## Publications

107 results found

Halliwell JJ, Mawby C, 2020, Conditions for macrorealism for systems described by many-valued variables, *Physical Review A: Atomic, Molecular and Optical Physics*, Vol: 102, Pages: 012209 – 1-012209 – 15, ISSN: 1050-2947

Macrorealism (MR) is the view that a system evolving in time possesses definite properties independent of past or future measurements and is traditionally tested for systems described at each time by a single dichotomic variable Q. A number of necessary and sufficient conditions for macrorealism have been derived for a dichotomic variable using sets of Leggett-Garg (LG) inequalities, or the stronger no signaling in time (NSIT) conditions, or a combination thereof. Here we extend this framework by establishing necessary and sufficient conditions for macrorealism for measurements made at two and three times for systems described by variables taking three or more values at each time. Our results include a generalization of Fine's theorem to many-valued variables for measurements at three pairs of times and we derive the corresponding complete set of LG inequalities. We find that LG inequalities and NSIT conditions for many-valued variables do not enjoy the simple hierarchical relationship exhibited by the dichotomic case. This sheds light on some recent experiments on three-level systems which exhibit a LG inequality violation even though certain NSIT conditions are satisfied. Under measurements of dichotomic variables using the Lüders projection rule the three-time LG inequalities cannot be violated beyond the Lüders bound (which coincides numerically with the Tsirelson bound obeyed by correlators in Bell experiments), but this bound can be violated in LG tests using degeneracy-breaking (von Neumann) measurements. We identify precisely which MR conditions are violated under these circumstances.

Halliwell JJ, Mawby C, 2019, Fine's theorem for Leggett-Garg tests with an arbitrary number of measurement times, *Physical Review A: Atomic, Molecular and Optical Physics*, Vol: 100, Pages: 042103 – 1-042103 – 12, ISSN: 1050-2947

If the time evolution of a quantum system can be understood classically, then there must exist an underlying probability distribution for the variables describing the system at a sequence of times. It is well known that for systems described by a single time-evolving dichotomic variable Q and for which a given set of temporal correlation functions are specified, a necessary set of conditions for the existence of such a probability are provided by the Leggett-Garg (LG) inequalities. Fine's theorem in this context is the nontrivial result that a suitably augmented set of LG inequalities are both necessary and sufficient conditions for the existence of an underlying probability. We present a proof of Fine's theorem for the case of measurements on a dichotomic variable at an arbitrary number of times, thereby generalizing the familiar proofs for three and four times. We demonstrate how the LG framework and Fine's theorem can be extended to the case in which all possible two-time correlation functions are measured (instead of the partial set of two-time correlators normally studied). We examine the limit of a large number of measurements for both of the above cases.

Janssen O, Halliwell JJ, Hertog T, 2019, No-boundary proposal in biaxial Bianchi IX minisuperspace, *Physical Review D: Particles, Fields, Gravitation and Cosmology*, Vol: 99, ISSN: 1550-2368

We implement the no-boundary proposal for the wave function of the Universe in an exactly solvable Bianchi IX minisuperspace model with two scale factors. We extend our earlier work [Phys. Rev. Lett. 121, 081302 (2018)] to include the contribution from the CP2\B4 topology. The resulting wave function yields normalizable probabilities and thus fits into a predictive framework for semiclassical quantum cosmology. We find that the amplitude is low for large anisotropies. In the isotropic limit, the usual Hartle-Hawking wave function for the de Sitter minisuperspace model is recovered. Inhomogeneous perturbations in an extended minisuperspace are shown to be initially in their ground state. We also demonstrate that the precise mathematical implementation of the no-boundary proposal as a functional integral in minisuperspace depends on detailed aspects of the model, including the choice of gauge fixing. This shows in particular that the choice of contour cannot be fundamental, adding weight to the recent proposal that the semiclassical no-boundary wave function should be defined solely in terms of a collection of saddle points. We adopt this approach in most of this paper. Finally, we show that the semiclassical tunneling wave function of the Universe is essentially equal to the no-boundary state in this particular minisuperspace model, at least in the subset of the classical domain where the former is known.

