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Metric adjusted skew informationFrank Hansen†

Department of Economics, University of Copenhagen, Studiestraede 6, DK-1455 Copenhagen, Denmark;

Edited by Richard V. Kadison, University of Pennsylvania, Philadelphia, PA, and approved April 17, 2008 (received for review January 17, 2007)

We extend the concept of Wigner–Yanase–Dyson skew informa-tion to something we call “metric adjusted skew information” (ofa state with respect to a conserved observable). This “skew infor-mation” is intended to be a non-negative quantity bounded by thevariance (of an observable in a state) that vanishes for observablescommuting with the state. We show that the skew information isa convex function on the manifold of states. It also satisfies otherrequirements, proposed by Wigner and Yanase, for an effectivemeasure-of-information content of a state relative to a conservedobservable. We establish a connection between the geometricalformulation of quantum statistics as proposed by Chentsov andMorozova and measures of quantum information as introducedby Wigner and Yanase and extended in this article. We show thatthe set of normalized Morozova–Chentsov functions describing thepossible quantum statistics is a Bauer simplex and determine itsextreme points. We determine a particularly simple skew informa-tion, the “λ-skew information,” parametrized by a λ ∈ (0, 1], andshow that the convex cone this family generates coincides with theset of all metric adjusted skew informations.

convexity | monotone metric | Morozova–Chentsov function | λ-skewinformation

I n the mathematical model for a quantum mechanical system,the physical observables are represented by self-adjoint oper-

ators on a Hilbert space. The “states” (that is, the “expectationfunctionals” associated with the states) of the physical system areoften “modeled” by the unit vectors in the underlying Hilbertspace. So, if A represents an observable and x ∈ H corresponds toa state of the system, the expectation of A in that state is (Ax | x).For what we shall be proving, it will suffice to assume that ourHilbert space is finite dimensional and that the observables areself-adjoint operators, or the matrices that represent them, onthat finite dimensional space. In this case, the states can be real-ized with the aid of the trace (functional) on matrices and anassociated “density matrix.” We denote by Tr(B) the usual traceof a matrix B [that is, Tr(B) is the sum of the diagonal entries of B].The expectation functional of a state can be expressed as Tr(ρA),where ρ is a matrix, the density matrix associated with the state,and “Tr(ρA)” is the trace of the product ρA of the two matricesρ and A. (Henceforth, we write “Tr ρA” omitting the parentheseswhen they are clearly understood.)

In ref. 1, Wigner noticed that in the presence of a conserva-tion law the obtainable accuracy of the measurement of a physicalobservable is limited if the operator representing the observabledoes not commute with (the operator representing) the conservedquantity (observable). Wigner proved it in the simple case wherethe physical observable is the x-component of the spin of a spinone-half particle and the z-component of the angular momentumis conserved. Araki and Yanase (2) demonstrated that this is a gen-eral phenomenon and pointed out, following Wigner’s example,that under fairly general conditions an approximate measurementmay be carried out.

Another difference is that observables that commute with a con-served additive quantity, like the energy, components of the linearor angular momenta, or the electrical charge, can be measuredeasily and accurately by microscopic apparatuses (the analysisis restricted to one conserved quantity), while other observablescan be only approximately measured by a macroscopic apparatus

large enough to superpose sufficiently many states with differentquantum numbers of the conserved quantity.

Wigner and Yanase (3) proposed finding a measure of ourknowledge of a difficult-to-measure observable with respect toa conserved quantity. The quantum mechanical entropy is a mea-sure of our ignorance of the state of a system, and minus theentropy can therefore be considered as an expression of our knowl-edge of the system. This measure has many attractive propertiesbut does not take into account the conserved quantity. In partic-ular, Wigner and Yanase wanted a measure that vanishes whenthe observable commutes with the conserved quantity. It shouldtherefore not measure the effect of mixing in the classical senseas long as the pure states taking part in the mixing commute withthe conserved quantity. Only transition probabilities of pure states“lying askew” (to borrow from the introduction of ref. 3) to theeigenvectors of the conserved quantity should give contributionsto the proposed measure.

Wigner and Yanase discussed a number of requirements thatsuch a measure should satisfy in order to be meaningful andsuggested, tentatively, the skew information defined by

I(ρ, A) = −12

Tr([ρ1/2, A]2) ,

where [C, D] is the usual “bracket notation” for operators or matri-ces: [C, D] = CD−DC, as a measure of the information containedin a state ρ with respect to a conserved observable A. It manifestlyvanishes when ρ commutes with A, and it is homogeneous in ρ.

The requirements Wigner and Yanase discussed, all reflectedproperties considered attractive or even essential. Since informa-tion is lost when separated systems are united such a measureshould be decreasing under the mixing of states, that is, be convexin ρ. The authors proved this for the skew information but notedthat other measures may enjoy the same properties; in particular,the expression

−12

Tr [ρp, A][ρ1−p, A

]0 < p < 1

proposed by Dyson. Convexity of this expression in ρ becamethe celebrated Wigner–Yanase–Dyson conjecture which was laterproved by Lieb (4). (See also ref. 5 for a truly elementary proof.)

