Geometry of density states
We reconsider the geometry of pure and mixed states in a finite quantum system. The ranges of eigenvalues of the density matrices delimit a regular symplex (Hypertetrahedron ) in any dimension ; the polytope isometry group is the symmetric group , and splits in chambers, the orbits of the states under the projective group . The type of states correlates with the vertices, edges, faces, etc. of the polytope, with the vertices making up a base of orthogonal pure states. The entropy function as a measure of the purity of these states is also easily calculable; we draw and consider some isentropic surfaces. The Casimir invariants acquire then also a more transparent interpretation.
pacs:03.65.Ca, 02.10.De, 02.40.Et
Pure states in Quantum Mechanics are described by rays in a Hilbert space , or equivalently as unidimensional subspaces. As such, they can be identified with orthogonal projectors of unit trace (so for -dimensional subspaces they will have trace ); denoting by End the linear operators of a vector space , we have
To include mixed states , it is enough (von Neumann) to relax idempotency ( , so Spectrum ) to mere positivity, Spec 0, which implies for Spec . Hence
So the pure states are extremal, as , and one can write the mixed states as incoherent addition of pure ones,
For finite quantum systems ( , dim ) the geometry of these states has been studied lately from different points of view, as it is pertinent to quantum information and other modern applications. For example, in 1 () the nature of the space of pure states, namely the projective spaces , is stressed, as they are rank-one symmetric spaces; in 2 () the mixed states are related to a coherence vector , lying in a subset of euclidean space, determined by the values of Casimir invariants for the group . 3 () studies the stratification of general states under the projective group
in relation to a (in general not regular) foliation by Khler manifolds, similar to the orbits of the (co-)adjoint representation of , that is, the Kirillov-Souriau method. The geometry of the case had been studied sometime ago also by Michel 4 () in relation to as a flavor group.
In this report we would like to complete the above descriptions in several ways. In particular, we elaborate an idea in 3 (), showing that the set of eigenvalues of the density operator , Spec , coincides precisely with the points of a solid simplex (the hypertetrahedron ), the simplest regular polytope. The natural isometry group, the full symmetric group , corresponds, of course, to a (finite) Coxeter group, that is, the Weyl group of the Lie algebra of the group : the unit trace in our case is traded for the traceless character of (anti-)hermitian matrices describing . The orbit space is still an (irregular) hypertetrahedron, of size of the previous regular one, behaves like the Weyl chambers, and describes precisely the orbits of our states under the projective group, each point being just an orbit. The type of orbits, as classified by the little groups, is in correspondence with the combinatorial elements of the simplices, namely vertices, edges, faces and so on, and also with the partitions of the number .
For example for the set of pure states is the 2-sphere , the set of all states is the three dimensional Bloch ball B of unit radius, with boundary being the pure states, the simplex is just the closed segment I: [-1, 1], the Weyl group Z is reflection in the middle, and the chamber is the half-segment I/Z =[0, 1] with 0 the most mixed states (called or henceforth), and with 1 indicating the (sphere of) pure states. In this simplest (and nonrepresentative) case there is only one stratum, i.e. one type of orbit (namely 2-spheres), besides of course the fix point or 0; the (single) Casimir just labels the radius of these spheres , . The state corresponds to partition , the rest to , as stratum orbits .
The Casimir invariants admit a double interpretation, as an homogeneous system of generators of the center of the enveloping algebra of Lie group, or in our case, as the symmetric functions (Newton) over the roots of the spectral equation for the density operator. We elaborate here the considerations of 2 (), based on the pioneer work of Biedenharn 5 ().
In this picture it is also clear how to compute the entropy of these mixed states, where = - Trlog(). This varies between = log for the state to log =0 for pure states. We include some graphs showing the entropy for some boundary lines on the ”chambers” of the orbit space, as well as some isentropic surfaces in the general case. Each case includes, in a precise sense, all the previous ones, .
We stress also the action of the projective isometry group as the group acting effectively in the space of states, in the sense of the characterization of geometries (F. Klein): in the transformation the effective group acting is : the kernel of the action is , and even acts with kernel : the explanative diagram is
The organization of the paper is as follows. In Sect. 2 we establish notation and show several properties of our objects needed later. In Sect. 3 we recall the situation for qubits (=2), qutris (=3), and =4, incorporating the new observations of above and suggesting generalizations to higher dimensions. Then Sect. 4 deals with the general theory for arbitrary : we exhibit explicitely the polytope, the Weyl quotient (”chamber”), the types of states, the Casimir invariants and discuss the entropy function, and also some cases of isentropic surfaces.
Other considerations (including mention of omissions) are in our final Sect. 5.