Halliwell JJ, 2019, Leggett-Garg tests of macrorealism: checks for noninvasiveness and generalizations to higher-order correlators, *Physical Review A*, Vol: 99, ISSN: 1050-2947

In the tests for macrorealism proposed by Leggett and Garg, the temporal correlation functions of a dichotomic variable Q must be measured in a noninvasive way to rule out alternative classical explanations of Leggett-Garg inequality violations. Ideal negative measurements in which a null result is argued to be a noninvasive determination of the system's state are often used. From a quantum-mechanical perspective, such a measurement collapses the wave function and will therefore typically be found to be invasive under any experimental check. Here, a simple modified ideal negative measurement protocol is described for measuring the correlation functions, which is argued to be noninvasive from both classical and quantum perspectives and hence the noninvasiveness can then be checked experimentally, thereby permitting a quantitative measure of the degree of clumsiness of the measurement. It is also shown how this procedure may be extended to measure higher-order correlation functions, and a number of higher-order conditions for macrorealism are derived.

Halliwell JJ, Hartle JB, Hertog T, 2019, What is the no-boundary wave function of the Universe?, *Physical Review D*, Vol: 99, ISSN: 2470-0010

We specify the semiclassical no-boundary wave function of the Universe without relying on a functional integral of any kind. The wave function is given as a sum of specific saddle points of the dynamical theory that satisfy conditions of regularity on the geometry and field and that together yield a time-neutral state that is normalizable in an appropriate inner product. This specifies a predictive framework of semiclassical quantum cosmology that is adequate to make probabilistic predictions, which are in agreement with observations in simple models. The use of holography to go beyond the semiclassical approximation is briefly discussed.

Halliwell JJ, 2019, Necessary and sufficient conditions for macrorealism using two- and three-time Leggett-Garg inequalities, 9th International Workshop on Decoherence, Information, Complexity and Entropy (DICE) - From Discrete Structures and Dynamics to Top-Down Causation, Publisher: IOP PUBLISHING LTD, Pages: 1-12, ISSN: 1742-6588

The Leggett-Garg (LG) inequalities were introduced, as a temporal parallel of the Bell inequalities, to test macroscopic realism – the view that a macroscopic system evolving in time possesses definite properties which can be determined without disturbing the future or past state. The original LG inequalities are only a necessary condition for macrorealism, and are therefore not a decisive test. We argue, for the case of measurements of a single dichotomic variable Q, that when the original four three-time LG inequalities are augmented with a set of twelve two-time inequalities also of the LG form, Fine's theorem applies and these augmented conditions are then both necessary and sufficient. A comparison is carried out with the alternative necessary and sufficient conditions for macrorealism based on no-signaling in time conditions which ensure that all probabilities for Q at one and two times are independent of whether earlier or intermediate measurements are made. We argue that the two tests differ in their implementation of the key requirement of non-invasive measurability so are testing different notions of macrorealism, and these notions are elucidated.

Halliwell JJ, Beck H, Lee BKB,
et al., 2019, Quasiprobability for the arrival-time problem with links to backflow and the Leggett-Garg inequalities, *Physical Review A*, Vol: 99, ISSN: 1050-2947

The arrival-time problem for the free particle in one dimension may be formulated as the problem of determining a joint probability for the particle being found on opposite sides of the x axis at two different times. We explore this problem using a two-time quasiprobability linear in the projection operators, a natural counterpart of the corresponding classical problem. We show that it can be measured either indirectly, by measuring its moments in different experiments, or directly, in a single experiment using a pair of sequential measurements in which the first measurement is weak (or, more generally, ambiguous). We argue that when positive, it corresponds to a measurement-independent arrival-time probability. For small time intervals it coincides approximately with the time-averaged current, in agreement with semiclassical expectations. The quasiprobability can be negative and we exhibit a number of situations in which this is the case. We interpret these situations as the presence of “quantumness,” in which the arrival-time probability is not properly defined in a measurement-independent manner. Backflow states, in which the current flows in the direction opposite to the momentum, are shown to provide an interesting class of examples such situations. We also show that the quasiprobability is closely linked to a set of two-time Leggett-Garg inequalities, which test for macroscopic realism.

