Geometry of the set of mixed quantum states: An apophatic approach
GGeometry of the set of mixed quantum states: An apophatic approach
Ingemar Bengtsson , Stephan Weis and Karol ˙Zyczkowski , Stockholms Universitet, Fysikum, Roslagstullsbacken 21, 106 91 Stockholm, Sweden Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103 Leipzig, Germany Institute of Physics, Jagiellonian University, ul. Reymonta 4, 30-059 Krak´ow, Poland and Center for Theoretical Physics, Polish Academy of Sciences,Aleja Lotnik´ow 32/46, PL-02-668 Warsaw, Poland (Dated: December 10, 2011)The set of quantum states consists of density matrices of order N , which are hermitian, positiveand normalized by the trace condition. We analyze the structure of this set in the framework ofthe Euclidean geometry naturally arising in the space of hermitian matrices. For N = 2 this set isthe Bloch ball, embedded in R . For N ≥ N − Dedicated to prof. Bogdan Mielnik on the occasion ofhis 75-th birthday
I. INTRODUCTION
Quantum information processing differs significantlyfrom processing of classical information. This is due tothe fact that the space of all states allowed in the quan-tum theory is much richer than the space of classicalstates [1–6]. Thus an author of a quantum algorithm,writing a screenplay designed specially for the quantumscene, can rely on states and transformations not admit-ted by the classical theory.For instance, in the theory of classical information thestandard operation of inversion of a bit, called the
NOT gate, cannot be represented as a concatenation of twoidentical operations on a bit. But the quantum theoryallows one to construct the gate called √ NOT, whichperformed twice is equivalent to the flip of a qubit.This simple example can be explained by comparingthe geometries of classical and quantum state spaces.Consider a system containing N perfectly distinguish-able states. In the classical case the set of classical states,equivalent to N –point probability distributions, forms aregular simplex ∆ N − in N − N isolated points.In a quantum set-up the set of states Q N , consisting ofhermitian, positive and normalized density matrices, has N − N = 3 the geometric structure of the eight dimen-sional set Q is not easy to analyse nor to describe [8, 9]. Therefore we are going to use an apophatic approach , inwhich one tries to describe the properties of a given ob-ject by specifying simple features it does not have. Thenwe use a more conventional [10–12] constructive approachand investigate two-dimensional cross-sections and pro-jections of the set Q [13–15]. Thereby a cross-section isdefined as the intersection of a given set with an affinespace. We happily recommend a very recent work for amore exhaustive discussion of the cross-sections [16]. II. CLASSICAL AND QUANTUM STATES
A classical state is a probability vector (cid:126)p =( p , p , . . . , p N ), such that (cid:80) i p i = 1 and p i ≥ i = 1 , . . . N . Assuming that a pure quantum state | ψ (cid:105) belongs to an N –dimensional Hilbert space H N , a gen-eral quantum state is a density matrix ρ of size N , whichis hermitian, ρ = ρ † , with positive eigenvalues, ρ ≥ ρ = 1. Note that any density matrixcan be diagonalised, and then it has a probability vec-tor along its diagonal. But clearly the space of all quan-tum states Q N is significantly larger than the space of allclassical states—there are N − N − convex set. By definition a convex set is a subset ofEuclidean space, such that given any two points in thesubset the line segment between the two points also be-longs to that subset. The points in the interior of the linesegment are said to be mixtures of the original points.Points that cannot be written as mixtures of two distinctpoints are called extremal or pure . Taking all mixturesof three pure points we get a triangle ∆ , mixtures offour pure points form a tetrahedron ∆ , etc.The individuality of a convex set is expressed on itsboundary. Each point on the boundary belongs to a face ,which is in itself a convex subset. To qualify as a facethis convex subset must also be such that for all possible a r X i v : . [ qu a n t - ph ] D ec ways of decomposing any of its points into pure states,these pure states themselves belong to the subset. Wewill see that the boundary of Q N is quite different fromthe boundary of the set of classical states. A. Classical case: the probability simplex
The simplest convex body one can think of is a simplex ∆ N − with N pure states at its corners. The set of allclassical states forms such a simplex, with the probabili-ties p i telling us how much of the i th pure state that hasbeen mixed in. The simplex is the only convex set whichis such that a given point can be written as a mixture ofpure states in one and only one way.The number r of non–zero components of the vector (cid:126)p is called the rank of the state. A state of rank oneis pure and corresponds to a corner of the simplex. Anypoint inside the simplex ∆ N − has full rank, r = N . Theboundary of the set of classical states is formed by stateswith rank smaller than N . Each face is itself a simplex∆ r − . Corners and edges are special cases of faces. Aface of dimension one less than that of the set itself iscalled a facet .It is natural to think of the simplex as a regular sim-plex, with all its edges having length one. This can al-ways be achieved, by defining the distance between twoprobability vectors (cid:126)p and (cid:126)q as D [ (cid:126)p, (cid:126)q ] = (cid:118)(cid:117)(cid:117)(cid:116) N (cid:88) i =1 ( p i − q i ) . (1)The geometry is that of Euclid. With this geometry inplace we can ask for the outsphere , the smallest spherethat surrounds the simplex, and the insphere , the largestsphere inscribed in it. Let the radius of the outspherebe R N and that of the insphere be r N . One finds that R N /r N = N − B. The Bloch ball
