New Research In
Physical Sciences
Social Sciences
Featured Portals
Articles by Topic
Biological Sciences
Featured Portals
Articles by Topic
 Agricultural Sciences
 Anthropology
 Applied Biological Sciences
 Biochemistry
 Biophysics and Computational Biology
 Cell Biology
 Developmental Biology
 Ecology
 Environmental Sciences
 Evolution
 Genetics
 Immunology and Inflammation
 Medical Sciences
 Microbiology
 Neuroscience
 Pharmacology
 Physiology
 Plant Biology
 Population Biology
 Psychological and Cognitive Sciences
 Sustainability Science
 Systems Biology
Metric adjusted skew information

Edited by Richard V. Kadison, University of Pennsylvania, Philadelphia, PA, and approved April 17, 2008 (received for review January 17, 2007)
Abstract
We extend the concept of Wigner–Yanase–Dyson skew information to something we call “metric adjusted skew information” (of a state with respect to a conserved observable). This “skew information” is intended to be a nonnegative quantity bounded by the variance (of an observable in a state) that vanishes for observables commuting with the state. We show that the skew information is a convex function on the manifold of states. It also satisfies other requirements, proposed by Wigner and Yanase, for an effective measureofinformation content of a state relative to a conserved observable. We establish a connection between the geometrical formulation of quantum statistics as proposed by Chentsov and Morozova and measures of quantum information as introduced by Wigner and Yanase and extended in this article. We show that the set of normalized Morozova–Chentsov functions describing the possible quantum statistics is a Bauer simplex and determine its extreme points. We determine a particularly simple skew information, the “λskew information,” parametrized by a λ ∈ (0, 1], and show that the convex cone this family generates coincides with the set of all metric adjusted skew informations.
In the mathematical model for a quantum mechanical system, the physical observables are represented by selfadjoint operators on a Hilbert space. The “states” (that is, the “expectation functionals” associated with the states) of the physical system arexs often “modeled” by the unit vectors in the underlying Hilbert space. So, if A represents an observable and x ∈ H corresponds to a 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 our Hilbert space is finite dimensional and that the observables are selfadjoint operators, or the matrices that represent them, on that finite dimensional space. In this case, the states can be realized with the aid of the trace (functional) on matrices and an associated “density matrix.” We denote by Tr(B) the usual trace of 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 matrices ρ and A. (Henceforth, we write “Tr ρA” omitting the parentheses when they are clearly understood.)
In ref. 1, Wigner noticed that in the presence of a conservation law the obtainable accuracy of the measurement of a physical observable is limited if the operator representing the observable does not commute with (the operator representing) the conserved quantity (observable). Wigner proved it in the simple case where the physical observable is the xcomponent of the spin of a spin onehalf particle and the zcomponent of the angular momentum is conserved. Araki and Yanase (2) demonstrated that this is a general phenomenon and pointed out, following Wigner's example, that under fairly general conditions an approximate measurement may be carried out.
Another difference is that observables that commute with a conserved additive quantity, like the energy, components of the linear or angular momenta, or the electrical charge, can be measured easily and accurately by microscopic apparatuses (the analysis is restricted to one conserved quantity), while other observables can be only approximately measured by a macroscopic apparatus large enough to superpose sufficiently many states with different quantum numbers of the conserved quantity.
Wigner and Yanase (3) proposed finding a measure of our knowledge of a difficulttomeasure observable with respect to a conserved quantity. The quantum mechanical entropy is a measure of our ignorance of the state of a system, and minus the entropy can therefore be considered as an expression of our knowledge of the system. This measure has many attractive properties but does not take into account the conserved quantity. In particular, Wigner and Yanase wanted a measure that vanishes when the observable commutes with the conserved quantity. It should therefore not measure the effect of mixing in the classical sense as long as the pure states taking part in the mixing commute with the conserved quantity. Only transition probabilities of pure states “lying askew” (to borrow from the introduction of ref. 3) to the eigenvectors of the conserved quantity should give contributions to the proposed measure.
