Universal freezing of quantum correlations within the geometric approach

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.

In quantum mechanics, the mathematical description of a composite quantum system is based on both the superposition principle and the tensorial structure of the Hilbert space associated with it. The coexistence of these two principles makes the properties of generic states of a composite quantum system particularly weird, in the sense that they cannot be reproduced by any state of a classical system. Some of the most striking non-classical properties exhibited by quantum states can be collected under the name of quantum correlations 1,2 . Nowadays, there is universal consensus on the fact that the quantum correlations shared by two subsystems in a global pure state are entirely captured by entanglement and can be quantified by any valid entanglement measure 1,3 . On the other hand, it is also now quite clear that there exist non-entangled mixed states still manifesting some non-classical features, such as an unavoidable disturbance due to local measurements, which embodies the concept of quantum discord 4,5 . Therefore, in realistic open quantum systems, entanglement may represent only a portion, sometimes negligible, of the quantumness of correlations, while more general figures of merit to quantify quantum correlations are provided by suitable measures of discord-type correlations 2,6 . Despite an intense recent activity in investigating interpretation, quantification, and applications of discord and related quantifiers of quantum correlations 2 , these quantities still remain less understood than entanglement.
In order to unveil the most profound signatures of quantumness in composite systems, it is essential to identify mathematically rigorous and physically meaningful properties that differentiate the notion of discord-type quantum correlations from that of entanglement (and of classical correlations), and are manifested by any valid measure thereof. One such property common to generic discord-type correlation measures is, for instance, the absence of monogamy 7 . Besides the fundamental implications, this area of investigation has a technological motivation 8,9 , since quantum correlations beyond and even without entanglement have been shown to play a resource role for certain schemes of quantum computation [10][11][12] , communication [13][14][15][16] , and metrology [17][18][19] . Finding valuable and general traits of these quantum correlation resources, in particular for what concerns their dynamical preservation during the unavoidable interaction of a principal quantum systems with the surrounding environments, constitutes an important aim with a clear potential to lead to useful recipes for their practical exploitation.
Numerous works have in fact investigated the dynamics of general quantum correlations in open quantum systems undergoing various types of Markovian or non-Markovian evolution, as reviewed e.g. in Refs. 2,20,21. Although different measures of quantum correlations can exhibit distinct features and impose inequivalent orderings on the set of quantum states, it has emerged as a general trait that discord-type quantum correlations are more robust than entanglement against noise [22][23][24][25][26] (see also 27,28 , for a critical assessment) and cannot generally vanish at a finite evolution time (due to the fact that zero-discord states are of null measure 25 , while entanglement can suffer so-called sudden death 29,30 . However, a fascinating and nontrivial phenomenon of extreme robustness to noise exhibited by general quantum correlations deserves special attention, and is the subject of our investigation. Namely, under local non-dissipative decoherence evolutions, it has been observed that a number of known discord-type measures all remain constant ('frozen') for a finite time interval in Markovian conditions 6,31,32 , and for multiple intervals [33][34][35] , or forever 36 , in non-Markovian conditions, when considering two non-interacting qubits initially in a specific class of Bell-diagonal states. This freezing phenomenon, not exhibited by any measure of entanglement, is quite appealing since it implies that every protocol relying on discord-type quantum correlations as a resource will run with a performance unaffected by noise in the specific dynamical conditions. Currently, the occurrence of freezing has been investigated by explicitly considering the evaluation of specific discord-type measures on a case by case basis 6,31,33 However, it is natural to ask whether this phenomenon is a mere mathematical accident due to the particular choices of quantum correlations quantifiers, or whether it must manifest independently of the adopted measure, thus having a universal character and promising to bear a deep physical meaning. This work addresses such an issue.
We prove that freezing occurs for any geometric measure of quantum correlations, whenever the distance defining the measure respects a minimal set of physical assumptions, namely dynamical contractivity under quantum channels, invariance under transposition, and convexity. The freezing phenomenon is therefore revealed as universal within the geometric approach to quantum correlations. Notice that our work differs from other complementary investigations of the freezing phenomenon 32,37 . In particular, in a recent work 37 , the authors provide necessary and sufficient conditions for a general state to exhibit freezing under non-dissipative decoherence, according to some specific measure of discord. Here, instead, we focus on a specific class of initial states, and we identify the minimal set of conditions that any general distance-based measure of discord needs to satisfy in order to freeze. On some random family of initial states, it is certainly possible to see freezing according to one discord-type measure but not to another. What we prove here is that, for the specific class of Bell-diagonal states identified in Refs. 6, 31, all bona fide geometric quantifiers of quantum correlations (respecting the three physical assumptions mentioned above) undergo the same dynamics, featuring the freezing phenomenon. In proving the main result, we also introduce and characterise a global quantum control channel which can completely invert decoherence on a subset of Bell-diagonal two-qubit states, and can be of independent interest.
