Wehrl-Type Coherent State Entropy Inequalities for $ SU (1, 1) $ and Its

Home , Wehrl entropy

arXiv:1906.00223v2 [math-ph] 1 Nov 2020 o uiifnto)a follows: as function) Husimi (or J054 S;email: USA; 08544, NJ iesakn5 K20 oehgnØ emr email: ; Denmark Ø, Copenhagen DK-2100 5, sitetsparken aie ucinin function malized hudb eadda oki progress. in work as regarded be should pc o atcei n-ieso.I sgvn(with given is It one-dimension. in particle a for space h trigpitfrti work this for point starting The Introduction 1 nrp n i ojcueaoti’ iiu au [ value minimum it’s about conjecture his and entropy ! er-yechrn tt nrp inequalities entropy state coherent Wehrl-type 2 ( 1 Here R ..Slvj MT,Dprmn fMteaia cecs University Sciences, Mathematical of Department QMATH, Solovej: J.P. Princeto Hall, Jadwin Physics, and Mathematics of Departments Lieb: E.H. hsppri oihdvrino ae naxv[ arxiv on paper a of version polished a is paper This ) on ntecs that case the in eue oa novdpolmaotaayi ucin nt on disc. functions unit analytic the about and problem unsolved an to reduces (* ! apstv prtrwoetaei )w en t lsia rbblt den probability classical its define we 1) is trace whose operator positive (a eiae oorfin n olau r atvo i 70 his on Laptev Ari colleague and friend our to Dedicated ? for nrso hs oeetsae as nw steRnientr Renyi the as known (also states coherent these of -norms ! ( | edsusteteWhltp nrp nqaiyconjectur inequality entropy Wehrl-type the the discuss We 1 ,@ ?, 2 , ( 1 R ) i + ) n o t subgroup its for and (* ste(crdne,Kadr lue)chrn tt,wihi nor- a is which state, coherent Glauber) Klauder, (Schrödinger, the is n hi oeetsae.For states. coherent their and , lit .Le a hlpSolovej Philip Jan Lieb H. Elliott [email protected] ? ( ! sa vnitgr eas hwhwtegeneral the how show also We integer. even an is 2 1 | ( ,@ ?, R , ) 1 i( aaerzdby parametrized ) d G cl ) n its and 1 itrcly a er’ ento fsemiclassical of definition Wehrl’s was historically, , ( = ,@ ?, c - − ) 1 4 Abstract + : exp = 1 h ,@ ?, [−( o ffiegop,terrepresentations their group), affine (or - - ,@ ?, G | d + 23 − [email protected] | ,@ ?, ∈ .I snttefia eso,hwvr and however, version, final the not is It ]. @ + ) h er-yecnetr for conjecture Wehrl-type the R 27 2 i 2 = \ + . hc stecasclphase classical the is which , .Gvnadniymatrix density a Given ]. 8?G )by 1) subgroup ] . euprhl plane half upper he pe)i proved is opies) o h group the for e fCpnae,Univer- Copenhagen, of nvriy Princeton, University, n th - birthday. + case (1.1) (1.2) d sity on 2 Elliott H. Lieb, Jan Philip Solovej

We use the usual Dirac notation in which is a vector and is the inner product conjugate linear in the ﬁrst variable|···i and linear in the second.h···|···i The von Neumann entropy of any quantum state with density matrix d is

(Q d := Trd log d ( ) − and the classical entropy of any continuous probability density d ?, @ is ( )

(cl d := d log d d? d@. e ( ) − ¹ The (Q entropy is non-negative,e while (cl de 0e for any d that is pointwise 1, as is the case for dcl. ( ) ≥ ≤ For the reader’s convenience we recalle some basic factse of, and the interest in, the Wehrl entropy. A classical probability does not always lead to a positive entropy and it often leads to an entropy that is . The quantum von Neumann entropy is always non-negative. Wehrl’s contribution−∞ was to derive a classical probability distribution from a quantum state that has several very desirable features. One is that it is always non-negative. Another is that it is monotone, meaning that the Wehrl entropy of a quantum state of a tensor product is always greater than or equal to the entropy of each subsystem obtained by taking partial traces. This monotonicity holds for marginals of classical probabilities, but not always for the quantum von Neumann entropy (the entropy of the universe may be smaller that the entropy of a peanut). In short the Wehrl entropy combines some of the desirable properties of the classical and quantum entropies. The minimum possible von Neumann entropy is zero and occurs when d is any pure state, while that of the classical (cl dcl is strictly positive; Wehrl’s conjecture was that the minimum is 1 and occurs when( )

