Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Cambridge University Press eBooks

Discrete Events from Repeatability: “Quantum Jumps” and Wavepacket Collapse

Wojciech Hubert Zurek

Published Mar 6, 2025• 0 citations
Publisher

Abstract

Chapter 2 shows how the discreteness that sets the stage for discontinuous quantum jumps between a restricted set of states is a consequence of the symmetry breaking that resolves the tension between the unitarity of quantum evolutions, and repeatable information transfer (the essence of quantum Darwinism, the subject of Chapters 7 and 8). Chapter 2 shows that, while the quantum superposition principle declares that every superposition is an equally legal quantum state, repeatability restricts states that can be recorded (found out) multiple times to an orthogonal set determined by the unitary dynamics of the process responsible for the repeated information transfer (i.e., for amplification). Such states persist and can imprint the evidence of their continued presence in other systems, e.g., on the subsystems of the environment. They become the elements of objective reality—e.g., outcomes of the measurements we perceive. Moreover, Chapter 2 motivates the need for the derivation of the probabilities of measurements (to be carried out in Chapter 3).

DOI
10.1017/9781009552868.003

Related papers

Wave-packet collapse and the core quantum postulates: Discreteness of quantum jumps from unitarity, repeatability, and actionable information

Wojciech Hubert Zurek

An unknown quantum state of a single system cannot be discovered, as a measured system is reprepared---it jumps into an eigenstate of the measured observable. This impossibility of finding the quantum state and other symptoms usually blamed on wave-packet collapse follow (as was recently demonstrated for pure states of measured systems) from unitarity (which does not, however, allow for a literal collapse) and from the repeatability of measurements: Continuous unitary evolution and repeatability suffice to establish the discreteness that underlies quantum jumps. Here we consider mixed states of a macroscopic, open system (such as an apparatus), and we allow its microscopic state to change when, e.g., measured by an observer, provided that its salient features remain unchanged and that observers regard macroscopic state of the pointer as representing the same record. We conclude that repeatably accessible states of macroscopic systems (such as the states of the apparatus pointer) must correspond to orthogonal subspaces in the Hilbert space. The symmetry breaking we exhibit defies the egalitarian quantum superposition principle and unitary symmetry of the Hilbert space, as it singles out preferred subspaces. We conclude that the resulting discreteness (which underlies quantum jumps) emerges from the continuity of the core quantum postulates plus repeatability also in macroscopic and open---but ultimately quantum---systems such as measuring devices accessed by observers, where (in contrast with pure states of microsystems) repeatability is paramount.

arXiv:1212.3245•Physical Review A•May 10, 2013•38 citations

Quantum origin of quantum jumps: Breaking of unitary symmetry induced by information transfer in the transition from quantum to classical

Wojciech Hubert Zurek

Measurements transfer information about a system to the apparatus and then, further on, to observers and (often inadvertently) to the environment. I show that even imperfect copying essential in such situations restricts possible unperturbed outcomes to an orthogonal subset of all possible states of the system, thus breaking the unitary symmetry of its Hilbert space implied by the quantum superposition principle. Preferred outcome states emerge as a result. They provide a framework for ``wave-packet collapse,'' designating terminal points of quantum jumps and defining the measured observable by specifying its eigenstates. In quantum Darwinism, they are the progenitors of multiple copies spread throughout the environment---the fittest quantum states that not only survive decoherence, but subvert the environment into carrying information about them---into becoming a witness.

arXiv:quant-ph/0703160•Physical Review A•Nov 16, 2007•102 citations

Quantum Theory of the Classical: Einselection, Envariance, Quantum Darwinism and Extantons

