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The standard model of quantum field theory, successful as it is, does not yet incorporate gravitation. The attempt to develop a theory that does justice both the quantum phenomena and to gravitational phenomena gives rise to serious conceptual issues see the entry on quantum gravity. For our purposes, the most important features of this equation is that it is deterministic and linear. The state vector at any time, together with the equation, uniquely determines the state vector at any other time. Textbook formulations of quantum mechanics usually include an additional postulate about how to assign a state vector after an experiment.

Thus after the first measurement has been made, there is no indeterminacy in the result of the second. This conclusion must still hold if the second measurement is not actually made. In this way we see that a measurement always causes the system to jump into an eigenstate of the dynamical variable that is being measured, the eigenvalue this eigenstate belongs to being equal to the result of the measurement Dirac Neither von Neumann nor Dirac, however, seem to think of it this way; it is treated by both as a physical process.

Though, in his extended discussion of the measurement process, von Neumann , , Ch.

## Quantum theory cannot consistently describe the use of itself

VI does discuss the act of observation, he emphasizes that the collapse postulate may be applied to interactions with quantum systems with measuring apparatus, before an observer is aware of the result. A formulation of a version of the collapse postulate according to which a measurement is not completed until the result is observed is found in London and Bauer They deny, however, that it represents a mysterious kind of interaction between the observer and the quantum system; for them, the replacement of the pre-observation state vector with a new one is a matter of the observer acquiring new information.

These two interpretations of the collapse postulate, as either a real change of the physical state of the system, or as a mere updating of information on the part of an observer, have persisted in the literature. As John S. In addition to the product states, the tensor product space contains linear combinations of product states, that is, state vectors of the form. The tensor product space can be defined as the smallest Hilbert space containing all of the product states. Any pure state represented by a state vector that is not a product vector is an entangled state. The state of the composite system assigns probabilities to outcomes of all experiments that can be performed on the composite system.

In general, any state, pure or mixed, that is neither a product state nor a mixture of product states, is called an entangled state. The existence of pure entangled states means that, if we consider a composite system consisting of spatially separated parts, then, even when the state of the system is a pure state, the state is not determined by the reduced states of its component parts.

Thus, quantum states exhibit a form of nonseparability. See the entry on holism and nonseparability in physics for more information. Quantum entanglement results in a form of nonlocality that is alien to classical physics. If quantum theory is meant to be in principle a universal theory, it should be applicable, in principle, to all physical systems, including systems as large and complicated as our experimental apparatus.

Consider, now, a schematized experiment. This is not an eigenstate of the instrument reading variable, but is, rather, a state in which the reading variable and the system variable are entangled with each other. The eigenstate-eigenvalue link, applied to a state like this, does not yield a definite result for the instrument reading.

It is sometimes said that this conflicts with our experience, according to which experimental outcome variables, such as pointer readings, always have definite values. Nonetheless, we are faced with an interpretational problem. If we take the quantum state to be a complete description of the system, then the state is, contrary to what would antecedently expect, not a state corresponding to a unique, definite outcome. This is what led J. This gives us a prima facie tidy way of classifying approaches to the measurement problem:. We include in the first category approaches that deny that a quantum state should be thought of as representing anything in reality at all.

These include variants of the Copenhagen interpretation, as well as pragmatic and other anti-realist approaches. Also in the first category are approaches that seek a completion of the quantum state description. These include hidden-variables approaches and modal interpretations. The second category of interpretation motivates a research programme of finding suitable indeterministic modifications of the quantum dynamics. From the early days of quantum mechanics, there has been a strain of thought that holds that the proper attitude to take towards quantum mechanics is an instrumentalist or pragmatic one.

On such a view, quantum mechanics is a tool for coordinating our experience and for forming expectations about the outcomes of experiments. Variants of this view include what has been called the Copenhagen Interpretation or Copenhagen Interpretations, as recent scholarship has emphasized differences between figures associated with this view ; see the entry on Copenhagen interpretation of quantum mechanics. More recently, views of this sort have been advocated by physicists, including QBists, who hold that quantum states represent subjective or epistemic probabilities see Fuchs et al.

