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 quantum information, complementarity of quantum mechanical observables plays a key role. The eigenstates of two complementary observables form a pair of mutually unbiased bases (MUBs). More generally, a set of MUBs consists of bases that are all pairwise unbiased. Except for specific dimensions of the Hilbert space, the maximal sets of MUBs are unknown in general. Even for a dimension as low as six, the identification of a maximal set of MUBs remains an open problem, although there is strong numerical evidence that no more than three simultaneous MUBs do exist. Here, by exploiting a newly developed holographic technique, we implement and test different sets of three MUBs for a single photon six-dimensional quantum state (a "qusix"), encoded exploiting polarization and orbital angular momentum of photons. A close agreement is observed between theory and experiments. Our results can find applications in state tomography, quantitative wave-particle duality, quantum key distribution.
We consider an unknown pure stabilizer state of
n qudits with a fixed prime local dimension
p, together with prescribed commuting Pauli observables. Rather than learning the whole state, the learner must recover only complementary stabilizers that commute with the prescribed observables, together with their eigenvalues. Once the prescribed measurement is performed, this information and the observed outcome determine the corresponding conditional stabilizer state. We compare the number
k of copies available to the learner with the number
n of qudits and refer to their ratio
k/n as the copy rate.
Suppose that the dimension
m of the requested complementary stabilizer subspace satisfies
m=βn+o(n) for a fixed
0<β≤1. If the copy rate converges to a value below one, the optimal exact recovery probabilities and verification scores converge to zero, both in the worst case and under the uniform prior. Conversely, if the number
k of copies minus the number
n of qudits tends to positive infinity, the entire stabilizer state can be identified with probability tending to one, and the requested complementary information can then be extracted.
Thus, restricting the learning target to the stabilizer information needed after the prescribed measurement yields no copy-rate discount.