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arXiv:0802.2391 · Journal of Physics Conference Series

Complementarity and the algebraic structure of finite quantum systems

Dėnes Petz

Published Jan 7, 2009• 6 citations
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Abstract

Complementarity is a very old concept in quantum mechanics, however the rigorous definition is not so old. Complementarity of orthogonal bases can be formulated in terms of maximal Abelian algebras and this may lead to avoid commutativity of the subalgebras. In some sense this means that quantum information is treated instead of classical (measurement) information. The subject is to extend to the quantum case some features from the classical case. This includes construction of complementary subalgebras. The Bell basis has also some relation. Several open questions are discussed.

DOI
10.1088/1742-6596/143/1/012011

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Complementarity is a very old concept in quantum mechanics, however the rigorous definition is not so old. Complementarity of orthogonal bases can be formulated in terms of maximal Abelian algebras and this may lead to avoid commutativity of the subalgebras. In some sense this means that quantum information is treated instead of classical (measurement) information. The subject is to extend to the quantum case some features from the classical case. This includes construction of complementary subalgebras. The Bell basis has also some relation. Several open questions are discussed.
Abstract: Reduction of a state of a quantum system to a subsystem gives partial quantum information about the true state of the total system. Two subalgebras A1 and A2 of B(H) are called complementary if the traceless subspaces of A1 and A2 are orthogonal (with respect to the Hilbert-Schmidt inner product). When both subalgebras are maximal Abelian, then the concept reduces to complementary observables or mutually unbiased bases. In the paper several characterizations of complementary subalgebras are given in the general case and several examples are presented. For a 4-level quantum system, the structure of complementary subalgebras can be described very well, the Cartan decomposition of unitaries plays a role. It turns out that a measurement corresponding to the Bell basis is complementary to any local measurement of the two-qubit-system.
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.