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Generalizing Bell nonlocality without global causal assumptions
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Abstract
Bell scenarios are multipartite scenarios that exclude any communication between parties. This constraint leads to a strict hierarchy of correlation sets in such scenarios, namely, classical, quantum, and nonsignaling. However, without any constraints on communication between the parties, they can realize arbitrary correlations by exchanging only classical systems. Here we consider a multipartite scenario where the parties can engage in at most a single round of communication, i.e., each party is allowed to receive a system once, implement any local intervention on it, and send out the resulting system once. While no global assumption about causal relations between parties is assumed in this scenario, we do make a causal assumption local to each party, i.e., the input received by it causally precedes the output it sends out. We then introduce antinomicity, a notion of nonclassicality for correlations in such scenarios, and prove the existence of a strict hierarchy of correlation sets classified by their antinomicity. Antinomicity serves as a generalization of Bell nonlocality: when all the parties discard their output systems (i.e., in a nonsignaling scenario), it is mathematically equivalent to Bell nonlocality. Like Bell nonlocality, it can be understood as an instance of fine-tuning, one that is necessary in any classical model of cyclic causation that avoids time-travel antinomies but allows antinomic correlations. Furthermore, antinomicity resolves a long-standing puzzle, i.e., the failure of causal inequality violations as device-independent witnesses of nonclassicality. Antinomicity implies causal inequality violations, but not conversely.
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Causal reasoning is essential to science, yet quantum theory challenges it. Quantum correlations violating Bell inequalities defy satisfactory causal explanations within the framework of classical causal models. What is more, a theory encompassing quantum systems and gravity is expected to allow causally nonseparable processes featuring operations in indefinite causal order, defying that events be causally ordered at all. The first challenge has been addressed through the recent development of intrinsically quantum causal models, allowing causal explanations of quantum processes – provided they admit a definite causal order, i.e. have an acyclic causal structure. This work addresses causally nonseparable processes and offers a causal perspective on them through extending quantum causal models to cyclic causal structures. Among other applications of the approach, it is shown that all unitarily extendible bipartite processes are causally separable and that for unitary processes, causal nonseparability and cyclicity of their causal structure are equivalent.
We develop an extension of the process matrix (PM) framework for correlations between quantum operations with no causal order that allows multiple rounds of information exchange for each party compatibly with the assumption of well-defined causal order of events locally. We characterise the higher-order process describing such correlations, which we name the multi-round process matrix (MPM), and formulate a notion of causal nonseparability for it that extends the one for standard PMs. We show that in the multi-round case there are novel manifestations of causal nonseparability that are not captured by a naive application of the standard PM formalism: we exhibit an instance of an operator that is both a valid PM and a valid MPM, but is causally separable in the first case and can violate causal inequalities in the second case due to the possibility of using a side channel.
I give a historical survey of the discussions about the existence of closed timelike curves in general relativistic models of the universe, opening the physical possibility of time travel in the past, as first recognized by K. Gödel in his rotating universe model of 1949. I emphasize that journeying into the past is intimately linked to spacetime models devoid of timelike singularities. Since such singularities arise as an inevitable consequence of the equations of general relativity given physically reasonable assumptions, time travel in the past becomes possible only when one or another of these assumptions is violated. It is the case with wormhole-type solutions. S. Hawking and other authors have tried to “save” the paradoxical consequences of time travel in the past by advocating physical mechanisms of chronological protection; however, such mechanisms remain presently unknown, even when quantum fluctuations near horizons are taken into account. I close the survey by a brief and pedestrian discussion of Causal Dynamical Triangulations, an approach to quantum gravity in which causality plays a seminal role.
We take a resource-theoretic approach to the problem of quantifying nonclassicality in Bell scenarios. The resources are conceptualized as probabilistic processes from the setting variables to the outcome variables having a particular causal structure, namely, one wherein the wings are only connected by a common cause. We term them "common-cause boxes". We define the distinction between classical and nonclassical resources in terms of whether or not a classical causal model can explain the correlations. One can then quantify the relative nonclassicality of resources by considering their interconvertibility relative to the set of operations that can be implemented using a classical common cause (which correspond to local operations and shared randomness). We prove that the set of free operations forms a polytope, which in turn allows us to derive an efficient algorithm for deciding whether one resource can be converted to another. We moreover define two distinct monotones with simple closed-form expressions in the two-party binary-setting binary-outcome scenario, and use these to reveal various properties of the pre-order of resources, including a lower bound on the cardinality of any complete set of monotones. In particular, we show that the information contained in the degrees of violation of facet-defining Bell inequalities is not sufficient for quantifying nonclassicality, even though it is sufficient for witnessing nonclassicality. Finally, we show that the continuous set of convexly extremal quantumly realizable correlations are all at the top of the pre-order of quantumly realizable correlations. In addition to providing new insights on Bell nonclassicality, our work also sets the stage for quantifying nonclassicality in more general causal networks.