The measure should also be additive with respect to the aggre-gation of isolated subsystems and, for an isolated system, indepen-dent of time. These requirements are discussed in more detail inConvexity Statements. They are easily seen to be satisfied by theskew information.

In the process that is the opposite of mixing, the informationcontent should decrease. This requirement comes from thermo-dynamics where it is satisfied for both classical and quantummechanical systems. It reflects the loss of information about sta-tistical correlations between two subsystems when they are onlyconsidered separately. Wigner and Yanase conjectured that the

Author contributions: F.H. performed research and wrote the paper.

This article is a PNAS Direct Submission.†E-mail: frank.hansen@econ.ku.dk.

© 2008 by The National Academy of Sciences of the USA

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skew information also possesses this property. They proved itwhen the state of the aggregated system is pure.‡

The aim of this article is to connect the subject of measuresof quantum information as laid out by Wigner and Yanase withthe geometrical formulation of quantum statistics by Chentsov,Morozova, and Petz.

The Fisher information measures the statistical distinguishabil-ity of probability distributions. Let Pn = {p = (p1, . . . , pn) | pi >0} be the (open) probability simplex with tangent space TPn. TheFisher–Rao metric is then given by

Mp(u, v) =n∑

i=1

uivi

piu, v ∈ TPn.

Note that u = (u1, . . . , un) ∈ TPn if and only if u1+· · ·+un = 0, butthat the metric is well defined also on Rn. Chentsov proved that theFisher–Rao metric is the unique Riemannian metric contractingunder Markov morphisms (7).

Since Markov morphisms represent coarse graining or random-ization, it means that the Fisher information is the only Riemann-ian metric possessing the attractive property that distinguishabilityof probability distributions becomes more difficult when they areobserved through a noisy channel.

Chentsov and Morozova extended the analysis to quantummechanics by replacing Riemannian metrics defined on the tan-gent space of the simplex of probability distributions with positivedefinite sesquilinear (originally bilinear) forms Kρ defined on thetangent space of a quantum system, where ρ is a positive defi-nite state. Customarily, Kρ is extended to all operators (matrices)supported by the underlying Hilbert space; cf. refs. 8 and 9 fordetails. Noisy channels are in this setting represented by stochas-tic (completely positive and trace preserving) mappings T , and thecontraction property by the monotonicity requirement

KT(ρ)(T(A), T(A)) ≤ Kρ(A, A)

is imposed for every stochastic mapping T : Mn(C) → Mm(C).Unlike the classical situation, it turned out that this require-ment no longer uniquely determines the metric. By the combinedefforts of Chentsov, Morozova, and Petz it is established that themonotone metrics are given on the form

Kρ(A, B) = Tr A∗c(Lρ , Rρ)B, [1]

where c is a so called Morozova–Chentsov function and c(Lρ , Rρ)is the function taken in the pair of commuting left and right mul-tiplication operators (denoted Lρ and Rρ , respectively) by ρ. TheMorozova–Chentsov function is of the form

c(x, y) = 1yf (xy−1)

x, y > 0,

where f is a positive operator monotone function defined in thepositive half-axis satisfying the functional equation

f (t) = t f (t−1) t > 0. [2]

The function

f (t) = t + 2√

t + 14

t > 0

is clearly operator monotone and satisfies Eq. 2. The associatedMorozova–Chentsov function

cWY (x, y) = 4(√

x + √y)2

x, y > 0

‡We subsequently demonstrated (6) that the conjecture fails for general mixed states.

therefore defines a monotone metric

KWYρ (A, B) = Tr A∗cWY (Lρ , Rρ)B,

which we shall call the Wigner–Yanase metric. The starting pointof our investigation is the observation by Gibilisco and Isola (10)that

I(ρ, A) = 18

Tr i[ρ, A]cWY (Lρ , Rρ)i[ρ, A].There is thus a relationship between the Wigner–Yanase measureof quantum information and the geometrical theory of quantumstatistics. It is the aim of the present article to explore this relation-ship in detail. The main result is that all well behaved measuresof quantum information—including the Wigner–Yanase–Dysonskew informations—are given in this way for a suitable subclassof monotone metrics.

1. Regular MetricsDefinition 1.1 (regular metric). We say that a symmetric monotonemetric (11, 12) on the state space of a quantum system is regu-lar, if the corresponding Morozova–Chentsov function c admits astrictly positive limit

m(c) = limt→0

c(t, 1)−1.

We call m(c) the metric constant.

We also say, more informally, that a Morozova–Chentsov func-tion c is regular if m(c) > 0. The function f (t) = c(t, 1)−1 ispositive and operator monotone on the positive half-line andmay be extended to the closed positive half-line. Thus the metricconstant m(c) = f (0).

Definition 1.2 (metric adjusted skew information). Let c be theMorozova–Chentsov function of a regular metric. We introducethe metric adjusted skew information Ic

ρ(A) by setting

Icρ(A) = m(c)

2Kc

ρ(i[ρ, A], i[ρ, A])

= m(c)2

Tr i[ρ, A]c(Lρ , Rρ)i[ρ, A] [3]

for every ρ ∈ Mn (the manifold of states) and every self-adjointA ∈ Mn(C).

