Complementary observables

STEPHEN D. BRIERLEY, Andrei Yu. Khrennikov

Paul Busch

Stanley Gudder

Markus Frembs, Giovanni Natale, Christopher S. P. Wever +1 more

Observables in quantum mechanics are generally complementary, that is, they reveal mutually incompatible pieces of information about a given system. This property is not only a fundamental tenet of the quantum formalism, but also a key component in quantum cryptographic protocols. As such it has fuelled much research into finding sets of highly complementary observables. In its strongest form -- the one we consider in this work -- the information between complementary observables is not merely incompatible but mutually exclusive: maximal information about one observable implies no information about the other, and vice versa. Maximal sets of non-degenerate complementary observables are known to exist in systems of prime power dimension such as n-qubit systems. Here, we study complementarity of degenerate observables, specifically we prove that observables associated with the n-qubit Pauli group also exhibit complementarity under this restriction: first, we obtain a criterion for two such observables to be complementary and, second, we relate maximal sets of complementary observables with informational pure state complementarity equalities, further studied in two companion papers. Equivalently, these results can be formulated in terms of maximal complementary sets of (not necessarily maximal) Abelian subgroups of the Pauli group linking with (possibly coarse-grained) mutually unbiased bases. Finally, we prove that these complementarity sets entail strong entropic uncertainty relations.

Markus Grassl, Daniel McNulty, Ladislav Mišta +1 more

The eigenstates of complementary observables play an elementary role in quantum physics. Here the concept of unextendible sets of complementary observables is analyzed and a practical application in witnessing physical properties, such as the degree of purity of a state, is described.

Franklin E. Schroeck