The philosopher Richard Healey defends a related view on which quantum states, though objective, do not represent physical reality see Healey ; Healey forthcoming. That a quantum state description cannot be regarded as a complete description of physical reality was argued for in a famous paper by Einstein, Podolsky and Rosen EPR and by Einstein in subsequent publications Einstein , , See the entry on the Einstein-Podolsky-Rosen argument in quantum theory.

There are a number of theorems that circumscribe the scope of possible hidden-variables theories. Theories of this sort are called noncontextual hidden-variables theory. It was shown by Bell and Kochen and Specker that there are no such theories for any system whose Hilbert space dimension is greater than three see the entry on the Kochen-Specker theorem. The Bell-Kochen-Specker Theorem does not rule out hidden-variables theories tout court. The most thoroughly worked-out theory of this type is the pilot wave theory developed by de Broglie and presented by him at the Fifth Solvay Conference held in Brussels in , revived by David Bohm in , and currently an active area of research by a small group of physicists and philosophers.

According to this theory, there are particles with definite trajectories, that are guided by the quantum wave function. For the history of the de Broglie theory, see the introductory chapters of Bacciagaluppi and Valentini For any overview of the de Broglie-Bohm theory and philosophical issues associated with it see the entry on Bohmian mechanics.

There have been other proposals for supplementing the quantum state with additional structure; these have come to be called modal interpretations ; see the entry on modal interpretations of quantum mechanics. As already mentioned, von Neumann and Dirac wrote as if the collapse of the quantum state vector precipitated by an experimental intervention on the system is a genuine physical change, distinct from the usual unitary evolution. If collapse is to be taken as a genuine physical process, then something more needs to be said about the circumstances under which it occurs than merely that it happens when an experiment is performed.

The only promising collapse theories are stochastic in nature; indeed, it can be shown that a deterministic collapse theory would permit superluminal signalling. See Allori et al. The resulting interpretation he called the relative state interpretation. The basic idea is this. After an experiment, the quantum state of the system plus apparatus is typically a superposition of terms corresponding to distinct outcomes. As the apparatus interacts with its environment, which may include observers, these systems become entangled with the apparatus and quantum system, the net result of which is a quantum state involving, for each of the possible experimental outcomes, a term in which the apparatus reading corresponds to that outcome, there are records of that outcome in the environment, observers observe that outcome, etc.

Everett proposed that each of these terms be taken to be equally real. As time goes on, there is a proliferation of these worlds, as situations arise that give rise to a further multiplicity of outcomes see the entry many-worlds interpretation of quantum mechanics , and Saunders , for overviews of recent discussions; Wallace is an extended defense of an Everettian interpretation of quantum mechanics. These views agree with Everett in attributing to a system definite values of dynamical variables only relative to the states of other systems; they differ in that, unlike Everett, they do not take the quantum state as their basic ontology see the entry on relational quantum mechanics for more detail.

The difference between a coherent superposition of two terms and a mixture has empirical consequences. To see this, consider the double-slit experiment, in which a beam of particles such as electrons, neutrons, or photons passes through two narrow slits and then impinges on a screen, where the particles are detected. The fact that the state is a superposition of these two alternatives is exhibited in interference fringes at the screen, alternating bands of high and low rates of absorption.

This is often expressed in terms of a difference between classical and quantum probabilities. The appearance of interference is an index of nonclassicality. That is, we can treat the particles as if they obeyed approximately definite trajectories, and apply probabilities in a classical manner. Now, macroscopic objects are typically in interaction with a large and complex environment—they are constantly being bombarded with air molecules, photons, and the like.

As a result, the reduced state of such a system quickly becomes a mixture of quasi-classical states, a phenomenon known as decoherence. A generalization of decoherence lies at the heart of an approach to the interpretation of quantum mechanics that goes by the name of decoherent histories approach see the entry on the consistent histories approach to quantum mechanics for an overview.