Wigner and Yanase discussed a number of requirements that such a measure should satisfy in order to be meaningful and suggested, tentatively, the skew information defined by where [C,D] is the usual “bracket notation” for operators or matrices: [C,D] = CD − DC, as a measure of the information contained in a state ρ with respect to a conserved observable A. It manifestly vanishes when ρ commutes with A, and it is homogeneous in ρ.
The requirements Wigner and Yanase discussed, all reflected properties considered attractive or even essential. Since information is lost when separated systems are united such a measure should be decreasing under the mixing of states, that is, be convex in ρ. The authors proved this for the skew information but noted that other measures may enjoy the same properties; in particular, the expression proposed by Dyson. Convexity of this expression in ρ became the celebrated Wigner–Yanase–Dyson conjecture which was later proved by Lieb (4). (See also ref. 5 for a truly elementary proof.)
The measure should also be additive with respect to the aggregation of isolated subsystems and, for an isolated system, independent of time. These requirements are discussed in more detail in Convexity Statements. They are easily seen to be satisfied by the skew information.
In the process that is the opposite of mixing, the information content should decrease. This requirement comes from thermodynamics where it is satisfied for both classical and quantum mechanical systems. It reflects the loss of information about statistical correlations between two subsystems when they are only considered separately. Wigner and Yanase conjectured that the skew information also possesses this property. They proved it when the state of the aggregated system is pure.^{‡}
The aim of this article is to connect the subject of measures of quantum information as laid out by Wigner and Yanase with the geometrical formulation of quantum statistics by Chentsov, Morozova, and Petz.
The Fisher information measures the statistical distinguishability of probability distributions. Let 𝒫_{n}={p=(p_{1},…,p_{n})p_{i}>0} be the (open) probability simplex with tangent space T𝒫_{n}. The Fisher–Rao metric is then given by Note that u=(u_{1},…,u_{n})∈T𝒫_{n} if and only if u_{1} + … + u_{n} = 0, but that the metric is well defined also on R^{n}. Chentsov proved that the Fisher—Rao metric is the unique Riemannian metric contracting under Markov morphisms (7).
Since Markov morphisms represent coarse graining or randomization, it means that the Fisher information is the only Riemannian metric possessing the attractive property that distinguishability of probability distributions becomes more difficult when they are observed through a noisy channel.
Chentsov and Morozova extended the analysis to quantum mechanics by replacing Riemannian metrics defined on the tangent space of the simplex of probability distributions with positive definite sesquilinear (originally bilinear) forms K_{ρ} defined on the tangent space of a quantum system, where ρ is a positive definite state. Customarily, K_{ρ} is extended to all operators (matrices) supported by the underlying Hilbert space; cf. refs. 8 and 9 for details. Noisy channels are in this setting represented by stochastic (completely positive and trace preserving) mappings T, and the contraction property by the monotonicity requirement is imposed for every stochastic mapping T : M_{n}(C) → M_{m}(C). Unlike the classical situation, it turned out that this requirement no longer uniquely determines the metric. By the combined efforts of Chentsov, Morozova, and Petz it is established that the monotone metrics are given on the form where c is a so called Morozova–Chentsov function and c(L_{ρ}, R_{ρ}) is the function taken in the pair of commuting left and right multiplication operators (denoted L_{ρ} and R_{ρ}, respectively) by ρ. The Morozova–Chentsov function is of the form where f is a positive operator monotone function defined in the positive halfaxis satisfying the functional equation The function is clearly operator monotone and satisfies Eq. 2. The associated Morozova—Chentsov function therefore defines a monotone metric which we shall call the Wigner–Yanase metric. The starting point of our investigation is the observation by Gibilisco and Isola (10) that There is thus a relationship between the Wigner–Yanase measure of quantum information and the geometrical theory of quantum statistics. It is the aim of the present article to explore this relationship in detail. The main result is that all well behaved measures of quantum information—including the Wigner–Yanase–Dyson skew informations—are given in this way for a suitable subclass of monotone metrics.
1. Regular Metrics
Definition 1.1(regular metric). We say that a symmetric monotone metric (11, 12) on the state space of a quantum system is regular, if the corresponding Morozova–Chentsov function c admits a strictly positive limit We call m(c) the metric constant.
