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 ...

Quantum correlations in a composite system can be measured by resorting to a geometric approach, according to which the distance from the state of the system to a suitable set of classically correlated states is considered. Here we show that all distance functions, which respect natural assumptions of invariance under transposition, convexity, and contractivity under quantum channels, give rise to geometric quantifiers of quantum correlations which exhibit the peculiar freezing phenomenon, i.e., remain constant during the evolution of a paradigmatic class of states of two qubits each independently interacting with a non-dissipative decohering environment. Our results demonstrate from first principles that freezing of geometric quantum correlations is independent of the adopted distance and therefore universal. This finding paves the way to a deeper physical interpretation and future practical exploitation of the phenomenon for noisy quantum technologies.

The presence of quantum correlations in a quantum state is related to the state's response to local unitary perturbations. Such response is quantified by the distance between the unperturbed and perturbed states, minimized with respect to suitably identified sets of local unitary operations. In order to be a bona fide measure of quantum correlations, the distance function must be chosen among those that are contractive under completely positive and trace preserving maps. The most relevant instances of such physically well-behaved metrics include the trace, the Bures, and the Hellinger distance. To each of these metrics one can associate the corresponding discord of response, namely the trace, or Hellinger, or Bures minimum distance from the set of unitarily perturbed states. All these three discords of response satisfy the basic axioms for a proper measure of quantum correlations. In the present work we focus in particular on the Bures distance, which enjoys the unique property of being both Riemannian and contractive under completely positive and trace preserving maps, and admits important operational interpretations in terms of state distinguishability. We compute analytically the Bures discord of response for two-qubit states with maximally mixed marginals and we compare it with the corresponding Bures geometric discord, namely the geometric measure of quantum correlations defined as the Bures distance from the set of classical-quantum states. Finally, we investigate and identify the maximally quantum correlated twoqubit states according to the Bures discord of response. These states exhibit a remarkable nonlinear dependence on the global state purity.

Abstract:We investigate the dynamics of an N -level quantum system weakly coupled to a thermal reservoir. For any fixed temperature of the bath there exists a natural reference state: the equilibrium state of the system. Among all quantum operations on the system one distinguishes Davies maps, they preserve the equilibrium state, satisfy the detailed balance condition and belong to a semi-group. A complete characterization of the three dimensional set of qubit Davies maps is given. We analyze these maps and find their minimum output entropy. A characterization of Davies maps for qutrits is also provided.

We introduce a necessary and sufficient criterion for the non-Markovianity of Gaussian quantum dynamical maps based on the violation of divisibility. The criterion is derived by defining a general vectorial representation of the covariance matrix which is then exploited to determine the condition for the complete positivity of partial maps associated to arbitrary time intervals. Such construction does not rely on the Choi-Jamiolkowski representation and does not require optimization over states. PACS numbers: 03.65.Yz, 03.65.Ta, 42.50.Lc In recent years much effort has been devoted to the characterization and quantification of non-Markovianity in the evolution of open quantum systems (see e.g. Ref.[1] for a recent review). Non-Markovian quantum evolutions may typically arise in the presence of structured environments, such as in quantum biological systems [2][3][4] and in squeezed baths of light with finite bandwidth [5]. Moreover, recent studies suggest that properly engineered non-Markovian channels can improve the efficiency of quantum technology protocols due to the backflow of information from the environment to the system [6][7][8][9][10][11][12]. Establishing whether noisy quantum evolutions are non-Markovian and therefore preserve some memory on the story of the system is of capital importance in the field of quantum cryptography [13].Various approaches to the characterization and quantification of quantum non-Markovianity have been introduced in recent years [1,14,15]. Most of them are witnesses, and thus rely on sufficient, but not necessary, conditions [1]. They usually are based on the non-monotonic behavior of certain quantities in the presence of memory effects [16][17][18][19][20][21]. Moreover, most of them rely on optimization over states.Proper measures of non-Markovianity have also been introduced for finite-dimensional systems. These measures include the amount of isotropic noise necessary to make the dynamics completely positive in every arbitrary short interval of time [22] and the negativity of the decay rates appearing in the generators of the time evolution, once the associated master equation is expressed in canonical form [23].A further necessary and sufficient criterion has been obtained by Rivas, Huelga, and Plenio (RHP) by considering the violation of the divisibility property, which expresses the possibility of decomposing the evolution on a generic time interval into two successive, independent completely positive maps. Non-Markovianity is then characterized by the extent that the intermediate map violates complete positivity (CP) [17]. These three necessary and sufficient criteria for * Corresponding author: illuminati@sa.infn.it finite-dimensional systems have been shown to be completely equivalent [23]; moreover, the criterion based on the isotropic noise and the RHP criterion rely on positivity of the ChoiJamiołkowski states corresponding to the channels [24,25].Addressing the general characterization and quantification of non-Markovianity in the infinite-dimensional case i...