Discrete structures in Hilbert space play a crucial role in finding optimal schemes for quantum measurements. We solve the problem whether a complete set of five iso-entangled mutually unbiased bases exists in dimension four, providing an explicit analytical construction. The reduced matrices of the 20 pure states forming this generalized quantum measurement form a regular dodecahedron inscribed in a sphere of radius 3/20 located inside the Bloch ball of radius 1/2. Such a set forms a mixed-state 2-design -a discrete set of quantum states with the property that the mean value of any quadratic function of density matrices is equal to the integral over the entire set of mixed states with respect to the flat Hilbert-Schmidt measure. We establish necessary and sufficient conditions mixed-state designs need to satisfy and present general methods to construct them. These peculiar constellations of density matrices constitute a class of generalized measurements with additional symmetries useful for the reconstruction of unknown quantum states. Furthermore, we show that partial traces of a projective design in a composite Hilbert space form a mixed-state design, while decoherence of elements of a projective design yields a design in the classical probability simplex. PACS numbers:Introduction.-Recent progress of the theory of quantum information triggered renewed interest in foundations of quantum mechanics. Problems related to measurements of an unknown quantum state attract particular interest. The powerful technique of state tomography [1,2], allowing one to recover all entries of a density matrix, can be considered as a generalized quantum measurement, determined by a suitable set of pure quantum states of a fixed size d. Notable examples include symmetric informationally complete measurements (SIC) [3,4] consisting of d 2 pure states, which form a regular simplex inscribed inside the set Ω d of all density matrices of size d, and complete sets of (d + 1) mutually unbiased bases (MUBs) [5] such that the overlap of any two vectors belonging to different bases is fixed.The above schemes are distinguished by the fact that they allow to maximize the information obtained from a measurement and minimize the uncertainty of the results obtained under the presence of errors in both state preparation and measurement stages [4,6]. Interestingly, it is still unknown, whether these configurations exist for an arbitrary dimension. In the case of SIC measurements analytical results were known in some dimensions up to d = 48, see [7] and references therein. More recently, a putative infinite family of SICs starting with dimensions d = 4, 8, 19, 48, 124, 323 has been constructed [8], while the general problem remains open. Nonetheless, numerical results suggest [7] that such configurations might exist in every finite dimension d. For MUBs explicit constructions are known in every prime power dimension d [5], and it is uncertain whether such a solution exists otherwise, in particular [9, 10] in dimension d = 6.When the dimension is a ...

We analyze tight informationally complete measurements for arbitrarily large multipartite systems and study their configurations of entanglement. We demonstrate that tight measurements cannot be exclusively composed neither of fully separable nor maximally entangled states. We establish an upper bound on the maximal number of fully separable states allowed by tight measurements and investigate the distinguished case, in which every measurement operator carries the same amount of entanglement. Furthermore, we introduce the notion of nested tight measurements, i.e. multipartite tight informationally complete measurements such that every reduction to a certain number of parties induces a lower dimensional tight measurement, proving that they exist for any number of parties and internal levels.

The optimal allocation of resources is a crucial task for their efficient use in a wide range of practical applications in science and engineering. This paper investigates the optimal allocation of resources in multipartite quantum systems. In particular, we show the relevance of proportional fairness and optimal reliability criteria for the application of quantum resources. Moreover, we present optimal allocation solutions for an arbitrary number of qudits using measurement incompatibility as an exemplary resource theory. Besides, we study the criterion of optimal equitability and demonstrate its relevance to scenarios involving several resource theories such as nonlocality vs local contextuality. Finally, we highlight the potential impact of our results for quantum networks and other multi-party quantum information processing, in particular to the future Quantum Internet.

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For a given Hamiltonian H on a multipartite quantum system, one is interested in finding the energy E 0 of its ground state. In the separability approximation, arising as a natural consequence of measurement in a separable basis, one looks for the minimal expectation value λ ⊗ min of H among all product states. For several concrete model Hamiltonians, we investigate the difference λ ⊗ min − E 0 , called separability gap, which vanishes if the ground state has a product structure. In the generic case of a random Hermitian matrix of the Gaussian orthogonal ensemble, we find explicit bounds for the size of the gap which depend on the number of subsystems and hold with probability one. This implies an effective entanglement criterion applicable for any multipartite quantum system: If an expectation value of a typical observable among a given state is sufficiently distant from the average value, the state is almost surely entangled.