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arXiv:1506.04788 · Entropy

Minimal Rényi–Ingarden–Urbanik Entropy of Multipartite Quantum States

Marco Enríquez, Zbigniew Puchała, Karol Życzkowski

Published Jul 20, 2015• 16 citations
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Abstract

We study the entanglement of a pure state of a composite quantum system consisting of several subsystems with d levels each. It can be described by the Rényi–Ingarden–Urbanik entropy Sq of a decomposition of the state in a product basis, minimized over all local unitary transformations. In the case q = 0, this quantity becomes a function of the rank of the tensor representing the state, while in the limit q → ∞, the entropy becomes related to the overlap with the closest separable state and the geometric measure of entanglement. For any bipartite system, the entropy S1 coincides with the standard entanglement entropy. We analyze the distribution of the minimal entropy for random states of three- and four-qubit systems. In the former case, the distribution of the three-tangle is studied and some of its moments are evaluated, while in the latter case, we analyze the distribution of the hyperdeterminant. The behavior of the maximum overlap of a three-qudit system with the closest separable state is also investigated in the asymptotic limit.

DOI
10.3390/e17075063

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We study the entanglement of a pure state of a composite quantum system consisting of several subsystems with d levels each. It can be described by the Rényi–Ingarden–Urbanik entropy Sq of a decomposition of the state in a product basis, minimized over all local unitary transformations. In the case q = 0, this quantity becomes a function of the rank of the tensor representing the state, while in the limit q → ∞, the entropy becomes related to the overlap with the closest separable state and the geometric measure of entanglement. For any bipartite system, the entropy S1 coincides with the standard entanglement entropy. We analyze the distribution of the minimal entropy for random states of three- and four-qubit systems. In the former case, the distribution of the three-tangle is studied and some of its moments are evaluated, while in the latter case, we analyze the distribution of the hyperdeterminant. The behavior of the maximum overlap of a three-qudit system with the closest separable state is also investigated in the asymptotic limit.
We study entanglement properties of generic three-qubit pure states. First, we obtain the distributions of both the coefficients and the only phase in the five-term decomposition of Acín et al. for an ensemble of random pure states generated by the Haar measure on U ( 8 ) . Furthermore, we analyze the probability distributions of two sets of polynomial invariants. One of these sets allows us to classify three-qubit pure states into four classes. Entanglement in each class is characterized using the minimal Rényi-Ingarden-Urbanik entropy. Besides, the fidelity of a three-qubit random state with the closest state in each entanglement class is investigated. We also present a characterization of these classes in terms of the corresponding entanglement polytope. The entanglement classes related to stochastic local operations and classical communication (SLOCC) are analyzed as well from this geometric perspective. The numerical findings suggest some conjectures relating some of those invariants with entanglement properties to be ground in future analytical work.
Understanding and classifying multipartite entanglement is fundamental to quantum information processing. This work focuses on absolutely maximally entangled (AME) states, a class of highly entangled states characterized by their maximal entanglement across any bipartitions. To analyze and classify AME states, we employ the minimal decomposition entropy, defined as the minimum Rényi entropy SqS_qSq​ associated with the state's decomposition over all local product bases. This quantity identifies the product bases in which the state is maximally localized, thereby yielding optimal representations for analyzing properties of AME states. We develop an efficient algorithm for computing the minimal decomposition entropy for finite q>1q>1q>1 and compare AME and Haar-random states for q=2 q = 2 q=2 and q=∞ q = \infty q=∞ in qubit, qutrit, and ququad systems. For q=2 q = 2 q=2, AME states of four qutrits and ququads show lower minimal entropy than generic states, indicating sparser optimal forms. For q=∞ q = \infty q=∞ -- related to the geometric measure of entanglement -- AME states exhibit higher entanglement. The algorithm also simplifies known AME states into sparser representations, aiding in distinguishing genuinely quantum AME states from those constructible from classical combinatorial designs. Our results advance the classification of AME states and demonstrate the utility of minimal decomposition entropy as both a local unitary invariant and a tool for state simplification.
We investigate the geometric characterization of pure state bipartite entanglement of $(2\ifmmode\times\else\texttimes\fi{}D)$- and $(3\ifmmode\times\else\texttimes\fi{}D)$-dimensional composite quantum systems. To this aim, we analyze the relationship between states and their images under the action of particular classes of local unitary operations. We find that invariance of states under the action of single-qubit and single-qutrit transformations is a necessary and sufficient condition for separability. We demonstrate that in the $(2\ifmmode\times\else\texttimes\fi{}D)$-dimensional case the von Neumann entropy of entanglement is a monotonic function of the minimum squared Euclidean distance between states and their images over the set of single qubit unitary transformations. Moreover, both in the $(2\ifmmode\times\else\texttimes\fi{}D)$- and in the $(3\ifmmode\times\else\texttimes\fi{}D)$-dimensional cases the minimum squared Euclidean distance exactly coincides with the linear entropy [and thus as well with the tangle measure of entanglement in the $(2\ifmmode\times\else\texttimes\fi{}D)$-dimensional case]. These results provide a geometric characterization of entanglement measures originally established in informational frameworks. Consequences and applications of the formalism to quantum critical phenomena in spin systems are discussed.
A general framework is developed for separating classical and quantum correlations in a multipartite system. Entanglement is defined as the difference in the correlation information encoded by the state of a system and a suitably defined separable state with the same marginals. A generalization of the Schmidt decomposition is developed to implement the separation of correlations for any pure, multipartite state. The measure based on this decomposition is a generalization of the entanglement of formation to multipartite systems, provides an upper bound for the relative entropy of entanglement, and is directly computable on pure states. The example of pure three-qubit states is analyzed in detail, and a classification based on minimal, four-term decompositions is developed.
We propose to quantify the entanglement of pure states of $N\ifmmode\times\else\texttimes\fi{}N$ bipartite quantum systems by defining its Husimi distribution with respect to $\mathrm{SU}(N)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(N)$ coherent states. The Wehrl entropy is minimal if and only if the analyzed pure state is separable. The excess of the Wehrl entropy is shown to be equal to the subentropy of the mixed state obtained by partial trace of the bipartite pure state. This quantity, as well as the generalized (R\'enyi) subentropies, are proved to be Schur concave, so they are entanglement monotones and may be used as alternative measures of entanglement.