The paper is organised as follows. We first summarise the properties that any valid measure of quantum correlations is expected to hold, and provide general definitions for distance-based measures, distinguishing between those for entanglement and those for discord-type correlations. In the Results section, we illustrate the freezing phenomenon in geometric terms, by considering for convenience the specific case of the Bures distance-based measure of quantum correlations. We then present the main result, by proving that freezing must happen for any bona fide distance-based discord-type measure. Finally, we offer our conclusions in the Discussion section. Some technical bits are deferred to the Methods section.

Measures of quantum correlations
Here we recall the requirements that a valid measure of quantum correlations is expected to have, later focusing specifically on geometric (distance-based) definitions. For more details on the quantification of quantum correlations, the reader is referred to recent review articles 1,2 .
In this paper, we consider a state AB ρ ρ ≡ of a two-qubit system, with subsystems tagged A and B. As is well known 1,38 , entanglement quantifiers capture the degree of non-separability of the state ρ of the global system; the corresponding distance-based measures of entanglement are calculated from the set  of separable states 39 , i.e., states σ which can be written as convex combination of product states, Quantifiers of quantum correlations other than entanglement, the so-called discord-type measures, capture instead the minimal degree of disturbance on the state ρ after local projective measurements on the system 2,4,5 . The projective measurements can be performed either on a subsystem only (which gives rise to one-way, asymmetric discord-type measures) or on both subsystems (which gives rise to two-way, symmetric discord-type measures). These two versions of discord-type measures have valuable operational meanings in different contexts 2 and the corresponding distance-based measures are calculated, respectively, from the set of classical-quantum (CQ) and classical-classical (CC) states 41 . Explicitly, a CQ state (with respect to measurements on subsystem A) is a particular type of separable state, which can be written in the form where p i is a probability distribution, { } where p ij is a joint probability distribution, while { } i A and { } j B denote orthonormal bases for subsystem A and B, respectively. Clearly, the set  of CC states is contained in the set of CQ states, which is a subset of the set  of separable states.
In general, if not explicitly written, by "classical" states we hereafter mean CC states, i.e., states χ which are diagonal in a product basis, as defined by Eq. (3); these states correspond merely to the embedding of a bipartite probability distribution { } p ij into the quantum formalism. We further specify that for Bell-diagonal states 42 , namely the specific class of two-qubit states considered in this work, the two notions of discord are completely equivalent, therefore our conclusions about the universality of the freezing will apply indifferently to both one-way and two-way geometric measures of discord-type correlations.
From a quantitative point of view, a valid entanglement measure, also known as an entanglement monotone 3 , is any real and nonnegative function E on the set of states ρ satisfying the following basic axioms 1,43 : ∈ , . Notice, however, that while convexity is physically desirable (as it would mean that entanglement cannnot increase by mixing states), it is not an essential property, since there are valid entanglement monotones which are not convex 44 .
The theory of quantum correlations other than entanglement is not completely developed yet 2,38 , but we can nonetheless identify some desiderata for any quantifier thereof. A (two-way) discord-type measure is any real and nonnegative function Q on the set of states ρ satisfying the following requirements: for arbitrary marginal states A B ρ ( ) and A B σ ( ) ; (Q.iv) Q reduces to an entanglement measure for pure states, i.e. Q E AB AB ψ ψ ( )= ( ) for any pure state AB ψ . We introduce property (Q.iii) in analogy with property (E.iii) for entanglement. Namely, it is known that local commutativity preserving channels cannot create discord-type correlations, as they leave the set of classical states invariant 45 . We thus require that any valid measure of discord-type correlations should be monotonically nonincreasing under such channels. Notice, in particular, that for two qubits these channels include local unital channels 22 .