d = ?, @ ?, @ . | ih | That is, when d is a pure state projector onto any coherent state. This conjecture was proved by one of us [19]. The uniqueness of this choice of d was shown by Carlen [12]. Recently De Palma [15] determined the minimal Wehrl entropy when the von Neumann entropy of d is fixed to some positive value. The Wehrl conjecture was generalized in [19] to the theorem that the operator cl norm d ?, ? > 1 was maximized for the same choice of d. This is equivalent to sayingk thatk the classical Renyi entropy of dcl is minimized for this same choice of d. Later, in our joint paper [20], the same upper bound result was shown to hold for the integral of any convex function of dcl, not just G G ?. Note that minus a convex function is a concave function so maximizing integrals→ of convex functions is the same as minimizing integrals of concave functions. We achieved the generalization to all convex functions in [20] by considering a generalization of the map from quantum states to the classical Husimi functions. Instead of only considering maps from quantum states to classical states we may Coherent state entropy inequalities for (* 1, 1 and its - subgroup 3 ( ) + consider maps between quantum states. In 1991 in [22] we defined a general formalism for defining such maps between different unitary representations of the same group. In [22] we named these maps quantum coherent operators. They are, in fact, examples of quantum channels, i.e., completely positive trace preserving maps. Moreover, they are covariant in the sense that applying a unitary transformation to the state prior to the map is the same as applying it after the map. Later such covariant quantum channels were studied in [1, 2, 8, 9, 18]. The possibility of using coherent states to associate a classical density to a quantum state d has a group-theoretic interpretation, which we will explain below. In the case of the classical coherent states discussed above this relates to the Heisenberg group and the coherent states are heighest weight vectors in the unitary irreducible representation. This suggests that we can look at other groups, their representations, and associated coherent states (heighest weight vectors) and ask whether the analog of the Wehrl conjecture holds. The first quesiton in this direction was raised in [19] for the group (* 2 where the representations are labeled by the quantum spin = 0, 1 2, 1, 3 2,.... The( spin) cases = 1 2, 3 2 were proved by Schupp in [25] and in full/ generality/ by us in [20]. / / The obvious next case is (* # for general # which has many kinds of representations. We showed the Wehrl hypothesis( ) for all the symmetric representations in [21]. Here we turn our attention to another group of some physical but also mathematical interest which is the group (* 1, 1 and its subgroup the - group or affine group. These groups, like the Heisenberg( ) group, are not compact a+nd have only infinite dimensional unitary representations. The affine group is not even unimodular which means that the left and right invariant Haar measures are different. The group (* 1, 1 is not simply connected. The purely mathematical interest in the affine group is in( sig-) nal processing (see [14]). The coherent states for the affine group are like continuous families of wavelets. The - (or affine) group is = R R with the composition rule + + ×

0, 1 0′, 1′ = 00′, 01′ 1 . ( )·( ) ( + ) This comes from thinking of the group acting on the real line as