Note that the metric adjusted skew information is proportionalto the square of the metric length, as it is calculated by the sym-metric monotone metric Kc

ρ with Morozova–Chentsov function c,of the commutator i[ρ, A], and that this commutator belongs tothe tangent space of the state manifold Mn. Metric adjusted skewinformation is thus a non-negative quantity. If we consider theWYD-metric with Morozova–Chentsov function

cWYD(x, y) = 1p(1 − p)

· (xp − yp)(x1−p − y1−p)(x − y)2 0 < p < 1,

then the metric constant m(cWYD) = p(1 − p) and the metricadjusted skew information

IcWYDρ (A) = p(1 − p)

2Tr i[ρ, A]cWYD(Lρ , Rρ)i[ρ, A]

= −12

Tr[ρp, A][ρ1−p, A]becomes the Dyson generalization of the Wigner–Yanase skewinformation.§ The choice of the factor m(c) therefore works

§Hasegawa and Petz proved in (13) that the function cWYD is a Morozova–Chentsov func-tion. They also proved that the Wigner–Yanase–Dyson skew information is proportionalto the (corresponding) quantum Fisher information of the commutator i[ρ, A].

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also for p �= 1/2. It is in fact a quite general construction, andthe metric constant is related to the topological properties ofthe metric adjusted skew information close to the border of thestate manifold. But it is difficult to ascertain these propertiesdirectly, so we postpone further investigation until having estab-lished that Ic

ρ(A) is a convex function in ρ. Since the commutatori[ρ, A] = i(Lρ − Rρ)A we may rewrite the metric adjusted skewinformation as

Icρ(A) = m(c)

2Tr A(i(Lρ − Rρ))∗c(Lρ , Rρ)i(Lρ − Rρ)A

= m(c)2

Tr Ac(Lρ , Rρ)A, [4]

where

c(x, y) = (x − y)2c(x, y) x, y > 0. [5]

Before we can address these questions in more detail, we have tostudy various characterizations of (symmetric) monotone metrics.

2. Characterizations of Monotone MetricsTheorem 2.1. A positive operator monotone decreasing function gdefined in the positive half-axis and satisfying the functional equation

g(t−1) = t · g(t) [6]

has a canonical representation

g(t) =∫ 1

0

(1

t + λ+ 1

1 + tλ

)dµ(λ), [7]

where µ is a finite Borel measure with support in [0, 1].

Proof . The function g is necessarily of the form

g(t) = β +∫ ∞

0

1t + λ

dµ(λ),

where β ≥ 0 is a constant and µ is a positive Borel measure suchthat the integrals

∫(1 + λ2)−1dµ(λ) and

∫λ(1 + λ2)−1dµ(λ) are

finite (cf. ref. 14 page 9). We denote by µ the measure obtainedfrom µ by removing a possible atom in zero. Then, by making thetransformation λ → λ−1, we may write

g(t) = β + µ(0)t

+∫ ∞

0

1t + λ

dµ(λ)

= β + µ(0)t

+∫ ∞

0

1t + λ−1 · 1

λ2 dµ(λ−1)

= β + µ(0)t

+∫ ∞

0

11 + tλ

dν(λ),

where ν is the Borel measure given by dν(λ) = λ−1dµ(λ−1). Sinceg satisfies the functional equation (6) we obtain

β + µ(0)t +∫ ∞

0

11 + t−1λ

dν(λ) = tβ + µ(0) +∫ ∞

0

tt + λ

dµ(λ).

By letting t → 0 and since ν and µ have no atoms in zero, weobtain β = µ(0) and consequently

∫ ∞

0

1t + λ

dν(λ) =∫ ∞

0

1t + λ

dµ(λ) t > 0.

By analytic continuation we realize that both measures ν and µappear as the representing measure of an analytic function withnegative imaginary part in the complex upper half plane. They

are therefore, by the representation theorem for this class offunctions, necessarily identical. We finally obtain

g(t) = β + β

t+

∫ ∞

0

1t + λ

dµ(λ)

= β + β

t+

∫ 1

0

1t + λ

dµ(λ) +∫ 1

0

1t + λ−1 · 1

λ2 dµ(λ−1)

= β + β

t+

∫ 1

0

1t + λ

dµ(λ) +∫ 1

0

11 + tλ

dν(λ)

= β + β

t+

∫ 1

0

(1

t + λ+ 1

1 + tλ

)dµ(λ)

=∫ 1

0

(1

t + λ+ 1

1 + tλ

)dµ(λ).

The statement follows since every function of this form obviously isoperator monotone decreasing and satisfy the functional equation(6). We also realize that the representing measure µ is uniquelydefined.

Remark 2.2. Inspection of the proof of Theorem 2.1 shows that thePick function −g(x) = −c(x, 1) has the canonical representation

−g(x) = −g(0) +∫ 0

−∞1

λ − tdµ(−λ).

The representing measure therefore appears as 1/π times the limitmeasure of the imaginary part of the analytic continuation −g(z)as z approaches the closed negative half-axis from above (cf., forexample, ref. 15). The measure µ in Eq. 7 therefore appears asthe image of the representing measure’s restriction to the interval[−1, 0] under the transformation λ → −λ.

