Uncertainty principle and quantum fisher information. 2.

Uncertainty principle and quantum Fisher information. II.

Paolo Gibiliscoa兲

Dipartimento SEFEMEQ, Facoltà di Economia, Università di Roma “Tor Vergata,” Via Columbia 2, 00133 Rome, Italy

Daniele Imparatob兲

Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Turin, Italy

Tommaso Isolac兲

Dipartimento di Matematica, Università di Roma “Tor Vergata,” Via della Ricerca Scientifica, 00133 Rome, Italy

共Received 16 April 2007; accepted 30 April 2007; published online 30 July 2007兲

Heisenberg and Schrödinger uncertainty principles give lower bounds for the prod-uct of variances Var_␳共A兲Var_␳共B兲 if the observables A,B are not compatible, namely, if the commutator关A,B兴 is not zero. In this paper, we prove an uncertainty prin-ciple in Schrödinger form where the bound for the product of variances Var_␳共A兲Var_␳共B兲 depends on the area spanned by the commutators i关␳, A兴 and i关␳, B兴 with respect to an arbitrary quantum version of the Fisher information. © 2007 American Institute of Physics.关DOI:10.1063/1.2748210兴

I. INTRODUCTION

Let X , Y be random variables on a probability space 共⍀,G,p兲 and consider the covariance Covp共X,Y兲ªEp共XY兲−Ep共X兲Ep共Y兲 and the variance Varp共X兲ªCovp共X,X兲. The best one can get

from the Cauchy-Schwartz inequality is the following inequality:

Varp共X兲Varp共Y兲 − 兩Covp共X,Y兲兩2艌 0. 共1.1兲

The uncertainty principle is one of the most striking consequences of noncommutativity in quan-tum mechanics and is a key point in which quanquan-tum probability differs from classical probability. We shall limit our discussion to the matrix case. If␳ is a state共i.e., density matrix兲, A,B observ-ables共i.e., self-adjoint matrices兲, set Cov_␳共A,B兲ªTr共␳AB兲−Tr共␳A兲Tr共␳B兲. Define also the sym-metrized covariance as Cov_␳s共A,B兲ª1₂关Cov_␳共A,B兲+Cov_␳共B,A兲兴 and the variance as Var共A兲 ªCov␳共A,A兲=Cov␳s共A,A兲. Again from the Cauchy-Schwartz inequality, one gets the following inequality:

Var_␳共A兲Var_␳共B兲 − 兩Cov_␳s共A,B兲兩2艌1

4兩Tr共␳关A,B兴兲兩

2_{, 共1.2兲}

which is known as the Schrödinger uncertainty principle. By omitting the covariance part, one gets the Heisenberg uncertainty principle共see Refs.12and27兲. Inequality共1.2兲states that the condi-tion关A,B兴⫽0 共i.e., A,B are not compatible兲 gives a limitation to the simultaneous “smallness” of both Var_␳共A兲 and Var_␳共B兲, and this has very important consequences in quantum mechanics.

a兲_{Electronic mail: gibilisco@volterra.uniroma2.it. URL: http://www.economia.uniroma2.it/sefemeq/professori/gibilisco} b兲_{Electronic mail: daniele.imparato@polito.it}

c兲_{Electronic mail: isola@mat.uniroma2.it. URL: http://www.mat.uniroma2.it/⬃isola}

48, 072109-1

However, noncommutativity can enter also from another side. One may naturally ask if there are similar bounds for the product Var_␳共A兲Var_␳共B兲 due to the fact that the observables A,B do not commute with the state␳. Indeed this is the case, and our main result will provide such a bound in terms of an “area” spanned by the commutators i关␳, A兴 and i关␳, B兴.

To state our result we have to introduce the notion of quantum Fisher information. Let us denote byFopthe class of normalized symmetric operator monotone functions on共0, +⬁兲. It is by now a classical result of Petz that to each function f苸Fop, one can associate a Riemannian metric 具A,B典_␳,f on the state manifold that is monotone and therefore a quantum version of the Fisher information共see Refs.2and23兲. For example, the functions,

f_␤共x兲 ª␤共1 −␤兲 共x − 1兲 2

共x␤_{− 1兲共x}1−␤_{− 1兲}, x⬎ 0, ␤苸 共0,1/2兴,

are associated with a quantum Fisher information that is related to the well known Wigner-Yanase-Dyson共WYD兲 skew information. Indeed, for the WYD information of parameter␤, one has

− 1 2Tr共关␳

␤_,A兴关␳1−␤_{,A兴兲 =}␤共1 −␤兲

2 具i关␳,A兴,i关␳,A兴典␳,f␤.

We denote by Area_␳f共u,v兲 the area spanned by the tangent vectors u,v with respect to the Rie-mannian monotone metric associated with f 共at the point ␳兲. For f 苸Fop, let us define f共0兲 ªlimx_→0f共x兲; f is said regular if f共0兲⬎0 共see Ref.9兲. The subset of regular elements of Fop is denoted byF_opr .

The goal of the present paper is to prove the following inequality:

Var_␳共A兲Var_␳共B兲 − 兩Cov_␳s共A,B兲兩2艌 1

4共f共0兲 Area␳

f_共i关␳

,A兴,i关␳,B兴兲兲2, ∀ f 苸 F_opr . 共1.3兲 Note that inequality 共1.3兲holds trivially in the nonregular case. Our result has been inspired by particular cases of the above theorem that have been proved recently. Luo and Zhang19 con-jectured inequality共1.3兲for the Wigner-Yanase metric, namely, for the function f_1/2. This conjec-ture was proved shortly after by Luo and Zhang.21The case of WYD metric关namely, the metric associated with f_␤for␤苸共0,1/2兲兴 was proved independently by Kosaki13and by Yanagi et al.28 In the paper of Gibilisco and Isola,6 they emphasized the geometric aspects of the question and they succeeded to formulate Eq.共1.3兲for a general quantum Fisher information.

It is worth to emphasize the dynamical meaning of inequality 共1.3兲. Indeed, each positive 共self-adjoint兲 operator H determines a time evolution of a state␳ according to the formula␳H共t兲

ªe−iHt_␳eiHt_{. If关␳, H兴=0, then there is no evolution. Therefore, we may say that the bound given}

by Eq.共1.3兲appears for those pairs of observables that are “dynamically incompatible,” that is, for pairs H , K such that the associated evolutions ␳H共t兲,␳K共t兲 are different and nontrivial 共this is

equivalent to the linear independence of关␳, H兴 and 关␳, K兴兲.

As a by-product of the work needed to prove our main result, we derive also other two inequalities interesting per se. Indeed, a crucial ingredient in the proof of Eq.共1.3兲is the following formula:

f˜共x兲 ª1

2

冋

共x + 1兲 − 共x − 1兲 2f共0兲

f共x兲

册

,

that associates to any element f苸F_opanother element f˜苸F_op. Let A₀ªA−Tr共␳A兲I and denote by L_␳, R_␳, respectively, the left and right multiplication by␳. Let mf be the mean associated with f

C_␳f_共A

0兲 ª Tr共mf共L␳,R␳兲共A0兲A0兲, I␳f共A兲 ª Var␳共A兲 − C␳f˜共A0兲.

Hansen introduced I_␳f共A兲 in the paper9with a different approach. We shall call it “metric adjusted skew information” or “f information” to stress the dependence on the function f. One may consider I_␳f共A兲 as a generalization of the WYD information. Let fRLD共x兲=2x/共x+1兲; we prove the following inequality:

Var_␳共A兲Var_␳共B兲 艌 关I_␳f共A兲 + C_␳fRLD共A

0兲兴关I␳f共B兲 + C␳fRLD共B0兲兴, ∀ f 苸 Fop r

. 共1.4兲

Moreover, we also prove that if f苸Fop,

Var_␳共A兲Var_␳共B兲 艌 C_␳f共A0兲C␳f共B0兲 + 1

4兩Tr共␳关A,B兴兲兩

2_{⇔ f共x兲 艋}

冑

_{x. 共1.5兲}

Inequality共1.4兲is a refinement of an inequality proved by Luo, for the Wigner-Yanase metric, and by Hansen, in the general case共see Refs.18and9兲. Inequality共1.5兲for the function

冑

x is due to Park and, independently, to Luo共see Refs. 22and16兲. Here we simply prove the optimality of their bound inFop.

The plan of the paper is as follows.

Sections II–IV contain preliminary notions. In Sec. II we recall the standard Heisenberg and Schrödinger uncertainty principles. In Sec. III we give the fundamental definitions and theorems for number and operator means. In Sec. IV we review the classification theorem for the quantum Fisher informations.