Another simple example of a convex set is a three di-mensional ball. The pure states sit on its surface, andeach such point is a zero dimensional face. There are nohigher dimensional faces (unless we count the entire ballas a face). Given a point that is not pure it is now possi-ble to decompose it in infinitely many ways as a mixtureof pure states.Remarkably this ball is the space of states Q of a sin-gle qubit , the simplest quantum mechanical state space.For concreteness introduce the Pauli matrices σ = (cid:0) (cid:1) , σ = (cid:0) − ii 0 (cid:1) , σ = (cid:0) − (cid:1) . These three matrices form anorthonormal basis for the set of traceless Hermitian ma-trices of size two, or in other words for the Lie algebra of SU (2). If we add the identity matrix σ = = (cid:0) (cid:1) , wecan expand an arbitrary state ρ in this basis as ρ = 12 + (cid:88) i =1 τ i σ i , (2)where the expansion coefficients are τ i = Tr ρσ i /
2. Thesethree numbers are real since the matrix ρ is Hermitian.The three dimensional vector (cid:126)τ = ( τ , τ , τ ) is called the Bloch vector (or coherence vector). If (cid:126)τ = 0 we have the maximally mixed state . Pure states are represented byprojectors, ρ = ρ . FIG. 1: The set of mixed states of a qubit forms the
Bloch ball with pure states at the boundary and the maximally mixedstate ρ ∗ = at its center: The Hilbert–Schmidt distancebetween any two states is the length of the difference betweentheir Bloch vectors, || (cid:126)τ a − (cid:126)τ b || . Since the Pauli matrices are traceless the coefficient standing in front of the identity matrix assures thatTr ρ = 1, but we must also ensure that all eigenvalues arenon-negative. By computing the determinant we findthat this is so if and only if the length of the Bloch vec-tor is bounded, || (cid:126)τ || ≤
1. Hence Q is indeed a solidball, with the pure states forming its surface—the Blochsphere .A simple but important point is that the set of classi-cal states ∆ , which is just a line segment in this case,sits inside the Bloch ball as one of its diameters. Thisgoes for any diameter, since we are free to regard anytwo commuting projectors as our classical bit. Two com-muting projectors sit at antipodal points on the Blochsphere. To ensure that the distance between any pair ofantipodal equals one we define the distance between twodensity matrices ρ A and ρ B to be D HS ( ρ A , ρ B ) = (cid:114)
12 Tr[( ρ A − ρ B ) ] . (3)This is known as the Hilbert-Schmidt distance . Let usexpress this in the Cartesian coordinate system providedby the Bloch vector, D HS [ ρ A , ρ B ] = (cid:118)(cid:117)(cid:117)(cid:116) (cid:88) i =1 ( τ Ai − τ Bi ) = || (cid:126)τ A − (cid:126)τ B || . (4)This is the Euclidean notion of distance. C. Quantum case: Q N When
N > ρ, σ ∈ Q N . It is then easy to see that any convexcombination of these two states, aρ +(1 − a ) σ ∈ Q N where a ∈ [0 , Q N . This shows that the set of quantum statesis convex. For all N the face structure of the boundarycan be discussed in a unified way. Moreover it remainstrue that Q N is swept out by rotating a classical proba-bility simplex ∆ N − in R N − , but for N > ρ = k (cid:88) i =1 p i | φ i (cid:105)(cid:104) φ i | , (5)where (cid:126)p = ( p , p , . . . , p k ) is a probability vector. In con-trast to the classical case there exist infinitely many de-compositions of any mixed state ρ (cid:54) = ρ . The number k can be arbitrarily large, and many different choices canbe made for the pure states | φ i (cid:105) . But there does exist adistinguished decomposition. Diagonalising the densitymatrix we find its eigenvalues λ i ≥ | ψ i (cid:105) . This allows us to write the eigendecomposition of astate, ρ = N (cid:88) j =1 λ j | ψ j (cid:105)(cid:104) ψ j | . (6)The number r of non-zero components of the probabilityvector (cid:126)λ is called the rank of the state ρ , and does notexceed N . This is the usual definition of the rank of amatrix, and by happy accident it agrees with the defini-tion of rank in convex set theory: the rank of a point in aconvex set is the smallest number of pure points neededto form the given point as a mixture.Consider now a general convex set in d dimensions.Any point belonging to it can be represented by a convexcombination of not more than d + 1 extremal states. In-terestingly, Q N has a peculiar geometric structure sinceany given density operator ρ can be represented by acombination of not more than N pure states, which ismuch smaller than d + 1 = N . In Hilbert space these N pure states are the orthogonal eigenvectors of ρ . If FIG. 2: The set Q of quantum states of a qutrit containspositive semi-definite matrices with spectrum from the sim-plex ∆ of classical states. The corners of the triangle be-come the 4 D set of pure states, the edges lead to the 7 D boundary ∂ Q , while interior of the triangle gives the inte-rior of the 8 D convex body. The set Q is inscribed inside a7–sphere of radius R = (cid:112) / r = 1 / √ we adopt the Hilbert-Schmidt definition of distance (3)they form a copy of the classical state space, the regularsimplex ∆ N − .Conversely, every density matrix can be reached froma diagonal density matrix by means of an SU ( N ) trans-formation. Such transformations form a subgroup of therotation group SO ( N − N > SU ( N )is a proper subgroup of SO ( N − Q N forms a solid ball only if N = 2. The relative sizesof the outsphere and the insphere are still related by R N /r N = N − Q N shows some similaritieswith that of its classical cousin. It consists of all matri-ces whose rank is smaller than N . There will be faces ofrank 1 (the pure states), of rank 2 (in themselves they arecopies of Q ), and so on up to faces of rank N − Q N − ). Note that there are no hard edges: the mini-mal non-extremal faces are solid three dimensional balls.The largest faces have a dimension much smaller thanthe dimension of the boundary of Q N . As in the classi-cal case, any face can be described as the intersection ofthe convex set with a bounding hyperplane in the con-tainer space. In technical language one says that all facesare exposed. Note also that every point on the boundarybelongs to a face that is tangent to the insphere. Thishas the interesting consequence that the area A of theboundary is related to the volume V of the body by rAV = d , (7)where r is the radius of the insphere and d is the dimen-sion of the body (in this case d = N −
1) [17]. Inciden-tally the volume of Q N is known explicitly [18].There are differences too. A typical state on theboundary has rank N −
