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In a recent paper, PNAS, 118, e1921529118 (2021), it was argued that while the standard definition of conservation laws in quantum mechanics, which is of a statistical character, is perfectly valid, it misses essential features of nature and it can and must be revisited to address the issue of conservation/non-conservation in individual cases. Specifically, in the above paper an experiment was presented in which it can be proven that in some individual cases energy is not conserved, despite being conserved statistically. It was felt however that this is worrisome, and that something must be wrong if there are individual instances in which conservation doesn't hold, even though this is not required by the standard conservation law. Here we revisit that experiment and show that although its results are correct, there is a way to circumvent them and ensure individual case conservation in that situation. The solution is however quite unusual, challenging one of the basic assumptions of quantum mechanics, namely that any quantum state can be prepared, and it involves a time-holistic, double non-conservation effect. Our results bring new light on the role of the preparation stage of the initial state of a particle and on the interplay of conservation laws and frames of reference. We also conjecture that when such a full analysis of any conservation experiment is performed, conservation is obeyed in every individual case.

Infants, adults, non-human primates and non-primates all learn patterns implicitly, and they do so across modalities. The biological evidence supports the hypothesis that the mechanism for this learning is general but computationally local. We hypothesize that the mechanism itself is predictive error-driven learning. We build on recent work that advanced a biologically plausible model of error backpropagation learning which proposes that higher order thalamic nuclei provide a locale for a temporal difference between top-down predictions and an actual event outcome. Our neural network based on that work also models the auditory cortex hierarchy of core, belt and parabelt and the caudal-rostral axis within regions. We simulated two studies showing statistical learning in infants, a seminal study using synthesized speech and a more recent study using human speech. Before simulating these studies the network was trained on spoken sentences from the TIMIT corpus to emulate infant's experience listening to random speech. The implemented neural network, learning only by predicting the next brief speech segment, learned in both simulations to predict in-word syllables better than next-word syllables showing that prediction could be the basis for word segmentation and thus statistical learning., Comment: 24 pages, 11 figures

We give an unconditional proof that self-dual Artin representations of $\mathbb{Q}$ of dimension $3$ have density $0$ among all Artin representations of $\mathbb{Q}$ of dimension $3$. Previously this was known under the assumption of Malle's Conjecture., Comment: Appendix by David E. Rohrlich

We revisit the "counterfactual quantum communication" of Salih et al. [1], who claim that an observer "Bob" can send one bit of information to a second observer "Alice" without any physical particle traveling between them. We show that a locally conserved, massless current - specifically, a current of modular angular momentum, $L_z$ mod 2$\hbar$ - carries the one bit of information. We integrate the flux of $L_z$ mod 2$\hbar$ from Bob to Alice and show that it equals one of the two eigenvalues of $L_z$ mod 2$\hbar$, either 0 or $\hbar$, thus precisely accounting for the one bit of information he sends her. We previously [2] obtained this result using weak values of $L_z$ mod 2$\hbar$; here we do not use weak values., Comment: Physical Review Letters, in press

The Stern-Gerlach effect, discovered a century ago, has become a paradigm of quantum mechanics. Surprisingly there has been little evidence that the original scheme with freely propagating atoms exposed to gradients from macroscopic magnets is a fully coherent quantum process. Specifically, no full-loop Stern-Gerlach interferometer has been realized with the scheme as envisioned decades ago. Furthermore, several theoretical studies have explained why such an interferometer is a formidable challenge. Here we provide a detailed account of the first full-loop Stern-Gerlach interferometer realization, based on highly accurate magnetic fields, originating from an atom chip, that ensure coherent operation within strict constraints described by previous theoretical analyses. Achieving this high level of control over magnetic gradients is expected to facilitate technological as well as fundamental applications, such as probing the interface of quantum mechanics and gravity. While the experimental realization described here is for a single atom, future challenges would benefit from utilizing macroscopic objects doped with a single spin. Specifically, we show that such an experiment is in principle feasible, opening the door to a new era of fundamental probes.