Wojciech Hubert Zurek

Core quantum postulates including the superposition principle and the unitarity of evolutions are natural and strikingly simple. I show that—when supplemented with a limited version of predictability (captured in the textbook accounts by the repeatability postulate)—these core postulates can account for all the symptoms of classicality. In particular, both objective classical reality and elusive information about reality arise, via quantum Darwinism, from the quantum substrate. This approach shares with the Relative State Interpretation of Everett the view that collapse of the wavepacket reflects perception of the state of the rest of the Universe relative to the state of observer’s records. However, our “let quantum be quantum” approach poses questions absent in Bohr’s Copenhagen Interpretation that relied on the preexisting classical domain. Thus, one is now forced to seek preferred, predictable, hence effectively classical but ultimately quantum states that allow observers keep reliable records. Without such (i) preferred basis relative states are simply “too relative”, and the ensuing basis ambiguity makes it difficult to identify events (e.g., measurement outcomes). Moreover, universal validity of quantum theory raises the issue of (ii) the origin of Born’s rule, pk=|ψk|2, relating probabilities and amplitudes (that is simply postulated in textbooks). Last not least, even preferred pointer states (defined by einselection—environment—induced superselection)—are still quantum. Therefore, unlike classical states that exist objectively, quantum states of an individual system cannot be found out by an initially ignorant observer through direct measurement without being disrupted. So, to complete the ‘quantum theory of the classical’ one must identify (iii) quantum origin of objective existence and explain how the information about objectively existing states can appear to be essentially inconsequential for them (as it does for states in Newtonian physics) and yet matter in other settings (e.g., thermodynamics). I show how the mathematical structure of quantum theory supplemented by the only uncontroversial measurement postulate (that demands immediate repeatability—hence, predictability) leads to preferred states. These (i) pointer states correspond to measurement outcomes. Their stability is a prerequisite for objective existence of effectively classical states and for events such as quantum jumps. Events at hand, one can now enquire about their probability—the probability of a pointer state (or of a measurement record). I show that the symmetry of entangled states—(ii) entanglement—assisted invariance or envariance—implies Born’s rule. Envariance also accounts for the loss of phase coherence between pointer states. Thus, decoherence can be traced to symmetries of entanglement and understood without its usual tool—reduced density matrices. A simple and manifestly noncircular derivation of pk=|ψk|2 follows. Monitoring of the system by its environment in course of decoherence typically leaves behind multiple copies of its pointer states in the environment. Only pointer states can survive decoherence and can spawn such plentiful information-theoretic progeny. This (iii) quantum Darwinism allows observers to use environment as a witness—to find out pointer states indirectly, leaving systems of interest untouched. Quantum Darwinism shows how epistemic and ontic (coexisting in epiontic quantum state) separate into robust objective existence of pointer states and detached information about them, giving rise to extantons—composite objects with system of interest in the core and multiple records of its pointer states in the halo comprising of environment subsystems (e.g., photons) which disseminates that information throughout the Universe.

arXiv:2208.09019•Entropy•Oct 24, 2022•56 citations

Fundamental Issues and Measurement Problem in Quantum Mechanics

Tadashi Nakajima

The foundations of quantum mechanics are reconsidered in relation to the measurement problem. We define two kinds of quantities. The type A quantities are observables obtained experimentally from a single measurement. A dynamical variable is a type A quantity whose measurement selects a set of eigenvalues (EVs) as \(c\)-numbers. The type B quantities represent theoretical quantities related to probabilities. State vectors, the wave function, and the probability distribution (PD) are type B quantities. They are derived from an eigenvalue equation such as the Schrödinger equation. A quantum jump (QJ) is the measurement of a dynamical variable to select a set of EVs. The wave function is a theoretical quantity that persists throughout an experiment and therefore does not collapse. After a QJ, a selected set of EVs persists until it becomes macroscopic. However, an eigenstate is a theoretical notion irrelevant to a single measurement and does not exist in reality after a QJ. One experiment is complete when an ensemble of EVs has been obtained and compared with the theoretically predicted PD. To obtain an ensemble, repeated measurements are required in general, but there are some exceptions, like a Bose-Einstein condensate system, for which a single measurement yields a real ensemble. We comment on the relation between the reality of the wave function and the derivation of non-relativistic quantum mechanics from relativistic quantum field theory.