We also say, more informally, that a Morozova–Chentsov function c is regular if m(c) < 0. The function f(t) = c(t,1)^{−1} is positive and operator monotone on the positive halfline and may be extended to the closed positive halfline. Thus the metric constant m(c) = f(0).
Definition 1.2 (metric adjusted skew information) Let c be the Morozova–Chentsov function of a regular metric. We introduce the metric adjusted skew information I_{ρ}^{c}(A) by setting for every ρ∈ℳ_{n} (the manifold of states) and every selfadjoint A ∈ M_{n}(C).
Note that the metric adjusted skew information is proportional to the square of the metric length, as it is calculated by the symmetric monotone metric K_{ρ}^{c} with Morozova–Chentsov function c, of the commutator i[ρ, A], and that this commutator belongs to the tangent space of the state manifold ℳ_{n}. Metric adjusted skew information is thus a nonnegative quantity. If we consider the WYDmetric with Morozova–Chentsov function then the metric constant m(c^{WYD}) = p(1 − p) and the metric adjusted skew information becomes the Dyson generalization of the Wigner–Yanase skew information.^{§} The choice of the factor m(c) therefore works also for p ≠ 1/2. It is in fact a quite general construction, and the metric constant is related to the topological properties of the metric adjusted skew information close to the border of the state manifold. But it is difficult to ascertain these properties directly, so we postpone further investigation until having established that I_{ρ}^{c}(A) is a convex function in ρ. Since the commutator i[ρ, A] = i(L_{ρ}  R_{ρ})A we may rewrite the metric adjusted skew information as where
Before we can address these questions in more detail, we have to study various characterizations of (symmetric) monotone metrics.
2. Characterizations of Monotone Metrics
Theorem 2.1. A positive operator monotone decreasing function g defined in the positive halfaxis and satisfying the functional equation has a canonical representation where μ is a finite Borel measure with support in [0,1].
Proof. The function g is necessarily of the form where β ≥ 0 is a constant and μ is a positive Borel measure such that the integrals ∫(1+λ^{2})^{−1}dμ(λ) and ∫λ(1+λ^{2})^{−1}dμ(λ) are finite (cf. ref. 14 page 9). We denote by μ̃ the measure obtained from μ by removing a possible atom in zero. Then, by making the transformation λ → λ^{−1}, we may write where v is the Borel measure given by dv(λ)=λ−^{1}dμ̃(λ^{−1}). Since g satisfies the functional equation (6) we obtain By letting t → 0 and since v and μ̃ have no atoms in zero, we obtain β = μ(0) and consequently By analytic continuation we realize that both measures v and μ̃ appear as the representing measure of an analytic function with negative imaginary part in the complex upper half plane. They are therefore, by the representation theorem for this class of functions, necessarily identical. We finally obtain The statement follows since every function of this form obviously is operator monotone decreasing and satisfy the functional equation (6). We also realize that the representing measure μ is uniquely defined.
Remark 2.2. Inspection of the proof of Theorem 2.1 shows that the Pick function g(x) = c(x,1) has the canonical representation The representing measure therefore appears as 1/π times the limit measure of the imaginary part of the analytic continuation g(z) as z approaches the closed negative halfaxis from above (cf., for example, ref. 15). The measure μ in Eq. 7 therefore appears as the 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 measure μ_{g} on the closed interval [0,1] by setting and obtain:
Corollary 2.3. A positive operator monotone decreasing function g defined in the positive halfaxis and satisfying the functional equation (6) has a canonical representation where μ_{g} is a finite Borel measure with support in [0,1]. The function g is normalized in the sense that g(1) = 1, if and only if μ_{g} is a probability measure.
Corollary 2.4. A Morozova–Chentsov function c allows a canonical representation of the form where μ_{c} is a finite Borel measure on [0,1] and The Morozova–Chentsov function c is normalized in the sense that c(1,1) = 1 (corresponding to a Fisher adjusted metric), if and only if μ_{c} is a probability measure.
Proof. A Morozova–Chentsov function is of the form c(x,y) = y^{−1}f(xy^{−1})^{−1}, where f is a positive operator monotone function defined in the positive halfaxis and satisfying the functional equation f(t) = tf(t^{−1}). The function g(t) = f(t)^{−1} is therefore operator monotone decreasing and satisfies the functional equation (6). It is consequently of the form of Eq. 9 for some finite Borel measure μ_{g}. Since also c(x,y) = y^{−1}g(xy^{−1}) the assertion follows by setting μ_{c} = μ_{g}.
We have shown that the set of normalized Morozova—Chentsov functions is a Bauer simplex, and that the extreme points exactly are the functions of the form of Eq. 11.
Theorem 2.5. We exhibit the measure μ_{c} in the canonical representation (10) for a number of Morozova–Chentsov functions.