Dorronsoro JD, Halliwell JJ, Hartle JB,
et al., 2018, Damped perturbations in the no-boundary state, *Physical Review Letters*, Vol: 121, ISSN: 0031-9007

We evaluate the no-boundary path integral exactly in a Bianchi type IX minisuperspace with two scale factors. In this model the no-boundary proposal can be implemented by requiring one scale factor to be zero initially together with a judiciously chosen regularity condition on the momentum conjugate to the second scale factor. Taking into account the nonlinear backreaction of the perturbations we recover the predictions of the original semiclassical no-boundary proposal. In particular we find that large perturbations are strongly damped, consistent with vacuum state wave functions.

Dorronsoro JD, Halliwell JJ, Hartle JB,
et al., 2017, Real no-boundary wave function in Lorentzian quantum cosmology, *Physical Review D: Particles, Fields, Gravitation and Cosmology*, Vol: 96, ISSN: 1550-2368

It is shown that the standard no-boundary wave function has a natural expression in terms of a Lorentzian path integral with its contour defined by Picard-Lefschetz theory. The wave function is real, satisfies the Wheeler-DeWitt equation and predicts an ensemble of asymptotically classical, inflationary universes with nearly-Gaussian fluctuations and with a smooth semiclassical origin.

Halliwell JJ, 2017, Incompatible multiple consistent sets of histories and measures of quantumness, *PHYSICAL REVIEW A*, Vol: 96, ISSN: 2469-9926

In the consistent histories approach to quantum theory probabilities are assigned to histories subject to a consistency condition of negligible interference. The approach has the feature that a given physical situation admits multiple sets of consistent histories that cannot in general be united into a single consistent set, leading to a number of counterintuitive or contrary properties if propositions from different consistent sets are combined indiscriminately. An alternative viewpoint is proposed in which multiple consistent sets are classified according to whether or not there exists any unifying probability for combinations of incompatible sets which replicates the consistent histories result when restricted to a single consistent set. A number of examples are exhibited in which this classification can be made, in some cases with the assistance of the Bell, Clauser-Horne-Shimony-Holt, or Leggett-Garg inequalities together with Fine's theorem. When a unifying probability exists logical deductions in different consistent sets can in fact be combined, an extension of the “single framework rule.” It is argued that this classification coincides with intuitive notions of the boundary between classical and quantum regimes and in particular, the absence of a unifying probability for certain combinations of consistent sets is regarded as a measure of the “quantumness” of the system. The proposed approach and results are closely related to recent work on the classification of quasiprobabilities and this connection is discussed.

Halliwell JJ, 2017, Comparing conditions for macrorealism: Leggett-Garg inequalities versus no-signaling in time, *PHYSICAL REVIEW A*, Vol: 96, ISSN: 2469-9926

We consider two different types of conditions which were proposed to test macrorealism in the context of a system described by a single dichotomic variable Q. This is the view that a macroscopic system evolving in time possesses definite properties which can be determined without disturbing the future or past state. The Leggett-Garg (LG) inequalities, the most commonly studied test, are only necessary conditions for macrorealism, but, building on earlier work J. J. Halliwell, Phys. Rev. A 93, 022123 (2016), it is shown that when the four three-time LG inequalities are augmented with a certain set of two-time inequalities also of the LG form, Fine's theorem applies and these augmented conditions are then both necessary and sufficient. A comparison is carried out with a very different set of necessary and sufficient conditions for macrorealism, namely the no-signaling in time (NSIT) conditions proposed by J. Kofler and C. Brukner, Phys. Rev. A 87, 052115 (2013) and L. Clemente and J. Kofler, Phys. Rev. A 91, 062103 (2015), which ensure that all probabilities for Q at one and two times are independent of whether earlier or intermediate measurements are made in a given run, and do not require (but imply) the LG inequalities. We argue that tests based on the LG inequalities have the form of very weak classicality conditions and can be satisfied in the face of moderate interference effects, but those based on NSIT conditions have the form of much stronger coherence witness conditions, satisfied only for zero interference. The two tests differ in their implementation of noninvasive measurability and so are testing different notions of macrorealism: the augmented LG tests are indirect, entailing a combination of the results of different experiments with only compatible quantities measured in each experimental run, in close analogy with Bell tests, and are primarily tests for macrorealism per se; in contrast, the NSIT tests entail sequential measurements of incompatible quan