G 0G 1, 7→ + hence the name 0G 1 group. It is not unimodular and the left Haar measure on the group is given by + 2 0− d0 d1, 1 (for reference, the right Haar measure is 0− d0 d1.) The group (* 1, 1 is the group of 2 2 complex matrices of determinant one, that leave the form( I 2) I 2 invariant, i.e.,× all matrices of the form | 1| − | 2| U V , (1.3) V U 4 Elliott H. Lieb, Jan Philip Solovej where U 2 V 2 = 1. It is easily seen that we may consider the affine - group as a subgroup| | − | of| (* 1, 1 through the injective group homomorphism + ( ) 1 0, 1 U, V = 0 1 81,0 1 81 . ( ) 7→ ( ) 2√0 ( + + − + ) Both the - group and (* 1, 1 act transitively on the open complex unit disk by + ( ) UI V I + . 7→ VI U + This is equivalent to a transitive action on the upper half-plane. For the - group the action on the upper half plane is particularly simple as it becomes + I 0I 1. 7→ + The coherent states for the two groups will naturally be parametrized by points on the unit disk or equivalently by points on the upper half plane. In Section 2 we introduce the unitary representations of the affine group and its coherent states and state the Wehrl entropy conjecture in this case. In Section 3 we generalize the conjecture to a larger class of functions, not just the entropy and we give an elementary proof in special cases. In Section 4 we formulate the generalized Wehrl inequality as a statement about analytic functions on the upper half plane or the unit disk. In Section 5 we consider (* 1, 1 and a discrete series of unitary representations and its corresponding coherent states.( ) We formulate the generalized Wehrl conjecture for (* 1, 1 and show that it is a consequence of the generalized Wehrl conjecture for the affine( -) group. In Section 6 we define for (* 1, 1 certain covariant quantum channels that+ map density matrices on one representation( space) to density matrices on another. These quantum channels are analogous to those (* 2 and (* # channels we introduced in [20, 21]. We conjecture that the majorization( ) results from( ) [20, 21] hold also in the case of (* 1, 1 . We finally prove that this majorization conjecture implies our generalized Wehrl( inequality) by taking an appropriate semi-classical limit.

2 Unitary representations and coherent states for the - group + We begin with the subgroup of (* 1, 1 since that is easier than the full (* 1, 1 group and where we have the most results.( ) ( ) The group has two irreducible faithful unitary representations [3, 17] which may be realized on !2 R , 3: either by ( + ) 1 2 * 0, 1 5 : = exp 2c81: 0 / 5 0: [ ( ) ]( ) (− ) ( ) 2c81: 2c81: or with 4− replaced by 4+ . For the representation above the coherent states we shall consider are given in terms of fiducial vectors U [U : = U : exp : ( ) ( ) (− ) Coherent state entropy inequalities for (* 1, 1 and its - subgroup 5 ( ) + where the parameter U is positive and which we will henceforth keep fixed. These functions are identified as affine coherent states (extremal weight vectors) for the representation of the affine group in [13, 26]. The constant U is chosen such that 2 ( ) ∞ [U : : d: = 1, i.e., 0 | ( )| / ∫ U 1 2 U = 2 Γ 2U − / . ( ) ( ) Then ∞ ∞ 2 * 0, 1 [U [U * 0, 1 ∗0− d0 d1 = . (2.1) 0 ( )| ih | ( ) ¹−∞ ¹ For any function 5 !2 R we may introduce the coherent state transform ∈ ( +) U 1 2 ∞ U ℎ 5 0, 1 = * 0, 1 [U 5 = U 0 + / exp :0 2c81: : 5 : d:. (2.2) ( ) h ( ) | i ( ) (− + ) ( ) ¹0 As in the case of the classical coherent states, we then have

∞ ∞ 2 2 ∞ 2 ℎ 5 0, 1 0− d0 d1 = 5 : d:, 0 | ( )| 0 | ( )| ¹−∞ ¹ ¹ 2 2 and if 5 is normalized in ! we may consider ℎ 5 0, 1 as a probability density 2 | ( )| relative to the Haar measure 0− d0 d1. Observe that in contrast to the classical case 2 we do not have ℎ 5 0, 1 1, but rather | ( )| ≤ 2 2 ℎ 5 0, 1 [U : d: = U. | ( )| ≤ | ( )| ¹ ∞ 2 = This is a consequence of the fact that (2.1) requires 0 [U : : d: 1, which 2 | ( )| / is diﬀerent from the ! -normalization. This diﬀerence∫ is due to the group not being uni-modular. The Wehrl entropy

W 2 2 2 ( 5 = ℎ 5 0, 1 ln ℎ 5 0, 1 0− d0 d1 (2.3) ( ) − | ( )| (| ( )| ) ¹ is, therefore, not necessarily non-negative, but by the deﬁnition, and by the normalization of ℎ 5 , it is bounded below by ln U. The natural generalization of Wehrl’s − W 1 2 conjecture in this case is that the entropy ( 5 is minimal for 5 = U− / [U, in which 1 ( ) case it is 1 2U − ln U. In the following+( ) section− we state a more general conjecture and prove it in special cases. In the last section of the paper we show that our conjecture and theorem are equivalent to ! ? estimates for analytic functions in the complex upper half plane.

