Stochastic Heisenberg limit: Optimal estimation of a fluctuating phase

The ultimate limits to estimating a fluctuating phase imposed on an optical beam can be found using the recently derived continuous quantum Cramer-Rao bound. For Gaussian stationary statistics, and a phase spectrum scaling asymptotically as 1/omega^p with p>1, the minimum mean-square error in any (single-time) phase estimate scales as N^{-2(p-1)/(p+1)}, where N is the photon flux. This gives the usual Heisenberg limit for a constant phase (as the limit p--> infinity) and provides a stochastic Heisenberg limit for fluctuating phases. For p=2 (Brownian motion), this limit can be attained by phase tracking.Comment: 5+4 pages, to appear in Physical Review Letter

The quantum Bell-Ziv-Zakai bounds and Heisenberg limits for waveform estimation

We propose quantum versions of the Bell-Ziv-Zakai lower bounds on the error in multiparameter estimation. As an application we consider measurement of a time-varying optical phase signal with stationary Gaussian prior statistics and a power law spectrum $\sim 1/|\omega|^p$, with $p>1$. With no other assumptions, we show that the mean-square error has a lower bound scaling as $1/{\cal N}^{2(p-1)/(p+1)}$, where ${\cal N}$ is the time-averaged mean photon flux. Moreover, we show that this accuracy is achievable by sampling and interpolation, for any $p>1$. This bound is thus a rigorous generalization of the Heisenberg limit, for measurement of a single unknown optical phase, to a stochastically varying optical phase.Comment: 18 pages, 6 figures, comments welcom

Optimal Heisenberg-style bounds for the average performance of arbitrary phase estimates

The ultimate bound to the accuracy of phase estimates is often assumed to be given by the Heisenberg limit. Recent work seemed to indicate that this bound can be violated, yielding measurements with much higher accuracy than was previously expected. The Heisenberg limit can be restored as a rigorous bound to the accuracy provided one considers the accuracy averaged over the possible values of the unknown phase, as we have recently shown [Phys. Rev. A 85, 041802(R) (2012)]. Here we present an expanded proof of this result together with a number of additional results, including the proof of a previously conjectured stronger bound in the asymptotic limit. Other measures of the accuracy are examined, as well as other restrictions on the generator of the phase shifts. We provide expanded numerical results for the minimum error and asymptotic expansions. The significance of the results claiming violation of the Heisenberg limit is assessed, followed by a detailed discussion of the limitations of the Cramer-Rao bound.Comment: 22 pages, 4 figure

Experimental optical phase measurement approaching the exact Heisenberg limit

The use of quantum resources can provide measurement precision beyond the shot-noise limit (SNL). The task of ab initio optical phase measurement---the estimation of a completely unknown phase---has been experimentally demonstrated with precision beyond the SNL, and even scaling like the ultimate bound, the Heisenberg limit (HL), but with an overhead factor. However, existing approaches have not been able---even in principle---to achieve the best possible precision, saturating the HL exactly. Here we demonstrate a scheme to achieve true HL phase measurement, using a combination of three techniques: entanglement, multiple samplings of the phase shift, and adaptive measurement. Our experimental demonstration of the scheme uses two photonic qubits, one double passed, so that, for a successful coincidence detection, the number of photon-passes is $N=3$. We achieve a precision that is within $4\%$ of the HL, surpassing the best precision theoretically achievable with simpler techniques with $N=3$. This work represents a fundamental achievement of the ultimate limits of metrology, and the scheme can be extended to higher $N$ and other physical systems.Comment: (12 pages, 6 figures), typos correcte

A Theory of Mind investigation into the appreciation of visual jokes in schizophrenia

BACKGROUND: There is evidence that groups of people with schizophrenia have deficits in Theory of Mind (ToM) capabilities. Previous studies have found these to be linked to psychotic symptoms (or psychotic symptom severity) particularly the presence of delusions and hallucinations. METHODS: A visual joke ToM paradigm was employed where subjects were asked to describe two types of cartoon images, those of a purely Physical nature and those requiring inferences of mental states for interpretation, and to grade them for humour and difficulty. Twenty individuals with a DSM-lV diagnosis of schizophrenia and 20 healthy matched controls were studied. Severity of current psychopathology was measured using the Krawiecka standardized scale of psychotic symptoms. IQ was estimated using the Ammons and Ammons quick test. RESULTS: Individuals with schizophrenia performed significantly worse than controls in both conditions, this difference being most marked in the ToM condition. No relationship was found for poor ToM performance and psychotic positive symptomatology, specifically delusions and hallucinations. CONCLUSION: There was evidence for a compromised ToM capability in the schizophrenia group on this visual joke task. In this instance this could not be linked to particular symptomatology

The Anne Boleyn Illusion is a six-fingered salute to sensory remapping

The Anne Boleyn Illusion exploits the somatotopic representation of touch to create the illusion of an extra digit and demonstrates the instantaneous remapping of relative touch location into body-based coordinates through visuo-tactile integration. Performed successfully on thousands, it is also a simple demonstration of the flexibility of body representations for use at public events, in schools or in the home and can be implemented anywhere by anyone with a mirror and some degree of bimanual coordination

Developing undergraduate practical skills and independence with ‘at home practical kits’

The Covid-19 pandemic posed significant challenges for practical teaching within the sciences. While many instructors adopted innovative alternatives to conventional practicals, many relied on digital approaches that did not give students hands-on experience. In this study we evaluate the use of ‘at home’ practical kits used in first year physics and biology teaching at a UK university as an alternative to laboratory classes. In particular we focus on the enforced independence over time, space and help-seeking inherent in the at-home model as a driver of student learning and confidence. Students reported the kits encouraged independence, problem solving and self-reliance. Students associated the at-home practical kits with higher level cognitive skills as defined by Bloom’s revised taxonomy. While most students enjoyed using the kits, those who did not enjoy them tended to have higher previous experience of practical work before university. Students saw potential value in the kits after the pandemic, so could be an alternative or supplement to in-person practicals. We recommend that practical organisers use our findings around the development of student self-reliance to reconsider practical design and incorporate more opportunities for students to solve problems independently to increase effectiveness of practical teaching

The Heisenberg limit for laser coherence

To quantify quantum optical coherence requires both the particle- and wave-natures of light. For an ideal laser beam [1,2,3], it can be thought of roughly as the number of photons emitted consecutively into the beam with the same phase. This number, $\mathfrak{C}$, can be much larger than $\mu$, the number of photons in the laser itself. The limit on $\mathfrak{C}$ for an ideal laser was thought to be of order $\mu^2$ [4,5]. Here, assuming nothing about the laser operation, only that it produces a beam with certain properties close to those of an ideal laser beam, and that it does not have external sources of coherence, we derive an upper bound: $\mathfrak{C} = O(\mu^4)$. Moreover, using the matrix product states (MPSs) method [6,7,8,9], we find a model that achieves this scaling, and show that it could in principle be realised using circuit quantum electrodynamics (QED) [10]. Thus $\mathfrak{C} = O(\mu^2)$ is only a standard quantum limit (SQL); the ultimate quantum limit, or Heisenberg limit, is quadratically better.Comment: 6 pages, 4 figures, and 31 pages of supplemental information. v2: This paper is now published [Nature Physics DOI:10.1038/s41567-020-01049-3 (26 October 2020)]. For copyright reasons, this arxiv paper is based on a version of the paper prior to the accepted (21 August 2020) versio