dorsal/arxiv
View SchemaInformation-disturbance tradeoff in quantum measurement on the uniform ensemble and on the mutually unbiased bases
| Authors | Howard Barnum |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0205155 |
| URL | https://arxiv.org/abs/quant-ph/0205155 |
Abstract
I consider the tradeoff between the information gained about an initially unknown quantum state, and the disturbance caused to that state by the measurement process. I show that for any distribution of initial states, the information-disturbance frontier is convex, and disturbance is nondecreasing with information gain. I consider the most general model of quantum measurements, and all post-measurement dynamics compatible with a given measurement. For the uniform initial distribution over states, I show that the least-disturbing way of making any measurement is with conditional dynamics satisfying a generalization of the projection postulate, the ``square-root dynamics.'' Thus, procedures for achieving a point on the information-disturbance frontier may be assumed to involve such conditional dynamics. Also, the information-disturbance frontier for the uniform ensemble may be achieved with ``isotropic'' (unitarily covariant) dynamics. This results in a significant simplification of the optimization problem for calculating the tradeoff in this case, giving hope for a closed-form solution. I also show that the discrete ensembles uniform on the d(d+1) vectors of a certain set of d+1 ``mutually unbiased'' or conjugate bases in d dimensions form spherical 2-designs in CP_{d-1} when d is a power of an odd prime. This implies that many of the results of the paper apply also to these discrete ensembles.
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"abstract": "I consider the tradeoff between the information gained about an initially\nunknown quantum state, and the disturbance caused to that state by the\nmeasurement process. I show that for any distribution of initial states, the\ninformation-disturbance frontier is convex, and disturbance is nondecreasing\nwith information gain. I consider the most general model of quantum\nmeasurements, and all post-measurement dynamics compatible with a given\nmeasurement. For the uniform initial distribution over states, I show that the\nleast-disturbing way of making any measurement is with conditional dynamics\nsatisfying a generalization of the projection postulate, the ``square-root\ndynamics.\u0027\u0027 Thus, procedures for achieving a point on the\ninformation-disturbance frontier may be assumed to involve such conditional\ndynamics. Also, the information-disturbance frontier for the uniform ensemble\nmay be achieved with ``isotropic\u0027\u0027 (unitarily covariant) dynamics. This results\nin a significant simplification of the optimization problem for calculating the\ntradeoff in this case, giving hope for a closed-form solution. I also show that\nthe discrete ensembles uniform on the d(d+1) vectors of a certain set of d+1\n``mutually unbiased\u0027\u0027 or conjugate bases in d dimensions form spherical\n2-designs in CP_{d-1} when d is a power of an odd prime. This implies that many\nof the results of the paper apply also to these discrete ensembles.",
"arxiv_id": "quant-ph/0205155",
"authors": [
"Howard Barnum"
],
"categories": [
"quant-ph"
],
"title": "Information-disturbance tradeoff in quantum measurement on the uniform ensemble and on the mutually unbiased bases",
"url": "https://arxiv.org/abs/quant-ph/0205155"
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