The above requirements need to be slightly modified if a one-way discord-type measure Q → , say with measurements on A, is considered. Specifically, property (Q.i) becomes: Q → (ρ)=0 if ρ is a CQ state as Scientific RepoRts | 5:10177 | DOi: 10.1038/srep10177 defined in Eq. (2). Furthermore, a stricter monotonicity requirement supplements (Q.iii) for all valid one-way discord-type measures 6,46,47 , namely (Q.iii.bis) Q → is monotonically nonincreasing under arbitrary local quantum channels on the unmeasured subsystem B, that is, Q → ((� A ⊗Λ B )(ρ)) ≤ Q → (ρ) for any state ρ and any completely positive trace-preserving (CPTP) map B Λ on subsystem B. Properties (Q.ii) and (Q.iv) apply equally to two-way and one-way discord-type measures. The latter property just signifies that, in pure bipartite states, there is a unique kind of quantum correlations, arising in all but tensor product states. Even correlations stronger than entanglement, such as steering and nonlocality, just collapse back to non-separability in the case of pure states.
In order to investigate the freezing phenomenon, in this paper we resort to a geometric approach to define a very general class of valid measures of quantum correlations. According to such an approach, the entanglement E and the discord-type correlations Q of a state ρ can be quantified as the minimal distance from ρ to the sets  and  of separable and classical states, respectively 40,48 . In formulae, where separable states σ are defined by Eq. (1), while classical states χ are defined respectively by Eqs. (3) and (2) depending on whether a two-way or one-way discord-type measure is considered. In these definitions, D can denote in principle any suitable distance on the set of quantum states.
In order for the geometric measures E D and Q D to respect the essential properties listed above, the distance D needs to satisfy certain mathematical requirements 49 . Here, we identify a minimal set of three such requirements, that will be said to characterise D as a bona fide distance. Given any states ρ, σ, τ, and ς, these are: for any q [0 1] ∈ , . Let us comment on the physical significance of these requirements. On the one hand, the contractivity property (D.i) of D, Eq. (6) is a fundamental requirement for a distance in quantum information theory 49 , and has two purposes. First, it makes D a statistically relevant distance, due to the fact that non-invertible CPTP maps are the mathematical counterparts of noise and the latter cannot lead to any increase in the information related to the distinguishability of quantum states 50,51 . Second, (D.i) makes the distance-based measures of entanglement E D and quantum correlations Q D [Eqs. (4) and (5)] physically meaningful, by implying the essential properties (E.ii), (E.iii), (Q.ii), (Q.iii), and also (Q.iii.bis), where the latter applies to the corresponding distance-based one-way discord-type measure → Q D defined by choosing the set of CQ (rather than CC) states in Eq. (5). Notice that (D.i) implies in particular the standard property of invariance of the distance D under unitary operations, since they can be seen as reversible CPTP maps; namely, , for any pair of states ρ, σ, and any unitary U.
On the other hand, the invariance of a distance D under transposition, Eq. (7), is not typically discussed in the literature. However, transposition of an N N × hermitian matrix, which amounts to complex conjugation in the computational basis, corresponds to a reflection in a N N [ 1 2 1] ( + )/ −dimensional hyperplane. Property (D.ii) thus means that a distance D on the set of quantum states is assumed to be invariant under reflections, which appears as a fairly natural requirement 50  , for any pair of states ρ, σ.
Finally, the joint convexity property (D.iii) of D, Eq. (8), is also quite intuitive and it makes the corresponding distance-based entanglement measure E D convex, implying the desirable property (E.iv).
Notice however that discord-type measures are, by contrast, neither convex nor concave 52 , as the set  of classical states is not a convex set 25 .
There are a number of known distances which satisfy the three physical assumptions listed above 50 , and have been employed to define valid geometric measures of quantum correlations. Suitable examples for D include in particular the relative entropy 40,53 , the squared Bures distance 6,54,55 , the squared Hellinger distance 18,56,57 , and the trace (or Schatten one-norm) distance [58][59][60] . Contrarily, the Hilbert-Schmidt distance does not respect the contractivity property (D.i), and earlier attempts to adopt it to define geometric measures of quantum correlations 48,61 have led to inconsistencies 62,63 .
