Geometric approach to non-relativistic Quantum Dynamics of mixed states

In this paper we propose a geometrization of the non-relativistic quantum mechanics for mixed states. Our geometric approach makes use of the Uhlmann's principal fibre bundle to describe the space of mixed states and as a novelty tool, to define a dynamic-dependent metric tensor on the principal manifold, such that the projection of the geodesic flow to the base manifold gives the temporal evolution predicted by the von Neumann equation. Using that approach we can describe every conserved quantum observable as a Killing vector field, and provide a geometric proof for the Poincare quantum recurrence in a physical system with finite energy levels.


I. INTRODUCTION
The geometrization of physical theories is a successful and challenging area in theoretical physics. The most well known examples are Hamiltonian mechanics based on symplectic geometry, General Relativity based on semi-Riemannian geometry and classical Yang-Mills theory which uses fibre bundles 11 .
Geometric ideas have also found a clear utility in non-relativistic quantum mechanics problems because quantum theory can be formulated in the language of Hamiltonian phasespace dynamics 12 . Hence, the quantum theory has an intrinsic mathematical structure equivalent to Hamiltonian phase-space dynamics. However, the underlying phase-space is not the same space of classical mechanics, but the space of quantum mechanics itself, i.e., the space of pure states or the space of mixed states.
Unlike General Relativity or Gauge Theory where the metric tensor or the connection are related with the physical interaction, the most usual geometric formulation of the geometry of non-relativistic quantum mechanics is not dynamic, in the sense that is insensitive to changes in the Hamiltonian of the system. Under these assumptions, that approach only makes use of the differential structure of the Hilbert space for quantum states and the Fubini-Study metric. See for example the geometric interpretation of Berry's phase 3 .
From a more dynamical point of view, A. Kryukov 16 has stated that the Schrödinger equation 19 for a pure state |α t (Plank's constant is set equal to 1) can be considered as a geodesic flow in a certain Riemannian manifold with an accurate metric which depends on the Hamiltonian of the system.
The goal of this paper is to generalize the work of Kryukov for mixed states. To this end, we provide an underlying differential manifold to describe mixed states and a dynamicdependent Riemannian metric tensor to analyze their temporal evolution.
The mixed states are characterized by density matrices and the equation which plays the role of the Schrödinger one is the von Neumann equation 23 To obtain the underlying differential manifold following the Uhlmann's geometrization for non-relativistic quantum mechanics [26][27][28][29] , we make use of a principal fibre bundle such that its base manifold is the space of mixed states. Finally, to provide the Riemannian metric we choose an appropriate metric in the principal bundle, in such a way that the projection of the geodesic flow in the principal manifold to the base manifold is just the temporal evolution given by the von Neumann equation.
Among the geometric properties that are observed due to the movement of this geodesic flow, in this paper we analyze the phase volume conservation according to the Liouville Theorem. That allow us to show a geometric proof of the Poincare recurrence theorem relating it with the recurrence principle for physical systems with discrete energy levels. Let us emphasize that our geometric proof for the quantum Poincare recurrence is closer to the classical mechanics proof 1 (that also uses the conservation of the volume in the phase-space evolution) than the previous given in the quantum setting 5,22,24 .

