End-to-end correlated subgap states in hybrid nanowires
G.L.R. Anselmetti, E.A. Martinez, G.C. Ménard, D.Puglia, F.K. Malinowski, J.S. Lee, S. Choi, M. Pendharkar, C.J. Palmstrøm, C.M. Marcus, L. Casparis, A.P. Higginbotham
EEnd-to-end correlated subgap states in hybrid nanowires
G. L. R. Anselmetti, E. A. Martinez, G. C. M´enard, D. Puglia, F. K. Malinowski, J.S. Lee, S. Choi, M. Pendharkar, C. J. Palmstrøm,
2, 3, 4
C. M. Marcus, L. Casparis, ∗ and A. P. Higginbotham
1, 5, † Center for Quantum Devices, Niels Bohr Institute,University of Copenhagen, and Microsoft Quantum - Copenhagen,Universitetsparken 5, 2100 Copenhagen, Denmark California NanoSystems Institute, University of California, Santa Barbara, California 93106, USA Department of Electrical Engineering, University of California, Santa Barbara, California 93106, USA Materials Department, University of California, Santa Barbara, California 93106, USA Institute of Science and Technology Austria, 3400 Klosterneuburg, Austria (Dated: September 18, 2019)End-to-end correlated bound states are investigated in superconductor-semiconductor hybridnanowires at zero magnetic field. Peaks in subgap conductance are independently identified fromeach wire end, and a cross-correlation function is computed that counts end-to-end coincidences,averaging over thousands of subgap features. Strong correlations in a short, 300 nm device arereduced by a factor of four in a long, 900 nm device. In addition, subgap conductance distributionsare investigated, and correlations between the left and right distributions are identified based ontheir mutual information.
Single Majorana bound states emerge at each end ofa one-dimensional topological superconductor [1], andpairs of Majorana bound states have been proposed tononlocally encode quantum information [2, 3]. Followingthe theoretical suggestion that hybrid superconductor-semiconductor nanowires can possess a topological phase[4, 5], bound states within the superconducting gap (sub-gap states), have been extensively studied using thetunneling-conductance from a single wire end, and theresults are broadly consistent with Majorana modes [6–10]. It has also been discovered, both in experiment[11, 12] and theory [13–19], that localized non-topologicalor quasi-Majorana bound states can mimic many sig-natures of well-separated Majorana bound states. Fur-ther, quantum-dot experiments can give information onthe spatial extent of subgap states [12, 20–22]. Probingboth ends of the Majorana wire has been proposed todistinguish local states from Majorana modes by reveal-ing end-to-end correlations between Majorana pairs, andbulk signatures of the topological transition [19, 23–27].Independent tunneling spectroscopy of both wire ends– which is fundamentally required to assess the pres-ence of end-to-end correlations – has so far been chal-lenging due to an inability to non-invasively ground thewire bulk. Top-down lithography can interfere with theproximity effect [28, 29], and semiconductor T-junctionshave a poorly understood effect on the topological phase.The recently-demonstrated selective area growth (SAG)of superconductor-semiconductor heterostructures offersan appealing solution to this problem, as it allows thelithographic processing on epitaxial contacts to be iso-lated from the delicate superconductor-semiconductorinterface [30, 31]. More broadly, SAG constitutes apotential platform for multi-terminal superconductor-semiconductor devices such as qubits [32–35], Cooper-pair splitters [36], and multi-terminal Josephson junc- tions [37–41].In this Letter, we demonstrate that many of the chal-lenges associated with studying end-to-end correlationsin Majorana nanowires have been overcome. We iden-tify subgap features at both ends of three-terminal SAG (a) (b)(c) short device long device V p V LB V Al V RB V R I L I R I L V L I L FIG. 1. (a) Scanning electron micrograph of the short devicewith 300 nm superconducting segment. (b) Scanning electronmicrograph of the long device with 900 nm superconductingsegment. (c) Conceptual sketch of the devices showing theInP substrate, SiOx mask, Al lead (blue), Ti/Au Ohmic