A Unified Approach to Local Quantum Uncertainty and Interferometric Power by Metric Adjusted Skew Information

Local quantum uncertainty and interferometric power were introduced by Girolami et al. as geometric quantifiers of quantum correlations. The aim of the present paper is to discuss their properties in a unified manner by means of the metric adjusted skew information defined by Hansen.


INTRODUCTION
One of the key traits of many-body quantum systems is that the full knowledge of their global configurations does not imply full knowledge of their constituents.The impossibility to reconstruct the local wave functions |ψ 1 , |ψ 2 (pure states) of two interacting quantum particles from the wave function of the whole system, |ψ 12 |ψ 1 ⊗ |ψ 2 , is due to the existence of entanglement [4].Investigating open quantum systems, whose (mixed) states are described by density matrices ρ 12 = i p i |ψ i 12 ψ i |, revealed that the boundary between the classical and quantum worlds is more blurred than we thought.There exists a genuinely quantum kind of correlation, quantum discord, which manifests even in absence of entanglement, i.e. in separable density matrices ρ 12 = i p i ρ 1,i ⊗ ρ 2,i [5,6].The discovery triggered theoretical and experimental studies to understand the physical meaning of quantum discord, and the potential use of it as a resource for quantum technologies [7].Relying on the known interplay between geometrical and physical properties of mixed states [8,9], a stream of works employed information geometry techniques to construct quantifiers of quantum discord [10][11][12][13][14][15].In particular, two of the most popular ones are the Local Quantum Uncertainty (LQU) and the Interferometric Power (IP) [1,2].A merit of these two measures is that they admit an analytical form for N qubit states across the 1 vs N − 1 qubit partition.Also, they have a clear-cut physical interpretation.The lack of certainty about quantum measurement outcomes is due to the fact that density matrices are changed by quantum operations.The LQU evaluates the minimum uncertainty about the outcome of a local quantum measurement, when performed on a bipartite system.It is proven that two-particle density matrices display quantum discord if and only if they are not "classical-quantum" states.That is, they are not (mixture of) eigenvalues of local observables, ρ 12 i p i |i 1 i| ⊗ ρ 2,i , or ρ 12 i p i ρ 1,i ⊗ |i 2 i|, in which {|i } is an orthonormal basis.Indeed, this is the only case in which one can identify a local measurement that does not change a bipartite quantum state, whose spectral decomposition reads A 1 = i λ i |i 1 i|, or A 2 = i λ i |i 2 i|.The LQU was built as the minimum of the Wigner-Yanase skew information, a well-known information geometry measure [16], between a density matrix and a finite-dimensional observable (Hermitian operator).It quantifies how much a density matrix ρ 12 is different from being a zero-discord state.The IP was concocted by following a similar line of thinking.Quantum discord implies a non-classical sensitivity to local perturbations.This feature of quantum particles, while apparently a limitation, translates into an advantage in the context of quantum metrology [17].It was theoretically proven and experimentally demonstrated that quantum systems sharing quantum discord are more sensitive probes for interferometric phase estimation.The figure of merit of such measurement protocols is the quantum Fisher information of the state under scrutiny with respect to a local Hamiltonian (in Information Geometry the QFI is known as the SLD or Bures-Uhlmann metric).The latter generates a unitary evolution that imprints information about a physical parameter on the quantum probe.The IP is the minimum quantum Fisher information over all the possible local Hamiltonians, being zero if and only if the probe states are classically correlated.
Here, we polish and extend the mathematical formalization of information-geometric quantum correlation measures.We build a class of parent quantities of the LQU (and consequently of the IP) in terms of the the metric adjusted skew informations [3].In Sections 2,3, we review definition and main properties of operator means.In Sections 4-6, we discuss information-geometric quantities that capture complementarity between quantum states and observables.In particular, we focus on the quantum fcovariances and the quantum Fisher information.They quantify the inherent uncertainty about quantum measurement outcomes.After having recalled the definition of metric adjusted skew information (Section 7), we build a new quantum discord measure, the metric adjusted local quantum uncertainty ( f -LQU), in Section 8. Finally we are able to show that LQU and IP are just two particular members of this family allowing a unified treatment of their fundamental properties.

