Hilbert-Schmidt quantum coherence in multi-qudit systems
Using Bloch’s parametrization for qudits (-level quantum systems), we write the Hilbert-Schmidt distance (HSD) between two generic -qudit states as an Euclidean distance between two vectors of observables mean values in , where is the dimension for qudit . Then, applying the generalized Gell Mann’s matrices to generate , we use that result to obtain the Hilbert-Schmidt quantum coherence (HSC) of -qudit systems. As examples, we consider in details one-qubit, one-qutrit, two-qubit, and two copies of one-qubit states. In this last case, the possibility for controlling local and non-local coherences by tuning local populations is studied and the contrasting behaviors of HSC, -norm coherence, and relative entropy of coherence in this regard are noticed. We also investigate the decoherent dynamics of these coherence functions under the action of qutrit dephasing and dissipation channels. At last, we analyze the non-monotonicity of HSD under tensor products and report the first instance of a consequence (for coherence quantification) of this kind of property of a quantum distance measure.
Coherent superposition states are an important aspect of quantum systems Feynman (); Wolf (), one which is believed to make possible their use for more efficient realization of energy and information manipulation tasks Adesso_CE (); Gu (); Allegra (); Scholes (); Uzdin (); Hillery (); Ma (); Lewenstein-1 (); Pati (); Fan (). In the last few years, much attention has been given for the development, analysis, and application of a resource theory for quantum coherence (RTC). For a recent review, see Ref. Plenio_QC_RMP (). It is certainly worthwhile recognizing the existence of several levels of incoherent operations (IO) in the RTC Gour (); Spekkens (). Naturally, depending on its properties, different coherence functions may or may not be a coherence monotone under a specific element of a given set of IO Plenio_QC (); Adesso_RoC (); Adesso_RoC-1 (); Mintert (). This kind of observation points for the need of further investigations concerning the similarities and differences among the properties enjoyed by varied quantumness functions.
Here we shall pay special attention to the geometric quantum coherence function which is obtained using Hilbert-Schmidt’s distance (HSD). Although the possible non-contractivity under general quantum operations of the HSD Ruskai (); Schirmer (); Hanggi () demands care when of its application for quantumnesses quantification Ozawa (); Piani (), it is a fact that the HSD has been a handy tool for a variety of investigations in quantum information science Walther (); Zhou (); Thirring (); Krammer (); Brukner (); Cohen (); Wunsche (); Sommers (); Winter (); Soto (). In addition to that, recently it was shown that the HSD leads to an observable upper bound for the robustness of coherence, which is a full quantum coherence monotone Adesso_RoC (). Besides, because of its friendliness for analytical calculations, the HSD can be a useful tool for starting on quantumnesses quantification, as happened, for instance, with quantum entanglement and quantum discord Trucks (); Moor (); Maziero_HSE (); Brukner_HSDisc (); Li (); Fu_HSDisc (); Azmi (); Fu_MIN (); Adesso_NegHSD (); Hou ().
Our main goal in this article is obtaining the Hilbert-Schmidt coherence (HSC) of multipartite qudit states and to use it to analyze the distribution and manipulation of quantum coherence in this kind of scenario. We shall then deal with two relevant-associated topics. We will look at the possibility of controlling the local and/or non-local coherences Maziero_NLQC (); Byrnes (); Jeong () of a system by manipulating its local populations. And we will investigate the issue of a possible non-monotonic behavior of quantum distance measures under tensor products Verstraete (); Maziero_NMuTP ().
The non-monotonicity under tensor products (NMuTP) of a distance measure, investigated recently for the trace distance Maziero_NMuTP (), is the possibility for an inversion in the dissimilarity relation between pairs of states when we consider their copies. That is to say, for four density matrices , it can happen that
Once there is no known analytical expression for the trace distance coherence for general states of systems with dimension greater or equal to three Lewenstein (); Li-1 (); Wang (), and with the aim of unveiling the possible implications of the NMuTP in quantum information science, here we shall analyze it using the HSD and look for its possible implications in coherence quantification.