Sections V–VIII contain the core of the paper. In Sec. V we show that to any operator monotone function f苸F_op, one may associate another element f˜苸F_opby formula共5.1兲; we study the properties of the mean mf˜and of an associated function Hf, discussing, in particular, how they

behave as functions of f. In Sec. VI we prove the main result, namely, inequality共1.3兲. Further-more, we study the right side of the above inequality as a function of f and relate it to quantum evolution of states, as said before. In Sec. VII we introduce the f correlation共a kind of generalized WYD correlation兲 and the f information and discuss their relation with the quantum Fisher infor-mation associated with f苸Fop. In this way, we are able to show how inequality共1.3兲generalizes the previously known results. Moreover, we prove that by choosing the SLD共Symmetric Loga-rithmic Derivative or Bures-Uhlmann兲 metric, the lower bound given in Eq. 共1.3兲is optimal and strictly greater than the previously known optimal bound共given by the Wigner-Yanase metric兲. In Sec. VIII we prove a necessary and sufficient condition to get the equality in Eq.共1.3兲.

In Sec. IX we prove inequality 共1.4兲. In Sec. X we produce counterexamples to prove the logical independence of the uncertainty principles studied in this paper, that is, inequalities共1.3兲 and共1.4兲, from the standard Heisenberg-Schrödinger uncertainty principles. In Sec. XI we discuss what happens for not faithful and pure states, also at the light of the notion of radial extension for quantum Fisher information. In Sec. XII we show the optimality of an improvement of the Heisenberg uncertainty principle recently proposed by Park and Luo, namely, we prove inequality 共1.5兲.

II. HEISENBERG AND SCHRöDINGER UNCERTAINTY PRINCIPLES

Let Mnª Mn共C兲 关Mn,saª Mn共C兲sa兴 be the set of all n⫻n complex matrices 共all n⫻n

self-adjoint matrices兲. We shall denote general matrices by X,Y ,..., while letters A,B,... will be used for self-adjoint matrices. The Hilbert-Schmidt scalar product is denoted by具A,B典=Tr共A*_{B兲. The} adjoint of a matrix X is denoted by X†, while the adjoint of a superoperator T :共Mn,具·, ·典兲 →共Mn,具·, ·典兲 is denoted by T*. LetDnbe the set of strictly positive elements of MnandDn

1_傺D n

Dn 1

=兵␳苸 Mn兩Tr␳= 1,␳⬎ 0其.

From now on, we shall treat the case of faithful states, namely,␳⬎0. We shall consider the general case␳艌0 at the end of the paper, in Sec. XI, where we shall also discuss in detail what happens for pure states.

Definition 2.1: Suppose that␳苸D_n1 is fixed. Define X_0ªX−Tr共␳X兲I.

Definition 2.2: For A , B苸Mn,saand␳苸Dn 1

define covariance and variance as Cov_␳共A,B兲 ª Tr共␳AB兲 − Tr共␳A兲Tr共␳B兲 = Tr共␳A0B0兲,

Var_␳共A兲 ª Tr共␳A2兲 − Tr共␳A兲2= Tr共␳A₀2兲. Proposition 2.1:

2 Re兵Cov_␳共A,B兲其 = Cov_␳共A,B兲 + Cov_␳共B,A兲 = Tr共␳兵A₀,B₀其兲, 2i Im兵Cov␳共A,B兲其 = Cov␳共A,B兲 − Cov␳共B,A兲 = Tr共␳关A,B兴兲,

where, for any X , Y苸Mn,关X,Y兴ªXY −YX, 兵X,Y其ªXY +YX.

We define the symmetrized covariance as Covs_␳共A,B兲ª1₂关Cov_␳共A,B兲+Cov_␳共B,A兲兴 = Re兵Cov␳共A,B兲其. The Cauchy-Schwartz inequality implies

兩Cov␳共A,B兲兩2艋 Var␳共A兲Var␳共B兲.

From this, one gets the Schrödinger and Heisenberg uncertainty principles which are stated in the following theorem.

Theorem 2.1:共See Ref.27兲 For A,B苸M_n,saand␳苸D_n1, one has Var_␳共A兲Var_␳共B兲 − 兩Cov_␳s共A,B兲兩2艌1

4兩Tr共␳关A,B兴兲兩 2 that implies Var_␳共A兲Var_␳共B兲 艌1 4兩Tr共␳关A,B兴兲兩 2_.

III. MEANS FOR POSITIVE NUMBERS AND MATRICES

For this section we refer to the exposition contained in Ref.26.

Definition 3.1: LetR+_{ª共0, +⬁兲. A mean for pairs of positive numbers is a function m:R}+ ⫻R+_→R+_{such that}

共i兲 m共x,x兲=x, 共ii兲 m共x,y兲=m共y,x兲,

共iii兲 x⬍y⇒x⬍m共x,y兲⬍y,

共iv兲 x⬍x⬘, y⬍y⬘⇒m共x,y兲⬍m共x⬘, y

⬘兲,

共v兲 m is continuous, and

共vi兲 for t⬎0, one has m共tx,ty兲=t·m共x,y兲. We denote by Mnuthe set of means.

Definition 3.2:Fnuis the class of functions f :R+→R+such that

共i兲 f共1兲=1, 共ii兲 tf共t−1_{兲= f共t兲,}

共iii兲 t苸共0,1兲⇒ f共t兲苸共0,1兲, 共iv兲 t苸共1,⬁兲⇒ f共t兲苸共1,⬁兲,

共v兲 f is continuous, and

共vi兲 f is monotone increasing.

Proposition 3.1: There is bijection betweenMnuandFnugiven by the formulas

mf共x,y兲 ª yf共xy−1兲, fm共t兲 ª m共1,t兲.

Remark 3.1:

f艋 g ⇔ mf艋 mg.

Here below we report the Kubo-Ando theory of matrix means共see Ref.14兲 as exposed in Ref. 26. In the sequel, for any pairs of matrices A, B, we shall write A⬍B whenever B−A is positive semidefinite.

Definition 3.3: Recall thatDnª兵A苸 Mn共C兲兩A⬎0其. A mean for pairs of positive matrices is a

function m共·, ·兲:Dn⫻Dn→Dnsuch that conditions共i兲–共v兲 of Definition 3.1 hold 共with the matrix

partial order defined above兲 and the transformer inequality,

Cm共A,B兲C*艋 m共CAC*,CBC*兲, ∀ C, replaces共vi兲. We denote by Mopthe set of matrix means.

Example 3.1: The arithmetic, geometric, and harmonic 共matrix兲 means are given,

respec-tively, by

Aⵜ B ª1₂共A + B兲,

A # Bª A1/2_共A−1/2_BA−1/2_兲1/2_A1/2_, A!Bª 2共A−1_{+ B}−1_兲−1_.

Let us recall that a function f :共0,⬁兲→R is said operator monotone if, for any n苸N, any A, B苸Mnsuch that 0艋A艋B, the inequalities 0艋 f共A兲艋 f共B兲 hold. An operator monotone function

is said symmetric if f共x兲=xf共x−1_{兲 and normalized if f共1兲=1.}

Definition 3.4: Fop is the class of operator monotone functions f :R+_→R+ _{such that} condi-tions共i兲–共v兲 of Definition 3.2 hold 共with the matrix partial order defined above兲.

Note that the above definition is redundant共see, for example, Ref.1兲; however, it emphasizes well the similarity with the number case. Indeed, one has the following result.

Proposition 3.2:Fop is the class of functions f :R+_→R+_{such that}

共i

⬘

兲 f共1兲=1,

共ii

⬘

兲 tf共t−1_{兲= f共t兲, and}

共iii

⬘

兲 f is operator monotone increasing.

Equivalently, f苸Fopif and only if f is a normalized, symmetric, operator monotone function.

The fundamental result, due to Kubo and Ando, is the following.

Theorem 3.1: There is bijection between Mopand Fopgiven by the formula mf共A,B兲 ª A1/2f共A−1/2BA−1/2兲A1/2.

When A and B commute, we have that

mf共A,B兲 ª A · f共BA−1兲.

Theorem 3.2: Among matrix means, arithmetic is the largest while harmonic is the smallest.

Proof: See Theorem 4.5 in Ref.14. 䊐

2x 1 + x艋 f共x兲 艋 1 + x 2 , 2xy x + y艋 mf共x,y兲 艋 x + y 2 .

