PHYSICAL REVIEW A 69, 022317 ͑2004͒ Wehrl entropy, Lieb conjecture, and entanglement monotones Florian Mintert Max Planck Institute for the Physics of Complex Systems, No¨thnitzerstrasse 38 01187 Dresden, Germany and Uniwersytet Jagiellon´ski, Instytut Fizyki im. M. Smoluchowskiego, ul. Reymonta 4, 30-059 Krako´w, Poland Karol Z˙ yczkowski Uniwersytet Jagiellon´ski, Instytut Fizyki im. M. Smoluchowskiego, ul. Reymonta 4, 30-059 Krako´w, Poland and Centrum Fizyki Teoretycznej, Polska Akademia Nauk, Al. Lotniko´w 32/44, 02-668 Warszawa, Poland ͑Received 2 September 2003; published 24 February 2004͒ We propose to quantify the entanglement of pure states of NϫN bipartite quantum systems by defining its Husimi distribution with respect to SU(N)ϫSU(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 ͑Re´nyi͒ subentropies, are proved to be Schur concave, so they are entanglement monotones and may be used as alternative measures of entanglement. DOI: 10.1103/PhysRevA.69.022317 PACS number͑s͒: 03.67.Mn, 03.65.Ud, 89.70.ϩc I. INTRODUCTION original statement and its proof was given in ͓12,13͔.Itis also straightforward to formulate the Lieb conjecture for We- Investigating the properties of a quantum state it is useful hrl entropy computed with respect to the SU(K)-coherent to analyze its phase-space representation. The Husimi distri- states ͓14,15͔, but this conjecture seems not to be simpler bution is often very convenient to work with: it exists for any than the original one. quantum state, is non-negative for any point ␣ of the classi- In this work we consider the Husimi function of an arbi- cal phase space ⍀, and may be normalized as a probability trary mixed quantum state ␳ of size N with respect to distribution ͓1͔. The Husimi distribution can be defined as SU(N)-coherent states and calculate both its statistical mo- the expectation value of the analyzed state % with respect to ments and the Wehrl entropy. The difference between the the coherent state ͉␣͘, localized at the corresponding point latter quantity and the minimal entropy attained for pure ␣෈⍀. states can be considered as a measure of the degree of mix- If the classical phase space is equivalent to the plane, one ing. uses the standard harmonic oscillator coherent states ͑CS’s͒, Analyzing pure states of a bipartite system, it is helpful to but in general one may apply the group-theoretical construc- define coherent states with respect to product groups. The tion of Perelomov ͓2͔. For instance, the group SU(2) leads Wehrl entropy of a given state with respect to the product to spin-coherent states parametrized by the sphere S2ϭCP1 states was first considered in the original paper of Wehrl ͓16͔ ͓3,4͔, while the simplest, degenerated representation of and later used in ͓17͔ to characterize correlations between SU(K) leads to the higher vector coherent states param- both subsystems. Recently Sugita proposed using moments Ϫ etrized by points on the complex projective manifold CPK 1 of the Husimi distribution defined with respect to the ͓5–7͔. One may define the Husimi distribution of the SU(N)ϫSU(N) product-coherent states as a way to charac- N-dimensional pure state ͉␾͘ with respect to SU(K)-CS’s for terize the entanglement of the analyzed state ͓18͔. any KрN, but it is important to note that if KϭN, any pure In this paper we follow his idea and compute explicitly state is by definition SU(N) coherent. the Wehrl entropy and the generalized Re´nyi-Wehrl entropies Localization properties of a state under consideration may for any pure state ͉⌿͘ of the NϫN composite system. The be characterized by the Wehrl entropy, defined as the con- Husimi function is computed with respect to SU(N) tinuous Boltzmann-Gibbs entropy of the Husimi function ϫSU(N)-CS’s, so the Wehrl entropies achieve their mini- ͓8͔. The Wehrl entropy admits the smallest values for coher- mum if and only if the state ͉⌿͘ is a product state. Hence the ent states, which are as localized in the phase space, as al- entropy excess defined as the difference with respect to the lowed by the Heisenberg uncertainty relation. Interestingly, minimal value quantifies, to what extend the analyzed state is this property, first conjectured by Wehrl ͓8͔ for the harmonic entangled. The Wehrl entropy excess is shown to be equal to oscillator coherent states, was proved soon afterwords by the subentropy defined by Jozsa et al. ͓19͔. Calculating the Lieb ͓9͔, but an analogous result