*Article* **Memory Effects in High-Dimensional Systems Faithfully Identified by Hilbert–Schmidt Speed-Based Witness**

**Kobra Mahdavipour 1,2, Mahshid Khazaei Shadfar 1,2, Hossein Rangani Jahromi 3,\*, Roberto Morandotti <sup>2</sup> and Rosario Lo Franco 1,\***


**Abstract:** A witness of non-Markovianity based on the Hilbert–Schmidt speed (HSS), a special type of quantum statistical speed, has been recently introduced for low-dimensional quantum systems. Such a non-Markovianity witness is particularly useful, being easily computable since no diagonalization of the system density matrix is required. We investigate the sensitivity of this HSS-based witness to detect non-Markovianity in various high-dimensional and multipartite open quantum systems with finite Hilbert spaces. We find that the time behaviors of the HSS-based witness are always in agreement with those of quantum negativity or quantum correlation measure. These results show that the HSS-based witness is a faithful identifier of the memory effects appearing in the quantum evolution of a high-dimensional system with a finite Hilbert space.

**Keywords:** non-Markovianity; Hilbert–Schmidt speed; high-dimensional system; multipartite open quantum systems; memory effects

#### **1. Introduction**

The unavoidable interaction of quantum systems with their environments induces decoherence and dissipation of energy. Recently, because of important developments in both theoretical and experimental branches of quantum information theory, studies of memory effects (non-Markovianity) during the evolution of quantum systems have attracted much attention (see Refs. [1–3] for some reviews). Some approaches used for a quantitative description of non-Markovian processes are either related to the presence of information backflows [4] or to the indivisibility of the dynamical map [5]. However, while well-defined for classical evolution, the notion of non-Markovianity appears to still lack a unique definition in the quantum scenario [6].

Non-Markovian processes, exhibiting quantum memory effects, have been characterized and observed in various realistic systems such as quantum optical systems [7–12], superconducting qubits [13,14], photonic crystals [15–17], light-harvesting complexes [18], and chemical compounds [19,20]. Moreover, it is known that non-Markovianity can be a resource for quantum information tasks [21–25]. Accordingly, various witnesses have been proposed to identify non-Markovianity based on, for example, distinguishability between evolved quantum states of the system [4], fidelity [26–28], quantum relative entropies [29,30], quantum Fisher information [31], capacity measure [32–34] and Bloch volume measure [35–37].

It has been shown that the nonmonotonic behavior of quantum resources such as entanglement [5], quantum coherence [38–41] and quantum mutual information [42] can be interpreted as a witness of quantum non-Markovianity. Using entanglement to witness non-Markovianity was first proposed in Ref. [5]. This proposal has been theoretically investigated for qubits coupled to bosonic environments [43–45], for a damped harmonic

**Citation:** Mahdavipour, K.; Khazaei Shadfar, M.; Rangani Jahromi, H.; Morandotti, R.; Lo Franco, R. Memory Effects in High-Dimensional Systems Faithfully Identified by Hilbert–Schmidt Speed-Based Witness. *Entropy* **2022**, *24*, 395. https://doi.org/10.3390/e24030395

Academic Editors: Bassano Vacchini, Andrea Smirne and Nina Megier

Received: 20 January 2022 Accepted: 10 March 2022 Published: 12 March 2022

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oscillator [46], and for random unitary dynamics and classical noise models [47–49]. It is also shown that entanglement cannot capture all the quantumness of correlations because there are some separable mixed states with vanishing entanglement, which can nevertheless have have nonzero quantum correlations [50]. Therefore, quantum correlations are more robust than entanglement [51–54], while entanglement may suffer sudden death [55,56]. Consequently, many methods to quantify quantum correlations have been provided, among which quantum discord [57,58] and measurement-induced disturbance [59] are proper for any bipartite state.

Recently, Hilbert–Schmidt speed (HSS) [60], a measure of quantum statistical speed which has the advantage of avoiding diagonalization of the evolved density matrix, has been proposed and employed as a faithful witness of non-Markovianity in Hermitian systems [61–64] and an efficient tool in quantum metrology [65,66]. These studies are so far especially limited to low-dimensional systems, while high-dimensional ones have not been investigated in detail. We know that high-dimensional systems play a crucial role in increasing the security in quantum cryptography [67,68], as well as in enhancing quantum logic gates, fault-tolerant quantum computation and quantum error correction [69]. This motivates us to check the sensitivity of HSS-based witness to detect non-Markovianity in high-dimensional and multipartite open quantum systems.

In this work, we analyze the validity of our HSS-based witness in various examples of high-dimensional open quantum systems with finite Hilbert spaces, such as qudits and hybrid qubit–qutrit systems. In particular, we consider a single qudit (spin-S systems) subject to a squeezed vacuum reservoir [70], and hybrid qubit–qutrit system coupled to quantum as well as classical noises [71]. We observe that the HSS-based witness is consistent with established non-Markovianity quantifiers based on dynamical breakdown of monotonicity for the quantum information resources.