con-tacts (yellow), and Ti/Au electrostatic gates (red) separatedby an HfO layer. Left bias voltage, V L , left measured cur-rent, I L , right bias voltage, V R , right measured current, I R ,aluminum voltage, V Al , left-barrier voltage, V LB , right barriervoltage, V RB , and plunger voltage, V P , labeled. a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p devices at zero magnetic field, and quantify the corre-lations by averaging over thousands of subgap features.Varying the length of the superconductor-semiconductorhybrid segment, we find that correlations are reduced bya factor of four in a long, 900 nm nanowire, as comparedto a short, 300 nm nanowire. We also study the mutualinformation between the left and right subgap conduc-tance distributions, finding subtle signatures of conduc-tance correlations which are especially pronounced for afamily of stable, zero-energy features. This work demon-strates an experimental protocol for quantifying end-to-end correlations and determining the associated lengthscales. Application of these results at non-zero magneticfield will enable searches for correlated Majorana pairswith statistical significance.The platform for observing end-to-end correlations isAl-InAs SAG, grown by chemical beam epitaxy (CBE) onan InP substrate with a graded InAsP buffer layer [42],with epitaxial Al deposited in situ immediately af-ter semiconductor growth. Following growth, Al (blueFig. 1) was selectively removed from the substrate andthe wire ends by a wet etch, and Ti/Au Ohmic contacts[yellow Fig. 1] were deposited on both ends of the wire.The device was then covered by a global HfOx dielec-tric before electrostatic Ti/Au gates [red Fig. 1] are de-posited. Three devices were cofabricated, nominally dif-fering only in the size of the superconducting region. Theshort [300 nm, Fig. 1(a)] and long [900 nm, Fig. 1(b)]devices were studied in detail. An intermediate-lengthdevice did not show evidence of superconductivity andwas damaged due to electrostatic discharge; it was notstudied further. All measurements were performed ina dilution refrigerator with base electron temperature <
100 mK. To focus solely on the identification of end-to-end correlations without reference to the topologicalregime, the magnetic field is fixed to B = 0; finite-fieldeffects are explored in a partner paper [43].Both devices have three terminals for electrical mea-surement [Fig. 1(c)]. Two normal-conducting contactswere used for tunneling spectroscopy, and were equippedwith Basel SP983 I-to-V converters ( <
100 Ω inputimpedance). The left normal contact sources a voltage V L and measures a current I L . Likewise, the right nor-mal contact sources a voltage V R and measure a current I R . The third terminal is formed from a selective etchof the epitaxial Al, and is set to a voltage V Al . Becausethe third terminal is fabricated from a subtractive pro-cess after growth, it can be formed without disruptingthe fragile Al-InAs interface. Electrostatic gates, V LB and V RB , tune the coupling to the left and right leads,and a plunger gate, V P , tunes the electrochemical poten-tial within the wire. This setup allows the left conduc-tance, g L = δI L /δ ( V L − V Al ), and the right conductance, g R = δI R /δ ( V R − V Al ) to be independently measuredusing lock-in techniques. The barriers are adjusted suchthat the above-gap conductance, averaged over gate volt- FIG. 2. (a) Measured right conductance, g R , versus plunger, V p , and right bias, V R . (b) Measured left conductance, g L ,versus plunger, V p , and left bias, V L . (c) Left and right con-ductance measured at fixed plunger voltage indicated by hori-zontal lines in Fig. 1(a,b). V R,L denotes V L for g L , and V R for g R . Data are from the short device. (d) Measured right con-ductance, g R , versus plunger, V p , and middle lead bias, V Al .(e) Measured right conductance, g R , versus plunger, V p , andmiddle lead bias, V Al . (f) Left and right conductance mea-sured at fixed plunger voltage indicated by horizontal lines inFig. 1(d,e). V R,L denotes V L for g L , and V R for g R . Data arefrom the long device. age, is < .