II. DENSITY MATRICES SPACE AS A BASE OF A PRINCIPAL FIBRE BUNDLE
The most general state, the so-called mixed state, is represented by a density operator in the Hilbert space H. In this paper we always assume that dim(H) = n < ∞, being H a vector space on the complex field (H = C n ). The density operator ρ is in fact a density matrix. Recall that a density matrix is a complex matrix ρ that satisfies the following properties: 1. ρ is a hermitian matrix, i.e, the matrix coincides with its conjugate transpose matrix: 2. ρ is positive ρ ≥ 0. It means that any eigenvalue of A is non-negative.
Let us denote by P the space of mixed quantum states. Note that the space of pure states P(H) is just Recall that the space of quantum pure states has an elegant interpretation as a U(1)-fibre bundle S(H) → P(H). Following Uhlmann [26][27][28][29] In quantum information theory 18 , the procedure of extension, H → H ext is known as attaching an ancilla living in H. Obviously, the space of squared matrices M n,n (C) (n rows, n columns) over C (that is a 2n 2 real dimensional manifold) can be identified with H ext M n,n (C) ∼ = H ext .
Since tr(W W † ) is a smooth function in the space of squared matrices, by the Regular Level Set Theorem 17 , the set is a smooth manifold of M n,n (C). Actually, it is not hard to see that S 0 is diffeomorphic to the sphere S 2n 2 −1 . If ρ is a mixed state in P, we shall denote an element W ∈ S 0 a purification of ρ if therefore, we get the space of density matrices P by the projection π : S 0 → P, where the projection is given by Observe that, if u is an unitary matrix (i.e, uu † = u † u = I n ) then Moreover, to fix notation recall that the Lie group U(n) is a Lie transformation group 13 acting on S 0 on the right. In general, a principal fibre bundle 13 will be denoted by P (M, G, π), being P the total space, M the base space, G the structure group and π the projection. For each x ∈ M, π −1 (x) is a closed submanifold of P , called the fibre over x.
If p is a point of π −1 (x), then π −1 (x) is the set of points {pa, a ∈ G}, and it is called fibre through p.
At this point, an important question to answer, is if S 0 (P, U(n), π) is a principal fibre bundle over the base manifold P of density matrices. Unfortunately the answer is no, because U(n) does not act freely on S 0 . In general, W u = W for W ∈ S 0 and u ∈ U(n) do not imply that u = I n , but observe that if det(W ) = 0 ( i.e, W is an invertible matrix ) U(n) would act freely on our space. This should be the way to describe the space of density matrices. Instead of starting with M n,n (C), we start with the subset of invertible matrices.
That subset has the differentiable structure of the Lie group GL(n, C). Then, we build a submanifold S of GL(n, C) given by Finally, we obtain the base manifold P + using the projection π : S → P + given by and therefore, S(P + , U(n), π) becomes a principal fibre bundle. Observe that contains only strictly positive (or faithful) density operators. But P can be recovered from P + by continuity arguments 3 .
In short, we describe the geometry of density matrices as a base manifold of a principal fibre bundle consisting of a submanifold S of the Lie group GL(n, C) diffeomorphic to the sphere S 2n 2 −1 as a total space and the Lie group U(n) as structure group.
Since S(P + , U(n), π) admits a global section 20 τ : therefore S(P + , U(n), π) is a trivial bundle from a topological point of view, that means that 3,13 III. HAMILTONIAN VECTOR FIELD, DYNAMIC RIEMANNIAN

METRIC, SHG-QUANTUM FIBRE BUNDLE AND MAIN THEOREM
In this section, we define a Riemannian metric for dynamics systems and we study how this metric acts within the tangent vector space of S. We also discuss its relationship with other metrics such as the Bures metric or the metric proposed by Kryukov 16 .