We define, in the above setting, an equivalent Borel measureµg on the closed interval [0, 1] by setting

dµg(λ) = 21 + λ

dµ(λ) [8]

and obtain:

Corollary 2.3. A positive operator monotone decreasing function gdefined in the positive half-axis and satisfying the functional equation(6) has a canonical representation

g(t) =∫ 1

0

1 + λ

2

(1

t + λ+ 1

1 + tλ

)dµg(λ), [9]

where µg is a finite Borel measure with support in [0, 1]. The func-tion g is normalized in the sense that g(1) = 1, if and only if µg is aprobability measure.

Corollary 2.4. A Morozova–Chentsov function c allows a canonicalrepresentation of the form

c(x, y) =∫ 1

0cλ(x, y) dµc(λ) x, y > 0, [10]

where µc is a finite Borel measure on [0, 1] and

cλ(x, y) = 1 + λ

2

(1

x + λy+ 1

λx + y

)λ ∈ [0, 1]. [11]

The Morozova–Chentsov function c is normalized in the sense thatc(1, 1) = 1 (corresponding to a Fisher adjusted metric), if and onlyif µc is a probability measure.

Proof . A Morozova–Chentsov function is of the form c(x, y) =y−1f (xy−1)−1, where f is a positive operator monotone function

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defined in the positive half-axis and satisfying the functional equa-tion f (t) = t f (t−1). The function g(t) = f (t)−1 is therefore oper-ator monotone decreasing and satisfies the functional equation(6). It is consequently of the form of Eq. 9 for some finite Borelmeasure µg . Since also c(x, y) = y−1g(xy−1) the assertion followsby setting µc = µg .

We have shown that the set of normalized Morozova–Chentsovfunctions is a Bauer simplex, and that the extreme points exactlyare the functions of the form of Eq. 11.

Theorem 2.5. We exhibit the measure µc in the canonical representa-tion (10) for a number of Morozova–Chentsov functions.

1. The Wigner–Yanase–Dyson metric with (normalized) Morozova–Chentsov function

c(x, y) = 1p(1 − p)

· (xp − yp)(x1−p − y1−p)(x − y)2

is represented by

dµc(λ) = 2 sin pπ

πp(1 − p)· λp + λ1−p

(1 + λ)3 dλ

for 0 < p < 1.The Wigner–Yanase metric is obtained by setting p = 1/2 and

it is represented by

dµc(λ) = 16λ1/2

π(1 + λ)3 dλ.

2. The Kubo metric with (normalized) Morozova–Chentsov function

c(x, y) = log x − log yx − y

is represented by

dµc(λ) = 2(1 + λ)2 dλ.

3. The increasing bridge with (normalized) Morozova–Chentsovfunctions

cγ (x, y) = x−γ y−γ

(x + y

2

)2γ−1

is represented by⎧⎪⎪⎪⎨⎪⎪⎪⎩

µc = δ(λ − 1) γ = 0

dµc(λ) = 2 sin γπ

(1 + λ)πλ−γ

(1 − λ

2

)2γ−1

dλ 0 < γ < 1

µc = δ(λ) γ = 1,

where δ is the Dirac measure with unit mass in zero.

Proof . We calculate the measures by the method outlined inRemark 2.2.

1. For the Wigner–Yanase–Dyson metric we therefore considerthe analytic continuation

−g(reiφ) = −c(reiφ , 1) = −1p(1 − p)

· (rpeipφ − 1)(r1−pei(1−p)φ − 1)(reiφ − 1)2

where r > 0 and 0 < φ < π . We calculate the imaginary partand note that r → −λ and φ → π for z → λ < 0. We make surethat the representing measure has no atom in zero and obtainthe desired expression by tedious but elementary calculations.

2. For the Kubo metric we consider

−g(x) = −c(x, 1) = − log xx − 1

and calculate the imaginary part

−g(reiφ) = 2r log r sin φ + φ − φr cos φ

r2 − 2r cos φ + 1

of the analytic continuation. It converges towards π/(1−λ) forz → λ < 0 and is bounded for z → 0. The representing mea-sure has therefore no atom in zero, and dµ(λ) = dλ/(1 + λ)which may be verified by direct calculation.

3. For the increasing bridge we consider

−gγ (x) = −cγ (x, 1) = −x−γ

(x + 1

2

)2γ−1

and calculate the imaginary part

−gγ (reiφ) = −r−γ r2γ−11 exp i(−γφ + (2γ − 1)θ)

of the analytic continuation, where

r1 = 12

(r2+2r cos φ+1)1/2 and θ = arctanr sin φ

1 + r cos φ.

We first note that θ = π/2 and r1 = (r sin φ)/2 for λ = −1, andthat θ → 0 and r1 → (1 + λ)/2 for −1 < λ ≤ 0. The statementnow follows by examination of the different cases.

In the reference (9) we proved the following exponentialrepresentation of the Morozova–Chentsov functions.

Theorem 2.6. A Morozova–Chentsov function c admits a canonicalrepresentation

c(x, y) = C0

x + yexp

∫ 1

0

1 − λ2

λ2 + 1· x2 + y2

(x + λy)(λx + y)h(λ) dλ [12]

where h : [0, 1] → [0, 1] is a measurable function and C0 is a pos-itive constant. Both C0 and the equivalence class containing h areuniquely determined by c. Any function c on the given form is aMorozova–Chentsov function.