The paper is organized as follows: In Section 2, we briefly review the definition of quantifiers. In Section 3, the sensitivity of HSS-based witness in high-dimensional and multipartite open quantum systems with finite Hilbert spaces through various examples is studied. Finally, Section 4 summarizes the main results and prospects.

#### **2. Preliminaries**

In this section, we briefly review the relevant quantifiers and concepts employed in this paper.

#### *2.1. Non-Markovinity Definition*

A classical *Markov process* is described by a family of random variables {*X*(*t*), *t* ∈ *I* ⊂ R}, for which the probability that *X* takes a value *xn* at any arbitrary time *tn* ∈ *I*, provided that it took value *xn*−<sup>1</sup> at some previous time *tn*−<sup>1</sup> < *tn*, can be determined uniquely and may not be influenced by the possible values of *X* at times prior to *tn*−1. It can be formulated in terms of conditional probabilities as follows: P(*xn*, *tn*|*xn*−1, *tn*−1; ... ; *x*0, *t*0) = P(*xn*, *tn*|*xn*−1, *tn*−1) for all {*tn* ≥ *tn*−<sup>1</sup> ≥ ... ≥ *t*0} ⊂ *I*. Roughly speaking, its concept is connected with the memorylessness of the process and informally encapsulated by the statement that "a Markov process has no memory of the history of past values of *X*, i.e., the future of the process is independent of its history".

To achieve a similar formulation in the quantum scenario we should find a way to define P(*xn*, *tn*|*xn*−1, *tn*−1; ... ; *x*0, *t*0) for quantum systems. In the classical realm, we may sample a stochastic variable without affecting its posterior statistics. However, 'sampling' a quantum system requires measuring process, and hence disturbs the state of the system, affecting the subsequent outcomes. Therefore, P(*xn*, *tn*|*xn*−1, *tn*−1; ... ; *x*0, *t*0) depends on not only the dynamics but also the measurement process. Since in such a case the Markovian character of a quantum dynamical system is dependent on the the measurement scheme, chosen to obtain P(*xn*, *tn*|*xn*−1, *tn*−1; ... ; *x*0, *t*0), a definition of quantum Markovianity in terms of which is a challenging task. In fact, a reliable definition of quantum Markovianity should be independent of what is required to verify it.

The aforesaid problem may be solved by adopting a different approach focusing on studying one-time probabilities P(*x*, *t*). For these, in *linear* quantum evolutions, the definition of Markovianity reduces to the concept of *divisibility* defined without any explicit reference to measurement processes in the quantum scenario [1]. To introduce the divisibility concept, let us assume that the inverse of a quantum dynamical map E*<sup>t</sup>* exists for all times *t* ≥ 0. Then it is possible to define a two-parameter family of maps by means of <sup>E</sup>*t*,*<sup>s</sup>* <sup>=</sup> <sup>E</sup>*t*E−<sup>1</sup> *<sup>s</sup>* (*<sup>t</sup>* <sup>≥</sup> *<sup>s</sup>* <sup>≥</sup> <sup>0</sup>) such that <sup>E</sup>*t*,0 <sup>=</sup> <sup>E</sup>*<sup>t</sup>* and <sup>E</sup>*t*,0 <sup>=</sup> <sup>E</sup>*t*,*s*E*s*,0. It should be noted that the existence of the inverse for all positive times guarantees the possibility of introducing the notion of divisibility, while E*t*,0 and E*s*,0 are required to be completely positive by construction, the map E*t*,*<sup>s</sup>* need not be completely positive and not even positive. It stems from the fact that the inverse <sup>E</sup>−<sup>1</sup> *<sup>s</sup>* of a completely positive map <sup>E</sup>*<sup>s</sup>* need not be positive. The family of dynamical maps is called (C)P divisible when E*t*,*<sup>s</sup>* is (completely) positive for all *t* ≥ *s* ≥ 0.

The trace norm given by *<sup>ρ</sup>* <sup>=</sup> Tr *ρ*†*ρ* = ∑ *k* <sup>√</sup>*ak*, in which *ak*'s represent the eigen-

