title: Bounding generalized relative entropies: Nonasymptotic quantum speed limits
authors: Diego Paiva Pires Kavan Modi Lucas Chibebe Céleri
text: 
  
  
  
  
  
  International Institute of Physics and Departamento de Física Teórica e Experimental, Universidade Federal do Rio Grande do Norte, Natal, RN, 59078-970, Brazil
  
  
  School of Physics & Astronomy, Monash University, Clayton, Victoria 3800, Australia
  
  
  Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, E-48080 Bilbao, Spain
  Institute of Physics, Federal University of Goiás, 74.690-900 Goiânia, Goiás, Brazil
  
  
  Information theory has become an increasingly important research field to better understand quantum mechanics. Noteworthy, it covers both foundational and applied perspectives, also offering a common technical language to study a variety of research areas. Remarkably, one of the key information-theoretic quantities is given by the relative entropy, which quantifies how difficult is to tell apart two probability distributions, or even two quantum states. Such a quantity rests at the core of fields like metrology, quantum thermodynamics, quantum communication and quantum information. Given this broadness of applications, it is desirable to understand how this quantity changes under a quantum process. By considering a general unitary channel, we establish a bound on the generalized relative entropies (Rényi and Tsallis) between the output and the input of the channel. As an application of our bounds, we derive a family of quantum speed limits based on relative entropies. Possible connections between this family with thermodynamics, quantum coherence, asymmetry and single-shot information theory are briefly discussed. 
  
  
  
  
  Bounding generalized relative entropies: Nonasymptotic quantum speed limits
      Lucas Chibebe Céleri
      December 30, 2023
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  § INTRODUCTION
  
  
  
  Since its formulation decades ago by Shannon <cit.>, information theory has played a major role in both applied and fundamental science, ranging from neuroscience <cit.> to quantum gravity <cit.>, and along the way has impacted thermodynamics <cit.>, finance <cit.>, and evolutionary biology <cit.>. A central e­le­ment in this theory is the Shannon entropy, which measure how much information is contained in a probability distribution. Shannon entropy also plays a role on the speed of evolution of classical stochastic processes <cit.>. However, when the basic assumptions of the theory do not hold, e.g., extensivity or very large data sets (nonasymptotic regime), other information measures appear as gene­ra­li­za­tions of the Shannon entropy. Indeed, such a family of information-theoretic measures include the paradigmatic cases of Tsallis <cit.> and Rényi <cit.> entropies.
  
  Each of these developments is based on the idea that a physical process could be understood as an information processing protocol. In such tasks, dis­tin­gui­shing classical probability distributions or quantum states plays a fundamental role. The relative entropy (RE) <cit.>, also called divergence <cit.>, stand as a remarkable information-theoretic distin­guishability metric, thus exhibiting distinct operational meanings in se­ve­ral fields. For ins­tance, relative entropy quantifies the dissipated work in a driven evolution <cit.>, the amount of entanglement <cit.> and quantum coherence in a given state <cit.>. Moreover, it unveils the role of entropy production in thermal relaxation processes <cit.>, and also the asymmetry of a state or process <cit.>. 
  
  Rényi relative entropy (RRE) determine an entire family of second laws of thermodynamics in the quantum regime <cit.>, which also applies to black hole physics <cit.>, and cut-off rate in the hypothesis testing theory <cit.>, and quantum Gaussian states <cit.>, to name only a few. Furthermore, RRE is linked to an entropic energy-time uncertainty relation for time-independent systems <cit.>, also being related to the concept of multiple quantum coherences <cit.>.
  
  Tsallis entropy is mainly considered in the field of nonextensive statistical mechanics <cit.>. However, important applications of this theory also appear in several other areas <cit.>. Interestingly, it has been shown that Tsallis relative entropies (TRE) define a bona fide quantum coherence quantifier <cit.>. Furthermore, TRE satisfies a class of bounds derived from Pinsker and Fannes-type inequalities <cit.>.
  
  Here we consider the fundamental pro­blem of boun­ding the chan­ge in the generalized relative entropies under an arbitrary unitary process. Specifically, we derive an upper bound on both asymmetric and symmetric versions of RRE and TRE, between the initial and final states. As an application of this result, we show that this upper bound implies an entirely family of quantum speed limits (QSLs). 
  
  The importance of our results is twofold. First, it establishes a bound on entropic quantifiers that are employed in distinct fields, from quantum communication to bio­logy <cit.>. In general, since the computation of relative entropies are usually difficult, our main result can directly be applied in all of these fields by providing bounds on central quantities. Second, our family of QSLs provide nonasymptotic bounds on the time evolution of quantum systems in the sense of the so-called single-shot information theory <cit.>. Furthermore, due to the broadness application of RRE, it provides a bridge among the speed of quantum evolution, thermodynamics <cit.> and quantum resources, e.g., entanglement, coherence and asymmetry <cit.>. Importantly, since our results apply to TRE, it also provides a nonextensive version of the QSL, which can found several applications, both on fundamental and practical aspects <cit.>.
  
  The paper is organized as follows. In Sec. <ref> we briefly review the main properties of Rényi and Tsallis relative entropies, which in turn can be recasted in terms of a generalized entropy. In Sec. <ref>, we introduce the physical setting and present an upper bound on those generalized entropies. Next, in Sec. <ref> we discuss an entire family of QSLs derived from the referred upper bound on Rényi and Tsallis relative entropies. In Sec. <ref> we illus­tra­te our findings via the prototypical case of the single-qubit state, thus presenting analytical results for the family of QSLs, and also discussing the tightness of the main bound on generalized entropies. Finally, in Sec. <ref> we close the paper discussing these results and comment on possible applications.
  
  
  
  
  
  
  
  
  
  
  § GENERALIZED RELATIVE ENTROPIES
  
  
  
  Let us start by de­fi­ning our physical system, which is described by a finite-dimensional Hilbert space ℋ, with d = dim(ℋ). In ge­ne­ral, the state of the system will be given by a density matrix ρ∈Ω, where Ω = {ρ∈ℋ|ρ^† = ρ, ρ≥ 0, Tr(ρ) = 1} defines the convex space of density operators. In this setting, given two states ρ, ω∈Ω, the Rényi (RRE) and Tsallis (TRE) relative entropies are defined, respectively, as <cit.>
  
      R_α(ρω) = 1/α - 1ln[g_α(ρ,ω) ]  ,
  
  and
  
      H_α(ρω) = 1/1 - α[ 1 - g_α(ρ,ω)]  ,
  
  where g_α(ρ,ω) = Tr(ρ^αω^1 - α) is the α-relative purity, also called Petz-Rényi relative quasi-entropy <cit.>, with the parameter α∈ (0,1)∪(1,+∞) labeling the fa­mi­ly of quantum relative entropies <cit.>. Equation (<ref>) is also called Petz-Rényi relative entropy <cit.>, standing as the first quantum extension of the classical RRE. Indeed, due to the noncommutativity of quantum states, the nonuniqueness of quantum information-theoretic quantifiers has triggered the search for a plethora of quantum entropies, e.g., sandwiched Rényi relative entropy <cit.>, and α-z-relative Rényi entropy <cit.>.
  
  Importantly, relative purity satisfies the property g_α(ω,ρ) = g_1 - α(ρ,ω), i.e., it is skew symmetric with respect to α. In particular, when ρ = ω we have g_α(ρ,ρ) = 1 for all α, and thus one gets R_α(ρρ) = H_α(ρρ) = 0. Noteworthy, for α = 1/2 one recovers the so-called quantum affinity, which is related to Hellinger angle <cit.>. In turn, Hellinger angle is associated to Wigner-Yanase skew information me­tric and cha­rac­te­rizes the length of the geodesic path connecting states ρ, ω∈Ω <cit.>.
  
  In the following we summarize the main properties of RRE and TRE. A more complete presentation can be found in Ref. <cit.>. Starting with RRE, the limit α→ 1 recovers the well-known quantum relative entropy R_1(ρω) = S(ρω) := Tr(ρlnρ - ρlnω). For α = 0, RRE reduces to the min-relative entropy R_0(ρω) = -lnTr(Π_ρ ω), with Π_ρ being the projector onto the support of the state ρ <cit.>. Noteworthy, for 0 ≤α≤ 2, RRE satisfies the data-processing ine­qua­li­ty, i.e., R_α(Λ(ρ)Λ(ω)) ≤R_α(ρω), thus being monotonic under any completely positive and trace pre­ser­ving map Λ(∙) <cit.>. This is a fundamental inequality not only within information theory, but also for physics (see, for instance, Ref. <cit.> where the second law of thermodynamics is obtained from such inequality). Moving to Tsallis relative entropy, it has been shown that, for 0 ≤α < 1, TRE is (i) nonnegative, i.e., H_α (ρω) ≥ 0 for all ρ,ω∈Ω, with the equality holding if and only if ρ = ω; (ii) jointly convex; (iii) nonaditive; and (iv) contractive under completely positive and trace preserving maps <cit.>. Importantly, TRE also recovers the standard quantum relative entropy in the limit α→ 1, i.e., H_1(ρω) = S(ρω).
  
  We shall stress that RRE and TRE are asymmetric with respect to states ρ,σ∈Ω. However, a bona fide distance measure within the information geometry theory is usually symmetric. For instance, the so-called quantum Jensen-Shannon divergence, i.e., the square-root of symmetrized quantum relative entropy, was proved to be a metric on the space of density matrices <cit.>. For the case at hand, the aforementioned entropies can be symmetrized as
  
      𝒪_α(ρ:ω) := 𝒪_α(ρω) + 𝒪_α(ωρ)  ,
  
  where index 𝒪≡{R , H} labels RRE and TRE, respectively. We are now ready to present our main result.
  
  
  
  
  
  
  
  
  
  
  
  § BOUNDS ON GENERALIZED RELATIVE ENTROPIES
  
  
  
  The dynamics of our system is governed by a time-dependent Hamiltonian H_t ∈ℬ(ℋ), with ℬ(ℋ) being the set of bounded operators acting on ℋ. In general, the Hamiltonian H_t is not self-commuting at different times, i.e., [H_s,H_t] ≠ 0 for s ≠ t. The initial state ρ_0 ∈Ω undergoes the unitary evolution ρ_t = U_t  ρ_0 U^†_t, for t ∈ [0,τ], where U_t = 𝒯e^- i ∫_0^t  d s H_s is the time-ordered unitary evolution operator satisfying the equation -i(dU_t/dt) = H_tU_t. From now on, we will work in natural units, ħ = k_B = 1.
  
