We show that the amount of complementarity that a quantum system can exhibit depends on which complementary properties one is considering. Consider a set of mutually unbiased bases (MUBs) which corresponds to the spectral decomposition of some maximally complementary properties: The perfect knowledge of one, i.e. the state of the system is one of the basis states, implies that all the others are completely unknown, namely the measurement of another property will find any of its possible outcomes with uniform probability. We show that, depending on which three of the MUBs we choose, a 5-dimensional system can have different degrees of complementarity, measured using the minimum over all the system states of the sum of the entropies and of the variances of the measurement outcomes related to the three chosen observables. This property was first found experimentally, and demonstrates that an experimental discovery can disclose quantum information effects.
Quantum uncertainty relations impose fundamental limits on the joint knowledge that can be acquired from complementary observables: Perfect knowledge of a quantum state in one basis implies maximal indetermination in all other mutually unbiased bases (MUBs). Uncertainty relations derived from joint properties of the MUBs are generally assumed to be uniform, irrespective of the specific observables chosen within a set. In this work, we demonstrate instead that the uncertainty relations can depend on the choice of observables. Through both experimental observation and numerical methods, we show that selecting different sets of three MUBs in a five-dimensional quantum system results in distinct uncertainty bounds, i.e., in varying degrees of complementarity, in terms of both entropy and variance.
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.
Mutually unbiased bases (MUBs) are a crucial ingredient for many protocols in quantum information processing. Measurements performed in these bases are unbiased to the maximally possible extent, which is used to prove the randomness or secrecy of measurement results. In this work we show that certain properties of sets of MUBs crucially depend on their specific choice, including, somewhat surprisingly, measurement outcome labeling. If measurements are chosen in a coherent way, the secrecy of the result can be completely lost for specific sets of MUB measurements but partially retained for others. This could potentially impact a broad spectrum of applications where MUBs are utilized.
The concept of quantum coherence, including various ways to quantify the degree of coherence with respect to the prescribed basis, is currently the subject of active research. The complementarity of quantum coherence in different bases was studied by deriving upper bounds on the sum of the corresponding measures. To obtain a two-sided estimate, lower bounds on the coherence quantifiers are also of interest. Such bounds are naturally referred to as uncertainty relations for quantum coherence. We obtain new uncertainty relations for coherence quantifiers averaged with respect to a set of mutually unbiased bases (MUBs). To quantify the degree of coherence, the relative entropy of coherence and the geometric coherence are used. Further, we also derive novel state-independent uncertainty relations for a set of MUBs in terms of the min-entropy.
In the framework of mutually unbiased bases (MUBs), a measurement in one basis gives no information about the outcomes of measurements in another basis. Here we relax the no-information condition by allowing the
d outcomes to be predicted according to a predefined probability distribution $q=({q}_{0},...,{q}_{d\ensuremath{-}1})$. The notion of mutual unbiasedness, however, is preserved by requiring that the extracted information is the same for any preparation and any measurement; regardless of which state from which basis is chosen to prepare the system, the outcomes of measuring the system with respect to the other basis generate the same probability distribution. In light of this, we define the notion of mutually equibiased bases (MEBs) such that within each basis the states are equibiased with respect to the states of the other basis and that the bases are mutually equibiased with respect to each other. For
d=2,3, we derive a set of
d+1 MEBs. The mutual equibiasedness imposes nontrivial constraints on the distribution
q, leading, for
d=3, to the restriction $1/3\ensuremath{\le}\ensuremath{\mu}\ensuremath{\le}1/2$, where $\ensuremath{\mu}={\ensuremath{\sum}}_{k=0}^{2}{q}_{k}^{2}$. To capture the incompatibility of the measurements in MEBs, we derive an inequality for the probabilities of projective measurements in a qudit system, which yields an associated entropic uncertainty inequality. Finally, we construct a class of positive maps and their associated entanglement witnesses based on MEBs. While an entanglement witness constructed from MUBs is generally finer than one based on MEBs when both use the same number of bases, for certain values of the index $\ensuremath{\mu}$, employing a larger set of MEBs can yield a finer witness. We illustrate this behavior using isotropic states of a $3\ifmmode\times\else\texttimes\fi{}3$ system. Our results reveal that not all bases in a set of
L MEBs can contribute to the entanglement detection. A constraint, dependent on the probability distribution
q, limits the number of MEBs that can be employed.
Mutually unbiased bases (MUBs) are highly symmetric bases on complex Hilbert spaces, and the corresponding rank-1 projective measurements are ubiquitous in quantum information theory. In this work, we study a recently introduced generalisation of MUBs called mutually unbiased measurements (MUMs). These measurements inherit the essential property of complementarity from MUBs, but the Hilbert space dimension is no longer required to match the number of outcomes. This operational complementarity property renders MUMs highly useful for device-independent quantum information processing. It has been shown that MUMs are strictly more general than MUBs. In this work we provide a complete proof of the characterisation of MUMs that are direct sums of MUBs. We then construct new examples of MUMs that are not direct sums of MUBs. A crucial technical tool for this construction is a correspondence with quaternionic Hadamard matrices, which allows us to map known examples of such matrices to MUMs that are not direct sums of MUBs. Furthermore, we show that—in stark contrast with MUBs—the number of MUMs for a fixed outcome number is unbounded. Next, we focus on the use of MUMs in quantum communication. We demonstrate how any pair of MUMs with
doutcomes defines a
d-dimensional superdense coding protocol. Using MUMs that are not direct sums of MUBs, we disprove a recent conjecture due to Nayak and Yuen on the rigidity of superdense coding, for infinitely many dimensions. The superdense coding protocols arising in the refutation reveal how shared entanglement may be used in a manner heretofore unknown.