Qeios•Dec 11, 2025•0 citations

Fundamental Issues and Measurement Problem in Quantum Mechanics

Tadashi Nakajima

The foundations of quantum mechanics are reconsidered in relation to the measurement problem. We define two kinds of quantities. The type A quantities are observables obtained experimentally from a single measurement. A dynamical variable is a type A quantity whose measurement selects a set of eigenvalues (EVs) as \(c\)-numbers. The type B quantities represent theoretical quantities related to probabilities. State vectors, the wave function, and the probability distribution (PD) are type B quantities. They are derived from an eigenvalue equation such as the Schrödinger equation. A quantum jump (QJ) is the measurement of a dynamical variable to select a set of EVs. The wave function is a theoretical quantity that persists throughout an experiment and therefore does not collapse. After a QJ, a selected set of EVs persists until it becomes macroscopic. However, an eigenstate is a theoretical notion irrelevant to a single measurement and does not exist in reality after a QJ. One experiment is complete when an ensemble of EVs has been obtained and compared with the theoretically predicted PD. To obtain an ensemble, repeated measurements are required in general, but there are some exceptions, like a Bose-Einstein condensate system, for which a single measurement yields a real ensemble.

Qeios•Dec 8, 2025•0 citations

Fundamental Issues and Measurement Problem in Quantum Mechanics

Tadashi Nakajima

Foundations of quantum mechanics are reconsidered in relation to the measurement problem. We define two kinds of quantities. The type A quantities are observables obtained experimentally from a single measurement. A dynamical variable is a type A quantity whose measurement selects a set of eigenvalues (EVs) as \(c\)-numbers. The type B quantities represent theoretical quantities related to probabilities. State vectors, wave function, and the probability distribution (PD) are type B quantities. They are derived from an eigenvalue equation such as Schrödinger equation. A quantum jump (QJ) is the measurement of a dynamical variable to select a set of EVs. The wave function is a theoretical quantity which persists throughout an experiment and therefore it does not collapse. After a QJ, a selected set of EVs persists until it becomes macroscopic. However, an eigenstate is a theoretical notion irrelevant to a single measurement and it does not exist in reality after a QJ. One experiment is complete when an ensemble of EVs has been obtained and compared with the theoretically predicted PD. To obtain an ensemble, repeated measurements are required in general, but there are some exceptions like a Bose-Einstein condensate system for which a single measurement yields a real ensemble. We discuss the relation between the reality of the wave function and the derivation of non-relativistic quantum mechanics from quantum field theory.