In this invited review in honor of 100 years since the Stern-Gerlach (SG) experiments, we describe a decade of SG interferometry on the atom chip. The SG effect has been a paradigm of quantum mechanics throughout the last century, but there has been surprisingly little evidence that the original scheme, with freely propagating atoms exposed to gradients from macroscopic magnets, is a fully coherent quantum process. Specifically, no full-loop SG interferometer (SGI) has been realized with the scheme as envisioned decades ago. Furthermore, several theoretical studies have explained why it is a formidable challenge. Here we provide a review of our SG experiments over the last decade. We describe several novel configurations such as that giving rise to the first SG spatial interference fringes, and the first full-loop SGI realization. These devices are based on highly accurate magnetic fields, originating from an atom chip, that ensure coherent operation within strict constraints described by previous theoretical analyses. Achieving this high level of control over magnetic gradients is expected to facilitate technological applications such as probing of surfaces and currents, as well as metrology. Fundamental applications include the probing of the foundations of quantum theory, gravity, and the interface of quantum mechanics and gravity. We end with an outlook describing possible future experiments., Comment: 20 pages, 10 figures, >100 references. Invited review from the September 2019 conference in Frankfurt celebrating the centennial of the Stern-Gerlach experiment

How do humans learn from raw sensory experience? Throughout life, but most obviously in infancy, we learn without explicit instruction. We propose a detailed biological mechanism for the widely-embraced idea that learning is based on the differences between predictions and actual outcomes (i.e., predictive error-driven learning). Specifically, numerous weak projections into the pulvinar nucleus of the thalamus generate top-down predictions, and sparse, focal driver inputs from lower areas supply the actual outcome, originating in layer 5 intrinsic bursting (5IB) neurons. Thus, the outcome is only briefly activated, roughly every 100 msec (i.e., 10 Hz, alpha), resulting in a temporal difference error signal, which drives local synaptic changes throughout the neocortex, resulting in a biologically-plausible form of error backpropagation learning. We implemented these mechanisms in a large-scale model of the visual system, and found that the simulated inferotemporal (IT) pathway learns to systematically categorize 3D objects according to invariant shape properties, based solely on predictive learning from raw visual inputs. These categories match human judgments on the same stimuli, and are consistent with neural representations in IT cortex in primates., Comment: 56 pages, 22 figures

Clocks play a special role at the interface of general relativity and quantum mechanics. We analyze a clock-interferometry thought experiment and go on to theoretically derive and experimentally test a complementarity relation for quantum clocks in the context of the gravitational time lag. We study this relation in detail and discuss its application to various types of quantum clocks., Comment: Supplementary Material included. Comments welcome!

The Stern-Gerlach (SG) effect, discovered almost a century ago, has become a paradigm of quantum mechanics. Surprisingly there is little evidence that the original scheme with freely propagating atoms exposed to gradients from macroscopic magnets is a fully coherent quantum process. Specifically, no high-visibility spatial interference pattern has been observed with such a scheme, and furthermore no full-loop SG interferometer has been realized with the scheme as envisioned decades ago. On the contrary, numerous theoretical studies explained why it is a near impossible endeavor. Here we demonstrate for the first time both a high-visibility spatial SG interference pattern and a full-loop SG interferometer, based on an accurate magnetic field, originating from an atom chip, that ensures coherent operation within strict constraints described by previous theoretical analyses. This also allows us to observe the gradual emergence of time-irreversibility as the splitting is increased. Finally, achieving this high level of control over magnetic gradients may facilitate technological applications such as large-momentum-transfer beam splitting for metrology with atom interferometry, ultra-sensitive probing of electron transport down to shot-noise and squeezed currents, as well as nuclear magnetic resonance and compact accelerators., Comment: 13 pages and 5 figures, and supplementary materials