1, and any two such states canbe connected with a curve of states such that all stateson the curve have the same rank. In this sense Q N ismore like an egg than a polytope [19].We can regard the set of N by N matrices as a vectorspace (called Hilbert-Schmidt space), endowed with thescalar product (cid:104) A | B (cid:105) HS = 12 Tr A † B . (8)The set of hermitian matrices with unit trace is not avector space as it stands, but it can be made into one byseparating out the traceless part. Thus we can representa density matrix as ρ = 1 N + u , (9)where u is traceless. The set of traceless matrices is anEuclidean subspace of Hilbert-Schmidt space, and theHilbert-Schmidt distance (3) arises from this scalar prod-uct. In close analogy to eq. (2) we can introduce a basisfor the set of traceless matrices, and write the densitymatrix in the generalized Bloch vector representation, ρ = 1 N + N − (cid:88) i =1 u i γ i . (10)Here γ i are hermitian basis vectors. The components u i must be chosen such that ρ is a positive definite matrix. D. Dual and self-dual convex sets
Both the classical and the quantum state spaces havethe remarkable property that they are self-dual . But theword duality has many meanings. In projective geometrythe dual of a point is a plane. If the point is representedby a vector (cid:126)x , we can define the dual plane as the set ofvectors (cid:126)y such that (cid:126)x · (cid:126)y = − . (11)The dual of a line is the intersection of a one-parameterfamily of planes dual to the points on the line. This isin itself a line. The dual of a plane is a point, while thedual of a curved surface is another curved surface—theenvelope of the planes that are dual to the points on the original surface. To define the dual of a convex body witha given boundary we change the definition slightly, andinclude all points on one side of the dual planes in thedual. Thus the dual X ∗ of a convex body X is definedto be X ∗ = { (cid:126)x | (cid:126)x · (cid:126)y ≥ ∀ (cid:126)y ∈ X } . (12)The dual of a convex body including the origin is theintersection of half-spaces { (cid:126)x | (cid:126)x · (cid:126)y ≥ } for extremalpoints (cid:126)y of X [20]. If we enlarge a convex body theconditions on the dual become more stringent, and hencethe dual shrinks. The dual of a sphere centred at theorigin is again a sphere, so a sphere (of suitable radius) isself-dual. The dual of a cube is an octahedron. The dualof a regular tetrahedron is another copy of the originaltetrahedron, possibly of a different size. Hence this is aself-dual body. Convex subsets F ⊂ X are mapped tosubsets of X ∗ by F (cid:55)→ (cid:98) F := { (cid:126)x ∈ X ∗ | (cid:126)x · (cid:126)y = 0 ∀ (cid:126)y ∈ F } . (13)Geometrically, (cid:98) F equals X ∗ intersected with the dualaffine space (11) of the affine span of F . If the originlies in the interior of the convex body X then F (cid:55)→ (cid:98) F isa one-to-one inclusion-reversing correspondence betweenthe exposed faces of X and of X ∗ [21]. If X is a tetrahe-dron, then vertices and faces are exchanged, while edgesgo to edges.What we need in order to prove the self-duality of Q N is the key fact that a hermitian and unit trace matrix σ is a density matrix if and only ifTr σρ ≥ ρ . It will be convenient to thinkof a density matrix ρ as represented by a “vector” u , asin eq. (9). As a direct consequence of eq. (14) the set ofquantum states Q N is self-dual in the precise sense that Q N − /N = { u | /N + Tr( uv ) ≥ ∀ v ∈ Q N − /N } . (15)In this equation the trace is to be interpreted as a scalarproduct in a vector space. Duality (13) exchanges facesof rank r (copies of Q r ) and faces of rank N − r (copiesof Q N − r ).Self-duality is a key property of state spaces [22, 23],and we will use it extensively when we discuss projectionsand cross-sections of Q N . This notion is often introducedin the larger vector space consisting of all hermitian ma-trices, with the origin at the zero matrix. The set of pos-itive semi-definite matrices forms a cone in this space,with its apex at the origin. It is a cone because any pos-itive semi-definite matrix remains positive semi-definiteif multiplied by a positive real number. This defines therays of the cone, and each ray intersects the set of unittrace matrices exactly once. The dual of this cone is theset of all matrices a such that Tr ab ≥ b within the cone—and indeed the dual cone is equal tothe original, so it is self-dual. III. AN APOPHATIC APPROACH TO THEQUTRIT
For N = 3 we are dealing with the states of the qutrit .The Gell-Mann matrices are a standard choice [16] for theeight matrices γ i , and the expansion coefficients are τ i = Tr ργ i . Unfortunately, although the sufficient conditionsfor (cid:126)τ to represent a state are known [9, 24, 25], they donot improve much our understanding of the geometry of Q .We know that the set of pure states has 4 real dimen-sions, and that the faces of Q are copies of the 3D Blochball, filling out the 7 dimensional boundary. The centresof these balls touch the largest inscribed sphere of Q .But what does it all really look like?We try to answer this question by presenting some 3Dobjects, and explaining why they cannot serve as modelsof Q . Apart from the fact that our objects are not eightdimensional, all of them lack some other features of theset of quantum states.Fig. 3 presents a hairy set which is nice but not convex.Fig. 4 shows a ball, and we know that Q is not a ball. Itis not a polytope either, so the polytope shown in Fig. 5cannot model the set of quantum states. FIG. 3: Apophatic approach: this object is not a good modelof the set Q as it is not a convex set. Let us have a look at the cylinder shown in Fig. 6, andlocate the extremal points of the convex body shown.This subset consists of the two circles surrounding bothbases. This is a disconnected set, in contrast to the con-nected set of pure quantum states. However, if one splitsthe cylinder into two halves and rotates one half by π/ FIG. 4: The set Q is not a ball...FIG. 5: The set Q is not a polytope... taking the convex hull of the seam of a tennis ball: theone dimensional seam contains the extremal points of thisset and forms a connected set.Thus the seam of the tennis ball (look again at Fig.4) corresponds to the 4 D connected set of pure statesof N = 3 quantum system. The convex hull of the