Geometric characteristics of quantum evolution: curvature and torsion

We study characteristics of quantum evolution which can be called curvature and torsion. The curvature shows a deviation of the state vector in quantum evolution from the geodesic line. The torsion shows a deviation of state vector from the plane of evolution (a two-dimensional subspace) at a given time.

In the classical case, the curvature and torsion are important geometric characteristics of the trajectory. The aim of the present paper is to answer the question: What is the quantum analogue of these classical geometrical notions? Partly, the answer to this question was given in [13] where the authors from the perspective different from this paper, namely, considering geometry of quantum statistical interference, derived an explicit expression for the curvature of quantum evolution.
It is convenient to put γ = 2. Then, in a two-dimensional case, g i j is a metric tensor of a sphere with the radius equal to one (the Bloch sphere). According to Schrödinger equation, one can introduce the velocity of quantum evolution [1] where ∆H = H − 〈H 〉.

Geodesic in the space of quantum state vectors
The geodesic line (one-parametric set of the quantum state vectors) that connects two state vectors |ψ 0 〉 and |ψ 1 〉 can be defined as their linear combination where ξ is a real parameter changing from 0 to 1. This definition is similar to the definition of a direct line connecting two points r 0 and r 1 in Euclidean space r = (1−ξ)r 0 +ξr 1 . However, in contrast to the classical case, in quantum case the states |ψ 0 〉 and e iφ 0 |ψ 0 〉 describe the same quantum state, similarly, |ψ 1 〉 and e iφ 1 |ψ 1 〉 describe the same quantum state. Therefore, we require that geodesic lines defined between the states |ψ 0 〉, |ψ 1 〉 and between the states e iφ 0 |ψ 0 〉, e iφ 1 |ψ 1 〉 coincide. This requirement is satisfied if we choose e iφ = 〈ψ 1 |ψ 0 〉 |〈ψ 1 |ψ 0 〉| . (2. 2) The normalization condition 〈ψ(ξ)|ψ(ξ)〉 = 1 gives Now, let us show that (2.1) is really a geodesic line. For this purpose, we calculate its length and show that it is a minimal possible length. The geodesic line (2.1) is a one-parametric set of states and there exist many possibilities to parameterize it. One can show that the length of the curve in quantum space does not depend on the way of its parametrization. To calculate the length of the geodesic line it is convenient to write its equation as follows: here, a new parameter θ changes in the range 0 θ π and the normalization constant reads Let us stress once more that (2.1) and (2.4) describe the same one-parametric family of quantum state vectors, namely, the geodesic line. Comparing (2.1) and (2.4) we find the relation between the parameters ξ and θ ξ = tan(θ/2) 1 + tan(θ/2) . (2.6) One can verify that substituting (2.6) into (2.1) we find (2.4).
Then, the length of the geodesic line connecting the states |ψ 0 〉 and |ψ 1 〉 is Thus, this length is equal to the Wootters distance, that is, the length of geodesic between two states (see figure 1). For γ = 2, the Wootters distance is equal to the angle between vectors a 0 and a 1 on Bloch sphere which correspond to |ψ 0 〉 and |ψ 1 〉, respectively. This angle is the minimal possible length of the curve on the Bloch sphere connecting the states |ψ 0 〉 and |ψ 1 〉.
In conclusion of this section, let us note that we can calculate the length of the curve (2.1) connecting the states |ψ 0 〉 and |ψ 1 〉 for an arbitrary phase φ. Then, the geodesic line can be defined as the one having a minimal length. One can find that the minimal length is achieved for φ given in (2.2) and is equal to the Wootters distance.

Curvature
The state vector of the quantum evolution belongs to a one-parametric set of state vectors |ψ(t )〉 = exp(−iH t )|ψ 0 〉 generated by the Hamiltonian of the system. The deviation of evolution state vector |ψ(t )〉 from the geodesic, connecting the same two state vectors, is related with the curvature of quantum evolution.
In order to introduce the curvature as well as the torsion, we consider the evolution in two stages. At first, we consider the evolution during the time ∆t from an initial state |ψ 0 〉 to and then the evolution during the time ∆t from |ψ 〉 to |ψ 1 〉 = e −iH ∆t /ħ |ψ 〉 = e −iH (∆t +∆t )/ħ |ψ 0 〉, (3.2) where H is a time independent Hamiltonian. In this section, without the loss of generality, we put ∆t = ∆t . A deviation of the quantum evolution from the geodesic line connecting |ψ 0 〉 and |ψ 1 〉 can be characterized by the minimal distance between the state |ψ 〉 and the geodesic line |ψ(ξ)〉 The minimal value of this expression is achieved at ξ = 1/2. Taking into account the terms of order (∆t ) 4 we find Here, the multiplier can be called the curvature coefficient or curvature. It is convenient to introduce a dimensionless curvature coefficientκ For the first time, this result was obtained within the framework of the study of the geometry of quantum statistical interference in [13]. Now, we show that the curvature of the quantum evolution can also be obtained using the geometric treatment. For a small time, the classical motion along a given curve can be treated as a motion along the circle with radius R for which we can write where s is the length of the curve between two neighboring points on it, which can be considered as an arc of the circle, and d is the distance between the middle point of an arc and the chord connecting these two points. Similarly to (3.7) we define the radius of the curvature for the quantum evolution. In our case, d is given by (3.4) and is the length that a quantum system passes during the time 2∆t of the evolution. Here, v is the velocity of quantum evolution given in (1.6). As a result, we have (3.9)

Torsion
Torsion is related with the deviation of the evolution state vector from the plane of evolution (a twodimensional subspace) at a given time.
In order to find torsion, we consider the evolution in the two stages given by (3.1) and (3.2). Two vectors |ψ 0 〉 and |ψ 〉 that form the first stage define the plane of evolution. Using these vectors we can construct the orthogonal ones  where a and φ are defined by 〈ψ 0 |ψ 〉 = ae iα . Then, the unit operator in a two-dimensional subspace spanned by |φ 1 〉 and |φ 2 〉 isÎ 2 = |φ 1 〉〈φ 1 | + |φ 2 〉〈φ 2 |. (4.3) Note that this is the projection operator of an arbitrary state vector on a two-dimensional subspace.

Discussion
In this paper, we have obtained the curvature and torsion coefficients (3.6) and (4.8) for the quantum evolution which is governed by a time independent Hamiltonian. In this case, the curvature and torsion coefficients are constant.
In conclusion, let us note an interesting fact which follows from (4.8). Namely, for symmetric states when 〈(∆H ) 3 〉 = 0, we find thatκ =τ. It means that the curvature and torsion during the evolution of symmetric states are strongly related.
Finally, we would like to note that the curvature and torsion, presented in this paper, are interesting on their own rights. They can be used for the study of evolution of different quantum systems. In this paper, we presented a simple example of quantum system, namely spin in magnetic field. For the system, we find curvature and torsion during evolution. Of course, this example can be considered as a simple demonstration. It is also interesting to study curvature and torsion for a many-spin system during the evolution, in particular for a two-spin system. We think that such characteristics of quantum evolution as curvature and torsion are useful for the study of brachistochrone problem and entanglement. Note also that in this paper we considered curvature and torsion for a time independent Hamiltonian. Of course, there appears a question regarding generalization of these characteristics on time dependent Hamiltonian. This question is worth to be studied separately.