15 e / h.To study subgap structures in the short device, theright and left conductance, g R and g L , were measured asa function of bias and plunger gate with V Al = 0 fixed.The data were obtained by sweeping V R while measur-ing g R with V L = 0 fixed, and then sweeping V L whilemeasuring g L with V R = 0 fixed. After these sequentialbias sweeps, V P was incremented and the process was re-peated. The resulting right-side tunneling conductance[Fig. 2(a)] exhibits a region of suppressed conductanceat low bias, a characteristic of a superconductor, withdiscrete features inside the gap that oscillate as a func-tion of plunger gate, a characteristic of Andreev boundstates. The left-side tunneling conductance [Fig. 2(b)] FIG. 3. (a) Extracted conductance-peak plunger voltage, V P ,and bias voltage, V R,L for the two lowest-energy states fromshort-device data in Fig. 2(a) [squares] and Fig. 2(b) [dia-monds]. Solid square and diamond markers denote pointsindicated in Fig. 2(c). (b) Extracted conductance-peakplunger voltage, V P , and bias voltage, V Al for the two lowest-energy states from long-device data in Fig. 2(d) [squares] andFig. 2(e) [diamonds]. Solid square and diamond markers de-note points indicated in Fig. 2(f). (c) Correlator C of thebinary peak masks for the data in Fig. 3a [dashed] and thefull 2353-peak dataset [solid] from the short device. (d) Cor-relator C of the binary peak masks for the data in Fig. 3b[dashed] and the full 2058-peak dataset [solid] from the longdevice. Shaded regions represent ± σ error bands derivedfrom plunger-shifted data. likewise exhibits a region of suppressed conductance atlow bias, with discrete oscillating features inside the gap.The subgap structure observed on the left and right sidesare similar. Directly comparing g L and g R at fixed gatevoltage, as in Fig. 2(c), emphasizes that subgap peaksgenerally occur at the same bias voltage, suggesting thatleft and right peaks originate from the same Andreevbound state. Although peaks are generally observed atthe same bias voltages, the conductance associated withthese peaks does not show a qualitatively clear corre-spondence. On both the left and right side, the subgapconductance is not a symmetric function of bias, whichis the focal point of a partner paper [43].It is interesting to note that the low-bias features inFigs. 2(a,b) are isolated from higher-lying states, stableat zero energy, and present on both ends of the device –signatures traditionally associated with Majorana modes.We emphasize that the data are taken at B = 0, wheretopological effects are not generally expected [4, 5].Turning to the long device, g R versus gate and bias ex-hibits a gapped low-bias conductance with discrete sub-gap features [Fig. 2(d)], qualitatively similar to the short NMINMI PNMIPNMI
FIG. 4. Sub-gap conductance pairs ( g R , g L ) for short devicebias V R,L < .
15 mV (a) and long device bias V Al < .
15 mV(b). Normalized mutual information, NMI, of the joint con-ductance distribution as a function bias-shift, δV , and plungershift, δV P for the short device (c) and long device (d). Point-wise normalized mutual information, PNMI, for short devicesubgap states (a) and for long device subgap states (b). device. The left-side conductance, shown in Fig. 2(e),also exhibits a gapped low-bias conductance and discretesubgap features. Unlike the short device, in the long de-vice the left and right subgap features exhibit markedlydifferent gate-voltage dependences. A direct comparisonof g L and g R for the long device, shown in Fig. 2(f), con-firms that subgap peaks on the left and right occur atdifferent bias voltages, suggesting that the subgap fea-tures originate from different Andreev bound states. Wenote that the long-device data are acquired differentlythan the short device data; by sweeping the aluminumbias V Al and simultaneously measuring the currents onthe left and right with V R,L = 0. This method has theadvantage of simultaneous data acquisition on the leftand right sides, but is disadvantageous in the short de-vice, where the left and right biases must be carefullytrimmed to zero to avoid complications from nonlocal ef-fects. We have directly compared the two methods on asmall subset of the data and found that they are in agree-ment [44]. Taken together, the observations in Fig. 2 sug-gest a picture where individual subgap states can extendacross the short device, but are associated with a singleend of the long device.To further analyze the conductance data, subgap peaksare identified using a peak finding algorithm over a widerange of gate voltages, of which the data in Fig. 2 isa small subset. A total of 2353 peaks are extractedfrom the full short device dataset, and 2058 peaks areextracted from the full long device dataset. A plot of theidentified peak locations for each plunger-gate value fromFig. 2 demonstrates that the peaks are identified accu-rately, and emphasizes that the sugap peaks are indeedgenerally correlated for the short device [Fig. 3(a)], butare uncorrelated for the long device [Fig. 3(b)].The correlations are quantified by introducing binarypeak masks for the left side, B L ( V, V P ), and the rightside, B R ( V, V P ) which take a value of 1 if a peak is iden-tified at bias voltage V and plunger voltage V P and avalue of 0 otherwise. The cross-covariance, C , betweenthe left and right binary masks is then computed, allow-ing for a bias offset, δV , between the two sides, C ( δV ) = (cid:68) B L ( V, V P ) B R ( V + δV, V P ) (cid:69) (1) − (cid:68) B L ( V, V P ) (cid:69)(cid:68) B R ( V, V P ) (cid:69) , where (cid:104)·(cid:105) denotes the expectation value with respect toboth V and V P . C serves as a correlation metric by count-ing the number of coincident peaks in each bias trace,averaged over plunger voltage.In the short device, C is peaked around δV = 0. In-cluding only the subset of data from Fig. 3(a) results ina correlator that is significantly less than unity [dashedtrace, Fig. 3(c)], indicating that on average less than onepeak is identified as being correlated at each gate volt-age. This is a result of the stringent definition of C ; itcounts peaks as correlated only if they occur at identicalbias voltages. Indeed, inspection of the data in Fig. 3(a)reveals that every peak has a nearby partner, while onlysome occur at identical parameters. Computing C forthe full short-device dataset [solid trace, Fig. 3(c)] in-creases the average number of coincident subgap peaksto C = 0 .