Decoherence plays important roles in the other approaches to quantum mechanics, though the role it plays varies with approach; see the entry on the role of decoherence in quantum mechanics for information on this. All of the above approaches take it that the goal is to provide an account of events in the world that recovers, at least in some approximation, something like our familiar world of ordinary objects behaving classically. None of the mainstream approaches accord any special physical role to conscious observers.

## Reality and Measurement in Algebraic Quantum Theory - John Templeton Foundation

There have, however, been proposals in that direction see the entry on quantum approaches to consciousness for discussion. All of the above-mentioned approaches are consistent with observation. Mere consistency, however, is not enough; the rules for connecting quantum theory with experimental results typically involve nontrivial that is, not equal to zero or one probabilities assigned to experimental outcomes. These calculated probabilities are confronted with empirical evidence in the form of statistical data from repeated experiments.

Extant hidden-variables theories reproduce the quantum probabilities, and collapse theories have the intriguing feature of reproducing very close approximations to quantum probabilities for all experiments that have been performed so far but departing from the quantum probabilities for other conceivable experiments.

This permits, in principle, an empirical discrimination between such theories and no-collapse theories. It has been the subject of much recent work on Everettian theories; see Saunders for an introduction and overview. If one accepts that Everettians have a solution to the evidential problem, then, among the major lines of approach, none is favored in a straightforward way by the empirical evidence. If one is to make a decision as to which, if any, one should accept, it is to be made on other grounds.

There will not be space here to give an in-depth overview of these ongoing discussions, but a few considerations can be mentioned, to give the reader a flavor of the discussions; see entries on particular approaches for more detail. Bohmians claim, in favor of the Bohmian approach, that a theory on these lines provides the most straightforward picture of events; ontological issues are less clear-cut when it comes to Everettian theories or collapse theories. Another consideration is compatibility with relativistic causal structure.

The de Broglie-Bohm theory requires a distinguished relation of distant simultaneity for its formulation, and, it can be argued, this is an ineliminable feature of any hidden-variables theory of this sort, that selects some observable to always have definite values see Berndl et al. On the other hand, there are collapse models that are fully relativistic. On such models, collapses are localized events.

### 1. Introduction

Though probabilities of collapses at spacelike separation from each other are not independent, this probabilistic dependence does not require us to single one out as earlier and the other later. Thus, such theories do not require a distinguished relation of distant simultaneity. See the entry on collapse theories and references therein; see also, for some recent contributions to the discussion, Fleming , Maudlin , and Myrvold In the case of Everettian theories, one must first think about how to formulate the question of relativistic locality.

Several authors have approached this issue in somewhat different ways, with a common conclusion that Everettian quantum mechanics is, indeed, local. As mentioned, a central question of interpretation of quantum mechanics concerns whether quantum states should be regarded as representing anything in physical reality. If this is answered in the affirmative, this gives rise to new questions, namely, what sort of physical reality is represented by the quantum state, and whether a quantum state could in principle give an exhaustive account of physical reality.

Harrigan and Spekkens have introduced a framework for discussing these issues. In their terminology, a complete specification of the physical properties is given by the ontic state of a system. An ontological model posits a space of ontic states and associates, with any preparation procedure, a probability distribution over ontic states. This gives a nice way of posing the question of quantum state realism: are there preparations corresponding to distinct pure quantum states that can give rise to the same ontic state, or, conversely, are there ontic states compatible with distinct quantum states?

The Pusey, Barrett and Rudolph PBR theorem does not close off all options for anti-realism about quantum states; an anti-realist about quantum states could reject the Preparation Independence assumption, or reject the framework within which the theorem is set; see discussion in Spekkens 92— See also Leifer for a careful and thorough overview of theorems relevant to quantum state realism. The major realist approaches to the measurement problem are all, in some sense, realist about quantum states.