Where does quantum mechanics part ways with classical mechanics? How does quantum randomness differ fundamentally from classical randomness? We cannot fully explain how the theories differ until we can derive them within a single axiomatic framework, allowing an unambiguous account of how one theory is the limit of the other. Here we derive nonrelativistic quantum mechanics and classical statistical mechanics within a common framework. The common axioms include conservation of average energy and conservation of probability current. But two axioms distinguish quantum mechanics from classical statistical mechanics: an "ontic extension" defines a nonseparable (global) random variable that generates physical correlations, and an "epistemic restriction" constrains allowed phase space distributions. The ontic extension and epistemic restriction, with strength on the order of Planck's constant, imply quantum entanglement and uncertainty relations. This framework suggests that the wave function is epistemic, yet it does not provide an ontic dynamics for individual systems., Comment: 12 pages; comments welcome

How does the neocortex learn and develop the foundations of all our high-level cognitive abilities? We present a comprehensive framework spanning biological, computational, and cognitive levels, with a clear theoretical continuity between levels, providing a coherent answer directly supported by extensive data at each level. Learning is based on making predictions about what the senses will report at 100 msec (alpha frequency) intervals, and adapting synaptic weights to improve prediction accuracy. The pulvinar nucleus of the thalamus serves as a projection screen upon which predictions are generated, through deep-layer 6 corticothalamic inputs from multiple brain areas and levels of abstraction. The sparse driving inputs from layer 5 intrinsic bursting neurons provide the target signal, and the temporal difference between it and the prediction reverberates throughout the cortex, driving synaptic changes that approximate error backpropagation, using only local activation signals in equations derived directly from a detailed biophysical model. In vision, predictive learning requires a carefully-organized developmental progression and anatomical organization of three pathways (What, Where, and What * Where), according to two central principles: top-down input from compact, high-level, abstract representations is essential for accurate prediction of low-level sensory inputs; and the collective, low-level prediction error must be progressively and opportunistically partitioned to enable extraction of separable factors that drive the learning of further high-level abstractions. Our model self-organized systematic invariant object representations of 100 different objects from simple movies, accounts for a wide range of data, and makes many testable predictions., Comment: 64 pages, 24 figures, 291 references. Submitted for publication

We raise fundamental questions about the very meaning of conservation laws in quantum mechanics and we argue that the standard way of defining conservation laws, while perfectly valid as far as it goes, misses essential features of nature and has to be revisited and extended.

A recent paper [1] argues that bipartite "PR-box" correlations, though designed to respect relativistic causality, in fact violate relativistic causality in the classical limit. As a test of Ref. [1], we consider GHZ correlations as a tripartite version of PR-box correlations, and ask whether the arguments of Ref. [1] extend to GHZ correlations. If they do - i.e. if they show that GHZ correlations violate relativistic causality in the classical limit - then Ref. [1] must be incorrect, since GHZ correlations are quantum correlations and respect relativistic causality. But the arguments fail. We also show that both PR-box correlations and GHZ correlations can be retrocausal, but the retrocausality of PR-box correlations leads to self-contradictory causal loops, while the retrocausality of GHZ correlations does not., Comment: 12 pages, 3 figures

Are the electromagnetic scalar and vector potentials dispensable? Lev Vaidman has suggested that local interactions of gauge-invariant quantities, e.g. magnetic torques, suffice for the description of all quantum electromagnetic phenomena. We analyze six thought experiments that challenge this suggestion. All of them have explanations in terms of $local$ interactions of gauge-dependent quantities, in addition, some have explanations in terms of nonlocal interactions of gauge-invariant quantities. We claim, however, that two of our examples have no gauge-invariant formal description and that, in general, no local description can dispense with electromagnetic potentials., Comment: 8 pages, 2 figures