The Wigner–Yanase–Dyson metric with (normalized) Morozova–Chentsov function is represented by for 0 > p > 1.
The Wigner–Yanase metric is obtained by setting p = 1/2 and it is represented by

The Kubo metric with (normalized) Morozova–Chentsov function is represented by

The increasing bridge with (normalized) Morozova–Chentsov functions is represented by where δ is the Dirac measure with unit mass in zero.
Proof. We calculate the measures by the method outlined in Remark 2.2.

For the Wigner–Yanase–Dyson metric we therefore consider the analytic continuation where r < 0 and 0 > ϕ > π. We calculate the imaginary part and note that r → −λ and ϕ → π for z → λ > 0. We make sure that the representing measure has no atom in zero and obtain the desired expression by tedious but elementary calculations.

For the Kubo metric we consider and calculate the imaginary part of the analytic continuation. It converges towards π/(1−λ) for z → λ < 0 and is bounded for z → 0. The representing measure has therefore no atom in zero, and dμ(λ) = dλ/(1+λ) which may be verified by direct calculation.

For the increasing bridge we consider and calculate the imaginary part of the analytic continuation, where We first note that θ = π/2 and r_{1} = (r sinϕ)/2 for λ = −1, and that θ → 0 and r_{1} → (1 + λ)/2 for −1 < λ ≤ 0. The statement now follows by examination of the different cases.
In the reference (9) we proved the following exponential representation of the Morozova—Chentsov functions.
Theorem 2.6. A Morozova–Chentsov function c admits a canonical representation where h : [0,1] → [0,1] is a measurable function and C_{0} is a positive constant. Both C_{0} and the equivalence class containing h are uniquely determined by c. Any function c on the given form is a Morozova–Chentsov function.
Theorem 2.7. We exhibit the constant C_{0} and the representing function h in the canonical representation (12) for a number of Morozova–Chentsov functions.

The Wigner–Yanase–Dyson metric with Morozova–Chentsov function is represented by and for 0 > p > 1. Note that 0 ≤ h ≤ 1/2.
The Wigner–Yanase metric is obtained by setting p = 1/2 and is represented by and

The Kubo metric with Morozova–Chentsov function is represented by Note that 0 ≤ h ≤ 1/2.

The increasing bridge with Morozova—Chentsov functions is represented by
Setting γ = 0, we obtain that the Bures metric with Morozova–Chentsov function c(x, y) = 2/(x + y) is represented by C_{0} = 2 and h(λ) = 0.
Proof. The analytic continuation of the operator monotone function g(x) = log f(x) into the upper complex plane, where f(x) = c(x,1)^{−1} is the operator monotone function representing (8) the Morozova–Chentsov function, has bounded imaginary part. The representing measure of the Pick function g is therefore absolutely continuous with respect to Lebesgue measure. Since f satisfies the functional equation f(t) = tf(t^{−1}) we only need to consider the restriction of the measure to the interval [−1, 0], and the function h appears (9) as the image under the transformation λ → −λ of the Radon—Nikodym derivative. In the same reference it is shown that the constant

For the Wigner–Yanase–Dyson metric the corresponding operator monotone function and we calculate by tedious but elementary calculations where and happens to be the square of the complex number with positive real part and negative imaginary part. Since H has modulus one we can therefore write where 0 < θ < π/2 and which implies the expression for h. The constant C_{0} is obtained by a simple calculation.

For the Kubo metric the corresponding operator monotone function and we obtain by setting z = re^{iϕ} and z − 1 = r_{1}e^{iϕ1} the expression Since for z → λ ∈ (−1,0) and we obtain Therefore which entails the desired result. The constant C_{0} is obtained by a straightforward calculation.