Theorem 2.7. We exhibit the constant C0 and the representing functionh in the canonical representation (12) for a number of Morozova–Chentsov functions.

1. The Wigner–Yanase–Dyson metric with Morozova–Chentsovfunction

c(x, y) = 1p(1 − p)

· (xp − yp)(x1−p − y1−p)(x − y)2

is represented by

C0 =√

2p(1 − p)

(1 − cos p

π

2

)1/2 (1 − cos(1 − p)

π

2

)1/2

and

h(λ) = 1π

arctan(λp + λ1−p) sin pπ

1 − λ − (λp − λ1−p) cos pπ0 < λ < 1,

for 0 < p < 1. Note that 0 ≤ h ≤ 1/2.

The Wigner–Yanase metric is obtained by setting p = 1/2 andis represented by

C0 = 4(√

2 − 1)

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and

h(λ) = 1π

arctan2λ1/2

1 − λ0 < λ < 1.

2. The Kubo metric with Morozova–Chentsov function

c(x, y) = log x − log yx − y

is represented by

C0 = π

2and h(λ) = 1

2− 1

πarctan

(− log λ

π

).

Note that 0 ≤ h ≤ 1/2.3. The increasing bridge with Morozova–Chentsov functions

cγ (x, y) = x−γ y−γ

(x + y

2

)2γ−1

is represented by

C0 = 21−γ and h(λ) = γ , 0 ≤ γ ≤ 1.

Setting γ = 0, we obtain that the Bures metric with Morozova–Chentsov function c(x, y) = 2/(x + y) is represented by C0 = 2and h(λ) = 0.

Proof . The analytic continuation of the operator monotone func-tion g(x) = log f (x) into the upper complex plane, where f (x) =c(x, 1)−1 is the operator monotone function representing (8) theMorozova–Chentsov function, has bounded imaginary part. Therepresenting measure of the Pick function g is therefore absolutelycontinuous with respect to Lebesgue measure. Since f satisfies thefunctional equation f (t) = t f (t−1) we only need to consider therestriction of the measure to the interval [−1, 0], and the functionh appears (9) as the image under the transformation λ → −λ ofthe Radon–Nikodym derivative. In the same reference it is shownthat the constant C0 = √

2e−β where β = log f (i).

1. For the Wigner–Yanase–Dyson metric the corresponding oper-ator monotone function

f (x) = 1c(x, 1)

= p(1 − p)(x − 1)2

(xp − 1)(x1−p − 1)

and we calculate by tedious but elementary calculations

limz→λ

log f (z) = − 12i

log H λ ∈ (−1, 0),

where

H = N((−λ)2p − 2(−λ)p cos pπ + 1)

· ((−λ)2(1−p) − 2(−λ)(1−p) cos(1 − p)π + 1)

and

N = (−λ)2 + 2(−λ)1+peipπ + (−λ)2pe2ipπ − 2(−λ)2−pe−ipπ

+ 4λ − 2(−λ)peipπ + (−λ)2(1−p)e−2ipπ + 2(−λ)1−pe−ipπ + 1

happens to be the square of the complex number

(1+λ)−((−λ)p −(−λ)1−p) cos pπ − i((−λ)p +(−λ)1−p) sin pπ

with positive real part and negative imaginary part. Since H hasmodulus one we can therefore write

H = e−2iθ λ ∈ (−1, 0),

where 0 < θ < π/2 and

tan θ = ((−λ)p + (−λ)1−p) sin pπ

1 + λ − ((−λ)p − (−λ)1−p) cos pπ

which implies the expression for h. The constant C0 is obtainedby a simple calculation.

2. For the Kubo metric the corresponding operator monotonefunction

f (x) = 1c(x, 1)

= x − 1log x

and we obtain by setting z = reiφ and z − 1 = r1eiφ1 theexpression

log f (z) = 12i

(log

log r − iφlog r + iφ

+ 2iφ1

)0 < φ < φ1 < π .

Since

loglog r − iφlog r + iφ

→ loglog(−λ) − iπlog(−λ) + iπ

for z → λ ∈ (−1, 0) and

log(−λ) − iπlog(−λ) + iπ

= e−2iθ where tan θ = π

log(−λ)

we obtain

limz→λ

log f (z) = π − θπ

2< θ < π .

Therefore

h(λ) = 1 − 1π

arctanπ

log λ

which entails the desired result. The constant C0 is obtained bya straightforward calculation.

3. The statement for the increasing bridge was proved in ref. 9.

3. Convexity StatementsProposition 3.1. Every Morozova–Chentsov function c is operatorconvex, and the mappings

(ρ, δ) → Tr A∗c(Lρ , Rδ)A

and

ρ → Kcρ(A, A)

defined on the state manifold are convex for arbitrary A ∈ Mn(C).

Proof . Let c be a Morozova–Chentsov function. Since inversion isoperator convex, it follows from the representation given in Eq. 10that c as a function of two variables is operator convex. The twoassertions now follow from ref. 5 theorem 1.1.