Halliwell JJ, 2016, Decoherent histories and measurement of temporal correlation functions for Leggett-Garg inequalities, *PHYSICAL REVIEW A*, Vol: 94, ISSN: 2469-9926

Halliwell JJ, 2016, Leggett-Garg correlation functions from a noninvasive velocity measurement continuous in time, *PHYSICAL REVIEW A*, Vol: 94, ISSN: 2469-9926

Halliwell JJ, 2016, Leggett-Garg inequalities and no-signaling in time: A quasiprobability approach, *Physical Review A*, Vol: 93, ISSN: 1050-2947

The Leggett-Garg (LG) inequalities were proposed in order to assess whether sets of pairs of sequential measurements on a single quantum system can be consistent with an underlying notion of macrorealism. Here, the LG inequalities are explored using a simple quasiprobability linear in the projection operators to describe the properties of the system at two times. We show that this quasiprobability is measurable, has the same correlation function as the usual two-time measurement probability (for the bivalent variables considered here) and has the key property that the probabilities for the later time are independent of whether an earlier measurement was made, a generalization of the no-signaling in time condition of Kofler and Brukner. We argue that this quasiprobability, appropriately measured, provides a noninvasive measure of macrorealism per se at the two-time level. This measure, when combined with the LG inequalities, provides a characterization of macrorealism more detailed than that provided by the LG inequalities alone. When the quasiprobability is non-negative, the LG system has a natural parallel with the Einstein-Podolsky-Rosen-Bohm system and Fine's theorem. A simple spin model illustrating key features of the approach is exhibited.

Halliwell JJ, Evaeus J, London J,
et al., 2015, A self-adjoint arrival time operator inspired by measurement models, *Physics Letters A*, Vol: 379, Pages: 2445-2451, ISSN: 0375-9601

We introduce an arrival time operator which is self-adjoint and, unlike previously proposed arrival time operators, has a close link to simple measurement models. Its spectrum leads to an arrival time distribution which is a variant of the Kijowski distribution (a re-ordering of the current) in the large momentum regime but is proportional to the kinetic energy density in the small momentum regime, in agreement with measurement models. A brief derivation of the latter distribution is given. We make some simple observations about the physical reasons for self-adjointness, or its absence, in both arrival time operators and the momentum operator on the half-line and we also compare our operator with the dwell time operator.

Halliwell JJ, 2014, Two proofs of Fine's theorem, *Physics Letters A*, Vol: 378, Pages: 2945-2950, ISSN: 0375-9601

Fine's theorem concerns the question of determining the conditions under which a certain set of probabilities for pairs of four bivalent quantities may be taken to be the marginals of an underlying probability distribution. The eight CHSH inequalities are well-known to be necessary conditions, but Fine's theorem is the striking result that they are also sufficient conditions. Here two transparent and self-contained proofs of Fine's theorem are presented. The first is a physically motivated proof using an explicit local hidden variables model. The second is an algebraic proof which uses a representation of the probabilities in terms of correlation functions.

Bedingham D, Halliwell JJ, 2014, Classical limit of the quantum Zeno effect by environmental decoherence, *Physical Review A*, Vol: 89, ISSN: 1050-2947