seamforms a 3 D object which is easy to visualize, and serves asour first rough model of the solid 8 D body Q of qutritstates. However, a characteristic feature of the latteris that each one of its points belongs to a cross-sectionwhich is an equilateral triangle ∆ . (This is the eigenvec-tor decomposition.) The convex set determined by theseam of the tennis ball, and the set shown in Fig. 7, donot have this property.As we have seen Q can be obtained if we take anequilateral triangle ∆ and subject it to SU (3) rotationsin eight dimensions. We can try to do something similarin three dimensions. If we rotate a triangle along oneof its bisections we obtain a cone, for which the set ofextremal states consists of a circle and an apex (see Fig.10 b)), a disconnected set. We obtain a better model if FIG. 6: The set of pure states in Q is connected, but for thecylinder the pure states form two circles.FIG. 7: This is now the convex hull of a single space curve,but one cannot inscribe copies of the classical set ∆ in it. we consider the space curve (cid:126)x ( t ) = (cid:0) cos( t ) cos(3 t ) , cos( t ) sin(3 t ) , − sin( t ) (cid:1) T . (16)Note that the curve is closed, (cid:126)x ( t ) = (cid:126)x ( t + 2 π ), and be-longs to the unit sphere, || (cid:126)x ( t ) || = 1. Moreover || (cid:126)x ( t ) − (cid:126)x ( t + π ) || = √ t . Hence every point (cid:126)x ( t ) belongs toan equilateral triangle with vertices at (cid:126)x ( t ) , (cid:126)x ( t + π ) , and (cid:126)x ( t + π ) . They span a plane including the z -axis for all times t .During the time ∆ t = π this plane makes a full turnabout the z -axis, while the triangle rotates by the angle2 π/ (cid:126)x ( t ) is shown in Fig. 8 a)together with exemplary positions of the rotating trian-gle, and Fig. 8 b) shows its convex hull C . This convexhull is symmetric under reflections in the ( x - y ) and ( x - z ) FIG. 8: a) The space curve (cid:126)x ( t ) modelling pure quantumstates is obtained by rotating an equilateral triangle accordingto Eq. (16) —three positions of the triangle are shown); b)The convex hull C of the curve models the set of all quantumstates. planes. Since the set of pure states is connected this isour best model so far of the set of quantum pure states,although the likeness is not perfect.It is interesting to think a bit more about the boundaryof C . There are three flat faces, two triangular ones andone rectangular. The remaining part of the boundaryconsists of ruled surfaces: they are curved, but containone dimensional faces (straight lines). The boundary ofthe set shown in Fig. 7 has similar properties. The ruledsurfaces of C have an analogue in the boundary of theset of quantum states Q , we have already noted that ageneric point in the boundary of Q belongs to a copy of Q (the Bloch ball), arising as the intersection of Q witha hyperplane. The flat pieces of C have no analogues inthe boundary of Q , apart from Bloch balls (rank two)and pure states (rank one) no other faces exist.Still this model is not perfect: Its set of pure stateshas self-intersections. Although it is created by rotatinga triangle, the triangles are not cross-sections of C . Itis not true that every point on the boundary belongsto a face that touches the largest inscribed sphere, asit happens for the set of quantum states [17]. Indeed itsboundary is not quite what we want it to be, in particularit has non-exposed faces—a point to which we will return.Above all this is not a self-dual body. IV. A CONSTRUCTIVE APPROACH
The properties of the eight-dimensional convex set Q might conflict if we try to realize them in dimension three.Instead of looking for an ideal three dimensional modelwe shall thus use a complementary approach. To re-duce the dimensionality of the problem we investigatecross-sections of the 8 D set Q with a plane of dimen-sion two or three, as well as its orthogonal projectionson these planes—the shadows cast by the body on theplanes, when illuminated by a very distant light source.Clearly the cross-sections will always be contained in theprojections, but in exceptional cases they may coincide.What kind of cross-sections arise? In the classical caseit is known that every convex polytope arises as a cross-section of a simplex ∆ N − of sufficiently high dimension[21]. It is also true that every convex polytope arisesas the projection of a simplex. But what are the cross-sections and the projections of Q N ? There has been con-siderable progress on this question recently. The convexset is said to be a spectrahedron if it is a cross-section of acone of semi-positive definite matrices of some given size.In the branch of mathematics known as convex algebraicgeometry one asks what kind of convex bodies that canbe obtained as projections of spectrahedra. Surprisingly,the convex hull of any trigonometric space curve in threedimensions can be so obtained [26]. This includes ourset C , which can be shown to be a projection of an 8-dimensional cross-section of the 35 D set Q of quantumstates of size N = 6. We do so in Appendix A. A. The duality between projections andcross-sections
In the vector space of traceless hermitian matrices wechoose a linear subspace U . The intersection of theconvex body Q N of quantum states with the subspace U + 1l /N through the maximally mixed state 1l /N is thecross-section S U , and the orthogonal projection of Q N down to U is the projection P U . There exists a beautifulrelation between projections and cross-sections, holdingfor self-dual convex bodies such as the classical and thequantum state spaces [14]. For them cross-sections andprojections are dual to each other, in the sense that S U − /N = { u | /N + Tr( uv ) ≥ ∀ v ∈ P U } (18)and P U = { u | /N + Tr( uv ) ≥ ∀ v ∈ S U − /N } . (19) FIG. 9: The triangle is self-dual. We intersect it with a one-dimensional subspace through the centre, U , and obtain across-section extending from a to b . The dual of this line inthe plane is a 2-dimensional strip, and when we project thisonto U we obtain a projection extending from A to B , whichis dual to the cross-section within U . This is best explained in a picture (namely Fig. 9). Aspecial case of these dualities is the self-duality of the fullstate-space, eq. (15).Let