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Journal ArticleDOI

Operational formulation of time reversal in quantum theory

01 Oct 2015-Nature Physics (Nature Research)-Vol. 11, Iss: 10, pp 853-858
TL;DR: The symmetry of quantum theory under time reversal has long been a subject of controversy because the transition probabilities given by Born's rule do not apply backward in time as mentioned in this paper, and it has been argued that reconciling time reversal with the probabilistic rules of the theory requires a notion of operation that permits realizations through both pre- and post-selection.
Abstract: The symmetry of quantum theory under time reversal has long been a subject of controversy because the transition probabilities given by Born’s rule do not apply backward in time. Here, we resolve this problem within a rigorous operational probabilistic framework. We argue that reconciling time reversal with the probabilistic rules of the theory requires a notion of operation that permits realizations through both pre- and post-selection. We develop the generalized formulation of quantum theory that stems from this approach and give a precise definition of time-reversal symmetry, emphasizing a previously overlooked distinction between states and effects. We prove an analogue of Wigner’s theorem, which characterizes all allowed symmetry transformations in this operationally time-symmetric quantum theory. Remarkably, we find larger classes of symmetry transformations than previously assumed, suggesting a possible direction in the search for extensions of known physics. A reformulation of quantum theory aims at reconciling transition probabilities with time reversal in connection to Wigner’s notion of symmetry, expanding the known classes of symmetry transformations.
Citations
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DOI
26 Jul 2010
TL;DR: In this article, the candidates are asked to write their CANDIDATE NUMBER clearly on each of the three answer books provided and then enter the number of each question attempted in the horizontal box on the front cover of its corresponding answer book.
Abstract: General Instructions Write your CANDIDATE NUMBER clearly on each of the THREE answer books provided. If an electronic calculator is used, write its serial number in the box at the top right hand corner of the front cover of each answer book. USE ONE ANSWER BOOK FOR EACH QUESTION. Enter the number of each question attempted in the horizontal box on the front cover of its corresponding answer book. Hand in THREE answer books even if they have not all been used. You are reminded that the Examiners attach great importance to legibility, accuracy and clarity of expression.

294 citations

Journal ArticleDOI
TL;DR: In this paper, a quantum version of Reichenbach's principle is proposed for quantum causal models, in which the causal relationships among variables constrain the form of their joint probability distribution.
Abstract: 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.

206 citations

Journal ArticleDOI
TL;DR: In this paper, a theory of causality compatible with a dynamical causal order has been developed in the tripartite case of quantum correlations in which the causal order of a set of events could be random.
Abstract: The idea that events are equipped with a partial causal order is central to our understanding of physics in the tested regimes: given two pointlike events A and B, either A is in the causal past of B, B is in the causal past of A, or A and B are space-like separated Operationally, the meaning of these order relations corresponds to constraints on the possible correlations between experiments performed in the vicinities of the respective events: if A is in the causal past of B, an experimenter at A could signal to an experimenter at B but not the other way around, while if A and B are space-like separated, no signaling is possible in either direction In the context of a concrete physical theory, the correlations compatible with a given causal configuration may obey further constraints For instance, space-like correlations in quantum mechanics arise from local measurements on joint quantum states, while time-like correlations are established via quantum channels Similarly to other variables, however, the causal order of a set of events could be random, and little is understood about the constraints that causality implies in this case A main difficulty concerns the fact that the order of events can now generally depend on the operations performed at the locations of these events, since, for instance, an operation at A could influence the order in which B and C occur in A's future So far, no formal theory of causality compatible with such dynamical causal order has been developed Apart from being of fundamental interest in the context of inferring causal relations, such a theory is imperative for understanding recent suggestions that the causal order of events in quantum mechanics can be indefinite Here, we develop such a theory in the general multipartite case Starting from a background-independent definition of causality, we derive an iteratively formulated canonical decomposition of multipartite causal correlations For a fixed number of settings and outcomes for each party, these correlations form a polytope whose facets define causal inequalities The case of quantum correlations in this paradigm is captured by the process matrix formalism We investigate the link between causality and the closely related notion of causal separability of quantum processes, which we here define rigorously in analogy with the link between Bell locality and separability of quantum states We show that causality and causal separability are not equivalent in general by giving an example of a physically admissible tripartite quantum process that is causal but not causally separable We also show that there are causally separable quantum processes that become non-causal if extended by supplying the parties with entangled ancillas This motivates the concepts of extensibly causal and extensibly causally separable (ECS) processes, for which the respective property remains invariant under extension We characterize the class of ECS quantum processes in the tripartite case via simple conditions on the form of the process matrix We show that the processes realizable by classically controlled quantum circuits are ECS and conjecture that the reverse also holds