3 Generalized conjecture for the - group and a partial result +

We make the following more general conjecture. 6 Elliott H. Lieb, Jan Philip Solovej

Conjecture 3.1 (Wehrl-type conjecture for the - group). Let U > 0 be ﬁxed. If : 0, U R is a convex function then + [ ] →

∞ ∞ 2 2 ℎ 5 0, 1 0− d0 d1 0 (| ( )| ) ¹−∞ ¹ is maximized among all normalized 5 !2 R if and only if (up to a phase) 5 has 1 2 ∈ ( +) the form 5 = U * 0, 1 [U for some 0 > 0 and 1 R. − / ( ) ∈ Remark. The maximal value above may be inﬁnite. If C = CB for B 1 the maximal value of the integral is conjectured to be ( ) ≥

2UB . 2U 1 B 1 ( + ) − The analog of Wehrl’s original entropy conjecture follows from this conjecture by taking minus a derivative at B = 1 as in [19] and gives the minimal entropy

, 1 ( = 1 2U − ln U, (3.1) min +( ) − which agrees with the lower bound ln U mentioned above. − B 1 For C = C with B = 1 2U 1 − the conjecture was proved in [4] by J. Bandyopadhyay.( ) The following+ theorem, ( + ) which has an elementary proof, yields the conjecture for integer B. As we explain in the next section this result also follows from an application of Theorem 3.1 of [11] and from [5] in their works on holomorphic functions. An alternative proof with a few generalizations of the result below was given recently in [6]. In [5] a conjecture is formulated that is equivalent to the above for C = CB for all B 1. ( ) ≥ Main Theorem 3.2. The statement of Conjecture 3.1 holds for the special cases of C = CB, for B being a positive integer. ( ) Proof. We want to prove that

∞ ∞ 2B 2 ℎ 5 0, 1 0− d0 d1 0 | ( )| ¹−∞ ¹ is maximized if and only if

5 : = : U exp : ( ) (− ) with , C such that 5 !2 R is normalized. The proof will rely only on a Schwarz inequality∈ and existence∈ ( and+) uniqueness will follow from the corresponding uniqueness of optimizers for Schwarz inequalities. If B is a positive integer we can write Coherent state entropy inequalities for (* 1, 1 and its - subgroup 7 ( ) +

B B B U 1 2 ∞ ∞ ℎ 5 0, 1 = U 0 ( + / ) exp 0 2c81 :1 :B ( ) ( ) 0 ··· 0 [(− + )( +···+ )] ¹ ¹ U : :B 5 : 5 :B d: d:B. ×( 1 ··· ) ( 1)··· ( ) 1 ··· Hence doing the 1 integration gives

∞ ∞ 2B 2 ℎ 5 0, 1 0− d0 d1 (3.2) 0 | ( )| ¹−∞ ¹ 2B ∞ ∞ B 2U 1 1 = U Γ B 2U 1 1 2 : :B − ( + )+ ( ) ( ( + )− ) ··· [ ( 1 +··· )] ¹0 ¹0 U U U U :1 5 :1 :B 5 :B :B 1 5 :B 1 :2B 5 :2B d:1 d:2B 1. × ( )··· ( ) + ( + )··· ( ) ··· − where :2B = :1 :B :B 1 :2B 1 . We now do the 0+···+integration−( and+ arrive+···+ at − )

∞ ∞ 2B 2 ℎ 5 0, 1 0− d0 d1 (3.3) 0 | ( )| ¹−∞ ¹ 2B ∞ ∞ B 2U 1 1 = U Γ B 2U 1 1 2 : :B − ( + )+ ( ) ( ( + )− ) ··· [ ( 1 +··· )] ¹0 ¹0 U U U U :1 5 :1 :B 5 :B :B 1 5 :B 1 :2B 5 :2B d:1 d:2B 1. × ( )··· ( ) + ( + )··· ( ) ··· − We now change variables to

