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Register a new userK-Nearest Neighbor Approximation Via the Friend-of-a-Friend Principle
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Suppose $V$ is an $n$-element set where for each $x in V$, the elements of $V setminus {x}$ are ranked by their similarity to $x$. The $K$-nearest neighbor graph is a directed graph including an arc from each $x$ to the $K$ points of $V setminus {x}$ most similar to $x$. Constructive approximation to this graph using far fewer than $n^2$ comparisons is important for the analysis of large high-dimensional data sets. $K$-Nearest Neighbor Descent is a parameter-free heuristic where a sequence of graph approximations is constructed, in which second neighbors in one approximation are proposed as neighbors in the next. Run times in a test case fit an $O(n K^2 log{n})$ pattern. This bound is rigorously justified for a similar algorithm, using range queries, when applied to a homogeneous Poisson process in suitable dimension. However the basic algorithm fails to achieve subquadratic complexity on sets whose similarity rankings arise from a ``generic linear order on the $binom{n}{2}$ inter-point distances in a metric space.
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The measurement problem is seen as an ambiguity of quantum mechanics, or, beyond that, as a contradiction within the theory: Quantum mechanics offers two conflicting descriptions of the Wigners-friend experiment. As we argue in this note there are, however, obstacles from within quantum mechanics and regarding our perspective onto doing physics towards fully describing a measurement. We conclude that the ability to exhaustively describe a measurement is an assumption necessary for the common framing of the measurement problem and ensuing suggested solutions.
In a joint paper Jeff Bub and Itamar Pitowski argued that the quantum state represents `the credence function of a rational agent [...] who is updating probabilities on the basis of events that occur. In the famous thought experiment designed by Wigner, Wigners friend performs a measurement in an isolated laboratory which in turn is measured by Wigner. Here we consider Wigners friend as a rational agent and ask what her `credence function is. We find experimental situations in which the friend can convince herself that updating the probabilities on the basis of events that happen solely inside her laboratory is not rational and that conditioning needs to be extended to the information that is available outside of her laboratory. Since the latter can be transmitted into her laboratory, we conclude that the friend is entitled to employ Wigners perspective on quantum theory when making predictions about the measurements performed on the entire laboratory, in addition to her own perspective, when making predictions about the measurements performed inside the laboratory.
Wigners friend thought experiment is intended to reveal the inherent tension between unitary evolution and measurement collapse. On the basis of Wigners friend experiment, Brukner derives a no-go theorem for observer-independent facts. We construct an extended Wigners friend scenario including three laboratories, namely, Alices laboratory, Bobs laboratory and Charlies laboratory, where Alice, Bob and Charlie are standing outside the laboratories while their friends are placed inside their own laboratories. We consider quantum simulation via Q# quantum programming and also realize the primary quantum circuits using IBM quantum computers. Then, we calculate the probabilities and corresponding statistical uncertainties. It has been shown that the results of quantum simulation are clearly consistent with theoretical values, while it has a slightly higher error rates for the experimental results of quantum computers mainly because of a series of quantum gates, especially CNOT gates.
The Wigners friend paradox concerns one of the most puzzling problems of quantum mechanics: the consistent description of multiple nested observers. Recently, a variation of Wigners gedankenexperiment, introduced by Frauchiger and Renner, has lead to new debates about the self-consistency of quantum mechanics. At the core of the paradox lies the description of an observer and the object it measures as a closed system obeying the Schrodinger equation. We revisit this assumption to derive a necessary condition on a quantum system to behave as an observer. We then propose a simple single-photon interferometric setup implementing Frauchiger and Renners scenario, and use the derived condition to shed a new light on the assumptions leading to their paradox. From our description, we argue that the three apparently incompatible properties used to question the consistency of quantum mechanics correspond to two logically distinct contexts: either one assumes that Wigner has full control over his friends lab, or conversely that some parts of the labs remain unaffected by Wigners subsequent measurements. The first context may be seen as the quantum erasure of the memory of Wigners friend. We further show these properties are associated with observables which do not commute, and therefore cannot take well-defined values simultaneously. Consequently, the three contradictory properties never hold simultaneously.
With this note, we remember our friend Maria Krawczyk, who passed away this year, on May 24th. We briefly outline some of her physics interests and main accomplishments, and her great human and moral qualities.
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