Geometric measure of mixing of quantum state
We define the geometric measure of mixing of quantum state as a minimal Hilbert-Schmidt distance between the mixed state and a set of pure states. An explicit expression for the geometric measure is obtained. It is interesting that this expression corresponds to the squared Euclidian distance betw...
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| Zitieren: | Geometric measure of mixing of quantum state / H.P. Laba, V.M. Tkachuk // Condensed Matter Physics. — 2018. — Т. 21, № 3. — С. 33003: 1–4. — Бібліогр.: 7 назв. — англ. |
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irk-123456789-1571112019-06-20T01:25:46Z Geometric measure of mixing of quantum state Laba, H.P. Tkachuk, V.M. We define the geometric measure of mixing of quantum state as a minimal Hilbert-Schmidt distance between the mixed state and a set of pure states. An explicit expression for the geometric measure is obtained. It is interesting that this expression corresponds to the squared Euclidian distance between the mixed state and the pure one in space of eigenvalues of the density matrix. As an example, geometric measure of mixing for spin-1/2 states is calculated. Ми означаємо геометричну мiру змiшаностi квантоваго стану як мiнiмальну вiдстань Гiльберта-Шмiдта мiж змiшаним станом та набором чистих станiв. Отримано явний вираз для геометричної мiри змiшаностi. Цiкавим є те, що цей вираз вiдповiдає квадрату евклiдової вiдстанi мiж змiшаним та чистим станами у просторi власних значень матрицi густини. Як приклад, обчислено геометричну мiру змiшаностi станiв спiна 1/2. 2018 Article Geometric measure of mixing of quantum state / H.P. Laba, V.M. Tkachuk // Condensed Matter Physics. — 2018. — Т. 21, № 3. — С. 33003: 1–4. — Бібліогр.: 7 назв. — англ. 1607-324X PACS: 03.65.-w, 03.67.-a DOI:10.5488/CMP.21.33003 arXiv:1809.09469 http://dspace.nbuv.gov.ua/handle/123456789/157111 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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We define the geometric measure of mixing of quantum state as a minimal Hilbert-Schmidt distance between
the mixed state and a set of pure states. An explicit expression for the geometric measure is obtained. It is
interesting that this expression corresponds to the squared Euclidian distance between the mixed state and
the pure one in space of eigenvalues of the density matrix. As an example, geometric measure of mixing for
spin-1/2 states is calculated. |
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Article |
| author |
Laba, H.P. Tkachuk, V.M. |
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Laba, H.P. Tkachuk, V.M. Geometric measure of mixing of quantum state Condensed Matter Physics |
| author_facet |
Laba, H.P. Tkachuk, V.M. |
| author_sort |
Laba, H.P. |
| title |
Geometric measure of mixing of quantum state |
| title_short |
Geometric measure of mixing of quantum state |
| title_full |
Geometric measure of mixing of quantum state |
| title_fullStr |
Geometric measure of mixing of quantum state |
| title_full_unstemmed |
Geometric measure of mixing of quantum state |
| title_sort |
geometric measure of mixing of quantum state |
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Інститут фізики конденсованих систем НАН України |
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2018 |
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http://dspace.nbuv.gov.ua/handle/123456789/157111 |
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Geometric measure of mixing of quantum state / H.P. Laba, V.M. Tkachuk // Condensed Matter Physics. — 2018. — Т. 21, № 3. — С. 33003: 1–4. — Бібліогр.: 7 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT labahp geometricmeasureofmixingofquantumstate AT tkachukvm geometricmeasureofmixingofquantumstate |
| first_indexed |
2025-07-14T09:26:03Z |
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2025-07-14T09:26:03Z |
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1837613882766524416 |
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Condensed Matter Physics, 2018, Vol. 21, No 3, 33003: 1–4
DOI: 10.5488/CMP.21.33003
http://www.icmp.lviv.ua/journal
Geometric measure of mixing of quantum state
H.P. Laba 1, V.M. Tkachuk 2
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 June 23, 2018
We define the geometric measure of mixing of quantum state as a minimal Hilbert-Schmidt distance between
the mixed state and a set of pure states. An explicit expression for the geometric measure is obtained. It is
interesting that this expression corresponds to the squared Euclidian distance between the mixed state and
the pure one in space of eigenvalues of the density matrix. As an example, geometric measure of mixing for
spin-1/2 states is calculated.
Key words:mixed states, density matrix, Hilbert-Schmidt distance
PACS: 03.65.-w, 03.67.-a
1. Introduction
Pure and mixed states are the key concept in quantum mechanics and in quantum information theory.