Qeios•Feb 20, 2026•0 citations
Chapter 2 shows how the discreteness that sets the stage for discontinuous quantum jumps between a restricted set of states is a consequence of the symmetry breaking that resolves the tension between the unitarity of quantum evolutions, and repeatable information transfer (the essence of quantum Darwinism, the subject of Chapters 7 and 8). Chapter 2 shows that, while the quantum superposition principle declares that every superposition is an equally legal quantum state, repeatability restricts states that can be recorded (found out) multiple times to an orthogonal set determined by the unitary dynamics of the process responsible for the repeated information transfer (i.e., for amplification). Such states persist and can imprint the evidence of their continued presence in other systems, e.g., on the subsystems of the environment. They become the elements of objective reality—e.g., outcomes of the measurements we perceive. Moreover, Chapter 2 motivates the need for the derivation of the probabilities of measurements (to be carried out in Chapter 3).
An unknown quantum state of a single system cannot be discovered, as a measured system is reprepared---it jumps into an eigenstate of the measured observable. This impossibility of finding the quantum state and other symptoms usually blamed on wave-packet collapse follow (as was recently demonstrated for pure states of measured systems) from unitarity (which does not, however, allow for a literal collapse) and from the repeatability of measurements: Continuous unitary evolution and repeatability suffice to establish the discreteness that underlies quantum jumps. Here we consider mixed states of a macroscopic, open system (such as an apparatus), and we allow its microscopic state to change when, e.g., measured by an observer, provided that its salient features remain unchanged and that observers regard macroscopic state of the pointer as representing the same record. We conclude that repeatably accessible states of macroscopic systems (such as the states of the apparatus pointer) must correspond to orthogonal subspaces in the Hilbert space. The symmetry breaking we exhibit defies the egalitarian quantum superposition principle and unitary symmetry of the Hilbert space, as it singles out preferred subspaces. We conclude that the resulting discreteness (which underlies quantum jumps) emerges from the continuity of the core quantum postulates plus repeatability also in macroscopic and open---but ultimately quantum---systems such as measuring devices accessed by observers, where (in contrast with pure states of microsystems) repeatability is paramount.
Measurements transfer information about a system to the apparatus and then, further on, to observers and (often inadvertently) to the environment. I show that even imperfect copying essential in such situations restricts possible unperturbed outcomes to an orthogonal subset of all possible states of the system, thus breaking the unitary symmetry of its Hilbert space implied by the quantum superposition principle. Preferred outcome states emerge as a result. They provide a framework for ``wave-packet collapse,'' designating terminal points of quantum jumps and defining the measured observable by specifying its eigenstates. In quantum Darwinism, they are the progenitors of multiple copies spread throughout the environment---the fittest quantum states that not only survive decoherence, but subvert the environment into carrying information about them---into becoming a witness.
Core quantum postulates including the superposition principle and the unitarity of evolutions are natural and strikingly simple. I show that—when supplemented with a limited version of predictability (captured in the textbook accounts by the repeatability postulate)—these core postulates can account for all the symptoms of classicality. In particular, both objective classical reality and elusive information about reality arise, via quantum Darwinism, from the quantum substrate. This approach shares with the Relative State Interpretation of Everett the view that collapse of the wavepacket reflects perception of the state of the rest of the Universe relative to the state of observer’s records. However, our “let quantum be quantum” approach poses questions absent in Bohr’s Copenhagen Interpretation that relied on the preexisting classical domain. Thus, one is now forced to seek preferred, predictable, hence effectively classical but ultimately quantum states that allow observers keep reliable records. Without such (i) preferred basis relative states are simply “too relative”, and the ensuing basis ambiguity makes it difficult to identify events (e.g., measurement outcomes). Moreover, universal validity of quantum theory raises the issue of (ii) the origin of Born’s rule, pk=|ψk|2, relating probabilities and amplitudes (that is simply postulated in textbooks). Last not least, even preferred pointer states (defined by einselection—environment—induced superselection)—are still quantum. Therefore, unlike classical states that exist objectively, quantum states of an individual system cannot be found out by an initially ignorant observer through direct measurement without being disrupted. So, to complete the ‘quantum theory of the classical’ one must identify (iii) quantum origin of objective existence and explain how the information about objectively existing states can appear to be essentially inconsequential for them (as it does for states in Newtonian physics) and yet matter in other settings (e.g., thermodynamics). I show how the mathematical structure of quantum theory supplemented by the only uncontroversial measurement postulate (that demands immediate repeatability—hence, predictability) leads