Multipartite maximally entangled states in symmetric scenarios

We consider the class of (N+1)-partite states suitable for protocols where there is a powerful party, the authority, and the other N parties play the same role, namely the state of their system live in the symmetric Hilbert space. We show that, within this scenario, there is a "maximally entangled state" that can be transform by a LOCC protocol into any other state. In addition, we show how to make the protocol efficiently including the construction of the state and discuss security issues for possible applications to cryptographic protocols. As an immediate consequence we recover a sequential protocol that implements the one to N symmetric cloning.Comment: 6 pages, 4 figure

Euclidean distance between Haar orthogonal and gaussian matrices

In this work we study a version of the general question of how well a Haar distributed orthogonal matrix can be approximated by a random gaussian matrix. Here, we consider a gaussian random matrix $Y_n$ of order $n$ and apply to it the Gram-Schmidt orthonormalization procedure by columns to obtain a Haar distributed orthogonal matrix $U_n$. If $F_i^m$ denotes the vector formed by the first $m$-coordinates of the $i$th row of $Y_n-\sqrt{n}U_n$ and $\alpha=\frac{m}{n}$, our main result shows that the euclidean norm of $F_i^m$ converges exponentially fast to $\sqrt{ \left(2-\frac{4}{3} \frac{(1-(1 -\alpha)^{3/2})}{\alpha}\right)m}$, up to negligible terms. To show the extent of this result, we use it to study the convergence of the supremum norm $\epsilon_n(m)=\sup_{1\leq i \leq n, 1\leq j \leq m} |y_{i,j}- \sqrt{n}u_{i,j}|$ and we find a coupling that improves by a factor $\sqrt{2}$ the recently proved best known upper bound of $\epsilon_n(m)$. Applications of our results to Quantum Information Theory are also explained.Comment: v2: minor modifications to match journal version, 26 pages, 0 figures, J Theor Probab (2016

A feature selection method based on Shapley values robust to concept shift in regression

Feature selection is one of the most relevant processes in any methodology for creating a statistical learning model. Generally, existing algorithms establish some criterion to select the most influential variables, discarding those that do not contribute any relevant information to the model. This methodology makes sense in a classical static situation where the joint distribution of the data does not vary over time. However, when dealing with real data, it is common to encounter the problem of the dataset shift and, specifically, changes in the relationships between variables (concept shift). In this case, the influence of a variable cannot be the only indicator of its quality as a regressor of the model, since the relationship learned in the traning phase may not correspond to the current situation. Thus, we propose a new feature selection methodology for regression problems that takes this fact into account, using Shapley values to study the effect that each variable has on the predictions. Five examples are analysed: four correspond to typical situations where the method matches the state of the art and one example related to electricity price forecasting where a concept shift phenomenon has occurred in the Iberian market. In this case the proposed algorithm improves the results significantly

Mixing and localization in random time-periodic quantum circuits of Clifford unitaries

How much do local and time-periodic dynamics resemble a random unitary? In the present work, we address this question by using the Clifford formalism from quantum computation. We analyze a Floquet model with disorder, characterized by a family of local, time-periodic, and random quantum circuits in one spatial dimension. We observe that the evolution operator enjoys an extra symmetry at times that are a half-integer multiple of the period. With this, we prove that after the scrambling time, namely, when any initial perturbation has propagated throughout the system, the evolution operator cannot be distinguished from a (Haar) random unitary when all qubits are measured with Pauli operators. This indistinguishability decreases as time goes on, which is in high contrast to the more studied case of (time-dependent) random circuits. We also prove that the evolution of Pauli operators displays a form of mixing. These results require the dimension of the local subsystem to be large. In the opposite regime, our system displays a novel form of localization, produced by the appearance of effective one-sided walls, which prevent perturbations from crossing the wall in one direction but not the other