When transforming pairs of independent quantum operations according to the fundamental rules of quantum theory, an intriguing phenomenon emerges: some such higher-order operations may act on the input operations in an indefinite causal order. Recently, the formalism of process matrices has been developed to investigate these noncausal properties of higher-order operations. This formalism predicts, in principle, statistics that ensure indefinite causal order even in a device-independent scenario, where the involved operations are not characterised. Nevertheless, all physical implementations of process matrices proposed so far require full characterisation of the involved operations in order to certify such phenomena. Here we consider a semi-device-independent scenario, which does not require all operations to be characterised. We introduce a framework for certifying noncausal properties of process matrices in this intermediate regime and use it to analyse the quantum switch, a well-known higher-order operation, to show that, although it can only lead to causal statistics in a device-independent scenario, it can exhibit noncausal properties in semi-device-independent scenarios. This proves that the quantum switch generates stronger noncausal correlations than it was previously known.
To study which are the most general causal structures which are compatible with local quantum mechanics, Oreshkov et al. introduced the notion of a process: a resource shared between some parties that allows for quantum communication between them without a predetermined causal order. These processes can be used to perform several tasks that are impossible in standard quantum mechanics: they allow for the violation of causal inequalities, and provide an advantage for computational and communication complexity. Nonetheless, no process that can be used to violate a causal inequality is known to be physically implementable. There is therefore considerable interest in determining which processes are physical and which are just mathematical artefacts of the framework. Here we make the first step in this direction, by proposing a purification postulate: processes are physical only if they are purifiable. We derive necessary conditions for a process to be purifiable, and show that several known processes do not satisfy them.
Reichenbach's principle asserts that if two observed variables are found to be correlated, then there should be a causal explanation of these correlations. Furthermore, if the explanation is in terms of a common cause, then the conditional probability distribution over the variables given the complete common cause should factorize. The principle is generalized by the formalism of causal models, in which the causal relationships among variables constrain the form of their joint probability distribution. In the quantum case, however, the observed correlations in Bell experiments cannot be explained in the manner Reichenbach's principle would seem to demand. Motivated by this, we introduce a quantum counterpart to the principle. We demonstrate that under the assumption that quantum dynamics is fundamentally unitary, if a quantum channel with input A and outputs B and C is compatible with A being a complete common cause of B and C, then it must factorize in a particular way. Finally, we show how to generalize our quantum version of Reichenbach's principle to a formalism for quantum causal models, and provide examples of how the formalism works.
Bell non-local correlations cannot be naturally explained in a fixed causal
structure. This serves as a motivation for considering models where no global
assumption is made beyond logical consistency. The assumption of a fixed causal
order between a set of parties, together with free randomness, implies
device-independent inequalities --- just as the assumption of locality does. It
is known that local validity of quantum theory is consistent with violating
such inequalities. Moreover, for three parties or more, even the (stronger)
assumption of local classical probability theory plus logical consistency
allows for violating causal inequalities. Here, we show that a classical
environment (with which the parties interact), possibly containing loops, is
logically consistent if and only if whatever the involved parties do, there is
exactly one fixed-point, the latter being representable as a mixture of
deterministic fixed-points. We further show that the non-causal view allows for
a model of computation strictly more powerful than computation in a world of
fixed causal orders.
In a scenario where two parties share, act on and exchange some physical
resource, the assumption that the parties' actions are ordered according to a
definite causal structure yields constraints on the possible correlations that
can be established. We show that the set of correlations that are compatible
with a definite causal order forms a polytope, whose facets define causal
inequalities. We fully characterize this causal polytope in the simplest case
of bipartite correlations with binary inputs and outputs. We find two families
of nonequivalent causal inequalities; both can be violated in the recently
introduced framework of process matrices, which extends the standard quantum
formalism by relaxing the implicit assumption of a fixed causal structure. Our
work paves the way to a more systematic investigation of causal inequalities in
a theory-independent way, and of their violation within the framework of
process matrices.
