Sensitivity and Robustness of Quantum Spin-½ Rings to Parameter Uncertainty

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1 Sensitivity and Robustness of Quantum Sin-½ Rings to Parameter Uncertainty Sean O Neil, Member, IEEE, Edmond Jonckheere, Life Fellow, IEEE, Sohie Schirmer, Member, IEEE, and Frank Langbein, Member, IEEE Abstract Selective transfer of information between sin-1/2 articles arranged in a ring is achieved by otimizing the transfer fidelity over a readout time window via shaing, externally alied, static bias fields. Such static control fields have roerties that clash with the exectations of classical control theory. Previous work has shown that there are cases in which the logarithmic differential sensitivity of the transfer fidelity to uncertainty in couling strength or sillage of the bias field to adjacent sins is minimized by controllers that roduce the best fidelity. Here we exand uon these examles and examine cases of both classical and non-classical behavior of logarithmic sensitivity to arameter uncertainty and robustness as measured by the μ function for quantum systems. In articular we examine these roerties in an 11-sin ring with a single uncertainty in couling strength or a single bias sillage. I. INTRODUCTION Information encoded in networks of couled sins can roagate without mass or change transort. For examle, sintronic devices using nuclear or electron sins confined to quantum dots in 2D electron gas (2DEG) controlled by surface electrodes could overcome limitations imosed by mass or charge transort and hold significant romise for on-chi communication [1]. Linear chains and rings can be used as the comonents of quantum wires and routers. Due to the comlex wave-like roagation of excitations in such networks, efficient transfer of information is non-trivial, necessitating effective control. One way this can be achieved is by energylandscae shaing via time indeendent controls, such as voltages alied to the gate electrodes, to alter the energy levels of the electrons confined to the quantum dots [2,3]. Such systems are interesting from a control theory ersective as they exhibit unusual robustness roerties. As demonstrated in [4] and [5], when examining the sensitivity of the system to uncertainty in sin coulings or leakage of the nominal bias field from the intended sin to adjacent sins or measuring the robustness of the system s erformance to these same uncertainties we observe trends that aear to contradict the classical control limitations imosed by the identity S + T = I where S is the sensitivity transfer matrix and T is the comlementary sensitivity transfer matrix. To be more recise, we observe cases in which the robability of successful transfer is maximal while the logarithmic sensitivity is nearly zero, in contradiction to the classical intuition [4]. Sean O Neil is with the Electrical Engineering and Comuter Science Deartment of the United States Military Academy, West Point, NY 10996, USA. (hone: ; sean.o neil@usma.edu). Dr. Edmond Jonckheere is with the Deartment of Electrical Engineering at the University of Southern California, Los Angeles, CA 90089, USA. ( jonckhee@usc.edu). Additionally, extending the analysis to larger, non-differential uncertainties through -analysis reveals instances of anticlassical behavior with the most otimal controllers also being the most robust in many cases [5]. In this aer, we aim to exand uon the results detailed in [5] by examining a larger data set and looking at cases of both classical as well as anticlassical behavior. II. BACKGROUND A. Problem Formulation and Structure As discussed in detail in [2], we consider a system comosed of N sin- 1 articles arranged in a ring with one 2 excitation resent between the N sins. We aim to find a control D that maximizes the robability of transfer of the single excitation from a articular sin n to a secific target sin m at a given time t f or over a time window [t f δt, t f + δt]. We can then identify the state of the quantum system with the excitation localized at the initial sin as the state IN and identify the desired final state with the excitation localized at sin m as OUT. Taking the control D as a diagonal N N matrix describing the bias alied to each sin to affect the desired transfer, we see that the system is governed by the equation = i(h + D) where H is the single excitation subsace Hamiltonian of the ring (here a constant circulant matrix). The robability of transfer at a given time t f is then equal to the squared fidelity OUT (t f ) 2 = OUT e i(h+d)t f IN 2 = rob. (1) Dearting from this nominal model, we consider two categories of erturbations as described in [5]. The first is an uncertainly in the assumed uniform couling strengths between sins. We model this erturbation as an element δ k,k+1 S k,k+1 aended to the nominal Hamiltonian. Here S k,k+1 is an N N matrix that rovides a secific structure to the erturbation with the only non-zero elements being 1 s in the (k, k + 1) and (k + 1, k) ositions for k < N and in the (N, 1) and (1, N) ositons for k = N. Additionally δ k,k+1 rovides the size of the erturbation to the nominal couling strength. The other category of erturbation we consider is that of a leakage of the bias field intended for sin k to its neighbors. We model this erturbation as a term δ k D k S k added to the Hamiltonian. As before δ k rovides the size of the erturbation with resect to sillage at sin k, and S k is a Dr. Sohie Schirmer is with the College of Science, Swansea University, Swansea Wales, UK. ( lw1660@gmail.com). Dr. Frank Langbein is with the School of Comuter Science & Informatics, Cardiff University, Cardiff, Wales, UK. ( LangbeinFC@cf.ac.uk). FCL and SGS are suorted by Sêr Cymru NRN AEM grant 82.