97. For large bias shifts, the correlator C hassmall fluctuations, consistent with the background levelinferred by introducing a gate-voltage shift between thedatasets.In contrast, for the long device, no significant correla-tions are observed from the subset of dataset in Fig. 3(b)[dashed trace, Fig. 3(d)], and a small correlation peak of C = 0 .
27 is resolved when averaging over the full dataset[solid trace, Fig. 3(d)]. The observed drop in correla-tions, by a factor of 3 .
7, presumably reflects a charac-teristic length-scale for subgap states in these structures.By examining subsets of the long device dataset, we have identified a small region in gate voltage that gives rise tothe correlations [44], demonstrating that C is a useful toolfor identifying rare features in the data.While the presence of subgap features in the shortdevice is strongly correlated between both ends, theconductance associated with these features fluctuatesstrongly. To study correlations in these fluctuations,each measurement of the right and left conductances isvisualized as a point in the ( g R , g L ) plane. The sub-gap ( g R , g L ) data for the short device [Fig. 4(a)] is in-deed widely distributed, with a long tail extending out to g R,L ∼ . / h. In the long device, the subgap ( g R , g L )data are also broadly scattered, but the ( g R , g L ) pairstend more towards the axes in the g R , g L plane [Fig. 4(b)].In other words, there are more points with large weighton both sides in the short device as compared to the longdevice, which gives a preliminary indication of positivecorrelations in conductance for the short device.To quantify this observation, joint distributions, P ( g R , g L ), and marginal distributions, P ( g R ) and P ( g L ),are estimated from the scatter-plot data. In general, forindependent variables one expects P ( a, b ) = P ( a ) P ( b ).Deviations from this relationship are quantified by thenormalized mutual information, NMI, which, in anal-ogy with a correlator, is calculated as a function of biasand plunger shift between the left and right datasets,NMI = NMI( δV, δV P ) [44].For the short device, the mutual information isstrongly peaked at δV = δV P = 0 [Fig. 4(c)], indicatingthe presence of correlations in the conductance distribu-tions. As a function of bias shift, δV , the mutual infor-mation decays with a 1 /e characteristic length that ap-proximately matches the observed width of subgap peaks,suggesting that the conductance correlations are associ-ated with subgap states. As a function of plunger shift, δV P , the mutual information also decays sharply, againconsistent with contributions from subgap states. In con-trast, for the long device, the mutual information is rel-atively flat and featureless [Fig. 4(d)], suggesting thatthere are no conductance correlations originating fromthe subgap states. The mutual information for the longdevice, however, still differs significantly from zero, in-dicating there are broad, gate and bias-voltage indepen-dent features that are mutually dependent between thetwo ends.A more granular view of the conductance correlationsis obtained by visualizing the pointwise normalized mu-tual information, PNMI, which is defined by the relationNMI = PNMI, where the overline denotes an expectationvalue over P ( g R , g L ). Whereas NMI is a property of theentire distribution, PNMI is associated with individual( g R , g L ) pairs, and can therefore be mapped as a functionof the measurement parameters. In the short device, amap of the PNMI as a function of gate and bias voltagereveals regions of elevated PNMI corresponding to thestable low-lying bound state identified in previous dis-cussion [Fig. 4(e)]. This bound state also has an elevatedPNMI compared to other correlated features throughoutthe entire dataset [44], suggesting that although manyfeatures in the short device are measurable from bothdevice ends, additional correlations in the conductanceof these features are relatively rare. In contrast to theshort device, for the long device the PNMI is smaller, al-though there is still some weak structure [Fig. 4(f)]. Theconductance correlations are not thoroughly understood,although, qualitatively, correlations in the conductancecould be expected as a result of states that span the en-tire device.In summary, we interpret the well-correlated subgapstates in the short device as a clear indication that boundstates extend several hundred nanometers into the hy-brid region, essentially spanning the entire device. If thezero-field bound states are attributed to an accidentalpotential well in the tunneling region [9], then their spa-tial extent would indicate a soft confinement, which willstrongly effect the behaviour at nonzero magnetic field[13, 14, 17]. Following this reasoning, we anticipate thatthese measurements could be used to refine the theoret-ical understanding of topologically trivial subgap states,including in numerical simulations, and possibly aid indistinguishing them from the topological case.Looking ahead, the techniques developed here can beapplied in the putative topological regime. 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G. L. R. Anselmetti, E. A. Martinez, G. C. M´enard, D. Puglia, F. K. Malinowski, J.S. Lee, S. Choi, M. Pendharkar, C. J. Palmstrøm,