  Based on this physical setting, our goal is to provide a class of nontrivial upper bounds for RRE and TRE. In Appendix <ref> we have proved that, for α∈(0,1), RRE and TRE satisfy the ine­qua­lity
  
      𝒪_α(ρ_τρ_0) ≤τ ⟨⟨𝒢^𝒪_α(t) ⟩⟩_τ/|1 - α| ,
  
  where ⟨⟨∙⟩⟩_τ = τ^-1 ∫_0^τ∙  d t stands for the time average, and
  
      𝒢^𝒪_α(t) :=  Φ_α^𝒪 ρ_0^1 - α_2  [ H_t, ρ_0^α ]  _2 ,
  
  with A _2 = √(Tr  (A^†A)) being the Schatten 2-norm. Here Φ_α^𝒪 is an auxiliary function which reads 
  
      Φ_α^𝒪 = 
      |1 + (1 - α) ln(λ_min(ρ_0)) |^-1 ,   
      
      1  ,    ,
  
  where λ_min(ρ_0) sets the smallest eigenvalue of the input state ρ_0. Noteworthy, Eq. (<ref>) is the first main result of this article. Remarkably, the bound mostly depends on ρ_0 and H_t. Naturally, a similar bound can be obtained for the case in which the arrangement of states ρ_0 and ρ_τ in Eq. (<ref>) is swapped, which is given by (see Appendix <ref>)
  
      𝒪_α(ρ_0ρ_τ) ≤τ ⟨⟨𝒢^𝒪_1 - α(t) ⟩⟩_τ/|1 - α| .
  
  Furthermore, the corres­pon­ding ine­qua­lity for symmetrized forms of RRE and TRE is then obtained, roughly speaking, by com­bi­ning the two nonsymmetric upper bounds, and reads (see details in Appendix <ref>)
  
      𝒪_α(ρ_τ : ρ_0) ≤τ ⟨⟨𝒢^𝒪_α(t) + 𝒢^𝒪_1 - α(t) ⟩⟩_τ/|1 - α| .
  
  Remarkably, such bounds do not depend on the time-ordered evolution operator U_t, neither on the evolved state of the system. Thus, for states ρ_0 and ρ_τ of a given closed quantum system, our results provide a route to estimate both RRE and TRE entropies which will depend mostly on the spectral properties of the initial state ρ_0 and the driving Hamiltonian.
  
  Importantly, Eq. (<ref>) can be recasted in terms of the Fröbenius norm of the initial state of the system, while being a function of the time average of the Schatten 2-norm of the Hamiltonian. To see this, we first point out that for two arbitrary complex matrices X and Y, it has been proved the Schatten 2-norm fulfills the ine­qua­lity [X,Y]_2≤√(2) X_2Y_2 <cit.>. Hence, from Eq. (<ref>) one readily obtains the upper bound 𝒢^𝒪_α(t) ≤√(2) ρ_0^1 - α_2 ρ_0^α_2 H_t_2. Therefore, the bound in Eq. (<ref>) can be recasted as
  
      𝒪_α(ρ_τρ_0) ≤√(2) τ Φ_α^𝒪 ρ_0^1 - α_2 ρ_0^α_2 ⟨⟨ H_t_2⟩⟩_τ/|1 - α| .
  
  In particular, note that ⟨⟨ H_t_2⟩⟩_τ =  H _2 for the case in which the Hamiltonian is time independent, i.e., H_t ≡ H. Overall, the bound in Eq. (<ref>) does requires minimal information about the system, e.g., its initial state ρ_0 and the e­ner­gy levels of the Hamiltonian H_t. Indeed, the latter comes from the fact that, given the spectral decomposition H_t = ∑_j = 1^d ϵ_j(t)|ϕ_j(t)⟩⟨ϕ_j(t)|, with d = ℋ the dimension of the Hilbert space, where the eigenvalues {ϵ_j(t) }_j = 1,…,d and eigenstates { |ϕ_j(t)⟩}_j = 1,…,d are time dependent, and thus one obtains H_t_2^2 = ∑_j = 1^d ϵ_j(t)^2.
  
  Finally, regardless of the simplicity and usefulness of the bound in Eq. (<ref>), we shall point out the original bound in Eq. (<ref>) might stand as the general and tighter one. Before discussing the physical significance of this bound, we will make use of it to obtain a family of QSLs.
  
  
  
  
  
  
  
  
  
  
  § QUANTUM SPEED LIMITS
  
  
  
  The quantum speed limit (QSL) signals the minimum time of evolution between two quantum states undergoing an arbitrary dynamics. Indeed, Mandelstam and Tamm addressed this question around 75 years ago for closed quantum systems, thus showing the QSL time for orthogonal states is given by τ_QSL = ħπ/(2ΔE), where (ΔE)^2 = ⟨ψ_0|H^2|ψ_0⟩ - ⟨ψ_0|H|ψ_0⟩^2 stands for the variance of the Hamiltonian with respect to the initial state |ψ_0⟩ of the system <cit.>. Later on, Margolus and Levitin derived the QSL time τ_QSL = ħπ/(2⟨ψ_0|H|ψ_0⟩) for the same physical se­tting, i.e., closed quantum systems evolving between orthogonal states <cit.>. Importantly, both lower bounds can be combined accordingly onto a tighter one as τ≥max{ħπ/(2ΔE), ħπ/(2⟨ψ_0|H|ψ_0⟩) } <cit.>. In the last decade, several bounds were introduced in the literature covering the QSL time for different physical settings, e.g., addressing pure and mixed states, for closed and open quantum systems <cit.>.
  
  Here we will present a family of QSLs by time a­ve­ra­ging the right-hand side of Eq. (<ref>), thus followed by a rearrangement of the resulting inequality. Indeed, the time τ required for an arbitrary unitary evolution driving a closed quantum system from ρ_0 to ρ_τ is lower bounded as 
  
      τ≥τ_α^𝒪 := max{τ^𝒪_α(ρ_τρ_0), τ^𝒪_α (ρ_0 ρ_τ), τ^𝒪_α (ρ_0:ρ_τ)} ,
  
  where
  
      τ^𝒪_α(ρ_τρ_0) := | 1 - α |  𝒪_α(ρ_τρ_0)/⟨⟨𝒢^𝒪_α(t) ⟩⟩_τ ,
  
  and
  
      τ^𝒪_α(ρ_0ρ_τ) := | 1 - α |  𝒪_α(ρ_0ρ_τ)/⟨⟨𝒢^𝒪_1 - α(t) ⟩⟩_τ ,
  
  while the QSL time due to symmetrized relative entropies reads [see Appendix <ref>]
  
      τ_α^𝒪(ρ_0:ρ_τ):= | 1 - α |  𝒪_α(ρ_0 : ρ_τ)/⟨⟨ 𝒢^𝒪_α(t) + 𝒢^𝒪_1 - α(t)  ⟩⟩_τ .
  
  
  Equation (<ref>) stands as our second main result, thus establishing a family of entropic QSLs, i.e., RRE and TRE provide lower bounds on the time of evolution between the initial and final states of the quantum system. Recently, a related family of QSLs, based on the relative entropy, were derived bounding the time it takes to generate or consume a given quantum resource such as entanglement, asymmetry, and athermality <cit.>. These bounds, dubbed as resource speed limits (RSL) were shown to be tighter than QSLs in several instances. However, as RSLs are constructed using the standard relative entropy (α→ 1), thus being only mea­ning­ful in the asymptotic limit. RSLs for single shot scenarios requires working with Rényi relative entropies. Here, we have taken the first step in this direction.
  
  We shall stress that the previous discussion is valid for α∈ (0,1). In the following, we will discuss the li­mi­ting cases α→ 1 and α→ 0, which crucially reduce to the standard relative entropy and the so-called min-relative entropy, respectively. Importantly, these results cannot be simply obtained from the results above. While the case α→ 1 is clearly delicate from the definition of RRE and TRE given in Eqs. (<ref>) and (<ref>), respectively, the case α→ 0 must be ca­re­fully con­si­de­red sin­ce sim­ply ta­king α=0 in Eq. (<ref>) would provide us a trivial bound, independently of the initial state and the dynamics.
  
  
  
  
  
  
  
   §.§ Limiting case of α→ 1
  
  
  Let us start by considering the limit α→ 1, in which both RRE and TRE recover the quantum relative entropy, i.e., lim_α→ 1𝒪_α(ϱω) = S(ϱω). In this case, it can be proved that the following upper bound applies (see details in Appendix <ref>)
  
      S(ρ_τρ_0) ≤τ lnρ_0_2 ⟨⟨  [ H_t , ρ_t ] _2⟩⟩_τ .
  
  In this case, the corresponding QSL family reads 
  
      τ≥τ_1^RE := max{τ_1^RE(ρ_τρ_0), τ_1^RE(ρ_0 ρ_τ), τ_1^RE(ρ_0 : ρ_τ) }
  
  where
  
      τ^RE_1(AB):= S(A B )/lnρ_0_2 ⟨⟨  [ H_t , A ] _2⟩⟩_τ ,
  
  and
  
      τ^RE_1(ρ_0 : ρ_τ) := S(ρ_τρ_0) + S(ρ_0ρ_τ)/lnρ_0_2 ⟨⟨  [ H_t , ρ_t ] _2 +  [ H_t , ρ_0 ] _2⟩⟩_τ .
  
  
  In particular, when the Hamiltonian is time-independent, i.e., H_t≡ H, one obtains [ H , ρ_t ] _2 =  [ H , ρ_0 ] _2 = 2 √(ℐ_L(ρ_0, H)), where we have used the fact that Scha­tten 2-norm is unitarily invariant. Here ℐ_L(ρ_0, H) = (1/4)  [ H , ρ_0 ] _2^2 = - (1/4)  Tr([ρ_0,H]^2) define a time-independent quantum coherence quantifier which sets a lower bound on Wigner-Yanase skew information <cit.>. Now, since ℐ_L(ρ_0,H) ≤ (Δ H)^2, where (Δ H)^2 = Tr(ρ_0H^2) - [Tr(ρ_0H)]^2 is the squared deviation of the Hamiltonian, thus Eq. (<ref>) implies the lower bounds τ_1^RE(ρ_0ρ_τ) ≥S(ρ_0ρ_τ ) / ( 2  ΔH lnρ_0_2), and τ_1^RE(ρ_τρ_0) ≥S(ρ_τρ_0 ) / ( 2  ΔH lnρ_0_2).
  