Merely saying this is insufficient to give an account of the ontology of a given interpretation. Among the questions to be addressed are: if quantum states represent something physically real, what sort of thing is it? This gives us a prima facie tidy way of classifying approaches to the measurement problem:. We include in the first category approaches that deny that a quantum state should be thought of as representing anything in reality at all. These include variants of the Copenhagen interpretation, as well as pragmatic and other anti-realist approaches.

Also in the first category are approaches that seek a completion of the quantum state description. These include hidden-variables approaches and modal interpretations.

The second category of interpretation motivates a research programme of finding suitable indeterministic modifications of the quantum dynamics. From the early days of quantum mechanics, there has been a strain of thought that holds that the proper attitude to take towards quantum mechanics is an instrumentalist or pragmatic one. On such a view, quantum mechanics is a tool for coordinating our experience and for forming expectations about the outcomes of experiments. Variants of this view include what has been called the Copenhagen Interpretation or Copenhagen Interpretations, as recent scholarship has emphasized differences between figures associated with this view ; see the entry on Copenhagen interpretation of quantum mechanics.

More recently, views of this sort have been advocated by physicists, including QBists, who hold that quantum states represent subjective or epistemic probabilities see Fuchs et al. The philosopher Richard Healey defends a related view on which quantum states, though objective, do not represent physical reality see Healey ; Healey forthcoming. That a quantum state description cannot be regarded as a complete description of physical reality was argued for in a famous paper by Einstein, Podolsky and Rosen EPR and by Einstein in subsequent publications Einstein , , See the entry on the Einstein-Podolsky-Rosen argument in quantum theory.

There are a number of theorems that circumscribe the scope of possible hidden-variables theories. Theories of this sort are called noncontextual hidden-variables theory. It was shown by Bell and Kochen and Specker that there are no such theories for any system whose Hilbert space dimension is greater than three see the entry on the Kochen-Specker theorem. The Bell-Kochen-Specker Theorem does not rule out hidden-variables theories tout court.

The most thoroughly worked-out theory of this type is the pilot wave theory developed by de Broglie and presented by him at the Fifth Solvay Conference held in Brussels in , revived by David Bohm in , and currently an active area of research by a small group of physicists and philosophers. According to this theory, there are particles with definite trajectories, that are guided by the quantum wave function.

For the history of the de Broglie theory, see the introductory chapters of Bacciagaluppi and Valentini For any overview of the de Broglie-Bohm theory and philosophical issues associated with it see the entry on Bohmian mechanics. There have been other proposals for supplementing the quantum state with additional structure; these have come to be called modal interpretations ; see the entry on modal interpretations of quantum mechanics. As already mentioned, von Neumann and Dirac wrote as if the collapse of the quantum state vector precipitated by an experimental intervention on the system is a genuine physical change, distinct from the usual unitary evolution.

If collapse is to be taken as a genuine physical process, then something more needs to be said about the circumstances under which it occurs than merely that it happens when an experiment is performed. The only promising collapse theories are stochastic in nature; indeed, it can be shown that a deterministic collapse theory would permit superluminal signalling. See Allori et al. The resulting interpretation he called the relative state interpretation.

The basic idea is this. After an experiment, the quantum state of the system plus apparatus is typically a superposition of terms corresponding to distinct outcomes. As the apparatus interacts with its environment, which may include observers, these systems become entangled with the apparatus and quantum system, the net result of which is a quantum state involving, for each of the possible experimental outcomes, a term in which the apparatus reading corresponds to that outcome, there are records of that outcome in the environment, observers observe that outcome, etc.

Everett proposed that each of these terms be taken to be equally real. As time goes on, there is a proliferation of these worlds, as situations arise that give rise to a further multiplicity of outcomes see the entry many-worlds interpretation of quantum mechanics , and Saunders , for overviews of recent discussions; Wallace is an extended defense of an Everettian interpretation of quantum mechanics.

These views agree with Everett in attributing to a system definite values of dynamical variables only relative to the states of other systems; they differ in that, unlike Everett, they do not take the quantum state as their basic ontology see the entry on relational quantum mechanics for more detail.

