perfect coherence.
Abstract: The wave nature of matter is a key ingredient of quantum physics and yet it defies our classical intuition. First proposed by Louis de Broglie a century ago, it has since been confirmed with a variety of particles from electrons up to molecules. Here we demonstrate new high-contrast quantum experiments with large and massive tailor-made organic molecules in a near-field interferometer. our experiments prove the quantum wave nature and delocalization of compounds composed of up to 430 atoms, with a maximal size of up to 60 A, masses up to m = 6,910 AMU and de Broglie wavelengths down to λdB = h/mv1 pm. We show that even complex systems, with more than 1,000 internal degrees of freedom, can be prepared in quantum states that are sufficiently well isolated from their environment to avoid decoherence and to show almost perfect coherence.
367 citations
TL;DR: In this article, the full formalism of quantum theory can be derived from five simple physical requirements, based on elementary assumptions regarding preparations, transformations and measurements, which is very similar to the usual formulation of special relativity.
Abstract: Quantum theory (QT) is usually formulated in terms of abstract mathematical postulates involving Hilbert spaces, state vectors and unitary operators. In this paper, we show that the full formalism of QT can instead be derived from five simple physical requirements, based on elementary assumptions regarding preparations, transformations and measurements. This is very similar to the usual formulation of special relativity, where two simple physical requirements—the principles of relativity and light speed invariance—are used to derive the mathematical structure of Minkowski space–time. Our derivation provides insights into the physical origin of the structure of quantum state spaces (including a group-theoretic explanation of the Bloch ball and its three dimensionality) and suggests several natural possibilities to construct consistent modifications of QT.
362 citations
TL;DR: This work generated motional entanglement between vibrational states of two spatially separated, millimeter-sized diamonds at room temperature and showed that the quantum state of the diamonds has positive concurrence with 98% probability.
Abstract: Quantum entanglement in the motion of macroscopic solid bodies has implications both for quantum technologies and foundational studies of the boundary between the quantum and classical worlds. Entanglement is usually fragile in room-temperature solids, owing to strong interactions both internally and with the noisy environment. We generated motional entanglement between vibrational states of two spatially separated, millimeter-sized diamonds at room temperature. By measuring strong nonclassical correlations between Raman-scattered photons, we showed that the quantum state of the diamonds has positive concurrence with 98% probability. Our results show that entanglement can persist in the classical context of moving macroscopic solids in ambient conditions.
358 citations
TL;DR: In this paper, the authors present a reliable algorithm to evaluate quantum discord for general two-qubit states, amending and extending an approach recently put forward for the subclass of $X$ states.
Abstract: We present a reliable algorithm to evaluate quantum discord for general two-qubit states, amending and extending an approach recently put forward for the subclass of $X$ states. A closed expression for the discord of arbitrary states of two qubits cannot be obtained, as the optimization problem for the conditional entropy requires the solution to a pair of transcendental equations in the state parameters. We apply our algorithm to run a numerical comparison between quantum discord and an alternative, computable measure of nonclassical correlations, namely, the geometric discord. We identify the extremally nonclassically correlated two-qubit states according to the (normalized) geometric discord, at a fixed value of the conventional quantum discord. The latter cannot exceed the square root of the former for systems of two qubits.
317 citations
Book•
01 Mar 2011TL;DR: In this paper, the authors propose a real-time formalism for quantum fields and Green's functions with path integrals and generating functionals, and analyze the properties of Green's function.
Abstract: Preface 1. Quantum fields 2. Operators on the multi-particle state space 3. Quantum dynamics and Green's functions 4. Non-equilibrium theory 5. Real-time formalism 6. Linear response theory 7. Quantum kinetic equations 8. Non-equilibrium superconductivity 9. Diagrammatics and generating functionals 10. Effective action 11. Disordered conductors 12. Classical statistical dynamics Appendices: A. Path integrals B. Retarded and advanced propagators C. Analytic properties of Green's functions Bibliography Index.