m is continuous.
6. m is positively homogeneous; that is m(tx, ty) = t • m(x, y) for t > 0.
We use the notation M num for the set of bivariate means described above.Definition 2. Let F num denote the class of functions f : R The following result is straightforward.
Proposition 1.There is a bijection f → m f betwen F nu and M nu given by m f (x, y) = y f (y −1 x) and in reverse f (t) = m(1, t) for positive numbers x, y and t.
In Table 1 we have some examples of means.

MEANS FOR POSITIVE OPERATORS IN THE SENSE OF KUBO-ANDO
The celebrated Kubo-Ando theory of operator means [18][19][20] may be viewed as the operator version of the results of Section .Definition 3. A bivariate mean m for pairs of positive operators is a function defined in and with values in positive definite operators on a Hilbert space, that satisfies, mutatis mutandis, conditions (1) to (5) in Definition 1.In addition, the transformer inequality should also hold for positive definite A, B and arbitrary C.
Note that the transformer inequality replaces condition (6) in Definition 1.We denote by M op the set of matrix means.
Example 1.The arithmetic, geometric and harmonic operator means are defined, respectively, by setting We recall that a function f : (0, ∞) → R is said to be operator monotone (increasing) if for positive definite matrices of arbitrary order.It then follows that the inequality also holds for positive operators on an arbitrary Hilbert space.An operator monotone function f is said to be symmetric if The fundamental result, due to Kubo and Ando, is the following.
Theorem.There is a bijection f → m f between M op and F op given by the formula Remark 1.The function in F op are (operator) concave which makes the operator case quite different from the numerical (commutative) case.For example, there exist convex functions in F num , see [21].
If ρ is a density matrix (a quantum state) and A is a self-adjoint matrix (a quantum observable), then the expectation of A in the state ρ is defined by setting E ρ (A) = Tr(ρA).

THE CORRESPONDENCE BETWEEN FISHER INFORMATION AND METRIC ADJUSTED SKEW INFORMATION
We introduce now a technical tool which is useful to establish some fundamental relations between quantum covariance, quantum Fisher information and the metric adjusted skew information.Definition 5.For f ∈ F op we define f (0) = lim x→0 f (x).We say that a function f ∈ F op is regular if f (0) 0, and non-regular if f (0) = 0, cf.[3,22].Definition 6.A quantum Fisher information is extendable if its radial limit exists and it is a Riemannian metric on the real projective space generated by the pure states.
For the definition of the radial limit see [22] where the following fundamental result is proved.
Theorem.An operator monotone function f ∈ F op is regular, if and only if •, • ρ, f is extendable.
Remark 2. The reader should be aware that there is no negative connotation associated with the qualification "non-regular".For example, a very important quantum Fisher information in quantum physics (see [23]), namely the Kubo-Mori metric related to the function f (x) = (x − 1)/ log x, is non-regular.
We introduce the sets of regular and non-regular functions and notice that trivially Definition 7. We introduce to f ∈ F r op the transform f given by for x > 0. We may also write f = G( f ), cf.[19,24].
The following result is taken from [19, Theorem 5.1].
Theorem.The correspondence f → f is a bijection between F r op and F n op .
The notion of quantum f -covariance has been introduced by Petz, see [25,26].Any Kubo-Ando function The operator m f (L ρ , R ρ ) is well-defined by the spectral theorem for any state, see [ The f -variance is a positive semi-definite sesquilinear form and Note that for the standard covariance we have Cov ρ (A, B) = Cov S LD ρ (A, B), where the SLD or Bures-Uhlmann metric is the one associated with the function Corollary.If ρ is a pure state and f is non-regular, then The theory of quantum Fisher information is due to Petz and we recall here the basic results.If N is a differentiable manifold we denote by T ρ N the tangent space to N at the point ρ ∈ N. Recall that there exists a natural identification of T ρ D 1 n with the space of self-adjoint traceless matrices; namely, for any ρ ∈ D A stochastic map is a completely positive and trace preserving operator holds for every stochastic map T : M n → M m , every faithful state ρ ∈ D 1 n , and every X ∈ T ρ D 1 n .Usually monotone metrics are normalized in such a way that [A, ρ] = 0 implies g ρ (A, A) = Tr(ρ −1 A 2 ).A monotone metric is also called (an example of) quantum Fisher information (QFI).This notation is inspired by Chentsov's uniqueness theorem for commutative monotone metrics [28].
Define L ρ (A) = ρA and R ρ (A) = Aρ, and observe that L ρ and R ρ are commuting positive superoperators on M n .For any f ∈ F op one may also define the positive (non-linear) superoperator m f (L ρ , R ρ ).The fundamental theorem of monotone metrics may be stated in the following way: Theorem.(See [29]).There exists a bijective correspondence between monotone metrics (quantum Fisher information(s)) on D 1 n and functions f ∈ F op .The correspondence is given by the formula for positive matrices A and B.