2 matrix that carries the structure of the erturbation. Here S k is a matrix of all zeros save for 1 in the (k, k) osition and 1 2 in the (k 1, k 1) and (k + 1, k + 1) ositions. D k is a scalar that reresents the bias intended for the kth sin (i.e. the (k, k) element of the diagonal matrix of controls D). B. Sensitivity Analysis In classical multivariable control, we see the tension between tracking error and logarithmic sensitivity to arameter variation in the identity S + T = I. This tension is evident through identification of the log-sensitivity with T through the relation S 1 (ds) = (dl)l 1 T [4]. To relate this to the quantum system of interest, we first define a tracking error in the sense of the difference between the achieved robability of transfer and unity (erfect state transfer). Matters are comlicated by the fact that in the case of maximum fidelity we have OUT (t f ) 2 = 1 which imlies that (t f ) = e i (t f) OUT for some global hase factor (t) that is hidden in the comutation of the fidelity squared. Therefore, if we take the tracking error as the closeness of the state achieved at time t f to the desired state OUT, we get from [4] OUT e i (t f) (t f ) 2 = 2(1 OUT (t f ) ) = 2 err 2 (2) Note that to minimize this error does not require that (t f ) aroach OUT in the sense of an ordinary signal, but that the norm in (2) be minimized with resect to (t f ), and as such is referred to as the rojective tracking error [4]. With this relation of the error to the reference signal OUT, we then see that comuting the logarithmic sensitivity of the system to arameter uncertainty is tantamount to taking the derivative (err) ( 1 ) which, with rob = (1 err) δ k err allows for concordance between the size of (err) ( 1 δ k err (rob) ( 1 ) = (rob) δ k err 1 ( ). δ k 1 (rob) Now we can evaluate (rob) δ k as er [2]: OUT e i(h )t f IN 2 δ k ) and for a single erturbation δ k = 2t f OUT Π ms k Π n IN sinc ( 1 2 t fω mn ) m,n IN Π l OUT sin ( 1 2 t f(ω nl + ω ml )) l (3) Here Π l and ω nl = n l are taken from the eigendecomosition of H = l l Π l where H is the erturbed Hamiltonian defined as H + D + N 1 k=1 δ k,k+1 S k,k+1 + δ N,1 S N,1 for the case of couling uncertainty or H + D + N k=1 δ k D k S k for bias sillage. An integral of (3) over the readout window centered on t f then yields a measure of the windowed fidelity s error to differential arameter variations and thence to the log-sensitivity. Fig. 1. Block diagram reresentation of the system with a single uncertainty in the couling strengths between sins modeled as an inverse additive uncertainty around the lant. Fig. 2. Block diagram reresentation of the system with a single uncertainty in the alied bias modeled as an additive uncertainty around the control. C. μ-analysis To analyze the robustness of the system via the μ-function we reresent the systems described by the two forms of erturbations as in Figs. 1 and 2 in accordance with [6, Chater 8] where the erturbations are limited to that of a single δ. Imortantly, note that although we have the controller id located in the feedback ath, there is no measurement erformed on the wavefunction to be comared against a reference signal in order to drive the dynamics. Rather, the controller alters the energy landscae of the system to modify the natural evolution of the system in a re-determined manner [2]. Here, we define the initial condition as a disturbance at the lant inut so that w = IN is our generalized disturbance. We assign the generalized error as in [5] as z = C ψ where C is an (N 1) N matrix with rows that form a basis for the orthogonal comlement of our desired outut OUT. In both cases Δ is an N N diagonal matrix that consists of δ k,k+1 or δ k times the identity matrix. η and v are signals used to close the loo around the uncertainty that s been ulled out of the system. Solving in terms of the generalized inuts and oututs yields the following for the couling uncertainty and bias sillage cases resectively: v is k,k+1 Φ is k,k+1 Φ is k,k+1 Φ η η ( z ) = ( CΦ CΦ CΦ ) ( w) = P 1 ( w) (4) ψ Φ Φ Φ u u v id k S k Φ id k S k Φ id k S k Φ η η ( z ) = ( CΦ CΦ CΦ ) ( w) = P 2 ( w) (5) ψ Φ Φ Φ u u where we use Φ = (si + ih) 1 to simlify the notation as in [4]. From this oint we erform a lower linear fractional transformation as er [6, Cha. 8] to ull the controller into the