  
  
  
  
  
  
   §.§ Limiting case of α→ 0
  
  
  Considering now the case α→ 0, in Appendix <ref> we have proved that Rényi min-relative entropy is upper bounded as
  
      |R_0(ρ_τρ_0)| ≤τ ⟨⟨𝒬_0^t(ρ_0,Π_ρ_0) ⟩⟩_τ ,
  
  with 
  
      𝒬_0^t(A,B) := A_2  [U_t^†H_tU_t,B] _2/ |Tr(AU_tBU_t^†) | .
  
  Here Π_ρ_0 is the projector onto the support of the initial state ρ_0. From Eq. (<ref>), we can derive the QSL time as 
  
      τ≥τ_0^R := max{τ_0^R(ρ_τρ_0), τ_0^R(ρ_0 ρ_τ), τ_0^R(ρ_0 : ρ_τ) } ,
  
  where 
  
      τ_0^R(ρ_τρ_0) := |R_0(ρ_τρ_0)|/⟨⟨𝒬_0^t(ρ_0,Π_ρ_0) ⟩⟩_τ ,
  
  and
  
      τ_0^R(ρ_0ρ_τ) := |R_0(ρ_0ρ_τ)|/⟨⟨𝒬_0^t(Π_ρ_0, ρ_0) ⟩⟩_τ ,
  
  while the QSL related to the symmetric min-entropy is given by
  
      τ_0^R(ρ_0 : ρ_τ) := |R_0(ρ_τρ_0) + R_0(ρ_0ρ_τ)|/⟨⟨𝒬_0^t(ρ_0,Π_ρ_0) + 𝒬_0^t(Π_ρ_0 , ρ_0) ⟩⟩_τ .
  
  
  Noteworthy, the “speed” contribution 𝒬_0^t is closely related to the QSL derived with respect to Euclidean distance in the Bloch sphere <cit.>. Importantly, when the density matrix ρ_0 has full-rank, i.e., dim(ρ_0) = supp(ρ_0), thus Rényi min-relative entropy vanishes and implies that τ_R^(0) = 0. To see this, let ρ_0 = ∑_ℓ = 1^d p_ℓ |ψ_ℓ⟩⟨ψ_ℓ| be the spectral decomposition of the input state, with d = dim(ℋ). Note the projector Π_ρ_0 onto the support of the full-rank state ρ_0 is equal to the identity, Π_ρ_0 = ∑_ℓ : p_ℓ≠ 0  |ψ_ℓ⟩⟨ψ_ℓ| =  𝕀. Hence, it is straightforward to verify the symmetric min-entropy is identically zero because R_0(ρ_τρ_0) = - ln[Tr(U_τ^†ρ_0U_τ)] = 0 and R_0(ρ_0ρ_τ) = - ln[Tr(U_τρ_0U_τ^†)] = 0. Furthermore, from Eq. (<ref>) one readily gets 𝒬_0^t(ρ_0,Π_ρ_0) = 0, while the functional 𝒬_0^t(Π_ρ_0,ρ_0) = √(d)  [U_t^†H_tU_t,ρ_0] _2 remains finite for t ∈ [0,τ].
  
  
  
  
  
  
  
  
  
  
  § EXAMPLE
  
  
  
  We now provide an analytical example in order to make clearer the physical implications of our results. Let us consider a single-qubit state, whose Bloch sphere representation is written as ρ_0 = (1/2)(𝕀 + r⃗·σ⃗), where σ⃗ = (σ_x,σ_y,σ_z) is the vector of Pauli matrices, r⃗ = r r̂ is the Bloch vector, with r̂ = {sinθcosϕ,sinθsinϕ,cosθ}, 0 < r < 1, θ∈ [0,π] and ϕ∈ [0,2π], while 𝕀 is the 2× 2 identity matrix. Particularly, for 0 < α < 1, the operator ρ_0^α respective to the initial single-qubit state is written as
  
      ρ_0^α = 1/2[ ξ_α^+ 𝕀 + ξ_α^-(r̂·σ⃗) ]  ,
  
  where
  
      ξ_α^± = 2^-α[(1 + r)^α±(1 - r)^α]  .
  
  
  
  
  
  
  
  
  
  The dynamics of the system is governed by the time-dependent Hamiltonian H_t = ϖ 𝕀 + n̂_t·σ⃗, where n̂_t = { n_t^x,n_t^y,n_t^z} is a time-dependent unit vector, |n̂_t| = 1, and ϖ∈ℝ. In this case, the time ordered evolution operator becomes U_t =  e^-itϖ[cos(|u⃗_t|) 𝕀 - i sin(|u⃗_t|)  (û_t·σ⃗)], where û_t = u⃗_t/|u⃗_t| is a unit vector, with u⃗_t := ∫_0^t  ds n̂_s. In particular, if H_t≡ H is time independent, i.e., n̂_t = n̂ is a constant unit vector, we directly obtain û_t = n̂. Next, by performing a lengthy but straightforward calculation, one may verify the evolved single-qubit state becomes
  
      ρ_t^α = U_t ρ_0^αU_t^† = 1/2[ξ_α^+ 𝕀 + ξ_α^- (ν̂_t·σ⃗) ]  ,
  
  with the unit vector ν̂_t := r̂   +  sin(2 |u⃗_t| )   (û_t×r̂) + 2 sin^2(|u⃗_t| )[  (û_t·r̂)û_t - r̂]. Hence, by considering the range 0 < α < 1, from Eqs. (<ref>) and (<ref>) we thus have that α-relative purity reads
  
      g_α(ρ_t,ρ_0) = 1 - ξ_α^-ξ_1 - α^-(1 - (û_t·r̂)^2) sin^2(|u⃗_t| )  .
  
  
  Interestingly, since relative purity is skew symmetric over the index α, Eq. (<ref>) implies that g_α(ρ_t,ρ_0) = g_1 - α(ρ_t,ρ_0) = g_α(ρ_0,ρ_t). In turn, both RRE and TRE will satisfy the constraint 𝒪_α(ρ_tρ_0) = 𝒪_α(ρ_0ρ_t) for single-qubit states. Furthermore, from Eq. (<ref>), note that α-relative purity is equal to 1 for |u⃗_t| = nπ, with n∈ℤ and t∈[0,τ]. Furthermore, g_α(ρ_t,ρ_0) = 1 if vectors û_t and r̂ are parallel. Conversely, α-relative purity becomes g_α(ρ_t,ρ_0) = 1 - ξ_α^-ξ_1 - α^-sin^2(|u⃗_t|) if vectors û_t and r̂ are orthogonal. In Table <ref> we summarize the quantities required to evaluate the QSL bounds τ_α^𝒪, τ^RE_1, and τ^R_0, for the case of a single-qubit state.
  
  For simplicity, let us focus on a Hamiltonian with ϖ = 0 and n̂_t = γ^-1{Δ,0,vt }, with γ := √(Δ^2 + (vt)^2), where v stands as a “level velocity” of the energies of the system, and Δ is the level splitting <cit.>. Figure <ref> shows the QSL τ_α^R and τ_α^H as function of time τ and the parameter α, for the initial single-qubit state with {r,θ,ϕ} = {1/4,π/4,π/4}, also setting the ratio Δ/v = 0.5. In Appendix <ref> we provide a complementary numerical study, thus exploiting in details the qualitative behavior depicted in Fig. <ref> for RRE and TRE.
  
  
  
  
  
  
  
  
  Finally, we will investigate the tightness of our bounds for nonsymmetric and symmetric relative entropies. Overall, the tightness of the bounds is related with the tightness of the QSL. To see this, we introduce the following figures of merit for nonsymmetric RRE and TRE
  
      δ^𝒪_1(α,τ) := τ ⟨⟨ 𝒢^𝒪_α(t) ⟩⟩_τ - |1 - α|𝒪_α(ρ_τρ_0)  ,
  
  and 
  
      δ^𝒪_2(α,τ) := τ ⟨⟨ 𝒢^𝒪_1 - α(t) ⟩⟩_τ - |1 - α|𝒪_α(ρ_0ρ_τ)  ,
  
  while for the symmetrized relative entropies one gets
  
      δ^𝒪_3(α,τ) := δ^𝒪_1(α,τ) + δ^𝒪_2(α,τ)  ,
  
  with 𝒪≡{R , H}. We note that Eqs. (<ref>), (<ref>), and (<ref>) will quantify how much the bounds deviate from the actual value of RRE and TRE entropies, for both nonsymmetric and symmetric cases. Hereafter, we will set the initial single-qubit state parameterized as {r,θ,ϕ} = {1/4,π/4,π/4}, and also fixing the ratio Δ/v = 0.5. Figure <ref> shows the plot of the normalized quantity δ^𝒪_l(α,τ) = δ^𝒪_l(α,τ)/max[δ^𝒪_l(α,τ)], with l = {1,2,3}. On the one hand, Figs. <ref>(a) and <ref>(b) show that, for 0 ≤α≤ 1, both quantities δ^R_1(α,τ) and δ^R_2(α,τ) approaches zero for short times (0 ≲τ≲ 2). On the other hand, Figs. <ref>(d) and <ref>(e) indicate that, as the time increases, the figure of merit δ^H_1(α,τ) will remain close to zero as long as α≲ 0.4, while δ^H_2(α,τ) approaches a small value for α≳ 0.4 during the range 0≲τ≲ 4. Finally, Figs. <ref>(c) and <ref>(f) suggest that, for the chosen initial parameters, both quantities δ^R_3(α,τ) and δ^H_3(α,τ) behave similarly, being slightly tight in the time window 0 ≲τ≲ 2, for 0 ≤α≤ 1. Nonetheless, we stress that a more ge­ne­ral analysis requires varying those parameters to include a larger class of initial states and Hamiltonians. Of course, this subject should deserve further investigation, also including a detailed study of Eqs. (<ref>), (<ref>) and (<ref>), as well as the family of QSLs bounds presented in Sec. <ref>, for higher-dimensional systems.
  