The remainder of the article is structured as follows. In Sec. II we make use of Bloch’s representation to write the Hilbert-Schmidt distance between two generic -qudit states as an Euclidean distance between the associated rescaled Bloch’s vectors. In Sec. III we use this result to obtain an analytical formula for the Hilbert-Schmidt coherence of -qudit states. We instantiate this general result by looking at one-qubit (Sec. III.1), one-qutrit (Sec. III.2), two-qubit (Sec. III.3), and two copies of one-qubit (Sec. III.4) states. After writing, in Sec. III.3, the total HSC of two-qubit states in terms of its local and nonlocal parts, in Sec. III.4 we study the possibility for controlling these quantities by manipulating the local populations of two copies of one-qubit states. The contrasting behaviors of HSC, -norm coherence, and relative entropy of coherence in this regard are noticed. We also investigate, in Sec. IV, the dynamical behavior of quantum coherence functions under the action of qutrit phase damping and amplitude damping channels. Finally, the non-monotonicity of the HSD under tensor products is investigated in Sec. V, where we report the first example of an important implication of this kind of property of a quantum distance function. We summarize our findings and discuss related open questions in Sec. VI.
Ii Hilbert-Schmidt distance between -qudit states
If the matrices satisfy the algebraic relations
then the unit trace of requires . Besides, the other parameters , which are the components of the so called Bloch’s vector, can be obtained as follows:
Now let’s regard the Hilbert-Schmidt distance (HSD) Cohen (),
between two generic -qudit states. Using , we shall have
Thus, if we define the components of the rescaled Bloch’s vector as
with the analogous holding for , the HSD between and can be written as the Euclidean distance between the vectors and :
We observe that if the generators are Hermitian, then the distinguishability of the two states gains a nice physical significance in terms of differences between the corresponding pairs of observables mean values. Notice that for this kind of generator we have, e.g., , with . For a generic two-qudit density matrix, Ref. Maziero_AMP () showed an optimized way to compute numerically the components of the associated Bloch’s vector (the Fortran code produced for that purpose can be accessed freely in LibForQ ()).
Iii Hilbert-Schmidt quantum coherence for -qudit states
Incoherent states, , are represented by density matrices which are diagonal in some reference, orthonormal, basis . That is to say Plenio_QC ()
with being a probability distribution. The Hilbert-Schmidt coherence (HSC) of a state is defined as its minimum Hilbert-Schmidt distance (HSD) to incoherent states:
The HSC is nonnegative and is equal to zero if and only if is an incoherent state. It is also worthwhile mentioning that although the HSC is not a “full coherence monotone” Plenio_QC (), it is a monotone (i.e. ) under incoherent operations of the dephasing type (in basis ): Cohen ().
For computing the HSC, one convenient choice for the generators is the generalized Gell Mann matrices Krammer-1 ():
These three sets of matrices are named the diagonal, symmetric, and antisymmetric sets, respectively. The particular cases with and give the well known Pauli and Gell Mann matrices, respectively.
If the reference basis is chosen to be the -qubit computational basis , with , then the presence of any non-null component of the Bloch’s vector corresponding to generators from the symmetric and/or antisymmetric sets implies in the presence of off diagonal elements in the density matrix. Thus, the more general -qudit incoherent state is obtained utilizing the identity matrix and the generators from the diagonal set:
As seen in Eq. (8), the HSD is given in terms of the sum of positive terms, which are equal to the square of the difference between corresponding components of the rescaled vectors. Therefore, the minimum in Eq. (10) is obtained if we set for all and . Thus we see that the optimal incoherent state under the HSD is
We shall define the coherence vector, , as the vector obtained using the components of the rescaled Bloch’s vector in Eq. (7) corresponding to one or more generators from the symmetric and/or antisymmetric sets. Hence the HSC of a -qudit state can be cast as the Euclidean norm of this dimension coherence vector:
In the next sub-sections, we exemplify the application of this general result to some particular cases.
iii.1 One-qubit state
In this example , , and . So, the one-qubit HSC is given by
In this section, and in the remainder of this article, when there is no chance for confusion, we omit the density matrix in the expression for the mean values.
iii.2 One-qutrit state
Here , , and . So,
iii.3 Two-qubit state
iii.4 Two copies of one-qubit states: Controlling coherences via the manipulation of populations
As a special case of the class of states regarded in the last sub-section, let us look to two copies of one-qubit states: . For this kind of state we have and
Considering that the local populations are equal to
with , we notice that although we cannot control the local HSC by manipulating , we can change the non-local HSC by modifying the difference between those populations: .