METRIC ADJUSTED SKEW INFORMATION
By using the general form of the quantum Fisher information it is possible to greatly generalize the Wigner-Yanase information measure.To f ∈ F op the so-called Morosova function c f (x, y) is defined by setting The corresponding monotone symmetric metric K ρ is given by where L ρ and R ρ denote left and right multiplication with ρ.Note that K f ρ (A) is increasing in c f and thus decreasing in f.If furthermore f is regular, the notion of metric adjusted skew information [3, Definition 1.2] is defined by setting where ρ > 0. We use the second notation, I f (ρ, A), when the expression of the state takes up too much space.We also tacitly extended the metric adjusted skew information to arbitrary (non-self-adjoint) operators A. It is convex [3,Theorem 3.7] in the state variable ρ and with equality if ρ is pure [3, Theorem 3.8], see also the summery with interpretations in [30,Theorem 1.2].Furthermore, the notion of unbounded metric adjusted skew information for non-regular functions in F op is introduced in [30, Theorem 5.1].For regular f ∈ F op the metric adjusted skew information may be written as se [31, equation (7)].We thus obtain that the metric adjusted skew information is decreasing in the transform f for arbitrary self-adjoint A, that is We may also write and obtain cf. [31, equation (10)].It follows that the metric adjusted skew information is increasing in f for arbitrary A. It may be derived from [24, Proposition 6.3, page 11], that the metric adjusted skew information can be expressed as the difference with extension to the sesquilinear form

Information inequalities
A function f : R + → R + is in F op if and only if it allows a representation of the form where the weight function h f : [0, 1] → [0, 1] is measurable.The equivalence class containing h f is uniquely determined by f, cf.[31,Theorem 2.1].This representation gives rise to an order relation in F op .
Definition 9. Let f, g ∈ F op .We say that f is majorized by g and write f g, if the function The partial order relation is stronger that the usual order relation ≤, and it renders (F op , ) into a lattice with as respectively minimal element and maximal element.Furthermore, cf. [31,Theorem 2.4].The restriction of to the regular part of F op induces a partial order relation on the set of metric adjusted skew informations.
Proposition 3. The restriction of the order relation renders the regular part of F op into a lattice.In addition, if one of two functions f, g ∈ F op is non-regular, then the minorant f ∧ g is also non-regular.
Proof.Take f ∈ F op with representative function h f as given in (7).Then it follows that f is regular if and only if the integral Take now regular functions f, g ∈ F op .We know that F op , is a lattice [31, bottom of page 141], and that the representative function in (7) for the minorant f ∧ g is given by showing that also h f ∧g satisfies the integrability condition (10) implying that f ∧ g is regular.Since it also follows that the majorant is regular.We now take functions f, g ∈ F op with representative functions h f and h g and assume that f is non-regular.Since we obtain that also the minorant f ∧ g is non-regular.QED The Wigner-Yanase-Dyson skew informations The Wigner-Yanase-Dyson skew information (with parameter p) is defined by setting It is an example of a metric adjusted skew information and reduces to the Wigner-Yanase skew information for p = 1/2 .The representing function f p of I p (ρ, A) is given by The weight-functions h p (λ) in equation ( 7) corresponding to the representing functions f p are given by It is a non-trivial result that the Wigner-Yanase-Dyson skew informations I p (ρ, A) are increasing in the parameter p for 0 < p ≤ 1/2 and decreasing in p for 1/2 ≤ p < 1 with respect to the order relation , cf. [31,Theorem 2.8].The Wigner-Yanase skew information is thus the maximal element among the Wigner-Yanase-Dyson skew informations with respect to the order relation .

