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Unsupervised classification of quantum data
https://ddd.uab.cat/record/214914
We introduce the problem of unsupervised classification of quantum data, namely, of systems whose quantum states are unknown. We derive the optimal single-shot protocol for the binary case, where the states in a disordered input array are of two types. Our protocol is universal and able to automatically sort the input under minimal assumptions, yet partially preserves information contained in the states. We quantify analytically its performance for an arbitrary size and dimension of the data. We contrast it with the performance of its classical counterpart, which clusters data that have been sampled from two unknown probability distributions. We find that the quantum protocol fully exploits the dimensionality of the quantum data to achieve a much higher performance, provided the data are at least three dimensional. For the sake of comparison, we discuss the optimal protocol when the classical and quantum states are known. Sentís Herrera, GaelThu, 14 Nov 2019 12:16:24 GMThttps://ddd.uab.cat/record/2149142019Quantum learning of classical stochastic processes :: The completely-positive realization problem
https://ddd.uab.cat/record/204038
Among several tasks in Machine Learning, is the problem of inferring the latent variables of a system and their causal relations with the observed behavior. A paradigmatic instance of such problem is the task of inferring the Hidden Markov Model underlying a given stochastic process. This is known as the positive realization problem (PRP) [3] and constitutes a central problem in machine learning. The PRP and its solutions have far-reaching consequences in many areas of systems and control theory, and is nowadays an important piece in the broad field of positive systems theory [21]. We consider the scenario where the latent variables are quantum (e. g. , quantum states of a finite-dimensional system), and the system dynamics is constrained only by physical transformations on the quantum system. The observable dynamics is then described by a quantum instrument, and the task is to determine which quantum instrument - if any - yields the process at hand by iterative application. We take as a starting point the theory of quasi-realizations, whence a description of the dynamics of the process is given in terms of linear maps on state vectors and probabilities are given by linear functionals on the state vectors. This description, despite its remarkable resemblance with the Hidden Markov Model, or the iterated quantum instrument, is however devoid from any stochastic or quantum mechanical interpretation, as said maps fail to satisfy any positivity conditions. The Completely-Positive realization problem then consists in determining whether an equivalent quantum mechanical description of the same process exists. We generalize some key results of stochastic realization theory, and show that the problem has deep connections with operator systems theory, giving possible insight to the lifting problem in quotient operator systems. Our results have potential applications in quantum machine learning, device-independent characterization and reverse-engineering of stochastic processes and quantum processors, and more generally, of dynamical processes with quantum memory [16, 17]. Monras Blasi, ÀlexTue, 23 Apr 2019 13:54:32 GMThttps://ddd.uab.cat/record/204038urn:ISBN:978-3-939897-73-6Leibniz-Zentrum fuer Informatik,2014Optimal full estimation of qubit mixed states
https://ddd.uab.cat/record/115665
We obtain the optimal scheme for estimating unknown qubit mixed states when an arbitrary number N of identically prepared copies is available. We discuss the case of states in the whole Bloch sphere as well as the restricted situation where these states are known to lie on the equatorial plane. For the former case we obtain that the optimal measurement does not depend on the prior probability distribution provided it is isotropic. Although the equatorial-plane case does not have this property for arbitrary N, we give a prior-independent scheme which becomes optimal in the asymptotic limit of large N. We compute the maximum mean fidelity in this asymptotic regime for the two cases. We show that within the pointwise estimation approach these limits can be obtained in a rather easy and rapid way. This derivation is based on heuristic arguments that are made rigorous by using van Trees inequalities. The interrelation between the estimation of the purity and the direction of the state is also discussed. In the general case we show that they correspond to independent estimations whereas for the equatorial-plane states this is only true asymptotically. Bagán Capella, EmiliTue, 11 Feb 2014 12:01:04 GMThttps://ddd.uab.cat/record/1156652006Phase variance of squeezed vacuum states
https://ddd.uab.cat/record/115635
We consider the problem of estimating the phase of squeezed vacuum states within a Bayesian framework. We derive bounds on the average Holevo variance for an arbitrary number N of uncorrelated copies. We find that it scales with the mean photon number n, as dictated by the Heisenberg limit, i. e. , as n−2, only for N>4. For N 4 this fundamental scaling breaks down and it becomes n−N/2. Thus, a single squeezed vacuum state performs worse than a single coherent state with the same energy. We find the optimal splitting of a fixed given energy among various copies. We also compute the variance for repeated individual measurements (without classical communication or adaptivity) and find that the standard Heisenberg-limited scaling n−2 is recovered for large samples. Bagán Capella, EmiliMon, 10 Feb 2014 15:31:18 GMThttps://ddd.uab.cat/record/1156352008