IV. QUANTUM FISHER INFORMATIONS

In what follows, given a differential manifoldN, we denote by T_␳N the tangent space to N at the point␳苸N. In the commutative case a Markov morphism is a stochastic map T:Rn_→Rm_{. Let}

Pnª 兵␳苸 Rn兩␳i⬎ 0其, Pn 1

ª

兵

␳苸 Pn兩

兺

␳i= 1

其

.

The natural representation for the tangent space is given by T_␳Pn

1

=

再

v苸 Rn兩

兺

i

vi= 0

冎

.

In this case a monotone metric is defined as a family of Riemannian metrics g =兵gn_{其 on 兵P} n 1_{其, n} 苸N, such that gT共␳兲 m _{共TX,TX兲 艋 g} ␳ n_共X,X兲

holds for every Markov morphism T :Rn_→Rm_{, for every␳苸P} n

1 _{and for every X苸T} ␳Pn

1_. The Fisher information is the Riemannian metric onP_n1 defined as

具u,v典␳,Fª

兺

i

uivi

␳i

, u,v苸 T_␳Pn1.

Theorem 4.1:共See Ref.2兲 There exists a unique monotone metric on Pn1(up to scalars) given by the Fisher information.

In the noncommutative case, a Markov morphism is a completely positive and trace preserv-ing operator T : Mn→Mm. Recall that there exists a natural identification of T␳Dn1with the space of

self-adjoint traceless matrices, namely, for any␳苸Dn1, T_␳Dn

1

=兵A 苸 Mn,sa兩Tr共A兲 = 0其.

In perfect analogy with the commutative case, a monotone metric in the noncommutative case is a family of Riemannian metrics g =兵gn其 on 兵Dn

1_{其, n苸N, such that} gT共␳兲

m _{共TX,TX兲 艋 g} ␳ n_共X,X兲

holds for every Markov morphism T : Mn→Mm, for every␳苸Dn 1

, and for every X苸T_␳Dn 1

. Mono-tone metrics are usually normalized in such a way that关A,␳兴=0 implies gf,␳共A,A兲=Tr共␳−1A2兲.

To a normalized symmetric operator monotone function f苸Fop, one associates the so-called Chentsov-Morozowa共CM兲 function

cf共x,y兲 ª

1

y f共xy−1兲= mf共x,y兲

−1 _{for x,y⬎ 0.}

Define L_␳共A兲ª␳A, and R_␳共A兲ªA␳; observe that they are self-adjoint operators on M_n,sa. Since L_␳ and R_␳commute we may define cf共L␳, R␳兲=mf共L␳, R␳兲−1. Since mf is a matrix mean one gets the

following result.

Proposition 4.1:共See Ref.23兲 mf共L␳, R␳兲 and cf共L␳, R␳兲 are positive and therefore self-adjoint.

Now we can state the fundamental theorems about noncommutative monotone metrics.

met-rics onDn1and normalized symmetric operator monotone functions f苸Fop. This correspondence is given by the formula

具A,B典␳,fª Tr共Acf共L␳,R␳兲共B兲兲 = Tr共Amf共L␳,R␳兲−1共B兲兲.

We set储A储_␳,f2 ª具A,A典␳,f. Because of the above theorems, we shall use the terms “monotone metrics” and “quantum Fisher informations”共QFI兲 with the same meaning.

For a symmetric operator monotone function, define f共0兲ªlimx_→0f共x兲=limx_→+⬁关f共x兲/x兴. Of

course, f共0兲艌0. The condition f共0兲⫽0 is relevant because it is a necessary and sufficient condi-tion for the existence of the so-called radial extension of a monotone metric to pure states 共see Refs.25and24or Sec. XI兲. Following Ref.9we say that a function f苸Fopis regular if and only if f共0兲⫽0. The corresponding operator mean, CM function, associated QFI, etc., are said regular too. The class of regular共nonregular兲 functions f 苸Fopis denoted by Fop

r _共F op n _兲.

As proved by Lesniewski and Ruskai, each QFI is the Hessian of a suitable relative entropy 共see Ref.15兲.

V. THE FUNCTION f˜ AND THE PROPERTIES OF THE ASSOCIATED MEAN

In Ref.9 the following result has been proved.

Proposition 5.1:共Proposition 3.4 in Ref.9兲

If f苸F_op is regular, define the representing function as

df共x,y兲 ª

共x + y兲

f共0兲 −共x − y兲 2_c

f共x,y兲, x,y ⬎ 0.

Then, the function df is positive and operator concave.

Definition 5.1: For f苸F_opand x⬎0, set f˜共x兲 ª1 2

冋

共x + 1兲 − 共x − 1兲 2f共0兲 f共x兲

册

. 共5.1兲 Proposition 5.2: f苸 Fop⇒ f˜ 苸 Fop.

Proof: Easy calculations show that f˜ is normalized and symmetric. To prove that f is operator monotone note that

共a兲 if f is not regular then f˜共x兲=1

2共1+x兲 and the conclusion follows;

共b兲 if f is regular then f˜共x兲=关f共0兲/2兴d共x,1兲. Since d is positive and operator concave so is f˜. We get the conclusion because operator concavity is equivalent to operator monotonicity

共see Ref.10兲. 䊐

Remark 5.1: Note that f regular⇒f˜ not regular.

Following the terminology of Sec. III, we associate to f˜ both a number and an operator mean by the formulas

mf˜共x,y兲 ª yf˜共xy−1兲,

mf˜共A,B兲 ª A1/2f˜共A−1/2BA−1/2兲A1/2.

Remark 5.2: Observe that mf˜共x,y兲=关共x+y兲/2兴−关f共0兲/2兴关共x−y兲2/ y f共x/y兲兴.

From Corollary 3.1 one obtains this result.

2x 1 + x艋 f˜共x兲 艋 1 + x 2 , 2xy x + y艋 mf˜共x,y兲 艋 x + y 2 .

Moreover we have the following result, whose proof is elementary.

Proposition 5.3: For every x⬎0 and f ,g苸Fop,

f˜共x兲 艋 g˜共x兲 ⇔f共0兲 f共x兲艌

g共0兲 g共x兲. We synthetize some results in Table I.

Example 5.1: Let x⬎0 and␤苸共0,1/2兲. If we set

fSLD共x兲 ª 1 + x 2 , fWY共x兲 ª

冉

1 +

冑

x 2

冊

2 , f_␤共x兲 ª␤共1 −␤兲 共x − 1兲 2 共x␤_{− 1兲共x}1−␤_{− 1兲}, fRLD共x兲 ª 2x 1 + x, one has共see TableI兲

f˜_SLD共x兲 = 2x 1 + x, f˜WY共x兲 =

冑

x, f˜␤共x兲 ª x␤+ x1−␤ 2 , f˜RLD共x兲 ª 1 + x 2 .

Note that if x⬎0 is fixed, the function ␤苸共0,1/2兲哫x␤+ x1−␤苸R+ is decreasing. This im-plies

f˜_SLD艋 f˜_WY艋 f˜_␤艋 f˜_RLD, and therefore

mf˜_SLD艋 mf˜_WY艋 mf˜_␤艋 mf˜_RLD,

that is, a refined arithmetic-geometric-harmonic inequality, 2xy x + y艋

冑

xy艋 1 2共x ␤_y1−␤_{+ x}1−␤_y␤_{兲 艋}1 + x 2 , x,y⬎ 0, ␤苸 共0,1/2兲.

Remark 5.3: The metrics associated with the functions f_␤ are equivalent to the metrics induced by noncommutative ␣ divergences, where ␤=共1−␣兲/2 共see Ref. 11兲. They are very important in information geometry and are related to the WYD information 共see, for example, Refs. 3,4,5 and 7兲. Defining ᐉ_␥共x兲ª共共1+x␥兲/2兲1/␥ for ␥苸关1/2,1兴, one has ᐉ_␥苸F_op. The two parametric families f_␤,ᐉ_␥give us a continuum of operator monotone functions from the smallest

TABLE I. We have, for some QFI the name, the function f, the mean mf, the value of f at 0, the function f˜, and

the mean mf˜. QFI f mf f共0兲 f˜ mf˜ RLD 2x /共x+1兲 2 /共1/x+1/y兲 0 共1+x兲/2 共x+y兲/2 WYD共␤兲 ␤苸共−1,0兲 ␤共1−␤兲共x−1兲 2_/ 共x␤_{− 1兲共x}1−␤_{− 1兲} ␤共1−␤兲共x−y兲 2_/ 共x␤_{− y}␤_兲共x1−␤_{− y}1−␤_兲 0 共1+x兲/2 共x+y兲2

BKM 共x−1兲/log x 共x−y兲/共log x−log y兲 0 共1+x兲/2 共x+y兲/2 WYD共␤兲 ␤苸共0,1/2兲 ␤共1−␤兲共x−1兲 2_/ 共x␤_{− 1兲共x}1−␤_{− 1兲} ␤共1−␤兲共x−y兲 2_/ 共x␤_{− y}␤_兲共x1−␤_{− y}1−␤_兲 ␤共1−␤兲 共x ␤_{+ x}1−␤_{兲/2 共x}␤_y1−␤_{+ x}1−␤_y␤_兲/2 WY 共共1+冑x兲/2兲2 _共共冑_{x +}冑_y兲/2兲2 _{1 / 4} 冑_x 冑_xy SLD 共1+x兲/2 共x+y兲/2 1 / 2 2x /共x+1兲 2 /共1/x+1/y兲

function 2x /共x+1兲 to the largest function 共1+x兲/2. Further examples of this kind of “bridges” can be found in. Refs.8 and9. Note that also g0共x兲ª

冑

x is an element ofFop.

