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 give...
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| Cite this: | Geometric characteristics of quantum evolution: curvature and torsion / H.P. Laba, V.M. Tkachuk // Condensed Matter Physics. — 2017. — Т. 20, № 1. — С. 13003: 1–7. — Бібліогр.: 19 назв. — англ. |
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| citation_txt | Geometric characteristics of quantum evolution: curvature and torsion / H.P. Laba, V.M. Tkachuk // Condensed Matter Physics. — 2017. — Т. 20, № 1. — С. 13003: 1–7. — Бібліогр.: 19 назв. — англ. |
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| description | 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.
Ми вивчаємо характеристики квантової еволюцiї, такi як кривизна та кручення. Кривизна показує вiдхилення вектора стану пiд час квантової еволюцiї вiд геодезичної лiнiї. Кручення визначає вiдхилення
вектора стану вiд площини еволюцiї (двовимiрний пiдпростiр) в заданий момент часу.
|
| first_indexed | 2025-12-02T05:42:53Z |
| format | Article |
| fulltext |
Condensed Matter Physics, 2017, Vol. 20, No 1, 13003: 1–7
DOI: 10.5488/CMP.20.13003
http://www.icmp.lviv.ua/journal
Geometric characteristics of quantum evolution:
curvature and torsion
H.P. Laba1, V.M. Tkachuk2
1 Department of Applied Physics and Nanomaterials Science, Lviv Polytechnic National University,
5 Ustiyanovych St., 79013 Lviv, Ukraine
2 Department for Theoretical Physics, Ivan Franko National University of Lviv,
12 Drahomanov St., 79005 Lviv, Ukraine
Received January 20, 2017, in final form March 2, 2017
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.
Key words: curvature, torsion, quantum evolution, geometry of quantum state space
PACS: 03.65.-w, 03.65.Aa
1. Introduction
Geometric ideas play an important role in quantum mechanics, in particular in the studies of quan-
tum evolution [1–4], quantum brachistochrone problem [5–8], entanglement of quantum states [9, 10],
quantum correlations [11], Berry phase that has geometric origin [12].
In the classical case, the curvature and torsion are important geometric characteristics of the trajec-
tory. 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 in-
terference, derived an explicit expression for the curvature of quantum evolution.
First, let us consider some facts of the geometry of the space of quantum states. A distance between
two quantum states |ψ1〉, |ψ2〉 which is normed to unity can be defined in different ways. In this paper,we will refer to the Fubiny-Study distance and the Wootters distance defined, respectively, as follows:
d (FS)(|ψ1〉, |ψ2〉) = γ
√
1−|〈ψ1|ψ2〉|2, (1.1)
d (W)(|ψ1〉, |ψ2〉) = γarccos |〈ψ1|ψ2〉|, (1.2)
where γ is an arbitrary constant (for a short review see, for instance, [14]).
These distances are equivalent for the neighboring states when |〈ψ1|ψ2〉|2 = 1−δ2, where δ is a small
value, namely, d (FS) = d (W) = γδ. As a result, the element of length for the family (set) of quantum state
vectors |ψ(ξ1,ξ2, . . . ,ξk )〉 parameterized by k parameters ξ1, ξ2, . . . , ξk is the same for different definitions
of the distance
ds2 = gi j dξi dξ j (1.3)
with metric tensor
gi j = γ2 Re(〈ψi |ψ j 〉−〈ψi |ψ〉〈ψ|ψ j 〉), (1.4)
© H.P. Laba, V.M. Tkachuk, 2017 13003-1
https://doi.org/10.5488/CMP.20.13003
http://www.icmp.lviv.ua/journal
H.P. Laba, V.M. Tkachuk
where
|ψi 〉 = ∂
∂ξi
|ψ(ξ1,ξ2, . . . ,ξk )〉. (1.5)
This form of metrics of quantum states was discussed by many authors (see, for example, [2, 15–19]).
It is convenient to put γ= 2. Then, in a two-dimensional case, gi j is a metric tensor of a sphere withthe radius equal to one (the Bloch sphere).