  
  
  
  
  
  
  
  
  
  § DISCUSSION
  
  
  
  The main contribution of this paper is to provide an upper bound on ge­ne­ra­li­zed entropies when the physical system undergoes a unitary transformation. As the first application of our bound, we derived a family of QSLs in Eqs. (<ref>), (<ref>), and (<ref>). From this result, the minimum time required for the state transformation is inversely proportional to a quantity that involves the average energy of the system [see Eq. (<ref>)]. In turn, this determines the evolution speed, while being directly proportional to 𝒪_α, and thus implying the entropies play the role of distances. Furthermore, our derivations of QSLs, based on RRE, is first step toward resource speed limits quantifying consumption of resource in single shot scenarios <cit.>.
  
  Another interesting connection can be built based on asymmetry monotones (AMs), which in turn characterize conservation laws for general quantum systems, in the sense of Noether's theorem <cit.>. In short, AM is a function f  : ℬ(ℋ) →ℝ that quantifies how much the state of the system breaks a given symmetry. Mathematically, the action of a symmetry group G on a quantum system is described by the operation 𝒰_g(ρ) = U_g ρ U_g^†, where the variable g labels the group elements. The key idea behind AMs is to recog­nize the orbit of each quantum state as an en­co­ding process, while the map 𝒰_g(ρ_0) →𝒰_g(ρ_τ) is viewed as data processing. This implies that we can employ any contractive information measure to characterize the orbit of each state, which leads to an AM satisfying f(𝒰_g(ρ)) ≤ f(ρ) <cit.>.
  
  Considering the range of α in which RRE and TRE are contractive under the action of a completely-positive and trace-preserving map, we immediately see the symmetrized relative entropies in Eq. (<ref>) (as well as its nonsymmetric versions) define an entire family of AMs. Indeed, the standard relative entropy (α→ 1) was previously considered as an asymmetry monotone <cit.>. 
  
  Next, since Eqs. (<ref>), (<ref>), and (<ref>) are valid for unitary transformations encoding any unknown parameter into the state of the system, i.e., it goes beyond the paradigmatic case of time evolutions, thus we can replace t by the group variable g. Indeed, this is due to the fact the unitary representation of the symmetry group leads to the evolution equation dρ_g/d g = -i[K,ρ_g], with K being the generator of the transformation. Therefore, our results provide upper bounds on how much the state breaks the symmetry generated by K. This sets upper bounds of how much the conservation of the associated physical quantity can be broken. In the specific case of QSLs, this implies the minimum evolution time is determined by the asymmetry measure respective to nonsymmetric and symmetric RRE and TRE, which in turn stands as a measure of how much the initial state breaks the time-translation symmetry [see Eqs. (<ref>), (<ref>), and (<ref>)].
  
  Finally, we will discuss the relation between our results and the concept of nonequilibrium entropy production. We begin by setting the initial state of the system as a thermal one, ρ_0 = ρ_β = exp-β H_0/𝒵, where 𝒵 is the partition function, and H_0 is the “bare” Hamiltonian of the system, i.e., [H_0,H_t] ≠ 0 for all t ≠ 0. From Eq. (<ref>), one may verify the lower bound on the time of evolution is proportional to S(ρ_τρ_β), which in turn stands as the entropy production associated with the process under consideration. Therefore, a natural question arises about the extension of this connection to ge­ne­ral entropies and systems. Indeed, such a general picture could be addressed by exploiting the entire family of second laws of thermodynamics based on RRE <cit.>. This may open an avenue for the comprehension of QSLs <cit.>, asymmetry monotones <cit.>, and quantum ther­mo­dy­na­mics <cit.>, based on a strictly geometric framework.
  
  The results presented here raise another questions. First, we could consider the extension of our results to open quantum systems. Moreover, given the link between quantum co­he­ren­ce and TRE <cit.>, we could investigate the trade-off among entropy production, QSL and quantum coherence in this scenario. Furthermore, since our results also apply for the min-entropy, i.e., the limit α→ 0 regarding to RRE, they can be employed in the single-shot information theo­ry, where the relations among asymmetry, QSL, and thermodynamics can be further de­ve­lo­ped into the nonasymptotic regime. 
  
  
  
  
  
  
  D. P. P. and L. C. C. would like to acknowledge the financial support from the Brazilian ministries MEC and MCTIC, funding agencies CAPES and CNPq, and the Bra­zi­lian National Institute of Science and Technology of Quantum Information (INCT-IQ). This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior, Brasil (CAPES), Finance Code 001. L. C. C. also acknowledges support from Spanish MCIU/AEI/FEDER (Grant No. PGC2018-095113-BI00), Basque Government IT986-16, Projects No. QMiCS (820505) and No. OpenSuperQ (820363) of the EU Flagship on Quantum Technologies, EU FET Open Grant Quromorphic (828826), the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research (ASCR) quantum algorithm teams program, under field work proposal number ERKJ333, and the Shanghai STCSM (Grant No. 2019SHZDZX01-ZX04).
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
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  gobbleappendixname#1thesubsectionsection.subsection
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  § APPENDIX
  
  
  
  
  
  
  
  
  
  
  
  
  
  § BOUND ON RELATIVE ENTROPIES
  
  
  
  In this Appendix we will present in details the derivation of the results discussed in Sec. <ref>.
  
  
  
   §.§ Bounding α-relative entropy
  
  
  
  Here we will derive a nontrivial lower bound on the α-relative purity, which will be useful throughout this supplemental material. Let ρ_1,ρ_2∈Ω be two arbitrary density matrices, with Ω⊂ℋ, where Ω = {ρ∈ℋ|ρ^† = ρ, ρ≥ 0, Tr(ρ) = 1} sets the convex space of quantum states, while ℋ is a d-dimensional Hilbert space, with d = dim(ℋ). The Tsallis relative entropy (TRE) is defined as H_α(ρ_1ρ_2) = (1 - α)^-1[ 1 - g_α(ρ_1,ρ_2) ] where g_α(ρ_1,ρ_2) = Tr(ρ_1^αρ_2^1 - α) stands for the α-relative purity <cit.>. In particular, for 0 ≤α≤ 1, it has been proved that TRE is upper bounded as follows <cit.>
  
      H_α(ρ_1ρ_2) ≤ S(ρ_1ρ_2)  ,
  
  where S(ρ_1ρ_2) = Tr[ρ_1(lnρ_1 - lnρ_2)] is the “standard” quantum relative entropy. Interestingly, from Eq. (<ref>), we readily derive the lower bound on α-relative purity,
  
      g_α(ρ_1,ρ_2) ≥ 1 - (1 - α)S(ρ_1ρ_2)  .
  
  Noteworthy, bound in Eq. (<ref>) exhibits two important features. On the one hand, for α = 1, it trivially saturates since g_1(ρ_1,ρ_2) = 1 for all states ρ_1,ρ_2∈Ω. On the other hand, for α = 0 one recovers the Klein's inequality, S(ρ_1ρ_2) ≥ 0, due to the fact that g_0(ρ_1,ρ_2) = 1 for all ρ_1,ρ_2∈Ω, thus stating the quantum relative entropy is nonnegative. Next, quantum relative entropy satisfies the following lower bound <cit.>
  
      S(ρ_1ρ_2) ≤ - ln(λ_min(ρ_2))  ,
  
  where λ_min(∙) sets the minimum eigenvalue of the referred density matrix. Importantly, authors in Refs. <cit.> have derived a plethora of lower and upper bounds on S(ρ_1ρ_2), which in turn depend on some distance measures as the operator norm, Schatten 1-norm, and Fröbenius norm. However, we shall stress the bound in Eq. (<ref>) is more suitable for our purposes, mostly because it stands as one of the simplest nontrivial lower bounds on quantum relative entropy. Therefore, by combining Eqs. (<ref>) and (<ref>), we thus have that
  
      g_α(ρ_1,ρ_2) ≥ 1 + (1 - α) ln(λ_min(ρ_2))   ,
  
  which implies the following upper bound on α-relative purity:
  
      [g_α(ρ_1,ρ_2)]^-1≤[1 + (1 - α) ln(λ_min(ρ_2))]^-1 .
  
  
  
  
  
  
  
  
  
   §.§ Bound on non-symmetric relative entropies
  
  
  
  Here we will address the upper bound for nonsymmetric quantum relative entropies. From now on, we will focus on both Rényi, R_α(ρ_tρ_0), and Tsallis, H_α(ρ_tρ_0), relative entropies, where ρ_0 is the initial state of the system, and ρ_t = U_tρ_0U_t^† its respective evolved state, with U_t = 𝒯e^-i ∫_0^t  ds  H_s being the unitary time-ordered evolution operator. The absolute value of the time derivative of Rényi relative entropy (RRE) of states ρ_0 and ρ_t is given by
  
      |d/dtR_α(ρ_tρ_0)| = 1/|1 - α| g_α(ρ_t,ρ_0)| d/dtg_α(ρ_t,ρ_0) |  .
  
  In Appendix <ref> we have derived an upper bound on the inverse of the α-relative purity, which in turn will exhbit a logarithmic dependence on the smallest eigenvalue of the initial state of the system. Indeed, by substituting Eq. (<ref>) into (<ref>), one obtains the following inequality
  
      |d/dtR_α(ρ_tρ_0)| ≤| d/dtg_α(ρ_t,ρ_0) |/|1 - α| |1 + (1 - α) ln(λ_min(ρ_0)) |  .
  
  For completeness, the absolute value of the time derivative of Tsallis relative entropy (TRE) of states ρ_0 and ρ_t is given by
  
      |d/dtH_α(ρ_tρ_0)| = 1/|1 - α|| d/dtg_α(ρ_t,ρ_0) |  .
  