In this paper, we label a generic distance D obeying properties (D.i), (D.ii), and (D.iii) as a bona fide one, and the associated distance-based quantities E D and Q D as bona fide measures of entanglement and discord-type correlations respectively. Therefore, the main result of this paper will be a proof of the universality of the freezing for all geometric measures of quantum correlations constructed via bona fide distances as formalised in this Section.

Results
Freezing of quantum correlations measured by Bures distance. We now present the freezing phenomenon from a geometric perspective, by employing a particular bona fide measure of quantum correlations, that is the Bures distance-based measure 6,55,64,65 . We first recall all the basic ingredients for the complete description of the phenomenon.
The Bures distance D u B between two states ρ and σ is defined as is the Uhlmann fidelity 66 . The Bures distance arises from a specific case of a general family of Riemannian contractive metrics on the set of density matrices, characterised by Petz 67 following the work by Morozova and Čencov 68 . It can be connected operationally to the success probability in ambiguous quantum state discrimination, and it has been successfully employed to define geometric measures of entanglement, quantum, classical, and total correlations 6

≡
, which is a bona fide one obeying properties (D.i), (D.ii), and (D.iii). Bell-diagonal (BD) states, also referred to as T-states or two-qubit states with maximally mixed marginals 42 , are structurally simple states which nonetheless remain of high relevance to theoretical and experimental research in quantum information, as they include the well-known Bell and Werner states 39 and can be employed as resources for operational tasks such as entanglement activation and distribution via discord-type correlations 47,[71][72][73] . BD states are by definition diagonal in the basis of the four maximally entangled Bell states, and their Bloch representation in the computational basis is , , .
A non-dissipative quantum channel acting on a qubit induces decoherence with no excitation exchange between the qubit and its environment. We consider the evolution of two non-interacting qubits undergoing local identical non-dissipative decoherence channels. The action of any such channel on each single qubit is characterised by the following Lindblad operator 6,31 , ρ is the reduced state of subsystem m (m A B = , ) and k {1 2 3} ∈ , , represents the direction of the noise. Namely, the choice of k 1 2 3 = , , respectively identifies decoherence in the Pauli x y z , , basis for each qubit, and the corresponding channels are known in the quantum computing language as bit flip (k 1 = ), bit-phase flip (k 2 = ), and phase flip (k 3 = ) channels 31,49 . It is worth noting that one can easily derive the dynamics of the composite two-qubit system from the dynamics of the single qubits, since each of the two qubits is locally interacting only with its own environments (and not with the other qubit), so that they have independent dynamical evolutions 21 .
Equivalently, the evolution of a two-qubit state ρ under local non-dissipative decoherence channels can be obtained in the operator-sum representation by the map we find that the Bures distance-based measure of discord Q u B stays constant (freezes) for t t 0 ≤ < ⁎ and then decays exponentially from t t > ⁎ onwards, as is shown in Fig. 2. This can be straightforwardly shown by exploiting the available closed formula for Bures discord-type correlations of BD states 6,55 . On the contrary, entanglement measured e.g. by E u B undergoes a typical sudden death at a finite time 29 . We stress again that this behaviour of quantum correlations, here illustrated for Q u B , has been independently observed (on a case by case basis) for several valid discord-type measures in the aforementioned dynamical conditions 6 : this paper will provide a rigorous basis to establish its universality within the bona fide geometric approach.
The freezing phenomenon can be understood in geometric terms by looking at Fig. 1, which represents the phase flip freezing surface containing BD states of the form c cc c { } . From Eq. (5) we know that the discord-type quantum correlations of t ρ ( ) for t t < ⁎ are given exactly by the (squared) Bures distance between the evolving state t t ρ ( < ) ⁎ and this closest classical state t t χ ρ ( < ) ⁎ . Interestingly, one can observe that this distance is constant for any t t < ⁎ , which indeed implies that the quantum correlations of t ρ ( ) are frozen for any t t < ⁎ , given arbitrary initial conditions on the freezing surface defined by (17).  is instead its Euclidean orthogonal projection onto the c 3 -axis, i.e. the BD classical state with triple , which is independent of time. Therefore the quantum correlations of the evolved state t ρ ( ) decrease for any t t > ⁎ , as the distance between the evolving state t t ρ ( > ) ⁎ and the steady closest classical state t t χ ρ ( > ) ⁎ decreases for any t t > ⁎ . For two-qubit BD states of the form (17)  The main result of this paper will be to show that these two properties are satisfied by any contractive, transposition invariant, and convex distance, thus implying the freezing phenomenon for any bona fide distance-based measure of discord-type quantum correlations as defined above. Contrarily, we remark that the non-contractive Hilbert-Schmidt distance satisfies only the first property (F.i), whereas it does not manifest the kind of translational invariance expressed in Eq. (19), due to the fact that the trajectory of the evolved state is not parallel to the c 1 -axis according to the Euclidean geometry, as is shown in Fig. 1. As a result, the Hilbert-Schmidt geometric discord 48 , which is not a bona fide measure 63 , does not manifest freezing in the considered dynamical conditions, as previously observed 6 . where the first equality is due to the fact that 3 3 and the final inequality is due to contractivity of the distance D.

