Nonlinear modifications of quantum mechanics have a troubled history.They were initially studied for many promising reasons: resolving the measurement problem, formulating a theory of quantum mechanics and gravity, and understanding the limits of standard quantum mechanics.However, certain non-linear theories have been experimentally tested and failed.More significantly, it has been shown that, in general, deterministic non-linear theories can be used for superluminal communication.We highlight another serious issue: the distribution of measurement results predicted by non-linear quantum mechanics depends on the formulation of quantum mechanics.In other words, Born's rule cannot be uniquely extended to non-linear quantum mechanics.We present these generalizations of Born's rule, and then examine whether some exclude superluminal communication.We determine that a large class do not allow for superluminal communication, but many lack a consistent definition.Nonetheless, we find a single extension of Born's rule with a sound operational definition, and that does not exhibit superluminal communication.The non-linear time-evolution leading to a certain measurement event is driven by the state conditioned on measurements that lie within the past light cone of that event.
Nonlinear modifications of quantum mechanics have a troubled history.They were initially studied for many promising reasons: resolving the measurement problem, formulating a theory of quantum mechanics and gravity, and understanding the limits of standard quantum mechanics.However, certain non-linear theories have been experimentally tested and failed.More significantly, it has been shown that, in general, deterministic non-linear theories can be used for superluminal communication.We highlight another serious issue: the distribution of measurement results predicted by non-linear quantum mechanics depends on the formulation of quantum mechanics.In other words, Born's rule cannot be uniquely extended to non-linear quantum mechanics.We present these generalizations of Born's rule, and then examine whether some exclude superluminal communication.We determine that a large class do not allow for superluminal communication, but many lack a consistent definition.Nonetheless, we find a single extension of Born's rule with a sound operational definition, and that does not exhibit superluminal communication.The non-linear time-evolution leading to a certain measurement event is driven by the state conditioned on measurements that lie within the past light cone of that event.
Nonlinear modifications of quantum mechanics have a troubled history. They were initially studied for many promising reasons: resolving the measurement problem, testing the limits of standard quantum mechanics, and reconciling it with gravity. Two results substantially undermined the credibility of non-linear theories. Some have been experimentally refuted, and more importantly, all deterministic non-linear theories can be used for superluminal communication. However, these results are unconvincing because they overlook the fact that the distribution of measurement results predicted by non-linear quantum mechanics depends on the interpretation of quantum mechanics that one uses. For instance, although the Everett and Copenhagen interpretations agree on the expression of Born's rule for the outcomes of multiple measurements in linear quantum mechanics, they disagree in non-linear quantum mechanics. We present the range of expressions of Born's rule that can be obtained by applying different formulations of quantum mechanics to a class of non-linear quantum theories. We then determine that many do not allow for superluminal communication but only two seem to have a reasonable justification. The first is the Everett interpretation, and the second, which we name causal-conditional, states that a measurement broadcasts its outcome to degrees of freedom in its future light-cone, who update the wavefunction that their non-linear Hamiltonian depends on according to this new information.
Quantum theory is compatible with special relativity. In particular, though measurements on entangled systems are correlated in a way that cannot be reproduced by local hidden variables, they cannot be used for superluminal signalling. As Czachor, Gisin and Polchinski pointed out, this is not true for general nonlinear modifications of the Schroedinger equation. Excluding superluminal signalling has thus been taken to rule out most nonlinear versions of quantum theory. The no superluminal signalling constraint has also been used for alternative derivations of the optimal fidelities attainable for imperfect quantum cloning and other operations. These results apply to theories satisfying the rule that their predictions for widely separated and slowly moving entangled systems can be approximated by non-relativistic equations of motion with respect to a preferred time coordinate. This paper describes a natural way in which this rule might fail to hold. In particular, it is shown that quantum readout devices which display the values of localised pure states need not allow superluminal signalling, provided that the devices display the values of the states of entangled subsystems as defined in a non-standard, but natural, way. It follows that any locally defined nonlinear evolution of pure states can be made consistent with Minkowski causality.
Quantum theory is compatible with special relativity. In particular, though measurements on entangled systems are correlated in a way that cannot be reproduced by local hidden variables, they cannot be used for superluminal signaling. As Czachor, Gisin, and Polchinski pointed out, this is not generally true of general nonlinear modifications of the Schr\"odinger equation. Excluding superluminal signaling has thus been taken to rule out most nonlinear versions of quantum theory. The no-superluminal-signaling constraint has also been used for alternative derivations of the optimal fidelities attainable for imperfect quantum cloning and other operations. These results apply to theories satisfying the rule that their predictions for widely separated and slowly moving entangled systems can be approximated by nonrelativistic equations of motion with respect to a preferred time coordinate. This paper describes a natural way in which this rule might fail to hold. In particular, it is shown that quantum readout devices which display the values of localized pure states need not allow superluminal signaling, provided that the devices display the values of the states of entangled subsystems as defined in a nonstandard, although natural, way. It follows that any locally defined nonlinear evolution of pure states can be made consistent with Minkowski causality.
A realist description of our universe requires a twofold concept of locality. On one hand, there are the strictly Einstein-local interactions which generate the time evolution. On the other hand, the quantum state space calls for a non-local description of multi-particle states. This article uses a behavioristic approach to argue, that an observer in a universe like this has to rely on local interactions to learn about its properties and behavior. Such an observer is fundamentally restricted in his ability to understand and structurally reconstruct the individual local physical universe. We argue, that this reconstruction based on dynamically available information is the defining process of observation in quantum theory. The observer-centric view of the global quantum dynamics is shown to be non-unitary and non-linear in general, even if the universe itself evolves unitarily. Interactions with massless free particles are found to have great influence on observation, because of their special role in the light-cone structure of an Einstein-local universe. For a specific scattering process with a photon of unknown state, the observed outcome can be subjectively random and follow the Born statistic, while the output state is really determined by the photon polarization. Based on this result, a theory of quantum measurement is formulated, which describes a measurement device as a cascade of scattering events.
Abstract A unique characteristic of quantum mechanics is entanglement describing correlations between particles irrespective of their locations. This property, called non-locality, has no classical analogue. Over the past few years, quantum physicists have reached a consensus that we lack a physical theory to account for a class of states whose non-local character exceeds the bounds allowed by quantum mechanics. Motivated by our observation that an extension of the Schrödinger equation with non-linear terms is directly linked to a relaxation of Born’s rule, an axiom of quantum mechanics, we derive a physical theory that accounts for such hyper-correlated states and modifies Born’s rule. We model correlated particles with a generalized probability theory whose dynamics are described with a non-linear version of Schrödinger’s equation and demonstrate how that deviates from the standard formulation of quantum mechanics in experimental probability-prediction. We show also that the violation of the Clauser-Horn-Shimony-Holt inequality, the amount of non-locality, is proportional to the degree of non-linearity, which can be experimentally tested.
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.