us look at two examples for Q , choosing the vectorspace U to be three dimensional. In Fig.10 a) we showthe cross-section containing all states of the form ρ = / x yx / zy z / , ρ ≥ . (20)They form an overfilled tetrapak cartoon [8], also knownas an elliptope [27] and an obese tetrahedron [16]. Likethe tetrahedron it has six straight edges. Its boundary isknown as Cayley’s cubic surface, and it is smooth every-where except at the four vertices. In the picture it is sur-rounded by its dual projection, which is the convex hullof a quartic surface known as Steiner’s Roman surface.To understand the shape of the dual, start with a pairof dual tetrahedra (one of them larger than the other).Then we “inflate” the small tetrahedron a little, so thatits facets turn into curved surfaces. It grows larger, soits dual must shrink—the vertices of the dual becomesmooth, while the facets of the dual will be containedwithin the original triangles. What we see in Fig.10 a)is a “critical” case, in which the facets of the dual haveshrunk to four circular disks that just touch each otherin six special points.In Fig.10 b) we see the cross-section containing allstates (positive matrices) of the form ρ = / z/ √ x − i y x + i y / z/ √ / − z/ √ . (21) FIG. 10: a) The cross-section S U − / Q is drawn inside the projection P U of Q . b) The cone is self-dual, it is a cross-section and aprojection of Q with S U − / P U . This cross-section is a self-dual set, meaning that theprojection to this 3-dimensional plane coincides with thecross-section. In itself it is the state space of a realsubalgebra of the qutrit obervables. There exist alsotwo-dimensional self-dual cross-sections, which are sim-ply copies of the classical simplex ∆ —the state space ofthe subalgebra of diagonal matrices. B. Two-dimensional projections and cross-sections
To appreciate what we see in cross-sections and pro-jections we will concentrate on 2-dimensional screens.We can compute 2D projections using the fact thatthey are dual to a cross-section. But we can also use thenotion of the numerical range W of a given operator A ,a subset of the complex plane [28–30] W ( A ) = { z ∈ C : z = Tr ρA, ρ ∈ Q N } . (22)If the matrix A is hermitian its numerical range reducesto a line segment, otherwise it is a convex region of thecomplex plane. To see the connection to projections, ob-serve that changing the trace of A gives rise to a trans-lation of the whole set, so we may as well fix the trace toequal unity. Then we can write for some λ ∈ C A = λ + u + i v , (23)where u and v are traceless hermitian matrices. It fol-lows that the set of all possible numerical ranges W ( A )of arbitrary matrices A of order N is affinely equivalentto the set of orthogonal projections of Q N on a 2-plane[15, 31]. Thus to understand the structure of projec-tions of Q N onto a plane it is sufficient to analyze thegeometry of numerical ranges of any operator of size N .For instance, in the simplest case of a matrix A of order N = 2, its numerical range forms an elliptical disk, whichmay reduce to an interval. These are just possible (notnecessarily orthogonal) projections of the Bloch ball Q onto a plane. FIG. 11: The drawings are dual pairs of planar cross-sections S U − / P U (bright) of the convexbody of qutrit quantum states Q . Drawing a is obtainedfrom the 3D dual pair in Fig. 10 a) and b)–d) are derivedfrom the self-dual cone in Fig. 10 b). The cross-sections in b)–d) have an elliptic, parabolic and hyperbolic boundary piece,respectively. In the case of a matrix A of order N = 3 the shape ofits numerical range was characterized algebraically in [32,33]. Regrouping this classification we divide the possibleshapes into four cases according to the number of flatboundary parts: The set W is compact and its boundary ∂W
1. has no flat parts. Then W is strictly convex, it isbounded by an ellipse or equals the convex hull ofa (irreducible) sextic space curve;2. has one flat part, then W is the convex hull of aquartic space curve – e.g. W is the convex hull ofa trigonometric curve known as the cardioid;3. has two flat parts, then W is the convex hull of anellipse and a point outside it;4. has three flat parts, then W is a triangle with cor-ners at eigenvalues of A .In case 4 the matrix A is normal, AA † = A † A , and thenumerical range is a projection of the simplex ∆ ontoa plane. Looking at the planar projections of Q shownin Fig.11 we recognize cases 2 and 3. All four cases areobtained as projections of the Roman surface in Fig. 10a) or the cone shown in Fig. 10 b). A rotund shape andone with two flats are obtained as a projection of both3D bodies. A triangle is obtained from the cone and ashape with one flat from the Roman surface.In order to actually calculate a 2 D projection P := { (Tr uρ, Tr vρ ) T ∈ R | ρ ∈ Q } of the set Q determined Exemplary sets disk a ) drop b ) truncateddisk c ) truncateddrop d )non-exposed points ( ∗ ) no yes no yesnon-polyhedralcorners ( o ) no no yes yesset is self-dual yes no no yesFIG. 12: Exemplary convex sets and their duals. Sym-bols: non-exposed point ( ∗ ), polyhedral corners (+) and non-polyhedral corners ( o ). Sets a) and d) are self-dual, while b)and c) is a dual pair. Sets a) and c) have properties like 2Dcross-sections of Q N , while sets a) and b) could be obtainedfrom Q N by projection. by two traceless hermitian matrices u and v one may pro-ceed as follows [28]. For every non-zero matrix F in thereal span of u and v we calculate the maximal eigenvalue λ and the corresponding normalized eigenvector | ψ (cid:105) with F | ψ (cid:105) = λ | ψ (cid:105) . Then ( (cid:104) ψ | u | ψ (cid:105) , (cid:104) ψ | v | ψ (cid:105) ) T belongs to theprojection P , and these points cover all exposed pointsof P . C. Exposed and non-exposed faces