2, 3, 4
C. M. Marcus, L. Casparis, ∗ and A. P. Higginbotham † Center for Quantum Devices, Niels Bohr Institute,University of Copenhagen, and Microsoft Quantum - Copenhagen,Universitetsparken 5, 2100 Copenhagen, Denmark California NanoSystems Institute, University of California, Santa Barbara, California 93106, USA Department of Electrical Engineering, University of California, Santa Barbara, California 93106, USA Materials Department, University of California, Santa Barbara, California 93106, USA
COMPARISON OF DIFFERENT BIASING CONFIGURATIONS
In the main text, the short-device and long-device datasets use different biasing configurations. Here we show thatthe two biasing methods give the same results by directly comparing them on the short device. Sweeping the right-biasvoltage, V R while measuring I R with V Al = 0 generates a map of subgap states [Fig. S1(a)], which is familiar fromthe main text. Sweeping the aluminum bias voltage, V Al while measuring I R with V R = 0 fixed reveals the sameconductance features [Fig. S1(b)], but with the dependence on bias voltage inverted, as one would expect from theother side of the right junction being biased. Note that the V R and V Al scans are interlaced so that switches and deviceinstabilities effect the two plots symmetrically. The strong similarity between Fig. S1(a) and Fig. S1(b) explicitlydemonstrates the equivalence of the two biasing configurations. FIG. S1. (a) Right conductance, dI R /dV R , measured as a function of right bias, V R and plunger, V P . In the main text, thisbiasing procedure was used for the short device. (b) Right conductance, dI R /dV R , measured as a function of aluminum bias, V Al , and plunger, V P . In the main text, this biasing procedure was used for the long device. CORRELATOR ERROR ESTIMATES AND STATISTICAL INFORMATION
It is interesting to consider the error in the correlation metric, C , that is computed in the main text Fig. 2. Onemethod to estimate the uncertainty is to apply a numerical plunger-voltage shift, δV P , to one dataset, compute thecorrelation metric C δV P , and then find the standard deviation of σ ( C δV P ) over a large ensemble of δV P . Error bandsin Fig. 3(c,d) are then given by C δV P ± σ ( C δV P ).The computed error, σ ( C δV P ), is compared with the data for the long-device correlator in Fig. S2. For large biasshifts, the correlator fluctuates but typically remains within the 1 σ error bars, confirming that the error has beenreasonably estimated. At zero bias shift, a 3 σ peak in C is observed, with 4 σ and 3 σ satellite peaks, confirming thatthe corrrelations observed in the long device are statistically significant. FIG. S2. Correlator, C , for the long-device data set. Shaded region indicates ± σ error bars, estimated from plunger-shifteddata. Colored circuls mark a 3 σ central peak, with 4 σ and 3 σ satellite peaks. When comparing the correlator between the short and long device it is useful to know that the properties of thetwo datasets are generally similar. The short-device data set has an average of 3.9 subgap peaks per bias scan, andthe long-device data set has an average of 5.1 subgap peaks per bias scan. To evaluate the typical autocorrelationof the datasets, we have computed the self-mutual information,
SN M I , of the conductances as a function of plungershift, for instance
SN M I ( δV P ) = N M I ( g R ( V P ) , g R ( V P + δV P )). The self-mutual information decays as a functionof plunger shift with a characteristic auto-correlation length. The extracted auto-correlation length is 4 mV for theshort device, and 1 . . . FULL DATASETS
For completeness, the full conductance datasets used in the main text are presented in Fig. S3. Most states in theshort device can be identified on both sides, albeit with fluctuating weights [Fig. S3(a,b)]. The conductance associatedlow-lying stable bound state discussed in the main text has an anomalously large point-wise mutual information,
P M I [Fig. S3 (c)]. For the long device, most states are uncorrelated, but there is a small region of correlated featuresresponsible for the correlator peak discussed in the main text [orange box Fig. S3(d,e)]. The conductance-
P M I forthe long device is lower than the short device, with weak structure visible corresponding to the uncorrelated features[Fig. S3(f)].