values of *ρ*†*ρ*, leads to an important measure called *trace distance*, *D*(*ρ*1, *ρ*2) = <sup>1</sup> <sup>2</sup> *<sup>ρ</sup>*<sup>1</sup> <sup>−</sup> *<sup>ρ</sup>*<sup>2</sup> , for the distance between two quantum states *ρ*<sup>1</sup> and *ρ*2. The trace distance *D*(*ρ*1, *ρ*2) is interpreted as the *distinguishability* between states *ρ*<sup>1</sup> and *ρ*2. Moreover, it is *contractive* for any completely positive and trace preserving (CPTP) map E affecting two arbitrary quantum states *ρ*1,2, i.e., *D* <sup>E</sup>(*ρ*1), <sup>E</sup>(*ρ*2) <sup>≤</sup> *<sup>D</sup>*(*ρ*1, *<sup>ρ</sup>*2) [3]. Because the dynamics of an open quantum system is described by a CPTP map E*t*, the trace distance between the initial states is always larger than the trace distance between the time-evolved quantum states. Nevertheless, this fact does *not* mean that *D ρ*1(*t*), *ρ*2(*t*) , in which *<sup>ρ</sup>*1,2(*t*) ≡ E*t*(*ρ*1,2(0)), exhibits a monotonically decreasing function versus time [72].

There are various ways to define and detect non-Markovianity or memory effects in quantum mechanics (see [1] for a review). In Refs. [4,29], Breuer–Laine–Piilo (BLP) proposed one of the most well-known approaches, based on the variation in distinguishability of quantum states, to characterize the non-Markovian feature of the system dynamics. This is the non-Markovianity definition which we mention in our paper. According to BLP measure, for a Markovian process, the distinguishability between any two initial states of the open system, continuously diminishes over time. In other words, a quantum evolution, mathematically described by a quantum dynamical map E*t*, is called Markovian if, for any arbitrary pair of initial quantum states *ρ*1(0) and *ρ*2(0), the evolved trace distance *D ρ*1(*t*), *ρ*2(*t*) monotonically decreases with time. Hence, quantum Markovian dynamics exhibits a continuous loss of information from the open system to the environment. Consequently, a non-Markovian evolution is defined as a process in which, for certain time intervals, d*D ρ*1(*t*), *ρ*2(*t*) /d*t* > 0, usually interpreted as the information flowing back into the system temporarily. Provided that E*<sup>t</sup>* is invertible, one can show that the quantum process is BLP Markovian if and only if E*<sup>t</sup>* is P-divisible [3,73].

#### *2.2. HSS-Based Witness of Non-Markovianity*

Considering the distance measure [60]

$$[\mathbf{d}(p,q)]^2 = \frac{1}{2} \sum\_{\mathbf{x}} |p\_{\mathbf{x}} - q\_{\mathbf{x}}|^2 \tag{1}$$

where *p* = {*px*}*<sup>x</sup>* and *q* = {*qx*}*<sup>x</sup>* denote the probability distributions, one can quantify the distance between infinitesimally close distributions taken from a one-parameter family *px*(*φ*) and then define the classical statistical speed as

$$\mathrm{s}\left[p(\phi\_0)\right] = \frac{d}{d\phi}d\left(p(\phi\_0 + \phi), p(\phi\_0)\right). \tag{2}$$

These classical notions can be generalized to the quantum case by taking a pair of quantum states *ρ* and *σ*, and writing *px* = Tr{*Exρ*} and *qx* = Tr{*Exσ*} which represent the measurement probabilities corresponding to the positive-operator-valued measure (POVM) defined by {*Ex* ≥ 0} satisfying ∑ *x Ex* = I.

The associated quantum distance, which is called Hilbert–Schmidt distance [74], can be achieved by maximizing the classical distance over all possible choices of POVMs [75]

$$\mathrm{D}(\rho,\sigma) \equiv \max\_{\{E\_x\}} \mathrm{d}(p,q) = \sqrt{\frac{1}{2}\mathrm{Tr}\left[\left(\rho-\sigma\right)^2\right]}.\tag{3}$$

Consequently the HSS, i.e. the corresponding quantum statistical speed, is defined as follows:

$$H\text{SS}(\rho\_{\phi}) \equiv H\text{SS}\_{\phi} \equiv \max\_{\{E\_x\}} \text{s} \left[ p(\phi) \right] = \sqrt{\frac{1}{2} \text{Tr} \left[ \left( \frac{d\rho\_{\phi}}{d\phi} \right)^2 \right]}\,\tag{4}$$

which can be easily computed without the diagonalization of *dρφ*/*dφ*.

The recently proposed protocol, completely consistent with the BLP witness and used to detect non-Markovianity based on the HSS, is now briefly recalled [61]. We consider an *n*-dimensional quantum system whose initial state is given by

$$|\psi\_0\rangle = \frac{1}{\sqrt{n}} (\mathbf{e}^{i\phi} |\psi\_1\rangle + \dots + |\psi\_n\rangle),\tag{5}$$

where *φ* is an unknown phase shift and {|*ψ*1, ... , |*ψn*} denotes a complete and orthonormal set (basis) for the corresponding Hilbert space H. Given this initial state, the HSS-based witness of non-Markovianity is defined by

$$\text{Non-Markovarity Witness}: \chi(t) \equiv \frac{\text{d}H \text{SS} \left(\rho\_{\phi}(t)\right)}{\text{d}t} > 0,\tag{6}$$

in which *ρφ*(*t*) is the evolved state of the system.