An exposed face of a convex set X is the intersectionof X with an affine hyperplane H such that X \ H isconvex, i.e. H intersects X only at the boundary. Ex-amples in the plane are the boundary points of the diskin Fig. 12 a) or the boundary segments in panels b) andd). A non-exposed face of X is a face of X that is notan exposed face. In dimension two non-exposed faces arenon-exposed points, they are the endpoints of boundarysegment of X which are not exposed faces by themselves.Examples are the lower endpoints of the boundary seg-ments in Fig. 12 b) or d).It is known that cross-sections of Q N have no non-exposed faces. On the other hand the twisted cylinder(see Fig. 7) and the convex hull C of the space curve(Fig. 8) do have non-exposed faces of dimension one. Incontrast to cross-sections, projections of Q N can havenon-exposed points, see e.g. the planar projections of Q in Fig. 11. They are related to discontinuities in certainentropy functionals (in use as information measures) [34].The dual concept to exposed face is normal cone [13].The normal cone of a two-dimensional convex set X ⊂ R at ( x , x ) ∈ X is { ( y , y ) T ∈ R | ( z − x ) y + ( z − x ) y ≤ ∀ ( z , z ) ∈ X } . The normal cone generalizes outward pointing normalvectors of a smooth boundary curve of X to points( x , x ) where this curve is not smooth. Then the dimen-sion of the normal cone is two and we call ( x , x ) a cor-ner. The examples in Fig. 12 have 0 , , , Q N is polyhedral [13]. Ananalogue property holds in higher dimensions but it cannot be formulated in terms of polyhedra. Fig. 11 showsthat two-dimensional cross-sections of Q can have non-polyhedral corners.Given a two-dimensional convex body including theorigin in the interior, the duality (13) maps non-exposedpoints onto the set of non-polyhedral corners of the dualconvex body. There will be one or two non-exposedpoints in each fiber depending on whether the corner doesor does not lie on a boundary segment of the dual body[35]. We conclude that a two-dimensional self-dual con-vex set has no non-exposed points if and only if all itscorners are polyhedral. V. WHEN THE DIMENSION MATTERS
So far we have discussed the qutrit, and properties ofthe qutrit that generalise to any dimension N . But whatis special about a quantum system whose Hilbert spacehas dimension N ? The question gains some relevancefrom recent attempts to find direct experimental signa-tures of the dimension,One obvious answer is that if and only if N is a compos-ite number, the system admits a description in terms ofentangled subsystems. But we can look for an answer inother directions too. We emphasised that a regular sim-plex ∆ N − can be inscribed in the quantum state space Q N . But in the Bloch ball we can clearly inscribe notonly ∆ (a line segment), but also ∆ (a triangle) and∆ (a tetrahedron). If we insist that the vertices of theinscribed simplex should lie on the outsphere of Q N , andalso that the simplex should be centred at the maximallymixed state, then this gives rise to a non-trivial problemonce the dimension N >
2. This is clear from our modelof the latter as the convex hull of the seam of a tennisball, or in other words because the set of pure states forma very small subset of the outsphere. Still we saw, in Fig.10 a), that not only ∆ but also ∆ can be inscribed in0 Q , and as a matter of fact so can ∆ and ∆ . But is italways possible to inscribe the regular simplex ∆ N − in Q N , in such a way that the N vertices are pure states?Although the answer is not obvious, it is perhaps sur-prising to learn that the answer is not known, despite aconsiderable amount of work in recent years.The inscribed regular simplices ∆ N − are known assymmetric informationally complete positive operatorvalued measures, or SIC-POVMs for short. Their ex-istence has been established, by explicit construction, inall dimensions N ≤
16 and in a handful of larger di-mensions. The conjecture is that they always exist [36].But the available constructions have so far not revealedany pattern allowing one to write down a solution for alldimensions N . Already here the quantum state spacebegins to show some N -dependent individuality.Another question where the dimension matters con-cerns complementary bases in Hilbert space. As we haveseen, given a basis in Hilbert space, there is an ( N − Q N in which these vectorsappear as the vertices of a regular simplex ∆ N − . Wecan—for instance for tomographic reasons [37]—decideto look for two such cross-sections placed in such a waythat they are totally orthogonal with respect to the traceinner product. If the two cross-sections are spanned bytwo regular simplices stemming from two Hilbert spacebases {| e i (cid:105)} N − i =0 and {| f i (cid:105)} N − i =0 , then the requirement onthe bases is that |(cid:104) e i | f j (cid:105)| = 1 N (24)for all i, j . Such bases are said to be complementary, andform a key element in the Copenhagen interpretation ofquantum mechanics [38]. But do they exist for all N ?The answer is yes. To see this, let one basis be the com-putational one, and let the other be expressed in termsof it as the column vectors of the Fourier matrix F N = 1 √ N . . . ω ω . . . ω N ω ω . . . ω N − ... ... ... ...1 ω N − ω N − . . . ω ( N − , (25)where ω = e πi/N is a primitive root of unity. The Fouriermatrix is an example of a complex Hadamard matrix , aunitary matrix all of whose matrix elements have thesame modulus.We are interested in finding all possible complementarypairs up to unitary equivalences. The latter are largelyfixed by requiring that one member of the pair is the com-putational basis, since the second member will then bedefined by a complex Hadamard matrix. The remainingfreedom is taken into account by declaring two complexHadamard matrices H and H (cid:48) to be equivalent if theycan be related by H (cid:48) = D P HP D , (26)where D i are diagonal unitary matrices and P i are per-mutation matrices.The task of classifying pairs of cross-sections of Q N forming simplices ∆ N − and sitting in totally orthogonal N -planes is therefore equivalent to the problem of classi-fying complementary pairs of bases in Hilbert space. Thisproblem in turn is equivalent to the problem of classify-ing complex Hadamard matrices of a given size. But thelatter problem has been open since it was first raised bySylvester and Hadamard, back in the nineteenth century.It has been completely solved only for N ≤