Classical correlations without predefined causal order arise from processes
where parties manipulate random variables, and where the order of these
interactions is not predefined. No assumption on the causal order of the
parties is made, but the processes are restricted to be logically consistent
under any choice of the parties' operations. It is known that for three parties
or more, this set of processes is larger than the set of processes achievable
in a predefined ordering of the parties. Here, we model all classical processes
without predefined causal order geometrically and find that the set of such
processes forms a polytope. Additionally, we model a smaller polytope --- the
deterministic-extrema polytope --- where all extremal points represent
deterministic processes. This polytope excludes probabilistic processes that
must be --- quite unnaturally --- fine-tuned, because any variation of the
weights in a decomposition into deterministic processes leads to a logical
inconsistency.
Many of the heated arguments about the meaning of "Bell's theorem" arise
because this phrase can refer to two different theorems that John Bell proved,
the first in 1964 and the second in 1976. His 1964 theorem is the
incompatibility of quantum phenomena with the dual assumptions of locality and
determinism. His 1976 theorem is the incompatibility of quantum phenomena with
the unitary property of local causality. This is contrary to Bell's own later
assertions, that his 1964 theorem began with that single, and indivisible,
assumption of local causality (even if not by that name). While there are other
forms of Bell's theorems --- which I present to explain the relation between
Jarrett-completeness, "weak locality", and EPR-completeness --- I maintain that
Bell's two versions are the essential ones. Although the two Bell's theorems
are logically equivalent, their assumptions are not, and the different versions
of the theorem suggest quite different conclusions, which are embraced by
different communities. For realists, the notion of local causality, ruled out
by Bell's 1976 theorem, is motivated implicitly by Reichenbach's Principle of
common cause and explicitly by Einstein's relativity Principle, and it is the
latter which must be forgone. Operationalists pay no heed to Reichenbach's
Principle, but wish to keep Einstein's relativity Principle, which, bolstered
by an implicit "Principle of causal efficacy", implies their notion of
locality. Thus for operationalists, Bell's theorem is the 1964 one, and implies
that it is determinism that must be forgone. I discuss why the two `camps' are
drawn to these different conclusions, and what can be done to increase mutual
understanding.
The idea that events obey a definite causal order is deeply rooted in our understanding of the world and at the basis of the very notion of time. But where does causal order come from, and is it a necessary property of nature? Here, we address these questions from the standpoint of quantum mechanics in a new framework for multipartite correlations that does not assume a pre-defined global causal structure but only the validity of quantum mechanics locally. All known situations that respect causal order, including space-like and time-like separated experiments, are captured by this framework in a unified way. Surprisingly, we find correlations that cannot be understood in terms of definite causal order. These correlations violate a 'causal inequality' that is satisfied by all space-like and time-like correlations. We further show that in a classical limit causal order always arises, which suggests that space-time may emerge from a more fundamental structure in a quantum-to-classical transition.
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. 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.
We challenge the view that there is a basic conflict between the fundamental principles of Quantum Theory and General Relativity and, in particular, the fact that a superposition of massive bodies would lead to a violation of the Equivalence Principle. It has been argued that this violation implies that such a superposition must inevitably spontaneously collapse (like in the Diósi–Penrose model). We identify the origin of such an assertion in the impossibility of finding a local and classical reference frame in which Einstein's Equivalence Principle would hold. In contrast, we argue that the formulation of the Equivalence Principle can be generalized so that it holds for reference frames that are associated with quantum systems in a superposition of spacetimes. The core of this new formulation is the introduction of a quantum diffeomorphism to such Quantum Reference Frames. This procedure reconciles the principle of linear superposition in Quantum Theory with the principle of general covariance and the Equivalence Principle of General Relativity. Hence, it is not necessary to invoke a gravity-induced spontaneous state reduction when a massive body is prepared in a spatial superposition.
Causal loops are loops in cause-effect relations, where, say for two events A , B , the event A is a cause of B and, vice versa, B is a cause of A . Such loops are traditionally ruled out due to potential logical problems, e. g. , where an effect suppresses its own cause. Motivated by our current physical theories, we show that not only causal loops exist that are logically consistent, but that these loops are computationally tame and help to further investigate on the theoretical foundations of time travel. Causal loops do not necessarily pose problems from a logics, computer-science, and physics point of view. This opens their potential applicability in various fields from philosophy of language to computer science and physics.
Bell's 1964 theorem, which states that the predictions of quantum theory
cannot be accounted for by any local theory, represents one of the most
profound developments in the foundations of physics. In the last two decades,
Bell's theorem has been a central theme of research from a variety of
perspectives, mainly motivated by quantum information science, where the
nonlocality of quantum theory underpins many of the advantages afforded by a
quantum processing of information. The focus of this review is to a large
extent oriented by these later developments. We review the main concepts and
tools which have been developed to describe and study the nonlocality of
quantum theory, and which have raised this topic to the status of a full
sub-field of quantum information science.