Therefore, an important question arises regarding the degree ofmixing of a quantum state. In the literature,
von Neumann entropy is often used to answer this question:
S = −Tr ρ̂ ln ρ̂ = −〈ln ρ̂〉 , (1.1)
which is zero for a pure state and has a maximal value for maximally mixed states. The entropy can be
used as a measure of the degree of mixing of a quantum state. To explicitly calculate the von Neumann
entropy, it is necessary to know the eigenvalue of density matrix which is a nontrivial problem. Therefore,
the linear entropy as approximation of von Neumann entropy is also used
ln ρ̂ = ln [1 − (1 − ρ̂)] ' −(1 − ρ̂) . (1.2)
In this approximation, the linear entropy reads
SL = Tr
(
ρ̂ − ρ̂2) = 1 − Tr ρ̂2. (1.3)
Linear entropy does not satisfy the properties of von Neumann entropy. However, to calculate the linear
entropy, it is not necessary to know the eigenvalues of a density matrix. In this case, we can directly
calculate the trace of ρ̂2. Note that Tr ρ̂2 is called purity and is used for quantifying the degree of the
purity of state. For pure state ρ̂2 = ρ̂, and purity takes a maximal value 1 and is less 1 for mixed states.
A review on entropy in quantum information can be found in book [1] (see also [2]).
Geometric ideas play an important role in quantum mechanics and in quantum information theory
(for review see, for instance, [3]). In our previous paper [4], we use the geometric characteristics such as
curvature and torsion to study the quantum evolution. The geometry of quantum states in the evolution of
a spin system was studied in [5, 6]. In [7], the distance between quantum states was used for quantifying
the entanglement of pure and mixed states.
This work is licensed under a Creative Commons Attribution 4.0 International License . Further distribution
of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
33003-1
https://doi.org/10.5488/CMP.21.33003
http://www.icmp.lviv.ua/journal
http://creativecommons.org/licenses/by/4.0/
Short authors list
In this paper, we use Hilbert-Schmidt distance in order to measure the degree of mixing of quantum
state. We define the geometric measure of mixing of quantum state as minimal Hilbert-Schmidt distance
between the mixed state and a set of pure states. In section 2, using this definition, we find an explicit
expression for the geometric measure of mixing of quantum state. Conclusions are presented in section 3.
2. Hilbert-Schmidt distance and degree of mixing of quantum state
To define the geometric measure of degree of mixing of quantum state, we use the Hilbert-Schmidt
distance between two mixed states. The squared Hilbert-Schmidt distance reads
d2(ρ̂1, ρ̂2) = Tr
(
ρ̂1 − ρ̂2
)2
, (2.1)
where ρ̂1 and ρ̂1 are density matrices of the first and the second mixed states. We define geometric
measure of mixing of quantum states as minimal squared Hilbert-Schmidt distance from the given mixed
state to a set of pure states
D = min
|ψ〉
Tr
(
ρ̂ − ρ̂pure
)2
, (2.2)
where ρ̂ is density matrix of the given mixed states,
ρ̂pure = |ψ〉〈ψ | (2.3)
is density matrix of a pure state described by the state vector |ψ〉, and minimization is done over all
possible pure states.
Let us rewrite the geometric measure of mixing of quantum states as follows:
D = min
|ψ〉
(
Tr ρ̂2 + Tr ρ̂2
pure − 2 Tr ρ̂ρ̂ pure
)
. (2.4)
Three terms in (2.4) can be calculated separately. For the first term, we find
Tr ρ̂2 =
∑
i
λ2
i , (2.5)
where λi are eigenvalues of density matrix ρ̂. For pure state ρ̂2
pure = ρ̂pure, so the second term reads
Tr ρ̂2
pure = Tr ρ̂pure = 1. (2.6)
Trace is invariant with respect to choosing the basic vectors. To calculate the third term, we use the
following orthogonal basic vectors |ψ〉, |ψ1〉, |ψ2〉, . . . , where the first vector is equal to the state of pure
state in (2.3), 〈ψ |ψi〉 = 0, 〈ψi |ψj〉 = 0, i = 1, 2, . . . , j = 1, 2, . . . . Then,
ρ̂pure |ψ〉 = |ψ〉, (2.7)
ρ̂pure |ψi〉 = |ψ〉〈ψ |ψi〉 = 0, i = 1, 2, . . . . (2.8)
As a result, for the third term we have
Tr ρ̂ρ̂pure = 〈ψ | ρ̂|ψ〉. (2.9)
Substituting (2.5), (2.6), (2.9) into (2.4), we find
D = min
|ψ〉
(∑
i
λ2
i + 1 − 2〈ψ | ρ̂|ψ〉
)
. (2.10)
33003-2
Geometric measure of mixing of quantum state
This expression reaches a minimal value when |ψ〉 is equal to the eigenvector of density matrix ρ̂ with
maximal eigenvalue. Thus, finally, for geometric measure of mixing of quantum state we have
D =
∑
i
λ2
i + 1 − 2λmax = (1 − λmax)
2 +
∑
λi<λmax
λ2
i . (2.11)
For the pure state λmax = 1 and all other eigenvalues are zero. Thus, for the pure state D = 0 as it
should really be. It is interesting to note that (2.11) is a squared Euclidian distance in the eigenvalue
space between the mixed state with eigenvalues of density matrix λmax, . . . λi . . . and pure state with
eigenvalues 1, . . . 0, . . . .
One can easily find that D is maximal when all eigenvalues of the density matrix are the same
λi = 1/n, i = 1, 2, . . . n, where n is a dimension of the quantum system. So, the maximal value of
geometric measure of mixing of quantum state in this case is D = 1 − 1/n and density matrix reads
ρ̂max =
1
n
1̂ (2.12)
and can be referred to as the maximally mixed state.
