{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:31:49Z","timestamp":1760243509112,"version":"build-2065373602"},"reference-count":17,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2013,8,2]],"date-time":"2013-08-02T00:00:00Z","timestamp":1375401600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>A quantum measurement can be regarded as a communication channel, in which the parameters of the state are expressed only in the probabilities of the outcomes of the measurement. We begin this paper by considering, in a non-quantum-mechanical setting, the problem of communicating through probabilities. For example, a sender, Alice, wants to convey to a receiver, Bob, the value of a continuous variable, \u03b8, but her only means of conveying this value is by sending Bob a coin in which the value of \u03b8 is encoded in the probability of heads. We ask what the optimal encoding is when Bob will be allowed to flip the coin only a finite number of times. As the number of tosses goes to infinity, we find that the optimal encoding is the same as what nature would do if we lived in a world governed by real-vector-space quantum theory. We then ask whether the problem might be modified, so that the optimal communication strategy would be consistent with standard, complex-vector-space quantum theory.<\/jats:p>","DOI":"10.3390\/e15083220","type":"journal-article","created":{"date-parts":[[2013,8,2]],"date-time":"2013-08-02T11:56:22Z","timestamp":1375444582000},"page":"3130-3147","update-policy":"https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Communicating through Probabilities: Does Quantum Theory Optimize the Transfer of Information?"],"prefix":"10.3390","volume":"15","author":[{"given":"William","family":"Wootters","sequence":"first","affiliation":[{"name":"Department of Physics, Williams College, Williamstown, MA 01267, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2013,8,2]]},"reference":[{"key":"ref_1","unstructured":"Bohm, D. (1951). 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