Quantifying Bell: the Resource Theory of Nonclassicality of Common-Cause Boxes (original) (raw)
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Causal Networks and Freedom of Choice in Bell’s Theorem
PRX quantum, 2021
Bell's theorem is typically understood as the proof that quantum theory is incompatible with localhidden-variable models. More generally, we can see the violation of a Bell inequality as witnessing the impossibility of explaining quantum correlations with classical causal models. The violation of a Bell inequality, however, does not exclude classical models where some level of measurement dependence is allowed, that is, the choice made by observers can be correlated with the source generating the systems to be measured. Here, we show that the level of measurement dependence can be quantitatively upper bounded if we arrange the Bell test within a network. Furthermore, we also prove that these results can be adapted in order to derive nonlinear Bell inequalities for a large class of causal networks and to identify quantumly realizable correlations that violate them.
Causation, decision theory, and Bell’s theorem: a quantum analogue of the Newcomb problem
2009
I apply some of the lessons from quantum theory, in particular from Bell’s theorem, to a debate on the foundations of decision theory and causation. By tracing a formal analogy between the basic assumptions of Causal Decision Theory (CDT)—which was developed partly in response to New-comb’s problem — and those of a Local Hidden Variable (LHV) theory in the context of quantum mechanics, I show that an agent who acts according to CDT and gives any nonzero credence to some possible causal interpretations underlying quantum phenomena should bet against quantum mechanics in some feasible game scenarios involving entangled systems, no matter what evidence they acquire. As a consequence, either the most accepted version of decision theory is wrong, or it provides a practical distinction, in terms of the prescribed behaviour of rational agents, between
Bell's Inequalities — Foundations and Quantum Communication
Handbook of Natural Computing, 2012
For individual events quantum mechanics makes only probabilistic predictions. Can one go beyond quantum mechanics in this respect? This question has been a subject of debate and research since the early days of the theory. Efforts to construct deeper, realistic, level of physical description, in which individual systems have, like in classical physics, preexisting properties revealed by measurements are known as hidden-variable programs. Demonstrations that a hiddenvariable program necessarily requires outcomes of certain experiments to disagree with the predictions of quantum theory are called "no-go theorems". The Bell theorem excludes local hidden variable theories. The Kochen-Specker theorem excludes noncontextual hidden variable theories. In local hidden-variable theories faster-thatlight-influences are forbidden, thus the results for a given measurement (actual, or just potentially possible) are independent of the settings of other measurement devices which are at space-like separation. In noncontextual hidden-variable theories the predetermined results of a (degenerate) observable are independent of any other observables that are measured jointly with it. It is a fundamental doctrine of quantum information science that quantum communication and quantum computation outperforms their classical counterparts. If this is to be true, some fundamental quantum characteristics must be behind betterthan-classical performance of information processing tasks. This chapter aims at establishing connections between certain quantum information protocols and foundational issues in quantum theory. After a brief discusion of the most common misinterpretations of Bell's theorem and a discussion of what its real me aning is, iť
New Journal of Physics, 2015
An active area of research in the fields of machine learning and statistics is the development of causal discovery algorithms, the purpose of which is to infer the causal relations that hold among a set of variables from the correlations that these exhibit. We apply some of these algorithms to the correlations that arise for entangled quantum systems. We show that they cannot distinguish correlations that satisfy Bell inequalities from correlations that violate Bell inequalities, and consequently that they cannot do justice to the challenges of explaining certain quantum correlations causally. Nonetheless, by adapting the conceptual tools of causal inference, we can show that any attempt to provide a causal explanation of nonsignalling correlations that violate a Bell inequality must contradict a core principle of these algorithms, namely, that an observed statistical independence between variables should not be explained by fine-tuning of the causal parameters. In particular, we demonstrate the need for such fine-tuning for most of the causal mechanisms that have been proposed to underlie Bell correlations, including superluminal causal influences, superdeterminism (that is, a denial of freedom of choice of settings), and retrocausal influences which do not introduce causal cycles.
The collapse of Bell determinism
The Bell-Kochen-Specker conditions (BKS) for a deterministic noncontextual hiddenvariable model are wonderfully simple to state, deal with just one-dimensional projectors on a Hilbert space H and make no reference to a probabilistic phase space or quantum system. They only ask for an assignment of zero or one to every projector such that the assignment respects orthogonal resolutions of the identity. Various nogo results in the literature show that the pair of statements {BKS is valid; dim H ≥ 3} are inconsistent. Here we show, more radically, that the pair actually contradicts the dimensionality of the space itself, by implying that there can exist at most a single onedimensional projector acting on H. Our derivation involves only elementary inner product spaces. It is non-probabilistic, inequality-free, state independent, does not use entanglement, and is simultaneously valid in all dimensions three or greater. of the BKS conditions, that is, in exploring different kinds of deterministic models, or indeed, exploration of contextual models or suitable hybrids.
