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Register a new userSchmidt decomposition and multivariate statistical analysis
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Added by
Yurii Ivanovich Bogdanov
Publication date
2017
fields
Physics
and research's language is
English
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The new method of multivariate data analysis based on the complements of classical probability distribution to quantum state and Schmidt decomposition is presented. We considered Schmidt formalism application to problems of statistical correlation analysis. Correlation of photons in the beam splitter output channels, when input photons statistics is given by compound Poisson distribution is examined. The developed formalism allows us to analyze multidimensional systems and we have obtained analytical formulas for Schmidt decomposition of multivariate Gaussian states. It is shown that mathematical tools of quantum mechanics can significantly improve the classical statistical analysis. The presented formalism is the natural approach for the analysis of both classical and quantum multivariate systems and can be applied in various tasks associated with research of dependences.
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In the standard geometric approach, the entanglement of a pure state is $sin^2theta$, where $theta$ is the angle between the entangled state and the closest separable state of products of normalised qubit states. We consider here a generalisation of this notion by considering separable states that consist of products of unnormalised states of different dimension. The distance between the target entangled state and the closest unnormalised product state can be interpreted as a measure of the entanglement of the target state. The components of the closest product state and its norm have an interpretation in terms of, respectively, the eigenvectors and eigenvalues of the reduced density matrices arising in the Schmidt decomposition of the state vector. For several cases where the target state has a large degree of symmetry, we solve the system of equations analytically, and look specifically at the limit where the number of qubits is large.
In the standard geometric approach to a measure of entanglement of a pure state, $sin^2theta$ is used, where $theta$ is the angle between the state to the closest separable state of products of normalized qubit states. We consider here a generalization of this notion to separable states consisting of products of unnormalized states of different dimension. In so doing, the entanglement measure $sin^2theta$ is found to have an interpretation as the distance between the state to the closest separable state. We also find the components of the closest separable state and its norm have an interpretation in terms of, respectively, the eigenvectors and eigenvalues of the reduced density matrices arising in the Schmidt decomposition of the state vector.
In a previous paper we examined a geometric measure of entanglement based on the minimum distance between the entangled target state of interest and the space of unnormalized product states. Here we present a detailed study of this entanglement measure for target states with a large degree of symmetry. We obtain analytic solutions for the extrema of the distance function and solve for the Hessian to show that, up to the action of trivial symmetries, the solutions correspond to local minima of the distance function. In addition, we show that the conditions that determine the extremal solutions for general target states can be obtained directly by parametrizing the product states via their Schmidt decomposition.
We study the mathematical structures and relations among some quantities in the theory of quantum entanglement, such as separability, weak Schmidt decompositions, Hadamard matrices etc.. We provide an operational method to identify the Schmidt-correlated states by using weak Schmidt decomposition. We show that a mixed state is Schmidt-correlated if and only if its spectral decomposition consists of a set of pure eigenstates which can be simultaneously diagonalized in weak Schmidt decomposition, i.e. allowing for complex-valued diagonal entries. For such states, the separability is reduced to the orthogonality conditions of the vectors consisting of diagonal entries associated to the eigenstates, which is surprisingly related to the so-called complex Hadamard matrices. Using the Hadamard matrices, we provide a variety of generalized maximal entangled Bell bases.
It is well known that the Schmidt decomposition exists for all pure states of a two-party quantum system. We demonstrate that there are two ways to obtain an analogous decomposition for arbitrary rank-1 operators acting on states of a bipartite finite-dimensional Hilbert space. These methods amount to joint Schmidt-type decompositions of two pure states where the two sets of coefficients and local bases depend on the properties of either state, however, at the expense of the local bases not all being orthonormal and in one case the complex-valuedness of the coefficients. With these results we derive several generally valid purity-type formulae for one-party reductions of rank-1 operators, and we point out relevant relations between the Schmidt decomposition and the Bloch representation of bipartite pure states.
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