A = : :B 1 +···+ D 9 = : 9 A, 9 = 1,...,B 1 (3.4) / − E 9 = :B 9 A, 9 = 1,...,B 1. + / − The Jacobian determinant for this change of variables is easily found to be

m :1,...,:2B 1 2B 2 det ( − ) = A − . m A, D1,...,D2B 1, E1,...,E2B 1 − − ( ) We arrive at

∞ ∞ 2B 2 ℎ 5 0, 1 0− 3031 0 | ( )| ¹−∞ ¹ 2B ∞ B 2U 1 1 B 1 = U Γ B 2U 1 1 2− ( + )+ A − ( ) ( ( + )− ) 0 ··· ··· ¹ D1¹ DB ¹1 1 E1¹ EB ¹1 1 +···+ − ≤ +···+ − ≤ U D1 DB 1 1 D1 DB 1 (3.5) − − ×[ ··· ( − −···− )]U E1 EB 1 1 E1 EB 1 ×[ ··· − ( − −···− − )] 5 D1A 5 DB 1A 5 A 1 D1 DB 1 × ( )··· ( − ) ( ( − −···− − )) 5 E1A 5 EB 1A 5 A 1 E1 EB 1 × ( )··· ( − ) ( ( − −···− − )) dD1 dDB 1 dE1 dEB 1 dA. × ··· − ··· − 8 Elliott H. Lieb, Jan Philip Solovej

Let us apply the Cauchy-Schwarz inequality for each ﬁxed A to conclude that

∞ ∞ 2B 2 ℎ 5 0, 1 0− d0 d1 0 | ( )| −∞ ¹ ¹B 2U 1 1 2B 2− ( + )+ U Γ B 2U 1 1 ≤ ( ) ( ( + )− ) 2U D1 DB 1 1 D1 DB 1 dD1 dDB 1 × ··· [ ··· − ( − −···− − )] ··· − D1¹ DB ¹1 1 +···+ − ≤ ∞ 2 5 D1A 5 DB 1A 5 A 1 D1 DB 1 × 0 ··· | ( )··· ( − ) ( ( − −···− − ))| ¹ D1¹ DB ¹1 1 +···+ − ≤ B 1 dD1 dDB 1 A − dA × ··· − B B 2U 1 1 2B ∞ 2 = 2− ( + )+ U Γ B 2U 1 1 5 A dA (3.6) ( ) ( ( + )− ) | ( )| ¹0 2U D1 DB 1 1 D1 DB 1 dD1 dDB 1 . × ··· [ ··· − ( − −···− − )] ··· − D1¹ DB ¹1 1 +···+ − ≤ If 5 is normalized this is a number depending on U and B. The important observation is that the upper bound is achieved if and only if there is a function A depending on A such that ( )

U 5 D1 5 DB 1 5 A D1 DB 1 = A D1 DB 1 A D1 DB 1 ( )··· ( − ) ( − −···− − ) ( )[ ··· − ( − −···− − )] for almost all 0 D1,...,DB 1,A satisfying D1 DB 1 A. If we define 6 D = U ≤ − +···+ − ≤ ( ) D− 5 D and introduce the variable DB = A D1 DB 1 we may rewrite this is as ( ) −( +···+ − ) 6 D 6 DB = D DB ( 1)··· ( ) ( 1 +···+ ) for almost all 0 D1,...,DB. It is not difficult to show that if a locally integrable function 6 satisfies≤ this it must be smooth and it follows easily that

6 D = exp D ( ) (− ) for complex numbers , which is exactly what we wanted to prove. The maximal value can be found by a straightforward computation.