The monotonous bridge
The family of metrics with representing functions decrease monotonously (with respect to ) from the largest monotone symmetric metric down to the Bures metric for α increasing from 0 to 1.They correspond the the constant weight functions h α (λ) = α in equation (7).However, the only regular metric in this bridge is the Bures metric (α = 1).It is however possible to construct a variant bridge by choosing the weight functions in equation ( 7) instead of the constant weight functions.It is non-trivial that these weight functions provide a monotonously decreasing bridge (with respect to ) of monotone symmetric metrics between the smallest and the largest (monotone symmetric) metric.The benefit of this variant bridge is that all the constituent metrics are regular except for p = 1.

METRIC ADJUSTED LOCAL QUANTUM UNCERTAINTY
We consider a bipartite system H = H 1 ⊗ H 2 of two finite dimensional Hilbert spaces.
Definition 10.Let f ∈ F op be regular and take a vector Λ ∈ R d .We define the Metric Adjusted Local Quantum Uncertainty (or f -LQU) by setting where ρ 12 is a bipartite state, and K 1 is the partial trace of an observable K on H.
The minimum in the above definition is thus taken over local observables K 1 ⊗ 1 2 ∈ B(H 1 ⊗ H 2 ) such that K 1 is unitarily equivalent with the diagonal matrix diag(Λ).
Remark 3. The metric adjusted LQU has been studied in the literature for specific choices of f.
Proposition 4. For f, g ∈ F r op with g ≤ f we have the inequality U Λ, f 1 (ρ 12 ) ≤ U Λ,g 1 (ρ 12 ).In particular the LQU is smaller than the IP.
Proof.Let K1 be the local observable with spectrum Λ minimizing the metric adjusted skew information.Then where we used the inequality in (6).QED Corollary.Let g 1 and g 2 be regular functions in F op and set f = g1 ∧ g2 with respect to the lattice structure in F op .Then there is a regular function g in F op such that g = f = g1 ∧ g2 and for arbitrary ρ 12 .
Proof.The functions g1 and g2 are non-regular by Theorem .By Proposition 3 we thus obtain that also the minorant f is nonregular.Therefore there exists, by the correspondence in Theorem , a (unique) regular function g in F op such that g = f.The assertion then follows by Proposition 4. QED Following [13] we prove that the metric adjusted LQU is a measure of non-classical correlations, i.e. it meets the criteria which identify discord-like quantifiers, see [7].
Theorem.If the state ρ is classical-quantum in the sense of [32], then the metric adjusted LQU vanishes, that is U Λ, f 1 (ρ) = 0. Conversely, if the coordinates of Λ are mutually different (thus rendering the operator K 1 non-degenerate) and U Λ 1 (ρ) = 0, then ρ is classical-quantum.
Proof.We note that the metric adjusted skew information I f ρ (A) for a faithful state ρ is vanishing if and only if ρ and A commute.If ρ is classical-quantum, then for some von Neumann measurement P given by a resolution (P i ) of the identity 1 1 in terms of one-dimensional projections.We may choose K 1 diagonal with respect to this resolution, so K 1 ⊗ 1 2 and ρ commute and thus U Λ, f 1 (ρ) = 0.If on the other hand the f -LQU U Λ, f 1 (ρ) = 0, then there exist a local observable K 1 ⊗ 1 2 such that [ρ, K 1 ⊗ 1 2 ] = 0.By the spectral theorem we write we obtain by multiplying with P 1,i ⊗ 1 2 from the left and P 1, j ⊗ 1 2 from the right the identity By summing over all j different from