Michael J. W. Hall

Griffith University, Brisbane, Queensland, Australia

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Publications (60)169.41 Total impact

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    Michael J. W. Hall
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    ABSTRACT: A local and deterministic model of quantum correlations is always possible, as shown explicitly by Brans in 1988: one simply needs the physical systems being measured to have a suitable statistical correlation with the physical systems performing the measurement, via some common cause. Hence, to derive no-go results such as Bell inequalities, an assumption of measurement independence is crucial. It is a surprisingly strong assumption --- less than 1/15 bits of prior correlation suffice for a local model of the singlet state of two qubits --- with ramifications for the security of quantum communication protocols. Indeed, without this assumption, any statistical correlations whatsoever --- even those which appear to allow explicit superluminal signalling --- have a corresponding local deterministic model. It is argued that `quantum nonlocality' is bad terminology, and that measurement independence does not equate to `experimental free will'. Brans' 1988 model is extended to show that no more than 2 log d bits of prior correlation are required for a local deterministic model of the correlations between any two d-dimensional quantum systems.
    Preview · Article · Nov 2015
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    Shuming Cheng · Michael J. W. Hall
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    ABSTRACT: Various measures have been suggested recently for quantifying the coherence of a quantum state with respect to a given basis. We first use two of these, the l_1-norm and relative entropy measures, to investigate tradeoffs between the coherences of mutually unbiased bases. Results include relations between coherence, uncertainty and purity; tight general bounds restricting the coherences of mutually unbiased bases; and an exact complementarity relation for qubit coherences. We further define the average coherence of a quantum state. For the l_1-norm measure this is related to a natural 'coherence radius' for the state, and leads to a conjecture for an l_2-norm measure of coherence. For relative entropy the average coherence is determined by the difference between the von Neumann entropy and the quantum subentropy of the state, and leads to upper bounds for the latter quantity. Finally, we point out that the relative entropy of coherence is a special case of G-asymmetry, which immediately yields several operational interpretations in contexts as diverse as frame-alignment, quantum communication and metrology.
    Preview · Article · Aug 2015 · Physical Review A
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    ABSTRACT: Information-theoretic definitions for noise and disturbance in quantum measurements were given in Phys. Rev. Lett. 112, 050401 (2014) and a state-independent noise-disturbance uncertainty relation was obtained. Here, we derive a tight noise-disturbance uncertainty relation for complementary qubit observables and carry out an experimental test. Successive projective measurements on the neutron's spin-1/2system, together with a correction procedure which reduces the disturbance, are performed. Our experimental results saturate the tight noise-disturbance uncertainty relation for qubits when an optimal correction procedure is applied.
    Full-text · Article · Apr 2015 · Physical Review Letters
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    ABSTRACT: 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 $\boldsymbol n/(\boldsymbol n^\intercal A\boldsymbol n)^2$ over arbitrary unit hemispheres for any positive matrix $A$.
    Preview · Article · Nov 2014 · Journal of the Optical Society of America B
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    Dominic W. Berry · Mankei Tsang · Michael J. W. Hall · Howard M. Wiseman
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    ABSTRACT: 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.
    Preview · Article · Sep 2014 · Physical Review X
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    Sacha Kocsis · Michael J. W. Hall · Adam J. Bennet · Geoff J Pryde
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    ABSTRACT: Bell nonlocality between distant quantum systems---i.e., joint correlations which violate a Bell inequality---can be verified without trusting the measurement devices used, nor those performing the measurements. This leads to unconditionally secure protocols for quantum information tasks such as cryptographic key distribution. However, complete verification of Bell nonlocality requires high detection efficiencies, and is not robust to the typical transmission losses that occur in long distance applications. In contrast, quantum steering, a weaker form of quantum correlation, can be verified for arbitrarily low detection efficiencies and high losses. The cost is that current steering-verification protocols require complete trust in one of the measurement devices and its operator, allowing only one-sided secure key distribution. We present device-independent steering protocols that remove this need for trust, even when Bell nonlocality is not present. We experimentally demonstrate this principle for singlet states and states that do not violate a Bell inequality.