The distance between the maximally mixed (2.12) state and the arbitrary pure (2.3) one is
d2(ρ̂max, ρ̂pure) = Tr
(
ρ̂max − ρ̂pure
)2
=
(
1 −
1
n
)2
+ (n − 1)
1
n2 = 1 −
1
n
. (2.13)
where to calculate Tr we use the orthogonal basic vectors |ψ〉, |ψ1〉, |ψ2〉, . . . , where the first vector
corresponds to the pure state in (2.3). Note that this distance is the same between the maximally mixed
state and the arbitrary pure one.
At the end of this section, let us consider an explicit example of using the obtained result for calculation
of geometric measure of mixing of quantum state presented by (2.11). We consider the mixed state of
spin-1/2 described by the density matrix
ρ̂ =
1
2
[1 + (aσ)] , (2.14)
where a is Bloch vector, σ = (σx, σy, σz) are Pauli matrices. Eigenvalues of this matrix are
λ1 =
1
2
(1 + a) , λ2 =
1
2
(1 − a) , (2.15)
where a = |a| ≤ 1 is the length of Bloch vector. Note that λ1 corresponds here to λmax. Then, according
to (2.11), the geometric measure in this case reads
D =
1
2
(1 − a)2. (2.16)
At a = 1, which corresponds to pure states (Bloch sphere) as we see D = 0 and the mixed state is
maximally mixed D = 1/2 at a = 0.
3. Conclusions
We define the geometric measure of mixing of quantum state as minimal Hilbert-Schmidt distance
between the givenmixed state and a set of pure states. Themain problem in this definition is the procedure
of minimization over pure states. It is important that it is possible to perform this procedure and get an
explicit expression for geometric measure of mixing of the quantum state presented by (2.11). This is the
main result of the present paper. It is interesting to note that (2.11) is the squared Euclidian distance in
space of eigenvalues of the density matrix between the mixed state and the pure one. Finally, we would
like to note that similarly to the calculation of von Neumann entropy of mixed states, to calculate the
geometric measure of mixing of state, it is necessary to know the eigenvalues of the density matrix. So,
from this point of view, the difficulties of calculation of geometric measure of mixing of states is similar
to the difficulties of calculation of the entropy measure of mixing of state. However, definition of degree
of mixing of state presented in this paper is of geometric origin and is intuitively understandable. We
hope that this result provides a new inside into the problem under consideration.
33003-3
Short authors list
Acknowledgements
We thank the Members of Editorial Board for the invitation to present our results in a special issue of
CondensedMatter Physics dedicated to Prof. Stasyuk’s 80th birthday. I (VMT) have known Prof. Stasyuk
since 1978 when he delivered the lectures on Green’s function method for students of theoretical physics
department. The lectures were very interesting and I thank Prof. Stasyuk for that. We wish Prof. Stasyuk
long scientific life and bright ideas in the future.
References
1. Nielsen M.A., Chuang I.L., Quantum Computation and Quantum Information, 10th Anniversary Edition, Cam-
bridge University Press, New York, 2010.
2. Witten E., arXiv:1805.11965, 2018.
3. Bengtsson I., Życzkowski K., Geometry of Quantum States: An Introduction to Quantum Entanglement, Cam-
bridge University Press, New York, 2017.
4. Laba H.P., Tkachuk V.M., Condens. Matter Phys., 2017, 20, 13003, doi:10.5488/CMP.20.13003.
5. Kuzmak A.R., Tkachuk V.M., J. Phys. A:Math. Theor., 2016, 49, 045301, doi:10.1088/1751-8113/49/4/045301.
6. Kuzmak A.R., Tkachuk V.M., Phys. Lett. A, 2015, 379, 1233, doi:10.1016/j.physleta.2015.03.003.
7. Frydryszak A.M., Samar M.I., Tkachuk V.M., Eur. Phys. J. D, 2017, 71, 233, doi:10.1140/epjd/e2017-70752-3.
Геометрична мiра змiшаностi квантового стану
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 означаємо геометричну мiру змiшаностi квантоваго стану як мiнiмальну вiдстань Гiльберта-Шмiдта
мiж змiшаним станом та набором чистих станiв. Отримано явний вираз для геометричної мiри змiшано-
стi. Цiкавим є те, що цей вираз вiдповiдає квадрату евклiдової вiдстанi мiж змiшаним та чистим станами
у просторi власних значень матрицi густини. Як приклад, обчислено геометричну мiру змiшаностi станiв
спiна 1/2.
Ключовi слова: змiшанi стани, матриця густини, вiдстань Гiльберта-Шмiдта
33003-4
http://arxiv.org/abs/1805.11965
https://doi.org/10.5488/CMP.20.13003
https://doi.org/10.1088/1751-8113/49/4/045301
https://doi.org/10.1016/j.physleta.2015.03.003
https://doi.org/10.1140/epjd/e2017-70752-3
Introduction
Hilbert-Schmidt distance and degree of mixing of quantum state
Conclusions
|