In the sequel we need to study the following function.

Definition 5.2: For any f苸Fop, set

Hf共x,y,w,z兲 ª 关共x + y兲 − mf˜共x,y兲兴mf˜共w,z兲 + 关共w + z兲 − mf˜共w,z兲兴mf˜共x,y兲, x,y,w,z ⬎ 0.

Proposition 5.4: For any f , g苸Fop,

f˜ 艋 g˜

⇓

Hf共x,y,w,z兲 艋 Hg共x,y,w,z兲, ∀ x,y,w,z ⬎ 0. Proof: Since

共x + y兲 − mf˜共x,y兲 = 共x + y兲 − x + y 2 + 共x − y兲2 2y f共0兲 f共x/y兲= x + y 2 + 共x − y兲2 2y f共0兲 f共x/y兲, we have

Hf共x,y,w,z兲 ª 关共x + y兲 − mf˜共x,y兲兴mf˜共w,z兲 + 关共w + z兲 − mf˜共w,z兲兴mf˜共x,y兲

=

冉

x + y 2 + 共x − y兲2 2y f共0兲 f共x/y兲

冊冉

w + z 2 − 共w − z兲2 2z f共0兲 f共w/z兲

冊

+

冉

w + z 2 + 共w − z兲2 2z f共0兲 f共w/z兲

冊冉

x + y 2 − 共x − y兲2 2y f共0兲 f共x/y兲

冊

=1 2

冋

共x + y兲共w + z兲 −

冉

共x − y兲2 y 共w − z兲2 z

冊冉

f共0兲 f共x/y兲 f共0兲 f共w/z兲

冊

册

. 共5.2兲 Since, from Proposition 5.3,

f˜艋 g˜ ⇒ f共0兲 f共t兲 艌

g共0兲

g共t兲 ⬎ 0, ∀ t ⬎ 0, we obtain

Hf共x,y,w,z兲 艋 Hg共x,y,w,z兲, ∀ x,y,w,z ⬎ 0

by elementary computations. 䊐

Note that for f nonregular, one has Hf共x,y,w,z兲 =

1

2共x + y兲共w + z兲. On the other hand, for the function fSLD=

1

2共1+x兲, one has from Eq. 共5.2兲, H_SLD共x,y,w,z兲 =1 2

冋

共x + y兲共w + z兲 − 1 4

冉

共x − y兲2_{共w − z兲}2 关共x + y兲/2兴关共w + z兲/2兴

冊

册

= 2 xy共w2_{+ z}2_{兲 + wz共x}2_{+ y}2_兲 共x + y兲共w + z兲 . Therefore, we have the following bounds.

Corollary 5.2: For any f苸Fop,

0⬍ 2

冋

xy共w

2_{+ z}2_{兲 + wz共x}2_{+ y}2_兲

共x + y兲共w + z兲

册

艋 Hf共x,y,w,z兲 艋

1

2共x + y兲共w + z兲, ∀ x,y,w,z ⬎ 0.

fSLD共0兲 fSLD共x兲 = 1/2 共1 + x兲/2= 1 1 + x⬎ 1 1 + x + 2

冑

x= 1 共1 +

冑

x兲2= 1/4 共1 +

冑

x兲2/4= fWY共0兲 fWY共x兲 , so that for every x , y , w , z⬎0,

H_SLD共x,y,w,z兲 ⬍ H_WY共x,y,w,z兲.

VI. THE MAIN RESULT

Proposition 6.1: Given f苸Fop, let⌬ªmf˜共L␳, R␳兲. Recall that B0ªB−Tr共␳B兲. One has

(i) Tr共B0⌬共I兲兲=0, (ii) Tr共I⌬共B0兲兲=0, and

(iii) Tr共⌬共I兲兲=1.

Proof.

共i兲 Since共L_␳− R_␳兲共I兲=0 and Tr共␳B0兲=Tr共␳B兲−Tr共␳B兲=0, we have 具B0,mf˜共L␳,R␳兲共I兲典 = Tr共B0mf˜共L␳,R␳兲共I兲兲 = 1 2Tr共B0共L␳+ R␳兲共I兲兲 − 1 2f共0兲Tr共B0cf共L␳,R␳兲 ⫻共L_␳− R_␳兲2共I兲兲 =1 2Tr共B0␳+␳B0兲 = Tr共␳B0兲 = 0. 共ii兲 It is a simple consequence of共i兲 and of Proposition 4.1 Indeed,

具I,mf˜共L␳,R␳兲共B0兲典 = 具mf˜共L␳,R␳兲共I兲,B0典 = 0. 共iii兲 Tr共⌬共I兲兲 = Tr共mf˜共L␳,R␳兲共I兲兲 = 1 2Tr共共L␳+ R␳兲共I兲兲 − 1 2f共0兲Tr共cf共L␳,R␳兲共L␳− R␳兲 2_{共I兲兲 = Tr共␳兲} = 1 . 䊐 Proposition 6.2:

f共0兲具i关␳,A兴,i关␳,B兴典_␳,f= Tr共␳AB兲 + Tr共␳BA兲 − 2 Tr共A⌬共B兲兲. Proof: Let us introduce the shorthand notation

cˆf共x,y兲 ª 共x − y兲2cf共x,y兲,

so that by definition

f共0兲cˆf共x,y兲 = 共x + y兲 − 2mf˜共x,y兲.

Therefore, we have

f共0兲具i关␳,A兴,i关␳,B兴典_␳,f= f共0兲Tr共共i关␳,A兴兲cf共L␳,R␳兲共i关␳,B兴兲兲

= f共0兲具共i关␳,A兴兲,cf共L␳,R␳兲共i关␳,B兴兲典

= f共0兲具i共L␳− R␳兲共A兲,cf共L␳,R␳兲 ⴰ 共i共L␳− R␳兲兲共B兲典

= f共0兲具A,共i共L_␳− R_␳兲兲*_{ⴰ c}

f共L␳,R␳兲 ⴰ 共i共L␳− R␳兲兲共B兲典

= f共0兲具A,− i共L␳− R␳兲 ⴰ cf共L␳,R␳兲 ⴰ 共i共L␳− R␳兲兲共B兲典

= f共0兲Tr共Acˆf共L␳,R␳兲共B兲兲 = Tr共A共f共0兲cˆf共L_␳,R_␳兲兲共B兲兲 = Tr共A共L␳+ R␳− 2mf˜共L␳,R␳兲兲共B兲兲 =共Tr共␳AB兲 + Tr共␳BA兲 − 2 Tr共Amf˜共L␳,R␳兲共B兲兲. 䊐 Proposition 6.3:

f共0兲具i关␳,A兴,i关␳,B兴典_␳,f= 2共Re兵Cov_␳共A,B兲其 − Tr共⌬共A₀兲B₀兲兲. Proof: We have that

f共0兲具i关␳,A兴,i关␳,B兴典_␳,f= Tr共␳AB兲 + Tr共␳BA兲 − 2 Tr共A⌬共B兲兲

= Cov_␳共A,B兲 + Cov_␳共B,A兲 + 2 Tr共␳A兲Tr共␳B兲 − 2 Tr共A⌬共B兲兲 = 2 Re兵Cov␳共A,B兲其 + 2共Tr共␳A兲Tr共␳B兲 − Tr共A⌬共B兲兲兲. Moreover, because of Proposition 6.1,

Tr共␳A兲Tr共␳B兲 − Tr共⌬共A兲B兲 = Tr共␳A兲Tr共␳B兲 − Tr共⌬共A0+ Tr共␳A兲I兲共B0+ Tr共␳B兲I兲兲 = Tr共␳A兲Tr共␳B兲 − 关Tr共⌬共A0兲B0兲 + Tr共␳A兲Tr共⌬共I兲B0兲

+ Tr共⌬共A₀兲I兲Tr共␳B兲 + Tr共␳A兲Tr共␳B兲Tr共⌬共I兲I兲兴 = Tr共␳A兲Tr共␳B兲 − Tr共⌬共A0兲B0兲 − Tr共␳A兲Tr共␳B兲

= − Tr共⌬共A₀兲B₀兲 . 䊐

Therefore, the conclusion follows.