In 1964, John Bell proved that quantum mechanics is "unreasonable" (to use Einstein's term): there are nonlocal bipartite quantum correlations. But they are not the most nonlocal bipartite correlations consistent with relativistic causality ("no superluminal signalling"): also maximally nonlocal "superquantum" (or "PR-box") correlations are consistent with relativistic causality. I show that---unlike quantum correlations---these correlations do not have a classical limit consistent with relativistic causality. The generalization of this result to all stronger-than-quantum nonlocal correlations is a derivation of Tsirelson's bound---a theorem of quantum mechanics---from the three axioms of relativistic causality, nonlocality, and the existence of a classical limit. But is it reasonable to derive (a part of) quantum mechanics from the unreasonable axiom of nonlocality?! I consider replacing the nonlocality axiom with an equivalent axiom that even Bell and Einstein might have considered reasonable: an axiom of local retrocausality., Comment: To appear in Quantum Nonlocality and Reality: 50 Years of Bell's theorem, eds. S. Gao and M. Bell (Cambridge U. Press), 2015, in press

We experimentally demonstrate a new interferometry paradigm: a self-interfering clock. We split a clock into two spatially separated wave packets, and observe an interference pattern with a stable phase showing that the splitting was coherent, i.e., the clock was in two places simultaneously. We then make the clock wave packets "tick" at different rates to simulate a proper time lag. The entanglement between the clock's time and its path yields "which path" information, which affects the visibility of the clock's self-interference. By contrast, in standard interferometry, time cannot yield "which path" information. As a clock we use an atom prepared in a superposition of two spin states. This first proof-of-principle experiment may have far-reaching implications for the study of time and general relativity and their impact on fundamental quantum effects such as decoherence and wave packet collapse., Comment: 22 pages, 8 figures

Although the Aharonov-Bohm and related effects are familiar in solid state and high energy physics, the nonlocality of these effects has been questioned. Here we show, for the first time, that the Aharonov-Bohm effect has two very different aspects. One aspect is instantaneous and nonlocal; the other aspect, which depends on entanglement, unfolds continuously over time. While local, gauge-invariant variables may occasionally suffice for explaining the continuous aspect, we argue that they cannot explain the instantaneous aspect. Thus the Aharonov-Bohm effect is, in general, nonlocal., Comment: 12 pages, 2 figures

The "anomalous" values of C. Ferrie and J. Combes in Phys. Rev. Lett. 113, 120404 (2014) say nothing about quantum - or even classical - physics. They are not analogues of the weak values that emerge when we describe the quantum world via an initial state evolving forwards in time and a final state evolving backwards in time, and couple this world weakly to realistic measuring devices.

Superquantum ("PR-box") correlations, though designed to respect relativistic causality, violate relativistic causality in the classical limit. Generalizing to all stronger-than-quantum bipartite correlations, I derive Tsirelson's bound from the axioms of nonlocality, relativistic causality and the existence of a classical limit. This derivation of Tsirelson's bound does not assume quantum mechanics yet suggests how Hilbert space is implicit in quantum correlations., Comment: arXiv admin note: text overlap with arXiv:1407.8530

One of Yakir Aharonov's endlessly captivating physics ideas is the conjecture that two axioms, namely relativistic causality ("no superluminal signalling") and nonlocality, so nearly contradict each other that a unique theory - quantum mechanics - reconciles them. But superquantum (or "PR-box") correlations imply that quantum mechanics is not the most nonlocal theory (in the sense of nonlocal correlations) consistent with relativistic causality. Let us consider supplementing these two axioms with a minimal third axiom: there exists a classical limit in which macroscopic observables commute. That is, just as quantum mechanics has a classical limit, so must any generalization of quantum mechanics. In this classical limit, PR-box correlations violate relativistic causality. Generalized to all stronger-than-quantum bipartite correlations, this result is a derivation of Tsirelson's bound without assuming quantum mechanics., Comment: for a video of this talk at the Aharonov-80 Conference in 2012 at Chapman University, see quantum.chapman.edu/talk-10, published in Quantum Theory: A Two-Time Success Story (Yakir Aharonov Festschrift), eds. D. C. Struppa and J. M. Tollaksen (New York: Springer), 2013, pp. 205-211