Unbounded Violation of Tripartite Bell Inequalities
Communications in Mathematical Physics, 2008
We prove that there are tripartite quantum states (constructed from random unitaries) that can lead to arbitrarily large violations of Bell inequalities for dichotomic observables. As a consequence these states can withstand an arbitrary amount of white noise before they admit a description within a local hidden variable model. This is in sharp contrast with the bipartite case, where all violations are bounded by Grothendieck's constant. We will discuss the possibility of determining the Hilbert space dimension from the obtained violation and comment on implications for communication complexity theory. Moreover, we show that the violation obtained from generalized GHZ states is always bounded so that, in contrast to many other contexts, GHZ states do in this case not lead to extremal quantum correlations. In order to derive all these physical consequences, we will have to obtain new mathematical results in the theories of operator spaces and tensor norms. In particular, we will prove the existence of bounded but not completely bounded trilinear forms from commutative C*-algebras. of Bell inequalities and the mathematical theories. In Sec. IV we will prove that the violation remains bounded for GHZ states. Finally, Sec. V provides the proof for the main theorem. II. MAIN RESULT AND IMPLICATIONS We begin by specifying the framework. For the convenience of the non-specialist reader we will give first a brief introduction to Bell Inequalities. For further information we refer the reader to [72]. Bell inequalities can be dated back to the famous critic of Quantum Mechanics due to Einstein, Podolski and Rosen [27]. This critic was made under their believe that on a fundamental level Nature was described by a local hidden variable (LHV) model, i.e., that it is classical (realistic or deterministic) and local (or non-signaling). The latter essentially means that no information can travel faster than a maximal speed (e.g. of light) which implies in particular that the probability distribution for the outcomes of some experiment made by Alice cannot depend on what other (spatially separated) physicist Bob does in his lab. Otherwise, by choosing one or the other experiment, Bob could influence instantly Alice's results and hence transmit information at any speed. On the other hand, saying that Nature is classical or deterministic means that the randomness in the outcomes that is observed in the experiments comes from our ignorance of Nature, instead of being an intrinsic property of it (as Quantum Mechanics postulates). That is, Nature can stay in different configurations s with some probability p(s) (s is usually called a hidden variable). But once it is in a fixed configuration s, then any experiment has deterministic outputs. We note that there are non-deterministic LHV models as well, but they can all be cast into deterministic models [72]. Let us formalize this a bit more. Consider correlation experiments where each of N spatially separated observers (Alice, Bob, Charlie,.. .) can measure M different observables with outcomes ±1: {A i1 } M i1=1 for Alice, {B i2 } M i2=1 for Bob and so on. By repeating the experiment several times, for each possible configuration of the observables (Alice measuring with the aparatus A i1 , Bob with the aparatus B i2 ,. . .), they can obtain a good approximation of the expected value of the product of the outcomes of such configuration A i1 B i2 C i3 • • •. If Nature is described by a LHV model, then A i1 B i2 C i3 • • • = A i1 B i2 C i3 • • • p = s p(s)A i1 (s)B i2 (s) • • • , (1) where A i1 (s) = ±1 is the deterministic outcome obtained by Alice if she does the experiment A i1 and Nature is in state s (notice that we are including also the locality condition when assuming that A i1 (s) is independent of i 2 , i 3 ,. . .). For a quantum mechanical system in a state ρ we have to set A i1 B i2 C i3 • • • = A i1 B i2 C i3 • • • ρ = tr(ρA i1 ⊗ B i2 ⊗ C i3 • • •) (2) where ρ is a density operator acting on a Hilbert space C d1 ⊗ • • • ⊗ C d N and the observables satisfy −1 ≤ A i1 , B i2 , C i3 ,. .. ≤ 1, describing measurements within the framework of positive operator valued measures (POVMs). Note the parallelism with (1). In fact the quantum mechanical expression coincides with the classical one if the matrices A i1 's, B j2 's,. .. commute with each other (and therefore can be taken diagonal in some basis |s), and we take the state ρ to be the separable state given by ρ = s p(s)|s s| ⊗ |s s| ⊗ • • •. How can one then know if Nature allows for a LHV description or follows Quantum Mechanics? That is, how to discriminate between (1) and (2)? The key idea of Bell [10] was to realize that this can be done by taking linear combinations of the expectation values A i1 B i2 C i3 • • •. So, given real coefficients T i1i2,... , if we maximize the expression M −1 i1,••• ,i N =0 T i1•••i N A i1 B i2 C i3 • • • (3) assuming (1) we get [79] T := sup ai 1 ,bi 2 ,ci 3 ,...=±1 i∈Z N M T i1•••i N a i1 b i2 c i3 • • • .
Quantum non-classicality in the simplest causal network
arXiv (Cornell University), 2024
Bell's theorem prompts us with a fundamental inquiry: what is the simplest scenario leading to the incompatibility between quantum correlations and the classical theory of causality? Here we demonstrate that quantum non-classicality is possible in a network consisting of only three dichotomic variables, without the need of the locality assumption neither external measurement choices. We also show that the use of interventions, a central tool in the field of causal inference, significantly improves the noise robustness of this new kind of non-classical behaviour, making it feasible for experimental tests with current technology.
On the logical structure of Bell theorems
New Journal of Physics, 2006
Bell theorems show how to experimentally falsify local realism. Conclusive falsification is highly desirable as it would provide support for the most profoundly counterintuitive feature of quantum theory-nonlocality. Despite the preponderance of evidence for quantum mechanics, practical limits on detector efficiency and the difficulty of coordinating space-like separated measurements have provided loopholes for a classical worldview; these loopholes have never been simultaneously closed. A number of new experiments have recently been proposed to close both loopholes at once. We show some of these novel designs fail in the most basic way, by not ruling out local hidden variable models, and we provide an explicit classical model to demonstrate this. They share a common flaw, which reveals a basic misunderstanding of how nonlocality proofs work. Given the time and resources now being devoted to such experiments, theoretical clarity is essential. Our explanation is presented in terms of simple logic and should serve to correct misconceptions and avoid future mistakes. We also show a nonlocality proof involving four participants which has interesting theoretical properties.