Observing geometry of quantum states in a three-level system

Jie Xie,1, 2, ∗ Aonan Zhang,1, 2, ∗ Ningping Cao,3, 4 Huichao Xu,1, 2 Kaimin Zheng,1, 2

Yiu-Tung Poon,5 Nung-Sing Sze,6 Ping Xu,1, 2, 7 Bei Zeng,8, † and Lijian Zhang1, 2, ‡

1National Laboratory of Solid State Microstructures,Key Laboratory of Intelligent Optical Sensing and Manipulation (Ministry of Education),

College of Engineering and Applied Sciences and School of Physics, Nanjing University, Nanjing 210093, China2Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China

3Department of Mathematics & Statistics, University of Guelph, Guelph N1G 2W1, Ontario, Canada4Institute for Quantum Computing, University of Waterloo, Waterloo N2L 3G1, Ontario, Canada

5Department of Mathematics, Iowa State University, Ames, Iowa, IA 50011, USA6Department of Applied Mathematics, The Hong Kong Polytechnic University, 999077 Hong Kong, China

7Institute for Quantum Information & State Key Laboratory of High Performance Computing,College of Computer, National University of Defense Technology, Changsha 410073, China

8Department of Physics, The Hong Kong University of Science and Technology,Clear Water Bay, Kowloon, 999077 Hong Kong, China

(Dated: January 19, 2021)

In quantum mechanics, geometry has been demonstrated as a useful tool for inferring non-classicalbehaviors and exotic properties of quantum systems. One standard approach to illustrate thegeometry of quantum systems is to project the quantum state space to the Euclidean space viameasurements of observables on the system. Despite the great success of this method in studyingtwo-level quantum systems (qubits) with the celebrated Bloch sphere representation, there is alwaysthe difficulty to reveal the geometry of multi-dimensional quantum systems. Here we report thefirst experiment measuring the geometry of such projections beyond the qubit. Specifically, weobserve the joint numerical ranges (JNRs) of a triple of observables in a three-level photonic system,providing complete classification of the JNRs. We further show that the geometry of different classesreveal ground-state degeneracies of a Hamiltonian as a linear combination of the observables, whichis related to quantum phases in the thermodynamic limit. Our results offer a versatile geometricapproach for exploring the properties of higher-dimensional quantum systems.

Arising from Euclid’s first attempt of establishingits axiomatic form, geometry has become an essentialmethod for understanding the physical world, especiallyin the field of quantum physics [1–4]. An exemplaryuse of geometric method is in the creation of statisti-cal mechanics in 1870s, when Gibbs introduced a ge-ometric means to infer thermodynamic properties (e.g.energy or entropy) of a system by considering a con-vex body constituted by all possible values of physicalquantities [5]. Shifting to systems that the behaviorsare governed by quantum physics, the possible expecta-tion values of physical quantities are instead acquiredover the entire space of quantum states, mathemati-cally the set of all semi-definite matrices of trace oneMd = {ρ : ρ ≥ 0, Tr(ρ) = 1} in a d-dimensional Hilbertspace. The convex body formed by joint expectation val-ues of different observables on all quantum states canbe used as a geometric representation of quantum statespace. One of the most successful example is the Blochsphere of qubit state space [6], which has become thefundamental model in quantum information [7]. Manyworks are devoted to studying the geometry of higher-dimensional systems, such as generalizing the Bloch vec-tor [6, 8–10] and visualizing single qutrit state [11]. How-ever, there is no satisfactory geometric ways to visual-ize the whole higher-dimensional state space, which pos-sess properties dissimilar to those of qubits and start to

play indispensable roles in quantum information process-ing [12–14].In this work, we investigate the geometry of quan-

tum states by projecting the state space Md onto n-dimensional Euclidean space Rn via measurements of nobservables on the system [15–18]. This geometric con-struction allows exploration of the complicated structureand physical properties of high-dimensional quantum sys-tems through their lower-dimensional projections. Be-hind this construction is the concept of numerical rangein mathematical terminology. Back in 1918, Toeplitz [15]introduced the numerical range of a d × d complex ma-trix F , which is defined as W (F ) = {z = 〈ψ|F |ψ〉 : |ψ〉 ∈Cd, 〈ψ|ψ〉 = 1}. Here F = F1 + iF2 involves two Her-mitian matrices F1 and F2. The conjecture by Toeplitzthat W (F ) is convex was later proved by Hausdorff in1919 [15, 16]. A natural extension is termed as jointnumerical range (JNR) [18] involving a collection of Her-mitian matrices F = {F1, ..., Fn},

W (F) = {(〈ψ|F1|ψ〉, ..., 〈ψ|Fn|ψ〉) : |ψ〉 ∈ Cd, 〈ψ|ψ〉 = 1},(1)

naturally forming a geometric object in Rn. Then thestate space projection via these matrices, which allowsstatistical mixture of pure states |ψ〉, is simply the convexhull of the JNR

L(F) = {(Tr(ρF1), ...,Tr(ρFn)) : ρ ∈Md}. (2)

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In the following, we are mainly concerned with the setL(F) and do not distinguish it from W (F) (see Supple-mental Material Sec. I A [19]).

In recent years, the topic of numerical range has beenreviewed in the study of quantum phase transition [20–23], deriving uncertainty relations [24] and entanglementcriterion [25] and error correcting codes in quantum com-puting [26, 27]. Yet many characteristics of JNR are stillunknown for dimension as low as 3. Recently, Szymański,Weis and Życzkowski made a crucial step towards thisproblem by proving complete classification of the JNRin the case d = n = 3 [28]. However, experimentallyrecovering the geometry of JNR, together with its clas-sification, demands sampling adequate data, which is anon-trivial problem requiring the ability to implementarbitrary unitary operations on a qudit system [29, 30].Here we perform the first experiment allowing completeobservation of state space projections beyond qubit sys-tems.

As mentioned above, a simple example in qubit systemis the JNR of three Pauli operators known as the Blochsphere [6], whereas the JNR of other Hermitian matricesis equivalent to a transformation of the Bloch sphere, asshown in Fig. 1(a). Extended to higher-dimensional sys-tems, the geometry of JNR is associated with a class ofsystem Hamiltonians H(~h) =

∑hiFi parameterized by

~h = (h1, ..., hn) in the basis F . It is intuitive that sur-face points (extreme points) of JNR are genetrated byground-states of H(~h) (see Supplemental Material Sec.I B [19]). Geometrically, each parameter vector ~h rep-resents a inward-pointing unit vector in Rn and corre-sponds a supporting hyperplane Π tangent to the surfaceof JNR, as depicted in Fig. 1(b). The tangent point isacquired by the ground-state |ψg(~h)〉 of H(~h), while thecorresponding ground-state energy Eg can be obtainedby projecting the point (〈F1〉ψg(~h), ..., 〈Fn〉ψg(~h)) onto the~h direction. When continuously varying the parametervector ~h, the envelope of all supporting hyperplane con-stitutes the surface of JNR, which can be generated by allground-states |ψg(~h)〉. If there exists a flat portion on thesurface of JNR at a particular direction ~h, the supportinghyperplane will be tangent to the whole portion insteadof a single point. Therefore, this indicates ground-statedegeneracy in the sense that different ground-states areassociated with one system Hamiltonian H(~h) [23].In the case d = n = 3, the image of the JNR forms

a three-dimensional oval that can be classified in termsof its one- or two-dimensional faces, that is, segmentsor filled ellipses. These faces are invariant under lineartransformation and translation. According to the num-ber of segments (s) and filled ellipses (e) on the surface,all the JNRs can be divided into ten possible categories[28], among which the eight unitarily irreducible cate-gories that we measured are as follows (the other two cat-egories correspond to linearly dependent operator sets,

Quantum State Space Convex Hull of JNR

E

GL(2)

Bloch Sphere Transformation of Bloch Sphere

a

b

h

Degeneracy

O

→ h

E

}{

FIG. 1: Quantum state space projection and the jointnumerical range (JNR). (a) The state space of a qubitsystemM2 can be represented as a Bloch sphere, whichis the JNR of the three Pauli operators {σx, σy, σz}.Undergoing a linear transformation GL(2), the Blochsphere becomes an ellipsoid, as the JNR of other threelinearly independent matrices. (b) The state space of ahigher-dimensional quantum system (d ≥ 3) is a convexset with a more complicated structure. Following thequbit case, these structures can be revealed byprojecting the state spaceMd to Rn through a set of nHermitian observables. This constitute the convex hullof the JNR, whose surface can be generated by theground-states of a set of system Hamiltonians H(~h),geometrically corresponding to a set of supportinghyperplanes Π (grey, solid lines). In general,ground-state degeneracy happens when a flat portionappears on the surface.

which can be derived by lower-dimensional JNRs, seeSupplemental Material Sec. II D [19]):

s = 0; e = 0, 1, 2, 3, 4 and s = 1; e = 0, 1, 2.

To measure the JNR, one can prepare identical copiesof quantum states and obtain a triple of expectationvalues of F by separately measuring each of the threeobservables on the same states. Experimentally, qutritstates can be encoded in photons’ different degrees offreedom [31–34]. Here we prepare a single photon in thesuperposition of three modes which is equivalent to athree-level system, such as a spin-one particle. The singlephoton, generated in a heralded manner by the paramet-ric downconversion process, is initially injected into oneof the three modes. Then after two sequential two-modeunitary operations (Fig. 2(a)), the system is prepared inany pure superposition of the three levels, of the form|ψ〉 = cos θ1e

iφ1 |0〉+ sin θ1 sin θ2eiφ2 |1〉+ sin θ1 cos θ2 |2〉.

3

HWP1

a

b

c

HWP2

HWP3

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|0＞

|0'＞

|1'＞U2

|2'＞U3

PBS

SPCM

E-HWP

PR

HWP

QWP

Single Photon Source

State Preparation

Measurement

HWP"

HWPB

KDP

BBO

BD1

BD2

BD3

BD4

BD5|0＞

UA

UB

|1＞

|2＞

U1

FIG. 2: Experimental scheme. (a) Preparation of single photonic qutrit state. A single photon propagating in threemodes represents a three-level quantum system. Sequential two-mode unitary operations can evolve the system toan arbitrary superposition of the three levels. (b) Measurement of a Hermitian observable Fi. The three-outcomemeasurement is constituted by three two-modes unitary transformations followed by single-photon detections, whichprojects input states onto the three eigenstates of the observable Fi. (c) Experimental set-up. Photon pairs aregenerated through the parametric downconversion process in a phase-matched potassium di-hydrogen phosphate(KDP) crystal pumped by frequency-doubled Ti:Sapphire laser pulses and then separated by a polarizing beamsplitter (PBS). A single photon in the transmitted path is heralded by detection of a reflected photon at theheralding detector DT . The state preparation module is composed of two electronically-controlled half-wave plates(E-HWPs), two phase retarders (PRs) and two calcite beam displacers (BDs). The measurement part is composedof wave plates, BDs and three single photon counting modules (SPCMs). For some observables, a quarter-wave plate(QWP) is inserted before BD3. Unlabelled HWPs are set to 45◦ or 0◦.

This operation is realized by the state preparation mod-ule in Fig. 2(c), which involves wave-plates togetherwith a calcite beam displacer (BD) to distribute the sin-gle photons among three optical modes and two variablephase retarders to manipulate phases between differentmodes. The three optical modes are defined as the hor-izontal and vertical polarizations of a single path mode(top) and the vertical polarization (the horizon polar-ization is not used) of another path mode (lower). Themeasurement of any Hermitian observable (Fig. 2(b)) isrealized by a three-outcome quantum measurement col-lapsing the state onto one of the three eigenstates of theobservable Fi. The state firstly undergoes a unitary evo-lution consisting of three sequential two-mode unitaries,so that the input state is transformed in the eigen-basisof the observable Fi [29]. This transformation is accom-plished by three cascaded interferometers formed by a setof wave-plates and BDs, as shown in the measurementpart of Fig. 2(c). At the end three single photon count-ing modules (SPCMs) are used for the detection of singlephotons. Clicks of the three detectors indicate measur-ing the corresponding eigenvalues λ(i)

j (j = 0, 1, 2) of Fi,so that the expectation values of Fi can be estimatedthrough the measured probability distribution {pj}. The

wave-plates in the measurement stage can be configuredin different settings, thus enabling measurements of dif-ferent observables with respect to the same set of states.

For each class of JNR we provide an example of the3-tuple of observables F (see Supplemental Material Sec.II B [19]) being measured in our experiment. Given theset of JNR is convex for n = 3 [35] and any interior pointcan be obtained by the mixture of surface states, measur-ing pure surface states is enough for the observation ofJNR. We randomly sample 300 ground-states |ψg(θ, φ)〉of system Hamiltonians

H(θ, φ) = sin θ cosφF1 + sin θ sinφF2 + cos θF3 (3)

for each class (see Supplemental Material Sec. II A [19]).Here (θ, φ) defines the unit vector ~h = (sin θ cosφ, sin θsinφ, cos θ). Then we measure the expectation valuesof the three observables with respect to these ground-states. Figure 3 illustrates the experimental resultsfor the exemplary JNRs of eight classes. The exper-imental results are in line with the theoretical predic-tions, as they are very closed to the surface of the the-oretical predictions. The average similarity S betweenexperimentally measured probability distributions {pj}

4

s=0,e=0 s=0,e=1 s=0,e=2 s=0,e=3

s=0,e=4 s=1,e=0 s=1,e=1 s=1,e=2

a b c d

e f g h

FIG. 3: Complete observation of the joint numerical range (JNR). (a)-(h) Experimentally measured data (red dots)and the theoretical predictions of JNR (chromatic convex bodies) for the examples of the eight possible classes.Each class is specified by the number of segments (s) and filled ellipses (e) on the surface (ploted with gray lines) ofthe convex body. The experimental data are obtained by measuring the surface states (ground-states) of the systemHamiltonians (Eq. (3)). For each class, 300 surface states are sampled. The theoretical bodies are plotted bysampling adequate ground-states (more than 3000 for each class) and then generating the convex hull of thecorresponding theoretical points of the JNR.

and the theoretical values {pthj } is above 0.994, withS = (

√p0pth0 +

√p1pth1 +

√p2pth2 )2. The convex hulls

of the experimental data also show the same geometricalfeatures (the number of s and e). Deviations betweenthe observed data and the theoretical values are mainlyattributed to systematic errors in the settings of exper-imental parameters (see Supplemental Material Sec. IIE [19]). Apart from the first class, the rest seven classesof the JNR are different from either a sphere or an el-lipsoid, showing distinct properties of qutrit state spacefrom those of qubits.

Following the complete observation of the geometricbodies, next we show how the geometry of JNRs in Fig.3 determines the ground-state energies and degeneraciesof the system Hamiltonians. In Fig. 4, by combining allthe experimental results (〈F1〉, 〈F2〉, 〈F3〉) of F with theircorresponding unit vector (θ, φ), we obtain the expecta-tion value E of the system Hamiltonian H(θ, φ) and givethe diagrams of E (red dots) versus θ and φ. The re-sulting energies are in line with theoretical prediction ofground-state energies (the lower colored surfaces) withinexperimental errors. In particular, there is a clear corre-spondence between the segments and ellipses in Fig. 3and the degeneracy points in Fig. 4. For example, thefirst class (Fig. 4(a)) corresponds to a gaped Hamilto-nian and cannot see any flat portions from the surface ofits JNR. As for the case s = 1, e = 2, there are three de-

generacy points in the band structure diagram. Two arecone-shaped which are non-analytic in all directions andcorresponds to the two ellipses on the JNR. The otheris Λ-shaped which is non-analytic only in one directionand corresponds to the segment. These various bandstructure diagrams, which can be geometrically revealedby the surfaces of JNRs, also show distinct features ofthree-level quantum systems compared to two-level sys-tems.For quantum matters at the zero temperature, the

ground-states associated to a class of Hamiltonians areregarded as ‘quantum phases’ of the matter. The degen-eracy of ground-states, usually indicating a gap closing, isan important indicator of different quantum phases [36],such as symmetry breaking, topological ordered and gap-less phases. As demonstrated in our experiment, the flatsurfaces (ellipses and segments in the case d = n = 3)on an image of JNR indicate ground-state degeneraciesof the Hamiltonian H(~h), thus represent different phasesof the system. Therefore, the surface of JNR can beviewed as an intuitive geometric representation of quan-tum phases, on which different regions represent differentphases.We experimentally identify the geometry of a three-

level quantum system by observing the complete clas-sification of joint numerical ranges (JNRs) of three ob-servables. The results highlight the distinct difference

5

E E

EE

E E

EE

s=0,e=0 s=0,e=1 s=0,e=2 s=0,e=3

s=0,e=4 s=1,e=0 s=1,e=1 s=1,e=2

a b c d

e f g h

FIG. 4: Determining ground-state energy and degeneracy. (a)-(h) The band structures of the lowest two bands forthe system Hamiltonians of the eight JNR classes. Colored surfaces represent the theoretical energy of theground-state and the first excited state of the Hamiltonian H(θ, φ) (Eq. (3)) for varied θ, φ. The experimentalresults (red dots) are computed by the experimental data of each class of JNR, as indirectly measured expectationvalues of H(θ, φ). The number and features of degeneracy points shown in the band structure diagrams have acorrespondence with the number of segments and ellipses on the surfaces of JNR.

between high-dimensional quantum systems and qubits.Furthermore, we elucidate the relation between the ge-ometric characters of the JNR and ground-state degen-eracies of the system Hamiltonians. Our work opens theavenue to experimentally explore fascinating phenomenaof quantum systems via its state space projections. Thisgeometric method is also applicable to many-body sys-tems, when the set of observables are happen to be lo-cal Hamiltonians of the system, which may be easier toimplement than the global ones [37]. Besides, the con-cept of JNR has shown a wide range of potential ap-plications. As convex sets, JNRs have been adopted inthe N -representability problem [21, 38] and the quantummarginal problem [39] for visualizing the set of reduceddensity matrices. In the field of quantum information, itfinds applications in the derivation of entanglement wit-nesses [25, 40] and uncertainty relations [24, 41], and pro-vides theoretical foundations for the self-characterizationof quantum devices [42, 43]. We expect this versatile con-cept to promote further investigations in understandingthe geometry of quantum systems, inferring intriguingphases and properties of quantum matters, as well as de-veloping novel technologies in quantum information sci-ence.

The authors thank Y. Zhao, H. Zhang, and T.Lan for enlightening discussions. This work was sup-ported by the National Key Research and Develop-ment Program of China (grant nos. 2017YFA0303703and 2018YFA030602) and the National Natural ScienceFoundation of China (grant nos. 11690032, 91836303,61490711, 11474159 and 11574145). N.C. was supportedby Natural Sciences and Engineering Research Council

(NSERC) of Canada. Research of Sze was supported bya HK RGC grant PolyU 15305719.

∗ These authors contributed equally to this work.† zengb@ust.hk‡ lijian.zhang@nju.edu.cn

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7

SUPPLEMENTAL MATERIAL

DETAILED THEORETICAL INFORMATION

Numerical range and its extensions

Mathematically, Numerical Range (NR)W (F ) of a d×d matrix F is{〈x|A |x〉 , |x〉 ∈ Cd, 〈x|x〉 = 1

}. Physicists are

generally more interested in the case of F being a Hermitian matrix, i.e. F = F †. It is straightforward to observe thatW (F ) is the set of expectation values by measuring the observable F with pure quantum states. The first importantresult of NR is called Toeplitz-Hausdorff theorem which states NRs are convex and compact[15, 16].

A natural generalization of Numerical Range is that, instead of measuring one observable, one may measure a setof linearly independent operators F = {F1, · · · , Fn}. This is so called Joint Numerical Range (JNR)

W (F) ={

(〈x|F1 |x〉 , 〈x|F2 |x〉 , · · · , 〈x|Fn |x〉), |x〉 ∈ Cd, 〈x|x〉 = 1}. (4)

The JNR of {F1, F2} can be merged into NR since it equals to the NR of A = F1 + iF2. Therefore, people usuallyconsider n ≥ 3 while studying JNR. In contrast with Numerical Range, JNR is usually not convex [35, 44]. In otherwords, the set of measurement outcomes that using only pure states to measure a set of observables may not beconvex. The next problem is how to characterize measurements according to mixed states.

In 1979, Au-Yeung and Poon [35] studied a generalized JNR

W (r)(F) ={(

r∑i=1〈xi|F1 |xi〉 ,

r∑i=1〈xi|F2 |xi〉 , · · · ,

r∑i=1〈xi|Fn |xi〉

)∣∣∣∣ r∑i=1〈xi|xi〉 = 1, |xi〉 ∈ Cd

}. (5)

When r = 1, it reduces to standard JNR where r denotes the largest degree of quantum states used to measure theoperator set F . Notice that |xi〉 in Eq. (5) is not normalized. The probability pi of each pure state component isabsorbed in |xi〉. Rewrite |xi〉 as

√pi |x′i〉, where 〈xi|xi〉 = 1 and

∑i pi = 1, We can reformulate W (r)(F) as

W (r)(F) ={

(Tr(F1ρ),Tr(F2ρ), · · · ,Tr(Fnρ))∣∣∣∣ r∑i=1

pi|x′i〉〈x′i| = ρ,

r∑i=1

pi = 1, 〈x′i|x′i〉 = 1, |x′i〉 ∈ Cd, pi ∈ [0, 1]}. (6)

In [35], a lower bound forW (r)(F) to be convex has been provided. For the scenario considered in this paper, whichis qutrit systems (d = 3) with three linearly independent observables (n = 3), the measurement results of pure states(JNR of F , W (F)) is convex. Moreover, in qutrit systems, if we allow the use of mixed states up to degree 2, thenW (r)(F) is convex whenever |F| < 8.

Another concept introduced in [45] is the joint algebraic numerical range (JANR)

L(F) ={

(Tr(F1ρ),Tr(F2ρ), · · · ,Tr(Fnρ))∣∣∣∣ρ is a quantum state

}. (7)

Clearly, L(F) = Cov(W (F)). Since W (F) is convex for the case discussed in this paper (d = n = 3), we do notdistinguish the difference between the JANR and JNR in the following sections.

The structure of joint algebraic numerical range

In this section, we discuss the structure of joint algebraic numerical range (or the convex hull of JNR).Firstly, we show that the boundary of JANR of a linearly independent set F of operators can be reached by ground-

states of Hamiltonians H =∑ni=1 hiFi, where (h1, · · · , hn) ∈ Rn. Firstly, it is obvious that the point (〈F1〉, ..., 〈Fn〉)

with minimum value of 〈F1〉 can be obtained by the ground-state of F1, which is equivalent to making a tangent(hyper)plane of L(F) with inward-pointing normal vector (1, ..., 0). The resulting is the wanted extreme point. Thistangent point also corresponds to the point obtained by the ground-state of H((1, ..., 0)) = F1. Therefore, In orderto get other surface points, one only need to rotate the (hyper)plane with inward-pointing normal vector (1, ..., 0) toother direction ~h and tangent to L(F). This new tangent point can be obtained by the ground-state of a rotatedoperator F ′ = ~h · {F1, ..., Fn}, which is exactly the system Hamiltonian H(~h). And we call the tangent (hyper)plane

8

2.5 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5F1

2.0

1.5

1.0

0.5

0.0

0.5

1.0

1.5

F 2

Ground States1st Excited States2nd Excited States

FIG. 5: The Joint numerical range of two 9× 9 random generated hermitian operators {F1, F2}. (d = 9 and n = 2)The colored area is the JNR W (F) of F = {F1, F2}. The blue dots are the measurements according to ground-statesof H = h1F1 + h2F2, which h1 and h2 are random generated. The cyan dots is formed by the first excited states ofH. The green dots correspond to the second excited states.

with normal vector ~h as the supporting hyperplane, denoted as Π. Thus the ground-state energy Eg can be writtenas Eg = ~h · (〈F1〉ψg

, ..., 〈Fn〉ψg).

Notice that H is fully parameterized by (h1, · · · , hn) in terms of the operator set F . We thus represent theHamiltonian by the parameter vector ~h = (h1, · · · , hn). Observe that ~h = (h1, · · · , hn) has the same eigenspaceswith −~h and the eigenvalues differ from one anther with a constant −1. In other words, the ground-state of ~hbecomes the highest excited states of −~h. Since ~h and −~h are all in Rn, the image of ground-states of such class ofH (H =

∑ni=1 hiFi) is the same as the image of highest excited states of H. The same discussion could be extended

to the second excited states and the second highest excited states etc. Therefore, one only need to study dd2e levels ofthe eigenspaces and points generated by excited states are generally inside the JANR, apart from the highest excitedstates. For example, Fig. 5 is the structure of the first three energy levels of a qudit system for which d = 9 and theset {F1, F2} was random generated. The different energy levels show a nested structure, which is simply due to theorder of eigen-energies. As for the mixed states, it is evident that points generated by them should inside the JANR.

In terms of the geometry of JNR, people are more interested in ground-states which form the surface. If the ground-state of H is not degenerate, there is only one point (〈F1〉ψ, 〈F2〉ψ, · · · , 〈Fn〉ψ) on the surface of JANR correspondingto the ground-state.

Whereas, as for excited states, this is generally not the case. For one particular excited energy level, it forms a"fish" shape structure inside the JANR (see Fig. 5). Most of the points on the orbit are identifiable from each other,only a few points are overlapping at the "fish tail". That means eigen-states for the same excited level of differentHamiltonians may give the same measurement results of F .

Ground-state degeneracy and the geometry of JNR surface

The surface of joint numerical rangeW (F) is determined by the ground-states of the class of Hamiltonians associatedwith F . We considered the geometry of JNRs of three 3×3 Hermitian matrices in this paper. Suppose F = {F1, F2, F3}where {F1, F2, F3, Id} is linearly independent. The analysis of the influences of ground-state degeneracy on the surfaceof JNRs is given as follows:

Consider a supporting plane Π of W (F) with inward-pointing normal vector ~h = (h1, h2, h3). Suppose H =∑3i=1 hiFi has degenerate ground-states. Then the degeneracy is 2-dimensional. Choose two ground-states, |x1〉 and

|x2〉, orthogonal to each other. Let X be the 3 × 2 matrix with columns |x1〉 and |x2〉. Let S be the linear span ofthe set of 2× 2 matrices {X∗FiX : 1 ≤ i ≤ 3} ∪ {I2}. Then 1 ≤ dim S ≤ 3. We have

1. If dim S = 1, then Π touches W (F) at a point.

2. If dim S = 2, then Π ∩W (F) is a non-degenerate line segment.

9

0.00 0.25 0.50 0.75 1.00F41

1.0

0.5

0.0

0.5

1.0F 4

2

(a) W (F41, F42)

1.0 0.5 0.0 0.5 1.0F42

1.0

0.5

0.0

0.5

1.0

F 43

(b) W (F42, F43)

0.00 0.25 0.50 0.75 1.00F41

1.0

0.5

0.0

0.5

1.0

F 43

(c) W (F41, F43)(d) W (F4)

0.00 0.25 0.50 0.75 1.00F81

1.0

0.5

0.0

0.5

1.0

F 82

(e) W (F81, F82)

1.0 0.5 0.0 0.5 1.0F82

1.0

0.5

0.0

0.5

1.0

F 83

(f) W (F82, F83)

0.00 0.25 0.50 0.75 1.00F81

1.0

0.5

0.0

0.5

1.0

F 83

(g) W (F81, F83) (h) W (F8)

FIG. 6: The Joint Numerical Ranges of Class 4 and Class 8 and their projections to three coordinate planes.

3. If dim S = 3, then Π ∩W (F) is a non-degenerate ellipse.

The examplary JNR of Class 4 (e = 3.s = 0) and Class 8 (e = 2, s = 1) used in this paper are appropriate examplesfor scenario 2 and scenario 3. Fig. 6 depicts their JNRs and 2-D projections. For Class 4, the set of F is (see also inTable II)

F41 =

0 0 00 0 00 0 1

, F42 =

0 1 01 0 00 0 0

, F43 =

0 i 1−i 0 01 0 0

.

The projection of W (F) to the x-y plane (Fig. 6a, W (F41, F42)) is a triangle with inward-pointing normal vectors(1, 0), (−1,−1) and (−1, 1) showing the degeneracy for ~h = (1, 0, 0), (−1,−1, 0) and (−1, 1, 0). The corresponding Xis given by 1 0

0 10 0

,1√2

1 01 00√

2

,1√2

1 0−1 00√

2

respectively. Direct calculation shows that dim S = 3 in all three degenerate points. Therefore, W (F4) has threeellipses on its surface.

The same discussion holds for W (F8). For Class 8, we have (see also in Table II)

F81 =

0 0 00 0 00 0 1

, F82 =

0 0 10 0 01 0 0

, F83 =

0 1 01 0 00 0 0

The projection of W (F) to the x-z plane (Fig. 6g, W (F81, F83)) is a triangle with inward-pointing normal vectors(1, 0), (−1,−1) and (−1, 1) showing the degeneracy for ~h = (1, 0, 0), (−1, 0,−1) and (−1, 0, 1). The corresponding X

10

A A1 =

(0 0 00 0 00 0 1

)A2 =

(0 0 10 0 01 0 0

)A3 =

(0 0 00 0 10 1 0

)

B B1 =

(0 0 10 0 01 0 0

)B2 =

( 0 0 i0 0 0−i 0 0

)B3 =

(0 0 00 0 10 1 0

)

TABLE I: Two sets of hermitian matrices of Class 1.

is given by 1 00 10 0

,1√2

1 01 00√

2

,1√2

1 0−1 00√

2

respectively. Direct calculation shows that dim S = 2 in the first degenerate point and dim S = 3 in last twodegenerate points. Therefore, W (F8) has one line segment and two ellipses on its surface.For scenario 1, we provide the operator set A in Table I as an example. That is, the ground-state degeneracy of

H =∑i hiAi appears as a point on the surface of JNR when ~h = (1, 0, 0) and it can be easily checked that dim S = 1

at this degenerate point. Apart from this point, there is no other degenerate point of this system Hamiltonian. SoA belong to the first JNR class - there are no lines and ellipses on the surfaces of its JNR. However, the operatorset that we measured in our experiment is not A but B in Table I (see also the first class in Table II), and operatorset B also belong to class 1 but with no degeneracy. The existence of such a special operator set A is an interestingphenomenon to study, but beyond the scope of this work.

EXPERIMENTAL DETAILS

State preparation of photonic qutrit

Single photon source.—Light pulses with 830nm central wavelength from a ultrafast Ti:Sapphire laser (76MHzrepetition rate; Coherent Mira-HP) are firstly frequency doubled in a beta barium borate (β-BBO) crystal. Thesecond harmonics are then used to pump a 10 mm bulk potassium dihydrogen phosphate (KDP) crystal phase-matched for type-II collinear degenerate downconversion to produce photon pairs denoted as the signal and the idler.After the PBS, the idler mode is detected by a SPCM (Excelitas Technologies, SPCM-AQRH-FC with a detectionefficiency about 49%) as the trigger of the heralded single photon source, whereas the signal mode is directed towardsthe following set-up (shown in Fig. 2(c) in the main text).Qutrit state preparation.—We use a calcite beam displacer (BD), wave plates and two phase retarders (PRs) to

prepare arbitrary pure qutrit states encoded in the polarization and spatial optical modes, depicted in the statepreparation module in Fig. 2(c) in the main text. Single photons with horizontal polarization |H〉 firstly beenscrambled in the superposition of horizontal and vertical polarization by the half-wave plate (HWP) HWPA withsetting angle θA,

|ψ1〉 = cos θA |H〉+ sin θA |V 〉 .

Then followed by a PR, a relative phase between horizontal and vertical polarization is included and the state can bewritten as

|ψ2〉 = eiφ1 cos θA |H〉+ sin θA |V 〉 .

After BD1, the state is in the superposition of two spatial mode |s1〉 (top) and |s2〉 (lower),

|ψ3〉 = eiφ1 cos θA |H〉 ⊗ |s1〉+ sin θA |V 〉 ⊗ |s2〉 .

In the spatial mode |s1〉, a HWP with angle 45◦ flip the polarization into |V 〉, while in spatial mode |s2〉, HWPB isset to θB , therefore, the state after HWPB is

|ψ4〉 = eiφ1 cos θA |V 〉 ⊗ |s1〉+ sin θA(sin θB |H〉 − cos θB |V 〉)⊗ |s2〉 .

11

To generate arbitrary pure qutrit state, another relative phase still needed. Here we use the second PR simultaneouslymanipulating the two spatial mode, introducing another phase factor φ2 between horizontal and vertical polarizationmodes, thus in the end, when the photon pass through BD2, we have prepared an arbitrary qutrit state in the formof

|ψ5〉 = eiφ1 cos θA |V 〉 ⊗ |s1〉+ eiφ2 sin θA sin θB |H〉 ⊗ |s1〉 − sin θA cos θB |V 〉 ⊗ |s2〉 .

By defining the three eigen-modes of the qutrit state as

|0〉 ≡ |H〉 ⊗ |s1〉 , |1〉 ≡ |V 〉 ⊗ |s1〉 , |2〉 ≡ |V 〉 ⊗ |s2〉 ,

the state of single photons go through the state preparation module can be written as

|ψ〉 = eiφ2 sin θA sin θB |0〉+ eiφ1 cos θA |1〉 − sin θA cos θB |2〉 .

For realistic implementation of the two PRs, we use different set-ups. The first PR is implemented by a liquidcrystal phase retarder (Thorlabs, LCC1113-B) with its optical axis parallel to the horizontal polarization. Differentphase φ1 can be realized by applying different voltage to this retarder. However, due to the tiny separating distance(4mm) of the two spatial modes and limited retardance uniformity of the liquid crystal phase retarder, the second PRis realized by a electronically-controlled HWP sandwiched by two quarter-wave plates (QWPs) (a QWP-HWP-QWPconfiguration) [46] with the HWP setting at φ2

4 an the two QWPs setting at 45◦, which equivalently performing theunitary operation below by using the Jones matrix notation [47] of wave plates,

UQHQ = ei(π−φ) |H〉 〈H|+ |V 〉 〈V | .

By electrically setting different angels to E-HWP, different φ2 can also be achieved. All the components in statepreparation module are electronically-controlled or fixed, which ensure high repeatability in state preparation stagewhen measuring different observables.Sampling surface states.—Given a triple of Hermitian observables F = (F1, F2, F3), the surface points of L(F) can

be derived by the ground-state of the system Hamiltonian

H = sin θ cosφF1 + sin θ sinφF2 + cos θF3,

where (θ, φ) defines the orientation of the supporting hyperplane. In our experiment, we randomly sample θ ∈ [0, π]and φ ∈ [0, 2π] to generate different H, and then calculate their ground-states as the target states we are going toprepare with the above mentioned set-ups. Totally, we sampled 300 this kind of surface states for each class of F andmeasured the expectation values of F1, F2 and F3 to obtain the data points of L(F).

Measurement of joint numerical ranges

For the experimental observation of the classification of qutrit JNR, we experimentally measured the expectationvalues of 8 classes of triple Hermitian observables F = (F1, F2, F3). For each classes, 300 surface points were measuredby sampling 300 surface states. The 8 classes of exemplary Hermitian observables that we measured are shown inTable II.

To measure the expectation value 〈Fi〉 of a Hermitian observable Fi with respect to an arbitrary state |ψ〉, we firstrewrite |ψ〉 in the eigen-basis of Fi,

|ψ〉 = √p0 |λ(i)0 〉+√p1 |λ(i)

1 〉+√p2 |λ(i)2 〉 ,

where |λ(i)j 〉 (j = 0, 1, 2) is the corresponding eigen-vector with eigen-value λ

(i)j of Fi and pj(j = 0, 1, 2) is the

probability |ψ〉 been projected into the eigen-mode |λ(i)j 〉. In order to measure the probability distribution {pj}, we

can apply a unitary transformation

UFi = |0〉 〈λ(i)0 |+ |1〉 〈λ

(i)1 |+ |2〉 〈λ

(i)2 |

to |ψ〉, which transforms any state from the eigen-basis of Fi into computational or experimental basis. Thus {pj}can be directly read out from the measurement statistics when projecting state UF |ψ〉 into the three experimental

12

TABLE II: 8 classes of F measured in our experiment. (1 denote −1 and i denote −i)

Class Feature F

1 s = 0, e = 0 F11 =

(0 0 10 0 01 0 0

), F12 =

(0 0 i0 0 0i 0 0

), F13 =

(0 0 00 0 10 1 0

)

2 s = 0, e = 1 F21 =

(0 0 00 0 00 0 1

), F22 =

(0 1 01 0 10 1 0

), F23 =

(0 i 1i 0 01 0 0

)

3 s = 0, e = 2 F31 =

(0 0 00 0 00 0 1

), F32 =

(0 1 11 0 01 0 0

), F33 =

(1 0 00 1 10 1 0

)

4 s = 0, e = 3 F41 =

(0 0 00 0 00 0 1

), F42 =

(0 1 01 0 00 0 0

), F43 =

(0 i 1i 0 01 0 0

)

5 s = 0, e = 4 F51 =

(0 0 00 0 00 0 1

), F52 =

(1 0 10 1 01 0 0

), F53 =

(0 1 01 0 00 0 0

)

6 s = 1, e = 0 F61 =

(0 i 0i 0 00 0 1

), F62 =

(0 1 01 0 00 0 0

), F63 =

(0 0 10 0 01 0 0

)

7 s = 1, e = 1 F71 =

(0 0 00 0 00 0 1

), F72 =

(0 0 10 0 11 1 0

), F73 =

(0 1 01 0 10 1 0

)

8 s = 1, e = 2 F81 =

(0 0 00 0 00 0 1

), F82 =

(0 0 10 0 01 0 0

), F83 =

(0 1 01 0 00 0 0

)

TABLE III: Wave plate setting angles for different non-diagonal observables.

Observable HWP1 QWP HWP2 HWP3F11, F63, F82 90 * 90 112.5

F12 90 90 (before HWP1) 90 112.5F13 45 * 90 67.5

F22, F73 45 * 67.5 112.5F23, F43 90 0 (after HWP1) -67.5 -67.5

F32 90 * 112.5 67.5F33 90 * 60.86 45

F42, F53, F62, F83 67.5 * 45 90F52 45 * 60.86 90F61 -67.5 0 (before HWP1) 90 90F72 67.5 * 90 67.5

basis. By defining detecting events in the experimental basis |j〉 (j = 0, 1, 2) as measuring the outcome λ(i)j , then

〈Fi〉 can be derived by using 〈Fi〉 =∑j p

(i)j λ

(i)j . To realize this unitary operation UFi , we implemented a three stage

interferometer formed by BDs and wave plates, as shown in Fig 2c in the main text. In each stage, BDs and HWPspermutate two of the qutrit eigen-modes into the same spatial mode with different polarization, then the two modeswere interfered by HWP and QWP, which equivalently performing a 2× 2 unitary on the two modes and leaving thethird mode unchanged. It has been shown that any 3× 3 unitary operation U can be written as U = U3U2U1, whereU1, U2, U3 are of the form

U1 =

m1 n1 0p1 q2 00 0 1

, U2 =

m2 0 n20 1 0p2 0 q2

, U3 =

1 0 00 m3 n30 p3 q3

,

and mk, nk, pk, qk(k = 1, 2, 3) form a 2 × 2 unitary block. Therefore, by using the QWP-HWP-QWP configuration[46], arbitrary 2 × 2 unitary block in Uk can be realized in each interference stage, and in principle, this three stageinterferometer can realize any 3 × 3 unitary U . In our experiments, the three stage interferometer from left to rightperform unitary operations in the form of U1, U2 and U3 sequentially. For most of the observables in Table II, only

13

HWP is needed for the realization of UFi, and for some observables, QWP is needed, the wave plate setting angles of

all the non-diagonal observables listed in Table II are show in Table III. As for diagonal observables, UFi become theidentity operator, and all the wave plates set to zero.

Experimental observations of joint numerical ranges in the case d = 3, n = 2

-1 0 1〈F

52〉

-1

-0.5

0

0.5

1

〈F5

3〉

-1 -0.5 0 0.5 1〈F

61〉

-1

-0.5

0

0.5

1

〈F6

3〉

0 0.5 1〈F

51〉

-1

-0.5

0

0.5

1

1.5

〈F5

2〉

0 0.5 1〈F

51〉

-1

-0.5

0

0.5

1

〈F5

3〉

FIG. 7: Experimentally observed four classes of JNR of two 3× 3 observables. Theoretical boundary(two-dimensional surface) curves of these two dimensional plane sets are plotted with black lines while blue points(300 for each class) represent the experiment results.

By projecting the three dimensional JNR in the main text into two dimensional coordinate plane, our experimentalresults also show an complete observation of the four classes of L(F) in the case d = 3, n = 2 [48]. Four exemplaryresults are shown in Fig. 7. From left to right, the four classes are: an oval (the convex hull of a sextic curve), theconvex hull of a quartic curve with a flat portion on the boundary, the convex hull of an ellipse and a point outsidethe ellipse, a triangle. As the boundary states of L(F) with d = 3, n = 2 no longer being the surface states in thecase n = 3, most of the experimental points are inside the range, but still show well agreements with the theoreticallypredicted ranges.

Two classes of unitarily reducible JNRs in the case d = n = 3

As mentioned in the main text, in the case d = n = 3, there are ten possible categories of JNRs according to thenumber of ellipse e and segment s. The experimental results of the eight unitarily irreducible classes are shown in themain text. The another two classes with s =∞, e = 1 and s =∞, e = 0, which correspond to linearly dependent setof F , can be obtained by lower dimensional JNRs. For example, the JNR with s =∞, e = 1 can be generated by theJNR of the following two matrices

X =

0 1 01 0 00 0 0

, Y =

0 −i 0i 0 00 0 0

,

which is a circle belong the first class in Fig. 7 and have the exact shape of the two-dimensional projection of a Blochsphere on the x−y plane. By simply adding a z component that orthogonal to the nonzero two-dimensional subspaceof X and Y

Z =

0 0 00 0 00 0 1

,

it is not hard to imagine that the resulting JNR is a cone, with one ellipse and infinite number of segments. For each〈Z〉 value, the cross section of the JNR is a circle and when 〈Z〉 get larger, the radius of the circle get smaller.

14

Experimental error analysis

As mentioned in the main text, deviations between the observed data and the theoretical values are mainly attributedto two kinds of systematic errors in the settings of experimental parameters. The first is the imperfection of the twoPRs. The QWP-HWP-QWP configuration involvs three wave plates which suffering misalignments of the opticsaxis (typically ∼ 0.1 degree), retardation errors (typically ∼ λ/300 where λ = 830nm) and inaccuracies of settingangles (typically ∼ 0.2 degree) while the liquid crystal phase retarder was pre-calibrated by a co-linear interferometerformed by four wave plates which may transfer the experimental errors. Both cause inaccuracy in manipulatingrelative phase between horizontal and vertical polarizations and also cause inaccuracy in calibrating the three stageinterferometers in the measurement part. The second kind of systematic errors come from the slowly drift and slightvibrating of the interferometers during the measurement progress, which cause a decreasing of interference visibility.Overall, the interference visibility during the whole measuring progress is above 98.7% for all classes. The averagesimilarity S of experimentally measured probability distributions of all the operators are shown in Table IV. All theaverage similarities are above 0.994, which indicates well overall performances of the experimental settings and highsimilarities between the experimental datas and theoretical distributions.

TABLE IV: Average similarity S of experimentally measured probability distributions.

Class 1 2 3 4 5 6 7 8S(F1) 0.9997 0.9998 0.9998 0.9997 0.9999 0.9979 0.9999 0.9997S(F2) 0.9990 0.9985 0.9988 0.9986 0.9992 0.9986 0.9964 0.9948S(F2) 0.9998 0.9992 0.9998 0.9991 0.9988 0.9996 0.9977 0.9955