Measuring non-Markovian effects of quantum processes through Hilbert-Schmidt speed Hossein Rangani Jahromi, 1, ∗ Kobra Mahdavipour, 2, 3 Mahshid Khazaei Shadfar, 2, 3 and Rosario Lo Franco 4, † 1 Physics Department, Faculty of Sciences, Jahrom University, P.B. 7413188941, Jahrom, Iran 2 Dipartimento di Ingegneria, Università di Palermo, Viale delle Scienze, Edificio 9, 90128 Palermo, Italy 3 INRS-EMT, 1650 Boulevard Lionel-Boulet, Varennes, Québec J3X 1S2, Canada 4 Dipartimento di Ingegneria, Università di Palermo, Viale delle Scienze, Edificio 6, 90128 Palermo, Italy (Dated: November 15, 2021) Non-Markovian effects can speed up the dynamics of quantum systems while the limits of the evolution time can be derived by quantifiers of quantum statistical speed. We introduce a measure for characterizing the non-Markovianity of quantum evolutions through the Hilbert-Schmidt speed (HSS), which is a special type of quantum statistical speed. This measure has the advantage of not requiring diagonalization of evolved density matrix. Its sensitivity is investigated by considering several paradigmatic instances of open quantum systems, such as one qubit subject to phase-covariant noise and Pauli channel, two independent qubits locally interacting with leaky cavities, V-type and Λ-type three-level atom (qutrit) in a dissipative cavity. We show that the proposed HSS-based non-Markovianity measure detects memory effects in perfect agreement with the well-established trace distance-based measure, being sensitive to system-environment information backflows. I. INTRODUCTION The interaction of quantum systems with the surrounding environment leads to dissipating energy and losing quantum coherence [1]. Nevertheless, the process does not need to be monotonic and the quantum system may recover temporar- ily some of the lost energy or information due to memory ef- fects during the evolution [2–17]. This dynamical behavior, named non-Markovianity, can then act as a resource in various quantum information tasks such as teleportation with mixed states [18], improvement of capacity for long quantum chan- nels [19], efficient entangling protocols [20–22], and work ex- traction from an Otto cycle [23]. Caharacterization and quantification of non-Markovianity has been a subject of intense study [3, 4, 24, 25]. One route is to investigate temporary increases of the entanglement shared by the open quantum system with an isolated ancilla, which amounts to measure the deviation from complete positivity (CP-divisibility) of the dynamical map describing the evolu- tion of the system [26]. Another approach [27, 28] relies on measuring the distinguishability of two optimal initial states evolving through the same quantum channel and detecting any non-monotonicity (information backflows). Further wit- nesses of non-Markovianity have been proposed, based on dif- ferent dynamical figures of merit, such as: channel capacities [19], quantum mutual information [29], local quantum uncer- tainty [30], quantum interferometric power [31–34], coher- ence [35, 36], state fidelity [34, 37, 38], change of volume of the set of accessible states of the evolved system [39], Fisher information flow [40, 41], spectral analysis [42], entropy pro- duction rates [43, 44], correlation measures [45], Choi state [46] and quantum evolution speedup [47–49]. This variety of witnesses and approaches highlight the multifaceted nature of non-Markovian behavior which hence cannot be attributed to a unique feature of the system-environment interaction, pre- ∗ Electronic address: h.ranganijahromi@jahromu.ac.ir † Electronic address: rosario.lofranco@unipa.it venting the characterization by means of a single tool for such a phenomenon. CP-divisibility is the most common definition for Marko- vianity in open quantum systems [1, 4]. A dynamical map {E t } t≥0 is defined as a family of completely positive (CP) and trace-preserving (TP) maps acting on the system Hilbert space H . Generally speaking, one calls a map k-positive if the com- posite map E t ⊗ I k is positive, where k, I k denote the dimen- sionality of the ancillary Hilbert space and its identity opera- tor, respectively [50]. Provided that E t ⊗ I k is positive for all k ≥ 0 and for all t, then the dynamical map is completely positive. One then says that the dynamical map E t is CP- divisible (P-divisible) when the propagator V t, s , defined by E t = V t, s ◦E s , is completely positive (positive) for all t ≥ s ≥ 0 [1]. According to the non-Markovianity measure introduced by Rivas-Huelga-Plenio (RHP) [26], the quantum evolution is considered Markovian if and only if the corresponding dy- namical map E t is CP-divisible. The non-Markovian character of the system dynamics can be identified through another well-known perspective pro- posed by Breuer-Laine-Piilo (BLP), namely the distinguisha- bility of two evolving quantum states of the same system [27, 28]. This distinguishability is quantified by the trace dis- tance, a commonly used distance measure for two arbitrary states ρ 1 and ρ 2 , defined as D(ρ 1 ,ρ 2 ) = 1 2 Tr|ρ 1 − ρ 2 |, where | A| = √ A † A for some operator A. The trace distance D(ρ 1 ,ρ 2 ) is contractive under CPTP maps, i.e. D(E t (ρ 1 ), E t (ρ 2 )) ≤ D(ρ 1 ,ρ 2 ). Nevertheless, this does not mean generally that D(E t (ρ 1 ), E t (ρ 2 )) is a monotonically decreasing function of time. In fact, d dt D(E t (ρ 1 ), E t (ρ 2 )) > 0 implies violation of P- divisibility and therefore of CP-divisibility [27, 51]. In other words, under any Markovian evolution of the quantum sys- tem, one gets dD(E t (ρ 1 ), E t (ρ 2 ))/dt ≤ 0, owing to the con- traction property. Therefore, its non-monotonicity can be un- derstood as a witness of non-Markovianity due to system- environment backflows of information. Studies on the role of typical figures of merit for quantum metrology, based on quantum Fisher information metric, to witness non-Markovianity have been also reported [40, 52]. arXiv:2003.12681v1 [quant-ph] 28 Mar 2020