We present a comprehensive, novel framework for understanding how the neocortex, including the thalamocortical loops through the deep layers, can support a temporal context representation in the service of predictive learning. Many have argued that predictive learning provides a compelling, powerful source of learning signals to drive the development of human intelligence: if we constantly predict what will happen next, and learn based on the discrepancies from our predictions (error-driven learning), then we can learn to improve our predictions by developing internal representations that capture the regularities of the environment (e.g., physical laws governing the time-evolution of object motions). Our version of this idea builds upon existing work with simple recurrent networks (SRN's), which have a discretely-updated temporal context representations that are a direct copy of the prior internal state representation. We argue that this discretization of temporal context updating has a number of important computational and functional advantages, and further show how the strong alpha-frequency (10hz, 100ms cycle time) oscillations in the posterior neocortex could reflect this temporal context updating. We examine a wide range of data from biology to behavior through the lens of this LeabraTI model, and find that it provides a unified account of a number of otherwise disconnected findings, all of which converge to support this new model of neocortical learning and processing. We describe an implemented model showing how predictive learning of tumbling object trajectories can facilitate object recognition with cluttered backgrounds., Comment: 37 pages, 11 figures. arXiv admin note: text overlap with arXiv:1112.0778 by other authors

Protective measurements illustrate how Yakir Aharonov's fundamental insights into quantum theory yield new experimental paradigms that allow us to test quantum mechanics in ways that were not possible before. As for quantum theory itself, protective measurements demonstrate that a quantum state describes a single system, not only an ensemble of systems, and reveal a rich ontology in the quantum state of a single system. We discuss in what sense protective measurements anticipate the theorem of Pusey, Barrett, and Rudolph (PBR), stating that, if quantum predictions are correct, then two distinct quantum states cannot represent the same physical reality., Comment: To appear in Protective Measurement: Towards a New Understanding of Quantum Mechanics, ed. Shan Gao

Atom chips made of superconducting material can generate magnetic traps with significantly reduced noise. Recently, several designs for superconducting chips have been theoretically analyzed and experimentally tested, for cases with many vortices considered as an average vortex density. Here we show theoretically, for the first time, how the magnetic field of a $\it single$ vortex, pinned by a superconducting nano-disc of radius $\sim$100 nm and combined with an external bias field parallel to the disc surface, yields a closed 3D trap for cold atoms. The size of the trap, and its height above the superconductor surface, are typically tens or hundreds of nanometers. We estimate the average lifetime $\tau$ of $^{87}$Rb (rubidium) atoms (subject to thermal escape and Majorana spin flips) in the range 0.05-1.0 ms. Next, we model the trap in a quantum adiabatic approximation and apply Fermi's rule to estimate the lifetime of $^{87}$Rb atoms in the ground state of this trap. We obtain similar lifetimes $\tau$ as in the semiclassical estimate, in the range 0.05-3.5 ms. We find that $\tau$ depends on the gradient $B_0$ of the vortex's magnetic field according to $\tau \sim(B_0)^{-2/3}$., Comment: 20 pages, 5 figures

In this paper we present a quantum Cheshire Cat. In a pre- and post-selected experiment we find the Cat in one place, and its grin in another. The Cat is a photon, while the grin is its circular polarization., Comment: v2: 5 pages, 1 figure. New author added; New section added; Published version

The axioms of nonrelativistic quantum mechanics lack clear physical meaning. In particular, they say nothing about nonlocality. Yet quantum mechanics is not only nonlocal, it is twice nonlocal: there are nonlocal quantum correlations, and there is the Aharonov-Bohm effect, which implies that an electric or magnetic field h e r e may act on an electron t h e r e. Can we invert the logical hierarchy? That is, can we adopt nonlocality as an axiom for quantum mechanics and derive quantum mechanics from this axiom and an additional axiom of causality? Three versions of these two axioms lead to three different theories, characterized by "maximal nonlocal correlations", "jamming" and "modular energy". Where is quantum mechanics in these theories?, Comment: to appear in Probability in Physics, dedicated to the memory of Prof. Itamar Pitowsky, eds. Y. Ben-Menahem and M. Hemmo, Springer, Frontiers Collection, in press