  
  Next, the time derivative of relative purity g_α(ρ_t,ρ_0) = Tr(ρ_t^α ρ_0^1 - α) is evaluated as follows. Because ρ_t evolves unitarily, it is possible to verify the operator ρ_t^α = U_t ρ_0^α U_t^† satisfies the von Neumann equation dρ_t^α/dt = - i [H_t,ρ_t^α], where we used the identity U_t(dU_t^†/dt) = - (dU_t/dt)U_t^† = - i H_t. Hence, we thus have that
  
      d/dtg_α(ρ_t,ρ_0) = -i Tr(ρ_t^α  [ρ_0^1 - α,H_t])  ,
  
  where we have used the cyclic property of trace. By taking the absolute value of Eq. (<ref>), and applying the Cauchy-Schwarz inequality, |Tr(𝒜_1𝒜_2)| ≤𝒜_1_2𝒜_2_2, with 𝒜_2 = √(Tr(𝒜^†𝒜)), one readily gets
  
      | d/dtg_α(ρ_t,ρ_0) | ≤ρ_0^α_2  [ H_t, ρ_0^1 - α ] _2 ,
  
  where we have used that ρ_t^α_2 = U_tρ_0^αU_t^†_2 = ρ_0^α_2, i.e., Schatten 2-norm is unitarily invariant. Hence, by substituting Eq. (<ref>) into Eqs. (<ref>) and (<ref>), one finds the generalized upper bound 
  
      |d/dt𝒪_α(ρ_tρ_0)| ≤|1 - α|^-1𝒢^𝒪_α(t)  ,
  
  where we define the functional
  
      𝒢^𝒪_α(t) := Φ_α^𝒪 ρ_0^α_2  [ H_t, ρ_0^1 - α ] _2 ,
  
  while the auxiliary function reads
  
      Φ_α^𝒪 = 
      |1 + (1 - α) ln(λ_min(ρ_0)) |^-1 ,   
      
      1  ,    .
  
  Just to clarify, in the remainder of the paper the index 𝒪≡{R,H} will label RRE and TRE, respectively. Finally, by integrating Eq. (<ref>) over the interval 0 ≤ t ≤τ, we thus obtain the upper bound
  
      𝒪_α(ρ_τρ_0) ≤|1 - α|^-1 ∫_0^τ dt  𝒢^𝒪_α(t)  ,
  
  where we have invoked the inequality |∫ dx f(x)| ≤∫dx |f(x)|, and used the fact that both RRE and TRE are nonnegative, real-valued, information-theoretic measures. 
  
  Importantly, an analogous bound can be derived for the case in which the states ρ_0 and ρ_τ are swapped. We will briefly sketch the proof since the calculations are similar to the previous discussion. 
  The property g_α(ρ_0,ρ_t) = g_1 - α(ρ_t,ρ_0) for the relative purity implicates the skew symmetry 
  
      (1 - α) 𝒪_α(ρ_0ρ_t) = α 𝒪_1 - α(ρ_tρ_0)  ,
   
  which holds for all ρ_0,ρ_t and 0 < α < 1. In turn, the identity in Eq. (<ref>) allow us to write down the time-derivative 
  
      |d/dt𝒪_α(ρ_0ρ_t)| = α/|1 - α|| d/dt𝒪_1 - α(ρ_tρ_0) |  .
  
  Invoking Eq. (<ref>) and mapping α→ 1 - α, one obtains |d𝒪_1 - α(ρ_tρ_0)/dt | ≤α^-1𝒢^𝒪_1 - α(t). Hence, by inserting this expression into the right-hand side of Eq. (<ref>), we thus have that
  
      |d/dt𝒪_α(ρ_0ρ_t)| ≤|1 - α|^-1𝒢^𝒪_1 - α(t)  ,
  
  which implies the upper bound
  
      𝒪_α(ρ_0ρ_τ) ≤|1 - α|^-1 ∫_0^τ dt  𝒢^𝒪_1 - α(t)  .
  
  
  
  
  
  
  
  
  
  
   §.§ Bound on symmetric relative entropies
  
  
  
  Here we will address the upper bound for sym­me­tric quantum relative entropies. We shall begin with the symmetric relative entropy
  
      𝒪_α(ρ_t:ρ_0) := 𝒪_α(ρ_tρ_0) + 𝒪_α(ρ_0ρ_t)  ,
  
  where ρ_0 is the initial state of the system, while ρ_t = U_tρ_0U_t^† is the evolved state. Let |d 𝒪_α(ρ_t:ρ_0)/dt | be the absolute value of the time-derivative of Eq. (<ref>). Hence, by applying the triangle ine­qua­li­ty |a_1 + a_2| ≤ |a_1| + |a_2|, one obtains
  
      |d/dt𝒪_α(ρ_t : ρ_0)| ≤|d/dt𝒪_α(ρ_tρ_0)| + |d/dt𝒪_α(ρ_0ρ_t)|  .
  
  Based on Eqs. (<ref>) and (<ref>), it follows that
  
      |d/dt𝒪_α(ρ_t : ρ_0)| ≤|1 - α|^-1[ 𝒢^𝒪_α(t) + 𝒢^𝒪_1 - α(t)]  .
  
  Finally, by integrating Eq. (<ref>) over the interval t ∈ [0,τ], and using the fact that |∫ dx f(x)| ≤∫dx |f(x)|, one finds the upper bound for symmetric relative entropies
  
      |𝒪_α(ρ_τ : ρ_0)  | ≤|α - 1|^-1∫_0^τ dt [ 𝒢^𝒪_α(t) + 𝒢^𝒪_1 - α(t) ]  .
  
  
  
   
  
  
  
  
  
  
  
  
  
  
  § RECOVERING RELATIVE ENTROPY
  
  
  
  In this Appendix we present the details in the calculation of the limit α→ 1. The idea here is going back few steps and pinpoint the main features needed to properly address such nontrivial limit. For the two states ρ_0 and ρ_t = U_tρ_0U_t^†, we shall begin evaluating the limiting case α→ 1 of the time derivative of symmetric relative entropy defined in Eq. (<ref>), i.e.,
  
      lim_α→ 1 d/dt𝒪_α(ρ_t:ρ_0) = lim_α→ 1 d/dt𝒪_α(ρ_tρ_0) + lim_α→ 1 α/(1 - α)d/dt𝒪_1 - α(ρ_tρ_0)  ,
  
  where we have used the fact that 𝒪_α(ρ_0ρ_t) is skew symmetric [see Eq. (<ref>)]. Next, by taking the absolute value of Eq. (<ref>), then applying the triangle inequality |a_1 + a_2| ≤ |a_1| + |a_2|, and finally integrating the resulting expression over interval t ∈ [0,τ], we thus have that
  
  
  
  
  
  
      ∫_0^τ dt  | lim_α→ 1 d/dt𝒪_α(ρ_t:ρ_0)| ≤∫_0^τ dt  |lim_α→ 1 d/dt𝒪_α(ρ_tρ_0)| + ∫_0^τ dt  | lim_α→ 1 α/(1 - α)d/dt𝒪_1 - α(ρ_tρ_0)|  .
  
  Based on the definition of the relative entropy, S(AB) = Tr[A(lnA - lnB)], note that one may write down
  
      | S(ρ_τρ_0 ) + S(ρ_0ρ_τ ) | 
      = | lim_α→ 1 𝒪_α(ρ_τ : ρ_0)  | 
      =  |  ∫_0^τ dt  lim_α→ 1 d/dt𝒪_α(ρ_t:ρ_0)| 
      ≤∫_0^τ dt  | lim_α→ 1 d/dt𝒪_α(ρ_t : ρ_0)|  .
  
  Just to clarify, here we assume that RRE and TRE are continuous real-valued functions over the set α∈ (0,1)∪(1,+∞) and t ∈ [0, τ]. In this sense, we are formally able to switch the limit on parameter α with the integration sign over variable t. Thus, by combining Eqs. (<ref>) and (<ref>), one readily obtains
  
      | S(ρ_τρ_0 ) + S(ρ_0ρ_τ) |  ≤∫_0^τ dt |lim_α→ 1 ℰ_α^𝒪(ρ_t,ρ_0)/(1 - α)d/dtg_α(ρ_t,ρ_0) | + ∫_0^τ dt  |lim_α→ 1 ℰ_1 - α^𝒪(ρ_t,ρ_0)/(1 - α)d/dtg_1 - α(ρ_t,ρ_0) |  ,
  
  where we have used that
  
      |lim_α→ 1 d/dt𝒪_α(ρ_tρ_0)| =  |lim_α→ 1 ℰ_α^𝒪(ρ_t,ρ_0)/(1 - α)d/dtg_α(ρ_t,ρ_0) |  ,
  
  and
  
      | lim_α→ 1 α/(1 - α)d/dt𝒪_1 - α(ρ_tρ_0)| =  |lim_α→ 1 ℰ_1 - α^𝒪(ρ_t,ρ_0)/(1 - α)d/dtg_1 - α(ρ_t,ρ_0) |  ,
  
  with the auxiliary functional ℰ_s^𝒪(A,B) defined as 
  
      ℰ_s^𝒪(A,B) = 
      [g_s(A,B)]^-1 ,   
      
      1  ,    .
  
  Interestingly, one may verify the right-hand side of Eq. (<ref>) exhibits an indeterminacy in the limit α→ 1. Indeed, since (dρ_t^s/dt) = - i [H_t,ρ_t^s], it follows that 
  
      lim_α→ 1  [dg_α(ρ_t,ρ_0)/dt] = (-i) lim_α→ 1 Tr(ρ_0^1 - α[H_t,ρ_t^α]) = 0,
  
  and also 
  
      lim_α→ 1  [dg_1 - α(ρ_t,ρ_0)/dt] = (-i) lim_α→ 1 Tr(ρ_0^α[H_t,ρ_t^1 - α]) = 0  .
  
