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Register a new userThe inconsistency of linear dynamics and Borns rule
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Modern experiments using nanoscale devices come ever closer to bridging the divide between the quantum and classical realms, bringing experimental tests of objective collapse theories that propose alterations to Schr{o}dingers equation within reach. Such objective collapse theories aim to explain the emergence of classical dynamics in the thermodynamic limit and hence resolve the inconsistency that exists within the axioms of quantum mechanics. Here, we show that requiring the emergence of Borns rule for relative frequencies of measurement outcomes without imposing them as part of any axiom, implies that such objective collapse theories cannot be linear. Previous suggestions for a proof of the emergence of Borns rule in classes of problems that include linear objective collapse theories are analysed and shown to include hidden assumptions.
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It is shown that Schrodingers equation and Borns rule are sufficient to ensure that the states of macroscopic collective coordinate subsystems are microscopically localized in phase space and that the localized state follows the classical trajectory with random quantum noise that is indistinguishable from the pseudo-random noise of classical Brownian motion. This happens because in realistic systems the localization rate determined by the coupling to the environment is greater than the Lyapunov exponent that governs chaotic spreading in phase space. For realistic systems, the trajectories of the collective coordinate subsystem are at the same time an unravelling and a set of consistent/decoherent histories. Different subsystems have their own stochastic dynamics that generally knit together to form a global dynamics, although in certain contrived thought experiments, most notably Wigners friend, in the contrary, there is observer complementarity.
I show how probabilities arise in quantum physics by exploring implications of {it environment - assisted invariance} or {it envariance}, a recently discovered symmetry exhibited by entangled quantum systems. Envariance of perfectly entangled ``Bell-like states can be used to rigorously justify complete ignorance of the observer about the outcome of any measurement on either of the members of the entangled pair. For more general states, envariance leads to Borns rule, $p_k propto |psi_k|^2$ for the outcomes associated with Schmidt states. Probabilities derived in this manner are an objective reflection of the underlying state of the system -- they represent experimentally verifiable symmetries, and not just a subjective ``state of knowledge of the observer. Envariance - based approach is compared with and found superior to pre-quantum definitions of probability including the {it standard definition} based on the `principle of indifference due to Laplace, and the {it relative frequency approach} advocated by von Mises. Implications of envariance for the interpretation of quantum theory go beyond the derivation of Borns rule: Envariance is enough to establish dynamical independence of preferred branches of the evolving state vector of the composite system, and, thus, to arrive at the {it environment - induced superselection (einselection) of pointer states}, that was usually derived by an appeal to decoherence. Envariant origin of Borns rule for probabilities sheds a new light on the relation between ignorance (and hence, information) and the nature of quantum states.
In Mod. Phys. Lett. A 9, 3119 (1994), one of us (R.D.S) investigated a formulation of quantum mechanics as a generalized measure theory. Quantum mechanics computes probabilities from the absolute squares of complex amplitudes, and the resulting interference violates the (Kolmogorov) sum rule expressing the additivity of probabilities of mutually exclusive events. However, there is a higher order sum rule that quantum mechanics does obey, involving the probabilities of three mutually exclusive possibilities. We could imagine a yet more general theory by assuming that it violates the next higher sum rule. In this paper, we report results from an ongoing experiment that sets out to test the validity of this second sum rule by measuring the interference patterns produced by three slits and all the possible combinations of those slits being open or closed. We use attenuated laser light combined with single photon counting to confirm the particle character of the measured light.
We present a new experimental approach using a three-path interferometer and find a tighter empirical upper bound on possible violations of Borns Rule. A deviation from Borns rule would result in multi-order interference. Among the potential systematic errors that could lead to an apparent violation we specifically study the nonlinear response of our detectors and present ways to calibrate this error in order to obtain an even better bound.
In 2017, Lienert and Tumulka proved Borns rule on arbitrary Cauchy surfaces in Minkowski space-time assuming Borns rule and a corresponding collapse rule on horizontal surfaces relative to a fixed Lorentz frame, as well as a given unitary time evolution between any two Cauchy surfaces. Here, we prove Borns rule on arbitrary Cauchy surfaces from a different, but equally reasonable, set of assumptions. The conclusion is that if detectors are placed along any Cauchy surface $Sigma$, then the observed particle configuration on $Sigma$ has distribution $|Psi_Sigma|^2$, suitably understood. The main different assumption is that the Born and collapse rules hold on any spacelike hyperplane, i.e., at any time coordinate in any Lorentz frame. Heuristically, this follows if the dynamics of the detectors is Lorentz invariant. In addition, we assume, as did Lienert and Tumulka, that there is no interaction faster than light and that there is no propagation faster than light.
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