Universal geometric approach to uncertainty, entropy, and information

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 59, Issue 4, 1999, Pages 2602-2615

  Hall, Michael J W ^a  

^a AUSTRALIAN NATIONAL UNIVERSITY   (Australia)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001361646     PISSN: 10502947     EISSN: 10941622     Source Type: Journal    
DOI: 10.1103/PhysRevA.59.2602     Document Type: Article

References (42)

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18. To derive Eq. (4), define (Formula presented) implying (Formula presented)
19. R. Ash, Information Theory (Wiley, New York, 1965), Chaps. 1, 2, and 6.
20. This follows from a straightforward variational calculation, and may be generalized to all Renyi lengths.
21. For the general case of nonoverlapping (Formula presented) and (Formula presented) with equal Renyi lengths (Formula presented) one finds for the mixture (Formula presented) that (Formula presented) differentiating with respect to (Formula presented) yields a maximum of 2, attained for (Formula presented)
22. For an example in the context of optical phase, where the rms deviation gives a rather misleading indication of the spread of the (near optimally phase concentrated) coherent phase states, see M.J.W. Hall, J. Mod. Opt. 40, 809 (1993).
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28. Of course, in special contexts, restricted interpretations of ensemble entropy are possible: for example, as the logarithm of the number of microstates in statistical mechanics 5, or as the amount of information obtainable by measurement on discrete classical and quantum ensembles 2636 [see also L. Brillouin, Science and Information Theory, 2nd ed. (Academic, New York, 1962), Chap. 1]. Such interpretations are not applicable more generally, however. For example, for continuous classical ensembles the entropy can take any negative value, which is not the case for numbers of microstates and information gain. In contrast, in the geometric interpretation such negative values merely correspond to the possibility of arbitrarily small ensemble volumes.
29. Units of bits correspond to dividing (Formula presented) in Eq. (31) by (Formula presented)
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34. The signals are, in general, represented by ensembles, to take into account any noise processes in the transmitter and channel medium prior to detection at the receiver.
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40. A.S. Holevo, e-print quant-ph/9611023.
41. A complete set of coherent states (Formula presented) corresponds to a joint position and momentum measurement described by the positive-operator-valued measure (Formula presented) (see, e.g., Ref. 3); the statistics of the corresponding observable (Formula presented) for state (Formula presented) are then given by the Husimi distribution (Formula presented)
42. M.J.W. Hall, Phys. Rev. Lett. 74, 3307 (1995).