5, and it wasrecently almost completely solved for N = 6 [39].More is known if we restrict ourselves to continuousfamilies of complex Hadamard matrices that include theFourier matrix. Then it has been known for some time[40] that the dimension of such a family is bounded fromabove by d F N = N − (cid:88) k =0 gcd( k, N ) − (2 N − , (27)where gcd denotes the largest common divisor, andgcd(0 , N ) = N . We subtracted the 2 N − N = p k is a power of prime number p this bound is saturated byfamilies that have been constructed explicitly. In partic-ular, if N is a prime number d F p = 0, and the Fouriermatrix is an isolated solution. For N = 4 on the otherhand there exists a one-parameter family of inequivalentcomplex Hadamard matrices.Further results on this question were presented inBia(cid:32)lowie˙za [41]. In particular the above bound is notachieved for any N not equal to a prime power and notequal to 6. It turns out that the answer depends criticallyon the nature of the prime number decomposition of N .Thus, if N is a product of two odd primes the answerwill look different from the case when N is twice an oddprime. However, at the moment, the largest non-primepower dimension for which the answer is known—evenfor this restricted form of the problem—is N = 12.At the moment then, both the SIC problem and theproblem of complementary pairs of bases highlight thefact that the choice of Hilbert space dimension N hassome dramatic consequences for the geometry of Q N .Now the basic intuition that drove Mielnik’s attempts togeneralize quantum mechanics was the feeling that thenature of the physical system should be reflected in thegeometry of its convex body of states [1]. Perhaps thisintuition will eventually be vindicated within quantummechanics itself, in such a way that the individuality ofthe system is expressed in the choice of N ?1 VI. CONCLUDING REMARKS
Let us try to summarize basic properties of the set Q N of mixed quantum states of size N ≥ Q N is a convex set of N − Q N is inscribed in a sphere of radius R N = (cid:112) ( N − / N , and it contains the maximal ball of radius r N = 1 / (cid:112) N ( N − Q N is neither a polytope nor a smooth body.d) The set of mixed states is self-dual (15).e) All cross-sections of Q N have no non-exposed faces.f) All corners of two-dimensional projetions of Q N arepolyhedral.g) The boundary ∂ Q N contains all states of less thanmaximal rank.h) The set of extremal (pure) states forms a connected2 N − N − ∂ Q N .i) Explicit formulae for the volume V and the area A of the d = N − Q N are known[18]. The ratio Ar/V is equal to the dimension d , whichimplies that Q N has a constant height [17] and can bedecomposed into pyramids of equal height having all theirapices at the centre of the inscribed sphere.It is a pleasure to thank Marek Ku´s and GniewomirSarbicki for fruitful discussions and helpful remarks. I.B.and K. ˙Z. are thankful for an invitation for the workshopto Bia(cid:32)lowie˙za, where this work was presented and im-proved. Financial support by the grant number N N202090239 of Polish Ministry of Science and Higher Educa-tion and by the Swedish Research Council under contractVR 621-2010-4060 is gratefully acknowledged. Appendix A: Trigonometric curves
We write the convex hull C of the trigonometric spacecurve in Section III as a projection of a cross-section ofthe 35-dimensional set Q of density matrices. Up to thetrace normalization, this problem is solved in [26] for theconvex hull of any trigonometric curve [0 , π ) → R n . Theassumptions are that each of the n coefficient functionsof the curve is a trigonometric polynomial of some finitedegree 2 d , t (cid:55)→ (cid:80) dk =1 ( α k cos( kt ) + β k sin( kt )) + γ for real coefficients α k , β k , γ .The space curve (16) lives in dimension n = 3, we de-note its coefficients by (cid:126)x = ( x , x , x ) T . Using trigono-metric formulas and the parametrization cos( t ) = y − y y + y and sin( t ) = y y y + y we have1 def. = ( y + y ) ,x = ( y − y ) [( y − y ) − y y ) ] ,x = ( y − y )(2 y y )[3( y − y ) − (2 y y ) ] ,x = − ( y + y ) (2 y y ) . A basis vector of m -variate forms of degree 2 d = 8 is givenby (cid:126)ξ = ( x , x x , x x , x x , x x , x x , x x , x x , x )(for the number m = 1 used in [26] for the degrees offreedom of the projective coordinates ( y : y ) in the cir-cle P ( R )) and we have(1 , x , x , x ) T = A(cid:126)ξ for the 4 × A = (cid:18) −
16 0 30 0 −
16 0 10 6 0 −
26 0 26 0 − − − (cid:19) . Let us denote by M (cid:23) M is positive semi-definite. The 5 × (cid:126)u = ( u , . . . , u ) is given by M ( (cid:126)u ) = (cid:32) u u u u u u u u u u u u u u u u u u u u u u u u u (cid:33) . Now [26] provides the convex hull representation C def. = conv { (cid:126)x ( t ) ∈ R | t ∈ [0 , π ) } = (A1) { (cid:16) v v v (cid:17) ∈ R | ∃ (cid:126)u ∈ R s.t. (cid:18) v v v (cid:19) = A(cid:126)u and M ( (cid:126)u ) (cid:23) } which we shall simplify by eliminating the variables u , . . . , u .A particular solution of (1 , v , v , v ) T = A(cid:126)u is (cid:101) u = (4 + v ) , (cid:101) u = (3 v − v ) , (cid:101) u = (1 − v ) , (cid:101) u = ( − v − v ) , (cid:101) u = (cid:101) u = (cid:101) u = (cid:101) u = (cid:101) u = 0. The reduced row echelonform of A being (cid:32) / /
11 0 2 /
11 00 0 1 0 − / − /
11 0 3 /
11 0 (cid:33) and regarding u , . . . , u as free variables we have u = (cid:101) u − u − u , u = (cid:101) u − u − u ,u = (cid:101) u + u − u , u = (cid:101) u + u − u . One problem remains, the matrix M parametrized by v , v , v and u , . . . , u has not trace one,Tr M = u + u + u + u + u = (17 − u + 3 v ) . M andby defining M = (cid:32) M u + (1 − v ) (cid:33) . If M (cid:23) u ≥ u is a diagonalelement of M and − ≤ v ≤ v , v , v ) ∈ C is included in the unit ball of R . This proves M (cid:23) ⇐⇒ M (cid:23) C = { (cid:16) v v v (cid:17) ∈ R | ∃ (cid:32) u ... u (cid:33) ∈ R s.t. M (cid:23) } . We conclude that C is a projection of the 8-dimensionalspectrahedron { ( v , v , v , u , . . . , u ) ∈ R | M (cid:23) } ,which is a cross-section of Q . [1] B. Mielnik, Geometry of quantum states Commun. Math.Phys. , 55–80 (1968).[2] M. Adelman, J. V. Corbett and C. A. Hurst, The geom-etry of state space, Found. Phys. , 211 (1993).[3] G. Mahler and V. A. Weberuss, Quantum Networks (Springer, Berlin, 1998).[4] E. M. Alfsen and F. W. Shultz,
Geometry of State Spacesof Operator Algebras , (Boston: Birkh¨auser 2003)[5] J. Grabowski, M. Ku´s, G. Marmo Geometry of quantumsystems: density states and entanglement
J.Phys.