We recall some consequences of the spectral theorem we need in the sequel. Let␳ be a state, ␭iits eigenvalues, and Eithe associated eigenprojectors. The spectral decompositions of L␳and R␳

are the following:

L_␳=

兺

i

␭iLE_i, R␳=

兺

i

␭iRE_i.

Therefore, from the spectral theorem for commuting self-adjoint operators, we get the following result.

Corollary 6.1: Let ␳ be a state,␭i its eigenvalues, and Ei the projectors of the associated

eigenspaces. If s :关0, +⬁兲⫻关0, +⬁兲→R is a continuous function, then

s共L_␳,R_␳兲 =

兺

i,j

s共␭i,␭j兲LE_iRE_j.

Let V be a finite dimensional real vector space with a scalar product g共·, ·兲. We define, for

v , w苸V,

Areag共v,w兲 ª

冑

g共v,v兲g共w,w兲 − 兩g共v,w兲兩2.

In the Euclidean plane, Areag共v,w兲 is the area of the parallelogram spanned by v and w. If we are dealing with a␳ point-depending Riemannian metric, we write Area_␳g. If f苸Fop, we denote by Area_␳f the area functional associated with the monotone metric 具·, ·典_␳,f.

We are now ready for the main results.

(i)

Var_␳共A兲Var_␳共B兲 − 兩Cov_␳s共A,B兲兩2艌

冉

f共0兲 2 Area␳ f_共i关 ␳,A兴,i关␳,B兴兲

冊

2. (ii) g ˜艌 f˜ ⇒g共0兲 2 Area␳ g_共i关 ␳,A兴,i关␳,B兴兲 艋 f共0兲 2 Area␳ f_共i关 ␳,A兴,i关␳,B兴兲.

Proof: Fix A , B苸Mn,sa. Let us introduce, for the sake of brevity,

F共f兲 ª 共Var_␳共A兲Var_␳共B兲 − 兩Cov_␳s共A,B兲兩2兲 −

冉

f共0兲 2 Area␳

f_{共i关␳,A兴,i关␳,B兴兲}

冊

2

. Then, we have to show that F共f兲艌0 and g˜艌 f˜⇒F共g兲艌F共f兲.

Let 兵␸i其 be a complete orthonormal base composed of eigenvectors of␳ and兵␭i其 the

corre-sponding eigenvalues. Set aij⬅具A0␸i兩␸j典 and bij⬅具B0␸i兩␸j典. Note that aij⫽Aijª the i, j entry of A. Then we calculate Var_␳共A兲 = Tr共␳A₀2兲 =1 2

兺

i,j 共␭i+␭j兲aijaji, Var_␳共B兲 = Tr共␳B₀2兲 =1 2

兺

_i,j 共␭i+␭j兲bijbji, Cov_␳s共A,B兲 = Re兵Cov_␳共A,B兲其 = Re兵Tr共␳A0B0兲其 =

1

2

兺

_i,j 共␭i+␭j兲Re兵aijbji其, f共0兲

2 储i关␳,A兴储␳,f 2 _{= Var}

␳共A兲 − Tr共A0mf˜共L␳,R␳兲A0兲 = 1

2

兺

_i,j 共␭i+␭j兲aijaji−

兺

_i,j mf˜共␭i,␭j兲aijaji, f共0兲

2 储i关␳,B兴储␳,f 2 ₌1

2

兺

_i,j 共␭i+␭j兲bijbji−

兺

_i,j mf˜共␭i,␭j兲bijbji, f共0兲

2 具i关␳,A兴,i关␳,B兴典␳,f= Re兵Cov␳共A,B兲其 − Re兵Tr共mf˜共L␳,R␳兲共A0兲B0兲其 = 1 2

兺

_i,j 共␭i+␭j兲Re兵aijbji其 −

兺

i,j mf˜共␭i,␭j兲Re兵aijbji其. Set ␰ª Var␳共A兲Var␳共B兲 − f共0兲 2 4 储i关␳,A兴储␳,f 2 _{储i关␳,B兴储} ␳,f 2 =1 2_i,j,k,l

兺

兵共␭i+␭j兲mf˜共␭k,␭l兲 + 共␭k+␭l兲mf˜共␭i,␭j兲 − 2mf˜共␭i,␭j兲mf˜共␭k,␭l兲其aijajibklblk =1 4i,j,k,l

兺

兵共␭i+␭j兲mf˜共␭k,␭l兲 + 共␭k+␭l兲mf˜共␭i,␭j兲 − 2mf˜共␭i,␭j兲mf˜共␭k,␭l兲其兵aijajibklblk+ aklalkbijbji其,

␩ª 兩Cov␳s共A,B兲兩2− f共0兲2 4 兩具i关␳,A兴,i关␳,B兴典␳,f 2 _兩2₌1 2_i,j,k,l

兺

兵共␭i+␭j兲mf˜共␭k,␭l兲 + 共␭k+␭l兲mf˜共␭i,␭j兲 − 2mf˜共␭i,␭j兲mf˜共␭k,␭l兲其Re兵aijbji其Re兵aklblk其,

K_i,j,k,lª Ki,j,k,l共␳,A,B兲 ª 兩aij兩2兩bkl兩2+兩akl兩2兩bij兩2− 2 Re兵aijbji其Re兵aklblk其.

Since

兩aij兩2兩bkl兩2+兩akl兩2兩bij兩2艌 2兩aijbji兩兩aklblk兩 艌 2兩Re兵aijbji其Re兵aklblk其兩,

we have that K_i,j,k,l艌0. Note that K_i,j,k,l does not depend on f. Then

F共f兲 =␰−␩=1

4_i,j,k,l

兺

兵共␭i+␭j兲mf˜共␭k,␭l兲 + 共␭k+␭l兲mf˜共␭i,␭j兲 − 2mf˜共␭i,␭j兲mf˜共␭k,␭l兲其 ⫻兵兩aij兩2兩bkl兩2+兩akl兩2兩bij兩2− 2 Re兵aijbji其Re兵aklblk其其 =

1 4i,j,k,l

兺

Hf共␭i,␭j,␭k,␭l兲Ki,j,k,l.

Because of Proposition 5.4 and Corollary 5.2, one has

f˜艋 g˜ ⇒ 0 艋 Hf共␭i,␭j,␭k,␭l兲 艋 Hg共␭i,␭j,␭k,␭l兲,

and therefore

f˜艋 g˜ ⇒ 0 艋 F共f兲 艋 F共g兲,

and we get the thesis. 䊐

The standard Schrödinger uncertainty principle reads as

Area_␳Covs共A,B兲 艌1

2兩Tr共␳关A,B兴兲兩, while the main result of the present paper can be expressed as

Area_␳Covs共A,B兲 艌f共0兲 2 Area␳

f_共i关

␳,A兴,i关␳,B兴兲.

Corollary 3.2: For any f苸Fop, A , B苸Mn,sa, one has

f_SLD共0兲 2 Area␳ f_{SLD共i关␳} ,A兴,i关␳,B兴兲 艌f共0兲 2 Area␳ f_共i关␳ ,A兴,i关␳,B兴兲.

Proof: Immediate consequence of Corollary 3.1 䊐

Remark 6.1: Setting

N_␳f共A,B兲 ª Area_␳Covs共A,B兲 − f共0兲 2 Area␳

f_{共i关␳,A兴,i关␳,B兴兲 艌 0,}

we may strengthen the main result to

Area_␳Covs共A,B兲 艌f共0兲 2 Area␳

f_共i关

The above geometric considerations take a particularly interesting form when considering the dynamics of quantum states. Suppose that we have a positive共self-adjoint兲 operator H determining a quantum evolution. The state␳evolves according to the formula

␳H共t兲 ª e−itH␳eitH.

We say that␳H共t兲 is the time evolution of␳=␳H共0兲 determined by H. For the evolution␳H共t兲, this

is equivalent to satisfy the quantum analog of the Liouville theorem in classical statistical me-chanics, namely, the Landau-von Neumann equation.