01 Dec 2011
TL;DR: In this paper, a review of density functional theory and quantum Monte Carlo methods applied to the phase diagram of hydrogen, with particular focus on the solid phases and the liquid-liquid transition are discussed.
Abstract: Hydrogen and helium are the most abundant elements in the Universe. They are also, in principle, the most simple. Nonetheless, they display remarkable properties under extreme conditions of pressure and temperature that have fascinated theoreticians and experimentalists for over a century. Advances in computational methods have made it possible to elucidate ever more of their properties. Some of these methods that have been applied in recent years, in particular, those that perform simulations directly from the physical picture of electrons and ions, such as density functional theory and quantum Monte Carlo are reviewed. The predictions from such methods as applied to the phase diagram of hydrogen, with particular focus on the solid phases and the liquid-liquid transition are discussed. The predictions of ordered quantum states, including the possibilities of a low- or zero-temperature quantum fluid and high-temperature superconductivity are also considered. Finally, pure helium and hydrogen-helium mixtures, the latter which has particular relevance to planetary physics, are discussed.
TL;DR: In this paper, a scheme to realize quantum state tomography, squeezing, and state purification of a mechanical resonator using short optical pulses is presented, allowing observation of mechanical quantum features despite preparation from a thermal state and is experimentally feasible using optical microcavities.
Abstract: Studying mechanical resonators via radiation pressure offers a rich avenue for the exploration of quantum mechanical behavior in a macroscopic regime. However, quantum state preparation and especially quantum state reconstruction of mechanical oscillators remains a significant challenge. Here we propose a scheme to realize quantum state tomography, squeezing, and state purification of a mechanical resonator using short optical pulses. The scheme presented allows observation of mechanical quantum features despite preparation from a thermal state and is shown to be experimentally feasible using optical microcavities. Our framework thus provides a promising means to explore the quantum nature of massive mechanical oscillators and can be applied to other systems such as trapped ions.
TL;DR: This work proposes and experimentally test a new criterion based on entropy functions, and shows that it is more powerful than the variance inequality for identifying EPR steering, and results open up the possibility of observing this type of nonlocality in a wider variety of quantum states.
Abstract: Steering is a form of quantum nonlocality that is intimately related to the famous Einstein-Podolsky-Rosen (EPR) paradox that ignited the ongoing discussion of quantum correlations. Within the hierarchy of nonlocal correlations appearing in nature, EPR steering occupies an intermediate position between Bell nonlocality and entanglement. In continuous variable systems, EPR steering correlations have been observed by violation of Reid's EPR inequality, which is based on inferred variances of complementary observables. Here we propose and experimentally test a new criterion based on entropy functions, and show that it is more powerful than the variance inequality for identifying EPR steering. Using the entropic criterion our experimental results show EPR steering, while the variance criterion does not. Our results open up the possibility of observing this type of nonlocality in a wider variety of quantum states.
TL;DR: This work considers the manifold of all quantum many-body states that can be generated by arbitrary time-dependent local Hamiltonians in a time that scales polynomially in the system size, and shows that it occupies an exponentially small volume in Hilbert space.
Abstract: We consider the manifold of all quantum many-body states that can be generated by arbitrary time-dependent local Hamiltonians in a time that scales polynomially in the system size, and show that it occupies an exponentially small volume in Hilbert space This implies that the overwhelming majority of states in Hilbert space are not physical as they can only be produced after an exponentially long time We establish this fact by making use of a time-dependent generalization of the Suzuki-Trotter expansion, followed by a well-known counting argument This also demonstrates that a computational model based on arbitrarily rapidly changing Hamiltonians is no more powerful than the standard quantum circuit model
TL;DR: In this article, the requirements to test some of the most paradigmatic collapse models with a protocol that prepares quantum superpositions of massive objects are analyzed in a general framework and taking into account only unavoidable sources of decoherence: blackbody radiation and scattering of environmental particles.