The difference between a coherent superposition of two terms and a mixture has empirical consequences.

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To see this, consider the double-slit experiment, in which a beam of particles such as electrons, neutrons, or photons passes through two narrow slits and then impinges on a screen, where the particles are detected. The fact that the state is a superposition of these two alternatives is exhibited in interference fringes at the screen, alternating bands of high and low rates of absorption.

This is often expressed in terms of a difference between classical and quantum probabilities. The appearance of interference is an index of nonclassicality. That is, we can treat the particles as if they obeyed approximately definite trajectories, and apply probabilities in a classical manner. Now, macroscopic objects are typically in interaction with a large and complex environment—they are constantly being bombarded with air molecules, photons, and the like. As a result, the reduced state of such a system quickly becomes a mixture of quasi-classical states, a phenomenon known as decoherence.

A generalization of decoherence lies at the heart of an approach to the interpretation of quantum mechanics that goes by the name of decoherent histories approach see the entry on the consistent histories approach to quantum mechanics for an overview. Decoherence plays important roles in the other approaches to quantum mechanics, though the role it plays varies with approach; see the entry on the role of decoherence in quantum mechanics for information on this. All of the above approaches take it that the goal is to provide an account of events in the world that recovers, at least in some approximation, something like our familiar world of ordinary objects behaving classically.

None of the mainstream approaches accord any special physical role to conscious observers. There have, however, been proposals in that direction see the entry on quantum approaches to consciousness for discussion. All of the above-mentioned approaches are consistent with observation. Mere consistency, however, is not enough; the rules for connecting quantum theory with experimental results typically involve nontrivial that is, not equal to zero or one probabilities assigned to experimental outcomes.

These calculated probabilities are confronted with empirical evidence in the form of statistical data from repeated experiments. Extant hidden-variables theories reproduce the quantum probabilities, and collapse theories have the intriguing feature of reproducing very close approximations to quantum probabilities for all experiments that have been performed so far but departing from the quantum probabilities for other conceivable experiments.

This permits, in principle, an empirical discrimination between such theories and no-collapse theories. It has been the subject of much recent work on Everettian theories; see Saunders for an introduction and overview. If one accepts that Everettians have a solution to the evidential problem, then, among the major lines of approach, none is favored in a straightforward way by the empirical evidence. If one is to make a decision as to which, if any, one should accept, it is to be made on other grounds.

There will not be space here to give an in-depth overview of these ongoing discussions, but a few considerations can be mentioned, to give the reader a flavor of the discussions; see entries on particular approaches for more detail. Bohmians claim, in favor of the Bohmian approach, that a theory on these lines provides the most straightforward picture of events; ontological issues are less clear-cut when it comes to Everettian theories or collapse theories. Another consideration is compatibility with relativistic causal structure. The de Broglie-Bohm theory requires a distinguished relation of distant simultaneity for its formulation, and, it can be argued, this is an ineliminable feature of any hidden-variables theory of this sort, that selects some observable to always have definite values see Berndl et al.

On the other hand, there are collapse models that are fully relativistic. On such models, collapses are localized events. Though probabilities of collapses at spacelike separation from each other are not independent, this probabilistic dependence does not require us to single one out as earlier and the other later. Thus, such theories do not require a distinguished relation of distant simultaneity. See the entry on collapse theories and references therein; see also, for some recent contributions to the discussion, Fleming , Maudlin , and Myrvold In the case of Everettian theories, one must first think about how to formulate the question of relativistic locality.

Several authors have approached this issue in somewhat different ways, with a common conclusion that Everettian quantum mechanics is, indeed, local. As mentioned, a central question of interpretation of quantum mechanics concerns whether quantum states should be regarded as representing anything in physical reality.

If this is answered in the affirmative, this gives rise to new questions, namely, what sort of physical reality is represented by the quantum state, and whether a quantum state could in principle give an exhaustive account of physical reality. Harrigan and Spekkens have introduced a framework for discussing these issues.