A. Hamiltonian vector field, dynamic metric and its relation with other metrics
In order to provide explicit expressions for tangent vectors to S and the metric tensor, we identify the tangent space T W M n,n (C) with M n,n (C) itself. Since our total space S is a submanifold of the manifold M n,n (C), where each point W ∈ S is a matrix, and the tangent space T W S to S in the point W is a subspace of the tangent space T W M n,n (C). We can use a matrix to describe a point W ∈ S and a matrix to describe a tangent vector X ∈ T W S too.
First of all, note that the Hamiltonian operator H induces a vector field h : S → T S on S given by where h W denotes the vector field in the point W ∈ S, i.e, h W = h(W ). That vector field h will be denoted as the Hamiltonian vector field.
For any point W ∈ S, and any two tangent vectors X, Y ∈ T W S, we define the dynamic It will be denoted by ∇ H the Levi-Civita connexion (the sole metric torsion free connexion) given by g H . In the definition (12) we use H −2 assuming that H is an invertible matrix, but that in fact makes no restriction on the Hamiltonian of the system because we can set H → H + I n without changing the underlying physics. It is not hard to see that g H defines a positive definite inner product in each tangent space T W S, being therefore g H a Riemannian metric.
With that metric tensor g H the (sub)manifold (S, g H ) becomes a Riemannian manifold.
In order to fix the notation we denote {S(P + , U(n), π), h, g H } the SHg-quantum fibre bundle of dimension n.
The rest of this section will examine the inherited metric in the base manifold (theorem 1) from the dynamic metric in the principal manifold and its relation with other metrics.
The tangent space T W S at the point W ∈ S can be decomposed in its horizontal H W and vertical V W subspaces: Observe moreover that the vertical subspaces V W are the vectors tangent to the fibres.
Therefore, any vertical vector X V ∈ T W S can be written as where A ∈ u(n) (i.e, A is an antihermitian matrix). Note that our metric g H defines a natural connexion as follows: A tangent vector X at W is horizontal if it is orthogonal to the fibre passing through W , i.e., if Therefore, we can define a metric g P + H in the base manifold for any point ρ ∈ P + , given by where Y, Z ∈ T ρ P + and Y Hor (respectively Z Hor ) are the horizontal lift of Y (respectively Z). Theorem 1. The metric g P + H in the base manifold at any point ρ ∈ P + can be obtained as where G Y is the unique hermitian matrix satisfying Note that matrix G Y exists and is unique by the existence and uniqueness of the solution of the Sylvester equation 2,25 . Observe moreover that when H is the identity matrix, then where G Y is the (unique) solution of and that is the Bures metric 10 .
Proof. Let W : R → S be a curve, such thatẆ is an horizontal vector, then It is easy to see that the latter condition is fulfilled iḟ where G is an Hermitian matrix. Thereforė Hence applying equation 4 Suppose that Applying equation (15) in π * (V ) Using the above expression (16) the theorem follows.
In the case of pure states, our Hilbert space is C n and the tangent space will be C n too.
Following Kryukov work 16 , we can define a metric g K (X, Y ) for any two tangent vectors X = (x, x * ), Y = (y, y * ), by When H is the identity, we recover the Fubini-Study metric.
B. Geometric structure of the SHg-quantum fibre bundle As we have previously seen in the SHg-quantum fibre bundle {S(P + , U(n), π), h, g H } of dimension n, S(P + , U(n), π) is a principal (and trivial) fibre bundle, S is diffeomophic to the sphere of dimension 2n 2 − 1, h is a vector field on S, and (S, g H ) is a Riemannian manifold.
But the SHg-quantum fibre bundle has more geometric properties : Theorem 2 (Main theorem). Let {S(P + , U(n), π), h, g H } be a SHg-quantum fibre bundle of dimension n. Then: 1. h is a Killing vector field of (S, g H ).
2. The integral curves γ : I ⊂ R → S of h are geodesics of (S, g H ).
3. The projection on the base manifold P + of the geodesic γ satisfies the von Neumann Proof. Condition (1): In order to proof that h is a Killing vector field, we only have to show that the flow ϕ t : S → S given by is an isometry, i.e, for any X, Y ∈ T W S g H (ϕ t * (X), ϕ t * (Y )) = g H (X, Y ) .
Note that and g H (ϕ t * (X), ϕ t * (Y )) =g H (e −iHt X, e −iHt Y ) Conditions (2) and (3): First of all observe that if γ is the integral curve of the vector The projection of γ satisfies Hence, the projection of the integral curves of the vector field h satisfies the von Neumann equation. So all we have to prove is that curves are actually geodesic curves Since h is a Killing vector field, we only have to proof that h is a unitary vector field (due any unitary Killing vector field is a geodesic). Namely, the equality =tr γ † HH −2 Hγ = tr γ † γ = 1 .
being the ϕ t the 1−parametric subgroup of transformations given by the integral curves of the Killing field h.
being the ϕ t the 1−parametric subgroup of transformations given by the integral curves of the Killing vector field h.
Using the above theorem, we can therefore state a similar theorem to the Poincare recurrence theorem 1 .
Theorem 5 (Poincare Type Theorem). Let {S(P + , U(n), π), h, g H } be a SHg-quantum principal bundle of dimension n. For any domain Ω ⊂ S and any time period T ∈ R + there exist a point x ∈ Ω and a positive integer k > 0 such that being ϕ t : S → S the 1−parametric subgroup of transformations given by the integral curves of the Killing field h.
Proof. Consider the following sequence of domains Ω, ϕ T (Ω), ϕ 2T (Ω), · · · , ϕ kT (Ω), · · · All domain in the sequence belongs to the same volume Vol(Ω). If the above domains never intersect S, an infinite volume would obtain, but S is compact, so Vol(S) < ∞. Then, there exist l ≥ 0 and m > l such that so Setting k = m − l the theorem is proven.
Joining the above theorem with the Insensitivity to the Initial Conditions we get Theorem 6 (Strong Poincare Type Theorem). Let {S(P + , U(n), π), h, g H } be a SHgquantum principal bundle of dimension n. Then, for any point W ∈ S, any ǫ > 0 and any T ∈ R + , there exist a positive integer k > 0 such that being ϕ t : S → S the 1−parametric subgroup of transformations given by the integral curves of the Killing vector field h.
A. Physical systems with discrete energy eigenvalues Using previously stated theorems we can give an alternative proof and more geometric sense of well-known 5,22,24 principle of recurrence for physical systems with discrete energy eigenvalues.
Thus, defining the length A of a matrix A as follows 22