4 An analytic formulation

Using the Bergman-Paley-Wiener Theorem in [16] we can rephrase our conjecture and theorem in terms of analytic functions on the complex upper half plane C = I C I > 0 . Introducing the weighted Bergman space, + { ∈ |ℑ } 2 C = 2 C V 2 V ! , I 3 I analytic A ( +) { ∈ ( + (ℑ ) ) | } Coherent state entropy inequalities for (* 1, 1 and its - subgroup 9 ( ) + for V > 1. The Paley-Wiener Theorem in this context says that there is a unitary map − 2 R 2 C ! 5 2U 1 , ( +) ∋ 7→ ∈ A − ( +) given by 2U I = ∞ 48:I : U 5 : 3:. ( ) 2cΓ 2U 0 ( ) ( ) ¹ If we write I = 2c1 80 we recognizep the coherent state transform in (2.2) to be + U 1 2 ℎ 5 0, 1 = √2c I + / I . ( ) (ℑ ) ( ) The action of the aﬃne group on R may be extended to the upper half plane C by I 0I 2c1. The representation of the - group on the Bergman space is then+ 7→ + + U 2 ∞ 8:I U U 1 2 * 0, 1 ∗ I = 4 : * 0, 1 ∗ 5 : d: = 0 + / 0I 2c1 ( ( ) )( ) 2cΓ 2U 0 ( ( ) )( ) ( + ) ( ) ¹ p In the analytic representation our conjecture states that if : 0, U 2c R is convex then [ /( )] → 2 2U 1 2 2 I I + I − d I (4.1) C (| ( )| (ℑ ) )|ℑ | ¹ + is maximized among normalized functions in 2 C if and only if is propor- A2U 1 ( +) tional to I I 2U 1 for some I in the lower half-plane.− ( − 0)−( + ) 0 We may also formulate the conjecture for analytic functions on the unit disk D. Here it states that if : 0, U 2c R is convex then [ /( )] →

2 2 2U 1 2 2 2 I 1 I + 1 I − d I (4.2) D (| ( )| ( − | | ) )( − | | ) ¹ is maximized among all analytic functions on the unit disk with

2 2 2U 1 2 I 1 I − d I = 1 D | ( )| ( − | | ) ¹ if and only if I is proportional to 1 ZI 2U 1 for some Z inside the unit disk. ( ) ( − )−( + ) Our theorem from the previous section is therefore equivalent to the following result.

Theorem 4.1. If C = CB, with B a positive integer, then the integrals (4.1) and (4.2) satisfy the maximization( ) properties stated above. 10 Elliott H. Lieb, Jan Philip Solovej 5 Some unitary (* 1, 1 representations and their coherent states ( )

We may represent (* 1, 1 unitarily on the bosonic Fock space over two modes ( )

C2 = ∞ = # C2 = C # , # Sym , 0 . F( ) = H H ⊗ H Ê# 0 On this space we have the action of the creation and annihilation operators 01, 02 and 0†, 0 annihilation and creation particles in the two modes represented by the standard 1 2† basis in C2. The unitary representation of (* 1, 1 on C2 is such that the element (1.3) in (* 1, 1 acts as the unitary that transforms( ) F( the creation) and annihilation operators according( ) to

01 U V 01 0 V0† = 1 + 2 , 0† → V U 0† V0 0† 2 2 1 + 2 i.e., a Bogolubov transformation. It is is clear that this deﬁnes a unitary representation of (* 1, 1 . It is however not irreducible on C2 . We see that the operator ( ) F( ) = 0†0 0†0 1 1 − 2 2 is left invariant by the action of theb Bogolubov transformation. Hence we may write

2 ∞ C = , F( ) D = Ê−∞ where is the subspace where is equal to . Each of the spaces are invariant D D and irreducible for the action of (* 1, 1 moreover the representation on is the ( ) D same as the representation asb this corresponds just to the interchanging 01, 02. D− We will thus focus on for 0. D ≥ The Lie algebra of (* 1, 1 is generated by the three elements 0, with the commutation relations (see( [24])) ± b b 0, = , , = 2 0. [ ±] ± ± [ − +] On the Fock space they areb representedb b as b b b 1 0 = 0†01 0†02 1 , = 0†0†, = 0102. 2 ( 1 + 2 + ) + 1 2 −