3 generalized lant yielding M k = F l (P k, id) for k = 1,2 and P k is artitioned aroriately to yield: M k 11 = P 11 ip 13 D(I + ip 33 D) 1 P 31 M k 12 = P 12 ip 13 D(I + ip 33 D) 1 P 32 M k 21 = P 21 ip 23 D(I + ip 33 D) 1 P 31 M k 22 = P 22 ip 23 D(I + ip 33 D) 1 P 32 Finally, we can absorb the structured uncertainty into the lant-controller system yielding T zw = F u (M k, Δ) = M k 22 + M k 21 Δ(I M k 11 Δ)M k 12. Closing the loo from z to w with a fictitious, full uncertainty matrix Δ with dimensions consistent with z and w we obtain the system in Fig. 3 where Δ = ( Δ 0 0 Δ ) and has an obvious block diagonal structure. This use of a full uncertainty matrix Δ arises from a constraint of MATLAB s mussv function which requires a full uncertainty matrix for calculation of μ when the erturbations are comlex. As Δ closes the loo from the generalized error z to the generalized disturbance w = IN = ψ(0) we should exect Δ to be structured to only ermit influences from an error in the initial state rearation to affect the generalized error, or should be rather sarse with the only nonzero columns corresonding to entries in the state vectors that carry the comlex uncertainty in the initial state rearation. As such, the results roduced by mussv may be overly conservative. From here we can examine the robust erformance of the system in seeking a bound β such that T zw < β for all Δ 1 which from [7, Cha. 10] amounts to finding a lower β bound on the function μ Δ (M) allowing us to leverage the tools of μ-analysis to determine a measure of robustness of the system. Before roceeding, however, we must state the caveat as in [5] that though classically, nominal and robust stability are rerequisites of robust erformance, in this study, the system is not asymtotically stable in the usual sense, as the control is state selective and time-sensitive. So while we resently use the tools of μ-analysis to study robustness of the excitation transfer over a finite time window, it must be ket in mind that other tools may be necessary to study robustness in such a non-classical system generally. A. Simulation Procedure III. RESULTS As in [5] we use the model of an 11-ring with nominal XXcouling as our system of interest. For this ring size, we consider first the controllers otimized to maximize the transfer fidelity over a window [t f 0.1, t f + 0.1] [2]. For each transfer from 1 1 through 1 6 the v Δ Δ η Δ Mk 11 Mk Mk 11 Mk Mk 21 Mk N = F 22 u M k, Δ = T zw Mk 21 Mk 22 z w z w z w Fig. 3. Transformation of system to allow for μ-analysis with structured erturbation a block-diagonal Δ. v Δ η reviously executed otimization algorithm roduces a data set of u to 2000 diagonal controllers D along with the timeaveraged robability [8]. With each of these variables ordered by decreasing value of robability, we then use the simulation to test the trend between both log-sensitivity and robustness and the robability of transfer. The log-sensitivity is calculated in accordance with II-B for each D(n) otimized for the six ossible transfers within the 11-ring, taking into account the 11 ossible cases of couling uncertainty and searate 11 cases of bias sillage for each ossible transfer. This roduces a total of 132 test cases to measure the relationshi between robability of transfer and log-sensitivity. For the calculation of μ Δ (M(D(n))) we begin with each set of 2000 controllers for each of the six ossible transfers and use the system set-u detailed in II-C while leveraging MATLAB s mussv function to evaluate the lower bound on each μ Δ (M(D(n))). As in [5] we evaluate Φ = (si + ih) 1 at s = 0 to reflect that with the inut as a constant in time, it is art of the exonential regime e 0 t and thus the outut attributable to the inut is also art of this regime. Finally, we take the matrix Δ as an element of C structured blockdiagonally with the uer-left block consisting of δi for the model uncertainty and the lower-right block comosed of a full matrix. This rocess is reeated for each test case described above to ermit a comarison of robustness and robability. In addition to the data set described above, and to allow for a continuation of the results detailed in [5] we also consider the set of 1000 controllers otimized to rovide maximum fidelity within a shortest time t f for a 1 3 transfer. These controllers are reordered in descending rank based on their time-averaged robability of transfer via a numerical integration over the eriod [t f 0.1, t f + 0.1]. Then for each case of couling uncertainty from 1-2 through 11-1 and bias sillage over all eleven sins we comute the log-sensitivity and structured singular value for each of the set of 1000 reordered controllers at our disosal. This rovides another 44 test cases for our study though all are limited to a 1 3 transfer for this data set. B. Hyothesis Test and Statistical Analysis As in [4] and [5], the data gathered in the study is extremely noisy making emirical calculation of trends nearly imossible. Thus, with the large number of test cases at our disosal, we turn to statistical analysis to determine the trend or lack thereof between the metrics of interest: robability vs. log-sensitivity and robability vs. robustness. We establish the hyothesis test using the Kendall τ to measure the level of concordance between metrics. We set the null hyothesis H 0 to align with the mean of τ = 0, indicating no rank correlation between robability and log-sensitivity or robustness. We take the alternative hyothesis H 1 as negative correlation between the same metrics. Thus failure to reject H 1 indicates results inconsistent with the exectations of classical control theory. To rovide bounds on the hyothesis test we first note that the samle size in each test case is either 2000 or With such samle sizes, the Kendall τ tends to a standard normal

