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Renyi-Wehrl entropies as measures of localization in phase space

Sven Gnutzmann, Karol Zyczkowski

Published Jan 1, 2001• 53 citations
OpenAlex

Abstract

Abstract. We generalize the concept of the Wehrl entropy of quantum states which gives a basis–independent measure of their localization in phase space. We discuss the minimal values and the typical values of these Rényi–Wehrl entropies for pure states for spin systems. According to Lieb’s conjecture the minimal values are provided by the spin coherent states. Though Lieb’s conjecture remains unproven, we give new proofs of partial results that may be generalized for other systems. We also investigate random pure states and calculate the mean Rényi–Wehrl entropies averaged over the natural measure in the space of pure quantum states. 1.

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We generalize the concept of the Wehrl entropy of quantum states which gives a basis-independent measure of their localization in phase space. We discuss the minimal values and the typical values of these Rényi-Wehrl entropies for pure states for spin systems. According to Lieb's conjecture the minimal values are provided by the spin coherent states. Though Lieb's conjecture remains unproven, we give new proofs of partial results that may be generalized for other systems. We also investigate random pure states and calculate the mean Rényi-Wehrl entropies averaged over the natural measure in the space of pure quantum states.

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The Wehrl entropy is the entropy of the probability distribution in phase space corresponding to the Husimi function in terms of coherent states. We explain the significance of the Wehrl entropy in quantum information theory, and present the theory behind the Lieb conjecture, which states that, in finite dimensions, the minimum Wehrl entropy occurs for Bloch coherent states. This was proven by Lieb and Solovej in 2012. We present the theory behind coherent states, with a particular emphasis on Bloch coherent states, and give a geometrical representation of quantum states as points on a sphere. Using this representation, we identify spherical arrangements of 2–9 points that maximize the Wehrl entropy locally. We conjecture that these maxima are in fact global. Furthermore, we investigate how the maximally entangled symmetric states are related to the states corresponding to maximal Wehrl entropy.

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Wehrl used Glauber coherent states to define a map from quantum density matrices to classical phase space densities and conjectured that for Glauber coherent states the mininimum classical entropy would occur for density matrices equal to projectors onto coherent states. This was proved by Lieb in 1978 who also extended the conjecture to Bloch SU(2) spin-coherent states for every angular momentum J. This conjecture is proved here. We also recall our 1991 extension of the Wehrl map to a quantum channel from J to K=J+12,J+1,..., with K=∞ corresponding to the Wehrl map to classical densities. These channels were later recognized as the optimal quantum cloning channels. For each J and J<K⩽∞ we show that the minimal output entropy for the channels occurs for a J coherent state. We also show that coherent states both Glauber and Bloch minimize any concave functional, not just entropy.

arXiv:1208.3632•Acta Mathematica•Jan 1, 2014•61 citations
Abstract. We generalize the concept of the Wehrl entropy of quantum states which gives a basis–independent measure of their localization in phase space. We discuss the minimal values and the typical values of these Rényi–Wehrl entropies for pure states for spin systems. According to Lieb’s conjecture the minimal values are provided by the spin coherent states. Though Lieb’s conjecture remains unproven, we give new proofs of partial results that may be generalized for other systems. We also investigate random pure states and calculate the mean Rényi–Wehrl entropies averaged over the natural measure in the space of pure quantum states. 1.
We generalize the concept of the Wehrl entropy of quantum states which gives a basis-independent measure of their localization in phase space. We discuss the minimal values and the typical values of these Rényi-Wehrl entropies for pure states for spin systems. According to Lieb's conjecture the minimal values are provided by the spin coherent states. Though Lieb's conjecture remains unproven, we give new proofs of partial results that may be generalized for other systems. We also investigate random pure states and calculate the mean Rényi-Wehrl entropies averaged over the natural measure in the space of pure quantum states.
Wehrl's entropy is the entropy of the probability distribution in the phase space corresponding to the Q representation (antinormal ordering of operators) of a quantum state in terms of coherent states. The Hilbert space of a system of N spins (or N two-level atoms) is of 2 N dimensions. The subspace with maximum total spin N/2 is of N+1 dimensions. All discussions are restricted within this subspace. The probability amplitude for an arbitrary pure state in this subspace is a polynomial of degree N which can be factorised into the product of N linear factors and each root can be identified with one point on the unit sphere. Hence, an arbitrary state in the subspace can be represented geometrically by N unit vectors. The general expression for the Wehrl entropy is obtained as a finite series expansion in terms of some symmetric functions of sin 2 ( omega ij /2), where omega ij is the angle between a pair of unit vectors. The special cases of coherent and almost coherent spin states are then considered.
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.
The Wehrl entropy is the entropy of the probability distribution in phase space corresponding to the Husimi function in terms of coherent states. We explain the significance of the Wehrl entropy in quantum information theory, and present the theory behind the Lieb conjecture, which states that, in finite dimensions, the minimum Wehrl entropy occurs for Bloch coherent states. This was proven by Lieb and Solovej in 2012. We present the theory behind coherent states, with a particular emphasis on Bloch coherent states, and give a geometrical representation of quantum states as points on a sphere. Using this representation, we identify spherical arrangements of 2–9 points that maximize the Wehrl entropy locally. We conjecture that these maxima are in fact global. Furthermore, we investigate how the maximally entangled symmetric states are related to the states corresponding to maximal Wehrl entropy.
Wehrl used Glauber coherent states to define a map from quantum density matrices to classical phase space densities and conjectured that for Glauber coherent states the mininimum classical entropy would occur for density matrices equal to projectors onto coherent states. This was proved by Lieb in 1978 who also extended the conjecture to Bloch SU(2) spin-coherent states for every angular momentum J. This conjecture is proved here. We also recall our 1991 extension of the Wehrl map to a quantum channel from J to K=J+12,J+1,..., with K=∞ corresponding to the Wehrl map to classical densities. These channels were later recognized as the optimal quantum cloning channels. For each J and J<K⩽∞ we show that the minimal output entropy for the channels occurs for a J coherent state. We also show that coherent states both Glauber and Bloch minimize any concave functional, not just entropy.