  Similarly, one readily verifies that lim_α→ 1  (1 - α)[ℰ_α^𝒪(ρ_t,ρ_0)]^-1 = 0 and lim_α→ 1  (1 - α)[ℰ_1 - α^𝒪(ρ_t,ρ_0)]^-1 = 0, where we used the fact that lim_α→ 1 [ℰ_α^𝒪(ρ_t,ρ_0)]^-1 = 1, and lim_α→ 1 [ℰ_1 - α^𝒪(ρ_t,ρ_0)]^-1 = 1, for all 𝒪≡{R,H}. In this case, one obtains
  
      lim_α→ 1 ℰ_α^𝒪(ρ_t,ρ_0)/(1 - α)d/dtg_α(ρ_t,ρ_0) ⟶0/0 ,
  
  and 
  
      lim_α→ 1 ℰ_1 - α^𝒪(ρ_t,ρ_0)/(1 - α)d/dtg_1 - α(ρ_t,ρ_0) ⟶0/0 ,
  
  which in turn suggests the right-hand side of Eq. (<ref>) is not well behaved. We note, however, this such issue is readily circumvented by applying the L'Hospital rule, leading us to
  
      lim_α→ 1 ℰ_α^𝒪(ρ_t,ρ_0)/(1 - α)d/dtg_α(ρ_t,ρ_0)  =   lim_α→ 1  d[ d g_α(ρ_t,ρ_0)/dt ] /dα/ d ( (1 - α)  [ℰ_α^𝒪(ρ_t,ρ_0)]^-1)/dα ,
  
  and 
  
      lim_α→ 1 ℰ_1 - α^𝒪(ρ_t,ρ_0)/(1 - α)d/dtg_1 - α(ρ_t,ρ_0)  =   lim_α→ 1  d[ d g_1 - α(ρ_t,ρ_0)/dt ] /dα/ d ( (1 - α)  [ℰ_1 - α^𝒪(ρ_t,ρ_0)]^-1)/dα .
  
  
  In the following, we will discuss in details each contribution in the right-hand side of Eqs. (<ref>) and (<ref>). Let us start by evaluating the following derivatives:
  
      lim_α→ 1 d/dα(d/dtg_α(ρ_t,ρ_0))    = -i lim_α→ 1 d/dα(Tr(ρ_0^1 - α[H_t,ρ_t^α]) ) 
         = -i lim_α→ 1 ∑_j,ℓ d/dα( p_j^1 - αp_ℓ^α) ⟨ψ_j|   [H_t,U_t|ψ_ℓ⟩⟨ψ_ℓ|U_t^†]   |ψ_j⟩
         =  -i ∑_j,ℓ p_ℓ (lnp_ℓ - lnp_j)  ⟨ψ_j|   [H_t,U_t|ψ_ℓ⟩⟨ψ_ℓ|U_t^†]   |ψ_j⟩
         =  i  Tr(lnρ_0  [H_t,ρ_t ] )  ,
  
  and 
  
      lim_α→ 1 d/dα(d/dtg_1 - α(ρ_t,ρ_0) )    = -i lim_α→ 1 d/dα(Tr(ρ_0^α[H_t,ρ_t^1 - α])) 
         = -i lim_α→ 1 ∑_j,ℓ d/dα( p_j^αp_ℓ^1 - α) ⟨ψ_j|   [H_t,U_t|ψ_ℓ⟩⟨ψ_ℓ|U_t^†]   |ψ_j⟩
         = i ∑_j,ℓ p_j (lnp_ℓ - lnp_j)  ⟨ψ_j|   [H_t,U_t|ψ_ℓ⟩⟨ψ_ℓ|U_t^†]   |ψ_j⟩
         =  - i Tr( U_tlnρ_0U_t^†  [H_t, ρ_0]  )  ,
  
  where we have used that d  (p_j^1 - αp_ℓ^α)/dα = (lnp_ℓ - lnp_j)p_j^1 - αp_ℓ^α. Moving forward, note that
  
      d/dα( (1 - α)  [ℰ_α^𝒪(ρ_t,ρ_0)]^-1) = - [ℰ_α^𝒪(ρ_t,ρ_0)]^-1 + (1 - α)  d/dα[ℰ_α^𝒪(ρ_t,ρ_0)]^-1 ,
  
  and
  
      d/dα( (1 - α)  [ℰ_1 - α^𝒪(ρ_t,ρ_0)]^-1) = - [ℰ_1 - α^𝒪(ρ_t,ρ_0)]^-1 + (1 - α)  d/dα[ℰ_1 - α^𝒪(ρ_t,ρ_0)]^-1 .
  
  From Eqs. (<ref>) and (<ref>), we point out that lim_α→ 1 [ℰ_α^𝒪(ρ_t,ρ_0)]^-1 = 1, and lim_α→ 1 [ℰ_1 - α^𝒪(ρ_t,ρ_0)]^-1 = 1, for all 𝒪≡{R,H}. Hence, from now on it suffices to proceed by showing the derivatives d [ℰ_α^𝒪(ρ_t,ρ_0)]^-1/dα, and d [ℰ_1 - α^𝒪(ρ_t,ρ_0)]^-1/dα, are indeed well-behaved for α→ 1. On the one hand, for Tsallis relative entropy the au­xi­li­ary functional becomes ℰ_α^H(ρ_t,ρ_0) = ℰ_1 - α^H(ρ_t,ρ_0) = 1, for all α, and the aforementioned derivatives are identically zero. In this case, from Eqs. (<ref>) and (<ref>), one obtains
  
      d/dα( (1 - α)  [ℰ_α^H(ρ_t,ρ_0)]^-1) =d/dα( (1 - α)  [ℰ_1 - α^H(ρ_t,ρ_0)]^-1) = -1  .
  
  On the other hand, for Rényi relative entropy the auxiliary functional behave as [ℰ_α^R(ρ_t,ρ_0)]^-1 = g_α(ρ_t,ρ_0) and [ℰ_1 - α^R(ρ_t,ρ_0)]^-1 = g_1 - α(ρ_t,ρ_0), and the calculation is more involved. To see this, let ρ_0 = ∑_ℓ p_ℓ|ψ_ℓ⟩⟨ψ_ℓ| be the spectral decomposition of the initial state, with 0 ≤p_ℓ≤ 1 and ∑_ℓ p_ℓ = 1. In this case, given the evolved state ρ_t = U_t ρ_0U_t^†, we thus have that ρ_t^α = ∑_ℓ p_ℓ^α U_t|ψ_ℓ⟩⟨ψ_ℓ|U_t^†, and the relative purity becomes g_α(ρ_t,ρ_0) = ∑_j,ℓ p_j^1 - αp_ℓ^α | ⟨ψ_j|U_t|ψ_ℓ⟩ |^2. Hence, the derivative with respect to the parameter α is simply given by
  
      lim_α→ 1 d/dα[ℰ_α^R(ρ_t,ρ_0)]^-1   = lim_α→ 1 d/dαTr(ρ_t^αρ_0^1 - α) 
         = lim_α→ 1 ∑_j,ℓ p_ℓ^α p_j^1 - α(lnp_ℓ - lnp_j)  | ⟨ψ_j|U_t|ψ_ℓ⟩ |^2
         = S(ρ_tρ_0)  ,
  
  and
  
      lim_α→ 1 d/dα[ℰ_1 - α^R(ρ_t,ρ_0)]^-1   = lim_α→ 1 d/dαTr(ρ_t^1 - αρ_0^α) 
         = - lim_α→ 1 ∑_j,ℓ p_ℓ^1 - α p_j^α(lnp_ℓ - lnp_j)  | ⟨ψ_j|U_t|ψ_ℓ⟩ |^2
         = S(ρ_0ρ_t)  .
  
  Hence, by combining Eqs. (<ref>), (<ref>), (<ref>), (<ref>), and (<ref>), we get the result
  
      lim_α→ 1 d/dα( (1 - α)  [ℰ_α^𝒪(ρ_t,ρ_0)]^-1) = lim_α→ 1 d/dα( (1 - α)  [ℰ_1 - α^𝒪(ρ_t,ρ_0)]^-1) = -1  .
  
  Therefore, by inserting Eqs. (<ref>), (<ref>), (<ref>) and (<ref>), into Eqs. (<ref>) and (<ref>), we conclude
  
      lim_α→ 1 ℰ_α^𝒪(ρ_t,ρ_0)/(1 - α)d/dtg_α(ρ_t,ρ_0) = - i  Tr(lnρ_0  [H_t,ρ_t ] )   ,
  
  and 
  
      lim_α→ 1 ℰ_1 - α^𝒪(ρ_t,ρ_0)/(1 - α)d/dtg_1 - α(ρ_t,ρ_0) =  i Tr( U_tlnρ_0U_t^†  [H_t, ρ_0]  )   .
  
  Finally, by substituting Eqs. (<ref>) and (<ref>) into Eq. (<ref>), and then applying the Cauchy-Schwarz inequality, |Tr(𝒜_1𝒜_2)| ≤𝒜_1_2𝒜_2_2, with 𝒜_2 = √(Tr(𝒜^†𝒜)), it yields the result
  
      | S(ρ_τρ_0 ) + S(ρ_0ρ_τ) | ≤lnρ_0_2 ∫_0^τ dt   (  [ H_t , ρ_t ] _2 + [ H_t , ρ_0 ] _2)  .
  
  
  
  
  
  
   
  
  
  
  
  
  
  
  
  
  § RECOVERING MIN-RELATIVE ENTROPY
  
  
  
  In this Appendix we will discuss the case α = 0 for symmetric Rényi relative entropy, which is related to the min-relative entropy 
  
      R_0(ρ : ω) := R_0(ρω) + R_0(ωρ)  ,
  
  where R_0(ρω) = -lnTr(Π_ρ ω), with Π_ρ being the projector onto the support of the state ρ. Here we will focus on the time-independent initial state ρ_0, and its evolved state ρ_t = U_tρ_0U_t^†. By using the triangle inequality |a_1 + a_2| ≤ |a_1| + |a_2|, the absolute value of the time-derivative of Eq. (<ref>) is written as
  
      | d/dt R_0(ρ_t : ρ_0) | ≤| d/dtR_0(ρ_tρ_0)| + |d/dtR_0(ρ_0ρ_t) |  .
  
