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Wehrl Entropy, Lieb Conjecture and Entanglement Monotones

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Wehrl Entropy, Lieb Conjecture and Entanglement Monotones Wehrl entropy, Lieb conjecture and entanglement monotones Florian Mintert1,2 and Karol Zyczkowski˙ 2,3 1Max Planck Institute for the physics of complex systems N¨othnitzerstr. 38 01187 Dresden 2Uniwersytet Jagiello´nski, Instytut Fizyki im. M. Smoluchowskiego, ul. Reymonta 4, 30-059 Krak´ow, Poland and 3Centrum Fizyki Teoretycznej, Polska Akademia Nauk, Al. Lotnik´ow 32/44, 02-668 Warszawa, Poland (Dated: October 23, 2003) We propose to quantify the entanglement of pure states of N × N bipartite quantum systems by defining its Husimi distribution with respect to SU(N) × 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. PACS numbers: 03.67 Mn, 89.70.+c, I. INTRODUCTION original statement and its proof was given in [11, 13]. It is also straightforward to formulate the Lieb conjecture Investigating properties of a quantum state it is useful for Wehrl entropy computed with respect to the SU(K)– to analyze its phase space representation. The Husimi coherent states [14, 15], but this conjecture seems not to distribution is often very convenient to work with: it ex- be simpler than the original one. ists for any quantum state, is non-negative for any point In this work we consider the Husimi function of an α of the classical phase space Ω, and may be normalized arbitrary mixed quantum state ρ of size N with respect to as a probability distribution [1]. The Husimi distribution SU(N) coherent states, and calculate both its statistical can be defined as the expectation value of the analyzed moments and the Wehrl entropy. The difference between state ̺ with respect to the coherent state α , localized the latter quantity and the minimal entropy attained for at the corresponding point α Ω. | i pure states, can be considered as a measure of the degree ∈ If the classical phase space is equivalent to the plane of mixing. one uses the standard harmonic oscillator coherent states Analyzing pure states of a bipartite system it is helpful (CS), but in general one may apply the group–theoretical to define coherent states with respect to product groups. construction of Perelomov [2]. For instance, the group The Wehrl entropy of a given state with respect to the SU(2) leads to the spin coherent states parametrized by SU(N) SU(N)–CS was first considered in the original 2 1 paper of× Wehrl [16]. Recently Sugita proposed to use the sphere S = CP [3, 4], while the simplest, degener- ated representation of SU(K) leads to the higher vector moments of the Husimi distribution defined with respect coherent states parametrized by points on the complex to such product coherent states as a way to characterize K−1 the entanglement of the analyzed state [17]. projective manifold CP [5, 6, 7]. One may define the Husimi distribution of the N–dimensional pure state In this paper we follow his idea and compute explic- φ with respect to SU(K)–CS for any K N, but it is itly the Wehrl entropy and the generalized R´enyi–Wehrl important| i to note that if K = N any pure≤ state is by entropies for any pure state Ψ of the N N composite arXiv:quant-ph/0307169v2 4 Nov 2003 | i × definition SU(N)–coherent. system. The Husimi function is computed with respect to SU(N) SU(N)–CS, so the Wehrl entropies achieve Localization properties of a state under consideration × Wehrl entropy their minimum if and only if the state Ψ is a prod- may be characterized by the , defined as | i the continuous Boltzmann–Gibbs entropy of the Husimi uct state. Hence the entropy excess defined as the dif- function [8]. The Wehrl entropy admits smallest values ference with respect to the minimal value quantifies, to for coherent states, which are as localized in the phase what extend the analyzed state is entangled. The Wehrl space, as allowed by the Heisenberg uncertainty relation. entropy excess is shown to be equal to the subentropy Interestingly, this property conjectured first by Wehrl [8] defined by Jozsa et al. [18]. Calculating the excess of the for the harmonic oscillator coherent states was proved R´enyi–Wehrl entropies we define the R´enyi subentropy – soon afterwords by Lieb [9], but the analogous result for a natural continuous generalization of the subentropy. the SU(2)–CS still waits for a rigorous proof. This un- Several different measures of quantum entanglement