to preferred states. These (i) pointer states correspond to measurement outcomes. Their stability is a prerequisite for objective existence of effectively classical states and for events such as quantum jumps. Events at hand, one can now enquire about their probability—the probability of a pointer state (or of a measurement record). I show that the symmetry of entangled states—(ii) entanglement—assisted invariance or envariance—implies Born’s rule. Envariance also accounts for the loss of phase coherence between pointer states. Thus, decoherence can be traced to symmetries of entanglement and understood without its usual tool—reduced density matrices. A simple and manifestly noncircular derivation of pk=|ψk|2 follows. Monitoring of the system by its environment in course of decoherence typically leaves behind multiple copies of its pointer states in the environment. Only pointer states can survive decoherence and can spawn such plentiful information-theoretic progeny. This (iii) quantum Darwinism allows observers to use environment as a witness—to find out pointer states indirectly, leaving systems of interest untouched. Quantum Darwinism shows how epistemic and ontic (coexisting in epiontic quantum state) separate into robust objective existence of pointer states and detached information about them, giving rise to extantons—composite objects with system of interest in the core and multiple records of its pointer states in the halo comprising of environment subsystems (e.g., photons) which disseminates that information throughout the Universe.
The foundations of quantum mechanics are reconsidered in relation to the measurement problem. We define two kinds of quantities. The type A quantities are observables obtained experimentally from a single measurement. A dynamical variable is a type A quantity whose measurement selects a set of eigenvalues (EVs) as ccc-numbers. The type B quantities represent theoretical quantities related to probabilities. State vectors, the wave function, and the probability distribution (PD) are type B quantities. They are derived from an eigenvalue equation such as the Schrödinger equation. A quantum jump (QJ) is the measurement of a dynamical variable to select a set of EVs. The wave function is a theoretical quantity that persists throughout an experiment and therefore does not collapse. After a QJ, a selected set of EVs persists until it becomes macroscopic. However, an eigenstate is a theoretical notion irrelevant to a single measurement and does not exist in reality after a QJ. One experiment is complete when an ensemble of EVs has been obtained and compared with the theoretically predicted PD. To obtain an ensemble, repeated measurements are required in general, but there are some exceptions, like a Bose-Einstein condensate system, for which a single measurement yields a real ensemble. We comment on the relation between the reality of the wave function and the derivation of non-relativistic quantum mechanics from relativistic quantum field theory.
The foundations of quantum mechanics are reconsidered in relation to the measurement problem. We define two kinds of quantities. The type A quantities are observables obtained experimentally from a single measurement. A dynamical variable is a type A quantity whose measurement selects a set of eigenvalues (EVs) as ccc-numbers. The type B quantities represent theoretical quantities related to probabilities. State vectors, the wave function, and the probability distribution (PD) are type B quantities. They are derived from an eigenvalue equation such as the Schrödinger equation. A quantum jump (QJ) is the measurement of a dynamical variable to select a set of EVs. The wave function is a theoretical quantity that persists throughout an experiment and therefore does not collapse. After a QJ, a selected set of EVs persists until it becomes macroscopic. However, an eigenstate is a theoretical notion irrelevant to a single measurement and does not exist in reality after a QJ. One experiment is complete when an ensemble of EVs has been obtained and compared with the theoretically predicted PD. To obtain an ensemble, repeated measurements are required in general, but there are some exceptions, like a Bose-Einstein condensate system, for which a single measurement yields a real ensemble.
Foundations of quantum mechanics are reconsidered in relation to the measurement problem. We define two kinds of quantities. The type A quantities are observables obtained experimentally from a single measurement. A dynamical variable is a type A quantity whose measurement selects a set of eigenvalues (EVs) as ccc-numbers. The type B quantities represent theoretical quantities related to probabilities. State vectors, wave function, and the probability distribution (PD) are type B quantities. They are derived from an eigenvalue equation such as Schrödinger equation. A quantum jump (QJ) is the measurement of a dynamical variable to select a set of EVs. The wave function is a theoretical quantity which persists throughout an experiment and therefore it does not collapse. After a QJ, a selected set of EVs persists until it becomes macroscopic. However, an eigenstate is a theoretical notion irrelevant to a single measurement and it does not exist in reality after a QJ. One experiment is complete when an ensemble of EVs has been obtained and compared with the theoretically predicted PD. To obtain an ensemble, repeated measurements are required in general, but there are some exceptions like a Bose-Einstein condensate system for which a single measurement yields a real ensemble. We discuss the relation between the reality of the wave function and the derivation of non-relativistic quantum mechanics from quantum field theory.