The result above is to be compared with the one for the -norm coherence (L1C), which, in contrast to the HSC, is a coherence monotone under the incoherent operations regarded in Ref. Plenio_QC (). The L1C is given by the sum of the absolute values of the off-diagonal elements of a density matrix. Although is present in the off-diagonal elements of , it cancels out when we perform that sum, and we get
with . So, we cannot control the non-local L1C by tuning local populations.
Because of its operational interpretation Yang_RTC (), the relative entropy of coherence (REC) may be considered to be one of the main coherence quantifiers introduced up today. The REC is defined and given by Plenio_QC ():
with being the density matrix obtained by setting the off-diagonal elements of to zero and is von Neumann’s entropy. Using the additivity of von Neumann’s entropy under tensor products, i.e., and the fact that , we see that the REC is also additive under tensor products, i.e., . So, we shall be able to change the REC of the two copies of only if we can change its own REC. An interesting result we report here in this direction is that, contrary to what happens with the HSC and L1C, we can modify the REC of a qubit by controlling its populations, a quantity usually considered to have a classical character. For a qubit state, ,
with The form of the equation for is an indicative that the term , related to the populations, does not cancel out. We verify that this is indeed the case via an example, which is shown in Fig. 1.
Iv Coherence dynamics under qutrit dephasing and dissipation
In this section, we study the dynamics of some coherence functions under decoherence. Once for one-qubit states the Hilbert-Schmidt coherence (HSC) and the -norm coherence (L1C) are proportional, their dynamic behavior shall be equivalent in this physically meaningful kind of process. For instance, for a qubit the freezing conditions Maziero_NLQC (); Adesso_frozenC () for and for are the same. However, even in this simplest context, the dynamic behavior of relative entropy of coherence (REC) is qualitatively disparate from those of HSC and L1C. And this is due to the dependence of the first on the populations. On the other hand, already for qutrits the proportionality between and does not hold in general. Here we have with . So, we can warrant that only if one single pair is non-null. In the following we shall regard the dynamics induced by qutrit phase damping (PD) and amplitude damping (AD) channels. The Kraus’ operators for these channels are (see e.g. Refs. Guo_qutrit (); Khan_qutrit () and references therein):
with being the identity matrix.
For a qutrit prepared in a general state and submitted to the action of the PD or AD channels, the states at parametrized time Soares-Pinto2011 (), , are given as follows:
with . As all off-diagonal elements of decay at the same rate, we see that for the PD channel (PDC) the following motion equation holds for the L1C: . Besides, as all Bloch’s vector components from the symmetric and anti-symmetric sets are proportional to , we get the following evolution equation for HSC: . Therefore, even if there is no simple-general proportionality relation between the initial values of L1C and HSC, the rate with which these values diminish with time for the PDC is once again observed to be equivalent. On the other hand, for this particular model of dissipation of three level systems, not all off-diagonal elements of change equally with time. Here the time dependence of the coherences for the AD channel are:
We illustrate the time dependence of these functions, and of REC, with the specific examples shown in Fig. 2.
V Non-monotonicity of the HSD under tensor products
One can verify that, similarly to what happens with the trace distance (TD) Maziero_NMuTP (), the Hilbert-Schmidt distance (HSD) does not suffer from the non-monotonicity under tensor product (NMuTP) issue for general pure states or for one-qubit collinear states. Actually, one can show that this holds true for all -norm distances Ruskai (). For general qudit states, the probability for random quartets of states to present the NMuTP behavior of HSD decays with increasing the system dimension, also similar to what was seen for the TD Maziero_NMuTP (). For completeness, we show such probabilities in Fig. 3.
Regarding the Hilbert-Schmidt coherence (HSC), once
we see that the NMuTP behavior of the HSD can be revealed on its associated HSC if we manipulate accordingly; what we will do by using an example. We will consider and , what gives us and so
In order to invert the dissimilarity relation, we fix and change , which can vary from zero up to . As , we can use e.g. to get , and consequently
These last two inequalities are a manifestation of the NMuTP of HSD:
In the last equation we used the fact that, under the HSD, .
Vi Concluding remarks
In this article, we started obtaining a closed formula for the Hilbert-Schmidt distance (HSD) between two generic -qudit states in terms of the Euclidean distance between the corresponding rescaled Bloch’s vectors. We used this result to derive an analytical expression for the Hilbert-Schmidt quantum coherence (HSC) of -qudit systems. This formula was then exemplified in simple cases and we applied it to investigated issues related to the distribution and transformation of quantum coherence in some composite systems.
