Preparations and Tests. A preparation is an experimental procedure which is completely and exactly specified ( like a recipe ). A test starts like a preparation, but includes a final step where previously unknown information is obtained by the observer. The distinction between the two involves a direction for the flow of time, hence the asymmetry between past and future is fundamental to the axioms of quantum theory, even though its dynamical laws are invariant under time reversal.
Scope. Quantum theory is a set of rules which allows us to calculate the probabilities of outcomes resulting from specified preparations.
Measurement. Measurement is not passive acquisition of knowledge, but an active process which alters the system in question. Quantum theory is fundamentally incompatible with the notion that measurements reveal some unknown, but previously existing reality. There is no objective reality distinct from the numbers obtained via measurement.
Quantum System. A quantum system as defined by an equivalence class of preparations.
Quantum State. A state is characterised by the probabilities of the various outcomes of every conceivable test.
Consistency. The probability of obtaining conflicting results from equivalent measurements must be arbitrarily low.
Repeatability. If identical tests are performed in sequence with negligible time in between them, and they yield identical outcomes, then they are called repeatable.
Maximal Tests. Let N be the maximum number of different outcomes in a test of a given system. Then, any test that has exactly N different outcomes is called maximal, or complete.
Pure States. If a system is prepared so that it certainly yields a predictable outcome in a specified maximal test, then the system is said to be in a pure state, and the various outcomes of any other tests performed have definite probabilities.
Random Mixtures. Systems with N states can be prepared in such a way that every unbiased maximal test has the same probability for each outcome. This is equivalent to a complete lack of knowledge of the past history of the quantum system. Random mixtures are unique states, and they are dynamically invariant – i.e. if allowed to evolve independently, they will remain random mixtures. This is an example of the conservation of entropy.
Law of Reciprocity. Let A and B denote pure states. Then the probability of observing outcome A in a maximal test following preparation of state B, is equal to the probability of observing outcome B from preparation of state A.
Principle of Interference. If a quantum system can follow several possible paths to a given test, the probability for each outcome of that test is not in general the sum of the separate probabilities for each path.
Transition Amplitudes. If several paths are available from the initial state to the final outcome, and if there is no dynamical process allowing to distinguish between the paths, then the phases of the transition amplitudes can be chosen in such a way that the complete amplitude for the final outcome is the sum of the amplitudes of the various paths. Transition amplitudes are fundamental objects from which observable probabilities derive.
Source : Asher Peres, Quantum Theory : Concepts and Methods, Kluwer Academic Publishers, 2002