We characterize minimal measurement setups for validating the quantum coherence of an unknown quantum state. We show that for a d -level system, the optimal strategy consists of measuring d orthonormal bases such that each measured basis is mutually unbiased with respect to the reference basis, and together with the reference basis they form an informationally complete set of measurements. We prove that, in general, any strategy capable of validating quantum coherence allows one to evaluate also the exact value of coherence. We then give an explicit construction of the optimal measurements for arbitrary dimensions, and we derive a reconstruction formula for the off-diagonal terms. We also demonstrate that the same measurement setup is optimal for the modified task of verifying if the coherence is above or below a given threshold value. Finally, we show that the certification of entanglement of bipartite maximally correlated states is intimately connected with the certification of coherence.
We study the verification of maximally entangled states by virtue of the simplest measurement settings: local projective measurements without adaption. We show that optimal protocols are in one-to-one correspondence with complex projective 2-designs constructed from orthonormal bases. Optimal protocols with minimal measurement settings are in one-to-one correspondence with complete sets of mutually unbiased bases. Based on this observation, optimal protocols are constructed explicitly for any local dimension, which can also be applied to estimating the fidelity with the target state and to detecting entanglement. In addition, we show that incomplete sets of mutually unbiased bases are optimal for verifying maximally entangled states when the number of measurement settings is restricted. Moreover, we construct optimal protocols for the adversarial scenario in which state preparation is not trusted. The number of tests has the same scaling behavior as the counterpart for the nonadversarial scenario; the overhead is no more than three times. We also show that the entanglement of the maximally entangled state can be certified with any given significance level using only one test as long as the local dimension is large enough.
Quantum coherence is an essential ingredient in quantum information processing and plays a central role in emergent fields such as nanoscale thermodynamics and quantum biology. However, our understanding and quantitative characterization of coherence as an operational resource are still very limited. Here we show that any degree of coherence with respect to some reference basis can be converted to entanglement via incoherent operations. This finding allows us to define a novel general class of measures of coherence for a quantum system of arbitrary dimension, in terms of the maximum bipartite entanglement that can be generated via incoherent operations applied to the system and an incoherent ancilla. The resulting measures are proven to be valid coherence monotones satisfying all the requirements dictated by the resource theory of quantum coherence. We demonstrate the usefulness of our approach by proving that the fidelity-based geometric measure of coherence is a full convex coherence monotone, and deriving a closed formula for it on arbitrary single-qubit states. Our work provides a clear quantitative and operational connection between coherence and entanglement, two landmark manifestations of quantum theory and both key enablers for quantum technologies.
The resource theoretic measure of quantum coherence is basis dependent, and the amount of coherence contained in a state is different in different bases. We obtained analytical solutions for the maximum coherence by optimizing the reference basis and highlighted the essential role of the mutually unbiased bases (MUBs) on attaining the maximum coherence. Apart from the relative entropy of coherence, we showed that the MUBs are optimal for the robustness of coherence, the coherence weight, and the modified skew information measure of coherence for any state. Moreover, the MUBs are optimal for all the faithful coherence measures if the state is pure or is of the single qubit. We also highlighted an upper bound for the
l1 norm of coherence and compared it with the other bounds as well as the maximum one attainable by optimizing the reference basis.
A pair of quantum observables diagonal in the same basis can be measured jointly, so some coherence is obviously needed for measurement incompatibility. It is equally clear that these quantum resources are not comparable for general observables. However, in the spirit of complementarity between path and superposition, one expects a precise measurement of the incoherent basis label to be incompatible with sufficiently coherent observables. Here we formalise this idea in a very general framework, involving pairs where one observable is incoherent (diagonal), and the coherence in the other is adjusted by damping via genuinely incoherent operations. An extreme case is a basis measurement mutually unbiased to the incoherent basis, leading to an interesting interplay between coherence and complementarity. In terms of measurement theory, we show that if coherence is insufficient to sustain incompatibility, the associated joint measurements have sequential implementations via incoherent instruments. This yields a concrete method for solving joint measurability problems, tractable even in large systems without resorting to numerical optimisation: we find analytical bounds for the coherence needed for incompatibility, in terms of a divergence measure characterising basis state distinguishability of incoherent observables. We verify the consistency of our theory by a quick proof of the known noise bound for the incompatibility of mutually unbiased bases, and then apply it to study emergent classicality in the spin-boson model of pure decoherence on an N-qubit open quantum system.
In this paper we consider the quantum resources required to maximize the mean values of any nontrivial quantum observable. We show that the task of maximizing the mean value of an observable is equivalent to maximizing some form of coherence, up to the application of an incoherent operation. For any nontrivial observable, there always exists a set of preferred basis states where the superposition between such states is always useful for optimizing the mean value of a quantum observable. The usefulness of such states is expressed in terms of an infinitely large family of valid coherence measures which is then shown to be efficiently computable via a semidefinite program. We also show that these coherence measures respect a hierarchy that gives the robustness of coherence and the l 1 norm of coherence additional operational significance in terms of such optimization tasks.
The next frontier in device-independent quantum information lies in the certification of scalable and parallel quantum resources, which underpin advanced quantum technologies. We put forth a simultaneous self-testing framework for maximally entangled two-qudit state of local dimension
m∗=2⌊n/2⌋ (equivalently
⌊n/2⌋ copies of maximally entangled two-qubit pairs), together with
n numbers of anti-commuting observables on one side. To this end, we employ an
n-settings Bell inequality comprising two space-like separated observers, Alice and Bob, having
2n−1 and
n number of measurement settings, respectively. We derive the local ontic bound of this inequality and, crucially, employ the Sum-of-Squares decomposition to determine the optimal quantum bound without presupposing the dimension of the state or observables. We then establish that any physical realisation achieving the maximal quantum violation must, up to local isometries and complex conjugation, correspond to a reference strategy consisting of a maximally entangled state of local dimension of at least
2⌊n/2⌋ and local observables forming an irreducible representation of the Clifford algebra. This construction thereby demonstrates that the minimal dimension compatible with
n mutually anticommuting observables is naturally self-tested by the maximal violation of the proposed Bell functional. Finally, we analyse the robustness of the protocol by establishing quantitative bounds relating deviations in the observed Bell value to the fidelity between the realised and the ideal strategies. Our results thus provide a scalable, dimension-independent route for the certification of high-dimensional entanglement and Clifford measurements in a fully device-independent framework.