for the SU(2)-CS’s still excess of the Re´nyi-Wehrl entropies we define the Re´nyi awaits rigorous proof. This unproven property is known in subentropy—a natural continuous generalization of the sub- the literature as the Lieb conjecture ͓9͔. It is known that the entropy. SU(2)-CS’s provide a local minimum of the Wehrl entropy Several different measures of quantum entanglement in- ͓10͔, while a proof of the global minimum was given for troduced in the literature ͑see, e.g., ͓20–22͔ and references pure states of dimension Nϭ3 and Nϭ4 only ͓11,12͔.A therein͒ satisfy the list of axioms formulated in ͓20͔. In par- generalized Lieb conjecture, concerning the Re´nyi-Wehrl en- ticular, quantum entanglement cannot increase under the ac- tropies of integer order qу2, occurred to be easier than the tion of any local operations, and a measure fulfilling this 1050-2947/2004/69͑2͒/022317͑11͒/$22.5069 022317-1 ©2004 The American Physical Society F. MINTERT AND K. Z˙ YCZKOWSKI PHYSICAL REVIEW A 69, 022317 ͑2004͒ property is called an entanglement monotone. To characterize quantitatively the localization of an ana- The nonlocal properties of a pure state of an NϫN system lyzed state % in the phase space we compute its Husimi Ϫ has to be characterized by N 1 independent monotones distribution H% with respect to SU(N)-CS’s and analyze the ͓ ͔ Ϫ 21 . In this work we demonstrate that N 1 statistical mo- moments mq of the distribution: ments of properly defined Husimi functions naturally provide such a set of parameters, since they are Schur-concave func- ͑ ͒ϭ ͵ ␮͑␣͒ ͑ ͑␣͒͒q ͑ ͒ mq % d H% . 4 ⍀ tions of the Schmidt coefficients and thus are nonincreasing N under local operations. Here d␮(␣) denotes the unique, unitarily invariant measure NϪ1 II. HUSIMI FUNCTION AND WEHRL ENTROPY on CP , also called the Fubini-Study measure. The mea- sure d␮ is normalized such that for any state % the first ෈ NϫN Any density matrix % C can be represented by its moment m1 is equal to unity, so the non-negative Husimi ␣ Husimi function H%( ), which is defined as its expectation distribution H% may be regarded as a phase-space probability value with respect to coherent states ͉␣͘: distribution. The definition of the moments m is not restricted to in- ͑␣͒ϭ͗␣͉ ͉␣͘ ͑ ͒ q H% % . 1 teger values of q. However, from a practical point of view it will be easier to perform the integration for integer values of In general one may define the Husimi distribution with q. But once m are known for all integer q, there is a unique р р q respect to SU(K)-coherent states with 2 K N, but most analytic extension to complex ͑and therefore also real͒ q,as often one computes the Husimi distribution with respect to integers are dense at infinity. the SU(2)-coherent states, also called spin-coherent states Another quantity of interest is the Wehrl entropy SW , de- ͓3,4͔. Let us recall that any family of coherent states ͕͉␣͖͘ fined as ͓8͔ has to satisfy the resolution of identity, ͑ ͒ϭϪ͵ ␮͑␣͒ ͑␣͒ ͑␣͒ ͑ ͒ SW % d H% ln H% . 5 ⍀ ͵ d␮͑␣͉͒␣͗͘␣͉ϭ1, ͑2͒ N ⍀ Again, performing the integration might be difficult. But if ␮ ␣ ϭ where d ( ) is a uniform measure on the classical manifold all the moments mq are known in the vicinity of q 1, one ϭ q ץ q ץ ⍀ . In the case of the degenerated representation of can derive SW quite easily. Using H / q H ln H, one gets ͓ ͔ ⍀ SU(K)-coherent states 5–7 this manifold K ͒%͑ mץ ϭ ͓ Ϫ ϫ ͔ϭ KϪ1 U(K)/ U(K 1) U(1) CP is just equivalent to the ͒ϭϪ q ͑ ͒ SW͑% lim . 6 qץ ! space of all pure states of size K. This complex projective H q 1 space arises from the K-dimensional Hilbert space K by taking into account normalized states and identifying all el- The moments m of the Husimi function allow us to write H q ements of K which differ by an overall phase only. In the the Re´nyi-Wehrl entropy simplest case of SU(2)-coherent states it is just the well- ⍀ ϭ 1ϭ 2 known Bloch sphere 2 CP S . 1 S ͑%͒ ln m ͑%͒, ͑7͒ Analyzing a mixed state of dimensionality N we are going W,q ª1Ϫq q to use SU(N)-coherent states, which will be denoted by ͉␣ ⍀ N͘. With this convention the space is equivalent to the which tends to the Wehrl entropy S for q!1. As a Husimi Ϫ W complex projective space CPN 1, so every pure state is function is related to a selected classical phase space, the SU(N) coherent and their Husimi distributions have the Wehrl entropy is also called classical entropy ͓8,16͔, in con- ⍀ ϭϪ same shape and differ only by the localization in .