4 distribution [9] under the null hyothesis with a Z-score of Z τ = τ where σ σ τ = 2(2l+5). Here we use l to denote the τ 9l(l 1) number of samles (controllers) within the given data set. As such we can set the value of Tye I error as α = 0.05 in a single-tailed, negative-tailed test using the value of Z τ as the test statistic. We then take Z τ < as the indication for rejection of the null hyothesis and a strong indication of nonclassical behavior since Φ(Z τ ) = < 0.05 under this condition. Furthermore, for each case in which we reject H 0 we can associate a ower to the test based on associating the true oulation Kendall τ with the observed samle Kendall τ, in which case Z τ < would indicate a ower of 0.80 or greater for the case of our 2000 controller data sets and Z τ < would rovide the same for the 1000 controller set. We aly this hyothesis to the trends of log-sensitivity versus robability and μ versus robability for each of the 308 test cases described above, allowing for a decision on the rejection or failure to reject non-classical behavior for each transfer and each tye of erturbation based on the -value calculated above. To get a better result for the overall trends, however, we look to combine the data in such a manner as to indicate the overall decision on the existence of non-classical behavior for the entire set of ossible erturbations within each excitation transfer. As such we use Stouffer s method to combine the 11 values of Z τ for the couling uncertainty and bias sillage test cases using Z s = 11 k=1 Z τk, allowing for 11 calculation of the overall -value for each distinct transfer based on a Stouffer -value of S = Φ(Z S ) [10 and 11]. As a recondition for using Stouffer s method to synthesize -values, however, it s necessary that each exeriment (test case) be indeendent. We justify the indeendence among all test cases by the randomness of the numerical otimization scheme [4]. C. Couling Uncertainty Results The results of the hyothesis test alied to the logsensitivity and robustness to couling uncertainty are summarized in Table I. Note that the relationshi between robability and both log-sensitivity and μ reject the null hyothesis with an overall -value of zero to four decimal laces, indicating a very strong negative correlation among the metrics for the transfers 1 1, 1 2, and 1 3. The transfers to the remaining sins then show highly classical behavior in resonse to the set of couling uncertainties with -values of unity to four decimals laces. Thus we see that using Stouffer s Method to allow for the combination of -values almost roduces a zero-one hyothesis test for the existence of anti-classical behavior with a shar change in the system behavior as we move from the 1 3 to 1 4 transfers. This agrees with the results of [4] in which we see the highest levels on nonclassical behavior to couling uncertainty in the transfers that are in hysical roximity to the initial sin. As the target sin is moved to the antiodal oint on the ring, however, we regain the classical relations between robability and logsensitivity and robustness that one would exect. In Fig. 4 we see an illustration of these non-classical trends as borne out by the statistical tests. In like manner, the relation in Fig. 5 between the robability and log-sensitivity shows that the ring exhibits almost zero sensitivity to arameter variations for those controllers that allow for nearly erfect fidelity, again in contradiction to classical exectations. On the other hand, in Fig. 6 we see an illustration of the classical behavior for couling uncertainty between sins 5 and 6 in a 1 6 transfer. Here the hyothesis test rejects Fig. 4. Plot of the logarithm of μ versus logarithm of robability for a 1 to 2 transfer with couling uncertainty between sins 11 and 1 illustrating the overall trend of decreasing robustness with decreasing robability. Table I: Results of hyothesis test alied to the case of couling uncertainty. Shaded boxes indicate rejection of the null hyothesis and strong non-classical behavior. Couling Uncertainty Summary - 11 Ring dt-data Transf Probability and Probabillity and Log Sensitivity Mean t Mean Z Stouffer Mean t Mean Z Stouffer 1-> > > > > > Fig. 5. Plot of the log-sensitivity versus robability for couling uncertainty between sins 11 and 1 in an excitation transfer from sin 1 to 2 illustrating the overall negative trend between the two metrics. Note that the controllers that allow for almost erfect fidelity also have vanishing sensitivity, in contradiction to the exectations of classical control.