QUANTIFYING CONDUCTANCE CORRELATIONS WITH MUTUAL INFORMATION
In Fig. 4(a,b) of the main text we show the simultaneously measured left and right conductances as a scatter plotof ( g L , g R ). Our goal is to quantify conductance correlations in this data. The conductances are correlated if theycannot be described by the product of two independent conductance distributions on the left and right sides. Thestatistical measure for how much the joint conductance distribution deviates from independence is given by the mutualinformation between the left and right conductances.In the first place we seek to estimate the joint conductance probability distribution. For this we bin the measuredconductance points to calculate a histogram and estimate a discrete probability distribution p ( g L , g R ). A uniformlyspaced grid is not suitable for our measured conductance data, since the density of points changes by orders ofmagnitude in different regions of the conductance space. Therefore we use the adaptive binning algorithm of [41],where the sample space is recursively divided in rectangular bins with an equal number of points. As a criterion forcontinuing the subdivision of a bin, we require there be a minimum number of points in the bin. For the analysis inFig. 4 we stop subdividing bins with fewer than 80 points, which yields an average of 36 points per bin for the finalsubdivision. The end result of the algorithm is a grid of M rectangles A i × B i . The probability density p ( A i × B i ) FIG. S3. (a) Short device right-conductance, g R , measured as a function of gate, V P and bias, V R . (b) Short device left-conductance, g L , measured as a function of gate, V P and bias, V L . (c) Short device conductance pointwise mutual information, P MI , mapped as a function of gate, V P , and bias V R,L for subgap states. (d) Long device right-conductance, g R , measured asa function of gate, V P and bias, V Al . Orange square indicates correlated region responsible for conductance peak in main textFig. 3(d). (e) Long device left-conductance, g L , measured as a function of gate, V P and bias, V Al . Orange squares indicatescorrelated region responsible for conductance peak in main text Fig. 3(d). (f) Long device conductance pointwise mutualinformation, P MI , mapped as a function of gate, V P , and bias V Al for subgap states. in each bin i is estimated as the number of samples in the bin divided by the total number of samples. The mutualinformation, MI, is then calculated as:MI = M X i =1 p ( A i × B i ) log (cid:18) p ( A i × B i ) p ( A i × R ) p ( R × B i ) (cid:19) , (S1)where p ( A i × R ) is the probability density in the rectangle projected onto the left conductance axis, and p ( R × B i )likewise projected onto the right conductance axis. The contribution of each bin in the sample space to the mutualinformation is known as the pointwise mutual information (PMI):PMI = log (cid:18) p ( A i × B i ) p ( A i × R ) p ( R × B i ) (cid:19) . (S2)The mutual information is the expected value of the pointwise mutual information over the entire distribution.In the main text we quote the normalized mutual information, NMI, which lies in the range [0 ,
1] and is given byNMI = MI
H , (S3) where H is the Shannon entropy of the joint probability distribution: H = − M X i =1 p ( A i × B i ) log p ( A i × B i ) . (S4)The main text also quotes the normalized pointwise mutual information, NPMI, given byNPMI = PMI H . (S5)Note that different normalization choices are possible, see e.g. [42].The advantage of our selected method for calculating the mutual information is that the contribution of each bincan be easily assessed by visualizing the pointwise mutual information. This allows us to determine the relativecontribution of the measured points to the total correlation, as shown by the maps of Fig. 4(e,f) in the main text. Thedrawback of the method, as with any method that relies on binning of the data, is the dependence of the calculatedmutual information on the details of the binning. An alternative method for estimating the mutual informationof the measured distribution is a non-parametric estimator, such as the one from [43]. This method relies only onthe nearest-neighbor distances for estimating the mutual information. In Fig. S4 we compare the results obtainedfrom the adaptive binning and the non-parametric algorithms, finding that they result in a similar numeric value.The calculations of the non-parametric estimator have been done using its implementation in the Python sklearnpackage[44].
50 100 150 200Minimum samples per bin0.260.280.300.32 M I FIG. S4. Blue dots: mutual information (unnormalized) calculated with the adaptive binning method of [41] as a function of theminimum samples per bin cutoff for the subdivision algorithm. Red line: non-parametric estimate of the mutual informationfrom [43]. ∗ Equal contribution, [email protected] ††