According to Schrödinger equation, one can introduce the velocity of quantum evolution [1]
v = ds
dt
= γ
ħ
√
〈(∆H)2〉, (1.6)
where ∆H = H −〈H〉.
2. Geodesic in the space of quantum state vectors
The geodesic line (one-parametric set of the quantum state vectors) that connects two state vectors
|ψ0〉 and |ψ1〉 can be defined as their linear combination
|ψ(ξ)〉 =C
[
(1−ξ)|ψ0〉+ξ|ψ1〉eiφ]
, (2.1)
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 r0 and r1 in Euclidean space r = (1−ξ)r0+ξr1. However, in contrast to the classicalcase, in quantum case the states |ψ0〉 and eiφ0 |ψ0〉 describe the same quantum state, similarly, |ψ1〉 and
eiφ1 |ψ1〉 describe the same quantum state. Therefore, we require that geodesic lines defined between thestates |ψ0〉, |ψ1〉 and between the states eiφ0 |ψ0〉, eiφ1 |ψ1〉 coincide. This requirement is satisfied if wechoose
eiφ = 〈ψ1|ψ0〉
|〈ψ1|ψ0〉|
. (2.2)
The normalization condition 〈ψ(ξ)|ψ(ξ)〉 = 1 gives
C = 1√
1−2ξ(1−ξ)(1−|〈ψ1|ψ0〉|)
. (2.3)
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:
|ψ(θ)〉 =C ′[sin(θ/2)|ψ0〉+cos(θ/2)|ψ1〉eiφ]
, (2.4)
here, a new parameter θ changes in the range 0 É θ Éπ and the normalization constant reads
C ′ = 1√
1+|〈ψ1|ψ0〉|sinθ
. (2.5)
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).
13003-2
Geometric characteristics of quantum evolution
Figure 1. Geodesic on Bloch sphere. Vectors a0 and a1 on Bloch sphere correspond to |ψ0〉 and |ψ1〉.
Using (1.4) for the one-parameter set of states (2.4) we obtain
ds = γ
2
√
1−|〈ψ1|ψ0〉|2
1+|〈ψ1|ψ0〉|sinθ
dθ. (2.7)
Then, the length of the geodesic line connecting the states |ψ0〉 and |ψ1〉 is
s =
∫
ds =
π∫
0
γ
2
√
1−|〈ψ1|ψ0〉|2
1+|〈ψ1|ψ0〉|sinθ
dθ = γarccos |〈ψ1|ψ0〉|. (2.8)
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 a0 and a1 on Bloch spherewhich correspond to |ψ0〉 and |ψ1〉, respectively. This angle is the minimal possible length of the curve onthe Bloch sphere connecting the states |ψ0〉 and |ψ1〉.In conclusion of this section, let us note that we can calculate the length of the curve (2.1) connecting
the states |ψ0〉 and |ψ1〉 for an arbitrary phase φ. Then, the geodesic line can be defined as the one havinga minimal length. One can find that the minimal length is achieved for φ given in (2.2) and is equal to the
Wootters distance.
3. Curvature
The state vector of the quantum evolution belongs to a one-parametric set of state vectors |ψ(t )〉 =
exp(−iH t )|ψ0〉 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 evo-
lution.