Abstract: We analyze the requirements to test some of the most paradigmatic collapse models with a protocol that prepares quantum superpositions of massive objects. This consists of coherently expanding the wave function of a ground-state-cooled mechanical resonator, performing a squared position measurement that acts as a double slit, and observing interference after further evolution. The analysis is performed in a general framework and takes into account only unavoidable sources of decoherence: blackbody radiation and scattering of environmental particles. We also discuss the limitations imposed by the experimental implementation of this protocol using cavity quantum optomechanics with levitating dielectric nanospheres.
TL;DR: In this article, the uncertainty principle limits the precision with which we can measure two complementary properties of a quantum system, and entanglement, it has previously been proposed, can relax these constraints.
Abstract: Heisenberg’s uncertainty principle limits the precision with which we can measure two complementary properties of a quantum system. Entanglement, it has previously been proposed, can relax these constraints. This idea is now demonstrated experimentally with the aid of polarization-entangled photons.
TL;DR: In this paper, the uncertainty principle is observed experimentally and it is shown that if a particle is entangled with a quantum memory, the uncertainty of a measurement is reduced, i.e., two associated properties of a particle cannot be simultaneously known with infinite precision.
Abstract: The uncertainty principle tells us that two associated properties of a particle cannot be simultaneously known with infinite precision. However, if the particle is entangled with a quantum memory, the uncertainty of a measurement is reduced. This concept is now observed experimentally.
TL;DR: A criterion for entanglement which can easily be calculated by using semidefinite programing and improves all existing approaches significantly is presented, leading to an exponential improvement compared with existing schemes.
Abstract: We present an approach to characterize genuine multiparticle entanglement by using appropriate approximations in the space of quantum states. This leads to a criterion for entanglement which can easily be calculated by using semidefinite programing and improves all existing approaches significantly. Experimentally, it can also be evaluated when only some observables are measured. Furthermore, it results in a computable entanglement monotone for genuine multiparticle entanglement. Based on this, we develop an analytical approach for the entanglement detection in cluster states, leading to an exponential improvement compared with existing schemes.
TL;DR: An n-partite inequality is presented that is satisfied by all correlations produced by measurements on biseparable quantum states, but which can be violated by n- partite entangled states, such as Greenberger-Horne-Zeilinger states.
Abstract: We consider the problem of determining whether genuine multipartite entanglement was produced in an experiment, without relying on a characterization of the systems observed or of the measurements performed. We present an n-partite inequality that is satisfied by all correlations produced by measurements on biseparable quantum states, but which can be violated by n-partite entangled states, such as Greenberger-Horne-Zeilinger states. In contrast to traditional entanglement witnesses, the violation of this inequality implies that the state is not biseparable independently of the Hilbert space dimension and of the measured operators. Violation of this inequality does not imply, however, genuine multipartite nonlocality. We show more generically how the problem of identifying genuine tripartite entanglement in a deviceindependent way can be addressed through semidefinite programming.
TL;DR: In this article, the level surfaces of quantum discord for a class of two-qubit states with parallel nonzero Bloch vectors were studied under decoherence and it was shown that the transition between classical and quantum correlations is abrupt.
Abstract: We study the level surfaces of quantum discord for a class of two-qubit states with parallel nonzero Bloch vectors The dynamic behavior of quantum discord under decoherence is investigated It is shown that a class of $X$ states has sudden transition between classical and quantum correlations under decoherence Our results include the ones in M D Lang and C M Caves [Phys Rev Lett 105, 150501 (2010)] as a special case and show new pictures and structures of quantum discord
TL;DR: In this paper, a general quantum theory is proposed to describe the coupling of light with the motion of a dielectric object inside a high-finesse optical cavity, which is applied to the recent proposal of using an optically levitating nanodielectric as a cavity optomechanical system.
Abstract: We provide a general quantum theory to describe the coupling of light with the motion of a dielectric object inside a high-finesse optical cavity. In particular, we derive the total Hamiltonian of the system as well as a master equation describing the state of the center-of-mass mode of the dielectric and the cavity-field mode. In addition, a quantum theory of elasticity is used to study the coupling of the center-of-mass motion with internal vibrational excitations of the dielectric. This general theory is applied to the recent proposal of using an optically levitating nanodielectric as a cavity optomechanical system [see Romero-Isart et al., New J. Phys. 12, 033015 (2010); Chang et al., Proc. Natl. Acad. Sci. USA 107, 1005 (2010)]. On this basis, we also design a light-mechanics interface to prepare non-Gaussian states of the mechanical motion, such as quantum superpositions of Fock states. Finally, we introduce a direct mechanical tomography scheme to probe these genuine quantum states by time-of- flight experiments.
TL;DR: An implementation in a PPLN waveguide, based on spectrally engineered sum frequency generation (SFG), that allows us to pick well-defined spectral broadband modes from an ultrafast multi-mode state for interconversion to a broadband mode at another frequency.
Abstract: We introduce the concept of a quantum pulse gate (QPG), a method for accessing the intrinsic broadband spectral mode structure of ultrafast quantum states of light. This mode structure can now be harnessed for applications in quantum information processing. We propose an implementation in a PPLN waveguide, based on spectrally engineered sum frequency generation (SFG). It allows us to pick well-defined spectral broadband modes from an ultrafast multi-mode state for interconversion to a broadband mode at another frequency. By pulse-shaping the bright SFG pump beam, different orthogonal broadband modes can be addressed individually and extracted with high fidelity.
TL;DR: In this paper, the optimal measurements to access classical correlations in arbitrary two-qubit states are analyzed and an upper bound of quantum discord based on the maximal-correlation-direction measurement (MCDM) is derived.
Abstract: We analyze the optimal measurements to access classical correlations in arbitrary two-qubit states. Two-qubit states can be transformed into the canonical forms via local unitary operations. For the canonical forms, we investigate the probability distribution of the optimal measurements. The probability distribution of the optimal measurements is found to be centralized in the vicinity of a specific von Neumann measurement, which we call the maximal-correlation-direction measurement (MCDM). We prove that, for the states with zero discord and maximally mixed marginals, the MCDM is the optimal measurement. Furthermore, we give an upper bound of quantum discord based on the MCDM, and investigate its performance for approximating the quantum discord.
22 Oct 2011
TL;DR: It is obtained that the general adversary bound characterizes the quantum query complexity of any function whatsoever, implying that discrete and continuous-time query models are equivalent in the bounded-error setting, even for the general state-conversion problem.
Abstract: State conversion generalizes query complexity to the problem of converting between two input-dependent quantum states by making queries to the input. We characterize the complexity of this problem by introducing a natural information-theoretic norm that extends the Schur product operator norm. The complexity of converting between two systems of states is given by the distance between them, as measured by this norm. In the special case of function evaluation, the norm is closely related to the general adversary bound, a semi-definite program that lower-bounds the number of input queries needed by a quantum algorithm to evaluate a function. We thus obtain that the general adversary bound characterizes the quantum query complexity of any function whatsoever. This generalizes and simplifies the proof of the same result in the case of boolean input and output. Also in the case of function evaluation, we show that our norm satisfies a remarkable composition property, implying that the quantum query complexity of the composition of two functions is at most the product of the query complexities of the functions, up to a constant. Finally, our result implies that discrete and continuous-time query models are equivalent in the bounded-error setting, even for the general state-conversion problem.
TL;DR: In this paper, the quantum state is stored in a vapour of rubidium atoms and then recalled with a fidelity in excess of 98% with the help of optical quantum memories.
Abstract: Optical quantum memories—storage devices for the data encoded in light pulses—will be vital for buffering the flow of quantum information. Researchers now demonstrate such a device that can operate at room temperature. The quantum state is stored in a vapour of rubidium atoms and then recalled with a fidelity in excess of 98%.
TL;DR: In this article, a mesoscopic mechanical oscillator is pre-cooled to an occupancy below 200 quanta by thermalizing it with a 600mK cold He-3 gas.
Abstract: Cooling a mesoscopic mechanical oscillator to its quantum ground state is elementary for the preparation and control of quantum states of mechanical objects. Here, we pre-cool a 70-MHz micromechanical silica oscillator to an occupancy below 200 quanta by thermalizing it with a 600-mK cold He-3 gas. Two-level-system induced damping via structural defect states is shown to be strongly reduced and simultaneously serves as a thermometry method to independently quantify excess heating due to the cooling laser. We demonstrate that dynamical back action optical sideband cooling can reduce the average occupancy to 9 +/- 1 quanta, implying that the mechanical oscillator can be found (10 +/- 1)% of the time in its quantum ground state.
TL;DR: In this article, three-dimensional topological insulators represent a new quantum state and were originally proposed by a theoretical approach, but subsequently observed experimentally, e.g., in Bi 1-x Sb x and...
Abstract: Three-dimensional topological insulators represent a new quantum state. They were originally proposed by a theoretical approach, but subsequently observed experimentally, e.g., in Bi 1- x Sb x and ...
TL;DR: In this paper, it was shown that the quantum f-divergences are monotonic under substochastic maps whenever the defining function is operator convex, which is a special case of Petz' quasi-entropies.
Abstract: Quantum f-divergences are a quantum generalization of the classical notion of f-divergences, and are a special case of Petz' quasi-entropies. Many well-known distinguishability measures of quantum states are given by, or derived from, f-divergences. Special examples include the quantum relative entropy, the Renyi relative entropies, and the Chernoff and Hoeffding measures. Here we show that the quantum f-divergences are monotonic under substochastic maps whenever the defining function is operator convex. This extends and unifies all previously known monotonicity results for this class of distinguishability measures. We also analyze the case where the monotonicity inequality holds with equality, and extend Petz' reversibility theorem for a large class of f-divergences and other distinguishability measures. We apply our findings to the problem of quantum error correction, and show that if a stochastic map preserves the pairwise distinguishability on a set of states, as measured by a suitable f-divergence, then its action can be reversed on that set by another stochastic map that can be constructed from the original one in a canonical way. We also provide an integral representation for operator convex functions on the positive half-line, which is the main ingredient in extending previously known results on the monotonicity inequality and the case of equality. We also consider some special cases where the convexity of f is sufficient for the monotonicity, and obtain the inverse Holder inequality for operators as an application. The presentation is completely self-contained and requires only standard knowledge of matrix analysis.
TL;DR: In this article, an experiment that probes the transitions between quantum states of neutrons in the Earth's gravitational field demonstrates an exotic variant of spectroscopy, and one that might lead to sensitive fundamental tests of gravity laws.
Abstract: Spectroscopic techniques are mostly used to study the interaction between matter and electromagnetic fields. Here, an experiment that probes the transitions between quantum states of neutrons in the Earth’s gravitational field demonstrates an exotic variant of spectroscopy, and one that might lead to sensitive fundamental tests of gravity laws. Spectroscopy is a method typically used to assess an unknown quantity of energy by means of a frequency measurement. In many problems, resonance techniques1,2 enable high-precision measurements, but the observables have generally been restricted to electromagnetic interactions. Here we report the application of resonance spectroscopy to gravity. In contrast to previous resonance methods, the quantum mechanical transition is driven by an oscillating field that does not directly couple an electromagnetic charge or moment to an electromagnetic field. Instead, we observe transitions between gravitational quantum states when the wave packet of an ultra-cold neutron couples to the modulation of a hard surface as the driving force. The experiments have the potential to test the equivalence principle3 and Newton’s gravity law at the micrometre scale4,5.