Unitary $t$-designs are a ubiquitous tool in many research areas, including randomized benchmarking, quantum process tomography, and scrambling. Despite the intensive efforts of researchers, little is known about unitary with $t\ensuremath{\ge}3$ literature. We show that multiqubit Clifford group any even prime-power dimension not only 2-design, but also 3-design. Moreover, it minimal 3-design except for 4. As an immediate consequence, orbit pure states forms complex projective 3-design;...

We establish a general operational one-to-one mapping between coherence measures and entanglement measures: Any measure of bipartite pure states is the minimum suitable over product bases. states, with extension to mixed by convex roof, maximum generated incoherent operations acting on system an ancilla. Remarkably, generalized cnot gate universal optimal operation. In this way, all convex-roof measures, including formation, are endowed (additional) interpretations. By virtue connection,...

Quantum-state reconstruction on a finite number of copies quantum system with informationally incomplete measurements, as rule, does not yield unique result. We derive scheme where both the likelihood and von Neumann entropy functionals are maximized in order to systematically select most-likely estimator largest entropy, that is, least-bias estimator, consistent given set measurement data. This is equivalent joint consideration our partial knowledge ignorance about ensemble reconstruct its...

We show that in prime dimensions not equal to 3, each group covariant symmetric informationally complete positive operator valued measure (SIC POVM) is with respect a unique Heisenberg–Weyl (HW) group. Moreover, the symmetry of SIC POVM subgroup Clifford Hence, two POVMs HW are unitarily or antiunitarily equivalent if and only they on same orbit extended In dimension may be three nine groups, at least one groups these respectively. There exist orbits for POVM, depending order its then...

Collective measurements on identically prepared quantum systems can extract more information than local measurements, thereby enhancing information-processing efficiency. Although this nonclassical phenomenon has been known for two decades, it remained a challenging task to demonstrate the advantage of collective in experiments. Here, we introduce general recipe performing deterministic qubits based walks. Using photonic walks, realize experimentally an optimized measurement with fidelity...

Efficient verification of pure quantum states in the adversarial scenario is crucial to many applications information processing, such as blind measurement-based computation and networks. However, little known about this topic so far. Here, we establish a general framework for verifying clarify resource cost. Moreover, propose simple recipe constructing efficient protocols from nonadversarial scenario. With recipe, arbitrary can be verified with almost same efficiency Many important using...

Graph and hypergraph states are of wide interest in quantum information processing as well fundamental physics, efficient verification these is key to various applications. The authors propose a simple recipe for verifying that requires only two distinct Pauli measurements each party, dramatically more than conventional protocols based on local measurements. This approach enables genuine multipartite entanglement thousands qubits, even an adversarial scenario.

A unitary t-design is a set of unitaries that "evenly distributed" in the sense average any t-th order polynomial over design equals entire group. In various fields -- e.g. quantum information theory one frequently encounters constructions rely on matrices drawn uniformly at random from Often, it suffices to sample these t-design, for sufficiently high t. This results more explicit, derandomized constructions. The most prominent considered multi-qubit Clifford It known be 3-design, but,...

We propose a general framework for constructing universal steering criteria that are applicable to arbitrary bipartite states and measurement settings of the party. The same is also useful studying joint problem. Based on data-processing inequality an extended R\'enyi relative entropy, we then introduce family inequalities, which detect much more efficiently than those inequalities known before. As illustrations, show unbounded violation assemblages constructed from mutually unbiased bases...

Bipartite and multipartite entangled states are of central interest in quantum information processing foundational studies. Efficient verification these states, especially the adversarial scenario, is a key to various applications, including computation, simulation, networks. However, little known about this topic scenario. Here we initiate systematic study pure-state In particular, introduce general method for determining minimal number tests required by given strategy achieve precision....

We introduce random-matrix theory to study the tomographic efficiency of a wide class measurements constructed out weighted 2-designs, including symmetric informationally complete (SIC) probability operator (POMs). In particular, we derive analytic formulas for mean Hilbert-Schmidt distance and trace between estimator true state, which clearly show difference scaling behaviors two error measures with dimension Hilbert space. then prove that product SIC POMs, multipartite analog are optimal...

The challenges of entanglement detection lead to the development several measures and criteria. Here a special type measurement is used attest in stronger way, i.e. it catches forms that are missed by other criteria effective single-shot fashion.

We reconstruct quantum theory starting from the premise that, as Asher Peres remarked, "Unperformed experiments have no results." The tools of information theory, and in particular symmetric informationally complete (SIC) measurements, provide a concise expression how exactly Peres's dictum holds true. That is constraint on probability distributions for outcomes different, hypothetical mutually exclusive ought to mesh together, type not foreseen classical thinking. Taking this our...

