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Review
. 2018 Jul 13;376(2123):20180107.
doi: 10.1098/rsta.2018.0107.

Quantum theory of the classical: quantum jumps, Born's Rule and objective classical reality via quantum Darwinism

Wojciech Hubert Zurek  1
Affiliations
Review

Quantum theory of the classical: quantum jumps, Born's Rule and objective classical reality via quantum Darwinism

Wojciech Hubert Zurek. Philos Trans A Math Phys Eng Sci. .

Abstract

The emergence of the classical world from the quantum substrate of our Universe is a long-standing conundrum. In this paper, I describe three insights into the transition from quantum to classical that are based on the recognition of the role of the environment. I begin with the derivation of preferred sets of states that help to define what exists-our everyday classical reality. They emerge as a result of the breaking of the unitary symmetry of the Hilbert space which happens when the unitarity of quantum evolutions encounters nonlinearities inherent in the process of amplification-of replicating information. This derivation is accomplished without the usual tools of decoherence, and accounts for the appearance of quantum jumps and the emergence of preferred pointer states consistent with those obtained via environment-induced superselection, or einselection The pointer states obtained in this way determine what can happen-define events-without appealing to Born's Rule for probabilities. Therefore, pk =|ψk |2 can now be deduced from the entanglement-assisted invariance, or envariance-a symmetry of entangled quantum states. With probabilities at hand, one also gains new insights into the foundations of quantum statistical physics. Moreover, one can now analyse the information flows responsible for decoherence. These information flows explain how the perception of objective classical reality arises from the quantum substrate: the effective amplification that they represent accounts for the objective existence of the einselected states of macroscopic quantum systems through the redundancy of pointer state records in their environment-through quantum DarwinismThis article is part of a discussion meeting issue 'Foundations of quantum mechanics and their impact on contemporary society'.

Keywords: Born’s Rule; decoherence; probabilities; quantum Darwinism; quantum jumps.

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Conflict of interest statement

I declare I have no competing interests.

Figures

Figure 1.
Figure 1.
The fundamental (pre-quantum) connection between distinguishability and repeatability of measurements. The two circles represent two states of the measured system. They correspond to two outcomes—e.g. two properties of the underlying states (represented by two cross-hatchings). A measurement that can result in either outcome—that can produce a record correlated with these two properties—can be repeatable only when the two corresponding states (the two circles) do not overlap (case illustrated at the top). Repeatability is impossible without distinguishability: when two states overlap (case illustrated at the bottom), repetition of the measurement can always result in a system switching the state (and, thus, defying repeatability). In the quantum setting this pre-quantum connection between repeatability and distinguishability leads to the derivation of orthogonality of repeatable measurement outcomes (and the two cross-hatchings can be thought of as two linear polarizations of a photon—orthogonal on the top, but not below), but the basic intuition demanding distinguishability as a prerequisite for repeatability does not rely on the quantum formalism.
Figure 2.
Figure 2.
(Opposite.) Envariance—entanglement-assisted invariance—is a symmetry of entangled states. Envariance allows one to demonstrate Born’s Rule [17,41,42] using a combination of an old intuition of Laplace [47] about invariance and the origins of probability and quantum symmetries of entanglement. (a) Laplace’s principle of indifference (illustrated with playing cards) aims to establish symmetry using invariance under swaps. A player who doesn’t know the face values of cards is indifferent—does not care—if they are swapped before he gets the one on the left. For Laplace, this indifference was evidence of a (subjective) symmetry: it implied equal likelihood—equal probabilities of the invariantly swappable alternatives. For the two cards, subjective probability formula image would be inferred by someone who doesn’t know their face value, but knows that one of them is a spade. When probabilities of a set of elementary events are provably equal, one can compute the probabilities of composite events and thus develop a theory of probability. Even the additivity of probabilities can be established (e.g. [48]). This is in contrast to Kolmogorov’s measure-theoretic axioms (which include additivity of probabilities). Above all, Kolmogorov’s theory does not assign probabilities to elementary events (physical or otherwise), while the envariant approach yields probabilities when the symmetries of elementary events under swaps are known. (b) The problem with Laplace’s principle of indifference is its subjectivity. The actual physical state of the system (the two cards) is altered by the swap. A related problem is that the assessment of indifference is based on ignorance. It was argued, e.g. by supporters of the relative frequency approach (regarded by many as a more ‘objective’ foundation of probability), that it is impossible to deduce anything (including probabilities) from ignorance. This (along with subjectivity) is the reason why the equal likelihood approach is regarded with suspicion as a basis of probability in physics. (c) In quantum physics, symmetries of entanglement can be used to deduce objective probabilities starting with a known state. Envariance is the relevant symmetry. When a pure entangled state of a system formula image and another system we call ‘an environment formula image’ (anticipating connections with decoherence) formula image can be transformed by formula image acting solely on formula image, but the effect of formula image can be undone by acting solely on formula image with an appropriately chosen formula image, formula image, it is envariant under formula image. For such composite states, one can rigorously establish that the local state of formula image remains unaffected by formula image. Thus, for example, the phases of the coefficients in the Schmidt expansion formula image are envariant, as the effect of formula image can be undone by a countertransformation formula image acting solely on the environment. This envariance of phases implies their irrelevance for the local states—in effect, it implies decoherence. Moreover, when the absolute values of the Schmidt coefficients are equal, a swap formula image in formula image can be undone by a ‘counterswap’ formula image in formula image. So, as can be established more carefully [42], formula image follows from the objective symmetry of such an entangled state. This proof of equal probabilities is based not on ignorance (as in Laplace’s subjective ‘indifference’) but on knowledge of the ‘wrong property’—of the global observable that rules out (via quantum indeterminacy) any information about complementary local observables. When supplemented by simple counting, envariance leads to Born’s Rule also for unequal Schmidt coefficients [17,41,42].
Figure 3.
Figure 3.
Quantum Darwinism recognizes that environments consist of many subsystems and that observers acquire information about the system of interest formula image by intercepting copies of its pointer states deposited in formula image as a result of decoherence. (a) Decoherence paradigm: universe is divided into systemand environment. (b,c) Quantum Darwinism: environment consists of elementary subsystems—subenvironments. The latter can be combined into fragments that each have nearly complete information about the system. Redundancy is the number of such fragments. (Online version in colour.)
Figure 4.
Figure 4.
Information about the system contained in a fraction f of the environment. Red plot (with plateau) shows a typical formula image established by decoherence. The rapid rise means that nearly all classically accessible information is revealed bya small fraction of formula image. It is followed by a plateau: additional fragments only confirm what is already known. Redundancy formula image is the number of such independent fractions. Green plot shows formula image for a random state in the composite system formula image. (Online version in colour.)
Figure 5.
Figure 5.
The quantum mutual information formula image versus fragment size f at different elapsed times for an object illuminated by point-source black-body radiation. Individual curves are labelled by the time t in units of the decoherence time τD. For t≤τD (red dashed lines), the information about the system available in the environment is low. The linearity in f means each piece of the environment contains new, independent information. For t>τD (blue solid lines), the shape of the partial information plot indicates redundancy; the first few pieces of the environment increase the information, but additional pieces only confirm what is already known. (Online version in colour.)
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