Operationally Invariant Quantum Information
In quantum measurements the observer may decide to measure different sets of complementary variables thus gaining certainty about one or more variables at the expense of loosing certainty about other(s). In the case of spin these could be the projections along rotated directions, e.g., where the uncertainty in one component is reduced but the one in another component is increased correspondingly. Intuitively one expects that the total uncertainty or, equivalently, the total information carried by the system is invariant under such transformation from one complete set of complementary variables to another.
In 1999 we introduced a new measure of information in quantum mechanics which takes into account that for quantum systems the only feature known before an experiment is performed are the probabilities for various events to occur. We showed that the sum of the individual measures of information for mutually complementary observations is invariant under the choice of the particular set of complementary observations and conserved if there is no information exchange with an environment. That operational quantum information invariant results in N bits of information for a system consisting of N qubits.
In a separate work we showed that because of the completely different root of a quantum measurement as compared to a classical measurement conceptual difficulties arise when we try to define the information gain in a quantum measurement using the notion of Shannon information. This is because the Shannon measure describes our ignorance about properties of a system, which can be considered to pre-exist prior to and independent of observation.
Fig.: Two different sets of mutually complementary spin measurements (the full sets include also the spin measurement along the x-axis which is not shown in the figures). Rotating the spin components the uncertainty in one component is reduced but the one in another component is increased correspondingly. The total uncertainty, or equivalently amount of information, remains, however, conserved.
C. Brukner and A. Zeilinger,
Operationally Invariant Information in Quantum Measurements
Phys. Rev. Lett. 83 (1999) 3354 / e-print)
C. Brukner, and A. Zeilinger,
Conceptual Inadequacy of the Shannon Information in Quantum Measurements
