Quantum coherence marks a deviation from classical physics, and has been studied as a resource for metrology and quantum computation.Finding reliable and effective methods for assessing its presence is then highly desirable.Coherence witnesses rely on measuring observables whose outcomes can guarantee that a state is not diagonal in a known reference basis.Here, we experimentally measure a type of coherence witness that uses pairwise state comparisons to identify superpositions in a basis-independent way.Our experiment uses a single interferometric setup to simultaneously measure the three pairwise overlaps among three single-photon states via Hong-Ou-Mandel tests.Aside from coherence witnesses, we show the measurements also serve as a Hilbert-space dimension witness.Our results attest to the effectiveness of pooling many two-state comparison tests to ascertain various relational properties of a set of quantum states.
Quantum theory is traditionally formulated using complex numbers. This imaginarity of quantum theory has been quantified as a resource with applications in discrimination tasks, pseudorandomness generation, and quantum metrology. In the standard formulation, a quantum state is said to have "imaginarity" if the associated density matrix is not real-valued in a given, fixed basis. If instead we consider a set of states, it is possible to devise tests that guarantee imaginarity of some state in the set, independently of the basis chosen. Here we propose such basis-independent witnesses for imaginarity that rely on measurements of unitary-invariant properties of sets of states. For three pure states, we completely characterize the invariant values attainable by quantum theory, and give a partial characterization for four pure states. We show that simple pairwise overlap measurements suffice to witness imaginarity of sets of four states, but not for sets of three. Our witnesses are experimentally friendly, opening up a new path for measuring and using imaginarity as a resource.
The detection and characterization of quantum coherence is of fundamental importance both in the foundations of quantum theory as well as for the rapidly developing field of quantum technologies, where coherence has been linked to quantum advantage. Typical approaches for detecting coherence employ {\it coherence witnesses} -- observable quantities whose expectation value can be used to certify the presence of coherence. By design, coherence witnesses are only able to detect coherence for some, but not all, possible states of a quantum system. In this work we construct protocols capable of detecting the presence of coherence in an {\it unknown} pure quantum state
∣ψ⟩. Having access to
m copies of an unknown pure state
∣ψ⟩ we show that the sample complexity of any experimental procedure for detecting coherence with constant probability of success
≥2/3 is
Θ(c(∣ψ⟩)−1), where
c(∣ψ⟩) is the geometric measure of coherence of
∣ψ⟩. However, assuming access to the unitary
Uψ which prepares the unknown state
∣ψ⟩, and its inverse
Uψ†, we devise a coherence detecting protocol that employs amplitude-amplification {\it a la} Grover, and uses a quadratically smaller number
O(c(∣ψ⟩)−1/2) of samples. Furthermore, by augmenting amplitude amplification with phase estimation we obtain an experimental estimation of upper bounds on the geometric measure of coherence within additive error
ε with a sample complexity that scales as
O(1/ε) as compared to the
O(1/ε2) sample complexity of Monte Carlo estimation methods. The average number of samples needed in our amplitude estimation protocol provides a new operational interpretation for the geometric measure of coherence. Finally, we also derive bounds on the amount of noise our protocols are able to tolerate.
An ideal system of
n qubits has
2n dimensions. This exponential grants power, but also hinders characterizing the system's state and dynamics. We study a new problem: the qubits in a physical system might not be independent. They can "overlap," in the sense that an operation on one qubit slightly affects the others. We show that allowing for slight overlaps,
n qubits can fit in just polynomially many dimensions. (Defined in a natural way, all pairwise overlaps can be
≤ε in
nO(1/ε2) dimensions.) Thus, even before considering issues like noise, a real system of
n qubits might inherently lack any potential for exponential power. On the other hand, we also provide an efficient test to certify exponential dimensionality. Unfortunately, the test is sensitive to noise. It is important to devise more robust tests on the arrangements of qubits in quantum devices.
Dimension witnesses allow one to test the dimension of an unknown physical system in a device-independent manner, that is, without placing assumptions about the functioning of the devices used in the experiment. Here we present simple and general dimension witnesses for quantum systems of arbitrary Hilbert space dimension. Our approach is deeply connected to the problem of quantum state discrimination, hence establishing a strong link between these two research topics. Finally, our dimension witnesses can distinguish between classical and quantum systems of the same dimension, making them potentially useful for quantum information processing.
We extend the three-state overlap analysis to determine the exact quantum maximum over finite-dimensional pure-state realizations of the
n-cycle overlap inequalities,
Snmax=ncos2(π/(2n))−1, for arbitrary cycle length
n≥3. The bound is saturated by an explicit family of coplanar qubit states equally spaced along a Fubini--Study geodesic, establishing dimensional saturation of the overlap-cycle hierarchy. Thus, the global optimum over all finite-dimensional pure-state realizations is already achieved in dimension two. We further show how, under ideal symmetric interferometric conditions, the overlap quantities can be inferred from pairwise fringe visibilities. The three-state case recovers the known maximum
5/4, while the exact
n-cycle result shows that the quantum--classical gap approaches one as
n→∞, with corresponding visibility thresholds. Within generalized noncontextuality frameworks, and subject to the required operational equivalences, such violations can witness preparation contextuality.