Definition 6.1: Let ␳共t兲 be a curve in D_n1 and let H苸M_n,sa. We say that ␳共t兲 satisfies the Landau-von Neumann equation with respect to H if

␳˙共t兲 = d

dt␳共t兲 = i关␳共t兲,H兴.

Satisfying the Landau-von Neumann equation is equivalent to␳共t兲=␳H共t兲=e−itH␳eitH.

From Theorem 6.1, we get the following inequality.

Proposition 6.4: Let␳⬎0 be a state and H,K苸Mn,sa. Suppose that␳=␳H共0兲=␳K共0兲. Then,

for any f苸Fop, one has

Area_␳Covs共H,K兲 艌 f共0兲 2 Area␳

f_共␳˙

H共0兲,␳˙K共0兲兲.

Therefore, as we said in the Introduction, the bound on the right side of our inequality appears when the evolutions␳H共t兲,␳K共t兲 are different and not trivial.

VII. THE f-CORRELATION ASSOCIATED WITH QUANTUM FISHER INFORMATIONS

Mainly to confront our result with previous results, we introduce the notions of f correlation and f information. Definition 7.1: C_␳f_{共A,B兲 = C} ␳f共B,A兲 ª Tr共mf共L␳,R␳兲共A兲B兲, C_␳f 共A兲 ª C␳f共A,A兲.

Definition 7.2: For A , B苸M_n,sa, ␳苸D_n1, and f苸F_op, the metric adjusted correlation 共or f correlation兲 and the metric adjusted skew information 共or f information兲 are defined as

Corr_␳f共A,B兲 ª Tr共␳AB兲 − C_␳f˜共A,B兲 = Tr共␳AB兲 − Tr共mf˜共L␳,R␳兲共A兲B兲, I_␳f共A兲 ª Corr_␳f共A,A兲.

The definition of Corr_␳f共A,B兲 appeared in Ref.9in a different form. For the f correlation there is an analog of Proposition 2.1 for covariance.

Lemma 7.1: For any A , B苸Mn,sa,␳苸Dn

1

, and f苸Fop, one has

2 Re兵Corr_␳f共A,B兲其 = Corr_␳f共A,B兲 + Corrf_␳共B,A兲 = f共0兲具i关␳,A兴,i关␳,B兴典_␳,f, 2i Im兵Corr_␳f_{共A,B兲其 = Corr}

␳f共A,B兲 − Corr␳f共B,A兲 = Tr共␳关A,B兴兲. Proof: We have that

Corr_␳f共A,B兲 − Corr_␳f共B,A兲 = Tr共␳关A,B兴兲, which is purely imaginary.

This implies

Re兵Corr_␳f_{共A,B兲其 = Re兵Corr} ␳f共B,A兲其, so that

2 Re兵Corr_␳f_{共A,B兲其 = Corr} ␳

f_{共A,B兲 + Corr} ␳ f_共B,A兲,

2i Im兵Corr_␳f_{共A,B兲其 = Corr}

␳f共A,B兲 − Corr␳f共B,A兲. Since

Corr_␳f共A,B兲 + Corr_␳f共B,A兲 = Tr共␳AB兲 + Tr共␳BA兲 − 2 Tr共A⌬共B兲兲,

the conclusion follows from Proposition 6.2. 䊐

Corollary 7.1:

I_␳f共A兲 = Corr_␳f共A,A兲 = f共0兲

2 具i关␳,A兴,i关␳,A兴典␳,f= f共0兲 2 储i关␳,A兴储␳,f 2 _. Remark 7.1: If f_␤共x兲 ª␤共1 −␤兲 共x − 1兲 2 共x␤_{− 1兲共x}1−␤_{− 1兲}, ␤苸

冉

0, 1 2

册

, then I_␳f␤_{共A兲 =} f␤共0兲 2 Tr共i关␳,A兴cf␤共L␳,R␳兲i关␳,A兴兲 = − 1 2Tr共关␳ ␤_,A兴关␳1−␤_,A兴兲, so I_␳f␤共A兲 coincides with the WYD skew information.

Let us reformulate the main result in terms of f correlation.

Proposition 7.1: For any f苸Fop, one has

冉

f共0兲 2 Area␳

f_共i关␳

,A兴,i关␳,B兴兲

冊

2

= I_␳f共A兲I_␳f共B兲 −

兩

Re兵Corr_␳f共A,B兲其兩2. Proof:

冉

f共0兲 2 Area␳ f_共i关␳ ,A兴,i关␳,B兴兲

冊

2 = f共0兲 2

4 共具i关␳,A兴,i关␳,A兴典␳,f具i关␳,B兴,i关␳,B兴典␳,f−具i关␳,A兴,i关␳,B兴典␳,f 2 _兲 =

冉

f共0兲 2 储i关␳,A兴储␳,f 2

冊冉

f共0兲 2 储i关␳,B兴储␳,f 2

冊

−

冉

f共0兲 2 具i关␳,A兴,i关␳,B兴典␳,f

冊

2 = I_␳f共A兲I_␳f共B兲 −

兩

Re兵Corr_␳f共A,B兲其兩2. 䊐

Therefore, our main result states that

Var_␳共A兲Var_␳共B兲 − 兩Re兵Cov_␳共A,B兲其兩2_{艌 I}

␳f共A兲I␳f共B兲 − 兩Re兵Corr␳f共A,B兲其兩2. Recall that we introduced, for fixed␳, A , B, the functional

F共f兲 = Var_␳共A兲Var_␳共B兲 − 兩Cov_␳s共A,B兲兩2−

冉

f共0兲 2 Area␳

f_{共i关␳,A兴,i关␳,B兴兲}

冊

2

= Var_␳共A兲Var_␳共B兲 − 兩Re兵Cov_␳共A,B兲其兩2− If_␳共A兲I_␳f共B兲 + 兩Re兵Corr_␳f共A,B兲其兩2. As the main result, we proved that, for any f , g苸F_op, F共f兲艌0 and f˜艋g˜⇒F共f兲艋F共g兲.

Corollary 7.2: Suppose that ␳, A , B are fixed. Then the function of␤given by

F共␤兲 ª F共f_␤兲

is decreasing on

共0 ,

1₂

兴

and F共1/2兲艌0; therefore F共␤兲艌0.

Proof: Given x⬎0, the function␤哫 f˜_␤共x兲=1₂共x␤+ x1−␤兲 is decreasing in

共0 ,

1₂

兴, so that

␤1艋␤2⇒ f˜␤₁艌 f˜␤₂⇒ F共␤1兲 艌 F共␤2兲. _䊐

Remark 7.2: The above corollary was the content of Theorem 5, the main result in Ref.13, and of Proposition IV.1 in Ref.28. Note that, because of Corollary 7.2 the optimal bound previ-ously known was given by fWY, namely, the bound of Wigner-Yanase metric 共this was due to Kosaki in Ref.13兲. Remark 5.4 implies that the bound given by the SLD area is strictly greater than that given by the WY area.

Proposition 7.2:

Cov_␳共A,B兲 = Corr_␳f共A,B兲 + C_␳f˜共A0,B0兲,

Var_␳共A兲 = I_␳f共A兲 + C_␳f˜共A0兲. Proof: The calculations of Proposition 6.3 imply that

Corr_␳f共A,B兲 − Cov_␳共A,B兲 = Tr共␳A兲Tr共␳B兲 − Tr共⌬共A兲B兲 = − Tr共mf˜共L␳,R␳兲共A0兲B0兲 = − C␳f˜共A0,B0兲. 䊐 Luo17 suggested that if one considers the variance as a measure of “uncertainty” of an ob-servable A in the state␳, then the above equality splits the variance in a “quantum” part共I_␳f共A兲兲 plus a “classical” part共C_␳f˜共A0兲兲.

VIII. CONDITIONS FOR EQUALITY

In this section we give a necessary and sufficient condition to have equality in our main result.

Proposition 8.1: The inequality of Theorem 6.1 is an equality if and only if A₀ and B₀ are

proportional.

Proof: If A₀=␭B₀, with␭苸R, then

Var_␳共A兲Var_␳共B兲 −兵兩Re兵Cov_␳共A,B兲其兩2其 = Tr共␳A₀2兲Tr共␳B2₀兲 − 兩Re兵Tr共␳A0B0兲其兩2= Tr共␳共␭B0兲2兲Tr共␳B02兲 −兩Re兵Tr共␳␭B0B0兲其兩2=␭2Tr共␳B0

2_兲2_−␭2_兩Tr共␳B 0 2_兲兩2_{= 0.} In this case the inequality is just the equality 0 = 0.

Now we suppose that A₀, B₀are not proportional and we prove that the inequality is strict. We use the same notations as in the proof of Theorem 6.1.