143 citations


Cites background from "Operational formulation of time rev..."

  • ...[15, 16], it was argued that there are two main ideas that underlie the concept of operation in the standard circuit framework for operational probabilistic theories [25–28]....

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Journal ArticleDOI
TL;DR: In this paper, Oreshkov, Costa, Brukner, and O'Neill developed rigorous notions of causality and causal separability in the process framework, which describes correlations between separate local experiments without a prior assumption of causal order between them.
Abstract: We develop rigorous notions of causality and causal separability in the process framework introduced in [Oreshkov, Costa, Brukner, Nat. Commun. 3, 1092 (2012)], which describes correlations between separate local experiments without a prior assumption of causal order between them. We consider the general multipartite case and take into account the possibility for dynamical causal order, where the order of a set of events can depend on other events in the past. Starting from a general definition of causality, we derive an iteratively formulated canonical decomposition of multipartite causal processes, and show that for a fixed number of settings and outcomes for each party, the respective correlations form a polytope whose facets define causal inequalities. In the case of quantum processes, we investigate the link between causality and the theory-dependent notion of causal separability, which we here extend to the multipartite case based on concrete principles. We show that causality and causal separability are not equivalent in general by giving an example of a physically admissible tripartite quantum process that is causal but not causally separable. We also show that there exist causally separable (and hence causal) quantum processes that become non-causal if extended by supplying the parties with entangled ancillas. This example of activation of non-causality motivates the concepts of extensibly causal and extensibly causally separable (ECS) processes, for which the respective property remains invariant under extension with arbitrary ancillas. We characterize the class of tripartite ECS processes in terms of simple conditions on the form of the process matrix, which generalize the form of bipartite causally separable process matrices. We show that the processes realizable by classically controlled quantum circuits are ECS and conjecture that the reverse also holds.

106 citations

Journal ArticleDOI
TL;DR: In this article, a generalized formulation of quantum theory without predefined time or causal structure is proposed, which is compatible with indefinite causal order, timelike loops, and other acausal structures.
Abstract: The standard formulation of quantum theory assumes a predefined notion of time. This is a major obstacle in the search for a quantum theory of gravity, where the causal structure of space-time is expected to be dynamical and fundamentally probabilistic in character. Here, we propose a generalized formulation of quantum theory without predefined time or causal structure, building upon a recently introduced operationally time-symmetric approach to quantum theory. The key idea is a novel isomorphism between transformations and states which depends on the symmetry transformation of time reversal. This allows us to express the time-symmetric formulation in a time-neutral form with a clear physical interpretation, and ultimately drop the assumption of time. In the resultant generalized formulation, operations are associated with regions that can be connected in networks with no directionality assumed for the connections, generalizing the standard circuit framework and the process matrix framework for operations without global causal order. The possible events in a given region are described by positive semidefinite operators on a Hilbert space at the boundary, while the connections between regions are described by entangled states that encode a nontrivial symmetry and could be tested in principle. We discuss how the causal structure of space-time could be understood as emergent from properties of the operators on the boundaries of compact space-time regions. The framework is compatible with indefinite causal order, timelike loops, and other acausal structures.