    Preview · Article · Aug 2014 · Nature Communications
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    Michael J W Hall · James D Cresser · Li Li · Erika Andersson
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    ABSTRACT: Master equations govern the time evolution of a quantum system interacting with an environment, and may be written in a variety of forms. Time-independent or memoryless master equations, in particular, can be cast in the well-known Lindblad form. Any time-local master equation, Markovian or non-Markovian, may in fact also be written in a Lindblad-like form. A diagonalisation procedure results in a unique, and in this sense canonical, representation of the equation, which may be used to fully characterize the non-Markovianity of the time evolution. Recently, several different measures of non-Markovianity have been presented which reflect, to varying degrees, the appearance of negative decoherence rates in the Lindblad-like form of the master equation. We therefore propose using the negative decoherence rates themselves, as they appear in the canonical form of the master equation, to completely characterize non-Markovianity. The advantages of this are especially apparent when more than one decoherence channel is present. We show that a measure proposed by Rivas et al. is a surprisingly simple function of the canonical decoherence rates, and give an example of a master equation that is non-Markovian for all times t > 0, but to which nearly all proposed measures are blind. We also give necessary and sufficient conditions for trace distance and volume measures to witness non-Markovianity, in terms of the Bloch damping matrix.
    Full-text · Article · Apr 2014 · Physical Review A
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    Michael J. W. Hall · D. -A. Deckert · Howard M. Wiseman
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    ABSTRACT: We investigate whether quantum theory can be understood as the continuum limit of a mechanical theory, in which there is a huge, but finite, number of classical 'worlds', and quantum effects arise solely from a universal interaction between these worlds, without reference to any wave function. Here a `world' means an entire universe with well-defined properties, determined by the classical configuration of its particles and fields. In our approach each world evolves deterministically; probabilities arise due to ignorance as to which world a given observer occupies; and we argue that in the limit of infinitely many worlds the wave function can be recovered (as a secondary object) from the motion of these worlds. We introduce a simple model of such a 'many interacting worlds' approach and show that it can reproduce some generic quantum phenomena---such as Ehrenfest's theorem, wavepacket spreading, barrier tunneling and zero point energy---as a direct consequence of mutual repulsion between worlds. Finally, we perform numerical simulations using our approach. We demonstrate, first, that it can be used to calculate quantum ground states, and second, that it is capable of reproducing, at least qualitatively, the double-slit interference phenomenon.
    Full-text · Article · Feb 2014 · Physical Review X
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    Francesco Buscemi · Michael J W Hall · Masanao Ozawa · Mark M Wilde
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    ABSTRACT: We introduce information-theoretic definitions for noise and disturbance in quantum measurements and prove a state-independent noise-disturbance tradeoff relation that these quantities have to satisfy in any conceivable setup. Contrary to previous approaches, the information-theoretic quantities we define are invariant under the relabelling of outcomes and allow for the possibility of using quantum or classical operations to "correct" for the disturbance. We also show how our bound implies strong tradeoff relations for mean square deviations.
    Full-text · Article · Feb 2014 · Physical Review Letters
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    Dominic W Berry · Michael J W Hall · Howard M Wiseman
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    ABSTRACT: The ultimate limits to estimating a fluctuating phase imposed on an optical beam can be found using the recently derived continuous quantum Cramér-Rao bound. For Gaussian stationary statistics, and a phase spectrum scaling asymptotically as ω^{-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→∞) and provides a stochastic Heisenberg limit for fluctuating phases. For p=2 (Brownian motion), this limit can be attained by phase tracking.
    Preview · Article · Sep 2013 · Physical Review Letters
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    Michael J. W. Hall
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    ABSTRACT: The correlation distance quantifies the statistical independence of two classical or quantum systems, via the distance from their joint state to the product of the marginal states. Tight lower bounds are given for the mutual information between pairs of two-valued classical variables and quantum qubits, in terms of the corresponding classical and quantum correlation distances. These bounds are stronger than the Pinsker inequality (and refinements thereof) for relative entropy. The classical lower bound may be used to quantify properties of statistical models that violate Bell inequalities. Entangled qubits can have a lower mutual information than can any two-valued classical variables having the same correlation distance. The qubit correlation distance also provides a direct entanglement criterion, related to the spin covariance matrix. Connections of results with classically-correlated quantum states are briefly discussed.