We see that an orthonormalb basis for , b 0 is given byb D ≥ 1 2 =! = ! / = = = ( + ) 0 , | i ! + | i Coherent state entropy inequalities for (* 1, 1 and its - subgroup 11 ( ) + where 0 is defined (up to a phase) by | i 0†0 0 = 0 , 0 0 = 0. 1 1| i | i 2| i This easily shows that define irreducible representation spaces for (* 1, 1 . D ( ) The spaces , > 0 correspond to one of the discrete series of the representations for (* 1, 1 . ThereD is another discrete series of representations. As for the affine - group( the other) discrete series is obtained by first mapping the matrix in (1.3) to+ the matrix with complex conjugate entries. In each of the representation spaces , 0 we define the family of coherent D ≥ states as the unitary transforms of the minimal weight vector 0 . They can be parametrized by point on the unit disk D in C. | i Definition 5.1 (Coherent states for (* 1, 1 ). For each I D we define the coherent ( ) ∈ state I by the action of | i 2 1 2 1 I 1 I − / ( − | | ) I 1 on 0 . | i The coherent state I can be described up to a phase by the property | i ∈ D I0† 0 I = 0. ( 1 + 2)| i From this equation we deduce the explicit form

2 1 2 ∞ 9 9 I = 1 I ( + )/ + I 9 . | i ( − | | ) = s | i Õ9 0 We see that 2 2 2 I I 1 I − d I = . c D | i h |( − | | ) D ¹ Using these coherent states we can calculate the coherent state transform of Z | i 2 1 2 2 1 2 1 I Z = 1 I ( + )/ 1 Z ( + )/ 1 IZ − − h | i ( − | | ) ( − | | ) ( − ) we see that these indeed agree with the maximizers we conjectured in Section 4 if = 2U.

Conjecture 5.2 (Generalized Wehrl for (* 1, 1 ). For any unit vector k and any convex function : 0, 1 R we have( that) ∈ D [ ] → 2 2 2 2 I k 1 I − d I D (| h | i | )( − | | ) ¹ is maximized if and only if k = Z (up to an overall phase) for some Z D. | i ∈ This conjecture is equivalent to the - conjecture for half-integer U. + 12 Elliott H. Lieb, Jan Philip Solovej 6 The (* 1, 1 quantum channels ( ) If and ! are non-negative integers we may identify the representation space ! D + as a subspace of !. Indeed, we can map 0 ! to 0 0 ! and then lift to a representation byD ⊗ D | i + | i ⊗ | i

= = 0 ! ! 0 0 ! . + | i + 7→ ( + ⊗ D + D ⊗ +) | i ⊗ | i

Let us denote by % the projection from ! onto ! considered as a subspace D ⊗ D D + of !. D ⊗ D We then have a covariant quantum channel (completely positive trace preserving map) ! Φ d = 2%d ! % ( ) ⊗ (for an appropriate 2) from density matrices on to density matrices on !. We conjecture that D D +

! Conjecture 6.1 (Majorization of Φ ). For all density matrices d on the eigenval- ! ! D ues of Φ d will be majorized by the eigenvalues of Φ I I . ( ) (| i h |) Theorem 6.2. Conjecture 6.1 implies Conjecture 5.2 except for the uniqueness of the maximizers.

We shall not give the proof of this theorem here, but it goes along the same lines as the proof of Theorem 2.1 in [20].

Acknowledgments. Many thanks to Rupert Frank and Antti Haimi for valuable suggestions about references and for suggesting the expansion of the inititial version of this paper from - to (* 1, 1 . Thanks also to John Klauder who helped us understand the - group. This+ work was( supported) by the Villum Centre of Excellence for the Mathematics+ of Quantum Theory (QMATH) and the ERC Advanced grant 321029.

