25 May 2005 Invariant quantum ensemble metrics
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Abstract
We recently discovered when two ensembles of N qubits are initialized to random phases, then the Cartesian distance metric between the ensemble states is approximately sqrt(2N), with a standard deviation of 1/2. This research relates inner product and quantum ensemble metrics to the “standard distance” metrics defined by correlithm object theory. Correlithm object theory describes how randomly selected points (or COs) in high dimensional bounded spaces can be used as soft tokens to represent states at the ensemble level. The initial CO research was performed using unit N-cubes, but under Air Force SBIR contract, we extended this theory for other bounded metric spaces such as binary N-spaces, complex N-spaces and Hilbert spaces. A quantum encoded CO can be created by initializing an ensemble of qubits based on a phase ensemble of uniformly distributed, randomly chosen phases with values from 0 to 2pi. If a quantum encoded CO token is created as a qubit ensemble Q initialized using a specific phase ensemble P, upon quantum measurement using another phase ensemble of basis states B it produces a binary ensemble A of measurement answers. Multiple trials using the exact same ensembles P and B generate answer ensembles that are correlated with each other, since their distance metric is 70.7% of the binary standard distance of sqrt(N/2). This means that quantum encoded CO tokens survive this measurement process. In contrast, choosing new random ensembles P or B for each trial generates uncorrelated answer ensembles. This paper describes and demonstrates how quantum measurement acts as a noise injection process from the correlithm object perspective.
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Douglas J. Matzke, P. Nick Lawrence, "Invariant quantum ensemble metrics", Proc. SPIE 5815, Quantum Information and Computation III, (25 May 2005); doi: 10.1117/12.603511; https://doi.org/10.1117/12.603511
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