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arXiv:2301.03233 · Entropy

Quantum State Reduction of General Initial States through Spontaneous Unitarity Violation

Aritro Mukherjee, Srinivas Gotur, Jelle Aalberts, Rosa van den Ende, Lotte Mertens, Jasper van Wezel

Published Jan 31, 2024• 8 citations
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Abstract

The inability of Schrödinger's unitary time evolution to describe the measurement of a quantum state remains a central foundational problem. It was recently suggested that the unitarity of Schrödinger dynamics can be spontaneously broken, resulting in measurement as an emergent phenomenon in the thermodynamic limit. Here, we introduce a family of models for spontaneous unitarity violation that apply to generic initial superpositions over arbitrarily many states, using either single or multiple state-independent stochastic components. Crucially, we show that Born's probability rule emerges spontaneously in all cases.

DOI
10.3390/e26020131

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The inability of Schrödinger's unitary time evolution to describe the measurement of a quantum state remains a central foundational problem. It was recently suggested that the unitarity of Schrödinger dynamics can be spontaneously broken, resulting in measurement as an emergent phenomenon in the thermodynamic limit. Here, we introduce a family of models for spontaneous unitarity violation that apply to generic initial superpositions over arbitrarily many states, using either single or multiple state-independent stochastic components. Crucially, we show that Born's probability rule emerges spontaneously in all cases.
One of the central foundational problems in quantum mechanics is the inability of Schrödinger's unitary time evolution to describe measurement of a quantum state. Recent developments in models of spontaneous unitarity violation (SUV) propose that quantum state reduction can emerge as a thermodynamic phenomenon, offering a natural resolution to the measurement problem. This paper investigates the philosophical implications of SUV in relation to the nature of probabilities in quantum mechanics and statistical mechanics. It provides an alternate interpretation of probabilities in quantum mechanics that aligns with (and improves upon) the arguments of the propensity interpretation of probabilities.
Objective collapse theories propose a solution to the quantum measurement problem by predicting deviations from Schrödinger’s equation that can be tested experimentally. A class of objective theories based on spontaneous unitarity violation was recently introduced, in which the stochastic field required for obtaining Born’s rule does not depend on the state of the system being measured. Here, we classify possible models for the stochastic field dynamics in theories of spontaneous unitarity violation. We show that for correlated stochastic dynamics, the field must be defined on a closed manifold. In two or more dimensions, it is then always possible to find stochastic dynamics yielding Born’s rule, independent of the state being measured or the correlation time of the stochastic field. We show that the models defined this way are all isomorphic to the definition on a two-sphere, which we propose to be a minimal physical model for the stochastic field in models of spontaneous unitarity violation.
The Born rule for computing probabilities of the outcomes of measurements is an indispensable ingredient of quantum mechanics. The standard textbook description of this rule gives the impression that it implies the unitarity of time evolution. This view relies on the argument that unless the dynamics is unitary, the probabilities of finding all possible outcomes of a measurement do not add up to 1, i.e., the total probability is not conserved. We show that this argument is flawed, and that the general expression for the Born rule ensures the conservation of total probabilities even when the dynamics of a quantum system is not unitary. This applies to the dynamics of ensembles of quantum systems in both pure and mixed states. We discuss the status of the local conservation of probabilities and the arguments against the plausibility of non-unitary time evolutions that are based on the identification of the Hamiltonian operator with the energy observable.
Quantum theory has evolved from a set of provisional rules to an indispensable framework that underlies much of modern technology and infrastructure. Yet, after a century, Born's probability postulate remains at odds with the theory's unitary character. The problem stems from the linearity of the Schrödinger equation, as linear systems are insensitive to the magnitudes of their solutions' coefficients. If measurement is unitary, and thus linear, how can the frequency of an outcome depend on the magnitude of its amplitude? And if not, at what scale does unitarity break down? This question remains pressing, as the assumption of unitarity underlies both the design of large-scale fault-tolerant quantum devices, as well as our understanding of fundamental aspects of our universe, for example, the black hole information problem. Proponents of the many-worlds interpretation have argued that Born's rule is observed because events that violate it have vanishing norms, and so must be unphysical. However, this argument has only been made explicit for special cases involving infinitely many identical states. In this paper we provide a generalized form of Born's rule applicable to measurements of arbitrarily-prepared factorizable states, and prove that such systems contain no histories that violate Born's rule. Our result therefore demonstrates that purely unitary evolution can co-exist with Born's rule under more relaxed conditions. In addition, we apply our generalized rule to the problem of quantum circuit fidelity, and provide a single-shot alternative to cross-entropy benchmarking based on self-information rate.
Spontaneous unitarity violations were recently proposed as a cause of objective quantum state reduction. This complements proposals based on stochastic modifications of Schr\"odinger's equation, but also differs from them in several aspects. Here, we formalize the description of models for spontaneous unitarity violation, and show that they generically imply models of dynamical quantum state reduction driven by colored noise. We present a formalism for exploring such models as well as a prescription for enforcing explicit norm-preservation, and we show that the resulting pure state dynamics is described by a modified von Neumann-Liouville equation which in a particular limit reduces to the Gorini-Kossakowski-Sudarshan-Lindblad master equations. We additionally show adherence to Born's rule emerging in the same limit from a physical constraint relating fluctuating and dissipating components of the model.