Var_␳共A兲Var_␳共B兲 − 兩Re兵Cov_␳共A,B兲其兩2− I_␳f共A兲I_␳f共B兲 + 兩Re兵Corr_␳f共A,B兲其兩2 =␰−␩=1

4_i,j,k,l

兺

Hf共␭i,␭j,␭k,␭l兲Ki,j,k,l共A,B兲 and

Hf共␭i,␭j,␭k,␭l兲 ⬎ 0, Ki,j,k,l共A,B兲 艌 0, ∀ i, j,k,l.

Therefore, the strict inequality is equivalent to␰−␩⬎0, which is, in turn, equivalent to Ki,j,k,l共A,B兲 ⬎ 0,

for some i , j , k , l.

From the fact that A0, B0are not proportional, one can derive that also the matrices兵aij其,兵bij其

are not proportional and this implies 共the other cases being trivial兲 that there exist 共complex兲 aij, bij, akl, bkl⫽0 and 共real兲 ␭,␮⫽0 such that

aij=␭bij, akl=␮bkl, ␭ ⫽␮.

We get

Ki,j,k,l共A,B兲 = 兩aij兩2兩bkl兩2+兩akl兩2兩bij兩2− 2 Re兵aijbji其Re兵aklblk其

=兩aij兩2兩bkl兩2+兩␮bkl兩2

冏

aij ␭

冏

2 − 2 Re

再

aij aij ␭

冎

Re兵␮bklblk其 =

冉

1 +␮ 2 ␭2

冊

兩aij兩2兩bkl兩2− 2 ␮ ␭兩aij兩2兩bkl兩2 =

冉

1 +␮ 2 ␭2 − 2 ␮ ␭

冊

兩aij兩2兩bkl兩2=

冉

1 − ␮ ␭

冊

2 兩aij兩2兩bkl兩2⬎ 0, because

冉

1 −␮ ␭

冊

⫽ 0. Therefore, ␰−␩⫽ 0,

and this ends the proof. 䊐

The particular case f = f_␤ 共where ␤苸

共0 ,

1₂

兴兲 of the above proposition has been proved in

Proposition 6 in Ref.13.

IX. ANOTHER INEQUALITY

The study of the mean mf˜allows us to get another inequality that can be seen as an uncertainty

principle in Heisenberg form. Recall that

fRLD共x兲 ª 2x x + 1. Proposition 9.1:

Var_␳共A兲 艌 I_␳f共A兲 + C_␳fRLD共A

0兲, ∀ f 苸 Fop.

Var_␳共A兲 = Tr共␳A₀2兲 =1

2

兺

_i,j 共␭i+␭j兲aijaji, I_␳f共A兲 = Var_␳共A兲 − Tr共A0mf˜共L_␳,R_␳兲A0兲 =

1

2

兺

_i,j 共␭i+␭j兲aijaji−

兺

_i,j mf˜共␭i,␭j兲aijaji, C_␳f_RLD共A

0兲 =

兺

i,j

mh₀共␭i,␭j兲aijaji,

using Corollary 5.1 we have

Var_␳共A兲 − I_␳f共A兲 − C_␳fRLD共A

0兲 =

兺

i,j

关mf˜共␭i,␭j兲 − mh₀共␭i,␭j兲兴兩aij兩2艌 0 . 䊐

From this we get the following inequality.

Theorem 9.1:

Var_␳共A兲Var_␳共B兲 艌 关I_␳f共A兲 + C_␳fRLD共A

0兲兴关I␳f共B兲 + C␳fRLD共B0兲兴, ∀ f 苸 Fop. 共9.1兲 Since C_␳fRLD共A

0兲艌0, we obtain, as a corollary, two results due to Luo, for the case f = fWY =1₄共1+

冑

x兲2_{, and to Hansen, for the general case共see Refs.18and9兲.}

Proposition 9.2:

Var_␳共A兲 艌 I_␳f共A兲, ∀ f 苸 Fop.

Theorem 9.2:

Var_␳共A兲Var_␳共B兲 艌 I_␳f共A兲I_␳f共B兲 = f共0兲 2 4 储i关␳,A兴储␳,f 2 _{储i关␳,B兴储} ␳,f 2 _{, ∀ f 苸 F} op. Let us study how the bound I_␳f共A兲I_␳f共B兲 depends on f.

Proposition 9.3: For any f , g苸Fop,

f艋 g ⇒ C_␳f共A0兲 艋 C␳g共A0兲,

f˜艋 g˜ ⇒ I_␳f共A兲 艌 I_␳g共A兲.

Proof: We still use notations of Theorem 6.1. Since mf艋mg,

C_␳g_共A 0兲 − C␳ f_共A 0兲 =

兺

i,j mg共␭i,␭j兲aijaji−

兺

i,j mf共␭i,␭j兲aijaji=

兺

i,j 关mg共␭i,␭j兲 − mf共␭i,␭j兲兴兩aij兩2艌 0.

The second inequality is an immediate consequence of the first one. 䊐

Corollary 9.1:

I_␳SLD共A兲 艌 I_␳f共A兲, ∀ f 苸 Fop.

Proof: Immediate consequence of Corollary 5.1. 䊐

Corollary 9.2:

f˜艋 g˜ ⇒ I_␳f共A兲I_␳f共B兲 艌 I_␳g共A兲I_␳g共B兲. We discuss, now, the equality in Theorem 9.2.

Proposition 9.4:

Proof: Because of Proposition 9.2, we have

Var_␳共A兲Var_␳共B兲 = I_␳f共A兲I_␳f共B兲 ⇔ Var_␳共A兲 = I_␳f共A兲, Var_␳共B兲 = I_␳f共B兲.

Hence, we need to show Var_␳共A兲=I_␳f共A兲⇔A0= 0. Indeed, using the same notations as in Theorem 6.1,

Var_␳共A兲 = I_␳f共A兲 ⇔ Tr共A0mf˜共L␳,R␳兲A0兲 = 0 ⇔

兺

i,j

mf˜共␭i,␭j兲aijaji= 0⇔ aij= 0, ∀ i, j ⇔ A0= 0. 䊐

X. RELATION WITH THE STANDARD UNCERTAINTY PRINCIPLES

Some authors tried to prove the following inequalities:

冉

f共0兲

2 Area共i关␳,A兴,i关␳,B兴兲

冊

2

= I_␳f共A兲I_␳f共B兲 − 兩Re共Corr_␳f共A,B兲兲兩2艌 1

4兩Tr共␳关A,B兴兲兩

2_{, 共10.1兲}

I_␳f共A兲I_␳f共B兲 艌1

4兩Tr共␳关A,B兴兲兩

2_{. 共10.2兲}

They wanted to obtain the standard Heisenberg-Schrödinger uncertainty principles as conse-quences of the uncertainty principles discussed in the present paper. Actually inequality共10.1兲has been proved false for f = f_␤, that is, for the WYD case共see pp. 632 and 642–644 in Ref. 13, p. 4404 in Ref. 28, and Ref. 20兲. However, the discussion of Secs. VI–IX shows that the upper bounds G共f兲 =f共0兲 2 Area共i关␳,A兴,i关␳,B兴兲, N共f兲 ª I␳ f_共A兲I ␳ f_共B兲

can be larger than those of the WYD metric共we showed it for the SLD metric in Remark 5.4兲. It is, therefore, natural to ask if the above inequalities, that are false for the WYD metric, can be true for some different QFI共for example, for the SLD metric兲. The following theorem shows that this is not the case, even on 2⫻2 matrices.

Theorem 10.1: There exist 2⫻2 self-adjoint matrices A and B and a density matrix␳ such that

I_␳f共A兲I_␳f共B兲 ⬍1

4兩Tr共␳关A,B兴兲兩

2_{, ∀ f 苸 F} op.

Therefore, for these␳, A , B, we also have

冉

f共0兲

2 Area共i关␳,A兴,i关␳,B兴兲

冊

2

= I_␳f共A兲I_␳f共B兲 − 兩Re共Corr_␳f共A,B兲兲兩2⬍1

4兩Tr共␳关A,B兴兲兩

2_{, ∀ f 苸 F} op.

Proof: We use notations of Theorem 6.1: let兵␸i其 be a complete orthonormal base composed of

eigenvectors of␳ and兵␭i其 the corresponding eigenvalues. Set aij⬅具A0␸i兩␸j典 and bij⬅具B0␸i兩␸j典.

