  This paper contributes to a burgeoning area of investigation, the ambiguity
inherent in mathematics and the implications for physics of this ambiguity. To
display the mathematical form of equations of quantum theory used to describe
experiments, we make explicit the knobs by which the devices of an experiment
are arranged and adjusted. A quantum description comes in two parts: (1) a
statement of results of an experiment, expressed by probabilities of detections
as functions of knob settings, and (2) an explanation of how we think these
results come about, expressed by linear operators, also as functions of knob
settings. Because quantum mechanics separates the two parts of any description,
it is known that between the statements of results and the explanations lurks a
logical gap: given any statement of results one has a choice of explanations.
  Here we work out some consequences of this openness to choice. We show how
quantum theory as mathematical language in which to describe experiments
necessarily involves multiple descriptions: multiple explanations of a given
result, as well as multiple statements of results and multiple arrangement of
knobs. Appreciating these multiplicities resolves what otherwise is a confusion
in the concept of invariance. Implications of multiplicity of description for
the security of quantum key distribution are noted.