Lemma 3.2. Let λ ≥ 0 be a constant. The functions of two variables

f (t, s) = t2

t + λsand g(t, s) = ts

t + λs

are operator convex respectively operator concave on (0, ∞)×(0, ∞).

Proof . The first statement is an application of the convexity, dueto Lieb and Ruskai, of the mapping (A, B) → AB−1A. Indeed,setting

C1 = A1 ⊗ I2 + λI1 ⊗ B1 and C2 = A2 ⊗ I2 + λI1 ⊗ B2

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we obtain

f (tA1 + (1 − t)A2, tB1 + (1 − t)B2)

= ((tA1 + (1 − t)A2) ⊗ I2)(tC1 + (1 − t)C2)−1

((tA1 + (1 − t)A2) ⊗ I2)

≤ t(A1 ⊗ I2)C−11 (A1 ⊗ I2) + (1 − t)(A2 ⊗ I2)C−1

2 (A2 ⊗ I2)= t f (A1, B1) + (1 − t)f (A2, B2) t ∈ [0, 1].

The second statement is a consequence of the concavity of theharmonic mean

H(A, B) = 2(A−1 + B−1)−1.

Indeed, we may assume λ > 0 and obtain

g(tA1 + (1 − t)A2, tB1 + (1 − t)B2)

= 12

H(t(λ−1A1 ⊗ I2) + (1 − t)(λ−1A2 ⊗ I2),

t(I1 ⊗ B1) + (1 − t)(I1 ⊗ B2))

≥ t12

H(λ−1A1 ⊗ I2, I1 ⊗ B1)+(1−t)12

H(λ−1A2 ⊗ I2, I1 ⊗ B2)

= t g(A1, B1) + (1 − t)g(A2, B2)

for t ∈ (0, 1].

Proposition 3.3. Let c be a Morozova–Chentsov function. The func-tion of two variables

c(x, y) = (x − y)2c(x, y) x, y > 0

is operator convex.

Proof . A Morozova–Chentsov function c allows the representa-tion in Eq. 10 where µ is some finite Borel measure with supportin [0, 1]. Since

(x − y)2

x + λy= x2 + y2 − 2xy

x + λy

by Lemma 3.2 is a sum of operator convex functions the assertionfollows.

Proposition 3.4. Let c be a regular Morozova–Chentson function. Wemay write c(x, y) = (x − y)2c(x, y) on the form

c(x, y) = x + ym(c)

− dc(x, y) x, y > 0, [13]

where the positive symmetric function

dc(x, y) =∫ 1

0xy · cλ(x, y)

(1 + λ)2

λdµc(λ) [14]

is operator concave in the first quadrant, and the finite Borel measureµc is the representing measure in Eq. 10 of the Morozova–Chentsovfunction c. In addition, we obtain the expression

Icρ(A) = m(c)

2Tr Ac(Lρ , Rρ)A

= Tr ρA2 − m(c)2

Tr Adc(Lρ , Rρ)A[15]

for the metric adjusted skew information.

Proof . We first notice that∫ 1

0

(1 + λ)2

2λdµc(λ) = lim

t→0c(t, 1) = 1

m(c)[16]

and obtain

dc(x, y) = x + ym(c)

− c(x, y)

= x + ym(c)

− (x − y)2c(x, y)

= (x + y)∫ 1

0

(1 + λ)2

2λdµc(λ) − (x − y)2

∫ 1

0cλ(x, y) dµc(λ)

=∫ 1

0

((x + y)

(1 + λ)2

2λ− (x − y)2cλ(x, y)

)dµc(λ).

The asserted expression of dc then follows by a simple calculationand the definition of cλ(x, y) as given in Eq. 11. The function dc isoperator concave in the first quadrant by Proposition 3.3.

Definition 3.5. We call the function dc defined in Eq. 14 the repre-senting function for the metric adjusted skew information Ic

ρ(A)with (regular) Morozova–Chentsov function c.

We introduce for 0 < λ ≤ 1 the λ-skew information Iλ(ρ, A) bysetting

Iλ(ρ, A) = Icλρ (A).

The metric is regular with metric constant m(cλ) = 2λ(1 + λ)−2

and the representing measure µcλ is the Dirac measure in λ. Therepresenting function for the metric adjusted skew information isthus given by

dcλ(x, y) = xy · cλ(x, y)(1 + λ)2

λ= m(cλ)

2xy · cλ(x, y).

If we set

fλ(x, y) = xy · cλ(x, y) = 1 + λ

2

(xy

x + λy+ xy

λx + y

)x, y > 0,

[17]we therefore obtain the expression

Iλ(ρ, A) = Tr ρA2 − Tr Afλ(Lρ , Rρ)A [18]

for the λ-skew information.

Corollary 3.6. Let c be a regular Morozova–Chentsov function. Themetric adjusted skew information may be written on the form

Icρ(A) = m(c)

2

∫ 1

0Iλ(ρ, A)

(1 + λ)2

λdµc(λ),

where µc is the representing measure and m(c) is the metric constant.