After splitting the total HSC of two-qubit states into its local and non-local parts, we analyzed the possibility of controlling these coherences by tuning the local populations of two copies of a generic one-qubit state. Related to that matter, we observed some interesting contrasting behaviors of the HSC, -norm coherence (L1C), and relative entropy of coherence (REC). We noticed that it is not possible to change the local HSC and L1C by tuning local populations. However, contrary the what was seen for the L1C, we can control the HSC by doing that kind of operation. The REC of two copies of a qubit state can be changed only if the local REC is changed, what we showed being possible by the control of local populations. This raises yet another bold difference among these coherence functions. In future works, it could be fruitful to investigate this kind of issue considering its possible physical implications and also another coherence quantifiers.
We also investigated the time evolution of coherence under qutrit dephasing and dissipation environments. The population-dependent character of the REC leads to its quite different dynamical behavior when compared to those of L1C and HSC. These last two coherence functions are seen to be equivalent for qubits and for some qutrit states. Besides, we have shown that even when HSC and L1C are not proportional, their dynamical behavior can be similar.
We have identified the first important implication of the non-monotonicity under tensor product (NMuTP) property of a quantum distance measure. We showed that the NMuTP of HSD, when applied to some states, can indicate that although a configuration is more coherent than another state , two independent copies of the first state possess less HSC than two copies of the last. We emphasize that we succeeded in doing that at this moment mainly because we used the HSD. This unexpected result, obtained for the HSD, calls for further investigations of the NMuTP issue regarding other quantum distance measures and also considering other of its possible consequences.
Acknowledgements.This work was supported by the Brazilian funding agencies: Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), processes 441875/2014-9 and 303496/2014-2, Coordenação de Desenvolvimento de Pessoal de Nível Superior (CAPES), process 6531/2014-08, and Instituto Nacional de Ciência e Tecnologia de Informação Quântica (INCT-IQ), process 2008/57856-6.
- (1) R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, Volume 3 (Addison-Wesley, Massachusetts, 1965).
- (2) L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, England, 2008).
- (3) A. Streltsov, U. Singh, H. S. Dhar, M. N. Bera, and G. Adesso, Measuring quantum coherence with entanglement, Phys. Rev. Lett. 115, 020403 (2015).
- (4) J. Ma, B. Yadin, D. Girolami, V. Vedral, and M. Gu, Converting coherence to quantum correlations, Phys. Rev. Lett. 116, 160407 (2016).
- (5) P. Giorda and M. Allegra, Coherence in quantum estimation, arXiv:1611.02519.
- (6) A. Chenu and G. D. Scholes, Coherence in energy transfer and photosynthesis, Annu. Rev. Phys. Chem. 66, 69 (2015).
- (7) R. Uzdin, Coherence-induced reversibility and collective operation of quantum heat machines via coherence recycling, Phys. Rev. Applied 6, 024004 (2016).
- (8) M. Hillery, Coherence as a resource in decision problems: The Deutsch-Jozsa algorithm and a variation, Phys. Rev. A 93, 012111 (2016).
- (9) X. Yuan, K. Liu, Y. Xu, W. Wang, Y. Ma, F. Zhang, Z. Yan, R. Vijay, L. Sun, and X. Ma, Experimental quantum randomness processing using superconducting qubits, Phys. Rev. Lett. 117, 010502 (2016).
- (10) A. Streltsov, E. Chitambar, S. Rana, M. N. Bera, A. Winter, and M. Lewenstein, Entanglement and coherence in quantum state merging, Phys. Rev. Lett. 116, 240405 (2016).
- (11) A. Misra, U. Singh, S. Bhattacharya, and A. K. Pati, Energy cost of creating quantum coherence, Phys. Rev. A 93, 052335 (2016).
- (12) H.-L. Shi, S.-Y. Liu, X.-H. Wang, W.-L. Yang, Z.-Y. Yang, and H. Fan, Coherence depletion in the Grover quantum search algorithm, Phys. Rev. A 95, 032307 (2017).
- (13) A. Streltsov, G. Adesso, and M. B. Plenio, Quantum coherence as a resource, arXiv:1609.02439.