  
  From now on we will discuss the evaluation of each contribution in right-hand side of Eq. (<ref>). In order to do so, let ρ_0 = ∑_j p_ℓ |ψ_ℓ⟩⟨ψ_ℓ| be the spectral decomposition of the initial state ρ_0 into the basis { |ψ_ℓ⟩}_ℓ = 1,…, d, with 0 ≤p_ℓ≤ 1 and ∑_ℓ p_ℓ = 1. By hypothesis, the support of ρ_0 has dimension d_ρ_0 := dim[supp(ρ_0)], and is given by supp(ρ_0) = span{ |ψ_ℓ⟩ : p_ℓ≠ 0}. Thus, the projector onto the support of state ρ_0 is defined as Π_ρ_0 := ∑_ℓ :  p_ℓ≠ 0  |ψ_ℓ⟩⟨ψ_ℓ|. The evolved state is given by ρ_t = U_tρ_0U_t^† = ∑_j p_ℓ |ψ^t_ℓ⟩⟨ψ^t_ℓ|, with |ψ^t_ℓ⟩ := U_t|ψ_ℓ⟩, and its support is defined as supp(ρ_t) = span{ |ψ^t_ℓ⟩ : p_ℓ≠ 0}. Noteworthy, since the unitary evolution does not change the purity of the initial state, i.e., both states ρ_0 and ρ_t share the same set of eigenvalues, we thus have dim[supp(ρ_t)] = dim[supp(ρ_0)]. The projector Π_ρ_t onto the support of ρ_t read as 
  
      Π_ρ_t   = ∑_ℓ :  p_ℓ≠ 0  |ψ^t_ℓ⟩⟨ψ^t_ℓ| 
         =  ∑_ℓ :  p_ℓ≠ 0 U_t|ψ_ℓ⟩⟨ψ_ℓ| U_t^†
         = U_t Π_ρ_0U_t^† .
  
  Interestingly, starting from Eq. (<ref>), the projector Π_ρ_t fulfills the von Neumann-like equation (d Π_ρ_t/dt) = - i [H_t,Π_ρ_t], where we applied the identity U_t(dU_t^†/dt) = - (dU_t/dt)U_t^†  = - i H_t. Thus, the time derivative of min-relative entropy R_0(ρ_tρ_0) = -lnTr(Π_ρ_tρ_0) read as
  
      d/dt R_0(ρ_tρ_0)    = i Tr(ρ_0[H_t,Π_ρ_t]) /Tr(Π_ρ_tρ_0)
         = i Tr(U_t^†ρ_0 U_t[U_t^†H_tU_t,Π_ρ_0]) /Tr(Π_ρ_0U_t^†ρ_0 U_t) ,
  
  where we have explicitly used the property obtained in Eq. (<ref>). By taking the absolute value of Eq. (<ref>), and thus applying the Cauchy-Schwarz inequality, |Tr(𝒜_1𝒜_2)| ≤𝒜_1_2𝒜_2_2, with 𝒜_2 = √(Tr(𝒜^†𝒜)), one obtains
  
      |d/dt R_0(ρ_tρ_0)| ≤ρ_0_2  [U_t^†H_tU_t,Π_ρ_0] _2/ |Tr(Π_ρ_0U_t^†ρ_0 U_t) | .
  
  
  Let us now move to the the second term in the right-hand side of Eq. (<ref>), which is related to the time-derivative of R_0(ρ_0ρ_t) = -lnTr(Π_ρ_0 ρ_t). In this case, one readily obtains
  
      d/dt R_0(ρ_0ρ_t)    = i Tr(Π_ρ_0 [H_t,ρ_t]) /Tr(Π_ρ_0ρ_t)
         = i  Tr(U_t^†Π_ρ_0U_t[U_t^†H_tU_t,ρ_0]) /Tr(Π_ρ_0U_tρ_0U_t^†)  ,
  
  where we used the fact that ρ_t fulfills the von Neumann equation (dρ_t/dt) = - i [H_t,ρ_t]. By taking the absolute value of Eq. (<ref>), and thus applying the aforementioned Cauchy-Schwarz inequality, one obtains
  
      | d/dt R_0(ρ_0ρ_t)| ≤Π_ρ_0_2 [U_t^†H_tU_t,ρ_0] _2/| Tr(Π_ρ_0U_tρ_0U_t^†) | .
  
  
  Hence, by substituting Eqs. (<ref>) and (<ref>) into Eq. (<ref>), we thus have that 
  
      | d/dt R_0(ρ_t : ρ_0) | ≤𝒬_0^t(ρ_0,Π_ρ_0) + 𝒬_0^t(Π_ρ_0,ρ_0)   ,
  
  where the functional 𝒬_0(ρ_t,ρ) is defined as follows
  
      𝒬_0^t(A,B) := A_2  [U_t^†H_tU_t,B] _2/ |Tr(A U_tBU_t^†) | .
  
  Finally, by integrating Eq. (<ref>) over the interval t ∈ [0,τ], and thus applying the inequality |∫ dx f(x)| ≤∫dx |f(x)|, one gets the inequality
  
      | R_0(ρ_τ : ρ_0) | ≤∫_0^τ  dt  [𝒬_0^t(ρ_0,Π_ρ_0) + 𝒬_0^t(Π_ρ_0,ρ_0) ]  .
  
  
  
  
   
  
  
  
  
  
  
  
  
  
  
  § EXAMPLE: SINGLE-QUBIT STATE
  
  
  
  
  
  
  
  
  
   §.§ Tsallis relative entropy (TRE)
  
  
  
  We shall begin showing that, for an initial single-qubit state evolving unitarily according the physical setting presented in Sec. <ref>, the Tsallis relative entropy is given by
  
      H_α(ρ_τρ_0)    = H_α(ρ_0ρ_τ)  
         = 1 - g_α(ρ_τ,ρ_0)/1 - α
         =  ξ_α^-ξ_1 - α^-  |1 - (û_τ·r̂)^2|  sin^2(|u⃗_τ| ) /1 - α ,
  
  where the first equality follows from the property g_α(ρ_0,ρ_τ) = g_1 - α(ρ_τ,ρ_0) = g_α(ρ_τ,ρ_0) which holds for single-qubit states ρ_0 and ρ_τ [see Eq. (<ref>)]. From Eqs. (<ref>) and (<ref>), we thus have that [see details in Table <ref>]
  
      ⟨⟨𝒢^H_α(t) ⟩⟩_τ   = Φ_α^H ρ_0^1 - α_2 ⟨⟨ [ H_t, ρ_0^α ] _2⟩⟩_τ
         = √(2 ξ_2 - 2α^+) ξ_α^-1/τ ∫_0^τ  dt √(1 - (n̂_t·r̂)^2) ,
  
  and
  
      ⟨⟨𝒢^H_1 - α(t)⟩⟩_τ   =  Φ_1 - α^H ρ_0^α_2 ⟨⟨ [ H_t, ρ_0^1 - α ] _2⟩⟩_τ
         = √(2ξ_2α^+) ξ_1 - α^-1/τ ∫_0^τ  dt √(1 - (n̂_t·r̂)^2) .
  
  Based on Eqs. (<ref>), (<ref>) and (<ref>), the QSL time for TRE in Eqs. (<ref>) and (<ref>) can be written, respectively, as
  
      τ_α^H(ρ_τρ_0) = ξ_1 - α^-  |1 - (û_τ·r̂)^2|  sin^2(|u⃗_τ| ) /√(2 ξ_2 - 2α^+)( 1/τ ∫_0^τ  dt √(1 - (n̂_t·r̂)^2) )  ,
  
  and also
  
      τ_α^H(ρ_0ρ_τ) = ξ_α^-  |1 - (û_τ·r̂)^2|  sin^2(|u⃗_τ| )/√(2 ξ_2α^+)(1/τ ∫_0^τ  dt √(1 - (n̂_t·r̂)^2) )  ,
  
  while the QSL time for symmetrized TRE in Eq. (<ref>) reads
  
      τ^H_α(ρ_0 : ρ_τ) = √(2) ξ_α^-ξ_1 - α^-  |1 - (û_τ·r̂)^2|  sin^2(|u⃗_τ| )/(√(ξ_2 - 2α^+) ξ_α^- + √(ξ_2α^+) ξ_1 - α^-)  (1/τ ∫_0^τ  dt √(1 - (n̂_t·r̂)^2) ) .
  
  In Fig. <ref> we plot the QSL time τ^H_α = max{τ^H_α(ρ_τρ_0), τ^H_α (ρ_0 ρ_τ), τ^H_α (ρ_0:ρ_τ)}, as a function of time τ and α, for the initial single qubit state ρ_0 = (1/2)(𝕀 + r⃗·σ⃗) with {r,θ,ϕ} = {1/4,π/4,π/4}, and also varying the ratio (a) Δ/v = 0.5, (b) Δ/v = 1, (c) Δ/v = 5, and (d) Δ/v = 10. 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
   §.§ Rényi relative entropy (RRE)
  
  
  
  We shall begin showing that, for an initial single-qubit state evolving unitarily according the physical setting presented in Sec. <ref>, the Rényi relative entropy is given by
  
      R_α(ρ_τρ_0)    = R_α(ρ_0ρ_τ)  
         = ln[g_α(ρ_τ,ρ_0)]/α - 1
         = ln(1 - ξ_α^-ξ_1 - α^-(1 - (û_τ·r̂)^2) sin^2(|u⃗_τ| )) /α - 1 ,
  
  where the first equality follows from the identity g_α(ρ_0,ρ_τ) = g_1 - α(ρ_τ,ρ_0) = g_α(ρ_τ,ρ_0), which holds for single-qubit states ρ_0 and ρ_τ [see Eq. (<ref>)]. From Eqs. (<ref>) and (<ref>), we thus have that [see details in Table <ref>]
  
      ⟨⟨𝒢^R_α(t) ⟩⟩_τ   = ρ_0^1 - α_2 ⟨⟨ [ H_t, ρ_0^α ] _2⟩⟩_τ/|1 + (1 - α) ln(λ_min(ρ_0)) |
         = τ^-1 √(2 ξ_2 - 2α^+) ξ_α^- ∫_0^τ  dt √(1 - (n̂_t·r̂)^2)/| 1 + (1 - α)ln(1 - r/2) | ,
  
  and
  
      ⟨⟨𝒢^R_1 - α(t)⟩⟩_τ   =  ρ_0^α_2 ⟨⟨ [ H_t, ρ_0^1 - α ] _2⟩⟩_τ/| 1 + αln(λ_min(ρ_0)) |
         = τ^-1 √(2ξ_2α^+) ξ_1 - α^- ∫_0^τ  dt √(1 - (n̂_t·r̂)^2)/| 1 + αln(1 - r/2) | .
  