Freezing of quantum correlations for all
We now introduce a global two-qubit rephasing channel R q G 3 Λ with operator-sum representation Λ , which is a physical CPTP map for all ∈ − , q [ 1 1], transforms any two-qubit state into a BD state lying on the phase flip freezing surface, i.e., ρ is a BD state with characteristic triple given by q qT T { } 33 33 , − , , with σ σ ρ = ( ⊗ ) T Tr [ ] 33 3 3 . Specifically, the action of Λ on a BD state ρ which already belongs to the phase flip freezing surface is This map is therefore able to restore the lost coherence for any (even completely dephased) BD state on the freezing surface, thus effectively reverting their decoherence process.
We have then the inequality where the first equality is due to the fact that , , , , , = , , , ( ) and the final inequality is again due to contractivity of the distance D. By putting together the two opposite inequalities (22) and (27), we immediately get the invariance of Eq. (20) for any contractive distance.
To prove now the claim of Eq. (21), we introduce the unitary U i i y y By exploiting the invariance under unitaries of any contractive distance, and the just proven invariance expressed by Eq. (20), we finally have for any bona fide distance as defined above. We divide this result in two main steps, represented by Theorem 2 and Theorem 3. To bridge between the two Theorems we need four Lemmas which are formulated and proved in the Methods.
Let us begin by the following powerful result, which applies to all two-qubit BD states. Theorem 2. According to any contractive, transposition invariant, and convex distance, one of the closest classical states χ ρ to a BD state ρ is always a BD classical state, ( ) ∈ , , and some coefficient ∈ − , s [ 1 1].
Proof. For an arbitrary two-qubit state ς, described in the Bloch representation We will now consider the distance from ρ to the (larger) set of two-qubit CQ states, and show that its minimum can be attained by a fully classical (CC) BD state, hence proving the main result of the theorem. Recall that any CQ two-qubit state is of the form is an orthonormal basis for qubit A and B Temporarily, we relax the restriction that e → is of unit length and consider the distance from the (even larger) set of states for which e 1 → ≤ . This is a convex set and so, due to the convexity of the distance, any local minimum will be a global one. We can now use a trick analogous to the one used for Eq. , where the index k sets the nonzero vector element. From the previous results, we notice in fact that minimisation only needs to be performed over e k and s k as the distance can only decrease under any variation in any other single element. Furthermore, e k and s k appear only as a product e s k k in the density matrix, never on their own. This means that minimising over both is equivalent to setting e 1 k = and minimising only over s k , thus allowing us to reimpose the restriction that e 1 → = , thus coming back to analyse the distance from ρ to CQ states. The remaining states over which the minimisation in the single parameter s needs to be performed amount exactly to the set of BD classical states (aligned on the axes in Fig. 1), hence finding the minimum among these will return the global minimum for the distance D from an arbitrary BD state ρ to the set of two-qubit classical states, proving the claim. , , defining any given BD state ρ. When we restrict ourselves to BD states ρ belonging to the phase flip freezing surface of Eq. (17), the solution is provided by Theorem 3, which makes use of the auxiliary results proven in the Methods. distance-based measures of correlations, our result means that freezing of quantum correlations occurs independently of the adopted distance and is therefore universal within a bona fide geometric approach. Frozen quantum correlations have been verified both theoretically 6,24,[31][32][33]37,75 and experimentally 76-81 by using specific measures of quantum correlations 2,6 , but until now it was an open problem whether all suitable discord-type quantifiers (including potentially new ones yet to be defined) would freeze in the same dynamical conditions. Our work rigorously contributes to the settling of this problem and provides elegant evidence strongly supporting the conclusion that freezing of quantum correlations is a natural physical phenomenon and not merely a mathematical accident. Notice that freezing in BD states, as described in this paper, has also been observed for some discord-type measures which do not manifestly enjoy a distance-based definition, such as the local quantum uncertainty 6,18 and the interferometric power 19 . This leaves some room for further research aimed to prove the occurrence of freezing from only the basic properties (Q.i), (Q.ii), and (Q.iii) of quantum correlations, possibly without the need to invoke a geometric approach as considered in this work. Alternatively, our result might suggest that all measures of discord could possibly be recast into a geometric form via some bona fide distance, at least when restricted to BD states of two qubits (this is the case, for instance, for the conventional entropic measure of discord 5 , which becomes equivalent to the relative entropy-based discord 31 for BD states); this would also be an interesting direction to explore, in a more mathematical context of information geometry.