    Preview · Article · Jul 2013 · Entropy
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    Michael J. W. Hall
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    ABSTRACT: There has been much interest in developing phase estimation schemes which beat the so-called Heisenberg limit, i.e., for which the phase resolution scales better than 1/n, where n is a measure of resources such as the average photon number or the number of atomic qubits. In particular, a number of nonlinear schemes have been proposed for which the resolution appears to scale as 1/n^k or even exp(-n), based on optimising the quantum Cramer-Rao bound. Such schemes include the use of entangled coherent states. However, it may be shown that the average root mean square errors of the proposed schemes (averaged over any prior distribution of phase shifts), cannot beat the Heisenberg limit, and that simple estimation schemes based on entangled coherent states cannot scale better than 1/n^{1/4}. This paradox is related to the role of 'bias' in Cramer-Rao bounds, and is only partially ameliorated via iterative implementations of the proposed schemes. The results are based on new information-theoretic bounds for the average information gain and error of any phase estimation scheme, and generalise to estimates of shifts generated by any operator having discrete eigenvalues.
    Preview · Article · Jul 2013
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    ABSTRACT: Complementarity restricts the accuracy with which incompatible quantum observables can be jointly measured. Despite popular conception, the Heisenberg uncertainty relation does not quantify this principle. We report the experimental verification of universally valid complementarity relations, including an improved relation derived here. We exploit Einstein-Poldolsky-Rosen correlations between two photonic qubits to jointly measure incompatible observables of one. The product of our measurement inaccuracies is low enough to violate the widely used, but not universally valid, Arthurs-Kelly relation.
    Preview · Article · May 2013 · Physical Review Letters
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    Eric G. Cavalcanti · Michael J. W. Hall · Howard M. Wiseman
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    ABSTRACT: Various protocols exist by which a referee can be convinced that two observers share an entangled resource. Such protocols typically specify the types of communication allowed, and the degrees of trust required, between the referee and each observer. Here it is shown that the need for any degree of trust of the observers by the referee can be completely removed, allowing device independent verification of entanglement, via the referee using classical and quantum communication channels appropriately. In particular, trust-free verification of Bell nonlocality, EPR-steering, and entanglement, respectively, requires two classical channels, one classical and one quantum channel, and two quantum channels. These channels correspond to suitable inputs of quantum randomness by the referee, which prevent the observers from mimicking entanglement using shared classical randomness. Our results generalize recent work by F. Buscemi [Phys. Rev. Lett. {\bf 108}, 200401 (2012)], and offer a perspective on the operational significance of that work. They also offer the possibility of simpler experimental demonstrations of the basic idea of quantum-refereed nonlocality tests.
    Full-text · Article · Oct 2012 · Physical Review A
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    ABSTRACT: With the advent of quantum information, the violation of a Bell inequality is used to witness the absence of an eavesdropper in cryptographic scenarios such as key distribution and randomness expansion. One of the key assumptions of Bell's theorem is the existence of experimental "free will," meaning that measurement settings can be chosen at random and independently by each party. The relaxation of this assumption potentially shifts the balance of power towards an eavesdropper. We consider a no-signaling model with reduced "free will" and bound the adversary’s capabilities in the task of randomness expansion.
    Full-text · Article · Oct 2012 · Physical Review Letters
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    Dominic W. Berry · Michael J. W. Hall · Marcin Zwierz · Howard M. Wiseman
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    ABSTRACT: 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.
    Full-text · Article · Sep 2012 · Physical Review A
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    Marcel Reginatto · Michael J. W. Hall
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    ABSTRACT: We consider a system with a discrete configuration space. We show that the geometrical structures associated with such a system provide the tools necessary for a reconstruction of discrete quantum mechanics once dynamics is brought into the picture. We do this in three steps. Our starting point is information geometry, the natural geometry of the space of probability distributions. Dynamics requires additional structure. To evolve the probabilities P^k, we introduce coordinates S^k canonically conjugate to the P^k and a symplectic structure. We then seek to extend the metric structure of information geometry, to define a geometry over the full space of the P^k and S^k. Consistency between the metric tensor and the symplectic form forces us to introduce a K\"ahler geometry. The construction has notable features. A complex structure is obtained in a natural way. The canonical coordinates of the K\"ahler space are precisely the wave functions of quantum mechanics. The full group of unitary transformations is obtained. Finally, one may associate a Hilbert space with the K\"ahler space, which leads to the standard version of quantum theory. We also show that the metric that we derive here using purely geometrical arguments is precisely the one that leads to Wootters' expression for the statistical distance for quantum systems.
    Full-text · Article · Jul 2012
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    Michael J. W. Hall · Marcel Reginatto · C. M. Savage
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    ABSTRACT: When interactions are turned off, the theory of interacting quantum and classical ensembles due to Hall and Reginatto is shown to suffer from a nonlocal signaling effect that is effectively action at a distance. This limits the possible applicability of the theory. In its present form, it is restricted to those situations in which interactions are always on, such as classical gravity interacting with quantized matter.
    Preview · Article · Jul 2012 · Physical Review A
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    Michael J. W. Hall · David T. Pegg
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    ABSTRACT: A construction of covariant quantum phase observables, for Hamiltonians with a finite number of energy eigenvalues, has been recently given by D. Arsenovic et al. [Phys. Rev. A 85, 044103 (2012)]. For Hamiltonians generating periodic evolution, we show that this construction is just a simple rescaling of the known canonical 'time' or 'age' observable, with the period T rescaled to 2\pi. Further, for Hamiltonians generating quasiperiodic evolution, we note that the construction leads to a phase observable having several undesirable features, including (i) having a trivially uniform probability density for any state of the system, (ii) not reducing to the periodic case in an appropriate limit, and (iii) not having any clear generalisation to an infinite energy spectrum. In contrast, we note that a covariant time observable has been previously defined for such Hamiltonians, which avoids these features. We also show how this 'quasiperiodic' time observable can be represented as the well-defined limit of a sequence of periodic time observables.
    Full-text · Article · Jun 2012 · Physical Review A
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    Michael J. W. Hall · Howard M. Wiseman
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    ABSTRACT: A number of authors have suggested that nonlinear interactions can enhance resolution of phase shifts beyond the usual Heisenberg scaling of 1/n, where n is a measure of resources such as the number of subsystems of the probe state or the mean photon number of the probe state. These suggestions are based on calculations of `local precision' for particular nonlinear schemes. However, we show that there is no simple connection between the local precision and the average estimation error for these schemes, leading to a scaling puzzle. This puzzle is partially resolved by a careful analysis of iterative implementations of the suggested nonlinear schemes. However, it is shown that the suggested nonlinear schemes are still limited to an exponential scaling in \sqrt{n}. (This scaling may be compared to the exponential scaling in n which is achievable if multiple passes are allowed, even for linear schemes.) The question of whether nonlinear schemes may have a scaling advantage in the presence of loss is left open. Our results are based on a new bound for average estimation error that depends on (i) an entropic measure of the degree to which the probe state can encode a reference phase value, called the G-asymmetry, and (ii) any prior information about the phase shift. This bound is asymptotically stronger than bounds based on the variance of the phase shift generator. The G-asymmetry is also shown to directly bound the average information gained per estimate. Our results hold for any prior distribution of the shift parameter, and generalise to estimates of any shift generated by an operator with discrete eigenvalues.
    Preview · Article · May 2012 · Physical Review X

Publication Stats

1k Citations
169.41 Total Impact Points

Institutions

  • 2011-2015
    • Griffith University
      • Centre for Quantum Dynamics
      Brisbane, Queensland, Australia
  • 1987-2011
    • Australian National University
      • Department of Theoretical Physics
      Canberra, Australian Capital Territory, Australia
  • 2002-2006
    • Institute for Advanced Study
      Princeton Junction, New Jersey, United States