i we obtain so P 1,i ⊗ 1 2 and ρ commute.It follows that so ρ is left invariant under the von Neumann measurement P given by (P i ).Therefore, ρ is classical-quantum.QED Recall that Luo and Zhang [33] proved that a state ρ is classical-quantum if and only if there exists a resolution (P i ) of the identity 1 1 such that where each ρ i is a state on H 2 and p i ≥ 0, and the sum i p i = 1.By [30, Lemma 3.1] the inequality is valid, where ρ 1 = Tr 2 ρ 12 .Consequently, we obtain that where the minimum is taken over states σ 1 on H 1 unitarily equivalent with ρ 1 .
Theorem.The metric adjusted LQU is invariant under local unitary transformations.
Proof.For the metric adjusted skew information and local unitary transformations we have , where we used the definition in (10).QED Theorem.The metric adjusted LQU is contractive under completely positive trace-preserving maps on the non-measured subsystem.
Proof.Let K1 be the local observable minimizing the metric adjusted skew information.A completely positive trace preserving map Φ 2 on system 2 is obtained as an amplification followed by a partial trace (Stinespring dilation): Tr 3 (U 23 ρ 23 U † 23 ) = Φ 2 ρ 2 .The metric adjusted LQU is invariant under local unitaries.Also, the metric adjusted skew information is contractive under partial trace.Calling d 3 the dimension of the Hilbert space of the ancillary system 3, one has as desired.QED Theorem.The metric adjusted LQU reduces to an entanglement monotone for pure states.
Proof.The metric adjusted f -LQU coincides with the standard variance on pure states, that is whenever ρ is pure [3,Theorem 3.8].But in [1] it has been proved that the minimum local variance is an entanglement monotone for pure states.QED

CONCLUSION
In this work, we have built a unifying information-geometric framework to quantify quantum correlations in terms of metric adjusted skew informations.We extended the physically meaningful definition of LQU to a more general class of information measures.Crucially, metric adjusted quantum correlation quantifiers enjoy, by construction, a set of desirable properties which make them robust information measures.An important open question is whether information geometry methods may help characterize many-body quantum correlations.In general, the very concept of multipartite statistical dependence is not fully grasped in the quantum scenario.In particular, we do not have axiomatically consistent and operationally meaningful measures of genuine multipartite quantum discord.Unfortunately, the LQU and IP cannot be straightforwardly generalized to capture joint properties of more than two quantum particles.A promising starting point could be to translate into the information-geometry language the entropic multipartite correlation measures developed in [34].We plan to investigate the issue in future studies.This research is supported by a Rita Levi Montalcini Fellowship of the Italian Ministry of Research and Education (MIUR), grant number 54 AI20GD01.
24, Proposition 11.1 page 11].To self-adjoint A we set A 0 = A − (TrρA)I, where I is the identity operator.Note that TrρA 0 = TrρA − (TrρA)Trρ = 0, if ρ is a state.Definition 8. Given a state ρ, a function f ∈ F op and self-adjoint A, B we define the quantum f -covariance by setting Cov f ρ (A, B) = TrB 0 m f (L ρ , R ρ )A 0 and the corresponding quantum f -variance by Var f ρ

TABLE I :
Name