Entanglement condition via su(2) and su(1,1) algebra using Schr{\"o}dinger-Robertson uncertainty relation

The Schr{\"o}dinger-Robertson inequality generally provides a stronger bound on the product of uncertainties for two noncommuting observables than the Heisenberg uncertainty relation, and as such, it can yield a stricter separability condition in conjunction with partial transposition. In this paper, using the Schr{\"o}dinger-Robertson uncertainty relation, the separability condition previously derived from the su(2) and the su(1,1) algebra is made stricter and refined to a form invariant with respect to local phase shifts. Furthermore, a linear optical scheme is proposed to test this invariant separability condition.Comment: published version, 3.5 pages, 1 figur

Uncertainty-principle noise in vacuum-tunneling transducers

The fundamental sources of noise in a vacuum-tunneling probe used as an electromechanical transducer to monitor the location of a test mass are examined using a first-quantization formalism. We show that a tunneling transducer enforces the Heisenberg uncertainty principle for the position and momentum of a test mass monitored by the transducer through the presence of two sources of noise: the shot noise of the tunneling current and the momentum fluctuations transferred by the tunneling electrons to the test mass. We analyze a number of cases including symmetric and asymmetric rectangular potential barriers and a barrier in which there is a constant electric field. Practical configurations for reaching the quantum limit in measurements of the position of macroscopic bodies with such a class of transducers are studied

Heisenberg uncertainty for qubit measurements

Reports on experiments recently performed in Vienna [Erhard et al, Nature Phys. 8, 185 (2012)] and Toronto [Rozema et al, Phys. Rev. Lett. 109, 100404 (2012)] include claims of a violation of Heisenberg's error-disturbance relation. In contrast, we have presented and proven a Heisenberg-type relation for joint measurements of position and momentum [Phys. Rev. Lett. 111, 160405 (2013)]. To resolve the apparent conflict, we formulate here a new general trade-off relation for errors in qubit measurements, using the same concepts as we did in the position-momentum case. We show that the combined errors in an approximate joint measurement of a pair of +/-1 valued observables A,B are tightly bounded from below by a quantity that measures the degree of incompatibility of A and B. The claim of a violation of Heisenberg is shown to fail as it is based on unsuitable measures of error and disturbance. Finally we show how the experiments mentioned may directly be used to test our error inequality.Comment: Version 3 contains further clarifications in our argument refuting the alleged violation of Heisenberg's error-disturbance relation. Some new material added on the connection between preparation uncertainty and approximation error relation

Jednostavna eksperimentalna provjera heisenbergovih relacija neodređenosti

We show that the quantum mechanical interpretation of the diffraction of light on a slit, when a wave function is assigned to a photon, can be used for a direct experimental study of Heisenberg\u27s position-momentum and equivalent position-wave vector uncertainty relation for the photon. Results of an experimental test of the position-wave vector uncertainty relation, where the wavelength is used as the input parameter, are given and they very well confirm our approach. The same experimental results can also be used for a test of the position-momentum uncertainty relation when the momentum p0 of a photon is known as the input parameter. We show that a measurement of p0, independent of the knowledge of the value of the Planck\u27s constant, is possible. Using that value of p0, a test of the position-momentum uncertainty relation could be regarded as a method for a direct measurement of the Planck\u27s constant. This is discussed, since the diffraction pattern is also well described by classical electrodynamics in the considered experimental conditions. This approach for testing the Heisenberg\u27s uncertainty relations is very simple and consequently suitable as a quantitative exercise in undergraduate experimental courses, as well as a visual and attractive demonstration of the Heisenberg\u27s uncertainty principle in courses of quantum mechanics.Pokazujemo da se kvantno-mehaničko tumačenje ogiba svjetlosti na pukotini, u kojem smo fotonu pridijelili uobičajenu valnu funkciju, može upotrijebiti za izravnu eksperimentalnu provjeru Heisenbergovih relacija neodređenosti položaj–impuls i ekvivalentnih položaj–valni vektor. Rezultati testiranja relacije neodređenosti položaj–valni vektor, u kojem smo uzeli valnu duljinu lasera kao ulazni parametar, dobro potvrđuju na pristup. Na istovjetan način se može napraviti provjera relacija neodređenosti položaj–impuls ako je impuls p0 laserskih fotona poznat kao ulazni parametar. Dokazujemo da je moguće mjeriti p0 neovisno o poznavanju vrijednosti Plankove konstante. S tako dobivenom vrijednošću p0 opisani eksperiment se može promatrati i kao način mjerenja Plankove konstante. To smo detaljnije pojasnili s obzirom da ogibnu sliku u danim eksperimentalnim uvjetima možemo također odlično opisati i klasičnom elektrodinamikom. Opisani eksperimentalni pristup za provjeru Heisenbergovih relacija neodređenosti je vrlo jednostavan, te je stoga pogodan kao vježba u studentskim praktikumima ali i kao vizualno atraktivna demonstracija na predavanjima iz kvantno-mehaničkih kolegija

Jednostavna eksperimentalna provjera heisenbergovih relacija neodređenosti

We show that the quantum mechanical interpretation of the diffraction of light on a slit, when a wave function is assigned to a photon, can be used for a direct experimental study of Heisenberg\u27s position-momentum and equivalent position-wave vector uncertainty relation for the photon. Results of an experimental test of the position-wave vector uncertainty relation, where the wavelength is used as the input parameter, are given and they very well confirm our approach. The same experimental results can also be used for a test of the position-momentum uncertainty relation when the momentum p0 of a photon is known as the input parameter. We show that a measurement of p0, independent of the knowledge of the value of the Planck\u27s constant, is possible. Using that value of p0, a test of the position-momentum uncertainty relation could be regarded as a method for a direct measurement of the Planck\u27s constant. This is discussed, since the diffraction pattern is also well described by classical electrodynamics in the considered experimental conditions. This approach for testing the Heisenberg\u27s uncertainty relations is very simple and consequently suitable as a quantitative exercise in undergraduate experimental courses, as well as a visual and attractive demonstration of the Heisenberg\u27s uncertainty principle in courses of quantum mechanics.Pokazujemo da se kvantno-mehaničko tumačenje ogiba svjetlosti na pukotini, u kojem smo fotonu pridijelili uobičajenu valnu funkciju, može upotrijebiti za izravnu eksperimentalnu provjeru Heisenbergovih relacija neodređenosti položaj–impuls i ekvivalentnih položaj–valni vektor. Rezultati testiranja relacije neodređenosti položaj–valni vektor, u kojem smo uzeli valnu duljinu lasera kao ulazni parametar, dobro potvrđuju na pristup. Na istovjetan način se može napraviti provjera relacija neodređenosti položaj–impuls ako je impuls p0 laserskih fotona poznat kao ulazni parametar. Dokazujemo da je moguće mjeriti p0 neovisno o poznavanju vrijednosti Plankove konstante. S tako dobivenom vrijednošću p0 opisani eksperiment se može promatrati i kao način mjerenja Plankove konstante. To smo detaljnije pojasnili s obzirom da ogibnu sliku u danim eksperimentalnim uvjetima možemo također odlično opisati i klasičnom elektrodinamikom. Opisani eksperimentalni pristup za provjeru Heisenbergovih relacija neodređenosti je vrlo jednostavan, te je stoga pogodan kao vježba u studentskim praktikumima ali i kao vizualno atraktivna demonstracija na predavanjima iz kvantno-mehaničkih kolegija

Towards the Fundamental Quantum Limit of Linear Measurements of Classical Signals

The quantum Cram\'er-Rao bound (QCRB) sets a fundamental limit for the measurement of classical signals with detectors operating in the quantum regime. Using linear-response theory and the Heisenberg uncertainty relation, we derive a general condition for achieving such a fundamental limit. When applied to classical displacement measurements with a test mass, this condition leads to an explicit connection between the QCRB and the Standard Quantum Limit which arises from a tradeoff between the measurement imprecision and quantum backaction; the QCRB can be viewed as an outcome of a quantum non-demolition measurement with the backaction evaded. Additionally, we show that the test mass is more a resource for improving measurement sensitivity than a victim of the quantum backaction, which suggests a new approach to enhancing the sensitivity of a broad class of sensors. We illustrate these points with laser interferometric gravitational wave detectors.Comment: revised version with supplemental materials adde

Is "entanglement" always entangled?

Entanglement, including ``quantum entanglement,'' is a consequence of correlation between objects. When the objects are subunits of pairs which in turn are members of an ensemble described by a wave function, a correlation among the subunits induces the mysterious properties of ``cat-states.'' However, correlation between subsystems can be present from purely non-quantum sources, thereby entailing no unfathomable behavior. Such entanglement arises whenever the so-called ``qubit space'' is not afflicted with Heisenberg Uncertainty. It turns out that all optical experimental realizations of EPR's \emph{Gedanken} experiment in fact do not suffer Heisenberg Uncertainty. Examples will be analyzed and non-quantum models for some of these described. The consequences for experiments that were to test EPR's contention in the form of Bell's Theorem are drawn: \emph{valid tests of EPR's hypothesis have yet to be done.}Comment: 5 p. LaTeX + 3 eps & 1 ps fig; v2:typos fixe