In their terminology, a complete specification of the physical properties is given by the ontic state of a system. An ontological model posits a space of ontic states and associates, with any preparation procedure, a probability distribution over ontic states. This gives a nice way of posing the question of quantum state realism: are there preparations corresponding to distinct pure quantum states that can give rise to the same ontic state, or, conversely, are there ontic states compatible with distinct quantum states?

The Pusey, Barrett and Rudolph PBR theorem does not close off all options for anti-realism about quantum states; an anti-realist about quantum states could reject the Preparation Independence assumption, or reject the framework within which the theorem is set; see discussion in Spekkens 92— See also Leifer for a careful and thorough overview of theorems relevant to quantum state realism.

The major realist approaches to the measurement problem are all, in some sense, realist about quantum states. Merely saying this is insufficient to give an account of the ontology of a given interpretation. Among the questions to be addressed are: if quantum states represent something physically real, what sort of thing is it? This is the question of the ontological construal of quantum states. Another question is the EPR question, whether a description in terms of quantum states can be taken as, in principle, complete, or whether it must be supplemented by different ontology.

The original conception was that each particle would have its own guiding wave. If quantum states represent something in physical reality, they are unlike anything familiar in classical physics. On this view, this high-dimensional space is thought of as more fundamental than the familiar three-dimensional space or four-dimensional spacetime that is usually taken to be the arena of physical events.

See Albert , , for the classic statement of the view; other proponents include Loewer , Lewis , Ney , a,b, , and North Most of the discussion of this proposal has taken place within the context of nonrelativistic quantum mechanics, which is not a fundamental theory. It has been argued that considerations of how the wave functions of nonrelativistic quantum mechanics arise from a quantum field theory undermines the idea that wave functions are relevantly like fields on configuration space, and also the idea that configuration spaces can be thought of as more fundamental than ordinary spacetime Myrvold Taking a wave function to be a multi-field, on the other hand, involves accepting nonseparability.

Another difference between taking wave-functions as multi-fields on ordinary space and taking them to be fields on a high-dimensional space is that, on the multi-field view, there is no question about the relation of ordinary three-dimensional space to some more fundamental space. It has been argued that, on the de Broglie-Bohm pilot wave theory and related pilot wave theories, the quantum state plays a role more similar to that of a law in classical mechanics; its role is to provide dynamics for the Bohmian corpuscles, which, according to the theory, compose ordinary objects.

The conception is extended to interpretations of collapse theories by Allori et al. Primitive ontology is to be distinguished from other ontology, such as the quantum state, that is introduced into the theory to account for the behavior of the primitive ontology. The distinction is meant to be a guide as to how to conceive of the nonprimitive ontology of the theory. Quantum mechanics has not only given rise to interpretational conundrums; it has given rise to new concepts in computing and in information theory.

Quantum information theory is the study of the possibilities for information processing and transmission opened up by quantum theory. Another area of active research in the foundations of quantum mechanics is the attempt to gain deeper insight into the structure of the theory, and the ways in which it differs from both classical physics and other theories that one might construct, by characterizing the structure of the theory in terms of very general principles, often with an information-theoretic flavour. This project has its roots in early work of Mackey , , Ludwig , and Piron aiming to characterize quantum mechanics in operational terms.

This has led to the development of a framework of generalized probabilistic model. It also has connections with the investigations into quantum logic initiated by Birkhoff and von Neumann see the entry quantum logic and probability theory for an overview. Interest in the project of deriving quantum theory from axioms with clear operational content was revived by the work of Hardy [], Other Internet Resources. See Chiribella and Spekkens for a snapshot of the state of the art of this endeavour.

Introduction 2. Quantum theory 2. Entanglement, nonlocality, and nonseparability 4.

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The measurement problem 4. Ontological Issues 5. Quantum computing and quantum information theory 7. Bibliography Albert, David Z. Cushing, A. Goldstein eds. Placzek and J.