#### *2.3. Quantum Entanglement Measure*

Quantum entanglement is a kind of quantum correlations which, from an operational point of view, can be defined as those correlations between different subsystems which cannot be generated by local operations and classical communication (LOCC) procedures. We use negativity [76] to quantify the quantum entanglement of the state, which is a reliable measure of entanglement in the case of qubit–qubit and qubit–qutrit systems [77].

For any bipartite state, *ρAB*, the negativity is defined as

$$\mathcal{N}(\rho\_{AB}) = \sum\_{i} |\lambda\_i|\_{\prime} \tag{7}$$

where *λ<sup>i</sup>* is the negative eigenvalue of *ρTk* , with *ρTk* denoting the partial transpose of the density matrix *ρAB* with respect to subsystem *k* = *A*, *B*. Negativity can also be computed by the formula [78]

$$\mathcal{N}(\rho\_{AB}) = \frac{1}{2} \left( \left\| \rho^{T\_k} \right\| - 1 \right), \tag{8}$$

in which the trace norm of *ρTk* is equal to the sum of the absolute values of its eigenvalues [79], that is

$$\left\| \left| \rho^{T\_k} \right| \right\| = \sum\_{i} |\mu\_i|\_{\prime} \tag{9}$$

where the spectral decomposition of *<sup>ρ</sup>Tk* is given by <sup>∑</sup>*<sup>i</sup> <sup>μ</sup><sup>i</sup>* <sup>|</sup>*i i*|.

#### *2.4. Quantum Correlation Quantifier: Measurement-Induced Disturbance*

We use measurement-induced disturbance MID [59] as an alternative nonclassicality indicator for quantifying the quantum correlations of the bipartite quantum systems. It is defined as the minimum disturbance caused by local projective measurements leaving the reduced states invariant.

Considering the spectral resolutions of the reduced density states *ρ<sup>A</sup>* = ∑*<sup>i</sup> p<sup>A</sup> <sup>i</sup>* <sup>Π</sup>*<sup>A</sup> <sup>i</sup>* and *ρ<sup>B</sup>* = ∑*<sup>j</sup> p<sup>B</sup> <sup>j</sup>* <sup>Π</sup>*<sup>B</sup> <sup>i</sup>* , one can compute the MID as follows:

$$\mathcal{M}(\rho\_{AB}) = \mathcal{Z}\rho\_{AB} - \mathcal{Z}(\Pi(\rho\_{AB})),\tag{10}$$

where I is the mutual quantum information given by

$$\mathcal{L}(\rho\_{AB}) = \mathcal{S}(\rho\_A) + \mathcal{S}(\rho\_B) - \mathcal{S}(\rho\_{AB}),\tag{11}$$

in which *S*(*ρ*) = −tr*ρ* log (*ρ*) denotes the von Neumann entropy and

$$\Pi(\rho\_{AB}) = \sum\_{i,j} \left( \Pi\_i^A \otimes \Pi\_j^B \right) \rho\_{AB} \left( \Pi\_i^A \otimes \Pi\_j^B \right). \tag{12}$$

#### **3. Analyzing the Efficiency of the HSS Witness in High-Dimensional Systems with Finite Hilbert Spaces**

In this section, we check the sanity of HSS-based witness through several paradigmatic high-dimensional quantum systems with finite Hilbert spaces. The analyses are based on the fact that for systems in which the corresponding subsystems are coupled to independent environments, the oscillations of quantum correlations with time are associated with the non-Markovian evolution of the system [12,47,80], resulting in the transfer of correlations back and forth among the various parts of the total system. Moreover, by comparing the results presented in Refs. [10,61,81,82], we can demonstrate that the BLP measure of non-Markovianity can be used as a valid definition of non-Markovianity, when we intend to detect non-Markovianity by revivals of quantum correlations.

In particular, we consider a single qudit subject to a quantum environment, and a hybrid qubit-qutrit system coupled to independent as well as common quantum and classical noises. We show that the oscillation of the HSS-based witness is in qualitative agreement with nonmonotonic variations of the quantum resources, and hence it can be introduced as a faithful identifier of non-Markovianity in such high-dimensional systems with finite Hilbert spaces.

It should be noted that the efficiency of the HSS-based witness in detecting the non-Markovian nature of the dynamics directly depends on adopting the correct parametrization of the initial state of Equation (5), as discussed in Ref. [61]. However, often choosing the computational basis as the complete orthonormal set {|*ψ*1, ... , |*ψn*} is enough to capture the non-Markovianity, as shown in this paper. In all examples discussed below, the HSS is computed for the pure initial states while the quantum correlations may be calculated for mixed ones to illustrate the general efficiency off the HSS-based witness.