Proof . By applying the expressions in Eqs. 15 and 14 together withthe observation in Eq. 16 we obtain

Icρ(A) = Tr ρA2 − m(c)

2

∫ 1

0Tr Afλ(Lρ , Rρ)A

(1 + λ)2

λdµc(λ)

= m(c)2

∫ 1

0(Tr ρA2 − Tr Afλ(Lρ , Rρ)A)

(1 + λ)2

λdµc(λ)

and the assertion follows.

3.1. Measures of Quantum Information. The next result is a directgeneralization of the Wigner–Yanase–Dyson–Lieb convexitytheorem.

Theorem 3.7. Let c be a regular Morozova–Chentsov function. Themetric adjusted skew information is a convex function, ρ → Ic

ρ(A),on the manifold of states for any self-adjoint A ∈ Mn(C).

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Proof . The function c(x, y) = (x − y)2c(x, y) is by Proposition3.3 operator convex. Applying the representation of the metricadjusted skew information given in Eq. 4, the assertion now followsfrom ref. 5, theorem 1.1.

The above proof is particularly transparent for the Wigner–Yanase–Dyson metric, since the function

cWYD(x, y) = 1p(1 − p)

(x p − y p)(x1−p − y1−p)

= 1p(1 − p)

(2 − x py1−p − x1−py p)

is operator convex by the simple argument given in ref. 5corollary 2.2.

Wigner and Yanase (3) discussed a number of other conditionsthat a good measure of the quantum information contained ina state with respect to a conserved observable should satisfy, butnoted that convexity was the most obvious but also the most restric-tive and difficult condition. In addition to the convexity require-ment an information measure should be additive with respect tothe aggregation of isolated systems. Since the state of the aggre-gated system is represented by ρ = ρ1 ⊗ ρ2 where ρ1 and ρ2 arethe states of the systems to be united, and the conserved quantityA = A1 ⊗ 1 + 1 ⊗ A2 is additive in its components, we obtain

[ρ, A] = [ρ1, A1] ⊗ ρ2 + ρ1 ⊗ [ρ2, A2].Inserting ρ and A, as above, in the definition of the metric adjustedskew information in Eq. 3, we obtain

Icρ(A) = m(c)

2Tr(i[ρ1, A1] ⊗ ρ2 + ρ1 ⊗ i[ρ2, A2])

c(Lρ1 , Rρ1 ) ⊗ c(Lρ2 , Rρ2 )(i[ρ1, A1] ⊗ ρ2 + ρ1 ⊗ i[ρ2, A2]).

The cross terms vanish because of the cyclicity of the trace, andsince ρ1 and ρ2 have unit trace we obtain

Icρ(A) = Ic

ρ1(A1) + Ic

ρ2(A2)

as desired.The metric adjusted skew information for an isolated system

should also be independent of time. But a conserved quantity A inan isolated system commutes with the Hamiltonian H , and sincethe time evolution of ρ is given by ρt = eitHρe−itH we readily obtain

Icρt

(A) = Icρ(A) t ≥ 0

by using the unitary invariance of the metric adjusted skewinformation.

The variance Varρ(A) of a conserved observable A with respectto a state ρ is defined by setting

Varρ(A) = Tr ρA2 − (Tr ρA)2.

It is a concave function in ρ.

Theorem 3.8. Let c be a regular Morozova–Chentsov function. Themetric adjusted skew information Ic

ρ(A) may for each conserved (self-adjoint) observable A be extended from the state manifold to the statespace. Furthermore,

Icρ(A) = Varρ(A)

if ρ is a pure state, and

0 ≤ Icρ(A) ≤ Varρ(A)

for any density matrix ρ.

Proof . We note that the representing function d in Eq. 14 may beextended to a continuous operator concave function defined in

the closed first quadrant with d(t, 0) = d(0, t) = 0 for every t ≥ 0,and that d(1, 1) = 2/m(c). Since a pure state is a one-dimensionalprojection P, it follows from the representation in Eq. 4 and theformula 13 that

IcP(A) = m(c)

2Tr

(APA + AAP

m(c)− d(1, 1)APAP

)

= Tr PA2 − Tr(PAP)2

= Tr PA2 − (Tr PA)2

= VarP(A).

An arbitrary state ρ is by the spectral theorem a convex combina-tion ρ = ∑

i λiPi of pure states. Hence

Icρ(A) ≤

∑i

λiIcPi

(A) =∑

i

λi VarPi (A) ≤ Varρ(A),

where we used the convexity of the metric adjusted skew informa-tion and the concavity of the variance.

3.2. The Metric Adjusted Correlation. We have developed thenotion of metric adjusted skew information, which is a generaliza-tion of the Wigner–Yanase–Dyson skew information. It is definedfor all regular metrics (symmetric and monotone), where the termregular means that the associated Morozova–Chentsov functionshave continuous extensions to the closed first quadrant with finitevalues everywhere except in the point (0, 0).

Definition 3.9. Let c be a regular Morozova–Chentsov function,and let d be the representing function 14. The metric adjustedcorrelation is defined by

Corrcρ(A, B) = Tr ρA∗B − m(c)

2Tr A∗d(Lρ , Rρ)B

for arbitrary matrices A and B.