- (14) E. Chitambar and G. Gour, Critical examination of incoherent operations and a physically consistent resource theory of quantum coherence, Phys. Rev. Lett. 117, 030401 (2016).
- (15) I. Marvian and R. W. Spekkens, How to quantify coherence: Distinguishing speakable and unspeakable notions, Phys. Rev. A 94, 052324 (2016).
- (16) T. Baumgratz, M. Cramer, and M. B. Plenio, Quantifying coherence, Phys. Rev. Lett. 113, 170401 (2014).
- (17) C. Napoli, T. R. Bromley, M. Cianciaruso, M. Piani, N. Johnston, and G. Adesso, Robustness of coherence: An operational and observable measure of quantum coherence, Phys. Rev. Lett. 116, 150502 (2016).
- (18) M. Piani, M. Cianciaruso, T. R. Bromley, C. Napoli, N. Johnston, and G. Adesso, Robustness of asymmetry and coherence of quantum states, Phys. Rev. A 93, 042107 (2016).
- (19) F. Levi and F. Mintert, A quantitative theory of coherent delocalization, New J. Phys. 16, 033007 (2014).
- (20) D. Pérez-García, M. M. Wolf, D. Petz, and M. B. Ruskai, Contractivity of positive and trace-preserving maps under Lp norms, J. Math. Phys. 47, 083506 (2006).
- (21) X. Wang and S. G. Schirmer, Contractivity of the Hilbert-Schmidt distance under open-system dynamics, Phys. Rev. A 79, 052326 (2009).
- (22) J. Dajka, J. Łuczka, and P. Hänggi, Distance between quantum states in the presence of initial qubit-environment correlations: A comparative study, Phys. Rev. A 84, 032120 (2011).
- (23) M. Ozawa, Entanglement measures and the Hilbert-Schmidt distance, Phys. Lett. A 268, 158 (2000).
- (24) M. Piani, The problem with the geometric discord, Phys. Rev. A 86, 034101 (2012).
- (25) B. Dakic, Y. O. Lipp, X. Ma, M. Ringbauer, S. Kropatschek, S. Barz, T. Paterek, V. Vedral, A. Zeilinger, C. Brukner, and P. Walther, Quantum discord as resource for remote state preparation, Nat. Phys. 8, 666 (2012).
- (26) X. Wu and T. Zhou, Geometric discord: A resource for increments of quantum key generation through twirling, Sci. Rep. 5, 13365 (2015).
- (27) R. A. Bertlmann, H. Narnhofer, and W. Thirring, A geometric picture of entanglement and Bell inequalities, Phys. Rev. A 66, 032319 (2002).
- (28) R. A. Bertlmann, K. Durstberger, B. C. Hiesmayr, and P. Krammer, Optimal entanglement witnesses for qubits and qutrits, Phys. Rev. A 72, 052331 (2005).
- (29) J. Lee, M. S. Kim, and C. Brukner, Operationally invariant measure of the distance between quantum states by complementary measurements, Phys. Rev. Lett. 91, 087902 (2003).
- (30) B. Tamir and E. Cohen, A Holevo-type bound for a Hilbert Schmidt distance measure, J. Quant. Inf. Science 05, 127 (2015).
- (31) V. V. Dodonov, O. V. Man’ko, V. I. Man’ko, and A. Wünsche, Hilbert-Schmidt distance and non-classicality of states in quantum optics, J. Mod. Opt. 47, 633 (2000).
- (32) K. Zyczkowski and H.-J. Sommers, Hilbert-Schmidt volume of the set of mixed quantum states, J. Phys. A: Math. Gen. 36, 10115 (2003).
- (33) S. Popescu, A. J. Short, and A. Winter, Entanglement and the foundations of statistical mechanics, Nat. Phys. 2, 754 (2006).
- (34) G. Björk, H. de Guise, A. B. Klimov, P. de la Hoz, and L. L. Sánchez-Soto, Classical distinguishability as an operational measure of polarization, Phys. Rev. A 90, 013830 (2014).
- (35) C. Witte and M. Trucks, A new entanglement measure induced by the Hilbert-Schmidt norm, Phys. Lett. A 257, 14 (1999).
- (36) F. Verstraete, J. Dehaene, and B. De Moor, On the geometry of entangled states, J. Mod. Opt. 49, 1277 (2002).
- (37) J. Maziero, Computing partial transposes and related entanglement functions, Braz. J. Phys. 46, 605 (2016).