Then
Theorem 7. Let ρ be a mixed state of a quantum system with discrete energy spectrum.
Then, ρ is almost periodic. Namely, for an arbitrarily small positive error ǫ the inequality is satisfied by infinitely many values of T , these values being spread over the whole range −∞ to ∞ so as not to leave arbitrarily long empty intervals.
Proof. Let ρ(t) be the density matrix of a system with a discrete set of stationary states, labeled n = 0, 1, 2, · · · , with energies E n , some of which may be equal if there are degeneracies.
In energy representation the matrix elements are Let T n = |n n| be the projection operator onto the nth stationary state, then is the matrix which energy representation has only one nonzero element, equal to ρ nn ′ (t) and in the location (n, n ′ ). These matrices are orthogonal in density space and where ω nn ′ = (E n ′ − E n ). Now, consider the finite sum as an approximation to ρ(t). The square of the error is The second equality follows from the orthogonality of the ρ nn ′ . Since the error is independent of the time, σ N N ′ (t) converges uniformly to ρ(t) (in the -norm sense). So, ρ(t) can be approximated by σ N N ′ (t). σ N N ′ (t) is a discrete density with finite energy levels, σ N N ′ (t) ∈ P + , and the set is an open precompact set in P + . Using the global section given in equation (9), τ (B ǫ ) will be an open precompact set of S. But applying the Strong Poincare Type Theorem for any time period T > 0 there exists k > 0 such that for any ε > 0. Namely, being B S ε (τ (σ N N ′ (t))) the geodesic ball in S centered at τ (σ N N ′ (t)) of radius ε. Now choosing ε small enough Therefore by (46) Projecting to the base manifold σ N N ′ (t + kT ) ∈ B P + ǫ (σ N N ′ (t)) .
By definition of B P + ǫ (σ N N ′ (t)) And the theorem is proven.

V. APPENDIX
In this section we recall several well known results about Killing vector fields on Riemannian manifolds(for a more detailed approximation see O'Neill 21 ). Proof. Here, we need ∇˙γγ = ∇ X X = 0 , but, since X is a Killing vector field, the Lie derivative of the metric is zero L X g = 0 and (see O'Neill 21 , proposition 25) ∇X is skew-adjoint relative to g, then g(∇ X X, W ) + g(∇ W X, X) = 0 , for any X ∈ T S. Therefore 0 = g(∇ X X, W ) + 1/2W (g(X, X)) = g(∇ X X, W ) , then ∇ X X = 0.
Then, the divergence is just (see Chavel 8 ) where dµ g H denotes the Riemannian density measure.
The divergence is defined as 6 div H = tr(Y → ∇ H Y X) .
Given an orthonormal base And the theorem follows.