In the conventional approach to quantum mechanics, indeterminism is an axiom and nonlocality is a theorem. We consider inverting the logical order, making nonlocality an axiom and indeterminism a theorem. Nonlocal superquantum correlations, preserving relativistic causality, can violate the CHSH inequality more strongly than any quantum correlations.
I introduce a framework in which a variety of probabilistic theories can be defined, including classical and quantum theories, and many others. From two simple assumptions, a tensor product rule for combining separate systems can be derived. Certain features, usually thought of as specifically quantum, turn out to be generic in this framework, meaning that they are present in all except classical theories. These include the non-unique decomposition of a mixed state into pure states, a theorem involving disturbance of a system on measurement (suggesting that the possibility of secure key distribution is generic), and a no-cloning theorem. Two particular theories are then investigated in detail, for the sake of comparison with the classical and quantum cases. One of these includes states that can give rise to arbitrary non-signalling correlations, including the super-quantum correlations that have become known in the literature as Nonlocal Machines or Popescu-Rohrlich boxes. By investigating these correlations in the context of a theory with well-defined dynamics, I hope to make further progress with a question raised by Popescu and Rohrlich, which is, why does quantum theory not allow these strongly nonlocal correlations? The existence of such correlations forces much of the dynamics in this theory to be, in a certain sense, classical, with consequences for teleportation, cryptography and computation. I also investigate another theory in which all states are local. Finally, I raise the question of what further axiom(s) could be added to the framework in order uniquely to identify quantum theory, and hypothesize that quantum theory is optimal for computation.
This article identifies a series of properties common to all theories that do not allow for superluminal signaling and predict the violation of Bell inequalities. Intrinsic randomness, uncertainty due to the incompatibility of two observables, monogamy of correlations, impossibility of perfect cloning, privacy of correlations, bounds in the shareability of some states; all these phenomena are solely a consequence of the no-signaling principle and nonlocality. In particular, it is shown that for any distribution, the properties of (i) nonlocal, (ii) no arbitrarily shareable and (iii) positive secrecy content are equivalent. Comment: 10 pages
Bell's Theorem assumes that hidden variables are not influenced by future measurement settings. The assumption has sometimes been questioned, but the suggestion has been thought outlandish, even by the taxed standards of the discipline. (Bell thought that it led to fatalism.) The case for this reaction turns out to be surprisingly weak, however. We show that QM easily evades the standard objections to advanced action. And the approach has striking advantages, especially in avoiding the apparent conflict between Bell's Theorem and special relativity. The second part of the paper considers the broader question as to why advanced action seems so counterintuitive. We investigate the origins of our ordinary intuitions about causal asymmetry. It is argued that the view that the past does not depend on the future is largely anthropocentric, a kind of projection of our own temporal asymmetry. Many physicists have also reached this conclusion, but have thought that if causation has no objective direction, there is no objective content to an advanced action interpretation of QM. This turns out to be a mistake. From the ordinary subjective perspective, we can distinguish two sorts of objective world: one "looks as if" it contains only forward causation, the other ``looks as if'' it involves a mix of backward and forward causation. This clarifies the objective core of an advanced action interpretation of QM, and shows that there is an independent symmetry argument in favour of the approach. Comment: 35 pages, LaTex (forthcoming in MIND, July 1994; written for a philosophical audience, but perhaps of some interest here)
General relativity is a deterministic theory with non-fixed causal structure. Quantum theory is a probabilistic theory with fixed causal structure. In this paper we build a framework for probabilistic theories with non-fixed causal structure. This combines the radical elements of general relativity and quantum theory. The key idea in the construction is physical compression. A physical theory relates quantities. Thus, if we specify a sufficiently large set of quantities (this is the compressed set), we can calculate all the others. We apply three levels of physical compression. First, we apply it locally to quantities (actually probabilities) that might be measured in a particular region of spacetime. Then we consider composite regions. We find that there is a second level of physical compression for the composite region over and above the first level physical compression for the component regions. Each application of first and second level physical compression is quantified by a matrix. We find that these matrices themselves are related by the physical theory and can therefore be subject to compression. This is the third level of physical compression. This third level of physical compression gives rise to a new mathematical object which we call the causaloid. From the causaloid for a particular physical theory we can calculate verything the physical theory can calculate. This approach allows us to set up a framework for calculating probabilistic correlations in data without imposing a fixed causal structure (such as a background time). We show how to put quantum theory in this framework (thus providing a new formulation of this theory). We indicate how general relativity might be put into this framework and how the framework might be used to construct a theory of quantum gravity.
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