  Based on Eqs. (<ref>), (<ref>) and (<ref>), the QSL time for RRE in Eqs. (<ref>) and (<ref>) can be written, respectively, as
  
      τ_α^R(ρ_τρ_0) = | ln(1 - ξ_α^-ξ_1 - α^-(1 - (û_τ·r̂)^2) sin^2(|u⃗_τ| )) | /√(2 ξ_2 - 2α^+) ξ_α^-( 1/τ∫_0^τ  dt √(1 - (n̂_t·r̂)^2) ) /|1 + (1 - α)ln(1 - r/2) | ,
  
  and
  
      τ_α^R(ρ_0ρ_τ) = | ln(1 - ξ_α^-ξ_1 - α^-(1 - (û_τ·r̂)^2) sin^2(|u⃗_τ| )) | /√(2ξ_2α^+) ξ_1 - α^-(1/τ∫_0^τ  dt √(1 - (n̂_t·r̂)^2) ) /| 1 + αln(1 - r/2) | ,
  
  while the QSL time for symmetrized RRE in Eq. (<ref>) reads
  
      τ^R_α(ρ_0 : ρ_τ) = √(2)   | ln(1 - ξ_α^-ξ_1 - α^-(1 - (û_τ·r̂)^2) sin^2(|u⃗_τ| )) | /( √(ξ_2 - 2α^+) ξ_α^-/| 1 + (1 - α)ln(1 - r/2)|  + √(ξ_2α^+) ξ_1 - α^-/| 1 + αln(1 - r/2)| ) (  1/τ ∫_0^τ  dt √(1 - (n̂_t·r̂)^2) )   .
  
  In Fig. <ref> we plot the QSL time τ^R_α = max{τ^R_α(ρ_τρ_0), τ^R_α (ρ_0 ρ_τ), τ^R_α (ρ_0:ρ_τ)}, as a function of time τ and α, for the initial single qubit state ρ_0 = (1/2)(𝕀 + r⃗·σ⃗) with {r,θ,ϕ} = {1/4,π/4,π/4}, and also varying the ratio (a) Δ/v = 0.5, (b) Δ/v = 1, (c) Δ/v = 5, and (d) Δ/v = 10.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
   §.§ Relative entropy (RE)
  
  
  
  
  
  
  
  
  
  
  For the case of single-qubit states ρ_0 and ρ_τ discussed in Sec. <ref>, it is possible to show that relative entropy fulfills the identity S(ρ_τρ_0) = S(ρ_0ρ_τ), with [see Table <ref>]
  
      S(ρ_τρ_0) =  r ln(1 + r/1 - r)(1 - (û_τ·r̂)^2)sin^2( |u⃗_τ| )   .
  
  Next, by using the results in Table <ref>, one readily obtains the time averages
  
      ⟨⟨  [ H_t , ρ_t ] _2 ⟩⟩_τ = √(2)  r  1/τ ∫_0^τdt √(1 - (μ̂_t·r̂)^2) ,
  
  and
  
      ⟨⟨ [ H_t , ρ_0 ] _2 ⟩⟩_τ = √(2)  r  1/τ ∫_0^τdt √(1 - (n̂_t·r̂)^2) .
  
  Based on Eqs. (<ref>), (<ref>) and (<ref>), the QSL time related to the relative entropy in Eq. (<ref>) implies that
  
      τ^RE_1(ρ_τρ_0) = ln(1 + r/1 - r)(1 - (û_τ·r̂)^2)sin^2( |u⃗_τ| )/√(2) √(ln^2(1 - r/2) + ln^2(1 + r/2))( 1/τ ∫_0^τdt √(1 - (μ̂_t·r̂)^2))   ,
  
  and by swapping the arrangement of states ρ_0 and ρ_τ, it follows
  
      τ^RE_1(ρ_0ρ_τ) = ln(1 + r/1 - r)(1 - (û_τ·r̂)^2)sin^2( |u⃗_τ| )/√(2) √(ln^2(1 - r/2) + ln^2(1 + r/2))( 1/τ ∫_0^τdt √(1 - (n̂_t·r̂)^2))  ,
  
  while the QSL time for the symmetrized relative entropy in Eq. (<ref>) reads
  
      τ^RE_1(ρ_0 : ρ_τ) = √(2) ln(1 + r/1 - r)(1 - (û_τ·r̂)^2)sin^2( |u⃗_τ| )/√(ln^2(1 - r/2) + ln^2(1 + r/2))[  1/τ ∫_0^τdt ( √(1 - (μ̂_t·r̂)^2) +  √(1 - (n̂_t·r̂)^2) ) ]   .
  
  In Fig. <ref> we plot the QSL time τ^RE_1 = max{τ^RE_1(ρ_τρ_0), τ^RE_1 (ρ_0 ρ_τ), τ^RE_1 (ρ_0:ρ_τ)}, as a function of time τ and Δ/v, for the initial single qubit state ρ_0 = (1/2)(𝕀 + r⃗·σ⃗) with (a) {r,θ,ϕ} = {1/4,π/4,π/4}, (b) {r,θ,ϕ} = {1/4,π/3,π/4}, (c) {r,θ,ϕ} = {1/2,π/4,π/4}, and (d) {r,θ,ϕ} = {1/2,π/3,π/4}.
  
  
  
  
  
  
  
  
  
   §.§ Min-relative entropy
  
  
  
  Here we will present the details of the QSL time related to the min-relative entropy, with the latter defined as R_0(ρω) = -lnTr(Π_ρ ω), with Π_ρ being the projector onto the support of the state ρ. From now on, we will choose the initial state ρ_0 being a pure one, i.e., a non -full-rank density matrix. In particular, for a single-qubit state such a condition is equivalent to imposing the purity value r = 1, i.e., ρ_0 = (1/2)(𝕀 + r̂·σ⃗), which in turn implies the spectral decomposition ρ_0 = ∑_ℓ = ± p_ℓ |ψ_ℓ⟩⟨ψ_ℓ|, with eigenvalues p_+ = 1 and p_- = 0, and eigenstates |ψ_+⟩ = |θ,ϕ⟩ and |ψ_-⟩ = |θ - π,ϕ⟩, with |θ,ϕ⟩ := cos(θ/2)|0⟩ + e^-iϕsin(θ/2)|1⟩. Just to clarify, here |0⟩ = ( 1    0 )^ and |1⟩ = ( 0    1 )^ define the standard states of the computational basis. In this case, the projector onto the support of ρ_0 read as Π_ρ_0 = |ψ_+⟩⟨ψ_+|. Moving forward, one may proceed the calculation as follows
  
      Tr(Π_ρ_0U_τρ_0U_τ^†)    = Tr(ρ_0U_τΠ_ρ_0U_τ^†) 
          = |⟨ψ_+|U_τ |ψ_+⟩|^2
         = 1 - (1 - (û_t·r̂)^2) sin^2(|u⃗_t|)  .
  
  Next, by using the result of Eq. (<ref>), the min-relative entropy becomes
  
      R_0(ρ_τρ_0)    = R_0(ρ_0ρ_τ) 
         = -  ln ( 1 - (1 - (û_t·r̂)^2) sin^2(|u⃗_t|) )  .
  
  Furthermore, given that ⟨ψ_+|(U_t^†H_tU_t)^2|ψ_+⟩ = 1 + ϖ^2 + 2ϖ (μ̂_t·r̂) and also ⟨ψ_+|U_t^†H_tU_t|ψ_+⟩ = ϖ + μ̂_t·r̂, one obtains
  
      [U_t^†H_tU_t,ρ_0] _2^2   =  [U_t^†H_tU_t,Π_ρ_0] _2^2
         = 2 ( ⟨ψ_+|(U_t^†H_tU_t)^2|ψ_+⟩ - ⟨ψ_+|U_t^†H_tU_t|ψ_+⟩^2) 
         = 2(1 - (μ̂_t·r̂)^2)  .
  
  
  
  
  
  
  
  
  From Eqs. (<ref>), and (<ref>), and also using the Schatten 2-norms ρ_0_2 = Π_ρ_0_2 = 1, we thus conclude that 𝒬_0^t(ρ_0,Π_ρ_0) = 𝒬_0^t(Π_ρ_0,ρ_0) [see Eq. (<ref>)], which implies the following
  
      ⟨⟨𝒬_0^t(ρ_0,Π_ρ_0) ⟩⟩_τ   = ⟨⟨𝒬_0^t(Π_ρ_0,ρ_0)  ⟩⟩_τ
         = √(2)/τ ∫_0^τ dt √(1 - (μ̂_t·r̂)^2)/| 1 - (1 - (û_t·r̂)^2) sin^2(|u⃗_t|) | .
  
  Based on Eqs. (<ref>) and (<ref>), the QSL time for min-entropy defined in Eqs. (<ref>) and (<ref>) satisfy the following constraint
  
      τ_0^R(ρ_τρ_0)     = τ_0^R(ρ_0ρ_τ) 
         =  | ln ( 1 - (1 - (û_t·r̂)^2) sin^2(|u⃗_t|) ) |/√(2)( 1/τ ∫_0^τ dt √(1 - (μ̂_t·r̂)^2)/| 1 - (1 - (û_t·r̂)^2) sin^2(|u⃗_t|) | ) ,
  
  while the QSL time for the symmetrized min-relative entropy reads [see Eq. (<ref>)]
  
      τ^R_0(ρ_0 : ρ_τ) =  | ln ( 1 - (1 - (û_t·r̂)^2) sin^2(|u⃗_t|) ) |/√(2)( 1/τ ∫_0^τ dt √(1 - (μ̂_t·r̂)^2)/| 1 - (1 - (û_t·r̂)^2) sin^2(|u⃗_t|) | ) .
  
  In Fig. <ref> we plot the QSL time τ^R_0 = max{τ_0^R(ρ_τρ_0), τ_0^R(ρ_0 ρ_τ), τ_0^R(ρ_0 : ρ_τ) }, as a function of time τ and Δ/v, for the initial pure single qubit state ρ_0 = (1/2)(𝕀 + r̂·σ⃗) with r = 1 and (a) {θ,ϕ} = {π/4,π/4}; (b) {θ,ϕ} = {π/3,π/4}; (c) {θ,ϕ} = {π/4,π/3}; and (d) {θ,ϕ} = {π/3,π/3}.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
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