The statement for the increasing bridge was proved in ref. 9.
3. Convexity Statements
Proposition 3.1. Every Morozova–Chentsov function c is operator convex, and the mappings and defined on the state manifold are convex for arbitrary A ∈ M_{n}(C).
Proof. Let c be a Morozova–Chentsov function. Since inversion is operator convex, it follows from the representation given in Eq. 10 that c as a function of two variables is operator convex. The two assertions now follow from ref. 5 theorem 1.1.
Lemma 3.2. Let λ ≥ 0 be a constant. The functions of two variables are operator convex respectively operator concave on (0,∞) × (0,∞).
Proof. The first statement is an application of the convexity, due to Lieb and Ruskai, of the mapping (A, B) → AB^{−1}A. Indeed, setting we obtain The second statement is a consequence of the concavity of the harmonic mean Indeed, we may assume λ > 0 and obtain for t ∈ (0,1].
Proposition 3.3. Let c be a Morozova–Chentsov function. The function of two variables is operator convex.
Proof. A Morozova–Chentsov function c allows the representation in Eq. 10 where μ is some finite Borel measure with support in [0,1]. Since by Lemma 3.2 is a sum of operator convex functions the assertion follows.
Proposition 3.4. Let c be a regular Morozova–Chentson function. We may write ĉ(x,y)=(x−y)^{2}c(x,y) on the form where the positive symmetric function is operator concave in the first quadrant, and the finite Borel measure μ_{c} is the representing measure in Eq. 10 of the Morozova–Chentsov function c. In addition, we obtain the expression for the metric adjusted skew information.
Proof. We first notice that and obtain The asserted expression of d_{c} then follows by a simple calculation and the definition of c_{λ}(x, y) as given in Eq. 11. The function d_{c} is operator concave in the first quadrant by Proposition 3.3.
Definition 3.5. We call the function d_{c} defined in Eq. 14 the representing function for the metric adjusted skew information I_{ρ}^{c}(A) with (regular) Morozova–Chentsov function c.
We introduce for 0 < λ ≤ 1 the λskew information I_{λ}(ρ, A) by setting The metric is regular with metric constant m(c_{λ}) = 2λ(1 + λ)^{−2} and the representing measure μ_{cλ} is the Dirac measure in λ. The representing function for the metric adjusted skew information is thus given by If we set we therefore obtain the expression for the λskew information.
Corollary 3.6. Let c be a regular Morozova–Chentsov function. The metric adjusted skew information may be written on the form where μ_{c} is the representing measure and m(c) is the metric constant.
Proof. By applying the expressions in Eqs. 15 and 14 together with the observation in Eq. 16 we obtain and the assertion follows.
3.1. Measures of Quantum Information.
The next result is a direct generalization of the Wigner–Yanase–Dyson–Lieb convexity theorem.
Theorem 3.7. Let c be a regular Morozova–Chentsov function. The metric adjusted skew information is a convex function, ρ→I_{ρ}^{c}(A), on the manifold of states for any selfadjoint A ∈ M_{n}(C).
Proof. The function ĉ(x,y)=(x−y)^{2}c(x,y) is by Proposition 3.3 operator convex. Applying the representation of the metric adjusted skew information given in Eq. 4, the assertion now follows from ref. 5, theorem 1.1.
The above proof is particularly transparent for the Wigner–Yanase–Dyson metric, since the function is operator convex by the simple argument given in ref.5 corollary 2.2.
Wigner and Yanase (3) discussed a number of other conditions that a good measure of the quantum information contained in a state with respect to a conserved observable should satisfy, but noted that convexity was the most obvious but also the most restrictive and difficult condition. In addition to the convexity requirement an information measure should be additive with respect to the aggregation of isolated systems. Since the state of the aggregated system is represented by ρ = ρ_{1} ⊗ ρ_{2} where ρ_{1} and ρ_{2} are the states of the systems to be united, and the conserved quantity A = A_{1} ⊗ 1 + 1 ⊗ A_{2} is additive in its components, we obtain Inserting ρ and A, as above, in the definition of the metric adjusted skew information in Eq. 3, we obtain The cross terms vanish because of the cyclicity of the trace, and since ρ_{1} and ρ_{2} have unit trace we obtain as desired.
The metric adjusted skew information for an isolated system should also be independent of time. But a conserved quantity A in an isolated system commutes with the Hamiltonian H, and since the time evolution of ρ is given by ρ_{t} = e^{itH} ρe^{−itH} we readily obtain by using the unitary invariance of the metric adjusted skew information.