Using physical arguments, I derive the physically correct equations of motion for a classical charged particle from the Lorentz-Abraham-Dirac equations (LAD) which are well known to be physically incorrect. Since a charged particle can classically not be a point particle because of the Coulomb field divergence, my derivation accounts for that by imposing a basic condition on the external force. That condition ensures that the particle's finite size charge distribution looks like a point charge to the external force. Finite radius charge distributions are known not to lead to differential equations of motion. The present work is in agreemnent with the results by Spohn and by others. An example, uniform acceleration, demonstrates what the above basic condition entails. For clarity of the argument, I discuss the non-relativistic case before the relativistic one., Comment: 11 pages

Berry's phase, entry in the Compendium of Quantum Physics: Concepts, Experiments, History and Philosophy, ed. F. Weinert, K. Hentschel, D. Greenberger and B. Falkenburg (Springer), to appear, Comment: 5 pages + 1 figure

The Aharonov-Casher effect, entry in the Compendium of Quantum Physics: Concepts, Experiments, History and Philosophy, ed. F. Weinert, K. Hentschel, D. Greenberger and B. Falkenburg (Springer), to appear, Comment: 3 pages + 1 figure

Errors and paradoxes in quantum mechanics, entry in the Compendium of Quantum Physics: Concepts, Experiments, History and Philosophy, ed. F. Weinert, K. Hentschel, D. Greenberger and B. Falkenburg (Springer), to appear, Comment: 8 pages + 2 figures

Shot noise measurements provide information on particles' charge and their correlations. We report on shot noise measurements in a ubiquitous quantum dot under a quantized magnetic field. The measured noise at the peaks of a sequence of conductance resonances was some nine times higher than expected; suggesting bunching of electrons as they traverse through the dot. This enhancement might be mediated by an additional weakly coupled level to the leads or an excited state. Note that in the absence of magnetic filed no bunching had been observed.

Resonant tunneling through identical potential barriers is a textbook problem in quantum mechanics. Its solution yields total transparency (100% tunneling) at discrete energies. This dramatic phenomenon results from coherent interference among many trajectories, and it is the basis of transport through periodic structures. Resonant tunneling of electrons is commonly seen in semiconducting 'quantum dots'. Here we demonstrate that detecting (distinguishing) electron trajectories in a quantum dot (QD) renders the QD nearly insulating. We couple trajectories in the QD to a 'detector' by employing edge channels in the integer quantum Hall regime. That is, we couple electrons tunneling through an inner channel to electrons in the neighboring outer, 'detector' channel. A small bias applied to the detector channel suffices to dephase (quench) the resonant tunneling completely. We derive a formula for dephasing that agrees well with our data and implies that just a few electrons passing through the detector channel suffice to dephase the QD completely. This basic experiment shows how path detection in a QD induces a transition from delocalization (due to coherent tunneling) to localization (sequential tunneling).

We describe a new and distinctive interferometry in which a probe particle scatters off a superposition of locations of a single free target particle. In one dimension, probe particles incident on superposed locations of a single "mirror" can interfere as if in a Fabry-Perot interferometer; in two dimensions, probe particles scattering off superposed locations of a single "slit" can interfere as if in a two-slit Young interferometer. The condition for interference is loss of orthogonality of the target states and reduces, in simple examples, to transfer of orthogonality from target to probe states. We analyze experimental parameters and conditions necessary for interference to be observed., Comment: 5 pages, 2 figures, RevTeX, submitted to PRL

The first step in quantum information theory is the identification of entanglement as a valuable resource. The next step is learning how to exploit this resource efficiently. We learn how to exploit entanglement efficiently by applying analogues of thermodynamical concepts. These concepts include reversibility, entropy, and the distinction between intensive and extensive quantities. We discuss some of these analogues and show how they lead to a measure of entanglement for pure states. We also ask whether these analogues are more than analogues, and note that, l o c a l l y, entropy of entanglement is thermodynamical entropy., Comment: RevTeX, 10 pages, no figures