- (38) B. Dakic, V. Vedral, and C. Brukner, Necessary and sufficient condition for non-zero quantum discord, Phys. Rev. Lett. 105, 190502 (2010).
- (39) L. Li, Q.-W. Wang, S.-Q. Shen, and M. Li, Geometric measure of quantum discord with weak measurements, Quantum Inf. Process. 15, 291 (2016).
- (40) S. Luo and S. Fu, Geometric measure of quantum discord, Phys. Rev. A 82, 034302 (2010).
- (41) S. J. Akhtarshenas, H. Mohammadi, S. Karimi, and Z. Azmi, Computable measure of quantum correlation, Quantum Inf. Process. 14, 247 (2015).
- (42) S. Luo and S. Fu, Measurement-induced nonlocality, Phys. Rev. Lett. 106, 120401 (2011).
- (43) D. Girolami and G. Adesso, Interplay between computable measures of entanglement and other quantum correlations, Phys. Rev. A 84, 052110 (2011).
- (44) Y.-L. Yuan and X.-W. Hou, Thermal geometric discords in a two-qutrit system, Int. J. Quantum Inform. 14, 1650016 (2016).
- (45) M. B. Pozzobom and J. Maziero, Environment-induced quantum coherence spreading of a qubit, Ann. Phys. 377, 243 (2017).
- (46) R. Chandrashekar, P. Manikandan, J. Segar, and T. Byrnes, Distribution of quantum coherence in multipartite systems, Phys. Rev. Lett. 116, 150504 (2016).
- (47) K. C. Tan, H. Kwon, C.-Y. Park, and H. Jeong, Unified view of quantum correlations and quantum coherence, Phys. Rev. A 94, 022329 (2016).
- (48) K. M. R. Audenaert, J. Calsamiglia, R. Munõz-Tapia, E. Bagan, Ll. Masanes, A. Acin, F. Verstraete, Discriminating states: The quantum Chernoff bound, Phys. Rev. Lett. 98, 160501 (2007).
- (49) J. Maziero, Non-monotonicity of trace distance under tensor products, Braz. J. Phys. 45, 560 (2015).
- (50) S. Rana, P. Parashar, and M. Lewenstein, Trace-distance measure of coherence, Phys. Rev. A 93, 012110 (2016).
- (51) L.-H. Shao, Z. Xi, H. Fan, and Y. Li, Fidelity and trace-norm distances for quantifying coherence, Phys. Rev. A 91, 042120 (2015).
- (52) Z. Wang, Y.-L. Wang, and Z.-X. Wang, Trace distance measure of coherence for a class of qudit states, Quantum Inf. Process. 15, 4641 (2016).
- (53) G. Kimura, The Bloch vector for N-level systems, Phys. Lett. A 314, 339 (2003).
- (54) R. A. Bertlmann and P. Krammer, Bloch vectors for qudits, J. Phys. A: Math. Theor. 41, 235303 (2008).
- (55) J. Maziero, Computing coherence vectors and correlation matrices with application to quantum discord quantification, Adv. Math. Phys. 2016, e6892178 (2016).
- (56) https://github.com/jonasmaziero/LibForQ.
- (57) A. Winter and D. Yang, Operational resource theory of coherence, Phys. Rev. Lett. 116, 120404 (2016).
- (58) T. R. Bromley, M. Cianciaruso, and G. Adesso, Frozen quantum coherence, Phys. Rev. Lett., 114, 210401 (2015).
- (59) J.-L. Guo, H. Li, and G.-L. Long, Decoherent dynamics of quantum correlations in qubit–qutrit systems, Quantum Inf. Proc. 12, 3421 (2013).
- (60) S. Khan and I. Ahmad, Environment generated quantum correlations in bipartite qubit-qutrit systems, Optik 127, 2448 (2016).
- (61) D. O. Soares-Pinto, M. H. Y. Moussa, J. Maziero, E. R. deAzevedo, T. J. Bonagamba, R. M. Serra, and L. C. Céleri, Equivalence between Redfield- and master-equation approaches for a time-dependent quantum system and coherence control, Phys. Rev. A 83, 062336 (2011).
- (62) J. Maziero, Fortran code for generating random probability vectors, unitaries, and quantum states, Front. ICT 3, 4 (2016).
- (63) https://github.com/jonasmaziero/LibForro.