Ii General Description and Properties of Mixed States
are respectively linear and affine, so the whole set of solutions of the equation is like a real vector space of dim -1. The positivity condition , though, is a convex condition, and selects a nonlinear submanifold of R (which is closed as the spectral restriction is really 1), of the same dimension, with as the extremals of the convex set (not the boundary, in general: we know of course that , so dim for large ). In the set of hermitian traceless matrices one introduces the definite scalar product
and therefore there exists an orthonormal base ,, …, with properties
So the general density matrix can be written as
where the factor guarantees that pure states, have norm ; see 2 () for details; is called the coherence vector.
We recall also two related properties of the group ; first, the square of the adjoint representation contains the adjoint BOTH in the symmetric and in the antisymmetric part (for =2 the first is missing), and second, consequently, this induces two algebras in the space of matrices (corresponding to the coefficients and of Gell-Mann; of course, the ’s correspond to the Lie algebra structure); for details see 4 ().
Now the group is acting in the set of states as . If the eigenvalues of are different, the little group is , and then the generic orbit is , of dimension (-1). This is a Khler manifold, as are all orbits of the adjoint of any simple Lie group (we notice that (-1) is even); in particular this space is called a (complex) flag manifold, with structure (as homogeneous space),
The other extreme contains the pure states, with little group ; these most critical orbits have dimension 2(-1), as we said above. There is always naturally a single most-mixed state, O (or ), with = 0: this is the fixed point under the action, and it is unique. Orbits in state space with conjugate little groups form what is called a stratum (see e.g. 4 ()), and in fact for a compact group (as is our case) acting ”nicely” in some space X the number of strata is finite; we shall see that it coincides with the number of types of states and also naturally with the partitions of .
We define the entropy of a state as the expectation value of the operator = -log ( (von Neumann; notice , so log() is negative); recalling that in the density formalism , we have, with Spec ,
This varies, as said, between log and log 1=0: the state is the most disordered, and the pure states are the most ordered: we shall see also the case reproduces the entropy function for all previous cases , and therefore there must be several manifolds of mixed states isentropic, with the same value for the entropy: we shall provide some examples.
As for the Casimir invariants, we define them from the spectral equation for the density operator as the coefficients of the powers, i. e. :
where = Tr , (the first Casimir). They determine the spectrum up to a permutation, e.g.
These operators can be expressed also as traces of powers of representative matrices, as for example
There are also several inequalities assuring all Casimirs are nonnegative, etc.
Iii The Simplest Cases
For N=1 we have just a single point, a pure state, as = point, with entropy log(1)= 0. For N=2 the density matrix can be written as
we have ball (radius 1) with the 2-dim boundary sphere of pure states. The set of eigenvalues or () with makes up a segment I in fig 1,
where is e.g. the state [0, -1], is (1/2, 1/2), and is [1, 0]. The crucial but trivial point now is that under the symmetry : () () the two states are equivalent, so the segment I becomes just the half-segment [0, 1]; that is, the state (0), the mixed states () and the (representative of) pure state(s) , = 1. In this case , so the orbits are just spheres, and x in the half-segment means just the radius of them. There are only two strata, the fixed point (little group ) and the rest, little group SO(2)=U(1)). The two strata correspond to the two partitions namely is  and is the partition .
The entropy is a smooth function from log(2)=0.693 (for the state) to log(1)=0 for the pure one (fig 2): we shall see that the entropy function of the case always contains that of the previous cases.
There is only a Casimir, the quadratic one, with
which lies between 1/4 and zero.
One can also use an ”angle” picture 2 (), namely with , , . The (quadratic) Casimir is now . The entropy in terms of angle variables is
For N=3 we have ), where the ’s are e.g. the Gell-Mann matrices. Choosing and diagonal, we get, with and
as . Hence positivity implies , and one gets a regular triangle ”upside down” inscribed in the circle of radius one (fig 3).
Notice the three vertices ; 3 edges , and the three interior lines (heights) with intersection points . The figure is identical to the so-called Fano plane, related (among other things) to the octonion multiplication rule 6 () and to the projective plane ,with automorphisms the simple group of 168 elements 7 (). This is another test of the relation of with octonions!
Under the action of the symmetry group, now with 6 elements, isomorphic to the dihedral group (the isometry of the regular n-gon is ), the fundamental domain or chamber is a little rectangular triangle (there are six of them), 1/6 in area of the big one; we select one of them as as in fig. 3. Here is the state, the pure, intermediate, i. e., =log(3), =log(2), = log(1)=0.
In terms of the eigenvalues of , is (1/3, 1/3, 1/3), is (1, 0, 0), is (1/2, 1/2, 0); recall that the order is immaterial. The line is type (y); the line is type (), and this line reproduces the =2 case, with seen as the state.
The line is type , with (). Note all the statements are order-invariant! In fact in the large triangle the line is a single line! Notice also the boundary lines in the big triangle unite pure states, so they have 0 in the spectrum. Each line represents the three embeddings of in , called by Gell-Mann and spin. It becomes a single line in the little triangle, line .
