Inequalities for Quantum Skew Information

Let us note that Axioms (iii) and (iv) imply that, when n A ≤ n B , D is maximum on maximally entangled pure states, i.e., if ̺ is a maximally entangled pure state then D(̺) = D max [16]. This follows from the facts that a function D on S AB satisfying (iii) is maximal on pure states if n A ≤ n B [33] and that any pure state can be obtained from a maximally entangled pure state via a LOCC [15]. Thus, if Axioms (i-iv) are satisfied, the additional requirement in Axiom (v) is essentially that D(̺) = D max holds only for the maximally entangled states ̺.It has been shown in previous works [11,15] that the geometric discord D G Bu and discord of response D R Bu satisfy Axioms (i)-(iv) for the Bures distance, and hence are bona fide measures of quantum correlations. In this paper, we will prove that this is also the case for the three measures D G He , D M He , and D R He based on the Hellinger distance, as well as for the Bures measurement-induced geometric discord D M Bu and trace discord of response D R Tr . In contrast, it is known that D G HS = D M HS and D R HS do not fulfill Axiom (iii) because of the lack of monotonicity of the Hilbert-Schmidt distance under CPTP maps (an explicit counter-example is given in Ref. [34] for D G HS and applies to D R HS as well, see below). Therefore, the use of the Hilbert-Schmidt distance in the definitions of Eqs. (5)-(7) can and does lead to unphysical predictions. Considering the distances d p associated to the p-norms X p ≡ (Tr |X| p ) 1/p , one has that for p > 1, d p is not contractive under CPTP maps [35] (see also Ref.[36] for a counter-example for p = 2, which also holds for any p > 1). This is why the distances d p cannot be used to define measures of quantumness apart from the case p = 1, corresponding to the contractive trace distance, while for p = 2 the non-contractive Hilbert-Schmidt distance is well tractable and thus used to establish bounds on the bona fide geometric measures.Regarding our last Axiom (v), the only result established so far in the literature concerns the Bures geometric discord [9,15]. We will demonstrate below that all the other measures based on the trace, Bures, and Hellinger distances also satisfy this axiom. Our proofs are valid for arbitrary (finite) space dimensions n A and n B of subsystems A and B, excepted for D G He , for which they are restricted to the special cases n A = 2, 3, and for D M He , D G Tr , and D M Tr , for which they are restricted to n A = 2.The paper is organized as follows. Given its length and the wealth of mathematical relations and bounds that we have determined, we begin by summarizing the main results in Section II. We first give general expressions of the geometric measures for the Bures and Hellinger distances, which are convenient starting points to compare them (see Sec. II A). We then report in some synoptic Tables the various relations and bounds satisfied by D G , D M , and D R for the trace, Hilbert-Schmidt, Bures, and Hellinger distances (see Sec. II B). Closed expressions for the Hellinger geometric ...

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“…In view of Eqs. (42) and (43), the Schmidt decomposition of |Ψ (see Eq. (53)), and the spectral decomposition of U A (see Eq.…”

Section: B Bures Discord Of Responsementioning

confidence: 99%

Geometric measures of quantum correlations: characterization, quantification, and comparison by distances and operations

Roga¹,

Spehner²,

Illuminati³

2016

J. Phys. A: Math. Theor.

108

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“…For proving the existence of such function we need some more sophisticated tool. Next theorem establishes a canonical bijection between elements in F op and probability measures on [0, 1] (see [2,10] Let µ be a probability measure on [0, 1] defined by µ({0}) = µ({1}) = 1 2 . For the function g ∈ F op corresponding to µ we have 1 g(x) = 1 2…”

Section: The Global Upper Boundmentioning

confidence: 99%

“…where L n,D and R n,D are superoperators acting on n×n matrices like L n,D (A) = DA, R n,D (A) = AD and f is a symmetric and normalized operator monotone function [5]. Quantum covariances of the form (2) are called quantum f -covariance. It has been proved [5] that the generalized form of dynamical uncertainty principle holds true for an arbitrary quantum f -covariance…”

Section: Introductionmentioning

confidence: 99%

Refinement of Robertson-type uncertainty principles with geometric interpretation

Lovas

Andai

2016

Int. J. Quantum Inform.

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A generalisation of the classical covariance for quantum mechanical observables has previously been presented by Gibilisco, Hiai and Petz. Gibilisco and Isola has proved that the usual quantum covariance gives the sharpest inequalities for the determinants of covariance matrices. We introduce a new generalisation of the classical covariance which gives better inequalities than the classical one, furthermore it has a direct geometric interpretation.

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“…The quantity I f ρ (A) is known as metric adjusted skew information [12]. It is notable that the metric adjusted correlation measure Corr c ρ (A, B) was firstly introduced in [12] for a regular Morozova-Chentsov function c. Recently the notation I c ρ (A, B) in [1] and the notation I f ρ (A, B) in [11] were used. In addition, it is useful for the readers to be noted that the correlation I f ρ (A, B) can be expressed as a difference of covariances [11].…”

Section: A Further Generalization By Metric Adjusted Correlation Measurementioning

confidence: 99%

Schrödinger uncertainty relation, Wigner–Yanase–Dyson skew information and metric adjusted correlation measure

Furuichi

Yanagi

2012

Journal of Mathematical Analysis and Applications

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In this paper, we give Schrödinger-type uncertainty relation using the Wigner-Yanase-Dyson skew information. In addition, we give Schrödinger-type uncertainty relation by use of a two-parameter extended correlation measure. We finally show the further generalization of Schrödinger-type uncertainty relation by use of the metric adjusted correlation measure. These results generalize our previous result in [Phys. Rev. A, Vol.82(2010), 034101].

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Inequalities for Quantum Skew Information

Cited by 18 publications

References 14 publications

Geometric measures of quantum correlations: characterization, quantification, and comparison by distances and operations

Geometric measures of quantum correlations: characterization, quantification, and comparison by distances and operations

Refinement of Robertson-type uncertainty principles with geometric interpretation

Schrödinger uncertainty relation, Wigner–Yanase–Dyson skew information and metric adjusted correlation measure

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