5 Fig. 6. Consolidated lot of metrics for the case of a 1 to 6 transfer and couling uncertainty between sins 5 and 6. Note the very close concordance between the log-sensitivity and the robability, esecially in the region for controllers between 1000 and the ossibility of non-classical behavior with -values of near unity for both log-sensitivity and μ versus robability. This classical trend is easily observed from the grah. For the case of the 1000 controller data set used in [5], Table II rovides the results of the hyothesis test alied to each case of couling uncertainty for the 1 3 transfer. For the 22 available test cases, we see that 20 resent nonclassical trends and of these 20, 18 cases reject the nullhyothesis with a ower of 0.80 or greater, indicating very strong non-classical behavior. We do note, however, that the only test cases that fail to reject the null hyothesis with this ower threshold are those with couling uncertainty or bias sillage on the sins in the shortest hysical ath between the initial and target sin, indicating a trend toward more classical behavior with erturbations in these locations. D. Bias Sillage Results As with the couling uncertainty results, the results of the hyothesis test when taken over bias sillage are summarized in Table III. Here we see rejection of the null hyothesis in only three situations: between both μ and log-sensitivity and robability for the case of localization about the initial sin and between log-sensitivity and robability for the case of a 1 2 transfer. Though the overall results of the hyothesis test indicate far more classical behavior for erturbations in the form of bias sillage, we do again see that the excitation transfers with a target sin closest to the initial sin exhibit the most non-classical behavior. As Fig. 7 reveals for the case of localization of the excitation about 1, both μ and log-sensitivity steadily increase as the robability of transfer decreases. This should be somewhat exected as these cases are indicative of Anderson localization, erhas the most non-classical behavior ossible in a quantum ring. Finally, as an illustration of the classical behavior indicated by accetance of the null-hyothesis, we can refer to the grah of Fig. 8. Here it is clear that both μ and logsensitivity decrease in concordance with the robability. Table II: Results of hyothesis test for the case of a 1 to 3 transfer with the 1000-controler data set and taken across all ossibilities of a single couling uncertainty. Note that the only test cases that fail to reject the null hyothesis with a ower of at least 0.80 are those in which the couling uncertainty is hysically located between the initial and target sins. (Here μ Actual denotes the true oulation mean of the metric.) t for Couling Prob and Z-Score Uncertainty Table IV summarizes the results of the hyothesis test alied to the 1000-controller set secific to a 1 3 transfer. We note the overall mixed results of the hyothesis test in these cases but also again see the trend of more classical behavior as either the bias sillage or couling uncertainty is in hysical roximity to the transfer ath. IV. CONCLUSION Accet or Reject Would Power > 0.80 Under t = Actual Reject No Reject No Reject Yes Reject Yes Reject Yes Reject Yes Reject Yes Reject Yes Reject Yes Reject Yes Reject Yes Stouffer Statistics: Reject t for Prob Couling and Log Z-Score Uncertainty Sensitivity Accet or Reject Would Power > 0.80 Under t = Actual Accet Accet Reject Yes Reject Yes Reject Yes Reject Yes Reject Yes Reject Yes Reject Yes Reject Yes Reject Yes Stouffer Statistics: Reject We have demonstrated that in examining the log-sensitivity and robustness of quantum rings controlled by static fields to maximize the robability of transfer of a single excitation that the limits imosed by classical control need not necessarily Table III: Results of hyothesis test alied to the case of bias sillage. Shaded boxes indicate rejection of the null hyothesis and strong nonclassical behavior. Bias Sillage Summary - 11 Ring dt-data Trans Probability and Probabillity and Log Sensitivity Mean t Mean Z Stouffer Mean t Mean Z Stouffer 1-> > > > > >