References

[1] Al Nuwairan, M., Potential examples for non-additivity of the minimal output entropy. Preprint, arXiv:1312.2200 (2013) [2] Al Nuwairan, M., The extreme points of (* 2 -irreducibly covariant channels, Internat. J. Math., 25, (2014) , 1450048, ( ) [3] Aslaksen, Erik W. and Klauder, John R., Unitary representations of the affine group. J. Mathematical Phys. 9, (1968), 206–211. [4] Bandyopadhyay, Jogia, optimal concentration for SU(1, 1) coherent state transforms and an analogue of the Lieb-Wehrl conjecture for SU(1, 1), Commun. Math. Phys., 285, (2009), 1065–1086. Coherent state entropy inequalities for (* 1, 1 and its - subgroup 13 ( ) + [5] Bayart, Frederic, Brevig, OleFredrik, Haimi, Antti, Ortega-Cerda, Joaquim, Perfekt, Karl- Mikael, Contractive inequalities for Bergman spaces and multiplicative Hankel forms, Trans. Amer. Math. Soc. 371, (2019), 681-707. [6] Békollè, David, Gonessa, Jocelyn, and Sehba, Benoît F., About a conjecture of Lieb- Solovej, preprint, arXiv2010.14809, (2020). [7] Bodmann, Bernhard G., A Lower Bound for the Wehrl Entropy of Quantum Spin with Sharp High-Spin Asymptotics, Commun. Math. Phys. 250, (2004) 250–287. [8] Brannan, Michael, Collins, Benoît, Highly entangled, non-random subspaces of tensor products from quantum groups. Comm. Math. Phys. 358 (2018) 1007<96>-1025. [9] Brannan, Michael and Collins, Benoît and Lee, Hun Hee and Youn, Sang-Gyun, Temperley-Lieb quantum channels, Comm. Math. Phys. 376, (2020), 795–839. [10] Brevig, Ole F., Ortega-Cerda, Joaquim, Seip, Kristian, Zhao, Jing, Contractive inequalities for Hardy spaces, Funct Approximatio Commentarii Math. 59, (2018), 41–56. [11] Burbea, Jacob, Sharp inequalities for holomorphic functions, Illinois J. Math. 31, (1987), no. 2, 248–264. [12] Carlen, E.A., Some integral identities and inequalities for entire functions and their application to the coherent state transform. J. Funct. Anal., 97, (1991), 231–249. [13] Daubechies, Ingrid, Klauder, John R. and Paul, Thierry, Wiener measures for path integrals with affine kinematic variables, J. Math. Phys., 28, (1987), 85–102. [14] Daubechies, Ingrid, Orthonormal bases of coherent states: The canonical case and the - group, in Coherent States, Past, Present and Future, 103–107, D.H. Feng, J.R.+ Klauder, M.R. Streyer eds., World Scientific, Singapore (1994). [15] De Palma, Giacomo, The Wehrl entropy has Gaussian optimizers. Lett. Math. Phys. 108 (2018), no. 1, 97–116. [16] Duren, Peter, Gallardo-Gutiérrez, Eva A., and Montes-Rodríguez, Alfonso, A Paley- Wiener theorem for Bergman spaces with application to invariant subspaces, Bull. Lond. Math. Soc., 39, 2007, 459–466 [17] Gelfand, I. and Neumark, M., Unitary representations of the group of linear transforma- tions of the straight line. C. R. (Doklady) Acad. Sci URSS (N.S.) 55, (1947). 567–570. [18] Holevo, A.S., Additivity Conjecture and Covariant Channels, Int. J. Quantum Inf. 3, (2005), 41<96>-48. [19] Lieb, Elliott H., Proof of an entropy conjecture of Wehrl. Comm. Math. Phys. 62 (1978), no. 1, 35–41. [20] Lieb, Elliott H. and Solovej, Jan Philip, Proof of an entropy conjecture for Bloch coherent spin states and its generalizations. Acta Math. 212 (2014), no. 2, 379–398. [21] Lieb, Elliott H. and Solovej, Jan Philip, Proof of the Wehrl-type entropy conjecture for symmetric SU(N) coherent states. Comm. Math. Phys. 348 (2016), no. 2, 567–578. [22] Lieb, Elliott H. and Solovej, Jan Philip, Quantum coherent operators: A generalization of coherent states. Lett. Math. Phys. 22 (1991), 145–154. [23] Lieb, Elliott H. and Solovej, Jan Philip, A Wehrl-type entropy inequality for the affine AX+B group, arXiv:1906.00223v1 14 Elliott H. Lieb, Jan Philip Solovej

[24] Perelomov, Askold, Generalized coherent states and their applications, Springer, Heidel- berg (1986). ISBN 3-540-19512-6. [25] Schupp, Peter, On Lieb’s Conjecture for the Wehrl Entropy of Bloch Coherent States. Comm. Math. Phys. 207, (1999), no. 2, 481–493. [26] Watson, Glenn and Klauder, John R., Generalized aﬃne coherent states: a natural frame- work for the quantization of metric-like variables, J. Math. Phys., 41, (2000), 8072–8082. [27] Wehrl, Alfred, General properties of entropy. Rev. Modern Phys. 50, (1978), no. 2, 221– 260.