proven property is known in the literature as the Lieb introduced in the literature (see e.g. [19, 20, 21] and conjecture [9]. It is known that the SU(2)–CS provide references therein) satisfy the list of axioms formulated local minimum of the Wehrl entropy [10], while a proof in [19]. In particular, quantum entanglement cannot in- of the global minimum was given for pure states of di- crease under the action of any local operations, and a mension N = 3 and N = 4 only [11, 12]. A generalized measure fulfilling this property is called entanglement Lieb conjecture, concerning the R´enyi -Wehrl entropies monotone. of integer order q 2 occurred to be easier then the The non-local properties of a pure state of an N N ≥ × 2 system has to be characterized by N 1 independent momentum’s z-component J with maximal eigenvalue, − z monotones [20]. In this work we demonstrate that N 1 while the lowering operator reads J1 = J− = Jx iJy. statistical moments of properly defined Husimi functions− To characterize quantitatively the localization of− an an- naturally provide such a set of parameters, since they are alyzed state ̺ in the phase space we compute its Husimi Schur–concave functions of the Schmidt coefficients and distribution H̺ with respect to SU(N)–CS and analyze thus are non-increasing under local operations. the moments mq of the distribution II. HUSIMI FUNCTION AND WEHRL q mq(̺)= dµ(α) (H̺(α)) . (4) ENTROPY ZΩN N×N Here dµ(α) denotes the unique, unitarily invariant mea- Any density matrix ̺ C can be represented by N−1 ∈ sure on CP , also called Fubini–Study measure. The its Husimi function H̺(α), that is defined as its expec- measure dµ is normalized such that for any state ̺ the tation value with respect to coherent states α , | i first moment m1 is equal to unity, so the non–negative Husimi distribution H may be regarded as a phase-space H (α)= α ̺ α (1) ̺ ̺ h | | i probability distribution. In general one may define the Husimi distribution with The definition of the moments mq is not restricted to respect to SU(K)-coherent states with 2 K N, but integer values of q. However, from a practical point of most often one computes the Husimi distribution≤ ≤ with view it will be easier to perform the integration for integer respect to the SU(2) coherent states, also called spin co- values of q. But once mq are known for all integer q, there herent states [3, 4]. Let us recall that any family of coher- is a unique analytic extension to complex (and therefore ent states α has to satisfy the resolution of identity also real) q, as integers are dense at infinity. {| i} Another quantity of interest is the Wehrl entropy SW , defined as [8] dµ(α) α α = ½ , (2) | ih | ZΩ SW (̺)= dµ(α) H̺(α) ln H̺(α) . (5) where dµ(α) is a uniform measure on the classical mani- − ΩN fold Ω. In the case of the degenerated representation Z Again, performing the integration might be difficult. of SU(K) coherent states [5, 6, 7] this manifold, ΩK = K−1 But if all the moments m are known in the vicin- U(K)/[U(K 1) U(1)] = CP , is just equivalent q to the space− of all× pure states of size K. This complex ity of q = 1, one can derive SW quite easily. Using q q projective space arises from the K–dimensional Hilbert ∂H /∂q = H ln H, one gets space K by taking into account normalized states, and H ∂mq(̺) identifying all elements of K , which differ by an overall SW (̺)= lim . (6) phase only. In the simplestH case of SU(2) coherent states − q→1 ∂q 1 2 it is just the well known Bloch sphere, Ω2 = CP = S . The moments mq of the Husimi function allow us to write Analyzing a mixed state of dimensionality N we are go- the R´enyi–Wehrl entropy ing to use SU(N) coherent states, which will be denoted by αN . With this convention the space Ω is equivalent 1 | i N−1 SW,q(̺) := ln mq(̺) , (7) to the complex projective space CP , so every pure 1 q state is SU(N)–coherent and their Husimi distributions − which tends to the Wehrl entropy S for q 1. As a have the same shape and differ only by the localization W → in Ω. Husimi function is related to a selected classical phase classical en- The SU(N) coherent states may be defined according space, the Wehrl entropy is also called tropy von Neumann entropy to the general group-theoretical approach by Perelomov, [8, 16], in contrast to the S = Tr̺ ln ̺, that has no immediate relation to clas- by a set of generators of SU(N) acting on the distin- N − guished reference state 0 .
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