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arXiv:1908.03828 · Journal of Stochastic Analysis

Pauli Matrices: A Triple of Accardi Complementary Observables

Stephen Bruce Sontz

Published Nov 6, 2020• 0 citations
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Abstract

The definition due to Accardi of a pair of complementary observables is adapted to the context of the Lie algebra su(2).We show that the pair of Pauli matrices A, B associated to the unit directions α and β in R 3 are Accardi complementary if and only if α and β are orthogonal if and only if A and B are orthogonal.In particular, any pair of the standard triple of Pauli matrices is complementary.

DOI
10.31390/josa.1.4.02

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The definition due to Accardi of a pair of complementary observables is adapted to the context of the Lie algebra su(2).We show that the pair of Pauli matrices A, B associated to the unit directions α and β in R 3 are Accardi complementary if and only if α and β are orthogonal if and only if A and B are orthogonal.In particular, any pair of the standard triple of Pauli matrices is complementary.
The link between a quantum spin 1/2 and its associated su(2) algebra of Pauli spin matrices with Clifford algebra and quaternions is well known. A pair of spins or qubits, which are important throughout the field of quantum information for describing logic gates and entangled states, has similarly an su(4) algebra. We develop connections between this algebra and its subalgebras with the projective plane of seven elements (also related to octonions) and other entities in projective geometry and design theory.
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.
In this note we show that the momentum and position operators of μμμ-deformed quantum mechanics for μ>0μ> 0μ>0 are not Accardi complementary in a sense that we will define. We conjecture that this is also true if −1/2<μ<0-1/2 < μ< 0−1/2<μ<0.
Representation of su(8)\mathfrak{su}(8)su(8) in Pauli Basis
Quantum computation started to become significant field of studies as it hold great promising towards the upgrade of our current computational power. Studying the evolution of quantum states serves as a good fundamental in understanding quantum information which lead to quantum computation. This was assisted with the respective mathematical tools such as Lie group and Lie algebra. In this study, the Lie algebra of su(8)\mathfrak{su}(8)su(8) is represented in tensor product between three Pauli matrices. This is done by constructing the generalized Gell-Mann matrices and compared to the Pauli basis. This study will explicitly shows the one-to-one correlation of Gell-Mann matrices with the Pauli basis resembled change of coordinates. This is particularly useful when dealing with quantum circuit problems.