Any charged particle moving faster than light through a medium emits Cherenkov radiation. We show that charged particles moving faster than light through the v a c u u m emit Cherenkov radiation. How can a particle move faster than light? The w e a k speed of a charged particle can exceed the speed of light. By definition, the weak velocity is / where v is the velocity operator and |a> and |b> are, respectively, the states of a particle before and after a velocity measurement. We discuss the consistency of weak values and show that superluminal weak speed is consistent with relativistic causality., Comment: RevTeX, 18 pages, no figures. Revised and expanded for clarity; to appear in Physical Review A

Approximately 500 undergraduate students, representing colleges and universities across the United States, who were about to embark on a semester-long credit-bearing sojourn sponsored by Syracuse University (New York) to one of four locations (London, Strasbourg, Madrid, and Florence) were surveyed about their aspirations and attitudes regarding language, food and health, and adjustment/homesickness. Findings indicated: only three percent of respondents stated that learning the host language was the prime reason for studying abroad; just half of the respondents had a positive attitude toward language acquisition; 59 percent had positive expectations about the host country's food, while 27 percent had negative attitudes; 64 percent had positive expectations concerning matters of health; 80 percent felt positively about adjusting to a new lifestyle and a different academic setting; and homesickness was not expected to be a serious concern. Findings prompted a revision of a 1989 chart by Cornelius Grove which was designed to guide programmers planning an immediate post-arrival orientation in choosing the most pertinent topics. The chart shows when certain concerns become most acute, the complexity of the network of expectations, and the adjustment pattern. Implications for planning student exchange programs are discussed. (JDD)

In an effort to simplify the classification of pure entangled states of multi (m) -partite quantum systems, we study exactly and asymptotically (in n) reversible transformations among n'th tensor powers of such states (ie n copies of the state shared among the same m parties) under local quantum operations and classical communication (LOCC). With regard to exact transformations, we show that two states whose 1-party entropies agree are either locally-unitarily (LU) equivalent or else LOCC-incomparable. In particular we show that two tripartite Greenberger-Horne-Zeilinger (GHZ) states are LOCC-incomparable to three bipartite Einstein-Podolsky-Rosen (EPR) states symmetrically shared among the three parties. Asymptotic transformations result in a simpler classification than exact transformations. We show that m-partite pure states having an m-way Schmidt decomposition are simply parameterizable, with the partial entropy across any nontrivial partition representing the number of standard ``Cat'' states (|0^m>+|1^m>) asymptotically interconvertible to the state in question. For general m-partite states, partial entropies across different partitions need not be equal, and since partial entropies are conserved by asymptotically reversible LOCC operations, a multicomponent entanglement measure is needed, with each scalar component representing a different kind of entanglement, not asymptotically interconvertible to the other kinds. In particular the m=4 Cat state is not isentropic to, and therefore not asymptotically interconvertible to, any combination of bipartite and tripartite states shared among the four parties. Thus, although the m=4 cat state can be prepared from bipartite EPR states, the preparation process is necessarily irreversible, and remains so even asymptotically., Comment: 13 pages including 3 PostScript figures. v3 has updated references and discussion, to appear Phys. Rev. A

Quantum mechanics permits nonlocality - both nonlocal correlations and nonlocal equations of motion - while respecting relativistic causality. Is quantum mechanics the unique theory that reconciles nonlocality and causality? We consider two models, going beyond quantum mechanics, of nonlocality: "superquantum" correlations, and nonlocal "jamming" of correlations. These models are consistent with some definitions of nonlocality and causality., Comment: 9 pages, RevTeX; for the proceedings of the symposium on Causality and Locality in Modern Physics and Astronomy: Open Questions and Possible Solutions (York University, Toronto, August 25-29, 1997). This version contains an additional affiliation and address for one author

We point out formal correspondences between thermodynamics and entanglement. By applying them to previous work, we show that entropy of entanglement is the unique measure of entanglement for pure states., Comment: 8 pages, RevTeX; edited for clarity, additional references, to appear as a Rapid Communication in Phys. Rev. A