We study informationally overcomplete measurements for quantum state estimation so as to clarify their tomographic significance compared with minimal complete measurements. show that can improve the efficiency significantly over when states of interest have high purities. Nevertheless, is still too limited be satisfactory respect figures merit based on monotone Riemannian metrics, such Bures metric and Chernoff metric. In this way, we also pinpoint limitation nonadaptive motivate more...

We study the verification of maximally entangled states by virtue simplest measurement settings: local projective measurements without adaption. show that optimal protocols are in one-to-one correspondence with complex 2-designs constructed from orthonormal bases. Optimal minimal settings complete sets mutually unbiased Based on this observation, explicitly for any dimension, which can also be applied to estimating fidelity target state and detecting entanglement. In addition, we incomplete...

Quasiprobability representations, such as the Wigner function, play an important role in various research areas. The inevitable appearance of negativity representations is often regarded a signature nonclassicality, which has profound implications for quantum computation. However, little known about minimal that necessary general quasiprobability representations. Here we focus on natural class distinguished by simplicity and economy. We introduce three measures concerning states, unitary...

Among various multipartite entangled states, Dicke states stand out because their entanglement is maximally persistent and robust under particle losses. Although much attention has been attracted for potential applications in quantum information processing foundational studies, the characterization of remains as a challenging task experiments. Here, we propose efficient practical protocols verifying arbitrary $n$-qubit both adaptive nonadaptive ways. Our require only two distinct settings...

We investigate the steerability of two-qubit Bell-diagonal states under projective measurements by steering party. In simplest nontrivial scenario two measurements, we solve this problem completely virtue connection between and joint-measurement problem. A necessary sufficient criterion is derived together with a simple geometrical interpretation. Our study shows that state steerable iff it violates Clauser-Horne-Shimony-Holt (CHSH) inequality, in sharp contrast strict hierarchy expected...

We propose practical and efficient protocols for verifying bipartite pure states any finite dimension, which can also be applied to fidelity estimation. Our are based on adaptive local projective measurements with either one-way or two-way communications, very easy implement in practice. They extract the key information much more efficiently than known tomography direct estimation, their efficiencies comparable best entangling measurements. These highlight significance of mutually unbiased...

We construct optimal protocols for verifying qubit and qudit GHZ states using local projective measurements. When the dimension is a prime, an protocol constructed from Pauli measurements only. Our provide highly efficient way estimating fidelity certifying genuine multipartite entanglement. In particular, they enable certification of entanglement only one test when sufficiently large. By virtue adaptive measurements, we then GHZ-like that are over all based on one-way communication. The...

We provide methods for computing the geometric measure of entanglement two families pure states with both experimental and theoretical interests: symmetric multiqubit non-negative amplitudes in Dicke basis three-qubit states. In addition, we study systematically virtue a canonical form their two-qubit reduced derive analytical formulas three-parameter family Based on this result, further show that W state is maximally entangled respect to measure.

The existence of observables that are incompatible or not jointly measurable is a characteristic feature quantum mechanics, which lies at the root number nonclassical phenomena, such as uncertainty relations, wave--particle dual behavior, Bell-inequality violation, and contextuality. However, no intuitive criterion available for determining compatibility even two (generalized) observables, despite overarching importance this problem intensive efforts many researchers. Here we introduce an...

Quantum coherence plays a central role in various research areas. The ${l}_{1}$-norm of is one the most important measures that are easily computable, but it not easy to find simple interpretation. We show uniquely characterized by few axioms, which demonstrates precise sense analog negativity entanglement theory and sum resource magic-state quantum computation. also provide an operational interpretation as maximum entanglement, measured negativity, produced incoherent operations acting on...

The Wigner function provides a useful quasiprobability representation of quantum mechanics, with applications in various branches physics. Many nice properties the are intimately connected high symmetry underlying operator basis composed phase point operators: any pair operators can be transformed to other by unitary transformation. We prove that, discrete scenario, this permutation is equivalent group being 2 design. Such highly symmetric only appear odd prime power dimensions besides and...

We study systematically resource measures of coherence and entanglement based on R\'enyi relative entropies, which include the logarithmic robustness coherence, geometric conventional entropy together with their analogues. First, we show that each is equal to corresponding for any maximally correlated state. By virtue this observation, establish a simple operational connection between entropies. then prove all these measures, including are additive. Accordingly, additive states. In addition,...