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 |ψ0〉 to
|ψ′〉 = e−iH∆t/ħ|ψ0〉 (3.1)
and then the evolution during the time ∆t ′ from |ψ′〉 to
|ψ1〉 = e−iH∆t ′/ħ|ψ′〉 = e−iH(∆t+∆t ′)/ħ|ψ0〉, (3.2)
13003-3
H.P. Laba, V.M. Tkachuk
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 |ψ0〉 and |ψ1〉 can be charac-terized by the minimal distance between the state |ψ′〉 and the geodesic line |ψ(ξ)〉
d 2 = mind 2(ξ) = minγ2 [
1−|〈ψ′|ψ(ξ)〉|2] . (3.3)
The minimal value of this expression is achieved at ξ= 1/2. Taking into account the terms of order (∆t )4
we find
d 2 = γ2
4
[〈(∆H)4〉−〈(∆H)2〉2] (∆t )4
ħ4 = γ2
4
κ
(∆t )4
ħ4 . (3.4)
Here, the multiplier
κ= 〈(∆H)4〉−〈(∆H)2〉2 (3.5)
can be called the curvature coefficient or curvature. It is convenient to introduce a dimensionless curva-
ture coefficient
κ̄= 〈(∆H)4〉−〈(∆H)2〉2
〈(∆H)2〉2 . (3.6)
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
1
R
= 2d
(s/2)2 , (3.7)
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
s = v2∆t = γ
√
〈(∆H)2〉
ħ 2∆t (3.8)
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
1
R2 = 1
γ2
〈(∆H)4〉−〈(∆H)2〉2
〈(∆H)2〉2 = κ̄
γ2 . (3.9)
4. Torsion
Torsion is related with the deviation of the evolution state vector from the plane of evolution (a two-
dimensional 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 |ψ0〉 and |ψ′〉 that form the first stage define the plane of evolution. Using these vectors we can
construct the orthogonal ones
|φ1〉 = 1p
2(1+a)
(|ψ0〉+e−iα|ψ′〉) , (4.1)
|φ2〉 = 1p
2(1+a)
(|ψ0〉−e−iα|ψ′〉) , (4.2)
13003-4
Geometric characteristics of quantum evolution
where a and φ are defined by 〈ψ0|ψ′〉 = aeiα. Then, the unit operator in a two-dimensional subspace
spanned by |φ1〉 and |φ2〉 is
Î2 = |φ1〉〈φ1|+ |φ2〉〈φ2|. (4.3)
Note that this is the projection operator of an arbitrary state vector on a two-dimensional subspace.
In order to find the deviation of the state vector |ψ1〉 obtained at the second stage from the plane ofevolution, we calculate the mean value of Î2
I2 = 〈ψ1|Î2|ψ1〉 = |〈φ1|ψ1〉|2 +|〈φ2|ψ1〉|2. (4.4)
When I2 = 1, then |ψ1〉 belongs to the subspace spanned by |φ1〉 and |φ2〉. It means that three state vectors
|ψ0〉, |ψ′〉, |ψ1〉 belong to the same plane (the two-dimensional subspace) of evolution and thus the torsionis zero. The expression 1− I2 gives the magnitude of the torsion. Considering small ∆t and ∆t ′ and taking
into account the terms up to the fourth order, we find
1− I2 =
[
〈(∆H)4〉−〈(∆H)2〉2 − 〈(∆H)3〉2
〈(∆H)2〉
]
∆t 2(∆t +∆t ′)2
4ħ4 . (4.5)
The coefficient
τ= 〈(∆H)4〉−〈(∆H)2〉2 − 〈(∆H)3〉2
〈(∆H)2〉 (4.6)
does not depend on ∆t and ∆t ′ and can be called the torsion coefficient. For simplicity, we put ∆t = ∆t ′
and then
1− I2 = τ∆t 4
ħ4 . (4.7)
Similarly to the dimensionless curvature coefficient, we introduce a dimensionless torsion coefficient
τ̄= τ
〈(∆H)2〉2 = 〈(∆H)4〉−〈(∆H)2〉2
〈(∆H)2〉2 − 〈(∆H)3〉2
〈(∆H)2〉3 = κ̄− 〈(∆H)3〉2
〈(∆H)2〉3 . (4.8)
Now, let us show that 1− I2 has a geometrical meaning, namely, it is proportional to the squared dis-tance of the state |ψ1〉 to the plane of quantum evolution spanned by |φ1〉 and |φ2〉. The distance betweena given state |ψ1〉 and the plane is equal to the distance between |ψ1〉 and the normalized projection ofthis vector onto the plane. The normalized projection of |ψ1〉 on the plane is
|ψ̃1〉 = c Î2|ψ1〉, (4.9)
where from the condition 〈ψ̃1|ψ̃1〉 = 1, we find c = (〈ψ1|Î2 Î2|ψ1〉)−1/2 = (〈ψ1|Î2|ψ1〉)−1/2. Here, we use
that (Î2)2 = Î2. Then, the squared distance between the state |ψ1〉 and the plane is
d 2 = γ2 (
1−|〈ψ1|ψ̃1〉|2
)= γ2 (
1−|〈ψ1|Î2|ψ1〉|
)= γ2(1− I2), (4.10)
where we use that |I2| = I2.
5. 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.
The evolution is going along the geodesic when κ̄= 0 or explicitly
〈(∆H)4〉−〈(∆H)2〉2 = 0. (5.1)
13003-5
H.P. Laba, V.M. Tkachuk
Introducing operator  = (∆H)2 we rewrite this condition in the form 〈(∆Â)2〉 = 0. Then, we find that
(5.1) is equivalent to the equation ∆Â|ψ〉 = 0 which explicitly reads
(∆H)2|ψ〉 = 〈(∆H)2〉|ψ〉. (5.2)
The solution of this equation is
|ψ〉 = 1p
2
(|E1〉+eiα|E2〉
)
, (5.3)
where α is an arbitrary phase, |E1〉 and |E1〉 are two eigenstates of the Hamiltonian H with eigenenergies
E1 and E2. Considering (5.3) as an initial state for the time dependent state that evolves along the geodesicwe find
|ψ(t )〉 = 1p
2
(
e−iE1t/ħ|E1〉+eiαe−iE2t/ħ|E2〉
)
. (5.4)
One can find that for an arbitrary time, the evolution state vector (5.4) satisfies the equation (5.2) and the
curvature for this evolution is zero. It is worth stressing that the state vector of the geodesic evolution
contains only two eigenstates of the Hamiltonian and lies in a two-dimensional subspace.
Let us show that the torsion of the geodesic is zero. Using (5.2) we have
〈(∆H)3〉 = 〈ψ|∆H(∆H)2|ψ〉 = 〈(∆H)2〉〈ψ|∆H |ψ〉 = 0. (5.5)
Then, according to (4.8) and taking into account that for the geodesic κ̄= 0 we find that the torsion τ̄= 0.
Let us verify that for two-dimensional space the torsion given by (4.8) is zero because this is abiding
by the definition. The most general Hamiltonian for a two-dimensional case reads
H =ω(σn)+ε, (5.6)
where n is a unit vector. Note that the curvature and torsion depend on ∆H , where ε is canceled. So,
without loss of generality we put ε= 0. Then, using the properties of Pauli matrices for Hamiltonian (5.6)
with ε = 0 we find the following results 〈(∆H)2〉 = ω2 − 〈H〉2, 〈(∆H)4〉 − 〈(∆H)2〉2 = 4〈H〉2〈(∆H)2〉 and
〈(∆H)3〉 = −2〈H〉〈(∆H)2〉. Then, one can find that the torsion (4.8) in the two-dimensional case is always
zero. For curvature in this case, we have κ̄= 4〈H〉2/〈(∆H)2〉. Thus, in two-dimensional case the quantum
evolution is going along the geodesic line when 〈H〉 =ω〈(σn)〉 = 0.
In conclusion, let us note an interesting fact which follows from (4.8). Namely, for symmetric states
when 〈(∆H)3〉 = 0, we find that κ̄ = τ̄. 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.
Acknowledgement
We would like to thank Dr. Yu. Krynytskyi for constructive discussions and useful comments. We
also thank the Members of Editorial Board for the possibility to present our results in a special issue
of Condensed Matter Physics dedicated to Prof. Holovatch’s 60th birthday. We admire his activities in
various fields of science and in scientific life and wish him bright ideas in future.
13003-6
Geometric characteristics of quantum evolution
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Геометричнi характеристики квантової еволюцiї: кривизна
та кручення
Г.П. Лаба 1, В.М. Ткачук 2
1 Кафедра прикладної фiзики i наноматерiалознавства, Нацiональний унiверситет “Львiвська полiтехнiка”,
вул. Устияновича, 5, 79013 Львiв, Україна
2 Кафедра теоретичної фiзики, Львiвський нацiональний унiверситет iменi Iвана Франка,
вул. Драгоманова, 12, 79005 Львiв, Україна
Ми вивчаємо характеристики квантової еволюцiї, такi як кривизна та кручення. Кривизна показує вiд-
хилення вектора стану пiд час квантової еволюцiї вiд геодезичної лiнiї. Кручення визначає вiдхилення
вектора стану вiд площини еволюцiї (двовимiрний пiдпростiр) в заданий момент часу.
Ключовi слова: кривизна, кручення, квантова еволюцiя, геометрiя простору квантових станiв
13003-7
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https://doi.org/10.1103/PhysRevLett.99.130502
https://doi.org/10.1007/978-3-642-03174-8_12
https://doi.org/10.1103/PhysRevA.77.014103
https://doi.org/10.1016/j.physleta.2015.03.003
https://doi.org/10.1016/S0393-0440(00)00052-8
https://doi.org/10.1088/1742-6596/67/1/012044
https://doi.org/10.1007/s11128-015-1227-2
https://doi.org/10.1103/PhysRevLett.77.2851
https://doi.org/10.1238/Physica.Regular.059a00081
https://doi.org/10.1007/BF02193559
https://doi.org/10.1103/PhysRevA.55.1695
https://doi.org/10.1007/s10773-007-9438-7
Introduction
Geodesic in the space of quantum state vectors
Curvature
Torsion
Discussion
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| id | nasplib_isofts_kiev_ua-123456789-156556 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-12-02T05:42:53Z |
| publishDate | 2017 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Laba, H.P. Tkachuk, V.M. 2019-06-18T16:31:47Z 2019-06-18T16:31:47Z 2017 Geometric characteristics of quantum evolution: curvature and torsion / H.P. Laba, V.M. Tkachuk // Condensed Matter Physics. — 2017. — Т. 20, № 1. — С. 13003: 1–7. — Бібліогр.: 19 назв. — англ. 1607-324X PACS: 03.65.-w, 03.65.Aa DOI:10.5488/CMP.20.13003 arXiv:1006.4447 https://nasplib.isofts.kiev.ua/handle/123456789/156556 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. Ми вивчаємо характеристики квантової еволюцiї, такi як кривизна та кручення. Кривизна показує вiдхилення вектора стану пiд час квантової еволюцiї вiд геодезичної лiнiї. Кручення визначає вiдхилення вектора стану вiд площини еволюцiї (двовимiрний пiдпростiр) в заданий момент часу. We would like to thank Dr. Yu. Krynytskyi for constructive discussions and useful comments. We also thank the Members of Editorial Board for the possibility to present our results in a special issue of Condensed Matter Physics dedicated to Prof. Holovatch’s 60th birthday. We admire his activities in various fields of science and in scientific life and wish him bright ideas in future. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Geometric characteristics of quantum evolution: curvature and torsion Геометричнi характеристики квантової еволюцiї: кривизна та кручення Article published earlier |
| spellingShingle | Geometric characteristics of quantum evolution: curvature and torsion Laba, H.P. Tkachuk, V.M. |
| title | Geometric characteristics of quantum evolution: curvature and torsion |
| title_alt | Геометричнi характеристики квантової еволюцiї: кривизна та кручення |
| title_full | Geometric characteristics of quantum evolution: curvature and torsion |
| title_fullStr | Geometric characteristics of quantum evolution: curvature and torsion |
| title_full_unstemmed | Geometric characteristics of quantum evolution: curvature and torsion |
| title_short | Geometric characteristics of quantum evolution: curvature and torsion |
| title_sort | geometric characteristics of quantum evolution: curvature and torsion |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/156556 |
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