We revisit the the large spin asymptotics of 15j symbols in terms of cosines of the 4d Euclidean Regge action, as derived by Barrett and collaborators using a saddle point approximation. We bring it closer to the perspective of area-angle Regge calculus and twisted geometries, and compute explicitly the Hessian and phase offsets. We then extend it to more general SU(2) graph invariants associated to nj-symbols. We find that saddle points still exist for special boundary configurations, and that these have a clear geometric interpretation, but there is a novelty: Configurations with two distinct saddle points admit a conformal shape-mismatch of the geometry, and the cosine asymptotic behaviour oscillates with a generalisation of the Regge action. The allowed mismatch correspond to angle-matched twisted geometries, 3d polyhedral tessellations with adjacent faces matching areas and 2d angles, but not their diagonals. We study these geometries, identify the relevant subsets corresponding to 3d Regge data and 4d flat polytope data, and discuss the corresponding Regge actions emerging in the asymptotics. Finally, we also provide the first numerical confirmation of the large spin asymptotics of the 15j symbol. We show that the agreement is accurate to the per cent level already at spins of order 10, and the next-to-leading order oscillates with the same frequency and same global phase.5 Higher valence and polytopes 23 5.

The Lorentzian Engle-Pereira-Rovelli-Livine spin foam amplitude for loop quantum gravity is a multidimensional noncompact integral of highly oscillating functions. Using a method based on the decomposition of Clebsch-Gordan coefficients for the unitary infinite-dimensional representations of SL(2,C) in terms of those of SU(2), we are able to provide for the first time numerical evaluations of the vertex amplitude. The values obtained support the asymptotic formula obtained by Barrett and collaborators with a saddle point approximation, showing, in particular, a power law decay and oscillations related to the Regge action. The comparison offers a test of the efficiency of the method. Truncating the decomposition to the first few terms provides a qualitative matching of the power law decay and oscillations. For vector and Euclidean Regge boundary data, a qualitative matching is obtained with just the first term, which corresponds to the simplified EPRL model. We comment on future developments for the numerics and extension to higher vertices. We complete our work with some analytic results: We provide an algorithm and explicit configurations for the different geometries that can arise as boundary data, and explain the geometric consequences of the decomposition used.

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We study the estimation of the overlap between two Haar-random pure quantum states in a finite-dimensional Hilbert space, given M and N copies of them. We compute the statistics of the optimal measurement, which is a projection onto permutation-invariant subspaces, and provide lower bounds for the mean square error for both local and Bayesian estimation. In the former case, the bound is asymptotically saturable by a maximum likelihood estimator, whereas in the latter we give a simple exact formula for the optimal value. Furthermore, we introduce two LOCC strategies, relying on the estimation of one or both the states, and show that, although they are suboptimal, they outperform the commonly-used swap test. In particular, the swap test is extremely inefficient for small values of the overlap, which become exponentially more likely as the dimension increases. Finally, we show that the optimal measurement is less invasive than the swap test and study the robustness to depolarizing noise for qubit states.

We consider the problem of correctly classifying a given quantum two-level system (qubit) which is known to be in one of two equally probable quantum states. We assume that this task should be performed by a quantum machine which does not have at its disposal a complete classical description of the two template states, but can only have partial prior information about their level of purity and mutual overlap. Moreover, similarly to the classical supervised learning paradigm, we assume that the machine can be trained by n qubits prepared in the first template state and by n more qubits prepared in the second template state. In this situation we are interested in the optimal process which correctly classifies the input qubit with the largest probability allowed by quantum mechanics. The problem is studied in its full generality for a number of different prior information scenarios and for an arbitrary size n of the training data. Finite size corrections around the asymptotic limit n → ∞ are derived. When the states are assumed to be pure, with known overlap, the problem is also solved in the case of d-level systems.