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arXiv:1212.3245 · Physical Review A

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

Wojciech Hubert Zurek

Published May 10, 2013• 38 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.87.052111

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Communications in Theoretical Physics•Feb 1, 2016•4 citations
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.
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).
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.
One of the fundamental conceptual difficulties in quantum mechanics arises from the fact that two of the axioms of the theory concerning the evolution of a microscopic system are contradictory. The first axiom assumes that the state of a quantum system evolving undisturbed by any observer obeys a linear deterministic law expressed by the SchrSdinger equation. The second one (wave-packet reduction principle) states that, when the quantum system interacts with an apparatus devised to measure some of its dynamical variables, its state changes abruptly and stochastically, jumping into a normalized eigenstate of the measured variable. This evolution is not deterministic nor linear and cannot be accounted for by the SchrSdinger equation. As a consequence the internal coherence of the theory becomes questionable. On one side the influence of the apparatus on the quantum system cannot be described as an interaction of the quantum constituents of the measuring macroscopic apparatus with the measured quantum system. On the other side, the impossibility of considering a macroscopic object as made by quantum constituents, i .e . the difficulty of quantum mechanics in describing the behaviour of macroscopic systems, is even more dramatic if we consider the evolution of the closed system consisting of the macroscopic apparatus and of the microscopic system. In such a case the appearance of superpositions of macr0scopically distinguishable states related to the apparatus cannot be avoided. These superpositions contradict both our classical experience and wave-packet reduction principle, which provides for statistical mixtures.
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.
Quantum mechanics is widely recognised as being incomplete. It is not consistent with the second law of thermodynamics and does not provide a scientifically credible physical account of the measurement process, the means by which coherence is broken and classically observable states are recorded. This has led to many ad hoc assumptions being used to account for various properties of quantum systems, among which is the coherence time of quantum devices that determines their ability to perform computations. Here, we show that all these properties can be accommodated naturally and consistently in the context of mixing quantum systems which exhibit continuous spectra, as arises in the thermodynamic limit in quantum statistical mechanics and quantum gravity. In particular, for isolated systems we show that the time-reversal symmetry associated with unitary time evolution of the quantum state gives rise to time-symmetry breaking and a semi-group evolution which attains thermodynamic equilibrium at long times. Moreover, the emergence of this non-unitary time-asymmetry leads to microcanonical equilibrium states in which all quantum coherence is lost and is accompanied by the transformation of pure states into mixtures, leading in turn to an increase in entropy. Inclusion of a macroscopic measurement apparatus shows how the outcome of a measurement corresponds to the von Neumann projection postulate, arising with probabilities in conformance with the Born rule. The mathematical structure of the theory which applies to quantum systems with continuous spectra is closely analogous to the classical ergodic theory of dynamical systems and the conditions under which they attain equilibrium states.