72 citations

References
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Journal ArticleDOI
TL;DR: Koestler's book The Sleepwalkers as discussed by the authors is an account of the Copernican revolution, with Copernicus, Kepler, and Galilei as heroes, and he concluded that they were not really aware of what they were doing.
Abstract: ‘… the history of cosmic theories may without exaggeration be called a history of collective obsessions and controlled schizophrenias; and the manner in which some of the most important individual discoveries were arrived at reminds one of a sleepwalker's performance …’ This is a quotation from A. Koestler's book The Sleepwalkers . It is an account of the Copernican revolution, with Copernicus, Kepler, and Galilei as heroes. Koestler was of course impressed by the magnitude of the step made by these men. He was also fascinated by the manner in which they made it. He saw them as motivated by irrational prejudice, obstinately adhered to, making mistakes which they did not discover, which somehow cancelled at the important points, and unable to recognize what was important in their results, among the mass of details. He concluded that they were not really aware of what they were doing … sleepwalkers. I thought it would be interesting to keep Koestler's thesis in mind as we hear at this meeting about contemporary theories from contemporary theorists. For many decades now our fundamental theories have rested on the two great pillars to which this meeting is dedicated: quantum theory and relativity. We will see that the lines of research opened up by these theories remain splendidly vital. We will see that order is brought into a vast and expanding array of experimental data. We will see even a continuing ability to get ahead of the experimental data … as with the existence and masses of the W and Z mesons.

3,750 citations

Journal ArticleDOI
TL;DR: The superposition principle of the wave function is defined in this article, which is the fundamental principle of quantum mechanics that the system of states forms a linear manifold, in which a unitary scalar product is defined.
Abstract: It is perhaps the most fundamental principle of Quantum Mechanics that the system of states forms a linear manifold,1 in which a unitary scalar product is defined.2 The states are generally represented by wave functions3 in such a way that φ and constant multiples of φ represent the same physical state. It is possible, therefore, to normalize the wave function, i.e., to multiply it by a constant factor such that its scalar product with itself becomes 1. Then, only a constant factor of modulus 1, the so-called phase, will be left undetermined in the wave function. The linear character of the wave function is called the superposition principle. The square of the modulus of the unitary scalar product (ψ,Φ) of two normalized wave functions ψ and Φ is called the transition probability from the state ψ into Φ, or conversely. This is supposed to give the probability that an experiment performed on a system in the state Φ, to see whether or not the state is ψ, gives the result that it is ψ. If there are two or more different experiments to decide this (e.g., essentially the same experiment, performed at different times) they are all supposed to give the same result, i.e., the transition probability has an invariant physical sense.

2,694 citations

Book
01 Jan 1968

1,949 citations

Journal ArticleDOI
TL;DR: In this article, the usual formula for transition probabilities in nonrelativistic quantum mechanics is generalized to yield conditional probabilities for selected sequences of events at several different times, called consistent histories, through a criterion which ensures that, within limits which are explicitly defined within the formalism, classical rules for probabilities are satisfied.
Abstract: The usual formula for transition probabilities in nonrelativistic quantum mechanics is generalized to yield conditional probabilities for selected sequences of events at several different times, called “consistent histories,” through a criterion which ensures that, within limits which are explicitly defined within the formalism, classical rules for probabilities are satisfied The interpretive scheme which results is applicable to closed (isolated) quantum systems, is explicitly independent of the sense of time (ie, past and future can be interchanged), has no need for wave function “collapse,” makes no reference to processes of measurement (though it can be used to analyze such processes), and can be applied to sequences of microscopic or macroscopic events, or both, as long as the mathematical condition of consistency is satisfied When applied to appropriate macroscopic events it appears to yield the same answers as other interpretative schemes for standard quantum mechanics, though from a different point of view which avoids the conceptual difficulties which are sometimes thought to require reference to conscious observers or classical apparatus

922 citations