1,400 research outputs found
Entanglement, randomness and chaos
Entanglement is not only the most intriguing feature of quantum mechanics,
but also a key resource in quantum information science. The entanglement
content of random pure quantum states is almost maximal; such states find
applications in various quantum information protocols. The preparation of a
random state or, equivalently, the implementation of a random unitary operator,
requires a number of elementary one- and two-qubit gates that is exponential in
the number n_q of qubits, thus becoming rapidly unfeasible when increasing n_q.
On the other hand, pseudo-random states approximating to the desired accuracy
the entanglement properties of true random states may be generated efficiently,
that is, polynomially in n_q. In particular, quantum chaotic maps are efficient
generators of multipartite entanglement among the qubits, close to that
expected for random states. This review discusses several aspects of the
relationship between entanglement, randomness and chaos. In particular, I will
focus on the following items: (i) the robustness of the entanglement generated
by quantum chaotic maps when taking into account the unavoidable noise sources
affecting a quantum computer; (ii) the detection of the entanglement of
high-dimensional (mixtures of) random states, an issue also related to the
question of the emergence of classicality in coarse grained quantum chaotic
dynamics; (iii) the decoherence induced by the coupling of a system to a
chaotic environment, that is, by the entanglement established between the
system and the environment.Comment: Review paper, 40 pages, 7 figures, added reference
Sample Complexity of Dictionary Learning and other Matrix Factorizations
Many modern tools in machine learning and signal processing, such as sparse
dictionary learning, principal component analysis (PCA), non-negative matrix
factorization (NMF), -means clustering, etc., rely on the factorization of a
matrix obtained by concatenating high-dimensional vectors from a training
collection. While the idealized task would be to optimize the expected quality
of the factors over the underlying distribution of training vectors, it is
achieved in practice by minimizing an empirical average over the considered
collection. The focus of this paper is to provide sample complexity estimates
to uniformly control how much the empirical average deviates from the expected
cost function. Standard arguments imply that the performance of the empirical
predictor also exhibit such guarantees. The level of genericity of the approach
encompasses several possible constraints on the factors (tensor product
structure, shift-invariance, sparsity \ldots), thus providing a unified
perspective on the sample complexity of several widely used matrix
factorization schemes. The derived generalization bounds behave proportional to
w.r.t.\ the number of samples for the considered matrix
factorization techniques.Comment: to appea
Certifying an irreducible 1024-dimensional photonic state using refined dimension witnesses
We report on a new class of dimension witnesses, based on quantum random
access codes, which are a function of the recorded statistics and that have
different bounds for all possible decompositions of a high-dimensional physical
system. Thus, it certifies the dimension of the system and has the new distinct
feature of identifying whether the high-dimensional system is decomposable in
terms of lower dimensional subsystems. To demonstrate the practicability of
this technique we used it to experimentally certify the generation of an
irreducible 1024-dimensional photonic quantum state. Therefore, certifying that
the state is not multipartite or encoded using non-coupled different degrees of
freedom of a single photon. Our protocol should find applications in a broad
class of modern quantum information experiments addressing the generation of
high-dimensional quantum systems, where quantum tomography may become
intractable.Comment: Journal version (except for small editorial modifications), 4+12
pages, 7 figure
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