Since d is symmetric, the metric adjusted correlation is asymmetric sesqui-linear form which by Eq. 15 satisfies

Corrcρ(A, A) = Ic

ρ(A) for self-adjoint A.

The metric adjusted correlation is not a real form on self-adjointmatrices, and it is not positive on arbitrary matrices. Therefore,Cauchy–Schwartz inequality only gives a bound

∣∣ Corrcρ(A, B)

∣∣ ≤ Icρ(A)1/2Ic

ρ(A)1/2 ≤ Varρ(A)1/2 · Varρ(B)1/2

[19]for the real part of the metric adjusted correlation. However, since

Corrcρ(A, B) − Corrc

ρ(B, A) = Tr ρ[A, B] A∗ = A, B∗ = B,

we obtain12| Tr ρ[A, B]| = ∣∣ Corrc

ρ(A, B)∣∣

for self-adjoint A and B. The estimate in Eq. 19 can therefore notbe used to improve Heisenberg’s uncertainty relations.¶

3.3. The Variant Bridge. The notion of a regular metric seems tobe very important. We note that the Wigner–Yanase–Dyson met-rics and the Bures metric are regular, whereas the Kubo metricand the maximal symmetric monotone metric are not.

¶In the first version of this article, which appeared on July 22, 2006, the estimation in Eq.19 was erroneously extended to the metric adjusted skew information itself and not onlyto the real part; cf. also Luo (16) and Kosaki (17). The author is indebted to Gibilisco andIsola for pointing out the mistake.

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The continuously increasing bridge with Morozova–Chentsovfunctions

cγ (x, y) = x−γ y−γ

(x + y

2

)2γ−1

0 ≤ γ ≤ 1

connects the Bures metric c0(x, y) = 2/(x + y) with the maximalsymmetric monotone metric c1(x, y) = 2xy/(x+y). Since the Buresmetric is regular and the maximal symmetric monotone metric isnot, any bridge connecting them must fail to be regular at somepoint. However, the above bridge fails to be regular at any pointγ �= 0. A look at the formula 12 shows that a symmetric monot-one metric is regular, if and only if λ−1 is integrable with respectto h(λ)dλ. We may obtain this by choosing for example

hp(λ) ={

0, λ < 1 − pp, λ ≥ 1 − p 0 ≤ p ≤ 1

instead of the constant weight functions. Since∫(λ2 − 1)(1 + t2)

(1 + λ2)(λ + t)(1 + λt)dλ = log

1 + λ2

(λ + t)(1 + λt)

we are by tedious calculations able to obtain the expression

fp(t) = 1 + t2

(4(1 − p + t)(1 + (1 − p)t)

(2 − p)2(1 + t)2

)p

t > 0

for the normalized operator monotone functions represented bythe hp(λ) weight functions (ref. 9, theorem 1). The correspondingMorozova–Chentsov functions are then given by

cp(x, y) = (2 − p)2p

(x + (1 − p)y)p((1 − p)x + y)p

(x + y

2

)2p−1

[20]

for 0 ≤ p ≤ 1. The weight functions hp(λ) provides a continu-ously increasing bridge from the zero function to the unit func-tion. But we cannot be sure that the corresponding Morozova–Chentsov functions are everywhere increasing, since we haveadjusted the multiplicative constants such that all the functionsfp(t) are normalized to fp(1) = 1. However, since by calculation

∂

∂pfp(t) = −2p2(1 − t)2

(2 − p)3(1 + t)

(4(1 − p + t)(1 + (1 − p)t)

(2 − p)2(1 + t)2

)p−1

< 0,

we realize that the representing operator monotone functions aredecreasing in p for every t > 0. In conclusion, we have shownthat the symmetric monotone metrics given by Eq. 20 provides acontinuously increasing bridge between the smallest and largest(symmetric and monotone) metrics, and that all the metrics in thebridge are regular except for p = 1.

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5. Hansen F (2006) Extensions of Lieb’s concavity theorem. J Stat Phys 124:87–101.6. Hansen F (2007) The Wigner–Yanase entropy is not subadditive. J Stat Phys 126:643–

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(Translated from Russian). J Soviet Math 56:2648–2669.12. Petz D, Sudár C (1996) Geometries of quantum states. J Math Phys 37:2662–2673.13. Hasegawa H, Petz D (1996) On the Riemannian metric of α-entropies of density

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Suggested ReadingsHasegawa H, Petz D (1997) Non-commutative extension of the information geometryII. Quantum Communication and Measurement, ed Hirota O (Plenum, New York), pp.109–118.Heisenberg W (1927) Über den anschaulichen Inhalt der quantummechanischen Kinematikund Mechanik. Z Phys 43:172–198.Lesniewski A, Ruskai MB (1999) Monotone Riemannian metrics and relative entropy onnon-commutative probability spaces. J Math Phys 40:5702–5724.Gibilisco P, Isola T (2007) Uncertainty principle and quantum Fisher information. Ann InstStat Mech 59:147–159.Petz D (1994) Geometry of canonical correlation on the state space of a quantum system.J Math Phys 35:780–795.Yanagi K, Furuichi S, Kuriyama K (2005) A generalized skew information and uncertaintyrelation. IEEE Trans Inf Theory 51:4401–4404.

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