The variance Var_{ρ}(A) of a conserved observable A with respect to a state ρ is defined by setting It is a concave function in ρ.
Theorem 3.8. Let c be a regular Morozova–Chentsov function. The metric adjusted skew information I_{ρ}^{c}(A) may for each conserved (selfadjoint) variable A be extended from the state manifold to the state space. Furthermore, if ρ is a pure state, and for any density matrix ρ.
Proof. We note that the representing function d in Eq. 14 may be extended 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 onedimensional projection P, it follows from the representation in Eq. 4 and the formula 13 that An arbitrary state ρ is by the spectral theorem a convex combination ρ = ∑_{i} λ_{i}P_{i} of pure states. Hence where we used the convexity of the metric adjusted skew information and the concavity of the variance.
3.2. The Metric Adjusted Correlation.
We have developed the notion of metric adjusted skew information, which is a generalization of the Wigner–Yanase–Dyson skew information. It is defined for all regular metrics (symmetric and monotone), where the term regular means that the associated Morozova–Chentsov functions have continuous extensions to the closed first quadrant with finite values 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 adjusted correlation is defined by for arbitrary matrices A and B.
Since d is symmetric, the metric adjusted correlation is a symmetric sesquilinear form which by Eq. 15 satisfies The metric adjusted correlation is not a real form on selfadjoint matrices, and it is not positive on arbitrary matrices. Therefore, Cauchy–Schwartz inequality only gives a bound for the real part of the metric adjusted correlation. However, since we obtain for selfadjoint A and B. The estimate in Eq. 19 can therefore not be used to improve Heisenberg's uncertainty relations.^{¶}
3.3. The Variant Bridge.
The notion of a regular metric seems to be very important. We note that the Wigner–Yanase–Dyson metrics and the Bures metric are regular, whereas the Kubo metric and the maximal symmetric monotone metric are not.
The continuously increasing bridge with Morozova–Chentsov functions connects the Bures metric c_{0}(x, y) = 2/(x + y) with the maximal symmetric monotone metric c_{1}(x, y) = 2xy/(x + y). Since the Bures metric is regular and the maximal symmetric monotone metric is not, any bridge connecting them must fail to be regular at some point. However, the above bridge fails to be regular at any point γ ≠ 0. A look at the formula 12 shows that a symmetric monotone metric is regular, if and only if λ^{−1} is integrable with respect to h(λ) dλ. We may obtain this by choosing for example instead of the constant weight functions. Since we are by tedious calculations able to obtain the expression for the normalized operator monotone functions represented by the h_{p}(λ) weight functions (ref. 9, theorem 1). The corresponding Morozova–Chentsov functions are then given by for 0 ≤ p ≤ 1. The weight functions h_{p}(λ) provides a continuously increasing bridge from the zero function to the unit function. But we cannot be sure that the corresponding Morozova–Chentsov functions are everywhere increasing, since we have adjusted the multiplicative constants such that all the functions f_{p}(t) are normalized to f_{p}(1) = 1. However, since by calculation we realize that the representing operator monotone functions are decreasing in p for every t > 0. In conclusion, we have shown that the symmetric monotone metrics given by Eq. 20 provides a continuously increasing bridge between the smallest and largest (symmetric and monotone) metrics, and that all the metrics in the bridge are regular except for p = 1.
Footnotes
 ^{†}Email: frank.hansen{at}econ.ku.dk.

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

This article is a PNAS Direct Submission.

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

↵§ Hasegawa and Petz proved in (13) that the function c^{WYD} is a Morozova_Chentsov function. They also proved that the Wigner_Yanase_Dyson skew information is proportional to the (corresponding) quantum Fisher information of the commutator i[ρ,A].

↵¶ 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 only to the real part; cf. also Luo (16) and Kosaki (17). The author is indebted to Gibilisco and Isola for pointing out the mistake.
 © 2008 by The National Academy of Sciences of the USA
References
 ↵
 ↵
 Araki H,
 Yanase MM
 ↵
 Wigner EP,
 Yanase MM
 ↵
 ↵
 ↵
 ↵
 Censov NN
 ↵
 ↵
 Hansen F
 ↵
 ↵
 ↵
 ↵
 ↵
 Hansen F
 ↵
 Donoghue W
 ↵
 ↵
Citation Manager Formats
Sign up for Article Alerts
Jump to section
You May Also be Interested in
More Articles of This Classification
Physical Sciences
Related Content
 No related articles found.