In what follows,␭₁⬎␭₂⬎0, ␭₁+␭₂= 1, and

␳=

冉

␭1 0 0 ␭2

冊

, A =

冉

0 i − i 0

冊

, B =

冉

0 1 1 0

冊

共in terms of Pauli matrices, A=−␴2and B =␴1兲. Simple calculations show that 兩aii兩=兩bii兩=0, while

Var_␳共A兲 = Tr共␳A0兲 = 1 2

兺

_i,j 共␭i+␭j兲aijaji= 1 2共共␭1+␭2兲 + 共␭2+␭1兲兲 = 1, C_␳f˜_共A 0兲 =

兺

i,j mf˜共␭i,␭j兲aijaji=共mf˜共␭1,␭2兲 + mf˜共␭1,␭2兲兲 = 2mf˜共␭1,␭2兲,

I_␳f共A兲 = Var_␳共A兲 − C_␳f˜共A0兲 = 1 − 2mf˜共␭1,␭2兲. By the same reasoning,

Var_␳共B兲 = 1,

C_␳f˜_共B

0兲 = 2mf˜共␭1,␭2兲, I_␳f共B兲 = 1 − 2mf˜共␭1,␭2兲. Moreover, by direct calculation, one has

1

4兩Tr共␳关A,B兴兲兩 2_=共␭

1−␭2兲2.

Now, recall that since mf˜ is a mean共and because of Corollary 3.1兲, one has for any f 苸Fop, ␭1⬎ mf˜共␭1,␭2兲 ⬎ ␭2⬎ 0,

1 − 2mf˜共␭1,␭2兲 = 共␭1+␭2兲 − 2mf˜共␭1,␭2兲 艌 0. Hence, the following inequalities are equivalent:

I_␳f共A兲I_␳f共B兲 ⬍1 4兩Tr共␳关A,B兴兲兩 2_, 共1 − 2mf˜共␭1,␭2兲兲2⬍ 共␭1−␭2兲2, 共␭1+␭2兲 − 2mf˜共␭1,␭2兲 ⬍ ␭1−␭2, 2␭2⬍ 2mf˜共␭1,␭2兲, ␭2⬍ mf˜共␭1,␭2兲,

and so we get the conclusion. 䊐

Note that

1

4兩Tr共␳关A,B兴兲兩 2_{艌 I}

␳f共A兲I␳f共B兲

is obviously false, in general: if one takes A = B, the left side is zero and the right side could be positive at the same time.

1

4兩Tr共␳关A,B兴兲兩 2_{艌 I}

␳f共A兲I␳f共B兲 − 兩Re兵Corr␳f共A,B兲其兩2=

冉

f共0兲

2 Areaf共i关␳,A兴,i关␳,B兴兲

冊

2

.

Indeed, one may choose␳, A , B such that关A,B兴=0 while 关␳, A兴,关␳, B兴 are not proportional, so that they span a positive area.

We may conclude that the Heisenberg and Schrödinger uncertainty principles,

Var_␳共A兲Var_␳共B兲 艌1

4兩Tr共␳关A,B兴兲兩 2_,

Area_␳Covs共A,B兲 艌1

2兩Tr共␳关A,B兴兲兩, cannot be deduced from the uncertainty principles,

Var_␳共A兲Var_␳共B兲 艌 I_␳f共A兲I_␳f共B兲,

Area_␳Covs共A,B兲 艌f共0兲 2 Area␳

f_{共i关␳,A兴,i关␳,B兴兲,}

and vice versa.

The above described mistake appeared several times in the literature 共see Theorem 2 in Ref. 18, Theorem 2 in Ref.19, Theorem 1 in Ref.21, and Note 1, Sec. 3.2 in Ref.9兲. It can be helpful to explain its origin, again along the lines of Ref.13共see also Ref.28兲.

We have seen that 1 2iTr共␳关A,B兴兲 = 1 2i共Corr␳ f_{共A,B兲 − Corr} ␳ f_{共B,A兲兲 = Im共Corr} ␳ f_{共A,B兲兲,} and therefore 1 4兩Tr共␳关A,B兴兲兩 2_{=兩Im共Corr}

␳f共A,B兲兲兩2艋 兩Corr␳f共A,B兲兩2. If there were a Cauchy-Schwartz-type estimate,

兩Corr_␳f_共A,B兲兩2_{艋 Corr}

␳f共A,A兲Corr␳f共B,B兲, 共10.3兲

using, for example, Theorem 9.2, one would get a refined Heisenberg uncertainty principle in the form

Var_␳共A兲Var_␳共B兲 艌 I_␳f共A兲I_␳f共B兲 艌1

4兩Tr共␳关A,B兴兲兩 2_.

By Theorem 10.1 we know that this is impossible. The wrong point is the Cauchy-Schwartz estimate关Eq.共10.3兲兴, which is false. This depends on the following facts. The sesquilinear form

Corr_␳f共X,Y兲 ª Tr共␳X†Y兲 − Tr共X†mf˜共L␳,R␳兲共Y兲兲

on the complex space Mnis not positive共see p. 632 in Ref.13兲. On the other hand, Corr_␳f共A,B兲 is

not a real form on the real space M_n,sa: also in this case, one cannot prove the desired Cauchy-Schwartz inequality. The best one can have is a Cauchy-Cauchy-Schwartz estimate only for the 共real兲 positive bilinear form Re兵Corr_␳f_{共A,B兲其 on M}

n,sa 关see Ref. 13共p. 643兲 and Ref. 20兴. This would

冉

f共0兲 2 Area␳

f_{共i关␳,A兴,i关␳,B兴兲}

冊

2

= I_␳f共A兲I_␳f共B兲 − 兩Re兵Corr_␳f共A,B兲其兩2艌 0.

XI. NOT FAITHFUL STATES AND PURE STATES

We discuss, now, the general case␳艌0.

Proposition 11.1: The function mf˜:共0,⬁兲⫻共0,⬁兲→共0,⬁兲 has a continuous extension to

关0,⬁兲⫻关0,⬁兲.

Proof: If f is regular then, for example, lim 共x,y兲→共0,y0兲 mf˜共x,y兲 = y0 2 − f共0兲y₀2 2y0f共0兲 = 0.

If f is not regular then mf˜共x,y兲=共x+y兲/2 and we are done 共see Ref.9兲. 䊐

The definition of f correlation still makes sense and the inequality of Theorem 6.1 Var_␳共A兲Var_␳共B兲 − 兩Re兵Cov_␳共A,B兲其兩2艌 I_␳f共A兲I_␳f共B兲 − 兩Re兵Corr_␳f共A,B兲其兩2, holds by continuity for arbitrary共not necessarily faithful兲 states.

In what follows, we study the pure state case.

Corollary 11.1: If s :关0, +⬁兲⫻关0, +⬁兲→R is a continuous function and␳is a pure state, then

s共L_␳,R_␳兲共A兲 =␳A␳.

Proof: Consequence of Corollary 6.1 䊐

Lemma 11.1: If ␳is pure, then Tr共共␳A␳兲共␳B␳兲兲=Tr共␳A␳兲Tr共␳B␳兲.

Proof: Suppose for simplicity that ␳= diag共1,0, ... ,0兲 共the general case follows easily from

this兲. Then ␳A␳= diag共A11, 0 , . . . , 0兲 and the same holds for B. Therefore 共␳A␳兲共␳B␳兲 = diag共A11B11, 0 , . . . , 0兲. This implies

Tr共共␳A␳兲共␳B␳兲兲 = A₁₁B₁₁= Tr共␳A␳兲Tr共␳B␳兲 . 䊐

Lemma 11.2: If ␳is pure, then

Tr共mf共L␳,R␳兲共A兲B兲 = Tr共␳A兲Tr共␳B兲. Proof: By Corollary 11.1, one has

mf共L␳,R␳兲共A兲 =␳A␳,

and therefore

Tr共mf共L␳,R␳兲共A兲B兲 = Tr共␳A␳B兲 = Tr共共␳A␳兲共␳B␳兲兲 = Tr共␳A␳兲Tr共␳B␳兲 = Tr共␳A兲Tr共␳B兲.

䊐

Corollary 11.2: If␳ is pure, then

C_␳f_共A

0,B0兲 = Tr共mf共L_␳,R_␳兲共A0兲B0兲 = Tr共␳A0兲Tr共␳B0兲 = 0.

Proposition 11.2: If ␳is pure, then

Corr_␳f共A,B兲 = Cov_␳共A,B兲, ∀ f 苸 Fop.

Proof: Immediate from the above corollary and Proposition 7.2. 䊐

The case I_␳f共A兲=Var_␳共A兲 was proved by Hansen in Theorem 3.8 共p. 16兲 in Ref.9. Therefore, on pure states we have the equalities