We further remark that, although we have explicitly considered Markovian evolutions in our analysis, the freezing of quantum correlations also occurs in the presence of non-Markovian channels which can be described by a master equation with a memory kernel, as in the case of pure dephasing or decoherence under classical random external fields [33][34][35][36]82 . Indeed, in these cases the dynamics of BD states can be formally written as in Eq. (13), but with t 2γ replaced by a more general time-dependent rate t Γ ( ). This can give rise to a dynamics with multiple intervals of constant discord 33,34,82 , or discord frozen forever 36 depending on the initial conditions. By our analysis, we conclude that those fascinating features, which might be observable e.g. in the dynamics of impurity atoms in Bose-Einstein condensates 36,83 are universal too and manifest when probed by any bona fide geometric discord-type measure Q D .
Within this paper we introduced an intriguing global rephasing channel, which is able to reverse the effects of decoherence for certain two-qubit BD states. This physical CPTP channel may be of interest for applications other than proving the universality of the freezing phenomenon, for example quantum error correction 49 , where it is desirable to combat the effects of noise, typically manifesting via local bit flip, phase flip, or bit-phase flip channels. For suitable BD states, all these errors can be corrected by global maps such as the one in Eq. (24). We also note that the action of this channel resembles (but is different from) the physical situation of refocusing by dynamical decoupling control on qubits undergoing low-frequency pure dephasing 84,85 . The further characterisation and experimental implementation of our global rephasing map for quantum information processing calls for an independent analysis which is beyond the scope of this paper.
From a fundamental perspective it is important to understand the deeper physical origin of frozen quantum correlations. There are reasons to reckon that the phenomenon is related to the complementary freezing of classical correlations. Typically, as observed so far using specific quantifiers, given particular dynamical and initial conditions as studied here, quantum correlations are initially frozen and classical correlations decay but, after a characteristic time t ⁎ , classical correlations freeze and quantum correlations decay 31,65,74 . This has been linked to the finite-time emergence of the classical pointer basis within the fundamental theory of decoherence 75,78,81,86 . Nevertheless, classical correlations are still inconsistently defined in geometric approaches 65,87 , and it remains unknown whether they exhibit freezing after t ⁎ for any bona fide distance. This is certainly an aspect deserving further investigation.
Very recently, some of us have shown that an even more fundamental property of quantum systems, namely coherence 88 in a reference basis, can also remain frozen under local nondissipative decoherence channels for the same class of initial states as studied here 89 . Such a result holds more generally for a class of N -qubit states with maximally mixed marginals (for any even N ), which include and extend the two-qubit set discussed in this work. This suggests that multiqubit and multipartite quantum correlations can freeze as well under the same dynamical conditions 37,90 , and the methods of this work can be readily employed to prove the universality of freezing within the geometric approach, in such a more general instance as well.
Our result has also an impact from an applicative point of view. The property of being unaffected by the noise for a given period of time makes quantum correlations other than entanglement important for emergent quantum technologies 8,9 . Despite numerous basic experimental investigations, this resilience has yet to be properly exploited as a resource for quantum enhanced protocols e.g. in communication, computation, sensing and metrology. The universality of the freezing phenomenon for geometric quantum correlations, in paradigmatic quantum states and dynamical evolutions as shown here, promises to motivate further research in this context.

Methods
Here we derive some technical results needed for the proof of Theorem 3.
Lemma A1. According to any contractive distance D, it holds that: