The Shape of Higher-Dimensional State Space: Bloch-Ball Analog for a Qutrit

The shape of higher-dimensional state space: Bloch-ball analog for a qutrit

Christopher Eltschka1, Marcus Huber2,3, Simon Morelli3,2, and Jens Siewert4,5

1Institut fur¨ Theoretische Physik, Universit¨at Regensburg, D-93040 Regensburg, Germany 2Atominstitut, Technische Universit¨at Wien, 1020 Vienna, Austria 3Institute for Quantum Optics and Quantum Information - IQOQI Vienna, Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria 4Departamento de Qu´ımica F´ısica, Universidad del Pa´ıs Vasco UPV/EHU, E-48080 Bilbao, Spain 5IKERBASQUE Basque Foundation for Science, E-48009 Bilbao, Spain 22 June, 2021

Geometric intuition is a crucial tool to aware of various shortcomings of the Bloch ball if obtain deeper insight into many concepts systems of higher dimension d > 2 are to be discussed. of physics. A paradigmatic example of its For example, the entire surface of the Bloch ball is power is the Bloch ball, the geometrical covered by pure states, whereas most of the boundary representation for the state space of the of higher-dimensional state spaces is formed by mixed simplest possible quantum system, a two-level states. Those parts of the boundary may either be system (or qubit). However, already for a flat or curved, however, their curvature is different three-level system (qutrit) the state space from that of the pure-state surface parts, which has eight dimensions, so that its complexity actually represents a single unitary orbit. Another exceeds the grasp of our three-dimensional important fact not shown by the Bloch ball is that space of experience. This is unfortunate, given not every orthogonal transformation can be applied that the geometric object describing the state to any quantum state: Whenever a state is not space of a qutrit has a much richer structure an element of the inscribed sphere of the state and is in many ways more representative for space of maximum radius there exist rotations in a general quantum system than a qubit. In Bloch space that take it outside the state space this work we demonstrate that, based on the and therefore are not allowed. Consequently, it is Bloch representation of quantum states, it a long-standing question whether it is possible to is possible to construct a three dimensional construct a consistent three-dimensional model for model for the qutrit state space that captures higher-dimensional state spaces Qd, d ≥ 3, that most of the essential geometric features of captures at least a part of these important geometric the latter. Besides being of indisputable features. theoretical value, this opens the door to a new type of representation, thus extending our geometric intuition beyond the simplest 2 Bloch ball for qubits quantum systems. Let us briefly recapitulate the properties of the Bloch ball for qubits, d = 2. According to Fano [3,4] 1 Introduction density operators of qubits are parametrized by using the Pauli matrices σx, σy, σz and the identity 12 Nowadays virtually every student of quantum arXiv:2012.00587v2 [quant-ph] 22 Jun 2021 mechanics learns about the Bloch sphere and the 1 ρ = (s012 + x σx + y σy + z σz) (1a) Bloch ball as the geometrical representations of pure 2 and mixed states of qubits, respectively. These s = Tr (σsρ) , (1b) objects have become indispensable for developing an intuition of elementary concepts such as with real numbers s ∈ {x, y, z}, |s| ≤ 1, and the basic quantum operations and the action of normalization s0 = 1. The state space Q2 of all qubit decoherence [1], or more advanced topics like the density operators is represented by a ball of radius 3 Majorana representation of symmetric multi-qubit 1 about the origin of R , that is, each point (x, y, z) states [2]. Nonetheless, the more experienced of this ball corresponds to exactly one state ρ. Pure practitioner in the field of quantum mechanics is states lie on the surface (forming a connected set), whereas mixed states are located inside the Bloch Jens Siewert: [email protected] sphere, with the fully mixed state at the center.

Accepted in Quantum 2021-06-17, click title to verify. Published under CC-BY 4.0. 1 Interestingly, the convex combinations of two states, as not all rotations are allowed. In fact, for d > 2 ρ = λρ1 + (1 − λ)ρ2 (0 ≤ λ ≤ 1), are given by the special unitary group SU(d) is a proper subgroup 2 the straight line connecting the points of ρ1 and ρ2. of the rotation group SO(d -1), meaning that the 2 A common choice of computational basis states is quantum state space Qd is a proper subset of the d −1 {|0i , |1i}, which correspond to north and south pole dimensional (Hilbert-Schmidt) ball. of the sphere and form the regular simplex ∆1, the For qutrits the density matrices are parametrized special case of ∆d−1 for d = 2. This shows that by the identity 13 and the normalized Gell-Mann 3 the vectors in R belonging to the basis states are matrices Xj, Yk (j, k = 1, 2, 3), and Z1, Z2 (see not orthogonal. These vectors are orthogonal in R4 Appendix), including the direction of s0, but their projections into 3  3 3 2  R lose this property. 1 X X X ρ = 1 + x X + y Y + z Z Every density matrix can be obtained by unitarily 3  3 j j k k l l rotating a diagonal density matrix, that resembles j=1 k=1 l=1 a classical probability distribution. Since for d = 2 (2a) the special unitary group SU(2) is the universal cover xj = Tr (Xj ρ) , yk = Tr (Yk ρ) , zl = Tr (Zl ρ) , group of the rotation group SO(3), every rotation of (2b) 3 the Bloch ball Q2 ⊂ R has a corresponding unitary rotation in the state space. From this perspective, the with real numbers xj, yk, and zl, hence their Bloch ball is obtained by all possible rotations of the space has eight dimensions. A Euclidean 3 simplex ∆1 in R . metric, corresponding to that of our everyday To the best of our knowledge, the idea that geometric experience, is induced in this space the parametrization in Eq. (1) entails a useful by the Hilbert-Schmidt norm, kρ1 − ρ2k2≡ p 2 visualization for the non-unitary dynamics of a spin 3 Tr [(ρ1 − ρ2) ]. 1 The geometric properties of Q3 we wish for a global 2 , e.g., in a situation of radiation damping was put forward by Feynman and co-workers [5]. model to reproduce were thoroughly discussed by Bengtsson et al. in Ref. [18]. i) Most importantly, Q3 is a convex set with the 3 A Bloch-ball analog for a qutrit topology of a ball, so the model should share this characteristics. There are no pieces of lower Over the years much work has been done to elucidate dimension attached to it (“no hair” condition). the geometry of the state space of higher level ii) The actual Bloch body is neither a polytope nor a systems, with special focus on the qutrit state space smooth object. √ Q [6–25]. To develop an intuition for the full 3 iii) The Bloch body has an outer√ sphere of radius 2 high-dimensional geometry of the qutrit state space, and an inner sphere of radius 1/ 2 [28]. subsets, cross sections and projections onto two iv) The pure states (rank 1) form a connected√ and three dimensions were extensively studied [6, set on the surface at maximum distance 2 from 7, 10, 12, 15, 17, 18, 20–23] and multi-parameter 1 1 the completely mixed state 3 3. Its measure is representations of qutrit states were developed [14, zero compared to that of Q3. In particular we 21, 24–26]. While these approaches can reproduce aim at prominently displaying the three pure states many geometric properties correctly, they do not give corresponding to the preferred basis for the model. a global view of the Bloch body. v) Density matrices on the surface of Q3 are of rank In this section we review known facts about 1 or 2, whereas states inside Q3 are of full rank (rank the higher dimensional state space, with focus on 3). dimension d = 3, collecting a list of requirements Finally, there are some additional properties we wish our model to reproduce. We then construct specifically related to the nature of Q3 as a convex a three dimensional global model of the state space, set: that reproduces astonishingly many properties of Q3. vi) The set of quantum states is self-dual. vii) All cross sections of Q3 do not have non-exposed 3.1 Qutrit geometry faces. All corners of two-dimensional projections of Q3 are polyhedral. In analogy to qubits, density matrices describing the state of a d-level quantum system can be parametrized 2 3.2 A three dimensional model for a qutrit by the identity 1d and d − 1 traceless Hermitian matrices [4,6, 27]. So the quantum state space Qd Surprisingly, it is indeed possible to find an object d2−1 3 can be represented by a subset in R , constrained in R representing Q3 that obeys most of the by inequalities that arise from the positivity of the requirements in this list. In fact, there are (at least) density operators [10, 11]. While it is still true that two solutions with slightly different advantages. First the quantum state space Qd is obtained by rotating we construct an object that fulfills properties i–iv) d2−1 the simplex ∆d−1 in R , it is no longer a sphere and also partially v). It represents a valid model

Accepted in Quantum 2021-06-17, click title to verify. Published under CC-BY 4.0. 2 (1) for Q3 that we call Q . Then we show that this 3 +3 +3 (2) | ih | first solution can be extended to a model Q3 that possesses in particular also the property vi). Yet that model does not obey vii). + + It is well known that the computational basis states | 2ih 2| in d dimensions form the corners of a regular simplex 2 2 ∆d−1 in Qd [6, 18], that is, for a qutrit a basis will be | ih | 1 1 represented by ∆2, an equilateral triangle. We insist w | ih | that our model for Q3 displays and emphasizes one 11 particular basis {|0i , |1i , |2i}, because the mapping 3 1( 0 0 + 1 1 ) between physical states and points in the model will 2 | ih | | ih | depend on this choice. Hence, we use the coordinates z z2 [cf. Eq. (2b)] 1 0 0 z1 = Tr (Z1 ρ) , z2 = Tr (Z2 ρ) , | ih | (1) Figure 1: The model Q3 of the qutrit Bloch body Q3 to faithfully represent the diagonal matrices as a according to Eq. (4). The location of the basis states |ji simplex ∆2 in the horizontal coordinate plane. In is specified. The semicircular surface in the foreground is particular,√ the basis state |2i corresponds to (z1, z2) = the image of the Bloch ball spanned by the states |0i and (0, − 2), whereas the states |0i and |1i are located |1i and, in this representation, has the “north pole” |+2i = 1 q 3 q 1 √ (|0i + |1i). The point with the largest w coordinate is 2 in (± 2 , 2 ), respectively. The completely mixed the image of the maximally coherent superposition |+3i = state lies at the origin (0, 0). 1 √ (|0i + |1i + |2i). Note that all the pure states with For the remaining six coordinates we are left with 3 |hj|ψi|2 = 1 get mapped to this point, in particular all the only one direction in 3. Here, we propose to use the 3 R bases which are mutually unbiased with {|0i , |1i , |2i}. coordinate v u 3 uX else attached to it. The flat surfaces of the cuts w = t x2 + y2 , (3) j j are connected with the smooth upper boundary by j=1 a sharp corner, so the object is neither smooth nor which assumes only non-negative values. Hence we a polytope. Since our model preserves the length of have our first model for Q3, the Bloch vector,√ it is still circumscribed by an outer sphere of radius 2 and center at the origin. The (1) 3 Q3 = {(z1, z2, w) ∈ R s.t. radius of the inner (hemi-)sphere coincides with that of the in-circle of the simplex ∆ in the horizontal Eqs. (2b), (3) ∀ρ ∈ Q3}. (4) √ 2 plane and equals 1/ 2. The interpretation for the coordinates of a point The spherical part of the surface above the ground P = (z1, z2, w) is simple. Imagine the state ρ plane corresponds to the set of pure states, hence corresponding to P written as a sum of its diagonal to density matrices of rank 1. They form a and offdiagonal parts, simply connected√ surface of measure zero at maximal distance 2 from the origin. The cuts are half ρ = ρdiag + ρoffdiag . circles, in fact, these flat surfaces are the images of the three two-dimensional Bloch balls corresponding While the distance of P from the origin in the plane to the pairs of states {|0i , |1i}, {|0i , |2i}, and q 2 equals the Hilbert-Schmidt length 3 Tr(ρdiag) − 1 {|1i , |2i} (cf. also Fig.1). This means, these of the diagonal part of the Bloch vector for surfaces correspond to states of at most rank 2. It ρ, the vertical distance of P from the plane is (1) is understood that the base of Q3 represents an the Hilbert-Schmidt length of this Bloch vector’s artificial cut – similar to the base of the sculpture q 2 offdiagonal part, 3 Tr(ρoffdiag). The result is shown of a bust – that does not represent a boundary of Q3. in Fig.1. The states on the other surfaces are of lower rank, all states of rank 3 reside in the interior of the model. However rank-2 states that have all of the basis 3.3 Global geometric properties vectors |0i, |1i, |2i in their span are located inside (1) Let us analyze the properties of this first model for Q3 , although in Q3 they are part of the boundary. Q3 with respect to our list of requirements. The But the rank-2 states do not cover the entire interior (1) surface√ of this object corresponds to a hemisphere of of Q3 . Evidently, all points inside the inner sphere radius 2, where the three spherical segments beyond are of rank 3, since rank-2 states have purity of at 1 the triangle in the plane are cut off. Evidently, it is least 2 . However the set of points corresponding only both convex and simply connected, without anything to rank-3 states is larger than that; see Sec. 3.4 for

Accepted in Quantum 2021-06-17, click title to verify. Published under CC-BY 4.0. 3 details. Conversely, the interior points outside that set correspond both to rank-2 and rank-3 states. In particular, in agreement with property v), the images of rank-3 states cover the complete interior. Thus, (1) we have established that our construction Q3 has all the desired properties i–iv) of our list, property v) is at least partially satisfied. (1) We may ask how the coordinates in Q3 are related to those of the actual state space Q3. Clearly our model is neither a projection nor a cross section. Instead, it resembles a representation in cylindrical coordinates: Our “diagonal coordinates” z1, z2 correspond to two longitudinal axes, whereas each q 2 2 rj = xj + yj (where j = 1, 2, 3) may be viewed as a radial coordinate belonging to mutually orthogonal Figure 2: The SU(3) orbit of the state ρ = 6 |0ih0| + directions. Our model does not display the polar 0 10 3 |1ih1| + 1 |2ih2| is represented by the blue spherical angles and shows only the total radial distance. 10 10 surface. The image of ρ0 together with the other five points in the z1-z2 plane forms the Birkhoff polytope (see text). The 3.4 Local algebraic properties eigenvalue vectors of the diagonal states inside the polytope 6 3 1 are majorized by that of ρ0, ( 10 , 10 , 10 ). We have mentioned that the simplex ∆2 represents the diagonal states faithfully, it is isomorphic to that set just as the Bloch ball is isomorphic to Q2. But be reached through a unitary transformation, see (1) our three-dimensional model Q3 cannot represent all Fig.2. If the eigenvalues are degenerate, the hexagon states of the eight-dimensional state space faithfully. becomes a triangle. A special case are the pure states, (1) the unitary orbit of a pure state coincides with the In fact, a general point of Q3 corresponds to (1) infinitely many states and the simplex representing upper surface of Q3 . the basis states |jihj|, j = 0, 1, 2 is the only set of This visualization establishes a direct connection to states with a unique preimage. States mapped onto the action of doubly stochastic matrices on classical the same point in the model belong to the equivalence probability distributions. A classical probability class of states with the same diagonal entries and distribution p majorizes exactly the probability purity, they form a five-dimensional manifold in distributions M.p, where M is a doubly stochastic the original state space Q3. These subspaces are matrix. The set of distributions majorized by p closed under the action of unitary operators that is called the Birkhoff polytope [6]. The action of commute with Z1 and Z2, in particular diagonal doubly stochastic matrices on classical probability unitary operators. As a consequence the model is distributions corresponds to the action of unitary invariant under these transformations. transformations on normalized diagonal matrices. The action of general unitary transformations, That is, for every doubly stochastic matrix M however, is displayed by our model. Since the purity there exists a unitary matrix U, such that M.p = remains unchanged, all points in the unitary orbit diag(U.D.U †), where D is a diagonal matrix with p = have the same distance from the origin. This orbit diag(D). In the model this is visualized by unitarily forms a subset of a sphere, its shape depends on the transforming a diagonal state and then projecting it eigenvalues of the transformed state. As previously onto the ground triangle. This exactly reproduces elucidated, a point in the interior represents a whole the Birkhoff polytope, but it also generalizes it in the subspace of states with possibly different eigenvalues, sense that the norm of the offdiagonal part remains so it makes little sense to talk about the orbit of visible. a point in our model. However, diagonal states The images of SU(3) orbits also allow us to identify in the ground triangle are depicted faithfully and the set of images of rank-2 states. The rank-2 diagonal therefore uniquely identify a unitary orbit. One states are exactly the sides of the base triangle. only needs to use the eigenbasis of the state to Therefore the rank-2 states are given by all the orbits investigate which points can be reached by unitary of those points. It is readily seen that the boundary transformations. The permutations of the diagonal arcs connecting two triangle sides form half-circular entries correspond to SU(3) rotations, therefore the cones that each have one of the basis states as apex, unitary orbit includes six points on the triangle (for and are tangential to the insphere at their base. The non-degenerate eigenvalues). Since the eigenvalues points inside those cones, as well as the points inside of a matrix majorize the diagonal entries, the orbit the inner sphere, are not covered by those orbits, and remains on a sphere above this hexagon. Vice versa, therefore correspond only to states of rank 3. This every point on the sphere above the hexagon can rank-3 only region is convex; indeed it is the convex

Accepted in Quantum 2021-06-17, click title to verify. Published under CC-BY 4.0. 4 hull of the interior of the insphere and the interior of the base triangle. Note that the mirror image of this region and its boundary forms the lower part of our second model (2) Q3 which will be described in Sec.4. The map that takes the qutrit state space to our model is decidedly nonlinear, but amazingly retains some linear properties of the state space for the image. Any convex combination of two diagonal qutrit states δ = λδ1 + (1 − λ)δ2 with λ ∈ (0, 1) is again a diagonal state and its image lies on the straight line connecting the image points of δ1 and δ2. But even the convex combinations ρ = λσ + (1 − λ)δ of an arbitrary qutrit state σ with a diagonal state δ are located on the straight line connecting σ and δ. The mixture of Figure 3: Constructing the dual of the surface of the model (1) the diagonal parts is faithfully represented, but in Q3 . Red: Cross section for determining the dual states ρ˜(p) this case, this holds also for the offdiagonal parts, of the mixtures ρ(p) in Eq. (8). Gray: We have included the part that has been cut from the full Bloch body (see Fig.4, because δoffdiag simply vanishes. Consequently, for (2) any problem involving two arbitrary states we can which shows the complete object Q3 from the same point find a visualization of their mixture by means of a of view). By rotating the section highlighted in red by the (2) straight line: It is enough to consider the problem in angle β (see text) one obtains the lower conical part of Q3 . the eigenbasis of one of the states. Another noteworthy aspect of our model Q(1) is (1) 3 4.1 Self-dual extension of Q the separate treatment of diagonal and offdiagonal 3 parts of the state. The diagonal matrices may be To achieve self-duality, consider first the pure states Π viewed as states in a classical probability space [18]. with Tr(Π2) = Tr(Π) = 1. If we denote the vector of They become nonclassical by adding coherences, i.e., Gell-Mann matrices by ~h and the Bloch vector of the 1 1 ~ an offdiagonal part. The w coordinate in our model state Π by ~π, we can write Π = 3 ( 3 + ~π · h). Recall is the 2-norm for the offdiagonal part of the state. that reversing the sign of all coordinates in the qubit As the 1-norm of the offdiagonal part is established Bloch sphere amounts to a point reflection operation as a coherence measure [29] and the 1-norm is at the maximally mixed state. This fact suggests that lower-bounded by the 2-norm, our model explicitly we might try reversing the sign of w in a process of shows a lower bound to the coherence of a given reversing all of the coordinate signs, for example in an quantum state. Correspondingly, the maximally operation of the kind ρ −→ α13 − ρ (where α > 0). coherent state |+ i = √1 (|0i + |1i + |2i) is located We have to make sure that the result is again positive. 3 3 at the north pole of the hemisphere. In Ref. [30] the so-called universal state inversion map S(ρ) = ν(1 − ρ) was introduced; it guarantees that the result actually is a state. We choose the prefactor 1 4 Self-dual qutrit geometry ν = 2 so as to normalize the resulting qutrit state and write The model we have discussed so far does not meet 1 S(ρ) ≡ ρ˜ = (1 − ρ) . (6) the requirements v) and vi) of our list. In particular 2 3 it is not self-dual, that is, our three-dimensional Bloch The inverted state ρ˜ is positive and for pure states we vectors do not fulfill the following condition. Let ξ~ be have Tr(ΠΠ)˜ = 0. Therefore, the Bloch vector belonging to a state on the boundary of the state space. Then for the set of Bloch vectors 0 = 1 + ~π · ~π˜ , (7) ~η defining the dual hyperplanes enveloping the state space on the side opposite to ξ~ we have [18] that is, the Bloch vector ~π˜ defines the dual plane for ~π (and vice versa). As the image of ~π determines a point ξ~ · ~η = −1 . (5) in the spherical part of the boundary of our model, also the image of ~π˜ lies on a spherical surface, however, 1 Our model so far is not self-dual simply because with half the radius because of the prefactor ν = 2 . for vectors ξ~ on the surface with w > 0 the dual The result is that the spherical part of our model (1) planes do not touch the lower boundary of our Bloch Q3 gets replicated in the region w < 0, whereat it body model. We will now demonstrate that the is point-reflected at the origin and scaled down by a 1 model can be extended by adding a “lower part” with factor 2 . coordinates w < 0 so that the entire object becomes This reasoning cannot be applied for the mixed self-dual. boundary states of the model. Rather, we apply

Accepted in Quantum 2021-06-17, click title to verify. Published under CC-BY 4.0. 5 state inversion xj → −xj, yk → −yk, zl → −zl and determine the boundary states by explicit calculation. Consider, for example, the mixtures (cf. Fig.3) 1 − p ρ(p) = p |+ ih+ | + (|0ih0| + |1ih1|) (8) 2 2 2 with 0 ≤ p ≤ 1. It is straightforward to show that the dual states are given by √ √ 2 2 1 2 6 p | ih | 1 1 ρ˜(p) = 1 − Z − X | ih | 3 3 1 + 3p 2 1 + 3p 1 √ √ 1 √ 2 6 p 0 0 = 1 − (1 − q) 2 Z −q Z + X | ih | 3 3 2 4 2 4 1 (9) with q = 4p . That is, also ρ˜(p) lies on 1+3p Figure 4: The self-dual model Q(2) of the qutrit state space a straight line connecting |2ih2| and the dual of 3 Q3. |+2ih+2|. An analogous calculation applies to all 1 mixtures of 2 (|0ih0| + |1ih1|) with pure states on the circumference of the {|0i , |1i} “Bloch sphere”. 0 1 1 together with ρ also all its mixtures with 3 3 belong Instead of considering the cross section in the z2-w (2) to the lower part of Q3 , ensuring its compactness. plane for the states ρ˜(p) we can rotate this plane The meaning of this definition is clear: In comparison about the z axis by an angle β (see Fig.3), so (1) (2) 2 to the structure of Q , in Q we redistribute the that the pure state on the circumference becomes 3 3 location of the rank-2 states ρ0. If the purity of ρ0 π−2β π−2β cos |1i + sin |0i. 0 0 4 4 is large enough for its diagonal coordinates (z1, z2) it The shape of the corresponding parts of the cross needs to remain in the upper part, w > 0, otherwise section remains the same as in the one shown in Fig.3. it goes to the lower part, w < 0. Consequently, the dual states of the qubit Bloch ball This concludes the construction of the self-dual (2) {|0i , |1i} are located on a half circular cone with apex model Q3 for the qutrit Bloch body, see Fig.4: in |2ih2| (cf. Fig.4). Due to the three-fold symmetry (1) (2) 3 of Q3 the regions near the other pure states |0ih0|, Q3 = (z1,z2, w) ∈ R s.t. |1ih1| can be constructed analogously. Eqs. (2b), (3), (11)∀ρ ∈ Q3 . (12) Once we have found the shape [i.e., the boundary coordinates wmax(z1, z2) and wmin(z1, z2)] of the We note that the extension to a self-dual object can self-dual object, we also need to provide a rule to (1) be applied as well to the half-circle model Q2 of the decide whether a given state ρ from inside Q3 is d = 2 Boch ball. Then, however, the rule given in mapped to the “upper” (w > 0) or the “lower” Eq. (11) to determine the sign of w (without any (w < 0) part of the object. In other words, from addendum) does not improve the model. This is ρ we can determine the coordinates z1, z2, |w| as well because in d = 2 all offdiagonal elements can be made as wmin(z1, z2), but how can we decide about the sign real and positive by applying a single appropriate of w? diagonal unitary to the state. In contrast, for d > 2 at In the following we describe a procedure to define most 2(d−1) offdiagonal matrix elements of a generic this mapping ρ −→ P (z1, z2, w). It is both state can simultaneously be made positive by applying straightforward and consistent (however, there might a diagonal unitary. be other choices). Let ρ be an arbitrary qutrit state with smallest eigenvalue λmin. First we subtract as much of the fully mixed state so that one eigenvalue 4.2 More convexity properties and relation to vanishes, i.e., previous work

0 1 Convexity properties of the state space in the context ρ = (ρ − λmin13) (10) 1 − 3λmin of the Bloch-ball representation for d = 2 (and even for d = 3) were discussed early on, e.g., by Bloore [7] is a rank-2 state with Bloch coordinates (z0 , z0 , |w0|). 1 2 who analyzed the stratiﬁcation of the state space with Then respect to the matrix rank. For a qutrit, all states of 0 0 0 ) ( |w | > |wmin(z1, z2)| w > 0 reduced rank r < 3 are part of the boundary. The =⇒ . (11) 0 0 0 four-dimensional set of rank-1 states |ψihψ| forms the |w | ≤ |wmin(z1, z2)| w < 0 extremal states of the convex state space, as they do 0 This implies that in the case of equality the state ρ not have a convex decomposition into other states ρ1 (2) is located at the lower boundary in Q3 . Moreover, and ρ2, |ψihψ| 6= λρ1 + (1 − λ)ρ2 with 0 ≤ λ ≤ 1.

Accepted in Quantum 2021-06-17, click title to verify. Published under CC-BY 4.0. 6 Geometrically they are part√ of the outer sphere, Figs.3 and4 and is guaranteed by construction via because they have distance 2 from the origin. the universal state inversion, Eq. (6), (7). As described in the previous section, the lower It is noteworthy that the universal state inversion (2) (or reduction [31]) map finds an explicit geometric part of our self-dual model Q3 consists of three half-circular cones, each with a basis state as application here. Until now it was mainly associated their apex, which end at a half-sized point mirror with entanglement properties of multi-party states, image of the pure-state spherical section, that is, although the relevance of geometrical concepts was at the insphere. This description matches exactly implicit also in that context (cf., e.g., [32]). the one of the internal boundary separating the One of the most notable qutrit three-sections is points corresponding to rank-2 states from those the so-called obese tetrahedron (or three-dimensional corresponding only to rank-3 states in the model elliptope). It was already noted by Bloore [7] and (1) thoroughly studied by Goyal and co-workers [20] (see Q3 , except that it is below the base triangle rather than above. Also in the previous section, it was Sec. 5.4 and Fig. 5 in Ref. [20]). Its barycenter is established that rank-2 states are mapped to the lower the completely mixed state and the corners are given by the four non-orthogonal states √1 (|0i ± |1i ± |2i). half exactly if doing so does not locate them outside 3 the model. The faces are bulgy, because the tetrahedron contains This allows us to determine the image of the set of with its corner points also their duals on the opposite (2) side. The image of the tetrahedron in Q(2) is the rank-2 states in Q3 . The only rank-2 states that fit 3 (1) part of the vertical axis belonging to the model: The into the lower part are exactly those which in Q lie 3 corner points are all mapped to the north pole, the on the internal boundary between rank-2 states and barycenter to the origin and the centers of the faces rank-3 only states. Therefore all rank-2 states that get to the point with z = z = 0 and w = − √1 . mapped into the lower half of the self-dual model are 1 2 2 mapped onto the model’s surface, while the interior The last example we mention here is the conical of the lower part consists only of rank-3 states, just three-section, Sec. 5.1 and Fig. 2 in Ref. [20]. as criterion v) requires. Instead of the [128] section we consider the (unitarily equivalent) [138] section which indeed has However, the same is not true in the upper part. (2) (1) a three-dimensional image in Q3 . The states of the Almost all the rank-2 states in the interior of Q3 are not moved to the lower part and therefore are also in cone are formed by all convex combinations of the (2) real states of the {|0i , |1i} Bloch ball (a circular disk) the interior of Q3 . Only the points on the internal (1) and the state |2ih2|. Therefore, one might expect rank boundary in Q3 are moved to the boundary the image to include for w > 0 the corresponding (2) 1 of Q3 , leaving only rank-3 states on the internal half-cone with the semicircular surface at z2 = √ and boundary. That is, criterion v) is still not perfectly √ 2 (2) the apex at z2 = − 2. However, this is only partially satisfied. However, the new model Q3 is self-dual by correct, because a part of the half-cone gets mapped construction and hence obeys the criterion vi). to values w < 0 according to our rule, Eq. (11). The last criterion vii) is critical because also This includes the complete half-cone attached to |2ih2| √ the self-dual model Q(2) cannot reproduce these 2 3 for values w < 0 and z2 ≤ − 4 . Moreover, also properties. This can be observed in the highlighted the half-cone√ with apex in√ the origin and base at 2 6 cross section in Fig.3. The point corresponding z2 = − with base radius gets mapped to w < 0. 1 4 4 to 2 (|0ih0| + |1ih1|) is non-exposed and counts as The union of both parts at w < 0 and w ≥ 0 a non-exposed face of the two-dimensional cross corresponds to the half-cone mentioned before, that section [18]. As these non-exposed faces occur at is, to one half of the actual conic three-section. points where the self-dual extension is attached to (1) Q3 one might speculate that their occurrence is a price to pay for having such an extension. Moreover, 5 Quantum channels the highlighted cross section in Fig.3 coincides with (2) the projection of Q3 in the direction of the z1 axis. In order to give yet another demonstration that Neither the corner corresponding to |2ih2| nor that of our method directly connects to the well-known |+2ih+2| is polyhedral. properties of the Bloch sphere/ball visualization (2) The self-duality of Q3 is particularly interesting, we show the action of three standard decohering because it enables us to analyze how findings from channels (cf. Ref. [1]) on a qutrit. The depolarizing previous work are featured in our model. As a first channel corresponds to driving a state towards the 1 1 example we consider the duality of boundary states completely mixed state 3 3, whereas phase-damping on outer and inner spheres. As was elucidated in amounts to mixing a state with its diagonal part. Refs. [11, 20], boundary states on the outer sphere Amplitude-damping describes the relaxation to one (1) have their dual counterparts on the inner sphere, of the basis states. Following the discussion of Q3 and vice versa. This property can be observed in the action of these channels on the qutrit Bloch body

Accepted in Quantum 2021-06-17, click title to verify. Published under CC-BY 4.0. 7 is evident and completely analogous to what is known equally. That is, for the d = 2 Bloch ball (cf. Fig.5). p p The channels are characterized by their Kraus K0 = |0ih0| + 1 − γ |1ih1| + 1 − γ |2ih2| † √ operators K that obey the relation P K K = K1 = γ |0ih1| (14) j j j j √ 13. For the depolarizing channel we have K0 = K = γ |0ih2| . p p 2 1 − 8γ/9 13, Kj = γ/9 hj, where hj (j = 1 ... 8) stands as a shortcut for all the Gell-Mann matrices, This results in non-equal scaling of the non-diagonal so that elements.√ Whereas ρ01 and ρ02 scale with a prefactor of 1 − γ, the element ρ12 gets a prefactor of (1 − γ). 0 This does not matter for diagonal states (w = 0), ρ (γ) = (1 − γ) ρ + γ 13 . (13) which are mapped into the model uniquely, and which are therefore again scaled linearly, by The parameter γ ∈ [0, 1] describes the strength of the ! channel action. r3 r3 z 7→ z0 = + z − (1 − γ) (15) The Kraus operators for phase damping (or 1 1 2 1 2 pure dephasing) in its simplest version are K0 = p p p 1 1 1 − 2γ/3 13, K1 = γ/3 Z1, K2 = γ/3 Z2. 0 z2 7→ z2 = √ + z2 − √ (1 − γ) (16) Under the action of the depolarizing and the 2 2 phase-damping channel, the shrinking of the Bloch Note that these two coordinates are scaled the same (1) body model Q3 occurs proportionally along straight way for all the states. lines, and the images of states that are represented by For pure states, the absolute value of the the same point will again be represented by the same oﬀdiagonal elements is uniquely determined by the point in the shrunken model. The same is, however, diagonal elements, and since only the absolute value not true for the amplitude-damping channel. enters into the w coordinate, again each point of the The amplitude damping channel considered is of the original model gets mapped to only one point of the form given by Grassl et al. [33], with γ01 = γ02 =: γ shrunken model. In particular, the w coordinate of and γ12 = 0, so that the states |1i and |2i are damped pure states gets mapped according to

" r !# 02 2 2 2 z2 2 2 w = (1 − γ) 1 − z − √ + z1 √ z2 − 3 2 2 3 3 √ " r ! 2 # 4 2 2 2 z2 2z2 + (1 − γ) − z + − √ z2 z1 − + . (17) 3 1 3 3 3 3

For non-diagonal mixed states, states with different of arbitrary higher-dimensional state space than the distribution of absolute values in the offdiagonal Bloch ball for qubits. matrix elements get mapped to the same point in the model. Therefore states that are represented by the same point will get mapped by the channel to states Thus we hope our results are helpful to develop represented by different points. It is obvious that the a more precise intuition for objects in higher value of w will always be mapped somewhere into the 2 dimensions, which are commonly encountered in range [(1 − γ) w, (1 − γ)w), but that range will not quantum information science. be exhausted for all source points.

6 Conclusion It is conceivable that there are more possibilities for such visualizations along the lines of this work. The We have presented two ways of visualizing the essential features of this visualization persist even if eight-dimensional state space of a qutrit in R3, we use it to represent state spaces of dimensions d > 3 thereby preserving a number of essential geometric (splitting the generalized Gell-Mann into a diagonal properties of Q3, cf. Ref. [18]. Depending on the and oﬀ-diagonal part in an identical fashion). Finding context one or the other representation may appear the most useful geometrical representation of those more useful. Our ﬁndings are relevant, because the and categorizing them is a direction that we believe (1) (2) properties of Q3 and Q3 are much closer to those to yield further fruitful insight in the future.

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(1) Figure 5: Action of quantum channels on the qutrit state space in Q3 representation: a) depolarizing, b) phase-damping, and c) amplitude-damping channel.

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