6 Table IV: Results of hyothesis test for the case of a 1 to 3 transfer with the 1000-controller data set and taken across all ossibilities of a single bias sillage. (Here μ Actual denotes the true oulation mean of the metric.) Fig. 7. Consolidated lot of metrics for localization at the initial sin and bias sillage on sin 6. Note the steady increase in the log sensitivity as the robability of transfer decreases from unity and the shar utick in μ. Fig. 8. Consolidated lot of metrics for a transfer from sin 1 to sin 5 and bias sillage on sin 3. Here we see a decrease of both log-sensitivity and μ in concert with decreasing robability, save for a sike in both metrics around controller index aly in all cases. In articular we see a general trend of greater non-classicality for transfers between sins in relatively close hysical roximity. We also note that for cases in which the hysical location of the uncertainty in couling strength or bias sillage is in roximity to the excitation transort ath, the results more closely follow those anticiated by classical control. Paradoxically, when the source of uncertainty is hysically located on the oosite side of the ring from the excitation transfer, we are more likely to see non-classical trends. Looking forward, it s necessary to extend these results beyond that of an 11-ring to see if these trends can be generalized to systems of arbitrary rings with arbitrary transfers. Finally, it still remains to formulate a model that exlains the change from non-classical to classical behavior as the target sins moves to the anti-odal oints of the ring. REFERENCES [1] D. Loss and D.DiVincenzo. Quantum Comuting with Quantum Dots. Physical Review Letters, A 57, 120, Sillage Sin t for Prob and Z-Score Accet or Reject Would Power > 0.80 Under t = Actual Accet Reject Accet Accet Reject Yes Reject Yes Reject Yes Reject Yes Reject Yes Reject Yes Accet Stouffer Statistics: Reject Sillage Sin t for Prob and Log Sensitivity Z-Score Accet or Reject Would Power > 0.80 Under t = Actual Accet Accet Accet Accet Accet Reject Yes Reject Reject Reject Yes Accet Accet [2] S. G. Schirmer, E. Jonckheere, and F.C. Langbein. Design of Feedback Control Laws for Information Transfer in Sintronic Networks, available at arxiv: [3] F. C. Langbein, S. G. Schirmer, and E. Jonckheere. Time Otimal Information Transfer in Sintronic Networks. In IEEE Conference on Decision and Control, , Osaka, Jaan, December [4] E. Jonckheere, S. G. Schirmer, and F.C. Langbein. Jonckheere-Terstra Test for Nonclassical Error Versus Log-Sensitivity Relationshi of Quantum Sin Network Controllers, available at arxiv: [5] E. A. Jonckheere, S. G. Schirmer, and F.C. Langbein. Structured Singular Value Analysis for Sintronics Network Information Transfer Control. IEEE Transactions on Automatic Control, in ress 2017, available at DOI: /TAC , arxiv: [6] S. Skogestad and I. Postlethwaite. Multivariable Feedback Control: Analysis and Design. John Wiley and Sons Ltd, West Sussex, England, [7] K. Zhou and J.C. Doyle. Essentials of Robust Control. Prentice Hall, Uer Saddle River, NJ, 1998 [8] Frank C Langbein, Sohie G Schirmer, Edmond Jonckheere. Static Bias Controllers for XX Sin-1/2 Rings. Data set, figshare, DOI: /m9.figshare v1 [9] Herve Abdi, The Kendall Rank Correlation Coefficeint. Encycloedia of Measurement and Statistics. Sage, Thousand Oaks, CA, [10] M.C. Whitlock. Combining Probability from Indeendent Tests: The Weighted Z-Methd is Suerior to Fisher s Aroach. Journal of Evolutionary Biology. 18, , [11] K. Tsuyuzaki. Stouffer.test: Stouffer s Weighted Z-Score (Inverse Normal Method). in metaseq Version Available at: htts:// ics/stouffer.test [accessed August 2017].