Note the set of generic states (little group ) is 6-dimensional, open and dense; therefore, the representatives are bi-dimensional (2+6=8=dim ). The critical states (two equal eigenvalues, little group are the two lines emanating from , but is the fixed point, with isotropy just itself. The three partitions of  are: is , and , which is the flag manifold. Notice also the little triangle is rectangular, NOT regular.
The Casimir invariants now are two, quadratic and cubic. They are
The quadratic Casimir is zero for the pure state, the cubic zero for pure state also and on the boundary line QP. We have: 1/3 1/27 P 1/4 0 Q 0 0
There is an inequality between and , because the cubic equation has to have real roots; see 4 ().
One can also parameterize the states with two angles 2 (), with e.g.
The entropy of the () state, with , is . Fig 4 shows the entropy surface. It varies from log(3) at to log(2) at to log(1)=0 at . Fig 5 shows the isentropic lines over the triangle. The point on line with is .768, .116, .116.
We have come now to the case. The general state with is
We take as
which gives the diagonal density matrix
in particular . The figure is now a regular tetrahedron inscribed in . Again, the action of the 24 elements of the group generates the little rectangular tetrahedron as quotient (fig 6).
In the eigenvalue notation , the coordinates of the vertices are:
is (1/4, 1/4, 1/4, 1/4); state=, partition .
is (1, 0, 0, 0); pure state=, partition [3,1].
is (1/2, 1/2, 0, 0); nongeneric mixed state=, partition [2,2].
is (1/3, 1/3, 1/3, 0); nongeneric mixed state=, partition [3,1].
There are six lines joining vertices, as follows Length 1 1/3 Spectrum type [3,1] [2,2] [3,1] [2,1,1] [2,1,1] [2,1,1]
There are four faces, we describe just two here: Spectrum Type [2,1,1] [1,1,1,1] .
Finally, there are five types of orbits: Spectrum 1/4, 1/4, 1/4, 1/4 Representative =O =1000 Interior dim 0 6 8 10 12 character Fixed point Pure states Generic (Flag manifold) G/H U(4)/U(4) U(4)/[U(1)xU(3)] U(4)/U(2) U(4)/[U(2)xU(1)] U(4)/U(1) The discussion of the Casimir invariants and the calculation of the entropy follows similar lines to the case, so we omit the details here. The face reproduces the previous case. This case is important as being the first case with correlation, for example entanglement.
The density matrix for an level system is given by
where the ’s are Hermitian traceless matrices with square 2; of them can be diagonalized simultaneously; let us call them . Then the density matrix is given by
So again, ; the ranges of eigenvalues always give us a regular simplex or solid hypertetrahedron , because there are orthogonal pure states in , symmetrically distributed in the sphere of unit radius. The hyperfaces of the polytope delimit the range. The center corresponds to the state, which remains invariant under or .
The (full) symmetry group is , of course (arbitrary permutation of vertices etc.); this divides in rectangular, irregular little hypertetrahedra. In any of them we have , the state; , the vertex; and ’s, from (center of the hyperface) to (center of the edge uniting, in , with another vertex ).
Notice in this picture some elements are interior in the original, regular polytope, others are at the boundary (extremal); for example, the cell which does not contain the = is in the boundary, but the others are interior; similar for other elements: faces, edges, etc.
The calculation of the Casimir invariants starting from eq. (23) is mechanical and trivial. Some general results are
because of a zero eigenvalue. Also
because of two zero eigenvalues, etc. Besides
because all the eigenvalues are equal to .
Let us discuss the entropy in this general situation; the formula
can be applied without any difficulty; again, we just include some results
for . Besides for = edge, = face, we have
where is in the boundary edge, in the boundary face, …, in the boundary cell, the -polytope. There are plenty of isentropic surfaces, which we refrain to make explicit, as we already established them in the case. See also 2 ().
V Final Remarks
Our purpose in this work has been to describe the eigenvalue set and the orbit space of density matrices in a finite quantum system. The picture is
where = set of density states/.
The actual geometry of mixed states, out of the cases, is rather involved 8 () and we have not pretended to improve on it. There is also the interesting problem of finding maps between density states, positive/strictly positive, untouched also here; see e.g. the recent work 9 ().
The geometry of the complex flag manifolds is very rich; we remark here only one result
where the twisted product is similar to the used in 10 () to express the homology of Lie groups.
The case reminds one of the 3-dim Jordan algebras: the action of on the 3 x 3 hermitian traceless matrices is similar to the action of the exceptional group on the exceptional Jordan octonionic algebra; indeed, the Moufang plane is (Borel, 1950), whereas our is . In fact, the two cases are related, as the following beautiful chain of groups show
where Aut leads to the fixes sets (i.e. , and ), and (trialitiy) and (duality) means to relate Spin(8) with and with . That is, plus the 3 8-dim representations and plus 3 and .
Acknowledgements.We wish to thank Prof. Sudarshan, Prof. Byrd and Dr. Modi for previous discussions. L. J. B. thanks M. E. C. (Spain) for grant FPA-2006-20315.
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