We consider a point particle in one dimension initially confined to a finite spatial region whose state isfrequently monitored by projection operators onto that region. In the limit of infinitely frequent monitoring, thestate never escapes from the region—this is the Zeno effect. In the corresponding classical problem, by contrast,the state diffuses out of the region with the frequent monitoring simply removing probability. The aim of thispaper is to show how the Zeno effect disappears in the classical limit in this and similar examples. We givea general argument showing that the Zeno effect is suppressed in the presence of a decoherence mechanismwhich suppresses interference between histories. We show how this works explicitly in two examples involvingprojections onto a one-dimensional subspace and identify the key time scales for the process. We extend thisunderstanding to our main problem of interest, the case of a particle in a spatial region, by coupling it to adecohering environment. Smoothed projectors are required to give the problem proper definition and this impliesthe existence of a momentum cutoff and minimum length scale. We show that the escape rate from the regionapproaches the classically expected result, and hence the Zeno effect is suppressed, as long as the environmentallyinduced fluctuations in momentum are sufficiently large. We establish the time scale on which an arbitrary initialstate develops sufficiently large fluctuations to satisfy this condition. We link our results to earlier work on the →0 limit of the Zeno effect. We illustrate our results by plotting the probability flux lines for the density matrix(which are equivalent to Bohm trajectories in the pure-state case). These illustrate both the Zeno and anti-Zenoeffects very clearly, and their suppression. Our results are closely related to our earlier paper [Phys.Rev.A88,022128(2013)], demonstrating the suppression of quantum-mechanical reflection by decoherence.

Halliwell JJ, Gillman E, Lennon O,
et al., 2013, Quantum backflow states from eigenstates of the regularized current operator, *JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL*, Vol: 46, ISSN: 1751-8113

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Bedingham DJ, Halliwell JJ, 2013, Suppression of quantum-mechanical reflection by environmental decoherence, *PHYSICAL REVIEW A*, Vol: 88, ISSN: 1050-2947

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- Citations: 5

Halliwell JJ, Yearsley JM, 2013, Negative probabilities, Fine's theorem, and linear positivity, *PHYSICAL REVIEW A*, Vol: 87, ISSN: 1050-2947

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- Citations: 26

Halliwell JJ, 2013, Exact phase space localized projectors from energy eigenstates, *PHYSICS LETTERS A*, Vol: 377, Pages: 222-227, ISSN: 0375-9601

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- Citations: 2

Halliwell JJ, Yearsley JM, 2013, Amplitudes for spacetime regions and the quantum Zeno effect: pitfalls of standard path integral constructions, 6th International Workshop on Decoherence, Information, Complexity and Entropy (DICE), Publisher: IOP PUBLISHING LTD, ISSN: 1742-6588

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- Citations: 5

Yearsley JM, Halliwell JJ, 2013, An introduction to the quantum backflow effect, 6th International Workshop on Decoherence, Information, Complexity and Entropy (DICE), Publisher: IOP PUBLISHING LTD, ISSN: 1742-6588

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- Citations: 16

Yearsley JM, Halliwell JJ, Hartshorn R,
et al., 2012, Analytical examples, measurement models, and classical limit of quantum backflow, *PHYSICAL REVIEW A*, Vol: 86, ISSN: 1050-2947

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- Citations: 25

Halliwell JJ, Yearsley JM, 2012, Pitfalls of path integrals: Amplitudes for spacetime regions and the quantum Zeno effect, *PHYSICAL REVIEW D*, Vol: 86, ISSN: 1550-7998

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- Citations: 13

Yearsley JM, Downs DA, Halliwell JJ,
et al., 2011, Quantum arrival and dwell times via idealized clocks, *PHYSICAL REVIEW A*, Vol: 84, ISSN: 1050-2947

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- Citations: 24

Yearsley JM, Downs DA, Halliwell JJ,
et al., 2011, Quantum arrival and dwell times via idealized clocks, *Physical Review A*, Vol: 84

Halliwell JJ, 2011, Decoherent Histories Analysis of Minisuperspace Quantum Cosmology, 2010 DICE Conference, Space-Time-Matter: Current Issues in Quantum Theory and Beyond

Halliwell JJ, Yearsley JM, 2010, On the relationship between complex potentials and strings of projection operators, *JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL*, Vol: 43, ISSN: 1751-8113

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- Citations: 16

Halliwell JJ, Yearsley JM, 2009, Quantum arrival time formula from decoherent histories, *PHYSICS LETTERS A*, Vol: 374, Pages: 154-157, ISSN: 0375-9601

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- Citations: 17

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