A 38 ,10217 (2005).[6] I. Bengtsson and K. ˙Zyczkowski,
Geometry of quantumstates: An introduction to quantum entanglement (Cam-bridge: Cambridge University Press 2006).[7] L. Hardy, Quantum Theory From Five Reasonable Ax-ioms, preprint quant-ph/0101012[8] F. J. Bloore, Geometrical description of the convex setsof states for systems with spin − / − J. Phys.
A 9 , 2059 (1976).[9] Arvind, K. S. Mallesh and N. Mukunda, A generalizedPancharatnam geometric phase formula for three-levelquantum systems,
J. Phys.
A 30 , 2417 (1997).[10] L. Jak´obczyk and M. Siennicki, Geometry of Bloch vec-tors in two–qubit system,
Phys. Lett.
A 286 , 383 (2001).[11] F. Verstraete, J. Dahaene and B. DeMoor, On the geom-etry of entangled states,
J. Mod. Opt. , 1277 (2002).[12] P. Ø. Sollid, Entanglement and geometry, PhD thesis,Univ. of Oslo 2011.[13] S. Weis, A Note on Touching Cones and Faces, Journalof Convex Analysis (2012). http://arxiv.org/abs/1010.2991 [14] S. Weis, Quantum Convex Support, Lin. Alg. Appl. ,3168 (2011).[15] C. F. Dunkl, P. Gawron, J. A. Holbrook, J. A. Miszczak,Z. Pucha(cid:32)la and K. ˙Zyczkowski, Numerical shadow and ge-ometry of quantum states,
J. Phys.
A44 , 335301 (2011).[16] S. K. Goyal, B. N. Simon, R. Singh, and S. Simon,Geometry of the generalized Bloch sphere for qutrit, http://arxiv.org/abs/1111.4427 [17] S. Szarek, I. Bengtsson and K. ˙Zyczkowski, On the struc-ture of the body of states with positive partial transpose,
J. Phys.
A 39 , L119–L126 (2006).[18] K. ˙Zyczkowski and H.–J. Sommers, Hilbert–Schmidt vol-ume of the set of mixed quantum states,
J. Phys.
A 36 ,10115–10130 (2003).[19] J. Grabowski, M. Ku´s, and G. Marmo, Geometry ofquantum systems: density states and entanglement,
J.Phys.
A38 , 10217 (2005).[20] R. T. Rockafellar,
Convex Analysis (Princeton: Prince- ton University Press 1970).[21] B. Gr¨unbaum,
Convex Polytopes , 2nd ed., (New York:Springer-Verlag, 2003)[22] A. Wilce, Four and a half axioms for finite dimensionalquantum mechanics, http://arxiv.org/abs/0912.5530 (2009).[23] M. P. M¨uller and C. Ududec, The power of reversiblecomputation determines the self-duality of quantum the-ory, http://arxiv.org/abs/1110.3516 (2011).[24] G. Kimura, The Bloch vector for N –level systems, Phys.Lett.
A 314 , 339 (2003).[25] G. Kimura and A. Kossakowski, The Bloch-vector spacefor N -level systems — the spherical-coordinate point ofview, Open Sys. Information Dyn. , 207 (2005).[26] D. Henrion, Semidefinite representation of convex hullsof rational varieties, http://arxiv.org/abs/0901.1821 (2009).[27] P. Rostalski and B. Sturmfels, Dualities in convex alge-braic geometry, http://arxiv.org/abs/1006.4894 (2010).[28] A. Horn and C. R. Johnson, Topics in Matrix Analysis (Cambridge: Cambridge University Press, 1994)[29] K. E. Gustafson and D. K. M. Rao,
Numerical Range:The Field of Values of Linear Operators and Matrices (New York: Springer-Verlag, 1997)[30] P. Gawron, Z. Pucha(cid:32)la, J. A. Miszczak, (cid:32)L. Skowronekand K. ˙Zyczkowski, Restricted numerical range: a versa-tile tool in the theory of quantum information,
J. Math.Phys. , 102204 (2010).[31] D. Henrion, Semidefinite geometry of the numericalrange, Electronic J. Lin. Alg. , 322 (2010).[32] R. Kippenhahn, ¨Uber den Wertevorrat einer Matrix, Math. Nachr. , 193–228 (1951).[33] D.S. Keeler, L. Rodman and I.M. Spitkovsky, The nu-merical range of 3 × Lin. Alg. Appl. http://arxiv.org/abs/1007.5464 (2010).[35] S. Weis, Duality of non-exposed faces, http://arxiv.org/abs/1107.2319 (2011).[36] A. J. Scott and M. Grassl, SIC-POVMs: A new computerstudy,
J. Math. Phys. , 042203 (2010).[37] W. K. Wootters and B. D. Fields, Optimal state-determination by mutually unbiased measurements, Ann.Phys. , 363 (1989).[38] J. Schwinger:
Quantum Mechanics. Symbolism ofAtomic Measurements , ed. by B.–G. Englert (Berlin: Springer-Verlag 2001).[39] F. Sz¨oll˝osi, Construction, classification and parametriza-tion of complex Hadamard matrices, PhD thesis, http://arxiv.org/abs/1150.5590 (2011).[40] W. Tadej and K. ˙Zyczkowski, Defect of a unitary matrix, Lin. Alg. Appl.429